Group #13246791 — subgroups

Subgroups — #13246791

Order 1792.

320 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 93 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	#11736489	C₂	1	1	yes	maximal, characteristic
2	#11728569	C₂	1	2	yes	maximal
3	#11728548	C₂	1	2	yes	maximal
4	C₇×#1307	C₂	1	1	yes	maximal, characteristic
5	#11725522	C₂	1	1	yes	maximal, characteristic
6	#90161	C₂²	1	1	yes	characteristic
7	C₇×#432	C₂²	1	2	yes	—
8	C₇×#423	C₂²	1	1	yes	characteristic
9	#89894	C₄	1	2	yes	—
10	#89891	C₄	1	2	yes	—
11	#89698	C₂²	1	2	yes	—
12	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
13	#89603	—	2	2	no	—
14	#89602	—	2	2	no	—
15	#89275	—	2	2	no	—
16	#89274	—	2	2	no	—
17	#89260	—	2	4	no	—
18	#8631	—	7	1	no	maximal, Sylow
19	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
21	#6489	C₂×C₄	1	2	yes	—
22	#6487	—	2	2	no	—
23	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	#6433	D₄	1	2	yes	—
26	#6432	Q₈	1	2	yes	—
27	C₄×Dic₁₄	—	2	2	no	—
28	C₇×C₄⋊C₈	—	2	2	no	—
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	#6389	—	2	2	no	—
34	Dic₇⋊C₈	—	2	4	no	—
35	C₈×Dic₇	—	2	2	no	—
36	C₄⋊(C₇⋊C₈)	—	2	2	no	—
37	#6377	—	2	2	no	—
38	C₄×C₇⋊C₈	D₄	1	2	yes	—
39	C₄×C₇⋊C₈	D₄	1	2	yes	—
40	#2960	—	7	1	no	—
41	#1568	—	7	2	no	—
42	#1537	—	7	2	no	—
43	#1307	—	7	1	no	—
44	#1307	D₇	1	1	yes	characteristic
45	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	#41	1	2	yes	—
47	C₂×Dic₁₄	—	2	2	no	—
48	C₇×M₁₆	—	4	2	no	—
49	C₂×C₇×C₈	—	2	2	no	—
50	C₂×C₇×C₈	—	4	2	no	—
51	C₄²×C₇	C₂×D₄	1	2	yes	—
52	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
54	C₂²⋊Dic₇	—	2	2	no	—
55	C₄⋊Dic₇	—	2	2	no	—
56	C₄⋊Dic₇	C₄⋊C₄	1	4	yes	—
57	Dic₇⋊C₄	—	2	4	no	—
58	C₄×Dic₇	—	2	2	no	—
59	C₂×C₇⋊C₈	—	2	2	no	—
60	C₂×C₇⋊C₈	—	4	1	no	—
61	C₂×C₇⋊C₈	—	4	2	no	—
62	C₂×C₇⋊C₈	C₂²⋊C₄	1	2	yes	—
63	C₂×C₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
65	#533	—	7	1	no	—
66	#446	—	14	2	no	—
67	C₈×Q₈	—	14	2	no	—
68	#432	—	7	2	no	—
69	#432	C₂×D₇	1	2	yes	—
70	#424	—	7	2	no	—
71	#423	C₂×D₇	1	1	yes	characteristic
72	#405	—	7	2	no	—
73	C₂×C₄×C₈	—	7	1	no	—
74	#335	—	14	2	no	—
75	#334	—	14	2	no	—
76	#330	—	14	4	no	—
77	C₂³×C₇	#117	1	1	yes	characteristic
78	C₂×C₄×C₇	—	2	1	no	—
79	C₂×C₄×C₇	—	2	1	no	—
80	C₂×C₄×C₇	—	4	2	no	—
81	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇	#121	1	2	yes	—
84	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
85	C₂×Dic₇	—	2	4	no	—
86	Dic₁₄	—	4	4	no	—
87	C₇×C₈	—	4	2	no	—
88	C₇×C₈	—	8	2	no	—
89	C₇⋊C₈	—	2	1	no	—
90	C₇⋊C₈	—	2	2	no	—
91	C₇⋊C₈	—	8	2	no	—
92	C₇⋊C₈	#101	1	2	yes	—
93	C₂×M₁₆	C₄×D₇	1	2	yes	—
94	C₂²×C₈	—	7	1	no	—
95	C₂²×C₈	—	7	1	no	—
96	C₄⋊Q₈	—	7	2	no	—
97	#125	—	7	2	no	—
98	C₂²⋊Q₈	—	14	2	no	—
99	C₄×Q₈	—	14	2	no	—
100	#117	—	7	2	no	—
101	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	14	4	no	—
105	C₄⋊C₈	—	14	2	no	—
106	C₂²⋊C₈	—	2	2	no	—
107	C₂²⋊C₈	—	14	2	no	—
108	#97	—	2	2	no	—
109	#97	—	14	2	no	—
110	#97	—	14	2	no	—
111	C₄×C₈	—	7	2	no	—
112	C₄×C₈	—	7	2	no	—
113	C₄×C₈	—	14	2	no	—
114	C₂²×C₇	—	2	1	no	—
115	C₂²×C₇	—	2	2	no	—
116	C₂²×C₇	#380	1	1	yes	characteristic
117	C₄×C₇	—	2	1	no	—
118	C₄×C₇	—	2	2	no	—
119	C₄×C₇	—	2	2	no	—
120	C₄×C₇	C₂×#101	1	1	yes	characteristic
121	C₄×C₇	#413	1	1	yes	characteristic
122	Dic₇	—	4	4	no	—
123	C₂×Q₈	—	14	2	no	—
124	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
125	C₂²×C₄	#1157	1	2	yes	—
126	M₁₆	—	4	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	7	1	no	—
130	C₂×C₈	—	7	1	no	—
131	C₂×C₈	—	7	2	no	—
132	C₂×C₈	—	14	2	no	—
133	C₂×C₈	—	28	1	no	—
134	C₂×C₈	—	28	2	no	—
135	C₄⋊C₄	—	7	4	no	—
136	C₄⋊C₄	—	14	4	no	—
137	C₄⋊C₄	—	14	2	no	—
138	C₂²⋊C₄	—	14	2	no	—
139	C₄²	—	14	2	no	—
140	C₄²	D₄×D₇	1	2	yes	—
141	C₄²	D₄×D₇	1	1	yes	characteristic
142	C₄²	Q₈×D₇	1	1	yes	characteristic
143	C₂×C₇	—	4	1	no	—
144	C₂×C₇	#1801	1	1	yes	characteristic
145	C₂×C₇	#1814	1	2	yes	—
146	C₂³	#6435	1	1	yes	characteristic
147	Q₈	—	28	4	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	14	4	no	—
152	C₂×C₄	D₇×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₄	#6447	1	2	yes	—
154	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄	#6459	1	2	yes	—
156	C₈	—	4	2	no	—
157	C₈	—	7	2	no	—
158	C₈	—	8	2	no	—
159	C₈	—	14	2	no	—
160	C₈	—	14	1	no	—
161	C₈	—	56	2	no	—
162	C₇	#8631	1	1	yes	characteristic, complemented, Sylow
163	C₂²	—	2	1	no	—
164	C₂²	—	2	2	no	—
165	C₂²	#89439	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	—	28	4	no	—
170	C₄	#89517	1	1	yes	characteristic
171	C₄	D₇×#101	1	1	yes	characteristic
172	C₂	—	4	1	no	—
173	C₂	#11726751	1	1	yes	characteristic
174	C₂	#11726786	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11736489

C₂

yes

maximal, characteristic

#11728569

C₂

yes

maximal

#11728548

C₂

yes

maximal

C₇×#1307

C₂

yes

maximal, characteristic

#11725522

C₂

yes

maximal, characteristic

#90161

C₂²

yes

characteristic

C₇×#432

C₂²

yes

—

C₇×#423

C₂²

yes

characteristic

#89894

C₄

yes

—

#89891

C₄

yes

—

#89698

C₂²

yes

—

C₂×C₄×C₇⋊C₈

C₂²

yes

—

—

—

—

—

—

C₂×C₇×M₁₆

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

#6489

C₂×C₄

yes

—

#6487

—

C₂²×C₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇⋊C₈

C₂×C₄

yes

characteristic

#6433

D₄

yes

—

#6432

Q₈

yes

—

C₄×Dic₁₄

—

C₇×C₄⋊C₈

—

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₇×#97

—

C₂²⋊(C₇⋊C₈)

—

#6389

—

Dic₇⋊C₈

—

C₈×Dic₇

—

C₄⋊(C₇⋊C₈)

—

#6377

—

C₄×C₇⋊C₈

D₄

yes

—

C₄×C₇⋊C₈

D₄

yes

—

—

—

—

—

D₇

yes

characteristic

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂×Dic₁₄

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄²×C₇

C₂×Q₈

yes

characteristic

C₂²⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

C₄⋊C₄

yes

—

Dic₇⋊C₄

—

C₄×Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

C₂²⋊C₄

yes

—

C₂×C₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₇⋊C₈

C₂²⋊C₄

yes

characteristic

#533

—

#446

—

C₈×Q₈

—

#432

—

#432

C₂×D₇

yes

—

#424

—

#423

C₂×D₇

yes

characteristic

#405

—

C₂×C₄×C₈

—

#335

—

#334

—

#330

—

C₂³×C₇

#117

yes

characteristic

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

—

C₂×Dic₇

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₇⋊C₈

#101

yes

—

C₂×M₁₆

C₄×D₇

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

100

#117

—

101

C₂×C₄²

C₂²×D₇

yes

characteristic

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₂²⋊C₈

—

107

C₂²⋊C₈

—

108

#97

—

109

#97

—

110

#97

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₂²×C₇

—

115

C₂²×C₇

—

116

C₂²×C₇

#380

yes

characteristic

117

C₄×C₇

—

118

C₄×C₇

—

119

C₄×C₇

—

120

C₄×C₇

C₂×#101

yes

characteristic

121

C₄×C₇

#413

yes

characteristic

122

Dic₇

—

123

C₂×Q₈

—

124

C₂²×C₄

C₂×C₄×D₇

yes

characteristic

125

C₂²×C₄

#1157

yes

—

126

M₁₆

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₂²⋊C₄

—

139

C₄²

—

140

C₄²

D₄×D₇

yes

—

141

C₄²

D₄×D₇

yes

characteristic

142

C₄²

Q₈×D₇

yes

characteristic

143

C₂×C₇

—

144

C₂×C₇

#1801

yes

characteristic

145

C₂×C₇

#1814

yes

—

146

C₂³

#6435

yes

characteristic

147

Q₈

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

D₇×C₂²⋊C₄

yes

characteristic

153

C₂×C₄

#6447

yes

—

154

C₂×C₄

D₇×C₄⋊C₄

yes

characteristic

155

C₂×C₄

#6459

yes

—

156

C₈

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₇

#8631

yes

characteristic, complemented, Sylow

163

C₂²

—

164

C₂²

—

165

C₂²

#89439

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

#89517

yes

characteristic

171

C₄

D₇×#101

yes

characteristic

172

C₂

—

173

C₂

#11726751

yes

characteristic

174

C₂

#11726786

yes

—

	subgroup	G/H
center	C₂²	#89439
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1307	C₂
socle	C₂²×C₇	#380
prime radical	C₂³×C₇	#117
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1307	D₇
hypocenter	C₇	#8631
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

subgroup

G/H

center

C₂²

#89439

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 ◁ #1307

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

H ≅ G/N

type

162. C₇

#8631

semidirect

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

subgroup	index
#11736489	2
#11728569	2
#11728548	2
C₇×#1307	2
#11725522	2
#8631	7

subgroup

index

Subgroups — #13246791

Conjugacy classes of subgroups

Complements

Maximal subgroups