Group #15749611 — subgroups

Subgroups — #15749611

Order 3328.

204 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 45 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	#13116159	C₂	1	1	yes	maximal, characteristic
2	#13112706	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1322	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₅₂	C₄	1	1	yes	characteristic
5	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
6	C₂×#88056	—	2	1	no	—
7	#11718590	—	2	1	no	—
8	#11718587	—	2	1	no	—
9	C₁₃×C₂²⋊C₁₆	—	2	1	no	—
10	#11718143	C₄	1	1	yes	characteristic
11	C₂²×Dic₂₆	—	2	1	no	—
12	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₃	—	2	1	no	—
16	C₂×Dic₅₂	—	4	1	no	—
17	C₂×Dic₅₂	—	4	2	no	—
18	C₂×C₁₃×C₁₆	—	4	1	no	—
19	C₁₃×C₄⋊C₈	—	4	1	no	—
20	#88071	—	4	1	no	—
21	#88071	—	4	1	no	—
22	C₂²⋊#1055	—	4	1	no	—
23	#88056	—	4	2	no	—
24	C₄⋊#1055	—	4	1	no	—
25	#88043	—	2	2	no	—
26	#7082	—	13	1	no	maximal, Sylow
27	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
30	C₂²×Dic₁₃	—	4	1	no	—
31	C₂²×Dic₁₃	—	4	1	no	—
32	C₂×Dic₂₆	—	2	2	no	—
33	C₂×Dic₂₆	—	4	1	no	—
34	C₂×Dic₂₆	—	4	1	no	—
35	C₂×Dic₂₆	—	4	1	no	—
36	C₂×Dic₂₆	—	4	1	no	—
37	C₂×Dic₂₆	—	4	1	no	—
38	C₂×C₈×C₁₃	—	2	1	no	—
39	C₂×C₈×C₁₃	—	2	1	no	—
40	C₂×C₈×C₁₃	—	4	1	no	—
41	C₂×C₈×C₁₃	—	4	1	no	—
42	C₄²×C₁₃	—	2	1	no	—
43	C₄²×C₁₃	—	2	1	no	—
44	C₄⋊Dic₁₃	—	4	1	no	—
45	C₄⋊Dic₁₃	—	4	1	no	—
46	C₂×#1055	—	4	2	no	—
47	Dic₅₂	—	4	2	no	—
48	Dic₅₂	—	4	2	no	—
49	C₁₃×C₁₆	—	8	1	no	—
50	#2056	—	13	1	no	—
51	#1322	D₁₃	1	1	yes	characteristic
52	#1290	—	13	1	no	—
53	C₂³×C₁₃	#102	1	1	yes	characteristic
54	C₂×C₄×C₁₃	—	2	1	no	—
55	C₂×C₄×C₁₃	—	2	1	no	—
56	C₂×C₄×C₁₃	—	2	1	no	—
57	C₂×C₄×C₁₃	—	2	1	no	—
58	C₂×C₄×C₁₃	—	2	1	no	—
59	C₂×C₄×C₁₃	—	2	1	no	—
60	C₂×C₄×C₁₃	—	4	1	no	—
61	C₂×C₄×C₁₃	#99	1	1	yes	characteristic
62	C₂×C₄×C₁₃	#104	1	1	yes	characteristic
63	C₂×Dic₁₃	—	4	2	no	—
64	C₂×Dic₁₃	—	8	1	no	—
65	C₂×Dic₁₃	—	8	1	no	—
66	C₂×Dic₁₃	—	8	1	no	—
67	C₂×Dic₁₃	—	8	1	no	—
68	C₂×Dic₁₃	—	8	1	no	—
69	Dic₂₆	—	4	2	no	—
70	Dic₂₆	—	4	2	no	—
71	Dic₂₆	—	8	1	no	—
72	Dic₂₆	—	8	1	no	—
73	C₈×C₁₃	—	4	1	no	—
74	C₈×C₁₃	—	4	1	no	—
75	#1055	—	8	2	no	—
76	C₂²×Q₁₆	—	13	1	no	—
77	#432	—	26	1	no	—
78	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
79	C₂×#103	—	26	1	no	—
80	#391	—	26	1	no	—
81	C₂²⋊C₁₆	—	2	1	no	—
82	#340	—	13	1	no	—
83	C₂²×C₁₃	—	2	1	no	—
84	C₂²×C₁₃	—	2	1	no	—
85	C₂²×C₁₃	#327	1	1	yes	characteristic
86	C₂²×C₁₃	#359	1	1	yes	characteristic
87	C₂²×C₁₃	#362	1	1	yes	characteristic
88	C₄×C₁₃	—	2	1	no	—
89	C₄×C₁₃	—	4	1	no	—
90	C₄×C₁₃	—	4	1	no	—
91	C₄×C₁₃	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
92	C₄×C₁₃	#355	1	1	yes	characteristic
93	Dic₁₃	—	8	2	no	—
94	Dic₁₃	—	16	1	no	—
95	C₂²×Q₈	—	26	1	no	—
96	C₂×Q₁₆	—	52	1	no	—
97	C₂×Q₁₆	—	52	2	no	—
98	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
99	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
100	C₂×C₄⋊C₄	—	26	1	no	—
101	C₂×C₄²	D₅₂	1	1	yes	characteristic
102	C₂×C₁₆	—	4	1	no	—
103	C₄⋊C₈	—	4	1	no	—
104	C₄⋊C₈	—	52	1	no	—
105	#103	—	52	2	no	—
106	C₂²⋊C₈	—	52	1	no	—
107	#97	—	26	2	no	—
108	#95	—	52	1	no	—
109	#95	—	52	1	no	—
110	C₂×C₁₃	—	2	1	no	—
111	C₂×C₁₃	—	2	1	no	—
112	C₂×C₁₃	#1340	1	1	yes	characteristic
113	C₂×C₁₃	#1345	1	1	yes	characteristic
114	C₂×C₁₃	#1350	1	1	yes	characteristic
115	C₂×Q₈	—	26	2	no	—
116	C₂×Q₈	—	52	1	no	—
117	C₂×Q₈	—	52	1	no	—
118	C₂×Q₈	—	52	1	no	—
119	C₂×Q₈	—	52	1	no	—
120	C₂×Q₈	—	52	1	no	—
121	C₂²×C₄	—	52	1	no	—
122	C₂²×C₄	—	52	1	no	—
123	C₂²×C₄	#6095	1	1	yes	characteristic
124	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
125	C₂²×C₄	#6103	1	1	yes	characteristic
126	Q₁₆	—	52	2	no	—
127	Q₁₆	—	52	2	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	52	2	no	—
133	C₄⋊C₄	—	52	1	no	—
134	C₄⋊C₄	—	52	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	1	no	—
137	C₁₆	—	8	1	no	—
138	C₁₃	#7082	1	1	yes	characteristic, complemented, Sylow
139	C₂³	#88061	1	1	yes	characteristic
140	Q₈	—	52	2	no	—
141	Q₈	—	52	2	no	—
142	Q₈	—	104	1	no	—
143	Q₈	—	104	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	52	2	no	—
152	C₂×C₄	—	104	1	no	—
153	C₂×C₄	—	104	1	no	—
154	C₂×C₄	—	104	1	no	—
155	C₂×C₄	—	104	1	no	—
156	C₂×C₄	—	104	1	no	—
157	C₂×C₄	#88046	1	1	yes	characteristic
158	C₂×C₄	#88065	1	1	yes	characteristic
159	C₈	—	4	1	no	—
160	C₈	—	4	1	no	—
161	C₈	—	104	2	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#11718062	1	1	yes	characteristic
165	C₂²	#11718102	1	1	yes	characteristic
166	C₂²	#11718109	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	104	2	no	—
171	C₄	—	208	1	no	—
172	C₄	#11718063	1	1	yes	characteristic
173	C₄	#11718069	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	#13112825	1	1	yes	characteristic
177	C₂	#13112840	1	1	yes	characteristic
178	C₂	#13112855	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116159

C₂

yes

maximal, characteristic

#13112706

C₂

yes

maximal, characteristic

C₁₃×#1322

C₂

yes

maximal, characteristic

C₂²×Dic₅₂

C₄

yes

characteristic

C₂×C₁₃×C₄⋊C₈

C₂²

yes

—

—

—

C₁₃×C₂²⋊C₁₆

—

#11718143

C₄

yes

characteristic

C₂²×Dic₂₆

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×C₁₃×C₁₆

—

C₁₃×C₄⋊C₈

—

#88071

—

#88071

—

C₂²⋊#1055

—

—

—

—

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₃

D₈

yes

characteristic

C₂²×C₄×C₁₃

SD₁₆

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

Dic₅₂

—

Dic₅₂

—

C₁₃×C₁₆

—

#2056

—

#1322

D₁₃

yes

characteristic

#1290

—

C₂³×C₁₃

#102

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

C₂²×Q₁₆

—

#432

—

C₂×C₄⋊C₈

C₂×D₁₃

yes

characteristic

C₂×#103

—

#391

—

C₂²⋊C₁₆

—

#340

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

AΣL(1,𝔽₁₆)

yes

characteristic

C₄×C₁₃

#355

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₄×D₁₃

yes

characteristic

C₂²×C₈

C₂²⋊D₁₃

yes

characteristic

100

C₂×C₄⋊C₄

—

101

C₂×C₄²

D₅₂

yes

characteristic

102

C₂×C₁₆

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#103

—

106

C₂²⋊C₈

—

107

#97

—

108

#95

—

109

#95

—

110

C₂×C₁₃

—

111

C₂×C₁₃

—

112

C₂×C₁₃

#1340

yes

characteristic

113

C₂×C₁₃

#1345

yes

characteristic

114

C₂×C₁₃

#1350

yes

characteristic

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

#6095

yes

characteristic

124

C₂²×C₄

D₁₀₄

yes

characteristic

125

C₂²×C₄

#6103

yes

characteristic

126

Q₁₆

—

127

Q₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄²

—

136

C₄²

—

137

C₁₆

—

138

C₁₃

#7082

yes

characteristic, complemented, Sylow

139

C₂³

#88061

yes

characteristic

140

Q₈

—

141

Q₈

—

142

Q₈

—

104

—

143

Q₈

—

104

—

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

—

104

—

153

C₂×C₄

—

104

—

154

C₂×C₄

—

104

—

155

C₂×C₄

—

104

—

156

C₂×C₄

—

104

—

157

C₂×C₄

#88046

yes

characteristic

158

C₂×C₄

#88065

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

104

—

162

C₂²

—

163

C₂²

—

164

C₂²

#11718062

yes

characteristic

165

C₂²

#11718102

yes

characteristic

166

C₂²

#11718109

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

104

—

171

C₄

—

208

—

172

C₄

#11718063

yes

characteristic

173

C₄

#11718069

yes

characteristic

174

C₂

—

175

C₂

—

176

C₂

#13112825

yes

characteristic

177

C₂

#13112840

yes

characteristic

178

C₂

#13112855

yes

characteristic

	subgroup	G/H
center	C₂²	#11718062
commutator	C₂²×C₈×C₁₃	C₂×C₄
Frattini	C₂×C₄⋊C₈	C₂×D₁₃
Fitting	C₁₃×#1322	C₂
socle	C₂²×C₁₃	#327
prime radical	C₂³×C₁₃	#102
cosocle	C₂×C₁₃×C₄⋊C₈	C₂²
hypercenter	#1322	D₁₃
hypocenter	C₁₃	#7082
LCS	G ▷ C₂²×C₈×C₁₃ ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1322

subgroup

G/H

center

C₂²

#11718062

commutator

C₂²×C₈×C₁₃

C₂×C₄

Frattini

C₂×C₄⋊C₈

C₂×D₁₃

Fitting

C₁₃×#1322

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₁₃×C₄⋊C₈

C₂²

D₁₃

C₁₃

G ▷ C₂²×C₈×C₁₃ ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1322

N	H ≅ G/N	type
138. C₁₃	#7082	semidirect

H ≅ G/N

type

138. C₁₃

#7082

semidirect

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

subgroup	index
#13116159	2
#13112706	2
C₁₃×#1322	2
#7082	13

subgroup

index

#13116159

#13112706

C₁₃×#1322

#7082

Subgroups — #15749611

Conjugacy classes of subgroups

Complements

Maximal subgroups