Subgroups — #28336075

Order 13696.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293004	C₂	1	1	yes	maximal, characteristic, complemented
2	#20293003	C₂	1	4	yes	maximal, complemented
3	C₈×C₁₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#493×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈₅₆	—	2	2	no	—
6	C₂×#13133803	—	2	8	no	—
7	#16812106	C₂²	1	2	yes	—
8	C₄×C₁₀₇⋊C₈	C₂²	1	2	yes	—
9	C₂×D₈×C₁₀₇	—	2	2	no	—
10	#127×C₁₀₇	C₂²	1	2	yes	—
11	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
12	C₂×D₄₂₈	—	2	4	no	—
13	C₂×D₄₂₈	—	2	2	no	—
14	D₈₅₆	—	4	4	no	—
15	#13133803	—	4	16	no	—
16	C₂×C₁₀₇⋊C₈	D₄	1	4	yes	—
17	C₂×D₄×C₁₀₇	—	2	4	no	—
18	C₂×D₄×C₁₀₇	—	2	2	no	—
19	D₈×C₁₀₇	—	4	4	no	—
20	C₂×C₈×C₁₀₇	D₄	1	2	yes	—
21	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
22	C₂³×C₁₀₇	—	4	2	no	—
23	C₂²×D₁₀₇	—	4	2	no	—
24	D₄×C₁₀₇	—	4	4	no	—
25	D₄×C₁₀₇	—	4	8	no	—
26	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
27	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
28	D₄₂₈	—	4	8	no	—
29	D₄₂₈	—	4	4	no	—
30	C₈×C₁₀₇	D₈	1	4	yes	complemented
31	C₁₀₇⋊C₈	D₈	1	8	yes	complemented
32	C₂²×C₁₀₇	—	8	4	no	—
33	C₂²×C₁₀₇	—	8	2	no	—
34	C₂²×C₁₀₇	#127	1	1	yes	characteristic
35	C₂×D₁₀₇	—	8	4	no	—
36	C₂×D₁₀₇	—	8	2	no	—
37	C₄×C₁₀₇	C₂×D₈	1	4	yes	—
38	C₄×C₁₀₇	C₂×D₈	1	2	yes	—
39	C₂×C₁₀₇	—	16	2	no	—
40	C₂×C₁₀₇	#493	1	2	yes	—
41	C₂×C₁₀₇	#493	1	1	yes	characteristic
42	D₁₀₇	—	16	2	no	—
43	#1712	—	107	1	no	maximal, Sylow
44	C₁₀₇	#1712	1	1	yes	characteristic, complemented, Sylow
45	#493	—	107	4	no	—
46	#493	—	107	1	no	—
47	#493	D₁₀₇	1	1	yes	characteristic, complemented
48	C₈²	—	107	1	no	—
49	C₂×D₈	—	2	2	no	—
50	C₂×D₈	—	214	8	no	—
51	C₂×D₈	—	214	2	no	—
52	#127	—	107	2	no	—
53	#127	C₂×D₁₀₇	1	2	yes	—
54	C₄×C₈	—	107	2	no	—
55	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
56	C₂×D₄	—	2	4	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	—	214	4	no	—
59	C₂×D₄	—	214	2	no	—
60	D₈	—	4	4	no	—
61	D₈	—	428	16	no	—
62	D₈	—	428	4	no	—
63	C₂×C₈	—	107	4	no	—
64	C₂×C₈	C₂²⋊D₁₀₇	1	2	yes	—
65	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
66	C₂³	—	4	2	no	—
67	C₂³	—	428	2	no	—
68	D₄	—	4	8	no	—
69	D₄	—	4	4	no	—
70	D₄	—	428	8	no	—
71	D₄	—	428	4	no	—
72	C₂×C₄	D₄×D₁₀₇	1	2	yes	—
73	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
74	C₈	—	107	8	no	—
75	C₈	#13133803	1	4	yes	complemented
76	C₂²	—	8	4	no	—
77	C₂²	—	8	2	no	—
78	C₂²	—	856	4	no	—
79	C₂²	—	856	2	no	—
80	C₂²	#16812105	1	1	yes	characteristic
81	C₄	D₈×D₁₀₇	1	4	yes	—
82	C₄	C₂×#13133803	1	2	yes	—
83	C₂	—	16	2	no	—
84	C₂	—	1712	2	no	—
85	C₂	#20293002	1	2	yes	—
86	C₂	#20293003	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16812105
commutator, cosocle	C₄²×C₁₀₇	C₂³
Frattini	C₄²	C₂²×D₁₀₇
Fitting	#493×C₁₀₇	C₂
socle	C₂²×C₁₀₇	#127
hypercenter	#493	D₁₀₇
hypocenter	C₁₀₇	#1712
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. #20293004	C₂	semidirect
2. #20293003	C₂	semidirect
3. C₈×C₁₀₇⋊C₈	C₂	semidirect
4. #493×C₁₀₇	C₂	semidirect
30. C₈×C₁₀₇	D₈	semidirect
31. C₁₀₇⋊C₈	D₈	semidirect
44. C₁₀₇	#1712	semidirect
47. #493	D₁₀₇	semidirect
75. C₈	#13133803	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20293004	2
#20293003	2
C₈×C₁₀₇⋊C₈	2
#493×C₁₀₇	2
#1712	107