Subgroups — C₂×C₅×Q₈×D₇ #11765428

Order 1120.

312 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, complemented
2	C₅×Q₈×D₇	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄×C₅×D₇	C₂	1	3	yes	maximal
4	C₂×C₅×Dic₁₄	C₂	1	3	yes	maximal, complemented
5	C₂²×C₅×D₇	C₂²	1	1	yes	characteristic
6	C₅×C₇×Q₈	C₂²	1	4	yes	complemented
7	C₂×C₄×C₅×C₇	C₂²	1	3	yes	—
8	C₂×C₅×Dic₇	C₂²	1	3	yes	—
9	C₄×C₅×D₇	C₂²	1	12	yes	—
10	C₅×Dic₁₄	C₂²	1	12	yes	complemented
11	C₂×Q₈×D₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂²×C₅×Q₈	—	7	1	no	maximal, Hall
13	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
14	C₂×C₅×D₇	Q₈	1	4	yes	direct product factor
15	C₂×C₅×D₇	C₂³	1	2	yes	—
16	C₄×C₅×C₇	C₂³	1	6	yes	—
17	C₅×Dic₇	C₂³	1	6	yes	—
18	C₂×C₇×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
19	Q₈×D₇	C₂×C₅	1	8	yes	direct product factor
20	C₂×C₄×D₇	C₂×C₅	1	3	yes	—
21	C₂×Dic₁₄	C₂×C₅	1	3	yes	complemented
22	C₂×C₅×Q₈	—	7	8	no	—
23	C₂×C₅×Q₈	—	7	3	no	—
24	C₂×C₅×Q₈	D₇	1	1	yes	characteristic, direct product factor
25	C₂²×C₄×C₅	—	7	3	no	—
26	C₂×C₅×C₇	C₂×Q₈	1	2	yes	complemented
27	C₂×C₅×C₇	C₂⁴	1	1	yes	characteristic
28	C₅×D₇	C₂×Q₈	1	4	yes	direct product factor
29	C₂²×D₇	C₂²×C₅	1	1	yes	characteristic
30	C₇×Q₈	C₂²×C₅	1	4	yes	complemented
31	C₂×C₄×C₇	C₂²×C₅	1	3	yes	—
32	C₂×Dic₇	C₂²×C₅	1	3	yes	—
33	C₄×D₇	C₂²×C₅	1	12	yes	—
34	Dic₁₄	C₂²×C₅	1	12	yes	complemented
35	C₂³×C₅	—	7	1	no	—
36	C₅×Q₈	—	7	12	no	—
37	C₅×Q₈	C₂×D₇	1	4	yes	direct product factor
38	C₂×C₄×C₅	—	7	3	no	—
39	C₂×C₄×C₅	—	7	12	no	—
40	C₂×C₄×C₅	C₂×D₇	1	3	yes	—
41	C₅×C₇	C₂²×Q₈	1	1	yes	characteristic, complemented, Hall
42	C₂²×Q₈	—	7	1	no	Sylow
43	C₂²×C₇	C₂³×C₅	1	1	yes	characteristic
44	C₂×D₇	C₅×Q₈	1	4	yes	direct product factor
45	C₂×D₇	C₂³×C₅	1	2	yes	—
46	C₄×C₇	C₂³×C₅	1	6	yes	—
47	Dic₇	C₂³×C₅	1	6	yes	—
48	C₂²×C₅	—	7	4	no	—
49	C₂²×C₅	—	7	2	no	—
50	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
51	C₄×C₅	—	7	6	no	—
52	C₄×C₅	C₂²×D₇	1	6	yes	—
53	C₂×Q₈	—	7	8	no	—
54	C₂×Q₈	—	7	3	no	—
55	C₂×Q₈	C₅×D₇	1	1	yes	characteristic, direct product factor
56	C₂²×C₄	—	7	3	no	—
57	C₂×C₇	C₂×C₅×Q₈	1	2	yes	complemented
58	C₂×C₇	C₂⁴×C₅	1	1	yes	characteristic
59	D₇	C₂×C₅×Q₈	1	4	yes	direct product factor
60	C₂×C₅	—	7	4	no	—
61	C₂×C₅	Q₈×D₇	1	2	yes	direct product factor
62	C₂×C₅	C₂³×D₇	1	1	yes	characteristic
63	C₂³	—	7	1	no	—
64	Q₈	—	7	12	no	—
65	Q₈	C₂×C₅×D₇	1	4	yes	direct product factor
66	C₂×C₄	—	7	3	no	—
67	C₂×C₄	—	7	12	no	—
68	C₂×C₄	C₂×C₅×D₇	1	3	yes	—
69	C₇	C₂²×C₅×Q₈	1	1	yes	characteristic, complemented, Sylow
70	C₅	C₂×Q₈×D₇	1	1	yes	characteristic, direct product factor, Sylow
71	C₂²	—	7	4	no	—
72	C₂²	—	7	2	no	—
73	C₂²	C₂²×C₅×D₇	1	1	yes	characteristic
74	C₄	—	7	6	no	—
75	C₄	C₂²×C₅×D₇	1	6	yes	—
76	C₂	—	7	4	no	—
77	C₂	C₅×Q₈×D₇	1	2	yes	direct product factor
78	C₂	C₂³×C₅×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂²×D₇
commutator, cosocle	C₂×C₇	C₂⁴×C₅
Frattini	C₂	C₂³×C₅×D₇
Fitting	C₂×C₅×C₇×Q₈	C₂
socle	C₂²×C₅×C₇	C₂³
prime radical	C₂²×C₅×D₇	C₂²
hypercenter	C₂×C₅×Q₈	D₇
hypocenter	C₇	C₂²×C₅×Q₈
LCS	G ▷ C₂×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
1. C₂×C₅×C₇×Q₈	C₂	semidirect
2. C₅×Q₈×D₇	C₂	direct (N × H)
4. C₂×C₅×Dic₁₄	C₂	semidirect
6. C₅×C₇×Q₈	C₂²	semidirect
10. C₅×Dic₁₄	C₂²	semidirect
11. C₂×Q₈×D₇	C₅	direct (N × H)
14. C₂×C₅×D₇	Q₈	direct (N × H)
18. C₂×C₇×Q₈	C₂×C₅	semidirect
19. Q₈×D₇	C₂×C₅	direct (N × H)
21. C₂×Dic₁₄	C₂×C₅	semidirect
24. C₂×C₅×Q₈	D₇	direct (N × H)
26. C₂×C₅×C₇	C₂×Q₈	semidirect
28. C₅×D₇	C₂×Q₈	direct (N × H)
30. C₇×Q₈	C₂²×C₅	semidirect
34. Dic₁₄	C₂²×C₅	semidirect
37. C₅×Q₈	C₂×D₇	direct (N × H)
41. C₅×C₇	C₂²×Q₈	semidirect
44. C₂×D₇	C₅×Q₈	direct (N × H)
55. C₂×Q₈	C₅×D₇	direct (N × H)
57. C₂×C₇	C₂×C₅×Q₈	semidirect
59. D₇	C₂×C₅×Q₈	direct (N × H)
61. C₂×C₅	Q₈×D₇	direct (N × H)
65. Q₈	C₂×C₅×D₇	direct (N × H)
69. C₇	C₂²×C₅×Q₈	semidirect
70. C₅	C₂×Q₈×D₇	direct (N × H)
77. C₂	C₅×Q₈×D₇	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
C₅×Q₈×D₇	2
C₂×C₄×C₅×D₇	2
C₂×C₅×Dic₁₄	2
C₂×Q₈×D₇	5
C₂²×C₅×Q₈	7

Subgroups — C2×C5×Q8×D7 #11765428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×Q₈×D₇ #11765428