Group #19132985 — subgroups

Subgroups — #19132985

Order 5504.

312 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 97 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×C₄×D₁₇₂	C₂	1	1	yes	maximal, characteristic
2	#14582952	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582626	C₂	1	4	yes	maximal, complemented
4	C₄₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₇₂	C₄	1	1	yes	characteristic
6	C₂²×C₄×D₄₃	—	2	1	no	—
7	C₄×D₁₇₂	—	2	1	no	—
8	C₄×D₁₇₂	—	2	1	no	—
9	C₄×D₁₇₂	C₄	1	2	yes	—
10	C₄×D₁₇₂	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₃	C₄	1	1	yes	characteristic
12	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
13	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
14	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
15	C₄×C₄₃⋊C₈	C₂²	1	2	yes	complemented
16	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₄₃	C₂²	1	2	yes	complemented
18	C₂²⋊C₈×C₄₃	—	2	1	no	—
19	C₄×C₈×C₄₃	—	2	1	no	—
20	C₂³×D₄₃	—	2	1	no	—
21	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
24	C₂²×Dic₄₃	—	2	1	no	—
25	C₂×D₁₇₂	—	2	1	no	—
26	C₂×D₁₇₂	—	2	2	no	—
27	C₂×D₁₇₂	C₈	1	4	yes	complemented
28	C₂×D₁₇₂	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₄₃	—	2	2	no	—
30	C₂×C₄×D₄₃	—	2	2	no	—
31	C₂×C₄×D₄₃	—	4	4	no	—
32	C₂×C₈×C₄₃	—	2	2	no	—
33	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₄₃	D₄	1	1	yes	characteristic
35	C₄²×C₄₃	D₄	1	1	yes	characteristic
36	C₄²×C₄₃	C₂³	1	1	yes	characteristic
37	(C₂×D₄₃)⋊C₄	—	2	2	no	—
38	(C₂×D₄₃)⋊C₄	—	2	2	no	—
39	C₄⋊Dic₄₃	—	2	1	no	—
40	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
41	C₂×C₄₃⋊C₈	—	2	2	no	—
42	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×D₄₃	—	2	4	no	—
44	C₂²×D₄₃	—	2	2	no	—
45	C₂²×D₄₃	—	4	2	no	—
46	C₂²×D₄₃	—	4	2	no	—
47	C₂×C₄×C₄₃	—	2	2	no	—
48	C₂×C₄×C₄₃	—	4	1	no	—
49	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₂×C₈	1	2	yes	—
53	C₂×C₄×C₄₃	M₁₆	1	2	yes	—
54	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₄₃	—	2	2	no	—
58	C₂×Dic₄₃	—	4	2	no	—
59	D₁₇₂	—	2	4	no	—
60	D₁₇₂	—	2	2	no	—
61	D₁₇₂	C₂×C₈	1	4	yes	—
62	C₄×D₄₃	—	4	4	no	—
63	C₄×D₄₃	—	4	4	no	—
64	C₈×C₄₃	—	4	2	no	—
65	C₄₃⋊C₈	—	4	2	no	—
66	C₂²×C₄₃	—	2	1	no	—
67	C₂²×C₄₃	—	2	1	no	—
68	C₂²×C₄₃	C₂²⋊C₈	1	2	yes	—
69	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×D₄₃	—	2	4	no	—
71	C₂×D₄₃	—	4	8	no	—
72	C₂×D₄₃	—	4	2	no	—
73	C₂×D₄₃	—	4	2	no	—
74	C₄×C₄₃	—	2	1	no	—
75	C₄×C₄₃	—	2	1	no	—
76	C₄×C₄₃	—	2	1	no	—
77	C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
78	C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
79	C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
80	C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
81	Dic₄₃	—	4	2	no	—
82	#1478	—	43	1	no	maximal, Sylow
83	C₂×C₄₃	—	2	2	no	—
84	C₂×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄₃	#418	1	1	yes	characteristic
86	C₂×C₄₃	#421	1	1	yes	characteristic
87	D₄₃	—	4	4	no	—
88	C₂×C₄×D₄	—	43	1	no	—
89	#431	—	43	1	no	—
90	#431	D₄₃	1	1	yes	characteristic, complemented
91	D₄⋊C₈	—	43	4	no	—
92	C₄₃	#1478	1	1	yes	characteristic, complemented, Sylow
93	C₂²×D₄	—	43	1	no	—
94	C₂³×C₄	—	86	1	no	—
95	C₄×D₄	—	43	2	no	—
96	C₄×D₄	—	43	2	no	—
97	C₄×D₄	—	86	1	no	—
98	C₄×D₄	—	86	1	no	—
99	C₂×C₄⋊C₄	—	43	1	no	—
100	C₂×C₂²⋊C₄	—	86	1	no	—
101	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
102	C₄⋊C₈	—	86	1	no	—
103	C₄⋊C₈	C₂×D₄₃	1	2	yes	complemented
104	C₂²⋊C₈	—	2	1	no	—
105	C₂²⋊C₈	—	86	1	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	—	43	2	no	—
108	C₂⁴	—	86	1	no	—
109	C₂×D₄	—	43	4	no	—
110	C₂×D₄	—	43	2	no	—
111	C₂×D₄	—	86	2	no	—
112	C₂×D₄	—	86	1	no	—
113	C₂²×C₄	—	86	1	no	—
114	C₂²×C₄	—	86	2	no	—
115	C₂²×C₄	—	86	2	no	—
116	C₂²×C₄	—	172	4	no	—
117	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
118	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
119	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	86	2	no	—
122	C₄⋊C₄	—	43	2	no	—
123	C₄⋊C₄	—	86	1	no	—
124	C₂²⋊C₄	—	86	2	no	—
125	C₂²⋊C₄	—	86	2	no	—
126	C₄²	C₄×D₄₃	1	1	yes	characteristic
127	C₄²	D₁₇₂	1	1	yes	characteristic
128	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
129	C₄²	C₂²×D₄₃	1	1	yes	characteristic
130	C₂³	—	86	4	no	—
131	C₂³	—	86	2	no	—
132	C₂³	—	172	2	no	—
133	C₂³	—	172	2	no	—
134	C₂³	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
135	D₄	—	43	4	no	—
136	D₄	—	86	4	no	—
137	D₄	—	86	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	86	2	no	—
141	C₂×C₄	—	172	4	no	—
142	C₂×C₄	—	172	2	no	—
143	C₂×C₄	—	172	4	no	—
144	C₂×C₄	C₈×D₄₃	1	2	yes	—
145	C₂×C₄	#10621872	1	2	yes	—
146	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
147	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
148	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
150	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
151	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
152	C₈	—	4	2	no	—
153	C₈	—	172	2	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	86	4	no	—
157	C₂²	—	172	8	no	—
158	C₂²	—	172	2	no	—
159	C₂²	—	172	2	no	—
160	C₂²	(C₂×D₄₃)⋊C₈	1	2	yes	—
161	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	172	2	no	—
166	C₄	(C₂×D₄₃)⋊C₈	1	2	yes	—
167	C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
168	C₄	C₂×#10621872	1	1	yes	characteristic
169	C₄	#13063350	1	2	yes	—
170	C₂	—	2	2	no	—
171	C₂	—	172	4	no	—
172	C₂	C₂×(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
173	C₂	#14582903	1	1	yes	characteristic
174	C₂	#14582915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₁₇₂

C₂

yes

maximal, characteristic

#14582952

C₂

yes

maximal, characteristic, complemented

#14582626

C₂

yes

maximal, complemented

C₄₃×#431

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₇₂

C₄

yes

characteristic

C₂²×C₄×D₄₃

—

C₄×D₁₇₂

—

C₄×D₁₇₂

—

C₄×D₁₇₂

C₄

yes

—

C₄×D₁₇₂

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₄

yes

characteristic

C₂×(C₂×D₄₃)⋊C₄

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

C₂²

yes

complemented

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

complemented

C₂²⋊C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₂³×D₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₈

yes

complemented

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×C₈

yes

—

C₂×C₄×C₄₃

M₁₆

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

C₂×C₈

yes

—

C₄×D₄₃

—

C₄×D₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂²⋊C₈

yes

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊C₈

yes

—

C₄×C₄₃

C₂²×C₈

yes

characteristic

C₄×C₄₃

C₂×M₁₆

yes

characteristic

C₄×C₄₃

AGL(1,ℤ/8)

yes

—

Dic₄₃

—

#1478

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

C₂×C₂²⋊C₈

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

C₂×C₄×D₄

—

#431

—

#431

D₄₃

yes

characteristic, complemented

D₄⋊C₈

—

C₄₃

#1478

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂³×C₄

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

100

C₂×C₂²⋊C₄

—

101

C₂×C₄²

C₂×D₄₃

yes

characteristic

102

C₄⋊C₈

—

103

C₄⋊C₈

C₂×D₄₃

yes

complemented

104

C₂²⋊C₈

—

105

C₂²⋊C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₂⁴

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

172

—

117

C₂²×C₄

C₄×D₄₃

yes

characteristic

118

C₂²×C₄

D₁₇₂

yes

characteristic

119

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

120

C₂×C₈

—

121

C₂×C₈

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₂²⋊C₄

—

126

C₄²

C₄×D₄₃

yes

characteristic

127

C₄²

D₁₇₂

yes

characteristic

128

C₄²

C₂²⋊D₄₃

yes

characteristic

129

C₄²

C₂²×D₄₃

yes

characteristic

130

C₂³

—

131

C₂³

—

132

C₂³

—

172

—

133

C₂³

—

172

—

134

C₂³

(C₂×D₄₃)⋊C₄

yes

characteristic

135

D₄

—

136

D₄

—

137

D₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

172

—

142

C₂×C₄

—

172

—

143

C₂×C₄

—

172

—

144

C₂×C₄

C₈×D₄₃

yes

—

145

C₂×C₄

#10621872

yes

—

146

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

147

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

148

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

149

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

150

C₂×C₄

C₂×D₁₇₂

yes

characteristic

151

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

152

C₈

—

153

C₈

—

172

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

172

—

158

C₂²

—

172

—

159

C₂²

—

172

—

160

C₂²

(C₂×D₄₃)⋊C₈

yes

—

161

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

172

—

166

C₄

(C₂×D₄₃)⋊C₈

yes

—

167

C₄

C₂×C₈×D₄₃

yes

characteristic

168

C₄

C₂×#10621872

yes

characteristic

169

C₄

#13063350

yes

—

170

C₂

—

171

C₂

—

172

—

172

C₂

C₂×(C₂×D₄₃)⋊C₈

yes

characteristic

173

C₂

#14582903

yes

characteristic

174

C₂

#14582915

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₄₃
commutator	C₄×C₄₃	C₂²×C₈
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#431	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	#431	D₄₃
hypocenter	C₄₃	#1478
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₄₃

commutator

C₄×C₄₃

C₂²×C₈

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#431

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
2. #14582952	C₂	semidirect
3. #14582626	C₂	semidirect
4. C₄₃×#431	C₂	semidirect
15. C₄×C₄₃⋊C₈	C₂²	semidirect
17. C₄⋊C₈×C₄₃	C₂²	semidirect
27. C₂×D₁₇₂	C₈	semidirect
90. #431	D₄₃	semidirect
92. C₄₃	#1478	semidirect
103. C₄⋊C₈	C₂×D₄₃	semidirect

H ≅ G/N

type

2. #14582952

C₂

semidirect

3. #14582626

C₂

semidirect

4. C₄₃×#431

C₂

semidirect

15. C₄×C₄₃⋊C₈

C₂²

semidirect

17. C₄⋊C₈×C₄₃

C₂²

semidirect

27. C₂×D₁₇₂

C₈

semidirect

90. #431

D₄₃

semidirect

92. C₄₃

#1478

semidirect

103. C₄⋊C₈

C₂×D₄₃

semidirect

Subgroups — #19132985

Conjugacy classes of subgroups

Complements