Group #20242551 — subgroups

Subgroups — #20242551

Order 6016.

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

normal subgroups only

This group has 228 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	#15689473	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689410	C₂	1	1	yes	maximal, characteristic
4	#15688870	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688850	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#477	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₄₇	—	2	1	no	—
9	#13094024	—	2	1	no	—
10	#13094024	C₂²	1	1	yes	characteristic
11	#13094004	—	2	1	no	—
12	#13094004	—	2	2	no	—
13	C₄×D₁₈₈	—	2	1	no	—
14	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
15	#13093955	—	2	1	no	—
16	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
17	#13093948	—	2	2	no	—
18	#13093944	—	2	1	no	—
19	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
22	#107×C₄₇	—	2	1	no	—
23	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
24	#103×C₄₇	—	2	1	no	—
25	C₂×Q₈×C₄₇	—	2	1	no	—
26	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
27	Q₈×D₄₇	—	4	2	no	—
28	Q₈×D₄₇	—	4	2	no	—
29	C₂×D₁₈₈	D₄	1	1	yes	characteristic
30	C₂×C₄×D₄₇	—	2	2	no	—
31	C₂×C₄×D₄₇	—	2	1	no	—
32	C₂×Dic₉₄	—	2	2	no	—
33	C₂×Dic₉₄	—	2	1	no	—
34	C₂×C₈×C₄₇	—	2	1	no	—
35	C₄⋊C₄×C₄₇	—	2	1	no	—
36	C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₄⋊C₄×C₄₇	—	2	1	no	—
38	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₄²×C₄₇	—	2	1	no	—
40	C₄²×C₄₇	C₂³	1	1	yes	characteristic
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	D₄	1	2	yes	—
46	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄₇⋊C₈	—	2	1	no	—
48	C₂×C₄₇⋊C₈	—	2	1	no	—
49	C₂²×D₄₇	—	2	1	no	—
50	Q₈×C₄₇	—	2	1	no	—
51	Q₈×C₄₇	—	4	2	no	—
52	Q₈×C₄₇	SD₁₆	1	2	yes	—
53	C₂×C₄×C₄₇	—	2	1	no	—
54	C₂×C₄×C₄₇	—	2	1	no	—
55	C₂×C₄×C₄₇	—	4	1	no	—
56	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂×Dic₄₇	—	2	1	no	—
60	C₂×Dic₄₇	—	2	2	no	—
61	D₁₈₈	—	2	1	no	—
62	D₁₈₈	Q₁₆	1	2	yes	—
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	C₈×C₄₇	—	4	1	no	—
70	C₄₇⋊C₈	—	4	1	no	—
71	C₄₇⋊C₈	—	4	1	no	—
72	C₂²×C₄₇	#120	1	1	yes	characteristic
73	C₂×D₄₇	—	2	2	no	—
74	C₂×D₄₇	—	4	1	no	—
75	C₂×D₄₇	—	4	1	no	—
76	C₄×C₄₇	—	2	2	no	—
77	C₄×C₄₇	—	2	1	no	—
78	C₄×C₄₇	—	4	1	no	—
79	C₄×C₄₇	—	8	1	no	—
80	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
81	C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
82	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₄×C₄₇	#137	1	1	yes	characteristic
84	Dic₄₇	—	4	1	no	—
85	Dic₄₇	—	4	2	no	—
86	#1637	—	47	1	no	maximal, Sylow
87	C₂×C₄₇	#450	1	1	yes	characteristic
88	C₂×C₄₇	#451	1	1	yes	characteristic
89	C₂×C₄₇	#454	1	1	yes	characteristic
90	D₄₇	—	4	2	no	—
91	D₄×Q₈	—	47	1	no	—
92	#487	—	47	1	no	—
93	#477	D₄₇	1	1	yes	characteristic, complemented
94	#476	—	47	1	no	—
95	#332	—	47	1	no	—
96	Q₈⋊C₈	—	47	1	no	—
97	D₄⋊C₈	—	47	1	no	—
98	C₄₇	#1637	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	—	94	1	no	—
100	C₄⋊Q₈	—	47	1	no	—
101	C₄⋊Q₈	—	94	1	no	—
102	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
103	C₂²⋊Q₈	—	94	2	no	—
104	C₂²⋊Q₈	—	94	1	no	—
105	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
106	C₄×D₄	—	47	1	no	—
107	C₄×D₄	—	94	1	no	—
108	#107	—	2	1	no	—
109	#106	—	94	1	no	—
110	C₄⋊C₈	—	47	1	no	—
111	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
112	#103	—	2	1	no	—
113	#103	—	94	2	no	—
114	#102	—	94	1	no	—
115	C₄×C₈	—	47	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	94	2	no	—
118	C₂×Q₈	—	94	1	no	—
119	C₂×Q₈	—	188	2	no	—
120	C₂×Q₈	—	188	2	no	—
121	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
122	C₂×D₄	—	47	1	no	—
123	C₂²×C₄	—	94	2	no	—
124	C₂²×C₄	—	94	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	94	1	no	—
127	C₂×C₈	—	94	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	47	2	no	—
132	C₄⋊C₄	—	47	1	no	—
133	C₄⋊C₄	—	94	2	no	—
134	C₄⋊C₄	—	94	1	no	—
135	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
136	C₂²⋊C₄	—	94	2	no	—
137	C₂²⋊C₄	—	94	1	no	—
138	C₄²	—	2	1	no	—
139	C₄²	C₂²×D₄₇	1	1	yes	characteristic
140	C₂³	—	94	1	no	—
141	Q₈	—	2	1	no	—
142	Q₈	—	4	2	no	—
143	Q₈	—	188	4	no	—
144	Q₈	—	188	1	no	—
145	Q₈	—	188	1	no	—
146	Q₈	#10625336	1	2	yes	—
147	D₄	—	47	2	no	—
148	D₄	—	94	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	94	2	no	—
153	C₂×C₄	—	94	1	no	—
154	C₂×C₄	—	188	4	no	—
155	C₂×C₄	—	188	1	no	—
156	C₂×C₄	—	188	1	no	—
157	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
158	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
159	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
160	C₈	—	4	1	no	—
161	C₈	—	188	1	no	—
162	C₈	—	188	1	no	—
163	C₂²	—	94	2	no	—
164	C₂²	—	188	1	no	—
165	C₂²	—	188	1	no	—
166	C₂²	#13093992	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	8	1	no	—
171	C₄	—	188	2	no	—
172	C₄	—	188	1	no	—
173	C₄	C₂×#10625336	1	1	yes	characteristic
174	C₄	Q₁₆×D₄₇	1	1	yes	characteristic
175	C₄	#13094049	1	1	yes	characteristic
176	C₄	#13094056	1	1	yes	characteristic
177	C₂	—	188	2	no	—
178	C₂	#15689273	1	1	yes	characteristic
179	C₂	#15689277	1	1	yes	characteristic
180	C₂	#15689294	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₁₈₈

C₂

yes

maximal, characteristic

#15689473

C₂

yes

maximal, characteristic, complemented

#15689410

C₂

yes

maximal, characteristic

#15688870

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

#15688850

C₂

yes

maximal, characteristic

C₄₇×#477

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₄₇

—

#13094024

—

#13094024

C₂²

yes

characteristic

#13094004

—

#13094004

—

C₄×D₁₈₈

—

C₄×D₁₈₈

C₂²

yes

characteristic

#13093955

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

#13093948

—

#13093944

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

C₂²

yes

characteristic

#107×C₄₇

—

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

#103×C₄₇

—

C₂×Q₈×C₄₇

—

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×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

—

C₄⋊Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

—

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

—

D₁₈₈

—

D₁₈₈

Q₁₆

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

#120

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₄×C₄₇

C₂×Q₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

D₄×Q₈

—

#487

—

#477

D₄₇

yes

characteristic, complemented

#476

—

#332

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₄₇

#1637

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

100

C₄⋊Q₈

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

103

C₂²⋊Q₈

—

104

C₂²⋊Q₈

—

105

C₄×Q₈

C₂×D₄₇

yes

characteristic

106

C₄×D₄

—

107

C₄×D₄

—

108

#107

—

109

#106

—

110

C₄⋊C₈

—

111

C₄⋊C₈

C₂×D₄₇

yes

characteristic

112

#103

—

113

#103

—

114

#102

—

115

C₄×C₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

188

—

120

C₂×Q₈

—

188

—

121

C₂×Q₈

C₂²⋊D₄₇

yes

characteristic

122

C₂×D₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

C₂²⋊D₄₇

yes

characteristic

136

C₂²⋊C₄

—

137

C₂²⋊C₄

—

138

C₄²

—

139

C₄²

C₂²×D₄₇

yes

characteristic

140

C₂³

—

141

Q₈

—

142

Q₈

—

143

Q₈

—

188

—

144

Q₈

—

188

—

145

Q₈

—

188

—

146

Q₈

#10625336

yes

—

147

D₄

—

148

D₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

188

—

155

C₂×C₄

—

188

—

156

C₂×C₄

—

188

—

157

C₂×C₄

D₄×D₄₇

yes

characteristic

158

C₂×C₄

D₄×D₄₇

yes

characteristic

159

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

160

C₈

—

161

C₈

—

188

—

162

C₈

—

188

—

163

C₂²

—

164

C₂²

—

188

—

165

C₂²

—

188

—

166

C₂²

#13093992

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

188

—

172

C₄

—

188

—

173

C₄

C₂×#10625336

yes

characteristic

174

C₄

Q₁₆×D₄₇

yes

characteristic

175

C₄

#13094049

yes

characteristic

176

C₄

#13094056

yes

characteristic

177

C₂

—

188

—

178

C₂

#15689273

yes

characteristic

179

C₂

#15689277

yes

characteristic

180

C₂

#15689294

yes

characteristic

	subgroup	G/H
center	C₂²	#13093992
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#477	C₂
socle	C₂²×C₄₇	#120
hypercenter	#477	D₄₇
hypocenter	C₄₇	#1637
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#13093992

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#477

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
2. #15689473	C₂	semidirect
4. #15688870	C₂	semidirect
5. Q₈⋊(C₄₇⋊C₈)	C₂	semidirect
7. C₄₇×#477	C₂	semidirect
93. #477	D₄₇	semidirect
98. C₄₇	#1637	semidirect

H ≅ G/N

type

2. #15689473

C₂

semidirect

4. #15688870

C₂

semidirect

5. Q₈⋊(C₄₇⋊C₈)

C₂

semidirect

7. C₄₇×#477

C₂

semidirect

93. #477

D₄₇

semidirect

98. C₄₇

#1637

semidirect

Subgroups — #20242551

Conjugacy classes of subgroups

Complements