Group #13246790 — subgroups

Subgroups — #13246790

Order 1792.

320 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 101 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	#11740773	C₂	1	1	yes	maximal, characteristic
2	C₇×#1537	C₂	1	1	yes	maximal, characteristic
3	#11729577	C₂	1	1	yes	maximal, characteristic
4	#11728548	C₂	1	2	yes	maximal
5	#11725564	C₂	1	2	yes	maximal
6	C₇×#533	C₂²	1	1	yes	characteristic
7	C₇×#432	C₂²	1	2	yes	—
8	#89796	—	2	2	no	—
9	Q₈×C₇⋊C₈	—	2	2	no	—
10	#89771	C₄	1	2	yes	—
11	#89769	C₄	1	2	yes	—
12	#89698	C₂²	1	2	yes	—
13	#89696	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₇×#330	—	2	2	no	—
16	#89338	—	2	2	no	—
17	#89275	—	2	2	no	—
18	#89274	—	2	2	no	—
19	#8631	—	7	1	no	maximal, Sylow
20	C₇×C₄⋊Q₈	Q₈	1	2	yes	—
21	C₇×#125	D₄	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	C₂×C₄	1	2	yes	—
27	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₇×C₄⋊C₈	—	2	2	no	—
30	C₇×C₂²⋊C₈	—	2	2	no	—
31	C₇×#97	—	2	2	no	—
32	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₇⋊C₈)	—	2	2	no	—
34	C₄⋊(C₇⋊C₈)	—	2	4	no	—
35	C₄⋊(C₇⋊C₈)	—	2	2	no	—
36	#6377	—	2	2	no	—
37	#6377	—	2	2	no	—
38	C₄×C₇⋊C₈	—	2	2	no	—
39	C₄×C₇⋊C₈	D₄	1	2	yes	—
40	C₄×C₇⋊C₈	D₄	1	2	yes	—
41	#2960	—	7	1	no	—
42	#1568	—	7	2	no	—
43	#1537	—	7	1	no	—
44	#1537	D₇	1	1	yes	characteristic
45	#1307	—	7	2	no	—
46	C₂×C₇×Q₈	—	2	2	no	—
47	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	#41	1	2	yes	—
49	C₂×C₇×C₈	—	4	2	no	—
50	C₇×C₄⋊C₄	—	2	4	no	—
51	C₇×C₄⋊C₄	—	2	2	no	—
52	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
53	C₇×C₂²⋊C₄	—	2	2	no	—
54	C₄²×C₇	—	2	2	no	—
55	C₄²×C₇	C₂×D₄	1	2	yes	—
56	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
58	#1136	—	4	2	no	—
59	C₂×C₇⋊C₈	—	2	2	no	—
60	C₂×C₇⋊C₈	—	2	2	no	—
61	C₂×C₇⋊C₈	—	4	1	no	—
62	C₂×C₇⋊C₈	—	4	2	no	—
63	C₂×C₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₇⋊C₈	C₂²⋊C₄	1	2	yes	—
65	C₂×C₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
66	#533	C₂×D₇	1	1	yes	characteristic
67	#446	—	14	2	no	—
68	C₈×Q₈	—	14	2	no	—
69	#432	—	7	2	no	—
70	#432	C₂×D₇	1	2	yes	—
71	#424	—	7	2	no	—
72	#423	—	7	1	no	—
73	#405	—	7	2	no	—
74	C₂×C₄×C₈	—	7	1	no	—
75	#335	—	14	2	no	—
76	#334	—	14	2	no	—
77	#330	—	2	2	no	—
78	#330	—	14	2	no	—
79	C₂³×C₇	#117	1	1	yes	characteristic
80	C₇×Q₈	—	4	4	no	—
81	C₂×C₄×C₇	—	2	1	no	—
82	C₂×C₄×C₇	—	2	1	no	—
83	C₂×C₄×C₇	—	2	4	no	—
84	C₂×C₄×C₇	—	4	2	no	—
85	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₇	#121	1	2	yes	—
88	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
89	C₇×C₈	—	8	2	no	—
90	C₇⋊C₈	—	2	1	no	—
91	C₇⋊C₈	—	2	2	no	—
92	C₇⋊C₈	—	4	2	no	—
93	C₇⋊C₈	—	8	2	no	—
94	C₇⋊C₈	#101	1	2	yes	—
95	C₂×M₁₆	—	7	2	no	—
96	C₂²×C₈	—	7	1	no	—
97	C₂²×C₈	—	7	1	no	—
98	C₄⋊Q₈	Dic₁₄	1	2	yes	—
99	#125	C₂²⋊D₇	1	2	yes	—
100	C₂²⋊Q₈	—	2	2	no	—
101	C₄×Q₈	—	2	2	no	—
102	#117	C₄×D₇	1	2	yes	—
103	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	14	4	no	—
106	C₄⋊C₈	—	14	2	no	—
107	C₄⋊C₈	—	14	2	no	—
108	C₂²⋊C₈	—	2	2	no	—
109	C₂²⋊C₈	—	14	2	no	—
110	#97	—	2	2	no	—
111	#97	—	14	2	no	—
112	#97	—	14	2	no	—
113	C₄×C₈	—	7	2	no	—
114	C₄×C₈	—	7	2	no	—
115	C₄×C₈	—	14	2	no	—
116	C₂²×C₇	—	2	1	no	—
117	C₂²×C₇	—	2	2	no	—
118	C₂²×C₇	#380	1	1	yes	characteristic
119	C₄×C₇	—	2	1	no	—
120	C₄×C₇	—	2	2	no	—
121	C₄×C₇	—	2	2	no	—
122	C₄×C₇	—	4	4	no	—
123	C₄×C₇	C₂×#101	1	1	yes	characteristic
124	C₄×C₇	#413	1	1	yes	characteristic
125	C₂×Q₈	—	2	2	no	—
126	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
127	C₂²×C₄	#1159	1	2	yes	—
128	M₁₆	—	28	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	7	1	no	—
131	C₂×C₈	—	7	1	no	—
132	C₂×C₈	—	7	2	no	—
133	C₂×C₈	—	14	2	no	—
134	C₂×C₈	—	14	2	no	—
135	C₂×C₈	—	28	1	no	—
136	C₂×C₈	—	28	2	no	—
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	Dic₇⋊C₄	1	4	yes	—
140	C₂²⋊C₄	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
143	C₄²	D₄×D₇	1	2	yes	—
144	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
145	C₂×C₇	—	4	1	no	—
146	C₂×C₇	#1801	1	1	yes	characteristic
147	C₂×C₇	#1814	1	2	yes	—
148	C₂³	#6440	1	1	yes	characteristic
149	Q₈	—	4	4	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	#6441	1	2	yes	—
155	C₂×C₄	D₇×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
157	C₂×C₄	#6502	1	2	yes	—
158	C₈	—	7	2	no	—
159	C₈	—	8	2	no	—
160	C₈	—	14	1	no	—
161	C₈	—	14	2	no	—
162	C₈	—	28	2	no	—
163	C₈	—	56	2	no	—
164	C₇	#8631	1	1	yes	characteristic, complemented, Sylow
165	C₂²	—	2	2	no	—
166	C₂²	—	2	1	no	—
167	C₂²	#89714	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	1	no	—
171	C₄	—	4	4	no	—
172	C₄	D₇×#101	1	1	yes	characteristic
173	C₄	#89523	1	1	yes	characteristic
174	C₂	—	4	1	no	—
175	C₂	#11727851	1	1	yes	characteristic
176	C₂	#11728150	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11740773

C₂

yes

maximal, characteristic

C₇×#1537

C₂

yes

maximal, characteristic

#11729577

C₂

yes

maximal, characteristic

#11728548

C₂

yes

maximal

#11725564

C₂

yes

maximal

C₇×#533

C₂²

yes

characteristic

C₇×#432

C₂²

yes

—

#89796

—

Q₈×C₇⋊C₈

—

#89771

C₄

yes

—

#89769

C₄

yes

—

#89698

C₂²

yes

—

#89696

C₂²

yes

characteristic

C₂×C₄×C₇⋊C₈

C₂²

yes

—

—

—

—

—

C₇×C₄⋊Q₈

Q₈

yes

—

C₇×#125

D₄

yes

—

C₇×C₂²⋊Q₈

—

C₄×C₇×Q₈

—

C₇×#117

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×#1136

C₂×C₄

yes

—

C₂²×C₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇⋊C₈

C₂×C₄

yes

characteristic

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₇×#97

—

C₂²⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

#6377

—

#6377

—

C₄×C₇⋊C₈

—

C₄×C₇⋊C₈

D₄

yes

—

C₄×C₇⋊C₈

D₄

yes

—

—

—

—

D₇

yes

characteristic

#1307

—

C₂×C₇×Q₈

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂×C₇×C₈

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

C₄⋊C₄

yes

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄²×C₇

C₂×Q₈

yes

characteristic

#1136

—

C₂×C₇⋊C₈

—

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

#533

C₂×D₇

yes

characteristic

#446

—

C₈×Q₈

—

#432

—

#432

C₂×D₇

yes

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

—

—

—

—

—

C₂³×C₇

#117

yes

characteristic

C₇×Q₈

—

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

—

C₇⋊C₈

—

C₇⋊C₈

#101

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₁₄

yes

—

#125

C₂²⋊D₇

yes

—

100

C₂²⋊Q₈

—

101

C₄×Q₈

—

102

#117

C₄×D₇

yes

—

103

C₂×C₄²

C₂²×D₇

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₂²⋊C₈

—

109

C₂²⋊C₈

—

110

#97

—

111

#97

—

112

#97

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₂²×C₇

—

117

C₂²×C₇

—

118

C₂²×C₇

#380

yes

characteristic

119

C₄×C₇

—

120

C₄×C₇

—

121

C₄×C₇

—

122

C₄×C₇

—

123

C₄×C₇

C₂×#101

yes

characteristic

124

C₄×C₇

#413

yes

characteristic

125

C₂×Q₈

—

126

C₂²×C₄

C₂×C₄×D₇

yes

characteristic

127

C₂²×C₄

#1159

yes

—

128

M₁₆

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

Dic₇⋊C₄

yes

—

140

C₂²⋊C₄

—

141

C₄²

—

142

C₄²

C₂×Dic₁₄

yes

characteristic

143

C₄²

D₄×D₇

yes

—

144

C₄²

C₂×C₂²⋊D₇

yes

characteristic

145

C₂×C₇

—

146

C₂×C₇

#1801

yes

characteristic

147

C₂×C₇

#1814

yes

—

148

C₂³

#6440

yes

characteristic

149

Q₈

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

#6441

yes

—

155

C₂×C₄

D₇×C₂²⋊C₄

yes

characteristic

156

C₂×C₄

C₂×Dic₇⋊C₄

yes

characteristic

157

C₂×C₄

#6502

yes

—

158

C₈

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₇

#8631

yes

characteristic, complemented, Sylow

165

C₂²

—

166

C₂²

—

167

C₂²

#89714

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

D₇×#101

yes

characteristic

173

C₄

#89523

yes

characteristic

174

C₂

—

175

C₂

#11727851

yes

characteristic

176

C₂

#11728150

yes

—

	subgroup	G/H
center	C₂²	#89714
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1537	C₂
socle	C₂²×C₇	#380
prime radical	C₂³×C₇	#117
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1537	D₇
hypocenter	C₇	#8631
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1537

subgroup

G/H

center

C₂²

#89714

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 ◁ #1537

N	H ≅ G/N	type
164. C₇	#8631	semidirect

H ≅ G/N

type

164. C₇

#8631

semidirect

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

subgroup	index
#11740773	2
C₇×#1537	2
#11729577	2
#11728548	2
#11725564	2
#8631	7

subgroup

index

Subgroups — #13246790

Conjugacy classes of subgroups

Complements

Maximal subgroups