Subgroups — #20298765

Order 7232.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813667	C₂	1	1	yes	maximal, characteristic, complemented
2	#16813648	C₂	1	2	yes	maximal, complemented
3	#16813550	C₂	1	2	yes	maximal, complemented
4	#16813539	C₂	1	1	yes	maximal, characteristic, complemented
5	#127×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14268647	C₂²	1	2	yes	—
7	#14268646	—	2	2	no	—
8	C₂×C₂²⋊D₁₁₃	—	2	2	no	—
9	D₄×D₁₁₃	—	2	2	no	—
10	#14268635	—	2	2	no	—
11	#14268632	—	2	2	no	—
12	#14268631	C₂²	1	2	yes	complemented
13	C₂×D₄×C₁₁₃	—	2	2	no	—
14	C₂×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂×D₄×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
16	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
17	C₂³×C₁₁₃	—	2	1	no	—
18	C₂³×C₁₁₃	D₄	1	2	yes	—
19	C₂²×D₁₁₃	—	2	2	no	—
20	D₄×C₁₁₃	—	2	2	no	—
21	D₄×C₁₁₃	—	2	2	no	—
22	D₄×C₁₁₃	—	4	4	no	—
23	C₂×C₄×C₁₁₃	—	2	1	no	—
24	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
25	C₂²⋊D₁₁₃	—	2	4	no	—
26	C₂²⋊D₁₁₃	—	2	2	no	—
27	C₂×Dic₁₁₃	—	2	2	no	—
28	D₄₅₂	D₄	1	2	yes	complemented
29	C₄×D₁₁₃	—	2	2	no	—
30	Dic₂₂₆	D₄	1	2	yes	complemented
31	#11744751	—	2	2	no	—
32	C₂²×C₁₁₃	—	2	2	no	—
33	C₂²×C₁₁₃	—	4	2	no	—
34	C₂²×C₁₁₃	—	4	2	no	—
35	C₂²×C₁₁₃	—	4	1	no	—
36	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×D₁₁₃	—	2	2	no	—
38	C₂×D₁₁₃	—	8	2	no	—
39	C₄×C₁₁₃	—	2	2	no	—
40	C₄×C₁₁₃	C₂×D₄	1	2	yes	—
41	Dic₁₁₃	—	2	2	no	—
42	C₂×C₁₁₃	—	2	1	no	—
43	C₂×C₁₁₃	—	4	2	no	—
44	C₂×C₁₁₃	—	8	1	no	—
45	C₂×C₁₁₃	#120	1	1	yes	characteristic
46	D₁₁₃	—	4	2	no	—
47	C₁₁₃	#453	1	1	yes	characteristic, complemented, Sylow
48	#453	—	113	1	no	maximal, Sylow
49	H(5,𝔽₂)	—	113	1	no	—
50	AGL(1,ℤ/8)	—	113	2	no	—
51	#127	D₁₁₃	1	1	yes	characteristic, complemented
52	#104	—	113	2	no	—
53	#100	—	113	1	no	—
54	#41	—	113	2	no	—
55	#41	—	226	2	no	—
56	C₂×D₄	—	2	2	no	—
57	C₂×D₄	—	226	2	no	—
58	C₂×D₄	—	226	2	no	—
59	C₂×D₄	C₂×D₁₁₃	1	1	yes	characteristic
60	C₂×D₄	C₂×D₁₁₃	1	1	yes	characteristic, complemented
61	SD₁₆	—	226	2	no	—
62	D₈	—	226	2	no	—
63	M₁₆	—	113	2	no	—
64	C₄²	C₂×D₁₁₃	1	1	yes	characteristic, complemented
65	C₂³	—	2	1	no	—
66	C₂³	—	226	2	no	—
67	C₂³	C₂²⋊D₁₁₃	1	2	yes	—
68	Q₈	—	113	2	no	—
69	D₄	—	2	2	no	—
70	D₄	—	2	2	no	—
71	D₄	—	4	4	no	—
72	D₄	—	113	2	no	—
73	D₄	—	226	4	no	—
74	D₄	—	226	2	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	226	2	no	—
77	C₂×C₄	—	226	2	no	—
78	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
79	C₈	—	226	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	4	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	904	2	no	—
85	C₂²	—	226	2	no	—
86	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	226	2	no	—
89	C₄	D₄×D₁₁₃	1	2	yes	—
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	8	1	no	—
93	C₂	—	452	2	no	—
94	C₂	#16813590	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16813590
commutator	C₂×C₄×C₁₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	#127×C₁₁₃	C₂
socle	C₂×C₁₁₃	#120
hypercenter	#127	D₁₁₃
hypocenter	C₁₁₃	#453
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #127

Complements

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

N	H ≅ G/N	type
1. #16813667	C₂	semidirect
2. #16813648	C₂	semidirect
3. #16813550	C₂	semidirect
4. #16813539	C₂	semidirect
5. #127×C₁₁₃	C₂	semidirect
12. #14268631	C₂²	semidirect
15. C₂×D₄×C₁₁₃	C₂²	semidirect
16. C₄²×C₁₁₃	C₂²	semidirect
28. D₄₅₂	D₄	semidirect
30. Dic₂₂₆	D₄	semidirect
47. C₁₁₃	#453	semidirect
51. #127	D₁₁₃	semidirect
60. C₂×D₄	C₂×D₁₁₃	semidirect
64. C₄²	C₂×D₁₁₃	semidirect