Subgroups — #28338779

Order 13696.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293059	C₂	1	1	yes	maximal, characteristic
2	#20292467	C₂	1	2	yes	maximal, complemented
3	#20292464	C₂	1	2	yes	maximal, complemented
4	#20292461	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₃₂×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812131	—	2	1	no	—
7	Dic₈₅₆	C₂²	1	1	yes	characteristic
8	#16812054	—	2	1	no	—
9	D₁₇₁₂	C₂²	1	1	yes	characteristic
10	C₁₀₇⋊C₃₂	C₂²	1	2	yes	—
11	C₂×Q₁₆×C₁₀₇	—	2	1	no	—
12	Q₃₂×C₁₀₇	—	2	1	no	—
13	Q₃₂×C₁₀₇	C₂²	1	2	yes	—
14	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
15	#13133818	—	4	1	no	—
16	Dic₄₂₈	—	2	1	no	—
17	#13133807	—	4	1	no	—
18	D₈₅₆	—	2	1	no	—
19	C₂×Q₈×C₁₀₇	—	4	1	no	—
20	Q₁₆×C₁₀₇	—	2	2	no	—
21	Q₁₆×C₁₀₇	—	4	1	no	—
22	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
23	C₁₆×C₁₀₇	D₄	1	1	yes	characteristic
24	C₁₆×C₁₀₇	C₂³	1	1	yes	characteristic
25	Q₈×C₁₀₇	—	4	2	no	—
26	Q₈×C₁₀₇	—	8	1	no	—
27	C₂×C₄×C₁₀₇	—	8	1	no	—
28	C₂×C₄×C₁₀₇	D₈	1	1	yes	characteristic
29	C₂²⋊D₁₀₇	—	8	1	no	—
30	D₄₂₈	—	4	1	no	—
31	C₄×D₁₀₇	—	8	1	no	—
32	Dic₂₁₄	—	4	1	no	—
33	C₈×C₁₀₇	D₈	1	1	yes	characteristic
34	C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₁₀₇	D₁₆	1	1	yes	characteristic
36	C₂×D₁₀₇	—	8	1	no	—
37	C₄×C₁₀₇	—	8	2	no	—
38	C₄×C₁₀₇	D₁₆	1	1	yes	characteristic
39	C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
40	Dic₁₀₇	—	8	1	no	—
41	C₂×C₁₀₇	—	2	1	no	—
42	C₂×C₁₀₇	C₂×D₁₆	1	1	yes	characteristic
43	D₁₀₇	—	16	1	no	—
44	#2264	—	107	1	no	maximal, Sylow
45	C₁₀₇	#2264	1	1	yes	characteristic, complemented, Sylow
46	#508	—	107	1	no	—
47	C₂×Q₃₂	D₁₀₇	1	1	yes	characteristic, complemented
48	Q₆₄	—	107	2	no	—
49	SD₆₄	—	107	2	no	—
50	M₆₄	—	107	1	no	—
51	#135	—	214	1	no	—
52	C₂×Q₁₆	—	2	1	no	—
53	Q₃₂	—	2	1	no	—
54	Q₃₂	—	107	1	no	—
55	Q₃₂	C₂×D₁₀₇	1	2	yes	—
56	SD₃₂	—	214	1	no	—
57	D₁₆	—	107	1	no	—
58	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
59	C₃₂	—	107	2	no	—
60	#41	—	428	1	no	—
61	C₂×Q₈	—	4	1	no	—
62	Q₁₆	—	2	2	no	—
63	Q₁₆	—	4	1	no	—
64	Q₁₆	—	214	1	no	—
65	SD₁₆	—	428	1	no	—
66	D₈	—	214	1	no	—
67	C₂×C₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
68	C₁₆	C₂²⋊D₁₀₇	1	1	yes	characteristic
69	C₁₆	C₂²×D₁₀₇	1	1	yes	characteristic
70	Q₈	—	4	2	no	—
71	Q₈	—	8	1	no	—
72	Q₈	—	428	1	no	—
73	D₄	—	856	1	no	—
74	D₄	—	428	1	no	—
75	C₂×C₄	—	8	1	no	—
76	C₂×C₄	—	856	1	no	—
77	C₂×C₄	#13133803	1	1	yes	characteristic
78	C₈	#13133803	1	1	yes	characteristic
79	C₈	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
80	C₂²	—	856	1	no	—
81	C₂²	#16812050	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	856	1	no	—
84	C₄	#16812050	1	1	yes	characteristic
85	C₄	C₂×#13133803	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	1712	1	no	—
88	C₂	C₂×#16812050	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16812050
commutator, cosocle	C₁₆×C₁₀₇	C₂³
Frattini	C₁₆	C₂²×D₁₀₇
Fitting	C₂×Q₃₂×C₁₀₇	C₂
socle	C₂×C₁₀₇	C₂×D₁₆
prime radical	#20293059	C₂
hypercenter	C₂×Q₃₂	D₁₀₇
hypocenter	C₁₀₇	#2264
LCS	G ▷ C₁₆×C₁₀₇ ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #20292467	C₂	semidirect
3. #20292464	C₂	semidirect
4. #20292461	C₂	semidirect
5. C₂×Q₃₂×C₁₀₇	C₂	semidirect
45. C₁₀₇	#2264	semidirect
47. C₂×Q₃₂	D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#20293059	2
#20292467	2
#20292464	2
#20292461	2
C₂×Q₃₂×C₁₀₇	2
#2264	107