Subgroups — C₅×#6467 #11765062

Order 1120.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1157	C₂	1	2	yes	maximal, complemented
2	C₂×C₅×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×Dic₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₅₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1132	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₅×C₂²⋊D₇	—	2	2	no	—
8	C₅×D₂₈	C₂²	1	2	yes	—
9	C₄×C₅×D₇	—	2	2	no	—
10	C₅×Dic₁₄	C₂²	1	2	yes	—
11	C₅×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₅×C₇×C₈	C₂²	1	1	yes	characteristic
13	#6467	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₅×#135	—	7	1	no	maximal, Hall
15	C₂²×C₅×C₇	D₄	1	1	yes	characteristic
16	C₂×C₅×D₇	—	2	2	no	—
17	C₄×C₅×C₇	D₄	1	1	yes	characteristic
18	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
19	C₅×Dic₇	—	2	2	no	—
20	#1157	C₂×C₅	1	2	yes	complemented
21	C₂×C₇×C₈	C₂×C₅	1	1	yes	characteristic, complemented
22	Dic₂₈	C₂×C₅	1	1	yes	characteristic, complemented
23	D₅₆	C₂×C₅	1	1	yes	characteristic, complemented
24	#1132	C₂×C₅	1	2	yes	complemented
25	C₅×#41	—	7	2	no	—
26	C₅×Q₁₆	—	7	1	no	—
27	C₅×SD₁₆	—	7	2	no	—
28	C₅×D₈	—	7	1	no	—
29	C₂×C₅×C₈	D₇	1	1	yes	characteristic, complemented
30	C₂×C₅×C₇	—	2	1	no	—
31	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
32	C₅×D₇	—	4	2	no	—
33	C₂×C₄×C₇	C₂²×C₅	1	1	yes	characteristic
34	C₂²⋊D₇	—	2	2	no	—
35	D₂₈	C₂²×C₅	1	2	yes	—
36	C₄×D₇	—	2	2	no	—
37	Dic₁₄	C₂²×C₅	1	2	yes	—
38	C₇×C₈	C₂²×C₅	1	1	yes	characteristic
39	C₇×C₈	C₂²×C₅	1	1	yes	characteristic
40	C₅×Q₈	—	7	2	no	—
41	C₅×D₄	—	7	2	no	—
42	C₅×D₄	—	14	2	no	—
43	C₂×C₄×C₅	—	14	2	no	—
44	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
45	C₅×C₈	C₂×D₇	1	1	yes	characteristic
46	C₅×C₈	C₂×D₇	1	1	yes	characteristic
47	C₅×C₇	#135	1	1	yes	characteristic, complemented, Hall
48	#135	—	7	1	no	Sylow
49	C₂²×C₇	C₅×D₄	1	1	yes	characteristic
50	C₂×D₇	—	2	2	no	—
51	C₄×C₇	C₅×D₄	1	1	yes	characteristic
52	C₄×C₇	C₂³×C₅	1	1	yes	characteristic
53	Dic₇	—	2	2	no	—
54	C₂²×C₅	—	14	2	no	—
55	C₂²×C₅	D₂₈	1	1	yes	characteristic
56	C₄×C₅	—	14	2	no	—
57	C₄×C₅	D₂₈	1	1	yes	characteristic
58	C₄×C₅	C₂²×D₇	1	1	yes	characteristic
59	#41	—	7	2	no	—
60	Q₁₆	—	7	1	no	—
61	SD₁₆	—	7	2	no	—
62	D₈	—	7	1	no	—
63	C₂×C₈	C₅×D₇	1	1	yes	characteristic, complemented
64	C₂×C₇	—	2	1	no	—
65	C₂×C₇	C₂×C₅×D₄	1	1	yes	characteristic
66	D₇	—	4	2	no	—
67	C₂×C₅	—	2	1	no	—
68	C₂×C₅	—	28	2	no	—
69	C₂×C₅	C₂×D₂₈	1	1	yes	characteristic
70	Q₈	—	7	2	no	—
71	D₄	—	7	2	no	—
72	D₄	—	14	2	no	—
73	C₂×C₄	—	14	2	no	—
74	C₂×C₄	C₂×C₅×D₇	1	1	yes	characteristic
75	C₈	C₂×C₅×D₇	1	1	yes	characteristic
76	C₈	C₂×C₅×D₇	1	1	yes	characteristic
77	C₇	C₅×#135	1	1	yes	characteristic, complemented, Sylow
78	C₅	#6467	1	1	yes	characteristic, direct product factor, Sylow
79	C₂²	—	14	2	no	—
80	C₂²	C₅×D₂₈	1	1	yes	characteristic
81	C₄	—	14	2	no	—
82	C₄	C₅×D₂₈	1	1	yes	characteristic
83	C₄	C₂²×C₅×D₇	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	28	2	no	—
86	C₂	C₂×C₅×D₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	D₂₈
commutator, cosocle	C₄×C₇	C₂³×C₅
Frattini	C₄	C₂²×C₅×D₇
Fitting	C₂×C₅×C₇×C₈	C₂
socle	C₂×C₅×C₇	C₂×D₄
hypercenter	C₂×C₅×C₈	D₇
hypocenter	C₇	C₅×#135
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈

Complements

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

N	H ≅ G/N	type
1. C₅×#1157	C₂	semidirect
2. C₂×C₅×C₇×C₈	C₂	semidirect
3. C₅×Dic₂₈	C₂	semidirect
4. C₅×D₅₆	C₂	semidirect
5. C₅×#1132	C₂	semidirect
13. #6467	C₅	direct (N × H)
20. #1157	C₂×C₅	semidirect
21. C₂×C₇×C₈	C₂×C₅	semidirect
22. Dic₂₈	C₂×C₅	semidirect
23. D₅₆	C₂×C₅	semidirect
24. #1132	C₂×C₅	semidirect
29. C₂×C₅×C₈	D₇	semidirect
47. C₅×C₇	#135	semidirect
63. C₂×C₈	C₅×D₇	semidirect
77. C₇	C₅×#135	semidirect
78. C₅	#6467	direct (N × H)

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×C₅×C₇×C₈	2
C₅×Dic₂₈	2
C₅×D₅₆	2
C₅×#1132	2
#6467	5
C₅×#135	7

Subgroups — C5×#6467 #11765062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#6467 #11765062