Subgroups — #20299365

Order 7232.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813629	C₂	1	8	yes	maximal, complemented
2	C₂×#14268631	C₂	1	6	yes	maximal, complemented
3	C₂×#41×C₁₁₃	C₂	1	1	yes	maximal, characteristic
4	#14268631	C₂²	1	4	yes	—
5	#14268631	C₂²	1	12	yes	complemented
6	C₂×#11744751	C₂²	1	12	yes	complemented
7	#41×C₁₁₃	C₄	1	4	yes	—
8	#41×C₁₁₃	C₂²	1	4	yes	—
9	C₂×Q₈×C₁₁₃	C₄	1	1	yes	characteristic
10	C₂×D₄×C₁₁₃	C₄	1	3	yes	—
11	C₂²×C₄×C₁₁₃	C₂²	1	3	yes	—
12	C₂³×C₁₁₃	C₂×C₄	1	3	yes	—
13	Q₈×C₁₁₃	C₂×C₄	1	4	yes	—
14	D₄×C₁₁₃	C₂×C₄	1	12	yes	—
15	C₂×C₄×C₁₁₃	C₂×C₄	1	3	yes	—
16	C₂×C₄×C₁₁₃	C₂×C₄	1	6	yes	—
17	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₄×C₁₁₃	C₂³	1	6	yes	—
19	#11744751	C₂³	1	8	yes	—
20	C₂²×C₁₁₃	—	4	3	no	—
21	C₂²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₁₁₃	C₂²×C₄	1	6	yes	—
23	C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₁₁₃	C₂²×C₄	1	6	yes	—
25	C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
26	C₂×C₁₁₃	—	2	1	no	—
27	C₂×C₁₁₃	—	2	6	no	—
28	C₂×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
29	C₁₁₃	#568	1	1	yes	characteristic, complemented, Sylow
30	#568	—	113	1	no	maximal, Sylow
31	C₂×#41	D₁₁₃	1	1	yes	characteristic
32	#131	—	113	8	no	—
33	C₂×M₁₆	—	113	6	no	—
34	#41	Dic₁₁₃	1	4	yes	—
35	#41	C₂×D₁₁₃	1	4	yes	—
36	C₂×Q₈	Dic₁₁₃	1	1	yes	characteristic
37	C₂×D₄	Dic₁₁₃	1	3	yes	—
38	C₂²×C₄	C₂×D₁₁₃	1	3	yes	—
39	M₁₆	—	113	4	no	—
40	M₁₆	—	113	12	no	—
41	C₂×C₈	—	113	12	no	—
42	C₂³	C₂×Dic₁₁₃	1	3	yes	—
43	Q₈	C₂×Dic₁₁₃	1	4	yes	—
44	D₄	C₂×Dic₁₁₃	1	12	yes	—
45	C₂×C₄	C₂×Dic₁₁₃	1	3	yes	—
46	C₂×C₄	C₂×Dic₁₁₃	1	6	yes	—
47	C₂×C₄	C₂²×D₁₁₃	1	6	yes	—
48	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
49	C₈	—	113	8	no	—
50	C₂²	—	4	3	no	—
51	C₂²	C₂²×Dic₁₁₃	1	1	yes	characteristic
52	C₂²	C₂²×Dic₁₁₃	1	6	yes	—
53	C₄	C₂²×Dic₁₁₃	1	1	yes	characteristic
54	C₄	C₂²×Dic₁₁₃	1	6	yes	—
55	C₄	C₂³×D₁₁₃	1	1	yes	characteristic
56	C₂	—	2	1	no	—
57	C₂	—	2	6	no	—
58	C₂	C₂³×Dic₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×D₁₁₃
commutator, socle	C₂×C₁₁₃	C₂³×C₄
Fitting	C₂×#41×C₁₁₃	C₂
hypercenter	C₂×#41	D₁₁₃
hypocenter	C₁₁₃	#568
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #16813629	C₂	semidirect
2. C₂×#14268631	C₂	semidirect
5. #14268631	C₂²	semidirect
6. C₂×#11744751	C₂²	semidirect
29. C₁₁₃	#568	semidirect