Subgroups — C₂×C₅×C₄⋊Dic₇ #11765083

Order 1120.

184 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
3. C₅×C₄⋊Dic₇	C₂	direct (N × H)
5. C₂×C₄×C₅×C₇	C₄	semidirect
9. C₂×C₄⋊Dic₇	C₅	direct (N × H)
15. C₄×C₅×C₇	C₂×C₄	semidirect
19. C₄⋊Dic₇	C₂×C₅	direct (N × H)
23. C₂×C₅×C₇	C₄⋊C₄	semidirect
28. C₂×C₄×C₇	C₄×C₅	semidirect
35. C₂×C₄×C₅	Dic₇	semidirect
37. C₅×C₇	C₂×C₄⋊C₄	semidirect
43. C₄×C₇	C₂×C₄×C₅	semidirect
50. C₄×C₅	C₂×Dic₇	semidirect
54. C₂×C₇	C₅×C₄⋊C₄	semidirect
58. C₂×C₅	C₄⋊Dic₇	direct (N × H)
65. C₂×C₄	C₅×Dic₇	semidirect
67. C₇	C₂×C₅×C₄⋊C₄	semidirect
68. C₅	C₂×C₄⋊Dic₇	direct (N × H)
74. C₄	C₂×C₅×Dic₇	semidirect
75. C₂	C₅×C₄⋊Dic₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×C₇	2
C₂²×C₅×Dic₇	2
C₅×C₄⋊Dic₇	2
C₂×C₄⋊Dic₇	5
C₂×C₅×C₄⋊C₄	7

Subgroups — C2×C5×C4⋊Dic7 #11765083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×C₄⋊Dic₇ #11765083