Group #13187884 — subgroups

Subgroups — C₇×#10668 #13187884

Order 1792.

316 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 114 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×#1995	C₂	1	1	yes	maximal, characteristic
2	C₇×#1843	C₂	1	1	yes	maximal, characteristic
3	C₇×#1571	C₂	1	2	yes	maximal
4	C₇×#1481	C₂	1	2	yes	maximal
5	C₇×#1274	C₂	1	1	yes	maximal, characteristic
6	C₇×#533	C₂²	1	1	yes	characteristic
7	C₇×#431	C₂²	1	2	yes	—
8	C₇×#423	C₂²	1	1	yes	characteristic
9	C₂×C₇×C₄⋊C₈	—	2	2	no	—
10	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
11	C₇×#376	C₂²	1	1	yes	characteristic
12	C₇×#339	—	2	2	no	—
13	C₇×#335	—	2	4	no	—
14	C₇×Q₈⋊C₈	—	2	4	no	—
15	#10668	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂×C₇×M₁₆	—	2	1	no	—
17	C₂²×C₇×C₈	—	2	2	no	—
18	C₇×C₄⋊Q₈	Q₈	1	2	yes	—
19	C₇×#125	—	2	1	no	—
20	C₇×C₂²⋊Q₈	—	2	2	no	—
21	C₄×C₇×Q₈	—	2	2	no	—
22	C₇×#117	—	2	1	no	—
23	C₂×C₄²×C₇	D₄	1	2	yes	—
24	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
25	C₇×C₄⋊C₈	—	2	2	no	—
26	C₇×C₄⋊C₈	—	2	2	no	—
27	C₇×C₄⋊C₈	—	2	2	no	—
28	C₇×C₄⋊C₈	—	2	2	no	—
29	C₇×C₄⋊C₈	—	4	4	no	—
30	C₇×C₂²⋊C₈	—	2	2	no	—
31	C₄×C₇×C₈	D₄	1	4	yes	—
32	C₇×#95	—	2	2	no	—
33	#1995	C₂×C₇	1	1	yes	characteristic
34	#1843	C₂×C₇	1	1	yes	characteristic
35	#1571	C₂×C₇	1	2	yes	—
36	#1481	C₂×C₇	1	2	yes	—
37	#1274	C₂×C₇	1	1	yes	characteristic
38	C₂×C₇×Q₈	—	2	2	no	—
39	C₂²×C₄×C₇	—	2	2	no	—
40	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₇	#41	1	2	yes	—
42	C₇×M₁₆	—	4	2	no	—
43	C₂×C₇×C₈	—	2	4	no	—
44	C₂×C₇×C₈	—	4	2	no	—
45	C₂×C₇×C₈	—	4	1	no	—
46	C₂×C₇×C₈	—	4	4	no	—
47	C₇×C₄⋊C₄	—	2	2	no	—
48	C₇×C₄⋊C₄	—	2	2	no	—
49	C₇×C₄⋊C₄	—	4	2	no	—
50	C₇×C₂²⋊C₄	—	4	1	no	—
51	C₄²×C₇	—	2	2	no	—
52	C₄²×C₇	—	4	1	no	—
53	C₄²×C₇	Q₁₆	1	4	yes	—
54	C₄²×C₇	C₂×D₄	1	2	yes	—
55	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₇	#41	1	1	yes	characteristic
57	#533	C₂²×C₇	1	1	yes	characteristic
58	#431	C₂²×C₇	1	2	yes	—
59	#423	C₂²×C₇	1	1	yes	characteristic
60	C₂×C₄⋊C₈	—	2	2	no	—
61	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
62	#376	C₂²×C₇	1	1	yes	characteristic
63	#339	—	2	2	no	—
64	#335	—	2	4	no	—
65	Q₈⋊C₈	—	2	4	no	—
66	C₂³×C₇	#123	1	1	yes	characteristic
67	C₇×Q₈	—	4	4	no	—
68	C₂×C₄×C₇	—	2	4	no	—
69	C₂×C₄×C₇	—	4	2	no	—
70	C₂×C₄×C₇	—	4	4	no	—
71	C₂×C₄×C₇	—	4	2	no	—
72	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₇	#123	1	2	yes	—
75	C₂×C₄×C₇	#127	1	1	yes	characteristic
76	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
78	C₇×C₈	—	4	4	no	—
79	C₇×C₈	—	8	2	no	—
80	C₇×C₈	—	8	1	no	—
81	C₂×M₁₆	—	2	1	no	—
82	C₂²×C₈	—	2	2	no	—
83	C₄⋊Q₈	C₇×Q₈	1	2	yes	—
84	#125	—	2	1	no	—
85	C₂²⋊Q₈	—	2	2	no	—
86	C₄×Q₈	—	2	2	no	—
87	#117	—	2	1	no	—
88	C₂×C₄²	C₇×D₄	1	2	yes	—
89	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
90	C₄⋊C₈	—	2	2	no	—
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	—	2	2	no	—
93	C₄⋊C₈	—	2	2	no	—
94	C₄⋊C₈	—	4	4	no	—
95	C₂²⋊C₈	—	2	2	no	—
96	C₄×C₈	C₇×D₄	1	4	yes	—
97	#95	—	2	2	no	—
98	C₂²×C₇	—	2	2	no	—
99	C₂²×C₇	#399	1	1	yes	characteristic
100	C₂²×C₇	#484	1	2	yes	—
101	C₄×C₇	—	2	2	no	—
102	C₄×C₇	—	2	2	no	—
103	C₄×C₇	—	4	4	no	—
104	C₄×C₇	—	8	2	no	—
105	C₄×C₇	#477	1	2	yes	—
106	C₄×C₇	#494	1	1	yes	characteristic
107	C₄×C₇	#496	1	1	yes	characteristic
108	C₂×Q₈	—	2	2	no	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
111	C₂²×C₄	C₇×#41	1	2	yes	—
112	M₁₆	—	4	2	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	4	2	no	—
120	C₂²⋊C₄	—	4	1	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	4	1	no	—
123	C₄²	C₇×Q₁₆	1	4	yes	—
124	C₄²	C₂×C₇×D₄	1	2	yes	—
125	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
126	C₄²	C₇×#41	1	1	yes	characteristic
127	C₂×C₇	—	2	2	no	—
128	C₂×C₇	#2095	1	1	yes	characteristic
129	C₂×C₇	#2098	1	2	yes	—
130	C₂³	C₇×#123	1	1	yes	characteristic
131	Q₈	—	4	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
137	C₂×C₄	C₇×C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄	C₇×#123	1	2	yes	—
139	C₂×C₄	C₇×#127	1	1	yes	characteristic
140	C₂×C₄	C₂×C₇×Q₁₆	1	2	yes	—
141	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
142	C₈	—	4	4	no	—
143	C₈	—	8	2	no	—
144	C₈	—	8	1	no	—
145	C₇	#10668	1	1	yes	characteristic, direct product factor, Sylow
146	C₂²	—	2	2	no	—
147	C₂²	C₇×#399	1	1	yes	characteristic
148	C₂²	C₇×#484	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	8	2	no	—
153	C₄	C₇×#477	1	2	yes	—
154	C₄	C₇×#494	1	1	yes	characteristic
155	C₄	C₇×#496	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	C₇×#2095	1	1	yes	characteristic
158	C₂	C₇×#2098	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1995

C₂

yes

maximal, characteristic

C₇×#1843

C₂

yes

maximal, characteristic

C₇×#1571

C₂

yes

maximal

C₇×#1481

C₂

yes

maximal

C₇×#1274

C₂

yes

maximal, characteristic

C₇×#533

C₂²

yes

characteristic

C₇×#431

C₂²

yes

—

C₇×#423

C₂²

yes

characteristic

C₂×C₇×C₄⋊C₈

—

C₂×C₇×C₄⋊C₈

C₂²

yes

—

C₇×#376

C₂²

yes

characteristic

C₇×#339

—

C₇×#335

—

C₇×Q₈⋊C₈

—

#10668

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇×M₁₆

—

C₂²×C₇×C₈

—

C₇×C₄⋊Q₈

Q₈

yes

—

C₇×#125

—

C₇×C₂²⋊Q₈

—

C₄×C₇×Q₈

—

C₇×#117

—

C₂×C₄²×C₇

D₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₇×C₄⋊C₈

—

C₇×C₄⋊C₈

—

C₇×C₄⋊C₈

—

C₇×C₄⋊C₈

—

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₄×C₇×C₈

D₄

yes

—

C₇×#95

—

#1995

C₂×C₇

yes

characteristic

#1843

C₂×C₇

yes

characteristic

#1571

C₂×C₇

yes

—

#1481

C₂×C₇

yes

—

#1274

C₂×C₇

yes

characteristic

C₂×C₇×Q₈

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

Q₁₆

yes

—

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

C₂×Q₈

yes

C₄²×C₇

yes

C₂²×C₇

yes

characteristic

#431

C₂²×C₇

yes

—

#423

C₂²×C₇

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²×C₇

yes

—

#376

C₂²×C₇

yes

characteristic

#339

—

#335

—

Q₈⋊C₈

—

C₂³×C₇

#123

yes

characteristic

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇

#123

yes

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×Q₁₆

yes

—

C₂×C₄×C₇

AGL(1,ℤ/8)

yes

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

C₇×Q₈

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₇×D₄

yes

—

C₂×C₄²

C₂³×C₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₇×D₄

yes

—

#95

—

C₂²×C₇

—

C₂²×C₇

#399

yes

characteristic

100

C₂²×C₇

#484

yes

—

101

C₄×C₇

—

102

C₄×C₇

—

103

C₄×C₇

—

104

C₄×C₇

—

105

C₄×C₇

#477

yes

—

106

C₄×C₇

#494

yes

characteristic

107

C₄×C₇

#496

yes

characteristic

108

C₂×Q₈

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂×C₇×D₄

yes

characteristic

111

C₂²×C₄

C₇×#41

yes

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₄²

C₇×Q₁₆

yes

—

124

C₄²

C₂×C₇×D₄

yes

—

125

C₄²

C₂×C₇×Q₈

yes

characteristic

126

C₄²

C₇×#41

yes

characteristic

127

C₂×C₇

—

128

C₂×C₇

#2095

yes

characteristic

129

C₂×C₇

#2098

yes

—

130

C₂³

C₇×#123

yes

characteristic

131

Q₈

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₇×C₂²⋊Q₈

yes

—

137

C₂×C₄

C₇×C₂²⋊Q₈

yes

characteristic

138

C₂×C₄

C₇×#123

yes

—

139

C₂×C₄

C₇×#127

yes

characteristic

140

C₂×C₄

C₂×C₇×Q₁₆

yes

—

141

C₂×C₄

C₇×AGL(1,ℤ/8)

yes

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₇

#10668

yes

characteristic, direct product factor, Sylow

146

C₂²

—

147

C₂²

C₇×#399

yes

characteristic

148

C₂²

C₇×#484

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

C₇×#477

yes

—

154

C₄

C₇×#494

yes

characteristic

155

C₄

C₇×#496

yes

characteristic

156

C₂

—

157

C₂

C₇×#2095

yes

characteristic

158

C₂

C₇×#2098

yes

—

	subgroup	G/H
center	C₂×C₄×C₇	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	#399
prime radical	C₂³×C₇	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₇

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₇

C₂²×C₇

C₂³×C₇

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

N	H ≅ G/N	type
15. #10668	C₇	direct (N × H)
145. C₇	#10668	direct (N × H)

H ≅ G/N

type

15. #10668

C₇

direct (N × H)

145. C₇

#10668

direct (N × H)

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

subgroup	index
C₇×#1995	2
C₇×#1843	2
C₇×#1571	2
C₇×#1481	2
C₇×#1274	2
#10668	7

subgroup

index

Subgroups — C₇×#10668 #13187884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#10668 #13187884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10668 #13187884