Group #19134807 — subgroups

Subgroups — #19134807

Order 5504.

324 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 97 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	#14583374	C₂	1	1	yes	maximal, characteristic
2	#14582941	C₂	1	2	yes	maximal, complemented
3	#14582802	C₂	1	1	yes	maximal, characteristic
4	#14582660	C₂	1	2	yes	maximal, complemented
5	C₂×#100×C₄₃	C₂	1	1	yes	maximal, characteristic
6	D₄×Dic₄₃	—	2	2	no	—
7	D₄×Dic₄₃	—	2	2	no	—
8	#13063284	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
11	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
12	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
13	#13063259	—	2	2	no	—
14	#13063257	—	2	2	no	—
15	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₄₃	C₂²	1	2	yes	—
17	#100×C₄₃	C₄	1	4	yes	complemented
18	C₂⁴×C₄₃	D₄	1	2	yes	—
19	C₂×D₄×C₄₃	—	2	2	no	—
20	C₂×D₄×C₄₃	—	2	2	no	—
21	C₂×D₄×C₄₃	C₂×C₄	1	2	yes	—
22	C₂×D₄×C₄₃	Q₈	1	2	yes	—
23	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₄₃	—	2	2	no	—
25	C₂²×Dic₄₃	—	2	2	no	—
26	C₂²×Dic₄₃	—	2	1	no	—
27	C₂²×Dic₄₃	—	2	1	no	—
28	M₁₆×C₄₃	—	2	2	no	—
29	M₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
30	C₂×C₈×C₄₃	—	2	2	no	—
31	C₂²⋊Dic₄₃	—	2	4	no	—
32	C₂²⋊Dic₄₃	—	2	2	no	—
33	C₂²⋊Dic₄₃	—	4	4	no	—
34	C₄⋊Dic₄₃	—	2	2	no	—
35	C₄⋊Dic₄₃	D₄	1	2	yes	—
36	C₄⋊Dic₄₃	D₄	1	2	yes	—
37	C₄×Dic₄₃	—	2	2	no	—
38	C₄×Dic₄₃	—	2	2	no	—
39	C₂³×C₄₃	—	2	2	no	—
40	C₂³×C₄₃	—	2	2	no	—
41	C₂³×C₄₃	—	4	2	no	—
42	C₂³×C₄₃	—	4	2	no	—
43	C₂³×C₄₃	C₄⋊C₄	1	4	yes	—
44	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
45	D₄×C₄₃	—	2	4	no	—
46	D₄×C₄₃	—	2	4	no	—
47	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₄₃	#41	1	2	yes	—
51	C₂×Dic₄₃	—	2	2	no	—
52	C₂×Dic₄₃	—	2	2	no	—
53	C₂×Dic₄₃	—	2	2	no	—
54	C₂×Dic₄₃	—	4	1	no	—
55	C₂×Dic₄₃	—	4	1	no	—
56	C₂×Dic₄₃	—	4	2	no	—
57	C₂×Dic₄₃	—	4	1	no	—
58	C₂×Dic₄₃	—	8	2	no	—
59	C₂×Dic₄₃	—	8	2	no	—
60	C₈×C₄₃	—	2	4	no	—
61	C₂²×C₄₃	—	2	1	no	—
62	C₂²×C₄₃	—	2	1	no	—
63	C₂²×C₄₃	—	2	4	no	—
64	C₂²×C₄₃	—	4	4	no	—
65	C₂²×C₄₃	—	4	2	no	—
66	C₂²×C₄₃	—	4	2	no	—
67	C₂²×C₄₃	—	8	2	no	—
68	C₂²×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄₃	#120	1	1	yes	characteristic
70	C₂²×C₄₃	#123	1	1	yes	characteristic
71	C₄×C₄₃	C₄×D₄	1	2	yes	—
72	C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
73	Dic₄₃	—	4	2	no	—
74	Dic₄₃	—	4	2	no	—
75	Dic₄₃	—	4	2	no	—
76	Dic₄₃	—	8	1	no	—
77	#1856	—	43	1	no	maximal, Sylow
78	C₂×C₄₃	—	2	1	no	—
79	C₂×C₄₃	—	2	1	no	—
80	C₂×C₄₃	—	4	4	no	—
81	C₂×C₄₃	#385	1	1	yes	characteristic
82	C₂×C₄₃	#453	1	1	yes	characteristic
83	C₂×C₄₃	#455	1	1	yes	characteristic
84	#518	—	43	1	no	—
85	#428	—	43	2	no	—
86	C₂×#100	D₄₃	1	1	yes	characteristic
87	#390	—	43	1	no	—
88	#339	—	43	2	no	—
89	C₄₃	#1856	1	1	yes	characteristic, complemented, Sylow
90	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic
91	C₂×M₁₆	C₂×D₄₃	1	2	yes	—
92	C₄×D₄	—	86	2	no	—
93	C₄×D₄	—	86	2	no	—
94	#117	—	43	2	no	—
95	C₂×C₄⋊C₄	—	43	1	no	—
96	C₂×C₂²⋊C₄	—	86	2	no	—
97	C₂×C₂²⋊C₄	—	86	2	no	—
98	C₂×C₄²	—	43	1	no	—
99	#107	—	86	2	no	—
100	#106	—	86	2	no	—
101	#100	Dic₄₃	1	4	yes	complemented
102	C₂⁴	D₁₇₂	1	2	yes	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	Dic₈₆	1	2	yes	—
106	C₂×D₄	C₂×Dic₄₃	1	2	yes	—
107	C₂²×C₄	—	86	2	no	—
108	C₂²×C₄	—	86	2	no	—
109	C₂²×C₄	—	86	1	no	—
110	C₂²×C₄	—	86	1	no	—
111	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
112	M₁₆	—	2	2	no	—
113	M₁₆	C₂×Dic₄₃	1	4	yes	complemented
114	C₂×C₈	—	2	2	no	—
115	C₄⋊C₄	—	43	2	no	—
116	C₄⋊C₄	—	43	2	no	—
117	C₄⋊C₄	—	86	2	no	—
118	C₂²⋊C₄	—	86	4	no	—
119	C₂²⋊C₄	—	86	2	no	—
120	C₂²⋊C₄	—	172	4	no	—
121	C₄²	—	86	2	no	—
122	C₄²	—	86	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	2	2	no	—
125	C₂³	—	4	2	no	—
126	C₂³	—	4	2	no	—
127	C₂³	C₄⋊Dic₄₃	1	4	yes	—
128	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
129	D₄	—	2	4	no	—
130	D₄	—	2	4	no	—
131	C₂×C₄	—	86	2	no	—
132	C₂×C₄	—	86	2	no	—
133	C₂×C₄	—	86	2	no	—
134	C₂×C₄	—	344	2	no	—
135	C₂×C₄	—	344	2	no	—
136	C₂×C₄	—	172	2	no	—
137	C₂×C₄	—	172	1	no	—
138	C₂×C₄	—	172	1	no	—
139	C₂×C₄	—	172	1	no	—
140	C₂×C₄	C₂×Dic₈₆	1	1	yes	characteristic
141	C₂×C₄	D₄×D₄₃	1	2	yes	—
142	C₂×C₄	#10621900	1	2	yes	—
143	C₂×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	4	4	no	—
149	C₂²	—	4	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	8	2	no	—
152	C₂²	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
153	C₂²	#13063295	1	1	yes	characteristic
154	C₂²	#13063309	1	1	yes	characteristic
155	C₄	—	344	1	no	—
156	C₄	—	172	2	no	—
157	C₄	—	172	2	no	—
158	C₄	—	172	2	no	—
159	C₄	D₄×Dic₄₃	1	2	yes	—
160	C₄	#13063302	1	2	yes	—
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	4	4	no	—
164	C₂	#14582774	1	1	yes	characteristic
165	C₂	#14583069	1	1	yes	characteristic
166	C₂	#14583079	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583374

C₂

yes

maximal, characteristic

#14582941

C₂

yes

maximal, complemented

#14582802

C₂

yes

maximal, characteristic

#14582660

C₂

yes

maximal, complemented

C₂×#100×C₄₃

C₂

yes

maximal, characteristic

D₄×Dic₄₃

—

D₄×Dic₄₃

—

#13063284

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₂²⋊Dic₄₃

—

C₂×C₂²⋊Dic₄₃

—

C₂×C₄×Dic₄₃

C₂²

yes

characteristic

#13063259

—

#13063257

—

C₂²×D₄×C₄₃

C₂²

yes

characteristic

C₂×M₁₆×C₄₃

C₂²

yes

—

#100×C₄₃

C₄

yes

complemented

C₂⁴×C₄₃

D₄

yes

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

C₂×C₄

yes

—

C₂×D₄×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

M₁₆×C₄₃

—

M₁₆×C₄₃

C₂×C₄

yes

complemented

C₂×C₈×C₄₃

—

C₂²⋊Dic₄₃

—

C₂²⋊Dic₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

D₄

yes

—

C₄⋊Dic₄₃

D₄

yes

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₄⋊C₄

yes

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₄⋊C₄

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

C₄×D₄

yes

—

C₄×C₄₃

C₂²⋊Q₈

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

#1856

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

—

D₄₃

yes

characteristic

#390

—

#339

—

C₄₃

#1856

yes

characteristic, complemented, Sylow

C₂²×D₄

C₂×D₄₃

yes

characteristic

C₂×M₁₆

C₂×D₄₃

yes

—

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#107

—

100

#106

—

101

#100

Dic₄₃

yes

complemented

102

C₂⁴

D₁₇₂

yes

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

Dic₈₆

yes

—

106

C₂×D₄

C₂×Dic₄₃

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²×D₄₃

yes

characteristic

112

M₁₆

—

113

M₁₆

C₂×Dic₄₃

yes

complemented

114

C₂×C₈

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

172

—

121

C₄²

—

122

C₄²

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₄⋊Dic₄₃

yes

—

128

C₂³

C₂×D₁₇₂

yes

characteristic

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

344

—

135

C₂×C₄

—

344

—

136

C₂×C₄

—

172

—

137

C₂×C₄

—

172

—

138

C₂×C₄

—

172

—

139

C₂×C₄

—

172

—

140

C₂×C₄

C₂×Dic₈₆

yes

characteristic

141

C₂×C₄

D₄×D₄₃

yes

—

142

C₂×C₄

#10621900

yes

—

143

C₂×C₄

C₂²×Dic₄₃

yes

characteristic

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×C₄⋊Dic₄₃

yes

characteristic

153

C₂²

#13063295

yes

characteristic

154

C₂²

#13063309

yes

characteristic

155

C₄

—

344

—

156

C₄

—

172

—

157

C₄

—

172

—

158

C₄

—

172

—

159

C₄

D₄×Dic₄₃

yes

—

160

C₄

#13063302

yes

—

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

#14582774

yes

characteristic

165

C₂

#14583069

yes

characteristic

166

C₂

#14583079

yes

characteristic

	subgroup	G/H
center	C₂²	#13063295
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×#100×C₄₃	C₂
socle	C₂²×C₄₃	#120
hypercenter	C₂×#100	D₄₃
hypocenter	C₄₃	#1856
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

subgroup

G/H

center

C₂²

#13063295

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×#100×C₄₃

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

N	H ≅ G/N	type
2. #14582941	C₂	semidirect
4. #14582660	C₂	semidirect
17. #100×C₄₃	C₄	semidirect
29. M₁₆×C₄₃	C₂×C₄	semidirect
89. C₄₃	#1856	semidirect
101. #100	Dic₄₃	semidirect
113. M₁₆	C₂×Dic₄₃	semidirect

H ≅ G/N

type

2. #14582941

C₂

semidirect

4. #14582660

C₂

semidirect

17. #100×C₄₃

C₄

semidirect

29. M₁₆×C₄₃

C₂×C₄

semidirect

89. C₄₃

#1856

semidirect

101. #100

Dic₄₃

semidirect

113. M₁₆

C₂×Dic₄₃

semidirect

Subgroups — #19134807

Conjugacy classes of subgroups

Complements