Subgroups — C₄₁×#20324 #23796525

Order 10496.

570 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3385	C₂	1	2	yes	maximal, complemented
2	C₄₁×#3357	C₂	1	2	yes	maximal
3	C₄₁×#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1696	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1695	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1634	C₂	1	1	yes	maximal, characteristic
7	C₄₁×#1633	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1632	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1590	C₂	1	2	yes	maximal, complemented
10	Q₈²×C₄₁	C₂²	1	1	yes	characteristic
11	C₄₁×#555	C₂²	1	1	yes	characteristic
12	C₄₁×#554	C₂²	1	2	yes	—
13	C₄₁×#501	—	2	2	no	—
14	C₄₁×#500	C₂²	1	2	yes	—
15	C₄₁×#500	C₂²	1	2	yes	—
16	C₄₁×#494	C₂²	1	1	yes	characteristic
17	C₄₁×#494	C₂²	1	1	yes	characteristic
18	C₄₁×#485	—	2	2	no	—
19	C₄₁×#480	—	2	2	no	—
20	C₄₁×#479	—	2	2	no	—
21	C₄₁×#477	C₂²	1	2	yes	—
22	C₄₁×#477	C₂²	1	2	yes	—
23	C₄₁×#476	C₂²	1	2	yes	—
24	C₄₁×#476	C₂²	1	2	yes	—
25	C₄₁×#474	—	2	2	no	—
26	C₄₁×#468	—	2	2	no	—
27	C₄₁×#446	C₂²	1	2	yes	—
28	C₄₁×#440	—	2	2	no	—
29	C₄×SD₁₆×C₄₁	C₂²	1	2	yes	—
30	C₄₁×#436	C₂²	1	2	yes	—
31	C₄₁×#428	—	2	2	no	—
32	C₄₁×#334	C₂²	1	2	yes	—
33	C₄₁×#333	C₂²	1	2	yes	—
34	C₄₁×#333	C₂²	1	1	yes	characteristic
35	C₄₁×#333	C₂²	1	1	yes	characteristic
36	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₄₁×D₄⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₄₁	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₄₁	—	4	1	no	—
40	C₂×Q₁₆×C₄₁	—	4	1	no	—
41	C₂×SD₁₆×C₄₁	D₄	1	2	yes	—
42	C₂×M₁₆×C₄₁	—	2	2	no	—
43	C₄⋊Q₈×C₄₁	—	2	2	no	—
44	C₄⋊Q₈×C₄₁	—	4	1	no	—
45	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
48	#125×C₄₁	—	2	2	no	—
49	#124×C₄₁	—	4	1	no	—
50	C₂²⋊Q₈×C₄₁	—	2	2	no	—
51	C₂²⋊Q₈×C₄₁	—	4	2	no	—
52	#121×C₄₁	—	2	2	no	—
53	C₄×Q₈×C₄₁	—	2	2	no	—
54	C₄×Q₈×C₄₁	C₂³	1	2	yes	—
55	C₄×D₄×C₄₁	—	2	2	no	—
56	C₄×D₄×C₄₁	C₂³	1	2	yes	—
57	#117×C₄₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
59	#107×C₄₁	—	2	2	no	—
60	#107×C₄₁	—	2	2	no	—
61	#107×C₄₁	—	2	2	no	—
62	#107×C₄₁	—	2	2	no	—
63	#107×C₄₁	—	2	2	no	—
64	#107×C₄₁	—	2	2	no	—
65	#106×C₄₁	—	2	2	no	—
66	#106×C₄₁	—	2	2	no	—
67	#106×C₄₁	D₄	1	2	yes	—
68	C₄⋊C₈×C₄₁	—	2	2	no	—
69	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
70	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
71	#103×C₄₁	—	2	2	no	—
72	#103×C₄₁	—	2	2	no	—
73	#103×C₄₁	D₄	1	2	yes	—
74	#102×C₄₁	—	2	2	no	—
75	#102×C₄₁	—	2	2	no	—
76	#102×C₄₁	D₄	1	2	yes	—
77	C₂²⋊C₈×C₄₁	—	2	2	no	—
78	#97×C₄₁	—	2	2	no	—
79	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₄₁	C₂³	1	2	yes	—
81	C₂×Q₈×C₄₁	—	2	2	no	—
82	C₂×Q₈×C₄₁	—	2	1	no	—
83	C₂×Q₈×C₄₁	—	2	1	no	—
84	C₂×Q₈×C₄₁	—	4	1	no	—
85	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
86	C₂×D₄×C₄₁	—	4	1	no	—
87	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₄₁	—	2	2	no	—
89	C₂²×C₄×C₄₁	—	4	2	no	—
90	Q₁₆×C₄₁	—	8	2	no	—
91	Q₁₆×C₄₁	—	8	2	no	—
92	SD₁₆×C₄₁	—	2	4	no	—
93	M₁₆×C₄₁	—	4	4	no	—
94	C₂×C₈×C₄₁	—	2	1	no	—
95	C₂×C₈×C₄₁	—	2	2	no	—
96	C₂×C₈×C₄₁	—	2	2	no	—
97	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
99	C₄⋊C₄×C₄₁	—	2	2	no	—
100	C₄⋊C₄×C₄₁	—	2	2	no	—
101	C₄⋊C₄×C₄₁	—	2	2	no	—
102	C₄⋊C₄×C₄₁	—	2	2	no	—
103	C₄⋊C₄×C₄₁	—	2	2	no	—
104	C₄⋊C₄×C₄₁	—	4	2	no	—
105	C₄⋊C₄×C₄₁	—	4	2	no	—
106	C₄⋊C₄×C₄₁	—	4	2	no	—
107	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
108	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
109	C₂²⋊C₄×C₄₁	—	2	2	no	—
110	C₂²⋊C₄×C₄₁	—	4	2	no	—
111	C₂²⋊C₄×C₄₁	—	4	2	no	—
112	C₄²×C₄₁	—	2	2	no	—
113	C₄²×C₄₁	—	4	1	no	—
114	C₄²×C₄₁	—	4	1	no	—
115	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
116	C₂³×C₄₁	—	2	2	no	—
117	Q₈×C₄₁	—	2	2	no	—
118	Q₈×C₄₁	—	2	2	no	—
119	Q₈×C₄₁	—	8	2	no	—
120	Q₈×C₄₁	—	8	1	no	—
121	Q₈×C₄₁	—	8	1	no	—
122	Q₈×C₄₁	—	16	1	no	—
123	D₄×C₄₁	—	2	2	no	—
124	D₄×C₄₁	—	2	2	no	—
125	D₄×C₄₁	—	16	1	no	—
126	C₂×C₄×C₄₁	—	2	2	no	—
127	C₂×C₄×C₄₁	—	2	2	no	—
128	C₂×C₄×C₄₁	—	2	2	no	—
129	C₂×C₄×C₄₁	—	4	2	no	—
130	C₂×C₄×C₄₁	—	4	2	no	—
131	C₂×C₄×C₄₁	—	4	1	no	—
132	C₂×C₄×C₄₁	—	4	1	no	—
133	C₂×C₄×C₄₁	—	4	2	no	—
134	C₂×C₄×C₄₁	—	16	2	no	—
135	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
136	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
137	C₈×C₄₁	—	2	2	no	—
138	C₈×C₄₁	—	2	2	no	—
139	C₈×C₄₁	—	4	2	no	—
140	C₈×C₄₁	—	4	2	no	—
141	C₈×C₄₁	#137	1	4	yes	—
142	#20324	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂²×C₄₁	—	4	2	no	—
144	C₂²×C₄₁	—	4	2	no	—
145	C₂²×C₄₁	—	4	2	no	—
146	C₂²×C₄₁	D₄²	1	1	yes	characteristic
147	C₄×C₄₁	—	2	1	no	—
148	C₄×C₄₁	—	4	2	no	—
149	C₄×C₄₁	—	4	2	no	—
150	C₄×C₄₁	—	4	2	no	—
151	C₄×C₄₁	—	8	2	no	—
152	C₄×C₄₁	—	8	1	no	—
153	C₄×C₄₁	—	8	1	no	—
154	C₄×C₄₁	C₂×#137	1	2	yes	—
155	C₄×C₄₁	#578	1	2	yes	—
156	#3385	C₂×C₄₁	1	2	yes	complemented
157	#3357	C₂×C₄₁	1	2	yes	—
158	#1713	C₂×C₄₁	1	1	yes	characteristic, complemented
159	#1696	C₂×C₄₁	1	2	yes	complemented
160	#1695	C₂×C₄₁	1	2	yes	complemented
161	#1634	C₂×C₄₁	1	1	yes	characteristic
162	#1633	C₂×C₄₁	1	1	yes	characteristic, complemented
163	#1632	C₂×C₄₁	1	2	yes	complemented
164	#1590	C₂×C₄₁	1	2	yes	complemented
165	C₂×C₄₁	—	8	2	no	—
166	C₂×C₄₁	#3276	1	2	yes	—
167	C₂×C₄₁	#3292	1	1	yes	characteristic
168	Q₈²	C₂²×C₄₁	1	1	yes	characteristic
169	#555	C₂²×C₄₁	1	1	yes	characteristic
170	#554	C₂²×C₄₁	1	2	yes	—
171	#501	—	2	2	no	—
172	#500	C₂²×C₄₁	1	2	yes	—
173	#500	C₂²×C₄₁	1	2	yes	—
174	#494	C₂²×C₄₁	1	1	yes	characteristic
175	#494	C₂²×C₄₁	1	1	yes	characteristic
176	#485	—	2	2	no	—
177	#480	—	2	2	no	—
178	#479	—	2	2	no	—
179	#477	C₂²×C₄₁	1	2	yes	—
180	#477	C₂²×C₄₁	1	2	yes	—
181	#476	C₂²×C₄₁	1	2	yes	—
182	#476	C₂²×C₄₁	1	2	yes	—
183	#474	—	2	2	no	—
184	#468	—	2	2	no	—
185	#446	C₂²×C₄₁	1	2	yes	—
186	#440	—	2	2	no	—
187	C₄×SD₁₆	C₂²×C₄₁	1	2	yes	—
188	#436	C₂²×C₄₁	1	2	yes	—
189	#428	—	2	2	no	—
190	#334	C₂²×C₄₁	1	2	yes	—
191	#333	C₂²×C₄₁	1	2	yes	—
192	#333	C₂²×C₄₁	1	1	yes	characteristic
193	#333	C₂²×C₄₁	1	1	yes	characteristic
194	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	—
195	D₄⋊C₈	C₂²×C₄₁	1	2	yes	—
196	C₈²	C₂²×C₄₁	1	1	yes	characteristic
197	C₄₁	#20324	1	1	yes	characteristic, direct product factor, Sylow
198	C₂×Q₁₆	—	4	1	no	—
199	C₂×Q₁₆	—	4	1	no	—
200	C₂×SD₁₆	D₄×C₄₁	1	2	yes	—
201	C₂×M₁₆	—	2	2	no	—
202	C₄⋊Q₈	—	2	2	no	—
203	C₄⋊Q₈	—	4	1	no	—
204	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
205	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
206	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
207	#125	—	2	2	no	—
208	#124	—	4	1	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₂²⋊Q₈	—	4	2	no	—
211	#121	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	C₂³×C₄₁	1	2	yes	—
214	C₄×D₄	—	2	2	no	—
215	C₄×D₄	C₂³×C₄₁	1	2	yes	—
216	#117	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	#107	—	2	2	no	—
219	#107	—	2	2	no	—
220	#107	—	2	2	no	—
221	#107	—	2	2	no	—
222	#107	—	2	2	no	—
223	#107	—	2	2	no	—
224	#106	—	2	2	no	—
225	#106	—	2	2	no	—
226	#106	D₄×C₄₁	1	2	yes	—
227	C₄⋊C₈	—	2	2	no	—
228	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
229	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
230	#103	—	2	2	no	—
231	#103	—	2	2	no	—
232	#103	D₄×C₄₁	1	2	yes	—
233	#102	—	2	2	no	—
234	#102	—	2	2	no	—
235	#102	D₄×C₄₁	1	2	yes	—
236	C₂²⋊C₈	—	2	2	no	—
237	#97	—	2	2	no	—
238	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₄₁	1	2	yes	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	4	2	no	—
249	Q₁₆	—	8	2	no	—
250	Q₁₆	—	8	2	no	—
251	SD₁₆	—	2	4	no	—
252	M₁₆	—	4	4	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
257	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
267	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	4	1	no	—
273	C₄²	—	4	1	no	—
274	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
275	C₂³	—	2	2	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	8	2	no	—
279	Q₈	—	8	1	no	—
280	Q₈	—	8	1	no	—
281	Q₈	—	16	1	no	—
282	D₄	—	2	2	no	—
283	D₄	—	2	2	no	—
284	D₄	—	16	1	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	16	2	no	—
294	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
295	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	2	2	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	2	no	—
300	C₈	#137×C₄₁	1	4	yes	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	D₄²×C₄₁	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	4	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	2	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	1	no	—
311	C₄	—	8	1	no	—
312	C₄	C₂×#137×C₄₁	1	2	yes	—
313	C₄	C₄₁×#578	1	2	yes	—
314	C₂	—	8	2	no	—
315	C₂	C₄₁×#3276	1	2	yes	—
316	C₂	C₄₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₄₁×#555	C₂²
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×#3385	C₂	semidirect
3. C₄₁×#1713	C₂	semidirect
4. C₄₁×#1696	C₂	semidirect
5. C₄₁×#1695	C₂	semidirect
7. C₄₁×#1633	C₂	semidirect
8. C₄₁×#1632	C₂	semidirect
9. C₄₁×#1590	C₂	semidirect
142. #20324	C₄₁	direct (N × H)
156. #3385	C₂×C₄₁	semidirect
158. #1713	C₂×C₄₁	semidirect
159. #1696	C₂×C₄₁	semidirect
160. #1695	C₂×C₄₁	semidirect
162. #1633	C₂×C₄₁	semidirect
163. #1632	C₂×C₄₁	semidirect
164. #1590	C₂×C₄₁	semidirect
197. C₄₁	#20324	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#3385	2
C₄₁×#3357	2
C₄₁×#1713	2
C₄₁×#1696	2
C₄₁×#1695	2
C₄₁×#1634	2
C₄₁×#1633	2
C₄₁×#1632	2
C₄₁×#1590	2
#20324	41

Subgroups — C41×#20324 #23796525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20324 #23796525