Subgroups — #28338756

Order 13696.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293043	C₂	1	1	yes	maximal, characteristic
2	#20292460	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₀₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₁₀₇	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812115	C₄	1	1	yes	characteristic
7	#16812049	C₄	1	1	yes	characteristic
8	#16812048	C₄	1	2	yes	—
9	C₁₆×D₁₀₇	C₂²	1	2	yes	—
10	C₁₀₇⋊C₃₂	C₂²	1	1	yes	characteristic
11	C₁₀₇⋊C₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
13	C₃₂×C₁₀₇	C₂²	1	2	yes	—
14	#13133818	C₈	1	1	yes	characteristic
15	#13133802	C₈	1	1	yes	characteristic
16	#13133801	C₈	1	2	yes	—
17	C₈×D₁₀₇	C₂×C₄	1	2	yes	—
18	C₁₀₇⋊C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₀₇	C₂×C₄	1	1	yes	characteristic
21	C₁₆×C₁₀₇	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₁₀₇	C₂×C₈	1	1	yes	characteristic
23	C₂²⋊D₁₀₇	C₁₆	1	2	yes	—
24	D₄₂₈	C₁₆	1	1	yes	characteristic
25	C₄×D₁₀₇	C₂×C₈	1	2	yes	—
26	Dic₂₁₄	C₁₆	1	1	yes	characteristic
27	C₈×C₁₀₇	C₂×C₈	1	1	yes	characteristic
28	C₈×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
29	C₁₀₇⋊C₈	C₂×C₈	1	2	yes	—
30	C₂²×C₁₀₇	C₂×C₁₆	1	1	yes	characteristic
31	C₂×D₁₀₇	C₂×C₁₆	1	2	yes	—
32	C₄×C₁₀₇	C₂×C₁₆	1	1	yes	characteristic
33	C₄×C₁₀₇	C₂²×C₈	1	1	yes	characteristic
34	Dic₁₀₇	C₂×C₁₆	1	2	yes	—
35	C₂×C₁₀₇	—	2	1	no	—
36	C₂×C₁₀₇	C₂²×C₁₆	1	1	yes	characteristic
37	D₁₀₇	—	2	2	no	—
38	#2258	—	107	1	no	maximal, Sylow
39	C₁₀₇	#2258	1	1	yes	characteristic, complemented, Sylow
40	#504	—	107	1	no	—
41	M₆₄	—	107	2	no	—
42	M₆₄	D₁₀₇	1	1	yes	characteristic, complemented
43	C₂×C₃₂	—	107	1	no	—
44	C₂×C₃₂	—	107	2	no	—
45	#131	—	107	1	no	—
46	M₃₂	—	107	1	no	—
47	M₃₂	—	107	2	no	—
48	C₂×C₁₆	—	107	2	no	—
49	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
50	C₃₂	—	107	1	no	—
51	C₃₂	—	107	1	no	—
52	C₃₂	C₂×D₁₀₇	1	2	yes	—
53	#41	—	107	1	no	—
54	M₁₆	—	107	1	no	—
55	M₁₆	—	107	2	no	—
56	C₂×C₈	—	107	2	no	—
57	C₂×C₈	C₄×D₁₀₇	1	1	yes	characteristic
58	C₁₆	—	107	2	no	—
59	C₁₆	C₄×D₁₀₇	1	1	yes	characteristic
60	C₁₆	C₂²×D₁₀₇	1	1	yes	characteristic
61	Q₈	—	107	1	no	—
62	D₄	—	107	2	no	—
63	D₄	—	107	1	no	—
64	C₂×C₄	—	107	2	no	—
65	C₂×C₄	C₈×D₁₀₇	1	1	yes	characteristic
66	C₈	—	107	2	no	—
67	C₈	C₈×D₁₀₇	1	1	yes	characteristic
68	C₈	C₂×C₄×D₁₀₇	1	1	yes	characteristic
69	C₂²	—	107	2	no	—
70	C₂²	C₁₆×D₁₀₇	1	1	yes	characteristic
71	C₄	—	107	2	no	—
72	C₄	C₁₆×D₁₀₇	1	1	yes	characteristic
73	C₄	C₂×C₈×D₁₀₇	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	214	2	no	—
76	C₂	C₂×C₁₆×D₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂²×D₁₀₇
commutator, socle	C₂×C₁₀₇	C₂²×C₁₆
Fitting	M₆₄×C₁₀₇	C₂
prime radical	#13133818	C₈
cosocle	C₁₆×C₁₀₇	C₂³
hypercenter	M₆₄	D₁₀₇
hypocenter	C₁₀₇	#2258
LCS	G ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #20292460	C₂	semidirect
3. C₂×C₁₀₇⋊C₃₂	C₂	semidirect
4. C₃₂×D₁₀₇	C₂	semidirect
5. M₆₄×C₁₀₇	C₂	semidirect
39. C₁₀₇	#2258	semidirect
42. M₆₄	D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#20293043	2
#20292460	2
C₂×C₁₀₇⋊C₃₂	2
C₃₂×D₁₀₇	2
M₆₄×C₁₀₇	2
#2258	107