Subgroups — C₅×#6507 #11765099

Order 1120.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×Dic₁₄	C₂	1	1	yes	maximal, characteristic
3	C₅×Dic₇⋊C₄	C₂	1	4	yes	maximal
4	C₄×C₅×Dic₇	C₂	1	1	yes	maximal, characteristic
5	C₅×C₇×Q₈	—	2	2	no	—
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	2	yes	—
10	C₅×Dic₁₄	—	2	2	no	—
11	#6507	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₅×C₄⋊Q₈	—	7	1	no	maximal, Hall
13	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
14	C₄×C₅×C₇	—	2	2	no	—
15	C₄×C₅×C₇	D₄	1	2	yes	—
16	C₅×Dic₇	—	2	2	no	—
17	C₅×Dic₇	Q₈	1	4	yes	complemented
18	C₂×C₇×Q₈	C₂×C₅	1	1	yes	characteristic
19	C₂×Dic₁₄	C₂×C₅	1	1	yes	characteristic
20	Dic₇⋊C₄	C₂×C₅	1	4	yes	—
21	C₄×Dic₇	C₂×C₅	1	1	yes	characteristic
22	C₂×C₅×Q₈	—	7	1	no	—
23	C₂×C₅×Q₈	D₇	1	1	yes	characteristic
24	C₅×C₄⋊C₄	—	7	4	no	—
25	C₄²×C₅	—	7	1	no	—
26	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
27	C₂×C₅×C₇	C₂×Q₈	1	2	yes	—
28	C₇×Q₈	—	2	2	no	—
29	C₂×C₄×C₇	C₂²×C₅	1	2	yes	—
30	C₂×C₄×C₇	C₂²×C₅	1	1	yes	characteristic
31	C₂×Dic₇	C₂²×C₅	1	2	yes	—
32	C₂×Dic₇	C₂²×C₅	1	2	yes	—
33	Dic₁₄	—	2	2	no	—
34	C₅×Q₈	—	2	2	no	—
35	C₅×Q₈	—	14	2	no	—
36	C₂×C₄×C₅	—	7	2	no	—
37	C₂×C₄×C₅	—	7	2	no	—
38	C₂×C₄×C₅	C₂×D₇	1	2	yes	—
39	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
40	C₅×C₇	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
41	C₄⋊Q₈	—	7	1	no	Sylow
42	C₂²×C₇	C₂³×C₅	1	1	yes	characteristic
43	C₄×C₇	—	2	2	no	—
44	C₄×C₇	C₅×D₄	1	2	yes	—
45	Dic₇	—	2	2	no	—
46	Dic₇	C₅×Q₈	1	4	yes	complemented
47	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
48	C₄×C₅	—	2	2	no	—
49	C₄×C₅	—	7	4	no	—
50	C₄×C₅	—	14	2	no	—
51	C₄×C₅	C₂²⋊D₇	1	2	yes	—
52	C₂×Q₈	—	7	1	no	—
53	C₂×Q₈	C₅×D₇	1	1	yes	characteristic
54	C₄⋊C₄	—	7	4	no	—
55	C₄²	—	7	1	no	—
56	C₂×C₇	C₂×C₅×D₄	1	1	yes	characteristic
57	C₂×C₇	C₂×C₅×Q₈	1	2	yes	—
58	C₂×C₅	Q₈×D₇	1	2	yes	—
59	C₂×C₅	C₂×C₂²⋊D₇	1	1	yes	characteristic
60	Q₈	—	2	2	no	—
61	Q₈	—	14	2	no	—
62	C₂×C₄	—	7	2	no	—
63	C₂×C₄	—	7	2	no	—
64	C₂×C₄	C₂×C₅×D₇	1	2	yes	—
65	C₂×C₄	C₂×C₅×D₇	1	1	yes	characteristic
66	C₇	C₅×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
67	C₅	#6507	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²	C₂²×C₅×D₇	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	7	4	no	—
71	C₄	—	14	2	no	—
72	C₄	C₅×C₂²⋊D₇	1	2	yes	—
73	C₂	C₅×Q₈×D₇	1	2	yes	—
74	C₂	C₂×C₅×C₂²⋊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	C₂×C₅×C₇×Q₈	C₂
socle, prime radical	C₂²×C₅×C₇	C₂³
hypercenter	C₂×C₅×Q₈	D₇
hypocenter	C₇	C₅×C₄⋊Q₈
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×Q₈

Complements

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

N	H ≅ G/N	type
11. #6507	C₅	direct (N × H)
17. C₅×Dic₇	Q₈	semidirect
40. C₅×C₇	C₄⋊Q₈	semidirect
46. Dic₇	C₅×Q₈	semidirect
66. C₇	C₅×C₄⋊Q₈	semidirect
67. C₅	#6507	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×C₇×Q₈	2
C₂×C₅×Dic₁₄	2
C₅×Dic₇⋊C₄	2
C₄×C₅×Dic₇	2
#6507	5
C₅×C₄⋊Q₈	7

Subgroups — C5×#6507 #11765099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#6507 #11765099