Subgroups — #20298410

Order 7232.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₉₀₄	C₂	1	1	yes	maximal, characteristic
2	#16813562	C₂	1	1	yes	maximal, characteristic, complemented
3	#108×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄₅₂	—	2	1	no	—
5	D₉₀₄	—	2	1	no	—
6	D₉₀₄	C₄	1	2	yes	—
7	#14268622	—	2	1	no	—
8	M₁₆×C₁₁₃	—	2	1	no	—
9	C₂×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
10	C₂²×D₁₁₃	—	4	1	no	—
11	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
12	D₄₅₂	—	2	2	no	—
13	D₄₅₂	—	4	1	no	—
14	C₈×C₁₁₃	—	4	1	no	—
15	C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
16	C₈×C₁₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₁₁₃	SD₁₆	1	1	yes	characteristic
18	C₂×D₁₁₃	—	4	2	no	—
19	C₂×D₁₁₃	—	8	2	no	—
20	C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₁₁₃	D₈	1	1	yes	characteristic
22	C₂×C₁₁₃	—	2	1	no	—
23	C₂×C₁₁₃	#102	1	1	yes	characteristic
24	D₁₁₃	—	8	2	no	—
25	C₁₁₃	#361	1	1	yes	characteristic, complemented, Sylow
26	#361	—	113	1	no	maximal, Sylow
27	C₂×D₈	—	113	1	no	—
28	M₃₂	—	113	1	no	—
29	#108	D₁₁₃	1	1	yes	characteristic, complemented
30	C₂×D₄	—	226	1	no	—
31	D₈	—	113	2	no	—
32	D₈	—	226	1	no	—
33	M₁₆	—	2	1	no	—
34	C₂×C₈	C₂×D₁₁₃	1	1	yes	characteristic
35	C₁₆	—	226	1	no	—
36	C₂³	—	452	1	no	—
37	D₄	—	452	1	no	—
38	D₄	—	226	2	no	—
39	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
40	C₈	—	4	1	no	—
41	C₈	C₄×D₁₁₃	1	1	yes	characteristic
42	C₈	D₄₅₂	1	1	yes	characteristic
43	C₂²	—	904	2	no	—
44	C₂²	—	452	2	no	—
45	C₂²	#14268634	1	1	yes	characteristic
46	C₄	#14268624	1	1	yes	characteristic
47	C₄	#14268632	1	1	yes	characteristic
48	C₂	—	2	1	no	—
49	C₂	—	904	2	no	—
50	C₂	#16813542	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16813542
commutator	C₈×C₁₁₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₁₃
Fitting	#108×C₁₁₃	C₂
socle	C₂×C₁₁₃	#102
hypercenter	#108	D₁₁₃
hypocenter	C₁₁₃	#361
LCS	G ▷ C₈×C₁₁₃ ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

Complements

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

N	H ≅ G/N	type
2. #16813562	C₂	semidirect
3. #108×C₁₁₃	C₂	semidirect
25. C₁₁₃	#361	semidirect
29. #108	D₁₁₃	semidirect