Subgroups — #28338707

Order 13696.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293029	C₂	1	1	yes	maximal, characteristic
2	#20293007	C₂	1	1	yes	maximal, characteristic
3	#20292432	C₂	1	4	yes	maximal
4	#346×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₂₈	—	2	1	no	—
6	C₂×Dic₄₂₈	C₂²	1	2	yes	—
7	#16812109	—	2	1	no	—
8	#16812101	—	2	1	no	—
9	#16812043	C₂²	1	2	yes	—
10	#16812041	—	2	1	no	—
11	C₂×C₁₆×C₁₀₇	C₂²	1	2	yes	—
12	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₁₄	—	2	2	no	—
14	Dic₄₂₈	—	2	4	no	—
15	Dic₄₂₈	—	4	2	no	—
16	C₄⋊Dic₁₀₇	—	2	2	no	—
17	C₄⋊Dic₁₀₇	—	4	2	no	—
18	Dic₁₀₇⋊C₄	—	4	2	no	—
19	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
21	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
22	C₁₆×C₁₀₇	—	2	2	no	—
23	C₂×C₄×C₁₀₇	D₈	1	2	yes	—
24	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
25	C₂×Dic₁₀₇	—	4	2	no	—
26	C₂×Dic₁₀₇	—	4	2	no	—
27	Dic₂₁₄	—	4	2	no	—
28	Dic₂₁₄	—	4	2	no	—
29	C₈×C₁₀₇	SD₁₆	1	2	yes	—
30	C₈×C₁₀₇	#41	1	2	yes	—
31	C₂²×C₁₀₇	C₂×D₈	1	1	yes	characteristic
32	C₄×C₁₀₇	—	2	2	no	—
33	C₄×C₁₀₇	#124	1	1	yes	characteristic
34	C₄×C₁₀₇	C₂×SD₁₆	1	1	yes	characteristic
35	Dic₁₀₇	—	8	2	no	—
36	Dic₁₀₇	—	8	2	no	—
37	C₂×C₁₀₇	#486	1	1	yes	characteristic
38	C₂×C₁₀₇	#510	1	2	yes	—
39	#2245	—	107	1	no	maximal, Sylow
40	C₁₀₇	#2245	1	1	yes	characteristic, complemented, Sylow
41	#499	—	107	1	no	—
42	#494	—	107	1	no	—
43	#358	—	107	4	no	—
44	#346	D₁₀₇	1	1	yes	characteristic
45	C₂×Q₁₆	—	107	2	no	—
46	C₂×Q₁₆	—	214	1	no	—
47	C₄⋊Q₈	—	214	1	no	—
48	#125	—	214	1	no	—
49	C₂×C₁₆	C₂×D₁₀₇	1	2	yes	—
50	#107	—	107	2	no	—
51	#106	—	214	1	no	—
52	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
53	C₂×Q₈	—	214	2	no	—
54	Q₁₆	—	428	2	no	—
55	Q₁₆	—	214	4	no	—
56	C₂×C₈	D₄₂₈	1	1	yes	characteristic
57	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
58	C₄⋊C₄	—	428	2	no	—
59	C₄⋊C₄	—	428	2	no	—
60	C₄⋊C₄	—	214	2	no	—
61	C₄²	D₄₂₈	1	1	yes	characteristic
62	C₁₆	—	2	2	no	—
63	Q₈	—	428	2	no	—
64	Q₈	—	428	2	no	—
65	C₂×C₄	—	428	2	no	—
66	C₂×C₄	—	428	2	no	—
67	C₂×C₄	D₈₅₆	1	2	yes	—
68	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
69	C₈	#13133807	1	2	yes	—
70	C₈	#13133818	1	2	yes	—
71	C₂²	C₂×D₈₅₆	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	856	2	no	—
74	C₄	—	856	2	no	—
75	C₄	#16812097	1	1	yes	characteristic
76	C₄	C₂×#13133807	1	1	yes	characteristic
77	C₂	#20292974	1	1	yes	characteristic
78	C₂	#20293070	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈₅₆
commutator, cosocle	C₂×C₈×C₁₀₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₇
Fitting	#346×C₁₀₇	C₂
socle, prime radical	C₂²×C₁₀₇	C₂×D₈
hypercenter	#346	D₁₀₇
hypocenter	C₁₀₇	#2245
LCS	G ▷ C₂×C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
40. C₁₀₇	#2245	semidirect

Maximal subgroups

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

subgroup	index
#20293029	2
#20293007	2
#20292432	2
#346×C₁₀₇	2
#2245	107