Subgroups — #20298323

Order 7232.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268626	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊#11744751	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊C₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₁₃	C₄	1	1	yes	characteristic
5	C₂²×Dic₁₁₃	C₄	1	1	yes	characteristic
6	C₂×#11744751	—	2	1	no	—
7	#14268626	—	2	1	no	—
8	#14268626	—	2	1	no	—
9	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₁₁₃	—	2	1	no	—
11	C₂³×C₁₁₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₁₁₃	—	4	1	no	—
13	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
14	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
15	C₂×Dic₁₁₃	—	2	1	no	—
16	C₂×Dic₁₁₃	—	2	1	no	—
17	C₂×Dic₁₁₃	—	2	1	no	—
18	C₂×Dic₁₁₃	—	4	1	no	—
19	C₂×Dic₁₁₃	C₈	1	2	yes	complemented
20	C₈×C₁₁₃	—	4	1	no	—
21	#11744751	—	4	1	no	—
22	C₂²×C₁₁₃	—	2	1	no	—
23	C₂²×C₁₁₃	—	2	1	no	—
24	C₂²×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₁₁₃	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₁₁₃	M₁₆	1	1	yes	characteristic
27	C₄×C₁₁₃	—	2	1	no	—
28	C₄×C₁₁₃	—	2	1	no	—
29	Dic₁₁₃	—	2	2	no	—
30	Dic₁₁₃	—	4	1	no	—
31	C₂×C₁₁₃	—	2	1	no	—
32	C₂×C₁₁₃	—	2	1	no	—
33	C₂×C₁₁₃	C₂²⋊C₈	1	1	yes	characteristic
34	C₂×C₁₁₃	#99	1	1	yes	characteristic
35	C₂×C₁₁₃	#101	1	1	yes	characteristic
36	C₁₁₃	#324	1	1	yes	characteristic, complemented, Sylow
37	#324	—	113	1	no	maximal, Sylow
38	C₂×C₄⋊C₄	—	113	1	no	—
39	C₂²⋊C₈	—	113	1	no	—
40	C₂²⋊C₈	D₁₁₃	1	1	yes	characteristic
41	C₂²×C₄	—	113	1	no	—
42	C₂²×C₄	—	113	1	no	—
43	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	226	1	no	—
46	C₄⋊C₄	—	226	1	no	—
47	C₄⋊C₄	—	226	1	no	—
48	C₂³	C₄×D₁₁₃	1	1	yes	characteristic
49	C₂×C₄	—	4	1	no	—
50	C₂×C₄	—	113	2	no	—
51	C₂×C₄	—	452	1	no	—
52	C₂×C₄	—	226	1	no	—
53	C₂×C₄	—	226	1	no	—
54	C₂×C₄	—	226	1	no	—
55	C₂×C₄	D₄₅₂	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
57	C₈	—	4	1	no	—
58	C₈	—	452	1	no	—
59	C₂²	—	2	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	#14268624	1	1	yes	characteristic
62	C₂²	C₈×D₁₁₃	1	1	yes	characteristic
63	C₂²	#14268630	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	2	1	no	—
66	C₄	—	452	1	no	—
67	C₄	—	226	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	2	1	no	—
70	C₂	#16813534	1	1	yes	characteristic
71	C₂	#16813536	1	1	yes	characteristic
72	C₂	#16813540	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14268624
commutator	C₂²×C₁₁₃	C₂×C₈
Frattini	C₂²×C₄	C₂×D₁₁₃
Fitting	C₂²⋊C₈×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂²⋊C₄
hypercenter	C₂²⋊C₈	D₁₁₃
hypocenter	C₁₁₃	#324
LCS	G ▷ C₂²×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
19. C₂×Dic₁₁₃	C₈	semidirect
36. C₁₁₃	#324	semidirect