Group #20254283 — subgroups

Subgroups — #20254283

Order 6016.

378 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 117 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	#15689694	C₂	1	2	yes	maximal
2	#15689465	C₂	1	4	yes	maximal, complemented
3	#15689381	C₂	1	2	yes	maximal, complemented
4	#15689364	C₂	1	2	yes	maximal, complemented
5	#15689281	C₂	1	4	yes	maximal, complemented
6	C₄₇×#408	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₉₄	C₂²	1	2	yes	—
8	C₂×Dic₁₈₈	C₂²	1	2	yes	—
9	C₂×Dic₁₈₈	C₂²	1	2	yes	—
10	#13094022	—	2	2	no	—
11	#13094011	—	2	2	no	—
12	#13094005	—	2	2	no	—
13	#13094005	—	2	2	no	—
14	#13094003	C₂²	1	4	yes	—
15	#13094003	C₂²	1	4	yes	—
16	#13093983	C₂²	1	2	yes	—
17	#13093958	C₂²	1	2	yes	—
18	#13093956	C₂²	1	2	yes	—
19	#13093949	C₂²	1	4	yes	—
20	#13093949	C₂²	1	4	yes	—
21	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²⋊C₈×C₄₇	C₂²	1	4	yes	—
25	#41×C₄₇	—	4	1	no	—
26	#41×C₄₇	—	4	1	no	—
27	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂×D₄×C₄₇	D₄	1	1	yes	characteristic
29	C₂×D₄×C₄₇	D₄	1	2	yes	—
30	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
32	C₂²×Dic₄₇	—	2	2	no	—
33	C₂×Dic₉₄	—	2	2	no	—
34	C₂×Dic₉₄	—	4	2	no	—
35	C₂×Dic₉₄	—	4	2	no	—
36	C₂×Dic₉₄	C₂³	1	4	yes	—
37	M₁₆×C₄₇	—	4	1	no	—
38	C₂×C₈×C₄₇	—	4	1	no	—
39	C₂×C₈×C₄₇	C₂³	1	2	yes	—
40	C₂×C₈×C₄₇	C₂³	1	2	yes	—
41	C₂²⋊Dic₄₇	—	2	2	no	—
42	C₂²⋊Dic₄₇	—	2	4	no	—
43	C₂²⋊Dic₄₇	—	2	4	no	—
44	C₄⋊Dic₄₇	C₂³	1	4	yes	—
45	Dic₄₇⋊C₄	—	2	4	no	—
46	Dic₄₇⋊C₄	—	2	4	no	—
47	C₄×Dic₄₇	—	2	2	no	—
48	Dic₁₈₈	—	4	2	no	—
49	Dic₁₈₈	—	4	2	no	—
50	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂³×C₄₇	C₂×D₄	1	2	yes	—
52	Q₈×C₄₇	—	4	1	no	—
53	D₄×C₄₇	—	4	2	no	—
54	D₄×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	—	2	2	no	—
59	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₄₇	—	2	4	no	—
63	C₂×Dic₄₇	—	2	4	no	—
64	C₂×Dic₄₇	—	4	4	no	—
65	Dic₉₄	—	2	4	no	—
66	Dic₉₄	—	2	4	no	—
67	Dic₉₄	—	4	4	no	—
68	C₈×C₄₇	—	2	2	no	—
69	C₈×C₄₇	—	2	2	no	—
70	C₂²×C₄₇	—	2	2	no	—
71	C₂²×C₄₇	—	2	1	no	—
72	C₂²×C₄₇	—	2	2	no	—
73	C₂²×C₄₇	—	2	4	no	—
74	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
75	C₄×C₄₇	—	2	2	no	—
76	C₄×C₄₇	—	2	1	no	—
77	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
78	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
79	Dic₄₇	—	4	4	no	—
80	Dic₄₇	—	4	4	no	—
81	#3202	—	47	1	no	maximal, Sylow
82	C₂×C₄₇	—	4	2	no	—
83	C₂×C₄₇	—	4	1	no	—
84	C₂×C₄₇	#534	1	1	yes	characteristic
85	C₂×C₄₇	#578	1	2	yes	—
86	#536	—	47	2	no	—
87	#485	—	47	4	no	—
88	#470	—	47	2	no	—
89	#467	—	47	2	no	—
90	#451	—	47	4	no	—
91	#408	D₄₇	1	1	yes	characteristic
92	C₄₇	#3202	1	1	yes	characteristic, complemented, Sylow
93	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
94	C₂²×Q₈	—	47	2	no	—
95	C₂×Q₁₆	—	47	2	no	—
96	C₂×Q₁₆	—	47	2	no	—
97	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
98	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
99	C₄⋊Q₈	—	94	2	no	—
100	#124	—	94	2	no	—
101	#123	—	94	2	no	—
102	#123	—	94	2	no	—
103	C₂²⋊Q₈	—	47	4	no	—
104	C₂²⋊Q₈	—	47	4	no	—
105	#117	—	47	2	no	—
106	#107	—	47	2	no	—
107	#106	—	47	2	no	—
108	#103	—	47	4	no	—
109	#103	—	47	4	no	—
110	C₂²⋊C₈	C₂×D₄₇	1	4	yes	—
111	#41	—	4	1	no	—
112	#41	—	4	1	no	—
113	C₂×Q₈	—	47	4	no	—
114	C₂×Q₈	—	94	2	no	—
115	C₂×Q₈	—	188	2	no	—
116	C₂×Q₈	—	188	2	no	—
117	C₂×Q₈	D₁₈₈	1	1	yes	characteristic
118	C₂×D₄	D₁₈₈	1	1	yes	characteristic
119	C₂×D₄	D₁₈₈	1	2	yes	—
120	C₂²×C₄	—	94	2	no	—
121	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
122	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
123	Q₁₆	—	188	2	no	—
124	Q₁₆	—	188	2	no	—
125	M₁₆	—	4	1	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	C₂²×D₄₇	1	2	yes	—
128	C₂×C₈	C₂²×D₄₇	1	2	yes	—
129	C₄⋊C₄	—	47	4	no	—
130	C₄⋊C₄	—	94	4	no	—
131	C₄⋊C₄	—	94	4	no	—
132	C₂²⋊C₄	—	94	2	no	—
133	C₂²⋊C₄	—	94	4	no	—
134	C₂²⋊C₄	—	94	4	no	—
135	C₄²	—	94	2	no	—
136	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
137	C₂³	C₂×D₁₈₈	1	2	yes	—
138	Q₈	—	4	1	no	—
139	Q₈	—	94	4	no	—
140	Q₈	—	94	4	no	—
141	Q₈	—	188	4	no	—
142	D₄	—	4	2	no	—
143	D₄	—	4	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	94	4	no	—
149	C₂×C₄	—	94	4	no	—
150	C₂×C₄	—	188	4	no	—
151	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
152	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
153	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	2	2	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	4	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	1	no	—
163	C₄	—	188	4	no	—
164	C₄	—	188	4	no	—
165	C₄	#13094069	1	1	yes	characteristic
166	C₄	#13094069	1	1	yes	characteristic
167	C₂	—	4	2	no	—
168	C₂	—	4	1	no	—
169	C₂	#15689685	1	1	yes	characteristic
170	C₂	#15689940	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689694

C₂

yes

maximal

#15689465

C₂

yes

maximal, complemented

#15689381

C₂

yes

maximal, complemented

#15689364

C₂

yes

maximal, complemented

#15689281

C₂

yes

maximal, complemented

C₄₇×#408

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

—

C₂×Dic₁₈₈

C₂²

yes

—

C₂×Dic₁₈₈

C₂²

yes

—

—

—

—

—

C₂²

yes

—

#13094003

C₂²

yes

—

#13093983

C₂²

yes

—

#13093958

C₂²

yes

—

#13093956

C₂²

yes

—

#13093949

C₂²

yes

—

#13093949

C₂²

yes

—

C₂×#41×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

characteristic

C₂²⋊C₈×C₄₇

C₂²

yes

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×D₄×C₄₇

D₄

yes

characteristic

C₂×D₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂³

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

—

C₂×C₈×C₄₇

C₂³

yes

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

Dic₁₈₈

—

Dic₁₈₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂³×C₄₇

C₂×D₄

yes

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

#3202

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#534

yes

characteristic

C₂×C₄₇

#578

yes

—

—

—

—

—

—

D₄₇

yes

characteristic

C₄₇

#3202

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₄₇

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

characteristic

C₂²×C₈

C₂×D₄₇

yes

characteristic

C₄⋊Q₈

—

100

#124

—

101

#123

—

102

#123

—

103

C₂²⋊Q₈

—

104

C₂²⋊Q₈

—

105

#117

—

106

#107

—

107

#106

—

108

#103

—

109

#103

—

110

C₂²⋊C₈

C₂×D₄₇

yes

—

111

#41

—

112

#41

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

188

—

116

C₂×Q₈

—

188

—

117

C₂×Q₈

D₁₈₈

yes

characteristic

118

C₂×D₄

D₁₈₈

yes

characteristic

119

C₂×D₄

D₁₈₈

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

C₂²×D₄₇

yes

—

122

C₂²×C₄

C₂²×D₄₇

yes

characteristic

123

Q₁₆

—

188

—

124

Q₁₆

—

188

—

125

M₁₆

—

126

C₂×C₈

—

127

C₂×C₈

C₂²×D₄₇

yes

—

128

C₂×C₈

C₂²×D₄₇

yes

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₂³

C₂×D₁₈₈

yes

characteristic

137

C₂³

C₂×D₁₈₈

yes

—

138

Q₈

—

139

Q₈

—

140

Q₈

—

141

Q₈

—

188

—

142

D₄

—

143

D₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

188

—

151

C₂×C₄

C₂×D₁₈₈

yes

—

152

C₂×C₄

C₂×D₁₈₈

yes

characteristic

153

C₂×C₄

C₂³×D₄₇

yes

characteristic

154

C₈

—

155

C₈

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

C₂²×D₁₈₈

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

188

—

164

C₄

—

188

—

165

C₄

#13094069

yes

characteristic

166

C₄

#13094069

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#15689685

yes

characteristic

170

C₂

#15689940

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#408	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#408	D₄₇
hypocenter	C₄₇	#3202
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #408

subgroup

G/H

center

C₂²

C₂²×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#408

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ #408

N	H ≅ G/N	type
2. #15689465	C₂	semidirect
3. #15689381	C₂	semidirect
4. #15689364	C₂	semidirect
5. #15689281	C₂	semidirect
92. C₄₇	#3202	semidirect

H ≅ G/N

type

2. #15689465

C₂

semidirect

3. #15689381

C₂

semidirect

4. #15689364

C₂

semidirect

5. #15689281

C₂

semidirect

92. C₄₇

#3202

semidirect

Subgroups — #20254283

Conjugacy classes of subgroups

Complements