Group #20242574 — subgroups

Subgroups — #20242574

Order 6016.

212 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 57 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	#15689469	C₂	1	1	yes	maximal, characteristic
2	#15689408	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689333	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688867	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#559	C₂	1	1	yes	maximal, characteristic
8	C₂×#10625337	—	2	1	no	—
9	#13094024	C₂²	1	1	yes	characteristic
10	#13093955	—	2	1	no	—
11	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	#13093948	—	2	1	no	—
14	#13093947	—	2	1	no	—
15	#13093945	—	2	2	no	—
16	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	#127×C₄₇	C₂²	1	1	yes	characteristic
18	#124×C₄₇	—	2	1	no	—
19	#124×C₄₇	—	2	1	no	—
20	#123×C₄₇	—	2	2	no	—
21	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₄₇	—	2	1	no	—
24	C₄×D₄×C₄₇	—	2	1	no	—
25	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
26	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
27	C₂×D₄×C₄₇	—	2	1	no	—
28	C₂×D₄×C₄₇	—	2	1	no	—
29	C₂×D₄×C₄₇	D₄	1	2	yes	—
30	C₂²×C₄×C₄₇	—	2	2	no	—
31	C₂×Dic₉₄	—	2	1	no	—
32	C₄⋊C₄×C₄₇	—	2	1	no	—
33	C₄⋊C₄×C₄₇	—	2	1	no	—
34	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
36	C₂²⋊C₄×C₄₇	—	2	2	no	—
37	C₂²⋊C₄×C₄₇	—	2	2	no	—
38	C₂²⋊C₄×C₄₇	—	2	2	no	—
39	C₄²×C₄₇	—	2	1	no	—
40	C₄²×C₄₇	—	2	1	no	—
41	C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	#10625337	—	4	2	no	—
43	C₄⋊Dic₄₇	—	2	1	no	—
44	C₄⋊Dic₄₇	—	2	1	no	—
45	C₂×C₄₇⋊C₈	—	2	1	no	—
46	C₂×C₄₇⋊C₈	—	2	1	no	—
47	C₂×C₄₇⋊C₈	—	2	1	no	—
48	C₂³×C₄₇	—	2	2	no	—
49	Q₈×C₄₇	—	2	1	no	—
50	Q₈×C₄₇	—	2	1	no	—
51	Q₈×C₄₇	—	2	1	no	—
52	Q₈×C₄₇	—	2	1	no	—
53	D₄×C₄₇	—	4	2	no	—
54	D₄×C₄₇	—	4	1	no	—
55	D₄×C₄₇	—	4	1	no	—
56	D₄×C₄₇	—	4	1	no	—
57	D₄×C₄₇	—	4	1	no	—
58	C₂×C₄×C₄₇	—	2	1	no	—
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	2	1	no	—
61	C₂×C₄×C₄₇	—	2	1	no	—
62	C₂×C₄×C₄₇	—	8	2	no	—
63	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂×Dic₄₇	—	4	1	no	—
67	Dic₉₄	—	4	1	no	—
68	Dic₉₄	—	4	1	no	—
69	C₄₇⋊C₈	—	4	1	no	—
70	C₄₇⋊C₈	—	4	1	no	—
71	C₄₇⋊C₈	—	4	1	no	—
72	C₂²×C₄₇	—	4	2	no	—
73	C₂²×C₄₇	—	4	2	no	—
74	C₂²×C₄₇	—	4	2	no	—
75	C₂²×C₄₇	#120	1	1	yes	characteristic
76	C₄×C₄₇	—	2	1	no	—
77	C₄×C₄₇	—	4	1	no	—
78	C₄×C₄₇	—	4	1	no	—
79	C₄×C₄₇	—	4	1	no	—
80	C₄×C₄₇	—	4	1	no	—
81	C₄×C₄₇	#135	1	1	yes	characteristic
82	C₄×C₄₇	#135	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	—	8	1	no	—
86	#1640	—	47	1	no	maximal, Sylow
87	C₂×C₄₇	—	8	2	no	—
88	C₂×C₄₇	#449	1	1	yes	characteristic
89	C₂×C₄₇	#452	1	1	yes	characteristic
90	C₂×C₄₇	#455	1	1	yes	characteristic
91	#559	D₄₇	1	1	yes	characteristic
92	#486	—	47	1	no	—
93	#475	—	47	1	no	—
94	#462	—	47	1	no	—
95	#331	—	47	1	no	—
96	Q₈⋊C₈	—	47	1	no	—
97	Q₈⋊C₈	—	47	1	no	—
98	C₄₇	#1640	1	1	yes	characteristic, complemented, Sylow
99	C₂×Q₁₆	—	94	1	no	—
100	C₄⋊Q₈	—	47	1	no	—
101	#127	C₂×D₄₇	1	1	yes	characteristic
102	#124	—	2	1	no	—
103	#124	—	2	1	no	—
104	#123	—	2	2	no	—
105	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
106	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
107	C₄×D₄	—	2	1	no	—
108	C₄×D₄	—	2	1	no	—
109	#106	—	94	1	no	—
110	C₄⋊C₈	—	47	1	no	—
111	C₄⋊C₈	—	47	1	no	—
112	#103	—	94	1	no	—
113	#103	—	94	1	no	—
114	#102	—	94	2	no	—
115	C₄×C₈	—	47	1	no	—
116	C₂×Q₈	—	94	1	no	—
117	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
118	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
119	C₂×D₄	—	2	1	no	—
120	C₂×D₄	—	2	1	no	—
121	C₂×D₄	C₂²⋊D₄₇	1	2	yes	—
122	C₂²×C₄	—	2	2	no	—
123	Q₁₆	—	188	2	no	—
124	C₂×C₈	—	94	1	no	—
125	C₂×C₈	—	94	1	no	—
126	C₂×C₈	—	94	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	94	1	no	—
130	C₄⋊C₄	—	94	1	no	—
131	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
132	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
133	C₂²⋊C₄	—	2	2	no	—
134	C₂²⋊C₄	—	2	2	no	—
135	C₂²⋊C₄	—	2	2	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	C₂²×D₄₇	1	1	yes	characteristic
139	C₂³	—	2	2	no	—
140	Q₈	—	2	1	no	—
141	Q₈	—	2	1	no	—
142	Q₈	—	2	1	no	—
143	Q₈	—	2	1	no	—
144	Q₈	—	188	1	no	—
145	Q₈	—	188	1	no	—
146	D₄	—	4	2	no	—
147	D₄	—	4	1	no	—
148	D₄	—	4	1	no	—
149	D₄	—	4	1	no	—
150	D₄	—	4	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	188	1	no	—
157	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
158	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
159	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
160	C₈	—	188	1	no	—
161	C₈	—	188	1	no	—
162	C₈	—	188	1	no	—
163	C₂²	—	4	2	no	—
164	C₂²	—	4	2	no	—
165	C₂²	—	4	2	no	—
166	C₂²	#13093993	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	4	1	no	—
171	C₄	—	4	1	no	—
172	C₄	—	376	1	no	—
173	C₄	#13094044	1	1	yes	characteristic
174	C₄	#13094044	1	1	yes	characteristic
175	C₄	#13094049	1	1	yes	characteristic
176	C₄	#13094054	1	1	yes	characteristic
177	C₂	—	8	2	no	—
178	C₂	#15689268	1	1	yes	characteristic
179	C₂	#15689285	1	1	yes	characteristic
180	C₂	#15689301	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689469

C₂

yes

maximal, characteristic

#15689408

C₂

yes

maximal, characteristic, complemented

#15689333

C₂

yes

maximal, characteristic, complemented

#15688867

C₂

yes

maximal, characteristic

Q₈⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₄₇×#559

C₂

yes

maximal, characteristic

C₂×#10625337

—

#13094024

C₂²

yes

characteristic

#13093955

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

—

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

#127×C₄₇

C₂²

yes

characteristic

#124×C₄₇

—

#124×C₄₇

—

#123×C₄₇

—

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₄×D₄×C₄₇

—

C₄×D₄×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625337

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

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

—

Dic₉₄

—

Dic₉₄

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#120

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#1640

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₄₇

#1640

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

100

C₄⋊Q₈

—

101

#127

C₂×D₄₇

yes

characteristic

102

#124

—

103

#124

—

104

#123

—

105

C₄×Q₈

C₂×D₄₇

yes

characteristic

106

C₄×Q₈

C₂×D₄₇

yes

characteristic

107

C₄×D₄

—

108

C₄×D₄

—

109

#106

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

#103

—

113

#103

—

114

#102

—

115

C₄×C₈

—

116

C₂×Q₈

—

117

C₂×Q₈

C₂²⋊D₄₇

yes

characteristic

118

C₂×Q₈

C₂²⋊D₄₇

yes

characteristic

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

C₂²⋊D₄₇

yes

—

122

C₂²×C₄

—

123

Q₁₆

—

188

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

C₂²⋊D₄₇

yes

characteristic

132

C₄⋊C₄

C₂²⋊D₄₇

yes

characteristic

133

C₂²⋊C₄

—

134

C₂²⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₄²

C₂²×D₄₇

yes

characteristic

139

C₂³

—

140

Q₈

—

141

Q₈

—

142

Q₈

—

143

Q₈

—

144

Q₈

—

188

—

145

Q₈

—

188

—

146

D₄

—

147

D₄

—

148

D₄

—

149

D₄

—

150

D₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

188

—

157

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

158

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

159

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

160

C₈

—

188

—

161

C₈

—

188

—

162

C₈

—

188

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

#13093993

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

376

—

173

C₄

#13094044

yes

characteristic

174

C₄

#13094044

yes

characteristic

175

C₄

#13094049

yes

characteristic

176

C₄

#13094054

yes

characteristic

177

C₂

—

178

C₂

#15689268

yes

characteristic

179

C₂

#15689285

yes

characteristic

180

C₂

#15689301

yes

characteristic

	subgroup	G/H
center	C₂²	#13093993
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#559	C₂
socle	C₂²×C₄₇	#120
hypercenter	#559	D₄₇
hypocenter	C₄₇	#1640
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #559

subgroup

G/H

center

C₂²

#13093993

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#559

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #15689408	C₂	semidirect
3. #15689333	C₂	semidirect
5. Q₈⋊(C₄₇⋊C₈)	C₂	semidirect
6. Q₈⋊(C₄₇⋊C₈)	C₂	semidirect
98. C₄₇	#1640	semidirect

H ≅ G/N

type

2. #15689408

C₂

semidirect

3. #15689333

C₂

semidirect

5. Q₈⋊(C₄₇⋊C₈)

C₂

semidirect

6. Q₈⋊(C₄₇⋊C₈)

C₂

semidirect

98. C₄₇

#1640

semidirect

Subgroups — #20242574

Conjugacy classes of subgroups

Complements