Group #20243501 — subgroups

Subgroups — #20243501

Order 6016.

236 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 73 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	#15689600	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093942	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093942	C₂	1	1	yes	maximal, characteristic
4	#15688895	C₂	1	1	yes	maximal, characteristic
5	#15688894	C₂	1	1	yes	maximal, characteristic
6	#15688893	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#344	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
9	C₂×#10625329	C₂²	1	1	yes	characteristic
10	C₂×#10625329	C₂²	1	1	yes	characteristic
11	C₄×Dic₉₄	—	2	2	no	—
12	#13093989	—	2	2	no	—
13	#13093983	C₂²	1	1	yes	characteristic
14	#13093980	C₄	1	4	yes	—
15	#13093942	—	2	2	no	—
16	#13093942	—	2	2	no	—
17	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
19	#117×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
21	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
22	C₂×Dic₉₄	—	2	1	no	—
23	C₂×Dic₉₄	—	2	1	no	—
24	C₂×Dic₉₄	—	2	1	no	—
25	C₂×Dic₉₄	—	2	1	no	—
26	C₂×Dic₉₄	D₄	1	2	yes	—
27	C₂×Dic₉₄	D₄	1	2	yes	—
28	M₁₆×C₄₇	—	2	1	no	—
29	M₁₆×C₄₇	—	2	1	no	—
30	M₁₆×C₄₇	—	2	1	no	—
31	M₁₆×C₄₇	—	2	1	no	—
32	C₂×C₈×C₄₇	—	2	1	no	—
33	C₂×C₈×C₄₇	—	2	1	no	—
34	C₄⋊C₄×C₄₇	—	2	1	no	—
35	C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
36	C₄²×C₄₇	—	2	1	no	—
37	C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₄₇	—	2	1	no	—
39	Dic₄₇⋊C₄	—	2	4	no	—
40	C₄×Dic₄₇	—	2	1	no	—
41	C₄×Dic₄₇	—	2	4	no	—
42	#10625329	—	2	1	no	—
43	#10625329	—	2	1	no	—
44	#10625329	—	2	1	no	—
45	#10625329	—	2	1	no	—
46	C₂×C₄₇⋊C₈	—	2	1	no	—
47	C₂×C₄₇⋊C₈	—	2	1	no	—
48	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
56	C₂×Dic₄₇	—	2	2	no	—
57	C₂×Dic₄₇	—	2	2	no	—
58	C₂×Dic₄₇	—	2	2	no	—
59	C₂×Dic₄₇	C₂²⋊C₄	1	4	yes	—
60	Dic₉₄	—	2	2	no	—
61	Dic₉₄	—	2	2	no	—
62	Dic₉₄	—	2	2	no	—
63	Dic₉₄	—	2	2	no	—
64	C₈×C₄₇	—	4	1	no	—
65	C₈×C₄₇	—	4	1	no	—
66	C₄₇⋊C₈	—	4	1	no	—
67	C₄₇⋊C₈	—	4	1	no	—
68	C₂²×C₄₇	—	4	1	no	—
69	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄₇	#117	1	1	yes	characteristic
71	C₂²×C₄₇	#117	1	1	yes	characteristic
72	C₄×C₄₇	—	4	2	no	—
73	C₄×C₄₇	#123	1	1	yes	characteristic
74	C₄×C₄₇	#123	1	1	yes	characteristic
75	C₄×C₄₇	#123	1	1	yes	characteristic
76	C₄×C₄₇	#123	1	1	yes	characteristic
77	Dic₄₇	—	2	4	no	—
78	Dic₄₇	—	4	2	no	—
79	#1830	—	47	1	no	maximal, Sylow
80	C₂×C₄₇	—	2	1	no	—
81	C₂×C₄₇	—	2	1	no	—
82	C₂×C₄₇	—	2	1	no	—
83	C₂×C₄₇	#381	1	1	yes	characteristic
84	#519	—	47	1	no	—
85	C₂×#101	—	47	1	no	—
86	C₂×#101	—	47	1	no	—
87	#344	—	47	1	no	—
88	#344	—	47	1	no	—
89	#344	—	47	1	no	—
90	#344	D₄₇	1	1	yes	characteristic
91	C₄₇	#1830	1	1	yes	characteristic, complemented, Sylow
92	C₂²×Q₈	—	47	1	no	—
93	C₂×M₁₆	—	47	1	no	—
94	C₂×M₁₆	—	47	1	no	—
95	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
96	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
97	C₄×Q₈	—	94	2	no	—
98	C₄×Q₈	—	94	2	no	—
99	#117	—	47	1	no	—
100	#117	—	47	4	no	—
101	#117	C₂×D₄₇	1	1	yes	characteristic
102	#101	—	94	2	no	—
103	#101	—	94	2	no	—
104	C₂×Q₈	—	47	2	no	—
105	C₂×Q₈	—	47	2	no	—
106	C₂×Q₈	—	94	1	no	—
107	C₂×Q₈	—	94	1	no	—
108	C₂×Q₈	—	94	1	no	—
109	C₂×Q₈	—	94	1	no	—
110	C₂²×C₄	—	47	2	no	—
111	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
112	M₁₆	—	2	1	no	—
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	2	1	no	—
116	M₁₆	—	94	1	no	—
117	M₁₆	—	94	1	no	—
118	M₁₆	—	94	1	no	—
119	M₁₆	—	94	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	94	1	no	—
123	C₂×C₈	—	94	1	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	94	4	no	—
126	C₄⋊C₄	—	94	1	no	—
127	C₂²⋊C₄	—	47	2	no	—
128	C₂²⋊C₄	C₄×D₄₇	1	2	yes	—
129	C₄²	—	2	1	no	—
130	C₄²	—	94	4	no	—
131	C₄²	—	94	1	no	—
132	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
133	Q₈	—	94	2	no	—
134	Q₈	—	94	2	no	—
135	Q₈	—	94	2	no	—
136	Q₈	—	94	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	47	4	no	—
139	C₂×C₄	—	94	2	no	—
140	C₂×C₄	—	94	2	no	—
141	C₂×C₄	—	94	2	no	—
142	C₂×C₄	#10625350	1	1	yes	characteristic
143	C₂×C₄	#10625350	1	1	yes	characteristic
144	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
145	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
146	C₂×C₄	#10625352	1	1	yes	characteristic
147	C₂×C₄	#10625354	1	1	yes	characteristic
148	C₈	—	4	1	no	—
149	C₈	—	4	1	no	—
150	C₈	—	188	1	no	—
151	C₈	—	188	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
154	C₂²	#13093981	1	1	yes	characteristic
155	C₂²	#13093982	1	1	yes	characteristic
156	C₄	—	4	2	no	—
157	C₄	—	94	4	no	—
158	C₄	—	188	2	no	—
159	C₄	#13094006	1	1	yes	characteristic
160	C₄	#13094006	1	1	yes	characteristic
161	C₄	#13094009	1	1	yes	characteristic
162	C₄	#13094009	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	#15688985	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689600

C₂

yes

maximal, characteristic

C₂×#13093942

C₂

yes

maximal, characteristic

C₂×#13093942

C₂

yes

maximal, characteristic

#15688895

C₂

yes

maximal, characteristic

#15688894

C₂

yes

maximal, characteristic

#15688893

C₂

yes

maximal, characteristic

C₄₇×#344

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

characteristic

C₄×Dic₉₄

—

#13093989

—

#13093983

C₂²

yes

characteristic

#13093980

C₄

yes

—

#13093942

—

#13093942

—

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

#117×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

D₄

yes

—

C₂×Dic₉₄

D₄

yes

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

C₂×C₄

yes

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

—

—

—

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²⋊C₄

yes

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

#1830

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

—

—

—

—

—

—

D₄₇

yes

characteristic

C₄₇

#1830

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

characteristic

C₂×M₁₆

C₂×D₄₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#117

—

100

#117

—

101

#117

C₂×D₄₇

yes

characteristic

102

#101

—

103

#101

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²×D₄₇

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

118

M₁₆

—

119

M₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

128

C₂²⋊C₄

C₄×D₄₇

yes

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₂³

C₂×C₄×D₄₇

yes

characteristic

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

#10625350

yes

characteristic

143

C₂×C₄

#10625350

yes

characteristic

144

C₂×C₄

D₄×D₄₇

yes

characteristic

145

C₂×C₄

D₄×D₄₇

yes

characteristic

146

C₂×C₄

#10625352

yes

characteristic

147

C₂×C₄

#10625354

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

188

—

151

C₈

—

188

—

152

C₂²

—

153

C₂²

C₂²⋊C₄×D₄₇

yes

characteristic

154

C₂²

#13093981

yes

characteristic

155

C₂²

#13093982

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

188

—

159

C₄

#13094006

yes

characteristic

160

C₄

#13094006

yes

characteristic

161

C₄

#13094009

yes

characteristic

162

C₄

#13094009

yes

characteristic

163

C₂

—

164

C₂

—

165

C₂

—

166

C₂

#15688985

yes

characteristic

	subgroup	G/H
center	C₂	#15688985
commutator	C₂³×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#344	C₂
socle	C₂×C₄₇	#381
hypercenter	#344	D₄₇
hypocenter	C₄₇	#1830
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₂

#15688985

commutator

C₂³×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#344

C₂

C₂×C₄₇

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
91. C₄₇	#1830	semidirect

H ≅ G/N

type

91. C₄₇

#1830

semidirect

Subgroups — #20243501

Conjugacy classes of subgroups

Complements