Group #19134002 — subgroups

Subgroups — #19134002

Order 5504.

176 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 55 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	#14583245	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583189	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582649	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582647	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582643	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#459	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄₃⋊C₈	—	2	1	no	—
9	#13063325	C₂²	1	1	yes	characteristic
10	#13063256	—	2	1	no	—
11	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
13	#13063246	—	2	1	no	—
14	#13063246	—	2	1	no	—
15	#13063246	—	2	1	no	—
16	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
17	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₄₃	—	2	1	no	—
19	#127×C₄₃	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
22	#102×C₄₃	—	2	1	no	—
23	C₂×D₄×C₄₃	—	2	1	no	—
24	C₂×D₄×C₄₃	—	2	1	no	—
25	C₂×D₄×C₄₃	—	2	1	no	—
26	C₂×D₄×C₄₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₃	—	2	1	no	—
28	C₂×Dic₈₆	—	2	1	no	—
29	D₈×C₄₃	—	4	2	no	—
30	C₂×C₈×C₄₃	—	2	1	no	—
31	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
32	C₂²⋊C₄×C₄₃	—	2	1	no	—
33	C₄²×C₄₃	C₂³	1	1	yes	characteristic
34	C₄⋊Dic₄₃	—	2	1	no	—
35	C₄⋊Dic₄₃	—	2	1	no	—
36	C₂×C₄₃⋊C₈	—	2	1	no	—
37	C₂×C₄₃⋊C₈	—	2	1	no	—
38	C₂×C₄₃⋊C₈	—	4	2	no	—
39	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
41	C₂³×C₄₃	—	2	1	no	—
42	C₂³×C₄₃	—	4	1	no	—
43	D₄×C₄₃	—	2	1	no	—
44	D₄×C₄₃	—	4	1	no	—
45	D₄×C₄₃	—	4	1	no	—
46	D₄×C₄₃	—	4	1	no	—
47	D₄×C₄₃	—	4	1	no	—
48	D₄×C₄₃	—	8	1	no	—
49	D₄×C₄₃	SD₁₆	1	2	yes	—
50	C₂×C₄×C₄₃	—	2	1	no	—
51	C₂×C₄×C₄₃	—	4	1	no	—
52	C₂×C₄×C₄₃	—	4	1	no	—
53	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
56	C₂×Dic₄₃	—	4	1	no	—
57	Dic₈₆	—	8	1	no	—
58	C₈×C₄₃	—	4	1	no	—
59	C₄₃⋊C₈	—	2	1	no	—
60	C₄₃⋊C₈	—	2	1	no	—
61	C₄₃⋊C₈	—	4	1	no	—
62	C₄₃⋊C₈	—	4	1	no	—
63	C₂²×C₄₃	—	2	2	no	—
64	C₂²×C₄₃	—	4	1	no	—
65	C₂²×C₄₃	—	4	1	no	—
66	C₂²×C₄₃	—	8	1	no	—
67	C₂²×C₄₃	—	8	1	no	—
68	C₂²×C₄₃	—	8	1	no	—
69	C₂²×C₄₃	#121	1	1	yes	characteristic
70	C₄×C₄₃	—	2	1	no	—
71	C₄×C₄₃	—	4	1	no	—
72	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₄×C₄₃	#135	1	1	yes	characteristic
74	C₄×C₄₃	#135	1	1	yes	characteristic
75	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	Dic₄₃	—	8	1	no	—
77	#1677	—	43	1	no	maximal, Sylow
78	C₂×C₄₃	—	4	2	no	—
79	C₂×C₄₃	—	16	1	no	—
80	C₂×C₄₃	#463	1	1	yes	characteristic
81	C₂×C₄₃	#465	1	1	yes	characteristic
82	C₂×C₄₃	#472	1	1	yes	characteristic
83	#486	—	43	1	no	—
84	#476	—	43	1	no	—
85	#459	D₄₃	1	1	yes	characteristic
86	C₈×D₄	—	43	1	no	—
87	#334	—	43	1	no	—
88	#333	—	43	1	no	—
89	#331	—	43	1	no	—
90	C₄₃	#1677	1	1	yes	characteristic, complemented, Sylow
91	C₂×D₈	—	2	1	no	—
92	C₂²×C₈	—	86	1	no	—
93	C₄⋊Q₈	—	43	1	no	—
94	#127	C₂×D₄₃	1	1	yes	characteristic
95	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
96	#106	—	86	1	no	—
97	C₄⋊C₈	—	43	1	no	—
98	C₄⋊C₈	—	43	1	no	—
99	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
100	#102	—	2	1	no	—
101	#102	—	86	1	no	—
102	#102	—	86	1	no	—
103	#102	—	86	1	no	—
104	C₂²⋊C₈	—	86	1	no	—
105	C₄×C₈	—	43	1	no	—
106	C₂×Q₈	—	86	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	C₂²⋊D₄₃	1	1	yes	characteristic
111	C₂²×C₄	—	2	1	no	—
112	D₈	—	4	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	43	1	no	—
115	C₂×C₈	—	43	1	no	—
116	C₂×C₈	—	86	1	no	—
117	C₂×C₈	—	86	1	no	—
118	C₂×C₈	—	172	2	no	—
119	C₄⋊C₄	—	86	1	no	—
120	C₄⋊C₄	—	86	1	no	—
121	C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	C₂²×D₄₃	1	1	yes	characteristic
124	C₂³	—	2	1	no	—
125	C₂³	—	4	1	no	—
126	Q₈	—	344	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	4	1	no	—
132	D₄	—	8	1	no	—
133	D₄	#10621883	1	2	yes	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	172	1	no	—
138	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
139	C₂×C₄	#10621900	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
141	C₈	—	4	1	no	—
142	C₈	—	86	1	no	—
143	C₈	—	86	1	no	—
144	C₈	—	172	1	no	—
145	C₈	—	172	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	—	8	1	no	—
150	C₂²	—	8	1	no	—
151	C₂²	—	8	1	no	—
152	C₂²	#13063297	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	344	1	no	—
156	C₄	C₂×#10621883	1	1	yes	characteristic
157	C₄	#13063344	1	1	yes	characteristic
158	C₄	#13063345	1	1	yes	characteristic
159	C₄	#13063349	1	1	yes	characteristic
160	C₂	—	4	2	no	—
161	C₂	—	16	1	no	—
162	C₂	#14583112	1	1	yes	characteristic
163	C₂	#14583126	1	1	yes	characteristic
164	C₂	#14583167	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583245

C₂

yes

maximal, characteristic, complemented

#14583189

C₂

yes

maximal, characteristic, complemented

D₄×C₄₃⋊C₈

C₂

yes

maximal, characteristic, complemented

#14582649

C₂

yes

maximal, characteristic, complemented

#14582647

C₂

yes

maximal, characteristic, complemented

#14582643

C₂

yes

maximal, characteristic, complemented

C₄₃×#459

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

—

#13063325

C₂²

yes

characteristic

#13063256

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

—

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₄₃

—

#127×C₄₃

C₂²

yes

characteristic

C₄×D₄×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

#102×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

—

C₂×Dic₈₆

—

D₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

characteristic

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

D₄

yes

characteristic

C₂×C₄₃⋊C₈

D₄

yes

characteristic

C₂³×C₄₃

—

C₂³×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

SD₁₆

yes

—

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

#41

yes

characteristic

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#121

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×SD₁₆

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

#1677

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

—

D₄₃

yes

characteristic

C₈×D₄

—

#334

—

#333

—

#331

—

C₄₃

#1677

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#127

C₂×D₄₃

yes

characteristic

C₄×D₄

C₂×D₄₃

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₃

yes

characteristic

100

#102

—

101

#102

—

102

#102

—

103

#102

—

104

C₂²⋊C₈

—

105

C₄×C₈

—

106

C₂×Q₈

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

C₂²⋊D₄₃

yes

characteristic

111

C₂²×C₄

—

112

D₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

172

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

C₂²⋊D₄₃

yes

characteristic

122

C₂²⋊C₄

—

123

C₄²

C₂²×D₄₃

yes

characteristic

124

C₂³

—

125

C₂³

—

126

Q₈

—

344

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

#10621883

yes

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

172

—

138

C₂×C₄

D₄×D₄₃

yes

characteristic

139

C₂×C₄

#10621900

yes

characteristic

140

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

172

—

145

C₈

—

172

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

#13063297

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

344

—

156

C₄

C₂×#10621883

yes

characteristic

157

C₄

#13063344

yes

characteristic

158

C₄

#13063345

yes

characteristic

159

C₄

#13063349

yes

characteristic

160

C₂

—

161

C₂

—

162

C₂

#14583112

yes

characteristic

163

C₂

#14583126

yes

characteristic

164

C₂

#14583167

yes

characteristic

	subgroup	G/H
center	C₂²	#13063297
commutator	C₄²×C₄₃	C₂³
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#459	C₂
socle	C₂²×C₄₃	#121
hypercenter	#459	D₄₃
hypocenter	C₄₃	#1677
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

subgroup

G/H

center

C₂²

#13063297

commutator

C₄²×C₄₃

C₂³

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#459

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #459

N	H ≅ G/N	type
1. #14583245	C₂	semidirect
2. #14583189	C₂	semidirect
3. D₄×C₄₃⋊C₈	C₂	semidirect
4. #14582649	C₂	semidirect
5. #14582647	C₂	semidirect
6. #14582643	C₂	semidirect
90. C₄₃	#1677	semidirect

H ≅ G/N

type

1. #14583245

C₂

semidirect

2. #14583189

C₂

semidirect

3. D₄×C₄₃⋊C₈

C₂

semidirect

4. #14582649

C₂

semidirect

5. #14582647

C₂

semidirect

6. #14582643

C₂

semidirect

90. C₄₃

#1677

semidirect

Subgroups — #19134002

Conjugacy classes of subgroups

Complements