Group #16790121 — subgroups

Subgroups — #16790121

Order 3328.

434 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 124 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	#13131063	C₂	1	2	yes	maximal
2	#13129883	C₂	1	4	yes	maximal, complemented
3	C₂×#11718658	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
5	C₂²×#6099	C₄	1	1	yes	characteristic
6	C₂³×#1055	C₂²	1	1	yes	characteristic
7	C₂×#88106	C₂²	1	4	yes	—
8	#11718658	—	2	4	no	—
9	C₂×#88043	C₂²	1	2	yes	—
10	C₂×C₄×#1055	C₄	1	2	yes	—
11	C₂²⋊#6092	—	2	4	no	—
12	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₁₃×#117	—	2	2	no	—
14	C₁₃×#117	—	2	2	no	—
15	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
16	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
18	C₂×#6099	—	2	2	no	—
19	C₂×#6099	—	2	4	no	—
20	C₂²×#1055	—	2	4	no	—
21	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
22	C₂²×#1055	C₂×C₄	1	2	yes	—
23	C₂²×#1055	D₄	1	4	yes	—
24	C₂²×#1055	C₂³	1	1	yes	characteristic
25	#88106	—	2	8	no	—
26	C₂×#6092	—	2	4	no	—
27	#88043	—	2	4	no	—
28	C₄×#1055	—	2	4	no	—
29	#12847	—	13	1	no	maximal, Sylow
30	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	—	2	2	no	—
32	C₂²×C₄×C₁₃	—	2	2	no	—
33	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₁₃×C₄⋊C₄	—	2	4	no	—
39	C₁₃×C₂²⋊C₄	—	2	4	no	—
40	C₄²×C₁₃	—	2	2	no	—
41	C₄²×C₁₃	—	2	2	no	—
42	#6099	—	2	8	no	—
43	C₂×#1055	—	2	2	no	—
44	C₂×#1055	—	2	2	no	—
45	C₂×#1055	—	2	2	no	—
46	C₂×#1055	—	2	8	no	—
47	C₂×#1055	C₂²⋊C₄	1	2	yes	—
48	C₂×#1055	M₁₆	1	4	yes	—
49	C₂×#1055	C₂×D₄	1	2	yes	—
50	C₂×#1055	#41	1	4	yes	—
51	#6092	—	4	4	no	—
52	C₂×#405	—	13	1	no	—
53	#2112	—	13	2	no	—
54	#1377	—	13	4	no	—
55	C₂³×C₁₃	—	2	2	no	—
56	C₂³×C₁₃	—	4	2	no	—
57	C₂³×C₁₃	C₂²⋊C₈	1	2	yes	—
58	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₃	—	2	2	no	—
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	4	no	—
63	C₂×C₄×C₁₃	—	2	4	no	—
64	C₂×C₄×C₁₃	—	4	4	no	—
65	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
66	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃	#117	1	2	yes	—
68	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
70	#1055	—	2	2	no	—
71	#1055	—	2	4	no	—
72	#1055	#123	1	4	yes	—
73	C₂²×M₁₆	—	13	1	no	—
74	C₂³×C₈	—	13	1	no	—
75	C₂×#117	#261	1	1	yes	characteristic
76	C₂×M₃₂	—	13	4	no	—
77	#405	—	26	4	no	—
78	C₂×#97	—	13	2	no	—
79	C₂×C₄×C₈	—	13	2	no	—
80	C₂²⋊C₁₆	—	26	4	no	—
81	C₂²×C₁₃	—	2	2	no	—
82	C₂²×C₁₃	—	2	2	no	—
83	C₂²×C₁₃	—	2	4	no	—
84	C₂²×C₁₃	—	4	4	no	—
85	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
86	C₂²×C₁₃	#431	1	2	yes	—
87	C₄×C₁₃	—	2	2	no	—
88	C₄×C₁₃	—	4	4	no	—
89	C₄×C₁₃	#381	1	1	yes	characteristic
90	C₄×C₁₃	#407	1	1	yes	characteristic
91	C₄×C₁₃	#432	1	2	yes	—
92	C₂³×C₄	C₂×#261	1	1	yes	characteristic
93	C₂×M₁₆	—	26	4	no	—
94	C₂×M₁₆	—	26	2	no	—
95	C₂²×C₈	—	13	1	no	—
96	C₂²×C₈	—	13	4	no	—
97	C₂²×C₈	—	13	1	no	—
98	C₂²×C₈	—	13	2	no	—
99	C₂²×C₈	—	26	4	no	—
100	#117	—	2	2	no	—
101	#117	—	2	2	no	—
102	C₂×C₄⋊C₄	#1057	1	2	yes	—
103	C₂×C₂²⋊C₄	#1057	1	2	yes	—
104	C₂×C₄²	C₂×#261	1	2	yes	—
105	M₃₂	—	26	8	no	—
106	C₂×C₁₆	—	26	4	no	—
107	#97	—	26	4	no	—
108	C₄×C₈	—	26	4	no	—
109	C₂×C₁₃	—	2	2	no	—
110	C₂×C₁₃	—	4	2	no	—
111	C₂×C₁₃	#1817	1	1	yes	characteristic
112	C₂×C₁₃	#2115	1	2	yes	—
113	C₂⁴	C₂×#1057	1	1	yes	characteristic
114	C₂²×C₄	—	2	2	no	—
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
117	C₂²×C₄	C₂×#1057	1	4	yes	—
118	C₂²×C₄	#6122	1	2	yes	—
119	C₂²×C₄	C₂²⋊#261	1	2	yes	—
120	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
121	M₁₆	—	26	8	no	—
122	C₂×C₈	—	13	4	no	—
123	C₂×C₈	—	13	2	no	—
124	C₂×C₈	—	13	4	no	—
125	C₂×C₈	—	13	2	no	—
126	C₂×C₈	—	26	8	no	—
127	C₂×C₈	—	26	2	no	—
128	C₂×C₈	—	26	2	no	—
129	C₂×C₈	—	26	2	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	—	2	4	no	—
132	C₄²	—	2	2	no	—
133	C₄²	—	2	2	no	—
134	C₁₆	—	52	4	no	—
135	C₁₃	#12847	1	1	yes	characteristic, complemented, Sylow
136	C₂³	—	2	2	no	—
137	C₂³	—	4	2	no	—
138	C₂³	C₂²⋊#1057	1	2	yes	—
139	C₂³	C₂²×#1057	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	C₂²⋊#1057	1	2	yes	—
147	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
148	C₂×C₄	#88237	1	2	yes	—
149	C₂×C₄	C₂×#6122	1	1	yes	characteristic
150	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
151	C₈	—	13	4	no	—
152	C₈	—	26	4	no	—
153	C₈	—	26	2	no	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	4	4	no	—
158	C₂²	#11719119	1	2	yes	—
159	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	#11719126	1	1	yes	characteristic
163	C₄	#11719128	1	2	yes	—
164	C₄	#11719135	1	1	yes	characteristic
165	C₂	—	2	2	no	—
166	C₂	—	4	2	no	—
167	C₂	#13130726	1	1	yes	characteristic
168	C₂	#13131072	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13131063

C₂

yes

maximal

#13129883

C₂

yes

maximal, complemented

C₂×#11718658

C₂

yes

maximal, characteristic

C₂×C₁₃×#117

C₄

yes

characteristic

C₂²×#6099

C₄

yes

characteristic

C₂³×#1055

C₂²

yes

characteristic

C₂×#88106

C₂²

yes

—

#11718658

—

C₂×#88043

C₂²

yes

—

C₂×C₄×#1055

C₄

yes

—

C₂²⋊#6092

—

C₂³×C₄×C₁₃

C₂×C₄

yes

characteristic

C₁₃×#117

—

C₁₃×#117

—

C₂×C₁₃×C₄⋊C₄

C₈

yes

—

C₂×C₁₃×C₂²⋊C₄

C₈

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×#6099

—

C₂×#6099

—

C₂²×#1055

—

C₂²×#1055

C₂×C₄

yes

characteristic

C₂²×#1055

C₂×C₄

yes

—

C₂²×#1055

D₄

yes

—

C₂²×#1055

C₂³

yes

—

—

—

—

—

C₂⁴×C₁₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₈

yes

—

C₂²×C₄×C₁₃

M₁₆

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

—

—

—

—

—

—

C₂²⋊C₄

yes

—

C₂×#1055

M₁₆

yes

—

C₂×#1055

C₂×D₄

yes

—

C₂×#1055

#41

yes

—

—

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂²⋊C₈

yes

—

C₂³×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

—

yes

—

C₂²×M₁₆

—

C₂³×C₈

—

C₂×#117

#261

yes

characteristic

C₂×M₃₂

—

#405

—

C₂×#97

—

C₂×C₄×C₈

—

C₂²⋊C₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₁₃

#431

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

C₂³×C₄

C₂×#261

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

100

#117

—

101

#117

—

102

C₂×C₄⋊C₄

#1057

yes

—

103

C₂×C₂²⋊C₄

#1057

yes

—

104

C₂×C₄²

C₂×#261

yes

—

105

M₃₂

—

106

C₂×C₁₆

—

107

#97

—

108

C₄×C₈

—

109

C₂×C₁₃

—

110

C₂×C₁₃

—

111

C₂×C₁₃

#1817

yes

characteristic

112

C₂×C₁₃

#2115

yes

—

113

C₂⁴

C₂×#1057

yes

characteristic

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂×#1057

yes

characteristic

117

C₂²×C₄

C₂×#1057

yes

—

118

C₂²×C₄

#6122

yes

—

119

C₂²×C₄

C₂²⋊#261

yes

—

120

C₂²×C₄

C₂²×#261

yes

characteristic

121

M₁₆

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₄⋊C₄

—

131

C₂²⋊C₄

—

132

C₄²

—

133

C₄²

—

134

C₁₆

—

135

C₁₃

#12847

yes

characteristic, complemented, Sylow

136

C₂³

—

137

C₂³

—

138

C₂³

C₂²⋊#1057

yes

—

139

C₂³

C₂²×#1057

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊#1057

yes

—

147

C₂×C₄

D₁₃⋊M₁₆

yes

—

148

C₂×C₄

#88237

yes

—

149

C₂×C₄

C₂×#6122

yes

characteristic

150

C₂×C₄

C₂×C₂²⋊#261

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

#11719119

yes

—

159

C₂²

C₂×C₂²⋊#1057

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

#11719126

yes

characteristic

163

C₄

#11719128

yes

—

164

C₄

#11719135

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

#13130726

yes

characteristic

168

C₂

#13131072

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#261
commutator	C₂³×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×#117	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×#117	#261
hypocenter	C₁₃	#12847
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊#261

commutator

C₂³×C₁₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#261

Fitting

C₂×C₁₃×#117

C₄

socle

C₂²×C₁₃

C₂×C₂²⋊C₈

prime radical

C₂⁴×C₁₃

C₂×C₈

cosocle

C₂²×#1055

C₂³

C₁₃

G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

N	H ≅ G/N	type
2. #13129883	C₂	semidirect
135. C₁₃	#12847	semidirect

H ≅ G/N

type

2. #13129883

C₂

semidirect

135. C₁₃

#12847

semidirect

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

subgroup	index
#13131063	2
#13129883	2
C₂×#11718658	2
#12847	13

subgroup

index

Subgroups — #16790121

Conjugacy classes of subgroups

Complements

Maximal subgroups