Group #20242850 — subgroups

Subgroups — #20242850

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	#15689400	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689315	C₂	1	1	yes	maximal, characteristic
3	#15689202	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688876	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688869	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688867	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#492	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625334	—	2	1	no	—
9	C₂×#10625324	—	2	1	no	—
10	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
11	#13093957	—	2	1	no	—
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
15	#13093945	—	2	1	no	—
16	#13093944	—	2	1	no	—
17	(C₂×D₄₇)⋊C₈	—	2	1	no	—
18	C₂×SD₁₆×C₄₇	—	2	1	no	—
19	C₂×SD₁₆×C₄₇	—	2	1	no	—
20	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
21	#127×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₄₇	—	2	1	no	—
24	C₂×D₄×C₄₇	—	2	1	no	—
25	C₂×D₄×C₄₇	—	2	1	no	—
26	C₂×D₄×C₄₇	—	2	1	no	—
27	C₂×D₁₈₈	D₄	1	1	yes	characteristic
28	C₂×C₄×D₄₇	—	2	1	no	—
29	SD₁₆×C₄₇	—	4	2	no	—
30	SD₁₆×C₄₇	—	4	2	no	—
31	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
33	C₄⋊C₄×C₄₇	—	2	1	no	—
34	C₄⋊C₄×C₄₇	—	2	1	no	—
35	C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	#10625334	—	4	2	no	—
37	(C₂×D₄₇)⋊C₄	—	2	1	no	—
38	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄₇⋊C₈	—	2	1	no	—
40	C₂×C₄₇⋊C₈	—	2	1	no	—
41	C₂×C₄₇⋊C₈	—	2	1	no	—
42	#10625324	—	4	2	no	—
43	C₂³×C₄₇	—	4	1	no	—
44	C₂²×D₄₇	—	2	1	no	—
45	Q₈×C₄₇	—	4	1	no	—
46	Q₈×C₄₇	—	4	1	no	—
47	D₄×C₄₇	—	4	1	no	—
48	D₄×C₄₇	—	4	1	no	—
49	D₄×C₄₇	—	4	1	no	—
50	D₄×C₄₇	—	4	1	no	—
51	D₄×C₄₇	—	8	1	no	—
52	C₂×C₄×C₄₇	—	4	1	no	—
53	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
56	C₂×Dic₄₇	—	2	1	no	—
57	D₁₈₈	—	2	1	no	—
58	D₁₈₈	—	2	1	no	—
59	C₄×D₄₇	—	4	1	no	—
60	C₄×D₄₇	—	4	1	no	—
61	C₈×C₄₇	—	2	1	no	—
62	C₈×C₄₇	D₈	1	2	yes	complemented
63	C₄₇⋊C₈	—	4	1	no	—
64	C₄₇⋊C₈	—	4	1	no	—
65	C₄₇⋊C₈	—	4	1	no	—
66	C₂²×C₄₇	—	8	1	no	—
67	C₂²×C₄₇	—	8	1	no	—
68	C₂²×C₄₇	—	8	1	no	—
69	C₂²×C₄₇	#121	1	1	yes	characteristic
70	C₂×D₄₇	—	4	1	no	—
71	C₂×D₄₇	—	4	1	no	—
72	C₂×D₄₇	—	4	1	no	—
73	C₄×C₄₇	—	2	1	no	—
74	C₄×C₄₇	—	8	1	no	—
75	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
76	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₄×C₄₇	#137	1	1	yes	characteristic
79	Dic₄₇	—	4	1	no	—
80	#1685	—	47	1	no	maximal, Sylow
81	C₂×C₄₇	—	16	1	no	—
82	C₂×C₄₇	#459	1	1	yes	characteristic
83	C₂×C₄₇	#468	1	1	yes	characteristic
84	C₂×C₄₇	#471	1	1	yes	characteristic
85	D₄₇	—	8	1	no	—
86	#492	D₄₇	1	1	yes	characteristic, complemented
87	#474	—	47	1	no	—
88	#459	—	47	1	no	—
89	#436	—	47	1	no	—
90	#335	—	47	1	no	—
91	#332	—	47	1	no	—
92	#331	—	47	1	no	—
93	C₄₇	#1685	1	1	yes	characteristic, complemented, Sylow
94	C₂×SD₁₆	—	2	1	no	—
95	C₂×SD₁₆	—	2	1	no	—
96	C₂×D₈	—	94	1	no	—
97	C₂×M₁₆	—	94	1	no	—
98	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
99	#127	C₂×D₄₇	1	1	yes	characteristic
100	C₄×D₄	—	47	1	no	—
101	#107	—	94	1	no	—
102	C₄⋊C₈	—	47	1	no	—
103	C₄⋊C₈	—	47	1	no	—
104	C₄⋊C₈	—	47	1	no	—
105	#102	—	94	1	no	—
106	#102	—	94	1	no	—
107	C₂²⋊C₈	—	94	1	no	—
108	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
109	C₂×Q₈	—	2	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	47	1	no	—
114	C₂²×C₄	—	94	1	no	—
115	SD₁₆	—	4	2	no	—
116	SD₁₆	—	4	2	no	—
117	D₈	—	188	2	no	—
118	M₁₆	—	188	2	no	—
119	C₂×C₈	—	94	1	no	—
120	C₂×C₈	—	94	1	no	—
121	C₂×C₈	—	94	1	no	—
122	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
123	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	47	1	no	—
127	C₂²⋊C₄	—	94	1	no	—
128	C₄²	C₂²×D₄₇	1	1	yes	characteristic
129	C₂³	—	4	1	no	—
130	C₂³	—	94	1	no	—
131	Q₈	—	4	1	no	—
132	Q₈	—	4	1	no	—
133	D₄	—	4	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	8	1	no	—
138	D₄	—	94	1	no	—
139	D₄	—	94	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	94	1	no	—
142	C₂×C₄	—	188	1	no	—
143	C₂×C₄	—	188	1	no	—
144	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
145	C₂×C₄	#10625352	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	188	1	no	—
149	C₈	—	188	1	no	—
150	C₈	—	188	1	no	—
151	C₈	#10625334	1	2	yes	complemented
152	C₂²	—	8	1	no	—
153	C₂²	—	8	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	188	1	no	—
156	C₂²	—	188	1	no	—
157	C₂²	—	188	1	no	—
158	C₂²	#13093995	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	8	1	no	—
161	C₄	—	188	1	no	—
162	C₄	C₂×#10625334	1	1	yes	characteristic
163	C₄	#13094050	1	1	yes	characteristic
164	C₄	#13094051	1	1	yes	characteristic
165	C₄	#13094053	1	1	yes	characteristic
166	C₂	—	16	1	no	—
167	C₂	—	376	1	no	—
168	C₂	#15689315	1	1	yes	characteristic
169	C₂	#15689366	1	1	yes	characteristic
170	C₂	#15689383	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689400

C₂

yes

maximal, characteristic, complemented

#15689315

C₂

yes

maximal, characteristic

#15689202

C₂

yes

maximal, characteristic, complemented

#15688876

C₂

yes

maximal, characteristic, complemented

#15688869

C₂

yes

maximal, characteristic, complemented

#15688867

C₂

yes

maximal, characteristic, complemented

C₄₇×#492

C₂

yes

maximal, characteristic, complemented

C₂×#10625334

—

C₂×#10625324

—

C₄×D₁₈₈

C₂²

yes

characteristic

#13093957

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

#13093945

—

#13093944

—

(C₂×D₄₇)⋊C₈

—

C₂×SD₁₆×C₄₇

—

C₂×SD₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

#127×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₁₈₈

D₄

yes

characteristic

C₂×C₄×D₄₇

—

SD₁₆×C₄₇

—

SD₁₆×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625334

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

#10625324

—

C₂³×C₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

D₈

yes

complemented

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

C₂×D₈

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#1685

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#492

D₄₇

yes

characteristic, complemented

—

—

—

—

—

—

C₄₇

#1685

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

#127

C₂×D₄₇

yes

characteristic

100

C₄×D₄

—

101

#107

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#102

—

106

#102

—

107

C₂²⋊C₈

—

108

C₄×C₈

C₂×D₄₇

yes

characteristic

109

C₂×Q₈

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

—

115

SD₁₆

—

116

SD₁₆

—

117

D₈

—

188

—

118

M₁₆

—

188

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

123

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

C₂²×D₄₇

yes

characteristic

129

C₂³

—

130

C₂³

—

131

Q₈

—

132

Q₈

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

188

—

143

C₂×C₄

—

188

—

144

C₂×C₄

D₄×D₄₇

yes

characteristic

145

C₂×C₄

#10625352

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

147

C₈

—

148

C₈

—

188

—

149

C₈

—

188

—

150

C₈

—

188

—

151

C₈

#10625334

yes

complemented

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

188

—

156

C₂²

—

188

—

157

C₂²

—

188

—

158

C₂²

#13093995

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

188

—

162

C₄

C₂×#10625334

yes

characteristic

163

C₄

#13094050

yes

characteristic

164

C₄

#13094051

yes

characteristic

165

C₄

#13094053

yes

characteristic

166

C₂

—

167

C₂

—

376

—

168

C₂

#15689315

yes

characteristic

169

C₂

#15689366

yes

characteristic

170

C₂

#15689383

yes

characteristic

	subgroup	G/H
center	C₂²	#13093995
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#492	C₂
socle	C₂²×C₄₇	#121
hypercenter	#492	D₄₇
hypocenter	C₄₇	#1685
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#13093995

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#492

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #15689400	C₂	semidirect
3. #15689202	C₂	semidirect
4. #15688876	C₂	semidirect
5. #15688869	C₂	semidirect
6. #15688867	C₂	semidirect
7. C₄₇×#492	C₂	semidirect
62. C₈×C₄₇	D₈	semidirect
86. #492	D₄₇	semidirect
93. C₄₇	#1685	semidirect
151. C₈	#10625334	semidirect

H ≅ G/N

type

1. #15689400

C₂

semidirect

3. #15689202

C₂

semidirect

4. #15688876

C₂

semidirect

5. #15688869

C₂

semidirect

6. #15688867

C₂

semidirect

7. C₄₇×#492

C₂

semidirect

62. C₈×C₄₇

D₈

semidirect

86. #492

D₄₇

semidirect

93. C₄₇

#1685

semidirect

151. C₈

#10625334

semidirect

Subgroups — #20242850

Conjugacy classes of subgroups

Complements