Group #15781028 — subgroups

Subgroups — #15781028

Order 3328.

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

normal subgroups only

This group has 294 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	#13116622	C₂	1	2	yes	maximal
2	C₂×#11718055	C₂	1	1	yes	maximal, characteristic
3	#13114715	C₂	1	1	yes	maximal, characteristic
4	#13112944	C₂	1	2	yes	maximal
5	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#107	C₂²	1	2	yes	—
7	C₂×C₁₃×#106	—	2	1	no	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#382	—	2	1	no	—
10	C₂×#88059	C₂²	1	1	yes	characteristic
11	#11718767	C₂²	1	1	yes	characteristic
12	#11718762	—	2	1	no	—
13	C₂×#88052	C₂²	1	2	yes	—
14	#11718115	—	2	1	no	—
15	#11718082	—	2	2	no	—
16	#11718081	—	2	2	no	—
17	#11718055	C₄	1	4	yes	complemented
18	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
21	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
22	#88180	—	2	1	no	—
23	C₂×#6099	—	2	1	no	—
24	C₁₃×#107	—	2	2	no	—
25	C₁₃×#107	—	2	2	no	—
26	C₁₃×#106	—	4	1	no	—
27	C₁₃×#106	—	4	1	no	—
28	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₃	Q₈	1	2	yes	—
30	C₁₃×#95	—	4	2	no	—
31	#88059	—	2	1	no	—
32	#88059	Q₈	1	2	yes	—
33	#88058	D₄	1	2	yes	—
34	#88057	—	2	1	no	—
35	#88055	—	4	1	no	—
36	C₈×Dic₁₃	—	4	1	no	—
37	#88052	—	2	2	no	—
38	#88052	C₂×C₄	1	4	yes	—
39	C₂×#6090	—	2	2	no	—
40	#13180	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	—	2	1	no	—
42	C₂²×C₄×C₁₃	—	4	2	no	—
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
46	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
47	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
48	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
53	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
54	C₁₃×C₄⋊C₄	—	4	2	no	—
55	C₁₃×C₄⋊C₄	—	4	2	no	—
56	C₄²×C₁₃	—	2	1	no	—
57	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
58	C₂²⋊Dic₁₃	—	4	1	no	—
59	C₄⋊Dic₁₃	—	2	2	no	—
60	C₄×Dic₁₃	—	4	1	no	—
61	#6099	—	2	2	no	—
62	#6099	—	4	1	no	—
63	C₂×#1055	—	4	1	no	—
64	#6090	—	2	4	no	—
65	#2159	—	13	2	no	—
66	C₂×#364	—	13	1	no	—
67	#1846	D₁₃	1	1	yes	characteristic
68	#1772	—	13	1	no	—
69	#1387	—	13	2	no	—
70	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	4	2	no	—
73	C₂×C₄×C₁₃	—	8	4	no	—
74	C₂×C₄×C₁₃	—	8	2	no	—
75	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
81	C₂×Dic₁₃	—	4	2	no	—
82	C₈×C₁₃	#107	1	2	yes	—
83	C₈×C₁₃	#107	1	2	yes	—
84	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
85	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₁₃	#125	1	1	yes	characteristic
87	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
88	#1055	—	4	2	no	—
89	C₂×M₃₂	—	13	2	no	—
90	C₂×#108	—	13	1	no	—
91	#427	—	13	1	no	—
92	C₂×#107	C₂×D₁₃	1	2	yes	—
93	C₂×#106	—	2	1	no	—
94	#405	—	26	1	no	—
95	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
96	#382	—	2	1	no	—
97	#367	—	26	2	no	—
98	#366	—	26	2	no	—
99	#364	—	13	4	no	—
100	#339	—	26	1	no	—
101	C₂²×C₁₃	—	2	2	no	—
102	C₂²×C₁₃	#428	1	1	yes	characteristic
103	C₂²×C₁₃	#459	1	1	yes	characteristic
104	C₂²×C₁₃	#477	1	1	yes	characteristic
105	C₄×C₁₃	—	4	2	no	—
106	C₄×C₁₃	—	16	2	no	—
107	C₄×C₁₃	#389	1	1	yes	characteristic
108	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
109	C₄×C₁₃	#462	1	1	yes	characteristic
110	C₄×C₁₃	#474	1	1	yes	characteristic
111	Dic₁₃	—	8	2	no	—
112	C₂×M₁₆	—	26	1	no	—
113	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
115	#117	—	26	1	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
118	M₃₂	—	13	4	no	—
119	M₃₂	—	26	2	no	—
120	C₂×C₁₆	—	26	2	no	—
121	#108	—	13	2	no	—
122	#108	—	26	1	no	—
123	#107	—	2	2	no	—
124	#107	—	2	2	no	—
125	#107	—	13	2	no	—
126	#106	—	4	1	no	—
127	#106	—	4	1	no	—
128	#106	—	26	1	no	—
129	#97	—	52	1	no	—
130	C₄×C₈	—	52	1	no	—
131	C₄×C₈	Dic₂₆	1	2	yes	—
132	C₄×C₈	C₄×D₁₃	1	2	yes	—
133	#95	—	4	2	no	—
134	C₂×C₁₃	—	2	1	no	—
135	C₂×C₁₃	—	2	1	no	—
136	C₂×C₁₃	#1947	1	1	yes	characteristic
137	C₂×C₁₃	#2212	1	2	yes	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
141	M₁₆	—	26	2	no	—
142	M₁₆	—	52	1	no	—
143	C₂×C₈	—	52	1	no	—
144	C₂×C₈	#6101	1	2	yes	—
145	C₂×C₈	#6104	1	1	yes	characteristic
146	C₂×C₈	#6104	1	1	yes	characteristic
147	C₂×C₈	#6107	1	2	yes	—
148	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
149	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
150	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
151	C₂×C₈	#6129	1	1	yes	characteristic
152	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
153	C₂×C₈	#6131	1	1	yes	characteristic
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	26	2	no	—
157	C₂²⋊C₄	—	52	1	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	52	1	no	—
160	C₄²	#6101	1	2	yes	—
161	C₁₆	—	26	4	no	—
162	C₁₃	#13180	1	1	yes	characteristic, complemented, Sylow
163	C₂³	#88186	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	8	4	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	52	2	no	—
169	C₂×C₄	C₂×#6101	1	1	yes	characteristic
170	C₂×C₄	C₂×#6104	1	1	yes	characteristic
171	C₂×C₄	#88192	1	1	yes	characteristic
172	C₂×C₄	#88196	1	1	yes	characteristic
173	C₂×C₄	C₂×#6107	1	1	yes	characteristic
174	C₂×C₄	#88200	1	1	yes	characteristic
175	C₈	—	52	2	no	—
176	C₈	#88047	1	2	yes	—
177	C₈	#88047	1	2	yes	—
178	C₈	#88141	1	1	yes	characteristic
179	C₈	#88142	1	1	yes	characteristic
180	C₈	#88144	1	1	yes	characteristic
181	C₈	#88147	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	#11718625	1	1	yes	characteristic
184	C₂²	#11718733	1	1	yes	characteristic
185	C₂²	#11718752	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	16	2	no	—
188	C₄	—	104	2	no	—
189	C₄	C₂×#88047	1	1	yes	characteristic
190	C₄	#11718634	1	1	yes	characteristic
191	C₄	#11718748	1	1	yes	characteristic
192	C₄	#11718751	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#13115592	1	1	yes	characteristic
196	C₂	#13116902	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13116622

C₂

yes

maximal

C₂×#11718055

C₂

yes

maximal, characteristic

#13114715

C₂

yes

maximal, characteristic

#13112944

C₂

yes

maximal

C₁₃×#1846

C₂

yes

maximal, characteristic

C₂×C₁₃×#107

C₂²

yes

—

C₂×C₁₃×#106

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#382

—

C₂×#88059

C₂²

yes

characteristic

#11718767

C₂²

yes

characteristic

#11718762

—

C₂×#88052

C₂²

yes

—

—

—

—

C₄

yes

complemented

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

D₄

yes

characteristic

#88180

—

C₂×#6099

—

C₁₃×#107

—

C₁₃×#107

—

C₁₃×#106

—

C₁₃×#106

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

C₁₃×#95

—

#88059

—

#88059

Q₈

yes

—

#88058

D₄

yes

—

#88057

—

#88055

—

C₈×Dic₁₃

—

#88052

—

#88052

C₂×C₄

yes

—

C₂×#6090

—

#13180

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

D₈

yes

characteristic

C₂×C₈×C₁₃

D₈

yes

characteristic

C₂×C₈×C₁₃

Q₁₆

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

C₂×C₈×C₁₃

yes

C₂×C₈×C₁₃

yes

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

C₄⋊C₄

yes

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

—

—

—

D₁₃

yes

characteristic

#1772

—

#1387

—

C₂³×C₁₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₃

#137

yes

characteristic

C₂×Dic₁₃

—

C₈×C₁₃

#107

yes

—

C₈×C₁₃

#107

yes

—

C₈×C₁₃

C₄×D₄

yes

characteristic

C₈×C₁₃

C₄×Q₈

yes

characteristic

C₈×C₁₃

#125

yes

characteristic

C₈×C₁₃

C₄⋊Q₈

yes

characteristic

#1055

—

C₂×M₃₂

—

C₂×#108

—

#427

—

C₂×#107

C₂×D₁₃

yes

—

C₂×#106

—

#405

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

—

—

—

100

#339

—

101

C₂²×C₁₃

—

102

C₂²×C₁₃

#428

yes

characteristic

103

C₂²×C₁₃

#459

yes

characteristic

104

C₂²×C₁₃

#477

yes

characteristic

105

C₄×C₁₃

—

106

C₄×C₁₃

—

107

C₄×C₁₃

#389

yes

characteristic

108

C₄×C₁₃

C₂×#107

yes

characteristic

109

C₄×C₁₃

#462

yes

characteristic

110

C₄×C₁₃

#474

yes

characteristic

111

Dic₁₃

—

112

C₂×M₁₆

—

113

C₂²×C₈

C₂²⋊D₁₃

yes

characteristic

114

C₂²×C₈

C₂²×D₁₃

yes

characteristic

115

#117

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄²

C₂²⋊D₁₃

yes

characteristic

118

M₃₂

—

119

M₃₂

—

120

C₂×C₁₆

—

121

#108

—

122

#108

—

123

#107

—

124

#107

—

125

#107

—

126

#106

—

127

#106

—

128

#106

—

129

#97

—

130

C₄×C₈

—

131

C₄×C₈

Dic₂₆

yes

—

132

C₄×C₈

C₄×D₁₃

yes

—

133

#95

—

134

C₂×C₁₃

—

135

C₂×C₁₃

—

136

C₂×C₁₃

#1947

yes

characteristic

137

C₂×C₁₃

#2212

yes

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

141

M₁₆

—

142

M₁₆

—

143

C₂×C₈

—

144

C₂×C₈

#6101

yes

—

145

C₂×C₈

#6104

yes

characteristic

146

C₂×C₈

#6104

yes

characteristic

147

C₂×C₈

#6107

yes

—

148

C₂×C₈

C₂×Dic₂₆

yes

characteristic

149

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

150

C₂×C₈

D₄×D₁₃

yes

characteristic

151

C₂×C₈

#6129

yes

characteristic

152

C₂×C₈

Q₈×D₁₃

yes

characteristic

153

C₂×C₈

#6131

yes

characteristic

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₂²⋊C₄

—

158

C₄²

—

159

C₄²

—

160

C₄²

#6101

yes

—

161

C₁₆

—

162

C₁₃

#13180

yes

characteristic, complemented, Sylow

163

C₂³

#88186

yes

characteristic

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₂×#6101

yes

characteristic

170

C₂×C₄

C₂×#6104

yes

characteristic

171

C₂×C₄

#88192

yes

characteristic

172

C₂×C₄

#88196

yes

characteristic

173

C₂×C₄

C₂×#6107

yes

characteristic

174

C₂×C₄

#88200

yes

characteristic

175

C₈

—

176

C₈

#88047

yes

—

177

C₈

#88047

yes

—

178

C₈

#88141

yes

characteristic

179

C₈

#88142

yes

characteristic

180

C₈

#88144

yes

characteristic

181

C₈

#88147

yes

characteristic

182

C₂²

—

183

C₂²

#11718625

yes

characteristic

184

C₂²

#11718733

yes

characteristic

185

C₂²

#11718752

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

104

—

189

C₄

C₂×#88047

yes

characteristic

190

C₄

#11718634

yes

characteristic

191

C₄

#11718748

yes

characteristic

192

C₄

#11718751

yes

characteristic

193

C₂

—

194

C₂

—

195

C₂

#13115592

yes

characteristic

196

C₂

#13116902

yes

—

	subgroup	G/H
center	C₂²	#11718733
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1846	C₂
socle	C₂²×C₁₃	#459
prime radical	C₂³×C₁₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1846	D₁₃
hypocenter	C₁₃	#13180
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂²

#11718733

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₁₃×#1846

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
17. #11718055	C₄	semidirect
162. C₁₃	#13180	semidirect

H ≅ G/N

type

17. #11718055

C₄

semidirect

162. C₁₃

#13180

semidirect

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

subgroup	index
#13116622	2
C₂×#11718055	2
#13114715	2
#13112944	2
C₁₃×#1846	2
#13180	13

subgroup

index

C₁₃×#1846

Subgroups — #15781028

Conjugacy classes of subgroups

Complements

Maximal subgroups