Subgroups — C₂×C₄₁×#2207 #23809777

Order 10496.

806 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SD₃₂×C₄₁	C₂	1	2	yes	maximal, complemented
2	C₂×C₄₁×#493	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2207	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₁×#357	C₂	1	2	yes	maximal, complemented
7	C₂²×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
8	C₂²×D₈×C₄₁	—	2	1	no	—
9	C₂²×D₈×C₄₁	C₂²	1	2	yes	—
10	C₂×#127×C₄₁	—	2	1	no	—
11	C₂×C₄×Q₈×C₄₁	—	2	1	no	—
12	C₂×SD₃₂×C₄₁	—	2	8	no	—
13	C₂×SD₃₂×C₄₁	C₂²	1	8	yes	complemented
14	C₂²×C₁₆×C₄₁	C₂²	1	2	yes	—
15	C₄₁×#493	—	2	2	no	—
16	C₄₁×#493	C₂²	1	4	yes	—
17	C₄×Q₁₆×C₄₁	—	2	2	no	—
18	C₄×Q₁₆×C₄₁	C₂²	1	4	yes	complemented
19	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×#103×C₄₁	—	2	1	no	—
21	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
22	C₄₁×C₄⋊C₁₆	C₂²	1	4	yes	complemented
23	C₄₁×#357	C₂²	1	8	yes	complemented
24	C₂²×Q₈×C₄₁	—	2	1	no	—
25	C₂²×D₄×C₄₁	—	2	2	no	—
26	C₂²×D₄×C₄₁	—	4	1	no	—
27	C₂²×D₄×C₄₁	—	4	1	no	—
28	C₂×Q₁₆×C₄₁	—	2	1	no	—
29	C₂×Q₁₆×C₄₁	—	2	2	no	—
30	C₂×Q₁₆×C₄₁	D₄	1	4	yes	complemented
31	C₂×Q₁₆×C₄₁	C₂³	1	2	yes	—
32	C₂×D₈×C₄₁	—	2	4	no	—
33	C₂×D₈×C₄₁	—	2	4	no	—
34	C₂×D₈×C₄₁	—	2	2	no	—
35	C₂×D₈×C₄₁	—	2	4	no	—
36	C₂×D₈×C₄₁	—	4	4	no	—
37	C₂×D₈×C₄₁	C₂³	1	4	yes	—
38	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
40	#127×C₄₁	—	2	4	no	—
41	#127×C₄₁	—	4	2	no	—
42	C₄×Q₈×C₄₁	—	2	4	no	—
43	C₄×Q₈×C₄₁	—	4	2	no	—
44	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
45	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
46	C₂×C₄²×C₄₁	—	4	1	no	—
47	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
48	SD₃₂×C₄₁	—	2	16	no	—
49	C₂×C₁₆×C₄₁	—	2	4	no	—
50	C₂×C₁₆×C₄₁	C₂³	1	4	yes	—
51	#107×C₄₁	—	2	1	no	—
52	#107×C₄₁	C₂³	1	2	yes	—
53	#103×C₄₁	—	2	4	no	—
54	C₄×C₈×C₄₁	D₄	1	2	yes	—
55	C₄×C₈×C₄₁	C₂³	1	2	yes	—
56	C₂⁴×C₄₁	—	4	2	no	—
57	C₂×Q₈×C₄₁	—	2	4	no	—
58	C₂×Q₈×C₄₁	—	2	2	no	—
59	C₂×Q₈×C₄₁	—	4	1	no	—
60	C₂×Q₈×C₄₁	—	4	2	no	—
61	C₂×D₄×C₄₁	—	2	4	no	—
62	C₂×D₄×C₄₁	—	4	4	no	—
63	C₂×D₄×C₄₁	—	4	4	no	—
64	C₂×D₄×C₄₁	—	4	4	no	—
65	C₂×D₄×C₄₁	—	4	2	no	—
66	C₂×D₄×C₄₁	—	4	4	no	—
67	C₂×D₄×C₄₁	—	8	4	no	—
68	C₂×D₄×C₄₁	—	8	4	no	—
69	C₂²×C₄×C₄₁	—	4	1	no	—
70	C₂²×C₄×C₄₁	—	4	1	no	—
71	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
74	Q₁₆×C₄₁	—	2	4	no	—
75	Q₁₆×C₄₁	—	2	2	no	—
76	Q₁₆×C₄₁	C₂×D₄	1	4	yes	complemented
77	D₈×C₄₁	—	2	4	no	—
78	D₈×C₄₁	—	2	4	no	—
79	D₈×C₄₁	—	4	8	no	—
80	D₈×C₄₁	—	4	4	no	—
81	C₂×C₈×C₄₁	—	2	2	no	—
82	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₄₁	#41	1	2	yes	—
86	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₄₁	—	2	2	no	—
88	C₄⋊C₄×C₄₁	—	4	1	no	—
89	C₄⋊C₄×C₄₁	—	4	4	no	—
90	C₄²×C₄₁	—	4	4	no	—
91	C₄²×C₄₁	D₈	1	2	yes	—
92	C₄²×C₄₁	C₂×D₄	1	2	yes	—
93	C₁₆×C₄₁	—	2	4	no	—
94	C₂³×C₄₁	—	4	4	no	—
95	C₂³×C₄₁	—	8	4	no	—
96	C₂³×C₄₁	—	8	4	no	—
97	C₂³×C₄₁	—	8	4	no	—
98	C₂³×C₄₁	C₂×D₈	1	1	yes	characteristic
99	Q₈×C₄₁	—	2	4	no	—
100	Q₈×C₄₁	—	4	2	no	—
101	Q₈×C₄₁	—	4	4	no	—
102	D₄×C₄₁	—	4	4	no	—
103	D₄×C₄₁	—	4	4	no	—
104	D₄×C₄₁	—	8	8	no	—
105	D₄×C₄₁	—	8	4	no	—
106	D₄×C₄₁	—	8	8	no	—
107	C₂×C₄×C₄₁	—	2	2	no	—
108	C₂×C₄×C₄₁	—	4	2	no	—
109	C₂×C₄×C₄₁	—	4	4	no	—
110	C₂×C₄×C₄₁	—	4	2	no	—
111	C₂×C₄×C₄₁	—	8	2	no	—
112	C₂×C₄×C₄₁	SD₃₂	1	4	yes	complemented
113	C₂×C₄×C₄₁	#121	1	2	yes	—
114	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
118	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
119	C₈×C₄₁	—	2	2	no	—
120	C₈×C₄₁	#121	1	2	yes	—
121	C₈×C₄₁	C₂²×D₄	1	1	yes	characteristic
122	C₈×C₄₁	C₂×#41	1	1	yes	characteristic
123	C₂×#2207	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₄₁	—	8	4	no	—
125	C₂²×C₄₁	—	8	4	no	—
126	C₂²×C₄₁	—	8	4	no	—
127	C₂²×C₄₁	—	16	8	no	—
128	C₂²×C₄₁	#459	1	2	yes	—
129	C₂²×C₄₁	C₂×SD₃₂	1	2	yes	—
130	C₂²×C₄₁	#509	1	2	yes	—
131	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₄₁	—	2	2	no	—
133	C₄×C₄₁	—	4	4	no	—
134	C₄×C₄₁	—	8	2	no	—
135	C₄×C₄₁	#459	1	2	yes	—
136	C₄×C₄₁	C₂×SD₃₂	1	4	yes	complemented
137	C₄×C₄₁	C₂×#121	1	1	yes	characteristic
138	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂²×SD₃₂	C₂×C₄₁	1	2	yes	complemented
140	C₂×#493	C₂×C₄₁	1	1	yes	characteristic
141	C₂×C₄×Q₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
142	#2207	C₂×C₄₁	1	8	yes	direct product factor
143	C₂×C₄⋊C₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
144	C₂×#357	C₂×C₄₁	1	2	yes	complemented
145	C₂×C₄₁	—	16	4	no	—
146	C₂×C₄₁	#2207	1	4	yes	direct product factor
147	C₂×C₄₁	C₂×#459	1	1	yes	characteristic
148	C₂×C₄₁	C₂²×SD₃₂	1	1	yes	characteristic
149	C₂×C₄₁	C₂×#509	1	1	yes	characteristic
150	C₂²×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
151	C₂²×D₈	—	2	1	no	—
152	C₂²×D₈	C₂²×C₄₁	1	2	yes	—
153	C₂×#127	—	2	1	no	—
154	C₂×C₄×Q₈	—	2	1	no	—
155	C₂×SD₃₂	—	2	8	no	—
156	C₂×SD₃₂	C₂²×C₄₁	1	8	yes	complemented
157	C₂²×C₁₆	C₂²×C₄₁	1	2	yes	—
158	#493	—	2	2	no	—
159	#493	C₂²×C₄₁	1	4	yes	—
160	C₄×Q₁₆	—	2	2	no	—
161	C₄×Q₁₆	C₂²×C₄₁	1	4	yes	complemented
162	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
163	C₂×#103	—	2	1	no	—
164	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
165	C₄⋊C₁₆	C₂²×C₄₁	1	4	yes	complemented
166	#357	C₂²×C₄₁	1	8	yes	complemented
167	C₄₁	C₂×#2207	1	1	yes	characteristic, direct product factor, Sylow
168	C₂²×Q₈	—	2	1	no	—
169	C₂²×D₄	—	2	2	no	—
170	C₂²×D₄	—	4	1	no	—
171	C₂²×D₄	—	4	1	no	—
172	C₂×Q₁₆	—	2	1	no	—
173	C₂×Q₁₆	—	2	2	no	—
174	C₂×Q₁₆	D₄×C₄₁	1	4	yes	complemented
175	C₂×Q₁₆	C₂³×C₄₁	1	2	yes	—
176	C₂×D₈	—	2	4	no	—
177	C₂×D₈	—	2	4	no	—
178	C₂×D₈	—	2	2	no	—
179	C₂×D₈	—	2	4	no	—
180	C₂×D₈	—	4	4	no	—
181	C₂×D₈	C₂³×C₄₁	1	4	yes	—
182	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
184	#127	—	2	4	no	—
185	#127	—	4	2	no	—
186	C₄×Q₈	—	2	4	no	—
187	C₄×Q₈	—	4	2	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	4	1	no	—
190	C₂×C₄²	—	4	1	no	—
191	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
192	SD₃₂	—	2	16	no	—
193	C₂×C₁₆	—	2	4	no	—
194	C₂×C₁₆	C₂³×C₄₁	1	4	yes	—
195	#107	—	2	1	no	—
196	#107	C₂³×C₄₁	1	2	yes	—
197	#103	—	2	4	no	—
198	C₄×C₈	D₄×C₄₁	1	2	yes	—
199	C₄×C₈	C₂³×C₄₁	1	2	yes	—
200	C₂⁴	—	4	2	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	—	4	1	no	—
204	C₂×Q₈	—	4	2	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	4	4	no	—
207	C₂×D₄	—	4	4	no	—
208	C₂×D₄	—	4	4	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	—	4	4	no	—
211	C₂×D₄	—	8	4	no	—
212	C₂×D₄	—	8	4	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
216	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
218	Q₁₆	—	2	4	no	—
219	Q₁₆	—	2	2	no	—
220	Q₁₆	C₂×D₄×C₄₁	1	4	yes	complemented
221	D₈	—	2	4	no	—
222	D₈	—	2	4	no	—
223	D₈	—	4	8	no	—
224	D₈	—	4	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
227	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
229	C₂×C₈	#41×C₄₁	1	2	yes	—
230	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄²	—	4	4	no	—
235	C₄²	D₈×C₄₁	1	2	yes	—
236	C₄²	C₂×D₄×C₄₁	1	2	yes	—
237	C₁₆	—	2	4	no	—
238	C₂³	—	4	4	no	—
239	C₂³	—	8	4	no	—
240	C₂³	—	8	4	no	—
241	C₂³	—	8	4	no	—
242	C₂³	C₂×D₈×C₄₁	1	1	yes	characteristic
243	Q₈	—	2	4	no	—
244	Q₈	—	4	2	no	—
245	Q₈	—	4	4	no	—
246	D₄	—	4	4	no	—
247	D₄	—	4	4	no	—
248	D₄	—	8	8	no	—
249	D₄	—	8	8	no	—
250	D₄	—	8	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	SD₃₂×C₄₁	1	4	yes	complemented
257	C₂×C₄	#121×C₄₁	1	2	yes	—
258	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
259	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
260	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
261	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
262	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	#121×C₄₁	1	2	yes	—
265	C₈	C₂²×D₄×C₄₁	1	1	yes	characteristic
266	C₈	C₂×#41×C₄₁	1	1	yes	characteristic
267	C₂²	—	8	4	no	—
268	C₂²	—	8	4	no	—
269	C₂²	—	8	4	no	—
270	C₂²	—	16	8	no	—
271	C₂²	C₄₁×#459	1	2	yes	—
272	C₂²	C₂×SD₃₂×C₄₁	1	2	yes	—
273	C₂²	C₄₁×#509	1	2	yes	—
274	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	8	2	no	—
278	C₄	C₄₁×#459	1	2	yes	—
279	C₄	C₂×SD₃₂×C₄₁	1	4	yes	complemented
280	C₄	C₂×#121×C₄₁	1	1	yes	characteristic
281	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
282	C₂	—	16	4	no	—
283	C₂	C₄₁×#2207	1	4	yes	direct product factor
284	C₂	C₂×C₄₁×#459	1	1	yes	characteristic
285	C₂	C₂²×SD₃₂×C₄₁	1	1	yes	characteristic
286	C₂	C₂×C₄₁×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
prime radical	C₂×C₄₁×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ C₂×C₄×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×SD₃₂×C₄₁	C₂	semidirect
3. C₂×C₄×Q₁₆×C₄₁	C₂	semidirect
4. C₄₁×#2207	C₂	direct (N × H)
5. C₂×C₄₁×C₄⋊C₁₆	C₂	semidirect
6. C₂×C₄₁×#357	C₂	semidirect
13. C₂×SD₃₂×C₄₁	C₂²	semidirect
18. C₄×Q₁₆×C₄₁	C₂²	semidirect
22. C₄₁×C₄⋊C₁₆	C₂²	semidirect
23. C₄₁×#357	C₂²	semidirect
30. C₂×Q₁₆×C₄₁	D₄	semidirect
76. Q₁₆×C₄₁	C₂×D₄	semidirect
112. C₂×C₄×C₄₁	SD₃₂	semidirect
123. C₂×#2207	C₄₁	direct (N × H)
136. C₄×C₄₁	C₂×SD₃₂	semidirect
139. C₂²×SD₃₂	C₂×C₄₁	semidirect
141. C₂×C₄×Q₁₆	C₂×C₄₁	semidirect
142. #2207	C₂×C₄₁	direct (N × H)
143. C₂×C₄⋊C₁₆	C₂×C₄₁	semidirect
144. C₂×#357	C₂×C₄₁	semidirect
146. C₂×C₄₁	#2207	direct (N × H)
156. C₂×SD₃₂	C₂²×C₄₁	semidirect
161. C₄×Q₁₆	C₂²×C₄₁	semidirect
165. C₄⋊C₁₆	C₂²×C₄₁	semidirect
166. #357	C₂²×C₄₁	semidirect
167. C₄₁	C₂×#2207	direct (N × H)
174. C₂×Q₁₆	D₄×C₄₁	semidirect
220. Q₁₆	C₂×D₄×C₄₁	semidirect
256. C₂×C₄	SD₃₂×C₄₁	semidirect
279. C₄	C₂×SD₃₂×C₄₁	semidirect
283. C₂	C₄₁×#2207	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×SD₃₂×C₄₁	2
C₂×C₄₁×#493	2
C₂×C₄×Q₁₆×C₄₁	2
C₄₁×#2207	2
C₂×C₄₁×C₄⋊C₁₆	2
C₂×C₄₁×#357	2
C₂×#2207	41

Subgroups — C2×C41×#2207 #23809777

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#2207 #23809777