Subgroups — C₇×#4023 #11765121

Order 1120.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₅×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₇×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₇×#683	C₂	1	2	yes	maximal, complemented
5	C₄×C₇×Dic₅	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×C₇	C₂²	1	2	yes	—
8	C₂×C₇×Dic₅	C₄	1	4	yes	complemented
9	C₂×C₇×Dic₅	C₂²	1	2	yes	—
10	C₂×C₇×Dic₅	C₂²	1	1	yes	characteristic
11	C₂×C₇×Dic₅	C₂²	1	1	yes	characteristic
12	C₇×#117	—	5	1	no	maximal, Hall
13	#4023	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₅×C₇	—	2	2	no	—
15	C₂²×C₅×C₇	C₂×C₄	1	2	yes	—
16	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
17	C₄×C₅×C₇	—	2	2	no	—
18	C₇×Dic₅	—	2	2	no	—
19	C₇×Dic₅	C₂×C₄	1	2	yes	—
20	C₇×Dic₅	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₇	—	5	1	no	—
22	C₇×C₄⋊C₄	—	5	2	no	—
23	C₇×C₂²⋊C₄	—	5	1	no	—
24	C₇×C₂²⋊C₄	D₅	1	1	yes	characteristic
25	C₄²×C₇	—	5	2	no	—
26	C₂²×Dic₅	C₂×C₇	1	1	yes	characteristic
27	C₅×C₂²⋊C₄	C₂×C₇	1	1	yes	characteristic
28	C₂²⋊Dic₅	C₂×C₇	1	1	yes	characteristic
29	#683	C₂×C₇	1	2	yes	complemented
30	C₄×Dic₅	C₂×C₇	1	2	yes	complemented
31	C₂×C₅×C₇	—	2	2	no	—
32	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₅×C₇	#41	1	2	yes	—
34	C₂³×C₇	C₂×D₅	1	1	yes	characteristic
35	C₂×C₄×C₇	—	5	4	no	—
36	C₂×C₄×C₇	—	5	1	no	—
37	C₂×C₄×C₇	—	5	1	no	—
38	C₂×C₄×C₇	—	5	2	no	—
39	C₂×C₄×C₇	C₂×D₅	1	2	yes	—
40	C₂³×C₅	C₂²×C₇	1	1	yes	characteristic
41	C₂×C₄×C₅	C₂²×C₇	1	2	yes	—
42	C₂×Dic₅	C₄×C₇	1	4	yes	complemented
43	C₂×Dic₅	C₂²×C₇	1	2	yes	—
44	C₂×Dic₅	C₂²×C₇	1	1	yes	characteristic
45	C₂×Dic₅	C₂²×C₇	1	1	yes	characteristic
46	C₅×C₇	#117	1	1	yes	characteristic, complemented, Hall
47	#117	—	5	1	no	Sylow
48	C₂²×C₇	—	2	2	no	—
49	C₂²×C₇	C₄×D₅	1	2	yes	—
50	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
51	C₄×C₇	—	2	2	no	—
52	C₄×C₇	—	5	2	no	—
53	C₄×C₇	—	5	2	no	—
54	C₄×C₇	—	10	2	no	—
55	C₂²×C₅	—	2	2	no	—
56	C₂²×C₅	C₂×C₄×C₇	1	2	yes	—
57	C₂²×C₅	C₂³×C₇	1	1	yes	characteristic
58	C₄×C₅	—	2	2	no	—
59	Dic₅	—	2	2	no	—
60	Dic₅	C₂×C₄×C₇	1	2	yes	—
61	Dic₅	C₂×C₄×C₇	1	2	yes	—
62	C₂²×C₄	—	5	1	no	—
63	C₄⋊C₄	—	5	2	no	—
64	C₂²⋊C₄	—	5	1	no	—
65	C₂²⋊C₄	C₇×D₅	1	1	yes	characteristic
66	C₄²	—	5	2	no	—
67	C₂×C₇	—	2	2	no	—
68	C₂×C₇	C₂×C₄×D₅	1	1	yes	characteristic
69	C₂×C₇	#711	1	2	yes	—
70	C₂×C₅	—	2	2	no	—
71	C₂×C₅	C₂²×C₄×C₇	1	1	yes	characteristic
72	C₂×C₅	C₇×#41	1	2	yes	—
73	C₂³	C₂×C₇×D₅	1	1	yes	characteristic
74	C₂×C₄	—	5	4	no	—
75	C₂×C₄	—	5	1	no	—
76	C₂×C₄	—	5	1	no	—
77	C₂×C₄	—	5	2	no	—
78	C₂×C₄	C₂×C₇×D₅	1	2	yes	—
79	C₇	#4023	1	1	yes	characteristic, direct product factor, Sylow
80	C₅	C₇×#117	1	1	yes	characteristic, complemented, Sylow
81	C₂²	—	2	2	no	—
82	C₂²	C₄×C₇×D₅	1	2	yes	—
83	C₂²	C₂²×C₇×D₅	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	—	5	2	no	—
86	C₄	—	5	2	no	—
87	C₄	—	10	2	no	—
88	C₂	—	2	2	no	—
89	C₂	C₂×C₄×C₇×D₅	1	1	yes	characteristic
90	C₂	C₇×#711	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₇×#683	C₂	semidirect
5. C₄×C₇×Dic₅	C₂	semidirect
8. C₂×C₇×Dic₅	C₄	semidirect
13. #4023	C₇	direct (N × H)
29. #683	C₂×C₇	semidirect
30. C₄×Dic₅	C₂×C₇	semidirect
42. C₂×Dic₅	C₄×C₇	semidirect
46. C₅×C₇	#117	semidirect
79. C₇	#4023	direct (N × H)
80. 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×C₇×Dic₅	2
C₅×C₇×C₂²⋊C₄	2
C₇×C₂²⋊Dic₅	2
C₇×#683	2
C₄×C₇×Dic₅	2
C₇×#117	5
#4023	7

Subgroups — C7×#4023 #11765121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#4023 #11765121