Subgroups — #28334435

Order 13696.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20292745	C₂	1	1	yes	maximal, characteristic
2	#20292448	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈²×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812131	C₄	1	1	yes	characteristic
5	#16812115	—	2	1	no	—
6	#16812049	—	2	1	no	—
7	#16812045	C₄	1	1	yes	characteristic
8	#16812037	—	2	1	no	—
9	C₄×C₈×C₁₀₇	—	2	1	no	—
10	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
11	#13133818	—	2	1	no	—
12	Dic₄₂₈	C₈	1	1	yes	characteristic, complemented
13	#13133807	—	2	1	no	—
14	D₈₅₆	C₈	1	1	yes	characteristic, complemented
15	#13133802	—	2	1	no	—
16	#13133801	—	4	1	no	—
17	C₈×D₁₀₇	—	4	1	no	—
18	C₁₀₇⋊C₁₆	—	4	1	no	—
19	C₂×C₈×C₁₀₇	—	2	2	no	—
20	C₂×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
22	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₀₇	—	2	1	no	—
24	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²⋊D₁₀₇	—	4	1	no	—
26	D₄₂₈	—	2	1	no	—
27	C₄×D₁₀₇	—	4	1	no	—
28	Dic₂₁₄	—	2	1	no	—
29	C₈×C₁₀₇	—	2	4	no	—
30	C₈×C₁₀₇	C₂×C₈	1	1	yes	characteristic
31	C₈×C₁₀₇	M₁₆	1	1	yes	characteristic
32	C₈×C₁₀₇	D₈	1	1	yes	characteristic
33	C₈×C₁₀₇	SD₁₆	1	1	yes	characteristic
34	C₁₀₇⋊C₈	—	4	1	no	—
35	C₂²×C₁₀₇	#104	1	1	yes	characteristic
36	C₂×D₁₀₇	—	4	1	no	—
37	C₄×C₁₀₇	—	2	2	no	—
38	C₄×C₁₀₇	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₀₇	#102	1	1	yes	characteristic
40	Dic₁₀₇	—	4	1	no	—
41	C₂×C₁₀₇	—	2	1	no	—
42	C₂×C₁₀₇	D₄⋊C₈	1	1	yes	characteristic
43	D₁₀₇	—	8	1	no	—
44	#1335	—	107	1	no	maximal, Sylow
45	C₁₀₇	#1335	1	1	yes	characteristic, complemented, Sylow
46	#443	—	107	1	no	—
47	#364	—	107	1	no	—
48	C₈²	D₁₀₇	1	1	yes	characteristic, complemented
49	#135	—	107	1	no	—
50	#131	—	214	1	no	—
51	M₃₂	—	214	1	no	—
52	#108	—	107	1	no	—
53	#104	—	214	1	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
56	#41	—	214	1	no	—
57	Q₁₆	—	107	1	no	—
58	SD₁₆	—	214	1	no	—
59	D₈	—	107	1	no	—
60	M₁₆	—	428	1	no	—
61	M₁₆	—	214	1	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	428	1	no	—
64	C₂×C₈	C₄×D₁₀₇	1	1	yes	characteristic
65	C₂×C₈	D₄₂₈	1	1	yes	characteristic
66	C₄²	C₂²⋊D₁₀₇	1	1	yes	characteristic
67	C₁₆	—	428	1	no	—
68	Q₈	—	214	1	no	—
69	D₄	—	428	1	no	—
70	D₄	—	214	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	428	1	no	—
73	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
74	C₈	—	2	4	no	—
75	C₈	—	428	1	no	—
76	C₈	C₈×D₁₀₇	1	1	yes	characteristic
77	C₈	#13133801	1	1	yes	characteristic
78	C₈	D₈₅₆	1	1	yes	characteristic
79	C₈	#13133807	1	1	yes	characteristic
80	C₂²	—	428	1	no	—
81	C₂²	#16812037	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	428	1	no	—
84	C₄	(C₂×D₁₀₇)⋊C₈	1	1	yes	characteristic
85	C₄	#16812031	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	856	1	no	—
88	C₂	#20292356	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₈₅₆
commutator	C₈×C₁₀₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₀₇
Fitting	C₈²×C₁₀₇	C₂
socle	C₂×C₁₀₇	D₄⋊C₈
prime radical	#16812131	C₄
cosocle	C₄×C₈×C₁₀₇	C₂²
hypercenter	C₈²	D₁₀₇
hypocenter	C₁₀₇	#1335
LCS	G ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #20292448	C₂	semidirect
3. C₈²×C₁₀₇	C₂	semidirect
12. Dic₄₂₈	C₈	semidirect
14. D₈₅₆	C₈	semidirect
45. C₁₀₇	#1335	semidirect
48. C₈²	D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#20292745	2
#20292448	2
C₈²×C₁₀₇	2
#1335	107