Subgroups — #20298404

Order 7232.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813644	C₂	1	1	yes	maximal, characteristic
2	C₂×#14268622	C₂	1	1	yes	maximal, characteristic, complemented
3	#108×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#14268647	—	2	1	no	—
5	Dic₄₅₂	C₄	1	1	yes	characteristic
6	#14268636	—	2	1	no	—
7	D₉₀₄	C₄	1	1	yes	characteristic
8	#14268622	—	2	1	no	—
9	M₁₆×C₁₁₃	—	2	1	no	—
10	C₂×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
12	C₂²⋊D₁₁₃	—	4	1	no	—
13	D₄₅₂	—	2	1	no	—
14	C₄×D₁₁₃	—	4	1	no	—
15	Dic₂₂₆	—	2	1	no	—
16	C₈×C₁₁₃	—	4	1	no	—
17	C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
18	C₈×C₁₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₁₁₃	SD₁₆	1	1	yes	characteristic
20	C₂×D₁₁₃	—	4	1	no	—
21	C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₁₁₃	D₈	1	1	yes	characteristic
23	Dic₁₁₃	—	4	1	no	—
24	C₂×C₁₁₃	—	2	1	no	—
25	C₂×C₁₁₃	#102	1	1	yes	characteristic
26	D₁₁₃	—	8	1	no	—
27	C₁₁₃	#359	1	1	yes	characteristic, complemented, Sylow
28	#359	—	113	1	no	maximal, Sylow
29	#135	—	113	1	no	—
30	C₂×C₁₆	—	113	1	no	—
31	#108	D₁₁₃	1	1	yes	characteristic, complemented
32	#41	—	226	1	no	—
33	Q₁₆	—	113	1	no	—
34	SD₁₆	—	226	1	no	—
35	D₈	—	113	1	no	—
36	M₁₆	—	2	1	no	—
37	C₂×C₈	C₂×D₁₁₃	1	1	yes	characteristic
38	C₁₆	—	226	1	no	—
39	Q₈	—	226	1	no	—
40	D₄	—	452	1	no	—
41	D₄	—	226	1	no	—
42	C₂×C₄	—	452	1	no	—
43	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
44	C₈	—	4	1	no	—
45	C₈	C₄×D₁₁₃	1	1	yes	characteristic
46	C₈	D₄₅₂	1	1	yes	characteristic
47	C₂²	—	452	1	no	—
48	C₂²	#14268634	1	1	yes	characteristic
49	C₄	—	452	1	no	—
50	C₄	#14268624	1	1	yes	characteristic
51	C₄	#14268632	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	904	1	no	—
54	C₂	#16813542	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14268632
commutator	C₈×C₁₁₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₁₃
Fitting	#108×C₁₁₃	C₂
socle	C₂×C₁₁₃	#102
hypercenter	#108	D₁₁₃
hypocenter	C₁₁₃	#359
LCS	G ▷ C₈×C₁₁₃ ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

Complements

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

N	H ≅ G/N	type
2. C₂×#14268622	C₂	semidirect
3. #108×C₁₁₃	C₂	semidirect
27. C₁₁₃	#359	semidirect
31. #108	D₁₁₃	semidirect