Group #20242639 — subgroups

Subgroups — #20242639

Order 6016.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic
2	#15689528	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689404	C₂	1	1	yes	maximal, characteristic
4	#15689333	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688868	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688852	C₂	1	1	yes	maximal, characteristic
7	C₄₇×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₄₇	—	2	1	no	—
9	C₂×#10625337	—	2	1	no	—
10	#13094024	—	2	1	no	—
11	#13094024	C₂²	1	1	yes	characteristic
12	#13094024	C₂²	1	1	yes	characteristic
13	#13094004	—	2	1	no	—
14	#13094004	—	2	2	no	—
15	C₄×D₁₈₈	—	2	1	no	—
16	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
17	#13093958	—	2	2	no	—
18	#13093958	—	2	1	no	—
19	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
20	#13093947	—	2	1	no	—
21	#13093946	—	2	1	no	—
22	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
26	Q₈×D₄₇	—	4	2	no	—
27	Q₈×D₄₇	—	4	2	no	—
28	C₂×D₁₈₈	D₄	1	1	yes	characteristic
29	C₂×C₄×D₄₇	—	2	2	no	—
30	C₂×C₄×D₄₇	—	2	1	no	—
31	C₂×Dic₉₄	—	2	2	no	—
32	C₂×Dic₉₄	—	2	1	no	—
33	C₂×Dic₉₄	—	2	1	no	—
34	C₂×C₈×C₄₇	—	2	1	no	—
35	C₂×C₈×C₄₇	—	2	1	no	—
36	C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
38	C₄²×C₄₇	—	2	1	no	—
39	C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	#10625337	—	4	2	no	—
41	(C₂×D₄₇)⋊C₄	—	2	2	no	—
42	(C₂×D₄₇)⋊C₄	—	2	1	no	—
43	C₄⋊Dic₄₇	—	2	2	no	—
44	C₄⋊Dic₄₇	—	2	1	no	—
45	C₄⋊Dic₄₇	—	2	1	no	—
46	C₄⋊Dic₄₇	—	2	1	no	—
47	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
48	C₄⋊Dic₄₇	D₄	1	2	yes	—
49	C₂×C₄₇⋊C₈	—	2	1	no	—
50	C₂²×D₄₇	—	2	1	no	—
51	Q₈×C₄₇	—	2	1	no	—
52	Q₈×C₄₇	D₈	1	2	yes	—
53	C₂×C₄×C₄₇	—	2	1	no	—
54	C₂×C₄×C₄₇	—	2	1	no	—
55	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₄₇	—	2	1	no	—
59	C₂×Dic₄₇	—	2	2	no	—
60	C₂×Dic₄₇	—	4	1	no	—
61	D₁₈₈	—	2	1	no	—
62	D₁₈₈	Q₁₆	1	2	yes	complemented
63	C₄×D₄₇	—	4	4	no	—
64	C₄×D₄₇	—	4	1	no	—
65	C₄×D₄₇	—	4	1	no	—
66	Dic₉₄	—	4	4	no	—
67	Dic₉₄	—	4	1	no	—
68	Dic₉₄	—	4	1	no	—
69	Dic₉₄	—	4	1	no	—
70	Dic₉₄	—	4	1	no	—
71	C₈×C₄₇	—	2	2	no	—
72	C₈×C₄₇	—	4	1	no	—
73	C₄₇⋊C₈	—	4	1	no	—
74	C₂²×C₄₇	#120	1	1	yes	characteristic
75	C₂×D₄₇	—	2	2	no	—
76	C₂×D₄₇	—	4	1	no	—
77	C₂×D₄₇	—	4	1	no	—
78	C₄×C₄₇	—	2	2	no	—
79	C₄×C₄₇	—	2	1	no	—
80	C₄×C₄₇	—	4	1	no	—
81	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
82	C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
83	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₄×C₄₇	#137	1	1	yes	characteristic
85	Dic₄₇	—	4	1	no	—
86	Dic₄₇	—	4	2	no	—
87	Dic₄₇	—	8	1	no	—
88	#1649	—	47	1	no	maximal, Sylow
89	C₂×C₄₇	#447	1	1	yes	characteristic
90	C₂×C₄₇	#451	1	1	yes	characteristic
91	C₂×C₄₇	#456	1	1	yes	characteristic
92	D₄₇	—	4	2	no	—
93	D₄×Q₈	—	47	1	no	—
94	#500	—	47	1	no	—
95	#474	—	47	1	no	—
96	#462	—	47	1	no	—
97	#332	—	47	1	no	—
98	Q₈⋊C₈	D₄₇	1	1	yes	characteristic, complemented
99	D₄⋊C₈	—	47	1	no	—
100	C₄₇	#1649	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	—	94	1	no	—
102	C₂×Q₁₆	—	94	1	no	—
103	C₄⋊Q₈	—	47	1	no	—
104	C₄⋊Q₈	—	47	1	no	—
105	C₄⋊Q₈	—	94	1	no	—
106	C₂²⋊Q₈	—	94	2	no	—
107	C₂²⋊Q₈	—	94	1	no	—
108	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
109	C₄×D₄	—	47	1	no	—
110	C₄×D₄	—	94	1	no	—
111	#107	—	94	1	no	—
112	#107	—	94	2	no	—
113	C₄⋊C₈	—	47	1	no	—
114	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
115	#103	—	94	1	no	—
116	#102	—	94	1	no	—
117	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
118	C₂×Q₈	—	94	2	no	—
119	C₂×Q₈	—	94	1	no	—
120	C₂×Q₈	—	94	1	no	—
121	C₂×Q₈	—	188	2	no	—
122	C₂×Q₈	—	188	2	no	—
123	C₂×Q₈	D₁₈₈	1	1	yes	characteristic
124	C₂×D₄	—	47	1	no	—
125	C₂²×C₄	—	94	2	no	—
126	C₂²×C₄	—	94	1	no	—
127	Q₁₆	—	188	2	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	94	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	47	1	no	—
133	C₄⋊C₄	—	47	2	no	—
134	C₄⋊C₄	—	94	2	no	—
135	C₄⋊C₄	—	94	1	no	—
136	C₄⋊C₄	—	94	1	no	—
137	C₄⋊C₄	—	94	1	no	—
138	C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
139	C₂²⋊C₄	—	94	2	no	—
140	C₂²⋊C₄	—	94	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	C₂²×D₄₇	1	1	yes	characteristic
143	C₂³	—	94	1	no	—
144	Q₈	—	2	1	no	—
145	Q₈	—	188	4	no	—
146	Q₈	—	188	1	no	—
147	Q₈	—	188	1	no	—
148	Q₈	—	188	1	no	—
149	Q₈	—	188	1	no	—
150	Q₈	D₃₇₆	1	2	yes	—
151	D₄	—	47	2	no	—
152	D₄	—	94	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	94	2	no	—
156	C₂×C₄	—	94	1	no	—
157	C₂×C₄	—	188	4	no	—
158	C₂×C₄	—	188	1	no	—
159	C₂×C₄	—	188	1	no	—
160	C₂×C₄	—	188	1	no	—
161	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
162	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
163	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	4	1	no	—
166	C₈	—	188	1	no	—
167	C₂²	—	94	2	no	—
168	C₂²	—	188	1	no	—
169	C₂²	—	188	1	no	—
170	C₂²	#13093994	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	2	1	no	—
173	C₄	—	4	1	no	—
174	C₄	—	376	1	no	—
175	C₄	—	188	2	no	—
176	C₄	—	188	1	no	—
177	C₄	C₂×D₃₇₆	1	1	yes	characteristic
178	C₄	Q₁₆×D₄₇	1	1	yes	characteristic
179	C₄	#13094052	1	1	yes	characteristic
180	C₄	#13094053	1	1	yes	characteristic
181	C₂	—	188	2	no	—
182	C₂	#15689259	1	1	yes	characteristic
183	C₂	#15689280	1	1	yes	characteristic
184	C₂	#15689307	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₁₈₈

C₂

yes

maximal, characteristic

#15689528

C₂

yes

maximal, characteristic, complemented

#15689404

C₂

yes

maximal, characteristic

#15689333

C₂

yes

maximal, characteristic, complemented

#15688868

C₂

yes

maximal, characteristic, complemented

#15688852

C₂

yes

maximal, characteristic

C₄₇×Q₈⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₄₇

—

C₂×#10625337

—

#13094024

—

#13094024

C₂²

yes

characteristic

#13094024

C₂²

yes

characteristic

#13094004

—

#13094004

—

C₄×D₁₈₈

—

C₄×D₁₈₈

C₂²

yes

characteristic

#13093958

—

#13093958

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

#13093947

—

#13093946

—

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

D₄

yes

characteristic

Q₈×D₄₇

—

Q₈×D₄₇

—

C₂×D₁₈₈

D₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625337

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

characteristic

C₄⋊Dic₄₇

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

D₈

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

Q₁₆

yes

complemented

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

#120

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

characteristic

C₄×C₄₇

C₂×Q₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

D₄×Q₈

—

—

—

—

—

Q₈⋊C₈

D₄₇

yes

characteristic, complemented

D₄⋊C₈

—

100

C₄₇

#1649

yes

characteristic, complemented, Sylow

101

C₂²×Q₈

—

102

C₂×Q₁₆

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

C₄⋊Q₈

—

106

C₂²⋊Q₈

—

107

C₂²⋊Q₈

—

108

C₄×Q₈

C₂×D₄₇

yes

characteristic

109

C₄×D₄

—

110

C₄×D₄

—

111

#107

—

112

#107

—

113

C₄⋊C₈

—

114

C₄⋊C₈

C₂×D₄₇

yes

characteristic

115

#103

—

116

#102

—

117

C₄×C₈

C₂×D₄₇

yes

characteristic

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

188

—

122

C₂×Q₈

—

188

—

123

C₂×Q₈

D₁₈₈

yes

characteristic

124

C₂×D₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

Q₁₆

—

188

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

D₁₈₈

yes

characteristic

139

C₂²⋊C₄

—

140

C₂²⋊C₄

—

141

C₄²

—

142

C₄²

C₂²×D₄₇

yes

characteristic

143

C₂³

—

144

Q₈

—

145

Q₈

—

188

—

146

Q₈

—

188

—

147

Q₈

—

188

—

148

Q₈

—

188

—

149

Q₈

—

188

—

150

Q₈

D₃₇₆

yes

—

151

D₄

—

152

D₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

188

—

158

C₂×C₄

—

188

—

159

C₂×C₄

—

188

—

160

C₂×C₄

—

188

—

161

C₂×C₄

C₂×D₁₈₈

yes

characteristic

162

C₂×C₄

D₄×D₄₇

yes

characteristic

163

C₂×C₄

D₄×D₄₇

yes

characteristic

164

C₈

—

165

C₈

—

166

C₈

—

188

—

167

C₂²

—

168

C₂²

—

188

—

169

C₂²

—

188

—

170

C₂²

#13093994

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

376

—

175

C₄

—

188

—

176

C₄

—

188

—

177

C₄

C₂×D₃₇₆

yes

characteristic

178

C₄

Q₁₆×D₄₇

yes

characteristic

179

C₄

#13094052

yes

characteristic

180

C₄

#13094053

yes

characteristic

181

C₂

—

188

—

182

C₂

#15689259

yes

characteristic

183

C₂

#15689280

yes

characteristic

184

C₂

#15689307

yes

characteristic

	subgroup	G/H
center	C₂²	#13093994
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×Q₈⋊C₈	C₂
socle	C₂²×C₄₇	#120
hypercenter	Q₈⋊C₈	D₄₇
hypocenter	C₄₇	#1649
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#13093994

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×Q₈⋊C₈

C₂

socle

C₂²×C₄₇

#120

hypercenter

Q₈⋊C₈

D₄₇

hypocenter

C₄₇

#1649

LCS

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
2. #15689528	C₂	semidirect
4. #15689333	C₂	semidirect
5. #15688868	C₂	semidirect
7. C₄₇×Q₈⋊C₈	C₂	semidirect
62. D₁₈₈	Q₁₆	semidirect
98. Q₈⋊C₈	D₄₇	semidirect
100. C₄₇	#1649	semidirect

H ≅ G/N

type

2. #15689528

C₂

semidirect

4. #15689333

C₂

semidirect

5. #15688868

C₂

semidirect

7. C₄₇×Q₈⋊C₈

C₂

semidirect

62. D₁₈₈

Q₁₆

semidirect

98. Q₈⋊C₈

D₄₇

semidirect

100. C₄₇

#1649

semidirect

Subgroups — #20242639

Conjugacy classes of subgroups

Complements