Group #13132932 — subgroups

Subgroups — D₇×C₅⋊S₄ #13132932

Order 1680.

132 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×A₄×D₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅⋊(C₇⋊S₄)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₅⋊S₄	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×C₂²⋊D₅	—	3	1	no	maximal, Hall
5	C₅×C₇×A₄	C₂²	1	1	yes	characteristic, complemented
6	D₇×D₁₅	—	4	1	no	maximal
7	D₇×S₄	—	5	1	no	maximal, Hall
8	C₂²×C₅×D₇	D₃	1	1	yes	characteristic, complemented
9	C₂×D₅×D₇	—	3	1	no	—
10	C₂²⋊D₃₅	—	3	1	no	—
11	C₇×C₂²⋊D₅	—	3	1	no	—
12	#63306	—	3	1	no	—
13	#63305	—	3	1	no	—
14	D₇×Dic₅	—	3	1	no	—
15	C₂×C₅⋊S₄	—	7	1	no	maximal, Hall
16	D₁₀₅	—	4	1	no	—
17	C₇×D₁₅	—	4	1	no	—
18	C₃×C₅×D₇	—	4	1	no	—
19	A₄×D₇	D₅	1	1	yes	characteristic, complemented
20	C₇⋊S₄	—	5	1	no	—
21	C₇×S₄	—	5	1	no	—
22	C₂²×C₅×C₇	C₂×D₃	1	1	yes	characteristic, complemented
23	C₂×D₃₅	—	3	1	no	—
24	C₂×C₇×D₅	—	3	1	no	—
25	C₂×C₅×D₇	—	3	1	no	—
26	C₂×C₅×D₇	—	3	1	no	—
27	D₅×D₇	—	6	1	no	—
28	D₅×D₇	—	6	1	no	—
29	Dic₃₅	—	3	1	no	—
30	C₇×Dic₅	—	3	1	no	—
31	C₂×C₅×A₄	—	7	1	no	—
32	C₅⋊S₄	—	7	1	no	—
33	C₅⋊S₄	D₇	1	1	yes	characteristic, direct product factor
34	D₄×D₇	—	15	1	no	Hall
35	C₃×C₅×C₇	—	4	1	no	Hall
36	C₇×A₄	C₂×D₅	1	1	yes	characteristic, complemented
37	D₃×D₇	—	20	1	no	—
38	C₂×C₂²⋊D₅	—	21	1	no	Hall
39	C₂×C₅×C₇	—	3	1	no	—
40	D₃₅	—	6	1	no	—
41	C₅×D₇	—	3	1	no	—
42	C₅×D₇	S₄	1	1	yes	characteristic, complemented
43	C₇×D₅	—	6	1	no	—
44	C₂×D₁₅	—	28	1	no	—
45	C₅×A₄	C₂×D₇	1	1	yes	characteristic, complemented
46	C₂²×D₇	—	15	1	no	—
47	C₂²×D₇	D₁₅	1	1	yes	characteristic, complemented
48	C₇×D₄	—	15	1	no	—
49	C₂²⋊D₇	—	15	1	no	—
50	C₂²⋊D₇	—	15	1	no	—
51	D₂₈	—	15	1	no	—
52	C₄×D₇	—	15	1	no	—
53	C₂×S₄	—	35	1	no	Hall
54	D₂₁	—	20	1	no	—
55	C₃×D₇	—	4	1	no	—
56	D₃×C₇	—	20	1	no	—
57	C₂³×C₅	—	7	1	no	—
58	C₂²×D₅	—	21	1	no	—
59	C₂²⋊D₅	—	3	1	no	—
60	C₂²⋊D₅	—	21	1	no	—
61	C₂²⋊D₅	—	21	1	no	—
62	C₂²⋊D₅	—	21	1	no	—
63	C₂×Dic₅	—	21	1	no	—
64	C₅×C₇	C₂×S₄	1	1	yes	characteristic, complemented, Hall
65	C₂×C₃×C₅	—	28	1	no	—
66	D₁₅	—	4	1	no	—
67	D₁₅	—	28	1	no	—
68	C₂²×C₇	—	15	1	no	—
69	C₂²×C₇	C₂×D₁₅	1	1	yes	characteristic, complemented
70	C₂×D₇	—	3	1	no	—
71	C₂×D₇	—	3	1	no	—
72	C₂×D₇	—	15	1	no	—
73	C₂×D₇	—	30	1	no	—
74	C₂×D₇	—	30	1	no	—
75	C₄×C₇	—	15	1	no	—
76	Dic₇	—	15	1	no	—
77	C₂×A₄	—	7	1	no	—
78	S₄	—	5	1	no	—
79	S₄	—	35	1	no	—
80	C₃×C₇	—	4	1	no	Hall
81	C₂²×C₅	—	21	1	no	—
82	C₂²×C₅	—	21	1	no	—
83	C₂²×C₅	D₃×D₇	1	1	yes	characteristic, complemented
84	C₂×D₅	—	3	1	no	—
85	C₂×D₅	—	21	1	no	—
86	C₂×D₅	—	42	1	no	—
87	C₂×D₅	—	42	1	no	—
88	Dic₅	—	3	1	no	—
89	Dic₅	—	21	1	no	—
90	C₂×D₄	—	105	1	no	Sylow
91	C₃×C₅	—	4	1	no	Hall
92	C₂×C₇	—	3	1	no	—
93	C₂×C₇	—	30	1	no	—
94	D₇	—	3	1	no	—
95	D₇	—	30	1	no	—
96	D₇	C₅⋊S₄	1	1	yes	characteristic, direct product factor
97	C₂×D₃	—	140	1	no	—
98	A₄	D₅×D₇	1	1	yes	characteristic, complemented
99	C₂×C₅	—	3	1	no	—
100	C₂×C₅	—	7	1	no	—
101	C₂×C₅	—	21	1	no	—
102	D₅	—	6	1	no	—
103	D₅	—	42	1	no	—
104	C₂³	—	7	1	no	—
105	C₂³	—	105	1	no	—
106	D₄	—	15	1	no	—
107	D₄	—	105	1	no	—
108	D₄	—	105	1	no	—
109	D₄	—	105	1	no	—
110	C₂×C₄	—	105	1	no	—
111	C₇	C₂×C₅⋊S₄	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₃	—	28	1	no	—
113	D₃	—	20	1	no	—
114	D₃	—	140	1	no	—
115	C₅	D₇×S₄	1	1	yes	characteristic, complemented, Sylow
116	C₂²	—	15	1	no	—
117	C₂²	—	21	1	no	—
118	C₂²	—	21	1	no	—
119	C₂²	—	105	1	no	—
120	C₂²	—	210	1	no	—
121	C₂²	—	210	1	no	—
122	C₂²	D₇×D₁₅	1	1	yes	characteristic, complemented
123	C₄	—	15	1	no	—
124	C₄	—	105	1	no	—
125	C₃	—	4	1	no	Sylow
126	C₂	—	3	1	no	—
127	C₂	—	7	1	no	—
128	C₂	—	21	1	no	—
129	C₂	—	30	1	no	—
130	C₂	—	210	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×A₄×D₇

C₂

yes

maximal, characteristic, complemented

C₅⋊(C₇⋊S₄)

C₂

yes

maximal, characteristic, complemented

C₇×C₅⋊S₄

C₂

yes

maximal, characteristic, complemented

D₇×C₂²⋊D₅

—

maximal, Hall

C₅×C₇×A₄

C₂²

yes

characteristic, complemented

D₇×D₁₅

—

maximal

D₇×S₄

—

maximal, Hall

C₂²×C₅×D₇

D₃

yes

characteristic, complemented

C₂×D₅×D₇

—

C₂²⋊D₃₅

—

C₇×C₂²⋊D₅

—

#63306

—

#63305

—

D₇×Dic₅

—

C₂×C₅⋊S₄

—

maximal, Hall

D₁₀₅

—

C₇×D₁₅

—

C₃×C₅×D₇

—

A₄×D₇

D₅

yes

characteristic, complemented

C₇⋊S₄

—

C₇×S₄

—

C₂²×C₅×C₇

C₂×D₃

yes

characteristic, complemented

C₂×D₃₅

—

C₂×C₇×D₅

—

C₂×C₅×D₇

—

C₂×C₅×D₇

—

D₅×D₇

—

D₅×D₇

—

Dic₃₅

—

C₇×Dic₅

—

C₂×C₅×A₄

—

C₅⋊S₄

—

C₅⋊S₄

D₇

yes

characteristic, direct product factor

D₄×D₇

—

Hall

C₃×C₅×C₇

—

Hall

C₇×A₄

C₂×D₅

yes

characteristic, complemented

D₃×D₇

—

C₂×C₂²⋊D₅

—

Hall

C₂×C₅×C₇

—

D₃₅

—

C₅×D₇

—

C₅×D₇

S₄

yes

characteristic, complemented

C₇×D₅

—

C₂×D₁₅

—

C₅×A₄

C₂×D₇

yes

characteristic, complemented

C₂²×D₇

—

C₂²×D₇

D₁₅

yes

characteristic, complemented

C₇×D₄

—

C₂²⋊D₇

—

C₂²⋊D₇

—

D₂₈

—

C₄×D₇

—

C₂×S₄

—

Hall

D₂₁

—

C₃×D₇

—

D₃×C₇

—

C₂³×C₅

—

C₂²×D₅

—

C₂²⋊D₅

—

C₂²⋊D₅

—

C₂²⋊D₅

—

C₂²⋊D₅

—

C₂×Dic₅

—

C₅×C₇

C₂×S₄

yes

characteristic, complemented, Hall

C₂×C₃×C₅

—

D₁₅

—

D₁₅

—

C₂²×C₇

—

C₂²×C₇

C₂×D₁₅

yes

characteristic, complemented

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

Dic₇

—

C₂×A₄

—

S₄

—

S₄

—

C₃×C₇

—

Hall

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

D₃×D₇

yes

characteristic, complemented

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

Dic₅

—

Dic₅

—

C₂×D₄

—

105

Sylow

C₃×C₅

—

Hall

C₂×C₇

—

C₂×C₇

—

D₇

—

D₇

—

D₇

C₅⋊S₄

yes

characteristic, direct product factor

C₂×D₃

—

140

—

A₄

D₅×D₇

yes

characteristic, complemented

C₂×C₅

—

100

C₂×C₅

—

101

C₂×C₅

—

102

D₅

—

103

D₅

—

104

C₂³

—

105

C₂³

—

105

—

106

D₄

—

107

D₄

—

105

—

108

D₄

—

105

—

109

D₄

—

105

—

110

C₂×C₄

—

105

—

111

C₇

C₂×C₅⋊S₄

yes

characteristic, complemented, Sylow

112

C₂×C₃

—

113

D₃

—

114

D₃

—

140

—

115

C₅

D₇×S₄

yes

characteristic, complemented, Sylow

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

105

—

120

C₂²

—

210

—

121

C₂²

—

210

—

122

C₂²

D₇×D₁₅

yes

characteristic, complemented

123

C₄

—

124

C₄

—

105

—

125

C₃

—

Sylow

126

C₂

—

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

—

210

—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₅×C₇×A₄	C₂²
Fitting, socle	C₂²×C₅×C₇	C₂×D₃
LCS	G ▷ C₅×C₇×A₄
UCS	C₁

subgroup

G/H

center, Frattini, hypercenter

C₁

commutator, cosocle, hypocenter

C₅×C₇×A₄

C₂²

Fitting, socle

C₂²×C₅×C₇

C₂×D₃

LCS

G ▷ C₅×C₇×A₄

UCS

C₁

N	H ≅ G/N	type
1. C₅×A₄×D₇	C₂	semidirect
2. C₅⋊(C₇⋊S₄)	C₂	semidirect
3. C₇×C₅⋊S₄	C₂	semidirect
5. C₅×C₇×A₄	C₂²	semidirect
8. C₂²×C₅×D₇	D₃	semidirect
19. A₄×D₇	D₅	semidirect
22. C₂²×C₅×C₇	C₂×D₃	semidirect
33. C₅⋊S₄	D₇	direct (N × H)
36. C₇×A₄	C₂×D₅	semidirect
42. C₅×D₇	S₄	semidirect
45. C₅×A₄	C₂×D₇	semidirect
47. C₂²×D₇	D₁₅	semidirect
64. C₅×C₇	C₂×S₄	semidirect
69. C₂²×C₇	C₂×D₁₅	semidirect
83. C₂²×C₅	D₃×D₇	semidirect
96. D₇	C₅⋊S₄	direct (N × H)
98. A₄	D₅×D₇	semidirect
111. C₇	C₂×C₅⋊S₄	semidirect
115. C₅	D₇×S₄	semidirect
122. C₂²	D₇×D₁₅	semidirect

H ≅ G/N

type

1. C₅×A₄×D₇

C₂

semidirect

2. C₅⋊(C₇⋊S₄)

C₂

semidirect

3. C₇×C₅⋊S₄

C₂

semidirect

5. C₅×C₇×A₄

C₂²

semidirect

8. C₂²×C₅×D₇

D₃

semidirect

19. A₄×D₇

D₅

semidirect

22. C₂²×C₅×C₇

C₂×D₃

semidirect

33. C₅⋊S₄

D₇

direct (N × H)

36. C₇×A₄

C₂×D₅

semidirect

42. C₅×D₇

S₄

semidirect

45. C₅×A₄

C₂×D₇

semidirect

47. C₂²×D₇

D₁₅

semidirect

64. C₅×C₇

C₂×S₄

semidirect

69. C₂²×C₇

C₂×D₁₅

semidirect

83. C₂²×C₅

D₃×D₇

semidirect

96. D₇

C₅⋊S₄

direct (N × H)

98. A₄

D₅×D₇

semidirect

111. C₇

C₂×C₅⋊S₄

semidirect

115. C₅

D₇×S₄

semidirect

122. C₂²

D₇×D₁₅

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×A₄×D₇	2
C₅⋊(C₇⋊S₄)	2
C₇×C₅⋊S₄	2
D₇×C₂²⋊D₅	3
D₇×D₁₅	4
D₇×S₄	5
C₂×C₅⋊S₄	7

subgroup

index

C₅×A₄×D₇

C₅⋊(C₇⋊S₄)

C₇×C₅⋊S₄

D₇×C₂²⋊D₅

D₇×D₁₅

D₇×S₄

C₂×C₅⋊S₄

Subgroups — D₇×C₅⋊S₄ #13132932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D7×C5⋊S4 #13132932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×C₅⋊S₄ #13132932