Group #19135522 — subgroups

Subgroups — #19135522

Order 5504.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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×#13063322	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13063245	C₂	1	4	yes	maximal, complemented
4	C₄×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₇₂	—	2	1	no	—
6	C₂²×D₁₇₂	—	2	1	no	—
7	C₂²×D₁₇₂	C₂²	1	2	yes	—
8	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
9	#13063322	—	2	2	no	—
10	#13063322	C₄	1	4	yes	complemented
11	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
12	#13063245	—	2	8	no	—
13	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₄₃	—	2	1	no	—
15	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
16	#95×C₄₃	—	2	1	no	—
17	C₂³×D₄₃	—	2	2	no	—
18	C₂²×C₄×C₄₃	—	2	2	no	—
19	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
22	C₂×D₁₇₂	—	2	4	no	—
23	C₂×D₁₇₂	—	2	4	no	—
24	C₂×D₁₇₂	—	2	2	no	—
25	C₂×D₁₇₂	—	2	4	no	—
26	C₂×D₁₇₂	—	2	4	no	—
27	C₂×D₁₇₂	—	4	4	no	—
28	C₂×D₁₇₂	—	4	2	no	—
29	C₂×D₁₇₂	—	4	2	no	—
30	C₂×D₁₇₂	C₂×C₄	1	4	yes	—
31	C₄⋊C₄×C₄₃	—	2	2	no	—
32	C₄⋊C₄×C₄₃	D₄	1	4	yes	complemented
33	C₄²×C₄₃	—	4	2	no	—
34	C₄²×C₄₃	C₂×C₄	1	2	yes	—
35	C₄²×C₄₃	D₄	1	2	yes	—
36	C₂×C₄₃⋊C₈	—	2	4	no	—
37	C₂×C₄₃⋊C₈	—	2	2	no	—
38	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×D₄₃	—	2	4	no	—
40	C₂²×D₄₃	—	4	4	no	—
41	C₂²×D₄₃	—	4	2	no	—
42	C₂²×D₄₃	—	4	2	no	—
43	C₂²×D₄₃	—	4	4	no	—
44	C₂×C₄×C₄₃	—	2	2	no	—
45	C₂×C₄×C₄₃	—	2	4	no	—
46	C₂×C₄×C₄₃	—	4	2	no	—
47	C₂×C₄×C₄₃	—	4	2	no	—
48	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₄₃	D₈	1	2	yes	—
51	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
52	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₄₃	#41	1	2	yes	—
56	D₁₇₂	—	2	8	no	—
57	D₁₇₂	—	4	8	no	—
58	D₁₇₂	—	4	8	no	—
59	D₁₇₂	—	4	2	no	—
60	D₁₇₂	—	4	2	no	—
61	C₄₃⋊C₈	—	4	2	no	—
62	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄₃	#121	1	2	yes	—
64	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
65	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
67	C₂×D₄₃	—	4	4	no	—
68	C₂×D₄₃	—	4	4	no	—
69	C₂×D₄₃	—	4	4	no	—
70	C₂×D₄₃	—	8	4	no	—
71	C₂×D₄₃	—	8	4	no	—
72	C₄×C₄₃	—	2	2	no	—
73	C₄×C₄₃	—	4	4	no	—
74	C₄×C₄₃	#102	1	4	yes	complemented
75	C₄×C₄₃	C₄×D₄	1	2	yes	—
76	C₄×C₄₃	#124	1	1	yes	characteristic
77	C₄×C₄₃	#127	1	1	yes	characteristic
78	#1982	—	43	1	no	maximal, Sylow
79	C₂×C₄₃	#390	1	1	yes	characteristic
80	C₂×C₄₃	C₂×#102	1	1	yes	characteristic
81	C₂×C₄₃	#418	1	1	yes	characteristic
82	C₂×C₄₃	#459	1	2	yes	—
83	C₂×C₄₃	#460	1	2	yes	—
84	D₄₃	—	8	4	no	—
85	C₂×#127	—	43	1	no	—
86	C₂×C₄⋊C₈	—	43	1	no	—
87	C₂×#102	—	43	4	no	—
88	C₄×C₄⋊C₄	D₄₃	1	1	yes	characteristic, complemented
89	C₄₃	#1982	1	1	yes	characteristic, complemented, Sylow
90	C₂²×D₄	—	43	2	no	—
91	C₂²×D₄	—	86	1	no	—
92	C₂²×D₄	—	86	1	no	—
93	C₂²×C₈	—	43	2	no	—
94	#127	—	43	4	no	—
95	#127	—	86	2	no	—
96	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
97	C₂×C₄²	—	2	1	no	—
98	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
99	C₄⋊C₈	—	86	2	no	—
100	#102	—	86	8	no	—
101	#95	—	2	1	no	—
102	C₂⁴	—	86	2	no	—
103	C₂×D₄	—	43	4	no	—
104	C₂×D₄	—	86	4	no	—
105	C₂×D₄	—	86	4	no	—
106	C₂×D₄	—	86	4	no	—
107	C₂×D₄	—	86	2	no	—
108	C₂×D₄	—	86	4	no	—
109	C₂×D₄	—	172	4	no	—
110	C₂×D₄	—	172	2	no	—
111	C₂×D₄	—	172	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
114	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
116	C₂×C₈	—	86	4	no	—
117	C₂×C₈	—	86	2	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	D₁₇₂	1	4	yes	complemented
120	C₄²	—	4	2	no	—
121	C₄²	C₄×D₄₃	1	2	yes	—
122	C₄²	D₁₇₂	1	2	yes	—
123	C₂³	—	86	4	no	—
124	C₂³	—	172	4	no	—
125	C₂³	—	172	2	no	—
126	C₂³	—	172	2	no	—
127	C₂³	—	172	4	no	—
128	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
129	D₄	—	86	8	no	—
130	D₄	—	172	8	no	—
131	D₄	—	172	8	no	—
132	D₄	—	172	2	no	—
133	D₄	—	172	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
139	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
140	C₂×C₄	#10621882	1	2	yes	—
141	C₂×C₄	#10621884	1	2	yes	—
142	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
143	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
144	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
145	C₂×C₄	#10621898	1	2	yes	—
146	C₈	—	172	2	no	—
147	C₂²	—	344	4	no	—
148	C₂²	—	344	4	no	—
149	C₂²	—	172	4	no	—
150	C₂²	—	172	4	no	—
151	C₂²	—	172	4	no	—
152	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
153	C₂²	#13063299	1	2	yes	—
154	C₂²	C₂×#10621882	1	1	yes	characteristic
155	C₂²	C₂×#10621884	1	1	yes	characteristic
156	C₂²	#13063350	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	4	4	no	—
159	C₄	#13063245	1	4	yes	complemented
160	C₄	C₄×D₁₇₂	1	2	yes	—
161	C₄	#13063313	1	1	yes	characteristic
162	C₄	#13063322	1	1	yes	characteristic
163	C₂	—	344	4	no	—
164	C₂	#14582801	1	1	yes	characteristic
165	C₂	C₂×#13063245	1	1	yes	characteristic
166	C₂	#14582903	1	1	yes	characteristic
167	C₂	#14583094	1	2	yes	—
168	C₂	#14583099	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063322

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₂×#13063245

C₂

yes

maximal, complemented

C₄×C₄⋊C₄×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₇₂

—

C₂²×D₁₇₂

—

C₂²×D₁₇₂

C₂²

yes

—

C₂²×C₄₃⋊C₈

C₂²

yes

—

#13063322

—

#13063322

C₄

yes

complemented

C₄⋊(C₄₃⋊C₈)

—

#13063245

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#95×C₄₃

—

C₂³×D₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂×C₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

complemented

C₄²×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

D₄

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

D₈

yes

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

#41

yes

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

#121

yes

—

C₂²×C₄₃

C₂×D₈

yes

characteristic

C₂²×C₄₃

C₂×SD₁₆

yes

characteristic

C₂²×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#102

yes

complemented

C₄×C₄₃

C₄×D₄

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#460

yes

—

D₄₃

—

C₂×#127

—

C₂×C₄⋊C₈

—

C₂×#102

—

C₄×C₄⋊C₄

D₄₃

yes

characteristic, complemented

C₄₃

#1982

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

100

#102

—

101

#95

—

102

C₂⁴

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

172

—

110

C₂×D₄

—

172

—

111

C₂×D₄

—

172

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

114

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

115

C₂²×C₄

C₂²×D₄₃

yes

characteristic

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

119

C₄⋊C₄

D₁₇₂

yes

complemented

120

C₄²

—

121

C₄²

C₄×D₄₃

yes

—

122

C₄²

D₁₇₂

yes

—

123

C₂³

—

124

C₂³

—

172

—

125

C₂³

—

172

—

126

C₂³

—

172

—

127

C₂³

—

172

—

128

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

129

D₄

—

130

D₄

—

172

—

131

D₄

—

172

—

132

D₄

—

172

—

133

D₄

—

172

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

139

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

140

C₂×C₄

#10621882

yes

—

141

C₂×C₄

#10621884

yes

—

142

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

143

C₂×C₄

C₂×D₁₇₂

yes

—

144

C₂×C₄

C₂×D₁₇₂

yes

characteristic

145

C₂×C₄

#10621898

yes

—

146

C₈

—

172

—

147

C₂²

—

344

—

148

C₂²

—

344

—

149

C₂²

—

172

—

150

C₂²

—

172

—

151

C₂²

—

172

—

152

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

153

C₂²

#13063299

yes

—

154

C₂²

C₂×#10621882

yes

characteristic

155

C₂²

C₂×#10621884

yes

characteristic

156

C₂²

#13063350

yes

—

157

C₄

—

158

C₄

—

159

C₄

#13063245

yes

complemented

160

C₄

C₄×D₁₇₂

yes

—

161

C₄

#13063313

yes

characteristic

162

C₄

#13063322

yes

characteristic

163

C₂

—

344

—

164

C₂

#14582801

yes

characteristic

165

C₂

C₂×#13063245

yes

characteristic

166

C₂

#14582903

yes

characteristic

167

C₂

#14583094

yes

—

168

C₂

#14583099

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄×C₄⋊C₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₄×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#1982
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄×C₄⋊C₄×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

hypercenter

C₄×C₄⋊C₄

D₄₃

hypocenter

C₄₃

#1982

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
2. C₂×C₄⋊(C₄₃⋊C₈)	C₂	semidirect
3. C₂×#13063245	C₂	semidirect
4. C₄×C₄⋊C₄×C₄₃	C₂	semidirect
10. #13063322	C₄	semidirect
32. C₄⋊C₄×C₄₃	D₄	semidirect
74. C₄×C₄₃	#102	semidirect
88. C₄×C₄⋊C₄	D₄₃	semidirect
89. C₄₃	#1982	semidirect
119. C₄⋊C₄	D₁₇₂	semidirect
159. C₄	#13063245	semidirect

H ≅ G/N

type

2. C₂×C₄⋊(C₄₃⋊C₈)

C₂

semidirect

3. C₂×#13063245

C₂

semidirect

4. C₄×C₄⋊C₄×C₄₃

C₂

semidirect

10. #13063322

C₄

semidirect

32. C₄⋊C₄×C₄₃

D₄

semidirect

74. C₄×C₄₃

#102

semidirect

88. C₄×C₄⋊C₄

D₄₃

semidirect

89. C₄₃

#1982

semidirect

119. C₄⋊C₄

D₁₇₂

semidirect

159. C₄

#13063245

semidirect

Subgroups — #19135522

Conjugacy classes of subgroups

Complements