Group #20305563 — subgroups

Subgroups — C₂₉×#10663 #20305563

Order 7424.

320 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 114 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	C₂₉×#1985	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#1849	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#1570	C₂	1	2	yes	maximal
4	C₂₉×#1487	C₂	1	2	yes	maximal
5	C₂₉×#1275	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
7	C₂₉×#431	C₂²	1	2	yes	—
8	C₂×C₂₉×#108	—	2	2	no	—
9	C₂₉×#423	C₂²	1	2	yes	—
10	C₂×C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₂₉×#97	C₂²	1	1	yes	characteristic
12	C₂₉×#336	—	2	2	no	—
13	C₂₉×#334	—	2	4	no	—
14	C₂₉×Q₈⋊C₈	—	2	4	no	—
15	C₂²×Q₈×C₂₉	Q₈	1	1	yes	characteristic
16	C₂×M₁₆×C₂₉	—	2	2	no	—
17	C₂²×C₈×C₂₉	—	2	1	no	—
18	C₂²×C₈×C₂₉	D₄	1	2	yes	—
19	C₄×Q₈×C₂₉	—	2	2	no	—
20	C₄×Q₈×C₂₉	—	2	2	no	—
21	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
22	C₂×C₄⋊C₄×C₂₉	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₂₉	—	2	1	no	—
24	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
25	C₂₉×#108	—	4	4	no	—
26	C₂₉×C₄⋊C₈	—	2	2	no	—
27	C₂₉×C₄⋊C₈	—	2	2	no	—
28	C₂₉×C₄⋊C₈	—	2	2	no	—
29	C₂₉×C₄⋊C₈	—	2	2	no	—
30	C₂₉×C₂²⋊C₈	—	2	2	no	—
31	C₂₉×#97	—	2	2	no	—
32	C₄×C₈×C₂₉	D₄	1	4	yes	—
33	C₂×Q₈×C₂₉	—	2	2	no	—
34	C₂×Q₈×C₂₉	—	4	2	no	—
35	C₂²×C₄×C₂₉	—	2	1	no	—
36	C₂²×C₄×C₂₉	—	2	1	no	—
37	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₂₉	#41	1	1	yes	characteristic
40	M₁₆×C₂₉	—	4	4	no	—
41	C₂×C₈×C₂₉	—	2	2	no	—
42	C₂×C₈×C₂₉	—	2	2	no	—
43	C₂×C₈×C₂₉	—	2	4	no	—
44	C₂×C₈×C₂₉	—	4	1	no	—
45	C₂×C₈×C₂₉	—	4	2	no	—
46	C₂×C₈×C₂₉	—	4	2	no	—
47	C₂×C₈×C₂₉	SD₁₆	1	4	yes	—
48	C₄⋊C₄×C₂₉	—	2	2	no	—
49	C₄⋊C₄×C₂₉	—	4	2	no	—
50	C₄²×C₂₉	—	4	1	no	—
51	C₄²×C₂₉	—	4	1	no	—
52	C₄²×C₂₉	C₂×D₄	1	2	yes	—
53	C₄²×C₂₉	#41	1	2	yes	—
54	#10663	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
56	Q₈×C₂₉	—	2	4	no	—
57	Q₈×C₂₉	—	4	2	no	—
58	C₂×C₄×C₂₉	—	2	2	no	—
59	C₂×C₄×C₂₉	—	4	1	no	—
60	C₂×C₄×C₂₉	—	4	1	no	—
61	C₂×C₄×C₂₉	—	4	1	no	—
62	C₂×C₄×C₂₉	—	4	1	no	—
63	C₂×C₄×C₂₉	—	4	2	no	—
64	C₂×C₄×C₂₉	—	4	1	no	—
65	C₂×C₄×C₂₉	C₂²⋊Q₈	1	2	yes	—
66	C₂×C₄×C₂₉	#123	1	2	yes	—
67	C₂×C₄×C₂₉	#123	1	1	yes	characteristic
68	C₂×C₄×C₂₉	#127	1	1	yes	characteristic
69	C₂×C₄×C₂₉	C₂×SD₁₆	1	2	yes	—
70	C₂×C₄×C₂₉	#137	1	2	yes	—
71	C₈×C₂₉	—	2	4	no	—
72	C₈×C₂₉	—	4	4	no	—
73	C₈×C₂₉	—	8	2	no	—
74	C₈×C₂₉	—	8	1	no	—
75	#1985	C₂×C₂₉	1	1	yes	characteristic
76	#1849	C₂×C₂₉	1	1	yes	characteristic
77	#1570	C₂×C₂₉	1	2	yes	—
78	#1487	C₂×C₂₉	1	2	yes	—
79	#1275	C₂×C₂₉	1	1	yes	characteristic
80	C₂²×C₂₉	—	2	1	no	—
81	C₂²×C₂₉	—	2	1	no	—
82	C₂²×C₂₉	#399	1	1	yes	characteristic
83	C₂²×C₂₉	#475	1	2	yes	—
84	C₄×C₂₉	—	2	2	no	—
85	C₄×C₂₉	—	2	2	no	—
86	C₄×C₂₉	—	4	2	no	—
87	C₄×C₂₉	—	8	1	no	—
88	C₄×C₂₉	#483	1	2	yes	—
89	C₄×C₂₉	#492	1	1	yes	characteristic
90	C₄×C₂₉	#497	1	1	yes	characteristic
91	C₂×C₄×Q₈	C₂²×C₂₉	1	1	yes	characteristic
92	#431	C₂²×C₂₉	1	2	yes	—
93	C₂×#108	—	2	2	no	—
94	#423	C₂²×C₂₉	1	2	yes	—
95	C₂×C₄⋊C₈	C₂²×C₂₉	1	1	yes	characteristic
96	C₂×#97	C₂²×C₂₉	1	1	yes	characteristic
97	#336	—	2	2	no	—
98	#334	—	2	4	no	—
99	Q₈⋊C₈	—	2	4	no	—
100	C₂×C₂₉	—	2	2	no	—
101	C₂×C₂₉	#2091	1	1	yes	characteristic
102	C₂×C₂₉	#2096	1	1	yes	characteristic
103	C₂×C₂₉	#2098	1	1	yes	characteristic
104	C₂²×Q₈	Q₈×C₂₉	1	1	yes	characteristic
105	C₂×M₁₆	—	2	2	no	—
106	C₂²×C₈	—	2	1	no	—
107	C₂²×C₈	D₄×C₂₉	1	2	yes	—
108	C₄×Q₈	—	2	2	no	—
109	C₄×Q₈	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	1	no	—
111	C₂×C₄⋊C₄	Q₈×C₂₉	1	1	yes	characteristic
112	C₂×C₄²	—	2	1	no	—
113	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
114	#108	—	4	4	no	—
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₂²⋊C₈	—	2	2	no	—
120	#97	—	2	2	no	—
121	C₄×C₈	D₄×C₂₉	1	4	yes	—
122	C₂₉	#10663	1	1	yes	characteristic, direct product factor, Sylow
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	—	4	2	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
128	C₂²×C₄	C₂×Q₈×C₂₉	1	1	yes	characteristic
129	C₂²×C₄	#41×C₂₉	1	1	yes	characteristic
130	M₁₆	—	4	4	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	SD₁₆×C₂₉	1	4	yes	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄²	—	4	1	no	—
141	C₄²	—	4	1	no	—
142	C₄²	C₂×D₄×C₂₉	1	2	yes	—
143	C₄²	#41×C₂₉	1	2	yes	—
144	C₂³	C₂₉×C₂²⋊Q₈	1	1	yes	characteristic
145	Q₈	—	2	4	no	—
146	Q₈	—	4	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	C₂₉×C₂²⋊Q₈	1	2	yes	—
155	C₂×C₄	C₂₉×#123	1	2	yes	—
156	C₂×C₄	C₂₉×#123	1	1	yes	characteristic
157	C₂×C₄	C₂₉×#127	1	1	yes	characteristic
158	C₂×C₄	C₂×SD₁₆×C₂₉	1	2	yes	—
159	C₂×C₄	C₂₉×#137	1	2	yes	—
160	C₈	—	2	4	no	—
161	C₈	—	4	4	no	—
162	C₈	—	8	2	no	—
163	C₈	—	8	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₂₉×#399	1	1	yes	characteristic
167	C₂²	C₂₉×#475	1	2	yes	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	8	1	no	—
172	C₄	C₂₉×#483	1	2	yes	—
173	C₄	C₂₉×#492	1	1	yes	characteristic
174	C₄	C₂₉×#497	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	C₂₉×#2091	1	1	yes	characteristic
177	C₂	C₂₉×#2096	1	1	yes	characteristic
178	C₂	C₂₉×#2098	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₉×#1985

C₂

yes

maximal, characteristic

C₂₉×#1849

C₂

yes

maximal, characteristic

C₂₉×#1570

C₂

yes

maximal

C₂₉×#1487

C₂

yes

maximal

C₂₉×#1275

C₂

yes

maximal, characteristic

C₂×C₄×Q₈×C₂₉

C₂²

yes

characteristic

C₂₉×#431

C₂²

yes

—

C₂×C₂₉×#108

—

C₂₉×#423

C₂²

yes

—

C₂×C₂₉×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₂₉×#97

C₂²

yes

characteristic

C₂₉×#336

—

C₂₉×#334

—

C₂₉×Q₈⋊C₈

—

C₂²×Q₈×C₂₉

Q₈

yes

characteristic

C₂×M₁₆×C₂₉

—

C₂²×C₈×C₂₉

—

C₂²×C₈×C₂₉

D₄

yes

—

C₄×Q₈×C₂₉

—

C₄×Q₈×C₂₉

—

C₂×C₄⋊C₄×C₂₉

—

C₂×C₄⋊C₄×C₂₉

Q₈

yes

characteristic

C₂×C₄²×C₂₉

—

C₂×C₄²×C₂₉

C₂³

yes

characteristic

C₂₉×#108

—

C₂₉×C₄⋊C₈

—

C₂₉×C₄⋊C₈

—

C₂₉×C₄⋊C₈

—

C₂₉×C₄⋊C₈

—

C₂₉×C₂²⋊C₈

—

C₂₉×#97

—

C₄×C₈×C₂₉

D₄

yes

—

C₂×Q₈×C₂₉

—

C₂×Q₈×C₂₉

—

C₂²×C₄×C₂₉

—

C₂²×C₄×C₂₉

—

C₂²×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₂₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₂₉

#41

yes

characteristic

M₁₆×C₂₉

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

SD₁₆

yes

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄²×C₂₉

—

C₄²×C₂₉

—

C₄²×C₂₉

C₂×D₄

yes

—

C₄²×C₂₉

#41

yes

—

#10663

C₂₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₉

C₂²⋊Q₈

yes

characteristic

Q₈×C₂₉

—

Q₈×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₂₉

#123

yes

—

C₂×C₄×C₂₉

yes

C₂×C₄×C₂₉

yes

C₂×C₄×C₂₉

C₂×SD₁₆

yes

—

C₂×C₄×C₂₉

#137

yes

—

C₈×C₂₉

—

C₈×C₂₉

—

C₈×C₂₉

—

C₈×C₂₉

—

#1985

C₂×C₂₉

yes

characteristic

#1849

C₂×C₂₉

yes

characteristic

#1570

C₂×C₂₉

yes

—

#1487

C₂×C₂₉

yes

—

#1275

C₂×C₂₉

yes

characteristic

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

#399

yes

characteristic

C₂²×C₂₉

#475

yes

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#483

yes

—

C₄×C₂₉

yes

C₄×C₂₉

yes

C₂×C₄×Q₈

C₂²×C₂₉

yes

characteristic

#431

C₂²×C₂₉

yes

—

C₂×#108

—

#423

C₂²×C₂₉

yes

—

C₂×C₄⋊C₈

C₂²×C₂₉

yes

characteristic

C₂×#97

C₂²×C₂₉

yes

characteristic

#336

—

#334

—

Q₈⋊C₈

—

100

C₂×C₂₉

—

101

C₂×C₂₉

#2091

yes

characteristic

102

C₂×C₂₉

#2096

yes

characteristic

103

C₂×C₂₉

#2098

yes

characteristic

104

C₂²×Q₈

Q₈×C₂₉

yes

characteristic

105

C₂×M₁₆

—

106

C₂²×C₈

—

107

C₂²×C₈

D₄×C₂₉

yes

—

108

C₄×Q₈

—

109

C₄×Q₈

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

Q₈×C₂₉

yes

characteristic

112

C₂×C₄²

—

113

C₂×C₄²

C₂³×C₂₉

yes

characteristic

114

#108

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₂²⋊C₈

—

120

#97

—

121

C₄×C₈

D₄×C₂₉

yes

—

122

C₂₉

#10663

yes

characteristic, direct product factor, Sylow

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂×D₄×C₂₉

yes

characteristic

128

C₂²×C₄

C₂×Q₈×C₂₉

yes

characteristic

129

C₂²×C₄

#41×C₂₉

yes

characteristic

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

SD₁₆×C₂₉

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄²

—

141

C₄²

—

142

C₄²

C₂×D₄×C₂₉

yes

—

143

C₄²

#41×C₂₉

yes

—

144

C₂³

C₂₉×C₂²⋊Q₈

yes

characteristic

145

Q₈

—

146

Q₈

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

C₂₉×C₂²⋊Q₈

yes

—

155

C₂×C₄

C₂₉×#123

yes

—

156

C₂×C₄

C₂₉×#123

yes

characteristic

157

C₂×C₄

C₂₉×#127

yes

characteristic

158

C₂×C₄

C₂×SD₁₆×C₂₉

yes

—

159

C₂×C₄

C₂₉×#137

yes

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₂²

—

165

C₂²

—

166

C₂²

C₂₉×#399

yes

characteristic

167

C₂²

C₂₉×#475

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

C₂₉×#483

yes

—

173

C₄

C₂₉×#492

yes

characteristic

174

C₄

C₂₉×#497

yes

characteristic

175

C₂

—

176

C₂

C₂₉×#2091

yes

characteristic

177

C₂

C₂₉×#2096

yes

characteristic

178

C₂

C₂₉×#2098

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₂₉	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₉
socle	C₂²×C₂₉	#399
prime radical	C₂³×C₂₉	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

subgroup

G/H

center

C₂×C₄×C₂₉

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₂₉

socle

C₂²×C₂₉

#399

prime radical

C₂³×C₂₉

C₂²⋊Q₈

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

N	H ≅ G/N	type
54. #10663	C₂₉	direct (N × H)
122. C₂₉	#10663	direct (N × H)

H ≅ G/N

type

54. #10663

C₂₉

direct (N × H)

122. C₂₉

#10663

direct (N × H)

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

subgroup	index
C₂₉×#1985	2
C₂₉×#1849	2
C₂₉×#1570	2
C₂₉×#1487	2
C₂₉×#1275	2
#10663	29

subgroup

index

C₂₉×#1985

C₂₉×#1849

C₂₉×#1570

C₂₉×#1487

C₂₉×#1275

#10663

Subgroups — C₂₉×#10663 #20305563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C29×#10663 #20305563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#10663 #20305563