Group #20249218 — subgroups

Subgroups — #20249218

Order 6016.

472 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13094016	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689056	C₂	1	4	yes	maximal, complemented
3	#15689046	C₂	1	4	yes	maximal
4	#15689021	C₂	1	2	yes	maximal, complemented
5	#15688993	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688983	C₂	1	2	yes	maximal
7	C₂×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
8	#13094016	—	2	4	no	—
9	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
12	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
13	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
14	C₂×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
15	C₄₇⋊#95	C₂²	1	4	yes	—
16	C₄₇⋊#95	C₂²	1	4	yes	—
17	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
18	#117×C₄₇	—	2	4	no	—
19	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	—	2	2	no	—
25	C₂²×C₄×C₄₇	—	2	2	no	—
26	C₂²×C₄×C₄₇	Q₈	1	4	yes	—
27	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
28	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
29	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂²×Dic₄₇	C₂³	1	4	yes	—
32	C₂²×Dic₄₇	C₂³	1	4	yes	—
33	C₄⋊C₄×C₄₇	—	2	4	no	—
34	C₂²⋊C₄×C₄₇	—	2	4	no	—
35	C₄²×C₄₇	—	2	2	no	—
36	C₄²×C₄₇	D₄	1	4	yes	—
37	C₂²⋊Dic₄₇	—	4	4	no	—
38	C₄⋊Dic₄₇	—	2	4	no	—
39	C₄⋊Dic₄₇	—	2	4	no	—
40	C₄⋊Dic₄₇	—	4	4	no	—
41	Dic₄₇⋊C₄	—	2	8	no	—
42	Dic₄₇⋊C₄	—	4	4	no	—
43	C₂³×C₄₇	—	2	1	no	—
44	C₂³×C₄₇	—	2	1	no	—
45	C₂³×C₄₇	—	2	2	no	—
46	C₂³×C₄₇	—	2	2	no	—
47	C₂³×C₄₇	C₂×Q₈	1	2	yes	—
48	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
49	C₂×C₄×C₄₇	—	2	1	no	—
50	C₂×C₄×C₄₇	—	2	1	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	—	2	8	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	—	2	2	no	—
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
58	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₄₇	#41	1	2	yes	—
62	C₂×Dic₄₇	—	2	4	no	—
63	C₂×Dic₄₇	—	2	4	no	—
64	C₂×Dic₄₇	—	2	4	no	—
65	C₂×Dic₄₇	—	2	4	no	—
66	C₂×Dic₄₇	—	2	4	no	—
67	C₂×Dic₄₇	—	2	4	no	—
68	C₂²×C₄₇	—	2	2	no	—
69	C₂²×C₄₇	—	2	4	no	—
70	C₂²×C₄₇	—	4	2	no	—
71	C₂²×C₄₇	—	4	1	no	—
72	C₂²×C₄₇	—	4	1	no	—
73	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
74	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
76	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
77	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
78	C₄×C₄₇	—	2	2	no	—
79	C₄×C₄₇	—	2	4	no	—
80	C₄×C₄₇	—	4	2	no	—
81	C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
82	Dic₄₇	—	4	4	no	—
83	Dic₄₇	—	4	4	no	—
84	#2629	—	47	1	no	maximal, Sylow
85	C₂×C₄₇	—	4	2	no	—
86	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄₇	#537	1	2	yes	—
88	C₂×C₄₇	#540	1	1	yes	characteristic
89	C₂×C₄₇	#541	1	1	yes	characteristic
90	C₂×C₄₇	#544	1	2	yes	—
91	C₂×#125	—	47	1	no	—
92	C₂×#117	D₄₇	1	1	yes	characteristic
93	#398	—	47	4	no	—
94	#396	—	47	4	no	—
95	#389	—	47	2	no	—
96	#384	—	47	1	no	—
97	#380	—	47	2	no	—
98	C₄₇	#2629	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
100	#125	—	94	4	no	—
101	#117	—	2	4	no	—
102	C₂×C₄⋊C₄	—	47	4	no	—
103	C₂×C₄⋊C₄	—	47	2	no	—
104	C₂×C₄⋊C₄	—	47	2	no	—
105	C₂×C₄⋊C₄	—	47	4	no	—
106	C₂×C₄⋊C₄	—	47	4	no	—
107	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
108	C₂×C₂²⋊C₄	—	47	4	no	—
109	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
110	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
111	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
112	#95	—	47	4	no	—
113	#95	—	47	4	no	—
114	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	47	4	no	—
118	C₂²×C₄	—	47	4	no	—
119	C₂²×C₄	Dic₉₄	1	4	yes	—
120	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
121	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
122	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
123	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
124	C₄⋊C₄	—	2	4	no	—
125	C₄⋊C₄	—	94	4	no	—
126	C₄⋊C₄	—	94	8	no	—
127	C₄⋊C₄	—	94	4	no	—
128	C₄⋊C₄	—	188	4	no	—
129	C₄⋊C₄	—	188	4	no	—
130	C₂²⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	—	188	4	no	—
132	C₄²	—	2	2	no	—
133	C₄²	C₂²⋊D₄₇	1	4	yes	—
134	C₂³	—	2	1	no	—
135	C₂³	—	2	1	no	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	2	no	—
138	C₂³	C₂×Dic₉₄	1	2	yes	—
139	C₂³	C₂³×D₄₇	1	1	yes	characteristic
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	94	4	no	—
149	C₂×C₄	—	94	4	no	—
150	C₂×C₄	—	94	4	no	—
151	C₂×C₄	—	94	4	no	—
152	C₂×C₄	—	94	4	no	—
153	C₂×C₄	—	94	4	no	—
154	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
155	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
156	C₂×C₄	#10625350	1	2	yes	—
157	C₂×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
158	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
159	C₂²	—	2	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	4	1	no	—
163	C₂²	—	4	2	no	—
164	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
165	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
166	C₂²	C₂×#10625350	1	1	yes	characteristic
167	C₂²	#13094070	1	2	yes	—
168	C₂²	#13094071	1	2	yes	—
169	C₄	—	2	2	no	—
170	C₄	—	2	4	no	—
171	C₄	—	4	2	no	—
172	C₄	—	188	4	no	—
173	C₄	—	188	4	no	—
174	C₄	#13094003	1	4	yes	—
175	C₂	—	4	2	no	—
176	C₂	C₂×#13094003	1	1	yes	characteristic
177	C₂	#15689701	1	2	yes	—
178	C₂	#15689728	1	1	yes	characteristic
179	C₂	#15689735	1	1	yes	characteristic
180	C₂	#15689751	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094016

C₂

yes

maximal, characteristic, complemented

#15689056

C₂

yes

maximal, complemented

#15689046

C₂

yes

maximal

#15689021

C₂

yes

maximal, complemented

#15688993

C₂

yes

maximal, characteristic, complemented

#15688983

C₂

yes

maximal

C₂×#117×C₄₇

C₂

yes

maximal, characteristic

#13094016

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

#117×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×Q₈

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

Dic₄₇

—

Dic₄₇

—

#2629

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₄₇

#537

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×#125

—

C₂×#117

D₄₇

yes

—

—

—

—

—

C₄₇

#2629

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₇

yes

characteristic

100

#125

—

101

#117

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

108

C₂×C₂²⋊C₄

—

109

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

110

C₂×C₄²

C₂×D₄₇

yes

characteristic

111

C₂×C₄²

C₂×D₄₇

yes

characteristic

112

#95

—

113

#95

—

114

C₂⁴

C₂²×D₄₇

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

Dic₉₄

yes

—

120

C₂²×C₄

C₂²×D₄₇

yes

—

121

C₂²×C₄

C₂²×D₄₇

yes

characteristic

122

C₂²×C₄

C₂²×D₄₇

yes

—

123

C₂²×C₄

C₂²×D₄₇

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

188

—

129

C₄⋊C₄

—

188

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

188

—

132

C₄²

—

133

C₄²

C₂²⋊D₄₇

yes

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₂×Dic₉₄

yes

—

139

C₂³

C₂³×D₄₇

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

C₂×Dic₉₄

yes

—

155

C₂×C₄

C₂×Dic₉₄

yes

—

156

C₂×C₄

#10625350

yes

—

157

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

158

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

165

C₂²

C₂²×Dic₉₄

yes

characteristic

166

C₂²

C₂×#10625350

yes

characteristic

167

C₂²

#13094070

yes

—

168

C₂²

#13094071

yes

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

188

—

173

C₄

—

188

—

174

C₄

#13094003

yes

—

175

C₂

—

176

C₂

C₂×#13094003

yes

characteristic

177

C₂

#15689701

yes

—

178

C₂

#15689728

yes

characteristic

179

C₂

#15689735

yes

characteristic

180

C₂

#15689751

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×#117×C₄₇	C₂
hypercenter	C₂×#117	D₄₇
hypocenter	C₄₇	#2629
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×#117×C₄₇

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×#117

N	H ≅ G/N	type
1. C₂×#13094016	C₂	semidirect
2. #15689056	C₂	semidirect
4. #15689021	C₂	semidirect
5. #15688993	C₂	semidirect
98. C₄₇	#2629	semidirect

H ≅ G/N

type

1. C₂×#13094016

C₂

semidirect

2. #15689056

C₂

semidirect

4. #15689021

C₂

semidirect

5. #15688993

C₂

semidirect

98. C₄₇

#2629

semidirect

Subgroups — #20249218

Conjugacy classes of subgroups

Complements