Group #19133143 — subgroups

Subgroups — #19133143

Order 5504.

252 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 69 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	#14582767	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582634	C₂	1	4	yes	maximal, complemented
4	C₄₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10621898	—	2	1	no	—
6	#13063325	C₄	1	1	yes	characteristic, complemented
7	#13063322	C₄	1	1	yes	characteristic, complemented
8	#13063313	—	2	1	no	—
9	#13063299	—	2	1	no	—
10	#13063299	C₂²	1	2	yes	—
11	C₄×D₁₇₂	—	2	1	no	—
12	C₄×D₁₇₂	C₄	1	2	yes	complemented
13	C₄₃⋊#95	—	2	1	no	—
14	C₄₃⋊#95	—	2	1	no	—
15	C₄₃⋊#95	C₂²	1	2	yes	—
16	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₄₃	—	2	1	no	—
18	C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
19	#97×C₄₃	—	2	1	no	—
20	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
23	C₂×C₂²⋊D₄₃	—	2	2	no	—
24	C₂²×Dic₄₃	—	2	2	no	—
25	#10621898	—	4	4	no	—
26	C₂×D₁₇₂	—	2	1	no	—
27	C₂×D₁₇₂	—	2	1	no	—
28	C₂×D₁₇₂	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₄₃	—	2	2	no	—
30	C₂×Dic₈₆	—	2	1	no	—
31	C₂×C₈×C₄₃	—	2	2	no	—
32	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₄₃	D₄	1	1	yes	characteristic
35	C₄²×C₄₃	D₄	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	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₄₃	—	2	2	no	—
42	C₂×C₄×C₄₃	—	2	2	no	—
43	C₂×C₄×C₄₃	—	4	1	no	—
44	C₂×C₄×C₄₃	—	4	1	no	—
45	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
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	C₂²⋊D₄₃	—	4	4	no	—
52	C₂×Dic₄₃	—	2	2	no	—
53	C₂×Dic₄₃	—	4	2	no	—
54	C₂×Dic₄₃	—	4	2	no	—
55	C₂×Dic₄₃	—	4	2	no	—
56	D₁₇₂	—	4	2	no	—
57	D₁₇₂	—	4	2	no	—
58	D₁₇₂	—	4	2	no	—
59	C₄×D₄₃	—	4	4	no	—
60	Dic₈₆	—	4	2	no	—
61	C₈×C₄₃	—	4	2	no	—
62	C₂²×C₄₃	—	2	1	no	—
63	C₂²×C₄₃	—	2	1	no	—
64	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
66	C₂×D₄₃	—	4	2	no	—
67	C₂×D₄₃	—	4	2	no	—
68	C₂×D₄₃	—	4	2	no	—
69	C₄×C₄₃	—	2	1	no	—
70	C₄×C₄₃	—	2	1	no	—
71	C₄×C₄₃	—	2	1	no	—
72	C₄×C₄₃	—	2	1	no	—
73	C₄×C₄₃	—	2	1	no	—
74	C₄×C₄₃	—	2	1	no	—
75	Dic₄₃	—	4	2	no	—
76	Dic₄₃	—	8	2	no	—
77	#1514	—	43	1	no	maximal, Sylow
78	C₂×C₄₃	—	2	2	no	—
79	C₂×C₄₃	#410	1	1	yes	characteristic
80	C₂×C₄₃	#418	1	1	yes	characteristic
81	C₂×C₄₃	#421	1	1	yes	characteristic
82	D₄₃	—	8	2	no	—
83	#532	—	43	1	no	—
84	#432	D₄₃	1	1	yes	characteristic, complemented
85	#383	—	43	1	no	—
86	#327	—	43	4	no	—
87	C₄₃	#1514	1	1	yes	characteristic, complemented, Sylow
88	C₂×#41	—	86	1	no	—
89	C₄⋊Q₈	—	43	1	no	—
90	#127	—	43	1	no	—
91	#124	—	86	1	no	—
92	#121	—	43	2	no	—
93	#121	—	86	1	no	—
94	C₄×D₄	—	43	2	no	—
95	C₄×D₄	—	86	1	no	—
96	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
97	C₄⋊C₈	—	2	1	no	—
98	C₂²⋊C₈	C₂×D₄₃	1	2	yes	—
99	#97	—	2	1	no	—
100	#95	—	43	2	no	—
101	#95	—	86	1	no	—
102	#95	—	86	1	no	—
103	#41	—	172	4	no	—
104	C₂×Q₈	—	86	1	no	—
105	C₂×D₄	—	43	2	no	—
106	C₂×D₄	—	86	2	no	—
107	C₂×D₄	—	86	1	no	—
108	C₂×D₄	—	86	1	no	—
109	C₂²×C₄	—	86	2	no	—
110	C₂²×C₄	—	86	2	no	—
111	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
112	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₄⋊C₄	—	43	2	no	—
116	C₄⋊C₄	—	86	1	no	—
117	C₂²⋊C₄	—	86	2	no	—
118	C₂²⋊C₄	—	86	2	no	—
119	C₄²	C₄×D₄₃	1	1	yes	characteristic
120	C₄²	C₄×D₄₃	1	1	yes	characteristic
121	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
122	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
123	C₂³	—	86	2	no	—
124	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
125	Q₈	—	172	2	no	—
126	D₄	—	172	2	no	—
127	D₄	—	172	2	no	—
128	D₄	—	172	2	no	—
129	D₄	—	172	4	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	86	2	no	—
134	C₂×C₄	—	172	4	no	—
135	C₂×C₄	—	172	2	no	—
136	C₂×C₄	—	172	2	no	—
137	C₂×C₄	—	172	2	no	—
138	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
139	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
140	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
141	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
144	C₈	—	4	2	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	172	2	no	—
148	C₂²	—	172	2	no	—
149	C₂²	—	172	2	no	—
150	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
151	C₂²	#13063353	1	2	yes	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	344	2	no	—
159	C₄	—	172	2	no	—
160	C₂	—	2	2	no	—
161	C₂	—	344	2	no	—
162	C₂	#14582872	1	1	yes	characteristic
163	C₂	#14582905	1	1	yes	characteristic
164	C₂	#14582915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583449

C₂

yes

maximal, characteristic

#14582767

C₂

yes

maximal, characteristic, complemented

#14582634

C₂

yes

maximal, complemented

C₄₃×#432

C₂

yes

maximal, characteristic, complemented

C₂×#10621898

—

#13063325

C₄

yes

characteristic, complemented

#13063322

C₄

yes

characteristic, complemented

#13063313

—

#13063299

—

#13063299

C₂²

yes

—

C₄×D₁₇₂

—

C₄×D₁₇₂

C₄

yes

complemented

C₄₃⋊#95

—

C₄₃⋊#95

—

C₄₃⋊#95

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

C₂²

yes

—

#97×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₂²⋊D₄₃

—

C₂²×Dic₄₃

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂³×C₄₃

C₂×D₄

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

Dic₄₃

—

Dic₄₃

—

#1514

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

#532

—

#432

D₄₃

yes

characteristic, complemented

#383

—

#327

—

C₄₃

#1514

yes

characteristic, complemented, Sylow

C₂×#41

—

C₄⋊Q₈

—

—

—

—

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

C₂×D₄₃

yes

—

#97

—

100

#95

—

101

#95

—

102

#95

—

103

#41

—

172

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

D₁₇₂

yes

characteristic

112

C₂²×C₄

D₁₇₂

yes

characteristic

113

C₂²×C₄

C₂²×D₄₃

yes

characteristic

114

C₂×C₈

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

C₄×D₄₃

yes

characteristic

120

C₄²

C₄×D₄₃

yes

characteristic

121

C₄²

C₂²⋊D₄₃

yes

characteristic

122

C₄²

C₂²⋊D₄₃

yes

characteristic

123

C₂³

—

124

C₂³

C₂×D₁₇₂

yes

characteristic

125

Q₈

—

172

—

126

D₄

—

172

—

127

D₄

—

172

—

128

D₄

—

172

—

129

D₄

—

172

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

172

—

135

C₂×C₄

—

172

—

136

C₂×C₄

—

172

—

137

C₂×C₄

—

172

—

138

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

139

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

140

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

141

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

142

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

172

—

148

C₂²

—

172

—

149

C₂²

—

172

—

150

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

151

C₂²

#13063353

yes

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

344

—

159

C₄

—

172

—

160

C₂

—

161

C₂

—

344

—

162

C₂

#14582872

yes

characteristic

163

C₂

#14582905

yes

characteristic

164

C₂

#14582915

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₄₃)⋊C₄
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#432	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	#432	D₄₃
hypocenter	C₄₃	#1514
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂×(C₂×D₄₃)⋊C₄

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#432

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
2. #14582767	C₂	semidirect
3. #14582634	C₂	semidirect
4. C₄₃×#432	C₂	semidirect
6. #13063325	C₄	semidirect
7. #13063322	C₄	semidirect
12. C₄×D₁₇₂	C₄	semidirect
84. #432	D₄₃	semidirect
87. C₄₃	#1514	semidirect

H ≅ G/N

type

2. #14582767

C₂

semidirect

3. #14582634

C₂

semidirect

4. C₄₃×#432

C₂

semidirect

6. #13063325

C₄

semidirect

7. #13063322

C₄

semidirect

12. C₄×D₁₇₂

C₄

semidirect

84. #432

D₄₃

semidirect

87. C₄₃

#1514

semidirect

Subgroups — #19133143

Conjugacy classes of subgroups

Complements