Subgroups — #28338776

Order 13696.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293059	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#20292466	C₂	1	2	yes	maximal, complemented
4	D₃₄₂₄	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812131	—	2	1	no	—
7	C₂×D₈₅₆	—	2	1	no	—
8	Dic₈₅₆	C₂²	1	1	yes	characteristic, complemented
9	#16812054	—	2	1	no	—
10	D₁₇₁₂	—	2	1	no	—
11	D₁₇₁₂	C₂²	1	2	yes	—
12	D₁₇₁₂	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₃₂×C₁₀₇	C₂²	1	2	yes	complemented
15	#13133818	—	4	1	no	—
16	C₂×D₄₂₈	—	4	1	no	—
17	Dic₄₂₈	—	2	1	no	—
18	#13133807	—	4	1	no	—
19	D₈₅₆	—	2	2	no	—
20	D₈₅₆	—	2	1	no	—
21	D₈₅₆	—	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	C₂²×D₁₀₇	—	8	1	no	—
26	C₂×C₄×C₁₀₇	D₈	1	1	yes	characteristic
27	C₂²⋊D₁₀₇	—	8	1	no	—
28	D₄₂₈	—	4	2	no	—
29	D₄₂₈	—	4	1	no	—
30	D₄₂₈	—	8	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	C₂×D₁₀₇	—	8	2	no	—
38	C₂×D₁₀₇	—	16	2	no	—
39	C₄×C₁₀₇	D₁₆	1	1	yes	characteristic
40	C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
41	Dic₁₀₇	—	8	1	no	—
42	C₂×C₁₀₇	—	2	1	no	—
43	C₂×C₁₀₇	C₂×D₁₆	1	1	yes	characteristic
44	D₁₀₇	—	16	1	no	—
45	D₁₀₇	—	16	2	no	—
46	#2263	—	107	1	no	maximal, Sylow
47	C₁₀₇	#2263	1	1	yes	characteristic, complemented, Sylow
48	#508	—	107	1	no	—
49	C₂×D₁₆	—	107	1	no	—
50	SD₆₄	—	107	2	no	—
51	D₃₂	—	107	2	no	—
52	M₆₄	D₁₀₇	1	1	yes	characteristic, complemented
53	#135	—	214	1	no	—
54	C₂×D₈	—	214	1	no	—
55	Q₃₂	—	107	1	no	—
56	SD₃₂	—	214	1	no	—
57	D₁₆	—	107	2	no	—
58	D₁₆	—	107	1	no	—
59	D₁₆	—	214	1	no	—
60	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
61	C₃₂	C₂×D₁₀₇	1	2	yes	complemented
62	#41	—	428	1	no	—
63	C₂×D₄	—	428	1	no	—
64	Q₁₆	—	214	1	no	—
65	SD₁₆	—	428	1	no	—
66	D₈	—	428	1	no	—
67	D₈	—	214	2	no	—
68	D₈	—	214	1	no	—
69	C₂×C₈	D₄₂₈	1	1	yes	characteristic
70	C₁₆	D₄₂₈	1	1	yes	characteristic
71	C₁₆	C₂²×D₁₀₇	1	1	yes	characteristic
72	C₂³	—	856	1	no	—
73	Q₈	—	428	1	no	—
74	D₄	—	856	1	no	—
75	D₄	—	856	1	no	—
76	D₄	—	428	2	no	—
77	D₄	—	428	1	no	—
78	C₂×C₄	—	856	1	no	—
79	C₂×C₄	D₈₅₆	1	1	yes	characteristic
80	C₈	D₈₅₆	1	1	yes	characteristic
81	C₈	C₂×D₄₂₈	1	1	yes	characteristic
82	C₂²	—	856	1	no	—
83	C₂²	—	856	2	no	—
84	C₂²	—	1712	2	no	—
85	C₂²	D₁₇₁₂	1	1	yes	characteristic
86	C₄	—	856	1	no	—
87	C₄	D₁₇₁₂	1	1	yes	characteristic
88	C₄	C₂×D₈₅₆	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	1712	2	no	—
91	C₂	—	1712	1	no	—
92	C₂	C₂×D₁₇₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₇₁₂
commutator, cosocle	C₁₆×C₁₀₇	C₂³
Frattini	C₁₆	C₂²×D₁₀₇
Fitting	M₆₄×C₁₀₇	C₂
socle	C₂×C₁₀₇	C₂×D₁₆
hypercenter	M₆₄	D₁₀₇
hypocenter	C₁₀₇	#2263
LCS	G ▷ C₁₆×C₁₀₇ ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
1. #20293059	C₂	semidirect
2. C₂×D₁₇₁₂	C₂	semidirect
3. #20292466	C₂	semidirect
4. D₃₄₂₄	C₂	semidirect
5. M₆₄×C₁₀₇	C₂	semidirect
8. Dic₈₅₆	C₂²	semidirect
12. D₁₇₁₂	C₂²	semidirect
14. C₃₂×C₁₀₇	C₂²	semidirect
47. C₁₀₇	#2263	semidirect
52. M₆₄	D₁₀₇	semidirect
61. C₃₂	C₂×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
C₂×D₁₇₁₂	2
#20292466	2
D₃₄₂₄	2
M₆₄×C₁₀₇	2
#2263	107