Subgroups — #20298409

Order 7232.

58 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	#16813554	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#14268647	—	2	1	no	—
5	Dic₄₅₂	C₄	1	1	yes	characteristic, complemented
6	#14268636	—	2	1	no	—
7	D₉₀₄	C₄	1	1	yes	characteristic, complemented
8	C₄⋊Dic₁₁₃	—	2	1	no	—
9	C₂×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
10	C₁₆×C₁₁₃	—	2	1	no	—
11	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
12	C₂²⋊D₁₁₃	—	4	1	no	—
13	C₂×Dic₁₁₃	—	4	1	no	—
14	D₄₅₂	—	2	1	no	—
15	C₄×D₁₁₃	—	4	1	no	—
16	Dic₂₂₆	—	2	1	no	—
17	C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
18	C₈×C₁₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₁₁₃	D₈	1	1	yes	characteristic
20	C₂×D₁₁₃	—	4	1	no	—
21	C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₁₁₃	SD₁₆	1	1	yes	characteristic
23	Dic₁₁₃	—	4	1	no	—
24	Dic₁₁₃	—	8	1	no	—
25	C₂×C₁₁₃	—	2	1	no	—
26	C₂×C₁₁₃	#102	1	1	yes	characteristic
27	D₁₁₃	—	8	1	no	—
28	C₁₁₃	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	113	1	no	maximal, Sylow
30	#135	—	113	1	no	—
31	M₃₂	D₁₁₃	1	1	yes	characteristic, complemented
32	#106	—	113	1	no	—
33	#41	—	226	1	no	—
34	Q₁₆	—	113	1	no	—
35	SD₁₆	—	226	1	no	—
36	D₈	—	113	1	no	—
37	C₂×C₈	C₂×D₁₁₃	1	1	yes	characteristic
38	C₄⋊C₄	—	226	1	no	—
39	C₁₆	—	2	1	no	—
40	Q₈	—	226	1	no	—
41	D₄	—	452	1	no	—
42	D₄	—	226	1	no	—
43	C₂×C₄	—	452	1	no	—
44	C₂×C₄	—	452	1	no	—
45	C₂×C₄	D₄₅₂	1	1	yes	characteristic
46	C₈	C₄×D₁₁₃	1	1	yes	characteristic
47	C₈	C₂²⋊D₁₁₃	1	1	yes	characteristic
48	C₂²	—	452	1	no	—
49	C₂²	D₉₀₄	1	1	yes	characteristic
50	C₄	—	904	1	no	—
51	C₄	—	452	1	no	—
52	C₄	#14268624	1	1	yes	characteristic
53	C₄	#14268636	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	904	1	no	—
56	C₂	#16813544	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16813544
commutator	C₈×C₁₁₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₁₃
Fitting	M₃₂×C₁₁₃	C₂
socle	C₂×C₁₁₃	#102
hypercenter	M₃₂	D₁₁₃
hypocenter	C₁₁₃	#360
LCS	G ▷ C₈×C₁₁₃ ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #16813554	C₂	semidirect
3. M₃₂×C₁₁₃	C₂	semidirect
5. Dic₄₅₂	C₄	semidirect
7. D₉₀₄	C₄	semidirect
28. C₁₁₃	#360	semidirect
31. M₃₂	D₁₁₃	semidirect