Subgroups — #20299008

Order 7232.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813622	C₂	1	1	yes	maximal, characteristic
2	#16813555	C₂	1	2	yes	maximal
3	#16813553	C₂	1	2	yes	maximal
4	#16813533	C₂	1	1	yes	maximal, characteristic
5	#125×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₂₆	—	2	1	no	—
7	C₂×#11744751	C₂²	1	2	yes	—
8	C₄⋊Dic₁₁₃	—	2	1	no	—
9	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
10	C₄⋊C₄×C₁₁₃	—	2	1	no	—
11	C₄⋊C₄×C₁₁₃	—	2	1	no	—
12	C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₁₃	—	2	1	no	—
16	C₂×C₄×C₁₁₃	—	2	1	no	—
17	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
20	C₂×Dic₁₁₃	—	2	2	no	—
21	Dic₂₂₆	—	4	2	no	—
22	#11744751	Q₈	1	4	yes	complemented
23	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
24	C₄×C₁₁₃	—	2	1	no	—
25	C₄×C₁₁₃	—	2	1	no	—
26	C₄×C₁₁₃	—	4	1	no	—
27	C₄×C₁₁₃	—	4	1	no	—
28	C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
29	Dic₁₁₃	—	4	2	no	—
30	C₂×C₁₁₃	C₄⋊Q₈	1	1	yes	characteristic
31	C₂×C₁₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
32	C₂×C₁₁₃	#137	1	1	yes	characteristic
33	C₁₁₃	#501	1	1	yes	characteristic, complemented, Sylow
34	#501	—	113	1	no	maximal, Sylow
35	C₄⋊Q₈	—	113	1	no	—
36	#125	D₁₁₃	1	1	yes	characteristic
37	#107	—	113	2	no	—
38	#106	—	113	2	no	—
39	#97	—	113	1	no	—
40	C₂×Q₈	—	226	1	no	—
41	C₂×C₈	—	113	2	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	—	113	2	no	—
45	C₄⋊C₄	—	226	1	no	—
46	C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
47	C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
48	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
49	Q₈	—	452	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	226	2	no	—
53	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
56	C₈	—	113	4	no	—
57	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	452	2	no	—
63	C₄	Q₈×D₁₁₃	1	2	yes	—
64	C₂	#16813620	1	1	yes	characteristic
65	C₂	#16813647	1	1	yes	characteristic
66	C₂	#16813652	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₁₃
commutator	C₂×C₄×C₁₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	#125×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₄
hypercenter	#125	D₁₁₃
hypocenter	C₁₁₃	#501
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
22. #11744751	Q₈	semidirect
33. C₁₁₃	#501	semidirect