Subgroups — #11765231

Order 1120.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10587987	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#10587898	C₂	1	1	yes	maximal, characteristic, complemented
4	#10587893	C₂	1	2	yes	maximal, complemented
5	C₈×D₃₅	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₂²⋊D₃₅	C₄	1	2	yes	—
8	D₁₄₀	C₄	1	1	yes	characteristic
9	C₄×D₃₅	C₂²	1	2	yes	—
10	Dic₇₀	C₄	1	1	yes	characteristic
11	C₅×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₅×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₇⋊#168	C₂²	1	2	yes	—
14	#6464	—	5	1	no	maximal, Hall
15	#4047	—	7	1	no	maximal, Hall
16	C₂²×C₅×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×D₃₅	C₂×C₄	1	2	yes	—
18	C₄×C₅×C₇	C₂×C₄	1	1	yes	characteristic
19	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
20	Dic₃₅	C₂×C₄	1	2	yes	—
21	#1157	—	5	1	no	—
22	C₂×C₇×C₈	D₅	1	1	yes	characteristic, complemented
23	#1136	—	5	1	no	—
24	#1131	—	5	2	no	—
25	C₈×D₇	—	5	2	no	—
26	#709	—	7	1	no	—
27	C₂×C₅×C₈	D₇	1	1	yes	characteristic, complemented
28	#681	—	7	1	no	—
29	#676	—	7	2	no	—
30	C₈×D₅	—	7	2	no	—
31	C₂×C₅×C₇	—	2	1	no	—
32	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
33	D₃₅	—	2	2	no	—
34	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
35	C₂²⋊D₇	—	5	2	no	—
36	D₂₈	—	5	1	no	—
37	C₄×D₇	—	5	2	no	—
38	Dic₁₄	—	5	1	no	—
39	C₇×C₈	C₂×D₅	1	1	yes	characteristic
40	C₇×C₈	C₂×D₅	1	1	yes	characteristic
41	C₇⋊C₈	—	5	2	no	—
42	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
43	C₂²⋊D₅	—	7	2	no	—
44	D₂₀	—	7	1	no	—
45	C₄×D₅	—	7	2	no	—
46	Dic₁₀	—	7	1	no	—
47	C₅×C₈	C₂×D₇	1	1	yes	characteristic
48	C₅×C₈	C₂×D₇	1	1	yes	characteristic
49	#168	—	7	2	no	—
50	C₅×C₇	#131	1	1	yes	characteristic, complemented, Hall
51	#131	—	35	1	no	Sylow
52	C₂²×C₇	C₄×D₅	1	1	yes	characteristic
53	C₂×D₇	—	5	2	no	—
54	C₄×C₇	C₄×D₅	1	1	yes	characteristic
55	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
56	Dic₇	—	5	2	no	—
57	C₂²×C₅	C₄×D₇	1	1	yes	characteristic
58	C₂×D₅	—	7	2	no	—
59	C₄×C₅	C₄×D₇	1	1	yes	characteristic
60	C₄×C₅	C₂²×D₇	1	1	yes	characteristic
61	Dic₅	—	7	2	no	—
62	#41	—	35	1	no	—
63	M₁₆	—	35	1	no	—
64	M₁₆	—	35	2	no	—
65	C₂×C₈	—	35	2	no	—
66	C₂×C₈	D₃₅	1	1	yes	characteristic, complemented
67	C₂×C₇	—	2	1	no	—
68	C₂×C₇	C₂×C₄×D₅	1	1	yes	characteristic
69	D₇	—	10	2	no	—
70	C₂×C₅	—	2	1	no	—
71	C₂×C₅	C₂×C₄×D₇	1	1	yes	characteristic
72	D₅	—	14	2	no	—
73	Q₈	—	35	1	no	—
74	D₄	—	35	2	no	—
75	D₄	—	35	1	no	—
76	C₂×C₄	—	35	2	no	—
77	C₂×C₄	C₂×D₃₅	1	1	yes	characteristic
78	C₈	—	35	2	no	—
79	C₈	C₂×D₃₅	1	1	yes	characteristic
80	C₈	C₂×D₃₅	1	1	yes	characteristic
81	C₇	#4047	1	1	yes	characteristic, complemented, Sylow
82	C₅	#6464	1	1	yes	characteristic, complemented, Sylow
83	C₂²	—	35	2	no	—
84	C₂²	C₄×D₃₅	1	1	yes	characteristic
85	C₄	—	35	2	no	—
86	C₄	C₄×D₃₅	1	1	yes	characteristic
87	C₄	C₂²×D₃₅	1	1	yes	characteristic
88	C₂	—	2	1	no	—
89	C₂	—	70	2	no	—
90	C₂	C₂×C₄×D₃₅	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₃₅
commutator, socle	C₂×C₅×C₇	C₂²×C₄
Frattini	C₄	C₂²×D₃₅
Fitting	C₂×C₅×C₇×C₈	C₂
prime radical	#10587987	C₂
cosocle	C₄×C₅×C₇	C₂³
hypercenter	C₂×C₈	D₃₅
hypocenter	C₅×C₇	#131
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₇×C₈	C₂	semidirect
3. #10587898	C₂	semidirect
4. #10587893	C₂	semidirect
5. C₈×D₃₅	C₂	semidirect
22. C₂×C₇×C₈	D₅	semidirect
27. C₂×C₅×C₈	D₇	semidirect
50. C₅×C₇	#131	semidirect
66. C₂×C₈	D₃₅	semidirect
81. C₇	#4047	semidirect
82. C₅	#6464	semidirect

Maximal subgroups

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

subgroup	index
#10587987	2
C₂×C₅×C₇×C₈	2
#10587898	2
#10587893	2
C₈×D₃₅	2
#6464	5
#4047	7