Subgroups — #28338735

Order 13696.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293031	C₂	1	2	yes	maximal
2	#20292453	C₂	1	1	yes	maximal, characteristic
3	#20292452	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₀₇	C₂	1	1	yes	maximal, characteristic
5	#366×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	#16812108	—	2	2	no	—
7	C₂×C₁₀₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16812043	C₂²	1	2	yes	—
9	#16812042	—	2	2	no	—
10	C₈×Dic₁₀₇	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
12	#107×C₁₀₇	C₂²	1	2	yes	—
13	C₂×Dic₂₁₄	—	4	2	no	—
14	C₂×C₁₀₇⋊C₈	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₀₇	—	2	2	no	—
16	Dic₁₀₇⋊C₄	—	4	2	no	—
17	C₄×Dic₁₀₇	D₄	1	1	yes	characteristic
18	C₁₀₇⋊C₁₆	Q₈	1	2	yes	complemented
19	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₀₇	—	2	2	no	—
21	C₁₆×C₁₀₇	Q₈	1	2	yes	complemented
22	C₂×C₄×C₁₀₇	—	4	2	no	—
23	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
24	C₂×Dic₁₀₇	—	4	2	no	—
25	C₂×Dic₁₀₇	D₈	1	1	yes	characteristic
26	C₂×Dic₁₀₇	D₈	1	1	yes	characteristic
27	Dic₂₁₄	—	8	4	no	—
28	C₈×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
29	C₈×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
30	C₁₀₇⋊C₈	Q₁₆	1	2	yes	—
31	C₂²×C₁₀₇	C₂×D₈	1	1	yes	characteristic
32	C₄×C₁₀₇	—	8	2	no	—
33	C₄×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
34	C₄×C₁₀₇	C₂×Q₁₆	1	1	yes	characteristic
35	Dic₁₀₇	—	8	2	no	—
36	Dic₁₀₇	D₁₆	1	2	yes	—
37	Dic₁₀₇	Q₃₂	1	2	yes	—
38	C₂×C₁₀₇	#500	1	1	yes	characteristic
39	C₂×C₁₀₇	C₂×D₁₆	1	1	yes	characteristic
40	C₂×C₁₀₇	C₂×Q₃₂	1	1	yes	characteristic
41	#2252	—	107	1	no	maximal, Sylow
42	C₁₀₇	#2252	1	1	yes	characteristic, complemented, Sylow
43	#500	—	107	2	no	—
44	#366	—	107	1	no	—
45	#366	—	107	2	no	—
46	#366	D₁₀₇	1	1	yes	characteristic
47	C₄×C₁₆	—	107	1	no	—
48	C₄⋊Q₈	—	214	2	no	—
49	C₂×C₁₆	—	107	1	no	—
50	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
51	#107	—	107	2	no	—
52	#107	—	214	2	no	—
53	#107	C₂×D₁₀₇	1	2	yes	—
54	C₄×C₈	—	107	1	no	—
55	C₂×Q₈	—	428	2	no	—
56	C₂×C₈	—	107	1	no	—
57	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	428	2	no	—
60	C₄⋊C₄	—	214	2	no	—
61	C₄²	—	107	1	no	—
62	C₁₆	—	107	2	no	—
63	C₁₆	Dic₂₁₄	1	2	yes	complemented
64	Q₈	—	856	4	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	107	1	no	—
67	C₂×C₄	—	107	1	no	—
68	C₂×C₄	—	428	2	no	—
69	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
70	C₈	—	107	2	no	—
71	C₈	Q₈×D₁₀₇	1	1	yes	characteristic
72	C₈	C₂×Dic₂₁₄	1	1	yes	characteristic
73	C₂²	D₈×D₁₀₇	1	1	yes	characteristic
74	C₄	—	8	2	no	—
75	C₄	—	856	2	no	—
76	C₄	—	107	2	no	—
77	C₄	—	107	2	no	—
78	C₄	#16812108	1	1	yes	characteristic
79	C₄	Q₁₆×D₁₀₇	1	1	yes	characteristic
80	C₂	#20293031	1	1	yes	characteristic
81	C₂	D₁₆×D₁₀₇	1	1	yes	characteristic
82	C₂	Q₃₂×D₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₀₇
commutator, cosocle	C₂×C₈×C₁₀₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₇
Fitting	#366×C₁₀₇	C₂
socle, prime radical	C₂²×C₁₀₇	C₂×D₈
hypercenter	#366	D₁₀₇
hypocenter	C₁₀₇	#2252
LCS	G ▷ C₂×C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
18. C₁₀₇⋊C₁₆	Q₈	semidirect
21. C₁₆×C₁₀₇	Q₈	semidirect
42. C₁₀₇	#2252	semidirect
63. C₁₆	Dic₂₁₄	semidirect

Maximal subgroups

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

subgroup	index
#20293031	2
#20292453	2
#20292452	2
C₁₆×Dic₁₀₇	2
#366×C₁₀₇	2
#2252	107