Group #20253206 — subgroups

Subgroups — #20253206

Order 6016.

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

normal subgroups only

This group has 384 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	C₄×#10625350	C₂	1	1	yes	maximal, characteristic
2	#15689427	C₂	1	4	yes	maximal, complemented
3	#15689420	C₂	1	4	yes	maximal, complemented
4	#15689163	C₂	1	2	yes	maximal, complemented
5	#15689147	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689122	C₂	1	2	yes	maximal
7	C₂×#125×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625350	C₂²	1	1	yes	characteristic
9	C₂×#10625329	C₂²	1	2	yes	—
10	C₄×Dic₉₄	C₂²	1	2	yes	—
11	C₄×D₁₈₈	C₂²	1	2	yes	—
12	C₄×C₂²⋊D₄₇	—	2	2	no	—
13	#13093983	C₂²	1	1	yes	characteristic
14	#13093981	—	2	1	no	—
15	C₄²×D₄₇	—	2	1	no	—
16	#13093957	C₂²	1	4	yes	—
17	#13093955	C₂²	1	4	yes	—
18	C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
19	#13093947	C₂²	1	4	yes	—
20	#13093944	C₂²	1	4	yes	—
21	#125×C₄₇	—	2	1	no	—
22	#125×C₄₇	—	2	1	no	—
23	#125×C₄₇	C₂²	1	4	yes	—
24	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
25	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	—	2	2	no	—
28	C₂²×C₄×C₄₇	D₄	1	2	yes	—
29	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×C₂²⋊D₄₇	—	2	1	no	—
31	#10625350	—	2	2	no	—
32	#10625350	Q₈	1	4	yes	—
33	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
34	C₂×C₄×D₄₇	—	2	1	no	—
35	C₂×C₄×D₄₇	—	2	2	no	—
36	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₄₇	—	2	4	no	—
38	C₄⋊C₄×C₄₇	—	2	4	no	—
39	C₄⋊C₄×C₄₇	—	2	2	no	—
40	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
41	C₄²×C₄₇	D₄	1	1	yes	characteristic
42	C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₄²×C₄₇	C₂³	1	2	yes	—
44	C₂²⋊Dic₄₇	—	2	1	no	—
45	(C₂×D₄₇)⋊C₄	—	2	2	no	—
46	C₄⋊Dic₄₇	C₂³	1	2	yes	—
47	Dic₄₇⋊C₄	—	2	2	no	—
48	C₄×Dic₄₇	—	2	1	no	—
49	C₄×Dic₄₇	—	2	2	no	—
50	#10625329	—	2	4	no	—
51	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
52	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×D₄₇	—	2	1	no	—
54	C₂×C₄×C₄₇	—	2	4	no	—
55	C₂×C₄×C₄₇	—	4	2	no	—
56	C₂×C₄×C₄₇	—	4	4	no	—
57	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₄₇	#41	1	2	yes	—
62	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
63	C₂²⋊D₄₇	—	2	4	no	—
64	C₂×Dic₄₇	—	2	1	no	—
65	C₂×Dic₄₇	—	2	2	no	—
66	D₁₈₈	—	2	1	no	—
67	D₁₈₈	C₂×Q₈	1	2	yes	—
68	C₄×D₄₇	—	2	4	no	—
69	C₄×D₄₇	—	4	4	no	—
70	Dic₉₄	—	2	1	no	—
71	Dic₉₄	C₂×Q₈	1	2	yes	—
72	C₄₇⋊C₈	—	2	4	no	—
73	C₂²×C₄₇	—	2	2	no	—
74	C₂²×C₄₇	C₂²⋊Q₈	1	2	yes	—
75	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
76	C₂×D₄₇	—	2	2	no	—
77	C₂×D₄₇	—	4	2	no	—
78	C₄×C₄₇	—	2	2	no	—
79	C₄×C₄₇	—	2	2	no	—
80	C₄×C₄₇	—	4	4	no	—
81	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
82	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
84	Dic₄₇	—	2	2	no	—
85	Dic₄₇	—	4	2	no	—
86	#3080	—	47	1	no	maximal, Sylow
87	C₂×C₄₇	—	2	2	no	—
88	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
89	C₂×C₄₇	#575	1	2	yes	—
90	D₄₇	—	4	2	no	—
91	C₂×#125	D₄₇	1	1	yes	characteristic, complemented
92	C₄×#41	—	47	1	no	—
93	#479	—	47	4	no	—
94	#478	—	47	4	no	—
95	#428	—	47	2	no	—
96	#423	—	47	1	no	—
97	#417	—	47	2	no	—
98	C₄₇	#3080	1	1	yes	characteristic, complemented, Sylow
99	C₂×#41	—	47	1	no	—
100	C₂×M₁₆	—	47	2	no	—
101	#125	—	2	1	no	—
102	#125	—	2	1	no	—
103	#125	C₂×D₄₇	1	4	yes	—
104	C₄×Q₈	—	47	2	no	—
105	C₄×D₄	—	47	2	no	—
106	C₄×D₄	—	94	2	no	—
107	#117	—	47	1	no	—
108	#117	—	94	1	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
111	C₂×C₄²	—	94	1	no	—
112	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
113	#107	—	47	4	no	—
114	#106	—	47	4	no	—
115	C₄⋊C₈	—	47	4	no	—
116	#103	—	47	4	no	—
117	#102	—	47	4	no	—
118	#41	—	47	4	no	—
119	#41	—	94	2	no	—
120	C₂×Q₈	—	47	1	no	—
121	C₂×D₄	—	47	1	no	—
122	C₂×D₄	—	94	1	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	94	1	no	—
125	C₂²×C₄	—	94	2	no	—
126	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
127	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
128	M₁₆	—	94	4	no	—
129	C₂×C₈	—	47	4	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	47	2	no	—
134	C₄⋊C₄	—	94	2	no	—
135	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
136	C₂²⋊C₄	—	94	1	no	—
137	C₂²⋊C₄	—	94	2	no	—
138	C₄²	—	94	1	no	—
139	C₄²	—	94	2	no	—
140	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
141	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
142	C₄²	C₂²×D₄₇	1	2	yes	—
143	C₂³	—	94	1	no	—
144	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
145	Q₈	—	47	2	no	—
146	Q₈	—	94	1	no	—
147	D₄	—	47	2	no	—
148	D₄	—	94	4	no	—
149	D₄	—	94	1	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	—	94	1	no	—
154	C₂×C₄	—	94	4	no	—
155	C₂×C₄	—	94	2	no	—
156	C₂×C₄	—	188	4	no	—
157	C₂×C₄	Q₈×D₄₇	1	2	yes	—
158	C₂×C₄	#10625354	1	2	yes	—
159	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
160	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
161	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
162	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
163	C₈	—	94	4	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	94	2	no	—
166	C₂²	—	188	2	no	—
167	C₂²	#13094000	1	2	yes	—
168	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	2	2	no	—
171	C₄	—	4	4	no	—
172	C₄	—	94	2	no	—
173	C₄	—	188	2	no	—
174	C₄	#13094000	1	2	yes	—
175	C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
176	C₄	C₂×#10625354	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	—	188	2	no	—
179	C₂	C₂×#13094000	1	1	yes	characteristic
180	C₂	#15689921	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10625350

C₂

yes

maximal, characteristic

#15689427

C₂

yes

maximal, complemented

#15689420

C₂

yes

maximal, complemented

#15689163

C₂

yes

maximal, complemented

#15689147

C₂

yes

maximal, characteristic, complemented

#15689122

C₂

yes

maximal

C₂×#125×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

—

C₄×Dic₉₄

C₂²

yes

—

C₄×D₁₈₈

C₂²

yes

—

C₄×C₂²⋊D₄₇

—

#13093983

C₂²

yes

characteristic

#13093981

—

C₄²×D₄₇

—

#13093957

C₂²

yes

—

#13093955

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093947

C₂²

yes

—

#13093944

C₂²

yes

—

#125×C₄₇

—

#125×C₄₇

—

#125×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

—

#10625350

—

#10625350

Q₈

yes

—

C₂×D₁₈₈

C₂³

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

—

C₂²⋊Dic₄₇

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

C₂×Q₈

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

C₂×Q₈

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²⋊Q₈

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

C₄×C₄₇

C₂²×Q₈

yes

characteristic

C₄×C₄₇

C₂×#41

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#3080

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₄₇

#575

yes

—

D₄₇

—

C₂×#125

D₄₇

yes

characteristic, complemented

C₄×#41

—

—

—

—

—

—

C₄₇

#3080

yes

characteristic, complemented, Sylow

C₂×#41

—

100

C₂×M₁₆

—

101

#125

—

102

#125

—

103

#125

C₂×D₄₇

yes

—

104

C₄×Q₈

—

105

C₄×D₄

—

106

C₄×D₄

—

107

#117

—

108

#117

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

111

C₂×C₄²

—

112

C₂×C₄²

C₂×D₄₇

yes

characteristic

113

#107

—

114

#106

—

115

C₄⋊C₈

—

116

#103

—

117

#102

—

118

#41

—

119

#41

—

120

C₂×Q₈

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂²⋊D₄₇

yes

—

127

C₂²×C₄

C₂²×D₄₇

yes

characteristic

128

M₁₆

—

129

C₂×C₈

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

C₂²×D₄₇

yes

—

136

C₂²⋊C₄

—

137

C₂²⋊C₄

—

138

C₄²

—

139

C₄²

—

140

C₄²

C₂²⋊D₄₇

yes

characteristic

141

C₄²

C₂²⋊D₄₇

yes

characteristic

142

C₄²

C₂²×D₄₇

yes

—

143

C₂³

—

144

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

145

Q₈

—

146

Q₈

—

147

D₄

—

148

D₄

—

149

D₄

—

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

Q₈×D₄₇

yes

—

158

C₂×C₄

#10625354

yes

—

159

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

160

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

161

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

162

C₂×C₄

C₂³×D₄₇

yes

characteristic

163

C₈

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

188

—

167

C₂²

#13094000

yes

—

168

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

188

—

174

C₄

#13094000

yes

—

175

C₄

C₂×Q₈×D₄₇

yes

characteristic

176

C₄

C₂×#10625354

yes

characteristic

177

C₂

—

178

C₂

—

188

—

179

C₂

C₂×#13094000

yes

characteristic

180

C₂

#15689921

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×#125×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	C₂×#125	D₄₇
hypocenter	C₄₇	#3080
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×#125×C₄₇

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #15689427	C₂	semidirect
3. #15689420	C₂	semidirect
4. #15689163	C₂	semidirect
5. #15689147	C₂	semidirect
7. C₂×#125×C₄₇	C₂	semidirect
91. C₂×#125	D₄₇	semidirect
98. C₄₇	#3080	semidirect

H ≅ G/N

type

2. #15689427

C₂

semidirect

3. #15689420

C₂

semidirect

4. #15689163

C₂

semidirect

5. #15689147

C₂

semidirect

7. C₂×#125×C₄₇

C₂

semidirect

91. C₂×#125

D₄₇

semidirect

98. C₄₇

#3080

semidirect

Subgroups — #20253206

Conjugacy classes of subgroups

Complements