Group #20244476 — subgroups

Subgroups — #20244476

Order 6016.

474 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂²×#10625335	C₂	1	2	yes	maximal, complemented
2	C₂×#13093945	C₂	1	2	yes	maximal, complemented
3	#15689029	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688878	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
7	C₂×#10625335	—	2	4	no	—
8	C₂×#10625335	—	2	4	no	—
9	C₂×#10625335	—	2	4	no	—
10	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
12	#13093945	—	2	4	no	—
13	C₄₇⋊#95	—	2	2	no	—
14	C₂²×D₄×C₄₇	—	2	2	no	—
15	C₂²×D₄×C₄₇	C₂²	1	2	yes	—
16	#127×C₄₇	—	2	4	no	—
17	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄₇	—	2	2	no	—
19	C₂×D₄×C₄₇	—	2	8	no	—
20	C₂×D₄×C₄₇	—	2	2	no	—
21	C₂×D₄×C₄₇	—	2	2	no	—
22	C₂×D₄×C₄₇	—	2	2	no	—
23	C₂×D₄×C₄₇	—	2	2	no	—
24	C₂×D₄×C₄₇	—	2	2	no	—
25	C₂×D₄×C₄₇	—	4	4	no	—
26	C₂×D₄×C₄₇	—	4	4	no	—
27	C₂×D₄×C₄₇	D₄	1	4	yes	—
28	C₂²×C₄×C₄₇	D₄	1	2	yes	—
29	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂²×Dic₄₇	—	2	1	no	—
31	C₂²×Dic₄₇	—	2	1	no	—
32	C₂×Dic₉₄	—	2	2	no	—
33	C₂×Dic₉₄	—	2	2	no	—
34	C₂×Dic₉₄	—	2	1	no	—
35	C₂×Dic₉₄	D₄	1	2	yes	complemented
36	C₄²×C₄₇	—	2	2	no	—
37	#10625335	—	2	16	no	—
38	C₄⋊Dic₄₇	—	2	2	no	—
39	C₂×C₄₇⋊C₈	—	2	2	no	—
40	C₂×C₄₇⋊C₈	—	2	2	no	—
41	C₂×C₄₇⋊C₈	D₄	1	4	yes	complemented
42	C₂³×C₄₇	—	2	4	no	—
43	C₂³×C₄₇	—	4	2	no	—
44	C₂³×C₄₇	—	4	2	no	—
45	C₂³×C₄₇	—	4	4	no	—
46	C₂³×C₄₇	—	4	4	no	—
47	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
48	D₄×C₄₇	—	2	4	no	—
49	D₄×C₄₇	—	2	4	no	—
50	D₄×C₄₇	—	4	16	no	—
51	D₄×C₄₇	—	4	2	no	—
52	D₄×C₄₇	—	4	2	no	—
53	C₂×C₄×C₄₇	—	2	2	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	SD₁₆	1	4	yes	complemented
56	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
60	C₂×Dic₄₇	—	2	2	no	—
61	C₂×Dic₄₇	—	4	1	no	—
62	C₂×Dic₄₇	—	4	1	no	—
63	C₂×Dic₄₇	—	4	1	no	—
64	C₂×Dic₄₇	—	4	2	no	—
65	Dic₉₄	—	2	4	no	—
66	Dic₉₄	—	4	2	no	—
67	C₄₇⋊C₈	—	2	4	no	—
68	C₂²×C₄₇	—	4	4	no	—
69	C₂²×C₄₇	—	4	8	no	—
70	C₂²×C₄₇	—	8	4	no	—
71	C₂²×C₄₇	—	8	2	no	—
72	C₂²×C₄₇	—	8	2	no	—
73	C₂²×C₄₇	#120	1	1	yes	characteristic
74	C₂²×C₄₇	#121	1	1	yes	characteristic
75	C₂²×C₄₇	#127	1	1	yes	characteristic
76	C₂²×C₄₇	C₂×SD₁₆	1	2	yes	—
77	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
78	C₄×C₄₇	—	2	4	no	—
79	C₄×C₄₇	#120	1	2	yes	—
80	C₄×C₄₇	#121	1	2	yes	—
81	Dic₄₇	—	4	2	no	—
82	Dic₄₇	—	8	1	no	—
83	#2014	—	47	1	no	maximal, Sylow
84	C₂×C₄₇	—	8	4	no	—
85	C₂×C₄₇	#392	1	1	yes	characteristic
86	C₂×C₄₇	#450	1	2	yes	—
87	C₂×C₄₇	#460	1	2	yes	—
88	C₂×C₄₇	#492	1	1	yes	characteristic
89	C₂×C₄₇	#496	1	1	yes	characteristic
90	C₂²×SD₁₆	—	47	2	no	—
91	C₂×#127	D₄₇	1	1	yes	characteristic
92	C₂×#102	—	47	2	no	—
93	#391	—	47	1	no	—
94	#336	—	47	1	no	—
95	C₄₇	#2014	1	1	yes	characteristic, complemented, Sylow
96	C₂²×Q₈	—	47	1	no	—
97	C₂²×D₄	—	2	2	no	—
98	C₂²×D₄	C₂×D₄₇	1	2	yes	—
99	C₂×SD₁₆	—	94	4	no	—
100	C₂×SD₁₆	—	94	4	no	—
101	C₂×SD₁₆	—	94	4	no	—
102	C₂²×C₈	—	47	2	no	—
103	#127	—	2	4	no	—
104	C₂×C₄⋊C₄	—	47	1	no	—
105	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
106	#102	—	94	4	no	—
107	#95	—	94	2	no	—
108	C₂⁴	—	2	2	no	—
109	C₂×Q₈	—	47	2	no	—
110	C₂×Q₈	—	94	1	no	—
111	C₂×Q₈	—	94	2	no	—
112	C₂×Q₈	—	94	2	no	—
113	C₂×D₄	—	2	8	no	—
114	C₂×D₄	—	2	2	no	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	—	2	2	no	—
117	C₂×D₄	—	2	2	no	—
118	C₂×D₄	—	2	2	no	—
119	C₂×D₄	—	4	4	no	—
120	C₂×D₄	—	4	4	no	—
121	C₂×D₄	C₂²⋊D₄₇	1	4	yes	—
122	C₂²×C₄	—	94	1	no	—
123	C₂²×C₄	—	94	1	no	—
124	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
125	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
126	SD₁₆	—	94	16	no	—
127	C₂×C₈	—	47	4	no	—
128	C₂×C₈	—	94	2	no	—
129	C₂×C₈	—	94	2	no	—
130	C₄⋊C₄	—	94	2	no	—
131	C₄²	—	2	2	no	—
132	C₂³	—	2	4	no	—
133	C₂³	—	4	2	no	—
134	C₂³	—	4	2	no	—
135	C₂³	—	4	4	no	—
136	C₂³	—	4	4	no	—
137	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
138	Q₈	—	94	4	no	—
139	Q₈	—	188	2	no	—
140	D₄	—	2	4	no	—
141	D₄	—	2	4	no	—
142	D₄	—	4	16	no	—
143	D₄	—	4	2	no	—
144	D₄	—	4	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	94	2	no	—
148	C₂×C₄	—	188	1	no	—
149	C₂×C₄	—	188	2	no	—
150	C₂×C₄	—	188	1	no	—
151	C₂×C₄	—	188	1	no	—
152	C₂×C₄	#10625335	1	4	yes	complemented
153	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
154	C₂×C₄	D₄×D₄₇	1	2	yes	—
155	C₂×C₄	#10625352	1	1	yes	characteristic
156	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
157	C₈	—	94	4	no	—
158	C₂²	—	4	8	no	—
159	C₂²	—	4	4	no	—
160	C₂²	—	8	2	no	—
161	C₂²	—	8	2	no	—
162	C₂²	—	8	4	no	—
163	C₂²	#13093993	1	1	yes	characteristic
164	C₂²	#13093995	1	1	yes	characteristic
165	C₂²	#13094020	1	1	yes	characteristic
166	C₂²	C₂×#10625335	1	2	yes	—
167	C₂²	#13094050	1	2	yes	—
168	C₄	—	2	4	no	—
169	C₄	—	376	1	no	—
170	C₄	—	188	2	no	—
171	C₄	#13093992	1	2	yes	—
172	C₄	#13093996	1	2	yes	—
173	C₂	—	8	4	no	—
174	C₂	#15689031	1	1	yes	characteristic
175	C₂	#15689274	1	2	yes	—
176	C₂	#15689320	1	2	yes	—
177	C₂	#15689495	1	1	yes	characteristic
178	C₂	#15689510	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10625335

C₂

yes

maximal, complemented

C₂×#13093945

C₂

yes

maximal, complemented

#15689029

C₂

yes

maximal, characteristic, complemented

#15688878

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

—

—

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

#13093945

—

C₄₇⋊#95

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

C₂²

yes

—

#127×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

D₄

yes

complemented

C₄²×C₄₇

—

#10625335

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

complemented

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

SD₁₆

yes

complemented

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

C₂×SD₁₆

yes

—

C₂²×C₄₇

AGL(1,ℤ/8)

yes

—

C₄×C₄₇

—

C₄×C₄₇

#120

yes

—

C₄×C₄₇

#121

yes

—

Dic₄₇

—

Dic₄₇

—

#2014

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#392

yes

characteristic

C₂×C₄₇

#450

yes

—

C₂×C₄₇

#460

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂²×SD₁₆

—

C₂×#127

D₄₇

yes

—

—

—

C₄₇

#2014

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂²×D₄

C₂×D₄₇

yes

—

C₂×SD₁₆

—

100

C₂×SD₁₆

—

101

C₂×SD₁₆

—

102

C₂²×C₈

—

103

#127

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄²

C₂×D₄₇

yes

characteristic

106

#102

—

107

#95

—

108

C₂⁴

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

C₂²⋊D₄₇

yes

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₂²⋊D₄₇

yes

—

125

C₂²×C₄

C₂²×D₄₇

yes

characteristic

126

SD₁₆

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₄⋊C₄

—

131

C₄²

—

132

C₂³

—

133

C₂³

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

138

Q₈

—

139

Q₈

—

188

—

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

144

D₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

188

—

149

C₂×C₄

—

188

—

150

C₂×C₄

—

188

—

151

C₂×C₄

—

188

—

152

C₂×C₄

#10625335

yes

complemented

153

C₂×C₄

D₄×D₄₇

yes

characteristic

154

C₂×C₄

D₄×D₄₇

yes

—

155

C₂×C₄

#10625352

yes

characteristic

156

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

157

C₈

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

#13093993

yes

characteristic

164

C₂²

#13093995

yes

characteristic

165

C₂²

#13094020

yes

characteristic

166

C₂²

C₂×#10625335

yes

—

167

C₂²

#13094050

yes

—

168

C₄

—

169

C₄

—

376

—

170

C₄

—

188

—

171

C₄

#13093992

yes

—

172

C₄

#13093996

yes

—

173

C₂

—

174

C₂

#15689031

yes

characteristic

175

C₂

#15689274

yes

—

176

C₂

#15689320

yes

—

177

C₂

#15689495

yes

characteristic

178

C₂

#15689510

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×#127×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×#127	D₄₇
hypocenter	C₄₇	#2014
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂²×C₄×C₄₇

C₂³

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×#127×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂²×#10625335	C₂	semidirect
2. C₂×#13093945	C₂	semidirect
3. #15689029	C₂	semidirect
4. #15688878	C₂	semidirect
35. C₂×Dic₉₄	D₄	semidirect
41. C₂×C₄₇⋊C₈	D₄	semidirect
55. C₂×C₄×C₄₇	SD₁₆	semidirect
95. C₄₇	#2014	semidirect
152. C₂×C₄	#10625335	semidirect

H ≅ G/N

type

1. C₂²×#10625335

C₂

semidirect

2. C₂×#13093945

C₂

semidirect

3. #15689029

C₂

semidirect

4. #15688878

C₂

semidirect

35. C₂×Dic₉₄

D₄

semidirect

41. C₂×C₄₇⋊C₈

D₄

semidirect

55. C₂×C₄×C₄₇

SD₁₆

semidirect

95. C₄₇

#2014

semidirect

152. C₂×C₄

#10625335

semidirect

Subgroups — #20244476

Conjugacy classes of subgroups

Complements