Group #19133394 — subgroups

Subgroups — #19133394

Order 5504.

180 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 73 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	#14583023	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#10621873	C₂	1	1	yes	maximal, characteristic
3	#14582964	C₂	1	1	yes	maximal, characteristic
4	#14582629	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582626	C₂	1	1	yes	maximal, characteristic
6	C₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10621873	C₄	1	1	yes	characteristic
9	C₂×C₈×D₄₃	—	2	1	no	—
10	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
11	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
12	#13063257	C₄	1	1	yes	characteristic
13	Dic₄₃⋊C₈	—	2	1	no	—
14	#13063250	C₄	1	1	yes	characteristic
15	#13063247	C₄	1	1	yes	characteristic
16	(C₂×D₄₃)⋊C₈	—	2	1	no	—
17	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×Dic₄₃	—	2	1	no	—
20	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×D₁₇₂	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₄₃	—	2	1	no	—
25	C₂×Dic₈₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₄₃	—	2	1	no	—
27	C₂×C₈×C₄₃	—	2	1	no	—
28	C₂×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₄₃	C₂³	1	1	yes	characteristic
31	(C₂×D₄₃)⋊C₄	—	2	1	no	—
32	C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
34	Dic₄₃⋊C₄	—	2	1	no	—
35	C₄×Dic₄₃	—	2	1	no	—
36	C₂×C₄₃⋊C₈	—	2	1	no	—
37	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
38	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
39	#10621873	C₈	1	4	yes	complemented
40	C₈×D₄₃	—	4	2	no	—
41	C₂²×D₄₃	—	2	1	no	—
42	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
45	C₂×Dic₄₃	—	2	1	no	—
46	C₂×Dic₄₃	—	2	1	no	—
47	C₂×Dic₄₃	—	2	1	no	—
48	D₁₇₂	—	2	1	no	—
49	D₁₇₂	C₂×C₈	1	2	yes	—
50	C₄×D₄₃	—	4	1	no	—
51	C₄×D₄₃	—	4	1	no	—
52	Dic₈₆	—	2	1	no	—
53	Dic₈₆	C₂×C₈	1	2	yes	complemented
54	C₈×C₄₃	—	2	1	no	—
55	C₈×C₄₃	—	4	1	no	—
56	C₈×C₄₃	—	4	1	no	—
57	C₈×C₄₃	C₂×C₈	1	2	yes	complemented
58	C₄₃⋊C₈	—	2	2	no	—
59	C₄₃⋊C₈	—	2	1	no	—
60	C₄₃⋊C₈	SD₁₆	1	2	yes	complemented
61	C₂²×C₄₃	C₄×D₄	1	1	yes	characteristic
62	C₂×D₄₃	—	2	2	no	—
63	C₂×D₄₃	—	4	1	no	—
64	C₂×D₄₃	—	4	1	no	—
65	C₄×C₄₃	—	2	1	no	—
66	C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
67	C₄×C₄₃	#131	1	1	yes	characteristic
68	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₄₃	#135	1	1	yes	characteristic
70	Dic₄₃	—	2	2	no	—
71	Dic₄₃	—	4	1	no	—
72	Dic₄₃	—	4	1	no	—
73	C₈×SD₁₆	—	43	1	no	maximal, Sylow
74	C₂×C₄₃	C₈×D₄	1	1	yes	characteristic
75	C₂×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
76	C₂×C₄₃	#443	1	1	yes	characteristic
77	D₄₃	—	4	2	no	—
78	C₈×Q₈	—	43	1	no	—
79	C₄×SD₁₆	—	43	1	no	—
80	C₈×D₄	—	43	1	no	—
81	#334	D₄₃	1	1	yes	characteristic, complemented
82	Q₈⋊C₈	—	43	1	no	—
83	D₄⋊C₈	—	43	1	no	—
84	C₈²	—	43	1	no	—
85	C₄₃	C₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	43	1	no	—
87	C₂²×C₈	—	86	1	no	—
88	C₄×Q₈	—	43	1	no	—
89	C₄×D₄	—	43	1	no	—
90	#106	—	43	1	no	—
91	C₄⋊C₈	—	86	1	no	—
92	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
93	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
94	#103	—	43	1	no	—
95	#102	—	43	1	no	—
96	C₂²⋊C₈	—	86	1	no	—
97	C₄×C₈	—	43	1	no	—
98	C₄×C₈	—	43	1	no	—
99	C₄×C₈	—	86	1	no	—
100	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
101	C₂×Q₈	—	43	1	no	—
102	C₂×D₄	—	43	1	no	—
103	C₂²×C₄	—	86	1	no	—
104	SD₁₆	—	43	4	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	43	1	no	—
108	C₂×C₈	—	43	1	no	—
109	C₂×C₈	—	86	1	no	—
110	C₂×C₈	—	172	2	no	—
111	C₂×C₈	C₄×D₄₃	1	1	yes	characteristic
112	C₂×C₈	C₄×D₄₃	1	1	yes	characteristic
113	C₄⋊C₄	—	43	1	no	—
114	C₄⋊C₄	—	43	1	no	—
115	C₄⋊C₄	—	86	1	no	—
116	C₂²⋊C₄	—	86	1	no	—
117	C₄²	—	86	1	no	—
118	C₄²	C₂²×D₄₃	1	1	yes	characteristic
119	C₂³	—	86	1	no	—
120	Q₈	—	43	2	no	—
121	Q₈	—	86	1	no	—
122	D₄	—	43	2	no	—
123	D₄	—	86	1	no	—
124	C₂×C₄	—	86	1	no	—
125	C₂×C₄	—	86	1	no	—
126	C₂×C₄	—	86	1	no	—
127	C₂×C₄	—	172	1	no	—
128	C₂×C₄	—	172	1	no	—
129	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
130	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
131	C₂×C₄	#10621902	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	4	1	no	—
135	C₈	—	43	2	no	—
136	C₈	—	86	2	no	—
137	C₈	—	86	1	no	—
138	C₈	C₈×D₄₃	1	2	yes	complemented
139	C₂²	—	86	2	no	—
140	C₂²	—	172	1	no	—
141	C₂²	—	172	1	no	—
142	C₂²	#13063287	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	86	2	no	—
145	C₄	—	172	1	no	—
146	C₄	—	172	1	no	—
147	C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
148	C₄	#13063333	1	1	yes	characteristic
149	C₄	SD₁₆×D₄₃	1	1	yes	characteristic
150	C₄	#13063346	1	1	yes	characteristic
151	C₂	—	172	2	no	—
152	C₂	#14582964	1	1	yes	characteristic
153	C₂	#14582986	1	1	yes	characteristic
154	C₂	#14583015	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583023

C₂

yes

maximal, characteristic, complemented

C₄×#10621873

C₂

yes

maximal, characteristic

#14582964

C₂

yes

maximal, characteristic

#14582629

C₂

yes

maximal, characteristic, complemented

#14582626

C₂

yes

maximal, characteristic

C₈×C₄₃⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄₃×#334

C₂

yes

maximal, characteristic, complemented

C₂×#10621873

C₄

yes

characteristic

C₂×C₈×D₄₃

—

C₄×Dic₈₆

C₂²

yes

characteristic

C₄×D₁₇₂

C₂²

yes

characteristic

#13063257

C₄

yes

characteristic

Dic₄₃⋊C₈

—

#13063250

C₄

yes

characteristic

#13063247

C₄

yes

characteristic

(C₂×D₄₃)⋊C₈

—

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₈×Dic₄₃

—

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×D₁₇₂

C₂×C₄

yes

characteristic

C₂×C₄×D₄₃

—

C₂×Dic₈₆

C₂×C₄

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

C₂×C₄

yes

characteristic

C₄⋊Dic₄₃

C₂×C₄

yes

characteristic

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

D₄

yes

characteristic

C₂×C₄₃⋊C₈

D₄

yes

characteristic

#10621873

C₈

yes

complemented

C₈×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

#41

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

C₂×C₈

yes

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

C₂×C₈

yes

complemented

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

C₂×C₈

yes

complemented

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

SD₁₆

yes

complemented

C₂²×C₄₃

C₄×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²×C₈

yes

characteristic

C₄×C₄₃

#131

yes

characteristic

C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₄×C₄₃

#135

yes

characteristic

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

C₈×SD₁₆

—

maximal, Sylow

C₂×C₄₃

C₈×D₄

yes

characteristic

C₂×C₄₃

C₄×SD₁₆

yes

characteristic

C₂×C₄₃

#443

yes

characteristic

D₄₃

—

C₈×Q₈

—

C₄×SD₁₆

—

C₈×D₄

—

#334

D₄₃

yes

characteristic, complemented

Q₈⋊C₈

—

D₄⋊C₈

—

C₈²

—

C₄₃

C₈×SD₁₆

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂²×C₈

—

C₄×Q₈

—

C₄×D₄

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₃

yes

characteristic

C₄⋊C₈

C₂×D₄₃

yes

characteristic

#103

—

#102

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×D₄₃

yes

characteristic

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂²×C₄

—

104

SD₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

172

—

111

C₂×C₈

C₄×D₄₃

yes

characteristic

112

C₂×C₈

C₄×D₄₃

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂²×D₄₃

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

172

—

128

C₂×C₄

—

172

—

129

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

130

C₂×C₄

D₄×D₄₃

yes

characteristic

131

C₂×C₄

#10621902

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

C₈×D₄₃

yes

complemented

139

C₂²

—

140

C₂²

—

172

—

141

C₂²

—

172

—

142

C₂²

#13063287

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

172

—

146

C₄

—

172

—

147

C₄

C₂×C₈×D₄₃

yes

characteristic

148

C₄

#13063333

yes

characteristic

149

C₄

SD₁₆×D₄₃

yes

characteristic

150

C₄

#13063346

yes

characteristic

151

C₂

—

172

—

152

C₂

#14582964

yes

characteristic

153

C₂

#14582986

yes

characteristic

154

C₂

#14583015

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₄₃
commutator	C₄×C₄₃	C₂²×C₈
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#334	C₂
socle	C₂²×C₄₃	C₄×D₄
hypercenter	#334	D₄₃
hypocenter	C₄₃	C₈×SD₁₆
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₄₃

commutator

C₄×C₄₃

C₂²×C₈

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#334

C₂

socle

C₂²×C₄₃

C₄×D₄

hypercenter

#334

D₄₃

hypocenter

C₄₃

C₈×SD₁₆

LCS

G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
1. #14583023	C₂	semidirect
4. #14582629	C₂	semidirect
6. C₈×C₄₃⋊C₈	C₂	semidirect
7. C₄₃×#334	C₂	semidirect
39. #10621873	C₈	semidirect
53. Dic₈₆	C₂×C₈	semidirect
57. C₈×C₄₃	C₂×C₈	semidirect
60. C₄₃⋊C₈	SD₁₆	semidirect
81. #334	D₄₃	semidirect
85. C₄₃	C₈×SD₁₆	semidirect
138. C₈	C₈×D₄₃	semidirect

H ≅ G/N

type

1. #14583023

C₂

semidirect

4. #14582629

C₂

semidirect

6. C₈×C₄₃⋊C₈

C₂

semidirect

7. C₄₃×#334

C₂

semidirect

39. #10621873

C₈

semidirect

53. Dic₈₆

C₂×C₈

semidirect

57. C₈×C₄₃

C₂×C₈

semidirect

60. C₄₃⋊C₈

SD₁₆

semidirect

81. #334

D₄₃

semidirect

85. C₄₃

C₈×SD₁₆

semidirect

138. C₈

C₈×D₄₃

semidirect

Subgroups — #19133394

Conjugacy classes of subgroups

Complements