Subgroups — #20299677

Order 7232.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813710	C₂	1	2	yes	maximal, complemented
2	#16813709	C₂	1	6	yes	maximal, complemented
3	C₂×#14268657	C₂	1	3	yes	maximal, complemented
4	D₁₁₃⋊M₁₆	C₂	1	3	yes	maximal, complemented
5	#41×D₁₁₃	C₂	1	1	yes	maximal, characteristic
6	#14268657	C₂²	1	1	yes	characteristic
7	#14268657	C₂²	1	3	yes	—
8	#14268657	C₂²	1	6	yes	complemented
9	#14268656	C₂²	1	6	yes	complemented
10	#14268655	C₂²	1	6	yes	complemented
11	C₂×#11744753	C₂²	1	6	yes	complemented
12	#14268648	C₂²	1	1	yes	characteristic
13	#14268647	C₄	1	3	yes	—
14	#14268646	C₂²	1	3	yes	—
15	Q₈×D₁₁₃	C₄	1	1	yes	characteristic
16	D₄×D₁₁₃	C₄	1	3	yes	—
17	C₂×C₄×D₁₁₃	C₂²	1	3	yes	—
18	#41×C₁₁₃	C₄	1	1	yes	characteristic
19	C₂²×D₁₁₃	C₂×C₄	1	3	yes	—
20	Q₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
21	D₄×C₁₁₃	C₂×C₄	1	3	yes	—
22	C₂×C₄×C₁₁₃	C₂×C₄	1	3	yes	—
23	C₂²⋊D₁₁₃	C₂×C₄	1	6	yes	—
24	C₂×Dic₁₁₃	C₂³	1	3	yes	—
25	D₄₅₂	C₂×C₄	1	3	yes	—
26	C₄×D₁₁₃	C₂×C₄	1	3	yes	—
27	C₄×D₁₁₃	C₂×C₄	1	3	yes	—
28	C₄×D₁₁₃	C₂³	1	1	yes	characteristic
29	C₄×D₁₁₃	C₂³	1	3	yes	—
30	Dic₂₂₆	C₂×C₄	1	3	yes	—
31	#11744753	C₂³	1	6	yes	—
32	#11744753	C₂³	1	2	yes	—
33	C₂²×C₁₁₃	C₂²×C₄	1	3	yes	—
34	C₂×D₁₁₃	—	4	3	no	—
35	C₂×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×D₁₁₃	C₂²×C₄	1	3	yes	—
37	C₄×C₁₁₃	C₂²×C₄	1	3	yes	—
38	C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
39	Dic₁₁₃	C₂²×C₄	1	3	yes	—
40	Dic₁₁₃	C₂⁴	1	1	yes	characteristic
41	C₂×C₁₁₃	—	2	3	no	—
42	C₂×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
43	D₁₁₃	—	2	1	no	—
44	D₁₁₃	—	2	3	no	—
45	C₁₁₃	#568	1	1	yes	characteristic, complemented, Sylow
46	#568	—	113	1	no	maximal, Sylow
47	C₂×#41	—	113	1	no	—
48	#131	—	113	6	no	—
49	#131	—	113	2	no	—
50	C₂×M₁₆	—	113	3	no	—
51	C₂×M₁₆	—	113	3	no	—
52	#41	—	113	1	no	—
53	#41	—	113	3	no	—
54	#41	—	113	3	no	—
55	#41	#90673	1	1	yes	characteristic
56	C₂×Q₈	—	113	1	no	—
57	C₂×D₄	—	113	3	no	—
58	C₂²×C₄	—	113	3	no	—
59	M₁₆	—	113	1	no	—
60	M₁₆	—	113	3	no	—
61	M₁₆	—	113	6	no	—
62	M₁₆	—	113	6	no	—
63	C₂×C₈	—	113	6	no	—
64	C₂×C₈	—	113	6	no	—
65	C₂³	—	113	3	no	—
66	Q₈	—	113	3	no	—
67	Q₈	C₂×#90673	1	1	yes	characteristic
68	D₄	—	113	6	no	—
69	D₄	—	113	3	no	—
70	D₄	C₂×#90673	1	3	yes	—
71	C₂×C₄	—	113	3	no	—
72	C₂×C₄	—	113	3	no	—
73	C₂×C₄	—	113	1	no	—
74	C₂×C₄	—	113	3	no	—
75	C₂×C₄	—	113	3	no	—
76	C₂×C₄	C₂×#90673	1	3	yes	—
77	C₈	—	113	6	no	—
78	C₈	—	113	2	no	—
79	C₂²	—	113	1	no	—
80	C₂²	—	113	3	no	—
81	C₂²	—	452	3	no	—
82	C₂²	C₂²×#90673	1	3	yes	—
83	C₄	—	113	1	no	—
84	C₄	—	113	3	no	—
85	C₄	C₂²×#90673	1	1	yes	characteristic
86	C₄	C₂²×#90673	1	3	yes	—
87	C₂	—	2	3	no	—
88	C₂	—	226	1	no	—
89	C₂	—	226	3	no	—
90	C₂	C₂³×#90673	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#90673
commutator, socle	C₂×C₁₁₃	C₂³×C₄
Fitting	#41×C₁₁₃	C₄
hypercenter	#41	#90673
hypocenter	C₁₁₃	#568
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #16813710	C₂	semidirect
2. #16813709	C₂	semidirect
3. C₂×#14268657	C₂	semidirect
4. D₁₁₃⋊M₁₆	C₂	semidirect
8. #14268657	C₂²	semidirect
9. #14268656	C₂²	semidirect
10. #14268655	C₂²	semidirect
11. C₂×#11744753	C₂²	semidirect
45. C₁₁₃	#568	semidirect