Group #19133274 — subgroups

Subgroups — #19133274

Order 5504.

240 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₄⋊Q₈×D₄₃	C₂	1	1	yes	maximal, characteristic
2	#14582951	C₂	1	1	yes	maximal, characteristic
3	#14582921	C₂	1	1	yes	maximal, characteristic
4	#14582648	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582647	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582647	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₄₃	C₄	1	2	yes	—
9	C₂×#10621872	—	2	1	no	—
10	#13063325	C₂²	1	1	yes	characteristic
11	#13063324	—	2	1	no	—
12	#13063324	—	2	1	no	—
13	#13063323	C₄	1	2	yes	—
14	C₄⋊C₄×D₄₃	—	2	1	no	—
15	C₄⋊C₄×D₄₃	—	2	1	no	—
16	C₄²×D₄₃	C₂²	1	1	yes	characteristic
17	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
19	Dic₄₃⋊C₈	—	2	1	no	—
20	(C₂×D₄₃)⋊C₈	—	2	1	no	—
21	C₈×Dic₄₃	—	2	1	no	—
22	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₄₃	C₂×C₄	1	2	yes	—
26	Q₈×D₄₃	—	4	4	no	—
27	C₂×C₄×D₄₃	—	2	2	no	—
28	C₂×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
30	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
31	C₂×Dic₈₆	—	2	2	no	—
32	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
33	C₂×C₈×C₄₃	—	2	1	no	—
34	C₂×C₈×C₄₃	—	2	1	no	—
35	C₄⋊C₄×C₄₃	—	2	1	no	—
36	C₄⋊C₄×C₄₃	—	2	1	no	—
37	C₄²×C₄₃	C₂³	1	1	yes	characteristic
38	C₄⋊Dic₄₃	—	2	1	no	—
39	C₄⋊Dic₄₃	—	2	1	no	—
40	Dic₄₃⋊C₄	—	2	2	no	—
41	Dic₄₃⋊C₄	—	2	2	no	—
42	C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₄₃	D₄	1	1	yes	characteristic
44	C₄×Dic₄₃	D₄	1	1	yes	characteristic
45	C₂×C₄₃⋊C₈	—	2	1	no	—
46	C₂×C₄₃⋊C₈	—	2	1	no	—
47	#10621872	—	4	2	no	—
48	C₂²×D₄₃	C₂²⋊C₄	1	1	yes	characteristic
49	Q₈×C₄₃	—	4	2	no	—
50	C₂×C₄×C₄₃	—	2	2	no	—
51	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₄₃	—	2	2	no	—
55	C₂×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₄×D₄₃	—	2	1	no	—
59	C₄×D₄₃	—	2	1	no	—
60	C₄×D₄₃	—	2	2	no	—
61	C₄×D₄₃	—	4	4	no	—
62	C₄×D₄₃	SD₁₆	1	2	yes	—
63	C₄×D₄₃	Q₁₆	1	2	yes	—
64	Dic₈₆	—	4	4	no	—
65	Dic₈₆	—	4	2	no	—
66	C₈×C₄₃	—	4	1	no	—
67	C₈×C₄₃	—	4	1	no	—
68	C₄₃⋊C₈	—	4	1	no	—
69	C₄₃⋊C₈	—	4	1	no	—
70	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×D₄₃	—	2	1	no	—
72	C₂×D₄₃	—	2	1	no	—
73	C₂×D₄₃	#103	1	2	yes	—
74	C₄×C₄₃	—	2	1	no	—
75	C₄×C₄₃	—	4	2	no	—
76	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₄×C₄₃	#137	1	1	yes	characteristic
79	C₄×C₄₃	#137	1	1	yes	characteristic
80	Dic₄₃	—	2	1	no	—
81	Dic₄₃	—	4	2	no	—
82	Dic₄₃	#100	1	2	yes	—
83	Dic₄₃	#103	1	2	yes	—
84	#1548	—	43	1	no	maximal, Sylow
85	C₂×C₄₃	C₂×#100	1	1	yes	characteristic
86	C₂×C₄₃	C₂×#103	1	1	yes	characteristic
87	C₂×C₄₃	#419	1	1	yes	characteristic
88	D₄₃	—	2	2	no	—
89	C₂×C₄⋊Q₈	—	43	1	no	—
90	#431	—	43	1	no	—
91	#423	—	43	1	no	—
92	#333	—	43	1	no	—
93	#333	—	43	1	no	—
94	#333	—	43	1	no	—
95	#333	D₄₃	1	1	yes	characteristic, complemented
96	C₄₃	#1548	1	1	yes	characteristic, complemented, Sylow
97	C₂²×Q₈	—	43	2	no	—
98	C₂×M₁₆	—	86	1	no	—
99	C₄⋊Q₈	—	43	2	no	—
100	C₄⋊Q₈	—	43	1	no	—
101	C₄⋊Q₈	—	86	1	no	—
102	C₄⋊Q₈	—	86	1	no	—
103	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	86	1	no	—
105	C₂×C₄⋊C₄	—	86	1	no	—
106	C₂×C₄²	—	43	1	no	—
107	C₄⋊C₈	—	43	1	no	—
108	C₄⋊C₈	—	43	1	no	—
109	C₄⋊C₈	—	86	1	no	—
110	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
111	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
112	C₂²⋊C₈	—	86	1	no	—
113	C₄×C₈	—	86	1	no	—
114	C₂×Q₈	—	43	2	no	—
115	C₂×Q₈	—	86	2	no	—
116	C₂×Q₈	—	172	4	no	—
117	C₂×Q₈	C₄×D₄₃	1	2	yes	—
118	C₂²×C₄	—	43	1	no	—
119	C₂²×C₄	—	43	1	no	—
120	C₂²×C₄	—	43	1	no	—
121	C₂²×C₄	—	86	2	no	—
122	M₁₆	—	172	2	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	86	1	no	—
126	C₂×C₈	—	86	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	86	2	no	—
130	C₄⋊C₄	—	86	2	no	—
131	C₄⋊C₄	—	86	1	no	—
132	C₄⋊C₄	—	86	1	no	—
133	C₄²	—	43	1	no	—
134	C₄²	—	43	1	no	—
135	C₄²	—	43	1	no	—
136	C₄²	C₂²×D₄₃	1	1	yes	characteristic
137	C₂³	—	43	1	no	—
138	Q₈	—	4	2	no	—
139	Q₈	—	172	2	no	—
140	Q₈	—	172	4	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	43	1	no	—
143	C₂×C₄	—	43	1	no	—
144	C₂×C₄	—	43	2	no	—
145	C₂×C₄	—	43	2	no	—
146	C₂×C₄	—	43	1	no	—
147	C₂×C₄	—	86	2	no	—
148	C₂×C₄	—	86	1	no	—
149	C₂×C₄	—	86	1	no	—
150	C₂×C₄	—	86	2	no	—
151	C₂×C₄	—	172	4	no	—
152	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
153	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
154	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
155	C₈	—	4	1	no	—
156	C₈	—	4	1	no	—
157	C₈	—	172	1	no	—
158	C₈	—	172	1	no	—
159	C₂²	—	43	2	no	—
160	C₂²	—	86	1	no	—
161	C₂²	—	86	1	no	—
162	C₂²	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	2	no	—
165	C₄	—	43	2	no	—
166	C₄	—	43	2	no	—
167	C₄	—	86	1	no	—
168	C₄	—	172	2	no	—
169	C₄	SD₁₆×D₄₃	1	1	yes	characteristic
170	C₄	Q₁₆×D₄₃	1	1	yes	characteristic
171	C₄	#13063354	1	1	yes	characteristic
172	C₄	#13063357	1	1	yes	characteristic
173	C₂	—	86	2	no	—
174	C₂	#100×D₄₃	1	1	yes	characteristic
175	C₂	#103×D₄₃	1	1	yes	characteristic
176	C₂	#14582906	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₄₃

C₂

yes

maximal, characteristic

#14582951

C₂

yes

maximal, characteristic

#14582921

C₂

yes

maximal, characteristic

#14582648

C₂

yes

maximal, characteristic, complemented

#14582647

C₂

yes

maximal, characteristic, complemented

#14582647

C₂

yes

maximal, characteristic, complemented

C₄₃×#333

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₄₃

C₄

yes

—

C₂×#10621872

—

#13063325

C₂²

yes

—

—

C₄

yes

—

C₄⋊C₄×D₄₃

—

C₄⋊C₄×D₄₃

—

C₄²×D₄₃

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

Dic₄₃⋊C₈

—

(C₂×D₄₃)⋊C₈

—

C₈×Dic₄₃

—

C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

C₂×C₄

yes

—

Q₈×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

C₂×C₄

yes

characteristic

C₂×C₄×D₄₃

D₄

yes

characteristic

C₂×C₄×D₄₃

D₄

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

C₂×C₄

yes

characteristic

C₄×Dic₄₃

D₄

yes

characteristic

C₄×Dic₄₃

D₄

yes

characteristic

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

#10621872

—

C₂²×D₄₃

C₂²⋊C₄

yes

characteristic

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

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

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₃

C₂²⋊C₄

yes

characteristic

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

SD₁₆

yes

—

C₄×D₄₃

Q₁₆

yes

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

#103

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₄×C₄₃

C₂×Q₁₆

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

Dic₄₃

#100

yes

—

Dic₄₃

#103

yes

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

C₂×C₄⋊Q₈

—

—

—

—

—

—

D₄₃

yes

characteristic, complemented

C₄₃

#1548

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

100

C₄⋊Q₈

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

C₂×D₄₃

yes

characteristic

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

C₂×D₄₃

yes

characteristic

111

C₄⋊C₈

C₂×D₄₃

yes

characteristic

112

C₂²⋊C₈

—

113

C₄×C₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

172

—

117

C₂×Q₈

C₄×D₄₃

yes

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

M₁₆

—

172

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

C₂²×D₄₃

yes

characteristic

137

C₂³

—

138

Q₈

—

139

Q₈

—

172

—

140

Q₈

—

172

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

172

—

152

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

153

C₂×C₄

D₄×D₄₃

yes

characteristic

154

C₂×C₄

D₄×D₄₃

yes

characteristic

155

C₈

—

156

C₈

—

157

C₈

—

172

—

158

C₈

—

172

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

C₂²⋊C₄×D₄₃

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

172

—

169

C₄

SD₁₆×D₄₃

yes

characteristic

170

C₄

Q₁₆×D₄₃

yes

characteristic

171

C₄

#13063354

yes

characteristic

172

C₄

#13063357

yes

characteristic

173

C₂

—

174

C₂

#100×D₄₃

yes

characteristic

175

C₂

#103×D₄₃

yes

characteristic

176

C₂

#14582906

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#333	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	#333	D₄₃
hypocenter	C₄₃	#1548
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#333

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
4. #14582648	C₂	semidirect
5. #14582647	C₂	semidirect
6. #14582647	C₂	semidirect
7. C₄₃×#333	C₂	semidirect
95. #333	D₄₃	semidirect
96. C₄₃	#1548	semidirect

H ≅ G/N

type

4. #14582648

C₂

semidirect

5. #14582647

C₂

semidirect

6. #14582647

C₂

semidirect

7. C₄₃×#333

C₂

semidirect

95. #333

D₄₃

semidirect

96. C₄₃

#1548

semidirect

Subgroups — #19133274

Conjugacy classes of subgroups

Complements