Subgroups — C₄₁×#61585 #23837786

Order 10496.

2378 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3496	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#547	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3330	C₂	1	8	yes	maximal, complemented
4	C₄₁×#3272	C₂	1	8	yes	maximal, complemented
5	C₄₁×#3206	C₂	1	2	yes	maximal, complemented
6	C₄₁×#3193	C₂	1	4	yes	maximal, complemented
7	C₄₁×#3100	C₂	1	2	yes	maximal, complemented
8	C₄₁×#3006	C₂	1	4	yes	maximal, complemented
9	C₄₁×#2975	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×H(5,𝔽₂)×C₄₁	—	2	1	no	—
11	C₂²×#41×C₄₁	C₂²	1	2	yes	complemented
12	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	8	yes	complemented
13	C₄₁×#547	—	2	4	no	—
14	C₄₁×#547	C₂²	1	8	yes	complemented
15	C₄₁×#547	C₂²	1	4	yes	complemented
16	C₄₁×#546	—	2	4	no	—
17	D₄²×C₄₁	C₂²	1	4	yes	complemented
18	C₄₁×#537	—	2	1	no	—
19	C₄₁×#534	—	2	2	no	—
20	C₄₁×#532	C₂²	1	2	yes	—
21	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
22	C₂×#123×C₄₁	—	2	2	no	—
23	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	complemented
24	C₂×#121×C₄₁	—	2	1	no	—
25	C₂×#121×C₄₁	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic, complemented
28	C₂²×C₄⋊C₄×C₄₁	—	2	1	no	—
29	C₄₁×#492	C₂²	1	4	yes	complemented
30	C₄₁×#487	C₂²	1	4	yes	complemented
31	C₄₁×#484	C₂²	1	8	yes	complemented
32	C₄₁×#480	C₂²	1	4	yes	complemented
33	C₄₁×#474	C₂²	1	4	yes	complemented
34	C₄₁×#468	C₂²	1	16	yes	complemented
35	C₄₁×#461	C₂²	1	8	yes	complemented
36	C₄₁×#459	C₂²	1	4	yes	complemented
37	C₄₁×#450	C₂²	1	8	yes	complemented
38	C₄₁×#449	C₂²	1	16	yes	complemented
39	C₄₁×#448	C₂²	1	8	yes	complemented
40	C₄₁×#447	C₂²	1	4	yes	complemented
41	C₄×D₈×C₄₁	C₂²	1	8	yes	complemented
42	C₄₁×#436	C₂²	1	8	yes	complemented
43	C₄₁×#431	C₂²	1	2	yes	complemented
44	C₄₁×#417	C₂²	1	8	yes	complemented
45	C₄₁×#407	C₂²	1	4	yes	complemented
46	H(5,𝔽₂)×C₄₁	—	4	8	no	—
47	C₂×#41×C₄₁	—	2	4	no	—
48	C₂×#41×C₄₁	—	2	2	no	—
49	C₂×#41×C₄₁	—	2	4	no	—
50	C₂×#41×C₄₁	—	4	8	no	—
51	C₂×#41×C₄₁	C₂³	1	8	yes	—
52	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
53	C₂²×D₄×C₄₁	—	2	2	no	—
54	C₂²×D₄×C₄₁	—	2	4	no	—
55	C₂²×D₄×C₄₁	—	2	4	no	—
56	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
57	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
59	C₂³×C₄×C₄₁	—	2	1	no	—
60	C₂³×C₄×C₄₁	—	2	2	no	—
61	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
62	AGL(1,ℤ/8)×C₄₁	—	4	16	no	—
63	C₂×SD₁₆×C₄₁	C₂³	1	16	yes	—
64	C₂×D₈×C₄₁	C₂³	1	8	yes	—
65	C₂×M₁₆×C₄₁	C₂³	1	8	yes	—
66	C₄⋊Q₈×C₄₁	—	2	2	no	—
67	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	—
68	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
69	#127×C₄₁	C₂³	1	2	yes	—
70	#124×C₄₁	—	2	2	no	—
71	#123×C₄₁	—	2	16	no	—
72	#123×C₄₁	—	2	8	no	—
73	C₂²⋊Q₈×C₄₁	—	2	4	no	—
74	C₂²⋊Q₈×C₄₁	—	2	4	no	—
75	C₂²⋊Q₈×C₄₁	C₂³	1	8	yes	—
76	#121×C₄₁	—	2	8	no	—
77	#121×C₄₁	—	2	4	no	—
78	#121×C₄₁	—	2	4	no	—
79	#121×C₄₁	—	2	2	no	—
80	#121×C₄₁	—	2	4	no	—
81	#121×C₄₁	C₂³	1	4	yes	—
82	#121×C₄₁	C₂³	1	8	yes	—
83	#120×C₄₁	—	2	8	no	—
84	C₄×D₄×C₄₁	—	2	2	no	—
85	C₄×D₄×C₄₁	—	2	2	no	—
86	C₄×D₄×C₄₁	D₄	1	4	yes	—
87	C₄×D₄×C₄₁	C₂³	1	4	yes	—
88	C₄×D₄×C₄₁	C₂³	1	4	yes	—
89	C₄×D₄×C₄₁	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
91	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
92	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
93	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
94	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
100	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
101	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
102	#107×C₄₁	C₂³	1	8	yes	—
103	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
104	#103×C₄₁	C₂³	1	16	yes	—
105	#102×C₄₁	C₂³	1	8	yes	—
106	#102×C₄₁	C₂³	1	8	yes	—
107	C₂²⋊C₈×C₄₁	C₂³	1	4	yes	—
108	C₂²⋊C₈×C₄₁	C₂³	1	8	yes	—
109	C₄×C₈×C₄₁	C₂³	1	4	yes	—
110	C₂⁴×C₄₁	—	2	1	no	—
111	C₂⁴×C₄₁	—	2	4	no	—
112	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
113	#41×C₄₁	—	2	16	no	—
114	#41×C₄₁	—	4	8	no	—
115	#41×C₄₁	—	4	16	no	—
116	#41×C₄₁	—	4	16	no	—
117	#41×C₄₁	—	4	8	no	—
118	C₂×Q₈×C₄₁	—	2	2	no	—
119	C₂×Q₈×C₄₁	—	2	2	no	—
120	C₂×Q₈×C₄₁	—	4	4	no	—
121	C₂×Q₈×C₄₁	C₂⁴	1	4	yes	—
122	C₂×D₄×C₄₁	—	2	4	no	—
123	C₂×D₄×C₄₁	—	2	8	no	—
124	C₂×D₄×C₄₁	—	2	4	no	—
125	C₂×D₄×C₄₁	—	2	8	no	—
126	C₂×D₄×C₄₁	—	2	4	no	—
127	C₂×D₄×C₄₁	—	2	1	no	—
128	C₂×D₄×C₄₁	—	2	2	no	—
129	C₂×D₄×C₄₁	—	2	2	no	—
130	C₂×D₄×C₄₁	—	2	4	no	—
131	C₂×D₄×C₄₁	—	4	8	no	—
132	C₂×D₄×C₄₁	—	4	16	no	—
133	C₂×D₄×C₄₁	—	4	8	no	—
134	C₂×D₄×C₄₁	—	4	16	no	—
135	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
136	C₂×D₄×C₄₁	C₂⁴	1	4	yes	—
137	C₂×D₄×C₄₁	C₂⁴	1	2	yes	—
138	C₂²×C₄×C₄₁	—	2	1	no	—
139	C₂²×C₄×C₄₁	—	2	2	no	—
140	C₂²×C₄×C₄₁	—	2	4	no	—
141	C₂²×C₄×C₄₁	—	2	4	no	—
142	C₂²×C₄×C₄₁	—	2	8	no	—
143	C₂²×C₄×C₄₁	—	2	4	no	—
144	C₂²×C₄×C₄₁	—	4	4	no	—
145	C₂²×C₄×C₄₁	—	4	8	no	—
146	C₂²×C₄×C₄₁	—	4	4	no	—
147	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
149	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
150	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
152	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
153	SD₁₆×C₄₁	—	4	16	no	—
154	D₈×C₄₁	—	2	16	no	—
155	M₁₆×C₄₁	—	4	8	no	—
156	C₂×C₈×C₄₁	C₂⁴	1	8	yes	—
157	C₄⋊C₄×C₄₁	—	2	4	no	—
158	C₄⋊C₄×C₄₁	—	2	8	no	—
159	C₄⋊C₄×C₄₁	—	2	4	no	—
160	C₄⋊C₄×C₄₁	—	2	1	no	—
161	C₄⋊C₄×C₄₁	—	2	4	no	—
162	C₄⋊C₄×C₄₁	—	2	2	no	—
163	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
164	C₄⋊C₄×C₄₁	C₂⁴	1	2	yes	—
165	C₄⋊C₄×C₄₁	C₂⁴	1	4	yes	—
166	C₂²⋊C₄×C₄₁	—	2	8	no	—
167	C₂²⋊C₄×C₄₁	—	2	4	no	—
168	C₂²⋊C₄×C₄₁	—	2	4	no	—
169	C₂²⋊C₄×C₄₁	—	2	8	no	—
170	C₂²⋊C₄×C₄₁	—	2	4	no	—
171	C₂²⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
172	C₄²×C₄₁	C₂×D₄	1	2	yes	—
173	C₄²×C₄₁	C₂⁴	1	2	yes	—
174	C₂³×C₄₁	—	2	4	no	—
175	C₂³×C₄₁	—	2	2	no	—
176	C₂³×C₄₁	—	2	4	no	—
177	C₂³×C₄₁	—	2	4	no	—
178	C₂³×C₄₁	—	4	2	no	—
179	C₂³×C₄₁	—	4	2	no	—
180	C₂³×C₄₁	—	4	2	no	—
181	C₂³×C₄₁	—	4	8	no	—
182	C₂³×C₄₁	—	4	2	no	—
183	C₂³×C₄₁	—	8	4	no	—
184	C₂³×C₄₁	—	8	4	no	—
185	C₂³×C₄₁	C₂²×D₄	1	4	yes	—
186	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
187	Q₈×C₄₁	—	2	8	no	—
188	Q₈×C₄₁	—	4	4	no	—
189	Q₈×C₄₁	—	4	4	no	—
190	D₄×C₄₁	—	2	4	no	—
191	D₄×C₄₁	—	2	8	no	—
192	D₄×C₄₁	—	2	2	no	—
193	D₄×C₄₁	—	2	8	no	—
194	D₄×C₄₁	—	4	16	no	—
195	D₄×C₄₁	—	4	16	no	—
196	D₄×C₄₁	—	4	8	no	—
197	D₄×C₄₁	—	4	4	no	—
198	D₄×C₄₁	—	8	2	no	—
199	D₄×C₄₁	—	8	4	no	—
200	D₄×C₄₁	H(5,𝔽₂)	1	4	yes	complemented
201	C₂×C₄×C₄₁	—	2	8	no	—
202	C₂×C₄×C₄₁	—	2	2	no	—
203	C₂×C₄×C₄₁	—	2	2	no	—
204	C₂×C₄×C₄₁	—	2	4	no	—
205	C₂×C₄×C₄₁	—	2	4	no	—
206	C₂×C₄×C₄₁	—	4	2	no	—
207	C₂×C₄×C₄₁	—	4	4	no	—
208	C₂×C₄×C₄₁	—	4	1	no	—
209	C₂×C₄×C₄₁	—	4	16	no	—
210	C₂×C₄×C₄₁	—	4	4	no	—
211	C₂×C₄×C₄₁	—	4	8	no	—
212	C₂×C₄×C₄₁	—	4	2	no	—
213	C₂×C₄×C₄₁	—	4	8	no	—
214	C₂×C₄×C₄₁	—	4	8	no	—
215	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	4	yes	complemented
216	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
218	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
219	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
220	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
221	C₈×C₄₁	—	2	8	no	—
222	#61585	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
223	C₂²×C₄₁	—	2	1	no	—
224	C₂²×C₄₁	—	2	1	no	—
225	C₂²×C₄₁	—	2	4	no	—
226	C₂²×C₄₁	—	2	4	no	—
227	C₂²×C₄₁	—	2	4	no	—
228	C₂²×C₄₁	—	2	4	no	—
229	C₂²×C₄₁	—	4	8	no	—
230	C₂²×C₄₁	—	4	2	no	—
231	C₂²×C₄₁	—	4	2	no	—
232	C₂²×C₄₁	—	4	4	no	—
233	C₂²×C₄₁	—	4	4	no	—
234	C₂²×C₄₁	—	4	4	no	—
235	C₂²×C₄₁	—	8	4	no	—
236	C₂²×C₄₁	—	16	8	no	—
237	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
238	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
239	C₄×C₄₁	—	2	2	no	—
240	C₄×C₄₁	—	2	4	no	—
241	C₄×C₄₁	—	2	1	no	—
242	C₄×C₄₁	—	4	4	no	—
243	C₄×C₄₁	—	4	2	no	—
244	C₄×C₄₁	—	4	4	no	—
245	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
246	C₄×C₄₁	C₂×H(5,𝔽₂)	1	2	yes	—
247	#3496	C₂×C₄₁	1	1	yes	characteristic, complemented
248	C₂×#547	C₂×C₄₁	1	1	yes	characteristic, complemented
249	#3330	C₂×C₄₁	1	8	yes	complemented
250	#3272	C₂×C₄₁	1	8	yes	complemented
251	#3206	C₂×C₄₁	1	2	yes	complemented
252	#3193	C₂×C₄₁	1	4	yes	complemented
253	#3100	C₂×C₄₁	1	2	yes	complemented
254	#3006	C₂×C₄₁	1	4	yes	complemented
255	#2975	C₂×C₄₁	1	1	yes	characteristic, complemented
256	C₂×C₄₁	—	2	2	no	—
257	C₂×C₄₁	—	4	4	no	—
258	C₂×C₄₁	—	4	4	no	—
259	C₂×C₄₁	—	8	4	no	—
260	C₂×C₄₁	#3484	1	1	yes	characteristic
261	C₂×C₄₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
262	C₂×C₄₁	#3586	1	1	yes	characteristic
263	C₂×H(5,𝔽₂)	—	2	1	no	—
264	C₂²×#41	C₂²×C₄₁	1	2	yes	complemented
265	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	8	yes	complemented
266	#547	—	2	4	no	—
267	#547	C₂²×C₄₁	1	8	yes	complemented
268	#547	C₂²×C₄₁	1	4	yes	complemented
269	#546	—	2	4	no	—
270	D₄²	C₂²×C₄₁	1	4	yes	complemented
271	#537	—	2	1	no	—
272	#534	—	2	2	no	—
273	#532	C₂²×C₄₁	1	2	yes	—
274	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
275	C₂×#123	—	2	2	no	—
276	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	complemented
277	C₂×#121	—	2	1	no	—
278	C₂×#121	C₂²×C₄₁	1	4	yes	—
279	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
280	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic, complemented
281	C₂²×C₄⋊C₄	—	2	1	no	—
282	#492	C₂²×C₄₁	1	4	yes	complemented
283	#487	C₂²×C₄₁	1	4	yes	complemented
284	#484	C₂²×C₄₁	1	8	yes	complemented
285	#480	C₂²×C₄₁	1	4	yes	complemented
286	#474	C₂²×C₄₁	1	4	yes	complemented
287	#468	C₂²×C₄₁	1	16	yes	complemented
288	#461	C₂²×C₄₁	1	8	yes	complemented
289	#459	C₂²×C₄₁	1	4	yes	complemented
290	#450	C₂²×C₄₁	1	8	yes	complemented
291	#449	C₂²×C₄₁	1	16	yes	complemented
292	#448	C₂²×C₄₁	1	8	yes	complemented
293	#447	C₂²×C₄₁	1	4	yes	complemented
294	C₄×D₈	C₂²×C₄₁	1	8	yes	complemented
295	#436	C₂²×C₄₁	1	8	yes	complemented
296	#431	C₂²×C₄₁	1	2	yes	complemented
297	#417	C₂²×C₄₁	1	8	yes	complemented
298	#407	C₂²×C₄₁	1	4	yes	complemented
299	C₄₁	#61585	1	1	yes	characteristic, direct product factor, Sylow
300	H(5,𝔽₂)	—	4	8	no	—
301	C₂×#41	—	2	4	no	—
302	C₂×#41	—	2	2	no	—
303	C₂×#41	—	2	4	no	—
304	C₂×#41	—	4	8	no	—
305	C₂×#41	C₂³×C₄₁	1	8	yes	—
306	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
307	C₂²×D₄	—	2	2	no	—
308	C₂²×D₄	—	2	4	no	—
309	C₂²×D₄	—	2	4	no	—
310	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
311	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
312	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
313	C₂³×C₄	—	2	1	no	—
314	C₂³×C₄	—	2	2	no	—
315	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
316	AGL(1,ℤ/8)	—	4	16	no	—
317	C₂×SD₁₆	C₂³×C₄₁	1	16	yes	—
318	C₂×D₈	C₂³×C₄₁	1	8	yes	—
319	C₂×M₁₆	C₂³×C₄₁	1	8	yes	—
320	C₄⋊Q₈	—	2	2	no	—
321	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	—
322	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
323	#127	C₂³×C₄₁	1	2	yes	—
324	#124	—	2	2	no	—
325	#123	—	2	16	no	—
326	#123	—	2	8	no	—
327	C₂²⋊Q₈	—	2	4	no	—
328	C₂²⋊Q₈	—	2	4	no	—
329	C₂²⋊Q₈	C₂³×C₄₁	1	8	yes	—
330	#121	—	2	8	no	—
331	#121	—	2	4	no	—
332	#121	—	2	4	no	—
333	#121	—	2	2	no	—
334	#121	—	2	4	no	—
335	#121	C₂³×C₄₁	1	4	yes	—
336	#121	C₂³×C₄₁	1	8	yes	—
337	#120	—	2	8	no	—
338	C₄×D₄	—	2	2	no	—
339	C₄×D₄	—	2	2	no	—
340	C₄×D₄	D₄×C₄₁	1	4	yes	—
341	C₄×D₄	C₂³×C₄₁	1	4	yes	—
342	C₄×D₄	C₂³×C₄₁	1	4	yes	—
343	C₄×D₄	C₂³×C₄₁	1	4	yes	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	—	2	2	no	—
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	—	2	8	no	—
348	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
349	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
350	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
351	C₂×C₂²⋊C₄	—	2	2	no	—
352	C₂×C₂²⋊C₄	—	2	1	no	—
353	C₂×C₂²⋊C₄	—	2	4	no	—
354	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
355	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
356	#107	C₂³×C₄₁	1	8	yes	—
357	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
358	#103	C₂³×C₄₁	1	16	yes	—
359	#102	C₂³×C₄₁	1	8	yes	—
360	#102	C₂³×C₄₁	1	8	yes	—
361	C₂²⋊C₈	C₂³×C₄₁	1	4	yes	—
362	C₂²⋊C₈	C₂³×C₄₁	1	8	yes	—
363	C₄×C₈	C₂³×C₄₁	1	4	yes	—
364	C₂⁴	—	2	1	no	—
365	C₂⁴	—	2	4	no	—
366	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
367	#41	—	2	16	no	—
368	#41	—	4	8	no	—
369	#41	—	4	16	no	—
370	#41	—	4	16	no	—
371	#41	—	4	8	no	—
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	4	4	no	—
375	C₂×Q₈	C₂⁴×C₄₁	1	4	yes	—
376	C₂×D₄	—	2	4	no	—
377	C₂×D₄	—	2	8	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	—	2	8	no	—
380	C₂×D₄	—	2	4	no	—
381	C₂×D₄	—	2	1	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	4	no	—
385	C₂×D₄	—	4	8	no	—
386	C₂×D₄	—	4	16	no	—
387	C₂×D₄	—	4	8	no	—
388	C₂×D₄	—	4	16	no	—
389	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
390	C₂×D₄	C₂⁴×C₄₁	1	4	yes	—
391	C₂×D₄	C₂⁴×C₄₁	1	2	yes	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	4	no	—
396	C₂²×C₄	—	2	8	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	4	4	no	—
399	C₂²×C₄	—	4	8	no	—
400	C₂²×C₄	—	4	4	no	—
401	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
403	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
404	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
405	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
406	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
407	SD₁₆	—	4	16	no	—
408	D₈	—	2	16	no	—
409	M₁₆	—	4	8	no	—
410	C₂×C₈	C₂⁴×C₄₁	1	8	yes	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	8	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	2	4	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
418	C₄⋊C₄	C₂⁴×C₄₁	1	2	yes	—
419	C₄⋊C₄	C₂⁴×C₄₁	1	4	yes	—
420	C₂²⋊C₄	—	2	8	no	—
421	C₂²⋊C₄	—	2	4	no	—
422	C₂²⋊C₄	—	2	4	no	—
423	C₂²⋊C₄	—	2	8	no	—
424	C₂²⋊C₄	—	2	4	no	—
425	C₂²⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
426	C₄²	C₂×D₄×C₄₁	1	2	yes	—
427	C₄²	C₂⁴×C₄₁	1	2	yes	—
428	C₂³	—	2	4	no	—
429	C₂³	—	2	2	no	—
430	C₂³	—	2	4	no	—
431	C₂³	—	2	4	no	—
432	C₂³	—	4	2	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	4	2	no	—
435	C₂³	—	4	8	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	8	4	no	—
438	C₂³	—	8	4	no	—
439	C₂³	C₂²×D₄×C₄₁	1	4	yes	—
440	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
441	Q₈	—	2	8	no	—
442	Q₈	—	4	4	no	—
443	Q₈	—	4	4	no	—
444	D₄	—	2	4	no	—
445	D₄	—	2	8	no	—
446	D₄	—	2	2	no	—
447	D₄	—	2	8	no	—
448	D₄	—	4	16	no	—
449	D₄	—	4	16	no	—
450	D₄	—	4	8	no	—
451	D₄	—	4	4	no	—
452	D₄	—	8	4	no	—
453	D₄	—	8	2	no	—
454	D₄	H(5,𝔽₂)×C₄₁	1	4	yes	complemented
455	C₂×C₄	—	2	8	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	4	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	16	no	—
464	C₂×C₄	—	4	4	no	—
465	C₂×C₄	—	4	8	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	8	no	—
468	C₂×C₄	—	4	8	no	—
469	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	4	yes	complemented
470	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
471	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
472	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
473	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
474	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
475	C₈	—	2	8	no	—
476	C₂²	—	2	1	no	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	4	no	—
479	C₂²	—	2	4	no	—
480	C₂²	—	2	4	no	—
481	C₂²	—	2	4	no	—
482	C₂²	—	4	8	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	4	no	—
486	C₂²	—	4	4	no	—
487	C₂²	—	4	4	no	—
488	C₂²	—	8	4	no	—
489	C₂²	—	16	8	no	—
490	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	2	yes	—
491	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
492	C₄	—	2	2	no	—
493	C₄	—	2	4	no	—
494	C₄	—	2	1	no	—
495	C₄	—	4	4	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	4	no	—
498	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	4	yes	—
499	C₄	C₂×H(5,𝔽₂)×C₄₁	1	2	yes	—
500	C₂	—	2	2	no	—
501	C₂	—	4	4	no	—
502	C₂	—	4	4	no	—
503	C₂	—	8	4	no	—
504	C₂	C₄₁×#3484	1	1	yes	characteristic
505	C₂	C₂²×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
506	C₂	C₄₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄×D₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3496	C₂	semidirect
2. C₂×C₄₁×#547	C₂	semidirect
3. C₄₁×#3330	C₂	semidirect
4. C₄₁×#3272	C₂	semidirect
5. C₄₁×#3206	C₂	semidirect
6. C₄₁×#3193	C₂	semidirect
7. C₄₁×#3100	C₂	semidirect
8. C₄₁×#3006	C₂	semidirect
9. C₄₁×#2975	C₂	semidirect
11. C₂²×#41×C₄₁	C₂²	semidirect
12. C₂×AGL(1,ℤ/8)×C₄₁	C₂²	semidirect
14. C₄₁×#547	C₂²	semidirect
15. C₄₁×#547	C₂²	semidirect
17. D₄²×C₄₁	C₂²	semidirect
21. C₂×C₄⋊Q₈×C₄₁	C₂²	semidirect
23. C₂×C₂²⋊Q₈×C₄₁	C₂²	semidirect
27. C₂×C₄×D₄×C₄₁	C₂²	semidirect
29. C₄₁×#492	C₂²	semidirect
30. C₄₁×#487	C₂²	semidirect
31. C₄₁×#484	C₂²	semidirect
32. C₄₁×#480	C₂²	semidirect
33. C₄₁×#474	C₂²	semidirect
34. C₄₁×#468	C₂²	semidirect
35. C₄₁×#461	C₂²	semidirect
36. C₄₁×#459	C₂²	semidirect
37. C₄₁×#450	C₂²	semidirect
38. C₄₁×#449	C₂²	semidirect
39. C₄₁×#448	C₂²	semidirect
40. C₄₁×#447	C₂²	semidirect
41. C₄×D₈×C₄₁	C₂²	semidirect
42. C₄₁×#436	C₂²	semidirect
43. C₄₁×#431	C₂²	semidirect
44. C₄₁×#417	C₂²	semidirect
45. C₄₁×#407	C₂²	semidirect
200. D₄×C₄₁	H(5,𝔽₂)	semidirect
215. C₂×C₄×C₄₁	AGL(1,ℤ/8)	semidirect
222. #61585	C₄₁	direct (N × H)
247. #3496	C₂×C₄₁	semidirect
248. C₂×#547	C₂×C₄₁	semidirect
249. #3330	C₂×C₄₁	semidirect
250. #3272	C₂×C₄₁	semidirect
251. #3206	C₂×C₄₁	semidirect
252. #3193	C₂×C₄₁	semidirect
253. #3100	C₂×C₄₁	semidirect
254. #3006	C₂×C₄₁	semidirect
255. #2975	C₂×C₄₁	semidirect
264. C₂²×#41	C₂²×C₄₁	semidirect
265. C₂×AGL(1,ℤ/8)	C₂²×C₄₁	semidirect
267. #547	C₂²×C₄₁	semidirect
268. #547	C₂²×C₄₁	semidirect
270. D₄²	C₂²×C₄₁	semidirect
274. C₂×C₄⋊Q₈	C₂²×C₄₁	semidirect
276. C₂×C₂²⋊Q₈	C₂²×C₄₁	semidirect
280. C₂×C₄×D₄	C₂²×C₄₁	semidirect
282. #492	C₂²×C₄₁	semidirect
283. #487	C₂²×C₄₁	semidirect
284. #484	C₂²×C₄₁	semidirect
285. #480	C₂²×C₄₁	semidirect
286. #474	C₂²×C₄₁	semidirect
287. #468	C₂²×C₄₁	semidirect
288. #461	C₂²×C₄₁	semidirect
289. #459	C₂²×C₄₁	semidirect
290. #450	C₂²×C₄₁	semidirect
291. #449	C₂²×C₄₁	semidirect
292. #448	C₂²×C₄₁	semidirect
293. #447	C₂²×C₄₁	semidirect
294. C₄×D₈	C₂²×C₄₁	semidirect
295. #436	C₂²×C₄₁	semidirect
296. #431	C₂²×C₄₁	semidirect
297. #417	C₂²×C₄₁	semidirect
298. #407	C₂²×C₄₁	semidirect
299. C₄₁	#61585	direct (N × H)
454. D₄	H(5,𝔽₂)×C₄₁	semidirect
469. C₂×C₄	AGL(1,ℤ/8)×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3496	2
C₂×C₄₁×#547	2
C₄₁×#3330	2
C₄₁×#3272	2
C₄₁×#3206	2
C₄₁×#3193	2
C₄₁×#3100	2
C₄₁×#3006	2
C₄₁×#2975	2
#61585	41

Subgroups — C41×#61585 #23837786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#61585 #23837786