Group #20242827 — subgroups

Subgroups — #20242827

Order 6016.

182 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 55 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	#15689471	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689318	C₂	1	1	yes	maximal, characteristic
3	D₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688875	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688869	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688866	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625334	—	2	1	no	—
9	C₂²×C₄₇⋊C₈	—	2	1	no	—
10	#13094021	C₂²	1	1	yes	characteristic
11	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	#13093944	—	2	1	no	—
14	#13093944	—	2	1	no	—
15	#13093944	—	2	1	no	—
16	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
17	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₄₇	—	2	1	no	—
19	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
22	#103×C₄₇	—	2	1	no	—
23	C₂×Q₈×C₄₇	—	2	1	no	—
24	C₂×D₄×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	—	2	1	no	—
26	C₂×D₁₈₈	—	2	1	no	—
27	C₂×D₁₈₈	—	2	1	no	—
28	C₂×D₁₈₈	—	2	1	no	—
29	SD₁₆×C₄₇	—	4	2	no	—
30	C₂×C₈×C₄₇	—	2	1	no	—
31	C₄⋊C₄×C₄₇	—	2	1	no	—
32	C₄⋊C₄×C₄₇	—	2	1	no	—
33	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
34	C₂²⋊C₄×C₄₇	—	2	1	no	—
35	C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	#10625334	—	4	2	no	—
37	C₂×C₄₇⋊C₈	—	2	1	no	—
38	C₂×C₄₇⋊C₈	—	2	1	no	—
39	C₂×C₄₇⋊C₈	—	4	2	no	—
40	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
42	C₂³×C₄₇	—	2	1	no	—
43	C₂²×D₄₇	—	4	1	no	—
44	Q₈×C₄₇	—	4	1	no	—
45	Q₈×C₄₇	—	4	1	no	—
46	D₄×C₄₇	—	2	1	no	—
47	D₄×C₄₇	D₈	1	2	yes	—
48	C₂×C₄×C₄₇	—	2	1	no	—
49	C₂×C₄×C₄₇	—	4	1	no	—
50	C₂×C₄×C₄₇	—	4	1	no	—
51	C₂×C₄×C₄₇	—	4	1	no	—
52	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
55	D₁₈₈	—	4	1	no	—
56	D₁₈₈	—	4	1	no	—
57	D₁₈₈	—	4	1	no	—
58	D₁₈₈	—	4	1	no	—
59	D₁₈₈	—	8	1	no	—
60	C₈×C₄₇	—	4	1	no	—
61	C₄₇⋊C₈	—	2	1	no	—
62	C₄₇⋊C₈	—	2	1	no	—
63	C₄₇⋊C₈	—	4	1	no	—
64	C₄₇⋊C₈	—	4	1	no	—
65	C₂²×C₄₇	—	2	2	no	—
66	C₂²×C₄₇	—	4	1	no	—
67	C₂²×C₄₇	—	4	1	no	—
68	C₂²×C₄₇	#121	1	1	yes	characteristic
69	C₂×D₄₇	—	8	1	no	—
70	C₂×D₄₇	—	8	1	no	—
71	C₂×D₄₇	—	8	1	no	—
72	C₄×C₄₇	—	2	1	no	—
73	C₄×C₄₇	—	4	1	no	—
74	C₄×C₄₇	—	8	1	no	—
75	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
76	C₄×C₄₇	#135	1	1	yes	characteristic
77	C₄×C₄₇	#135	1	1	yes	characteristic
78	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
79	#1681	—	47	1	no	maximal, Sylow
80	C₂×C₄₇	—	4	2	no	—
81	C₂×C₄₇	#463	1	1	yes	characteristic
82	C₂×C₄₇	#466	1	1	yes	characteristic
83	C₂×C₄₇	#471	1	1	yes	characteristic
84	D₄₇	—	16	1	no	—
85	#486	—	47	1	no	—
86	#461	D₄₇	1	1	yes	characteristic, complemented
87	#459	—	47	1	no	—
88	C₈×D₄	—	47	1	no	—
89	#335	—	47	1	no	—
90	#332	—	47	1	no	—
91	#331	—	47	1	no	—
92	C₄₇	#1681	1	1	yes	characteristic, complemented, Sylow
93	C₂×SD₁₆	—	2	1	no	—
94	C₂×D₈	—	94	1	no	—
95	C₂²×C₈	—	94	1	no	—
96	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
97	#127	—	47	1	no	—
98	C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
99	C₄⋊C₈	—	47	1	no	—
100	C₄⋊C₈	—	47	1	no	—
101	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
102	#103	—	2	1	no	—
103	#102	—	94	1	no	—
104	#102	—	94	1	no	—
105	#102	—	94	1	no	—
106	C₂²⋊C₈	—	94	1	no	—
107	C₄×C₈	—	47	1	no	—
108	C₂×Q₈	—	2	1	no	—
109	C₂×D₄	—	94	1	no	—
110	C₂×D₄	—	94	1	no	—
111	C₂×D₄	—	94	1	no	—
112	C₂×D₄	C₂²⋊D₄₇	1	1	yes	characteristic
113	C₂²×C₄	—	2	1	no	—
114	SD₁₆	—	4	2	no	—
115	D₈	—	188	2	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	47	1	no	—
118	C₂×C₈	—	47	1	no	—
119	C₂×C₈	—	94	1	no	—
120	C₂×C₈	—	94	1	no	—
121	C₂×C₈	—	188	2	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
125	C₂²⋊C₄	—	2	1	no	—
126	C₄²	C₂²×D₄₇	1	1	yes	characteristic
127	C₂³	—	2	1	no	—
128	C₂³	—	188	1	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	D₄	—	2	1	no	—
132	D₄	—	376	1	no	—
133	D₄	—	188	1	no	—
134	D₄	—	188	1	no	—
135	D₄	—	188	1	no	—
136	D₄	—	188	1	no	—
137	D₄	#10625334	1	2	yes	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
143	C₂×C₄	#10625352	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
145	C₈	—	4	1	no	—
146	C₈	—	94	1	no	—
147	C₈	—	94	1	no	—
148	C₈	—	188	1	no	—
149	C₈	—	188	1	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	4	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	376	1	no	—
154	C₂²	—	376	1	no	—
155	C₂²	—	376	1	no	—
156	C₂²	#13093996	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	1	no	—
159	C₄	—	8	1	no	—
160	C₄	C₂×#10625334	1	1	yes	characteristic
161	C₄	#13094044	1	1	yes	characteristic
162	C₄	#13094045	1	1	yes	characteristic
163	C₄	#13094051	1	1	yes	characteristic
164	C₂	—	4	2	no	—
165	C₂	—	752	1	no	—
166	C₂	#15689340	1	1	yes	characteristic
167	C₂	#15689357	1	1	yes	characteristic
168	C₂	#15689388	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689471

C₂

yes

maximal, characteristic, complemented

#15689318

C₂

yes

maximal, characteristic

D₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

#15688875

C₂

yes

maximal, characteristic, complemented

#15688869

C₂

yes

maximal, characteristic, complemented

#15688866

C₂

yes

maximal, characteristic, complemented

C₄₇×#461

C₂

yes

maximal, characteristic, complemented

C₂×#10625334

—

C₂²×C₄₇⋊C₈

—

#13094021

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

—

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄×D₄×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

#103×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

SD₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625334

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂³×C₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

D₈

yes

—

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

#41

yes

characteristic

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#121

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#486

—

#461

D₄₇

yes

characteristic, complemented

#459

—

C₈×D₄

—

#335

—

#332

—

#331

—

C₄₇

#1681

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×D₈

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

#127

—

C₄×D₄

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

C₂×D₄₇

yes

characteristic

102

#103

—

103

#102

—

104

#102

—

105

#102

—

106

C₂²⋊C₈

—

107

C₄×C₈

—

108

C₂×Q₈

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

C₂²⋊D₄₇

yes

characteristic

113

C₂²×C₄

—

114

SD₁₆

—

115

D₈

—

188

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

188

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

C₂²⋊D₄₇

yes

characteristic

125

C₂²⋊C₄

—

126

C₄²

C₂²×D₄₇

yes

characteristic

127

C₂³

—

128

C₂³

—

188

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

376

—

133

D₄

—

188

—

134

D₄

—

188

—

135

D₄

—

188

—

136

D₄

—

188

—

137

D₄

#10625334

yes

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

D₄×D₄₇

yes

characteristic

143

C₂×C₄

#10625352

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

188

—

149

C₈

—

188

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

376

—

154

C₂²

—

376

—

155

C₂²

—

376

—

156

C₂²

#13093996

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

C₂×#10625334

yes

characteristic

161

C₄

#13094044

yes

characteristic

162

C₄

#13094045

yes

characteristic

163

C₄

#13094051

yes

characteristic

164

C₂

—

165

C₂

—

752

—

166

C₂

#15689340

yes

characteristic

167

C₂

#15689357

yes

characteristic

168

C₂

#15689388

yes

characteristic

	subgroup	G/H
center	C₂²	#13093996
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#461	C₂
socle	C₂²×C₄₇	#121
hypercenter	#461	D₄₇
hypocenter	C₄₇	#1681
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

subgroup

G/H

center

C₂²

#13093996

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#461

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #461

N	H ≅ G/N	type
1. #15689471	C₂	semidirect
3. D₄×C₄₇⋊C₈	C₂	semidirect
4. #15688875	C₂	semidirect
5. #15688869	C₂	semidirect
6. #15688866	C₂	semidirect
7. C₄₇×#461	C₂	semidirect
86. #461	D₄₇	semidirect
92. C₄₇	#1681	semidirect

H ≅ G/N

type

1. #15689471

C₂

semidirect

3. D₄×C₄₇⋊C₈

C₂

semidirect

4. #15688875

C₂

semidirect

5. #15688869

C₂

semidirect

6. #15688866

C₂

semidirect

7. C₄₇×#461

C₂

semidirect

86. #461

D₄₇

semidirect

92. C₄₇

#1681

semidirect

Subgroups — #20242827

Conjugacy classes of subgroups

Complements