Group #16829147 — subgroups

Subgroups — #16829147

Order 3712.

460 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₂²×Dic₁₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#14271734	C₂	1	2	yes	maximal
3	#14271421	C₂	1	4	yes	maximal, complemented
4	#14271404	C₂	1	4	yes	maximal, complemented
5	#14271201	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271173	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₉×#131	C₂	1	1	yes	maximal, characteristic
8	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
9	C₂²×Dic₅₈	C₂²	1	2	yes	—
10	C₂₉×#131	—	2	4	no	—
11	C₂×M₁₆×C₂₉	C₂²	1	2	yes	—
12	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
15	#11745306	—	2	2	no	—
16	#11745298	—	2	2	no	—
17	#11745297	—	2	4	no	—
18	#11745288	C₂²	1	2	yes	—
19	#11745286	C₂²	1	8	yes	—
20	C₂×Dic₁₁₆	—	2	4	no	—
21	C₂×Dic₁₁₆	C₂²	1	2	yes	—
22	C₂×Dic₁₁₆	C₂²	1	2	yes	—
23	#11745166	C₂²	1	2	yes	—
24	#11745165	C₂²	1	2	yes	—
25	#11745164	C₂²	1	8	yes	—
26	#41×C₂₉	—	2	2	no	—
27	#41×C₂₉	—	2	2	no	—
28	C₂×Q₈×C₂₉	D₄	1	1	yes	characteristic
29	C₂×D₄×C₂₉	D₄	1	2	yes	—
30	C₂×D₄×C₂₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
32	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
33	C₂²×Dic₂₉	—	2	2	no	—
34	C₂×Dic₅₈	—	2	4	no	—
35	C₂×Dic₅₈	—	2	4	no	—
36	C₂×Dic₅₈	—	2	2	no	—
37	C₂×Dic₅₈	C₂³	1	4	yes	—
38	M₁₆×C₂₉	—	2	2	no	—
39	M₁₆×C₂₉	—	2	4	no	—
40	C₂×C₈×C₂₉	—	2	4	no	—
41	C₂×C₈×C₂₉	D₄	1	4	yes	—
42	C₂×C₈×C₂₉	C₂³	1	2	yes	—
43	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
45	C₂²⋊Dic₂₉	—	2	8	no	—
46	C₂²⋊Dic₂₉	—	2	2	no	—
47	C₄⋊Dic₂₉	C₂³	1	4	yes	—
48	#90779	—	2	8	no	—
49	C₄×Dic₂₉	—	2	2	no	—
50	Dic₁₁₆	—	2	4	no	—
51	Dic₁₁₆	—	2	4	no	—
52	C₂³×C₂₉	C₂×D₄	1	2	yes	—
53	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
54	Q₈×C₂₉	—	2	2	no	—
55	D₄×C₂₉	—	2	4	no	—
56	D₄×C₂₉	—	2	2	no	—
57	C₂×C₄×C₂₉	—	2	2	no	—
58	C₂×C₄×C₂₉	—	2	2	no	—
59	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₂₉	#41	1	2	yes	—
63	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
64	C₂×Dic₂₉	—	2	4	no	—
65	C₂×Dic₂₉	—	2	4	no	—
66	C₂×Dic₂₉	—	4	4	no	—
67	Dic₅₈	—	2	4	no	—
68	Dic₅₈	—	2	4	no	—
69	Dic₅₈	—	4	4	no	—
70	C₈×C₂₉	—	2	2	no	—
71	C₈×C₂₉	C₂×D₄	1	2	yes	—
72	C₈×C₂₉	C₂×D₄	1	2	yes	—
73	#3059	—	29	1	no	maximal, Sylow
74	C₂²×C₂₉	—	2	2	no	—
75	C₂²×C₂₉	—	2	2	no	—
76	C₂²×C₂₉	—	2	4	no	—
77	C₂²×C₂₉	#121	1	2	yes	—
78	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₂₉	—	2	2	no	—
80	C₄×C₂₉	#121	1	2	yes	—
81	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
82	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
83	Dic₂₉	—	4	4	no	—
84	Dic₂₉	—	4	4	no	—
85	C₂²×Q₁₆	—	29	1	no	—
86	C₂×#131	D₂₉	1	1	yes	characteristic
87	#536	—	29	2	no	—
88	#470	—	29	4	no	—
89	#467	—	29	4	no	—
90	#427	—	29	1	no	—
91	#419	—	29	2	no	—
92	C₂×C₂₉	—	2	2	no	—
93	C₂×C₂₉	—	4	2	no	—
94	C₂×C₂₉	C₂×#121	1	1	yes	characteristic
95	C₂×C₂₉	#578	1	2	yes	—
96	C₂×#41	C₂×D₂₉	1	1	yes	characteristic
97	C₂²×Q₈	—	29	2	no	—
98	C₂×Q₁₆	—	29	2	no	—
99	C₂×Q₁₆	—	29	2	no	—
100	C₂×Q₁₆	—	58	4	no	—
101	#131	—	2	4	no	—
102	C₂×M₁₆	C₂×D₂₉	1	1	yes	characteristic
103	C₂×M₁₆	C₂×D₂₉	1	2	yes	—
104	C₂²×C₈	C₂×D₂₉	1	2	yes	—
105	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
106	C₄⋊Q₈	—	58	2	no	—
107	#124	—	58	2	no	—
108	#123	—	58	4	no	—
109	C₂²⋊Q₈	—	29	8	no	—
110	#117	—	29	2	no	—
111	#107	—	29	2	no	—
112	#106	—	29	2	no	—
113	#103	—	29	8	no	—
114	C₂₉	#3059	1	1	yes	characteristic, complemented, Sylow
115	#41	—	2	2	no	—
116	#41	—	2	2	no	—
117	C₂×Q₈	—	29	4	no	—
118	C₂×Q₈	—	58	2	no	—
119	C₂×Q₈	—	58	4	no	—
120	C₂×Q₈	—	58	4	no	—
121	C₂×Q₈	D₁₁₆	1	1	yes	characteristic
122	C₂×D₄	D₁₁₆	1	2	yes	—
123	C₂×D₄	D₁₁₆	1	1	yes	characteristic
124	C₂²×C₄	—	58	2	no	—
125	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
126	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
127	Q₁₆	—	58	4	no	—
128	Q₁₆	—	58	4	no	—
129	M₁₆	—	2	2	no	—
130	M₁₆	—	2	4	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	C₂²⋊D₂₉	1	4	yes	—
133	C₂×C₈	C₂²×D₂₉	1	2	yes	—
134	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
135	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
136	C₄⋊C₄	—	29	4	no	—
137	C₄⋊C₄	—	58	8	no	—
138	C₂²⋊C₄	—	58	2	no	—
139	C₂²⋊C₄	—	58	8	no	—
140	C₄²	—	58	2	no	—
141	C₂³	C₂×D₁₁₆	1	2	yes	—
142	C₂³	C₂×D₁₁₆	1	1	yes	characteristic
143	Q₈	—	2	2	no	—
144	Q₈	—	58	4	no	—
145	Q₈	—	58	4	no	—
146	Q₈	—	116	4	no	—
147	D₄	—	2	4	no	—
148	D₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	58	4	no	—
152	C₂×C₄	—	58	4	no	—
153	C₂×C₄	—	116	4	no	—
154	C₂×C₄	C₂×D₁₁₆	1	2	yes	—
155	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
156	C₂×C₄	#90805	1	2	yes	—
157	C₂×C₄	C₂×C₂²⋊D₂₉	1	2	yes	—
158	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	C₂×C₂²⋊D₂₉	1	2	yes	—
161	C₈	C₂×C₂²⋊D₂₉	1	2	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	4	no	—
165	C₂²	#11745292	1	2	yes	—
166	C₂²	C₂²×D₁₁₆	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	116	4	no	—
169	C₄	—	116	4	no	—
170	C₄	#11745292	1	2	yes	—
171	C₄	C₂×#90805	1	1	yes	characteristic
172	C₄	C₂²×C₂²⋊D₂₉	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	—	4	2	no	—
175	C₂	C₂×#11745292	1	1	yes	characteristic
176	C₂	#14271980	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₁₆

C₂

yes

maximal, characteristic, complemented

#14271734

C₂

yes

maximal

#14271421

C₂

yes

maximal, complemented

#14271404

C₂

yes

maximal, complemented

#14271201

C₂

yes

maximal, characteristic, complemented

#14271173

C₂

yes

maximal, complemented

C₂×C₂₉×#131

C₂

yes

maximal, characteristic

C₂×#41×C₂₉

C₂²

yes

characteristic

C₂²×Dic₅₈

C₂²

yes

—

C₂₉×#131

—

C₂×M₁₆×C₂₉

C₂²

yes

—

C₂×M₁₆×C₂₉

C₂²

yes

characteristic

C₂²×C₈×C₂₉

C₂²

yes

—

C₂²×C₈×C₂₉

C₂²

yes

—

—

—

C₂²

yes

—

#11745286

C₂²

yes

—

C₂×Dic₁₁₆

—

C₂×Dic₁₁₆

C₂²

yes

—

C₂×Dic₁₁₆

C₂²

yes

—

#11745166

C₂²

yes

—

#11745165

C₂²

yes

—

#11745164

C₂²

yes

—

#41×C₂₉

—

#41×C₂₉

—

C₂×Q₈×C₂₉

D₄

yes

characteristic

C₂×D₄×C₂₉

D₄

yes

—

C₂×D₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

C₂³

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

C₂³

yes

—

M₁₆×C₂₉

—

M₁₆×C₂₉

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

D₄

yes

—

C₂×C₈×C₂₉

C₂³

yes

—

C₂×C₈×C₂₉

C₂³

yes

characteristic

C₂×C₈×C₂₉

C₂³

yes

characteristic

C₂²⋊Dic₂₉

—

C₂²⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂³

yes

—

#90779

—

C₄×Dic₂₉

—

Dic₁₁₆

—

Dic₁₁₆

—

C₂³×C₂₉

C₂×D₄

yes

—

C₂³×C₂₉

C₂×D₄

yes

characteristic

Q₈×C₂₉

—

D₄×C₂₉

—

D₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

#41

yes

—

C₂×C₄×C₂₉

C₂⁴

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

—

C₈×C₂₉

—

C₈×C₂₉

C₂×D₄

yes

—

C₈×C₂₉

C₂×D₄

yes

—

#3059

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

#121

yes

—

C₂²×C₂₉

C₂²×D₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

#121

yes

—

C₄×C₂₉

C₂²×D₄

yes

characteristic

C₄×C₂₉

C₂×#41

yes

characteristic

Dic₂₉

—

Dic₂₉

—

C₂²×Q₁₆

—

C₂×#131

D₂₉

yes

—

—

—

—

—

C₂×C₂₉

—

C₂×C₂₉

—

C₂×C₂₉

C₂×#121

yes

characteristic

C₂×C₂₉

#578

yes

—

C₂×#41

C₂×D₂₉

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

100

C₂×Q₁₆

—

101

#131

—

102

C₂×M₁₆

C₂×D₂₉

yes

characteristic

103

C₂×M₁₆

C₂×D₂₉

yes

—

104

C₂²×C₈

C₂×D₂₉

yes

—

105

C₂²×C₈

C₂×D₂₉

yes

characteristic

106

C₄⋊Q₈

—

107

#124

—

108

#123

—

109

C₂²⋊Q₈

—

110

#117

—

111

#107

—

112

#106

—

113

#103

—

114

C₂₉

#3059

yes

characteristic, complemented, Sylow

115

#41

—

116

#41

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

D₁₁₆

yes

characteristic

122

C₂×D₄

D₁₁₆

yes

—

123

C₂×D₄

D₁₁₆

yes

characteristic

124

C₂²×C₄

—

125

C₂²×C₄

C₂²×D₂₉

yes

—

126

C₂²×C₄

C₂²×D₂₉

yes

characteristic

127

Q₁₆

—

128

Q₁₆

—

129

M₁₆

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

C₂²⋊D₂₉

yes

—

133

C₂×C₈

C₂²×D₂₉

yes

—

134

C₂×C₈

C₂²×D₂₉

yes

characteristic

135

C₂×C₈

C₂²×D₂₉

yes

characteristic

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₂²⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

—

141

C₂³

C₂×D₁₁₆

yes

—

142

C₂³

C₂×D₁₁₆

yes

characteristic

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

Q₈

—

116

—

147

D₄

—

148

D₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

116

—

154

C₂×C₄

C₂×D₁₁₆

yes

—

155

C₂×C₄

C₂×D₁₁₆

yes

characteristic

156

C₂×C₄

#90805

yes

—

157

C₂×C₄

C₂×C₂²⋊D₂₉

yes

—

158

C₂×C₄

C₂³×D₂₉

yes

characteristic

159

C₈

—

160

C₈

C₂×C₂²⋊D₂₉

yes

—

161

C₈

C₂×C₂²⋊D₂₉

yes

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#11745292

yes

—

166

C₂²

C₂²×D₁₁₆

yes

characteristic

167

C₄

—

168

C₄

—

116

—

169

C₄

—

116

—

170

C₄

#11745292

yes

—

171

C₄

C₂×#90805

yes

characteristic

172

C₄

C₂²×C₂²⋊D₂₉

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

C₂×#11745292

yes

characteristic

176

C₂

#14271980

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₁₆
commutator, cosocle	C₂×C₄×C₂₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂×C₂₉×#131	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂×#41×C₂₉	C₂²
hypercenter	C₂×#131	D₂₉
hypocenter	C₂₉	#3059
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

subgroup

G/H

center

C₂²

C₂²×D₁₁₆

commutator, cosocle

C₂×C₄×C₂₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₉

Fitting

C₂×C₂₉×#131

C₂

socle

C₂²×C₂₉

C₂²×D₄

prime radical

C₂×#41×C₂₉

C₂²

D₂₉

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

N	H ≅ G/N	type
1. C₂²×Dic₁₁₆	C₂	semidirect
3. #14271421	C₂	semidirect
4. #14271404	C₂	semidirect
5. #14271201	C₂	semidirect
6. #14271173	C₂	semidirect
114. C₂₉	#3059	semidirect

H ≅ G/N

type

1. C₂²×Dic₁₁₆

C₂

semidirect

3. #14271421

C₂

semidirect

4. #14271404

C₂

semidirect

5. #14271201

C₂

semidirect

6. #14271173

C₂

semidirect

114. C₂₉

#3059

semidirect

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

subgroup	index
C₂²×Dic₁₁₆	2
#14271734	2
#14271421	2
#14271404	2
#14271201	2
#14271173	2
C₂×C₂₉×#131	2
#3059	29

subgroup

index

C₂²×Dic₁₁₆

C₂×C₂₉×#131

Subgroups — #16829147

Conjugacy classes of subgroups

Complements

Maximal subgroups