Group #15749521 — subgroups

Subgroups — #15749521

Order 3328.

352 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 107 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	#13114816	C₂	1	2	yes	maximal
2	C₁₃×#1817	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
4	C₂×C₁₃×C₂²⋊C₈	C₄	1	2	yes	—
5	C₂×C₂²⋊#1055	—	2	2	no	—
6	C₂×C₄⋊#1055	—	2	2	no	—
7	C₁₃×#336	—	2	2	no	—
8	#11718116	C₄	1	4	yes	—
9	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
10	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
11	C₂³×C₄×C₁₃	Q₈	1	1	yes	characteristic
12	C₂²×C₈×C₁₃	—	2	2	no	—
13	C₂×C₄²×C₁₃	—	4	2	no	—
14	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₁₃	D₄	1	2	yes	—
16	C₂²×#1055	—	2	4	no	—
17	C₁₃×C₂²⋊C₈	—	4	2	no	—
18	C₂²⋊#1055	—	4	2	no	—
19	C₂²⋊#1055	—	4	2	no	—
20	C₄⋊#1055	—	4	4	no	—
21	#7044	—	13	1	no	maximal, Sylow
22	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₃	—	2	2	no	—
24	C₂²×C₄×C₁₃	—	2	4	no	—
25	C₂²×C₄×C₁₃	—	4	1	no	—
26	C₂²×C₄×C₁₃	—	4	1	no	—
27	C₂²×C₄×C₁₃	—	4	2	no	—
28	C₂²×C₄×C₁₃	—	4	1	no	—
29	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
31	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
35	C₂²×C₄×C₁₃	Q₁₆	1	1	yes	characteristic
36	C₂×C₈×C₁₃	—	4	4	no	—
37	C₂×C₈×C₁₃	—	4	2	no	—
38	C₄²×C₁₃	—	2	4	no	—
39	C₄²×C₁₃	—	4	2	no	—
40	C₂×#1055	—	4	4	no	—
41	C₂×#1055	—	4	4	no	—
42	C₂×#1055	—	4	4	no	—
43	#1817	D₁₃	1	1	yes	characteristic
44	#1797	—	13	2	no	—
45	C₂³×C₁₃	—	2	1	no	—
46	C₂³×C₁₃	—	2	2	no	—
47	C₂³×C₁₃	—	2	1	no	—
48	C₂³×C₁₃	—	2	2	no	—
49	C₂³×C₁₃	#95	1	1	yes	characteristic
50	C₂³×C₁₃	#106	1	1	yes	characteristic
51	C₂³×C₁₃	#107	1	1	yes	characteristic
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	2	no	—
57	C₂×C₄×C₁₃	—	2	1	no	—
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	4	1	no	—
60	C₂×C₄×C₁₃	—	4	2	no	—
61	C₂×C₄×C₁₃	—	4	1	no	—
62	C₂×C₄×C₁₃	—	4	4	no	—
63	C₂×C₄×C₁₃	—	4	4	no	—
64	C₂×C₄×C₁₃	#99	1	2	yes	—
65	C₂×C₄×C₁₃	#102	1	2	yes	—
66	C₂×C₄×C₁₃	#103	1	2	yes	—
67	C₂×C₄×C₁₃	#108	1	4	yes	—
68	C₈×C₁₃	—	8	2	no	—
69	#1055	—	8	4	no	—
70	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
71	C₂×C₄⋊C₈	—	26	2	no	—
72	C₂×C₂²⋊C₈	—	26	2	no	—
73	C₂×C₂²⋊C₈	Dic₁₃	1	2	yes	—
74	#336	—	2	2	no	—
75	#336	—	13	4	no	—
76	C₂²×C₁₃	—	2	1	no	—
77	C₂²×C₁₃	—	2	1	no	—
78	C₂²×C₁₃	—	2	2	no	—
79	C₂²×C₁₃	—	2	2	no	—
80	C₂²×C₁₃	—	4	2	no	—
81	C₂²×C₁₃	—	4	1	no	—
82	C₂²×C₁₃	—	4	1	no	—
83	C₂²×C₁₃	H(3,ℤ/4)	1	1	yes	characteristic
84	C₂²×C₁₃	#340	1	1	yes	characteristic
85	C₂²×C₁₃	#341	1	2	yes	—
86	C₂²×C₁₃	#342	1	1	yes	characteristic
87	C₂²×C₁₃	#344	1	2	yes	—
88	C₄×C₁₃	—	2	2	no	—
89	C₄×C₁₃	—	2	2	no	—
90	C₄×C₁₃	—	2	4	no	—
91	C₄×C₁₃	—	2	2	no	—
92	C₄×C₁₃	—	4	1	no	—
93	C₂³×C₄	Dic₂₆	1	1	yes	characteristic
94	C₂³×C₄	D₅₂	1	1	yes	characteristic
95	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
96	C₂²×C₈	—	2	2	no	—
97	C₂²×C₈	—	26	4	no	—
98	C₂×C₄²	—	4	2	no	—
99	C₂×C₄²	C₄×D₁₃	1	2	yes	—
100	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
101	C₄⋊C₈	—	52	4	no	—
102	C₂²⋊C₈	—	4	2	no	—
103	C₂²⋊C₈	—	52	2	no	—
104	C₂²⋊C₈	—	52	2	no	—
105	C₂×C₁₃	—	4	1	no	—
106	C₂×C₁₃	—	4	1	no	—
107	C₂×C₁₃	#1287	1	2	yes	—
108	C₂×C₁₃	#1289	1	1	yes	characteristic
109	C₂×C₁₃	#1298	1	1	yes	characteristic
110	C₂×C₁₃	#1301	1	2	yes	—
111	C₂×C₁₃	#1306	1	1	yes	characteristic
112	C₂⁴	C₄⋊Dic₁₃	1	1	yes	characteristic
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	—	2	4	no	—
115	C₂²×C₄	—	4	1	no	—
116	C₂²×C₄	—	4	1	no	—
117	C₂²×C₄	—	4	1	no	—
118	C₂²×C₄	—	4	2	no	—
119	C₂²×C₄	#6095	1	2	yes	—
120	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
121	C₂²×C₄	Dic₅₂	1	1	yes	characteristic
122	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
123	C₂²×C₄	#6101	1	2	yes	—
124	C₂²×C₄	#6103	1	2	yes	—
125	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
126	C₂×C₈	—	4	4	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	—	52	4	no	—
129	C₂×C₈	—	52	4	no	—
130	C₂×C₈	—	52	4	no	—
131	C₄²	—	2	4	no	—
132	C₄²	—	4	2	no	—
133	C₁₃	#7044	1	1	yes	characteristic, complemented, Sylow
134	C₂³	—	2	1	no	—
135	C₂³	—	2	2	no	—
136	C₂³	—	2	1	no	—
137	C₂³	—	2	2	no	—
138	C₂³	#88057	1	1	yes	characteristic
139	C₂³	#88058	1	1	yes	characteristic
140	C₂³	#88071	1	1	yes	characteristic
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	#88056	1	2	yes	—
154	C₂×C₄	#88061	1	2	yes	—
155	C₂×C₄	#88062	1	4	yes	—
156	C₂×C₄	#88074	1	2	yes	—
157	C₈	—	8	2	no	—
158	C₈	—	104	4	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	2	no	—
164	C₂²	—	4	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	#11718118	1	1	yes	characteristic
167	C₂²	#11718125	1	2	yes	—
168	C₂²	#11718143	1	1	yes	characteristic
169	C₂²	#11718149	1	1	yes	characteristic
170	C₂²	#11718150	1	2	yes	—
171	C₄	—	2	2	no	—
172	C₄	—	2	2	no	—
173	C₄	—	2	4	no	—
174	C₄	—	2	2	no	—
175	C₄	—	4	1	no	—
176	C₂	—	4	1	no	—
177	C₂	—	4	1	no	—
178	C₂	#13112699	1	2	yes	—
179	C₂	#13112704	1	1	yes	characteristic
180	C₂	#13112723	1	1	yes	characteristic
181	C₂	#13112732	1	2	yes	—
182	C₂	#13112744	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114816

C₂

yes

maximal

C₁₃×#1817

C₂

yes

maximal, characteristic

C₂²×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×C₂²⋊C₈

C₄

yes

—

C₂×C₂²⋊#1055

—

C₂×C₄⋊#1055

—

C₁₃×#336

—

#11718116

C₄

yes

—

C₂³×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₁₃

D₄

yes

characteristic

C₂³×C₄×C₁₃

Q₈

yes

characteristic

C₂²×C₈×C₁₃

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂²×#1055

—

C₁₃×C₂²⋊C₈

—

C₂²⋊#1055

—

C₂²⋊#1055

—

C₄⋊#1055

—

#7044

—

maximal, Sylow

C₂⁴×C₁₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₄²

yes

characteristic

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

D₈

yes

characteristic

C₂²×C₄×C₁₃

SD₁₆

yes

—

C₂²×C₄×C₁₃

Q₁₆

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

—

—

—

D₁₃

yes

characteristic

#1797

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

yes

C₂³×C₁₃

yes

C₂³×C₁₃

yes

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#99

yes

—

C₂×C₄×C₁₃

#102

yes

—

C₂×C₄×C₁₃

#103

yes

—

C₂×C₄×C₁₃

#108

yes

—

C₈×C₁₃

—

#1055

—

C₂²×C₄²

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

Dic₁₃

yes

—

#336

—

#336

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

#342

yes

characteristic

C₂²×C₁₃

#344

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₂³×C₄

Dic₂₆

yes

characteristic

C₂³×C₄

D₅₂

yes

characteristic

C₂³×C₄

C₂×Dic₁₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

C₄×D₁₃

yes

—

100

C₂×C₄²

C₂²⋊D₁₃

yes

—

101

C₄⋊C₈

—

102

C₂²⋊C₈

—

103

C₂²⋊C₈

—

104

C₂²⋊C₈

—

105

C₂×C₁₃

—

106

C₂×C₁₃

—

107

C₂×C₁₃

#1287

yes

—

108

C₂×C₁₃

#1289

yes

characteristic

109

C₂×C₁₃

#1298

yes

characteristic

110

C₂×C₁₃

#1301

yes

—

111

C₂×C₁₃

#1306

yes

characteristic

112

C₂⁴

C₄⋊Dic₁₃

yes

characteristic

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

#6095

yes

—

120

C₂²×C₄

D₁₀₄

yes

characteristic

121

C₂²×C₄

Dic₅₂

yes

characteristic

122

C₂²×C₄

C₄×Dic₁₃

yes

characteristic

123

C₂²×C₄

#6101

yes

—

124

C₂²×C₄

#6103

yes

—

125

C₂²×C₄

C₂²⋊Dic₁₃

yes

characteristic

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₄²

—

132

C₄²

—

133

C₁₃

#7044

yes

characteristic, complemented, Sylow

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

#88057

yes

characteristic

139

C₂³

#88058

yes

characteristic

140

C₂³

#88071

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

#88056

yes

—

154

C₂×C₄

#88061

yes

—

155

C₂×C₄

#88062

yes

—

156

C₂×C₄

#88074

yes

—

157

C₈

—

158

C₈

—

104

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

#11718118

yes

characteristic

167

C₂²

#11718125

yes

—

168

C₂²

#11718143

yes

characteristic

169

C₂²

#11718149

yes

characteristic

170

C₂²

#11718150

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₂

—

177

C₂

—

178

C₂

#13112699

yes

—

179

C₂

#13112704

yes

characteristic

180

C₂

#13112723

yes

characteristic

181

C₂

#13112732

yes

—

182

C₂

#13112744

yes

characteristic

	subgroup	G/H
center	C₂³	#88071
commutator	C₂²×C₄×C₁₃	C₄²
Frattini	C₂²×C₄²	C₂×D₁₃
Fitting	C₁₃×#1817	C₂
socle	C₂³×C₁₃	#95
prime radical	C₂⁴×C₁₃	C₄⋊C₄
cosocle	C₂²×C₄²×C₁₃	C₂²
hypercenter	#1817	D₁₃
hypocenter	C₁₃	#7044
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1817

subgroup

G/H

center

C₂³

#88071

commutator

C₂²×C₄×C₁₃

C₄²

Frattini

C₂²×C₄²

C₂×D₁₃

Fitting

C₁₃×#1817

C₂

socle

C₂³×C₁₃

#95

prime radical

C₂⁴×C₁₃

C₄⋊C₄

cosocle

C₂²×C₄²×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1817

N	H ≅ G/N	type
133. C₁₃	#7044	semidirect

H ≅ G/N

type

133. C₁₃

#7044

semidirect

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

subgroup	index
#13114816	2
C₁₃×#1817	2
#7044	13

subgroup

index

#13114816

C₁₃×#1817

#7044

Subgroups — #15749521

Conjugacy classes of subgroups

Complements

Maximal subgroups