Subgroups — C₄₁×#19755 #23795956

Order 10496.

842 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3294	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×D₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#3272	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1721	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1687	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1681	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1667	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1665	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1639	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1625	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₁×#1619	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄₁×#1619	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄₁×#1581	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₁×#1580	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×AGL(1,ℤ/8)×C₄₁	—	2	1	no	—
17	C₂²×D₈×C₄₁	—	2	1	no	—
18	C₂²×D₈×C₄₁	—	2	1	no	—
19	C₄₁×#547	C₂²	1	1	yes	characteristic, complemented
20	D₄²×C₄₁	C₂²	1	1	yes	characteristic
21	D₄²×C₄₁	C₂²	1	1	yes	characteristic, complemented
22	D₄²×C₄₁	C₂²	1	1	yes	characteristic, complemented
23	C₄₁×#493	C₂²	1	1	yes	characteristic, complemented
24	C₄₁×#493	C₂²	1	1	yes	characteristic, complemented
25	C₄₁×#493	C₂²	1	1	yes	characteristic, complemented
26	C₄₁×#492	C₂²	1	1	yes	characteristic, complemented
27	C₄₁×#486	C₂²	1	1	yes	characteristic, complemented
28	C₄₁×#486	C₂²	1	1	yes	characteristic, complemented
29	C₄₁×#480	—	2	1	no	—
30	C₄₁×#474	C₂²	1	1	yes	characteristic, complemented
31	C₄₁×#468	—	2	1	no	—
32	C₄₁×#466	—	2	1	no	—
33	C₄₁×#466	—	2	1	no	—
34	C₄₁×#466	—	2	1	no	—
35	C₄₁×#461	C₂²	1	1	yes	characteristic, complemented
36	C₄₁×#459	C₂²	1	1	yes	characteristic
37	C₄₁×#459	C₂²	1	1	yes	characteristic
38	C₄₁×#459	C₂²	1	1	yes	characteristic, complemented
39	C₄₁×#459	C₂²	1	1	yes	characteristic, complemented
40	C₄₁×#459	C₂²	1	1	yes	characteristic, complemented
41	C₄₁×#459	C₂²	1	1	yes	characteristic, complemented
42	C₄₁×#450	—	2	1	no	—
43	C₄₁×#449	—	2	1	no	—
44	C₄₁×#447	—	2	1	no	—
45	C₄₁×#447	—	2	1	no	—
46	C₄₁×#447	—	2	1	no	—
47	C₄₁×#447	—	2	1	no	—
48	C₄₁×#447	—	2	1	no	—
49	C₄×D₈×C₄₁	C₂²	1	1	yes	characteristic
50	C₄×D₈×C₄₁	C₂²	1	1	yes	characteristic
51	C₄₁×#436	C₂²	1	1	yes	characteristic, complemented
52	C₈×D₄×C₄₁	C₂²	1	1	yes	characteristic, complemented
53	C₈×D₄×C₄₁	C₂²	1	1	yes	characteristic, complemented
54	C₈×D₄×C₄₁	C₂²	1	1	yes	characteristic, complemented
55	C₂×#102×C₄₁	—	2	1	no	—
56	C₄₁×#335	C₂²	1	1	yes	characteristic, complemented
57	C₄₁×#334	C₂²	1	1	yes	characteristic, complemented
58	C₄₁×#332	C₂²	1	1	yes	characteristic, complemented
59	C₄₁×#331	C₂²	1	1	yes	characteristic, complemented
60	C₄₁×#331	C₂²	1	1	yes	characteristic, complemented
61	C₄₁×#331	C₂²	1	1	yes	characteristic, complemented
62	C₄₁×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
63	C₄₁×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
64	C₄₁×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
65	C₄₁×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
66	C₄₁×#322	C₂²	1	1	yes	characteristic, complemented
67	C₂×#41×C₄₁	—	2	1	no	—
68	C₂²×D₄×C₄₁	—	2	1	no	—
69	C₂²×D₄×C₄₁	—	2	1	no	—
70	C₂²×D₄×C₄₁	—	2	1	no	—
71	C₂²×D₄×C₄₁	—	2	1	no	—
72	C₂²×D₄×C₄₁	—	2	1	no	—
73	C₂²×D₄×C₄₁	—	2	1	no	—
74	AGL(1,ℤ/8)×C₄₁	—	4	4	no	—
75	C₂×SD₁₆×C₄₁	—	2	1	no	—
76	C₂×SD₁₆×C₄₁	—	2	1	no	—
77	C₂×SD₁₆×C₄₁	—	2	1	no	—
78	C₂×D₈×C₄₁	—	2	1	no	—
79	C₂×D₈×C₄₁	—	2	1	no	—
80	C₂×D₈×C₄₁	—	2	1	no	—
81	C₂×D₈×C₄₁	—	2	1	no	—
82	C₂×D₈×C₄₁	—	2	1	no	—
83	C₂×D₈×C₄₁	—	2	1	no	—
84	C₂×D₈×C₄₁	—	2	1	no	—
85	C₂×D₈×C₄₁	—	2	1	no	—
86	C₂×D₈×C₄₁	—	2	1	no	—
87	C₂×D₈×C₄₁	—	2	1	no	—
88	C₂×D₈×C₄₁	—	4	1	no	—
89	C₂×D₈×C₄₁	—	4	4	no	—
90	C₂×D₈×C₄₁	—	4	4	no	—
91	C₂×D₈×C₄₁	D₄	1	1	yes	characteristic
92	C₂×D₈×C₄₁	D₄	1	1	yes	characteristic
93	C₂×M₁₆×C₄₁	—	2	1	no	—
94	C₂²×C₈×C₄₁	—	2	1	no	—
95	C₂²×C₈×C₄₁	—	2	1	no	—
96	C₂²×C₈×C₄₁	—	2	1	no	—
97	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
98	#127×C₄₁	C₂³	1	1	yes	characteristic
99	#127×C₄₁	C₂³	1	1	yes	characteristic
100	#127×C₄₁	C₂³	1	1	yes	characteristic
101	#123×C₄₁	—	4	1	no	—
102	C₂²⋊Q₈×C₄₁	—	2	1	no	—
103	#121×C₄₁	—	2	1	no	—
104	#121×C₄₁	—	2	1	no	—
105	#121×C₄₁	—	2	1	no	—
106	#121×C₄₁	—	2	1	no	—
107	#121×C₄₁	—	2	1	no	—
108	#121×C₄₁	—	2	1	no	—
109	#121×C₄₁	—	2	1	no	—
110	#120×C₄₁	—	4	1	no	—
111	#120×C₄₁	—	4	1	no	—
112	#120×C₄₁	—	4	1	no	—
113	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
114	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
115	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
116	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
118	#107×C₄₁	—	2	1	no	—
119	#107×C₄₁	D₄	1	1	yes	characteristic
120	#107×C₄₁	D₄	1	1	yes	characteristic
121	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
122	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
123	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
124	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
125	#103×C₄₁	—	2	1	no	—
126	#102×C₄₁	—	2	1	no	—
127	#102×C₄₁	—	2	1	no	—
128	#102×C₄₁	—	2	1	no	—
129	#102×C₄₁	—	2	1	no	—
130	#102×C₄₁	—	2	1	no	—
131	#102×C₄₁	—	2	1	no	—
132	#102×C₄₁	—	2	1	no	—
133	#102×C₄₁	—	2	1	no	—
134	#102×C₄₁	—	2	1	no	—
135	#102×C₄₁	—	4	1	no	—
136	#102×C₄₁	D₄	1	1	yes	characteristic
137	#102×C₄₁	D₄	1	1	yes	characteristic
138	#102×C₄₁	D₄	1	1	yes	characteristic
139	#102×C₄₁	D₄	1	1	yes	characteristic
140	C₂²⋊C₈×C₄₁	—	2	1	no	—
141	C₂²⋊C₈×C₄₁	—	2	1	no	—
142	C₂²⋊C₈×C₄₁	—	2	1	no	—
143	C₂²⋊C₈×C₄₁	—	2	1	no	—
144	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
145	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
146	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
147	C₂⁴×C₄₁	—	4	1	no	—
148	C₂⁴×C₄₁	—	4	1	no	—
149	C₂⁴×C₄₁	—	4	1	no	—
150	#41×C₄₁	—	4	2	no	—
151	#41×C₄₁	—	8	1	no	—
152	C₂×Q₈×C₄₁	—	2	1	no	—
153	C₂×D₄×C₄₁	—	2	1	no	—
154	C₂×D₄×C₄₁	—	2	1	no	—
155	C₂×D₄×C₄₁	—	2	1	no	—
156	C₂×D₄×C₄₁	—	2	1	no	—
157	C₂×D₄×C₄₁	—	2	1	no	—
158	C₂×D₄×C₄₁	—	2	1	no	—
159	C₂×D₄×C₄₁	—	2	1	no	—
160	C₂×D₄×C₄₁	—	2	1	no	—
161	C₂×D₄×C₄₁	—	2	1	no	—
162	C₂×D₄×C₄₁	—	4	1	no	—
163	C₂×D₄×C₄₁	—	4	1	no	—
164	C₂×D₄×C₄₁	—	4	1	no	—
165	C₂×D₄×C₄₁	—	4	1	no	—
166	C₂×D₄×C₄₁	—	4	1	no	—
167	C₂×D₄×C₄₁	—	4	1	no	—
168	C₂×D₄×C₄₁	—	4	2	no	—
169	C₂×D₄×C₄₁	—	4	2	no	—
170	C₂×D₄×C₄₁	—	4	2	no	—
171	C₂×D₄×C₄₁	—	4	2	no	—
172	C₂×D₄×C₄₁	—	4	2	no	—
173	C₂×D₄×C₄₁	—	4	2	no	—
174	C₂×D₄×C₄₁	—	4	1	no	—
175	C₂×D₄×C₄₁	—	8	1	no	—
176	C₂×D₄×C₄₁	—	8	1	no	—
177	C₂×D₄×C₄₁	—	8	1	no	—
178	C₂×D₄×C₄₁	—	8	1	no	—
179	C₂×D₄×C₄₁	—	8	1	no	—
180	C₂×D₄×C₄₁	—	8	1	no	—
181	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
182	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
183	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
184	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₄₁	—	2	1	no	—
186	C₂²×C₄×C₄₁	—	2	1	no	—
187	C₂²×C₄×C₄₁	—	2	1	no	—
188	C₂²×C₄×C₄₁	—	2	1	no	—
189	C₂²×C₄×C₄₁	—	4	1	no	—
190	SD₁₆×C₄₁	—	4	2	no	—
191	SD₁₆×C₄₁	—	4	2	no	—
192	SD₁₆×C₄₁	—	8	1	no	—
193	D₈×C₄₁	—	2	2	no	—
194	D₈×C₄₁	—	4	2	no	—
195	D₈×C₄₁	—	4	2	no	—
196	D₈×C₄₁	—	4	2	no	—
197	D₈×C₄₁	—	4	2	no	—
198	D₈×C₄₁	—	4	2	no	—
199	D₈×C₄₁	—	4	2	no	—
200	D₈×C₄₁	—	4	2	no	—
201	D₈×C₄₁	—	8	1	no	—
202	D₈×C₄₁	—	8	1	no	—
203	D₈×C₄₁	—	8	1	no	—
204	D₈×C₄₁	—	8	2	no	—
205	D₈×C₄₁	D₈	1	4	yes	complemented
206	M₁₆×C₄₁	—	4	2	no	—
207	C₂×C₈×C₄₁	—	2	1	no	—
208	C₂×C₈×C₄₁	—	2	1	no	—
209	C₂×C₈×C₄₁	—	2	1	no	—
210	C₂×C₈×C₄₁	—	2	1	no	—
211	C₂×C₈×C₄₁	—	2	1	no	—
212	C₂×C₈×C₄₁	—	4	2	no	—
213	C₂×C₈×C₄₁	—	4	2	no	—
214	C₂×C₈×C₄₁	—	8	1	no	—
215	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
216	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
217	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
218	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
219	C₄⋊C₄×C₄₁	—	2	1	no	—
220	C₄⋊C₄×C₄₁	—	2	1	no	—
221	C₄⋊C₄×C₄₁	—	4	1	no	—
222	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
223	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
224	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
225	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
226	C₂²⋊C₄×C₄₁	—	2	1	no	—
227	C₂²⋊C₄×C₄₁	—	2	1	no	—
228	C₂²⋊C₄×C₄₁	—	2	1	no	—
229	C₂²⋊C₄×C₄₁	—	2	1	no	—
230	C₂²⋊C₄×C₄₁	—	4	1	no	—
231	C₂²⋊C₄×C₄₁	—	4	1	no	—
232	C₂²⋊C₄×C₄₁	—	4	1	no	—
233	C₂²⋊C₄×C₄₁	—	4	1	no	—
234	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
235	C₂³×C₄₁	—	2	1	no	—
236	C₂³×C₄₁	—	2	1	no	—
237	C₂³×C₄₁	—	2	1	no	—
238	C₂³×C₄₁	—	2	1	no	—
239	C₂³×C₄₁	—	4	1	no	—
240	C₂³×C₄₁	—	4	1	no	—
241	C₂³×C₄₁	—	4	1	no	—
242	C₂³×C₄₁	—	8	2	no	—
243	C₂³×C₄₁	—	8	2	no	—
244	C₂³×C₄₁	—	8	2	no	—
245	C₂³×C₄₁	—	8	2	no	—
246	C₂³×C₄₁	—	8	2	no	—
247	C₂³×C₄₁	—	8	2	no	—
248	C₂³×C₄₁	—	16	1	no	—
249	C₂³×C₄₁	—	16	1	no	—
250	C₂³×C₄₁	—	16	1	no	—
251	Q₈×C₄₁	—	4	1	no	—
252	Q₈×C₄₁	—	4	1	no	—
253	D₄×C₄₁	—	2	1	no	—
254	D₄×C₄₁	—	2	1	no	—
255	D₄×C₄₁	—	2	1	no	—
256	D₄×C₄₁	—	2	1	no	—
257	D₄×C₄₁	—	2	1	no	—
258	D₄×C₄₁	—	4	1	no	—
259	D₄×C₄₁	—	4	1	no	—
260	D₄×C₄₁	—	4	1	no	—
261	D₄×C₄₁	—	4	1	no	—
262	D₄×C₄₁	—	4	1	no	—
263	D₄×C₄₁	—	4	1	no	—
264	D₄×C₄₁	—	4	1	no	—
265	D₄×C₄₁	—	4	1	no	—
266	D₄×C₄₁	—	4	1	no	—
267	D₄×C₄₁	—	4	1	no	—
268	D₄×C₄₁	—	4	1	no	—
269	D₄×C₄₁	—	4	1	no	—
270	D₄×C₄₁	—	8	1	no	—
271	D₄×C₄₁	—	8	1	no	—
272	D₄×C₄₁	—	8	1	no	—
273	D₄×C₄₁	—	8	2	no	—
274	D₄×C₄₁	—	8	2	no	—
275	D₄×C₄₁	—	8	2	no	—
276	D₄×C₄₁	—	8	2	no	—
277	D₄×C₄₁	—	8	2	no	—
278	D₄×C₄₁	—	8	2	no	—
279	D₄×C₄₁	—	8	2	no	—
280	D₄×C₄₁	C₂×D₈	1	2	yes	—
281	D₄×C₄₁	C₂×D₈	1	2	yes	complemented
282	D₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	complemented
283	C₂×C₄×C₄₁	—	2	1	no	—
284	C₂×C₄×C₄₁	—	2	1	no	—
285	C₂×C₄×C₄₁	—	2	1	no	—
286	C₂×C₄×C₄₁	—	2	1	no	—
287	C₂×C₄×C₄₁	—	4	1	no	—
288	C₂×C₄×C₄₁	—	4	1	no	—
289	C₂×C₄×C₄₁	—	4	1	no	—
290	C₂×C₄×C₄₁	—	4	1	no	—
291	C₂×C₄×C₄₁	—	4	1	no	—
292	C₂×C₄×C₄₁	—	4	1	no	—
293	C₂×C₄×C₄₁	—	4	1	no	—
294	C₂×C₄×C₄₁	—	4	1	no	—
295	C₂×C₄×C₄₁	—	4	1	no	—
296	C₂×C₄×C₄₁	—	8	2	no	—
297	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
298	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
300	C₈×C₄₁	—	2	1	no	—
301	C₈×C₄₁	—	2	1	no	—
302	C₈×C₄₁	—	2	1	no	—
303	C₈×C₄₁	—	4	1	no	—
304	C₈×C₄₁	—	4	1	no	—
305	C₈×C₄₁	—	4	1	no	—
306	C₈×C₄₁	—	4	1	no	—
307	C₈×C₄₁	—	4	1	no	—
308	C₈×C₄₁	C₂×D₈	1	2	yes	complemented
309	#19755	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
310	C₂²×C₄₁	—	2	2	no	—
311	C₂²×C₄₁	—	2	2	no	—
312	C₂²×C₄₁	—	2	2	no	—
313	C₂²×C₄₁	—	4	1	no	—
314	C₂²×C₄₁	—	4	1	no	—
315	C₂²×C₄₁	—	4	1	no	—
316	C₂²×C₄₁	—	4	1	no	—
317	C₂²×C₄₁	—	4	1	no	—
318	C₂²×C₄₁	—	4	1	no	—
319	C₂²×C₄₁	—	4	1	no	—
320	C₂²×C₄₁	—	4	1	no	—
321	C₂²×C₄₁	—	4	1	no	—
322	C₂²×C₄₁	—	8	1	no	—
323	C₂²×C₄₁	—	8	1	no	—
324	C₂²×C₄₁	—	8	1	no	—
325	C₂²×C₄₁	—	8	1	no	—
326	C₂²×C₄₁	—	8	1	no	—
327	C₂²×C₄₁	—	8	1	no	—
328	C₂²×C₄₁	—	8	1	no	—
329	C₂²×C₄₁	—	8	1	no	—
330	C₂²×C₄₁	—	8	1	no	—
331	C₂²×C₄₁	—	16	4	no	—
332	C₂²×C₄₁	—	16	4	no	—
333	C₂²×C₄₁	—	16	4	no	—
334	C₂²×C₄₁	D₄²	1	1	yes	characteristic
335	C₄×C₄₁	—	2	1	no	—
336	C₄×C₄₁	—	4	1	no	—
337	C₄×C₄₁	—	4	1	no	—
338	C₄×C₄₁	—	4	1	no	—
339	C₄×C₄₁	—	4	1	no	—
340	C₄×C₄₁	—	8	1	no	—
341	C₄×C₄₁	C₂²×D₈	1	1	yes	characteristic
342	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
343	C₄×C₄₁	#576	1	1	yes	characteristic
344	C₄×C₄₁	#576	1	1	yes	characteristic
345	#3294	C₂×C₄₁	1	1	yes	characteristic, complemented
346	D₄×D₈	C₂×C₄₁	1	1	yes	characteristic, complemented
347	D₄×D₈	C₂×C₄₁	1	1	yes	characteristic, complemented
348	#3272	C₂×C₄₁	1	1	yes	characteristic, complemented
349	#1721	C₂×C₄₁	1	1	yes	characteristic, complemented
350	#1687	C₂×C₄₁	1	1	yes	characteristic, complemented
351	#1681	C₂×C₄₁	1	1	yes	characteristic, complemented
352	#1667	C₂×C₄₁	1	1	yes	characteristic, complemented
353	#1665	C₂×C₄₁	1	1	yes	characteristic, complemented
354	#1639	C₂×C₄₁	1	1	yes	characteristic, complemented
355	#1625	C₂×C₄₁	1	1	yes	characteristic, complemented
356	#1619	C₂×C₄₁	1	1	yes	characteristic, complemented
357	#1619	C₂×C₄₁	1	1	yes	characteristic, complemented
358	#1581	C₂×C₄₁	1	1	yes	characteristic, complemented
359	#1580	C₂×C₄₁	1	1	yes	characteristic, complemented
360	C₂×C₄₁	—	4	2	no	—
361	C₂×C₄₁	—	4	2	no	—
362	C₂×C₄₁	—	4	2	no	—
363	C₂×C₄₁	—	8	1	no	—
364	C₂×C₄₁	—	16	1	no	—
365	C₂×C₄₁	—	16	1	no	—
366	C₂×C₄₁	—	16	1	no	—
367	C₂×C₄₁	#3264	1	1	yes	characteristic
368	C₂×C₄₁	D₄×D₈	1	1	yes	characteristic
369	C₂×C₄₁	#3288	1	1	yes	characteristic
370	C₂×AGL(1,ℤ/8)	—	2	1	no	—
371	C₂²×D₈	—	2	1	no	—
372	C₂²×D₈	—	2	1	no	—
373	#547	C₂²×C₄₁	1	1	yes	characteristic, complemented
374	D₄²	C₂²×C₄₁	1	1	yes	characteristic
375	D₄²	C₂²×C₄₁	1	1	yes	characteristic, complemented
376	D₄²	C₂²×C₄₁	1	1	yes	characteristic, complemented
377	#493	C₂²×C₄₁	1	1	yes	characteristic, complemented
378	#493	C₂²×C₄₁	1	1	yes	characteristic, complemented
379	#493	C₂²×C₄₁	1	1	yes	characteristic, complemented
380	#492	C₂²×C₄₁	1	1	yes	characteristic, complemented
381	#486	C₂²×C₄₁	1	1	yes	characteristic, complemented
382	#486	C₂²×C₄₁	1	1	yes	characteristic, complemented
383	#480	—	2	1	no	—
384	#474	C₂²×C₄₁	1	1	yes	characteristic, complemented
385	#468	—	2	1	no	—
386	#466	—	2	1	no	—
387	#466	—	2	1	no	—
388	#466	—	2	1	no	—
389	#461	C₂²×C₄₁	1	1	yes	characteristic, complemented
390	#459	C₂²×C₄₁	1	1	yes	characteristic
391	#459	C₂²×C₄₁	1	1	yes	characteristic
392	#459	C₂²×C₄₁	1	1	yes	characteristic, complemented
393	#459	C₂²×C₄₁	1	1	yes	characteristic, complemented
394	#459	C₂²×C₄₁	1	1	yes	characteristic, complemented
395	#459	C₂²×C₄₁	1	1	yes	characteristic, complemented
396	#450	—	2	1	no	—
397	#449	—	2	1	no	—
398	#447	—	2	1	no	—
399	#447	—	2	1	no	—
400	#447	—	2	1	no	—
401	#447	—	2	1	no	—
402	#447	—	2	1	no	—
403	C₄×D₈	C₂²×C₄₁	1	1	yes	characteristic
404	C₄×D₈	C₂²×C₄₁	1	1	yes	characteristic
405	#436	C₂²×C₄₁	1	1	yes	characteristic, complemented
406	C₈×D₄	C₂²×C₄₁	1	1	yes	characteristic, complemented
407	C₈×D₄	C₂²×C₄₁	1	1	yes	characteristic, complemented
408	C₈×D₄	C₂²×C₄₁	1	1	yes	characteristic, complemented
409	C₂×#102	—	2	1	no	—
410	#335	C₂²×C₄₁	1	1	yes	characteristic, complemented
411	#334	C₂²×C₄₁	1	1	yes	characteristic, complemented
412	#332	C₂²×C₄₁	1	1	yes	characteristic, complemented
413	#331	C₂²×C₄₁	1	1	yes	characteristic, complemented
414	#331	C₂²×C₄₁	1	1	yes	characteristic, complemented
415	#331	C₂²×C₄₁	1	1	yes	characteristic, complemented
416	D₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
417	D₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
418	D₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
419	D₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
420	#322	C₂²×C₄₁	1	1	yes	characteristic, complemented
421	C₄₁	#19755	1	1	yes	characteristic, direct product factor, Sylow
422	C₂×#41	—	2	1	no	—
423	C₂²×D₄	—	2	1	no	—
424	C₂²×D₄	—	2	1	no	—
425	C₂²×D₄	—	2	1	no	—
426	C₂²×D₄	—	2	1	no	—
427	C₂²×D₄	—	2	1	no	—
428	C₂²×D₄	—	2	1	no	—
429	AGL(1,ℤ/8)	—	4	4	no	—
430	C₂×SD₁₆	—	2	1	no	—
431	C₂×SD₁₆	—	2	1	no	—
432	C₂×SD₁₆	—	2	1	no	—
433	C₂×D₈	—	2	1	no	—
434	C₂×D₈	—	2	1	no	—
435	C₂×D₈	—	2	1	no	—
436	C₂×D₈	—	2	1	no	—
437	C₂×D₈	—	2	1	no	—
438	C₂×D₈	—	2	1	no	—
439	C₂×D₈	—	2	1	no	—
440	C₂×D₈	—	2	1	no	—
441	C₂×D₈	—	2	1	no	—
442	C₂×D₈	—	2	1	no	—
443	C₂×D₈	—	4	1	no	—
444	C₂×D₈	—	4	4	no	—
445	C₂×D₈	—	4	4	no	—
446	C₂×D₈	D₄×C₄₁	1	1	yes	characteristic
447	C₂×D₈	D₄×C₄₁	1	1	yes	characteristic
448	C₂×M₁₆	—	2	1	no	—
449	C₂²×C₈	—	2	1	no	—
450	C₂²×C₈	—	2	1	no	—
451	C₂²×C₈	—	2	1	no	—
452	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
453	#127	C₂³×C₄₁	1	1	yes	characteristic
454	#127	C₂³×C₄₁	1	1	yes	characteristic
455	#127	C₂³×C₄₁	1	1	yes	characteristic
456	#123	—	4	1	no	—
457	C₂²⋊Q₈	—	2	1	no	—
458	#121	—	2	1	no	—
459	#121	—	2	1	no	—
460	#121	—	2	1	no	—
461	#121	—	2	1	no	—
462	#121	—	2	1	no	—
463	#121	—	2	1	no	—
464	#121	—	2	1	no	—
465	#120	—	4	1	no	—
466	#120	—	4	1	no	—
467	#120	—	4	1	no	—
468	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
469	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
470	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
471	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
472	C₂×C₄⋊C₄	—	2	1	no	—
473	#107	—	2	1	no	—
474	#107	D₄×C₄₁	1	1	yes	characteristic
475	#107	D₄×C₄₁	1	1	yes	characteristic
476	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
477	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
478	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
479	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
480	#103	—	2	1	no	—
481	#102	—	2	1	no	—
482	#102	—	2	1	no	—
483	#102	—	2	1	no	—
484	#102	—	2	1	no	—
485	#102	—	2	1	no	—
486	#102	—	2	1	no	—
487	#102	—	2	1	no	—
488	#102	—	2	1	no	—
489	#102	—	2	1	no	—
490	#102	—	4	1	no	—
491	#102	D₄×C₄₁	1	1	yes	characteristic
492	#102	D₄×C₄₁	1	1	yes	characteristic
493	#102	D₄×C₄₁	1	1	yes	characteristic
494	#102	D₄×C₄₁	1	1	yes	characteristic
495	C₂²⋊C₈	—	2	1	no	—
496	C₂²⋊C₈	—	2	1	no	—
497	C₂²⋊C₈	—	2	1	no	—
498	C₂²⋊C₈	—	2	1	no	—
499	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
500	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
501	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
502	C₂⁴	—	4	1	no	—
503	C₂⁴	—	4	1	no	—
504	C₂⁴	—	4	1	no	—
505	#41	—	4	2	no	—
506	#41	—	8	1	no	—
507	C₂×Q₈	—	2	1	no	—
508	C₂×D₄	—	2	1	no	—
509	C₂×D₄	—	2	1	no	—
510	C₂×D₄	—	2	1	no	—
511	C₂×D₄	—	2	1	no	—
512	C₂×D₄	—	2	1	no	—
513	C₂×D₄	—	2	1	no	—
514	C₂×D₄	—	2	1	no	—
515	C₂×D₄	—	2	1	no	—
516	C₂×D₄	—	2	1	no	—
517	C₂×D₄	—	4	1	no	—
518	C₂×D₄	—	4	1	no	—
519	C₂×D₄	—	4	1	no	—
520	C₂×D₄	—	4	1	no	—
521	C₂×D₄	—	4	1	no	—
522	C₂×D₄	—	4	1	no	—
523	C₂×D₄	—	4	2	no	—
524	C₂×D₄	—	4	2	no	—
525	C₂×D₄	—	4	2	no	—
526	C₂×D₄	—	4	2	no	—
527	C₂×D₄	—	4	2	no	—
528	C₂×D₄	—	4	2	no	—
529	C₂×D₄	—	4	1	no	—
530	C₂×D₄	—	8	1	no	—
531	C₂×D₄	—	8	1	no	—
532	C₂×D₄	—	8	1	no	—
533	C₂×D₄	—	8	1	no	—
534	C₂×D₄	—	8	1	no	—
535	C₂×D₄	—	8	1	no	—
536	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
537	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
538	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
539	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
540	C₂²×C₄	—	2	1	no	—
541	C₂²×C₄	—	2	1	no	—
542	C₂²×C₄	—	2	1	no	—
543	C₂²×C₄	—	2	1	no	—
544	C₂²×C₄	—	4	1	no	—
545	SD₁₆	—	4	2	no	—
546	SD₁₆	—	4	2	no	—
547	SD₁₆	—	8	1	no	—
548	D₈	—	2	2	no	—
549	D₈	—	4	2	no	—
550	D₈	—	4	2	no	—
551	D₈	—	4	2	no	—
552	D₈	—	4	2	no	—
553	D₈	—	4	2	no	—
554	D₈	—	4	2	no	—
555	D₈	—	4	2	no	—
556	D₈	—	8	1	no	—
557	D₈	—	8	1	no	—
558	D₈	—	8	1	no	—
559	D₈	—	8	2	no	—
560	D₈	D₈×C₄₁	1	4	yes	complemented
561	M₁₆	—	4	2	no	—
562	C₂×C₈	—	2	1	no	—
563	C₂×C₈	—	2	1	no	—
564	C₂×C₈	—	2	1	no	—
565	C₂×C₈	—	2	1	no	—
566	C₂×C₈	—	2	1	no	—
567	C₂×C₈	—	4	2	no	—
568	C₂×C₈	—	4	2	no	—
569	C₂×C₈	—	8	1	no	—
570	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
571	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
572	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
573	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	2	1	no	—
576	C₄⋊C₄	—	4	1	no	—
577	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
578	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
579	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
580	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
581	C₂²⋊C₄	—	2	1	no	—
582	C₂²⋊C₄	—	2	1	no	—
583	C₂²⋊C₄	—	2	1	no	—
584	C₂²⋊C₄	—	2	1	no	—
585	C₂²⋊C₄	—	4	1	no	—
586	C₂²⋊C₄	—	4	1	no	—
587	C₂²⋊C₄	—	4	1	no	—
588	C₂²⋊C₄	—	4	1	no	—
589	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
590	C₂³	—	2	1	no	—
591	C₂³	—	2	1	no	—
592	C₂³	—	2	1	no	—
593	C₂³	—	2	1	no	—
594	C₂³	—	4	1	no	—
595	C₂³	—	4	1	no	—
596	C₂³	—	4	1	no	—
597	C₂³	—	8	2	no	—
598	C₂³	—	8	2	no	—
599	C₂³	—	8	2	no	—
600	C₂³	—	8	2	no	—
601	C₂³	—	8	2	no	—
602	C₂³	—	8	2	no	—
603	C₂³	—	16	1	no	—
604	C₂³	—	16	1	no	—
605	C₂³	—	16	1	no	—
606	Q₈	—	4	1	no	—
607	Q₈	—	4	1	no	—
608	D₄	—	2	1	no	—
609	D₄	—	2	1	no	—
610	D₄	—	2	1	no	—
611	D₄	—	2	1	no	—
612	D₄	—	2	1	no	—
613	D₄	—	4	1	no	—
614	D₄	—	4	1	no	—
615	D₄	—	4	1	no	—
616	D₄	—	4	1	no	—
617	D₄	—	4	1	no	—
618	D₄	—	4	1	no	—
619	D₄	—	4	1	no	—
620	D₄	—	4	1	no	—
621	D₄	—	4	1	no	—
622	D₄	—	4	1	no	—
623	D₄	—	4	1	no	—
624	D₄	—	4	1	no	—
625	D₄	—	8	1	no	—
626	D₄	—	8	1	no	—
627	D₄	—	8	1	no	—
628	D₄	—	8	2	no	—
629	D₄	—	8	2	no	—
630	D₄	—	8	2	no	—
631	D₄	—	8	2	no	—
632	D₄	—	8	2	no	—
633	D₄	—	8	2	no	—
634	D₄	—	8	2	no	—
635	D₄	C₂×D₈×C₄₁	1	2	yes	—
636	D₄	C₂×D₈×C₄₁	1	2	yes	complemented
637	D₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	complemented
638	C₂×C₄	—	2	1	no	—
639	C₂×C₄	—	2	1	no	—
640	C₂×C₄	—	2	1	no	—
641	C₂×C₄	—	2	1	no	—
642	C₂×C₄	—	4	1	no	—
643	C₂×C₄	—	4	1	no	—
644	C₂×C₄	—	4	1	no	—
645	C₂×C₄	—	4	1	no	—
646	C₂×C₄	—	4	1	no	—
647	C₂×C₄	—	4	1	no	—
648	C₂×C₄	—	4	1	no	—
649	C₂×C₄	—	4	1	no	—
650	C₂×C₄	—	4	1	no	—
651	C₂×C₄	—	8	2	no	—
652	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
653	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
654	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
655	C₈	—	2	1	no	—
656	C₈	—	2	1	no	—
657	C₈	—	2	1	no	—
658	C₈	—	4	1	no	—
659	C₈	—	4	1	no	—
660	C₈	—	4	1	no	—
661	C₈	—	4	1	no	—
662	C₈	—	4	1	no	—
663	C₈	C₂×D₈×C₄₁	1	2	yes	complemented
664	C₂²	—	2	2	no	—
665	C₂²	—	2	2	no	—
666	C₂²	—	2	2	no	—
667	C₂²	—	4	1	no	—
668	C₂²	—	4	1	no	—
669	C₂²	—	4	1	no	—
670	C₂²	—	4	1	no	—
671	C₂²	—	4	1	no	—
672	C₂²	—	4	1	no	—
673	C₂²	—	4	1	no	—
674	C₂²	—	4	1	no	—
675	C₂²	—	4	1	no	—
676	C₂²	—	8	1	no	—
677	C₂²	—	8	1	no	—
678	C₂²	—	8	1	no	—
679	C₂²	—	8	1	no	—
680	C₂²	—	8	1	no	—
681	C₂²	—	8	1	no	—
682	C₂²	—	8	1	no	—
683	C₂²	—	8	1	no	—
684	C₂²	—	8	1	no	—
685	C₂²	—	16	4	no	—
686	C₂²	—	16	4	no	—
687	C₂²	—	16	4	no	—
688	C₂²	D₄²×C₄₁	1	1	yes	characteristic
689	C₄	—	2	1	no	—
690	C₄	—	4	1	no	—
691	C₄	—	4	1	no	—
692	C₄	—	4	1	no	—
693	C₄	—	4	1	no	—
694	C₄	—	8	1	no	—
695	C₄	C₂²×D₈×C₄₁	1	1	yes	characteristic
696	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
697	C₄	C₄₁×#576	1	1	yes	characteristic
698	C₄	C₄₁×#576	1	1	yes	characteristic
699	C₂	—	4	2	no	—
700	C₂	—	4	2	no	—
701	C₂	—	4	2	no	—
702	C₂	—	8	1	no	—
703	C₂	—	16	1	no	—
704	C₂	—	16	1	no	—
705	C₂	—	16	1	no	—
706	C₂	C₄₁×#3264	1	1	yes	characteristic
707	C₂	D₄×D₈×C₄₁	1	1	yes	characteristic
708	C₂	C₄₁×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄²×C₄₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₄₁×#3294	C₂	semidirect
2. D₄×D₈×C₄₁	C₂	semidirect
3. D₄×D₈×C₄₁	C₂	semidirect
4. C₄₁×#3272	C₂	semidirect
5. C₄₁×#1721	C₂	semidirect
6. C₄₁×#1687	C₂	semidirect
7. C₄₁×#1681	C₂	semidirect
8. C₄₁×#1667	C₂	semidirect
9. C₄₁×#1665	C₂	semidirect
10. C₄₁×#1639	C₂	semidirect
11. C₄₁×#1625	C₂	semidirect
12. C₄₁×#1619	C₂	semidirect
13. C₄₁×#1619	C₂	semidirect
14. C₄₁×#1581	C₂	semidirect
15. C₄₁×#1580	C₂	semidirect
19. C₄₁×#547	C₂²	semidirect
21. D₄²×C₄₁	C₂²	semidirect
22. D₄²×C₄₁	C₂²	semidirect
23. C₄₁×#493	C₂²	semidirect
24. C₄₁×#493	C₂²	semidirect
25. C₄₁×#493	C₂²	semidirect
26. C₄₁×#492	C₂²	semidirect
27. C₄₁×#486	C₂²	semidirect
28. C₄₁×#486	C₂²	semidirect
30. C₄₁×#474	C₂²	semidirect
35. C₄₁×#461	C₂²	semidirect
38. C₄₁×#459	C₂²	semidirect
39. C₄₁×#459	C₂²	semidirect
40. C₄₁×#459	C₂²	semidirect
41. C₄₁×#459	C₂²	semidirect
51. C₄₁×#436	C₂²	semidirect
52. C₈×D₄×C₄₁	C₂²	semidirect
53. C₈×D₄×C₄₁	C₂²	semidirect
54. C₈×D₄×C₄₁	C₂²	semidirect
56. C₄₁×#335	C₂²	semidirect
57. C₄₁×#334	C₂²	semidirect
58. C₄₁×#332	C₂²	semidirect
59. C₄₁×#331	C₂²	semidirect
60. C₄₁×#331	C₂²	semidirect
61. C₄₁×#331	C₂²	semidirect
62. C₄₁×D₄⋊C₈	C₂²	semidirect
63. C₄₁×D₄⋊C₈	C₂²	semidirect
64. C₄₁×D₄⋊C₈	C₂²	semidirect
65. C₄₁×D₄⋊C₈	C₂²	semidirect
66. C₄₁×#322	C₂²	semidirect
205. D₈×C₄₁	D₈	semidirect
281. D₄×C₄₁	C₂×D₈	semidirect
282. D₄×C₄₁	AGL(1,ℤ/8)	semidirect
308. C₈×C₄₁	C₂×D₈	semidirect
309. #19755	C₄₁	direct (N × H)
345. #3294	C₂×C₄₁	semidirect
346. D₄×D₈	C₂×C₄₁	semidirect
347. D₄×D₈	C₂×C₄₁	semidirect
348. #3272	C₂×C₄₁	semidirect
349. #1721	C₂×C₄₁	semidirect
350. #1687	C₂×C₄₁	semidirect
351. #1681	C₂×C₄₁	semidirect
352. #1667	C₂×C₄₁	semidirect
353. #1665	C₂×C₄₁	semidirect
354. #1639	C₂×C₄₁	semidirect
355. #1625	C₂×C₄₁	semidirect
356. #1619	C₂×C₄₁	semidirect
357. #1619	C₂×C₄₁	semidirect
358. #1581	C₂×C₄₁	semidirect
359. #1580	C₂×C₄₁	semidirect
373. #547	C₂²×C₄₁	semidirect
375. D₄²	C₂²×C₄₁	semidirect
376. D₄²	C₂²×C₄₁	semidirect
377. #493	C₂²×C₄₁	semidirect
378. #493	C₂²×C₄₁	semidirect
379. #493	C₂²×C₄₁	semidirect
380. #492	C₂²×C₄₁	semidirect
381. #486	C₂²×C₄₁	semidirect
382. #486	C₂²×C₄₁	semidirect
384. #474	C₂²×C₄₁	semidirect
389. #461	C₂²×C₄₁	semidirect
392. #459	C₂²×C₄₁	semidirect
393. #459	C₂²×C₄₁	semidirect
394. #459	C₂²×C₄₁	semidirect
395. #459	C₂²×C₄₁	semidirect
405. #436	C₂²×C₄₁	semidirect
406. C₈×D₄	C₂²×C₄₁	semidirect
407. C₈×D₄	C₂²×C₄₁	semidirect
408. C₈×D₄	C₂²×C₄₁	semidirect
410. #335	C₂²×C₄₁	semidirect
411. #334	C₂²×C₄₁	semidirect
412. #332	C₂²×C₄₁	semidirect
413. #331	C₂²×C₄₁	semidirect
414. #331	C₂²×C₄₁	semidirect
415. #331	C₂²×C₄₁	semidirect
416. D₄⋊C₈	C₂²×C₄₁	semidirect
417. D₄⋊C₈	C₂²×C₄₁	semidirect
418. D₄⋊C₈	C₂²×C₄₁	semidirect
419. D₄⋊C₈	C₂²×C₄₁	semidirect
420. #322	C₂²×C₄₁	semidirect
421. C₄₁	#19755	direct (N × H)
560. D₈	D₈×C₄₁	semidirect
636. D₄	C₂×D₈×C₄₁	semidirect
637. D₄	AGL(1,ℤ/8)×C₄₁	semidirect
663. C₈	C₂×D₈×C₄₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄₁×#3294	2
D₄×D₈×C₄₁	2
C₄₁×#3272	2
C₄₁×#1721	2
C₄₁×#1687	2
C₄₁×#1681	2
C₄₁×#1667	2
C₄₁×#1665	2
C₄₁×#1639	2
C₄₁×#1625	2
C₄₁×#1619	2
C₄₁×#1581	2
C₄₁×#1580	2
#19755	41

Subgroups — C41×#19755 #23795956

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#19755 #23795956