Subgroups — #28334439

Order 13696.

94 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	C₄×C₁₀₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₂₁₄	C₄	1	1	yes	characteristic
5	C₂×C₁₀₇⋊C₁₆	—	2	1	no	—
6	C₄⋊(C₁₀₇⋊C₈)	C₄	1	1	yes	characteristic
7	Dic₁₀₇⋊C₈	—	2	1	no	—
8	C₈×Dic₁₀₇	—	2	1	no	—
9	C₂×C₁₆×C₁₀₇	—	2	1	no	—
10	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₁₄	C₈	1	1	yes	characteristic
12	C₂×C₁₀₇⋊C₈	—	2	1	no	—
13	C₄⋊Dic₁₀₇	C₈	1	1	yes	characteristic
14	Dic₁₀₇⋊C₄	—	2	1	no	—
15	C₄×Dic₁₀₇	—	2	1	no	—
16	C₁₀₇⋊C₁₆	—	2	2	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	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₀₇	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₁₀₇	M₁₆	1	1	yes	characteristic
24	C₂×Dic₁₀₇	—	2	1	no	—
25	C₂×Dic₁₀₇	—	2	1	no	—
26	Dic₂₁₄	—	2	1	no	—
27	Dic₂₁₄	C₁₆	1	2	yes	complemented
28	C₈×C₁₀₇	—	2	1	no	—
29	C₈×C₁₀₇	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₀₇	Q₁₆	1	1	yes	characteristic
31	C₁₀₇⋊C₈	—	4	1	no	—
32	C₂²×C₁₀₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₁₀₇	—	2	1	no	—
34	C₄×C₁₀₇	#103	1	1	yes	characteristic
35	C₄×C₁₀₇	#104	1	1	yes	characteristic
36	C₄×C₁₀₇	C₂×C₁₆	1	1	yes	characteristic
37	C₄×C₁₀₇	M₃₂	1	1	yes	characteristic
38	Dic₁₀₇	—	2	2	no	—
39	Dic₁₀₇	—	4	1	no	—
40	C₂×C₁₀₇	Q₈⋊C₈	1	1	yes	characteristic
41	C₂×C₁₀₇	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₂×C₁₀₇	#350	1	1	yes	characteristic
43	Q₈⋊C₁₆	—	107	1	no	maximal, Sylow
44	C₁₀₇	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
45	C₈×Q₈	—	107	1	no	—
46	C₄⋊C₁₆	D₁₀₇	1	1	yes	characteristic
47	C₄×C₁₆	—	107	1	no	—
48	C₄×Q₈	—	107	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	214	1	no	—
51	C₄⋊C₈	—	107	1	no	—
52	C₄⋊C₈	—	214	1	no	—
53	C₄×C₈	—	214	1	no	—
54	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
55	C₂×Q₈	—	107	1	no	—
56	C₂×C₈	—	214	1	no	—
57	C₂×C₈	D₄₂₈	1	1	yes	characteristic
58	C₂×C₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
59	C₄⋊C₄	—	107	1	no	—
60	C₄⋊C₄	—	214	1	no	—
61	C₄²	—	214	1	no	—
62	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
63	C₁₆	—	4	1	no	—
64	C₁₆	—	214	2	no	—
65	Q₈	—	107	2	no	—
66	Q₈	—	214	1	no	—
67	C₂×C₄	—	214	1	no	—
68	C₂×C₄	—	214	1	no	—
69	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
70	C₂×C₄	C₈×D₁₀₇	1	1	yes	characteristic
71	C₂×C₄	#13133801	1	1	yes	characteristic
72	C₈	—	2	1	no	—
73	C₈	—	428	1	no	—
74	C₈	#13133806	1	1	yes	characteristic
75	C₈	#13133808	1	1	yes	characteristic
76	C₂²	(C₂×D₁₀₇)⋊C₈	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	428	1	no	—
79	C₄	—	214	2	no	—
80	C₄	#16812032	1	1	yes	characteristic
81	C₄	#16812035	1	1	yes	characteristic
82	C₄	C₁₆×D₁₀₇	1	1	yes	characteristic
83	C₄	#16812048	1	1	yes	characteristic
84	C₂	#20292357	1	1	yes	characteristic
85	C₂	(C₂×D₁₀₇)⋊C₁₆	1	1	yes	characteristic
86	C₂	#20292412	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
27. Dic₂₁₄	C₁₆	semidirect
44. 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
C₄×C₁₀₇⋊C₁₆	2
C₄⋊C₁₆×C₁₀₇	2
Q₈⋊C₁₆	107