Group #19133753 — subgroups

Subgroups — #19133753

Order 5504.

188 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 61 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	#14583251	C₂	1	1	yes	maximal, characteristic
2	#14583194	C₂	1	1	yes	maximal, characteristic
3	#14583194	C₂	1	1	yes	maximal, characteristic
4	#14582645	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	Q₈⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#557	C₂	1	1	yes	maximal, characteristic
8	#13063325	C₂²	1	1	yes	characteristic
9	#13063258	—	2	1	no	—
10	#13063258	—	2	1	no	—
11	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
13	#13063249	—	2	1	no	—
14	#13063249	—	2	1	no	—
15	#13063248	—	2	2	no	—
16	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₄₃	—	2	1	no	—
18	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
19	#125×C₄₃	—	2	2	no	—
20	#125×C₄₃	—	2	1	no	—
21	C₄×Q₈×C₄₃	—	2	1	no	—
22	C₄×Q₈×C₄₃	—	2	1	no	—
23	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₄₃	—	2	1	no	—
26	C₂×Q₈×C₄₃	D₄	1	1	yes	characteristic
27	C₂×Q₈×C₄₃	D₄	1	1	yes	characteristic
28	C₂×Dic₈₆	—	2	1	no	—
29	C₄⋊C₄×C₄₃	—	2	2	no	—
30	C₄⋊C₄×C₄₃	—	2	2	no	—
31	C₄⋊C₄×C₄₃	—	2	2	no	—
32	C₄⋊C₄×C₄₃	—	2	1	no	—
33	C₄⋊C₄×C₄₃	—	2	1	no	—
34	C₄⋊C₄×C₄₃	—	2	1	no	—
35	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
37	C₄⋊C₄×C₄₃	D₄	1	2	yes	—
38	C₄²×C₄₃	—	2	2	no	—
39	C₄²×C₄₃	—	2	1	no	—
40	C₄²×C₄₃	—	2	1	no	—
41	C₄²×C₄₃	C₂³	1	1	yes	characteristic
42	C₄⋊Dic₄₃	—	2	1	no	—
43	C₄⋊Dic₄₃	—	2	1	no	—
44	C₂×C₄₃⋊C₈	—	2	1	no	—
45	C₂×C₄₃⋊C₈	—	2	1	no	—
46	C₂×C₄₃⋊C₈	—	2	1	no	—
47	Q₈×C₄₃	—	2	1	no	—
48	Q₈×C₄₃	—	2	1	no	—
49	Q₈×C₄₃	—	2	1	no	—
50	Q₈×C₄₃	—	4	1	no	—
51	Q₈×C₄₃	—	4	1	no	—
52	Q₈×C₄₃	Q₁₆	1	2	yes	—
53	C₂×C₄×C₄₃	—	2	1	no	—
54	C₂×C₄×C₄₃	—	2	1	no	—
55	C₂×C₄×C₄₃	—	2	1	no	—
56	C₂×C₄×C₄₃	—	2	1	no	—
57	C₂×C₄×C₄₃	—	2	2	no	—
58	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₄₃	—	4	1	no	—
62	Dic₈₆	—	4	2	no	—
63	C₄₃⋊C₈	—	4	1	no	—
64	C₄₃⋊C₈	—	4	1	no	—
65	C₄₃⋊C₈	—	4	1	no	—
66	C₂²×C₄₃	#120	1	1	yes	characteristic
67	C₄×C₄₃	—	2	2	no	—
68	C₄×C₄₃	—	2	1	no	—
69	C₄×C₄₃	—	4	1	no	—
70	C₄×C₄₃	—	4	2	no	—
71	C₄×C₄₃	—	4	1	no	—
72	C₄×C₄₃	—	4	1	no	—
73	C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₄₃	#135	1	1	yes	characteristic
75	C₄×C₄₃	#137	1	1	yes	characteristic
76	C₄×C₄₃	#137	1	1	yes	characteristic
77	Dic₄₃	—	8	1	no	—
78	#1636	—	43	1	no	maximal, Sylow
79	C₂×C₄₃	#451	1	1	yes	characteristic
80	C₂×C₄₃	#452	1	1	yes	characteristic
81	C₂×C₄₃	#454	1	1	yes	characteristic
82	#557	D₄₃	1	1	yes	characteristic
83	#487	—	43	1	no	—
84	#477	—	43	1	no	—
85	#477	—	43	1	no	—
86	#332	—	43	1	no	—
87	Q₈⋊C₈	—	43	1	no	—
88	Q₈⋊C₈	—	43	1	no	—
89	C₄₃	#1636	1	1	yes	characteristic, complemented, Sylow
90	C₄⋊Q₈	—	2	1	no	—
91	C₄⋊Q₈	—	43	1	no	—
92	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
93	#125	—	2	2	no	—
94	#125	—	2	1	no	—
95	C₄×Q₈	—	2	1	no	—
96	C₄×Q₈	—	2	1	no	—
97	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
98	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
99	#107	—	86	1	no	—
100	#107	—	86	1	no	—
101	C₄⋊C₈	—	43	1	no	—
102	C₄⋊C₈	—	43	1	no	—
103	#103	—	86	1	no	—
104	#103	—	86	1	no	—
105	#103	—	86	2	no	—
106	C₄×C₈	—	43	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	86	1	no	—
109	C₂×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
110	C₂×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
111	C₂×C₈	—	86	1	no	—
112	C₂×C₈	—	86	1	no	—
113	C₂×C₈	—	86	1	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	86	1	no	—
121	C₄⋊C₄	—	86	1	no	—
122	C₄⋊C₄	C₂²⋊D₄₃	1	2	yes	—
123	C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
124	C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
125	C₄²	—	2	2	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	2	1	no	—
128	C₄²	C₂²×D₄₃	1	1	yes	characteristic
129	Q₈	—	2	1	no	—
130	Q₈	—	2	1	no	—
131	Q₈	—	2	1	no	—
132	Q₈	—	4	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	172	2	no	—
135	Q₈	#10621885	1	2	yes	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	172	1	no	—
142	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
145	C₈	—	172	1	no	—
146	C₈	—	172	1	no	—
147	C₈	—	172	1	no	—
148	C₂²	#13063294	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	344	1	no	—
156	C₄	C₂×#10621885	1	1	yes	characteristic
157	C₄	#13063345	1	1	yes	characteristic
158	C₄	#13063355	1	1	yes	characteristic
159	C₄	#13063356	1	1	yes	characteristic
160	C₂	#14583055	1	1	yes	characteristic
161	C₂	#14583061	1	1	yes	characteristic
162	C₂	#14583072	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583251

C₂

yes

maximal, characteristic

#14583194

C₂

yes

maximal, characteristic

#14583194

C₂

yes

maximal, characteristic

#14582645

C₂

yes

maximal, characteristic

Q₈⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic

Q₈⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic

C₄₃×#557

C₂

yes

maximal, characteristic

#13063325

C₂²

yes

characteristic

#13063258

—

#13063258

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

—

—

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

#125×C₄₃

—

#125×C₄₃

—

C₄×Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

C₄×Q₈×C₄₃

C₂²

yes

characteristic

C₄×Q₈×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

D₄

yes

characteristic

C₂×Q₈×C₄₃

D₄

yes

characteristic

C₂×Dic₈₆

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

characteristic

C₄⋊C₄×C₄₃

D₄

yes

characteristic

C₄⋊C₄×C₄₃

D₄

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

Q₁₆

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

—

Dic₈₆

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

#120

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×Q₁₆

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

yes

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₄₃

#1636

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₃

yes

characteristic

#125

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₄₃

yes

characteristic

C₄×Q₈

C₂×D₄₃

yes

characteristic

#107

—

100

#107

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

#103

—

104

#103

—

105

#103

—

106

C₄×C₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

C₂²⋊D₄₃

yes

characteristic

110

C₂×Q₈

C₂²⋊D₄₃

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

C₂²⋊D₄₃

yes

—

123

C₄⋊C₄

C₂²⋊D₄₃

yes

characteristic

124

C₄⋊C₄

C₂²⋊D₄₃

yes

characteristic

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂²×D₄₃

yes

characteristic

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

172

—

135

Q₈

#10621885

yes

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

172

—

142

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

145

C₈

—

172

—

146

C₈

—

172

—

147

C₈

—

172

—

148

C₂²

#13063294

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

344

—

156

C₄

C₂×#10621885

yes

characteristic

157

C₄

#13063345

yes

characteristic

158

C₄

#13063355

yes

characteristic

159

C₄

#13063356

yes

characteristic

160

C₂

#14583055

yes

characteristic

161

C₂

#14583061

yes

characteristic

162

C₂

#14583072

yes

characteristic

	subgroup	G/H
center	C₂²	#13063294
commutator	C₄²×C₄₃	C₂³
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#557	C₂
socle	C₂²×C₄₃	#120
hypercenter	#557	D₄₃
hypocenter	C₄₃	#1636
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #557

subgroup

G/H

center

C₂²

#13063294

commutator

C₄²×C₄₃

C₂³

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#557

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #557

N	H ≅ G/N	type
89. C₄₃	#1636	semidirect

H ≅ G/N

type

89. C₄₃

#1636

semidirect

Subgroups — #19133753

Conjugacy classes of subgroups

Complements