Group #19134802 — subgroups

Subgroups — #19134802

Order 5504.

278 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂²×#10621877	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063261	C₂	1	2	yes	maximal, complemented
3	#14582863	C₂	1	1	yes	maximal, characteristic
4	#14582664	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#10621877	—	2	1	no	—
7	C₂×#10621877	—	2	4	no	—
8	C₂×#10621877	C₂²	1	1	yes	characteristic
9	C₂×#10621877	C₂²	1	1	yes	characteristic
10	C₂×#10621877	C₂²	1	2	yes	—
11	C₂²×C₄₃⋊C₈	—	2	1	no	—
12	#13063261	—	2	4	no	—
13	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
14	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
15	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₄₃	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₄₃	C₄	1	2	yes	—
18	C₂²⋊C₈×C₄₃	C₄	1	2	yes	—
19	C₂⁴×C₄₃	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₄₃	—	2	2	no	—
21	C₂²×C₄×C₄₃	—	2	2	no	—
22	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₃	Q₈	1	1	yes	characteristic
27	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₄₃	—	2	2	no	—
29	C₂×C₈×C₄₃	—	2	2	no	—
30	C₂×C₈×C₄₃	C₂×C₄	1	4	yes	—
31	#10621877	—	2	4	no	—
32	#10621877	—	2	2	no	—
33	#10621877	—	2	2	no	—
34	#10621877	—	2	2	no	—
35	#10621877	D₄	1	4	yes	—
36	C₂×C₄₃⋊C₈	—	2	4	no	—
37	C₂×C₄₃⋊C₈	—	2	2	no	—
38	C₂×C₄₃⋊C₈	—	2	1	no	—
39	C₂×C₄₃⋊C₈	—	2	1	no	—
40	C₂³×C₄₃	—	2	2	no	—
41	C₂³×C₄₃	—	2	2	no	—
42	C₂³×C₄₃	—	4	1	no	—
43	C₂³×C₄₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂³×C₄₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂³×C₄₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₄₃	—	2	2	no	—
47	C₂×C₄×C₄₃	—	2	2	no	—
48	C₂×C₄×C₄₃	—	2	2	no	—
49	C₂×C₄×C₄₃	—	2	1	no	—
50	C₂×C₄×C₄₃	—	2	1	no	—
51	C₂×C₄×C₄₃	—	2	1	no	—
52	C₂×C₄×C₄₃	—	2	1	no	—
53	C₂×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₄₃	#41	1	2	yes	—
59	C₈×C₄₃	—	2	4	no	—
60	C₄₃⋊C₈	—	2	4	no	—
61	C₄₃⋊C₈	—	4	1	no	—
62	C₄₃⋊C₈	—	4	1	no	—
63	C₂²×C₄₃	—	2	1	no	—
64	C₂²×C₄₃	—	2	1	no	—
65	C₂²×C₄₃	—	2	1	no	—
66	C₂²×C₄₃	—	2	1	no	—
67	C₂²×C₄₃	—	2	1	no	—
68	C₂²×C₄₃	—	2	1	no	—
69	C₂²×C₄₃	—	2	1	no	—
70	C₂²×C₄₃	—	4	4	no	—
71	C₂²×C₄₃	#108	1	2	yes	complemented
72	C₂²×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄₃	C₂²⋊Q₈	1	2	yes	—
74	C₄×C₄₃	—	2	1	no	—
75	C₄×C₄₃	—	2	1	no	—
76	C₄×C₄₃	C₄×D₄	1	2	yes	—
77	C₄×C₄₃	#120	1	1	yes	characteristic
78	C₄×C₄₃	#123	1	1	yes	characteristic
79	#1855	—	43	1	no	maximal, Sylow
80	C₂×C₄₃	—	2	2	no	—
81	C₂×C₄₃	—	2	2	no	—
82	C₂×C₄₃	—	4	1	no	—
83	C₂×C₄₃	#385	1	1	yes	characteristic
84	C₂×C₄₃	C₂×#108	1	1	yes	characteristic
85	C₂×C₄₃	#430	1	1	yes	characteristic
86	C₂²×M₁₆	—	43	1	no	—
87	C₂×#108	—	43	2	no	—
88	#407	—	43	1	no	—
89	C₂×C₂²⋊C₈	D₄₃	1	1	yes	characteristic
90	#341	—	43	2	no	—
91	C₄₃	#1855	1	1	yes	characteristic, complemented, Sylow
92	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
93	C₂×M₁₆	—	43	2	no	—
94	C₂×M₁₆	—	43	1	no	—
95	C₂×M₁₆	—	43	1	no	—
96	C₂×M₁₆	—	86	4	no	—
97	C₂×M₁₆	—	86	1	no	—
98	C₂²×C₈	—	86	1	no	—
99	C₂²×C₈	C₂×D₄₃	1	2	yes	—
100	#108	—	86	4	no	—
101	C₂²⋊C₈	—	86	1	no	—
102	C₂²⋊C₈	—	86	1	no	—
103	C₂²⋊C₈	Dic₄₃	1	2	yes	—
104	C₂²⋊C₈	Dic₄₃	1	2	yes	—
105	C₂⁴	Dic₈₆	1	1	yes	characteristic
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	Dic₈₆	1	1	yes	characteristic
109	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
110	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
111	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
112	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
114	M₁₆	—	43	4	no	—
115	M₁₆	—	86	4	no	—
116	M₁₆	—	86	2	no	—
117	M₁₆	—	86	2	no	—
118	M₁₆	—	86	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	86	4	no	—
122	C₂×C₈	—	86	2	no	—
123	C₂×C₈	—	86	1	no	—
124	C₂×C₈	—	86	1	no	—
125	C₂×C₈	C₂×Dic₄₃	1	4	yes	—
126	C₂³	—	2	2	no	—
127	C₂³	—	2	2	no	—
128	C₂³	—	4	1	no	—
129	C₂³	C₄⋊Dic₄₃	1	1	yes	characteristic
130	C₂³	C₄⋊Dic₄₃	1	1	yes	characteristic
131	C₂³	C₂×Dic₈₆	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
140	C₂×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
141	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
142	C₂×C₄	D₄×D₄₃	1	2	yes	—
143	C₂×C₄	#10621900	1	2	yes	—
144	C₂×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
145	C₈	—	2	4	no	—
146	C₈	—	86	4	no	—
147	C₈	—	172	1	no	—
148	C₈	—	172	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	4	4	no	—
157	C₂²	#13063261	1	2	yes	complemented
158	C₂²	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
159	C₂²	#13063302	1	2	yes	—
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	D₄×Dic₄₃	1	2	yes	—
163	C₄	#13063295	1	1	yes	characteristic
164	C₄	#13063309	1	1	yes	characteristic
165	C₂	—	2	2	no	—
166	C₂	—	2	2	no	—
167	C₂	—	4	1	no	—
168	C₂	#14582774	1	1	yes	characteristic
169	C₂	C₂×#13063261	1	1	yes	characteristic
170	C₂	#14582948	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10621877

C₂

yes

maximal, characteristic

C₂×#13063261

C₂

yes

maximal, complemented

#14582863

C₂

yes

maximal, characteristic

#14582664

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₄₃

C₂

yes

maximal, characteristic

C₂×#10621877

—

C₂×#10621877

—

C₂×#10621877

C₂²

yes

characteristic

C₂×#10621877

C₂²

yes

characteristic

C₂×#10621877

C₂²

yes

—

C₂²×C₄₃⋊C₈

—

#13063261

—

C₂²⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

C₂³×C₄×C₄₃

C₂²

yes

characteristic

C₂²×C₈×C₄₃

C₂²

yes

—

C₂²⋊C₈×C₄₃

C₄

yes

—

C₂²⋊C₈×C₄₃

C₄

yes

—

C₂⁴×C₄₃

Q₈

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

Q₈

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₄⋊C₄

yes

characteristic

C₂³×C₄₃

C₄⋊C₄

yes

characteristic

C₂³×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#108

yes

complemented

C₂²×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₃

C₂²⋊Q₈

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₄×D₄

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂²×M₁₆

—

C₂×#108

—

#407

—

C₂×C₂²⋊C₈

D₄₃

yes

characteristic

#341

—

C₄₃

#1855

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄₃

yes

—

100

#108

—

101

C₂²⋊C₈

—

102

C₂²⋊C₈

—

103

C₂²⋊C₈

Dic₄₃

yes

—

104

C₂²⋊C₈

Dic₄₃

yes

—

105

C₂⁴

Dic₈₆

yes

characteristic

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

Dic₈₆

yes

characteristic

109

C₂²×C₄

D₁₇₂

yes

characteristic

110

C₂²×C₄

D₁₇₂

yes

characteristic

111

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

112

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

113

C₂²×C₄

C₂²×D₄₃

yes

characteristic

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

118

M₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

C₂×Dic₄₃

yes

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₄⋊Dic₄₃

yes

characteristic

130

C₂³

C₄⋊Dic₄₃

yes

characteristic

131

C₂³

C₂×Dic₈₆

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₄⋊Dic₄₃

yes

characteristic

140

C₂×C₄

C₄⋊Dic₄₃

yes

characteristic

141

C₂×C₄

C₂×D₁₇₂

yes

characteristic

142

C₂×C₄

D₄×D₄₃

yes

—

143

C₂×C₄

#10621900

yes

—

144

C₂×C₄

C₂²×Dic₄₃

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

172

—

148

C₈

—

172

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

#13063261

yes

complemented

158

C₂²

C₂×C₄⋊Dic₄₃

yes

characteristic

159

C₂²

#13063302

yes

—

160

C₄

—

161

C₄

—

162

C₄

D₄×Dic₄₃

yes

—

163

C₄

#13063295

yes

characteristic

164

C₄

#13063309

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

—

168

C₂

#14582774

yes

characteristic

169

C₂

C₂×#13063261

yes

characteristic

170

C₂

#14582948

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×C₂²⋊C₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₄⋊C₄
hypercenter	C₂×C₂²⋊C₈	D₄₃
hypocenter	C₄₃	#1855
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×C₂²⋊C₈×C₄₃

C₂

socle

C₂²×C₄₃

C₂×C₄⋊C₄

hypercenter

C₂×C₂²⋊C₈

D₄₃

hypocenter

C₄₃

#1855

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×#13063261	C₂	semidirect
4. #14582664	C₂	semidirect
71. C₂²×C₄₃	#108	semidirect
91. C₄₃	#1855	semidirect
157. C₂²	#13063261	semidirect

H ≅ G/N

type

2. C₂×#13063261

C₂

semidirect

4. #14582664

C₂

semidirect

71. C₂²×C₄₃

#108

semidirect

91. C₄₃

#1855

semidirect

157. C₂²

#13063261

semidirect

Subgroups — #19134802

Conjugacy classes of subgroups

Complements