Subgroups — #11764783

Order 1120.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1157	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#681	C₂	1	1	yes	maximal, characteristic, complemented
4	#10587838	C₂	1	2	yes	maximal, complemented
5	D₇×#168	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₅×C₂²⋊D₇	C₄	1	2	yes	—
8	C₅×D₂₈	C₄	1	1	yes	characteristic
9	C₄×C₅×D₇	C₂²	1	2	yes	—
10	C₅×Dic₁₄	C₄	1	1	yes	characteristic
11	C₇⋊#168	C₂²	1	1	yes	characteristic
12	C₇⋊#168	C₂²	1	1	yes	characteristic
13	C₇×#168	C₂²	1	2	yes	—
14	#6470	—	5	1	no	maximal, Hall
15	#4094	—	7	1	no	maximal, Hall
16	C₂²×C₅×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₅×D₇	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	C₅×Dic₇	C₂×C₄	1	2	yes	—
21	#1157	D₅	1	1	yes	characteristic
22	C₇×M₁₆	—	5	1	no	—
23	C₂×C₇⋊C₈	—	5	1	no	—
24	#1131	—	5	2	no	—
25	C₈×D₇	—	5	2	no	—
26	C₅×#41	—	7	1	no	—
27	#681	—	7	2	no	—
28	#681	D₇	1	1	yes	characteristic, complemented
29	C₂×#168	—	7	1	no	—
30	C₂×#168	—	7	2	no	—
31	C₂×C₅×C₇	—	2	1	no	—
32	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
33	C₅×D₇	—	2	2	no	—
34	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
35	C₂²⋊D₇	Dic₅	1	2	yes	—
36	D₂₈	Dic₅	1	1	yes	characteristic
37	C₄×D₇	C₂×D₅	1	2	yes	—
38	Dic₁₄	Dic₅	1	1	yes	characteristic
39	C₇×C₈	—	5	2	no	—
40	C₇⋊C₈	—	5	1	no	—
41	C₇⋊C₈	—	5	1	no	—
42	C₅×Q₈	—	7	1	no	—
43	C₅×D₄	—	7	2	no	—
44	C₅×D₄	—	7	1	no	—
45	C₂×C₄×C₅	—	7	2	no	—
46	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
47	#168	—	7	1	no	—
48	#168	—	7	1	no	—
49	#168	C₂×D₇	1	2	yes	—
50	C₅×C₇	#131	1	1	yes	characteristic, complemented, Hall
51	#131	—	35	1	no	Sylow
52	C₂²×C₇	C₂×Dic₅	1	1	yes	characteristic
53	C₂×D₇	C₂×Dic₅	1	2	yes	—
54	C₄×C₇	C₂×Dic₅	1	1	yes	characteristic
55	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
56	Dic₇	C₂×Dic₅	1	2	yes	—
57	C₂²×C₅	—	7	2	no	—
58	C₂²×C₅	C₄×D₇	1	1	yes	characteristic
59	C₄×C₅	—	7	2	no	—
60	C₄×C₅	C₄×D₇	1	1	yes	characteristic
61	C₄×C₅	C₂²×D₇	1	1	yes	characteristic
62	#41	—	7	1	no	—
63	M₁₆	—	5	1	no	—
64	M₁₆	—	35	2	no	—
65	C₂×C₈	—	35	2	no	—
66	C₂×C₈	—	35	1	no	—
67	C₂×C₇	—	2	1	no	—
68	C₂×C₇	C₂²×Dic₅	1	1	yes	characteristic
69	D₇	—	2	2	no	—
70	C₂×C₅	—	2	1	no	—
71	C₂×C₅	—	14	2	no	—
72	C₂×C₅	C₂×C₄×D₇	1	1	yes	characteristic
73	Q₈	—	7	1	no	—
74	D₄	—	7	2	no	—
75	D₄	—	7	1	no	—
76	C₂×C₄	—	7	2	no	—
77	C₂×C₄	D₅×D₇	1	1	yes	characteristic
78	C₈	—	5	2	no	—
79	C₈	—	35	1	no	—
80	C₈	—	35	1	no	—
81	C₇	#4094	1	1	yes	characteristic, complemented, Sylow
82	C₅	#6470	1	1	yes	characteristic, complemented, Sylow
83	C₂²	—	7	2	no	—
84	C₂²	D₇×Dic₅	1	1	yes	characteristic
85	C₄	—	7	2	no	—
86	C₄	D₇×Dic₅	1	1	yes	characteristic
87	C₄	C₂×D₅×D₇	1	1	yes	characteristic
88	C₂	—	2	1	no	—
89	C₂	—	14	2	no	—
90	C₂	C₂×D₇×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×D₅×D₇
commutator, socle	C₂×C₅×C₇	C₂²×C₄
Fitting	C₂×C₄×C₅×C₇	C₂²
prime radical	C₅×#1157	C₂
cosocle	C₄×C₅×C₇	C₂³
hypercenter	C₂×C₄	D₅×D₇
hypocenter	C₅×C₇	#131
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₇⋊#168	C₂	semidirect
3. C₇×#681	C₂	semidirect
4. #10587838	C₂	semidirect
5. D₇×#168	C₂	semidirect
28. #681	D₇	semidirect
50. C₅×C₇	#131	semidirect
81. C₇	#4094	semidirect
82. C₅	#6470	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1157	2
C₂×C₇⋊#168	2
C₇×#681	2
#10587838	2
D₇×#168	2
#6470	5
#4094	7