Subgroups — #28338363

Order 13696.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₂₁₄	C₂	1	1	yes	maximal, characteristic
2	Dic₁₀₇⋊C₁₆	C₂	1	2	yes	maximal
3	C₄×C₁₀₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₁₀₇	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₂₁₄	C₄	1	1	yes	characteristic
7	C₂×C₁₀₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₀₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₄⋊(C₁₀₇⋊C₈)	C₄	1	1	yes	characteristic
10	Dic₁₀₇⋊C₈	C₄	1	2	yes	—
11	C₈×Dic₁₀₇	C₂²	1	2	yes	—
12	C₂×C₁₆×C₁₀₇	C₂²	1	2	yes	—
13	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₁₄	C₈	1	1	yes	characteristic
15	C₂×C₁₀₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₁₀₇	C₈	1	1	yes	characteristic
17	Dic₁₀₇⋊C₄	C₈	1	2	yes	—
18	C₄×Dic₁₀₇	C₂×C₄	1	2	yes	—
19	C₁₀₇⋊C₁₆	—	2	1	no	—
20	C₁₀₇⋊C₁₆	Q₈	1	2	yes	complemented
21	C₂×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
23	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
24	C₁₆×C₁₀₇	—	2	2	no	—
25	C₂×C₄×C₁₀₇	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₁₀₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₁₀₇	C₂×C₈	1	2	yes	—
29	C₂×Dic₁₀₇	C₂×C₈	1	2	yes	—
30	Dic₂₁₄	C₁₆	1	4	yes	complemented
31	C₈×C₁₀₇	—	2	1	no	—
32	C₈×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
33	C₈×C₁₀₇	#41	1	1	yes	characteristic
34	C₁₀₇⋊C₈	—	2	2	no	—
35	C₂²×C₁₀₇	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₁₀₇	—	2	1	no	—
37	C₄×C₁₀₇	C₂×C₁₆	1	2	yes	—
38	C₄×C₁₀₇	C₄×Q₈	1	1	yes	characteristic
39	C₄×C₁₀₇	#131	1	1	yes	characteristic
40	Dic₁₀₇	—	2	2	no	—
41	Dic₁₀₇	C₂×C₁₆	1	4	yes	—
42	C₂×C₁₀₇	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₁₀₇	C₂²×C₁₆	1	1	yes	characteristic
44	C₂×C₁₀₇	#504	1	1	yes	characteristic
45	Q₈×C₁₆	—	107	1	no	maximal, Sylow
46	C₁₀₇	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
47	C₈×Q₈	—	107	1	no	—
48	C₄⋊C₁₆	—	107	2	no	—
49	C₄⋊C₁₆	D₁₀₇	1	1	yes	characteristic
50	C₄×C₁₆	—	107	2	no	—
51	C₄×C₁₆	—	107	1	no	—
52	C₄×Q₈	—	107	1	no	—
53	C₂×C₁₆	—	107	1	no	—
54	C₂×C₁₆	—	107	1	no	—
55	C₂×C₁₆	C₂×D₁₀₇	1	2	yes	—
56	C₄⋊C₈	—	107	2	no	—
57	C₄⋊C₈	—	107	1	no	—
58	C₄×C₈	—	107	2	no	—
59	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
60	C₂×Q₈	—	107	1	no	—
61	C₂×C₈	—	107	2	no	—
62	C₂×C₈	C₄×D₁₀₇	1	1	yes	characteristic
63	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
64	C₄⋊C₄	—	107	2	no	—
65	C₄⋊C₄	—	107	1	no	—
66	C₄²	—	107	2	no	—
67	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
68	C₁₆	—	2	2	no	—
69	C₁₆	—	107	2	no	—
70	C₁₆	—	214	1	no	—
71	Q₈	—	107	4	no	—
72	C₂×C₄	—	107	2	no	—
73	C₂×C₄	—	107	2	no	—
74	C₂×C₄	C₈×D₁₀₇	1	1	yes	characteristic
75	C₂×C₄	C₈×D₁₀₇	1	1	yes	characteristic
76	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	214	2	no	—
79	C₈	Q₈×D₁₀₇	1	1	yes	characteristic
80	C₈	#13133817	1	1	yes	characteristic
81	C₂²	C₂×C₈×D₁₀₇	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	107	4	no	—
84	C₄	—	214	2	no	—
85	C₄	C₁₆×D₁₀₇	1	2	yes	—
86	C₄	#16812074	1	1	yes	characteristic
87	C₄	#16812117	1	1	yes	characteristic
88	C₂	#20292751	1	1	yes	characteristic
89	C₂	C₂×C₁₆×D₁₀₇	1	1	yes	characteristic
90	C₂	#20293045	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₇
commutator	C₂×C₁₀₇	C₂²×C₁₆
Fitting	C₄⋊C₁₆×C₁₀₇	C₂
socle, prime radical	C₂²×C₁₀₇	C₂²×C₈
cosocle	C₂×C₈×C₁₀₇	C₂³
hypercenter	C₄⋊C₁₆	D₁₀₇
hypocenter	C₁₀₇	Q₈×C₁₆
LCS	G ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
20. C₁₀₇⋊C₁₆	Q₈	semidirect
30. Dic₂₁₄	C₁₆	semidirect
46. C₁₀₇	Q₈×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₂₁₄	2
Dic₁₀₇⋊C₁₆	2
C₄×C₁₀₇⋊C₁₆	2
C₁₆×Dic₁₀₇	2
C₄⋊C₁₆×C₁₀₇	2
Q₈×C₁₆	107