Group #20245997 — subgroups

Subgroups — #20245997

Order 6016.

484 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 229 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	#15689016	C₂	1	4	yes	maximal
2	#15689009	C₂	1	4	yes	maximal, complemented
3	#15688993	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688989	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688983	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₄₇	C₂	1	2	yes	maximal, complemented
7	C₂×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
11	C₂×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
12	C₂×C₄×Dic₄₇	C₂²	1	4	yes	—
13	C₄₇⋊#95	C₂²	1	4	yes	—
14	C₄₇⋊#95	C₂²	1	4	yes	—
15	C₄₇⋊#95	C₂²	1	4	yes	—
16	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
17	#117×C₄₇	C₄	1	8	yes	complemented
18	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	—	2	2	no	—
24	C₂²×C₄×C₄₇	—	2	2	no	—
25	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
27	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
28	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂²×Dic₄₇	C₂³	1	4	yes	—
31	C₂²×Dic₄₇	C₂³	1	4	yes	—
32	C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
33	C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
34	C₄²×C₄₇	C₂×C₄	1	4	yes	—
35	C₄²×C₄₇	C₂×C₄	1	4	yes	—
36	C₂²⋊Dic₄₇	—	4	4	no	—
37	C₄⋊Dic₄₇	—	2	4	no	—
38	C₄⋊Dic₄₇	—	2	4	no	—
39	C₄⋊Dic₄₇	—	4	4	no	—
40	C₄×Dic₄₇	—	4	4	no	—
41	C₂³×C₄₇	—	2	1	no	—
42	C₂³×C₄₇	—	2	1	no	—
43	C₂³×C₄₇	—	2	2	no	—
44	C₂³×C₄₇	—	2	2	no	—
45	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
46	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
47	C₂×C₄×C₄₇	—	2	1	no	—
48	C₂×C₄×C₄₇	—	2	1	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	—	2	8	no	—
54	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
55	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₄₇	#41	1	2	yes	—
59	C₂×C₄×C₄₇	#41	1	2	yes	—
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	—	2	4	no	—
65	C₂×Dic₄₇	—	2	4	no	—
66	C₂²×C₄₇	—	2	2	no	—
67	C₂²×C₄₇	—	2	4	no	—
68	C₂²×C₄₇	—	4	1	no	—
69	C₂²×C₄₇	—	4	1	no	—
70	C₂²×C₄₇	—	4	2	no	—
71	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
72	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
73	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
74	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
75	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
76	C₄×C₄₇	—	2	2	no	—
77	C₄×C₄₇	—	2	4	no	—
78	C₄×C₄₇	—	2	4	no	—
79	C₄×C₄₇	#117	1	4	yes	—
80	Dic₄₇	—	4	4	no	—
81	Dic₄₇	—	4	4	no	—
82	#2314	—	47	1	no	maximal, Sylow
83	C₂×C₄₇	—	4	2	no	—
84	C₂×C₄₇	C₂×#117	1	1	yes	characteristic
85	C₂×C₄₇	#520	1	2	yes	—
86	C₂×C₄₇	#540	1	1	yes	characteristic
87	C₂×C₄₇	#541	1	1	yes	characteristic
88	C₂×C₄₇	#542	1	2	yes	—
89	C₂×#117	D₄₇	1	1	yes	characteristic
90	#388	—	47	4	no	—
91	#387	—	47	4	no	—
92	#384	—	47	1	no	—
93	#382	—	47	1	no	—
94	#380	—	47	2	no	—
95	C₄×C₄⋊C₄	—	47	2	no	—
96	C₄₇	#2314	1	1	yes	characteristic, complemented, Sylow
97	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
98	#117	Dic₄₇	1	8	yes	complemented
99	C₂×C₄⋊C₄	—	47	2	no	—
100	C₂×C₄⋊C₄	—	47	2	no	—
101	C₂×C₄⋊C₄	—	47	4	no	—
102	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
103	C₂×C₂²⋊C₄	—	47	4	no	—
104	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
105	C₂×C₄²	—	47	4	no	—
106	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
107	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
108	#95	—	47	4	no	—
109	#95	—	47	4	no	—
110	#95	—	47	4	no	—
111	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	—	47	4	no	—
115	C₂²×C₄	—	47	4	no	—
116	C₂²×C₄	C₂×Dic₄₇	1	4	yes	—
117	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
118	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
119	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
120	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
121	C₄⋊C₄	—	94	4	no	—
122	C₄⋊C₄	—	94	4	no	—
123	C₄⋊C₄	—	188	4	no	—
124	C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
125	C₂²⋊C₄	—	188	4	no	—
126	C₂²⋊C₄	C₂×Dic₄₇	1	8	yes	—
127	C₄²	—	188	4	no	—
128	C₄²	C₂×Dic₄₇	1	4	yes	—
129	C₄²	C₂×Dic₄₇	1	4	yes	—
130	C₂³	—	2	1	no	—
131	C₂³	—	2	1	no	—
132	C₂³	—	2	2	no	—
133	C₂³	—	2	2	no	—
134	C₂³	C₂²×Dic₄₇	1	2	yes	—
135	C₂³	C₂³×D₄₇	1	1	yes	characteristic
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	8	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	#10625350	1	2	yes	—
150	C₂×C₄	#10625350	1	2	yes	—
151	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
152	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
153	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
154	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	4	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	4	1	no	—
159	C₂²	—	4	2	no	—
160	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
161	C₂²	C₂×#10625350	1	1	yes	characteristic
162	C₂²	C₂×#10625350	1	1	yes	characteristic
163	C₂²	#13094070	1	2	yes	—
164	C₂²	#13094071	1	2	yes	—
165	C₄	—	2	2	no	—
166	C₄	—	2	4	no	—
167	C₄	—	2	4	no	—
168	C₄	—	188	4	no	—
169	C₄	—	188	4	no	—
170	C₄	#13093983	1	4	yes	—
171	C₂	—	4	2	no	—
172	C₂	C₂×#13093983	1	1	yes	characteristic
173	C₂	#15689606	1	2	yes	—
174	C₂	#15689728	1	1	yes	characteristic
175	C₂	#15689735	1	1	yes	characteristic
176	C₂	#15689744	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689016

C₂

yes

maximal

#15689009

C₂

yes

maximal, complemented

#15688993

C₂

yes

maximal, characteristic, complemented

#15688989

C₂

yes

maximal, characteristic, complemented

#15688983

C₂

yes

maximal

C₄×C₄⋊Dic₄₇

C₂

yes

maximal, complemented

C₂×#117×C₄₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

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

#117×C₄₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²×C₄

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

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

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

—

C₄×C₄₇

#117

yes

—

Dic₄₇

—

Dic₄₇

—

#2314

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×#117

yes

characteristic

C₂×C₄₇

#520

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×#117

D₄₇

yes

—

—

—

—

—

C₄×C₄⋊C₄

—

C₄₇

#2314

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₇

yes

characteristic

#117

Dic₄₇

yes

complemented

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

103

C₂×C₂²⋊C₄

—

104

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

105

C₂×C₄²

—

106

C₂×C₄²

C₂×D₄₇

yes

characteristic

107

C₂×C₄²

C₂×D₄₇

yes

characteristic

108

#95

—

109

#95

—

110

#95

—

111

C₂⁴

C₂²×D₄₇

yes

characteristic

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂×Dic₄₇

yes

—

117

C₂²×C₄

C₂²×D₄₇

yes

characteristic

118

C₂²×C₄

C₂²×D₄₇

yes

—

119

C₂²×C₄

C₂²×D₄₇

yes

—

120

C₂²×C₄

C₂²×D₄₇

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

188

—

124

C₄⋊C₄

C₂×Dic₄₇

yes

—

125

C₂²⋊C₄

—

188

—

126

C₂²⋊C₄

C₂×Dic₄₇

yes

—

127

C₄²

—

188

—

128

C₄²

C₂×Dic₄₇

yes

—

129

C₄²

C₂×Dic₄₇

yes

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

C₂²×Dic₄₇

yes

—

135

C₂³

C₂³×D₄₇

yes

characteristic

136

C₂×C₄

—

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

#10625350

yes

—

150

C₂×C₄

#10625350

yes

—

151

C₂×C₄

C₂²×Dic₄₇

yes

—

152

C₂×C₄

C₂²×Dic₄₇

yes

—

153

C₂×C₄

C₂²×Dic₄₇

yes

—

154

C₂×C₄

C₂²×Dic₄₇

yes

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

C₂³×Dic₄₇

yes

characteristic

161

C₂²

C₂×#10625350

yes

characteristic

162

C₂²

C₂×#10625350

yes

characteristic

163

C₂²

#13094070

yes

—

164

C₂²

#13094071

yes

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

188

—

169

C₄

—

188

—

170

C₄

#13093983

yes

—

171

C₂

—

172

C₂

C₂×#13093983

yes

characteristic

173

C₂

#15689606

yes

—

174

C₂

#15689728

yes

characteristic

175

C₂

#15689735

yes

characteristic

176

C₂

#15689744

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₂×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×#117	D₄₇
hypocenter	C₄₇	#2314
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₂×#117×C₄₇

C₂

socle

C₂³×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×#117

N	H ≅ G/N	type
2. #15689009	C₂	semidirect
3. #15688993	C₂	semidirect
4. #15688989	C₂	semidirect
6. C₄×C₄⋊Dic₄₇	C₂	semidirect
17. #117×C₄₇	C₄	semidirect
96. C₄₇	#2314	semidirect
98. #117	Dic₄₇	semidirect

H ≅ G/N

type

2. #15689009

C₂

semidirect

3. #15688993

C₂

semidirect

4. #15688989

C₂

semidirect

6. C₄×C₄⋊Dic₄₇

C₂

semidirect

17. #117×C₄₇

C₄

semidirect

96. C₄₇

#2314

semidirect

98. #117

Dic₄₇

semidirect

Subgroups — #20245997

Conjugacy classes of subgroups

Complements