Subgroups — C₄₁×#19937 #23796138

Order 10496.

660 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3366	C₂	1	2	yes	maximal, complemented
2	C₄₁×#3338	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3287	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1698	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1675	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1631	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1622	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1591	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₄₁	—	2	1	no	—
13	C₄₁×#554	C₂²	1	2	yes	—
14	C₄₁×#550	C₂²	1	2	yes	—
15	C₄₁×#500	C₂²	1	1	yes	characteristic
16	C₄₁×#493	C₂²	1	1	yes	characteristic
17	C₄₁×#492	C₂²	1	2	yes	—
18	C₄₁×#490	—	2	2	no	—
19	C₄₁×#486	C₂²	1	2	yes	—
20	C₄₁×#484	—	2	2	no	—
21	C₄₁×#478	—	2	2	no	—
22	C₄₁×#477	C₂²	1	2	yes	—
23	C₄₁×#476	C₂²	1	2	yes	—
24	C₄₁×#467	—	2	1	no	—
25	C₄₁×#466	—	2	1	no	—
26	C₄₁×#460	C₂²	1	2	yes	—
27	C₄₁×#459	—	2	2	no	—
28	C₄₁×#459	C₂²	1	2	yes	—
29	C₄₁×#449	—	2	2	no	—
30	C₄₁×#446	C₂²	1	2	yes	—
31	C₄₁×#442	—	2	2	no	—
32	C₄×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
33	C₄×D₈×C₄₁	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₄₁	C₂²	1	1	yes	characteristic
35	C₈×D₄×C₄₁	C₂²	1	1	yes	characteristic
36	C₂×#107×C₄₁	—	2	1	no	—
37	C₄₁×#335	C₂²	1	1	yes	characteristic
38	C₄₁×#335	C₂²	1	1	yes	characteristic
39	C₄₁×#333	C₂²	1	2	yes	—
40	C₄₁×#331	C₂²	1	2	yes	—
41	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	—
42	C₄₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₄₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₄₁	C₂²	1	1	yes	characteristic
45	C₂×#41×C₄₁	—	2	2	no	—
46	#135×C₄₁	—	4	4	no	—
47	C₂×Q₁₆×C₄₁	D₄	1	1	yes	characteristic
48	C₂×SD₁₆×C₄₁	—	2	2	no	—
49	C₂×SD₁₆×C₄₁	—	4	2	no	—
50	C₂×D₈×C₄₁	—	2	2	no	—
51	C₂×D₈×C₄₁	—	2	1	no	—
52	C₂×D₈×C₄₁	—	2	1	no	—
53	C₂×D₈×C₄₁	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₄₁	—	2	1	no	—
55	C₂²×C₈×C₄₁	—	2	1	no	—
56	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
57	#127×C₄₁	—	2	2	no	—
58	#127×C₄₁	C₂³	1	2	yes	—
59	#125×C₄₁	—	2	2	no	—
60	C₂²⋊Q₈×C₄₁	—	2	2	no	—
61	C₂²⋊Q₈×C₄₁	—	4	2	no	—
62	#121×C₄₁	—	2	2	no	—
63	#121×C₄₁	—	4	2	no	—
64	C₄×Q₈×C₄₁	C₂³	1	2	yes	—
65	C₄×D₄×C₄₁	—	2	2	no	—
66	C₄×D₄×C₄₁	—	2	2	no	—
67	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
70	#107×C₄₁	—	2	1	no	—
71	#107×C₄₁	—	2	1	no	—
72	#107×C₄₁	—	2	2	no	—
73	#107×C₄₁	D₄	1	1	yes	characteristic
74	#107×C₄₁	D₄	1	1	yes	characteristic
75	#106×C₄₁	—	2	2	no	—
76	C₄⋊C₈×C₄₁	—	2	2	no	—
77	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
80	#103×C₄₁	—	2	2	no	—
81	#103×C₄₁	D₄	1	2	yes	—
82	#102×C₄₁	—	2	2	no	—
83	#102×C₄₁	—	2	2	no	—
84	#102×C₄₁	—	2	2	no	—
85	#102×C₄₁	—	2	2	no	—
86	#102×C₄₁	—	2	2	no	—
87	#102×C₄₁	D₄	1	1	yes	characteristic
88	#102×C₄₁	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₄₁	—	2	1	no	—
90	C₂²⋊C₈×C₄₁	—	2	1	no	—
91	#97×C₄₁	—	2	2	no	—
92	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
95	#41×C₄₁	—	4	4	no	—
96	#41×C₄₁	—	8	2	no	—
97	C₂×Q₈×C₄₁	—	2	2	no	—
98	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
99	C₂×D₄×C₄₁	—	2	2	no	—
100	C₂×D₄×C₄₁	—	2	2	no	—
101	C₂×D₄×C₄₁	—	2	2	no	—
102	C₂×D₄×C₄₁	—	4	2	no	—
103	C₂×D₄×C₄₁	—	4	2	no	—
104	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
105	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₄₁	—	2	1	no	—
107	C₂²×C₄×C₄₁	—	2	1	no	—
108	C₂²×C₄×C₄₁	—	4	2	no	—
109	C₂²×C₄×C₄₁	—	4	2	no	—
110	Q₁₆×C₄₁	—	2	2	no	—
111	SD₁₆×C₄₁	—	4	4	no	—
112	SD₁₆×C₄₁	—	8	4	no	—
113	D₈×C₄₁	—	2	2	no	—
114	D₈×C₄₁	—	4	4	no	—
115	D₈×C₄₁	—	4	2	no	—
116	D₈×C₄₁	—	8	1	no	—
117	C₂×C₈×C₄₁	—	2	1	no	—
118	C₂×C₈×C₄₁	—	2	2	no	—
119	C₂×C₈×C₄₁	—	2	1	no	—
120	C₂×C₈×C₄₁	—	2	1	no	—
121	C₂×C₈×C₄₁	—	4	2	no	—
122	C₂×C₈×C₄₁	—	4	2	no	—
123	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
127	C₄⋊C₄×C₄₁	—	2	2	no	—
128	C₄⋊C₄×C₄₁	—	2	2	no	—
129	C₄⋊C₄×C₄₁	—	2	2	no	—
130	C₄⋊C₄×C₄₁	—	4	2	no	—
131	C₄⋊C₄×C₄₁	—	4	2	no	—
132	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
133	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
134	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
135	C₂²⋊C₄×C₄₁	—	2	1	no	—
136	C₂²⋊C₄×C₄₁	—	2	1	no	—
137	C₂²⋊C₄×C₄₁	—	4	2	no	—
138	C₂²⋊C₄×C₄₁	—	4	2	no	—
139	C₄²×C₄₁	—	2	2	no	—
140	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
141	C₂³×C₄₁	—	2	1	no	—
142	C₂³×C₄₁	—	2	1	no	—
143	C₂³×C₄₁	—	4	2	no	—
144	Q₈×C₄₁	—	2	2	no	—
145	Q₈×C₄₁	—	2	2	no	—
146	Q₈×C₄₁	—	8	2	no	—
147	D₄×C₄₁	—	2	1	no	—
148	D₄×C₄₁	—	2	1	no	—
149	D₄×C₄₁	—	2	1	no	—
150	D₄×C₄₁	—	4	2	no	—
151	D₄×C₄₁	—	4	2	no	—
152	D₄×C₄₁	—	4	2	no	—
153	D₄×C₄₁	—	4	2	no	—
154	D₄×C₄₁	—	8	2	no	—
155	D₄×C₄₁	—	8	4	no	—
156	D₄×C₄₁	—	16	2	no	—
157	D₄×C₄₁	#135	1	2	yes	complemented
158	C₂×C₄×C₄₁	—	2	2	no	—
159	C₂×C₄×C₄₁	—	2	2	no	—
160	C₂×C₄×C₄₁	—	2	1	no	—
161	C₂×C₄×C₄₁	—	2	1	no	—
162	C₂×C₄×C₄₁	—	4	1	no	—
163	C₂×C₄×C₄₁	—	4	1	no	—
164	C₂×C₄×C₄₁	—	4	1	no	—
165	C₂×C₄×C₄₁	—	4	1	no	—
166	C₂×C₄×C₄₁	—	4	2	no	—
167	C₂×C₄×C₄₁	—	8	4	no	—
168	C₂×C₄×C₄₁	—	16	2	no	—
169	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
172	C₈×C₄₁	—	2	2	no	—
173	C₈×C₄₁	—	2	1	no	—
174	C₈×C₄₁	—	2	1	no	—
175	C₈×C₄₁	—	2	1	no	—
176	C₈×C₄₁	—	4	2	no	—
177	C₈×C₄₁	—	4	1	no	—
178	C₈×C₄₁	—	4	1	no	—
179	C₈×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
180	#19937	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂²×C₄₁	—	2	2	no	—
182	C₂²×C₄₁	—	4	1	no	—
183	C₂²×C₄₁	—	4	1	no	—
184	C₂²×C₄₁	—	4	1	no	—
185	C₂²×C₄₁	—	4	1	no	—
186	C₂²×C₄₁	—	4	1	no	—
187	C₂²×C₄₁	—	8	2	no	—
188	C₂²×C₄₁	—	8	2	no	—
189	C₂²×C₄₁	—	8	2	no	—
190	C₂²×C₄₁	D₄²	1	1	yes	characteristic
191	C₄×C₄₁	—	2	1	no	—
192	C₄×C₄₁	—	4	1	no	—
193	C₄×C₄₁	—	4	1	no	—
194	C₄×C₄₁	—	4	2	no	—
195	C₄×C₄₁	—	4	2	no	—
196	C₄×C₄₁	—	8	2	no	—
197	C₄×C₄₁	C₂×#135	1	1	yes	characteristic
198	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
199	C₄×C₄₁	#577	1	1	yes	characteristic
200	C₄×C₄₁	#578	1	1	yes	characteristic
201	#3366	C₂×C₄₁	1	2	yes	complemented
202	#3338	C₂×C₄₁	1	1	yes	characteristic, complemented
203	#3287	C₂×C₄₁	1	1	yes	characteristic, complemented
204	#1706	C₂×C₄₁	1	1	yes	characteristic, complemented
205	#1698	C₂×C₄₁	1	2	yes	complemented
206	#1675	C₂×C₄₁	1	1	yes	characteristic, complemented
207	#1667	C₂×C₄₁	1	1	yes	characteristic, complemented
208	#1631	C₂×C₄₁	1	2	yes	complemented
209	#1622	C₂×C₄₁	1	2	yes	complemented
210	#1591	C₂×C₄₁	1	1	yes	characteristic, complemented
211	C₈×D₈	C₂×C₄₁	1	1	yes	characteristic, complemented
212	C₂×C₄₁	—	4	2	no	—
213	C₂×C₄₁	—	8	1	no	—
214	C₂×C₄₁	—	16	2	no	—
215	C₂×C₄₁	#3272	1	1	yes	characteristic
216	C₂×C₄₁	#3285	1	1	yes	characteristic
217	C₂×C₄₁	#3292	1	1	yes	characteristic
218	C₂×#135	—	2	1	no	—
219	#554	C₂²×C₄₁	1	2	yes	—
220	#550	C₂²×C₄₁	1	2	yes	—
221	#500	C₂²×C₄₁	1	1	yes	characteristic
222	#493	C₂²×C₄₁	1	1	yes	characteristic
223	#492	C₂²×C₄₁	1	2	yes	—
224	#490	—	2	2	no	—
225	#486	C₂²×C₄₁	1	2	yes	—
226	#484	—	2	2	no	—
227	#478	—	2	2	no	—
228	#477	C₂²×C₄₁	1	2	yes	—
229	#476	C₂²×C₄₁	1	2	yes	—
230	#467	—	2	1	no	—
231	#466	—	2	1	no	—
232	#460	C₂²×C₄₁	1	2	yes	—
233	#459	—	2	2	no	—
234	#459	C₂²×C₄₁	1	2	yes	—
235	#449	—	2	2	no	—
236	#446	C₂²×C₄₁	1	2	yes	—
237	#442	—	2	2	no	—
238	C₄×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
239	C₄×D₈	C₂²×C₄₁	1	1	yes	characteristic
240	C₈×D₄	C₂²×C₄₁	1	1	yes	characteristic
241	C₈×D₄	C₂²×C₄₁	1	1	yes	characteristic
242	C₂×#107	—	2	1	no	—
243	#335	C₂²×C₄₁	1	1	yes	characteristic
244	#335	C₂²×C₄₁	1	1	yes	characteristic
245	#333	C₂²×C₄₁	1	2	yes	—
246	#331	C₂²×C₄₁	1	2	yes	—
247	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	—
248	D₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
249	D₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
250	C₈²	C₂²×C₄₁	1	1	yes	characteristic
251	C₄₁	#19937	1	1	yes	characteristic, direct product factor, Sylow
252	C₂×#41	—	2	2	no	—
253	#135	—	4	4	no	—
254	C₂×Q₁₆	D₄×C₄₁	1	1	yes	characteristic
255	C₂×SD₁₆	—	2	2	no	—
256	C₂×SD₁₆	—	4	2	no	—
257	C₂×D₈	—	2	2	no	—
258	C₂×D₈	—	2	1	no	—
259	C₂×D₈	—	2	1	no	—
260	C₂×D₈	D₄×C₄₁	1	1	yes	characteristic
261	C₂²×C₈	—	2	1	no	—
262	C₂²×C₈	—	2	1	no	—
263	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
264	#127	—	2	2	no	—
265	#127	C₂³×C₄₁	1	2	yes	—
266	#125	—	2	2	no	—
267	C₂²⋊Q₈	—	2	2	no	—
268	C₂²⋊Q₈	—	4	2	no	—
269	#121	—	2	2	no	—
270	#121	—	4	2	no	—
271	C₄×Q₈	C₂³×C₄₁	1	2	yes	—
272	C₄×D₄	—	2	2	no	—
273	C₄×D₄	—	2	2	no	—
274	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
275	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
276	C₂×C₄⋊C₄	—	2	2	no	—
277	#107	—	2	1	no	—
278	#107	—	2	1	no	—
279	#107	—	2	2	no	—
280	#107	D₄×C₄₁	1	1	yes	characteristic
281	#107	D₄×C₄₁	1	1	yes	characteristic
282	#106	—	2	2	no	—
283	C₄⋊C₈	—	2	2	no	—
284	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
285	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
286	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
287	#103	—	2	2	no	—
288	#103	D₄×C₄₁	1	2	yes	—
289	#102	—	2	2	no	—
290	#102	—	2	2	no	—
291	#102	—	2	2	no	—
292	#102	—	2	2	no	—
293	#102	—	2	2	no	—
294	#102	D₄×C₄₁	1	1	yes	characteristic
295	#102	D₄×C₄₁	1	1	yes	characteristic
296	C₂²⋊C₈	—	2	1	no	—
297	C₂²⋊C₈	—	2	1	no	—
298	#97	—	2	2	no	—
299	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
300	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
301	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
302	#41	—	4	4	no	—
303	#41	—	8	2	no	—
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	4	2	no	—
310	C₂×D₄	—	4	2	no	—
311	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
312	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	4	2	no	—
316	C₂²×C₄	—	4	2	no	—
317	Q₁₆	—	2	2	no	—
318	SD₁₆	—	4	4	no	—
319	SD₁₆	—	8	4	no	—
320	D₈	—	2	2	no	—
321	D₈	—	4	4	no	—
322	D₈	—	4	2	no	—
323	D₈	—	8	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	4	2	no	—
329	C₂×C₈	—	4	2	no	—
330	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
331	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
332	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
333	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	4	2	no	—
338	C₄⋊C₄	—	4	2	no	—
339	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
340	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
341	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
342	C₂²⋊C₄	—	2	1	no	—
343	C₂²⋊C₄	—	2	1	no	—
344	C₂²⋊C₄	—	4	2	no	—
345	C₂²⋊C₄	—	4	2	no	—
346	C₄²	—	2	2	no	—
347	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
348	C₂³	—	2	1	no	—
349	C₂³	—	2	1	no	—
350	C₂³	—	4	2	no	—
351	Q₈	—	2	2	no	—
352	Q₈	—	2	2	no	—
353	Q₈	—	8	2	no	—
354	D₄	—	2	1	no	—
355	D₄	—	2	1	no	—
356	D₄	—	2	1	no	—
357	D₄	—	4	2	no	—
358	D₄	—	4	2	no	—
359	D₄	—	4	2	no	—
360	D₄	—	4	2	no	—
361	D₄	—	8	2	no	—
362	D₄	—	8	4	no	—
363	D₄	—	16	2	no	—
364	D₄	#135×C₄₁	1	2	yes	complemented
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	8	4	no	—
375	C₂×C₄	—	16	2	no	—
376	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
377	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
378	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
379	C₈	—	2	2	no	—
380	C₈	—	2	1	no	—
381	C₈	—	2	1	no	—
382	C₈	—	2	1	no	—
383	C₈	—	4	2	no	—
384	C₈	—	4	1	no	—
385	C₈	—	4	1	no	—
386	C₈	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
387	C₂²	—	2	2	no	—
388	C₂²	—	4	1	no	—
389	C₂²	—	4	1	no	—
390	C₂²	—	4	1	no	—
391	C₂²	—	4	1	no	—
392	C₂²	—	4	1	no	—
393	C₂²	—	8	2	no	—
394	C₂²	—	8	2	no	—
395	C₂²	—	8	2	no	—
396	C₂²	D₄²×C₄₁	1	1	yes	characteristic
397	C₄	—	2	1	no	—
398	C₄	—	4	2	no	—
399	C₄	—	4	1	no	—
400	C₄	—	4	1	no	—
401	C₄	—	4	2	no	—
402	C₄	—	8	2	no	—
403	C₄	C₂×#135×C₄₁	1	1	yes	characteristic
404	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
405	C₄	C₄₁×#577	1	1	yes	characteristic
406	C₄	C₄₁×#578	1	1	yes	characteristic
407	C₂	—	4	2	no	—
408	C₂	—	8	1	no	—
409	C₂	—	16	2	no	—
410	C₂	C₄₁×#3272	1	1	yes	characteristic
411	C₂	C₄₁×#3285	1	1	yes	characteristic
412	C₂	C₄₁×#3292	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×#3366	C₂	semidirect
2. C₄₁×#3338	C₂	semidirect
3. C₄₁×#3287	C₂	semidirect
4. C₄₁×#1706	C₂	semidirect
5. C₄₁×#1698	C₂	semidirect
6. C₄₁×#1675	C₂	semidirect
7. C₄₁×#1667	C₂	semidirect
8. C₄₁×#1631	C₂	semidirect
9. C₄₁×#1622	C₂	semidirect
10. C₄₁×#1591	C₂	semidirect
11. C₈×D₈×C₄₁	C₂	semidirect
157. D₄×C₄₁	#135	semidirect
180. #19937	C₄₁	direct (N × H)
201. #3366	C₂×C₄₁	semidirect
202. #3338	C₂×C₄₁	semidirect
203. #3287	C₂×C₄₁	semidirect
204. #1706	C₂×C₄₁	semidirect
205. #1698	C₂×C₄₁	semidirect
206. #1675	C₂×C₄₁	semidirect
207. #1667	C₂×C₄₁	semidirect
208. #1631	C₂×C₄₁	semidirect
209. #1622	C₂×C₄₁	semidirect
210. #1591	C₂×C₄₁	semidirect
211. C₈×D₈	C₂×C₄₁	semidirect
251. C₄₁	#19937	direct (N × H)
364. D₄	#135×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3366	2
C₄₁×#3338	2
C₄₁×#3287	2
C₄₁×#1706	2
C₄₁×#1698	2
C₄₁×#1675	2
C₄₁×#1667	2
C₄₁×#1631	2
C₄₁×#1622	2
C₄₁×#1591	2
C₈×D₈×C₄₁	2
#19937	41

Subgroups — C41×#19937 #23796138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#19937 #23796138