Group #19134386 — subgroups

Subgroups — #19134386

Order 5504.

360 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 107 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×#13063287	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063241	C₂	1	2	yes	maximal
3	M₁₆×Dic₄₃	C₂	1	2	yes	maximal, complemented
4	#14582668	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#343	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
7	C₂²×C₄×D₄₃	C₄	1	2	yes	—
8	C₂×#10621877	C₂²	1	2	yes	—
9	#13063287	—	2	2	no	—
10	#13063287	—	2	2	no	—
11	C₂×(C₂×D₄₃)⋊C₄	C₄	1	2	yes	—
12	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
13	#13063241	C₄	1	8	yes	complemented
14	#13063235	—	2	2	no	—
15	C₈×Dic₄₃	—	2	2	no	—
16	C₂×M₁₆×C₄₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
18	C₂³×D₄₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₄₃	C₂×C₄	1	1	yes	characteristic
23	C₂×D₁₇₂	—	2	2	no	—
24	C₂×D₁₇₂	—	2	1	no	—
25	C₂×D₁₇₂	—	2	1	no	—
26	C₂×D₁₇₂	C₂×C₄	1	4	yes	—
27	C₂×C₄×D₄₃	—	2	2	no	—
28	C₂×C₄×D₄₃	—	2	2	no	—
29	C₂×C₄×D₄₃	—	4	4	no	—
30	M₁₆×C₄₃	—	2	2	no	—
31	M₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
32	C₂×C₈×C₄₃	—	2	2	no	—
33	C₄⋊C₄×C₄₃	—	2	2	no	—
34	(C₂×D₄₃)⋊C₄	—	2	4	no	—
35	C₄×Dic₄₃	D₄	1	2	yes	—
36	C₄×Dic₄₃	D₄	1	2	yes	—
37	#10621877	—	2	2	no	—
38	#10621877	C₂×C₄	1	4	yes	complemented
39	C₂×C₄₃⋊C₈	—	2	2	no	—
40	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×D₄₃	—	2	2	no	—
42	C₂²×D₄₃	—	2	2	no	—
43	C₂²×D₄₃	—	4	2	no	—
44	C₂²×D₄₃	—	4	2	no	—
45	C₂²×D₄₃	C₄²	1	4	yes	—
46	C₂×C₄×C₄₃	—	2	2	no	—
47	C₂×C₄×C₄₃	—	4	2	no	—
48	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
51	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
52	C₂×Dic₄₃	—	2	2	no	—
53	C₂×Dic₄₃	—	4	1	no	—
54	C₂×Dic₄₃	C₂²⋊C₄	1	2	yes	—
55	C₂×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
57	D₁₇₂	—	2	4	no	—
58	D₁₇₂	—	2	4	no	—
59	C₄×D₄₃	—	4	4	no	—
60	C₄×D₄₃	—	4	4	no	—
61	C₈×C₄₃	—	2	4	no	—
62	C₄₃⋊C₈	—	2	4	no	—
63	C₂²×C₄₃	—	2	1	no	—
64	C₂²×C₄₃	—	2	1	no	—
65	C₂²×C₄₃	C₂×C₄²	1	1	yes	characteristic
66	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄₃	#117	1	1	yes	characteristic
68	C₂×D₄₃	—	2	4	no	—
69	C₂×D₄₃	—	4	4	no	—
70	C₂×D₄₃	—	4	2	no	—
71	C₂×D₄₃	—	4	2	no	—
72	C₂×D₄₃	—	4	4	no	—
73	C₄×C₄₃	—	4	2	no	—
74	C₄×C₄₃	C₄×D₄	1	2	yes	—
75	C₄×C₄₃	C₄×D₄	1	2	yes	—
76	Dic₄₃	—	2	1	no	—
77	Dic₄₃	—	2	2	no	—
78	Dic₄₃	#100	1	2	yes	complemented
79	C₄×#100	—	43	1	no	maximal, Sylow
80	C₂×C₄₃	—	2	1	no	—
81	C₂×C₄₃	—	2	1	no	—
82	C₂×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄₃	C₂×#100	1	1	yes	characteristic
84	C₂×C₄₃	#413	1	1	yes	characteristic
85	D₄₃	—	4	4	no	—
86	C₂×C₄×D₄	—	43	1	no	—
87	C₂×#100	—	43	2	no	—
88	C₄×M₁₆	—	43	2	no	—
89	#343	—	43	1	no	—
90	#343	D₄₃	1	1	yes	characteristic, complemented
91	C₄₃	C₄×#100	1	1	yes	characteristic, complemented, Sylow
92	C₂²×D₄	—	43	1	no	—
93	C₂³×C₄	—	43	2	no	—
94	C₂×M₁₆	—	43	2	no	—
95	C₂×M₁₆	C₂×D₄₃	1	2	yes	—
96	C₄×D₄	—	86	2	no	—
97	C₄×D₄	—	86	2	no	—
98	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
99	C₂×C₂²⋊C₄	—	43	2	no	—
100	C₂×C₄²	—	43	1	no	—
101	#100	—	43	8	no	—
102	#97	—	86	2	no	—
103	C₄×C₈	—	86	2	no	—
104	C₂⁴	—	43	2	no	—
105	C₂×D₄	—	43	4	no	—
106	C₂×D₄	—	86	2	no	—
107	C₂×D₄	—	86	1	no	—
108	C₂×D₄	—	86	1	no	—
109	C₂²×C₄	—	43	1	no	—
110	C₂²×C₄	—	43	1	no	—
111	C₂²×C₄	—	86	2	no	—
112	C₂²×C₄	—	86	2	no	—
113	C₂²×C₄	—	172	4	no	—
114	C₂²×C₄	C₄×D₄₃	1	2	yes	—
115	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
116	M₁₆	—	2	2	no	—
117	M₁₆	—	43	4	no	—
118	M₁₆	—	86	2	no	—
119	M₁₆	C₄×D₄₃	1	4	yes	complemented
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	86	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₂²⋊C₄	—	86	4	no	—
124	C₄²	—	43	2	no	—
125	C₄²	—	43	2	no	—
126	C₂³	—	43	4	no	—
127	C₂³	—	86	2	no	—
128	C₂³	—	86	2	no	—
129	C₂³	—	172	2	no	—
130	C₂³	—	172	2	no	—
131	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
132	D₄	—	86	4	no	—
133	D₄	—	86	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	43	1	no	—
137	C₂×C₄	—	43	2	no	—
138	C₂×C₄	—	43	1	no	—
139	C₂×C₄	—	86	2	no	—
140	C₂×C₄	—	172	4	no	—
141	C₂×C₄	—	172	4	no	—
142	C₂×C₄	—	172	1	no	—
143	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
144	C₂×C₄	D₄×D₄₃	1	2	yes	—
145	C₂×C₄	#10621900	1	1	yes	characteristic
146	C₂×C₄	#10621902	1	1	yes	characteristic
147	C₈	—	2	4	no	—
148	C₈	—	86	4	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	86	4	no	—
152	C₂²	—	172	4	no	—
153	C₂²	—	172	4	no	—
154	C₂²	—	172	2	no	—
155	C₂²	—	172	2	no	—
156	C₂²	C₄²×D₄₃	1	1	yes	characteristic
157	C₂²	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
158	C₂²	#13063283	1	1	yes	characteristic
159	C₄	—	4	2	no	—
160	C₄	—	43	2	no	—
161	C₄	—	86	1	no	—
162	C₄	—	86	2	no	—
163	C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
164	C₄	#13063287	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	—	172	4	no	—
168	C₂	#14582746	1	1	yes	characteristic
169	C₂	#100×D₄₃	1	1	yes	characteristic
170	C₂	#14582882	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063287

C₂

yes

maximal, characteristic

C₂×#13063241

C₂

yes

maximal

M₁₆×Dic₄₃

C₂

yes

maximal, complemented

#14582668

C₂

yes

maximal, characteristic, complemented

C₄₃×#343

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₇₂

C₂²

yes

characteristic

C₂²×C₄×D₄₃

C₄

yes

—

C₂×#10621877

C₂²

yes

—

#13063287

—

#13063287

—

C₂×(C₂×D₄₃)⋊C₄

C₄

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

characteristic

#13063241

C₄

yes

complemented

#13063235

—

C₈×Dic₄₃

—

C₂×M₁₆×C₄₃

C₂²

yes

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂³×D₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂×C₄

yes

characteristic

C₂²×Dic₄₃

C₂×C₄

yes

characteristic

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

M₁₆×C₄₃

—

M₁₆×C₄₃

C₂×C₄

yes

complemented

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

(C₂×D₄₃)⋊C₄

—

C₄×Dic₄₃

D₄

yes

—

C₄×Dic₄₃

D₄

yes

—

#10621877

—

#10621877

C₂×C₄

yes

complemented

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²×C₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

C₄²

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂²⋊C₄

yes

—

C₂×Dic₄₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₃

C₂²⋊C₄

yes

characteristic

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₄²

yes

characteristic

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

#117

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₄×D₄

yes

—

C₄×C₄₃

C₄×D₄

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

yes

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

C₄×C₂²⋊C₄

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

C₂×C₄×D₄

—

C₂×#100

—

C₄×M₁₆

—

#343

—

#343

D₄₃

yes

characteristic, complemented

C₄₃

C₄×#100

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₃

yes

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

C₂×C₂²⋊C₄

—

100

C₂×C₄²

—

101

#100

—

102

#97

—

103

C₄×C₈

—

104

C₂⁴

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

172

—

114

C₂²×C₄

C₄×D₄₃

yes

—

115

C₂²×C₄

C₂²×D₄₃

yes

characteristic

116

M₁₆

—

117

M₁₆

—

118

M₁₆

—

119

M₁₆

C₄×D₄₃

yes

complemented

120

C₂×C₈

—

121

C₂×C₈

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₄²

—

125

C₄²

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

172

—

130

C₂³

—

172

—

131

C₂³

C₂×C₄×D₄₃

yes

characteristic

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

172

—

141

C₂×C₄

—

172

—

142

C₂×C₄

—

172

—

143

C₂×C₄

C₂×C₄×D₄₃

yes

—

144

C₂×C₄

D₄×D₄₃

yes

—

145

C₂×C₄

#10621900

yes

characteristic

146

C₂×C₄

#10621902

yes

characteristic

147

C₈

—

148

C₈

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

172

—

153

C₂²

—

172

—

154

C₂²

—

172

—

155

C₂²

—

172

—

156

C₂²

C₄²×D₄₃

yes

characteristic

157

C₂²

C₂²⋊C₄×D₄₃

yes

characteristic

158

C₂²

#13063283

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

(C₂²⋊D₄₃)⋊C₄

yes

—

164

C₄

#13063287

yes

—

165

C₂

—

166

C₂

—

167

C₂

—

172

—

168

C₂

#14582746

yes

characteristic

169

C₂

#100×D₄₃

yes

characteristic

170

C₂

#14582882

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₄₃
commutator	C₂²×C₄₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#343	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	#343	D₄₃
hypocenter	C₄₃	C₄×#100
LCS	G ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₄₃

commutator

C₂²×C₄₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#343

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
3. M₁₆×Dic₄₃	C₂	semidirect
4. #14582668	C₂	semidirect
5. C₄₃×#343	C₂	semidirect
13. #13063241	C₄	semidirect
31. M₁₆×C₄₃	C₂×C₄	semidirect
38. #10621877	C₂×C₄	semidirect
78. Dic₄₃	#100	semidirect
90. #343	D₄₃	semidirect
91. C₄₃	C₄×#100	semidirect
119. M₁₆	C₄×D₄₃	semidirect

H ≅ G/N

type

3. M₁₆×Dic₄₃

C₂

semidirect

4. #14582668

C₂

semidirect

5. C₄₃×#343

C₂

semidirect

13. #13063241

C₄

semidirect

31. M₁₆×C₄₃

C₂×C₄

semidirect

38. #10621877

C₂×C₄

semidirect

78. Dic₄₃

#100

semidirect

90. #343

D₄₃

semidirect

91. C₄₃

C₄×#100

semidirect

119. M₁₆

C₄×D₄₃

semidirect

Subgroups — #19134386

Conjugacy classes of subgroups

Complements