Subgroups — #20298405

Order 7232.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268622	C₂	1	1	yes	maximal, characteristic, complemented
2	#16813558	C₂	1	1	yes	maximal, characteristic, complemented
3	#135×C₁₁₃	C₂	1	1	yes	maximal, characteristic
4	#14268631	—	2	1	no	—
5	#14268622	—	2	1	no	—
6	#41×C₁₁₃	—	2	1	no	—
7	Q₁₆×C₁₁₃	C₄	1	1	yes	characteristic
8	SD₁₆×C₁₁₃	—	2	1	no	—
9	D₈×C₁₁₃	C₄	1	1	yes	characteristic
10	C₂×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
11	Q₈×C₁₁₃	—	2	1	no	—
12	D₄×C₁₁₃	—	2	1	no	—
13	D₄×C₁₁₃	—	4	1	no	—
14	C₂×C₄×C₁₁₃	—	4	1	no	—
15	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
16	C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
17	C₈×C₁₁₃	D₄	1	1	yes	characteristic
18	#11744751	—	4	1	no	—
19	C₂²×C₁₁₃	—	4	1	no	—
20	C₂²×C₁₁₃	SD₁₆	1	1	yes	characteristic
21	C₄×C₁₁₃	—	4	1	no	—
22	C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₁₁₃	D₈	1	1	yes	characteristic
24	C₂×C₁₁₃	—	2	1	no	—
25	C₂×C₁₁₃	—	8	1	no	—
26	C₂×C₁₁₃	#102	1	1	yes	characteristic
27	C₁₁₃	#359	1	1	yes	characteristic, complemented, Sylow
28	#359	—	113	1	no	maximal, Sylow
29	#135	D₁₁₃	1	1	yes	characteristic
30	C₂×C₁₆	—	113	1	no	—
31	#108	—	113	1	no	—
32	#41	—	2	1	no	—
33	Q₁₆	Dic₁₁₃	1	1	yes	characteristic
34	SD₁₆	—	2	1	no	—
35	D₈	Dic₁₁₃	1	1	yes	characteristic
36	M₁₆	—	226	1	no	—
37	C₂×C₈	C₂×D₁₁₃	1	1	yes	characteristic
38	C₁₆	—	226	1	no	—
39	Q₈	—	2	1	no	—
40	D₄	—	2	1	no	—
41	D₄	—	4	1	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
44	C₈	—	452	1	no	—
45	C₈	C₂×Dic₁₁₃	1	1	yes	characteristic
46	C₈	C₂²⋊D₁₁₃	1	1	yes	characteristic
47	C₂²	—	4	1	no	—
48	C₂²	#14268635	1	1	yes	characteristic
49	C₄	—	4	1	no	—
50	C₄	C₂²⋊Dic₁₁₃	1	1	yes	characteristic
51	C₄	#14268632	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	8	1	no	—
54	C₂	#16813543	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14268632
commutator	C₈×C₁₁₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₁₃
Fitting	#135×C₁₁₃	C₂
socle	C₂×C₁₁₃	#102
hypercenter	#135	D₁₁₃
hypocenter	C₁₁₃	#359
LCS	G ▷ C₈×C₁₁₃ ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. C₂×#14268622	C₂	semidirect
2. #16813558	C₂	semidirect
27. C₁₁₃	#359	semidirect