Subgroups — #20298783

Order 7232.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813670	C₂	1	1	yes	maximal, characteristic
2	#16813655	C₂	1	2	yes	maximal, complemented
3	#16813620	C₂	1	1	yes	maximal, characteristic, complemented
4	#16813548	C₂	1	2	yes	maximal, complemented
5	#101×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14268648	—	2	2	no	—
7	#14268647	—	2	2	no	—
8	#14268647	C₂²	1	2	yes	—
9	C₂×Dic₂₂₆	C₂²	1	1	yes	characteristic
10	Q₈×D₁₁₃	—	2	2	no	—
11	Dic₄₅₂	—	2	2	no	—
12	#14268636	—	2	2	no	—
13	#14268626	—	2	2	no	—
14	C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
16	M₁₆×C₁₁₃	C₂²	1	2	yes	—
17	Q₈×C₁₁₃	—	2	2	no	—
18	C₂×C₄×C₁₁₃	D₄	1	2	yes	—
19	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
20	C₂²⋊D₁₁₃	—	2	2	no	—
21	C₂×Dic₁₁₃	—	2	1	no	—
22	C₂×Dic₁₁₃	—	2	1	no	—
23	D₄₅₂	—	2	2	no	—
24	D₄₅₂	D₄	1	2	yes	—
25	C₄×D₁₁₃	—	2	4	no	—
26	C₄×D₁₁₃	—	2	2	no	—
27	Dic₂₂₆	—	2	2	no	—
28	Dic₂₂₆	—	2	2	no	—
29	Dic₂₂₆	D₄	1	2	yes	—
30	C₈×C₁₁₃	—	2	2	no	—
31	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
32	C₂×D₁₁₃	—	2	2	no	—
33	C₄×C₁₁₃	—	2	2	no	—
34	C₄×C₁₁₃	C₂×D₄	1	2	yes	—
35	Dic₁₁₃	—	2	2	no	—
36	Dic₁₁₃	—	2	2	no	—
37	Dic₁₁₃	—	4	1	no	—
38	C₂×C₁₁₃	—	2	1	no	—
39	C₂×C₁₁₃	#120	1	1	yes	characteristic
40	D₁₁₃	—	4	2	no	—
41	C₁₁₃	#456	1	1	yes	characteristic, complemented, Sylow
42	#456	—	113	1	no	maximal, Sylow
43	2₋¹⁺⁴	—	113	1	no	—
44	#137	—	113	2	no	—
45	C₄⋊Q₈	—	113	1	no	—
46	#104	—	113	2	no	—
47	#101	D₁₁₃	1	1	yes	characteristic, complemented
48	#41	—	113	2	no	—
49	#41	—	226	2	no	—
50	#41	—	226	2	no	—
51	C₂×Q₈	—	113	1	no	—
52	C₂×Q₈	—	226	2	no	—
53	C₂×Q₈	C₂×D₁₁₃	1	1	yes	characteristic
54	Q₁₆	—	226	2	no	—
55	SD₁₆	—	226	2	no	—
56	M₁₆	C₂×D₁₁₃	1	2	yes	—
57	C₄⋊C₄	—	226	2	no	—
58	C₄²	—	113	1	no	—
59	Q₈	—	2	2	no	—
60	Q₈	—	113	2	no	—
61	Q₈	—	226	2	no	—
62	Q₈	—	226	2	no	—
63	D₄	—	113	2	no	—
64	D₄	—	226	2	no	—
65	D₄	—	226	2	no	—
66	C₂×C₄	—	226	4	no	—
67	C₂×C₄	—	226	1	no	—
68	C₂×C₄	—	226	1	no	—
69	C₂×C₄	—	226	2	no	—
70	C₂×C₄	D₄₅₂	1	2	yes	—
71	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₂²	—	226	2	no	—
74	C₂²	C₂×D₄₅₂	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	452	1	no	—
77	C₄	—	226	2	no	—
78	C₄	—	226	2	no	—
79	C₄	D₄×D₁₁₃	1	2	yes	—
80	C₂	—	2	1	no	—
81	C₂	—	452	2	no	—
82	C₂	#16813592	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16813592
commutator	C₂×C₄×C₁₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	#101×C₁₁₃	C₂
socle	C₂×C₁₁₃	#120
hypercenter	#101	D₁₁₃
hypocenter	C₁₁₃	#456
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

Complements

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

N	H ≅ G/N	type
2. #16813655	C₂	semidirect
3. #16813620	C₂	semidirect
4. #16813548	C₂	semidirect
5. #101×C₁₁₃	C₂	semidirect
41. C₁₁₃	#456	semidirect
47. #101	D₁₁₃	semidirect