Group #19133487 — subgroups

Subgroups — #19133487

Order 5504.

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

normal subgroups only

This group has 166 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	#14583029	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#10621873	C₂	1	1	yes	maximal, characteristic
3	#14582978	C₂	1	1	yes	maximal, characteristic
4	#14582650	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582631	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582628	C₂	1	1	yes	maximal, characteristic
7	C₈²×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10621873	C₄	1	1	yes	characteristic
9	C₂×#10621872	—	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	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₄₃⋊C₈	—	2	1	no	—
16	#13063250	C₄	1	1	yes	characteristic
17	#13063247	C₄	1	1	yes	characteristic
18	(C₂×D₄₃)⋊C₈	—	2	1	no	—
19	#13063235	—	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	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
30	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
31	C₄²×C₄₃	C₂³	1	1	yes	characteristic
32	(C₂×D₄₃)⋊C₄	—	2	1	no	—
33	C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
34	C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
35	Dic₄₃⋊C₄	—	2	1	no	—
36	C₄×Dic₄₃	—	2	1	no	—
37	C₂×C₄₃⋊C₈	—	2	1	no	—
38	C₂×C₄₃⋊C₈	—	2	1	no	—
39	#10621873	—	2	2	no	—
40	#10621872	—	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	—	2	1	no	—
50	C₄×D₄₃	—	4	1	no	—
51	C₄×D₄₃	—	4	1	no	—
52	Dic₈₆	—	2	1	no	—
53	Dic₈₆	—	2	1	no	—
54	C₈×C₄₃	—	2	2	no	—
55	C₈×C₄₃	—	2	1	no	—
56	C₈×C₄₃	—	2	1	no	—
57	C₈×C₄₃	M₁₆	1	2	yes	complemented
58	C₈×C₄₃	SD₁₆	1	2	yes	complemented
59	C₄₃⋊C₈	—	4	1	no	—
60	C₄₃⋊C₈	—	4	1	no	—
61	C₂²×C₄₃	C₄×D₄	1	1	yes	characteristic
62	C₂×D₄₃	—	4	1	no	—
63	C₂×D₄₃	—	4	1	no	—
64	C₂×D₄₃	—	4	1	no	—
65	C₄×C₄₃	—	2	1	no	—
66	C₄×C₄₃	C₂×M₁₆	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	—	4	1	no	—
71	Dic₄₃	—	4	1	no	—
72	Dic₄₃	—	4	1	no	—
73	#1590	—	43	1	no	maximal, Sylow
74	C₂×C₄₃	#436	1	1	yes	characteristic
75	C₂×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
76	C₂×C₄₃	#443	1	1	yes	characteristic
77	D₄₃	—	8	1	no	—
78	#446	—	43	1	no	—
79	C₄×SD₁₆	—	43	1	no	—
80	#436	—	43	1	no	—
81	#334	—	43	1	no	—
82	Q₈⋊C₈	—	43	1	no	—
83	D₄⋊C₈	—	43	1	no	—
84	C₈²	D₄₃	1	1	yes	characteristic, complemented
85	C₄₃	#1590	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	43	1	no	—
87	C₂×M₁₆	—	86	1	no	—
88	C₄×Q₈	—	43	1	no	—
89	C₄×D₄	—	43	1	no	—
90	#106	—	43	1	no	—
91	C₄⋊C₈	—	43	1	no	—
92	C₄⋊C₈	—	43	1	no	—
93	C₄⋊C₈	—	86	1	no	—
94	#103	—	43	1	no	—
95	#102	—	43	1	no	—
96	C₂²⋊C₈	—	86	1	no	—
97	#97	—	86	1	no	—
98	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
99	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
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	—	86	2	no	—
105	M₁₆	—	172	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	86	1	no	—
108	C₂×C₈	—	86	1	no	—
109	C₂×C₈	C₄×D₄₃	1	1	yes	characteristic
110	C₂×C₈	C₄×D₄₃	1	1	yes	characteristic
111	C₂×C₈	D₁₇₂	1	1	yes	characteristic
112	C₂×C₈	D₁₇₂	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	—	86	1	no	—
121	Q₈	—	86	1	no	—
122	D₄	—	86	1	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	C₂×D₁₇₂	1	1	yes	characteristic
131	C₂×C₄	#10621898	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	172	1	no	—
136	C₈	—	172	1	no	—
137	C₈	#10621872	1	2	yes	complemented
138	C₈	#10621873	1	2	yes	complemented
139	C₂²	—	172	1	no	—
140	C₂²	—	172	1	no	—
141	C₂²	—	172	1	no	—
142	C₂²	C₄×D₁₇₂	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	172	1	no	—
145	C₄	—	172	1	no	—
146	C₄	—	172	1	no	—
147	C₄	C₂×#10621872	1	1	yes	characteristic
148	C₄	#13063331	1	1	yes	characteristic
149	C₄	C₂×#10621873	1	1	yes	characteristic
150	C₄	#13063347	1	1	yes	characteristic
151	C₂	—	344	1	no	—
152	C₂	#14582978	1	1	yes	characteristic
153	C₂	C₄×#10621873	1	1	yes	characteristic
154	C₂	#14583017	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583029

C₂

yes

maximal, characteristic, complemented

C₄×#10621873

C₂

yes

maximal, characteristic

#14582978

C₂

yes

maximal, characteristic

#14582650

C₂

yes

maximal, characteristic, complemented

#14582631

C₂

yes

maximal, characteristic, complemented

#14582628

C₂

yes

maximal, characteristic

C₈²×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×#10621873

C₄

yes

characteristic

C₂×#10621872

—

C₄×Dic₈₆

C₂²

yes

characteristic

C₄×D₁₇₂

C₂²

yes

characteristic

#13063257

C₄

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

Dic₄₃⋊C₈

—

#13063250

C₄

yes

characteristic

#13063247

C₄

yes

characteristic

(C₂×D₄₃)⋊C₈

—

#13063235

—

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×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂×C₈×C₄₃

D₄

yes

characteristic

C₂×C₈×C₄₃

D₄

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

—

#10621873

—

#10621872

—

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

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

M₁₆

yes

complemented

C₈×C₄₃

SD₁₆

yes

complemented

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₄×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×M₁₆

yes

characteristic

C₄×C₄₃

#131

yes

characteristic

C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₄×C₄₃

#135

yes

characteristic

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

C₄×SD₁₆

yes

characteristic

C₂×C₄₃

#443

yes

characteristic

D₄₃

—

#446

—

C₄×SD₁₆

—

#436

—

#334

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₈²

D₄₃

yes

characteristic, complemented

C₄₃

#1590

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×D₄

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#102

—

C₂²⋊C₈

—

#97

—

C₄×C₈

C₂×D₄₃

yes

characteristic

C₄×C₈

C₂×D₄₃

yes

characteristic

100

C₄×C₈

C₂×D₄₃

yes

characteristic

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂²×C₄

—

104

SD₁₆

—

105

M₁₆

—

172

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₄×D₄₃

yes

characteristic

110

C₂×C₈

C₄×D₄₃

yes

characteristic

111

C₂×C₈

D₁₇₂

yes

characteristic

112

C₂×C₈

D₁₇₂

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

C₂×D₁₇₂

yes

characteristic

131

C₂×C₄

#10621898

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

172

—

136

C₈

—

172

—

137

C₈

#10621872

yes

complemented

138

C₈

#10621873

yes

complemented

139

C₂²

—

172

—

140

C₂²

—

172

—

141

C₂²

—

172

—

142

C₂²

C₄×D₁₇₂

yes

characteristic

143

C₄

—

144

C₄

—

172

—

145

C₄

—

172

—

146

C₄

—

172

—

147

C₄

C₂×#10621872

yes

characteristic

148

C₄

#13063331

yes

characteristic

149

C₄

C₂×#10621873

yes

characteristic

150

C₄

#13063347

yes

characteristic

151

C₂

—

344

—

152

C₂

#14582978

yes

characteristic

153

C₂

C₄×#10621873

yes

characteristic

154

C₂

#14583017

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₄²	C₂²×D₄₃
Fitting	C₈²×C₄₃	C₂
socle	C₂²×C₄₃	C₄×D₄
hypercenter	C₈²	D₄₃
hypocenter	C₄₃	#1590
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈²

subgroup

G/H

center

C₂×C₄

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₄²

C₂²×D₄₃

Fitting

C₈²×C₄₃

C₂

socle

C₂²×C₄₃

C₄×D₄

hypercenter

C₈²

D₄₃

hypocenter

C₄₃

#1590

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈²

N	H ≅ G/N	type
1. #14583029	C₂	semidirect
4. #14582650	C₂	semidirect
5. #14582631	C₂	semidirect
7. C₈²×C₄₃	C₂	semidirect
57. C₈×C₄₃	M₁₆	semidirect
58. C₈×C₄₃	SD₁₆	semidirect
84. C₈²	D₄₃	semidirect
85. C₄₃	#1590	semidirect
137. C₈	#10621872	semidirect
138. C₈	#10621873	semidirect

H ≅ G/N

type

1. #14583029

C₂

semidirect

4. #14582650

C₂

semidirect

5. #14582631

C₂

semidirect

7. C₈²×C₄₃

C₂

semidirect

57. C₈×C₄₃

M₁₆

semidirect

58. C₈×C₄₃

SD₁₆

semidirect

84. C₈²

D₄₃

semidirect

85. C₄₃

#1590

semidirect

137. C₈

#10621872

semidirect

138. C₈

#10621873

semidirect

Subgroups — #19133487

Conjugacy classes of subgroups

Complements