Subgroups — C₂×C₇×C₈×D₅ #11765143

Order 1120.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×C₈×D₅	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₇×D₅	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₇×Dic₅	C₄	1	1	yes	characteristic
8	C₄×C₇×D₅	C₄	1	2	yes	—
9	C₄×C₇×D₅	C₂²	1	2	yes	—
10	C₅×C₇×C₈	C₂²	1	2	yes	complemented
11	C₇×#168	C₂²	1	2	yes	complemented
12	C₂²×C₇×C₈	—	5	1	no	maximal, Hall
13	C₂×C₈×D₅	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₅×C₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₇×D₅	C₈	1	4	yes	direct product factor
16	C₂×C₇×D₅	C₂×C₄	1	2	yes	—
17	C₄×C₅×C₇	C₂×C₄	1	1	yes	characteristic
18	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
19	C₇×Dic₅	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₇	—	5	1	no	—
21	C₂×C₇×C₈	—	5	1	no	—
22	C₂×C₇×C₈	—	5	4	no	—
23	C₂×C₇×C₈	D₅	1	1	yes	characteristic, direct product factor
24	C₂×C₄×D₅	C₂×C₇	1	1	yes	characteristic
25	C₂×C₅×C₈	C₂×C₇	1	1	yes	characteristic, complemented
26	C₂×#168	C₂×C₇	1	1	yes	characteristic, complemented
27	C₈×D₅	C₂×C₇	1	4	yes	direct product factor
28	C₂×C₅×C₇	C₂×C₈	1	2	yes	complemented
29	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
30	C₇×D₅	C₂×C₈	1	4	yes	direct product factor
31	C₂³×C₇	—	5	1	no	—
32	C₂×C₄×C₇	—	5	1	no	—
33	C₂×C₄×C₇	—	5	2	no	—
34	C₂×C₄×C₇	—	5	2	no	—
35	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
36	C₇×C₈	—	5	2	no	—
37	C₇×C₈	C₂×D₅	1	2	yes	direct product factor
38	C₂²×D₅	C₄×C₇	1	1	yes	characteristic
39	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
40	C₂×Dic₅	C₄×C₇	1	1	yes	characteristic
41	C₄×D₅	C₄×C₇	1	2	yes	—
42	C₄×D₅	C₂²×C₇	1	2	yes	—
43	C₅×C₈	C₂²×C₇	1	2	yes	complemented
44	#168	C₂²×C₇	1	2	yes	complemented
45	C₅×C₇	C₂²×C₈	1	1	yes	characteristic, complemented, Hall
46	C₂²×C₈	—	5	1	no	Sylow
47	C₂²×C₇	—	5	4	no	—
48	C₂²×C₇	—	5	2	no	—
49	C₂²×C₇	C₄×D₅	1	1	yes	characteristic
50	C₄×C₇	—	5	2	no	—
51	C₄×C₇	C₄×D₅	1	1	yes	characteristic
52	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
53	C₂²×C₅	C₂×C₄×C₇	1	1	yes	characteristic
54	C₂×D₅	C₇×C₈	1	4	yes	direct product factor
55	C₂×D₅	C₂×C₄×C₇	1	2	yes	—
56	C₄×C₅	C₂×C₄×C₇	1	1	yes	characteristic
57	C₄×C₅	C₂³×C₇	1	1	yes	characteristic
58	Dic₅	C₂×C₄×C₇	1	2	yes	—
59	C₂²×C₄	—	5	1	no	—
60	C₂×C₈	—	5	1	no	—
61	C₂×C₈	—	5	4	no	—
62	C₂×C₈	C₇×D₅	1	1	yes	characteristic, direct product factor
63	C₂×C₇	—	5	4	no	—
64	C₂×C₇	C₈×D₅	1	2	yes	direct product factor
65	C₂×C₇	C₂×C₄×D₅	1	1	yes	characteristic
66	C₂×C₅	C₂×C₇×C₈	1	2	yes	complemented
67	C₂×C₅	C₂²×C₄×C₇	1	1	yes	characteristic
68	D₅	C₂×C₇×C₈	1	4	yes	direct product factor
69	C₂³	—	5	1	no	—
70	C₂×C₄	—	5	1	no	—
71	C₂×C₄	—	5	2	no	—
72	C₂×C₄	—	5	2	no	—
73	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
74	C₈	—	5	2	no	—
75	C₈	C₂×C₇×D₅	1	2	yes	direct product factor
76	C₇	C₂×C₈×D₅	1	1	yes	characteristic, direct product factor, Sylow
77	C₅	C₂²×C₇×C₈	1	1	yes	characteristic, complemented, Sylow
78	C₂²	—	5	2	no	—
79	C₂²	—	5	4	no	—
80	C₂²	C₄×C₇×D₅	1	1	yes	characteristic
81	C₄	—	5	2	no	—
82	C₄	C₄×C₇×D₅	1	1	yes	characteristic
83	C₄	C₂²×C₇×D₅	1	1	yes	characteristic
84	C₂	—	5	4	no	—
85	C₂	C₇×C₈×D₅	1	2	yes	direct product factor
86	C₂	C₂×C₄×C₇×D₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₇×C₈	D₅
commutator, hypocenter	C₅	C₂²×C₇×C₈
Frattini	C₄	C₂²×C₇×D₅
Fitting	C₂×C₅×C₇×C₈	C₂
socle	C₂²×C₅×C₇	C₂×C₄
prime radical	C₂²×C₇×D₅	C₄
cosocle	C₄×C₅	C₂³×C₇
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₇×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. C₂×C₇×#168	C₂	semidirect
4. C₇×C₈×D₅	C₂	direct (N × H)
10. C₅×C₇×C₈	C₂²	semidirect
11. C₇×#168	C₂²	semidirect
13. C₂×C₈×D₅	C₇	direct (N × H)
15. C₂×C₇×D₅	C₈	direct (N × H)
23. C₂×C₇×C₈	D₅	direct (N × H)
25. C₂×C₅×C₈	C₂×C₇	semidirect
26. C₂×#168	C₂×C₇	semidirect
27. C₈×D₅	C₂×C₇	direct (N × H)
28. C₂×C₅×C₇	C₂×C₈	semidirect
30. C₇×D₅	C₂×C₈	direct (N × H)
37. C₇×C₈	C₂×D₅	direct (N × H)
43. C₅×C₈	C₂²×C₇	semidirect
44. #168	C₂²×C₇	semidirect
45. C₅×C₇	C₂²×C₈	semidirect
54. C₂×D₅	C₇×C₈	direct (N × H)
62. C₂×C₈	C₇×D₅	direct (N × H)
64. C₂×C₇	C₈×D₅	direct (N × H)
66. C₂×C₅	C₂×C₇×C₈	semidirect
68. D₅	C₂×C₇×C₈	direct (N × H)
75. C₈	C₂×C₇×D₅	direct (N × H)
76. C₇	C₂×C₈×D₅	direct (N × H)
77. C₅	C₂²×C₇×C₈	semidirect
85. C₂	C₇×C₈×D₅	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×C₄×C₇×D₅	2
C₂×C₅×C₇×C₈	2
C₂×C₇×#168	2
C₇×C₈×D₅	2
C₂²×C₇×C₈	5
C₂×C₈×D₅	7

Subgroups — C2×C7×C8×D5 #11765143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×C₈×D₅ #11765143