Subgroups — C₄₁×#33600 #23809801

Order 10496.

710 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3050	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2217	C₂	1	4	yes	maximal, complemented
5	C₄₁×#2216	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2215	C₂	1	2	yes	maximal, complemented
7	C₄₁×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#2138	C₂	1	2	yes	maximal, complemented
9	C₂×#135×C₄₁	C₂²	1	2	yes	—
10	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
11	C₄₁×#525	—	2	2	no	—
12	C₄₁×#508	—	2	4	no	—
13	C₂×Q₃₂×C₄₁	C₂²	1	1	yes	characteristic, complemented
14	C₂×SD₃₂×C₄₁	C₂²	1	2	yes	complemented
15	C₂×D₁₆×C₄₁	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₁₆×C₄₁	—	2	4	no	—
17	C₂²×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₄₁	C₂²	1	2	yes	—
19	C₂²×C₁₆×C₄₁	C₂²	1	2	yes	—
20	C₂²×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
21	C₄₁×#467	C₂²	1	4	yes	—
22	C₄₁×#466	C₂²	1	4	yes	—
23	C₄₁×#465	—	2	4	no	—
24	C₄₁×#427	C₂²	1	2	yes	—
25	C₄₁×#416	—	2	2	no	—
26	C₄₁×#367	C₂²	1	2	yes	complemented
27	C₄₁×#366	C₂²	1	2	yes	complemented
28	C₄₁×#358	C₂²	1	4	yes	complemented
29	C₄₁×#357	C₂²	1	4	yes	complemented
30	C₂×#41×C₄₁	—	2	2	no	—
31	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
32	#135×C₄₁	—	2	4	no	—
33	#135×C₄₁	—	2	4	no	—
34	C₂×Q₁₆×C₄₁	C₂³	1	2	yes	—
35	C₂×SD₁₆×C₄₁	—	2	2	no	—
36	C₂×D₈×C₄₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₄₁	—	2	2	no	—
38	C₂²×C₈×C₄₁	—	2	2	no	—
39	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₁	D₄	1	2	yes	—
41	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
43	#123×C₄₁	—	4	2	no	—
44	C₂²⋊Q₈×C₄₁	—	2	4	no	—
45	#121×C₄₁	—	2	4	no	—
46	#120×C₄₁	—	4	2	no	—
47	C₄×D₄×C₄₁	—	4	4	no	—
48	#117×C₄₁	—	2	2	no	—
49	Q₃₂×C₄₁	—	2	2	no	—
50	SD₃₂×C₄₁	—	2	4	no	—
51	D₁₆×C₄₁	—	2	2	no	—
52	C₂×C₁₆×C₄₁	—	2	2	no	—
53	C₂×C₁₆×C₄₁	—	2	4	no	—
54	C₂×C₁₆×C₄₁	—	2	4	no	—
55	C₂×C₁₆×C₄₁	D₄	1	4	yes	complemented
56	C₂×C₁₆×C₄₁	C₂³	1	2	yes	—
57	C₂×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
59	#107×C₄₁	C₂³	1	4	yes	—
60	#106×C₄₁	—	2	2	no	—
61	#103×C₄₁	—	2	4	no	—
62	#102×C₄₁	—	2	4	no	—
63	C₂⁴×C₄₁	D₈	1	1	yes	characteristic
64	#41×C₄₁	—	4	4	no	—
65	#41×C₄₁	—	4	4	no	—
66	C₂×Q₈×C₄₁	—	2	2	no	—
67	C₂×D₄×C₄₁	—	2	2	no	—
68	C₂×D₄×C₄₁	—	4	4	no	—
69	C₂×D₄×C₄₁	—	4	2	no	—
70	C₂²×C₄×C₄₁	—	2	2	no	—
71	C₂²×C₄×C₄₁	—	2	2	no	—
72	C₂²×C₄×C₄₁	—	4	2	no	—
73	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₄₁	D₈	1	2	yes	—
75	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₄₁	—	2	2	no	—
78	Q₁₆×C₄₁	—	2	2	no	—
79	SD₁₆×C₄₁	—	4	4	no	—
80	D₈×C₄₁	—	2	2	no	—
81	D₈×C₄₁	—	2	2	no	—
82	C₂×C₈×C₄₁	—	2	2	no	—
83	C₂×C₈×C₄₁	—	2	4	no	—
84	C₂×C₈×C₄₁	—	2	2	no	—
85	C₂×C₈×C₄₁	—	2	2	no	—
86	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₄₁	#41	1	2	yes	—
90	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₄₁	—	2	4	no	—
92	C₄⋊C₄×C₄₁	—	4	4	no	—
93	C₂²⋊C₄×C₄₁	—	4	2	no	—
94	C₂²⋊C₄×C₄₁	—	4	4	no	—
95	C₂²⋊C₄×C₄₁	—	4	4	no	—
96	C₄²×C₄₁	—	4	2	no	—
97	C₁₆×C₄₁	—	2	2	no	—
98	C₁₆×C₄₁	C₂×D₄	1	2	yes	—
99	C₁₆×C₄₁	C₂×D₄	1	2	yes	—
100	C₂³×C₄₁	—	2	2	no	—
101	C₂³×C₄₁	—	2	4	no	—
102	C₂³×C₄₁	—	4	2	no	—
103	C₂³×C₄₁	C₂×D₈	1	1	yes	characteristic
104	C₂³×C₄₁	C₂×D₈	1	2	yes	—
105	Q₈×C₄₁	—	4	2	no	—
106	Q₈×C₄₁	—	4	2	no	—
107	D₄×C₄₁	—	4	2	no	—
108	D₄×C₄₁	—	4	2	no	—
109	D₄×C₄₁	—	8	4	no	—
110	D₄×C₄₁	—	8	8	no	—
111	C₂×C₄×C₄₁	—	2	2	no	—
112	C₂×C₄×C₄₁	—	2	2	no	—
113	C₂×C₄×C₄₁	—	2	2	no	—
114	C₂×C₄×C₄₁	—	2	4	no	—
115	C₂×C₄×C₄₁	—	4	4	no	—
116	C₂×C₄×C₄₁	—	4	2	no	—
117	C₂×C₄×C₄₁	—	8	4	no	—
118	C₂×C₄×C₄₁	#121	1	2	yes	—
119	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₄₁	#135	1	2	yes	—
122	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₄₁	—	2	2	no	—
124	C₈×C₄₁	#121	1	2	yes	—
125	C₈×C₄₁	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₄₁	C₂×#41	1	1	yes	characteristic
127	#33600	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₄₁	—	2	8	no	—
129	C₂²×C₄₁	—	2	2	no	—
130	C₂²×C₄₁	—	2	2	no	—
131	C₂²×C₄₁	—	2	2	no	—
132	C₂²×C₄₁	—	8	4	no	—
133	C₂²×C₄₁	—	8	2	no	—
134	C₂²×C₄₁	#466	1	2	yes	—
135	C₂²×C₄₁	#508	1	4	yes	complemented
136	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
137	C₄×C₄₁	—	2	2	no	—
138	C₄×C₄₁	—	8	4	no	—
139	C₄×C₄₁	—	8	2	no	—
140	C₄×C₄₁	#466	1	2	yes	—
141	C₄×C₄₁	C₂×#121	1	1	yes	characteristic
142	C₄×C₄₁	C₂×#135	1	1	yes	characteristic
143	C₂×#508	C₂×C₄₁	1	1	yes	characteristic, complemented
144	C₂³×C₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
145	#3050	C₂×C₄₁	1	2	yes	complemented
146	#2217	C₂×C₄₁	1	4	yes	complemented
147	#2216	C₂×C₄₁	1	2	yes	complemented
148	#2215	C₂×C₄₁	1	2	yes	complemented
149	#2158	C₂×C₄₁	1	1	yes	characteristic, complemented
150	#2138	C₂×C₄₁	1	2	yes	complemented
151	C₂×C₄₁	—	2	2	no	—
152	C₂×C₄₁	—	2	4	no	—
153	C₂×C₄₁	—	16	2	no	—
154	C₂×C₄₁	C₂×#466	1	1	yes	characteristic
155	C₂×C₄₁	C₂×#508	1	2	yes	—
156	C₂×#135	C₂²×C₄₁	1	2	yes	—
157	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
158	#525	—	2	2	no	—
159	#508	—	2	4	no	—
160	C₂×Q₃₂	C₂²×C₄₁	1	1	yes	characteristic, complemented
161	C₂×SD₃₂	C₂²×C₄₁	1	2	yes	complemented
162	C₂×D₁₆	C₂²×C₄₁	1	1	yes	characteristic, complemented
163	C₂²×C₁₆	—	2	4	no	—
164	C₂²×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
165	C₂²×C₁₆	C₂²×C₄₁	1	2	yes	—
166	C₂²×C₁₆	C₂²×C₄₁	1	2	yes	—
167	C₂²×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
168	#467	C₂²×C₄₁	1	4	yes	—
169	#466	C₂²×C₄₁	1	4	yes	—
170	#465	—	2	4	no	—
171	#427	C₂²×C₄₁	1	2	yes	—
172	#416	—	2	2	no	—
173	#367	C₂²×C₄₁	1	2	yes	complemented
174	#366	C₂²×C₄₁	1	2	yes	complemented
175	#358	C₂²×C₄₁	1	4	yes	complemented
176	#357	C₂²×C₄₁	1	4	yes	complemented
177	C₄₁	#33600	1	1	yes	characteristic, direct product factor, Sylow
178	C₂×#41	—	2	2	no	—
179	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
180	#135	—	2	4	no	—
181	#135	—	2	4	no	—
182	C₂×Q₁₆	C₂³×C₄₁	1	2	yes	—
183	C₂×SD₁₆	—	2	2	no	—
184	C₂×D₈	C₂³×C₄₁	1	2	yes	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
188	C₂²×C₈	D₄×C₄₁	1	2	yes	—
189	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
190	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
191	#123	—	4	2	no	—
192	C₂²⋊Q₈	—	2	4	no	—
193	#121	—	2	4	no	—
194	#120	—	4	2	no	—
195	C₄×D₄	—	4	4	no	—
196	#117	—	2	2	no	—
197	Q₃₂	—	2	2	no	—
198	SD₃₂	—	2	4	no	—
199	D₁₆	—	2	2	no	—
200	C₂×C₁₆	—	2	2	no	—
201	C₂×C₁₆	—	2	4	no	—
202	C₂×C₁₆	—	2	4	no	—
203	C₂×C₁₆	D₄×C₄₁	1	4	yes	complemented
204	C₂×C₁₆	C₂³×C₄₁	1	2	yes	—
205	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
206	C₂×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
207	#107	C₂³×C₄₁	1	4	yes	—
208	#106	—	2	2	no	—
209	#103	—	2	4	no	—
210	#102	—	2	4	no	—
211	C₂⁴	D₈×C₄₁	1	1	yes	characteristic
212	#41	—	4	4	no	—
213	#41	—	4	4	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	4	4	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	D₈×C₄₁	1	2	yes	—
222	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
225	Q₁₆	—	2	2	no	—
226	Q₁₆	—	2	2	no	—
227	SD₁₆	—	4	4	no	—
228	D₈	—	2	2	no	—
229	D₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
235	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
236	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
237	C₂×C₈	#41×C₄₁	1	2	yes	—
238	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₂²⋊C₄	—	4	4	no	—
244	C₄²	—	4	2	no	—
245	C₁₆	—	2	2	no	—
246	C₁₆	C₂×D₄×C₄₁	1	2	yes	—
247	C₁₆	C₂×D₄×C₄₁	1	2	yes	—
248	C₂³	—	2	4	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	4	2	no	—
251	C₂³	C₂×D₈×C₄₁	1	2	yes	—
252	C₂³	C₂×D₈×C₄₁	1	1	yes	characteristic
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	D₄	—	8	8	no	—
258	D₄	—	8	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	8	4	no	—
266	C₂×C₄	#121×C₄₁	1	2	yes	—
267	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
268	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
269	C₂×C₄	#135×C₄₁	1	2	yes	—
270	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	#121×C₄₁	1	2	yes	—
273	C₈	C₂²×D₄×C₄₁	1	1	yes	characteristic
274	C₈	C₂×#41×C₄₁	1	1	yes	characteristic
275	C₂²	—	2	8	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	4	no	—
281	C₂²	C₄₁×#466	1	2	yes	—
282	C₂²	C₄₁×#508	1	4	yes	complemented
283	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	8	2	no	—
286	C₄	—	8	4	no	—
287	C₄	C₄₁×#466	1	2	yes	—
288	C₄	C₂×#121×C₄₁	1	1	yes	characteristic
289	C₄	C₂×#135×C₄₁	1	1	yes	characteristic
290	C₂	—	2	4	no	—
291	C₂	—	2	2	no	—
292	C₂	—	16	2	no	—
293	C₂	C₂×C₄₁×#466	1	1	yes	characteristic
294	C₂	C₂×C₄₁×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
socle	C₂²×C₄₁	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ C₂³×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#508	C₂	semidirect
2. C₂³×C₁₆×C₄₁	C₂	semidirect
3. C₄₁×#3050	C₂	semidirect
4. C₄₁×#2217	C₂	semidirect
5. C₄₁×#2216	C₂	semidirect
6. C₄₁×#2215	C₂	semidirect
7. C₄₁×#2158	C₂	semidirect
8. C₄₁×#2138	C₂	semidirect
13. C₂×Q₃₂×C₄₁	C₂²	semidirect
14. C₂×SD₃₂×C₄₁	C₂²	semidirect
15. C₂×D₁₆×C₄₁	C₂²	semidirect
26. C₄₁×#367	C₂²	semidirect
27. C₄₁×#366	C₂²	semidirect
28. C₄₁×#358	C₂²	semidirect
29. C₄₁×#357	C₂²	semidirect
55. C₂×C₁₆×C₄₁	D₄	semidirect
127. #33600	C₄₁	direct (N × H)
135. C₂²×C₄₁	#508	semidirect
143. C₂×#508	C₂×C₄₁	semidirect
144. C₂³×C₁₆	C₂×C₄₁	semidirect
145. #3050	C₂×C₄₁	semidirect
146. #2217	C₂×C₄₁	semidirect
147. #2216	C₂×C₄₁	semidirect
148. #2215	C₂×C₄₁	semidirect
149. #2158	C₂×C₄₁	semidirect
150. #2138	C₂×C₄₁	semidirect
160. C₂×Q₃₂	C₂²×C₄₁	semidirect
161. C₂×SD₃₂	C₂²×C₄₁	semidirect
162. C₂×D₁₆	C₂²×C₄₁	semidirect
173. #367	C₂²×C₄₁	semidirect
174. #366	C₂²×C₄₁	semidirect
175. #358	C₂²×C₄₁	semidirect
176. #357	C₂²×C₄₁	semidirect
177. C₄₁	#33600	direct (N × H)
203. C₂×C₁₆	D₄×C₄₁	semidirect
282. C₂²	C₄₁×#508	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#508	2
C₂³×C₁₆×C₄₁	2
C₄₁×#3050	2
C₄₁×#2217	2
C₄₁×#2216	2
C₄₁×#2215	2
C₄₁×#2158	2
C₄₁×#2138	2
#33600	41

Subgroups — C41×#33600 #23809801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33600 #23809801