Subgroups — C₄₁×#25431 #23801632

Order 10496.

946 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#471	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3057	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#3043	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#3042	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#2949	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#2046	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#2046	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#2005	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#2002	C₂	1	1	yes	maximal, characteristic
11	C₄₁×#1977	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₁×#1945	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄₁×#1887	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄₁×#1886	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₁×#1764	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#137×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	1	yes	characteristic
18	C₂²×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
19	C₂²×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
20	C₂²×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
21	C₂²×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×#131×C₄₁	C₂²	1	1	yes	characteristic
23	C₄₁×#536	C₂²	1	1	yes	characteristic
24	C₄₁×#535	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#519	C₂²	1	1	yes	characteristic
28	C₄₁×#518	C₂²	1	1	yes	characteristic
29	C₄₁×#492	—	2	4	no	—
30	C₄₁×#471	—	2	4	no	—
31	C₄₁×#468	—	2	2	no	—
32	C₄₁×#465	—	2	2	no	—
33	C₄₁×#464	—	2	2	no	—
34	C₄₁×#463	—	2	2	no	—
35	C₄₁×#461	—	2	2	no	—
36	C₄₁×#460	—	2	2	no	—
37	C₄₁×#440	—	2	2	no	—
38	C₄×SD₁₆×C₄₁	—	2	2	no	—
39	C₂×#108×C₄₁	C₂²	1	1	yes	characteristic
40	C₄₁×#427	C₂²	1	1	yes	characteristic
41	C₄₁×#424	C₂²	1	1	yes	characteristic
42	C₄₁×#424	C₂²	1	1	yes	characteristic
43	C₂×#104×C₄₁	C₂²	1	1	yes	characteristic
44	C₄₁×#419	C₂²	1	1	yes	characteristic
45	C₄₁×#419	C₂²	1	1	yes	characteristic
46	C₄₁×#419	C₂²	1	1	yes	characteristic
47	C₄₁×#418	C₂²	1	1	yes	characteristic
48	C₄₁×#418	C₂²	1	1	yes	characteristic
49	C₄₁×#418	C₂²	1	1	yes	characteristic
50	C₄₁×#416	C₂²	1	1	yes	characteristic
51	C₂×#101×C₄₁	C₂²	1	1	yes	characteristic
52	C₂×#101×C₄₁	C₂²	1	1	yes	characteristic
53	C₂×#100×C₄₁	C₂²	1	1	yes	characteristic
54	C₂×#100×C₄₁	C₂²	1	1	yes	characteristic
55	C₄₁×#405	C₂²	1	1	yes	characteristic
56	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
57	C₄₁×#341	C₂²	1	1	yes	characteristic
58	C₄₁×#339	C₂²	1	1	yes	characteristic
59	C₄₁×#339	C₂²	1	1	yes	characteristic
60	C₄₁×#339	C₂²	1	1	yes	characteristic
61	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₄₁	—	4	1	no	—
65	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
67	#137×C₄₁	—	2	2	no	—
68	#137×C₄₁	—	4	1	no	—
69	AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
70	AGL(1,ℤ/8)×C₄₁	—	4	1	no	—
71	C₂×Q₁₆×C₄₁	—	2	1	no	—
72	C₂×SD₁₆×C₄₁	—	2	2	no	—
73	C₂×SD₁₆×C₄₁	—	2	1	no	—
74	C₂×SD₁₆×C₄₁	—	2	2	no	—
75	C₂×SD₁₆×C₄₁	—	2	1	no	—
76	C₂×SD₁₆×C₄₁	—	2	1	no	—
77	C₂×SD₁₆×C₄₁	—	2	1	no	—
78	C₂×SD₁₆×C₄₁	—	2	2	no	—
79	C₂×SD₁₆×C₄₁	—	2	2	no	—
80	C₂×SD₁₆×C₄₁	—	2	1	no	—
81	C₂×SD₁₆×C₄₁	—	2	1	no	—
82	C₂×SD₁₆×C₄₁	—	2	2	no	—
83	C₂×SD₁₆×C₄₁	—	2	2	no	—
84	C₂×SD₁₆×C₄₁	—	2	2	no	—
85	C₂×SD₁₆×C₄₁	—	4	1	no	—
86	C₂×SD₁₆×C₄₁	—	4	1	no	—
87	C₂×SD₁₆×C₄₁	D₄	1	4	yes	complemented
88	C₂×D₈×C₄₁	—	2	1	no	—
89	#131×C₄₁	—	2	2	no	—
90	#131×C₄₁	—	2	2	no	—
91	C₂×M₁₆×C₄₁	—	2	1	no	—
92	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
93	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
94	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
95	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
96	C₂²×C₈×C₄₁	—	2	1	no	—
97	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
98	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
99	C₄⋊Q₈×C₄₁	—	2	2	no	—
100	C₄⋊Q₈×C₄₁	—	2	1	no	—
101	C₄⋊Q₈×C₄₁	—	2	2	no	—
102	#127×C₄₁	—	2	2	no	—
103	#127×C₄₁	—	2	1	no	—
104	#127×C₄₁	—	2	2	no	—
105	#124×C₄₁	—	2	1	no	—
106	#124×C₄₁	—	2	1	no	—
107	#123×C₄₁	—	4	1	no	—
108	C₂²⋊Q₈×C₄₁	—	2	2	no	—
109	#121×C₄₁	—	2	2	no	—
110	#120×C₄₁	—	4	1	no	—
111	C₄×Q₈×C₄₁	—	2	2	no	—
112	C₄×Q₈×C₄₁	—	2	2	no	—
113	C₄×D₄×C₄₁	—	2	2	no	—
114	C₄×D₄×C₄₁	—	2	2	no	—
115	#117×C₄₁	—	4	1	no	—
116	#117×C₄₁	C₂³	1	1	yes	characteristic
117	#117×C₄₁	C₂³	1	1	yes	characteristic
118	#117×C₄₁	C₂³	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
120	C₂×C₂²⋊C₄×C₄₁	—	4	1	no	—
121	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
122	#108×C₄₁	—	2	2	no	—
123	#107×C₄₁	D₄	1	2	yes	complemented
124	#106×C₄₁	D₄	1	2	yes	complemented
125	C₄⋊C₈×C₄₁	—	2	2	no	—
126	C₄⋊C₈×C₄₁	—	2	2	no	—
127	#104×C₄₁	—	2	2	no	—
128	#103×C₄₁	—	2	2	no	—
129	#103×C₄₁	—	2	2	no	—
130	#103×C₄₁	—	2	2	no	—
131	#103×C₄₁	D₄	1	2	yes	complemented
132	#102×C₄₁	—	2	2	no	—
133	#102×C₄₁	—	2	2	no	—
134	#102×C₄₁	—	2	2	no	—
135	#102×C₄₁	D₄	1	2	yes	complemented
136	#101×C₄₁	—	2	2	no	—
137	#101×C₄₁	—	4	1	no	—
138	#100×C₄₁	—	2	2	no	—
139	#100×C₄₁	—	4	1	no	—
140	#97×C₄₁	—	2	1	no	—
141	C₄×C₈×C₄₁	—	2	2	no	—
142	C₄×C₈×C₄₁	—	2	1	no	—
143	C₂⁴×C₄₁	—	2	1	no	—
144	C₂⁴×C₄₁	—	2	1	no	—
145	#41×C₄₁	—	2	1	no	—
146	#41×C₄₁	—	2	1	no	—
147	#41×C₄₁	—	2	1	no	—
148	#41×C₄₁	—	2	1	no	—
149	C₂×Q₈×C₄₁	—	2	1	no	—
150	C₂×Q₈×C₄₁	—	2	1	no	—
151	C₂×Q₈×C₄₁	—	2	1	no	—
152	C₂×Q₈×C₄₁	—	2	1	no	—
153	C₂×Q₈×C₄₁	—	2	1	no	—
154	C₂×Q₈×C₄₁	—	2	1	no	—
155	C₂×Q₈×C₄₁	—	2	1	no	—
156	C₂×Q₈×C₄₁	—	2	1	no	—
157	C₂×Q₈×C₄₁	—	2	1	no	—
158	C₂×Q₈×C₄₁	—	2	1	no	—
159	C₂×Q₈×C₄₁	—	2	1	no	—
160	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
161	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
162	C₂×D₄×C₄₁	—	2	1	no	—
163	C₂×D₄×C₄₁	—	2	1	no	—
164	C₂×D₄×C₄₁	—	2	1	no	—
165	C₂×D₄×C₄₁	—	2	1	no	—
166	C₂×D₄×C₄₁	—	2	1	no	—
167	C₂×D₄×C₄₁	—	2	1	no	—
168	C₂×D₄×C₄₁	—	2	1	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	1	no	—
174	C₂×D₄×C₄₁	—	4	4	no	—
175	C₂×D₄×C₄₁	—	4	2	no	—
176	C₂×D₄×C₄₁	—	8	4	no	—
177	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
178	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₄₁	—	2	1	no	—
180	C₂²×C₄×C₄₁	—	2	1	no	—
181	C₂²×C₄×C₄₁	—	2	1	no	—
182	C₂²×C₄×C₄₁	—	2	1	no	—
183	C₂²×C₄×C₄₁	—	4	2	no	—
184	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
185	Q₁₆×C₄₁	—	4	2	no	—
186	SD₁₆×C₄₁	—	2	4	no	—
187	SD₁₆×C₄₁	—	2	4	no	—
188	SD₁₆×C₄₁	—	4	4	no	—
189	SD₁₆×C₄₁	—	4	4	no	—
190	SD₁₆×C₄₁	—	4	2	no	—
191	SD₁₆×C₄₁	—	4	2	no	—
192	SD₁₆×C₄₁	—	4	2	no	—
193	SD₁₆×C₄₁	—	4	2	no	—
194	D₈×C₄₁	—	4	2	no	—
195	M₁₆×C₄₁	—	2	1	no	—
196	M₁₆×C₄₁	—	2	1	no	—
197	M₁₆×C₄₁	—	2	1	no	—
198	M₁₆×C₄₁	—	2	1	no	—
199	M₁₆×C₄₁	—	2	1	no	—
200	M₁₆×C₄₁	—	2	1	no	—
201	M₁₆×C₄₁	—	2	1	no	—
202	M₁₆×C₄₁	—	2	1	no	—
203	M₁₆×C₄₁	—	4	2	no	—
204	C₂×C₈×C₄₁	—	2	1	no	—
205	C₂×C₈×C₄₁	—	2	1	no	—
206	C₂×C₈×C₄₁	—	2	1	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	—	4	2	no	—
211	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
212	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
213	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
214	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
215	C₂×C₈×C₄₁	#41	1	2	yes	—
216	C₄⋊C₄×C₄₁	—	4	4	no	—
217	C₄⋊C₄×C₄₁	—	4	2	no	—
218	C₄⋊C₄×C₄₁	—	4	2	no	—
219	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
220	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
221	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
222	C₂²⋊C₄×C₄₁	—	2	1	no	—
223	C₂²⋊C₄×C₄₁	—	2	1	no	—
224	C₂²⋊C₄×C₄₁	—	2	1	no	—
225	C₂²⋊C₄×C₄₁	—	4	2	no	—
226	C₂²⋊C₄×C₄₁	—	4	2	no	—
227	C₂²⋊C₄×C₄₁	—	4	2	no	—
228	C₄²×C₄₁	—	2	1	no	—
229	C₄²×C₄₁	—	2	1	no	—
230	C₄²×C₄₁	—	2	1	no	—
231	C₄²×C₄₁	—	2	1	no	—
232	C₄²×C₄₁	—	4	2	no	—
233	C₄²×C₄₁	#41	1	2	yes	—
234	C₂³×C₄₁	—	2	1	no	—
235	C₂³×C₄₁	—	2	2	no	—
236	C₂³×C₄₁	—	4	1	no	—
237	C₂³×C₄₁	—	4	1	no	—
238	C₂³×C₄₁	—	4	1	no	—
239	C₂³×C₄₁	—	4	1	no	—
240	C₂³×C₄₁	—	4	1	no	—
241	C₂³×C₄₁	—	8	2	no	—
242	C₂³×C₄₁	—	8	2	no	—
243	C₂³×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
244	Q₈×C₄₁	—	2	2	no	—
245	Q₈×C₄₁	—	2	1	no	—
246	Q₈×C₄₁	—	2	2	no	—
247	Q₈×C₄₁	—	2	1	no	—
248	Q₈×C₄₁	—	4	1	no	—
249	Q₈×C₄₁	—	4	1	no	—
250	Q₈×C₄₁	—	4	1	no	—
251	Q₈×C₄₁	—	4	1	no	—
252	Q₈×C₄₁	—	4	1	no	—
253	Q₈×C₄₁	—	4	1	no	—
254	D₄×C₄₁	—	2	2	no	—
255	D₄×C₄₁	—	2	1	no	—
256	D₄×C₄₁	—	2	1	no	—
257	D₄×C₄₁	—	2	2	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	—	8	8	no	—
267	D₄×C₄₁	—	16	2	no	—
268	C₂×C₄×C₄₁	—	2	4	no	—
269	C₂×C₄×C₄₁	—	2	2	no	—
270	C₂×C₄×C₄₁	—	2	2	no	—
271	C₂×C₄×C₄₁	—	2	2	no	—
272	C₂×C₄×C₄₁	—	2	2	no	—
273	C₂×C₄×C₄₁	—	2	1	no	—
274	C₂×C₄×C₄₁	—	2	2	no	—
275	C₂×C₄×C₄₁	—	4	1	no	—
276	C₂×C₄×C₄₁	—	4	1	no	—
277	C₂×C₄×C₄₁	—	4	1	no	—
278	C₂×C₄×C₄₁	—	4	1	no	—
279	C₂×C₄×C₄₁	—	4	1	no	—
280	C₂×C₄×C₄₁	—	4	1	no	—
281	C₂×C₄×C₄₁	—	4	1	no	—
282	C₂×C₄×C₄₁	—	16	2	no	—
283	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
285	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
286	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
287	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
288	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
289	C₈×C₄₁	—	2	2	no	—
290	C₈×C₄₁	—	4	1	no	—
291	C₈×C₄₁	—	4	1	no	—
292	C₈×C₄₁	—	4	1	no	—
293	C₈×C₄₁	—	4	1	no	—
294	C₈×C₄₁	#121	1	2	yes	—
295	C₈×C₄₁	#121	1	2	yes	—
296	#25431	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
297	C₂²×C₄₁	—	2	2	no	—
298	C₂²×C₄₁	—	4	1	no	—
299	C₂²×C₄₁	—	4	2	no	—
300	C₂²×C₄₁	—	4	2	no	—
301	C₂²×C₄₁	—	8	2	no	—
302	C₂²×C₄₁	—	8	2	no	—
303	C₂²×C₄₁	—	8	2	no	—
304	C₂²×C₄₁	—	8	1	no	—
305	C₂²×C₄₁	—	8	2	no	—
306	C₂²×C₄₁	—	16	1	no	—
307	C₂²×C₄₁	—	16	1	no	—
308	C₂²×C₄₁	#540	1	1	yes	characteristic
309	C₂²×C₄₁	D₄²	1	1	yes	characteristic
310	C₂²×C₄₁	#546	1	1	yes	characteristic
311	C₄×C₄₁	—	2	4	no	—
312	C₄×C₄₁	—	4	1	no	—
313	C₄×C₄₁	—	4	2	no	—
314	C₄×C₄₁	—	4	2	no	—
315	C₄×C₄₁	—	4	2	no	—
316	C₄×C₄₁	—	4	2	no	—
317	C₄×C₄₁	—	8	1	no	—
318	C₄×C₄₁	C₂×#121	1	1	yes	characteristic
319	C₄×C₄₁	#532	1	1	yes	characteristic
320	C₄×C₄₁	D₄²	1	1	yes	characteristic
321	C₄×C₄₁	#550	1	1	yes	characteristic
322	C₂×#492	C₂×C₄₁	1	1	yes	characteristic, complemented
323	C₂×#471	C₂×C₄₁	1	1	yes	characteristic, complemented
324	#3057	C₂×C₄₁	1	1	yes	characteristic, complemented
325	#3043	C₂×C₄₁	1	1	yes	characteristic, complemented
326	#3042	C₂×C₄₁	1	1	yes	characteristic, complemented
327	#2949	C₂×C₄₁	1	1	yes	characteristic, complemented
328	#2046	C₂×C₄₁	1	1	yes	characteristic, complemented
329	#2046	C₂×C₄₁	1	1	yes	characteristic, complemented
330	#2005	C₂×C₄₁	1	1	yes	characteristic, complemented
331	#2002	C₂×C₄₁	1	1	yes	characteristic
332	#1977	C₂×C₄₁	1	1	yes	characteristic, complemented
333	#1945	C₂×C₄₁	1	1	yes	characteristic, complemented
334	#1887	C₂×C₄₁	1	1	yes	characteristic, complemented
335	#1886	C₂×C₄₁	1	1	yes	characteristic, complemented
336	#1764	C₂×C₄₁	1	1	yes	characteristic, complemented
337	C₂×C₄₁	—	2	1	no	—
338	C₂×C₄₁	—	2	1	no	—
339	C₂×C₄₁	—	8	1	no	—
340	C₂×C₄₁	—	8	2	no	—
341	C₂×C₄₁	—	16	1	no	—
342	C₂×C₄₁	#2428	1	1	yes	characteristic
343	C₂×C₄₁	#3291	1	2	yes	—
344	C₂×#137	C₂²×C₄₁	1	1	yes	characteristic
345	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	1	yes	characteristic
346	C₂²×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
347	C₂²×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
348	C₂²×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
349	C₂²×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
350	C₂×#131	C₂²×C₄₁	1	1	yes	characteristic
351	#536	C₂²×C₄₁	1	1	yes	characteristic
352	#535	C₂²×C₄₁	1	1	yes	characteristic
353	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
354	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
355	#519	C₂²×C₄₁	1	1	yes	characteristic
356	#518	C₂²×C₄₁	1	1	yes	characteristic
357	#492	—	2	4	no	—
358	#471	—	2	4	no	—
359	#468	—	2	2	no	—
360	#465	—	2	2	no	—
361	#464	—	2	2	no	—
362	#463	—	2	2	no	—
363	#461	—	2	2	no	—
364	#460	—	2	2	no	—
365	#440	—	2	2	no	—
366	C₄×SD₁₆	—	2	2	no	—
367	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic
368	#427	C₂²×C₄₁	1	1	yes	characteristic
369	#424	C₂²×C₄₁	1	1	yes	characteristic
370	#424	C₂²×C₄₁	1	1	yes	characteristic
371	C₂×#104	C₂²×C₄₁	1	1	yes	characteristic
372	#419	C₂²×C₄₁	1	1	yes	characteristic
373	#419	C₂²×C₄₁	1	1	yes	characteristic
374	#419	C₂²×C₄₁	1	1	yes	characteristic
375	#418	C₂²×C₄₁	1	1	yes	characteristic
376	#418	C₂²×C₄₁	1	1	yes	characteristic
377	#418	C₂²×C₄₁	1	1	yes	characteristic
378	#416	C₂²×C₄₁	1	1	yes	characteristic
379	C₂×#101	C₂²×C₄₁	1	1	yes	characteristic
380	C₂×#101	C₂²×C₄₁	1	1	yes	characteristic
381	C₂×#100	C₂²×C₄₁	1	1	yes	characteristic
382	C₂×#100	C₂²×C₄₁	1	1	yes	characteristic
383	#405	C₂²×C₄₁	1	1	yes	characteristic
384	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
385	#341	C₂²×C₄₁	1	1	yes	characteristic
386	#339	C₂²×C₄₁	1	1	yes	characteristic
387	#339	C₂²×C₄₁	1	1	yes	characteristic
388	#339	C₂²×C₄₁	1	1	yes	characteristic
389	C₄₁	#25431	1	1	yes	characteristic, direct product factor, Sylow
390	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
391	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
392	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
393	C₂²×D₄	—	4	1	no	—
394	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
395	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
396	#137	—	2	2	no	—
397	#137	—	4	1	no	—
398	AGL(1,ℤ/8)	—	2	2	no	—
399	AGL(1,ℤ/8)	—	4	1	no	—
400	C₂×Q₁₆	—	2	1	no	—
401	C₂×SD₁₆	—	2	2	no	—
402	C₂×SD₁₆	—	2	1	no	—
403	C₂×SD₁₆	—	2	2	no	—
404	C₂×SD₁₆	—	2	1	no	—
405	C₂×SD₁₆	—	2	1	no	—
406	C₂×SD₁₆	—	2	1	no	—
407	C₂×SD₁₆	—	2	2	no	—
408	C₂×SD₁₆	—	2	2	no	—
409	C₂×SD₁₆	—	2	1	no	—
410	C₂×SD₁₆	—	2	1	no	—
411	C₂×SD₁₆	—	2	2	no	—
412	C₂×SD₁₆	—	2	2	no	—
413	C₂×SD₁₆	—	2	2	no	—
414	C₂×SD₁₆	—	4	1	no	—
415	C₂×SD₁₆	—	4	1	no	—
416	C₂×SD₁₆	D₄×C₄₁	1	4	yes	complemented
417	C₂×D₈	—	2	1	no	—
418	#131	—	2	2	no	—
419	#131	—	2	2	no	—
420	C₂×M₁₆	—	2	1	no	—
421	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
422	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
423	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
424	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
425	C₂²×C₈	—	2	1	no	—
426	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
427	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
428	C₄⋊Q₈	—	2	2	no	—
429	C₄⋊Q₈	—	2	1	no	—
430	C₄⋊Q₈	—	2	2	no	—
431	#127	—	2	2	no	—
432	#127	—	2	1	no	—
433	#127	—	2	2	no	—
434	#124	—	2	1	no	—
435	#124	—	2	1	no	—
436	#123	—	4	1	no	—
437	C₂²⋊Q₈	—	2	2	no	—
438	#121	—	2	2	no	—
439	#120	—	4	1	no	—
440	C₄×Q₈	—	2	2	no	—
441	C₄×Q₈	—	2	2	no	—
442	C₄×D₄	—	2	2	no	—
443	C₄×D₄	—	2	2	no	—
444	#117	—	4	1	no	—
445	#117	C₂³×C₄₁	1	1	yes	characteristic
446	#117	C₂³×C₄₁	1	1	yes	characteristic
447	#117	C₂³×C₄₁	1	1	yes	characteristic
448	C₂×C₄⋊C₄	—	4	1	no	—
449	C₂×C₂²⋊C₄	—	4	1	no	—
450	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
451	#108	—	2	2	no	—
452	#107	D₄×C₄₁	1	2	yes	complemented
453	#106	D₄×C₄₁	1	2	yes	complemented
454	C₄⋊C₈	—	2	2	no	—
455	C₄⋊C₈	—	2	2	no	—
456	#104	—	2	2	no	—
457	#103	—	2	2	no	—
458	#103	—	2	2	no	—
459	#103	—	2	2	no	—
460	#103	D₄×C₄₁	1	2	yes	complemented
461	#102	—	2	2	no	—
462	#102	—	2	2	no	—
463	#102	—	2	2	no	—
464	#102	D₄×C₄₁	1	2	yes	complemented
465	#101	—	2	2	no	—
466	#101	—	4	1	no	—
467	#100	—	2	2	no	—
468	#100	—	4	1	no	—
469	#97	—	2	1	no	—
470	C₄×C₈	—	2	2	no	—
471	C₄×C₈	—	2	1	no	—
472	C₂⁴	—	2	1	no	—
473	C₂⁴	—	2	1	no	—
474	#41	—	2	1	no	—
475	#41	—	2	1	no	—
476	#41	—	2	1	no	—
477	#41	—	2	1	no	—
478	C₂×Q₈	—	2	1	no	—
479	C₂×Q₈	—	2	1	no	—
480	C₂×Q₈	—	2	1	no	—
481	C₂×Q₈	—	2	1	no	—
482	C₂×Q₈	—	2	1	no	—
483	C₂×Q₈	—	2	1	no	—
484	C₂×Q₈	—	2	1	no	—
485	C₂×Q₈	—	2	1	no	—
486	C₂×Q₈	—	2	1	no	—
487	C₂×Q₈	—	2	1	no	—
488	C₂×Q₈	—	2	1	no	—
489	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
490	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
491	C₂×D₄	—	2	1	no	—
492	C₂×D₄	—	2	1	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	—	2	1	no	—
495	C₂×D₄	—	2	1	no	—
496	C₂×D₄	—	2	1	no	—
497	C₂×D₄	—	2	1	no	—
498	C₂×D₄	—	2	1	no	—
499	C₂×D₄	—	2	1	no	—
500	C₂×D₄	—	2	1	no	—
501	C₂×D₄	—	2	1	no	—
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	4	4	no	—
504	C₂×D₄	—	4	2	no	—
505	C₂×D₄	—	8	4	no	—
506	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
507	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
508	C₂²×C₄	—	2	1	no	—
509	C₂²×C₄	—	2	1	no	—
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	—	2	1	no	—
512	C₂²×C₄	—	4	2	no	—
513	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
514	Q₁₆	—	4	2	no	—
515	SD₁₆	—	2	4	no	—
516	SD₁₆	—	2	4	no	—
517	SD₁₆	—	4	4	no	—
518	SD₁₆	—	4	4	no	—
519	SD₁₆	—	4	2	no	—
520	SD₁₆	—	4	2	no	—
521	SD₁₆	—	4	2	no	—
522	SD₁₆	—	4	2	no	—
523	D₈	—	4	2	no	—
524	M₁₆	—	2	1	no	—
525	M₁₆	—	2	1	no	—
526	M₁₆	—	2	1	no	—
527	M₁₆	—	2	1	no	—
528	M₁₆	—	2	1	no	—
529	M₁₆	—	2	1	no	—
530	M₁₆	—	2	1	no	—
531	M₁₆	—	2	1	no	—
532	M₁₆	—	4	2	no	—
533	C₂×C₈	—	2	1	no	—
534	C₂×C₈	—	2	1	no	—
535	C₂×C₈	—	2	1	no	—
536	C₂×C₈	—	2	1	no	—
537	C₂×C₈	—	2	1	no	—
538	C₂×C₈	—	2	1	no	—
539	C₂×C₈	—	4	2	no	—
540	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
541	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
542	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
543	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
544	C₂×C₈	#41×C₄₁	1	2	yes	—
545	C₄⋊C₄	—	4	4	no	—
546	C₄⋊C₄	—	4	2	no	—
547	C₄⋊C₄	—	4	2	no	—
548	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
549	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
550	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
551	C₂²⋊C₄	—	2	1	no	—
552	C₂²⋊C₄	—	2	1	no	—
553	C₂²⋊C₄	—	2	1	no	—
554	C₂²⋊C₄	—	4	2	no	—
555	C₂²⋊C₄	—	4	2	no	—
556	C₂²⋊C₄	—	4	2	no	—
557	C₄²	—	2	1	no	—
558	C₄²	—	2	1	no	—
559	C₄²	—	2	1	no	—
560	C₄²	—	2	1	no	—
561	C₄²	—	4	2	no	—
562	C₄²	#41×C₄₁	1	2	yes	—
563	C₂³	—	2	1	no	—
564	C₂³	—	2	2	no	—
565	C₂³	—	4	1	no	—
566	C₂³	—	4	1	no	—
567	C₂³	—	4	1	no	—
568	C₂³	—	4	1	no	—
569	C₂³	—	4	1	no	—
570	C₂³	—	8	2	no	—
571	C₂³	—	8	2	no	—
572	C₂³	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
573	Q₈	—	2	2	no	—
574	Q₈	—	2	1	no	—
575	Q₈	—	2	2	no	—
576	Q₈	—	2	1	no	—
577	Q₈	—	4	1	no	—
578	Q₈	—	4	1	no	—
579	Q₈	—	4	1	no	—
580	Q₈	—	4	1	no	—
581	Q₈	—	4	1	no	—
582	Q₈	—	4	1	no	—
583	D₄	—	2	2	no	—
584	D₄	—	2	1	no	—
585	D₄	—	2	1	no	—
586	D₄	—	2	2	no	—
587	D₄	—	4	1	no	—
588	D₄	—	4	1	no	—
589	D₄	—	4	1	no	—
590	D₄	—	4	1	no	—
591	D₄	—	4	1	no	—
592	D₄	—	4	1	no	—
593	D₄	—	4	1	no	—
594	D₄	—	4	1	no	—
595	D₄	—	8	8	no	—
596	D₄	—	16	2	no	—
597	C₂×C₄	—	2	4	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	—	16	2	no	—
612	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
613	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
614	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
615	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
616	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
617	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
618	C₈	—	2	2	no	—
619	C₈	—	4	1	no	—
620	C₈	—	4	1	no	—
621	C₈	—	4	1	no	—
622	C₈	—	4	1	no	—
623	C₈	#121×C₄₁	1	2	yes	—
624	C₈	#121×C₄₁	1	2	yes	—
625	C₂²	—	2	2	no	—
626	C₂²	—	4	1	no	—
627	C₂²	—	4	2	no	—
628	C₂²	—	4	2	no	—
629	C₂²	—	8	2	no	—
630	C₂²	—	8	2	no	—
631	C₂²	—	8	2	no	—
632	C₂²	—	8	1	no	—
633	C₂²	—	8	2	no	—
634	C₂²	—	16	1	no	—
635	C₂²	—	16	1	no	—
636	C₂²	C₄₁×#540	1	1	yes	characteristic
637	C₂²	D₄²×C₄₁	1	1	yes	characteristic
638	C₂²	C₄₁×#546	1	1	yes	characteristic
639	C₄	—	2	4	no	—
640	C₄	—	4	1	no	—
641	C₄	—	4	2	no	—
642	C₄	—	4	2	no	—
643	C₄	—	4	2	no	—
644	C₄	—	4	2	no	—
645	C₄	—	8	1	no	—
646	C₄	C₂×#121×C₄₁	1	1	yes	characteristic
647	C₄	C₄₁×#532	1	1	yes	characteristic
648	C₄	D₄²×C₄₁	1	1	yes	characteristic
649	C₄	C₄₁×#550	1	1	yes	characteristic
650	C₂	—	2	1	no	—
651	C₂	—	2	1	no	—
652	C₂	—	8	1	no	—
653	C₂	—	8	2	no	—
654	C₂	—	16	1	no	—
655	C₂	C₄₁×#2428	1	1	yes	characteristic
656	C₂	C₄₁×#3291	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#2002	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#492	C₂	semidirect
2. C₂×C₄₁×#471	C₂	semidirect
3. C₄₁×#3057	C₂	semidirect
4. C₄₁×#3043	C₂	semidirect
5. C₄₁×#3042	C₂	semidirect
6. C₄₁×#2949	C₂	semidirect
7. C₄₁×#2046	C₂	semidirect
8. C₄₁×#2046	C₂	semidirect
9. C₄₁×#2005	C₂	semidirect
11. C₄₁×#1977	C₂	semidirect
12. C₄₁×#1945	C₂	semidirect
13. C₄₁×#1887	C₂	semidirect
14. C₄₁×#1886	C₂	semidirect
15. C₄₁×#1764	C₂	semidirect
87. C₂×SD₁₆×C₄₁	D₄	semidirect
123. #107×C₄₁	D₄	semidirect
124. #106×C₄₁	D₄	semidirect
131. #103×C₄₁	D₄	semidirect
135. #102×C₄₁	D₄	semidirect
296. #25431	C₄₁	direct (N × H)
322. C₂×#492	C₂×C₄₁	semidirect
323. C₂×#471	C₂×C₄₁	semidirect
324. #3057	C₂×C₄₁	semidirect
325. #3043	C₂×C₄₁	semidirect
326. #3042	C₂×C₄₁	semidirect
327. #2949	C₂×C₄₁	semidirect
328. #2046	C₂×C₄₁	semidirect
329. #2046	C₂×C₄₁	semidirect
330. #2005	C₂×C₄₁	semidirect
332. #1977	C₂×C₄₁	semidirect
333. #1945	C₂×C₄₁	semidirect
334. #1887	C₂×C₄₁	semidirect
335. #1886	C₂×C₄₁	semidirect
336. #1764	C₂×C₄₁	semidirect
389. C₄₁	#25431	direct (N × H)
416. C₂×SD₁₆	D₄×C₄₁	semidirect
452. #107	D₄×C₄₁	semidirect
453. #106	D₄×C₄₁	semidirect
460. #103	D₄×C₄₁	semidirect
464. #102	D₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#492	2
C₂×C₄₁×#471	2
C₄₁×#3057	2
C₄₁×#3043	2
C₄₁×#3042	2
C₄₁×#2949	2
C₄₁×#2046	2
C₄₁×#2005	2
C₄₁×#2002	2
C₄₁×#1977	2
C₄₁×#1945	2
C₄₁×#1887	2
C₄₁×#1886	2
C₄₁×#1764	2
#25431	41

Subgroups — C41×#25431 #23801632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#25431 #23801632