Subgroups — #28338256

Order 13696.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812045	C₂	1	2	yes	maximal
2	#20292457	C₂	1	4	yes	maximal, complemented
3	C₂×M₃₂×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133802	—	2	2	no	—
5	#16812045	—	2	2	no	—
6	#16812045	C₂²	1	4	yes	—
7	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
8	M₃₂×C₁₀₇	C₄	1	4	yes	—
9	C₂×C₁₆×C₁₀₇	C₂²	1	2	yes	—
10	#13133802	—	2	4	no	—
11	#13133802	—	4	2	no	—
12	C₂×C₁₀₇⋊C₈	—	4	2	no	—
13	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
15	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
16	C₂×C₈×C₁₀₇	Q₈	1	2	yes	—
17	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
18	C₁₆×C₁₀₇	C₂×C₄	1	4	yes	—
19	C₂³×C₁₀₇	Q₁₆	1	1	yes	characteristic
20	C₂×C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
21	C₂×C₄×C₁₀₇	D₈	1	2	yes	—
22	C₂×C₄×C₁₀₇	Q₁₆	1	1	yes	characteristic
23	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
24	C₈×C₁₀₇	C₄⋊C₄	1	2	yes	—
25	C₈×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
26	C₈×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
27	C₁₀₇⋊C₈	—	4	4	no	—
28	C₂²×C₁₀₇	—	4	1	no	—
29	C₂²×C₁₀₇	#107	1	2	yes	—
30	C₂²×C₁₀₇	C₂×Q₁₆	1	1	yes	characteristic
31	C₄×C₁₀₇	#107	1	2	yes	—
32	C₄×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
34	C₂×C₁₀₇	—	2	2	no	—
35	C₂×C₁₀₇	—	2	1	no	—
36	C₂×C₁₀₇	C₂×#107	1	1	yes	characteristic
37	#2161	—	107	1	no	maximal, Sylow
38	C₁₀₇	#2161	1	1	yes	characteristic, complemented, Sylow
39	C₂×M₃₂	D₁₀₇	1	1	yes	characteristic
40	C₂×#108	—	107	2	no	—
41	#368	—	107	4	no	—
42	C₂×M₁₆	—	214	2	no	—
43	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
44	M₃₂	Dic₁₀₇	1	4	yes	—
45	C₂×C₁₆	C₂×D₁₀₇	1	2	yes	—
46	#108	—	107	4	no	—
47	#108	—	214	2	no	—
48	C₂²×C₄	D₄₂₈	1	1	yes	characteristic
49	M₁₆	—	428	2	no	—
50	M₁₆	—	214	4	no	—
51	C₂×C₈	—	428	2	no	—
52	C₂×C₈	Dic₂₁₄	1	2	yes	—
53	C₂×C₈	D₄₂₈	1	1	yes	characteristic
54	C₂×C₈	C₂×Dic₁₀₇	1	2	yes	—
55	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
56	C₁₆	C₂×Dic₁₀₇	1	4	yes	—
57	C₂³	Dic₄₂₈	1	1	yes	characteristic
58	C₂×C₄	C₄⋊Dic₁₀₇	1	2	yes	—
59	C₂×C₄	D₈₅₆	1	2	yes	—
60	C₂×C₄	Dic₄₂₈	1	1	yes	characteristic
61	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
62	C₈	—	428	4	no	—
63	C₈	C₄⋊Dic₁₀₇	1	2	yes	—
64	C₈	C₂²×Dic₁₀₇	1	1	yes	characteristic
65	C₈	C₂×Dic₂₁₄	1	1	yes	characteristic
66	C₂²	—	4	1	no	—
67	C₂²	#16812043	1	2	yes	—
68	C₂²	C₂×Dic₄₂₈	1	1	yes	characteristic
69	C₄	#16812043	1	2	yes	—
70	C₄	C₂×C₄⋊Dic₁₀₇	1	1	yes	characteristic
71	C₄	C₂×D₈₅₆	1	1	yes	characteristic
72	C₂	—	2	2	no	—
73	C₂	—	2	1	no	—
74	C₂	C₂×#16812043	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈₅₆
commutator	C₈×C₁₀₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₇
Fitting	C₂×M₃₂×C₁₀₇	C₂
socle	C₂×C₁₀₇	C₂×#107
prime radical	C₂³×C₁₀₇	Q₁₆
cosocle	C₂×C₈×C₁₀₇	C₂³
hypercenter	C₂×M₃₂	D₁₀₇
hypocenter	C₁₀₇	#2161
LCS	G ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. #20292457	C₂	semidirect
38. C₁₀₇	#2161	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812045	2
#20292457	2
C₂×M₃₂×C₁₀₇	2
#2161	107