Subgroups — #11765249

Order 1120.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10587991	C₂	1	2	yes	maximal, complemented
2	C₅×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#10587905	C₂	1	2	yes	maximal, complemented
4	D₂₈₀	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×D₃₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×C₇×Q₈	C₂²	1	2	yes	—
7	D₁₄₀	—	2	2	no	—
8	D₁₄₀	C₂²	1	2	yes	—
9	C₄×D₃₅	—	2	2	no	—
10	C₄×D₃₅	C₂²	1	1	yes	characteristic
11	C₅×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇⋊#168	C₂²	1	1	yes	characteristic
13	#6482	—	5	1	no	maximal, Hall
14	#4065	—	7	1	no	maximal, Hall
15	C₂×D₃₅	—	2	2	no	—
16	C₂×D₃₅	D₄	1	1	yes	characteristic
17	C₄×C₅×C₇	—	2	2	no	—
18	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
19	Dic₃₅	D₄	1	1	yes	characteristic
20	#1161	—	5	2	no	—
21	C₇×Q₁₆	D₅	1	1	yes	characteristic, complemented
22	#1143	—	5	2	no	—
23	D₅₆	—	5	1	no	—
24	C₈×D₇	—	5	1	no	—
25	#713	—	7	2	no	—
26	C₅×Q₁₆	D₇	1	1	yes	characteristic, complemented
27	#688	—	7	2	no	—
28	D₄₀	—	7	1	no	—
29	C₈×D₅	—	7	1	no	—
30	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
31	D₃₅	—	2	1	no	—
32	D₃₅	—	4	2	no	—
33	C₇×Q₈	C₂×D₅	1	2	yes	—
34	D₂₈	—	5	2	no	—
35	D₂₈	—	10	2	no	—
36	C₄×D₇	—	5	1	no	—
37	C₄×D₇	—	10	2	no	—
38	C₇×C₈	C₂×D₅	1	1	yes	characteristic
39	C₇⋊C₈	—	5	1	no	—
40	C₅×Q₈	C₂×D₇	1	2	yes	—
41	D₂₀	—	7	2	no	—
42	D₂₀	—	14	2	no	—
43	C₄×D₅	—	7	1	no	—
44	C₄×D₅	—	14	2	no	—
45	C₅×C₈	C₂×D₇	1	1	yes	characteristic
46	#168	—	7	1	no	—
47	C₅×C₇	#135	1	1	yes	characteristic, complemented, Hall
48	#135	—	35	1	no	Sylow
49	C₂×D₇	—	5	1	no	—
50	C₂×D₇	—	10	2	no	—
51	C₄×C₇	—	2	2	no	—
52	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
53	Dic₇	—	5	1	no	—
54	C₂×D₅	—	7	1	no	—
55	C₂×D₅	—	14	2	no	—
56	C₄×C₅	—	2	2	no	—
57	C₄×C₅	C₂²×D₇	1	1	yes	characteristic
58	Dic₅	—	7	1	no	—
59	#41	—	35	2	no	—
60	Q₁₆	D₃₅	1	1	yes	characteristic, complemented
61	SD₁₆	—	35	2	no	—
62	D₈	—	35	1	no	—
63	C₂×C₈	—	35	1	no	—
64	C₂×C₇	D₄×D₅	1	1	yes	characteristic
65	D₇	—	10	1	no	—
66	D₇	—	20	2	no	—
67	C₂×C₅	D₄×D₇	1	1	yes	characteristic
68	D₅	—	14	1	no	—
69	D₅	—	28	2	no	—
70	Q₈	C₂×D₃₅	1	2	yes	—
71	D₄	—	35	2	no	—
72	D₄	—	70	2	no	—
73	C₂×C₄	—	35	1	no	—
74	C₂×C₄	—	70	2	no	—
75	C₈	—	35	1	no	—
76	C₈	C₂×D₃₅	1	1	yes	characteristic
77	C₇	#4065	1	1	yes	characteristic, complemented, Sylow
78	C₅	#6482	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	35	1	no	—
80	C₂²	—	70	2	no	—
81	C₄	—	2	2	no	—
82	C₄	—	35	1	no	—
83	C₄	C₂²×D₃₅	1	1	yes	characteristic
84	C₂	—	70	1	no	—
85	C₂	—	140	2	no	—
86	C₂	D₄×D₃₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₃₅
commutator, cosocle	C₄×C₅×C₇	C₂³
Frattini	C₄	C₂²×D₃₅
Fitting	C₅×C₇×Q₁₆	C₂
socle	C₂×C₅×C₇	C₂×D₄
hypercenter	Q₁₆	D₃₅
hypocenter	C₅×C₇	#135
LCS	G ▷ C₄×C₅×C₇ ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. #10587991	C₂	semidirect
2. C₅×C₇×Q₁₆	C₂	semidirect
3. #10587905	C₂	semidirect
4. D₂₈₀	C₂	semidirect
5. C₈×D₃₅	C₂	semidirect
21. C₇×Q₁₆	D₅	semidirect
26. C₅×Q₁₆	D₇	semidirect
47. C₅×C₇	#135	semidirect
60. Q₁₆	D₃₅	semidirect
77. C₇	#4065	semidirect
78. C₅	#6482	semidirect

Maximal subgroups

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

subgroup	index
#10587991	2
C₅×C₇×Q₁₆	2
#10587905	2
D₂₈₀	2
C₈×D₃₅	2
#6482	5
#4065	7