Subgroups — #20299072

Order 7232.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄₅₂	C₂	1	1	yes	maximal, characteristic
2	C₄×D₄₅₂	C₂	1	4	yes	maximal, complemented
3	#16813584	C₂	1	4	yes	maximal, complemented
4	#16813581	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊C₄×D₁₁₃	C₂	1	4	yes	maximal, complemented
6	#117×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₁₃	C₂²	1	2	yes	—
8	C₂×D₄₅₂	C₄	1	8	yes	complemented
9	C₂×D₄₅₂	C₂²	1	2	yes	—
10	C₂×D₄₅₂	C₂²	1	2	yes	—
11	C₂×C₄×D₁₁₃	C₂²	1	8	yes	complemented
12	C₄⋊Dic₁₁₃	C₂²	1	2	yes	complemented
13	C₂²⋊Dic₁₁₃	C₂²	1	2	yes	—
14	#14268624	C₂²	1	8	yes	complemented
15	C₄×Dic₁₁₃	C₂²	1	2	yes	complemented
16	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	complemented
18	C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
19	C₄²×C₁₁₃	C₂²	1	2	yes	complemented
20	C₂³×C₁₁₃	C₂³	1	1	yes	characteristic
21	C₂²×D₁₁₃	—	4	4	no	—
22	C₂²×D₁₁₃	C₂×C₄	1	8	yes	complemented
23	C₂²×D₁₁₃	C₂³	1	4	yes	—
24	C₂×C₄×C₁₁₃	C₂×C₄	1	4	yes	complemented
25	C₂×C₄×C₁₁₃	C₂³	1	4	yes	—
26	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₁₃	C₂³	1	4	yes	—
29	D₄₅₂	C₂×C₄	1	8	yes	complemented
30	D₄₅₂	C₂×C₄	1	8	yes	complemented
31	C₄×D₁₁₃	—	4	8	no	—
32	C₂²×C₁₁₃	—	2	2	no	—
33	C₂²×C₁₁₃	C₂²×C₄	1	2	yes	—
34	C₂²×C₁₁₃	C₂⁴	1	1	yes	characteristic
35	C₂×D₁₁₃	—	2	8	no	—
36	C₂×D₁₁₃	—	4	8	no	—
37	C₂×D₁₁₃	C₂²×C₄	1	8	yes	—
38	C₄×C₁₁₃	—	2	4	no	—
39	C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
40	C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
41	Dic₁₁₃	—	2	4	no	—
42	C₂×C₁₁₃	—	2	2	no	—
43	C₂×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
44	C₂×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
45	D₁₁₃	—	2	8	no	—
46	C₁₁₃	#518	1	1	yes	characteristic, complemented, Sylow
47	#518	—	113	1	no	maximal, Sylow
48	C₂²×D₄	—	113	1	no	—
49	C₄×D₄	—	113	4	no	—
50	C₄×D₄	—	113	4	no	—
51	#117	—	113	1	no	—
52	#117	D₁₁₃	1	1	yes	characteristic, complemented
53	C₂×C₂²⋊C₄	—	113	4	no	—
54	C₂⁴	—	113	2	no	—
55	C₂×D₄	—	113	8	no	—
56	C₂×D₄	—	113	2	no	—
57	C₂×D₄	—	113	2	no	—
58	C₂²×C₄	—	113	8	no	—
59	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
60	C₄⋊C₄	—	113	2	no	—
61	C₄⋊C₄	C₂×D₁₁₃	1	2	yes	complemented
62	C₂²⋊C₄	—	113	2	no	—
63	C₂²⋊C₄	—	113	8	no	—
64	C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
65	C₄²	—	113	2	no	—
66	C₄²	C₂×D₁₁₃	1	2	yes	complemented
67	C₂³	—	113	8	no	—
68	C₂³	—	113	4	no	—
69	C₂³	—	452	4	no	—
70	C₂³	C₂²×D₁₁₃	1	1	yes	characteristic
71	D₄	—	113	8	no	—
72	D₄	—	113	8	no	—
73	C₂×C₄	—	113	4	no	—
74	C₂×C₄	—	452	8	no	—
75	C₂×C₄	C₄×D₁₁₃	1	4	yes	complemented
76	C₂×C₄	C₂²×D₁₁₃	1	4	yes	—
77	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
78	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
79	C₂²	—	2	2	no	—
80	C₂²	—	113	8	no	—
81	C₂²	—	452	8	no	—
82	C₂²	—	226	8	no	—
83	C₂²	C₂×C₄×D₁₁₃	1	2	yes	—
84	C₂²	C₂³×D₁₁₃	1	1	yes	characteristic
85	C₄	—	2	4	no	—
86	C₄	—	226	4	no	—
87	C₄	C₂×C₄×D₁₁₃	1	2	yes	—
88	C₄	C₂×C₄×D₁₁₃	1	2	yes	—
89	C₂	—	2	2	no	—
90	C₂	—	226	8	no	—
91	C₂	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
92	C₂	#16813668	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₁₃
commutator	C₂×C₁₁₃	C₂³×C₄
Fitting	#117×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂⁴
hypercenter	#117	D₁₁₃
hypocenter	C₁₁₃	#518
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₄×D₄₅₂	C₂	semidirect
3. #16813584	C₂	semidirect
4. #16813581	C₂	semidirect
5. C₂²⋊C₄×D₁₁₃	C₂	semidirect
6. #117×C₁₁₃	C₂	semidirect
8. C₂×D₄₅₂	C₄	semidirect
11. C₂×C₄×D₁₁₃	C₂²	semidirect
12. C₄⋊Dic₁₁₃	C₂²	semidirect
14. #14268624	C₂²	semidirect
15. C₄×Dic₁₁₃	C₂²	semidirect
17. C₄⋊C₄×C₁₁₃	C₂²	semidirect
19. C₄²×C₁₁₃	C₂²	semidirect
22. C₂²×D₁₁₃	C₂×C₄	semidirect
24. C₂×C₄×C₁₁₃	C₂×C₄	semidirect
29. D₄₅₂	C₂×C₄	semidirect
30. D₄₅₂	C₂×C₄	semidirect
46. C₁₁₃	#518	semidirect
52. #117	D₁₁₃	semidirect
61. C₄⋊C₄	C₂×D₁₁₃	semidirect
66. C₄²	C₂×D₁₁₃	semidirect
75. C₂×C₄	C₄×D₁₁₃	semidirect