Subgroups — C₂×C₄×C₅×Dic₇ #11765080

Order 1120.

216 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	C₄×C₅×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	C₄	1	8	yes	direct product factor
8	C₂×C₅×Dic₇	C₂²	1	4	yes	—
9	C₂×C₄×Dic₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₄²×C₅	—	7	1	no	maximal, Hall
11	C₂²×C₅×C₇	C₂×C₄	1	4	yes	—
12	C₂²×C₅×C₇	C₂×C₄	1	2	yes	—
13	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
14	C₄×C₅×C₇	C₂×C₄	1	4	yes	complemented
15	C₅×Dic₇	C₂×C₄	1	8	yes	direct product factor
16	C₂²×C₄×C₇	C₂×C₅	1	1	yes	characteristic
17	C₂²×Dic₇	C₂×C₅	1	2	yes	—
18	C₄×Dic₇	C₂×C₅	1	4	yes	direct product factor
19	C₂²×C₄×C₅	—	7	2	no	—
20	C₂²×C₄×C₅	D₇	1	1	yes	characteristic
21	C₄²×C₅	—	7	4	no	—
22	C₂×C₅×C₇	C₄²	1	4	yes	complemented
23	C₂×C₅×C₇	C₂²×C₄	1	2	yes	—
24	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
25	C₂³×C₇	C₂²×C₅	1	1	yes	characteristic
26	C₂×C₄×C₇	C₄×C₅	1	4	yes	complemented
27	C₂×C₄×C₇	C₂²×C₅	1	2	yes	—
28	C₂×Dic₇	C₄×C₅	1	8	yes	direct product factor
29	C₂×Dic₇	C₂²×C₅	1	4	yes	—
30	C₂³×C₅	C₂×D₇	1	1	yes	characteristic
31	C₂×C₄×C₅	—	7	8	no	—
32	C₂×C₄×C₅	—	7	4	no	—
33	C₂×C₄×C₅	Dic₇	1	4	yes	direct product factor
34	C₂×C₄×C₅	C₂×D₇	1	2	yes	—
35	C₅×C₇	C₂×C₄²	1	1	yes	characteristic, complemented, Hall
36	C₂×C₄²	—	7	1	no	Sylow
37	C₂²×C₇	C₂×C₄×C₅	1	4	yes	—
38	C₂²×C₇	C₂×C₄×C₅	1	2	yes	—
39	C₂²×C₇	C₂³×C₅	1	1	yes	characteristic
40	C₄×C₇	C₂×C₄×C₅	1	4	yes	complemented
41	Dic₇	C₂×C₄×C₅	1	8	yes	direct product factor
42	C₂²×C₅	C₄×D₇	1	4	yes	—
43	C₂²×C₅	C₂×Dic₇	1	2	yes	—
44	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
45	C₄×C₅	—	7	8	no	—
46	C₄×C₅	C₂×Dic₇	1	4	yes	direct product factor
47	C₂²×C₄	—	7	2	no	—
48	C₂²×C₄	C₅×D₇	1	1	yes	characteristic
49	C₄²	—	7	4	no	—
50	C₂×C₇	C₄²×C₅	1	4	yes	complemented
51	C₂×C₇	C₂²×C₄×C₅	1	2	yes	—
52	C₂×C₇	C₂²×C₄×C₅	1	1	yes	characteristic
53	C₂×C₅	C₄×Dic₇	1	4	yes	direct product factor
54	C₂×C₅	C₂×C₄×D₇	1	2	yes	—
55	C₂×C₅	C₂²×Dic₇	1	1	yes	characteristic
56	C₂³	C₂×C₅×D₇	1	1	yes	characteristic
57	C₂×C₄	—	7	8	no	—
58	C₂×C₄	—	7	4	no	—
59	C₂×C₄	C₅×Dic₇	1	4	yes	direct product factor
60	C₂×C₄	C₂×C₅×D₇	1	2	yes	—
61	C₇	C₂×C₄²×C₅	1	1	yes	characteristic, complemented, Sylow
62	C₅	C₂×C₄×Dic₇	1	1	yes	characteristic, direct product factor, Sylow
63	C₂²	C₄×C₅×D₇	1	4	yes	—
64	C₂²	C₂×C₅×Dic₇	1	2	yes	—
65	C₂²	C₂²×C₅×D₇	1	1	yes	characteristic
66	C₄	—	7	8	no	—
67	C₄	C₂×C₅×Dic₇	1	4	yes	direct product factor
68	C₂	C₄×C₅×Dic₇	1	4	yes	direct product factor
69	C₂	C₂×C₄×C₅×D₇	1	2	yes	—
70	C₂	C₂²×C₅×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄×C₅	D₇
commutator, hypocenter	C₇	C₂×C₄²×C₅
Frattini	C₂²	C₂²×C₅×D₇
Fitting	C₂²×C₄×C₅×C₇	C₂
socle, prime radical	C₂³×C₅×C₇	C₂²
cosocle	C₂²×C₇	C₂³×C₅
LCS	G ▷ C₇
UCS	C₁ ◁ C₂²×C₄×C₅

Complements

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

N	H ≅ G/N	type
3. C₄×C₅×Dic₇	C₂	direct (N × H)
5. C₂×C₄×C₅×C₇	C₄	semidirect
7. C₂×C₅×Dic₇	C₄	direct (N × H)
9. C₂×C₄×Dic₇	C₅	direct (N × H)
14. C₄×C₅×C₇	C₂×C₄	semidirect
15. C₅×Dic₇	C₂×C₄	direct (N × H)
18. C₄×Dic₇	C₂×C₅	direct (N × H)
22. C₂×C₅×C₇	C₄²	semidirect
26. C₂×C₄×C₇	C₄×C₅	semidirect
28. C₂×Dic₇	C₄×C₅	direct (N × H)
33. C₂×C₄×C₅	Dic₇	direct (N × H)
35. C₅×C₇	C₂×C₄²	semidirect
40. C₄×C₇	C₂×C₄×C₅	semidirect
41. Dic₇	C₂×C₄×C₅	direct (N × H)
46. C₄×C₅	C₂×Dic₇	direct (N × H)
50. C₂×C₇	C₄²×C₅	semidirect
53. C₂×C₅	C₄×Dic₇	direct (N × H)
59. C₂×C₄	C₅×Dic₇	direct (N × H)
61. C₇	C₂×C₄²×C₅	semidirect
62. C₅	C₂×C₄×Dic₇	direct (N × H)
67. C₄	C₂×C₅×Dic₇	direct (N × H)
68. C₂	C₄×C₅×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
C₄×C₅×Dic₇	2
C₂×C₄×Dic₇	5
C₂×C₄²×C₅	7

Subgroups — C2×C4×C5×Dic7 #11765080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₅×Dic₇ #11765080