Group #13246826 — subgroups

Subgroups — #13246826

Order 1792.

336 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 127 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×#2960	C₂	1	1	yes	maximal, characteristic
2	#11729588	C₂	1	2	yes	maximal
3	#11727485	C₂	1	2	yes	maximal
4	C₇⋊#1309	C₂	1	2	yes	maximal
5	C₇×#533	C₂²	1	1	yes	characteristic
6	C₇×#446	—	2	2	no	—
7	C₇×C₈×Q₈	—	2	2	no	—
8	C₇×#424	C₄	1	2	yes	—
9	C₇×#423	C₂²	1	1	yes	characteristic
10	C₇×#405	C₄	1	2	yes	—
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	#89696	C₂²	1	4	yes	—
13	#89342	—	2	4	no	—
14	#89253	—	2	2	no	—
15	#89253	—	2	2	no	—
16	#8639	—	7	1	no	maximal, Sylow
17	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
18	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
20	C₇×C₄⋊Q₈	D₄	1	2	yes	—
21	C₇×#125	Q₈	1	2	yes	—
22	C₇×C₂²⋊Q₈	—	2	2	no	—
23	C₄×C₇×Q₈	—	2	2	no	—
24	C₇×#117	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
26	C₂×#1136	—	2	4	no	—
27	C₇×C₄⋊C₈	—	2	4	no	—
28	C₇×C₄⋊C₈	—	2	2	no	—
29	C₇×#97	—	2	2	no	—
30	C₄×C₇×C₈	—	2	2	no	—
31	C₄×C₇×C₈	D₄	1	2	yes	—
32	C₄×C₇×C₈	D₄	1	2	yes	—
33	C₄⋊(C₇⋊C₈)	—	2	4	no	—
34	C₄⋊(C₇⋊C₈)	—	2	4	no	—
35	#2960	D₇	1	1	yes	characteristic
36	#1567	—	7	2	no	—
37	#1561	—	7	2	no	—
38	#1309	—	7	2	no	—
39	C₂×C₇×Q₈	—	2	2	no	—
40	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇	#41	1	2	yes	—
42	C₇×M₁₆	—	4	2	no	—
43	C₂×C₇×C₈	—	2	2	no	—
44	C₂×C₇×C₈	—	2	2	no	—
45	C₂×C₇×C₈	—	4	1	no	—
46	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
48	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₇×C₄⋊C₄	—	2	2	no	—
50	C₇×C₄⋊C₄	—	2	4	no	—
51	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
52	C₇×C₂²⋊C₄	—	2	2	no	—
53	C₄²×C₇	—	2	2	no	—
54	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₇	C₂×D₄	1	2	yes	—
56	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
57	#1136	—	8	4	no	—
58	C₂×C₇⋊C₈	—	4	4	no	—
59	#533	C₂×D₇	1	1	yes	characteristic
60	#446	—	2	2	no	—
61	C₈×Q₈	—	2	2	no	—
62	#424	Dic₇	1	2	yes	—
63	#423	—	7	4	no	—
64	#423	C₂×D₇	1	1	yes	characteristic
65	#405	Dic₇	1	2	yes	—
66	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
67	#335	—	14	2	no	—
68	#335	—	14	2	no	—
69	#332	—	14	4	no	—
70	C₂³×C₇	#117	1	1	yes	characteristic
71	C₇×Q₈	—	4	2	no	—
72	C₇×Q₈	—	4	2	no	—
73	C₂×C₄×C₇	—	2	1	no	—
74	C₂×C₄×C₇	—	2	1	no	—
75	C₂×C₄×C₇	—	2	2	no	—
76	C₂×C₄×C₇	—	2	2	no	—
77	C₂×C₄×C₇	—	2	4	no	—
78	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₇	#121	1	2	yes	—
81	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
82	C₇×C₈	—	2	1	no	—
83	C₇×C₈	—	2	2	no	—
84	C₇×C₈	—	4	2	no	—
85	C₇×C₈	#101	1	2	yes	—
86	C₇⋊C₈	—	8	4	no	—
87	C₂×M₁₆	—	14	4	no	—
88	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
89	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
90	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
91	C₄⋊Q₈	D₂₈	1	2	yes	—
92	#125	Dic₁₄	1	2	yes	—
93	C₂²⋊Q₈	—	2	2	no	—
94	C₄×Q₈	—	2	2	no	—
95	#117	C₂×Dic₇	1	2	yes	—
96	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
97	C₄⋊C₈	—	2	4	no	—
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	—	14	4	no	—
100	C₄⋊C₈	—	14	4	no	—
101	#97	—	2	2	no	—
102	C₄×C₈	—	2	2	no	—
103	C₄×C₈	C₂²⋊D₇	1	2	yes	—
104	C₄×C₈	C₂²⋊D₇	1	2	yes	—
105	C₂²×C₇	—	2	1	no	—
106	C₂²×C₇	—	2	2	no	—
107	C₂²×C₇	#380	1	1	yes	characteristic
108	C₄×C₇	—	2	1	no	—
109	C₄×C₇	—	2	2	no	—
110	C₄×C₇	—	4	4	no	—
111	C₄×C₇	C₂×#101	1	1	yes	characteristic
112	C₄×C₇	#413	1	1	yes	characteristic
113	C₄×C₇	#472	1	2	yes	—
114	C₄×C₇	#473	1	2	yes	—
115	C₂×Q₈	—	2	2	no	—
116	C₂²×C₄	#1157	1	2	yes	—
117	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
118	M₁₆	—	4	2	no	—
119	M₁₆	—	56	4	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	28	4	no	—
124	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
126	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	C₄⋊Dic₇	1	4	yes	—
130	C₂²⋊C₄	—	2	2	no	—
131	C₄²	—	2	2	no	—
132	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
133	C₄²	C₂×D₂₈	1	1	yes	characteristic
134	C₄²	C₂×C₂²⋊D₇	1	2	yes	—
135	C₂×C₇	—	4	1	no	—
136	C₂×C₇	#1801	1	1	yes	characteristic
137	C₂×C₇	#1816	1	2	yes	—
138	C₂³	#6489	1	1	yes	characteristic
139	Q₈	—	4	2	no	—
140	Q₈	—	4	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	#6487	1	2	yes	—
147	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
148	C₂×C₄	#6493	1	2	yes	—
149	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	2	2	no	—
152	C₈	—	4	2	no	—
153	C₈	—	56	4	no	—
154	C₈	#6410	1	2	yes	—
155	C₇	#8639	1	1	yes	characteristic, complemented, Sylow
156	C₂²	—	2	2	no	—
157	C₂²	—	2	1	no	—
158	C₂²	#89984	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	1	no	—
161	C₄	—	4	4	no	—
162	C₄	#89914	1	2	yes	—
163	C₄	#89915	1	2	yes	—
164	C₄	C₂×#6410	1	1	yes	characteristic
165	C₄	#90009	1	1	yes	characteristic
166	C₂	—	4	1	no	—
167	C₂	#11729391	1	1	yes	characteristic
168	C₂	#11729770	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#2960

C₂

yes

maximal, characteristic

#11729588

C₂

yes

maximal

#11727485

C₂

yes

maximal

C₇⋊#1309

C₂

yes

maximal

C₇×#533

C₂²

yes

characteristic

C₇×#446

—

C₇×C₈×Q₈

—

C₇×#424

C₄

yes

—

C₇×#423

C₂²

yes

characteristic

C₇×#405

C₄

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

#89696

C₂²

yes

—

—

—

—

—

C₂×C₇×M₁₆

C₂×C₄

yes

—

C₂²×C₇×C₈

C₂×C₄

yes

characteristic

C₂²×C₇×C₈

C₂×C₄

yes

characteristic

C₇×C₄⋊Q₈

D₄

yes

—

C₇×#125

Q₈

yes

—

C₇×C₂²⋊Q₈

—

C₄×C₇×Q₈

—

C₇×#117

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×#1136

—

C₇×C₄⋊C₈

—

C₇×C₄⋊C₈

—

C₇×#97

—

C₄×C₇×C₈

—

C₄×C₇×C₈

D₄

yes

—

C₄×C₇×C₈

D₄

yes

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

#2960

D₇

yes

—

—

—

C₂×C₇×Q₈

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₇×C₈

C₂²⋊C₄

yes

—

C₂×C₇×C₈

C₂²⋊C₄

yes

characteristic

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

C₄⋊C₄

yes

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

C₂×Q₈

yes

characteristic

#1136

—

C₂×C₇⋊C₈

—

#533

C₂×D₇

yes

characteristic

#446

—

C₈×Q₈

—

#424

Dic₇

yes

—

#423

—

#423

C₂×D₇

yes

characteristic

#405

Dic₇

yes

—

C₂×C₄×C₈

C₂×D₇

yes

—

—

—

C₂³×C₇

#117

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇

#121

yes

—

C₂×C₄×C₇

C₂²⋊Q₈

yes

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

#101

yes

—

C₇⋊C₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×Dic₇

yes

—

C₂²×C₈

C₂×Dic₇

yes

characteristic

C₂²×C₈

C₂×Dic₇

yes

characteristic

C₄⋊Q₈

D₂₈

yes

—

#125

Dic₁₄

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×Dic₇

yes

—

C₂×C₄²

C₂²×D₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

#97

—

102

C₄×C₈

—

103

C₄×C₈

C₂²⋊D₇

yes

—

104

C₄×C₈

C₂²⋊D₇

yes

—

105

C₂²×C₇

—

106

C₂²×C₇

—

107

C₂²×C₇

#380

yes

characteristic

108

C₄×C₇

—

109

C₄×C₇

—

110

C₄×C₇

—

111

C₄×C₇

C₂×#101

yes

characteristic

112

C₄×C₇

#413

yes

characteristic

113

C₄×C₇

#472

yes

—

114

C₄×C₇

#473

yes

—

115

C₂×Q₈

—

116

C₂²×C₄

#1157

yes

—

117

C₂²×C₄

C₂²×Dic₇

yes

characteristic

118

M₁₆

—

119

M₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

C₂²⋊Dic₇

yes

characteristic

125

C₂×C₈

C₂²⋊Dic₇

yes

—

126

C₂×C₈

C₂²⋊Dic₇

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

C₄⋊Dic₇

yes

—

130

C₂²⋊C₄

—

131

C₄²

—

132

C₄²

C₂×Dic₁₄

yes

characteristic

133

C₄²

C₂×D₂₈

yes

characteristic

134

C₄²

C₂×C₂²⋊D₇

yes

—

135

C₂×C₇

—

136

C₂×C₇

#1801

yes

characteristic

137

C₂×C₇

#1816

yes

—

138

C₂³

#6489

yes

characteristic

139

Q₈

—

140

Q₈

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

#6487

yes

—

147

C₂×C₄

C₂×C₄⋊Dic₇

yes

characteristic

148

C₂×C₄

#6493

yes

—

149

C₂×C₄

C₂×C₂²⋊Dic₇

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

#6410

yes

—

155

C₇

#8639

yes

characteristic, complemented, Sylow

156

C₂²

—

157

C₂²

—

158

C₂²

#89984

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

#89914

yes

—

163

C₄

#89915

yes

—

164

C₄

C₂×#6410

yes

characteristic

165

C₄

#90009

yes

characteristic

166

C₂

—

167

C₂

#11729391

yes

characteristic

168

C₂

#11729770

yes

—

	subgroup	G/H
center	C₂²	#89984
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#2960	C₂
socle	C₂²×C₇	#380
prime radical	C₂³×C₇	#117
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#2960	D₇
hypocenter	C₇	#8639
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

subgroup

G/H

center

C₂²

#89984

commutator

C₂²×C₄×C₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂³×C₇

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

N	H ≅ G/N	type
155. C₇	#8639	semidirect

H ≅ G/N

type

155. C₇

#8639

semidirect

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

subgroup	index
C₇×#2960	2
#11729588	2
#11727485	2
C₇⋊#1309	2
#8639	7

subgroup

index

Subgroups — #13246826

Conjugacy classes of subgroups

Complements

Maximal subgroups