Group #15767724 — subgroups

Subgroups — #15767724

Order 3328.

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

normal subgroups only

This group has 346 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	#13114977	C₂	1	1	yes	maximal, characteristic, complemented
2	#13113504	C₂	1	2	yes	maximal
3	#13112694	C₂	1	2	yes	maximal, complemented
4	#13112681	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1961	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#102	—	2	2	no	—
8	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
9	C₁₃×#382	C₂²	1	1	yes	characteristic
10	#11718586	C₂²	1	2	yes	—
11	#11718585	C₂²	1	2	yes	—
12	#11718144	—	2	2	no	—
13	#11718136	—	2	4	no	—
14	C₂²×D₄×C₁₃	—	2	1	no	—
15	C₂²×D₄×C₁₃	—	2	2	no	—
16	C₂²×C₈×C₁₃	D₄	1	2	yes	—
17	C₁₃×#127	—	2	1	no	—
18	C₁₃×#127	—	2	2	no	—
19	C₁₃×#127	D₄	1	2	yes	—
20	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×#6099	—	2	2	no	—
23	C₁₃×#102	—	4	4	no	—
24	C₁₃×#97	—	2	2	no	—
25	C₁₃×#95	—	2	2	no	—
26	C₂²⋊#1055	—	2	2	no	—
27	C₄⋊#1055	—	2	2	no	—
28	C₄⋊#1055	—	2	2	no	—
29	C₄⋊#1055	—	2	2	no	—
30	C₄×#1055	—	2	2	no	—
31	#10634	—	13	1	no	maximal, Sylow
32	C₂⁴×C₁₃	—	4	1	no	—
33	C₂×D₄×C₁₃	—	2	2	no	—
34	C₂×D₄×C₁₃	—	2	4	no	—
35	C₂×D₄×C₁₃	—	4	1	no	—
36	C₂×D₄×C₁₃	—	4	2	no	—
37	C₂×D₄×C₁₃	—	4	2	no	—
38	C₂×D₄×C₁₃	—	4	2	no	—
39	C₂×D₄×C₁₃	—	8	4	no	—
40	C₂²×C₄×C₁₃	—	4	1	no	—
41	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	#41	1	2	yes	—
43	C₂×C₈×C₁₃	—	2	2	no	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	2	2	no	—
46	C₁₃×C₄⋊C₄	—	4	2	no	—
47	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₁₃	#41	1	1	yes	characteristic
49	C₄²×C₁₃	#41	1	2	yes	—
50	#6099	—	4	2	no	—
51	#6099	—	4	2	no	—
52	C₂×#1055	—	4	2	no	—
53	C₂×#1055	—	4	2	no	—
54	#1961	D₁₃	1	1	yes	characteristic
55	#1838	—	13	1	no	—
56	#1545	—	13	2	no	—
57	#1286	—	13	2	no	—
58	#1281	—	13	1	no	—
59	C₂³×C₁₃	—	4	2	no	—
60	C₂³×C₁₃	—	8	4	no	—
61	C₂³×C₁₃	—	8	1	no	—
62	C₂³×C₁₃	—	8	1	no	—
63	C₂³×C₁₃	#124	1	1	yes	characteristic
64	D₄×C₁₃	—	4	4	no	—
65	D₄×C₁₃	—	8	4	no	—
66	D₄×C₁₃	—	8	4	no	—
67	D₄×C₁₃	—	8	2	no	—
68	C₂×C₄×C₁₃	—	4	2	no	—
69	C₂×C₄×C₁₃	—	8	2	no	—
70	C₂×C₄×C₁₃	—	8	1	no	—
71	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
72	C₂×C₄×C₁₃	#123	1	2	yes	—
73	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
74	C₂×C₄×C₁₃	#126	1	2	yes	—
75	C₂×C₄×C₁₃	#135	1	2	yes	—
76	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
77	C₈×C₁₃	—	4	2	no	—
78	#1055	—	8	2	no	—
79	#1055	—	8	2	no	—
80	C₂×#127	C₂×D₁₃	1	1	yes	characteristic
81	#431	—	13	2	no	—
82	#423	—	13	2	no	—
83	C₂×#102	—	2	2	no	—
84	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
85	#382	C₂×D₁₃	1	1	yes	characteristic
86	#341	—	26	2	no	—
87	#331	—	26	4	no	—
88	C₂²×C₁₃	—	2	2	no	—
89	C₂²×C₁₃	—	8	2	no	—
90	C₂²×C₁₃	—	8	4	no	—
91	C₂²×C₁₃	—	16	2	no	—
92	C₂²×C₁₃	—	16	2	no	—
93	C₂²×C₁₃	#397	1	1	yes	characteristic
94	C₂²×C₁₃	#488	1	1	yes	characteristic
95	C₂²×C₁₃	#490	1	1	yes	characteristic
96	C₄×C₁₃	—	2	2	no	—
97	C₄×C₁₃	—	2	2	no	—
98	C₄×C₁₃	—	16	1	no	—
99	C₄×C₁₃	#463	1	2	yes	—
100	C₄×C₁₃	#482	1	2	yes	—
101	C₂²×D₄	—	2	2	no	—
102	C₂²×D₄	—	2	1	no	—
103	C₂×M₁₆	—	26	2	no	—
104	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
105	#127	—	2	1	no	—
106	#127	—	2	2	no	—
107	#127	D₅₂	1	2	yes	—
108	C₂×C₄⋊C₄	—	2	1	no	—
109	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
110	C₄⋊C₈	—	26	2	no	—
111	C₄⋊C₈	—	26	2	no	—
112	C₄⋊C₈	—	26	2	no	—
113	#102	—	4	4	no	—
114	C₂²⋊C₈	—	26	2	no	—
115	#97	—	2	2	no	—
116	C₄×C₈	—	26	2	no	—
117	#95	—	2	2	no	—
118	C₂×C₁₃	—	2	1	no	—
119	C₂×C₁₃	—	2	1	no	—
120	C₂×C₁₃	—	16	2	no	—
121	C₂×C₁₃	#2073	1	1	yes	characteristic
122	C₂×C₁₃	#2081	1	2	yes	—
123	C₂⁴	—	4	1	no	—
124	C₂×D₄	—	2	4	no	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	—	4	2	no	—
127	C₂×D₄	—	4	2	no	—
128	C₂×D₄	—	4	1	no	—
129	C₂×D₄	—	4	2	no	—
130	C₂×D₄	—	8	4	no	—
131	C₂²×C₄	—	4	1	no	—
132	C₂²×C₄	#6129	1	2	yes	—
133	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
134	M₁₆	—	52	2	no	—
135	M₁₆	—	52	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	52	2	no	—
140	C₂×C₈	—	52	2	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄²	C₂×D₅₂	1	1	yes	characteristic
143	C₄²	#6127	1	1	yes	characteristic
144	C₄²	#6129	1	2	yes	—
145	C₁₃	#10634	1	1	yes	characteristic, complemented, Sylow
146	C₂³	—	4	2	no	—
147	C₂³	—	8	4	no	—
148	C₂³	—	8	1	no	—
149	C₂³	—	8	1	no	—
150	C₂³	#88190	1	1	yes	characteristic
151	D₄	—	4	4	no	—
152	D₄	—	8	4	no	—
153	D₄	—	8	4	no	—
154	D₄	—	8	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	1	no	—
158	C₂×C₄	#88133	1	2	yes	—
159	C₂×C₄	#88140	1	2	yes	—
160	C₂×C₄	#88184	1	1	yes	characteristic
161	C₂×C₄	#88186	1	2	yes	—
162	C₂×C₄	#88189	1	1	yes	characteristic
163	C₂×C₄	#88204	1	2	yes	—
164	C₈	—	4	2	no	—
165	C₈	—	104	2	no	—
166	C₈	—	104	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	8	2	no	—
169	C₂²	—	8	4	no	—
170	C₂²	—	16	2	no	—
171	C₂²	—	16	2	no	—
172	C₂²	#11718614	1	1	yes	characteristic
173	C₂²	#11718717	1	1	yes	characteristic
174	C₂²	#11718732	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	16	1	no	—
178	C₄	#11718677	1	2	yes	—
179	C₄	#11718696	1	2	yes	—
180	C₂	—	2	1	no	—
181	C₂	—	2	1	no	—
182	C₂	—	16	2	no	—
183	C₂	#13116249	1	1	yes	characteristic
184	C₂	#13116293	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13114977

C₂

yes

maximal, characteristic, complemented

#13113504

C₂

yes

maximal

#13112694

C₂

yes

maximal, complemented

#13112681

C₂

yes

maximal, characteristic, complemented

C₁₃×#1961

C₂

yes

maximal, characteristic

C₂×C₁₃×#127

C₂²

yes

characteristic

C₂×C₁₃×#102

—

C₂×C₁₃×#97

C₂²

yes

characteristic

C₁₃×#382

C₂²

yes

characteristic

#11718586

C₂²

yes

—

#11718585

C₂²

yes

—

#11718144

—

#11718136

—

C₂²×D₄×C₁₃

—

C₂²×D₄×C₁₃

—

C₂²×C₈×C₁₃

D₄

yes

—

C₁₃×#127

—

C₁₃×#127

—

C₁₃×#127

D₄

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6099

—

C₁₃×#102

—

C₁₃×#97

—

C₁₃×#95

—

C₂²⋊#1055

—

—

—

—

—

—

C₂⁴×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

C₄²×C₁₃

yes

C₄²×C₁₃

yes

—

—

—

—

—

D₁₃

yes

—

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

#124

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

#123

yes

characteristic

C₂×C₄×C₁₃

#126

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₈×C₁₃

—

#1055

—

#1055

—

C₂×#127

C₂×D₁₃

yes

—

—

—

C₂×#97

C₂×D₁₃

yes

characteristic

#382

C₂×D₁₃

yes

characteristic

#341

—

#331

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#463

yes

—

100

C₄×C₁₃

#482

yes

—

101

C₂²×D₄

—

102

C₂²×D₄

—

103

C₂×M₁₆

—

104

C₂²×C₈

C₂²⋊D₁₃

yes

—

105

#127

—

106

#127

—

107

#127

D₅₂

yes

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄²

C₂²×D₁₃

yes

characteristic

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

#102

—

114

C₂²⋊C₈

—

115

#97

—

116

C₄×C₈

—

117

#95

—

118

C₂×C₁₃

—

119

C₂×C₁₃

—

120

C₂×C₁₃

—

121

C₂×C₁₃

#2073

yes

characteristic

122

C₂×C₁₃

#2081

yes

—

123

C₂⁴

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂²×C₄

—

132

C₂²×C₄

#6129

yes

—

133

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

134

M₁₆

—

135

M₁₆

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₄⋊C₄

—

142

C₄²

C₂×D₅₂

yes

characteristic

143

C₄²

#6127

yes

characteristic

144

C₄²

#6129

yes

—

145

C₁₃

#10634

yes

characteristic, complemented, Sylow

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

—

150

C₂³

#88190

yes

characteristic

151

D₄

—

152

D₄

—

153

D₄

—

154

D₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

#88133

yes

—

159

C₂×C₄

#88140

yes

—

160

C₂×C₄

#88184

yes

characteristic

161

C₂×C₄

#88186

yes

—

162

C₂×C₄

#88189

yes

characteristic

163

C₂×C₄

#88204

yes

—

164

C₈

—

165

C₈

—

104

—

166

C₈

—

104

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

#11718614

yes

characteristic

173

C₂²

#11718717

yes

characteristic

174

C₂²

#11718732

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

#11718677

yes

—

179

C₄

#11718696

yes

—

180

C₂

—

181

C₂

—

182

C₂

—

183

C₂

#13116249

yes

characteristic

184

C₂

#13116293

yes

—

	subgroup	G/H
center	C₂²	#11718614
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1961	C₂
socle	C₂²×C₁₃	#397
prime radical	C₂×C₁₃×#127	C₂²
hypercenter	#1961	D₁₃
hypocenter	C₁₃	#10634
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1961

subgroup

G/H

center

C₂²

#11718614

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1961

C₂

socle

C₂²×C₁₃

#397

prime radical

C₂×C₁₃×#127

C₂²

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1961

N	H ≅ G/N	type
1. #13114977	C₂	semidirect
3. #13112694	C₂	semidirect
4. #13112681	C₂	semidirect
145. C₁₃	#10634	semidirect

H ≅ G/N

type

1. #13114977

C₂

semidirect

3. #13112694

C₂

semidirect

4. #13112681

C₂

semidirect

145. C₁₃

#10634

semidirect

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

subgroup	index
#13114977	2
#13113504	2
#13112694	2
#13112681	2
C₁₃×#1961	2
#10634	13

subgroup

index

C₁₃×#1961

Subgroups — #15767724

Conjugacy classes of subgroups

Complements

Maximal subgroups