Subgroups — #28334641

Order 13696.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₀₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	#366×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₂₈	—	2	1	no	—
5	Dic₈₅₆	—	2	1	no	—
6	Dic₈₅₆	C₄	1	2	yes	complemented
7	C₁₀₇⋊C₃₂	—	2	1	no	—
8	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
9	#107×C₁₀₇	—	2	1	no	—
10	C₂×Dic₂₁₄	—	4	1	no	—
11	Dic₄₂₈	—	2	2	no	—
12	Dic₄₂₈	—	4	1	no	—
13	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
14	C₄⋊C₄×C₁₀₇	—	4	1	no	—
15	C₁₆×C₁₀₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₀₇	D₄	1	1	yes	characteristic
17	C₂×C₄×C₁₀₇	—	8	1	no	—
18	C₂×C₄×C₁₀₇	D₈	1	1	yes	characteristic
19	C₂×Dic₁₀₇	—	8	1	no	—
20	Dic₂₁₄	—	4	2	no	—
21	Dic₂₁₄	—	8	1	no	—
22	C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₀₇	SD₁₆	1	1	yes	characteristic
24	C₂²×C₁₀₇	D₁₆	1	1	yes	characteristic
25	C₄×C₁₀₇	—	16	1	no	—
26	C₄×C₁₀₇	#102	1	1	yes	characteristic
27	C₄×C₁₀₇	SD₃₂	1	1	yes	characteristic
28	Dic₁₀₇	—	8	2	no	—
29	C₂×C₁₀₇	#357	1	1	yes	characteristic
30	C₂×C₁₀₇	SD₆₄	1	1	yes	characteristic
31	C₂×C₁₀₇	Q₆₄	1	1	yes	characteristic
32	#1416	—	107	1	no	maximal, Sylow
33	C₁₀₇	#1416	1	1	yes	characteristic, complemented, Sylow
34	C₂×Q₃₂	—	107	1	no	—
35	C₂×C₃₂	—	107	1	no	—
36	#366	D₁₀₇	1	1	yes	characteristic
37	C₂×Q₁₆	—	214	1	no	—
38	Q₃₂	—	107	2	no	—
39	Q₃₂	—	214	1	no	—
40	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
41	#107	—	2	1	no	—
42	C₃₂	—	214	1	no	—
43	C₂×Q₈	—	428	1	no	—
44	Q₁₆	—	428	1	no	—
45	Q₁₆	—	214	2	no	—
46	C₂×C₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
47	C₄⋊C₄	—	4	1	no	—
48	C₁₆	C₄×D₁₀₇	1	1	yes	characteristic
49	C₁₆	D₄₂₈	1	1	yes	characteristic
50	Q₈	—	856	1	no	—
51	Q₈	—	428	2	no	—
52	C₂×C₄	—	8	1	no	—
53	C₂×C₄	—	856	1	no	—
54	C₂×C₄	#13133803	1	1	yes	characteristic
55	C₈	(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
56	C₈	#13133805	1	1	yes	characteristic
57	C₂²	#16812050	1	1	yes	characteristic
58	C₄	—	16	1	no	—
59	C₄	—	856	2	no	—
60	C₄	#16812029	1	1	yes	characteristic
61	C₄	#16812052	1	1	yes	characteristic
62	C₂	#20292427	1	1	yes	characteristic
63	C₂	#20292465	1	1	yes	characteristic
64	C₂	#20292467	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16812050
commutator	C₁₆×C₁₀₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₀₇
Fitting	#366×C₁₀₇	C₂
socle, prime radical	C₂²×C₁₀₇	D₁₆
cosocle	C₂×C₁₆×C₁₀₇	C₂²
hypercenter	#366	D₁₀₇
hypocenter	C₁₀₇	#1416
LCS	G ▷ C₁₆×C₁₀₇ ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
6. Dic₈₅₆	C₄	semidirect
33. C₁₀₇	#1416	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₅₆	2
C₂×C₁₀₇⋊C₃₂	2
#366×C₁₀₇	2
#1416	107