Group #13187516 — subgroups

Subgroups — C₇×#10300 #13187516

Order 1792.

592 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 106 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	C₇×#3253	C₂	1	3	yes	maximal, complemented
2	C₇×#1962	C₂	1	3	yes	maximal, complemented
3	C₇×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#532	C₂²	1	3	yes	—
5	C₇×#476	—	2	6	no	—
6	C₇×#469	—	2	6	no	—
7	C₇×#459	—	2	6	no	—
8	C₇×#423	C₂²	1	3	yes	—
9	C₂×C₇×#104	—	2	6	no	—
10	C₄³×C₇	C₂²	1	1	yes	characteristic
11	#10300	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂×C₇×#41	—	2	3	no	—
13	C₂×C₇×D₈	—	2	6	no	—
14	C₂×C₇×M₁₆	D₄	1	6	yes	—
15	C₇×C₄⋊Q₈	D₄	1	3	yes	—
16	C₇×#127	D₄	1	3	yes	—
17	C₇×#124	—	2	3	no	—
18	C₇×#121	—	2	6	no	—
19	C₄×C₇×D₄	—	2	6	no	—
20	C₂×C₄²×C₇	—	2	3	no	—
21	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
22	C₇×#106	—	2	6	no	—
23	C₇×C₄⋊C₈	—	2	6	no	—
24	C₇×#104	—	4	12	no	—
25	C₇×#102	—	2	12	no	—
26	#3253	C₂×C₇	1	3	yes	complemented
27	#1962	C₂×C₇	1	3	yes	complemented
28	#1309	C₂×C₇	1	1	yes	characteristic, complemented
29	C₇×#41	—	4	6	no	—
30	C₇×#41	—	4	6	no	—
31	C₂×C₇×Q₈	—	2	3	no	—
32	C₂×C₇×D₄	—	2	6	no	—
33	C₂×C₇×D₄	—	2	3	no	—
34	C₂×C₇×D₄	—	4	3	no	—
35	C₂²×C₄×C₇	—	4	1	no	—
36	C₂²×C₄×C₇	—	4	3	no	—
37	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
38	C₇×D₈	—	8	6	no	—
39	C₇×M₁₆	—	4	6	no	—
40	C₂×C₇×C₈	—	2	6	no	—
41	C₇×C₄⋊C₄	—	2	6	no	—
42	C₇×C₂²⋊C₄	—	4	6	no	—
43	C₄²×C₇	—	2	6	no	—
44	C₄²×C₇	—	4	3	no	—
45	C₄²×C₇	C₂×D₄	1	3	yes	—
46	C₄²×C₇	#41	1	1	yes	characteristic
47	#532	C₂²×C₇	1	3	yes	—
48	#476	—	2	6	no	—
49	#469	—	2	6	no	—
50	#459	—	2	6	no	—
51	#423	C₂²×C₇	1	3	yes	—
52	C₂×#104	—	2	6	no	—
53	C₄³	C₂²×C₇	1	1	yes	characteristic
54	C₂³×C₇	—	4	3	no	—
55	C₂³×C₇	#127	1	1	yes	characteristic
56	C₇×Q₈	—	4	6	no	—
57	C₇×D₄	—	4	6	no	—
58	C₇×D₄	—	4	6	no	—
59	C₇×D₄	—	4	6	no	—
60	C₇×D₄	—	8	6	no	—
61	C₂×C₄×C₇	—	2	3	no	—
62	C₂×C₄×C₇	—	2	3	no	—
63	C₂×C₄×C₇	—	4	6	no	—
64	C₂×C₄×C₇	—	4	3	no	—
65	C₂×C₄×C₇	—	8	3	no	—
66	C₂×C₄×C₇	—	8	3	no	—
67	C₂×C₄×C₇	#120	1	3	yes	—
68	C₂×C₄×C₇	#121	1	3	yes	—
69	C₇×C₈	—	4	6	no	—
70	C₂×#41	—	2	3	no	—
71	C₂×D₈	—	2	6	no	—
72	C₂×M₁₆	C₇×D₄	1	6	yes	—
73	C₄⋊Q₈	C₇×D₄	1	3	yes	—
74	#127	C₇×D₄	1	3	yes	—
75	#124	—	2	3	no	—
76	#121	—	2	6	no	—
77	C₄×D₄	—	2	6	no	—
78	C₂×C₄²	—	2	3	no	—
79	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
80	#106	—	2	6	no	—
81	C₄⋊C₈	—	2	6	no	—
82	#104	—	4	12	no	—
83	#102	—	2	12	no	—
84	C₂²×C₇	—	2	3	no	—
85	C₂²×C₇	—	8	3	no	—
86	C₂²×C₇	—	8	6	no	—
87	C₂²×C₇	#392	1	1	yes	characteristic
88	C₄×C₇	—	2	3	no	—
89	C₄×C₇	—	4	4	no	—
90	C₄×C₇	—	8	3	no	—
91	C₄×C₇	#453	1	3	yes	—
92	C₄×C₇	#454	1	3	yes	—
93	#41	—	4	6	no	—
94	#41	—	4	6	no	—
95	C₂×Q₈	—	2	3	no	—
96	C₂×D₄	—	2	6	no	—
97	C₂×D₄	—	2	3	no	—
98	C₂×D₄	—	4	3	no	—
99	C₂²×C₄	—	4	1	no	—
100	C₂²×C₄	—	4	3	no	—
101	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
102	D₈	—	8	6	no	—
103	M₁₆	—	4	6	no	—
104	C₂×C₈	—	2	6	no	—
105	C₄⋊C₄	—	2	6	no	—
106	C₂²⋊C₄	—	4	6	no	—
107	C₄²	—	2	6	no	—
108	C₄²	—	4	3	no	—
109	C₄²	C₂×C₇×D₄	1	3	yes	—
110	C₄²	C₇×#41	1	1	yes	characteristic
111	C₂×C₇	—	4	1	no	—
112	C₂×C₇	—	16	3	no	—
113	C₂×C₇	#2006	1	3	yes	—
114	C₂³	—	4	3	no	—
115	C₂³	C₇×#127	1	1	yes	characteristic
116	Q₈	—	4	6	no	—
117	D₄	—	4	6	no	—
118	D₄	—	4	6	no	—
119	D₄	—	4	6	no	—
120	D₄	—	8	6	no	—
121	C₂×C₄	—	2	3	no	—
122	C₂×C₄	—	2	3	no	—
123	C₂×C₄	—	4	6	no	—
124	C₂×C₄	—	4	3	no	—
125	C₂×C₄	—	8	3	no	—
126	C₂×C₄	—	8	3	no	—
127	C₂×C₄	C₇×#120	1	3	yes	—
128	C₂×C₄	C₇×#121	1	3	yes	—
129	C₈	—	4	6	no	—
130	C₇	#10300	1	1	yes	characteristic, direct product factor, Sylow
131	C₂²	—	2	3	no	—
132	C₂²	—	8	3	no	—
133	C₂²	—	8	6	no	—
134	C₂²	C₇×#392	1	1	yes	characteristic
135	C₄	—	2	3	no	—
136	C₄	—	4	4	no	—
137	C₄	—	8	3	no	—
138	C₄	C₇×#453	1	3	yes	—
139	C₄	C₇×#454	1	3	yes	—
140	C₂	—	4	1	no	—
141	C₂	—	16	3	no	—
142	C₂	C₇×#2006	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#3253

C₂

yes

maximal, complemented

C₇×#1962

C₂

yes

maximal, complemented

C₇×#1309

C₂

yes

maximal, characteristic, complemented

C₇×#532

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×C₇×#104

—

C₄³×C₇

C₂²

yes

characteristic

#10300

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇×#41

—

C₂×C₇×D₈

—

C₂×C₇×M₁₆

D₄

yes

—

C₇×C₄⋊Q₈

D₄

yes

—

C₇×#127

D₄

yes

—

C₇×#124

—

C₇×#121

—

C₄×C₇×D₄

—

C₂×C₄²×C₇

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₇×#106

—

C₇×C₄⋊C₈

—

C₇×#104

—

C₇×#102

—

#3253

C₂×C₇

yes

complemented

#1962

C₂×C₇

yes

complemented

#1309

C₂×C₇

yes

characteristic, complemented

C₇×#41

—

C₇×#41

—

C₂×C₇×Q₈

—

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

—

C₇×D₈

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₇×C₄⋊C₄

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

#41

yes

characteristic

#532

C₂²×C₇

yes

—

—

—

—

C₂²×C₇

yes

—

C₂×#104

—

C₄³

C₂²×C₇

yes

characteristic

C₂³×C₇

—

C₂³×C₇

#127

yes

characteristic

C₇×Q₈

—

C₇×D₄

—

C₇×D₄

—

C₇×D₄

—

C₇×D₄

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#120

yes

—

C₂×C₄×C₇

#121

yes

—

C₇×C₈

—

C₂×#41

—

C₂×D₈

—

C₂×M₁₆

C₇×D₄

yes

—

C₄⋊Q₈

C₇×D₄

yes

—

#127

C₇×D₄

yes

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

C₂³×C₇

yes

characteristic

#106

—

C₄⋊C₈

—

#104

—

#102

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

#392

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#453

yes

—

C₄×C₇

#454

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂×C₇×D₄

yes

—

102

D₈

—

103

M₁₆

—

104

C₂×C₈

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

C₂×C₇×D₄

yes

—

110

C₄²

C₇×#41

yes

characteristic

111

C₂×C₇

—

112

C₂×C₇

—

113

C₂×C₇

#2006

yes

—

114

C₂³

—

115

C₂³

C₇×#127

yes

characteristic

116

Q₈

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₇×#120

yes

—

128

C₂×C₄

C₇×#121

yes

—

129

C₈

—

130

C₇

#10300

yes

characteristic, direct product factor, Sylow

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₇×#392

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

C₇×#453

yes

—

139

C₄

C₇×#454

yes

—

140

C₂

—

141

C₂

—

142

C₂

C₇×#2006

yes

—

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₇

#392

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₇

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

N	H ≅ G/N	type
1. C₇×#3253	C₂	semidirect
2. C₇×#1962	C₂	semidirect
3. C₇×#1309	C₂	semidirect
11. #10300	C₇	direct (N × H)
26. #3253	C₂×C₇	semidirect
27. #1962	C₂×C₇	semidirect
28. #1309	C₂×C₇	semidirect
130. C₇	#10300	direct (N × H)

H ≅ G/N

type

1. C₇×#3253

C₂

semidirect

2. C₇×#1962

C₂

semidirect

3. C₇×#1309

C₂

semidirect

11. #10300

C₇

direct (N × H)

26. #3253

C₂×C₇

semidirect

27. #1962

C₂×C₇

semidirect

28. #1309

C₂×C₇

semidirect

130. C₇

#10300

direct (N × H)

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

subgroup	index
C₇×#3253	2
C₇×#1962	2
C₇×#1309	2
#10300	7

subgroup

index

Subgroups — C₇×#10300 #13187516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#10300 #13187516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10300 #13187516