Subgroups — C₄₁×#18698 #23794899

Order 10496.

834 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3254	C₂	1	2	yes	maximal
3	C₄₁×#3131	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1636	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1626	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1561	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1493	C₂	1	2	yes	maximal, complemented
8	C₄₁×#564	—	2	2	no	—
9	C₄₁×#557	—	2	2	no	—
10	C₄₁×#557	—	2	2	no	—
11	C₄₁×#557	C₂²	1	4	yes	—
12	C₄₁×#541	—	2	4	no	—
13	C₄₁×#541	—	2	2	no	—
14	C₄₁×#533	C₂²	1	2	yes	—
15	C₄₁×#533	C₂²	1	1	yes	characteristic
16	C₄₁×#532	C₂²	1	1	yes	characteristic
17	C₄₁×#488	—	2	2	no	—
18	C₄₁×#487	C₂²	1	2	yes	—
19	C₄₁×#486	C₂²	1	2	yes	—
20	C₄₁×#477	C₂²	1	4	yes	—
21	C₄₁×#474	—	2	2	no	—
22	C₄₁×#468	—	2	4	no	—
23	C₄₁×#461	—	2	2	no	—
24	C₄₁×#460	C₂²	1	4	yes	—
25	C₄₁×#423	C₂²	1	2	yes	—
26	C₄₁×#416	—	2	2	no	—
27	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
28	C₄₁×#332	C₂²	1	4	yes	—
29	C₄₁×Q₈⋊C₈	C₂²	1	8	yes	—
30	C₂×#41×C₄₁	—	2	1	no	—
31	C₂×SD₁₆×C₄₁	—	2	4	no	—
32	C₂×M₁₆×C₄₁	—	2	2	no	—
33	C₂²×C₈×C₄₁	—	2	1	no	—
34	C₄⋊Q₈×C₄₁	—	2	2	no	—
35	C₄⋊Q₈×C₄₁	—	2	4	no	—
36	C₄⋊Q₈×C₄₁	D₄	1	2	yes	—
37	C₄⋊Q₈×C₄₁	D₄	1	2	yes	—
38	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
40	#127×C₄₁	C₂³	1	1	yes	characteristic
41	#126×C₄₁	—	2	8	no	—
42	#125×C₄₁	—	2	2	no	—
43	#125×C₄₁	—	2	8	no	—
44	#125×C₄₁	—	2	8	no	—
45	#125×C₄₁	—	2	4	no	—
46	#125×C₄₁	—	2	8	no	—
47	#125×C₄₁	—	2	2	no	—
48	#125×C₄₁	D₄	1	2	yes	—
49	#124×C₄₁	—	2	1	no	—
50	C₂²⋊Q₈×C₄₁	—	2	2	no	—
51	C₂²⋊Q₈×C₄₁	—	2	2	no	—
52	C₂²⋊Q₈×C₄₁	D₄	1	4	yes	—
53	#121×C₄₁	—	2	2	no	—
54	C₄×Q₈×C₄₁	—	2	4	no	—
55	C₄×Q₈×C₄₁	—	2	4	no	—
56	C₄×Q₈×C₄₁	—	2	4	no	—
57	C₄×Q₈×C₄₁	—	2	2	no	—
58	C₄×Q₈×C₄₁	—	2	2	no	—
59	C₄×Q₈×C₄₁	C₂³	1	4	yes	—
60	C₄×D₄×C₄₁	—	2	2	no	—
61	#117×C₄₁	—	2	2	no	—
62	#117×C₄₁	D₄	1	2	yes	—
63	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
64	#107×C₄₁	—	2	4	no	—
65	C₄⋊C₈×C₄₁	—	2	2	no	—
66	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
67	#103×C₄₁	—	2	4	no	—
68	#103×C₄₁	—	2	4	no	—
69	#102×C₄₁	—	2	4	no	—
70	#102×C₄₁	—	2	4	no	—
71	C₄×C₈×C₄₁	—	2	1	no	—
72	C₄×C₈×C₄₁	C₂³	1	2	yes	—
73	#41×C₄₁	—	4	4	no	—
74	C₂×Q₈×C₄₁	—	2	2	no	—
75	C₂×Q₈×C₄₁	—	2	2	no	—
76	C₂×Q₈×C₄₁	—	2	1	no	—
77	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
78	C₂×D₄×C₄₁	—	2	1	no	—
79	C₂×D₄×C₄₁	—	2	2	no	—
80	C₂×D₄×C₄₁	—	4	1	no	—
81	C₂²×C₄×C₄₁	—	4	1	no	—
82	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
84	SD₁₆×C₄₁	—	8	4	no	—
85	M₁₆×C₄₁	—	8	2	no	—
86	C₂×C₈×C₄₁	—	2	2	no	—
87	C₂×C₈×C₄₁	—	2	4	no	—
88	C₂×C₈×C₄₁	—	4	2	no	—
89	C₄⋊C₄×C₄₁	—	2	8	no	—
90	C₄⋊C₄×C₄₁	—	2	8	no	—
91	C₄⋊C₄×C₄₁	—	2	8	no	—
92	C₄⋊C₄×C₄₁	—	2	2	no	—
93	C₄⋊C₄×C₄₁	—	2	2	no	—
94	C₄⋊C₄×C₄₁	—	2	2	no	—
95	C₄⋊C₄×C₄₁	—	2	4	no	—
96	C₄⋊C₄×C₄₁	—	2	4	no	—
97	C₄⋊C₄×C₄₁	—	2	2	no	—
98	C₄⋊C₄×C₄₁	—	2	4	no	—
99	C₄⋊C₄×C₄₁	—	2	2	no	—
100	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
101	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
102	C₂²⋊C₄×C₄₁	—	2	4	no	—
103	C₂²⋊C₄×C₄₁	—	2	2	no	—
104	C₂²⋊C₄×C₄₁	—	4	2	no	—
105	C₄²×C₄₁	—	2	8	no	—
106	C₄²×C₄₁	—	2	2	no	—
107	C₄²×C₄₁	—	2	4	no	—
108	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₄₁	C₂×D₄	1	2	yes	—
110	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
111	C₂³×C₄₁	—	4	1	no	—
112	C₂³×C₄₁	#120	1	1	yes	characteristic
113	Q₈×C₄₁	—	2	4	no	—
114	Q₈×C₄₁	—	2	4	no	—
115	Q₈×C₄₁	—	4	2	no	—
116	Q₈×C₄₁	—	4	2	no	—
117	Q₈×C₄₁	—	4	2	no	—
118	Q₈×C₄₁	—	4	4	no	—
119	D₄×C₄₁	—	4	2	no	—
120	D₄×C₄₁	—	4	2	no	—
121	D₄×C₄₁	—	4	2	no	—
122	D₄×C₄₁	—	8	2	no	—
123	C₂×C₄×C₄₁	—	2	2	no	—
124	C₂×C₄×C₄₁	—	2	2	no	—
125	C₂×C₄×C₄₁	—	2	2	no	—
126	C₂×C₄×C₄₁	—	2	4	no	—
127	C₂×C₄×C₄₁	—	2	4	no	—
128	C₂×C₄×C₄₁	—	2	4	no	—
129	C₂×C₄×C₄₁	—	4	1	no	—
130	C₂×C₄×C₄₁	—	8	2	no	—
131	C₂×C₄×C₄₁	#120	1	2	yes	—
132	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
133	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
135	C₈×C₄₁	—	4	2	no	—
136	C₈×C₄₁	—	4	4	no	—
137	#18698	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂²×C₄₁	—	2	2	no	—
139	C₂²×C₄₁	—	2	1	no	—
140	C₂²×C₄₁	—	8	2	no	—
141	C₂²×C₄₁	—	8	1	no	—
142	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
143	C₄×C₄₁	—	2	2	no	—
144	C₄×C₄₁	—	2	1	no	—
145	C₄×C₄₁	—	4	4	no	—
146	C₄×C₄₁	—	4	4	no	—
147	C₄×C₄₁	—	4	4	no	—
148	C₄×C₄₁	—	8	1	no	—
149	C₄×C₄₁	#454	1	2	yes	—
150	C₄×C₄₁	#577	1	2	yes	—
151	C₄×C₄₁	#578	1	2	yes	—
152	#3556	C₂×C₄₁	1	1	yes	characteristic, complemented
153	#3254	C₂×C₄₁	1	2	yes	—
154	#3131	C₂×C₄₁	1	1	yes	characteristic
155	#1636	C₂×C₄₁	1	4	yes	complemented
156	#1626	C₂×C₄₁	1	4	yes	complemented
157	#1561	C₂×C₄₁	1	1	yes	characteristic, complemented
158	#1493	C₂×C₄₁	1	2	yes	complemented
159	C₂×C₄₁	—	4	1	no	—
160	C₂×C₄₁	—	16	1	no	—
161	C₂×C₄₁	#3013	1	2	yes	—
162	C₂×C₄₁	C₂×#454	1	1	yes	characteristic
163	#564	—	2	2	no	—
164	#557	—	2	2	no	—
165	#557	—	2	2	no	—
166	#557	C₂²×C₄₁	1	4	yes	—
167	#541	—	2	4	no	—
168	#541	—	2	2	no	—
169	#533	C₂²×C₄₁	1	2	yes	—
170	#533	C₂²×C₄₁	1	1	yes	characteristic
171	#532	C₂²×C₄₁	1	1	yes	characteristic
172	#488	—	2	2	no	—
173	#487	C₂²×C₄₁	1	2	yes	—
174	#486	C₂²×C₄₁	1	2	yes	—
175	#477	C₂²×C₄₁	1	4	yes	—
176	#474	—	2	2	no	—
177	#468	—	2	4	no	—
178	#461	—	2	2	no	—
179	#460	C₂²×C₄₁	1	4	yes	—
180	#423	C₂²×C₄₁	1	2	yes	—
181	#416	—	2	2	no	—
182	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
183	#332	C₂²×C₄₁	1	4	yes	—
184	Q₈⋊C₈	C₂²×C₄₁	1	8	yes	—
185	C₄₁	#18698	1	1	yes	characteristic, direct product factor, Sylow
186	C₂×#41	—	2	1	no	—
187	C₂×SD₁₆	—	2	4	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₄⋊Q₈	—	2	2	no	—
191	C₄⋊Q₈	—	2	4	no	—
192	C₄⋊Q₈	D₄×C₄₁	1	2	yes	—
193	C₄⋊Q₈	D₄×C₄₁	1	2	yes	—
194	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
195	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
196	#127	C₂³×C₄₁	1	1	yes	characteristic
197	#126	—	2	8	no	—
198	#125	—	2	2	no	—
199	#125	—	2	8	no	—
200	#125	—	2	8	no	—
201	#125	—	2	4	no	—
202	#125	—	2	8	no	—
203	#125	—	2	2	no	—
204	#125	D₄×C₄₁	1	2	yes	—
205	#124	—	2	1	no	—
206	C₂²⋊Q₈	—	2	2	no	—
207	C₂²⋊Q₈	—	2	2	no	—
208	C₂²⋊Q₈	D₄×C₄₁	1	4	yes	—
209	#121	—	2	2	no	—
210	C₄×Q₈	—	2	4	no	—
211	C₄×Q₈	—	2	4	no	—
212	C₄×Q₈	—	2	4	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	C₂³×C₄₁	1	4	yes	—
216	C₄×D₄	—	2	2	no	—
217	#117	—	2	2	no	—
218	#117	D₄×C₄₁	1	2	yes	—
219	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
220	#107	—	2	4	no	—
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
223	#103	—	2	4	no	—
224	#103	—	2	4	no	—
225	#102	—	2	4	no	—
226	#102	—	2	4	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	C₂³×C₄₁	1	2	yes	—
229	#41	—	4	4	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	1	no	—
233	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
240	SD₁₆	—	8	4	no	—
241	M₁₆	—	8	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	2	no	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	2	8	no	—
247	C₄⋊C₄	—	2	8	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
257	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
258	C₂²⋊C₄	—	2	4	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₄²	—	2	8	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	4	no	—
264	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
265	C₄²	C₂×D₄×C₄₁	1	2	yes	—
266	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
267	C₂³	—	4	1	no	—
268	C₂³	#120×C₄₁	1	1	yes	characteristic
269	Q₈	—	2	4	no	—
270	Q₈	—	2	4	no	—
271	Q₈	—	4	2	no	—
272	Q₈	—	4	2	no	—
273	Q₈	—	4	2	no	—
274	Q₈	—	4	4	no	—
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	8	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	#120×C₄₁	1	2	yes	—
288	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
289	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
290	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
291	C₈	—	4	2	no	—
292	C₈	—	4	4	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	1	no	—
297	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	4	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	4	no	—
303	C₄	—	8	1	no	—
304	C₄	C₄₁×#454	1	2	yes	—
305	C₄	C₄₁×#577	1	2	yes	—
306	C₄	C₄₁×#578	1	2	yes	—
307	C₂	—	4	1	no	—
308	C₂	—	16	1	no	—
309	C₂	C₄₁×#3013	1	2	yes	—
310	C₂	C₂×C₄₁×#454	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₄₁×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3556	C₂	semidirect
4. C₄₁×#1636	C₂	semidirect
5. C₄₁×#1626	C₂	semidirect
6. C₄₁×#1561	C₂	semidirect
7. C₄₁×#1493	C₂	semidirect
137. #18698	C₄₁	direct (N × H)
152. #3556	C₂×C₄₁	semidirect
155. #1636	C₂×C₄₁	semidirect
156. #1626	C₂×C₄₁	semidirect
157. #1561	C₂×C₄₁	semidirect
158. #1493	C₂×C₄₁	semidirect
185. C₄₁	#18698	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×#3556	2
C₄₁×#3254	2
C₄₁×#3131	2
C₄₁×#1636	2
C₄₁×#1626	2
C₄₁×#1561	2
C₄₁×#1493	2
#18698	41

Subgroups — C41×#18698 #23794899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#18698 #23794899