Group #19133013 — subgroups

Subgroups — #19133013

Order 5504.

262 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 77 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	#14583449	C₂	1	1	yes	maximal, characteristic
2	#14582952	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582628	C₂	1	4	yes	maximal, complemented
4	C₄×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10621898	—	2	1	no	—
6	#13063325	C₄	1	1	yes	characteristic
7	#13063322	C₄	1	1	yes	characteristic
8	#13063313	—	2	1	no	—
9	#13063299	—	2	1	no	—
10	#13063299	C₄	1	2	yes	—
11	C₄×D₁₇₂	—	2	1	no	—
12	C₄×D₁₇₂	C₂²	1	2	yes	—
13	C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
14	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
15	C₄×C₄₃⋊C₈	—	2	1	no	—
16	C₂×M₁₆×C₄₃	—	2	1	no	—
17	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	#97×C₄₃	—	2	1	no	—
19	C₄×C₈×C₄₃	C₂²	1	2	yes	—
20	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
23	C₂×C₂²⋊D₄₃	—	2	2	no	—
24	#10621898	—	4	4	no	—
25	C₂×D₁₇₂	—	2	1	no	—
26	C₂×D₁₇₂	—	2	1	no	—
27	C₂×D₁₇₂	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₄₃	—	2	2	no	—
29	C₂×Dic₈₆	—	2	1	no	—
30	M₁₆×C₄₃	—	2	4	no	—
31	C₂×C₈×C₄₃	—	2	2	no	—
32	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₄₃	D₄	1	1	yes	characteristic
34	C₄²×C₄₃	D₄	1	1	yes	characteristic
35	C₄²×C₄₃	C₂³	1	1	yes	characteristic
36	(C₂×D₄₃)⋊C₄	—	2	2	no	—
37	(C₂×D₄₃)⋊C₄	—	2	2	no	—
38	C₄⋊Dic₄₃	—	2	1	no	—
39	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
40	C₂×C₄₃⋊C₈	—	2	2	no	—
41	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×D₄₃	—	2	2	no	—
43	C₂×C₄×C₄₃	—	2	2	no	—
44	C₂×C₄×C₄₃	—	2	2	no	—
45	C₂×C₄×C₄₃	—	2	2	no	—
46	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂²⋊D₄₃	—	4	4	no	—
53	C₂×Dic₄₃	—	2	2	no	—
54	D₁₇₂	—	2	4	no	—
55	D₁₇₂	—	4	2	no	—
56	D₁₇₂	—	4	2	no	—
57	C₄×D₄₃	—	4	4	no	—
58	Dic₈₆	—	4	2	no	—
59	C₈×C₄₃	—	2	4	no	—
60	C₄₃⋊C₈	—	4	2	no	—
61	C₂²×C₄₃	—	2	1	no	—
62	C₂²×C₄₃	—	2	1	no	—
63	C₂²×C₄₃	—	2	1	no	—
64	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×D₄₃	—	4	2	no	—
66	C₂×D₄₃	—	4	2	no	—
67	C₂×D₄₃	—	4	2	no	—
68	C₄×C₄₃	—	2	1	no	—
69	C₄×C₄₃	—	2	1	no	—
70	C₄×C₄₃	—	2	1	no	—
71	C₄×C₄₃	#104	1	2	yes	—
72	C₄×C₄₃	#131	1	2	yes	—
73	C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
74	Dic₄₃	—	4	2	no	—
75	#1484	—	43	1	no	maximal, Sylow
76	C₂×C₄₃	—	4	1	no	—
77	C₂×C₄₃	#408	1	1	yes	characteristic
78	C₂×C₄₃	#418	1	1	yes	characteristic
79	C₂×C₄₃	C₂×#104	1	1	yes	characteristic
80	D₄₃	—	8	2	no	—
81	#532	—	43	1	no	—
82	#431	—	43	1	no	—
83	C₄×M₁₆	D₄₃	1	1	yes	characteristic, complemented
84	D₄⋊C₈	—	43	4	no	—
85	C₄₃	#1484	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	—	86	1	no	—
87	C₂×M₁₆	—	2	1	no	—
88	C₄⋊Q₈	—	43	1	no	—
89	#127	—	43	1	no	—
90	#124	—	86	1	no	—
91	#121	—	43	2	no	—
92	#121	—	86	1	no	—
93	C₄×D₄	—	43	2	no	—
94	C₄×D₄	—	86	1	no	—
95	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
96	C₄⋊C₈	—	43	2	no	—
97	C₂²⋊C₈	—	86	1	no	—
98	#97	—	2	1	no	—
99	C₄×C₈	—	86	1	no	—
100	C₄×C₈	C₂×D₄₃	1	2	yes	—
101	#41	—	172	4	no	—
102	C₂×Q₈	—	86	1	no	—
103	C₂×D₄	—	43	2	no	—
104	C₂×D₄	—	86	2	no	—
105	C₂×D₄	—	86	1	no	—
106	C₂×D₄	—	86	1	no	—
107	C₂²×C₄	—	86	2	no	—
108	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
109	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
110	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
111	M₁₆	—	2	4	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	86	2	no	—
114	C₄⋊C₄	—	43	2	no	—
115	C₄⋊C₄	—	86	1	no	—
116	C₂²⋊C₄	—	86	2	no	—
117	C₂²⋊C₄	—	86	2	no	—
118	C₄²	C₄×D₄₃	1	1	yes	characteristic
119	C₄²	D₁₇₂	1	1	yes	characteristic
120	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
121	C₄²	C₂²×D₄₃	1	1	yes	characteristic
122	C₂³	—	86	2	no	—
123	C₂³	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
124	Q₈	—	172	2	no	—
125	D₄	—	86	4	no	—
126	D₄	—	172	2	no	—
127	D₄	—	172	4	no	—
128	D₄	—	172	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	86	2	no	—
133	C₂×C₄	—	172	4	no	—
134	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
135	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
138	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
140	C₈	—	2	4	no	—
141	C₈	—	172	2	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	172	2	no	—
146	C₂²	—	172	2	no	—
147	C₂²	—	172	2	no	—
148	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	172	2	no	—
153	C₄	#13063253	1	2	yes	—
154	C₄	#13063333	1	2	yes	—
155	C₄	#13063353	1	2	yes	—
156	C₂	—	4	1	no	—
157	C₂	—	344	2	no	—
158	C₂	#14582867	1	1	yes	characteristic
159	C₂	#14582905	1	1	yes	characteristic
160	C₂	C₂×#13063253	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583449

C₂

yes

maximal, characteristic

#14582952

C₂

yes

maximal, characteristic, complemented

#14582628

C₂

yes

maximal, complemented

C₄×M₁₆×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×#10621898

—

#13063325

C₄

yes

characteristic

#13063322

C₄

yes

—

—

C₄

yes

—

C₄×D₁₇₂

—

C₄×D₁₇₂

C₂²

yes

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₂×M₁₆×C₄₃

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#97×C₄₃

—

C₄×C₈×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂×C₂²⋊D₄₃

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#104

yes

—

C₄×C₄₃

#131

yes

—

C₄×C₄₃

AGL(1,ℤ/8)

yes

—

Dic₄₃

—

#1484

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

#532

—

#431

—

C₄×M₁₆

D₄₃

yes

characteristic, complemented

D₄⋊C₈

—

C₄₃

#1484

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

—

—

—

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

100

C₄×C₈

C₂×D₄₃

yes

—

101

#41

—

172

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₄₃

yes

characteristic

109

C₂²×C₄

D₁₇₂

yes

characteristic

110

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

C₄×D₄₃

yes

characteristic

119

C₄²

D₁₇₂

yes

characteristic

120

C₄²

C₂²⋊D₄₃

yes

characteristic

121

C₄²

C₂²×D₄₃

yes

characteristic

122

C₂³

—

123

C₂³

(C₂×D₄₃)⋊C₄

yes

characteristic

124

Q₈

—

172

—

125

D₄

—

126

D₄

—

172

—

127

D₄

—

172

—

128

D₄

—

172

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

172

—

134

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

135

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

136

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

137

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

138

C₂×C₄

C₂×D₁₇₂

yes

characteristic

139

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

140

C₈

—

141

C₈

—

172

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

172

—

146

C₂²

—

172

—

147

C₂²

—

172

—

148

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

172

—

153

C₄

#13063253

yes

—

154

C₄

#13063333

yes

—

155

C₄

#13063353

yes

—

156

C₂

—

157

C₂

—

344

—

158

C₂

#14582867

yes

characteristic

159

C₂

#14582905

yes

characteristic

160

C₂

C₂×#13063253

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄×M₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	C₄×M₁₆	D₄₃
hypocenter	C₄₃	#1484
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄×M₁₆×C₄₃

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

hypercenter

C₄×M₁₆

D₄₃

hypocenter

C₄₃

#1484

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #14582952	C₂	semidirect
3. #14582628	C₂	semidirect
4. C₄×M₁₆×C₄₃	C₂	semidirect
83. C₄×M₁₆	D₄₃	semidirect
85. C₄₃	#1484	semidirect

H ≅ G/N

type

2. #14582952

C₂

semidirect

3. #14582628

C₂

semidirect

4. C₄×M₁₆×C₄₃

C₂

semidirect

83. C₄×M₁₆

D₄₃

semidirect

85. C₄₃

#1484

semidirect

Subgroups — #19133013

Conjugacy classes of subgroups

Complements