Subgroups — C₅×#6440 #11765035

Order 1120.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×Dic₇	C₂	1	1	yes	maximal, characteristic
2	C₅×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₅×C₂²⋊Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₅×Dic₇⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₄×C₅×Dic₇	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×C₅×Dic₇	C₄	1	4	yes	complemented
9	C₂×C₅×Dic₇	C₂²	1	2	yes	—
10	C₂×C₅×Dic₇	C₂²	1	1	yes	characteristic
11	C₂×C₅×Dic₇	C₂²	1	1	yes	characteristic
12	#6440	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₅×#117	—	7	1	no	maximal, 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	C₅×Dic₇	—	2	2	no	—
19	C₅×Dic₇	C₂×C₄	1	2	yes	—
20	C₅×Dic₇	C₂×C₄	1	2	yes	—
21	C₂²×Dic₇	C₂×C₅	1	1	yes	characteristic
22	C₇×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic
23	C₂²⋊Dic₇	C₂×C₅	1	1	yes	characteristic
24	Dic₇⋊C₄	C₂×C₅	1	2	yes	complemented
25	C₄×Dic₇	C₂×C₅	1	2	yes	complemented
26	C₂²×C₄×C₅	—	7	1	no	—
27	C₅×C₄⋊C₄	—	7	2	no	—
28	C₅×C₂²⋊C₄	—	7	1	no	—
29	C₅×C₂²⋊C₄	D₇	1	1	yes	characteristic
30	C₄²×C₅	—	7	2	no	—
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	C₂²×C₅	1	1	yes	characteristic
35	C₂×C₄×C₇	C₂²×C₅	1	2	yes	—
36	C₂×Dic₇	C₄×C₅	1	4	yes	complemented
37	C₂×Dic₇	C₂²×C₅	1	2	yes	—
38	C₂×Dic₇	C₂²×C₅	1	1	yes	characteristic
39	C₂×Dic₇	C₂²×C₅	1	1	yes	characteristic
40	C₂³×C₅	C₂×D₇	1	1	yes	characteristic
41	C₂×C₄×C₅	—	7	4	no	—
42	C₂×C₄×C₅	—	7	1	no	—
43	C₂×C₄×C₅	—	7	1	no	—
44	C₂×C₄×C₅	—	7	2	no	—
45	C₂×C₄×C₅	C₂×D₇	1	2	yes	—
46	C₅×C₇	#117	1	1	yes	characteristic, complemented, Hall
47	#117	—	7	1	no	Sylow
48	C₂²×C₇	—	2	2	no	—
49	C₂²×C₇	C₂×C₄×C₅	1	2	yes	—
50	C₂²×C₇	C₂³×C₅	1	1	yes	characteristic
51	C₄×C₇	—	2	2	no	—
52	Dic₇	—	2	2	no	—
53	Dic₇	C₂×C₄×C₅	1	2	yes	—
54	Dic₇	C₂×C₄×C₅	1	2	yes	—
55	C₂²×C₅	—	2	2	no	—
56	C₂²×C₅	C₄×D₇	1	2	yes	—
57	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
58	C₄×C₅	—	2	2	no	—
59	C₄×C₅	—	7	2	no	—
60	C₄×C₅	—	7	2	no	—
61	C₄×C₅	—	14	2	no	—
62	C₂²×C₄	—	7	1	no	—
63	C₄⋊C₄	—	7	2	no	—
64	C₂²⋊C₄	—	7	1	no	—
65	C₂²⋊C₄	C₅×D₇	1	1	yes	characteristic
66	C₄²	—	7	2	no	—
67	C₂×C₇	—	2	2	no	—
68	C₂×C₇	C₂²×C₄×C₅	1	1	yes	characteristic
69	C₂×C₇	C₅×#41	1	2	yes	—
70	C₂×C₅	—	2	2	no	—
71	C₂×C₅	C₂×C₄×D₇	1	1	yes	characteristic
72	C₂×C₅	#1159	1	2	yes	—
73	C₂³	C₂×C₅×D₇	1	1	yes	characteristic
74	C₂×C₄	—	7	4	no	—
75	C₂×C₄	—	7	1	no	—
76	C₂×C₄	—	7	1	no	—
77	C₂×C₄	—	7	2	no	—
78	C₂×C₄	C₂×C₅×D₇	1	2	yes	—
79	C₇	C₅×#117	1	1	yes	characteristic, complemented, Sylow
80	C₅	#6440	1	1	yes	characteristic, direct product factor, Sylow
81	C₂²	—	2	2	no	—
82	C₂²	C₄×C₅×D₇	1	2	yes	—
83	C₂²	C₂²×C₅×D₇	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	—	7	2	no	—
86	C₄	—	7	2	no	—
87	C₄	—	14	2	no	—
88	C₂	—	2	2	no	—
89	C₂	C₂×C₄×C₅×D₇	1	1	yes	characteristic
90	C₂	C₅×#1159	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂²×D₇
commutator	C₂×C₇	C₂²×C₄×C₅
Frattini	C₂²	C₂²×C₅×D₇
Fitting	C₅×C₇×C₂²⋊C₄	C₂
socle	C₂²×C₅×C₇	C₂³
prime radical	C₂³×C₅×C₇	C₂²
cosocle	C₂²×C₇	C₂³×C₅
hypercenter	C₅×C₂²⋊C₄	D₇
hypocenter	C₇	C₅×#117
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₅×Dic₇⋊C₄	C₂	semidirect
5. C₄×C₅×Dic₇	C₂	semidirect
8. C₂×C₅×Dic₇	C₄	semidirect
12. #6440	C₅	direct (N × H)
24. Dic₇⋊C₄	C₂×C₅	semidirect
25. C₄×Dic₇	C₂×C₅	semidirect
36. C₂×Dic₇	C₄×C₅	semidirect
46. C₅×C₇	#117	semidirect
79. C₇	C₅×#117	semidirect
80. C₅	#6440	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×C₅×Dic₇	2
C₅×C₇×C₂²⋊C₄	2
C₅×C₂²⋊Dic₇	2
C₅×Dic₇⋊C₄	2
C₄×C₅×Dic₇	2
#6440	5
C₅×#117	7

Subgroups — C5×#6440 #11765035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#6440 #11765035