Subgroups — C₅×#6451 #11765049

Order 1120.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×Dic₁₄	C₂	1	2	yes	maximal
2	C₅×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₅×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₅×Dic₇⋊C₄	C₂	1	2	yes	maximal
5	C₄×C₅×Dic₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₅×C₇	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₅×Dic₇	C₂²	1	2	yes	—
9	C₂×C₅×Dic₇	C₂²	1	1	yes	characteristic
10	C₂×C₅×Dic₇	C₂²	1	1	yes	characteristic
11	C₅×Dic₁₄	—	2	4	no	—
12	#6451	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₅×C₄⋊Q₈	—	7	1	no	maximal, Hall
14	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
15	C₄×C₅×C₇	—	2	2	no	—
16	C₄×C₅×C₇	Q₈	1	2	yes	complemented
17	C₅×Dic₇	—	2	2	no	—
18	C₅×Dic₇	D₄	1	2	yes	—
19	C₅×Dic₇	Q₈	1	2	yes	complemented
20	C₂×Dic₁₄	C₂×C₅	1	2	yes	—
21	C₇×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
22	C₄⋊Dic₇	C₂×C₅	1	1	yes	characteristic
23	Dic₇⋊C₄	C₂×C₅	1	2	yes	—
24	C₄×Dic₇	C₂×C₅	1	1	yes	characteristic
25	C₂×C₅×Q₈	—	7	2	no	—
26	C₅×C₄⋊C₄	—	7	2	no	—
27	C₅×C₄⋊C₄	—	7	1	no	—
28	C₅×C₄⋊C₄	D₇	1	1	yes	characteristic
29	C₄²×C₅	—	7	1	no	—
30	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₅×C₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₅×C₇	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₇	C₂²×C₅	1	2	yes	—
34	C₂×C₄×C₇	C₂²×C₅	1	1	yes	characteristic
35	C₂×Dic₇	C₂²×C₅	1	2	yes	—
36	C₂×Dic₇	C₂²×C₅	1	1	yes	characteristic
37	C₂×Dic₇	C₂²×C₅	1	1	yes	characteristic
38	Dic₁₄	—	2	4	no	—
39	C₅×Q₈	—	14	4	no	—
40	C₂×C₄×C₅	—	7	1	no	—
41	C₂×C₄×C₅	—	7	1	no	—
42	C₂×C₄×C₅	—	7	2	no	—
43	C₂×C₄×C₅	C₂×D₇	1	2	yes	—
44	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
45	C₅×C₇	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
46	C₄⋊Q₈	—	7	1	no	Sylow
47	C₂²×C₇	C₂³×C₅	1	1	yes	characteristic
48	C₄×C₇	—	2	2	no	—
49	C₄×C₇	C₅×Q₈	1	2	yes	complemented
50	Dic₇	—	2	2	no	—
51	Dic₇	C₅×D₄	1	2	yes	—
52	Dic₇	C₅×Q₈	1	2	yes	complemented
53	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
54	C₄×C₅	—	2	2	no	—
55	C₄×C₅	—	7	2	no	—
56	C₄×C₅	—	7	2	no	—
57	C₄×C₅	—	14	2	no	—
58	C₄×C₅	Dic₁₄	1	2	yes	complemented
59	C₂×Q₈	—	7	2	no	—
60	C₄⋊C₄	—	7	2	no	—
61	C₄⋊C₄	—	7	1	no	—
62	C₄⋊C₄	C₅×D₇	1	1	yes	characteristic
63	C₄²	—	7	1	no	—
64	C₂×C₇	C₂×C₅×D₄	1	1	yes	characteristic
65	C₂×C₇	C₂×C₅×Q₈	1	1	yes	characteristic
66	C₂×C₇	C₂×C₅×Q₈	1	1	yes	characteristic
67	C₂×C₅	C₂×Dic₁₄	1	1	yes	characteristic
68	C₂×C₅	D₄×D₇	1	1	yes	characteristic
69	C₂×C₅	Q₈×D₇	1	1	yes	characteristic
70	Q₈	—	14	4	no	—
71	C₂×C₄	—	7	1	no	—
72	C₂×C₄	—	7	1	no	—
73	C₂×C₄	—	7	2	no	—
74	C₂×C₄	C₂×C₅×D₇	1	2	yes	—
75	C₂×C₄	C₂×C₅×D₇	1	1	yes	characteristic
76	C₇	C₅×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
77	C₅	#6451	1	1	yes	characteristic, direct product factor, Sylow
78	C₂²	C₂²×C₅×D₇	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	7	2	no	—
81	C₄	—	7	2	no	—
82	C₄	—	14	2	no	—
83	C₄	C₅×Dic₁₄	1	2	yes	complemented
84	C₂	C₂×C₅×Dic₁₄	1	1	yes	characteristic
85	C₂	C₅×D₄×D₇	1	1	yes	characteristic
86	C₂	C₅×Q₈×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂²×D₇
commutator, cosocle	C₂²×C₇	C₂³×C₅
Frattini	C₂²	C₂²×C₅×D₇
Fitting	C₅×C₇×C₄⋊C₄	C₂
socle, prime radical	C₂²×C₅×C₇	C₂³
hypercenter	C₅×C₄⋊C₄	D₇
hypocenter	C₇	C₅×C₄⋊Q₈
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
12. #6451	C₅	direct (N × H)
16. C₄×C₅×C₇	Q₈	semidirect
19. C₅×Dic₇	Q₈	semidirect
45. C₅×C₇	C₄⋊Q₈	semidirect
49. C₄×C₇	C₅×Q₈	semidirect
52. Dic₇	C₅×Q₈	semidirect
58. C₄×C₅	Dic₁₄	semidirect
76. C₇	C₅×C₄⋊Q₈	semidirect
77. C₅	#6451	direct (N × H)
83. C₄	C₅×Dic₁₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×Dic₁₄	2
C₅×C₇×C₄⋊C₄	2
C₅×C₄⋊Dic₇	2
C₅×Dic₇⋊C₄	2
C₄×C₅×Dic₇	2
#6451	5
C₅×C₄⋊Q₈	7

Subgroups — C5×#6451 #11765049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#6451 #11765049