Group #16820739 — subgroups

Subgroups — #16820739

Order 3712.

298 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 91 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×#11745231	C₂	1	1	yes	maximal, characteristic
2	C₂×#11745164	C₂	1	2	yes	maximal
3	C₂×#11745164	C₂	1	2	yes	maximal
4	#14271061	C₂	1	1	yes	maximal, characteristic
5	C₂₉×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₈	C₂²	1	2	yes	—
7	C₂²×C₈×C₂₉	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₉	—	2	1	no	—
10	C₂×C₄⋊Dic₂₉	—	2	1	no	—
11	C₂×C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
13	C₂×#90779	—	2	1	no	—
14	#11745231	—	2	4	no	—
15	#11745179	—	2	1	no	—
16	#11745164	—	2	4	no	—
17	#11745164	—	2	4	no	—
18	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
21	C₂²×Dic₂₉	—	2	1	no	—
22	C₂²×Dic₂₉	—	2	2	no	—
23	C₂²×Dic₂₉	—	2	1	no	—
24	C₂×Dic₅₈	—	2	2	no	—
25	C₂×Dic₅₈	—	2	2	no	—
26	C₂×Dic₅₈	—	2	2	no	—
27	C₂×Dic₅₈	—	2	2	no	—
28	C₂×Dic₅₈	—	2	2	no	—
29	C₂×Dic₅₈	D₄	1	4	yes	—
30	C₂×C₈×C₂₉	—	2	2	no	—
31	C₂×C₈×C₂₉	—	2	2	no	—
32	C₂×C₈×C₂₉	—	2	2	no	—
33	C₄²×C₂₉	—	2	2	no	—
34	C₄⋊Dic₂₉	—	2	4	no	—
35	C₄⋊Dic₂₉	—	2	4	no	—
36	C₄⋊Dic₂₉	—	2	1	no	—
37	C₄⋊Dic₂₉	—	2	2	no	—
38	C₄⋊Dic₂₉	D₄	1	2	yes	—
39	#90779	—	4	2	no	—
40	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₉	—	2	1	no	—
42	C₂×C₄×C₂₉	—	2	1	no	—
43	C₂×C₄×C₂₉	—	2	1	no	—
44	C₂×C₄×C₂₉	—	2	1	no	—
45	C₂×C₄×C₂₉	SD₁₆	1	2	yes	—
46	C₂×C₄×C₂₉	Q₁₆	1	2	yes	—
47	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₂₉	#41	1	2	yes	—
50	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
51	C₂×Dic₂₉	—	2	4	no	—
52	C₂×Dic₂₉	—	2	2	no	—
53	C₂×Dic₂₉	—	4	2	no	—
54	C₂×Dic₂₉	—	4	2	no	—
55	C₂×Dic₂₉	—	4	1	no	—
56	C₂×Dic₂₉	—	4	1	no	—
57	C₂×Dic₂₉	—	4	1	no	—
58	C₂×Dic₂₉	—	4	1	no	—
59	C₂×Dic₂₉	—	4	1	no	—
60	Dic₅₈	—	2	4	no	—
61	Dic₅₈	—	2	4	no	—
62	Dic₅₈	—	4	2	no	—
63	Dic₅₈	—	4	2	no	—
64	C₈×C₂₉	—	4	2	no	—
65	#2043	—	29	1	no	maximal, Sylow
66	C₂²×C₂₉	#120	1	1	yes	characteristic
67	C₂²×C₂₉	#121	1	1	yes	characteristic
68	C₂²×C₂₉	#124	1	1	yes	characteristic
69	C₂²×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
70	C₂²×C₂₉	C₂×Q₁₆	1	1	yes	characteristic
71	C₂²×C₂₉	#137	1	1	yes	characteristic
72	C₂²×C₂₉	#137	1	1	yes	characteristic
73	C₄×C₂₉	—	2	2	no	—
74	C₄×C₂₉	—	2	2	no	—
75	C₄×C₂₉	#121	1	2	yes	—
76	C₄×C₂₉	#123	1	2	yes	—
77	Dic₂₉	—	4	4	no	—
78	Dic₂₉	—	4	2	no	—
79	Dic₂₉	—	8	1	no	—
80	C₂×C₄⋊Q₈	—	29	1	no	—
81	C₂×#103	—	29	2	no	—
82	C₂×#103	—	29	2	no	—
83	#389	—	29	1	no	—
84	#336	D₂₉	1	1	yes	characteristic
85	C₂×C₂₉	#394	1	1	yes	characteristic
86	C₂×C₂₉	#448	1	1	yes	characteristic
87	C₂×C₂₉	#451	1	1	yes	characteristic
88	C₂×C₂₉	#461	1	1	yes	characteristic
89	C₂×C₂₉	#462	1	1	yes	characteristic
90	C₂×C₂₉	#487	1	1	yes	characteristic
91	C₂×C₂₉	#491	1	1	yes	characteristic
92	C₂²×Q₈	—	29	2	no	—
93	C₂²×C₈	C₂×D₂₉	1	2	yes	—
94	C₄⋊Q₈	—	58	4	no	—
95	C₂×C₄⋊C₄	—	29	1	no	—
96	C₂×C₄⋊C₄	—	29	1	no	—
97	C₂×C₄⋊C₄	—	58	1	no	—
98	C₂×C₄⋊C₄	—	58	1	no	—
99	C₂×C₄⋊C₄	—	58	1	no	—
100	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
101	#103	—	58	4	no	—
102	#103	—	58	4	no	—
103	#95	—	58	1	no	—
104	C₂₉	#2043	1	1	yes	characteristic, complemented, Sylow
105	C₂×Q₈	—	29	4	no	—
106	C₂×Q₈	—	58	2	no	—
107	C₂×Q₈	—	58	2	no	—
108	C₂×Q₈	—	58	2	no	—
109	C₂×Q₈	—	58	2	no	—
110	C₂×Q₈	—	58	2	no	—
111	C₂²×C₄	—	58	1	no	—
112	C₂²×C₄	—	58	2	no	—
113	C₂²×C₄	—	58	1	no	—
114	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
115	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₄⋊C₄	—	29	2	no	—
121	C₄⋊C₄	—	58	4	no	—
122	C₄⋊C₄	—	58	4	no	—
123	C₄⋊C₄	—	58	1	no	—
124	C₄⋊C₄	—	58	2	no	—
125	C₄⋊C₄	—	116	2	no	—
126	C₄²	—	2	2	no	—
127	C₂³	C₂×D₁₁₆	1	1	yes	characteristic
128	Q₈	—	58	4	no	—
129	Q₈	—	58	4	no	—
130	Q₈	—	116	2	no	—
131	Q₈	—	116	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	58	4	no	—
137	C₂×C₄	—	58	2	no	—
138	C₂×C₄	—	116	2	no	—
139	C₂×C₄	—	116	2	no	—
140	C₂×C₄	—	116	1	no	—
141	C₂×C₄	—	116	1	no	—
142	C₂×C₄	—	116	1	no	—
143	C₂×C₄	—	116	1	no	—
144	C₂×C₄	—	116	1	no	—
145	C₂×C₄	#90773	1	2	yes	—
146	C₂×C₄	Dic₁₁₆	1	2	yes	—
147	C₂×C₄	#90805	1	2	yes	—
148	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
149	C₂×C₄	D₄×D₂₉	1	2	yes	—
150	C₂×C₄	#90809	1	1	yes	characteristic
151	C₈	—	4	2	no	—
152	C₂²	#11745237	1	1	yes	characteristic
153	C₂²	#11745244	1	1	yes	characteristic
154	C₂²	#11745257	1	1	yes	characteristic
155	C₂²	C₂×#90773	1	1	yes	characteristic
156	C₂²	C₂×Dic₁₁₆	1	1	yes	characteristic
157	C₂²	#11745271	1	1	yes	characteristic
158	C₂²	#11745271	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	116	4	no	—
162	C₄	—	116	2	no	—
163	C₄	—	232	1	no	—
164	C₄	#11745246	1	2	yes	—
165	C₄	#11745256	1	2	yes	—
166	C₂	#14271081	1	1	yes	characteristic
167	C₂	#14271304	1	1	yes	characteristic
168	C₂	#14271321	1	1	yes	characteristic
169	C₂	#14271369	1	1	yes	characteristic
170	C₂	#14271374	1	1	yes	characteristic
171	C₂	#14271514	1	1	yes	characteristic
172	C₂	#14271532	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11745231

C₂

yes

maximal, characteristic

C₂×#11745164

C₂

yes

maximal

C₂×#11745164

C₂

yes

maximal

#14271061

C₂

yes

maximal, characteristic

C₂₉×#336

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

C₂²

yes

—

C₂²×C₈×C₂₉

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₉

—

C₂×C₄⋊Dic₂₉

—

C₂×C₄⋊Dic₂₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₉

C₂²

yes

—

—

—

—

—

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

—

C₂²×Dic₂₉

—

C₂²×Dic₂₉

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

D₄

yes

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₄²×C₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

D₄

yes

—

#90779

—

C₂³×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

SD₁₆

yes

—

C₂×C₄×C₂₉

Q₁₆

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

#41

yes

—

C₂×C₄×C₂₉

#41

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

—

C₈×C₂₉

—

—

C₂²×C₂₉

yes

C₂²×C₂₉

yes

C₂²×C₂₉

yes

C₂²×C₂₉

C₂×SD₁₆

yes

characteristic

C₂²×C₂₉

C₂×Q₁₆

yes

C₂²×C₂₉

yes

C₂²×C₂₉

yes

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#121

yes

—

C₄×C₂₉

#123

yes

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

C₂×C₄⋊Q₈

—

—

—

—

D₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂²×Q₈

—

C₂²×C₈

C₂×D₂₉

yes

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄²

C₂×D₂₉

yes

characteristic

101

#103

—

102

#103

—

103

#95

—

104

C₂₉

#2043

yes

characteristic, complemented, Sylow

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

D₁₁₆

yes

characteristic

115

C₂²×C₄

D₁₁₆

yes

characteristic

116

C₂²×C₄

C₂²×D₂₉

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

116

—

126

C₄²

—

127

C₂³

C₂×D₁₁₆

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

116

—

131

Q₈

—

116

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

116

—

139

C₂×C₄

—

116

—

140

C₂×C₄

—

116

—

141

C₂×C₄

—

116

—

142

C₂×C₄

—

116

—

143

C₂×C₄

—

116

—

144

C₂×C₄

—

116

—

145

C₂×C₄

#90773

yes

—

146

C₂×C₄

Dic₁₁₆

yes

—

147

C₂×C₄

#90805

yes

—

148

C₂×C₄

D₄×D₂₉

yes

characteristic

149

C₂×C₄

D₄×D₂₉

yes

—

150

C₂×C₄

#90809

yes

characteristic

151

C₈

—

152

C₂²

#11745237

yes

characteristic

153

C₂²

#11745244

yes

characteristic

154

C₂²

#11745257

yes

characteristic

155

C₂²

C₂×#90773

yes

characteristic

156

C₂²

C₂×Dic₁₁₆

yes

characteristic

157

C₂²

#11745271

yes

characteristic

158

C₂²

#11745271

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

116

—

162

C₄

—

116

—

163

C₄

—

232

—

164

C₄

#11745246

yes

—

165

C₄

#11745256

yes

—

166

C₂

#14271081

yes

characteristic

167

C₂

#14271304

yes

characteristic

168

C₂

#14271321

yes

characteristic

169

C₂

#14271369

yes

characteristic

170

C₂

#14271374

yes

characteristic

171

C₂

#14271514

yes

characteristic

172

C₂

#14271532

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₁₆
commutator, cosocle	C₂²×C₄×C₂₉	C₂³
Frattini	C₂²×C₄	C₂²×D₂₉
Fitting	C₂₉×#336	C₂
socle, prime radical	C₂³×C₂₉	C₂×D₄
hypercenter	#336	D₂₉
hypocenter	C₂₉	#2043
LCS	G ▷ C₂²×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₁₁₆

commutator, cosocle

C₂²×C₄×C₂₉

C₂³

Frattini

C₂²×C₄

C₂²×D₂₉

Fitting

C₂₉×#336

C₂

socle, prime radical

C₂³×C₂₉

C₂×D₄

D₂₉

C₂₉

G ▷ C₂²×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
104. C₂₉	#2043	semidirect

H ≅ G/N

type

104. C₂₉

#2043

semidirect

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

subgroup	index
C₂×#11745231	2
C₂×#11745164	2
#14271061	2
C₂₉×#336	2
#2043	29

subgroup

index

C₂₉×#336

Subgroups — #16820739

Conjugacy classes of subgroups

Complements

Maximal subgroups