Group #20243918 — subgroups

Subgroups — #20243918

Order 6016.

362 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 85 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	#15689690	C₂	1	1	yes	maximal, characteristic
2	#15689598	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689139	C₂	1	2	yes	maximal, complemented
4	#15688880	C₂	1	2	yes	maximal, complemented
5	C₂×#100×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625350	C₂²	1	1	yes	characteristic
7	#13094020	C₄	1	2	yes	complemented
8	#13094011	C₄	1	2	yes	complemented
9	#13093995	—	2	2	no	—
10	#13093993	—	2	2	no	—
11	D₄×Dic₄₇	—	2	4	no	—
12	#13093983	C₂²	1	1	yes	characteristic
13	#13093983	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
15	#13093950	—	2	4	no	—
16	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
18	#100×C₄₇	—	2	2	no	—
19	C₂⁴×C₄₇	D₄	1	2	yes	—
20	C₂×D₄×C₄₇	—	2	2	no	—
21	C₂×D₄×C₄₇	—	2	2	no	—
22	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
23	C₂×D₄×C₄₇	D₄	1	2	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂×C₂²⋊D₄₇	—	2	1	no	—
26	C₂×C₂²⋊D₄₇	—	2	4	no	—
27	C₂²×Dic₄₇	—	2	4	no	—
28	#10625350	—	2	2	no	—
29	#10625350	—	2	2	no	—
30	C₂×D₁₈₈	C₂×C₄	1	1	yes	characteristic, complemented
31	C₂×C₄×D₄₇	—	2	1	no	—
32	C₂×Dic₉₄	C₂×C₄	1	1	yes	characteristic, complemented
33	M₁₆×C₄₇	—	2	2	no	—
34	M₁₆×C₄₇	—	2	2	no	—
35	C₂×C₈×C₄₇	—	2	2	no	—
36	C₂²⋊Dic₄₇	—	2	4	no	—
37	C₂²⋊Dic₄₇	—	2	4	no	—
38	C₂²⋊Dic₄₇	—	2	2	no	—
39	C₂²⋊Dic₄₇	—	2	1	no	—
40	C₄⋊Dic₄₇	—	2	2	no	—
41	C₄⋊Dic₄₇	—	2	1	no	—
42	C₄×Dic₄₇	C₂×C₄	1	2	yes	complemented
43	C₄×Dic₄₇	D₄	1	4	yes	complemented
44	C₂³×C₄₇	—	2	2	no	—
45	C₂³×C₄₇	—	2	2	no	—
46	C₂³×C₄₇	—	8	2	no	—
47	C₂³×C₄₇	C₂²⋊C₄	1	4	yes	—
48	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×D₄₇	—	2	1	no	—
50	D₄×C₄₇	—	2	4	no	—
51	D₄×C₄₇	—	2	4	no	—
52	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#41	1	2	yes	—
56	C₂²⋊D₄₇	—	4	1	no	—
57	C₂²⋊D₄₇	—	4	1	no	—
58	C₂²⋊D₄₇	—	8	4	no	—
59	C₂×Dic₄₇	—	2	2	no	—
60	C₂×Dic₄₇	—	2	1	no	—
61	C₂×Dic₄₇	—	2	4	no	—
62	C₂×Dic₄₇	—	8	4	no	—
63	D₁₈₈	—	2	2	no	—
64	C₄×D₄₇	—	4	2	no	—
65	Dic₉₄	—	2	2	no	—
66	C₈×C₄₇	—	4	2	no	—
67	C₂²×C₄₇	—	2	4	no	—
68	C₂²×C₄₇	—	4	1	no	—
69	C₂²×C₄₇	—	4	4	no	—
70	C₂²×C₄₇	—	4	4	no	—
71	C₂²×C₄₇	—	8	2	no	—
72	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄₇	#120	1	1	yes	characteristic
74	C₂²×C₄₇	#123	1	1	yes	characteristic
75	C₂×D₄₇	—	4	2	no	—
76	C₂×D₄₇	—	4	1	no	—
77	C₄×C₄₇	C₄×D₄	1	2	yes	—
78	C₄×C₄₇	#121	1	2	yes	—
79	Dic₄₇	—	4	4	no	—
80	Dic₄₇	—	4	2	no	—
81	Dic₄₇	—	4	1	no	—
82	#1911	—	47	1	no	maximal, Sylow
83	C₂×C₄₇	—	2	1	no	—
84	C₂×C₄₇	—	2	1	no	—
85	C₂×C₄₇	—	2	1	no	—
86	C₂×C₄₇	—	4	4	no	—
87	C₂×C₄₇	#386	1	1	yes	characteristic
88	D₄₇	—	8	1	no	—
89	#535	—	47	1	no	—
90	#518	—	47	1	no	—
91	#421	—	47	2	no	—
92	C₂×#100	D₄₇	1	1	yes	characteristic, complemented
93	H(3,ℤ/4)	—	47	2	no	—
94	C₄₇	#1911	1	1	yes	characteristic, complemented, Sylow
95	C₂×#41	—	47	1	no	—
96	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
97	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
98	#127	—	47	2	no	—
99	#124	—	47	2	no	—
100	#121	—	94	2	no	—
101	#120	—	94	2	no	—
102	C₄×D₄	—	94	4	no	—
103	#117	—	47	2	no	—
104	#117	—	47	1	no	—
105	C₂×C₂²⋊C₄	—	94	2	no	—
106	#104	—	94	4	no	—
107	#100	—	2	2	no	—
108	C₂⁴	D₁₈₈	1	2	yes	—
109	#41	—	94	2	no	—
110	#41	—	94	2	no	—
111	C₂×Q₈	—	47	1	no	—
112	C₂×D₄	—	2	2	no	—
113	C₂×D₄	—	2	2	no	—
114	C₂×D₄	—	47	1	no	—
115	C₂×D₄	—	94	4	no	—
116	C₂×D₄	—	94	1	no	—
117	C₂×D₄	C₄×D₄₇	1	2	yes	—
118	C₂×D₄	C₂²⋊D₄₇	1	2	yes	—
119	C₂²×C₄	—	94	4	no	—
120	C₂²×C₄	—	94	1	no	—
121	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
122	M₁₆	—	2	2	no	—
123	M₁₆	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₄⋊C₄	—	94	2	no	—
126	C₄⋊C₄	—	94	1	no	—
127	C₂²⋊C₄	—	94	4	no	—
128	C₂²⋊C₄	—	94	4	no	—
129	C₂²⋊C₄	—	94	2	no	—
130	C₂²⋊C₄	—	94	1	no	—
131	C₄²	—	47	2	no	—
132	C₄²	—	47	4	no	—
133	C₂³	—	2	2	no	—
134	C₂³	—	2	2	no	—
135	C₂³	—	8	2	no	—
136	C₂³	—	94	1	no	—
137	C₂³	(C₂×D₄₇)⋊C₄	1	4	yes	—
138	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
139	Q₈	—	94	2	no	—
140	D₄	—	2	4	no	—
141	D₄	—	2	4	no	—
142	D₄	—	94	2	no	—
143	D₄	—	376	4	no	—
144	D₄	—	188	1	no	—
145	D₄	—	188	1	no	—
146	C₂×C₄	—	94	2	no	—
147	C₂×C₄	—	94	1	no	—
148	C₂×C₄	—	94	4	no	—
149	C₂×C₄	—	376	4	no	—
150	C₂×C₄	—	188	2	no	—
151	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
152	C₂×C₄	D₄×D₄₇	1	2	yes	—
153	C₂×C₄	#10625352	1	2	yes	—
154	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
155	C₈	—	4	2	no	—
156	C₂²	—	2	4	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	4	4	no	—
159	C₂²	—	4	4	no	—
160	C₂²	—	8	2	no	—
161	C₂²	—	188	2	no	—
162	C₂²	—	188	1	no	—
163	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
164	C₂²	#13093994	1	1	yes	characteristic
165	C₂²	#13094008	1	1	yes	characteristic
166	C₄	—	188	4	no	—
167	C₄	—	188	2	no	—
168	C₄	—	188	1	no	—
169	C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
170	C₄	#13093996	1	2	yes	—
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	—	4	4	no	—
175	C₂	—	376	1	no	—
176	C₂	#15689000	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689690

C₂

yes

maximal, characteristic

#15689598

C₂

yes

maximal, characteristic, complemented

#15689139

C₂

yes

maximal, complemented

#15688880

C₂

yes

maximal, complemented

C₂×#100×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

C₂²

yes

characteristic

#13094020

C₄

yes

complemented

#13094011

C₄

yes

complemented

#13093995

—

#13093993

—

D₄×Dic₄₇

—

#13093983

C₂²

yes

characteristic

#13093983

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

—

#13093950

—

C₂²×D₄×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

C₂²

yes

—

#100×C₄₇

—

C₂⁴×C₄₇

D₄

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂×D₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

—

C₂×C₂²⋊D₄₇

—

C₂²×Dic₄₇

—

#10625350

—

#10625350

—

C₂×D₁₈₈

C₂×C₄

yes

characteristic, complemented

C₂×C₄×D₄₇

—

C₂×Dic₉₄

C₂×C₄

yes

characteristic, complemented

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

C₂×C₄

yes

complemented

C₄×Dic₄₇

D₄

yes

complemented

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

#121

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#1911

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#386

yes

characteristic

D₄₇

—

—

—

—

D₄₇

yes

characteristic, complemented

H(3,ℤ/4)

—

C₄₇

#1911

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×D₄

C₂×D₄₇

yes

characteristic

C₂×M₁₆

C₂×D₄₇

yes

—

#127

—

#124

—

100

#121

—

101

#120

—

102

C₄×D₄

—

103

#117

—

104

#117

—

105

C₂×C₂²⋊C₄

—

106

#104

—

107

#100

—

108

C₂⁴

D₁₈₈

yes

—

109

#41

—

110

#41

—

111

C₂×Q₈

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

C₄×D₄₇

yes

—

118

C₂×D₄

C₂²⋊D₄₇

yes

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

C₂²×D₄₇

yes

characteristic

122

M₁₆

—

123

M₁₆

—

124

C₂×C₈

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

128

C₂²⋊C₄

—

129

C₂²⋊C₄

—

130

C₂²⋊C₄

—

131

C₄²

—

132

C₄²

—

133

C₂³

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

(C₂×D₄₇)⋊C₄

yes

—

138

C₂³

C₂×D₁₈₈

yes

characteristic

139

Q₈

—

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

376

—

144

D₄

—

188

—

145

D₄

—

188

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

376

—

150

C₂×C₄

—

188

—

151

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

152

C₂×C₄

D₄×D₄₇

yes

—

153

C₂×C₄

#10625352

yes

—

154

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

155

C₈

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

188

—

162

C₂²

—

188

—

163

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

164

C₂²

#13093994

yes

characteristic

165

C₂²

#13094008

yes

characteristic

166

C₄

—

188

—

167

C₄

—

188

—

168

C₄

—

188

—

169

C₄

(C₂²⋊D₄₇)⋊C₄

yes

—

170

C₄

#13093996

yes

—

171

C₂

—

172

C₂

—

173

C₂

—

174

C₂

—

175

C₂

—

376

—

176

C₂

#15689000

yes

characteristic

	subgroup	G/H
center	C₂	#15689000
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×#100×C₄₇	C₂
socle	C₂×C₄₇	#386
hypercenter	C₂×#100	D₄₇
hypocenter	C₄₇	#1911
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#100

subgroup

G/H

center

C₂

#15689000

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×#100×C₄₇

C₂

C₂×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#100

N	H ≅ G/N	type
2. #15689598	C₂	semidirect
3. #15689139	C₂	semidirect
4. #15688880	C₂	semidirect
5. C₂×#100×C₄₇	C₂	semidirect
7. #13094020	C₄	semidirect
8. #13094011	C₄	semidirect
30. C₂×D₁₈₈	C₂×C₄	semidirect
32. C₂×Dic₉₄	C₂×C₄	semidirect
42. C₄×Dic₄₇	C₂×C₄	semidirect
43. C₄×Dic₄₇	D₄	semidirect
92. C₂×#100	D₄₇	semidirect
94. C₄₇	#1911	semidirect

H ≅ G/N

type

2. #15689598

C₂

semidirect

3. #15689139

C₂

semidirect

4. #15688880

C₂

semidirect

5. C₂×#100×C₄₇

C₂

semidirect

7. #13094020

C₄

semidirect

8. #13094011

C₄

semidirect

30. C₂×D₁₈₈

C₂×C₄

semidirect

32. C₂×Dic₉₄

C₂×C₄

semidirect

42. C₄×Dic₄₇

C₂×C₄

semidirect

43. C₄×Dic₄₇

D₄

semidirect

92. C₂×#100

D₄₇

semidirect

94. C₄₇

#1911

semidirect

Subgroups — #20243918

Conjugacy classes of subgroups

Complements