Subgroups — #20298995

Order 7232.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813621	C₂	1	2	yes	maximal
2	#16813554	C₂	1	1	yes	maximal, characteristic
3	#16813553	C₂	1	2	yes	maximal
4	C₈×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic
5	#106×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₂₆	—	2	2	no	—
7	C₂×#11744751	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
9	#14268626	—	2	2	no	—
10	C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
13	C₂×C₄×C₁₁₃	—	2	2	no	—
14	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
15	C₂×Dic₁₁₃	—	2	2	no	—
16	C₂×Dic₁₁₃	D₄	1	2	yes	—
17	Dic₂₂₆	—	4	4	no	—
18	C₈×C₁₁₃	Q₈	1	2	yes	complemented
19	#11744751	Q₈	1	2	yes	complemented
20	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
21	C₄×C₁₁₃	—	4	2	no	—
22	C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
24	Dic₁₁₃	—	4	2	no	—
25	Dic₁₁₃	SD₁₆	1	4	yes	—
26	C₂×C₁₁₃	C₄⋊Q₈	1	1	yes	characteristic
27	C₂×C₁₁₃	C₂×SD₁₆	1	2	yes	—
28	C₁₁₃	#498	1	1	yes	characteristic, complemented, Sylow
29	#498	—	113	1	no	maximal, Sylow
30	C₄⋊Q₈	—	113	2	no	—
31	#106	—	113	1	no	—
32	#106	—	113	2	no	—
33	#106	D₁₁₃	1	1	yes	characteristic
34	C₄×C₈	—	113	1	no	—
35	C₂×Q₈	—	226	2	no	—
36	C₂×C₈	—	113	1	no	—
37	C₂×C₈	C₂×D₁₁₃	1	1	yes	characteristic
38	C₄⋊C₄	—	113	2	no	—
39	C₄⋊C₄	—	226	2	no	—
40	C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
41	C₄²	—	113	1	no	—
42	Q₈	—	452	4	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	113	2	no	—
45	C₂×C₄	—	226	2	no	—
46	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
47	C₈	—	113	2	no	—
48	C₈	Dic₂₂₆	1	2	yes	complemented
49	C₂²	D₄×D₁₁₃	1	1	yes	characteristic
50	C₄	—	4	2	no	—
51	C₄	—	113	4	no	—
52	C₄	—	452	2	no	—
53	C₄	Q₈×D₁₁₃	1	1	yes	characteristic
54	C₄	C₂×Dic₂₂₆	1	1	yes	characteristic
55	C₂	#16813621	1	1	yes	characteristic
56	C₂	SD₁₆×D₁₁₃	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₁₃
commutator	C₂×C₄×C₁₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	#106×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₄
hypercenter	#106	D₁₁₃
hypocenter	C₁₁₃	#498
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
18. C₈×C₁₁₃	Q₈	semidirect
19. #11744751	Q₈	semidirect
28. C₁₁₃	#498	semidirect
48. C₈	Dic₂₂₆	semidirect