Subgroups — C₅×Q₈×Dic₇ #11765100

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	C₅×C₄⋊Dic₇	C₂	1	3	yes	maximal
3	C₄×C₅×Dic₇	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×C₅×Dic₇	C₂²	1	3	yes	—
7	C₂×C₅×Dic₇	C₂²	1	1	yes	characteristic
8	Q₈×Dic₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₄×C₅×Q₈	—	7	1	no	maximal, Hall
10	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
11	C₄×C₅×C₇	C₂×C₄	1	6	yes	—
12	C₅×Dic₇	—	2	3	no	—
13	C₅×Dic₇	Q₈	1	2	yes	direct product factor
14	C₂×C₇×Q₈	C₂×C₅	1	1	yes	characteristic
15	C₄⋊Dic₇	C₂×C₅	1	3	yes	—
16	C₄×Dic₇	C₂×C₅	1	3	yes	—
17	C₂×C₅×Q₈	D₇	1	1	yes	characteristic
18	C₅×C₄⋊C₄	—	7	3	no	—
19	C₄²×C₅	—	7	3	no	—
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	C₄×C₅	1	4	yes	complemented
24	C₂×C₄×C₇	C₂²×C₅	1	3	yes	—
25	C₂×Dic₇	C₂²×C₅	1	3	yes	—
26	C₂×Dic₇	C₂²×C₅	1	1	yes	characteristic
27	C₅×Q₈	Dic₇	1	4	yes	direct product factor
28	C₂×C₄×C₅	—	7	1	no	—
29	C₂×C₄×C₅	—	7	3	no	—
30	C₂×C₄×C₅	C₂×D₇	1	3	yes	—
31	C₅×C₇	C₄×Q₈	1	1	yes	characteristic, complemented, Hall
32	C₄×Q₈	—	7	1	no	Sylow
33	C₂²×C₇	C₂³×C₅	1	1	yes	characteristic
34	C₄×C₇	C₂×C₄×C₅	1	6	yes	—
35	Dic₇	—	2	3	no	—
36	Dic₇	C₅×Q₈	1	2	yes	direct product factor
37	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
38	C₄×C₅	—	7	2	no	—
39	C₄×C₅	—	14	3	no	—
40	C₄×C₅	C₂×Dic₇	1	6	yes	—
41	C₂×Q₈	C₅×D₇	1	1	yes	characteristic
42	C₄⋊C₄	—	7	3	no	—
43	C₄²	—	7	3	no	—
44	C₂×C₇	C₂²×C₄×C₅	1	1	yes	characteristic
45	C₂×C₇	C₂×C₅×Q₈	1	1	yes	characteristic
46	C₂×C₇	C₅×#41	1	1	yes	characteristic
47	C₂×C₅	Q₈×D₇	1	1	yes	characteristic
48	C₂×C₅	#1161	1	1	yes	characteristic
49	C₂×C₅	C₂²×Dic₇	1	1	yes	characteristic
50	Q₈	C₅×Dic₇	1	4	yes	direct product factor
51	C₂×C₄	—	7	1	no	—
52	C₂×C₄	—	7	3	no	—
53	C₂×C₄	C₂×C₅×D₇	1	3	yes	—
54	C₇	C₄×C₅×Q₈	1	1	yes	characteristic, complemented, Sylow
55	C₅	Q₈×Dic₇	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²	C₂²×C₅×D₇	1	1	yes	characteristic
57	C₄	—	7	2	no	—
58	C₄	—	14	3	no	—
59	C₄	C₂×C₅×Dic₇	1	6	yes	—
60	C₂	C₅×Q₈×D₇	1	1	yes	characteristic
61	C₂	C₅×#1161	1	1	yes	characteristic
62	C₂	C₂²×C₅×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂²×D₇
commutator	C₂×C₇	C₂²×C₄×C₅
Frattini	C₂²	C₂²×C₅×D₇
Fitting	C₂×C₅×C₇×Q₈	C₂
socle, prime radical	C₂²×C₅×C₇	C₂³
cosocle	C₂²×C₇	C₂³×C₅
hypercenter	C₂×C₅×Q₈	D₇
hypocenter	C₇	C₄×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
4. C₅×C₇×Q₈	C₄	semidirect
8. Q₈×Dic₇	C₅	direct (N × H)
13. C₅×Dic₇	Q₈	direct (N × H)
23. C₇×Q₈	C₄×C₅	semidirect
27. C₅×Q₈	Dic₇	direct (N × H)
31. C₅×C₇	C₄×Q₈	semidirect
36. Dic₇	C₅×Q₈	direct (N × H)
50. Q₈	C₅×Dic₇	direct (N × H)
54. C₇	C₄×C₅×Q₈	semidirect
55. C₅	Q₈×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
C₂×C₅×C₇×Q₈	2
C₅×C₄⋊Dic₇	2
C₄×C₅×Dic₇	2
Q₈×Dic₇	5
C₄×C₅×Q₈	7

Subgroups — C5×Q8×Dic7 #11765100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×Q₈×Dic₇ #11765100