Group #20244796 — subgroups

Subgroups — #20244796

Order 6016.

238 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 89 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×#13093948	C₂	1	2	yes	maximal
2	#15688993	C₂	1	1	yes	maximal, characteristic
3	#15688885	C₂	1	2	yes	maximal
4	#15688878	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#391	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₄₇	—	2	1	no	—
8	C₂×C₄⋊Dic₄₇	—	2	1	no	—
9	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	#13093948	—	2	4	no	—
11	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	#95×C₄₇	—	2	1	no	—
15	#95×C₄₇	—	2	1	no	—
16	C₂×Q₈×C₄₇	—	2	1	no	—
17	C₂×Q₈×C₄₇	—	2	2	no	—
18	C₂×Q₈×C₄₇	—	2	2	no	—
19	C₂×Q₈×C₄₇	D₄	1	2	yes	—
20	C₂²×C₄×C₄₇	—	2	1	no	—
21	C₂²×C₄×C₄₇	—	2	1	no	—
22	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₇	—	2	2	no	—
26	C₄⋊C₄×C₄₇	—	2	2	no	—
27	C₄²×C₄₇	—	2	2	no	—
28	C₄⋊Dic₄₇	—	2	2	no	—
29	C₄⋊Dic₄₇	—	2	2	no	—
30	C₄⋊Dic₄₇	—	4	2	no	—
31	C₄⋊Dic₄₇	—	4	2	no	—
32	C₂×C₄₇⋊C₈	—	2	2	no	—
33	C₂×C₄₇⋊C₈	—	2	2	no	—
34	C₂×C₄₇⋊C₈	—	2	2	no	—
35	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
36	Q₈×C₄₇	—	2	4	no	—
37	Q₈×C₄₇	—	4	2	no	—
38	C₂×C₄×C₄₇	—	2	1	no	—
39	C₂×C₄×C₄₇	—	2	1	no	—
40	C₂×C₄×C₄₇	—	2	1	no	—
41	C₂×C₄×C₄₇	—	2	1	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	4	1	no	—
44	C₂×C₄×C₄₇	—	4	1	no	—
45	C₂×C₄×C₄₇	—	4	1	no	—
46	C₂×C₄×C₄₇	—	4	1	no	—
47	C₂×C₄×C₄₇	—	4	1	no	—
48	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
49	C₂×C₄×C₄₇	Q₁₆	1	2	yes	—
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#41	1	2	yes	—
53	C₂×C₄×C₄₇	#41	1	2	yes	—
54	C₂×Dic₄₇	—	4	2	no	—
55	C₂×Dic₄₇	—	4	2	no	—
56	C₂×Dic₄₇	—	4	2	no	—
57	C₄₇⋊C₈	—	4	2	no	—
58	C₂²×C₄₇	#121	1	1	yes	characteristic
59	C₂²×C₄₇	#123	1	1	yes	characteristic
60	C₂²×C₄₇	#124	1	1	yes	characteristic
61	C₂²×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
62	C₂²×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂²×C₄₇	#137	1	1	yes	characteristic
65	C₄×C₄₇	—	2	2	no	—
66	C₄×C₄₇	—	2	2	no	—
67	C₄×C₄₇	—	4	2	no	—
68	C₄×C₄₇	—	8	1	no	—
69	C₄×C₄₇	#123	1	2	yes	—
70	C₄×C₄₇	#126	1	2	yes	—
71	Dic₄₇	—	8	2	no	—
72	#2072	—	47	1	no	maximal, Sylow
73	C₂×C₄₇	#397	1	1	yes	characteristic
74	C₂×C₄₇	#460	1	1	yes	characteristic
75	C₂×C₄₇	#462	1	1	yes	characteristic
76	C₂×C₄₇	#483	1	1	yes	characteristic
77	C₂×C₄₇	#484	1	1	yes	characteristic
78	C₂×C₄₇	#487	1	1	yes	characteristic
79	C₂×C₄₇	#489	1	1	yes	characteristic
80	C₂×#103	—	47	2	no	—
81	#391	D₄₇	1	1	yes	characteristic
82	#384	—	47	1	no	—
83	#340	—	47	2	no	—
84	#336	—	47	1	no	—
85	C₄₇	#2072	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
87	C₂²×C₈	—	47	2	no	—
88	C₂×C₄⋊C₄	—	47	2	no	—
89	C₂×C₄⋊C₄	—	94	1	no	—
90	C₂×C₄⋊C₄	—	94	1	no	—
91	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
93	#103	—	94	4	no	—
94	#95	—	2	1	no	—
95	#95	—	2	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	D₁₈₈	1	2	yes	—
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	94	2	no	—
103	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
104	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
106	C₂×C₈	—	94	2	no	—
107	C₂×C₈	—	94	2	no	—
108	C₂×C₈	—	94	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	94	2	no	—
111	C₄⋊C₄	—	94	2	no	—
112	C₄⋊C₄	—	188	2	no	—
113	C₄⋊C₄	—	188	2	no	—
114	C₄²	—	2	2	no	—
115	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
116	Q₈	—	2	4	no	—
117	Q₈	—	4	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	188	2	no	—
129	C₂×C₄	—	188	2	no	—
130	C₂×C₄	—	188	2	no	—
131	C₂×C₄	#10625336	1	2	yes	—
132	C₂×C₄	#10625337	1	2	yes	—
133	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
134	C₂×C₄	#10625350	1	1	yes	characteristic
135	C₂×C₄	#10625352	1	2	yes	—
136	C₂×C₄	#10625352	1	2	yes	—
137	C₈	—	188	2	no	—
138	C₂²	#13093998	1	1	yes	characteristic
139	C₂²	#13094005	1	1	yes	characteristic
140	C₂²	#13094011	1	1	yes	characteristic
141	C₂²	C₂×#10625336	1	1	yes	characteristic
142	C₂²	C₂×#10625337	1	1	yes	characteristic
143	C₂²	#13094049	1	1	yes	characteristic
144	C₂²	#13094054	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	2	no	—
148	C₄	—	8	1	no	—
149	C₄	—	376	2	no	—
150	C₄	#13094008	1	2	yes	—
151	C₄	#13094017	1	2	yes	—
152	C₂	#15689048	1	1	yes	characteristic
153	C₂	#15689323	1	1	yes	characteristic
154	C₂	#15689333	1	1	yes	characteristic
155	C₂	#15689451	1	1	yes	characteristic
156	C₂	#15689456	1	1	yes	characteristic
157	C₂	#15689473	1	1	yes	characteristic
158	C₂	#15689482	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093948

C₂

yes

maximal

#15688993

C₂

yes

maximal, characteristic

#15688885

C₂

yes

maximal

#15688878

C₂

yes

maximal, characteristic

C₄₇×#391

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

#13093948

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

—

#95×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

Q₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

C₂×SD₁₆

yes

characteristic

C₂²×C₄₇

C₂×Q₁₆

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#123

yes

—

C₄×C₄₇

#126

yes

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

yes

—

—

—

C₄₇

#2072

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂×C₄²

C₂×D₄₇

yes

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₁₈₈

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

104

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

105

C₂²×C₄

C₂²×D₄₇

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

188

—

113

C₄⋊C₄

—

188

—

114

C₄²

—

115

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

188

—

129

C₂×C₄

—

188

—

130

C₂×C₄

—

188

—

131

C₂×C₄

#10625336

yes

—

132

C₂×C₄

#10625337

yes

—

133

C₂×C₄

C₂×D₁₈₈

yes

characteristic

134

C₂×C₄

#10625350

yes

characteristic

135

C₂×C₄

#10625352

yes

—

136

C₂×C₄

#10625352

yes

—

137

C₈

—

188

—

138

C₂²

#13093998

yes

characteristic

139

C₂²

#13094005

yes

characteristic

140

C₂²

#13094011

yes

characteristic

141

C₂²

C₂×#10625336

yes

characteristic

142

C₂²

C₂×#10625337

yes

characteristic

143

C₂²

#13094049

yes

characteristic

144

C₂²

#13094054

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

376

—

150

C₄

#13094008

yes

—

151

C₄

#13094017

yes

—

152

C₂

#15689048

yes

characteristic

153

C₂

#15689323

yes

characteristic

154

C₂

#15689333

yes

characteristic

155

C₂

#15689451

yes

characteristic

156

C₂

#15689456

yes

characteristic

157

C₂

#15689473

yes

characteristic

158

C₂

#15689482

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#391	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	#391	D₄₇
hypocenter	C₄₇	#2072
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂²×C₄×C₄₇

C₂³

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#391

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
85. C₄₇	#2072	semidirect

H ≅ G/N

type

85. C₄₇

#2072

semidirect

Subgroups — #20244796

Conjugacy classes of subgroups

Complements