Subgroups — C₄₁×#20323 #23796524

Order 10496.

572 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3400	C₂	1	2	yes	maximal
2	C₄₁×#3387	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1705	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1692	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1686	C₂	1	2	yes	maximal
6	C₄₁×#1653	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1630	C₂	1	1	yes	maximal, characteristic
8	C₄₁×#1623	C₂	1	1	yes	maximal, characteristic
9	C₄₁×#1591	C₂	1	2	yes	maximal, complemented
10	C₄₁×#559	C₂²	1	1	yes	characteristic
11	Q₈²×C₄₁	C₂²	1	1	yes	characteristic
12	C₄₁×#557	C₂²	1	2	yes	—
13	C₄₁×#501	—	2	2	no	—
14	C₄₁×#498	C₂²	1	2	yes	—
15	C₄₁×#498	C₂²	1	1	yes	characteristic
16	C₄₁×#497	—	2	2	no	—
17	C₄₁×#492	C₂²	1	2	yes	—
18	C₄₁×#492	C₂²	1	1	yes	characteristic
19	C₄₁×#479	—	2	2	no	—
20	C₄₁×#477	C₂²	1	2	yes	—
21	C₄₁×#475	—	2	2	no	—
22	C₄₁×#475	C₂²	1	2	yes	—
23	C₄₁×#464	—	2	2	no	—
24	C₄₁×#462	C₂²	1	2	yes	—
25	C₄₁×#461	—	2	2	no	—
26	C₄₁×#460	C₂²	1	2	yes	—
27	C₄₁×#446	C₂²	1	2	yes	—
28	C₄₁×#446	C₂²	1	2	yes	—
29	C₄₁×#441	—	2	2	no	—
30	C₄₁×#440	—	2	2	no	—
31	C₄×Q₁₆×C₄₁	C₂²	1	2	yes	—
32	C₄₁×#335	C₂²	1	2	yes	—
33	C₄₁×#333	C₂²	1	1	yes	characteristic
34	C₄₁×#332	C₂²	1	2	yes	—
35	C₄₁×#331	C₂²	1	1	yes	characteristic
36	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₄₁	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₄₁	—	2	2	no	—
40	C₂×Q₁₆×C₄₁	D₄	1	2	yes	—
41	C₂×SD₁₆×C₄₁	—	2	2	no	—
42	C₂×SD₁₆×C₄₁	—	2	2	no	—
43	C₂×SD₁₆×C₄₁	—	2	2	no	—
44	C₂×SD₁₆×C₄₁	—	4	1	no	—
45	C₄⋊Q₈×C₄₁	—	2	2	no	—
46	C₄⋊Q₈×C₄₁	—	2	2	no	—
47	C₄⋊Q₈×C₄₁	—	4	1	no	—
48	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
50	#127×C₄₁	C₂³	1	1	yes	characteristic
51	#125×C₄₁	—	2	2	no	—
52	#125×C₄₁	—	4	2	no	—
53	#124×C₄₁	—	2	2	no	—
54	#123×C₄₁	—	4	1	no	—
55	C₄×Q₈×C₄₁	—	2	2	no	—
56	C₄×Q₈×C₄₁	—	2	2	no	—
57	C₄×Q₈×C₄₁	—	2	2	no	—
58	C₄×Q₈×C₄₁	C₂³	1	2	yes	—
59	C₄×Q₈×C₄₁	C₂³	1	2	yes	—
60	C₄×D₄×C₄₁	—	2	2	no	—
61	#107×C₄₁	—	2	2	no	—
62	#107×C₄₁	D₄	1	2	yes	—
63	#106×C₄₁	—	2	2	no	—
64	#106×C₄₁	—	2	2	no	—
65	#106×C₄₁	—	2	2	no	—
66	#106×C₄₁	—	4	1	no	—
67	C₄⋊C₈×C₄₁	—	2	2	no	—
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	—	2	2	no	—
74	#103×C₄₁	D₄	1	2	yes	—
75	#103×C₄₁	D₄	1	2	yes	—
76	#102×C₄₁	—	2	2	no	—
77	#97×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	—	4	1	no	—
84	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
85	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
86	C₂×D₄×C₄₁	—	2	2	no	—
87	C₂×D₄×C₄₁	—	2	1	no	—
88	C₂²×C₄×C₄₁	—	4	1	no	—
89	Q₁₆×C₄₁	—	2	4	no	—
90	Q₁₆×C₄₁	—	8	2	no	—
91	SD₁₆×C₄₁	—	4	4	no	—
92	SD₁₆×C₄₁	—	8	2	no	—
93	SD₁₆×C₄₁	—	8	2	no	—
94	SD₁₆×C₄₁	—	8	2	no	—
95	C₂×C₈×C₄₁	—	2	1	no	—
96	C₂×C₈×C₄₁	—	2	2	no	—
97	C₂×C₈×C₄₁	—	2	2	no	—
98	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
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	—	2	2	no	—
105	C₄⋊C₄×C₄₁	—	4	2	no	—
106	C₄⋊C₄×C₄₁	—	4	2	no	—
107	C₄⋊C₄×C₄₁	—	4	2	no	—
108	C₄⋊C₄×C₄₁	—	4	2	no	—
109	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
110	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
111	C₂²⋊C₄×C₄₁	—	4	2	no	—
112	C₂²⋊C₄×C₄₁	—	4	1	no	—
113	C₄²×C₄₁	—	2	2	no	—
114	C₄²×C₄₁	—	2	2	no	—
115	C₄²×C₄₁	—	4	2	no	—
116	C₄²×C₄₁	—	4	1	no	—
117	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
118	C₂³×C₄₁	—	4	1	no	—
119	Q₈×C₄₁	—	2	2	no	—
120	Q₈×C₄₁	—	2	2	no	—
121	Q₈×C₄₁	—	2	2	no	—
122	Q₈×C₄₁	—	2	2	no	—
123	Q₈×C₄₁	—	4	2	no	—
124	Q₈×C₄₁	—	4	2	no	—
125	Q₈×C₄₁	—	8	1	no	—
126	Q₈×C₄₁	—	16	1	no	—
127	D₄×C₄₁	—	4	2	no	—
128	D₄×C₄₁	—	4	2	no	—
129	D₄×C₄₁	—	8	1	no	—
130	C₂×C₄×C₄₁	—	2	2	no	—
131	C₂×C₄×C₄₁	—	2	2	no	—
132	C₂×C₄×C₄₁	—	2	2	no	—
133	C₂×C₄×C₄₁	—	2	2	no	—
134	C₂×C₄×C₄₁	—	4	2	no	—
135	C₂×C₄×C₄₁	—	4	1	no	—
136	C₂×C₄×C₄₁	—	16	1	no	—
137	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
138	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
139	C₈×C₄₁	—	2	2	no	—
140	C₈×C₄₁	—	2	2	no	—
141	C₈×C₄₁	—	4	2	no	—
142	C₈×C₄₁	—	4	2	no	—
143	C₈×C₄₁	#137	1	4	yes	—
144	#20323	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂²×C₄₁	—	8	2	no	—
146	C₂²×C₄₁	—	8	1	no	—
147	C₂²×C₄₁	D₄²	1	1	yes	characteristic
148	C₄×C₄₁	—	2	1	no	—
149	C₄×C₄₁	—	4	2	no	—
150	C₄×C₄₁	—	4	2	no	—
151	C₄×C₄₁	—	4	2	no	—
152	C₄×C₄₁	—	4	2	no	—
153	C₄×C₄₁	—	8	2	no	—
154	C₄×C₄₁	—	8	1	no	—
155	C₄×C₄₁	C₂×#137	1	2	yes	—
156	C₄×C₄₁	#577	1	2	yes	—
157	#3400	C₂×C₄₁	1	2	yes	—
158	#3387	C₂×C₄₁	1	2	yes	complemented
159	#1705	C₂×C₄₁	1	1	yes	characteristic
160	#1692	C₂×C₄₁	1	2	yes	complemented
161	#1686	C₂×C₄₁	1	2	yes	—
162	#1653	C₂×C₄₁	1	2	yes	complemented
163	#1630	C₂×C₄₁	1	1	yes	characteristic
164	#1623	C₂×C₄₁	1	1	yes	characteristic
165	#1591	C₂×C₄₁	1	2	yes	complemented
166	C₂×C₄₁	—	16	1	no	—
167	C₂×C₄₁	#3278	1	2	yes	—
168	C₂×C₄₁	#3291	1	1	yes	characteristic
169	#559	C₂²×C₄₁	1	1	yes	characteristic
170	Q₈²	C₂²×C₄₁	1	1	yes	characteristic
171	#557	C₂²×C₄₁	1	2	yes	—
172	#501	—	2	2	no	—
173	#498	C₂²×C₄₁	1	2	yes	—
174	#498	C₂²×C₄₁	1	1	yes	characteristic
175	#497	—	2	2	no	—
176	#492	C₂²×C₄₁	1	2	yes	—
177	#492	C₂²×C₄₁	1	1	yes	characteristic
178	#479	—	2	2	no	—
179	#477	C₂²×C₄₁	1	2	yes	—
180	#475	—	2	2	no	—
181	#475	C₂²×C₄₁	1	2	yes	—
182	#464	—	2	2	no	—
183	#462	C₂²×C₄₁	1	2	yes	—
184	#461	—	2	2	no	—
185	#460	C₂²×C₄₁	1	2	yes	—
186	#446	C₂²×C₄₁	1	2	yes	—
187	#446	C₂²×C₄₁	1	2	yes	—
188	#441	—	2	2	no	—
189	#440	—	2	2	no	—
190	C₄×Q₁₆	C₂²×C₄₁	1	2	yes	—
191	#335	C₂²×C₄₁	1	2	yes	—
192	#333	C₂²×C₄₁	1	1	yes	characteristic
193	#332	C₂²×C₄₁	1	2	yes	—
194	#331	C₂²×C₄₁	1	1	yes	characteristic
195	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	—
196	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	—
197	C₈²	C₂²×C₄₁	1	1	yes	characteristic
198	C₄₁	#20323	1	1	yes	characteristic, direct product factor, Sylow
199	C₂×Q₁₆	—	2	2	no	—
200	C₂×Q₁₆	D₄×C₄₁	1	2	yes	—
201	C₂×SD₁₆	—	2	2	no	—
202	C₂×SD₁₆	—	2	2	no	—
203	C₂×SD₁₆	—	2	2	no	—
204	C₂×SD₁₆	—	4	1	no	—
205	C₄⋊Q₈	—	2	2	no	—
206	C₄⋊Q₈	—	2	2	no	—
207	C₄⋊Q₈	—	4	1	no	—
208	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
209	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
210	#127	C₂³×C₄₁	1	1	yes	characteristic
211	#125	—	2	2	no	—
212	#125	—	4	2	no	—
213	#124	—	2	2	no	—
214	#123	—	4	1	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	C₂³×C₄₁	1	2	yes	—
219	C₄×Q₈	C₂³×C₄₁	1	2	yes	—
220	C₄×D₄	—	2	2	no	—
221	#107	—	2	2	no	—
222	#107	D₄×C₄₁	1	2	yes	—
223	#106	—	2	2	no	—
224	#106	—	2	2	no	—
225	#106	—	2	2	no	—
226	#106	—	4	1	no	—
227	C₄⋊C₈	—	2	2	no	—
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
230	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
231	#103	—	2	2	no	—
232	#103	—	2	2	no	—
233	#103	—	2	2	no	—
234	#103	D₄×C₄₁	1	2	yes	—
235	#103	D₄×C₄₁	1	2	yes	—
236	#102	—	2	2	no	—
237	#97	—	2	2	no	—
238	#97	—	2	2	no	—
239	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
240	C₄×C₈	C₂³×C₄₁	1	2	yes	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
245	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	Q₁₆	—	2	4	no	—
250	Q₁₆	—	8	2	no	—
251	SD₁₆	—	4	4	no	—
252	SD₁₆	—	8	2	no	—
253	SD₁₆	—	8	2	no	—
254	SD₁₆	—	8	2	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
259	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
270	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	4	2	no	—
276	C₄²	—	4	1	no	—
277	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
278	C₂³	—	4	1	no	—
279	Q₈	—	2	2	no	—
280	Q₈	—	2	2	no	—
281	Q₈	—	2	2	no	—
282	Q₈	—	2	2	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	8	1	no	—
286	Q₈	—	16	1	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	8	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	16	1	no	—
297	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
298	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
299	C₈	—	2	2	no	—
300	C₈	—	2	2	no	—
301	C₈	—	4	2	no	—
302	C₈	—	4	2	no	—
303	C₈	#137×C₄₁	1	4	yes	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	1	no	—
306	C₂²	D₄²×C₄₁	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	4	2	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	1	no	—
314	C₄	C₂×#137×C₄₁	1	2	yes	—
315	C₄	C₄₁×#577	1	2	yes	—
316	C₂	—	16	1	no	—
317	C₂	C₄₁×#3278	1	2	yes	—
318	C₂	C₄₁×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	#127×C₄₁	C₂³
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
2. C₄₁×#3387	C₂	semidirect
4. C₄₁×#1692	C₂	semidirect
6. C₄₁×#1653	C₂	semidirect
9. C₄₁×#1591	C₂	semidirect
144. #20323	C₄₁	direct (N × H)
158. #3387	C₂×C₄₁	semidirect
160. #1692	C₂×C₄₁	semidirect
162. #1653	C₂×C₄₁	semidirect
165. #1591	C₂×C₄₁	semidirect
198. C₄₁	#20323	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×#3400	2
C₄₁×#3387	2
C₄₁×#1705	2
C₄₁×#1692	2
C₄₁×#1686	2
C₄₁×#1653	2
C₄₁×#1630	2
C₄₁×#1623	2
C₄₁×#1591	2
#20323	41

Subgroups — C41×#20323 #23796524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20323 #23796524