Subgroups — #28334497

Order 13696.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293004	C₂	1	1	yes	maximal, characteristic
2	#20292448	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₅₆	—	2	1	no	—
5	C₂×D₈₅₆	—	2	1	no	—
6	#16812106	C₄	1	2	yes	—
7	#16812049	—	2	1	no	—
8	M₃₂×C₁₀₇	—	2	1	no	—
9	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
10	C₂×D₄₂₈	—	2	2	no	—
11	C₂×D₄₂₈	—	4	2	no	—
12	D₈₅₆	—	4	1	no	—
13	D₈₅₆	—	4	1	no	—
14	D₈₅₆	—	4	1	no	—
15	D₈₅₆	—	4	1	no	—
16	C₁₀₇⋊C₁₆	—	4	1	no	—
17	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
19	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₀₇	—	4	1	no	—
21	C₂²×D₁₀₇	—	4	2	no	—
22	C₂×C₄×C₁₀₇	—	2	1	no	—
23	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
24	D₄₂₈	—	4	2	no	—
25	D₄₂₈	—	4	2	no	—
26	D₄₂₈	—	8	4	no	—
27	C₈×C₁₀₇	D₈	1	1	yes	characteristic
28	C₈×C₁₀₇	D₈	1	1	yes	characteristic
29	C₈×C₁₀₇	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₀₇	SD₁₆	1	1	yes	characteristic
31	C₂²×C₁₀₇	#100	1	1	yes	characteristic
32	C₂×D₁₀₇	—	8	4	no	—
33	C₂×D₁₀₇	—	8	2	no	—
34	C₄×C₁₀₇	—	2	2	no	—
35	C₄×C₁₀₇	#102	1	1	yes	characteristic
36	C₄×C₁₀₇	#102	1	1	yes	characteristic
37	C₂×C₁₀₇	—	2	1	no	—
38	C₂×C₁₀₇	#331	1	1	yes	characteristic
39	D₁₀₇	—	16	2	no	—
40	#1357	—	107	1	no	maximal, Sylow
41	C₁₀₇	#1357	1	1	yes	characteristic, complemented, Sylow
42	#493	—	107	1	no	—
43	#364	—	107	1	no	—
44	#364	D₁₀₇	1	1	yes	characteristic, complemented
45	C₂×D₈	—	214	1	no	—
46	C₂×D₈	—	214	1	no	—
47	#127	—	107	2	no	—
48	M₃₂	—	2	1	no	—
49	M₃₂	—	214	1	no	—
50	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
51	C₂×D₄	—	428	2	no	—
52	C₂×D₄	—	214	2	no	—
53	D₈	—	428	1	no	—
54	D₈	—	428	1	no	—
55	D₈	—	428	1	no	—
56	D₈	—	428	1	no	—
57	C₂×C₈	D₄₂₈	1	1	yes	characteristic
58	C₂×C₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
59	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
60	C₁₆	—	4	1	no	—
61	C₁₆	—	428	1	no	—
62	C₂³	—	428	2	no	—
63	D₄	—	856	4	no	—
64	D₄	—	428	2	no	—
65	D₄	—	428	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
68	C₈	#13133803	1	1	yes	characteristic
69	C₈	D₈₅₆	1	1	yes	characteristic
70	C₈	#13133805	1	1	yes	characteristic
71	C₈	#13133807	1	1	yes	characteristic
72	C₂²	—	856	4	no	—
73	C₂²	—	856	2	no	—
74	C₂²	#16812025	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	#16812029	1	1	yes	characteristic
77	C₄	#16812031	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	1712	2	no	—
80	C₂	#20292370	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20292370
commutator	C₄²×C₁₀₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₀₇
Fitting	#364×C₁₀₇	C₂
socle	C₂×C₁₀₇	#331
prime radical	#20293004	C₂
cosocle	C₄×C₈×C₁₀₇	C₂²
hypercenter	#364	D₁₀₇
hypocenter	C₁₀₇	#1357
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. #20292448	C₂	semidirect
3. #364×C₁₀₇	C₂	semidirect
41. C₁₀₇	#1357	semidirect
44. #364	D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#20293004	2
#20292448	2
#364×C₁₀₇	2
#1357	107