Subgroups — C₇×Q₈×Dic₅ #11765186

Order 1120.

140 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
2	C₇×C₄⋊Dic₅	C₂	1	3	yes	maximal
3	C₄×C₇×Dic₅	C₂	1	3	yes	maximal
4	C₅×C₇×Q₈	C₄	1	4	yes	complemented
5	C₂×C₄×C₅×C₇	C₂²	1	3	yes	—
6	C₂×C₇×Dic₅	C₂²	1	3	yes	—
7	C₂×C₇×Dic₅	C₂²	1	1	yes	characteristic
8	C₄×C₇×Q₈	—	5	1	no	maximal, Hall
9	Q₈×Dic₅	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
11	C₄×C₅×C₇	C₂×C₄	1	6	yes	—
12	C₇×Dic₅	—	2	3	no	—
13	C₇×Dic₅	Q₈	1	2	yes	direct product factor
14	C₂×C₇×Q₈	D₅	1	1	yes	characteristic
15	C₇×C₄⋊C₄	—	5	3	no	—
16	C₄²×C₇	—	5	3	no	—
17	C₂×C₅×Q₈	C₂×C₇	1	1	yes	characteristic
18	C₄⋊Dic₅	C₂×C₇	1	3	yes	—
19	C₄×Dic₅	C₂×C₇	1	3	yes	—
20	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₅×C₇	C₂×Q₈	1	1	yes	characteristic
22	C₂×C₅×C₇	#41	1	1	yes	characteristic
23	C₇×Q₈	Dic₅	1	4	yes	direct product factor
24	C₂×C₄×C₇	—	5	1	no	—
25	C₂×C₄×C₇	—	5	3	no	—
26	C₂×C₄×C₇	C₂×D₅	1	3	yes	—
27	C₅×Q₈	C₄×C₇	1	4	yes	complemented
28	C₂×C₄×C₅	C₂²×C₇	1	3	yes	—
29	C₂×Dic₅	C₂²×C₇	1	3	yes	—
30	C₂×Dic₅	C₂²×C₇	1	1	yes	characteristic
31	C₅×C₇	C₄×Q₈	1	1	yes	characteristic, complemented, Hall
32	C₄×Q₈	—	5	1	no	Sylow
33	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
34	C₄×C₇	—	5	2	no	—
35	C₄×C₇	—	10	3	no	—
36	C₄×C₇	C₂×Dic₅	1	6	yes	—
37	C₂²×C₅	C₂³×C₇	1	1	yes	characteristic
38	C₄×C₅	C₂×C₄×C₇	1	6	yes	—
39	Dic₅	—	2	3	no	—
40	Dic₅	C₇×Q₈	1	2	yes	direct product factor
41	C₂×Q₈	C₇×D₅	1	1	yes	characteristic
42	C₄⋊C₄	—	5	3	no	—
43	C₄²	—	5	3	no	—
44	C₂×C₇	Q₈×D₅	1	1	yes	characteristic
45	C₂×C₇	#713	1	1	yes	characteristic
46	C₂×C₇	C₂²×Dic₅	1	1	yes	characteristic
47	C₂×C₅	C₂²×C₄×C₇	1	1	yes	characteristic
48	C₂×C₅	C₂×C₇×Q₈	1	1	yes	characteristic
49	C₂×C₅	C₇×#41	1	1	yes	characteristic
50	Q₈	C₇×Dic₅	1	4	yes	direct product factor
51	C₂×C₄	—	5	1	no	—
52	C₂×C₄	—	5	3	no	—
53	C₂×C₄	C₂×C₇×D₅	1	3	yes	—
54	C₇	Q₈×Dic₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₅	C₄×C₇×Q₈	1	1	yes	characteristic, complemented, Sylow
56	C₂²	C₂²×C₇×D₅	1	1	yes	characteristic
57	C₄	—	5	2	no	—
58	C₄	—	10	3	no	—
59	C₄	C₂×C₇×Dic₅	1	6	yes	—
60	C₂	C₇×Q₈×D₅	1	1	yes	characteristic
61	C₂	C₇×#713	1	1	yes	characteristic
62	C₂	C₂²×C₇×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	C₂²×D₅
commutator	C₂×C₅	C₂²×C₄×C₇
Frattini	C₂²	C₂²×C₇×D₅
Fitting	C₂×C₅×C₇×Q₈	C₂
socle, prime radical	C₂²×C₅×C₇	C₂³
cosocle	C₂²×C₅	C₂³×C₇
hypercenter	C₂×C₇×Q₈	D₅
hypocenter	C₅	C₄×C₇×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₇×Q₈

Complements

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

N	H ≅ G/N	type
4. C₅×C₇×Q₈	C₄	semidirect
9. Q₈×Dic₅	C₇	direct (N × H)
13. C₇×Dic₅	Q₈	direct (N × H)
23. C₇×Q₈	Dic₅	direct (N × H)
27. C₅×Q₈	C₄×C₇	semidirect
31. C₅×C₇	C₄×Q₈	semidirect
40. Dic₅	C₇×Q₈	direct (N × H)
50. Q₈	C₇×Dic₅	direct (N × H)
54. C₇	Q₈×Dic₅	direct (N × H)
55. C₅	C₄×C₇×Q₈	semidirect

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
C₇×C₄⋊Dic₅	2
C₄×C₇×Dic₅	2
C₄×C₇×Q₈	5
Q₈×Dic₅	7

Subgroups — C7×Q8×Dic5 #11765186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×Q₈×Dic₅ #11765186