Group #13273692 — subgroups

Subgroups — #13273692

Order 1792.

276 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 83 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×#2161	C₂	1	1	yes	maximal, characteristic, complemented
2	#11729186	C₂	1	1	yes	maximal, characteristic
3	#11728473	C₂	1	2	yes	maximal
4	#11725624	C₂	1	1	yes	maximal, characteristic, complemented
5	#11725535	C₂	1	2	yes	maximal, complemented
6	C₂×#6464	C₂²	1	1	yes	characteristic
7	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₇×#108	C₂²	1	2	yes	—
9	#89911	—	2	2	no	—
10	#89878	C₂²	1	2	yes	—
11	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₇×#368	—	2	2	no	—
13	#89351	—	2	2	no	—
14	(C₂×D₇)⋊C₁₆	—	2	2	no	—
15	#13499	—	7	1	no	maximal, Sylow
16	C₂×#1157	D₄	1	1	yes	characteristic
17	C₂×C₇×M₁₆	—	2	2	no	—
18	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
19	#6489	—	2	2	no	—
20	C₂×#1136	D₄	1	1	yes	characteristic
21	#6464	—	2	2	no	—
22	#6464	—	2	2	no	—
23	C₂×#1131	—	2	1	no	—
24	C₂×C₈×D₇	—	2	1	no	—
25	C₇×M₃₂	—	2	2	no	—
26	C₂×C₇×C₁₆	D₄	1	2	yes	—
27	C₇×#108	—	2	2	no	—
28	C₇×#108	—	2	2	no	—
29	#6399	—	4	4	no	—
30	#6392	—	2	2	no	—
31	#6391	—	2	2	no	—
32	C₂×C₇⋊C₁₆	—	2	1	no	—
33	C₂×C₇⋊C₁₆	—	2	2	no	—
34	#2161	D₇	1	1	yes	characteristic, complemented
35	#2113	—	7	1	no	—
36	#1814	—	7	2	no	—
37	#1384	—	7	1	no	—
38	#1382	—	7	2	no	—
39	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₂²⋊D₇	—	2	1	no	—
41	#1157	—	2	2	no	—
42	#1157	—	2	2	no	—
43	C₂×D₂₈	Q₁₆	1	1	yes	characteristic
44	C₂×C₄×D₇	—	2	1	no	—
45	C₂×Dic₁₄	Q₁₆	1	1	yes	characteristic
46	C₇×M₁₆	—	4	2	no	—
47	C₇×M₁₆	—	4	2	no	—
48	C₂×C₇×C₈	—	4	2	no	—
49	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂×C₇×C₈	#41	1	2	yes	—
51	C₂×C₇×C₈	#41	1	2	yes	—
52	C₂×C₇×C₈	#41	1	1	yes	characteristic
53	C₂²⋊Dic₇	—	4	2	no	—
54	C₄⋊Dic₇	—	4	2	no	—
55	C₄×Dic₇	—	4	2	no	—
56	#1136	—	2	2	no	—
57	C₂×C₇⋊C₈	—	2	1	no	—
58	#1131	—	4	2	no	—
59	C₈×D₇	—	4	1	no	—
60	C₈×D₇	—	4	1	no	—
61	C₇×C₁₆	—	2	2	no	—
62	C₇⋊C₁₆	—	4	1	no	—
63	C₂×#131	—	7	1	no	—
64	C₂×M₃₂	C₂×D₇	1	1	yes	characteristic
65	C₂²×C₁₆	—	7	1	no	—
66	C₂×#108	C₂×D₇	1	2	yes	—
67	#427	—	7	2	no	—
68	#421	—	14	2	no	—
69	#368	—	2	2	no	—
70	C₂²⋊C₁₆	—	14	2	no	—
71	#344	—	14	2	no	—
72	C₂³×C₇	#135	1	1	yes	characteristic
73	C₂²×D₇	—	2	1	no	—
74	C₂×C₄×C₇	#121	1	1	yes	characteristic
75	C₂×C₄×C₇	#123	1	2	yes	—
76	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
78	C₂²⋊D₇	—	4	2	no	—
79	C₂×Dic₇	—	2	1	no	—
80	C₂×Dic₇	—	8	2	no	—
81	D₂₈	—	2	2	no	—
82	C₄×D₇	—	4	1	no	—
83	C₄×D₇	—	4	1	no	—
84	Dic₁₄	—	2	2	no	—
85	C₇×C₈	—	8	2	no	—
86	C₇×C₈	#123	1	1	yes	characteristic
87	C₇×C₈	#124	1	1	yes	characteristic
88	C₇×C₈	#126	1	2	yes	—
89	C₇⋊C₈	—	4	1	no	—
90	C₂×#41	—	7	1	no	—
91	#131	—	14	2	no	—
92	#131	—	14	2	no	—
93	C₂×M₁₆	—	2	2	no	—
94	C₂×M₁₆	—	7	1	no	—
95	C₂×M₁₆	—	14	1	no	—
96	C₂²×C₈	—	14	1	no	—
97	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
98	#117	—	14	2	no	—
99	M₃₂	—	2	2	no	—
100	C₂×C₁₆	—	14	2	no	—
101	C₂×C₁₆	—	14	1	no	—
102	C₂×C₁₆	C₂²⋊D₇	1	2	yes	—
103	#108	—	2	2	no	—
104	#108	—	2	2	no	—
105	#107	—	14	2	no	—
106	#106	—	14	2	no	—
107	#104	—	28	4	no	—
108	C₂²×C₇	—	4	1	no	—
109	C₂²×C₇	#467	1	1	yes	characteristic
110	C₂²×C₇	#482	1	2	yes	—
111	C₂×D₇	—	4	2	no	—
112	C₂×D₇	—	4	1	no	—
113	C₄×C₇	#397	1	1	yes	characteristic
114	C₄×C₇	#469	1	1	yes	characteristic
115	C₄×C₇	#484	1	2	yes	—
116	Dic₇	—	4	1	no	—
117	Dic₇	—	16	2	no	—
118	#41	—	14	2	no	—
119	#41	—	14	2	no	—
120	C₂×Q₈	—	7	1	no	—
121	C₂×D₄	—	7	1	no	—
122	C₂×D₄	—	14	1	no	—
123	C₂²×C₄	—	14	1	no	—
124	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
125	M₁₆	—	4	2	no	—
126	M₁₆	—	4	2	no	—
127	M₁₆	—	14	2	no	—
128	M₁₆	—	28	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	14	1	no	—
131	C₂×C₈	—	28	1	no	—
132	C₂×C₈	—	28	1	no	—
133	C₂×C₈	#1157	1	2	yes	—
134	C₂×C₈	#1159	1	1	yes	characteristic
135	C₂×C₈	#1161	1	2	yes	—
136	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
137	C₄⋊C₄	—	28	2	no	—
138	C₂²⋊C₄	—	28	2	no	—
139	C₄²	—	28	2	no	—
140	C₁₆	—	2	2	no	—
141	C₁₆	—	28	1	no	—
142	C₂×C₇	—	2	2	no	—
143	C₂×C₇	—	2	1	no	—
144	C₂×C₇	#2076	1	1	yes	characteristic
145	D₇	—	8	1	no	—
146	C₂³	—	14	1	no	—
147	C₂³	#6482	1	1	yes	characteristic
148	Q₈	—	14	2	no	—
149	D₄	—	14	2	no	—
150	D₄	—	28	2	no	—
151	C₂×C₄	—	14	1	no	—
152	C₂×C₄	—	28	1	no	—
153	C₂×C₄	—	28	1	no	—
154	C₂×C₄	—	56	2	no	—
155	C₂×C₄	#6457	1	2	yes	—
156	C₂×C₄	#6474	1	2	yes	—
157	C₂×C₄	D₇×Q₁₆	1	1	yes	characteristic
158	C₂×C₄	#6501	1	1	yes	characteristic
159	C₈	—	8	2	no	—
160	C₈	—	28	1	no	—
161	C₈	#6461	1	2	yes	—
162	C₈	#6492	1	1	yes	characteristic
163	C₈	#6510	1	1	yes	characteristic
164	C₇	#13499	1	1	yes	characteristic, complemented, Sylow
165	C₂²	—	4	1	no	—
166	C₂²	—	28	2	no	—
167	C₂²	—	28	1	no	—
168	C₂²	#89657	1	2	yes	—
169	C₂²	#89960	1	1	yes	characteristic
170	C₄	—	28	1	no	—
171	C₄	—	112	2	no	—
172	C₄	#89656	1	2	yes	—
173	C₄	#89765	1	1	yes	characteristic
174	C₄	#89931	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	—	2	1	no	—
177	C₂	—	56	1	no	—
178	C₂	#11728391	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#2161

C₂

yes

maximal, characteristic, complemented

#11729186

C₂

yes

maximal, characteristic

#11728473

C₂

yes

maximal

#11725624

C₂

yes

maximal, characteristic, complemented

#11725535

C₂

yes

maximal, complemented

C₂×#6464

C₂²

yes

characteristic

C₂×C₇×M₃₂

C₂²

yes

characteristic

C₂×C₇×#108

C₂²

yes

—

#89911

—

#89878

C₂²

yes

—

C₂²×C₇⋊C₁₆

C₂²

yes

characteristic

C₇×#368

—

#89351

—

(C₂×D₇)⋊C₁₆

—

#13499

—

maximal, Sylow

C₂×#1157

D₄

yes

characteristic

C₂×C₇×M₁₆

—

C₂²×C₇×C₈

C₂³

yes

characteristic

#6489

—

C₂×#1136

D₄

yes

—

—

—

C₂×C₈×D₇

—

C₇×M₃₂

—

C₂×C₇×C₁₆

D₄

yes

—

—

—

—

—

—

C₂×C₇⋊C₁₆

—

C₂×C₇⋊C₁₆

—

#2161

D₇

yes

characteristic, complemented

—

—

—

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₇

—

#1157

—

#1157

—

C₂×D₂₈

Q₁₆

yes

characteristic

C₂×C₄×D₇

—

C₂×Dic₁₄

Q₁₆

yes

characteristic

C₇×M₁₆

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

#41

yes

—

C₂×C₇×C₈

#41

yes

—

C₂×C₇×C₈

#41

yes

characteristic

C₂²⋊Dic₇

—

C₄⋊Dic₇

—

C₄×Dic₇

—

#1136

—

C₂×C₇⋊C₈

—

#1131

—

C₈×D₇

—

C₈×D₇

—

C₇×C₁₆

—

C₇⋊C₁₆

—

C₂×#131

—

C₂×M₃₂

C₂×D₇

yes

characteristic

C₂²×C₁₆

—

C₂×#108

C₂×D₇

yes

—

#427

—

#421

—

#368

—

C₂²⋊C₁₆

—

#344

—

C₂³×C₇

#135

yes

characteristic

C₂²×D₇

—

C₂×C₄×C₇

#121

yes

characteristic

C₂×C₄×C₇

#123

yes

—

C₂×C₄×C₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇

AGL(1,ℤ/8)

yes

—

C₂²⋊D₇

—

C₂×Dic₇

—

C₂×Dic₇

—

D₂₈

—

C₄×D₇

—

C₄×D₇

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

yes

C₇×C₈

yes

C₇×C₈

yes

—

C₇⋊C₈

—

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₇

yes

characteristic

#117

—

M₃₂

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

C₂²⋊D₇

yes

—

103

#108

—

104

#108

—

105

#107

—

106

#106

—

107

#104

—

108

C₂²×C₇

—

109

C₂²×C₇

#467

yes

characteristic

110

C₂²×C₇

#482

yes

—

111

C₂×D₇

—

112

C₂×D₇

—

113

C₄×C₇

#397

yes

characteristic

114

C₄×C₇

#469

yes

characteristic

115

C₄×C₇

#484

yes

—

116

Dic₇

—

117

Dic₇

—

118

#41

—

119

#41

—

120

C₂×Q₈

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂²×C₄

—

124

C₂²×C₄

D₄×D₇

yes

characteristic

125

M₁₆

—

126

M₁₆

—

127

M₁₆

—

128

M₁₆

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

#1157

yes

—

134

C₂×C₈

#1159

yes

characteristic

135

C₂×C₈

#1161

yes

—

136

C₂×C₈

C₂×C₂²⋊D₇

yes

characteristic

137

C₄⋊C₄

—

138

C₂²⋊C₄

—

139

C₄²

—

140

C₁₆

—

141

C₁₆

—

142

C₂×C₇

—

143

C₂×C₇

—

144

C₂×C₇

#2076

yes

characteristic

145

D₇

—

146

C₂³

—

147

C₂³

#6482

yes

characteristic

148

Q₈

—

149

D₄

—

150

D₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

#6457

yes

—

156

C₂×C₄

#6474

yes

—

157

C₂×C₄

D₇×Q₁₆

yes

characteristic

158

C₂×C₄

#6501

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

#6461

yes

—

162

C₈

#6492

yes

characteristic

163

C₈

#6510

yes

characteristic

164

C₇

#13499

yes

characteristic, complemented, Sylow

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

#89657

yes

—

169

C₂²

#89960

yes

characteristic

170

C₄

—

171

C₄

—

112

—

172

C₄

#89656

yes

—

173

C₄

#89765

yes

characteristic

174

C₄

#89931

yes

characteristic

175

C₂

—

176

C₂

—

177

C₂

—

178

C₂

#11728391

yes

characteristic

	subgroup	G/H
center	C₄	#89931
commutator, cosocle	C₂²×C₇×C₈	C₂³
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#2161	C₂
socle	C₂×C₇	#2076
prime radical	C₂×#1157	D₄
hypercenter	#2161	D₇
hypocenter	C₇	#13499
LCS	G ▷ C₂²×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

subgroup

G/H

center

C₄

#89931

commutator, cosocle

C₂²×C₇×C₈

C₂³

Frattini

C₂²×C₈

C₂²×D₇

C₂

C₂×C₇

D₄

D₇

C₇

G ▷ C₂²×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

N	H ≅ G/N	type
1. C₇×#2161	C₂	semidirect
4. #11725624	C₂	semidirect
5. #11725535	C₂	semidirect
34. #2161	D₇	semidirect
164. C₇	#13499	semidirect

H ≅ G/N

type

1. C₇×#2161

C₂

semidirect

4. #11725624

C₂

semidirect

5. #11725535

C₂

semidirect

34. #2161

D₇

semidirect

164. C₇

#13499

semidirect

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

subgroup	index
C₇×#2161	2
#11729186	2
#11728473	2
#11725624	2
#11725535	2
#13499	7

subgroup

index

Subgroups — #13273692

Conjugacy classes of subgroups

Complements

Maximal subgroups