Subgroups — #20298869

Order 7232.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813650	C₂	1	2	yes	maximal, complemented
2	C₂×D₉₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#16813628	C₂	1	1	yes	maximal, characteristic, complemented
4	#16813538	C₂	1	2	yes	maximal, complemented
5	#108×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14268647	C₂²	1	1	yes	characteristic
7	C₂×D₄₅₂	C₂²	1	2	yes	—
8	#14268636	—	2	2	no	—
9	D₉₀₄	—	2	2	no	—
10	D₉₀₄	—	2	1	no	—
11	D₉₀₄	—	2	1	no	—
12	#14268631	C₂²	1	1	yes	characteristic
13	#14268630	—	2	1	no	—
14	C₈×D₁₁₃	—	2	1	no	—
15	M₁₆×C₁₁₃	C₂²	1	2	yes	—
16	C₂×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
17	C₂²×D₁₁₃	—	2	2	no	—
18	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
19	C₂²⋊D₁₁₃	—	2	1	no	—
20	D₄₅₂	—	2	2	no	—
21	D₄₅₂	—	2	2	no	—
22	D₄₅₂	D₄	1	1	yes	characteristic
23	C₄×D₁₁₃	—	2	1	no	—
24	Dic₂₂₆	D₄	1	1	yes	characteristic
25	C₈×C₁₁₃	—	2	2	no	—
26	C₈×C₁₁₃	D₄	1	1	yes	characteristic
27	C₈×C₁₁₃	D₄	1	1	yes	characteristic
28	#11744751	—	2	1	no	—
29	C₂²×C₁₁₃	#41	1	1	yes	characteristic
30	C₂×D₁₁₃	—	2	1	no	—
31	C₂×D₁₁₃	—	4	2	no	—
32	C₂×D₁₁₃	—	8	2	no	—
33	C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
34	C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
35	Dic₁₁₃	—	2	1	no	—
36	C₂×C₁₁₃	—	2	1	no	—
37	C₂×C₁₁₃	#121	1	1	yes	characteristic
38	D₁₁₃	—	4	1	no	—
39	D₁₁₃	—	8	2	no	—
40	C₁₁₃	#472	1	1	yes	characteristic, complemented, Sylow
41	#472	—	113	1	no	maximal, Sylow
42	AGL(1,ℤ/8)	—	113	2	no	—
43	C₂×D₈	—	113	1	no	—
44	#131	—	113	1	no	—
45	#108	D₁₁₃	1	1	yes	characteristic, complemented
46	#100	—	113	2	no	—
47	#41	—	113	1	no	—
48	C₂×D₄	—	113	2	no	—
49	SD₁₆	—	226	2	no	—
50	D₈	—	226	2	no	—
51	D₈	—	226	1	no	—
52	D₈	—	226	1	no	—
53	M₁₆	—	113	1	no	—
54	M₁₆	—	226	1	no	—
55	M₁₆	C₂×D₁₁₃	1	2	yes	—
56	C₂×C₈	—	226	1	no	—
57	C₂×C₈	C₂×D₁₁₃	1	1	yes	characteristic
58	C₂³	—	226	2	no	—
59	Q₈	—	113	1	no	—
60	D₄	—	113	1	no	—
61	D₄	—	226	1	no	—
62	D₄	—	226	2	no	—
63	D₄	—	226	2	no	—
64	C₂×C₄	—	226	1	no	—
65	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₈	—	226	1	no	—
68	C₈	D₄₅₂	1	1	yes	characteristic
69	C₈	D₄₅₂	1	1	yes	characteristic
70	C₂²	—	904	2	no	—
71	C₂²	—	452	2	no	—
72	C₂²	—	226	1	no	—
73	C₂²	#14268648	1	1	yes	characteristic
74	C₄	—	226	1	no	—
75	C₄	D₄×D₁₁₃	1	1	yes	characteristic
76	C₄	C₂×D₄₅₂	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	904	2	no	—
79	C₂	—	452	1	no	—
80	C₂	#16813597	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16813597
commutator	C₂×C₄×C₁₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	#108×C₁₁₃	C₂
socle	C₂×C₁₁₃	#121
hypercenter	#108	D₁₁₃
hypocenter	C₁₁₃	#472
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
1. #16813650	C₂	semidirect
2. C₂×D₉₀₄	C₂	semidirect
3. #16813628	C₂	semidirect
4. #16813538	C₂	semidirect
5. #108×C₁₁₃	C₂	semidirect
40. C₁₁₃	#472	semidirect
45. #108	D₁₁₃	semidirect