Group #20244700 — subgroups

Subgroups — #20244700

Order 6016.

360 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 85 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×D₃₇₆	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13093944	C₂	1	2	yes	maximal, complemented
4	#15689024	C₂	1	2	yes	maximal, complemented
5	C₄₇×#340	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₈₈	C₂²	1	2	yes	—
7	C₂×D₃₇₆	—	2	4	no	—
8	C₂×D₃₇₆	—	2	2	no	—
9	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
11	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
12	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
13	Dic₄₇⋊C₈	—	2	2	no	—
14	#13093944	—	2	4	no	—
15	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
17	C₂³×D₄₇	—	2	2	no	—
18	C₂²×C₄×C₄₇	—	2	2	no	—
19	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
20	C₂²×Dic₄₇	D₄	1	1	yes	characteristic
21	C₂²×Dic₄₇	D₄	1	1	yes	characteristic
22	C₂×D₁₈₈	—	2	2	no	—
23	C₂×D₁₈₈	—	2	2	no	—
24	C₂×D₁₈₈	—	2	2	no	—
25	C₂×D₁₈₈	—	2	2	no	—
26	C₂×D₁₈₈	—	2	2	no	—
27	C₂×D₁₈₈	D₄	1	4	yes	—
28	C₂×C₈×C₄₇	—	2	2	no	—
29	C₂×C₈×C₄₇	D₄	1	2	yes	—
30	C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₄⋊C₄×C₄₇	—	2	2	no	—
32	(C₂×D₄₇)⋊C₄	—	4	4	no	—
33	(C₂×D₄₇)⋊C₄	—	4	4	no	—
34	C₄×Dic₄₇	—	2	1	no	—
35	C₄×Dic₄₇	—	2	1	no	—
36	C₂×C₄₇⋊C₈	—	2	2	no	—
37	C₂×C₄₇⋊C₈	—	2	1	no	—
38	D₃₇₆	—	2	8	no	—
39	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×D₄₇	—	2	4	no	—
41	C₂²×D₄₇	—	4	2	no	—
42	C₂²×D₄₇	—	4	2	no	—
43	C₂²×D₄₇	—	4	2	no	—
44	C₂²×D₄₇	—	4	2	no	—
45	C₂²×D₄₇	—	4	2	no	—
46	C₂²×D₄₇	—	4	2	no	—
47	C₂×C₄×C₄₇	—	4	2	no	—
48	C₂×C₄×C₄₇	—	4	2	no	—
49	C₂×C₄×C₄₇	—	4	2	no	—
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₄₇	#41	1	2	yes	—
53	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
54	C₂×Dic₄₇	—	2	1	no	—
55	C₂×Dic₄₇	—	2	1	no	—
56	C₂×Dic₄₇	—	2	1	no	—
57	C₂×Dic₄₇	—	2	2	no	—
58	C₂×Dic₄₇	D₈	1	2	yes	complemented
59	D₁₈₈	—	2	4	no	—
60	D₁₈₈	—	2	4	no	—
61	D₁₈₈	—	4	2	no	—
62	D₁₈₈	—	4	2	no	—
63	C₈×C₄₇	—	2	2	no	—
64	C₄₇⋊C₈	—	4	1	no	—
65	C₂²×C₄₇	#121	1	2	yes	—
66	C₂²×C₄₇	#121	1	1	yes	characteristic
67	C₂²×C₄₇	C₂×D₈	1	1	yes	characteristic
68	C₂²×C₄₇	#135	1	1	yes	characteristic
69	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
70	C₂×D₄₇	—	4	4	no	—
71	C₂×D₄₇	—	4	4	no	—
72	C₂×D₄₇	—	4	4	no	—
73	C₂×D₄₇	—	8	2	no	—
74	C₂×D₄₇	—	8	2	no	—
75	C₂×D₄₇	—	8	2	no	—
76	C₂×D₄₇	—	8	2	no	—
77	C₄×C₄₇	—	8	2	no	—
78	C₄×C₄₇	#120	1	1	yes	characteristic
79	C₄×C₄₇	#123	1	2	yes	—
80	C₄×C₄₇	#124	1	1	yes	characteristic
81	Dic₄₇	—	2	2	no	—
82	Dic₄₇	—	4	1	no	—
83	#2054	—	47	1	no	maximal, Sylow
84	C₂×C₄₇	#394	1	1	yes	characteristic
85	C₂×C₄₇	#459	1	2	yes	—
86	C₂×C₄₇	#463	1	2	yes	—
87	C₂×C₄₇	#466	1	1	yes	characteristic
88	C₂×C₄₇	#469	1	1	yes	characteristic
89	D₄₇	—	8	4	no	—
90	C₂²×D₈	—	47	1	no	—
91	C₂×C₄⋊C₈	—	47	1	no	—
92	C₂×#102	—	47	2	no	—
93	#390	—	47	2	no	—
94	#340	D₄₇	1	1	yes	characteristic, complemented
95	C₄₇	#2054	1	1	yes	characteristic, complemented, Sylow
96	C₂²×D₄	—	47	2	no	—
97	C₂×D₈	—	94	4	no	—
98	C₂×D₈	—	94	2	no	—
99	C₂²×C₈	—	47	1	no	—
100	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
101	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
102	C₂×C₂²⋊C₄	—	94	2	no	—
103	C₂×C₂²⋊C₄	—	94	2	no	—
104	C₂×C₄²	—	47	1	no	—
105	C₄⋊C₈	—	94	2	no	—
106	#102	—	94	4	no	—
107	C₂⁴	—	94	2	no	—
108	C₂×D₄	—	47	4	no	—
109	C₂×D₄	—	94	2	no	—
110	C₂×D₄	—	94	2	no	—
111	C₂×D₄	—	94	2	no	—
112	C₂×D₄	—	94	2	no	—
113	C₂×D₄	—	94	2	no	—
114	C₂²×C₄	—	2	2	no	—
115	C₂²×C₄	—	47	1	no	—
116	C₂²×C₄	—	47	1	no	—
117	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
118	D₈	—	94	8	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	94	1	no	—
121	C₂×C₈	—	94	2	no	—
122	C₂×C₈	D₁₈₈	1	2	yes	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	2	no	—
125	C₂²⋊C₄	—	188	4	no	—
126	C₂²⋊C₄	—	188	4	no	—
127	C₄²	—	94	1	no	—
128	C₄²	—	94	1	no	—
129	C₂³	—	94	4	no	—
130	C₂³	—	188	2	no	—
131	C₂³	—	188	2	no	—
132	C₂³	—	188	2	no	—
133	C₂³	—	188	2	no	—
134	C₂³	—	188	2	no	—
135	C₂³	—	188	2	no	—
136	C₂³	D₄×D₄₇	1	1	yes	characteristic
137	D₄	—	94	4	no	—
138	D₄	—	94	4	no	—
139	D₄	—	188	2	no	—
140	D₄	—	188	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	47	2	no	—
145	C₂×C₄	—	94	1	no	—
146	C₂×C₄	—	94	1	no	—
147	C₂×C₄	—	94	1	no	—
148	C₂×C₄	—	94	2	no	—
149	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
150	C₂×C₄	#10625350	1	2	yes	—
151	C₂×C₄	D₄×D₄₇	1	2	yes	—
152	C₂×C₄	#10625354	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	—	188	1	no	—
155	C₂²	—	376	2	no	—
156	C₂²	—	376	2	no	—
157	C₂²	—	376	2	no	—
158	C₂²	—	376	2	no	—
159	C₂²	—	188	4	no	—
160	C₂²	—	188	4	no	—
161	C₂²	—	188	4	no	—
162	C₂²	#13093997	1	2	yes	—
163	C₂²	#13093999	1	1	yes	characteristic
164	C₂²	D₈×D₄₇	1	1	yes	characteristic
165	C₂²	#13094047	1	1	yes	characteristic
166	C₂²	#13094051	1	2	yes	—
167	C₄	—	8	2	no	—
168	C₄	—	94	2	no	—
169	C₄	—	188	1	no	—
170	C₄	#13093994	1	1	yes	characteristic
171	C₄	#13094009	1	2	yes	—
172	C₄	#13094012	1	1	yes	characteristic
173	C₂	—	376	4	no	—
174	C₂	#15689041	1	1	yes	characteristic
175	C₂	#15689317	1	2	yes	—
176	C₂	#15689341	1	2	yes	—
177	C₂	#15689360	1	1	yes	characteristic
178	C₂	#15689377	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₃₇₆

C₂

yes

maximal, characteristic

C₂×Dic₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×#13093944

C₂

yes

maximal, complemented

#15689024

C₂

yes

maximal, complemented

C₄₇×#340

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×(C₂×D₄₇)⋊C₄

—

C₂×(C₂×D₄₇)⋊C₄

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

Dic₄₇⋊C₈

—

#13093944

—

C₂²×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

D₄

yes

characteristic

C₂²×Dic₄₇

D₄

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

D₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

D₃₇₆

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

D₈

yes

complemented

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

#121

yes

—

C₂²×C₄₇

#121

yes

characteristic

C₂²×C₄₇

C₂×D₈

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#120

yes

characteristic

C₄×C₄₇

#123

yes

—

C₄×C₄₇

#124

yes

characteristic

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#463

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

C₂²×D₈

—

C₂×C₄⋊C₈

—

C₂×#102

—

#390

—

#340

D₄₇

yes

characteristic, complemented

C₄₇

#2054

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

—

100

C₂²×C₈

C₂×D₄₇

yes

characteristic

101

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

102

C₂×C₂²⋊C₄

—

103

C₂×C₂²⋊C₄

—

104

C₂×C₄²

—

105

C₄⋊C₈

—

106

#102

—

107

C₂⁴

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂²×D₄₇

yes

characteristic

118

D₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

D₁₈₈

yes

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

188

—

126

C₂²⋊C₄

—

188

—

127

C₄²

—

128

C₄²

—

129

C₂³

—

130

C₂³

—

188

—

131

C₂³

—

188

—

132

C₂³

—

188

—

133

C₂³

—

188

—

134

C₂³

—

188

—

135

C₂³

—

188

—

136

C₂³

D₄×D₄₇

yes

characteristic

137

D₄

—

138

D₄

—

139

D₄

—

188

—

140

D₄

—

188

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

C₂×D₁₈₈

yes

characteristic

150

C₂×C₄

#10625350

yes

—

151

C₂×C₄

D₄×D₄₇

yes

—

152

C₂×C₄

#10625354

yes

characteristic

153

C₈

—

154

C₈

—

188

—

155

C₂²

—

376

—

156

C₂²

—

376

—

157

C₂²

—

376

—

158

C₂²

—

376

—

159

C₂²

—

188

—

160

C₂²

—

188

—

161

C₂²

—

188

—

162

C₂²

#13093997

yes

—

163

C₂²

#13093999

yes

characteristic

164

C₂²

D₈×D₄₇

yes

characteristic

165

C₂²

#13094047

yes

characteristic

166

C₂²

#13094051

yes

—

167

C₄

—

168

C₄

—

169

C₄

—

188

—

170

C₄

#13093994

yes

characteristic

171

C₄

#13094009

yes

—

172

C₄

#13094012

yes

characteristic

173

C₂

—

376

—

174

C₂

#15689041

yes

characteristic

175

C₂

#15689317

yes

—

176

C₂

#15689341

yes

—

177

C₂

#15689360

yes

characteristic

178

C₂

#15689377

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₄₇
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#340	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	#340	D₄₇
hypocenter	C₄₇	#2054
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₄₇

commutator

C₂²×C₄×C₄₇

C₂³

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#340

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
2. C₂×Dic₄₇⋊C₈	C₂	semidirect
3. C₂×#13093944	C₂	semidirect
4. #15689024	C₂	semidirect
5. C₄₇×#340	C₂	semidirect
58. C₂×Dic₄₇	D₈	semidirect
94. #340	D₄₇	semidirect
95. C₄₇	#2054	semidirect

H ≅ G/N

type

2. C₂×Dic₄₇⋊C₈

C₂

semidirect

3. C₂×#13093944

C₂

semidirect

4. #15689024

C₂

semidirect

5. C₄₇×#340

C₂

semidirect

58. C₂×Dic₄₇

D₈

semidirect

94. #340

D₄₇

semidirect

95. C₄₇

#2054

semidirect

Subgroups — #20244700

Conjugacy classes of subgroups

Complements