Group #20242103 — subgroups

Subgroups — #20242103

Order 6016.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₄⋊Q₈×D₄₇	C₂	1	1	yes	maximal, characteristic
2	#15689179	C₂	1	1	yes	maximal, characteristic
3	#15689179	C₂	1	1	yes	maximal, characteristic
4	#15688870	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688869	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688868	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₄₇	—	2	1	no	—
9	#13094024	C₂²	1	1	yes	characteristic
10	#13094023	—	2	1	no	—
11	#13094023	C₄	1	2	yes	—
12	#13094022	—	2	1	no	—
13	C₄⋊C₄×D₄₇	—	2	1	no	—
14	C₄⋊C₄×D₄₇	C₄	1	2	yes	—
15	C₄²×D₄₇	C₂²	1	1	yes	characteristic
16	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
18	(C₂×D₄₇)⋊C₈	—	2	1	no	—
19	(C₂×D₄₇)⋊C₈	—	2	1	no	—
20	#13093934	—	2	1	no	—
21	#13093934	—	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	—	2	1	no	—
26	Q₈×D₄₇	—	4	2	no	—
27	Q₈×D₄₇	—	4	2	no	—
28	C₂×C₄×D₄₇	—	2	2	no	—
29	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₄₇	D₄	1	1	yes	characteristic
31	C₂×C₄×D₄₇	D₄	1	1	yes	characteristic
32	C₂×Dic₉₄	—	2	1	no	—
33	C₂×Dic₉₄	—	2	2	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	C₂×C₄	1	2	yes	—
38	C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₄⋊Dic₄₇	—	2	1	no	—
40	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
41	Dic₄₇⋊C₄	—	2	2	no	—
42	Dic₄₇⋊C₄	—	2	2	no	—
43	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₄₇	D₄	1	1	yes	characteristic
45	C₄×Dic₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄₇⋊C₈	—	2	1	no	—
47	C₂×C₄₇⋊C₈	—	2	1	no	—
48	C₂²×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	Q₈×C₄₇	—	4	1	no	—
50	Q₈×C₄₇	—	4	1	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₄₇	—	2	2	no	—
56	C₂×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₄×D₄₇	—	2	2	no	—
60	C₄×D₄₇	—	2	1	no	—
61	C₄×D₄₇	—	2	1	no	—
62	C₄×D₄₇	—	2	1	no	—
63	C₄×D₄₇	—	2	1	no	—
64	C₄×D₄₇	—	4	4	no	—
65	Dic₉₄	—	4	4	no	—
66	Dic₉₄	—	4	1	no	—
67	Dic₉₄	—	4	1	no	—
68	C₈×C₄₇	—	4	1	no	—
69	C₈×C₄₇	—	4	1	no	—
70	C₄₇⋊C₈	—	4	1	no	—
71	C₄₇⋊C₈	—	4	1	no	—
72	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×D₄₇	—	2	1	no	—
74	C₂×D₄₇	—	2	1	no	—
75	C₂×D₄₇	#101	1	2	yes	—
76	C₄×C₄₇	—	2	1	no	—
77	C₄×C₄₇	—	4	2	no	—
78	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₄×C₄₇	#137	1	1	yes	characteristic
81	C₄×C₄₇	#137	1	1	yes	characteristic
82	Dic₄₇	—	2	1	no	—
83	Dic₄₇	—	2	1	no	—
84	Dic₄₇	—	2	1	no	—
85	Dic₄₇	—	4	2	no	—
86	#1556	—	47	1	no	maximal, Sylow
87	C₂×C₄₇	C₂×#101	1	1	yes	characteristic
88	C₂×C₄₇	#418	1	1	yes	characteristic
89	C₂×C₄₇	#419	1	1	yes	characteristic
90	D₄₇	—	2	2	no	—
91	C₂×C₄⋊Q₈	—	47	1	no	—
92	#432	—	47	1	no	—
93	#432	—	47	1	no	—
94	#332	—	47	1	no	—
95	#332	—	47	1	no	—
96	#332	—	47	1	no	—
97	#332	D₄₇	1	1	yes	characteristic, complemented
98	C₄₇	#1556	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	—	94	1	no	—
100	C₄⋊Q₈	—	47	2	no	—
101	C₄⋊Q₈	—	47	1	no	—
102	C₄⋊Q₈	—	94	1	no	—
103	C₄⋊Q₈	—	94	1	no	—
104	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	47	2	no	—
106	C₂×C₄⋊C₄	—	94	1	no	—
107	C₂×C₄²	—	47	1	no	—
108	C₄⋊C₈	—	47	1	no	—
109	C₄⋊C₈	—	47	1	no	—
110	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
111	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
112	C₂²⋊C₈	—	94	1	no	—
113	C₂²⋊C₈	—	94	1	no	—
114	#97	—	94	1	no	—
115	#97	—	94	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	C₂²×C₄	—	47	1	no	—
122	C₂²×C₄	—	47	1	no	—
123	C₂²×C₄	—	47	1	no	—
124	C₂²×C₄	—	94	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	94	1	no	—
128	C₂×C₈	—	94	1	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	47	2	no	—
131	C₄⋊C₄	—	94	2	no	—
132	C₄⋊C₄	—	94	2	no	—
133	C₄⋊C₄	—	94	1	no	—
134	C₄⋊C₄	C₄×D₄₇	1	2	yes	—
135	C₄²	—	47	1	no	—
136	C₄²	—	47	1	no	—
137	C₄²	—	47	1	no	—
138	C₄²	C₂²×D₄₇	1	1	yes	characteristic
139	C₂³	—	47	1	no	—
140	Q₈	—	4	1	no	—
141	Q₈	—	4	1	no	—
142	Q₈	—	188	4	no	—
143	Q₈	—	188	1	no	—
144	Q₈	—	188	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	47	1	no	—
147	C₂×C₄	—	47	1	no	—
148	C₂×C₄	—	47	1	no	—
149	C₂×C₄	—	94	2	no	—
150	C₂×C₄	—	94	1	no	—
151	C₂×C₄	—	94	1	no	—
152	C₂×C₄	—	94	1	no	—
153	C₂×C₄	—	94	1	no	—
154	C₂×C₄	—	94	2	no	—
155	C₂×C₄	—	188	4	no	—
156	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
157	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
158	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
159	C₈	—	4	1	no	—
160	C₈	—	4	1	no	—
161	C₈	—	188	1	no	—
162	C₈	—	188	1	no	—
163	C₂²	—	47	2	no	—
164	C₂²	—	94	1	no	—
165	C₂²	—	94	1	no	—
166	C₂²	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	4	2	no	—
169	C₄	—	94	1	no	—
170	C₄	—	94	1	no	—
171	C₄	—	94	1	no	—
172	C₄	—	188	2	no	—
173	C₄	#13094048	1	1	yes	characteristic
174	C₄	#13094051	1	1	yes	characteristic
175	C₄	#13094053	1	1	yes	characteristic
176	C₄	#13094056	1	1	yes	characteristic
177	C₂	—	94	2	no	—
178	C₂	#101×D₄₇	1	1	yes	characteristic
179	C₂	#15689126	1	1	yes	characteristic
180	C₂	#15689130	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₄₇

C₂

yes

maximal, characteristic

#15689179

C₂

yes

maximal, characteristic

#15689179

C₂

yes

maximal, characteristic

#15688870

C₂

yes

maximal, characteristic, complemented

#15688869

C₂

yes

maximal, characteristic, complemented

#15688868

C₂

yes

maximal, characteristic, complemented

C₄₇×#332

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₄₇

—

#13094024

C₂²

yes

characteristic

#13094023

—

#13094023

C₄

yes

—

#13094022

—

C₄⋊C₄×D₄₇

—

C₄⋊C₄×D₄₇

C₄

yes

—

C₄²×D₄₇

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

(C₂×D₄₇)⋊C₈

—

(C₂×D₄₇)⋊C₈

—

#13093934

—

#13093934

—

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

Q₈×D₄₇

—

Q₈×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

D₄

yes

characteristic

C₂×C₄×D₄₇

D₄

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₄×Dic₄₇

D₄

yes

characteristic

C₄×Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂²×D₄₇

C₂²⋊C₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₇

C₂²⋊C₄

yes

characteristic

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

#101

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

C₂×C₄⋊Q₈

—

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₄₇

#1556

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

100

C₄⋊Q₈

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄²

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

C₂×D₄₇

yes

characteristic

111

C₄⋊C₈

C₂×D₄₇

yes

characteristic

112

C₂²⋊C₈

—

113

C₂²⋊C₈

—

114

#97

—

115

#97

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

188

—

120

C₂×Q₈

—

188

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

C₄×D₄₇

yes

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

C₂²×D₄₇

yes

characteristic

139

C₂³

—

140

Q₈

—

141

Q₈

—

142

Q₈

—

188

—

143

Q₈

—

188

—

144

Q₈

—

188

—

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

—

155

C₂×C₄

—

188

—

156

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

157

C₂×C₄

D₄×D₄₇

yes

characteristic

158

C₂×C₄

D₄×D₄₇

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

188

—

162

C₈

—

188

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

C₂²⋊C₄×D₄₇

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

188

—

173

C₄

#13094048

yes

characteristic

174

C₄

#13094051

yes

characteristic

175

C₄

#13094053

yes

characteristic

176

C₄

#13094056

yes

characteristic

177

C₂

—

178

C₂

#101×D₄₇

yes

characteristic

179

C₂

#15689126

yes

characteristic

180

C₂

#15689130

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#332	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#332	D₄₇
hypocenter	C₄₇	#1556
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#332

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
4. #15688870	C₂	semidirect
5. #15688869	C₂	semidirect
6. #15688868	C₂	semidirect
7. C₄₇×#332	C₂	semidirect
97. #332	D₄₇	semidirect
98. C₄₇	#1556	semidirect

H ≅ G/N

type

4. #15688870

C₂

semidirect

5. #15688869

C₂

semidirect

6. #15688868

C₂

semidirect

7. C₄₇×#332

C₂

semidirect

97. #332

D₄₇

semidirect

98. C₄₇

#1556

semidirect

Subgroups — #20242103

Conjugacy classes of subgroups

Complements