Group #16015508 — subgroups

Subgroups — #16015508

Order 3328.

806 conjugacy classes of subgroups; only the 267 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 267 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	C₂×#11718725	C₂	1	4	yes	maximal
2	#13123199	C₂	1	1	yes	maximal, characteristic
3	#13115005	C₂	1	4	yes	maximal
4	#13115002	C₂	1	4	yes	maximal
5	C₄×#88043	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#2859	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#125	C₂²	1	2	yes	—
8	C₂×#88124	C₂²	1	2	yes	—
9	C₁₃×#389	—	2	4	no	—
10	C₁₃×#384	C₂²	1	2	yes	—
11	C₁₃×#382	C₂²	1	2	yes	—
12	C₄³×C₁₃	C₂²	1	1	yes	characteristic
13	#11718725	—	2	16	no	—
14	C₂×#88048	C₂²	1	8	yes	—
15	C₂×#88047	C₂²	1	8	yes	—
16	#11718597	C₂²	1	2	yes	—
17	#11718596	C₂²	1	2	yes	—
18	#11718589	—	2	4	no	—
19	C₂×#88043	C₂²	1	2	yes	—
20	C₂×#88043	C₂²	1	2	yes	—
21	C₂×C₄×#1055	C₂²	1	2	yes	—
22	C₁₃×#125	—	2	8	no	—
23	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
25	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
26	C₂×C₄²×C₁₃	D₄	1	4	yes	—
27	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₃	—	2	4	no	—
31	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
32	C₂×#6101	—	2	4	no	—
33	C₂²×#1055	C₂³	1	4	yes	—
34	#88124	—	2	8	no	—
35	C₁₃×#95	—	2	4	no	—
36	#88071	—	2	4	no	—
37	#88048	—	2	16	no	—
38	#88047	—	2	16	no	—
39	#88043	Q₈	1	8	yes	—
40	#88043	Q₈	1	8	yes	—
41	C₄×#1055	D₄	1	8	yes	—
42	#33315	—	13	1	no	maximal, Sylow
43	C₂²×C₄×C₁₃	—	2	4	no	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
48	C₂²×Dic₁₃	—	2	4	no	—
49	C₁₃×C₄⋊C₄	—	2	8	no	—
50	C₁₃×C₄⋊C₄	—	4	8	no	—
51	C₁₃×C₄⋊C₄	—	4	8	no	—
52	C₄²×C₁₃	—	4	4	no	—
53	C₄²×C₁₃	C₂×D₄	1	4	yes	—
54	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
55	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
56	C₄⋊Dic₁₃	—	2	8	no	—
57	C₄⋊Dic₁₃	—	4	8	no	—
58	#6101	—	4	8	no	—
59	C₂×#1055	C₂×D₄	1	8	yes	—
60	C₂×#1055	C₂×Q₈	1	8	yes	—
61	C₂×#1055	C₂×Q₈	1	8	yes	—
62	C₂×#499	—	13	4	no	—
63	#2859	—	13	1	no	—
64	#2859	D₁₃	1	1	yes	characteristic
65	#1847	—	13	4	no	—
66	#1847	—	13	4	no	—
67	C₄×#97	—	13	1	no	—
68	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	—	2	4	no	—
70	C₂×C₄×C₁₃	—	2	4	no	—
71	C₂×C₄×C₁₃	—	2	4	no	—
72	C₂×C₄×C₁₃	—	4	4	no	—
73	C₂×C₄×C₁₃	—	4	8	no	—
74	C₂×C₄×C₁₃	#127	1	4	yes	—
75	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
76	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
77	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
78	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
79	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
80	C₂×Dic₁₃	—	4	4	no	—
81	C₂×Dic₁₃	—	4	8	no	—
82	#1055	C₄⋊Q₈	1	16	yes	—
83	C₂×#125	—	13	2	no	—
84	C₂×#125	C₂×D₁₃	1	2	yes	—
85	#499	—	26	16	no	—
86	C₂×#107	—	13	8	no	—
87	C₂×#106	—	13	8	no	—
88	C₂×#97	—	13	2	no	—
89	C₂×#97	—	13	2	no	—
90	C₂×C₄×C₈	—	13	2	no	—
91	#389	—	2	4	no	—
92	#389	—	26	4	no	—
93	#384	—	13	2	no	—
94	#384	C₂×D₁₃	1	2	yes	—
95	#382	—	13	2	no	—
96	#382	C₂×D₁₃	1	2	yes	—
97	C₄³	C₂×D₁₃	1	1	yes	characteristic
98	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
99	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
100	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₂²×C₁₃	#575	1	4	yes	—
102	C₄×C₁₃	—	2	4	no	—
103	C₄×C₁₃	—	2	4	no	—
104	C₄×C₁₃	—	2	4	no	—
105	C₄×C₁₃	—	8	4	no	—
106	C₄×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
107	Dic₁₃	—	8	4	no	—
108	C₂²×C₈	—	13	4	no	—
109	#125	—	2	8	no	—
110	#125	—	26	8	no	—
111	C₂×C₄⋊C₄	—	2	4	no	—
112	C₂×C₄⋊C₄	—	2	4	no	—
113	C₂×C₄⋊C₄	—	13	4	no	—
114	C₂×C₄⋊C₄	—	26	4	no	—
115	C₂×C₄⋊C₄	—	26	4	no	—
116	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
117	C₂×C₄²	C₂²⋊D₁₃	1	4	yes	—
118	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
119	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
120	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
121	#107	—	26	16	no	—
122	#106	—	26	16	no	—
123	#97	—	13	8	no	—
124	#97	—	13	8	no	—
125	C₄×C₈	—	13	8	no	—
126	#95	—	2	4	no	—
127	#95	—	26	4	no	—
128	C₂×C₁₃	#2868	1	1	yes	characteristic
129	C₂×C₁₃	#3150	1	2	yes	—
130	C₂×C₁₃	#3166	1	2	yes	—
131	C₂×C₁₃	#3166	1	2	yes	—
132	C₂²×C₄	—	2	4	no	—
133	C₂²×C₄	—	26	4	no	—
134	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
135	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
136	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
137	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
138	C₂×C₈	—	13	8	no	—
139	C₂×C₈	—	13	8	no	—
140	C₂×C₈	—	13	8	no	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	4	8	no	—
143	C₄⋊C₄	—	4	8	no	—
144	C₄⋊C₄	—	26	8	no	—
145	C₄⋊C₄	—	52	8	no	—
146	C₄⋊C₄	—	52	8	no	—
147	C₄²	—	4	4	no	—
148	C₄²	D₄×D₁₃	1	4	yes	—
149	C₄²	Q₈×D₁₃	1	4	yes	—
150	C₄²	Q₈×D₁₃	1	4	yes	—
151	C₁₃	#33315	1	1	yes	characteristic, complemented, Sylow
152	C₂³	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	4	4	no	—
157	C₂×C₄	—	4	8	no	—
158	C₂×C₄	—	52	4	no	—
159	C₂×C₄	—	52	8	no	—
160	C₂×C₄	#88194	1	4	yes	—
161	C₂×C₄	#88198	1	4	yes	—
162	C₂×C₄	#88198	1	4	yes	—
163	C₂×C₄	C₂×D₄×D₁₃	1	2	yes	—
164	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
165	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
166	C₈	—	13	16	no	—
167	C₂²	C₂×#88194	1	1	yes	characteristic
168	C₂²	C₂×#88198	1	1	yes	characteristic
169	C₂²	C₂×#88198	1	1	yes	characteristic
170	C₂²	#11719527	1	4	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	4	no	—
173	C₄	—	2	4	no	—
174	C₄	—	8	4	no	—
175	C₄	—	104	4	no	—
176	C₄	D₁₃×C₄⋊Q₈	1	4	yes	—
177	C₂	#13123232	1	1	yes	characteristic
178	C₂	#13125281	1	2	yes	—
179	C₂	#13125385	1	2	yes	—
180	C₂	#13125389	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718725

C₂

yes

maximal

#13123199

C₂

yes

maximal, characteristic

#13115005

C₂

yes

maximal

#13115002

C₂

yes

maximal

C₄×#88043

C₂

yes

maximal, characteristic

C₁₃×#2859

C₂

yes

maximal, characteristic

C₂×C₁₃×#125

C₂²

yes

—

C₂×#88124

C₂²

yes

—

C₁₃×#389

—

C₁₃×#384

C₂²

yes

—

C₁₃×#382

C₂²

yes

—

C₄³×C₁₃

C₂²

yes

characteristic

#11718725

—

C₂×#88048

C₂²

yes

—

C₂×#88047

C₂²

yes

—

#11718597

C₂²

yes

—

#11718596

C₂²

yes

—

#11718589

—

C₂×#88043

C₂²

yes

—

C₂×#88043

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

—

C₁₃×#125

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂³

yes

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂³

yes

—

C₂×#6101

—

C₂²×#1055

C₂³

yes

—

#88124

—

C₁₃×#95

—

—

—

—

Q₈

yes

—

#88043

Q₈

yes

—

C₄×#1055

D₄

yes

—

#33315

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂⁴

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×Q₈

yes

—

C₄²×C₁₃

C₂×Q₈

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₂×#1055

C₂×D₄

yes

—

C₂×#1055

C₂×Q₈

yes

—

C₂×#1055

C₂×Q₈

yes

—

C₂×#499

—

#2859

—

#2859

D₁₃

yes

characteristic

#1847

—

#1847

—

C₄×#97

—

C₂³×C₁₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

—

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂²×D₄

yes

—

C₂×C₄×C₁₃

C₂²×Q₈

yes

—

C₂×C₄×C₁₃

C₂²×Q₈

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

#1055

C₄⋊Q₈

yes

—

C₂×#125

—

C₂×#125

C₂×D₁₃

yes

—

#499

—

C₂×#107

—

C₂×#106

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

—

—

—

C₂×D₁₃

yes

—

#382

—

#382

C₂×D₁₃

yes

—

C₄³

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

C₂×#127

yes

characteristic

C₂²×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

100

C₂²×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

101

C₂²×C₁₃

#575

yes

—

102

C₄×C₁₃

—

103

C₄×C₁₃

—

104

C₄×C₁₃

—

105

C₄×C₁₃

—

106

C₄×C₁₃

C₂×C₄⋊Q₈

yes

—

107

Dic₁₃

—

108

C₂²×C₈

—

109

#125

—

110

#125

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

C₂²×D₁₃

yes

—

117

C₂×C₄²

C₂²⋊D₁₃

yes

—

118

C₂×C₄²

C₂²×D₁₃

yes

characteristic

119

C₂×C₄²

C₂²×D₁₃

yes

characteristic

120

C₂×C₄²

C₂²×D₁₃

yes

characteristic

121

#107

—

122

#106

—

123

#97

—

124

#97

—

125

C₄×C₈

—

126

#95

—

127

#95

—

128

C₂×C₁₃

#2868

yes

characteristic

129

C₂×C₁₃

#3150

yes

—

130

C₂×C₁₃

#3166

yes

—

131

C₂×C₁₃

#3166

yes

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

135

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

136

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

137

C₂²×C₄

C₂³×D₁₃

yes

characteristic

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄²

—

148

C₄²

D₄×D₁₃

yes

—

149

C₄²

Q₈×D₁₃

yes

—

150

C₄²

Q₈×D₁₃

yes

—

151

C₁₃

#33315

yes

characteristic, complemented, Sylow

152

C₂³

C₂²×C₂²⋊D₁₃

yes

characteristic

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

#88194

yes

—

161

C₂×C₄

#88198

yes

—

162

C₂×C₄

#88198

yes

—

163

C₂×C₄

C₂×D₄×D₁₃

yes

—

164

C₂×C₄

C₂×Q₈×D₁₃

yes

—

165

C₂×C₄

C₂×Q₈×D₁₃

yes

—

166

C₈

—

167

C₂²

C₂×#88194

yes

characteristic

168

C₂²

C₂×#88198

yes

characteristic

169

C₂²

C₂×#88198

yes

characteristic

170

C₂²

#11719527

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

104

—

176

C₄

D₁₃×C₄⋊Q₈

yes

—

177

C₂

#13123232

yes

characteristic

178

C₂

#13125281

yes

—

179

C₂

#13125385

yes

—

180

C₂

#13125389

yes

—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#2859	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	#2859	D₁₃
hypocenter	C₁₃	#33315
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂²×C₂²⋊D₁₃

commutator, cosocle

C₂²×C₄×C₁₃

C₂⁴

Frattini

C₂²×C₄

C₂³×D₁₃

Fitting

C₁₃×#2859

C₂

socle, prime radical

C₂³×C₁₃

C₂²×D₄

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

N	H ≅ G/N	type
151. C₁₃	#33315	semidirect

H ≅ G/N

type

151. C₁₃

#33315

semidirect

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

subgroup	index
C₂×#11718725	2
#13123199	2
#13115005	2
#13115002	2
C₄×#88043	2
C₁₃×#2859	2
#33315	13

subgroup

index

C₁₃×#2859

Subgroups — #16015508

Conjugacy classes of subgroups

Complements

Maximal subgroups