Group #16777586 — subgroups

Subgroups — #16777586

Order 3328.

198 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 68 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	#13131200	C₂	1	1	yes	maximal, characteristic
2	#13130391	C₂	1	1	yes	maximal, characteristic
3	D₁₃×#335	C₂	1	1	yes	maximal, characteristic
4	#11719052	—	2	1	no	—
5	D₁₃×C₄⋊C₈	—	2	1	no	—
6	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
7	#11718259	C₄	1	2	yes	—
8	#11718258	—	2	1	no	—
9	#11718253	—	2	1	no	—
10	#11718253	C₄	1	2	yes	—
11	C₄⋊#6092	—	2	1	no	—
12	C₁₃×#335	C₄	1	1	yes	characteristic
13	#11718071	—	2	1	no	—
14	#11718050	C₄	1	1	yes	characteristic
15	C₂×#6118	—	4	1	no	—
16	C₂×C₈×D₁₃	—	4	1	no	—
17	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
19	C₄²×D₁₃	D₄	1	1	yes	characteristic
20	C₄⋊#1057	—	2	2	no	—
21	C₄⋊#1057	—	4	1	no	—
22	C₂×#6092	—	2	2	no	—
23	C₁₃×C₄⋊C₈	—	2	1	no	—
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	#88054	—	4	1	no	—
26	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₁₃	D₄	1	1	yes	characteristic
28	C₄⋊#1055	—	2	1	no	—
29	C₄×#1055	D₄	1	1	yes	characteristic
30	#7292	—	13	1	no	maximal, Sylow
31	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×D₁₃	SD₁₆	1	1	yes	characteristic
33	C₂×C₄×D₁₃	SD₁₆	1	1	yes	characteristic
34	C₂×#1057	—	4	2	no	—
35	#6118	—	8	2	no	—
36	C₂×C₈×C₁₃	—	4	1	no	—
37	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×Dic₁₃	D₈	1	1	yes	characteristic
42	C₄×Dic₁₃	Q₁₆	1	1	yes	characteristic
43	C₂×#1055	—	4	1	no	—
44	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
46	C₈×D₁₃	—	2	1	no	—
47	C₈×D₁₃	—	2	1	no	—
48	C₈×D₁₃	—	4	1	no	—
49	C₈×D₁₃	—	8	2	no	—
50	#6092	—	4	2	no	—
51	#2163	—	13	1	no	—
52	#1566	—	13	1	no	—
53	C₂×#335	—	13	1	no	—
54	C₂²×D₁₃	#106	1	1	yes	characteristic
55	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
57	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
58	C₂×Dic₁₃	#102	1	1	yes	characteristic
59	C₂×Dic₁₃	#103	1	1	yes	characteristic
60	C₂×Dic₁₃	#107	1	1	yes	characteristic
61	C₄×D₁₃	—	2	2	no	—
62	C₄×D₁₃	—	2	1	no	—
63	C₄×D₁₃	—	2	1	no	—
64	C₄×D₁₃	—	2	1	no	—
65	C₄×D₁₃	—	2	1	no	—
66	#1057	—	8	2	no	—
67	C₈×C₁₃	—	2	1	no	—
68	C₈×C₁₃	—	8	1	no	—
69	C₈×C₁₃	#108	1	1	yes	characteristic
70	C₈×C₁₃	#108	1	1	yes	characteristic
71	#1055	—	2	1	no	—
72	#1055	—	2	1	no	—
73	#1055	—	8	1	no	—
74	#423	—	26	1	no	—
75	C₂×C₄⋊C₈	—	26	1	no	—
76	C₂×C₄×C₈	—	13	1	no	—
77	C₄⋊C₁₆	—	26	1	no	—
78	C₂²⋊C₁₆	—	13	2	no	—
79	#346	—	26	1	no	—
80	#335	—	13	1	no	—
81	#335	—	26	1	no	—
82	#335	#261	1	1	yes	characteristic
83	#334	—	13	2	no	—
84	#334	—	26	1	no	—
85	C₂²×C₁₃	#340	1	1	yes	characteristic
86	C₂×D₁₃	—	2	1	no	—
87	C₂×D₁₃	—	2	1	no	—
88	C₂×D₁₃	#365	1	2	yes	—
89	C₄×C₁₃	—	2	1	no	—
90	C₄×C₁₃	#341	1	1	yes	characteristic
91	C₄×C₁₃	#344	1	1	yes	characteristic
92	C₄×C₁₃	#361	1	1	yes	characteristic
93	C₄×C₁₃	#362	1	1	yes	characteristic
94	Dic₁₃	—	2	1	no	—
95	Dic₁₃	—	2	1	no	—
96	Dic₁₃	—	2	1	no	—
97	C₂×M₁₆	—	52	1	no	—
98	C₂²×C₈	—	13	1	no	—
99	C₂²×C₈	—	13	1	no	—
100	C₂²×C₈	—	52	1	no	—
101	C₂×C₄²	—	13	1	no	—
102	C₂×C₁₆	—	26	2	no	—
103	C₄⋊C₈	—	2	1	no	—
104	C₄⋊C₈	—	26	2	no	—
105	C₄⋊C₈	—	26	1	no	—
106	C₄⋊C₈	—	52	1	no	—
107	C₄⋊C₈	—	52	1	no	—
108	C₄×C₈	—	13	1	no	—
109	C₄×C₈	—	13	1	no	—
110	C₄×C₈	—	13	1	no	—
111	C₄×C₈	C₂×#261	1	1	yes	characteristic
112	C₂×C₁₃	#1301	1	1	yes	characteristic
113	C₂×C₁₃	#1383	1	1	yes	characteristic
114	C₂×C₁₃	#1387	1	1	yes	characteristic
115	D₁₃	—	2	2	no	—
116	C₂²×C₄	—	13	1	no	—
117	C₂²×C₄	—	13	1	no	—
118	C₂²×C₄	—	13	1	no	—
119	M₁₆	—	104	2	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	13	1	no	—
122	C₂×C₈	—	13	1	no	—
123	C₂×C₈	—	26	1	no	—
124	C₂×C₈	—	26	1	no	—
125	C₂×C₈	—	52	1	no	—
126	C₂×C₈	—	52	2	no	—
127	C₂×C₈	—	52	1	no	—
128	C₂×C₈	—	104	2	no	—
129	C₂×C₈	C₄×#261	1	1	yes	characteristic
130	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
131	C₄²	—	13	1	no	—
132	C₄²	—	13	1	no	—
133	C₄²	—	13	1	no	—
134	C₄²	C₂²⋊#261	1	1	yes	characteristic
135	C₁₆	—	52	2	no	—
136	C₁₃	#7292	1	1	yes	characteristic, complemented, Sylow
137	C₂³	—	13	1	no	—
138	C₂×C₄	—	13	1	no	—
139	C₂×C₄	—	13	1	no	—
140	C₂×C₄	—	13	1	no	—
141	C₂×C₄	—	26	2	no	—
142	C₂×C₄	—	26	1	no	—
143	C₂×C₄	—	26	1	no	—
144	C₂×C₄	—	26	1	no	—
145	C₂×C₄	—	26	1	no	—
146	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
147	C₂×C₄	#88115	1	1	yes	characteristic
148	C₂×C₄	#88117	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	8	1	no	—
151	C₈	—	26	1	no	—
152	C₈	—	26	1	no	—
153	C₈	—	104	2	no	—
154	C₈	—	104	1	no	—
155	C₈	#88103	1	1	yes	characteristic
156	C₈	#88104	1	1	yes	characteristic
157	C₂²	—	13	2	no	—
158	C₂²	—	26	1	no	—
159	C₂²	—	26	1	no	—
160	C₂²	#11718246	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	26	1	no	—
163	C₄	—	26	1	no	—
164	C₄	—	26	1	no	—
165	C₄	#11718268	1	1	yes	characteristic
166	C₄	#11718274	1	1	yes	characteristic
167	C₄	#11718278	1	1	yes	characteristic
168	C₄	#11718282	1	1	yes	characteristic
169	C₂	—	26	2	no	—
170	C₂	#13129728	1	1	yes	characteristic
171	C₂	#13129902	1	1	yes	characteristic
172	C₂	#13129910	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131200

C₂

yes

maximal, characteristic

#13130391

C₂

yes

maximal, characteristic

D₁₃×#335

C₂

yes

maximal, characteristic

#11719052

—

D₁₃×C₄⋊C₈

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718259

C₄

yes

—

#11718258

—

#11718253

—

#11718253

C₄

yes

—

C₄⋊#6092

—

C₁₃×#335

C₄

yes

characteristic

#11718071

—

#11718050

C₄

yes

characteristic

C₂×#6118

—

C₂×C₈×D₁₃

—

C₂×C₈×D₁₃

C₂×C₄

yes

characteristic

C₂×C₈×D₁₃

Q₈

yes

characteristic

C₄²×D₁₃

D₄

yes

—

—

—

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

#88054

—

C₈×Dic₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

D₄

yes

characteristic

C₄⋊#1055

—

C₄×#1055

D₄

yes

characteristic

#7292

—

maximal, Sylow

C₂×C₄×D₁₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₁₃

SD₁₆

yes

characteristic

C₂×C₄×D₁₃

SD₁₆

yes

characteristic

C₂×#1057

—

#6118

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄²

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₄²×C₁₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₃

D₈

yes

characteristic

C₄×Dic₁₃

Q₁₆

yes

characteristic

C₂×#1055

—

C₂×#1055

C₂²⋊C₄

yes

characteristic

C₂×#1055

C₄⋊C₄

yes

characteristic

C₈×D₁₃

—

C₈×D₁₃

—

C₈×D₁₃

—

C₈×D₁₃

—

—

—

—

—

C₂²×D₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

yes

C₂×Dic₁₃

yes

C₂×Dic₁₃

yes

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

#1057

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

yes

C₈×C₁₃

yes

—

—

—

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

yes

—

—

C₂²×C₁₃

#340

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

#365

yes

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂×C₄²

—

102

C₂×C₁₆

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

C₂×#261

yes

characteristic

112

C₂×C₁₃

#1301

yes

characteristic

113

C₂×C₁₃

#1383

yes

characteristic

114

C₂×C₁₃

#1387

yes

characteristic

115

D₁₃

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

M₁₆

—

104

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

104

—

129

C₂×C₈

C₄×#261

yes

characteristic

130

C₂×C₈

C₄⋊#261

yes

characteristic

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

C₂²⋊#261

yes

characteristic

135

C₁₆

—

136

C₁₃

#7292

yes

characteristic, complemented, Sylow

137

C₂³

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

(C₂×#261)⋊C₄

yes

characteristic

147

C₂×C₄

#88115

yes

characteristic

148

C₂×C₄

#88117

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

104

—

154

C₈

—

104

—

155

C₈

#88103

yes

characteristic

156

C₈

#88104

yes

characteristic

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

#11718246

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

#11718268

yes

characteristic

166

C₄

#11718274

yes

characteristic

167

C₄

#11718278

yes

characteristic

168

C₄

#11718282

yes

characteristic

169

C₂

—

170

C₂

#13129728

yes

characteristic

171

C₂

#13129902

yes

characteristic

172

C₂

#13129910

yes

characteristic

	subgroup	G/H
center	C₂²	#11718246
commutator	C₂×C₈×C₁₃	C₄²
Frattini	C₄×C₈	C₂×#261
Fitting	C₁₃×#335	C₄
socle	C₂²×C₁₃	#340
prime radical	C₂²×D₁₃	#106
cosocle	C₄×C₈×D₁₃	C₂²
hypercenter	#335	#261
hypocenter	C₁₃	#7292
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

subgroup

G/H

center

C₂²

#11718246

commutator

C₂×C₈×C₁₃

C₄²

Frattini

C₄×C₈

C₂×#261

Fitting

C₁₃×#335

C₄

C₂²×C₁₃

C₂²×D₁₃

C₄×C₈×D₁₃

C₂²

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

N	H ≅ G/N	type
136. C₁₃	#7292	semidirect

H ≅ G/N

type

136. C₁₃

#7292

semidirect

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

subgroup	index
#13131200	2
#13130391	2
D₁₃×#335	2
#7292	13

subgroup

index

#13131200

#13130391

D₁₃×#335

#7292

Subgroups — #16777586

Conjugacy classes of subgroups

Complements

Maximal subgroups