Group #19147017 — subgroups

Subgroups — #19147017

Order 5504.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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	#14583615	C₂	1	2	yes	maximal, complemented
2	#14583226	C₂	1	4	yes	maximal, complemented
3	#14583186	C₂	1	2	yes	maximal, complemented
4	#14583136	C₂	1	2	yes	maximal, complemented
5	C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×D₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#14582936	C₂	1	2	yes	maximal, complemented
8	C₄₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₄₃	C₂²	1	2	yes	—
11	#13063314	—	2	4	no	—
12	#13063300	—	2	2	no	—
13	#13063300	C₂²	1	4	yes	—
14	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
15	C₄×D₁₇₂	C₂²	1	2	yes	—
16	#13063283	—	2	2	no	—
17	#13063283	C₂²	1	4	yes	—
18	#13063259	C₂²	1	1	yes	characteristic
19	#13063256	C₂²	1	4	yes	—
20	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
21	#13063247	C₂²	1	2	yes	—
22	#13063245	C₂²	1	4	yes	—
23	(C₂×D₄₃)⋊C₈	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
25	#107×C₄₃	C₂²	1	2	yes	—
26	#107×C₄₃	C₂²	1	2	yes	—
27	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₄₃	—	2	2	no	—
29	C₂×D₁₇₂	—	2	4	no	—
30	C₂×D₁₇₂	D₄	1	1	yes	characteristic
31	C₂×D₁₇₂	C₂³	1	2	yes	—
32	C₂×C₄×D₄₃	—	2	2	no	—
33	C₂×C₄×D₄₃	C₂³	1	2	yes	—
34	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₄₃	—	2	2	no	—
37	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
38	C₄²×C₄₃	C₂³	1	1	yes	characteristic
39	(C₂×D₄₃)⋊C₄	—	2	2	no	—
40	(C₂×D₄₃)⋊C₄	—	2	4	no	—
41	(C₂×D₄₃)⋊C₄	—	2	4	no	—
42	(C₂×D₄₃)⋊C₄	D₄	1	2	yes	—
43	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
44	C₄⋊Dic₄₃	C₂³	1	2	yes	—
45	C₄×Dic₄₃	—	2	4	no	—
46	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
47	D₃₄₄	—	2	2	no	—
48	C₈×D₄₃	—	2	4	no	—
49	C₂²×D₄₃	—	2	2	no	—
50	C₂²×D₄₃	C₂×D₄	1	2	yes	—
51	Q₈×C₄₃	—	8	2	no	—
52	C₂×C₄×C₄₃	—	2	4	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	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₄₃	—	2	2	no	—
57	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
58	D₁₇₂	—	2	2	no	—
59	D₁₇₂	—	2	2	no	—
60	D₁₇₂	—	2	2	no	—
61	D₁₇₂	—	8	4	no	—
62	C₄×D₄₃	—	2	2	no	—
63	C₄×D₄₃	—	2	2	no	—
64	C₄×D₄₃	—	4	4	no	—
65	C₈×C₄₃	—	2	1	no	—
66	C₈×C₄₃	#41	1	2	yes	—
67	C₄₃⋊C₈	—	2	2	no	—
68	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
69	C₂×D₄₃	—	2	2	no	—
70	C₂×D₄₃	—	2	2	no	—
71	C₂×D₄₃	—	2	2	no	—
72	C₂×D₄₃	—	4	2	no	—
73	C₂×D₄₃	—	4	2	no	—
74	C₂×D₄₃	—	4	2	no	—
75	C₄×C₄₃	—	2	1	no	—
76	C₄×C₄₃	—	4	4	no	—
77	C₄×C₄₃	#135	1	2	yes	—
78	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
79	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
80	Dic₄₃	—	2	2	no	—
81	Dic₄₃	—	4	2	no	—
82	#3341	—	43	1	no	maximal, Sylow
83	C₂×C₄₃	#547	1	1	yes	characteristic
84	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
85	C₂×C₄₃	#576	1	1	yes	characteristic
86	D₄₃	—	4	2	no	—
87	D₄₃	—	8	2	no	—
88	#555	—	43	2	no	—
89	#500	D₄₃	1	1	yes	characteristic, complemented
90	#482	—	43	4	no	—
91	#476	—	43	2	no	—
92	#466	—	43	2	no	—
93	C₄×D₈	—	43	1	no	—
94	C₈×D₄	—	43	1	no	—
95	#427	—	43	2	no	—
96	C₄₃	#3341	1	1	yes	characteristic, complemented, Sylow
97	C₂×D₈	—	43	1	no	—
98	C₂²×C₈	—	43	2	no	—
99	C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
100	#124	—	86	4	no	—
101	#121	—	43	4	no	—
102	#121	—	86	2	no	—
103	C₄×D₄	—	43	1	no	—
104	C₄×D₄	—	43	2	no	—
105	#117	—	43	4	no	—
106	#117	—	86	2	no	—
107	#107	—	43	1	no	—
108	#107	C₂×D₄₃	1	2	yes	—
109	#107	C₂×D₄₃	1	2	yes	—
110	#106	—	43	4	no	—
111	C₄⋊C₈	—	43	1	no	—
112	#102	—	43	2	no	—
113	#102	—	43	4	no	—
114	C₂²⋊C₈	—	43	2	no	—
115	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
116	C₂×Q₈	—	2	2	no	—
117	C₂×D₄	—	43	1	no	—
118	C₂×D₄	—	43	2	no	—
119	C₂×D₄	—	86	4	no	—
120	C₂²×C₄	—	43	2	no	—
121	C₂²×C₄	—	86	2	no	—
122	D₈	—	86	2	no	—
123	C₂×C₈	—	43	2	no	—
124	C₂×C₈	—	86	4	no	—
125	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
126	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	43	1	no	—
129	C₄⋊C₄	—	43	2	no	—
130	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
131	C₂²⋊C₄	—	43	2	no	—
132	C₂²⋊C₄	—	86	4	no	—
133	C₂²⋊C₄	—	86	4	no	—
134	C₂²⋊C₄	—	86	2	no	—
135	C₄²	—	86	4	no	—
136	C₄²	C₂²×D₄₃	1	1	yes	characteristic
137	C₂³	—	43	2	no	—
138	C₂³	—	86	2	no	—
139	Q₈	—	8	2	no	—
140	D₄	—	86	2	no	—
141	D₄	—	86	2	no	—
142	D₄	—	86	2	no	—
143	D₄	—	344	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	43	2	no	—
146	C₂×C₄	—	86	2	no	—
147	C₂×C₄	—	86	2	no	—
148	C₂×C₄	—	86	2	no	—
149	C₂×C₄	—	172	4	no	—
150	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
151	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
152	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	86	2	no	—
155	C₈	#10621902	1	2	yes	—
156	C₂²	—	86	2	no	—
157	C₂²	—	86	2	no	—
158	C₂²	—	86	2	no	—
159	C₂²	—	172	2	no	—
160	C₂²	—	172	2	no	—
161	C₂²	—	172	2	no	—
162	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	4	no	—
165	C₄	—	86	2	no	—
166	C₄	—	172	2	no	—
167	C₄	#13063348	1	2	yes	—
168	C₄	C₂×#10621902	1	1	yes	characteristic
169	C₄	#13063373	1	1	yes	characteristic
170	C₂	—	344	2	no	—
171	C₂	—	172	2	no	—
172	C₂	#14583552	1	1	yes	characteristic
173	C₂	C₂×#13063348	1	1	yes	characteristic
174	C₂	#14583700	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583615

C₂

yes

maximal, complemented

#14583226

C₂

yes

maximal, complemented

#14583186

C₂

yes

maximal, complemented

#14583136

C₂

yes

maximal, complemented

C₄×D₃₄₄

C₂

yes

maximal, characteristic, complemented

C₈×D₁₇₂

C₂

yes

maximal, characteristic, complemented

#14582936

C₂

yes

maximal, complemented

C₄₃×#500

C₂

yes

maximal, characteristic, complemented

C₂×D₃₄₄

C₂²

yes

characteristic

C₂×C₈×D₄₃

C₂²

yes

—

#13063314

—

#13063300

—

#13063300

C₂²

yes

—

C₄×D₁₇₂

C₂²

yes

characteristic

C₄×D₁₇₂

C₂²

yes

—

#13063283

—

#13063283

C₂²

yes

—

#13063259

C₂²

yes

characteristic

#13063256

C₂²

yes

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

#13063247

C₂²

yes

—

#13063245

C₂²

yes

—

(C₂×D₄₃)⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₄₃

C₂²

yes

—

#107×C₄₃

C₂²

yes

—

#107×C₄₃

C₂²

yes

—

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

D₄

yes

characteristic

C₂×D₁₇₂

C₂³

yes

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

C₂³

yes

—

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂³

yes

—

C₄²×C₄₃

C₂³

yes

characteristic

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

D₄

yes

—

C₄⋊Dic₄₃

D₄

yes

characteristic

C₄⋊Dic₄₃

C₂³

yes

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

C₂³

yes

—

D₃₄₄

—

C₈×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

C₂×D₄

yes

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₄

yes

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#41

yes

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#135

yes

—

C₄×C₄₃

C₂×#41

yes

characteristic

C₄×C₄₃

2₋¹⁺⁴

yes

characteristic

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

D₄₃

—

#555

—

#500

D₄₃

yes

characteristic, complemented

#482

—

#476

—

#466

—

C₄×D₈

—

C₈×D₄

—

#427

—

C₄₃

#3341

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₄₃

yes

—

100

#124

—

101

#121

—

102

#121

—

103

C₄×D₄

—

104

C₄×D₄

—

105

#117

—

106

#117

—

107

#107

—

108

#107

C₂×D₄₃

yes

—

109

#107

C₂×D₄₃

yes

—

110

#106

—

111

C₄⋊C₈

—

112

#102

—

113

#102

—

114

C₂²⋊C₈

—

115

C₄×C₈

C₂×D₄₃

yes

characteristic

116

C₂×Q₈

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

D₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

C₂²×D₄₃

yes

characteristic

126

C₂×C₈

C₂²×D₄₃

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

C₂²×D₄₃

yes

—

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

C₂²×D₄₃

yes

characteristic

137

C₂³

—

138

C₂³

—

139

Q₈

—

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

344

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

172

—

150

C₂×C₄

D₄×D₄₃

yes

characteristic

151

C₂×C₄

D₄×D₄₃

yes

characteristic

152

C₂×C₄

C₂³×D₄₃

yes

characteristic

153

C₈

—

154

C₈

—

155

C₈

#10621902

yes

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

172

—

160

C₂²

—

172

—

161

C₂²

—

172

—

162

C₂²

C₂×D₄×D₄₃

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

172

—

167

C₄

#13063348

yes

—

168

C₄

C₂×#10621902

yes

characteristic

169

C₄

#13063373

yes

characteristic

170

C₂

—

344

—

171

C₂

—

172

—

172

C₂

#14583552

yes

characteristic

173

C₂

C₂×#13063348

yes

characteristic

174

C₂

#14583700

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#500	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#500	D₄₃
hypocenter	C₄₃	#3341
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₄₃×#500

C₂

socle

C₂²×C₄₃

C₂²×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
1. #14583615	C₂	semidirect
2. #14583226	C₂	semidirect
3. #14583186	C₂	semidirect
4. #14583136	C₂	semidirect
5. C₄×D₃₄₄	C₂	semidirect
6. C₈×D₁₇₂	C₂	semidirect
7. #14582936	C₂	semidirect
8. C₄₃×#500	C₂	semidirect
89. #500	D₄₃	semidirect
96. C₄₃	#3341	semidirect

H ≅ G/N

type

1. #14583615

C₂

semidirect

2. #14583226

C₂

semidirect

3. #14583186

C₂

semidirect

4. #14583136

C₂

semidirect

5. C₄×D₃₄₄

C₂

semidirect

6. C₈×D₁₇₂

C₂

semidirect

7. #14582936

C₂

semidirect

8. C₄₃×#500

C₂

semidirect

89. #500

D₄₃

semidirect

96. C₄₃

#3341

semidirect

Subgroups — #19147017

Conjugacy classes of subgroups

Complements