Subgroups — C₇×#4072 #11765170

Order 1120.

152 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	C₇×C₂²⋊Dic₅	C₂	1	2	yes	maximal
3	C₇×C₄⋊Dic₅	C₂	1	2	yes	maximal, complemented
4	C₄×C₇×Dic₅	C₂	1	2	yes	maximal, complemented
5	C₂³×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₅×C₇	C₄	1	4	yes	complemented
7	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₇×Dic₅	C₂²	1	4	yes	—
10	C₇×#117	—	5	1	no	maximal, Hall
11	#4072	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂²×C₅×C₇	—	2	2	no	—
13	C₂²×C₅×C₇	C₂×C₄	1	2	yes	—
14	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
15	C₄×C₅×C₇	C₂×C₄	1	2	yes	—
16	C₄×C₅×C₇	C₂×C₄	1	2	yes	—
17	C₇×Dic₅	—	2	4	no	—
18	C₂²×C₄×C₇	D₅	1	1	yes	characteristic
19	C₇×C₄⋊C₄	—	5	2	no	—
20	C₇×C₂²⋊C₄	—	5	2	no	—
21	C₄²×C₇	—	5	2	no	—
22	C₂²×C₄×C₅	C₂×C₇	1	1	yes	characteristic
23	C₂²⋊Dic₅	C₂×C₇	1	2	yes	—
24	C₄⋊Dic₅	C₂×C₇	1	2	yes	complemented
25	C₄×Dic₅	C₂×C₇	1	2	yes	complemented
26	C₂×C₅×C₇	—	2	2	no	—
27	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₅×C₇	#41	1	2	yes	—
29	C₂³×C₇	C₂×D₅	1	1	yes	characteristic
30	C₂×C₄×C₇	—	5	4	no	—
31	C₂×C₄×C₇	Dic₅	1	4	yes	complemented
32	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
33	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
34	C₂³×C₅	C₂²×C₇	1	1	yes	characteristic
35	C₂×C₄×C₅	C₄×C₇	1	4	yes	complemented
36	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
37	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
38	C₂×Dic₅	C₂²×C₇	1	4	yes	—
39	C₅×C₇	#117	1	1	yes	characteristic, complemented, Hall
40	#117	—	5	1	no	Sylow
41	C₂²×C₇	—	2	2	no	—
42	C₂²×C₇	C₂×Dic₅	1	2	yes	—
43	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
44	C₄×C₇	—	10	4	no	—
45	C₄×C₇	C₂×Dic₅	1	2	yes	—
46	C₄×C₇	C₂×Dic₅	1	2	yes	—
47	C₂²×C₅	—	2	2	no	—
48	C₂²×C₅	C₂×C₄×C₇	1	2	yes	—
49	C₂²×C₅	C₂³×C₇	1	1	yes	characteristic
50	C₄×C₅	C₂×C₄×C₇	1	2	yes	—
51	C₄×C₅	C₂×C₄×C₇	1	2	yes	—
52	Dic₅	—	2	4	no	—
53	C₂²×C₄	C₇×D₅	1	1	yes	characteristic
54	C₄⋊C₄	—	5	2	no	—
55	C₂²⋊C₄	—	5	2	no	—
56	C₄²	—	5	2	no	—
57	C₂×C₇	—	2	2	no	—
58	C₂×C₇	#709	1	2	yes	—
59	C₂×C₇	C₂²×Dic₅	1	1	yes	characteristic
60	C₂×C₅	—	2	2	no	—
61	C₂×C₅	C₂²×C₄×C₇	1	1	yes	characteristic
62	C₂×C₅	C₇×#41	1	2	yes	—
63	C₂³	C₂×C₇×D₅	1	1	yes	characteristic
64	C₂×C₄	—	5	4	no	—
65	C₂×C₄	C₇×Dic₅	1	4	yes	complemented
66	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
67	C₂×C₄	C₂×C₇×D₅	1	1	yes	characteristic
68	C₇	#4072	1	1	yes	characteristic, direct product factor, Sylow
69	C₅	C₇×#117	1	1	yes	characteristic, complemented, Sylow
70	C₂²	—	2	2	no	—
71	C₂²	C₂×C₇×Dic₅	1	2	yes	—
72	C₂²	C₂²×C₇×D₅	1	1	yes	characteristic
73	C₄	—	10	4	no	—
74	C₄	C₂×C₇×Dic₅	1	2	yes	—
75	C₄	C₂×C₇×Dic₅	1	2	yes	—
76	C₂	—	2	2	no	—
77	C₂	C₇×#709	1	2	yes	—
78	C₂	C₂²×C₇×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₅
commutator	C₂×C₅	C₂²×C₄×C₇
Frattini	C₂²	C₂²×C₇×D₅
Fitting	C₂²×C₄×C₅×C₇	C₂
socle	C₂²×C₅×C₇	C₂³
prime radical	C₂³×C₅×C₇	C₂²
cosocle	C₂²×C₅	C₂³×C₇
hypercenter	C₂²×C₄×C₇	D₅
hypocenter	C₅	C₇×#117
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇

Complements

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

N	H ≅ G/N	type
3. C₇×C₄⋊Dic₅	C₂	semidirect
4. C₄×C₇×Dic₅	C₂	semidirect
6. C₂×C₄×C₅×C₇	C₄	semidirect
11. #4072	C₇	direct (N × H)
24. C₄⋊Dic₅	C₂×C₇	semidirect
25. C₄×Dic₅	C₂×C₇	semidirect
31. C₂×C₄×C₇	Dic₅	semidirect
35. C₂×C₄×C₅	C₄×C₇	semidirect
39. C₅×C₇	#117	semidirect
65. C₂×C₄	C₇×Dic₅	semidirect
68. C₇	#4072	direct (N × H)
69. C₅	C₇×#117	semidirect

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
C₇×C₂²⋊Dic₅	2
C₇×C₄⋊Dic₅	2
C₄×C₇×Dic₅	2
C₇×#117	5
#4072	7

Subgroups — C7×#4072 #11765170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#4072 #11765170