Subgroups — C₄₁×#24401 #23800602

Order 10496.

1348 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#137×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#471	C₂	1	2	yes	maximal, complemented
3	C₄₁×#3006	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₁×#448	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2880	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#2052	C₂	1	2	yes	maximal, complemented
7	C₄₁×#2019	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1908	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1900	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1855	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1784	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×#137×C₄₁	—	2	1	no	—
15	C₂×#137×C₄₁	—	2	1	no	—
16	C₂×#137×C₄₁	—	2	8	no	—
17	C₂×#137×C₄₁	C₂²	1	2	yes	—
18	C₂×#137×C₄₁	C₂²	1	2	yes	—
19	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	1	yes	characteristic
21	C₂²×Q₁₆×C₄₁	—	2	1	no	—
22	C₂²×SD₁₆×C₄₁	—	2	1	no	—
23	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	complemented
24	C₂×#131×C₄₁	C₂²	1	2	yes	—
25	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×#121×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#471	—	2	8	no	—
28	C₄₁×#449	—	2	1	no	—
29	C₄₁×#449	—	2	1	no	—
30	C₄₁×#448	—	2	1	no	—
31	C₄₁×#448	—	2	1	no	—
32	C₄₁×#448	—	2	1	no	—
33	C₄₁×#448	—	2	2	no	—
34	C₄₁×#447	—	2	1	no	—
35	C₂×#108×C₄₁	C₂²	1	2	yes	complemented
36	C₄₁×#417	C₂²	1	1	yes	characteristic
37	C₄₁×#417	C₂²	1	1	yes	characteristic
38	C₂×#103×C₄₁	C₂²	1	2	yes	complemented
39	C₂×#101×C₄₁	C₂²	1	2	yes	complemented
40	C₂×#101×C₄₁	C₂²	1	2	yes	complemented
41	C₂×#100×C₄₁	C₂²	1	2	yes	complemented
42	C₄₁×#408	—	2	2	no	—
43	C₄₁×#408	C₂²	1	2	yes	—
44	C₄₁×#407	—	2	1	no	—
45	C₄₁×#407	C₂²	1	1	yes	characteristic
46	C₄₁×#407	C₂²	1	1	yes	characteristic
47	C₂×C₂²⋊C₈×C₄₁	—	2	1	no	—
48	C₂×C₂²⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
49	C₄₁×#343	C₂²	1	2	yes	complemented
50	C₄₁×#341	C₂²	1	2	yes	complemented
51	C₂×#41×C₄₁	—	2	1	no	—
52	C₂×#41×C₄₁	—	2	1	no	—
53	C₂×#41×C₄₁	—	2	4	no	—
54	C₂×#41×C₄₁	—	4	2	no	—
55	C₂×#41×C₄₁	D₄	1	1	yes	characteristic
56	C₂×#41×C₄₁	D₄	1	1	yes	characteristic
57	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
58	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
59	C₂²×Q₈×C₄₁	—	2	1	no	—
60	C₂²×Q₈×C₄₁	—	2	4	no	—
61	C₂²×Q₈×C₄₁	—	2	1	no	—
62	C₂²×Q₈×C₄₁	—	2	1	no	—
63	C₂²×Q₈×C₄₁	—	2	1	no	—
64	C₂²×Q₈×C₄₁	—	2	1	no	—
65	C₂²×Q₈×C₄₁	—	4	2	no	—
66	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
67	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₄₁	—	2	1	no	—
69	C₂²×D₄×C₄₁	—	2	1	no	—
70	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
71	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₄₁	—	2	1	no	—
73	C₂³×C₄×C₄₁	—	2	1	no	—
74	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
75	#137×C₄₁	—	2	8	no	—
76	#137×C₄₁	—	2	8	no	—
77	#137×C₄₁	—	2	4	no	—
78	#137×C₄₁	—	2	4	no	—
79	#137×C₄₁	—	2	4	no	—
80	#137×C₄₁	D₄	1	8	yes	complemented
81	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
82	AGL(1,ℤ/8)×C₄₁	—	4	2	no	—
83	C₂×Q₁₆×C₄₁	—	2	8	no	—
84	C₂×Q₁₆×C₄₁	—	2	2	no	—
85	C₂×Q₁₆×C₄₁	—	2	2	no	—
86	C₂×SD₁₆×C₄₁	—	2	8	no	—
87	C₂×SD₁₆×C₄₁	—	2	2	no	—
88	C₂×SD₁₆×C₄₁	—	2	2	no	—
89	C₂×SD₁₆×C₄₁	—	2	4	no	—
90	C₂×SD₁₆×C₄₁	—	2	1	no	—
91	C₂×SD₁₆×C₄₁	—	2	1	no	—
92	C₂×SD₁₆×C₄₁	—	2	2	no	—
93	C₂×SD₁₆×C₄₁	—	2	2	no	—
94	C₂×SD₁₆×C₄₁	—	4	2	no	—
95	C₂×D₈×C₄₁	—	2	1	no	—
96	C₂×D₈×C₄₁	—	2	1	no	—
97	#131×C₄₁	—	2	4	no	—
98	#131×C₄₁	—	4	2	no	—
99	C₂×M₁₆×C₄₁	—	2	1	no	—
100	C₂×M₁₆×C₄₁	—	2	4	no	—
101	C₂×M₁₆×C₄₁	—	2	2	no	—
102	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
103	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
104	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
105	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
106	C₂²×C₈×C₄₁	—	2	1	no	—
107	C₂²×C₈×C₄₁	—	2	2	no	—
108	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
109	#121×C₄₁	—	2	1	no	—
110	#121×C₄₁	—	2	1	no	—
111	#121×C₄₁	—	2	1	no	—
112	#121×C₄₁	—	2	1	no	—
113	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
114	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
115	#108×C₄₁	—	2	4	no	—
116	#103×C₄₁	—	2	1	no	—
117	#103×C₄₁	—	2	1	no	—
118	#103×C₄₁	—	2	2	no	—
119	#103×C₄₁	—	2	2	no	—
120	#102×C₄₁	—	2	1	no	—
121	#102×C₄₁	—	2	1	no	—
122	#101×C₄₁	—	2	4	no	—
123	#101×C₄₁	—	4	2	no	—
124	#100×C₄₁	—	2	4	no	—
125	C₂²⋊C₈×C₄₁	—	2	1	no	—
126	C₂²⋊C₈×C₄₁	—	2	1	no	—
127	C₂²⋊C₈×C₄₁	—	2	1	no	—
128	C₂²⋊C₈×C₄₁	—	2	1	no	—
129	C₂²⋊C₈×C₄₁	—	2	2	no	—
130	C₂²⋊C₈×C₄₁	—	2	2	no	—
131	C₂²⋊C₈×C₄₁	D₄	1	1	yes	characteristic
132	C₂²⋊C₈×C₄₁	D₄	1	1	yes	characteristic
133	C₂²⋊C₈×C₄₁	D₄	1	2	yes	—
134	C₂⁴×C₄₁	—	2	1	no	—
135	C₂⁴×C₄₁	—	2	1	no	—
136	C₂⁴×C₄₁	#41	1	1	yes	characteristic
137	#41×C₄₁	—	2	2	no	—
138	#41×C₄₁	—	2	2	no	—
139	#41×C₄₁	—	2	2	no	—
140	#41×C₄₁	—	2	2	no	—
141	#41×C₄₁	—	2	2	no	—
142	#41×C₄₁	—	4	4	no	—
143	#41×C₄₁	—	4	1	no	—
144	#41×C₄₁	—	4	1	no	—
145	#41×C₄₁	—	4	4	no	—
146	#41×C₄₁	C₂×D₄	1	4	yes	—
147	C₂×Q₈×C₄₁	—	2	1	no	—
148	C₂×Q₈×C₄₁	—	2	2	no	—
149	C₂×Q₈×C₄₁	—	2	4	no	—
150	C₂×Q₈×C₄₁	—	2	4	no	—
151	C₂×Q₈×C₄₁	—	2	2	no	—
152	C₂×Q₈×C₄₁	—	2	2	no	—
153	C₂×Q₈×C₄₁	—	2	2	no	—
154	C₂×Q₈×C₄₁	—	2	2	no	—
155	C₂×Q₈×C₄₁	—	2	2	no	—
156	C₂×Q₈×C₄₁	—	4	4	no	—
157	C₂×Q₈×C₄₁	—	4	1	no	—
158	C₂×Q₈×C₄₁	—	4	4	no	—
159	C₂×Q₈×C₄₁	—	4	4	no	—
160	C₂×Q₈×C₄₁	—	4	2	no	—
161	C₂×Q₈×C₄₁	—	4	2	no	—
162	C₂×Q₈×C₄₁	—	4	1	no	—
163	C₂×Q₈×C₄₁	—	4	1	no	—
164	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
165	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
166	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
167	C₂×D₄×C₄₁	—	2	1	no	—
168	C₂×D₄×C₄₁	—	2	2	no	—
169	C₂×D₄×C₄₁	—	2	1	no	—
170	C₂×D₄×C₄₁	—	2	1	no	—
171	C₂×D₄×C₄₁	—	2	1	no	—
172	C₂×D₄×C₄₁	—	2	1	no	—
173	C₂×D₄×C₄₁	—	2	2	no	—
174	C₂×D₄×C₄₁	—	2	1	no	—
175	C₂×D₄×C₄₁	—	2	1	no	—
176	C₂×D₄×C₄₁	—	2	2	no	—
177	C₂×D₄×C₄₁	—	4	1	no	—
178	C₂×D₄×C₄₁	—	4	4	no	—
179	C₂×D₄×C₄₁	—	4	1	no	—
180	C₂×D₄×C₄₁	—	4	1	no	—
181	C₂×D₄×C₄₁	—	8	2	no	—
182	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
183	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₄₁	—	2	1	no	—
185	C₂²×C₄×C₄₁	—	2	2	no	—
186	C₂²×C₄×C₄₁	—	2	2	no	—
187	C₂²×C₄×C₄₁	—	2	1	no	—
188	C₂²×C₄×C₄₁	—	2	2	no	—
189	C₂²×C₄×C₄₁	—	2	1	no	—
190	C₂²×C₄×C₄₁	—	2	1	no	—
191	C₂²×C₄×C₄₁	—	2	1	no	—
192	C₂²×C₄×C₄₁	—	2	1	no	—
193	C₂²×C₄×C₄₁	—	4	4	no	—
194	C₂²×C₄×C₄₁	—	4	4	no	—
195	C₂²×C₄×C₄₁	—	4	1	no	—
196	C₂²×C₄×C₄₁	—	4	1	no	—
197	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
198	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
199	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
200	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
201	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
202	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
203	Q₁₆×C₄₁	—	2	8	no	—
204	Q₁₆×C₄₁	—	2	8	no	—
205	SD₁₆×C₄₁	—	2	8	no	—
206	SD₁₆×C₄₁	—	2	8	no	—
207	SD₁₆×C₄₁	—	4	4	no	—
208	SD₁₆×C₄₁	—	4	4	no	—
209	SD₁₆×C₄₁	—	4	2	no	—
210	SD₁₆×C₄₁	—	4	2	no	—
211	D₈×C₄₁	—	4	2	no	—
212	D₈×C₄₁	—	4	2	no	—
213	M₁₆×C₄₁	—	2	4	no	—
214	M₁₆×C₄₁	—	2	2	no	—
215	M₁₆×C₄₁	—	2	2	no	—
216	M₁₆×C₄₁	—	2	2	no	—
217	M₁₆×C₄₁	—	2	2	no	—
218	M₁₆×C₄₁	—	2	2	no	—
219	M₁₆×C₄₁	—	4	4	no	—
220	M₁₆×C₄₁	C₂×D₄	1	4	yes	—
221	C₂×C₈×C₄₁	—	2	4	no	—
222	C₂×C₈×C₄₁	—	2	2	no	—
223	C₂×C₈×C₄₁	—	2	2	no	—
224	C₂×C₈×C₄₁	—	2	2	no	—
225	C₂×C₈×C₄₁	—	2	2	no	—
226	C₂×C₈×C₄₁	—	2	1	no	—
227	C₂×C₈×C₄₁	—	2	1	no	—
228	C₂×C₈×C₄₁	—	4	4	no	—
229	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
230	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
231	C₄⋊C₄×C₄₁	—	2	1	no	—
232	C₄⋊C₄×C₄₁	—	2	1	no	—
233	C₂²⋊C₄×C₄₁	—	4	1	no	—
234	C₂²⋊C₄×C₄₁	—	4	1	no	—
235	C₂³×C₄₁	—	2	1	no	—
236	C₂³×C₄₁	—	2	2	no	—
237	C₂³×C₄₁	—	2	2	no	—
238	C₂³×C₄₁	—	2	1	no	—
239	C₂³×C₄₁	—	4	1	no	—
240	C₂³×C₄₁	—	4	2	no	—
241	C₂³×C₄₁	—	4	2	no	—
242	C₂³×C₄₁	—	4	1	no	—
243	C₂³×C₄₁	—	4	2	no	—
244	C₂³×C₄₁	—	8	1	no	—
245	C₂³×C₄₁	—	8	1	no	—
246	C₂³×C₄₁	—	8	1	no	—
247	C₂³×C₄₁	#121	1	1	yes	characteristic
248	C₂³×C₄₁	#121	1	1	yes	characteristic
249	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
250	Q₈×C₄₁	—	2	1	no	—
251	Q₈×C₄₁	—	2	4	no	—
252	Q₈×C₄₁	—	2	1	no	—
253	Q₈×C₄₁	—	2	1	no	—
254	Q₈×C₄₁	—	2	4	no	—
255	Q₈×C₄₁	—	4	2	no	—
256	Q₈×C₄₁	—	4	4	no	—
257	Q₈×C₄₁	—	4	4	no	—
258	Q₈×C₄₁	—	4	2	no	—
259	Q₈×C₄₁	—	4	2	no	—
260	Q₈×C₄₁	#120	1	2	yes	—
261	D₄×C₄₁	—	2	4	no	—
262	D₄×C₄₁	—	2	1	no	—
263	D₄×C₄₁	—	2	1	no	—
264	D₄×C₄₁	—	2	1	no	—
265	D₄×C₄₁	—	4	2	no	—
266	D₄×C₄₁	—	4	2	no	—
267	D₄×C₄₁	—	4	2	no	—
268	D₄×C₄₁	—	4	2	no	—
269	D₄×C₄₁	—	4	1	no	—
270	D₄×C₄₁	—	4	1	no	—
271	D₄×C₄₁	—	4	2	no	—
272	D₄×C₄₁	—	8	2	no	—
273	D₄×C₄₁	—	8	2	no	—
274	D₄×C₄₁	#120	1	2	yes	—
275	C₂×C₄×C₄₁	—	2	1	no	—
276	C₂×C₄×C₄₁	—	2	2	no	—
277	C₂×C₄×C₄₁	—	2	1	no	—
278	C₂×C₄×C₄₁	—	2	1	no	—
279	C₂×C₄×C₄₁	—	2	1	no	—
280	C₂×C₄×C₄₁	—	2	1	no	—
281	C₂×C₄×C₄₁	—	2	4	no	—
282	C₂×C₄×C₄₁	—	2	2	no	—
283	C₂×C₄×C₄₁	—	2	2	no	—
284	C₂×C₄×C₄₁	—	2	1	no	—
285	C₂×C₄×C₄₁	—	2	4	no	—
286	C₂×C₄×C₄₁	—	4	2	no	—
287	C₂×C₄×C₄₁	—	4	2	no	—
288	C₂×C₄×C₄₁	—	4	2	no	—
289	C₂×C₄×C₄₁	—	4	4	no	—
290	C₂×C₄×C₄₁	—	4	4	no	—
291	C₂×C₄×C₄₁	—	4	2	no	—
292	C₂×C₄×C₄₁	—	4	2	no	—
293	C₂×C₄×C₄₁	—	4	2	no	—
294	C₂×C₄×C₄₁	—	4	1	no	—
295	C₂×C₄×C₄₁	—	8	1	no	—
296	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
297	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
298	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
300	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
301	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
302	C₈×C₄₁	—	2	4	no	—
303	C₈×C₄₁	—	2	2	no	—
304	C₈×C₄₁	—	2	2	no	—
305	C₈×C₄₁	—	4	1	no	—
306	C₈×C₄₁	—	4	1	no	—
307	C₈×C₄₁	—	4	2	no	—
308	#24401	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
309	C₂²×C₄₁	—	2	1	no	—
310	C₂²×C₄₁	—	2	1	no	—
311	C₂²×C₄₁	—	2	1	no	—
312	C₂²×C₄₁	—	2	1	no	—
313	C₂²×C₄₁	—	2	1	no	—
314	C₂²×C₄₁	—	2	2	no	—
315	C₂²×C₄₁	—	2	1	no	—
316	C₂²×C₄₁	—	2	1	no	—
317	C₂²×C₄₁	—	4	4	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	—	4	1	no	—
323	C₂²×C₄₁	—	8	4	no	—
324	C₂²×C₄₁	—	8	1	no	—
325	C₂²×C₄₁	—	8	1	no	—
326	C₂²×C₄₁	—	8	1	no	—
327	C₂²×C₄₁	—	16	2	no	—
328	C₂²×C₄₁	#471	1	2	yes	complemented
329	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
330	C₂²×C₄₁	#546	1	2	yes	—
331	C₄×C₄₁	—	2	1	no	—
332	C₄×C₄₁	—	2	1	no	—
333	C₄×C₄₁	—	2	2	no	—
334	C₄×C₄₁	—	4	1	no	—
335	C₄×C₄₁	—	4	4	no	—
336	C₄×C₄₁	—	8	1	no	—
337	C₄×C₄₁	C₂×#120	1	1	yes	characteristic
338	C₄×C₄₁	#534	1	1	yes	characteristic
339	C₄×C₄₁	D₄²	1	2	yes	—
340	C₂²×#137	C₂×C₄₁	1	1	yes	characteristic, complemented
341	C₂×#471	C₂×C₄₁	1	2	yes	complemented
342	#3006	C₂×C₄₁	1	1	yes	characteristic
343	C₂×#448	C₂×C₄₁	1	1	yes	characteristic, complemented
344	#2880	C₂×C₄₁	1	1	yes	characteristic, complemented
345	#2052	C₂×C₄₁	1	2	yes	complemented
346	#2019	C₂×C₄₁	1	2	yes	complemented
347	#1908	C₂×C₄₁	1	2	yes	complemented
348	#1900	C₂×C₄₁	1	1	yes	characteristic, complemented
349	#1855	C₂×C₄₁	1	1	yes	characteristic, complemented
350	#1784	C₂×C₄₁	1	1	yes	characteristic, complemented
351	C₂×C₄₁	—	2	2	no	—
352	C₂×C₄₁	—	2	2	no	—
353	C₂×C₄₁	—	4	1	no	—
354	C₂×C₄₁	—	4	2	no	—
355	C₂×C₄₁	—	8	1	no	—
356	C₂×C₄₁	—	16	1	no	—
357	C₂×C₄₁	#2410	1	1	yes	characteristic
358	C₂×C₄₁	C₂×#471	1	1	yes	characteristic
359	C₂×C₄₁	#3069	1	1	yes	characteristic
360	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
361	C₂³×Q₈	C₂²×C₄₁	1	1	yes	characteristic
362	C₂×#137	—	2	1	no	—
363	C₂×#137	—	2	1	no	—
364	C₂×#137	—	2	8	no	—
365	C₂×#137	C₂²×C₄₁	1	2	yes	—
366	C₂×#137	C₂²×C₄₁	1	2	yes	—
367	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	1	yes	characteristic
368	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	1	yes	characteristic
369	C₂²×Q₁₆	—	2	1	no	—
370	C₂²×SD₁₆	—	2	1	no	—
371	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	complemented
372	C₂×#131	C₂²×C₄₁	1	2	yes	—
373	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
374	C₂×#121	C₂²×C₄₁	1	1	yes	characteristic
375	#471	—	2	8	no	—
376	#449	—	2	1	no	—
377	#449	—	2	1	no	—
378	#448	—	2	1	no	—
379	#448	—	2	1	no	—
380	#448	—	2	1	no	—
381	#448	—	2	2	no	—
382	#447	—	2	1	no	—
383	C₂×#108	C₂²×C₄₁	1	2	yes	complemented
384	#417	C₂²×C₄₁	1	1	yes	characteristic
385	#417	C₂²×C₄₁	1	1	yes	characteristic
386	C₂×#103	C₂²×C₄₁	1	2	yes	complemented
387	C₂×#101	C₂²×C₄₁	1	2	yes	complemented
388	C₂×#101	C₂²×C₄₁	1	2	yes	complemented
389	C₂×#100	C₂²×C₄₁	1	2	yes	complemented
390	#408	—	2	2	no	—
391	#408	C₂²×C₄₁	1	2	yes	—
392	#407	—	2	1	no	—
393	#407	C₂²×C₄₁	1	1	yes	characteristic
394	#407	C₂²×C₄₁	1	1	yes	characteristic
395	C₂×C₂²⋊C₈	—	2	1	no	—
396	C₂×C₂²⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
397	#343	C₂²×C₄₁	1	2	yes	complemented
398	#341	C₂²×C₄₁	1	2	yes	complemented
399	C₄₁	#24401	1	1	yes	characteristic, direct product factor, Sylow
400	C₂×#41	—	2	1	no	—
401	C₂×#41	—	2	1	no	—
402	C₂×#41	—	2	4	no	—
403	C₂×#41	—	4	2	no	—
404	C₂×#41	D₄×C₄₁	1	1	yes	characteristic
405	C₂×#41	D₄×C₄₁	1	1	yes	characteristic
406	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
407	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
408	C₂²×Q₈	—	2	1	no	—
409	C₂²×Q₈	—	2	4	no	—
410	C₂²×Q₈	—	2	1	no	—
411	C₂²×Q₈	—	2	1	no	—
412	C₂²×Q₈	—	2	1	no	—
413	C₂²×Q₈	—	2	1	no	—
414	C₂²×Q₈	—	4	2	no	—
415	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
416	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
417	C₂²×D₄	—	2	1	no	—
418	C₂²×D₄	—	2	1	no	—
419	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
420	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
421	C₂³×C₄	—	2	1	no	—
422	C₂³×C₄	—	2	1	no	—
423	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
424	#137	—	2	8	no	—
425	#137	—	2	8	no	—
426	#137	—	2	4	no	—
427	#137	—	2	4	no	—
428	#137	—	2	4	no	—
429	#137	D₄×C₄₁	1	8	yes	complemented
430	AGL(1,ℤ/8)	—	2	4	no	—
431	AGL(1,ℤ/8)	—	4	2	no	—
432	C₂×Q₁₆	—	2	8	no	—
433	C₂×Q₁₆	—	2	2	no	—
434	C₂×Q₁₆	—	2	2	no	—
435	C₂×SD₁₆	—	2	8	no	—
436	C₂×SD₁₆	—	2	2	no	—
437	C₂×SD₁₆	—	2	2	no	—
438	C₂×SD₁₆	—	2	4	no	—
439	C₂×SD₁₆	—	2	1	no	—
440	C₂×SD₁₆	—	2	1	no	—
441	C₂×SD₁₆	—	2	2	no	—
442	C₂×SD₁₆	—	2	2	no	—
443	C₂×SD₁₆	—	4	2	no	—
444	C₂×D₈	—	2	1	no	—
445	C₂×D₈	—	2	1	no	—
446	#131	—	2	4	no	—
447	#131	—	4	2	no	—
448	C₂×M₁₆	—	2	1	no	—
449	C₂×M₁₆	—	2	4	no	—
450	C₂×M₁₆	—	2	2	no	—
451	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
452	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
453	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
454	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
455	C₂²×C₈	—	2	1	no	—
456	C₂²×C₈	—	2	2	no	—
457	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
458	#121	—	2	1	no	—
459	#121	—	2	1	no	—
460	#121	—	2	1	no	—
461	#121	—	2	1	no	—
462	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
463	C₂×C₂²⋊C₄	—	2	1	no	—
464	#108	—	2	4	no	—
465	#103	—	2	1	no	—
466	#103	—	2	1	no	—
467	#103	—	2	2	no	—
468	#103	—	2	2	no	—
469	#102	—	2	1	no	—
470	#102	—	2	1	no	—
471	#101	—	2	4	no	—
472	#101	—	4	2	no	—
473	#100	—	2	4	no	—
474	C₂²⋊C₈	—	2	1	no	—
475	C₂²⋊C₈	—	2	1	no	—
476	C₂²⋊C₈	—	2	1	no	—
477	C₂²⋊C₈	—	2	1	no	—
478	C₂²⋊C₈	—	2	2	no	—
479	C₂²⋊C₈	—	2	2	no	—
480	C₂²⋊C₈	D₄×C₄₁	1	1	yes	characteristic
481	C₂²⋊C₈	D₄×C₄₁	1	1	yes	characteristic
482	C₂²⋊C₈	D₄×C₄₁	1	2	yes	—
483	C₂⁴	—	2	1	no	—
484	C₂⁴	—	2	1	no	—
485	C₂⁴	#41×C₄₁	1	1	yes	characteristic
486	#41	—	2	2	no	—
487	#41	—	2	2	no	—
488	#41	—	2	2	no	—
489	#41	—	2	2	no	—
490	#41	—	2	2	no	—
491	#41	—	4	4	no	—
492	#41	—	4	1	no	—
493	#41	—	4	1	no	—
494	#41	—	4	4	no	—
495	#41	C₂×D₄×C₄₁	1	4	yes	—
496	C₂×Q₈	—	2	1	no	—
497	C₂×Q₈	—	2	2	no	—
498	C₂×Q₈	—	2	4	no	—
499	C₂×Q₈	—	2	4	no	—
500	C₂×Q₈	—	2	2	no	—
501	C₂×Q₈	—	2	2	no	—
502	C₂×Q₈	—	2	2	no	—
503	C₂×Q₈	—	2	2	no	—
504	C₂×Q₈	—	2	2	no	—
505	C₂×Q₈	—	4	4	no	—
506	C₂×Q₈	—	4	1	no	—
507	C₂×Q₈	—	4	4	no	—
508	C₂×Q₈	—	4	4	no	—
509	C₂×Q₈	—	4	2	no	—
510	C₂×Q₈	—	4	2	no	—
511	C₂×Q₈	—	4	1	no	—
512	C₂×Q₈	—	4	1	no	—
513	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
514	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
515	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
516	C₂×D₄	—	2	1	no	—
517	C₂×D₄	—	2	2	no	—
518	C₂×D₄	—	2	1	no	—
519	C₂×D₄	—	2	1	no	—
520	C₂×D₄	—	2	1	no	—
521	C₂×D₄	—	2	1	no	—
522	C₂×D₄	—	2	2	no	—
523	C₂×D₄	—	2	1	no	—
524	C₂×D₄	—	2	1	no	—
525	C₂×D₄	—	2	2	no	—
526	C₂×D₄	—	4	1	no	—
527	C₂×D₄	—	4	4	no	—
528	C₂×D₄	—	4	1	no	—
529	C₂×D₄	—	4	1	no	—
530	C₂×D₄	—	8	2	no	—
531	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
532	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
533	C₂²×C₄	—	2	1	no	—
534	C₂²×C₄	—	2	2	no	—
535	C₂²×C₄	—	2	2	no	—
536	C₂²×C₄	—	2	1	no	—
537	C₂²×C₄	—	2	2	no	—
538	C₂²×C₄	—	2	1	no	—
539	C₂²×C₄	—	2	1	no	—
540	C₂²×C₄	—	2	1	no	—
541	C₂²×C₄	—	2	1	no	—
542	C₂²×C₄	—	4	4	no	—
543	C₂²×C₄	—	4	4	no	—
544	C₂²×C₄	—	4	1	no	—
545	C₂²×C₄	—	4	1	no	—
546	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
547	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
548	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
549	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
550	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
551	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
552	Q₁₆	—	2	8	no	—
553	Q₁₆	—	2	8	no	—
554	SD₁₆	—	2	8	no	—
555	SD₁₆	—	2	8	no	—
556	SD₁₆	—	4	4	no	—
557	SD₁₆	—	4	4	no	—
558	SD₁₆	—	4	2	no	—
559	SD₁₆	—	4	2	no	—
560	D₈	—	4	2	no	—
561	D₈	—	4	2	no	—
562	M₁₆	—	2	4	no	—
563	M₁₆	—	2	2	no	—
564	M₁₆	—	2	2	no	—
565	M₁₆	—	2	2	no	—
566	M₁₆	—	2	2	no	—
567	M₁₆	—	2	2	no	—
568	M₁₆	—	4	4	no	—
569	M₁₆	C₂×D₄×C₄₁	1	4	yes	—
570	C₂×C₈	—	2	4	no	—
571	C₂×C₈	—	2	2	no	—
572	C₂×C₈	—	2	2	no	—
573	C₂×C₈	—	2	2	no	—
574	C₂×C₈	—	2	2	no	—
575	C₂×C₈	—	2	1	no	—
576	C₂×C₈	—	2	1	no	—
577	C₂×C₈	—	4	4	no	—
578	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
579	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
580	C₄⋊C₄	—	2	1	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₂²⋊C₄	—	4	1	no	—
583	C₂²⋊C₄	—	4	1	no	—
584	C₂³	—	2	1	no	—
585	C₂³	—	2	2	no	—
586	C₂³	—	2	2	no	—
587	C₂³	—	2	1	no	—
588	C₂³	—	4	1	no	—
589	C₂³	—	4	2	no	—
590	C₂³	—	4	2	no	—
591	C₂³	—	4	1	no	—
592	C₂³	—	4	2	no	—
593	C₂³	—	8	1	no	—
594	C₂³	—	8	1	no	—
595	C₂³	—	8	1	no	—
596	C₂³	#121×C₄₁	1	1	yes	characteristic
597	C₂³	#121×C₄₁	1	1	yes	characteristic
598	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
599	Q₈	—	2	1	no	—
600	Q₈	—	2	4	no	—
601	Q₈	—	2	1	no	—
602	Q₈	—	2	1	no	—
603	Q₈	—	2	4	no	—
604	Q₈	—	4	2	no	—
605	Q₈	—	4	4	no	—
606	Q₈	—	4	4	no	—
607	Q₈	—	4	2	no	—
608	Q₈	—	4	2	no	—
609	Q₈	#120×C₄₁	1	2	yes	—
610	D₄	—	2	4	no	—
611	D₄	—	2	1	no	—
612	D₄	—	2	1	no	—
613	D₄	—	2	1	no	—
614	D₄	—	4	2	no	—
615	D₄	—	4	2	no	—
616	D₄	—	4	2	no	—
617	D₄	—	4	2	no	—
618	D₄	—	4	1	no	—
619	D₄	—	4	1	no	—
620	D₄	—	4	2	no	—
621	D₄	—	8	2	no	—
622	D₄	—	8	2	no	—
623	D₄	#120×C₄₁	1	2	yes	—
624	C₂×C₄	—	2	1	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	1	no	—
628	C₂×C₄	—	2	1	no	—
629	C₂×C₄	—	2	1	no	—
630	C₂×C₄	—	2	4	no	—
631	C₂×C₄	—	2	2	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	1	no	—
634	C₂×C₄	—	2	4	no	—
635	C₂×C₄	—	4	2	no	—
636	C₂×C₄	—	4	2	no	—
637	C₂×C₄	—	4	2	no	—
638	C₂×C₄	—	4	4	no	—
639	C₂×C₄	—	4	4	no	—
640	C₂×C₄	—	4	2	no	—
641	C₂×C₄	—	4	2	no	—
642	C₂×C₄	—	4	2	no	—
643	C₂×C₄	—	4	1	no	—
644	C₂×C₄	—	8	1	no	—
645	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
646	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
647	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
648	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
649	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
650	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
651	C₈	—	2	4	no	—
652	C₈	—	2	2	no	—
653	C₈	—	2	2	no	—
654	C₈	—	4	1	no	—
655	C₈	—	4	1	no	—
656	C₈	—	4	2	no	—
657	C₂²	—	2	1	no	—
658	C₂²	—	2	1	no	—
659	C₂²	—	2	1	no	—
660	C₂²	—	2	1	no	—
661	C₂²	—	2	1	no	—
662	C₂²	—	2	1	no	—
663	C₂²	—	2	1	no	—
664	C₂²	—	2	2	no	—
665	C₂²	—	4	4	no	—
666	C₂²	—	4	1	no	—
667	C₂²	—	4	1	no	—
668	C₂²	—	4	1	no	—
669	C₂²	—	4	1	no	—
670	C₂²	—	4	1	no	—
671	C₂²	—	8	4	no	—
672	C₂²	—	8	1	no	—
673	C₂²	—	8	1	no	—
674	C₂²	—	8	1	no	—
675	C₂²	—	16	2	no	—
676	C₂²	C₄₁×#471	1	2	yes	complemented
677	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
678	C₂²	C₄₁×#546	1	2	yes	—
679	C₄	—	2	1	no	—
680	C₄	—	2	1	no	—
681	C₄	—	2	2	no	—
682	C₄	—	4	1	no	—
683	C₄	—	4	4	no	—
684	C₄	—	8	1	no	—
685	C₄	C₂×#120×C₄₁	1	1	yes	characteristic
686	C₄	C₄₁×#534	1	1	yes	characteristic
687	C₄	D₄²×C₄₁	1	2	yes	—
688	C₂	—	2	2	no	—
689	C₂	—	2	2	no	—
690	C₂	—	4	1	no	—
691	C₂	—	4	2	no	—
692	C₂	—	8	1	no	—
693	C₂	—	16	1	no	—
694	C₂	C₄₁×#2410	1	1	yes	characteristic
695	C₂	C₂×C₄₁×#471	1	1	yes	characteristic
696	C₂	C₄₁×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#3006	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#137×C₄₁	C₂	semidirect
2. C₂×C₄₁×#471	C₂	semidirect
4. C₂×C₄₁×#448	C₂	semidirect
5. C₄₁×#2880	C₂	semidirect
6. C₄₁×#2052	C₂	semidirect
7. C₄₁×#2019	C₂	semidirect
8. C₄₁×#1908	C₂	semidirect
9. C₄₁×#1900	C₂	semidirect
10. C₄₁×#1855	C₂	semidirect
11. C₄₁×#1784	C₂	semidirect
23. C₂²×SD₁₆×C₄₁	C₂²	semidirect
35. C₂×#108×C₄₁	C₂²	semidirect
38. C₂×#103×C₄₁	C₂²	semidirect
39. C₂×#101×C₄₁	C₂²	semidirect
40. C₂×#101×C₄₁	C₂²	semidirect
41. C₂×#100×C₄₁	C₂²	semidirect
49. C₄₁×#343	C₂²	semidirect
50. C₄₁×#341	C₂²	semidirect
80. #137×C₄₁	D₄	semidirect
308. #24401	C₄₁	direct (N × H)
328. C₂²×C₄₁	#471	semidirect
340. C₂²×#137	C₂×C₄₁	semidirect
341. C₂×#471	C₂×C₄₁	semidirect
343. C₂×#448	C₂×C₄₁	semidirect
344. #2880	C₂×C₄₁	semidirect
345. #2052	C₂×C₄₁	semidirect
346. #2019	C₂×C₄₁	semidirect
347. #1908	C₂×C₄₁	semidirect
348. #1900	C₂×C₄₁	semidirect
349. #1855	C₂×C₄₁	semidirect
350. #1784	C₂×C₄₁	semidirect
371. C₂²×SD₁₆	C₂²×C₄₁	semidirect
383. C₂×#108	C₂²×C₄₁	semidirect
386. C₂×#103	C₂²×C₄₁	semidirect
387. C₂×#101	C₂²×C₄₁	semidirect
388. C₂×#101	C₂²×C₄₁	semidirect
389. C₂×#100	C₂²×C₄₁	semidirect
397. #343	C₂²×C₄₁	semidirect
398. #341	C₂²×C₄₁	semidirect
399. C₄₁	#24401	direct (N × H)
429. #137	D₄×C₄₁	semidirect
676. C₂²	C₄₁×#471	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#137×C₄₁	2
C₂×C₄₁×#471	2
C₄₁×#3006	2
C₂×C₄₁×#448	2
C₄₁×#2880	2
C₄₁×#2052	2
C₄₁×#2019	2
C₄₁×#1908	2
C₄₁×#1900	2
C₄₁×#1855	2
C₄₁×#1784	2
#24401	41

Subgroups — C41×#24401 #23800602

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24401 #23800602