Group #20250318 — subgroups

Subgroups — #20250318

Order 6016.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂×#13094005	C₂	1	2	yes	maximal
2	#15689053	C₂	1	2	yes	maximal, complemented
3	#15689048	C₂	1	4	yes	maximal, complemented
4	#15689021	C₂	1	2	yes	maximal, complemented
5	#15688998	C₂	1	4	yes	maximal, complemented
6	C₄₇×#390	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₄₇	C₂²	1	2	yes	—
8	#13094005	—	2	8	no	—
9	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
13	C₂×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
14	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
15	C₄₇⋊#95	C₂²	1	4	yes	—
16	C₄₇⋊#95	C₂²	1	4	yes	—
17	C₄₇⋊#95	C₂²	1	2	yes	—
18	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
21	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂⁴×C₄₇	C₂³	1	2	yes	—
23	C₂×D₄×C₄₇	—	4	1	no	—
24	C₂×D₄×C₄₇	—	4	1	no	—
25	C₂×D₄×C₄₇	D₄	1	4	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
27	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
29	C₂²×Dic₄₇	—	2	4	no	—
30	C₂²×Dic₄₇	—	4	2	no	—
31	C₂²×Dic₄₇	—	4	2	no	—
32	C₂²×Dic₄₇	C₂³	1	4	yes	—
33	C₂²×Dic₄₇	C₂³	1	4	yes	—
34	C₄⋊C₄×C₄₇	—	4	1	no	—
35	C₂²⋊C₄×C₄₇	—	2	8	no	—
36	C₄²×C₄₇	—	4	1	no	—
37	C₂²⋊Dic₄₇	—	2	4	no	—
38	C₂²⋊Dic₄₇	—	2	8	no	—
39	C₄⋊Dic₄₇	—	2	4	no	—
40	C₄⋊Dic₄₇	—	4	2	no	—
41	C₄⋊Dic₄₇	—	4	2	no	—
42	Dic₄₇⋊C₄	—	2	8	no	—
43	C₂³×C₄₇	—	2	4	no	—
44	C₂³×C₄₇	—	2	2	no	—
45	C₂³×C₄₇	—	2	2	no	—
46	C₂³×C₄₇	—	2	4	no	—
47	C₂³×C₄₇	C₂×D₄	1	4	yes	—
48	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
49	D₄×C₄₇	—	4	4	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	1	no	—
52	C₂×C₄×C₄₇	—	2	1	no	—
53	C₂×C₄×C₄₇	—	2	2	no	—
54	C₂×C₄×C₄₇	—	2	4	no	—
55	C₂×C₄×C₄₇	—	2	2	no	—
56	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₄₇	#41	1	4	yes	—
58	C₂×Dic₄₇	—	2	4	no	—
59	C₂×Dic₄₇	—	2	4	no	—
60	C₂×Dic₄₇	—	2	4	no	—
61	C₂×Dic₄₇	—	2	4	no	—
62	C₂×Dic₄₇	—	2	4	no	—
63	C₂×Dic₄₇	—	2	4	no	—
64	C₂×Dic₄₇	—	4	8	no	—
65	C₂²×C₄₇	—	2	4	no	—
66	C₂²×C₄₇	—	2	8	no	—
67	C₂²×C₄₇	—	4	4	no	—
68	C₂²×C₄₇	—	4	2	no	—
69	C₂²×C₄₇	—	4	2	no	—
70	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
71	C₂²×C₄₇	C₂×#41	1	2	yes	—
72	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
73	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
74	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
75	C₄×C₄₇	—	2	2	no	—
76	C₄×C₄₇	—	2	4	no	—
77	C₄×C₄₇	—	4	2	no	—
78	Dic₄₇	—	4	4	no	—
79	Dic₄₇	—	4	4	no	—
80	#2724	—	47	1	no	maximal, Sylow
81	C₂×C₄₇	—	4	4	no	—
82	C₂×C₄₇	#534	1	1	yes	characteristic
83	C₂×C₄₇	#539	1	2	yes	—
84	C₂×C₄₇	#547	1	2	yes	—
85	C₂×C₄₇	#559	1	1	yes	characteristic
86	C₂×C₄₇	#563	1	1	yes	characteristic
87	C₂×#123	—	47	2	no	—
88	#398	—	47	2	no	—
89	#397	—	47	4	no	—
90	#390	D₄₇	1	1	yes	characteristic
91	#389	—	47	2	no	—
92	#385	—	47	4	no	—
93	C₄₇	#2724	1	1	yes	characteristic, complemented, Sylow
94	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
95	C₂³×C₄	—	47	2	no	—
96	#123	—	94	8	no	—
97	C₂×C₄⋊C₄	—	47	4	no	—
98	C₂×C₄⋊C₄	—	47	2	no	—
99	C₂×C₄⋊C₄	—	47	2	no	—
100	C₂×C₄⋊C₄	—	47	2	no	—
101	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
102	C₂×C₂²⋊C₄	—	47	4	no	—
103	C₂×C₂²⋊C₄	—	47	2	no	—
104	C₂×C₂²⋊C₄	C₂×D₄₇	1	4	yes	—
105	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
106	#95	—	47	4	no	—
107	#95	—	47	4	no	—
108	#95	—	47	2	no	—
109	C₂⁴	C₂²×D₄₇	1	2	yes	—
110	C₂×D₄	—	4	1	no	—
111	C₂×D₄	—	4	1	no	—
112	C₂×D₄	D₁₈₈	1	4	yes	—
113	C₂²×C₄	—	47	4	no	—
114	C₂²×C₄	—	47	4	no	—
115	C₂²×C₄	—	94	4	no	—
116	C₂²×C₄	—	188	2	no	—
117	C₂²×C₄	—	188	2	no	—
118	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
119	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
120	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	94	8	no	—
123	C₄⋊C₄	—	94	4	no	—
124	C₄⋊C₄	—	188	2	no	—
125	C₄⋊C₄	—	188	2	no	—
126	C₂²⋊C₄	—	2	8	no	—
127	C₂²⋊C₄	—	94	4	no	—
128	C₂²⋊C₄	—	94	8	no	—
129	C₄²	—	4	1	no	—
130	C₂³	—	2	4	no	—
131	C₂³	—	2	2	no	—
132	C₂³	—	2	2	no	—
133	C₂³	—	2	4	no	—
134	C₂³	C₂×D₁₈₈	1	4	yes	—
135	C₂³	C₂³×D₄₇	1	1	yes	characteristic
136	D₄	—	4	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	94	4	no	—
144	C₂×C₄	—	94	4	no	—
145	C₂×C₄	—	94	4	no	—
146	C₂×C₄	—	94	4	no	—
147	C₂×C₄	—	94	4	no	—
148	C₂×C₄	—	94	4	no	—
149	C₂×C₄	—	188	8	no	—
150	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
151	C₂×C₄	#10625352	1	4	yes	—
152	C₂²	—	2	8	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	4	4	no	—
157	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
158	C₂²	C₂×#10625352	1	2	yes	—
159	C₂²	#13094069	1	1	yes	characteristic
160	C₂²	#13094069	1	1	yes	characteristic
161	C₂²	#13094071	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	4	no	—
164	C₄	—	4	2	no	—
165	C₄	—	188	4	no	—
166	C₄	—	188	4	no	—
167	C₂	—	4	4	no	—
168	C₂	#15689685	1	1	yes	characteristic
169	C₂	#15689718	1	2	yes	—
170	C₂	#15689777	1	2	yes	—
171	C₂	#15689859	1	1	yes	characteristic
172	C₂	#15689873	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094005

C₂

yes

maximal

#15689053

C₂

yes

maximal, complemented

#15689048

C₂

yes

maximal, complemented

#15689021

C₂

yes

maximal, complemented

#15688998

C₂

yes

maximal, complemented

C₄₇×#390

C₂

yes

maximal, characteristic

C₂³×Dic₄₇

C₂²

yes

—

#13094005

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×D₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

C₂³

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

#2724

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#534

yes

characteristic

C₂×C₄₇

#539

yes

—

C₂×C₄₇

#547

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

D₄₇

yes

characteristic

#389

—

#385

—

C₄₇

#2724

yes

characteristic, complemented, Sylow

C₂²×D₄

C₂×D₄₇

yes

characteristic

C₂³×C₄

—

#123

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

102

C₂×C₂²⋊C₄

—

103

C₂×C₂²⋊C₄

—

104

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

105

C₂×C₄²

C₂×D₄₇

yes

characteristic

106

#95

—

107

#95

—

108

#95

—

109

C₂⁴

C₂²×D₄₇

yes

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

D₁₈₈

yes

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

188

—

117

C₂²×C₄

—

188

—

118

C₂²×C₄

C₂²×D₄₇

yes

—

119

C₂²×C₄

C₂²×D₄₇

yes

characteristic

120

C₂²×C₄

C₂²×D₄₇

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

188

—

125

C₄⋊C₄

—

188

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

C₂×D₁₈₈

yes

—

135

C₂³

C₂³×D₄₇

yes

characteristic

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

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

—

188

—

150

C₂×C₄

C₂×D₁₈₈

yes

—

151

C₂×C₄

#10625352

yes

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₂²×D₁₈₈

yes

characteristic

158

C₂²

C₂×#10625352

yes

—

159

C₂²

#13094069

yes

characteristic

160

C₂²

#13094069

yes

characteristic

161

C₂²

#13094071

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

188

—

166

C₄

—

188

—

167

C₂

—

168

C₂

#15689685

yes

characteristic

169

C₂

#15689718

yes

—

170

C₂

#15689777

yes

—

171

C₂

#15689859

yes

characteristic

172

C₂

#15689873

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄₇×#390	C₂
hypercenter	#390	D₄₇
hypocenter	C₄₇	#2724
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #390

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₄₇×#390

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #390

N	H ≅ G/N	type
2. #15689053	C₂	semidirect
3. #15689048	C₂	semidirect
4. #15689021	C₂	semidirect
5. #15688998	C₂	semidirect
93. C₄₇	#2724	semidirect

H ≅ G/N

type

2. #15689053

C₂

semidirect

3. #15689048

C₂

semidirect

4. #15689021

C₂

semidirect

5. #15688998

C₂

semidirect

93. C₄₇

#2724

semidirect

Subgroups — #20250318

Conjugacy classes of subgroups

Complements