Group #13273676 — subgroups

Subgroups — #13273676

Order 1792.

284 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 87 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	#11729163	C₂	1	1	yes	maximal, characteristic
2	C₂×#89300	C₂	1	1	yes	maximal, characteristic, complemented
3	#11728475	C₂	1	2	yes	maximal
4	C₇×#1381	C₂	1	1	yes	maximal, characteristic, complemented
5	#11725538	C₂	1	2	yes	maximal, complemented
6	C₂×#6464	C₂²	1	1	yes	characteristic
7	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₇×#108	C₂²	1	2	yes	—
9	#89905	—	2	2	no	—
10	C₂×#6392	C₂²	1	2	yes	—
11	C₂×#6386	C₂²	1	1	yes	characteristic
12	#89349	—	2	2	no	—
13	(C₂×D₇)⋊C₁₆	—	2	2	no	—
14	#89300	—	2	2	no	—
15	#13496	—	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	C₂×C₄⋊Dic₇	—	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	C₂×C₇×C₁₆	—	2	2	no	—
26	C₂×C₇×C₁₆	D₄	1	2	yes	—
27	C₇×#108	—	2	2	no	—
28	C₇×#108	—	2	2	no	—
29	#6395	—	4	2	no	—
30	#6392	—	2	2	no	—
31	#6392	—	2	2	no	—
32	#6390	—	4	2	no	—
33	#6386	—	2	2	no	—
34	C₂×C₇⋊C₁₆	—	2	1	no	—
35	C₂×#367	—	7	1	no	—
36	#2113	—	7	1	no	—
37	#1816	—	7	2	no	—
38	#1387	—	7	2	no	—
39	#1381	D₇	1	1	yes	characteristic, complemented
40	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₂²⋊D₇	—	2	1	no	—
42	C₂²×Dic₇	—	4	2	no	—
43	#1157	—	2	2	no	—
44	#1157	—	2	2	no	—
45	C₂×D₂₈	D₈	1	1	yes	characteristic
46	C₂×C₄×D₇	—	2	1	no	—
47	C₂×Dic₁₄	D₈	1	1	yes	characteristic
48	C₇×M₁₆	—	4	2	no	—
49	C₇×M₁₆	—	4	2	no	—
50	C₂×C₇×C₈	—	4	2	no	—
51	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
52	C₂×C₇×C₈	#41	1	2	yes	—
53	C₂×C₇×C₈	#41	1	2	yes	—
54	C₂×C₇×C₈	#41	1	1	yes	characteristic
55	C₄⋊Dic₇	—	4	2	no	—
56	C₄⋊Dic₇	—	4	2	no	—
57	#1136	—	2	2	no	—
58	C₂×C₇⋊C₈	—	2	1	no	—
59	#1131	—	4	2	no	—
60	C₈×D₇	—	4	1	no	—
61	C₈×D₇	—	4	1	no	—
62	C₇×C₁₆	—	2	2	no	—
63	C₇⋊C₁₆	—	4	1	no	—
64	C₂×#131	—	7	1	no	—
65	C₂×M₃₂	—	7	1	no	—
66	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
67	C₂×#108	C₂×D₇	1	2	yes	—
68	C₂×#107	—	7	2	no	—
69	#417	—	14	2	no	—
70	#367	—	14	2	no	—
71	C₂²⋊C₁₆	—	14	2	no	—
72	#343	—	14	2	no	—
73	C₂³×C₇	#135	1	1	yes	characteristic
74	C₂²×D₇	—	2	1	no	—
75	C₂×C₄×C₇	#121	1	1	yes	characteristic
76	C₂×C₄×C₇	#123	1	2	yes	—
77	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₇	#137	1	2	yes	—
79	C₂²⋊D₇	—	4	2	no	—
80	C₂×Dic₇	—	2	1	no	—
81	C₂×Dic₇	—	8	2	no	—
82	C₂×Dic₇	—	16	2	no	—
83	D₂₈	—	2	2	no	—
84	C₄×D₇	—	4	1	no	—
85	C₄×D₇	—	4	1	no	—
86	Dic₁₄	—	2	2	no	—
87	C₇×C₈	—	8	2	no	—
88	C₇×C₈	#123	1	1	yes	characteristic
89	C₇×C₈	#124	1	1	yes	characteristic
90	C₇×C₈	#126	1	2	yes	—
91	C₇⋊C₈	—	4	1	no	—
92	C₂×#41	—	7	1	no	—
93	#131	—	14	2	no	—
94	#131	—	14	2	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	7	1	no	—
97	C₂×M₁₆	—	14	1	no	—
98	C₂²×C₈	—	14	1	no	—
99	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
100	C₂×C₄⋊C₄	—	14	2	no	—
101	M₃₂	—	14	2	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	14	1	no	—
104	C₂×C₁₆	D₂₈	1	2	yes	—
105	#108	—	2	2	no	—
106	#108	—	2	2	no	—
107	#107	—	14	2	no	—
108	#107	—	14	2	no	—
109	#103	—	28	2	no	—
110	#102	—	28	2	no	—
111	C₂²×C₇	—	2	2	no	—
112	C₂²×C₇	#466	1	1	yes	characteristic
113	C₂²×C₇	#485	1	2	yes	—
114	C₂×D₇	—	4	1	no	—
115	C₂×D₇	—	4	2	no	—
116	C₄×C₇	#397	1	1	yes	characteristic
117	C₄×C₇	#470	1	1	yes	characteristic
118	C₄×C₇	#480	1	2	yes	—
119	Dic₇	—	4	1	no	—
120	Dic₇	—	16	2	no	—
121	#41	—	14	2	no	—
122	#41	—	14	2	no	—
123	C₂×Q₈	—	7	1	no	—
124	C₂×D₄	—	7	1	no	—
125	C₂×D₄	—	14	1	no	—
126	C₂²×C₄	—	14	1	no	—
127	C₂²×C₄	—	28	2	no	—
128	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
129	M₁₆	—	4	2	no	—
130	M₁₆	—	4	2	no	—
131	M₁₆	—	14	2	no	—
132	M₁₆	—	28	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	14	1	no	—
135	C₂×C₈	—	28	1	no	—
136	C₂×C₈	—	28	1	no	—
137	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
138	C₂×C₈	#1157	1	2	yes	—
139	C₂×C₈	#1159	1	2	yes	—
140	C₂×C₈	#1161	1	1	yes	characteristic
141	C₄⋊C₄	—	28	2	no	—
142	C₄⋊C₄	—	28	2	no	—
143	C₁₆	—	2	2	no	—
144	C₁₆	—	28	1	no	—
145	C₂×C₇	—	2	2	no	—
146	C₂×C₇	#2075	1	1	yes	characteristic
147	C₂×C₇	#2223	1	2	yes	—
148	D₇	—	8	1	no	—
149	C₂³	—	14	1	no	—
150	C₂³	#6482	1	1	yes	characteristic
151	Q₈	—	14	2	no	—
152	D₄	—	14	2	no	—
153	D₄	—	28	2	no	—
154	C₂×C₄	—	14	1	no	—
155	C₂×C₄	—	28	1	no	—
156	C₂×C₄	—	28	1	no	—
157	C₂×C₄	—	56	2	no	—
158	C₂×C₄	—	112	2	no	—
159	C₂×C₄	#6446	1	2	yes	—
160	C₂×C₄	#6458	1	1	yes	characteristic
161	C₂×C₄	D₇×D₈	1	1	yes	characteristic
162	C₂×C₄	#6478	1	2	yes	—
163	C₈	—	8	2	no	—
164	C₈	—	28	1	no	—
165	C₈	#6438	1	1	yes	characteristic
166	C₈	#6449	1	1	yes	characteristic
167	C₈	#6461	1	2	yes	—
168	C₇	#13496	1	1	yes	characteristic, complemented, Sylow
169	C₂²	—	2	2	no	—
170	C₂²	—	28	2	no	—
171	C₂²	—	28	1	no	—
172	C₂²	#89586	1	2	yes	—
173	C₂²	#89655	1	1	yes	characteristic
174	C₄	—	28	1	no	—
175	C₄	—	112	2	no	—
176	C₄	#89446	1	1	yes	characteristic
177	C₄	#89546	1	2	yes	—
178	C₄	#89640	1	1	yes	characteristic
179	C₂	—	2	2	no	—
180	C₂	—	56	1	no	—
181	C₂	#11726584	1	1	yes	characteristic
182	C₂	#11727117	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11729163

C₂

yes

maximal, characteristic

C₂×#89300

C₂

yes

maximal, characteristic, complemented

#11728475

C₂

yes

maximal

C₇×#1381

C₂

yes

maximal, characteristic, complemented

#11725538

C₂

yes

maximal, complemented

C₂×#6464

C₂²

yes

characteristic

C₂²×C₇×C₁₆

C₂²

yes

characteristic

C₂×C₇×#108

C₂²

yes

—

#89905

—

C₂×#6392

C₂²

yes

—

C₂×#6386

C₂²

yes

characteristic

#89349

—

(C₂×D₇)⋊C₁₆

—

—

—

D₄

yes

characteristic

C₂×C₇×M₁₆

—

C₂²×C₇×C₈

C₂³

yes

characteristic

C₂×C₄⋊Dic₇

—

C₂×#1136

D₄

yes

—

—

—

C₂×C₈×D₇

—

C₂×C₇×C₁₆

—

C₂×C₇×C₁₆

D₄

yes

—

—

—

—

—

—

—

—

C₂×C₇⋊C₁₆

—

—

—

—

—

D₇

yes

characteristic, complemented

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₇

—

C₂²×Dic₇

—

#1157

—

#1157

—

C₂×D₂₈

D₈

yes

characteristic

C₂×C₄×D₇

—

C₂×Dic₁₄

D₈

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

C₄⋊Dic₇

—

C₄⋊Dic₇

—

#1136

—

C₂×C₇⋊C₈

—

#1131

—

C₈×D₇

—

C₈×D₇

—

C₇×C₁₆

—

C₇⋊C₁₆

—

C₂×#131

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₇

yes

characteristic

C₂×#108

C₂×D₇

yes

—

C₂×#107

—

#417

—

#367

—

C₂²⋊C₁₆

—

#343

—

C₂³×C₇

#135

yes

characteristic

C₂²×D₇

—

C₂×C₄×C₇

#121

yes

characteristic

C₂×C₄×C₇

#123

yes

—

C₂×C₄×C₇

C₂×D₈

yes

characteristic

C₂×C₄×C₇

#137

yes

—

C₂²⋊D₇

—

C₂×Dic₇

—

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

100

C₂×C₄⋊C₄

—

101

M₃₂

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

C₂×C₁₆

D₂₈

yes

—

105

#108

—

106

#108

—

107

#107

—

108

#107

—

109

#103

—

110

#102

—

111

C₂²×C₇

—

112

C₂²×C₇

#466

yes

characteristic

113

C₂²×C₇

#485

yes

—

114

C₂×D₇

—

115

C₂×D₇

—

116

C₄×C₇

#397

yes

characteristic

117

C₄×C₇

#470

yes

characteristic

118

C₄×C₇

#480

yes

—

119

Dic₇

—

120

Dic₇

—

121

#41

—

122

#41

—

123

C₂×Q₈

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

D₄×D₇

yes

characteristic

129

M₁₆

—

130

M₁₆

—

131

M₁₆

—

132

M₁₆

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

C₂×D₂₈

yes

characteristic

138

C₂×C₈

#1157

yes

—

139

C₂×C₈

#1159

yes

—

140

C₂×C₈

#1161

yes

characteristic

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₁₆

—

144

C₁₆

—

145

C₂×C₇

—

146

C₂×C₇

#2075

yes

characteristic

147

C₂×C₇

#2223

yes

—

148

D₇

—

149

C₂³

—

150

C₂³

#6482

yes

characteristic

151

Q₈

—

152

D₄

—

153

D₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

112

—

159

C₂×C₄

#6446

yes

—

160

C₂×C₄

#6458

yes

characteristic

161

C₂×C₄

D₇×D₈

yes

characteristic

162

C₂×C₄

#6478

yes

—

163

C₈

—

164

C₈

—

165

C₈

#6438

yes

characteristic

166

C₈

#6449

yes

characteristic

167

C₈

#6461

yes

—

168

C₇

#13496

yes

characteristic, complemented, Sylow

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

#89586

yes

—

173

C₂²

#89655

yes

characteristic

174

C₄

—

175

C₄

—

112

—

176

C₄

#89446

yes

characteristic

177

C₄

#89546

yes

—

178

C₄

#89640

yes

characteristic

179

C₂

—

180

C₂

—

181

C₂

#11726584

yes

characteristic

182

C₂

#11727117

yes

—

	subgroup	G/H
center	C₂²	#89655
commutator, cosocle	C₂²×C₇×C₈	C₂³
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1381	C₂
socle	C₂²×C₇	#466
prime radical	C₂×#1157	D₄
hypercenter	#1381	D₇
hypocenter	C₇	#13496
LCS	G ▷ C₂²×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂²

#89655

commutator, cosocle

C₂²×C₇×C₈

C₂³

Frattini

C₂²×C₈

C₂²×D₇

C₂

C₂²×C₇

D₄

D₇

C₇

G ▷ C₂²×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
2. C₂×#89300	C₂	semidirect
4. C₇×#1381	C₂	semidirect
5. #11725538	C₂	semidirect
39. #1381	D₇	semidirect
168. C₇	#13496	semidirect

H ≅ G/N

type

2. C₂×#89300

C₂

semidirect

4. C₇×#1381

C₂

semidirect

5. #11725538

C₂

semidirect

39. #1381

D₇

semidirect

168. C₇

#13496

semidirect

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

subgroup	index
#11729163	2
C₂×#89300	2
#11728475	2
C₇×#1381	2
#11725538	2
#13496	7

subgroup

index

Subgroups — #13273676

Conjugacy classes of subgroups

Complements

Maximal subgroups