Subgroups — #28338771

Order 13696.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293060	C₂	1	2	yes	maximal, complemented
2	#20292464	C₂	1	2	yes	maximal, complemented
3	D₃₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₆₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812132	—	2	2	no	—
7	#16812052	—	2	2	no	—
8	D₁₇₁₂	C₂²	1	2	yes	—
9	C₁₆×D₁₀₇	C₂²	1	1	yes	characteristic
10	C₁₀₇⋊C₃₂	C₂²	1	1	yes	characteristic
11	Q₃₂×C₁₀₇	C₂²	1	2	yes	—
12	C₃₂×C₁₀₇	C₂²	1	1	yes	characteristic
13	#13133819	—	4	2	no	—
14	#13133805	—	4	2	no	—
15	D₈₅₆	—	2	2	no	—
16	C₈×D₁₀₇	D₄	1	1	yes	characteristic
17	C₁₀₇⋊C₁₆	D₄	1	1	yes	characteristic
18	Q₁₆×C₁₀₇	—	2	2	no	—
19	C₁₆×C₁₀₇	C₂³	1	1	yes	characteristic
20	Q₈×C₁₀₇	—	4	2	no	—
21	D₄₂₈	—	4	2	no	—
22	D₄₂₈	—	8	2	no	—
23	C₄×D₁₀₇	—	8	2	no	—
24	C₄×D₁₀₇	D₈	1	1	yes	characteristic
25	C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
26	C₁₀₇⋊C₈	D₈	1	1	yes	characteristic
27	C₂×D₁₀₇	—	8	2	no	—
28	C₂×D₁₀₇	D₁₆	1	1	yes	characteristic
29	C₄×C₁₀₇	—	8	2	no	—
30	C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
31	Dic₁₀₇	D₁₆	1	1	yes	characteristic
32	C₂×C₁₀₇	C₂×D₁₆	1	1	yes	characteristic
33	D₁₀₇	—	2	1	no	—
34	D₁₀₇	—	16	2	no	—
35	#2262	—	107	1	no	maximal, Sylow
36	C₁₀₇	#2262	1	1	yes	characteristic, complemented, Sylow
37	#508	—	107	2	no	—
38	Q₆₄	D₁₀₇	1	1	yes	characteristic, complemented
39	SD₆₄	—	107	2	no	—
40	D₃₂	—	107	1	no	—
41	C₂×C₃₂	—	107	1	no	—
42	#135	—	214	2	no	—
43	Q₃₂	C₂×D₁₀₇	1	2	yes	—
44	SD₃₂	—	214	2	no	—
45	D₁₆	—	107	2	no	—
46	C₂×C₁₆	—	107	1	no	—
47	C₃₂	—	107	1	no	—
48	C₃₂	C₂×D₁₀₇	1	1	yes	characteristic
49	#41	—	428	2	no	—
50	Q₁₆	—	2	2	no	—
51	SD₁₆	—	428	2	no	—
52	D₈	—	214	2	no	—
53	C₂×C₈	—	107	1	no	—
54	C₁₆	—	107	1	no	—
55	C₁₆	C₂²×D₁₀₇	1	1	yes	characteristic
56	Q₈	—	4	2	no	—
57	D₄	—	856	2	no	—
58	D₄	—	428	2	no	—
59	C₂×C₄	—	856	2	no	—
60	C₂×C₄	—	107	1	no	—
61	C₈	—	107	1	no	—
62	C₈	D₄×D₁₀₇	1	1	yes	characteristic
63	C₂²	—	856	2	no	—
64	C₂²	—	107	1	no	—
65	C₄	—	8	2	no	—
66	C₄	—	107	1	no	—
67	C₄	D₈×D₁₀₇	1	1	yes	characteristic
68	C₂	—	1712	2	no	—
69	C₂	—	214	1	no	—
70	C₂	D₁₆×D₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆×D₁₀₇
commutator, cosocle	C₁₆×C₁₀₇	C₂³
Frattini	C₁₆	C₂²×D₁₀₇
Fitting	Q₆₄×C₁₀₇	C₂
socle	C₂×C₁₀₇	C₂×D₁₆
hypercenter	Q₆₄	D₁₀₇
hypocenter	C₁₀₇	#2262
LCS	G ▷ C₁₆×C₁₀₇ ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
1. #20293060	C₂	semidirect
2. #20292464	C₂	semidirect
3. D₃₄₂₄	C₂	semidirect
4. C₃₂×D₁₀₇	C₂	semidirect
5. Q₆₄×C₁₀₇	C₂	semidirect
36. C₁₀₇	#2262	semidirect
38. Q₆₄	D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#20293060	2
#20292464	2
D₃₄₂₄	2
C₃₂×D₁₀₇	2
Q₆₄×C₁₀₇	2
#2262	107