Group #16807363 — subgroups

Subgroups — #16807363

Order 3328.

526 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 270 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	#13131250	C₂	1	8	yes	maximal, complemented
2	#13131194	C₂	1	4	yes	maximal
3	#13131193	C₂	1	2	yes	maximal
4	C₂×#11718491	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₂₆	C₄	1	1	yes	characteristic
6	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
7	C₂×#88054	C₄	1	2	yes	—
8	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
9	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
10	#11718491	C₄	1	4	yes	—
11	#11718491	C₂²	1	4	yes	—
12	C₂²⋊#6092	C₂²	1	4	yes	—
13	#11718260	C₂²	1	8	yes	—
14	#11718257	C₂²	1	8	yes	—
15	C₄⋊#6092	C₂²	1	4	yes	—
16	C₄×#6092	C₂²	1	4	yes	—
17	C₂²×Dic₂₆	C₈	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₃	C₈	1	1	yes	characteristic
21	C₂×#6101	C₈	1	2	yes	—
22	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
23	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
24	C₂²×#1055	C₂³	1	1	yes	characteristic
25	C₄×Dic₂₆	C₈	1	4	yes	—
26	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
27	C₂×#6092	C₂³	1	8	yes	—
28	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
29	#88054	C₂×C₄	1	8	yes	—
30	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
31	C₈×Dic₁₃	C₂³	1	4	yes	—
32	C₄×#1055	C₂×C₄	1	2	yes	—
33	C₄×#1055	C₂³	1	2	yes	—
34	#33496	—	13	1	no	maximal, Sylow
35	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
39	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
40	C₂×Dic₂₆	C₁₆	1	8	yes	complemented
41	C₂×Dic₂₆	C₂×C₈	1	4	yes	—
42	C₂×C₈×C₁₃	—	4	2	no	—
43	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
44	C₄²×C₁₃	C₂×C₈	1	2	yes	—
45	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
46	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
47	#6101	C₂×C₈	1	8	yes	—
48	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
49	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
50	C₂×#1055	—	2	2	no	—
51	C₂×#1055	—	4	1	no	—
52	C₂×#1055	C₂²×C₄	1	2	yes	—
53	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
54	C₂×#1055	C₂⁴	1	1	yes	characteristic
55	#6092	—	2	8	no	—
56	C₂×C₈×Q₈	—	13	1	no	—
57	Q₈×C₁₆	—	13	8	no	—
58	#2162	—	13	2	no	—
59	#2162	—	13	4	no	—
60	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₃	—	2	1	no	—
62	C₂×C₄×C₁₃	—	2	1	no	—
63	C₂×C₄×C₁₃	—	4	1	no	—
64	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	—
65	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
67	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
68	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
69	C₂×Dic₁₃	—	4	2	no	—
70	C₂×Dic₁₃	C₂×C₁₆	1	8	yes	—
71	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
72	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
73	Dic₂₆	C₂×C₁₆	1	16	yes	—
74	C₈×C₁₃	—	2	4	no	—
75	#1055	—	2	1	no	—
76	#1055	—	2	2	no	—
77	#1055	2₋¹⁺⁴	1	2	yes	—
78	C₂×C₄×Q₈	—	13	1	no	—
79	C₈×Q₈	—	13	4	no	—
80	C₈×Q₈	—	13	4	no	—
81	C₂×C₄⋊C₈	—	13	2	no	—
82	C₂×C₄⋊C₈	#261	1	1	yes	characteristic
83	C₂×C₄×C₈	—	13	2	no	—
84	C₂×C₄×C₈	—	13	1	no	—
85	C₄⋊C₁₆	—	13	4	no	—
86	C₄⋊C₁₆	—	13	8	no	—
87	C₂²⋊C₁₆	—	13	4	no	—
88	C₄×C₁₆	—	13	4	no	—
89	C₄×C₁₆	—	13	8	no	—
90	C₂²×C₁₃	—	2	1	no	—
91	C₂²×C₁₃	—	2	1	no	—
92	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
93	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
94	C₄×C₁₃	—	2	2	no	—
95	C₄×C₁₃	—	2	1	no	—
96	C₄×C₁₃	C₂²×C₁₆	1	4	yes	—
97	C₄×C₁₃	#519	1	1	yes	characteristic
98	C₄×C₁₃	#568	1	1	yes	characteristic
99	Dic₁₃	—	2	4	no	—
100	Dic₁₃	C₂²×C₁₆	1	8	yes	—
101	C₂²×Q₈	—	13	1	no	—
102	C₂²×C₈	—	13	1	no	—
103	C₂²×C₈	—	13	1	no	—
104	C₂²×C₈	C₂×#261	1	2	yes	—
105	C₄×Q₈	—	13	4	no	—
106	C₄×Q₈	—	13	4	no	—
107	C₂×C₄⋊C₄	—	13	2	no	—
108	C₂×C₄⋊C₄	—	13	1	no	—
109	C₂×C₄²	—	13	2	no	—
110	C₂×C₄²	C₂×#261	1	1	yes	characteristic
111	C₂×C₁₆	—	13	8	no	—
112	C₄⋊C₈	—	13	8	no	—
113	C₄⋊C₈	C₂×#261	1	4	yes	—
114	C₄×C₈	—	13	4	no	—
115	C₄×C₈	—	13	4	no	—
116	C₄×C₈	—	13	2	no	—
117	C₄×C₈	—	13	2	no	—
118	C₂×C₁₃	—	2	2	no	—
119	C₂×C₁₃	#2982	1	1	yes	characteristic
120	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
121	C₂×C₁₃	#3407	1	1	yes	characteristic
122	C₂×Q₈	—	13	8	no	—
123	C₂×Q₈	—	13	4	no	—
124	C₂²×C₄	—	13	2	no	—
125	C₂²×C₄	—	13	2	no	—
126	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
127	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
128	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	13	1	no	—
131	C₂×C₈	—	13	1	no	—
132	C₂×C₈	—	13	2	no	—
133	C₂×C₈	—	26	2	no	—
134	C₂×C₈	—	52	1	no	—
135	C₂×C₈	C₂²×#261	1	4	yes	—
136	C₄⋊C₄	—	13	8	no	—
137	C₄⋊C₄	—	13	4	no	—
138	C₄²	—	13	4	no	—
139	C₄²	—	13	4	no	—
140	C₄²	D₁₃⋊C₈	1	2	yes	—
141	C₄²	C₂²×#261	1	2	yes	—
142	C₁₆	—	26	8	no	—
143	C₁₃	#33496	1	1	yes	characteristic, complemented, Sylow
144	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
145	Q₈	—	13	16	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	13	8	no	—
150	C₂×C₄	—	13	4	no	—
151	C₂×C₄	—	13	4	no	—
152	C₂×C₄	—	52	2	no	—
153	C₂×C₄	D₁₃⋊C₁₆	1	4	yes	—
154	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
155	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
156	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
157	C₂×C₄	C₂³×#261	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	13	2	no	—
160	C₈	—	26	2	no	—
161	C₈	—	26	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	C₂×D₁₃⋊C₁₆	1	2	yes	—
165	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	2	1	no	—
168	C₄	—	13	8	no	—
169	C₄	—	26	4	no	—
170	C₄	C₂×D₁₃⋊C₁₆	1	4	yes	—
171	C₄	#11719621	1	1	yes	characteristic
172	C₄	#11719627	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	#13131731	1	1	yes	characteristic
175	C₂	C₂²×D₁₃⋊C₁₆	1	1	yes	characteristic
176	C₂	#13131805	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131250

C₂

yes

maximal, complemented

#13131194

C₂

yes

maximal

#13131193

C₂

yes

maximal

C₂×#11718491

C₂

yes

maximal, characteristic

C₂×C₄×Dic₂₆

C₄

yes

characteristic

C₂×C₁₃×C₄⋊C₈

C₄

yes

characteristic

C₂×#88054

C₄

yes

—

C₂×C₈×Dic₁₃

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

characteristic

#11718491

C₄

yes

—

#11718491

C₂²

yes

—

C₂²⋊#6092

C₂²

yes

—

#11718260

C₂²

yes

—

#11718257

C₂²

yes

—

C₄⋊#6092

C₂²

yes

—

C₄×#6092

C₂²

yes

—

C₂²×Dic₂₆

C₈

yes

characteristic

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₁₃

C₈

yes

characteristic

C₂×#6101

C₈

yes

—

C₂×C₄×Dic₁₃

C₂×C₄

yes

—

C₂²×#1055

C₂×C₄

yes

characteristic

C₂²×#1055

C₂³

yes

characteristic

C₄×Dic₂₆

C₈

yes

—

C₄×Dic₂₆

C₂×C₄

yes

—

C₂×#6092

C₂³

yes

—

C₁₃×C₄⋊C₈

C₂×C₄

yes

—

#88054

C₂×C₄

yes

—

C₈×Dic₁₃

C₂×C₄

yes

—

C₈×Dic₁₃

C₂³

yes

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

C₂³

yes

—

#33496

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×Dic₁₃

C₂×C₈

yes

—

C₂²×Dic₁₃

C₂×C₈

yes

—

C₂×Dic₂₆

C₁₆

yes

complemented

C₂×Dic₂₆

C₂×C₈

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²×C₄

yes

—

C₄²×C₁₃

C₂×C₈

yes

—

C₄²×C₁₃

C₂²×C₄

yes

—

C₄⋊Dic₁₃

C₂×C₈

yes

—

#6101

C₂×C₈

yes

—

C₄×Dic₁₃

C₂×C₈

yes

—

C₄×Dic₁₃

C₂²×C₄

yes

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

C₂²×C₄

yes

—

C₂×#1055

C₂²×C₄

yes

characteristic

C₂×#1055

C₂⁴

yes

characteristic

#6092

—

C₂×C₈×Q₈

—

Q₈×C₁₆

—

#2162

—

#2162

—

C₂³×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₁₆

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₈

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

—

C₂×C₄×C₁₃

C₂³×C₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×C₁₆

yes

—

C₂×Dic₁₃

C₂²×C₈

yes

—

C₂×Dic₁₃

C₂²×C₈

yes

—

Dic₂₆

C₂×C₁₆

yes

—

C₈×C₁₃

—

#1055

—

#1055

—

#1055

2₋¹⁺⁴

yes

—

C₂×C₄×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

#261

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²×C₁₆

yes

—

C₂²×C₁₃

C₂³×C₈

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂²×C₁₆

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

100

Dic₁₃

C₂²×C₁₆

yes

—

101

C₂²×Q₈

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂²×C₈

C₂×#261

yes

—

105

C₄×Q₈

—

106

C₄×Q₈

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄²

—

110

C₂×C₄²

C₂×#261

yes

characteristic

111

C₂×C₁₆

—

112

C₄⋊C₈

—

113

C₄⋊C₈

C₂×#261

yes

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₂×C₁₃

—

119

C₂×C₁₃

#2982

yes

characteristic

120

C₂×C₁₃

C₂³×C₁₆

yes

characteristic

121

C₂×C₁₃

#3407

yes

characteristic

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

D₁₃⋊C₈

yes

characteristic

127

C₂²×C₄

D₁₃⋊C₈

yes

characteristic

128

C₂²×C₄

C₂²×#261

yes

characteristic

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

C₂²×#261

yes

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄²

—

139

C₄²

—

140

C₄²

D₁₃⋊C₈

yes

—

141

C₄²

C₂²×#261

yes

—

142

C₁₆

—

143

C₁₃

#33496

yes

characteristic, complemented, Sylow

144

C₂³

C₂×D₁₃⋊C₈

yes

characteristic

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

D₁₃⋊C₁₆

yes

—

154

C₂×C₄

C₂×D₁₃⋊C₈

yes

characteristic

155

C₂×C₄

C₂×D₁₃⋊C₈

yes

—

156

C₂×C₄

C₂×D₁₃⋊C₈

yes

—

157

C₂×C₄

C₂³×#261

yes

characteristic

158

C₈

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₂²

—

163

C₂²

—

164

C₂²

C₂×D₁₃⋊C₁₆

yes

—

165

C₂²

C₂²×D₁₃⋊C₈

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

C₂×D₁₃⋊C₁₆

yes

—

171

C₄

#11719621

yes

characteristic

172

C₄

#11719627

yes

characteristic

173

C₂

—

174

C₂

#13131731

yes

characteristic

175

C₂

C₂²×D₁₃⋊C₁₆

yes

characteristic

176

C₂

#13131805

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator	C₂×C₁₃	C₂³×C₁₆
Fitting	C₂×C₁₃×C₄⋊C₈	C₄
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×C₄⋊C₈	#261
hypocenter	C₁₃	#33496
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×#261

commutator

C₂×C₁₃

C₂³×C₁₆

Fitting

C₂×C₁₃×C₄⋊C₈

C₄

socle

C₂²×C₁₃

C₂³×C₈

prime radical

C₂³×C₁₃

C₂²×C₈

C₂⁴

C₂×C₄⋊C₈

C₁₃

G ▷ C₂×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
1. #13131250	C₂	semidirect
40. C₂×Dic₂₆	C₁₆	semidirect
143. C₁₃	#33496	semidirect

H ≅ G/N

type

1. #13131250

C₂

semidirect

40. C₂×Dic₂₆

C₁₆

semidirect

143. C₁₃

#33496

semidirect

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

subgroup	index
#13131250	2
#13131194	2
#13131193	2
C₂×#11718491	2
#33496	13

subgroup

index

Subgroups — #16807363

Conjugacy classes of subgroups

Complements

Maximal subgroups