Subgroups — #28334652

Order 13696.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₇₁₂	C₂	1	1	yes	maximal, characteristic
2	#20292457	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₅₆	—	2	1	no	—
5	D₁₇₁₂	—	2	1	no	—
6	D₁₇₁₂	C₄	1	2	yes	—
7	#16812045	—	2	1	no	—
8	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
9	C₃₂×C₁₀₇	—	2	1	no	—
10	C₂×D₄₂₈	—	4	1	no	—
11	D₈₅₆	—	2	2	no	—
12	D₈₅₆	—	4	1	no	—
13	#13133802	—	4	1	no	—
14	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
15	C₁₆×C₁₀₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₀₇	D₄	1	1	yes	characteristic
17	C₂²×D₁₀₇	—	8	1	no	—
18	C₂×C₄×C₁₀₇	D₈	1	1	yes	characteristic
19	D₄₂₈	—	4	2	no	—
20	D₄₂₈	—	8	1	no	—
21	C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₁₀₇	SD₁₆	1	1	yes	characteristic
23	C₁₀₇⋊C₈	—	8	1	no	—
24	C₂²×C₁₀₇	SD₃₂	1	1	yes	characteristic
25	C₂×D₁₀₇	—	8	2	no	—
26	C₂×D₁₀₇	—	16	2	no	—
27	C₄×C₁₀₇	#102	1	1	yes	characteristic
28	C₄×C₁₀₇	D₁₆	1	1	yes	characteristic
29	C₂×C₁₀₇	—	2	1	no	—
30	C₂×C₁₀₇	#357	1	1	yes	characteristic
31	D₁₀₇	—	16	2	no	—
32	#1419	—	107	1	no	maximal, Sylow
33	C₁₀₇	#1419	1	1	yes	characteristic, complemented, Sylow
34	C₂×D₁₆	—	107	1	no	—
35	M₆₄	D₁₀₇	1	1	yes	characteristic, complemented
36	#368	—	107	1	no	—
37	C₂×D₈	—	214	1	no	—
38	D₁₆	—	107	2	no	—
39	D₁₆	—	214	1	no	—
40	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
41	#108	—	214	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×D₄	—	428	1	no	—
44	D₈	—	428	1	no	—
45	D₈	—	214	2	no	—
46	M₁₆	—	428	1	no	—
47	C₂×C₈	D₄₂₈	1	1	yes	characteristic
48	C₁₆	C₄×D₁₀₇	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₁₀₇	1	1	yes	characteristic
50	C₂³	—	856	1	no	—
51	D₄	—	856	1	no	—
52	D₄	—	428	2	no	—
53	C₂×C₄	D₈₅₆	1	1	yes	characteristic
54	C₈	—	856	1	no	—
55	C₈	(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
56	C₈	#13133807	1	1	yes	characteristic
57	C₂²	—	856	2	no	—
58	C₂²	—	1712	2	no	—
59	C₂²	#16812054	1	1	yes	characteristic
60	C₄	#16812031	1	1	yes	characteristic
61	C₄	D₁₇₁₂	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	1712	2	no	—
64	C₂	#20292429	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20292429
commutator	C₁₆×C₁₀₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₀₇
Fitting	M₆₄×C₁₀₇	C₂
socle	C₂×C₁₀₇	#357
prime radical	C₂×D₁₇₁₂	C₂
cosocle	C₂×C₁₆×C₁₀₇	C₂²
hypercenter	M₆₄	D₁₀₇
hypocenter	C₁₀₇	#1419
LCS	G ▷ C₁₆×C₁₀₇ ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #20292457	C₂	semidirect
3. M₆₄×C₁₀₇	C₂	semidirect
33. C₁₀₇	#1419	semidirect
35. M₆₄	D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₇₁₂	2
#20292457	2
M₆₄×C₁₀₇	2
#1419	107