Group #20244617 — subgroups

Subgroups — #20244617

Order 6016.

290 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 69 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	#15689704	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093952	C₂	1	2	yes	maximal, complemented
3	#15689122	C₂	1	2	yes	maximal, complemented
4	#15688892	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#382	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625350	C₂²	1	2	yes	—
7	C₂×#10625329	C₂²	1	2	yes	—
8	#13094004	—	2	2	no	—
9	#13093999	—	2	2	no	—
10	#13093998	—	2	2	no	—
11	#13093952	—	2	4	no	—
12	#13093947	—	2	2	no	—
13	#13093944	—	2	2	no	—
14	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	#95×C₄₇	—	2	2	no	—
18	C₂²×C₄×C₄₇	—	2	1	no	—
19	C₂²×C₄×C₄₇	—	2	1	no	—
20	C₂²×C₄×C₄₇	D₄	1	2	yes	—
21	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₂×C₂²⋊D₄₇	—	2	2	no	—
23	#10625350	—	2	2	no	—
24	#10625350	—	2	2	no	—
25	#10625350	—	2	2	no	—
26	#10625350	—	2	2	no	—
27	C₂×D₁₈₈	—	2	2	no	—
28	C₂×D₁₈₈	D₄	1	2	yes	—
29	C₂×C₄×D₄₇	—	2	2	no	—
30	C₂×Dic₉₄	D₄	1	2	yes	—
31	C₄⋊C₄×C₄₇	—	2	2	no	—
32	C₄⋊C₄×C₄₇	—	2	2	no	—
33	C₄²×C₄₇	—	2	1	no	—
34	C₄²×C₄₇	D₄	1	2	yes	complemented
35	(C₂×D₄₇)⋊C₄	—	2	4	no	—
36	C₄⋊Dic₄₇	—	2	2	no	—
37	#10625329	—	2	2	no	—
38	#10625329	—	2	2	no	—
39	C₂×C₄₇⋊C₈	—	2	2	no	—
40	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₄₇	—	2	2	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	4	1	no	—
45	C₂×C₄×C₄₇	—	4	1	no	—
46	C₂×C₄×C₄₇	—	4	2	no	—
47	C₂×C₄×C₄₇	—	4	2	no	—
48	C₂×C₄×C₄₇	—	4	1	no	—
49	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#41	1	2	yes	—
53	C₂²⋊D₄₇	—	4	2	no	—
54	C₂²⋊D₄₇	—	4	2	no	—
55	C₂×Dic₄₇	—	2	2	no	—
56	D₁₈₈	—	2	2	no	—
57	D₁₈₈	—	2	2	no	—
58	D₁₈₈	—	8	2	no	—
59	C₄×D₄₇	—	4	2	no	—
60	C₄×D₄₇	—	4	2	no	—
61	Dic₉₄	—	2	2	no	—
62	Dic₉₄	—	2	2	no	—
63	C₄₇⋊C₈	—	4	2	no	—
64	C₂²×C₄₇	—	2	1	no	—
65	C₂²×C₄₇	—	2	1	no	—
66	C₂²×C₄₇	#120	1	1	yes	characteristic
67	C₂²×C₄₇	#121	1	1	yes	characteristic
68	C₂²×C₄₇	#124	1	1	yes	characteristic
69	C₂×D₄₇	—	4	2	no	—
70	C₂×D₄₇	—	4	2	no	—
71	C₂×D₄₇	—	4	2	no	—
72	C₄×C₄₇	—	4	2	no	—
73	C₄×C₄₇	—	4	2	no	—
74	C₄×C₄₇	—	8	1	no	—
75	C₄×C₄₇	#121	1	2	yes	—
76	C₄×C₄₇	#123	1	2	yes	—
77	Dic₄₇	—	4	2	no	—
78	#2040	—	47	1	no	maximal, Sylow
79	C₂×C₄₇	—	2	1	no	—
80	C₂×C₄₇	—	2	1	no	—
81	C₂×C₄₇	#394	1	1	yes	characteristic
82	C₂×C₄₇	#454	1	1	yes	characteristic
83	C₂×C₄₇	#455	1	1	yes	characteristic
84	D₄₇	—	8	2	no	—
85	#537	—	47	1	no	—
86	C₂×#104	—	47	2	no	—
87	#417	—	47	2	no	—
88	#382	D₄₇	1	1	yes	characteristic, complemented
89	#343	—	47	1	no	—
90	C₄₇	#2040	1	1	yes	characteristic, complemented, Sylow
91	C₂×#41	—	47	2	no	—
92	C₂×M₁₆	—	47	2	no	—
93	C₂²⋊Q₈	—	94	2	no	—
94	#121	—	94	2	no	—
95	#121	—	94	2	no	—
96	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
97	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
98	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
99	#104	—	94	4	no	—
100	#103	—	94	2	no	—
101	#102	—	94	2	no	—
102	#95	—	2	2	no	—
103	#41	—	94	2	no	—
104	#41	—	94	2	no	—
105	#41	—	94	2	no	—
106	#41	—	94	2	no	—
107	C₂×Q₈	—	47	2	no	—
108	C₂×D₄	—	47	2	no	—
109	C₂×D₄	—	94	2	no	—
110	C₂×D₄	—	94	2	no	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	94	2	no	—
114	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
115	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
116	M₁₆	—	94	2	no	—
117	M₁₆	—	94	2	no	—
118	C₂×C₈	—	94	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	94	2	no	—
122	C₂²⋊C₄	—	94	4	no	—
123	C₄²	—	2	1	no	—
124	C₄²	D₁₈₈	1	2	yes	complemented
125	C₂³	—	94	2	no	—
126	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
127	Q₈	—	94	2	no	—
128	Q₈	—	94	2	no	—
129	D₄	—	94	2	no	—
130	D₄	—	94	2	no	—
131	D₄	—	376	2	no	—
132	D₄	—	188	2	no	—
133	D₄	—	188	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	94	2	no	—
142	C₂×C₄	—	188	2	no	—
143	C₂×C₄	—	188	2	no	—
144	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
145	C₂×C₄	#10625350	1	1	yes	characteristic
146	C₂×C₄	D₄×D₄₇	1	2	yes	—
147	C₂×C₄	#10625354	1	2	yes	—
148	C₈	—	188	2	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	188	2	no	—
152	C₂²	—	188	2	no	—
153	C₂²	—	188	2	no	—
154	C₂²	#13093992	1	1	yes	characteristic
155	C₂²	#13093998	1	1	yes	characteristic
156	C₂²	#13094013	1	1	yes	characteristic
157	C₄	—	4	2	no	—
158	C₄	—	4	2	no	—
159	C₄	—	8	1	no	—
160	C₄	—	188	2	no	—
161	C₄	#13093999	1	2	yes	—
162	C₄	#13094009	1	2	yes	—
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	376	2	no	—
166	C₂	#15689040	1	1	yes	characteristic
167	C₂	#15689295	1	1	yes	characteristic
168	C₂	#15689300	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689704

C₂

yes

maximal, characteristic

C₂×#13093952

C₂

yes

maximal, complemented

#15689122

C₂

yes

maximal, complemented

#15688892

C₂

yes

maximal, characteristic, complemented

C₄₇×#382

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

C₂²

yes

—

C₂×#10625329

C₂²

yes

—

—

—

—

—

—

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

—

—

—

—

—

C₂×D₁₈₈

—

C₂×D₁₈₈

D₄

yes

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₄

yes

complemented

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

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×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#121

yes

—

C₄×C₄₇

#123

yes

—

Dic₄₇

—

#2040

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

—

—

—

D₄₇

yes

characteristic, complemented

#343

—

C₄₇

#2040

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₂²⋊Q₈

—

#121

—

#121

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂×C₄²

C₂×D₄₇

yes

characteristic

#104

—

100

#103

—

101

#102

—

102

#95

—

103

#41

—

104

#41

—

105

#41

—

106

#41

—

107

C₂×Q₈

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂²⋊D₄₇

yes

—

115

C₂²×C₄

C₂²×D₄₇

yes

characteristic

116

M₁₆

—

117

M₁₆

—

118

C₂×C₈

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

—

124

C₄²

D₁₈₈

yes

complemented

125

C₂³

—

126

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

127

Q₈

—

128

Q₈

—

129

D₄

—

130

D₄

—

131

D₄

—

376

—

132

D₄

—

188

—

133

D₄

—

188

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

188

—

143

C₂×C₄

—

188

—

144

C₂×C₄

C₂×D₁₈₈

yes

characteristic

145

C₂×C₄

#10625350

yes

characteristic

146

C₂×C₄

D₄×D₄₇

yes

—

147

C₂×C₄

#10625354

yes

—

148

C₈

—

188

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

188

—

152

C₂²

—

188

—

153

C₂²

—

188

—

154

C₂²

#13093992

yes

characteristic

155

C₂²

#13093998

yes

characteristic

156

C₂²

#13094013

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

188

—

161

C₄

#13093999

yes

—

162

C₄

#13094009

yes

—

163

C₂

—

164

C₂

—

165

C₂

—

376

—

166

C₂

#15689040

yes

characteristic

167

C₂

#15689295

yes

characteristic

168

C₂

#15689300

yes

characteristic

	subgroup	G/H
center	C₂²	#13093992
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#382	C₂
socle	C₂²×C₄₇	#120
hypercenter	#382	D₄₇
hypocenter	C₄₇	#2040
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #382

subgroup

G/H

center

C₂²

#13093992

commutator

C₂²×C₄×C₄₇

C₂³

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#382

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #382

N	H ≅ G/N	type
2. C₂×#13093952	C₂	semidirect
3. #15689122	C₂	semidirect
4. #15688892	C₂	semidirect
5. C₄₇×#382	C₂	semidirect
34. C₄²×C₄₇	D₄	semidirect
88. #382	D₄₇	semidirect
90. C₄₇	#2040	semidirect
124. C₄²	D₁₈₈	semidirect

H ≅ G/N

type

2. C₂×#13093952

C₂

semidirect

3. #15689122

C₂

semidirect

4. #15688892

C₂

semidirect

5. C₄₇×#382

C₂

semidirect

34. C₄²×C₄₇

D₄

semidirect

88. #382

D₄₇

semidirect

90. C₄₇

#2040

semidirect

124. C₄²

D₁₈₈

semidirect

Subgroups — #20244617

Conjugacy classes of subgroups

Complements