Group #20245990 — subgroups

Subgroups — #20245990

Order 6016.

512 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 189 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×Dic₄₇⋊C₄	C₂	1	1	yes	maximal, characteristic
2	#15689010	C₂	1	4	yes	maximal, complemented
3	#15689004	C₂	1	4	yes	maximal, complemented
4	#15688984	C₂	1	2	yes	maximal
5	#15688983	C₂	1	1	yes	maximal, characteristic
6	C₂²⋊C₄×Dic₄₇	C₂	1	2	yes	maximal
7	C₄₇×#380	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₄₇	C₂²	1	1	yes	characteristic
9	C₂³×Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂×Dic₄₇⋊C₄	C₄	1	8	yes	complemented
12	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
13	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
15	C₂×C₄×Dic₄₇	C₂²	1	4	yes	—
16	C₄₇⋊#95	C₂²	1	4	yes	—
17	C₄₇⋊#95	C₂²	1	2	yes	—
18	C₄₇⋊#95	C₂²	1	8	yes	—
19	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
22	#95×C₄₇	C₂²	1	2	yes	—
23	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	—	4	2	no	—
25	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
27	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	C₂²×Dic₄₇	—	2	2	no	—
30	C₂²×Dic₄₇	—	2	2	no	—
31	C₂²×Dic₄₇	—	4	2	no	—
32	C₂²×Dic₄₇	C₂×C₄	1	4	yes	—
33	C₂²×Dic₄₇	C₂×C₄	1	4	yes	—
34	C₂²×Dic₄₇	C₂³	1	4	yes	—
35	C₂²×Dic₄₇	C₂³	1	2	yes	—
36	C₂²×Dic₄₇	C₂³	1	2	yes	—
37	C₄⋊C₄×C₄₇	—	4	2	no	—
38	C₂²⋊C₄×C₄₇	—	2	4	no	—
39	C₂²⋊Dic₄₇	—	2	4	no	—
40	C₄⋊Dic₄₇	—	4	2	no	—
41	Dic₄₇⋊C₄	C₂×C₄	1	8	yes	—
42	Dic₄₇⋊C₄	C₂×C₄	1	8	yes	—
43	C₄×Dic₄₇	—	4	4	no	—
44	C₂³×C₄₇	—	2	2	no	—
45	C₂³×C₄₇	—	2	2	no	—
46	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
47	C₂³×C₄₇	#41	1	2	yes	—
48	C₂³×C₄₇	#41	1	2	yes	—
49	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	4	no	—
53	C₂×C₄×C₄₇	—	2	4	no	—
54	C₂×C₄×C₄₇	—	2	4	no	—
55	C₂×C₄×C₄₇	—	4	4	no	—
56	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂×Dic₄₇	—	2	8	no	—
59	C₂×Dic₄₇	—	2	2	no	—
60	C₂×Dic₄₇	—	2	2	no	—
61	C₂×Dic₄₇	—	2	4	no	—
62	C₂×Dic₄₇	—	2	4	no	—
63	C₂×Dic₄₇	—	2	4	no	—
64	C₂×Dic₄₇	—	2	4	no	—
65	C₂×Dic₄₇	—	4	4	no	—
66	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
67	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
68	C₂²×C₄₇	—	2	2	no	—
69	C₂²×C₄₇	—	2	2	no	—
70	C₂²×C₄₇	—	2	4	no	—
71	C₂²×C₄₇	—	2	4	no	—
72	C₂²×C₄₇	#117	1	4	yes	—
73	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
74	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
75	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
76	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
77	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
78	C₄×C₄₇	—	2	2	no	—
79	C₄×C₄₇	—	2	2	no	—
80	C₄×C₄₇	—	4	4	no	—
81	Dic₄₇	—	2	4	no	—
82	Dic₄₇	—	2	4	no	—
83	Dic₄₇	—	4	4	no	—
84	#2313	—	47	1	no	maximal, Sylow
85	C₂×C₄₇	—	2	4	no	—
86	C₂×C₄₇	C₂×#117	1	1	yes	characteristic
87	C₂×C₄₇	#518	1	1	yes	characteristic
88	C₂×C₄₇	#519	1	1	yes	characteristic
89	C₂×C₄₇	#539	1	2	yes	—
90	C₂×C₄₇	#539	1	2	yes	—
91	C₂²×C₄⋊C₄	—	47	1	no	—
92	#387	—	47	4	no	—
93	#387	—	47	4	no	—
94	#381	—	47	2	no	—
95	#380	—	47	1	no	—
96	#380	D₄₇	1	1	yes	characteristic
97	C₄×C₂²⋊C₄	—	47	2	no	—
98	C₄₇	#2313	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₄	—	47	1	no	—
100	C₂³×C₄	—	47	1	no	—
101	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
102	C₂×C₄⋊C₄	—	47	8	no	—
103	C₂×C₄⋊C₄	—	47	2	no	—
104	C₂×C₄⋊C₄	—	47	2	no	—
105	C₂×C₄⋊C₄	—	47	2	no	—
106	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
107	C₂×C₂²⋊C₄	—	47	2	no	—
108	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
109	C₂×C₄²	—	47	4	no	—
110	#95	—	47	4	no	—
111	#95	—	47	2	no	—
112	#95	—	47	8	no	—
113	#95	C₂×D₄₇	1	2	yes	—
114	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	—	47	4	no	—
117	C₂²×C₄	—	47	2	no	—
118	C₂²×C₄	—	47	2	no	—
119	C₂²×C₄	—	47	4	no	—
120	C₂²×C₄	—	47	4	no	—
121	C₂²×C₄	—	94	2	no	—
122	C₂²×C₄	—	94	2	no	—
123	C₂²×C₄	—	188	2	no	—
124	C₂²×C₄	C₄×D₄₇	1	4	yes	—
125	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
126	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	47	8	no	—
130	C₄⋊C₄	—	47	8	no	—
131	C₄⋊C₄	—	188	2	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₂²⋊C₄	—	94	4	no	—
134	C₄²	—	188	4	no	—
135	C₂³	—	2	2	no	—
136	C₂³	—	2	2	no	—
137	C₂³	C₂×C₄×D₄₇	1	2	yes	—
138	C₂³	#10625352	1	2	yes	—
139	C₂³	#10625354	1	2	yes	—
140	C₂³	C₂³×D₄₇	1	1	yes	characteristic
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	4	4	no	—
147	C₂×C₄	—	47	4	no	—
148	C₂×C₄	—	47	4	no	—
149	C₂×C₄	—	94	8	no	—
150	C₂×C₄	—	94	2	no	—
151	C₂×C₄	—	94	2	no	—
152	C₂×C₄	—	94	4	no	—
153	C₂×C₄	—	94	4	no	—
154	C₂×C₄	—	94	4	no	—
155	C₂×C₄	—	94	4	no	—
156	C₂×C₄	—	188	4	no	—
157	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
158	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	4	no	—
162	C₂²	—	2	4	no	—
163	C₂²	#13093982	1	4	yes	—
164	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
165	C₂²	C₂×#10625352	1	1	yes	characteristic
166	C₂²	C₂×#10625354	1	1	yes	characteristic
167	C₂²	#13094069	1	2	yes	—
168	C₂²	#13094071	1	2	yes	—
169	C₄	—	2	2	no	—
170	C₄	—	2	2	no	—
171	C₄	—	4	4	no	—
172	C₄	—	94	4	no	—
173	C₄	—	94	4	no	—
174	C₄	—	188	4	no	—
175	C₂	—	2	4	no	—
176	C₂	C₂×#13093982	1	1	yes	characteristic
177	C₂	#15689596	1	1	yes	characteristic
178	C₂	#15689600	1	1	yes	characteristic
179	C₂	#15689718	1	2	yes	—
180	C₂	#15689722	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₄₇⋊C₄

C₂

yes

maximal, characteristic

#15689010

C₂

yes

maximal, complemented

#15689004

C₂

yes

maximal, complemented

#15688984

C₂

yes

maximal

#15688983

C₂

yes

maximal, characteristic

C₂²⋊C₄×Dic₄₇

C₂

yes

maximal

C₄₇×#380

C₂

yes

maximal, characteristic

C₂³×Dic₄₇

C₂²

yes

characteristic

C₂³×Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₄

yes

complemented

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

#95×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

C₂×C₄

yes

—

Dic₄₇⋊C₄

C₂×C₄

yes

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²×C₄

yes

—

C₂³×C₄₇

#41

yes

—

C₂³×C₄₇

#41

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#117

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2313

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#539

yes

—

C₂²×C₄⋊C₄

—

—

—

—

—

D₄₇

yes

characteristic

C₄×C₂²⋊C₄

—

C₄₇

#2313

yes

characteristic, complemented, Sylow

C₂³×C₄

—

100

C₂³×C₄

—

101

C₂³×C₄

C₂×D₄₇

yes

characteristic

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

107

C₂×C₂²⋊C₄

—

108

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

109

C₂×C₄²

—

110

#95

—

111

#95

—

112

#95

—

113

#95

C₂×D₄₇

yes

—

114

C₂⁴

C₂²×D₄₇

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

188

—

124

C₂²×C₄

C₄×D₄₇

yes

—

125

C₂²×C₄

C₂²×D₄₇

yes

—

126

C₂²×C₄

C₂²×D₄₇

yes

characteristic

127

C₂²×C₄

C₂²×D₄₇

yes

characteristic

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

188

—

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

—

188

—

135

C₂³

—

136

C₂³

—

137

C₂³

C₂×C₄×D₄₇

yes

—

138

C₂³

#10625352

yes

—

139

C₂³

#10625354

yes

—

140

C₂³

C₂³×D₄₇

yes

characteristic

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

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

188

—

157

C₂×C₄

C₂×C₄×D₄₇

yes

—

158

C₂×C₄

C₂×C₄×D₄₇

yes

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

#13093982

yes

—

164

C₂²

C₂²×C₄×D₄₇

yes

characteristic

165

C₂²

C₂×#10625352

yes

characteristic

166

C₂²

C₂×#10625354

yes

characteristic

167

C₂²

#13094069

yes

—

168

C₂²

#13094071

yes

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

188

—

175

C₂

—

176

C₂

C₂×#13093982

yes

characteristic

177

C₂

#15689596

yes

characteristic

178

C₂

#15689600

yes

characteristic

179

C₂

#15689718

yes

—

180

C₂

#15689722

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#380	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	#380	D₄₇
hypocenter	C₄₇	#2313
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #380

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₄₇×#380

C₂

socle

C₂³×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #380

N	H ≅ G/N	type
2. #15689010	C₂	semidirect
3. #15689004	C₂	semidirect
11. C₂×Dic₄₇⋊C₄	C₄	semidirect
98. C₄₇	#2313	semidirect

H ≅ G/N

type

2. #15689010

C₂

semidirect

3. #15689004

C₂

semidirect

11. C₂×Dic₄₇⋊C₄

C₄

semidirect

98. C₄₇

#2313

semidirect

Subgroups — #20245990

Conjugacy classes of subgroups

Complements