Group #16807380 — subgroups

Subgroups — #16807380

Order 3328.

460 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 218 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	#13131252	C₂	1	4	yes	maximal, complemented
2	#13131250	C₂	1	4	yes	maximal, complemented
3	#13131202	C₂	1	4	yes	maximal
4	#13131193	C₂	1	2	yes	maximal
5	#13123766	C₂	1	1	yes	maximal, characteristic
6	#11719175	C₄	1	1	yes	characteristic
7	C₁₃×#423	C₄	1	1	yes	characteristic
8	#11718783	C₂²	1	2	yes	—
9	#11718780	C₄	1	2	yes	—
10	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
11	#11718492	C₄	1	4	yes	—
12	#11718491	C₂²	1	4	yes	—
13	C₂²⋊#6092	C₂²	1	4	yes	—
14	#11718260	C₂²	1	8	yes	—
15	#11718258	C₂²	1	4	yes	—
16	#11718257	C₂²	1	4	yes	—
17	C₄⋊#6092	C₂²	1	2	yes	—
18	C₄⋊#6092	C₂²	1	2	yes	—
19	C₄×#6092	C₂²	1	4	yes	—
20	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	#88180	C₂×C₄	1	2	yes	—
23	#88178	C₈	1	4	yes	—
24	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂³	1	1	yes	characteristic
26	#88124	C₈	1	2	yes	—
27	#88123	C₈	1	2	yes	—
28	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
29	C₂×#6092	C₂³	1	4	yes	—
30	C₂×#6092	C₂³	1	4	yes	—
31	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
32	#88055	C₂×C₄	1	4	yes	—
33	#88054	C₂×C₄	1	8	yes	—
34	C₈×Dic₁₃	C₂³	1	4	yes	—
35	C₄×#1055	C₂×C₄	1	2	yes	—
36	C₄×#1055	C₂³	1	2	yes	—
37	#33500	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₂₆	C₂×C₈	1	4	yes	—
41	C₁₃×M₁₆	—	4	2	no	—
42	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
43	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
44	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
46	C₂²⋊Dic₁₃	C₂×C₈	1	4	yes	—
47	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
48	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
49	#6101	C₂×C₈	1	8	yes	—
50	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
51	C₂×#1055	—	2	2	no	—
52	C₂×#1055	—	4	1	no	—
53	C₂×#1055	C₂²×C₄	1	2	yes	—
54	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
55	C₂×#1055	C₂⁴	1	1	yes	characteristic
56	#6092	—	2	4	no	—
57	#6092	—	2	4	no	—
58	#2960	—	13	1	no	—
59	#2183	—	13	4	no	—
60	Q₈×C₁₆	—	13	4	no	—
61	#2164	—	13	4	no	—
62	#2162	—	13	2	no	—
63	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₃	—	2	1	no	—
65	C₂×C₄×C₁₃	—	2	1	no	—
66	C₂×C₄×C₁₃	—	2	4	no	—
67	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
68	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
71	C₂×Dic₁₃	C₂²×C₈	1	8	yes	—
72	Dic₂₆	—	2	8	no	—
73	C₈×C₁₃	—	2	4	no	—
74	#1055	—	2	1	no	—
75	#1055	—	2	2	no	—
76	#1055	2₋¹⁺⁴	1	2	yes	—
77	#533	—	13	1	no	—
78	#446	—	13	4	no	—
79	C₈×Q₈	—	13	4	no	—
80	#424	—	13	2	no	—
81	#423	#261	1	1	yes	characteristic
82	#405	—	13	2	no	—
83	C₂×C₄×C₈	—	13	1	no	—
84	C₄⋊C₁₆	—	13	2	no	—
85	C₄⋊C₁₆	—	13	2	no	—
86	C₄⋊C₁₆	—	13	8	no	—
87	C₂²⋊C₁₆	—	13	4	no	—
88	#346	—	13	4	no	—
89	C₄×C₁₆	—	13	4	no	—
90	C₄×C₁₆	—	13	4	no	—
91	C₂²×C₁₃	—	2	2	no	—
92	C₂²×C₁₃	—	2	1	no	—
93	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
94	C₄×C₁₃	—	2	2	no	—
95	C₄×C₁₃	—	2	1	no	—
96	C₄×C₁₃	#504	1	4	yes	—
97	C₄×C₁₃	#519	1	1	yes	characteristic
98	C₄×C₁₃	#568	1	1	yes	characteristic
99	Dic₁₃	—	2	8	no	—
100	C₂×M₁₆	C₂×#261	1	2	yes	—
101	C₂²×C₈	—	13	1	no	—
102	C₂²×C₈	—	13	1	no	—
103	C₄⋊Q₈	—	13	2	no	—
104	#125	—	13	2	no	—
105	C₂²⋊Q₈	—	13	4	no	—
106	C₄×Q₈	—	13	4	no	—
107	#117	—	13	2	no	—
108	C₂×C₄²	C₂×#261	1	1	yes	characteristic
109	C₂×C₁₆	—	13	4	no	—
110	C₂×C₁₆	—	13	4	no	—
111	C₄⋊C₈	—	13	8	no	—
112	C₄⋊C₈	C₂×#261	1	4	yes	—
113	#97	—	13	4	no	—
114	C₄×C₈	—	13	4	no	—
115	C₄×C₈	—	13	2	no	—
116	C₄×C₈	—	13	2	no	—
117	C₂×C₁₃	—	4	1	no	—
118	C₂×C₁₃	#2982	1	1	yes	characteristic
119	C₂×C₁₃	C₂×#504	1	2	yes	—
120	C₂×Q₈	—	13	4	no	—
121	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
122	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
123	M₁₆	—	4	2	no	—
124	C₂×C₈	—	13	1	no	—
125	C₂×C₈	—	13	1	no	—
126	C₂×C₈	—	13	2	no	—
127	C₂×C₈	—	26	2	no	—
128	C₂×C₈	—	52	1	no	—
129	C₂×C₈	C₂²×#261	1	4	yes	—
130	C₄⋊C₄	—	13	4	no	—
131	C₄⋊C₄	—	13	8	no	—
132	C₄⋊C₄	—	13	4	no	—
133	C₂²⋊C₄	—	13	4	no	—
134	C₄²	—	13	4	no	—
135	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
136	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
137	C₄²	C₂²×#261	1	2	yes	—
138	C₁₆	—	26	4	no	—
139	C₁₆	—	26	4	no	—
140	C₁₃	#33500	1	1	yes	characteristic, complemented, Sylow
141	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
142	Q₈	—	26	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	13	8	no	—
147	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
148	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
149	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
150	C₂×C₄	C₂³×#261	1	1	yes	characteristic
151	C₈	—	2	4	no	—
152	C₈	—	13	2	no	—
153	C₈	—	26	2	no	—
154	C₈	—	26	1	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	1	no	—
157	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	1	no	—
160	C₄	—	26	8	no	—
161	C₄	#11719096	1	4	yes	—
162	C₄	#11719621	1	1	yes	characteristic
163	C₄	#11719627	1	1	yes	characteristic
164	C₂	—	4	1	no	—
165	C₂	#13131731	1	1	yes	characteristic
166	C₂	C₂×#11719096	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13131252

C₂

yes

maximal, complemented

#13131250

C₂

yes

maximal, complemented

#13131202

C₂

yes

maximal

#13131193

C₂

yes

maximal

#13123766

C₂

yes

maximal, characteristic

#11719175

C₄

yes

characteristic

C₁₃×#423

C₄

yes

characteristic

#11718783

C₂²

yes

—

#11718780

C₄

yes

—

C₂×C₄×#1055

C₂²

yes

characteristic

#11718492

C₄

yes

—

#11718491

C₂²

yes

—

C₂²⋊#6092

C₂²

yes

—

#11718260

C₂²

yes

—

#11718258

C₂²

yes

—

#11718257

C₂²

yes

—

C₄⋊#6092

C₂²

yes

—

C₄⋊#6092

C₂²

yes

—

C₄×#6092

C₂²

yes

—

C₂×C₁₃×M₁₆

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

#88180

C₂×C₄

yes

—

#88178

C₈

yes

—

C₂²×#1055

C₂×C₄

yes

characteristic

C₂²×#1055

C₂³

yes

characteristic

#88124

C₈

yes

—

#88123

C₈

yes

—

C₄×Dic₂₆

C₂×C₄

yes

—

C₂×#6092

C₂³

yes

—

C₂×#6092

C₂³

yes

—

C₁₃×C₄⋊C₈

C₂×C₄

yes

—

#88055

C₂×C₄

yes

—

#88054

C₂×C₄

yes

—

C₈×Dic₁₃

C₂³

yes

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

C₂³

yes

—

#33500

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×C₈

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×Dic₂₆

C₂×C₈

yes

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

C₂²×C₄

yes

—

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

C₂²×C₄

yes

—

C₂²⋊Dic₁₃

C₂×C₈

yes

—

C₄⋊Dic₁₃

C₂×C₈

yes

—

C₄⋊Dic₁₃

C₂×C₈

yes

—

#6101

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

—

—

—

—

Q₈×C₁₆

—

#2164

—

#2162

—

C₂³×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₈

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₈

yes

—

C₂×C₄×C₁₃

C₂³×C₄

yes

characteristic

C₂×Dic₁₃

C₂²×C₈

yes

—

Dic₂₆

—

C₈×C₁₃

—

#1055

—

#1055

—

#1055

2₋¹⁺⁴

yes

—

#533

—

#446

—

C₈×Q₈

—

—

yes

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂³×C₈

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#504

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

100

C₂×M₁₆

C₂×#261

yes

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₄⋊Q₈

—

104

#125

—

105

C₂²⋊Q₈

—

106

C₄×Q₈

—

107

#117

—

108

C₂×C₄²

C₂×#261

yes

characteristic

109

C₂×C₁₆

—

110

C₂×C₁₆

—

111

C₄⋊C₈

—

112

C₄⋊C₈

C₂×#261

yes

—

113

#97

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₂×C₁₃

—

118

C₂×C₁₃

#2982

yes

characteristic

119

C₂×C₁₃

C₂×#504

yes

—

120

C₂×Q₈

—

121

C₂²×C₄

D₁₃⋊C₈

yes

—

122

C₂²×C₄

C₂²×#261

yes

characteristic

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₂²×#261

yes

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

—

135

C₄²

D₁₃⋊C₈

yes

characteristic

136

C₄²

D₁₃⋊C₈

yes

characteristic

137

C₄²

C₂²×#261

yes

—

138

C₁₆

—

139

C₁₆

—

140

C₁₃

#33500

yes

characteristic, complemented, Sylow

141

C₂³

C₂×D₁₃⋊C₈

yes

characteristic

142

Q₈

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

C₂×D₁₃⋊C₈

yes

characteristic

148

C₂×C₄

C₂×D₁₃⋊C₈

yes

—

149

C₂×C₄

C₂×D₁₃⋊C₈

yes

—

150

C₂×C₄

C₂³×#261

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₂²×D₁₃⋊C₈

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#11719096

yes

—

162

C₄

#11719621

yes

characteristic

163

C₄

#11719627

yes

characteristic

164

C₂

—

165

C₂

#13131731

yes

characteristic

166

C₂

C₂×#11719096

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#423	C₄
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	#423	#261
hypocenter	C₁₃	#33500
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×#261

commutator, socle

C₂²×C₁₃

C₂³×C₈

Fitting

C₁₃×#423

C₄

prime radical

C₂³×C₁₃

C₂²×C₈

C₂⁴

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
1. #13131252	C₂	semidirect
2. #13131250	C₂	semidirect
140. C₁₃	#33500	semidirect

H ≅ G/N

type

1. #13131252

C₂

semidirect

2. #13131250

C₂

semidirect

140. C₁₃

#33500

semidirect

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

subgroup	index
#13131252	2
#13131250	2
#13131202	2
#13131193	2
#13123766	2
#33500	13

subgroup

index

Subgroups — #16807380

Conjugacy classes of subgroups

Complements

Maximal subgroups