Group #16820634 — subgroups

Subgroups — #16820634

Order 3712.

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

normal subgroups only

This group has 290 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	#14271069	C₂	1	1	yes	maximal, characteristic
4	#14270926	C₂	1	2	yes	maximal
5	C₂₉×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₈	—	2	1	no	—
7	C₂²×Dic₅₈	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₂₉	C₂²	1	2	yes	—
9	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₂₉	—	2	1	no	—
11	C₂×C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₂₉	C₂²	1	2	yes	—
13	#11745231	—	2	4	no	—
14	#11745179	—	2	1	no	—
15	#11745179	—	2	1	no	—
16	#11745164	—	2	4	no	—
17	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
20	C₂²×Dic₂₉	—	2	2	no	—
21	C₂²×Dic₂₉	—	2	1	no	—
22	C₂²×Dic₂₉	—	2	1	no	—
23	C₂×Dic₅₈	—	2	2	no	—
24	C₂×Dic₅₈	—	2	2	no	—
25	C₂×Dic₅₈	—	2	4	no	—
26	C₂×Dic₅₈	—	2	1	no	—
27	C₂×Dic₅₈	—	4	2	no	—
28	C₂×Dic₅₈	—	4	2	no	—
29	C₂×Dic₅₈	Q₈	1	2	yes	complemented
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	2	no	—
36	C₄⋊Dic₂₉	—	2	2	no	—
37	C₄⋊Dic₂₉	D₄	1	4	yes	—
38	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₉	—	2	1	no	—
40	C₂×C₄×C₂₉	—	2	1	no	—
41	C₂×C₄×C₂₉	—	2	1	no	—
42	C₂×C₄×C₂₉	—	2	1	no	—
43	C₂×C₄×C₂₉	SD₁₆	1	2	yes	—
44	C₂×C₄×C₂₉	Q₁₆	1	2	yes	—
45	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₂₉	#41	1	2	yes	—
48	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
49	C₂×Dic₂₉	—	2	4	no	—
50	C₂×Dic₂₉	—	2	2	no	—
51	C₂×Dic₂₉	—	4	1	no	—
52	C₂×Dic₂₉	—	4	1	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	Dic₅₈	—	2	4	no	—
59	Dic₅₈	—	4	8	no	—
60	Dic₅₈	—	4	2	no	—
61	C₈×C₂₉	—	4	2	no	—
62	#2024	—	29	1	no	maximal, Sylow
63	C₂²×C₂₉	#120	1	1	yes	characteristic
64	C₂²×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂²×C₂₉	#124	1	1	yes	characteristic
66	C₂²×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
67	C₂²×C₂₉	C₂×Q₁₆	1	1	yes	characteristic
68	C₂²×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₂²×C₂₉	#137	1	1	yes	characteristic
70	C₄×C₂₉	—	2	2	no	—
71	C₄×C₂₉	—	2	2	no	—
72	C₄×C₂₉	C₂²⋊Q₈	1	2	yes	—
73	C₄×C₂₉	#124	1	2	yes	—
74	Dic₂₉	—	4	2	no	—
75	Dic₂₉	—	4	4	no	—
76	Dic₂₉	—	8	1	no	—
77	C₂×C₄⋊Q₈	—	29	1	no	—
78	C₂×#103	—	29	2	no	—
79	#391	—	29	1	no	—
80	#340	—	29	2	no	—
81	#336	D₂₉	1	1	yes	characteristic
82	C₂×C₂₉	#393	1	1	yes	characteristic
83	C₂×C₂₉	#450	1	1	yes	characteristic
84	C₂×C₂₉	#451	1	1	yes	characteristic
85	C₂×C₂₉	#475	1	1	yes	characteristic
86	C₂×C₂₉	#477	1	1	yes	characteristic
87	C₂×C₂₉	#487	1	1	yes	characteristic
88	C₂×C₂₉	#489	1	1	yes	characteristic
89	C₂²×Q₈	—	29	1	no	—
90	C₂²×Q₈	—	58	1	no	—
91	C₂²×C₈	C₂×D₂₉	1	2	yes	—
92	C₄⋊Q₈	—	58	4	no	—
93	C₂×C₄⋊C₄	—	29	2	no	—
94	C₂×C₄⋊C₄	—	29	1	no	—
95	C₂×C₄⋊C₄	—	58	1	no	—
96	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
97	#103	—	58	4	no	—
98	#95	—	58	1	no	—
99	#95	—	58	1	no	—
100	C₂₉	#2024	1	1	yes	characteristic, complemented, Sylow
101	C₂×Q₈	—	29	2	no	—
102	C₂×Q₈	—	58	4	no	—
103	C₂×Q₈	—	58	1	no	—
104	C₂×Q₈	—	58	2	no	—
105	C₂×Q₈	—	58	2	no	—
106	C₂×Q₈	—	116	2	no	—
107	C₂×Q₈	—	116	2	no	—
108	C₂²×C₄	—	58	1	no	—
109	C₂²×C₄	—	58	2	no	—
110	C₂²×C₄	—	58	1	no	—
111	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
112	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₄⋊C₄	—	29	4	no	—
118	C₄⋊C₄	—	58	4	no	—
119	C₄⋊C₄	—	58	2	no	—
120	C₄⋊C₄	—	58	2	no	—
121	C₄²	—	2	2	no	—
122	C₂³	C₂×D₁₁₆	1	1	yes	characteristic
123	Q₈	—	58	4	no	—
124	Q₈	—	116	8	no	—
125	Q₈	—	116	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	58	4	no	—
131	C₂×C₄	—	58	2	no	—
132	C₂×C₄	—	116	1	no	—
133	C₂×C₄	—	116	1	no	—
134	C₂×C₄	—	116	2	no	—
135	C₂×C₄	—	116	2	no	—
136	C₂×C₄	—	116	1	no	—
137	C₂×C₄	—	116	1	no	—
138	C₂×C₄	—	116	1	no	—
139	C₂×C₄	#90773	1	2	yes	—
140	C₂×C₄	Dic₁₁₆	1	2	yes	—
141	C₂×C₄	#90805	1	2	yes	—
142	C₂×C₄	D₄×D₂₉	1	2	yes	—
143	C₂×C₄	#90807	1	1	yes	characteristic
144	C₂×C₄	Q₈×D₂₉	1	1	yes	characteristic
145	C₈	—	4	2	no	—
146	C₂²	#11745237	1	1	yes	characteristic
147	C₂²	#11745244	1	1	yes	characteristic
148	C₂²	#11745259	1	1	yes	characteristic
149	C₂²	C₂×#90773	1	1	yes	characteristic
150	C₂²	C₂×Dic₁₁₆	1	1	yes	characteristic
151	C₂²	#11745270	1	1	yes	characteristic
152	C₂²	#11745271	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	116	4	no	—
156	C₄	—	116	2	no	—
157	C₄	—	232	1	no	—
158	C₄	#11745247	1	2	yes	—
159	C₄	#11745258	1	2	yes	—
160	C₂	#14271076	1	1	yes	characteristic
161	C₂	#14271316	1	1	yes	characteristic
162	C₂	#14271321	1	1	yes	characteristic
163	C₂	#14271449	1	1	yes	characteristic
164	C₂	#14271459	1	1	yes	characteristic
165	C₂	#14271514	1	1	yes	characteristic
166	C₂	#14271523	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11745231

C₂

yes

maximal, characteristic

C₂×#11745164

C₂

yes

maximal

#14271069

C₂

yes

maximal, characteristic

#14270926

C₂

yes

maximal

C₂₉×#336

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

—

C₂²×Dic₅₈

C₂²

yes

characteristic

C₂²×C₈×C₂₉

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

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

—

C₂×Dic₅₈

Q₈

yes

complemented

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₄²×C₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

D₄

yes

—

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

—

C₂×C₄×C₂₉

C₂×Q₈

yes

characteristic

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

—

C₈×C₂₉

—

—

C₂²×C₂₉

yes

C₂²×C₂₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₉

#124

yes

characteristic

C₂²×C₂₉

C₂×SD₁₆

yes

characteristic

C₂²×C₂₉

C₂×Q₁₆

yes

C₂²×C₂₉

yes

C₂²×C₂₉

yes

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₂²⋊Q₈

yes

—

C₄×C₂₉

#124

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×Q₈

—

C₂²×C₈

C₂×D₂₉

yes

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₉

yes

—

—

—

100

C₂₉

#2024

yes

characteristic, complemented, Sylow

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

116

—

107

C₂×Q₈

—

116

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

D₁₁₆

yes

characteristic

112

C₂²×C₄

D₁₁₆

yes

characteristic

113

C₂²×C₄

C₂²×D₂₉

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄²

—

122

C₂³

C₂×D₁₁₆

yes

characteristic

123

Q₈

—

124

Q₈

—

116

—

125

Q₈

—

116

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

116

—

133

C₂×C₄

—

116

—

134

C₂×C₄

—

116

—

135

C₂×C₄

—

116

—

136

C₂×C₄

—

116

—

137

C₂×C₄

—

116

—

138

C₂×C₄

—

116

—

139

C₂×C₄

#90773

yes

—

140

C₂×C₄

Dic₁₁₆

yes

—

141

C₂×C₄

#90805

yes

—

142

C₂×C₄

D₄×D₂₉

yes

—

143

C₂×C₄

#90807

yes

characteristic

144

C₂×C₄

Q₈×D₂₉

yes

characteristic

145

C₈

—

146

C₂²

#11745237

yes

characteristic

147

C₂²

#11745244

yes

characteristic

148

C₂²

#11745259

yes

characteristic

149

C₂²

C₂×#90773

yes

characteristic

150

C₂²

C₂×Dic₁₁₆

yes

characteristic

151

C₂²

#11745270

yes

characteristic

152

C₂²

#11745271

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

116

—

156

C₄

—

116

—

157

C₄

—

232

—

158

C₄

#11745247

yes

—

159

C₄

#11745258

yes

—

160

C₂

#14271076

yes

characteristic

161

C₂

#14271316

yes

characteristic

162

C₂

#14271321

yes

characteristic

163

C₂

#14271449

yes

characteristic

164

C₂

#14271459

yes

characteristic

165

C₂

#14271514

yes

characteristic

166

C₂

#14271523

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	#2024
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
29. C₂×Dic₅₈	Q₈	semidirect
100. C₂₉	#2024	semidirect

H ≅ G/N

type

29. C₂×Dic₅₈

Q₈

semidirect

100. C₂₉

#2024

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
#14271069	2
#14270926	2
C₂₉×#336	2
#2024	29

subgroup

index

C₂₉×#336

Subgroups — #16820634

Conjugacy classes of subgroups

Complements

Maximal subgroups