Group #20244755 — subgroups

Subgroups — #20244755

Order 6016.

208 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 69 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×#13093959	C₂	1	1	yes	maximal, characteristic
2	#15689166	C₂	1	1	yes	maximal, characteristic
3	#15689150	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093948	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093943	C₂	1	1	yes	maximal, characteristic
6	#15688891	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#419	C₂	1	1	yes	maximal, characteristic
8	C₂×#10625329	C₂²	1	1	yes	characteristic
9	C₂×#10625329	C₂²	1	1	yes	characteristic
10	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
12	#13093959	—	2	2	no	—
13	#13093957	—	2	1	no	—
14	#13093955	—	2	1	no	—
15	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
16	#13093948	—	2	2	no	—
17	#13093943	—	2	2	no	—
18	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	#117×C₄₇	C₂²	1	1	yes	characteristic
21	#103×C₄₇	—	2	1	no	—
22	#103×C₄₇	—	2	1	no	—
23	C₂×Q₈×C₄₇	—	2	1	no	—
24	C₂×Q₈×C₄₇	—	2	1	no	—
25	C₂×Q₈×C₄₇	—	2	1	no	—
26	C₂×Q₈×C₄₇	—	2	1	no	—
27	C₂×Q₈×C₄₇	—	2	1	no	—
28	C₂×Q₈×C₄₇	Q₈	1	2	yes	—
29	C₂²×C₄×C₄₇	—	2	1	no	—
30	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂²×Dic₄₇	—	2	1	no	—
32	M₁₆×C₄₇	—	2	1	no	—
33	M₁₆×C₄₇	—	2	1	no	—
34	C₂×C₈×C₄₇	—	2	1	no	—
35	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₂²⋊C₄×C₄₇	—	2	1	no	—
38	C₄²×C₄₇	—	2	1	no	—
39	C₄⋊Dic₄₇	—	2	1	no	—
40	C₄⋊Dic₄₇	—	2	1	no	—
41	#10625329	—	2	1	no	—
42	#10625329	—	2	1	no	—
43	#10625329	—	2	1	no	—
44	#10625329	D₄	1	2	yes	—
45	C₂×C₄₇⋊C₈	—	2	1	no	—
46	C₂×C₄₇⋊C₈	—	2	1	no	—
47	C₂×C₄₇⋊C₈	—	2	1	no	—
48	C₂×C₄₇⋊C₈	—	2	1	no	—
49	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
50	C₂³×C₄₇	#41	1	1	yes	characteristic
51	Q₈×C₄₇	—	2	2	no	—
52	Q₈×C₄₇	—	2	2	no	—
53	Q₈×C₄₇	—	4	1	no	—
54	Q₈×C₄₇	—	4	1	no	—
55	C₂×C₄×C₄₇	—	2	1	no	—
56	C₂×C₄×C₄₇	—	2	1	no	—
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	—	4	1	no	—
59	C₂×C₄×C₄₇	—	4	1	no	—
60	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
65	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
66	C₂×Dic₄₇	—	4	1	no	—
67	C₂×Dic₄₇	—	8	1	no	—
68	C₈×C₄₇	—	4	1	no	—
69	C₄₇⋊C₈	—	2	2	no	—
70	C₄₇⋊C₈	—	2	2	no	—
71	C₄₇⋊C₈	—	4	1	no	—
72	C₂²×C₄₇	—	2	1	no	—
73	C₂²×C₄₇	—	2	1	no	—
74	C₂²×C₄₇	#121	1	1	yes	characteristic
75	C₂²×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₄₇	#126	1	1	yes	characteristic
77	C₄×C₄₇	—	4	2	no	—
78	C₄×C₄₇	—	4	1	no	—
79	C₄×C₄₇	—	4	1	no	—
80	C₄×C₄₇	#121	1	1	yes	characteristic
81	C₄×C₄₇	#121	1	1	yes	characteristic
82	C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
83	C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
84	Dic₄₇	—	8	1	no	—
85	#2064	—	47	1	no	maximal, Sylow
86	C₂×C₄₇	—	2	1	no	—
87	C₂×C₄₇	—	2	1	no	—
88	C₂×C₄₇	#396	1	1	yes	characteristic
89	C₂×C₄₇	#471	1	1	yes	characteristic
90	C₂×C₄₇	#473	1	1	yes	characteristic
91	C₂×#108	—	47	1	no	—
92	#428	—	47	1	no	—
93	#424	—	47	1	no	—
94	#419	D₄₇	1	1	yes	characteristic
95	C₂×#103	—	47	1	no	—
96	C₂×#101	—	47	1	no	—
97	#343	—	47	1	no	—
98	C₄₇	#2064	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
100	C₂×M₁₆	—	47	1	no	—
101	C₂×M₁₆	—	47	1	no	—
102	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
103	C₂²×C₈	—	47	1	no	—
104	#117	C₂×D₄₇	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	47	1	no	—
106	#108	—	94	2	no	—
107	#107	—	94	1	no	—
108	#106	—	94	1	no	—
109	C₄⋊C₈	—	94	2	no	—
110	#103	—	2	1	no	—
111	#103	—	2	1	no	—
112	#103	—	94	2	no	—
113	#101	—	94	2	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	2	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	2	1	no	—
118	C₂×Q₈	—	2	1	no	—
119	C₂×Q₈	Dic₉₄	1	2	yes	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	94	1	no	—
122	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
123	M₁₆	—	2	1	no	—
124	M₁₆	—	2	1	no	—
125	M₁₆	—	47	2	no	—
126	M₁₆	—	94	1	no	—
127	M₁₆	—	94	1	no	—
128	M₁₆	—	94	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	47	2	no	—
131	C₂×C₈	—	94	1	no	—
132	C₂×C₈	—	94	1	no	—
133	C₂×C₈	—	94	1	no	—
134	C₂×C₈	—	94	1	no	—
135	C₄⋊C₄	—	94	1	no	—
136	C₄⋊C₄	—	94	1	no	—
137	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
138	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
139	C₂²⋊C₄	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₂³	#10625352	1	1	yes	characteristic
142	Q₈	—	2	2	no	—
143	Q₈	—	2	2	no	—
144	Q₈	—	4	1	no	—
145	Q₈	—	4	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	376	1	no	—
152	C₂×C₄	—	188	1	no	—
153	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
154	C₂×C₄	#10625350	1	1	yes	characteristic
155	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
156	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
157	C₂×C₄	#10625352	1	1	yes	characteristic
158	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
159	C₈	—	4	1	no	—
160	C₈	—	94	2	no	—
161	C₈	—	94	2	no	—
162	C₈	—	188	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#13093996	1	1	yes	characteristic
166	C₂²	#13094001	1	1	yes	characteristic
167	C₂²	#13094017	1	1	yes	characteristic
168	C₄	—	4	2	no	—
169	C₄	—	4	1	no	—
170	C₄	—	4	1	no	—
171	C₄	—	376	1	no	—
172	C₄	#13093995	1	1	yes	characteristic
173	C₄	#13093997	1	1	yes	characteristic
174	C₄	#13094001	1	1	yes	characteristic
175	C₄	#13094003	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#15689046	1	1	yes	characteristic
179	C₂	#15689388	1	1	yes	characteristic
180	C₂	#15689396	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093959

C₂

yes

maximal, characteristic

#15689166

C₂

yes

maximal, characteristic

#15689150

C₂

yes

maximal, characteristic

C₂×#13093948

C₂

yes

maximal, characteristic

C₂×#13093943

C₂

yes

maximal, characteristic

#15688891

C₂

yes

maximal, characteristic

C₄₇×#419

C₂

yes

maximal, characteristic

C₂×#10625329

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

—

—

C₄⋊(C₄₇⋊C₈)

—

#13093948

—

#13093943

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

#117×C₄₇

C₂²

yes

characteristic

#103×C₄₇

—

#103×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

—

—

—

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂³×C₄₇

#41

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#121

yes

characteristic

C₂²×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₇

#126

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

Dic₄₇

—

#2064

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

D₄₇

yes

—

—

—

C₄₇

#2064

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₇

yes

characteristic

100

C₂×M₁₆

—

101

C₂×M₁₆

—

102

C₂×M₁₆

C₂×D₄₇

yes

characteristic

103

C₂²×C₈

—

104

#117

C₂×D₄₇

yes

characteristic

105

C₂×C₄⋊C₄

—

106

#108

—

107

#107

—

108

#106

—

109

C₄⋊C₈

—

110

#103

—

111

#103

—

112

#103

—

113

#101

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

Dic₉₄

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₂²×D₄₇

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

126

M₁₆

—

127

M₁₆

—

128

M₁₆

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

C₂²⋊D₄₇

yes

characteristic

138

C₄⋊C₄

C₂²⋊D₄₇

yes

characteristic

139

C₂²⋊C₄

—

140

C₄²

—

141

C₂³

#10625352

yes

characteristic

142

Q₈

—

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

376

—

152

C₂×C₄

—

188

—

153

C₂×C₄

C₂×Dic₉₄

yes

characteristic

154

C₂×C₄

#10625350

yes

characteristic

155

C₂×C₄

D₄×D₄₇

yes

characteristic

156

C₂×C₄

D₄×D₄₇

yes

characteristic

157

C₂×C₄

#10625352

yes

characteristic

158

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

188

—

163

C₂²

—

164

C₂²

—

165

C₂²

#13093996

yes

characteristic

166

C₂²

#13094001

yes

characteristic

167

C₂²

#13094017

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

376

—

172

C₄

#13093995

yes

characteristic

173

C₄

#13093997

yes

characteristic

174

C₄

#13094001

yes

characteristic

175

C₄

#13094003

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#15689046

yes

characteristic

179

C₂

#15689388

yes

characteristic

180

C₂

#15689396

yes

characteristic

	subgroup	G/H
center	C₂²	#13093996
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#419	C₂
socle	C₂²×C₄₇	#121
hypercenter	#419	D₄₇
hypocenter	C₄₇	#2064
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

#13093996

commutator

C₂²×C₄×C₄₇

C₂³

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#419

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

N	H ≅ G/N	type
98. C₄₇	#2064	semidirect

H ≅ G/N

type

98. C₄₇

#2064

semidirect

Subgroups — #20244755

Conjugacy classes of subgroups

Complements