Group #19147025 — subgroups

Subgroups — #19147025

Order 5504.

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

normal subgroups only

This group has 340 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	#14583623	C₂	1	2	yes	maximal
2	#14583242	C₂	1	4	yes	maximal, complemented
3	#14583181	C₂	1	2	yes	maximal, complemented
4	#14583141	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×D₁₇₂	C₂	1	1	yes	maximal, characteristic
7	#14582936	C₂	1	2	yes	maximal, complemented
8	C₄₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₄₃	C₂²	1	2	yes	—
11	#13063320	—	2	4	no	—
12	#13063314	—	2	2	no	—
13	#13063305	C₂²	1	4	yes	—
14	Q₈×Dic₄₃	—	2	2	no	—
15	C₄×Dic₈₆	C₂²	1	2	yes	—
16	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
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	#13063250	C₂²	1	2	yes	—
22	#13063248	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	D₄	1	1	yes	characteristic
30	C₂×C₄×D₄₃	C₂³	1	2	yes	—
31	C₂×Dic₈₆	C₂³	1	2	yes	—
32	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₄₃	—	2	2	no	—
35	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
36	C₄²×C₄₃	C₂³	1	1	yes	characteristic
37	(C₂×D₄₃)⋊C₄	—	2	4	no	—
38	(C₂×D₄₃)⋊C₄	—	2	4	no	—
39	(C₂×D₄₃)⋊C₄	D₄	1	2	yes	—
40	C₄⋊Dic₄₃	—	2	4	no	—
41	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
42	C₄⋊Dic₄₃	C₂³	1	2	yes	—
43	Dic₄₃⋊C₄	—	2	2	no	—
44	C₄×Dic₄₃	—	2	2	no	—
45	C₄×Dic₄₃	—	2	4	no	—
46	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
47	Dic₁₇₂	—	2	2	no	—
48	C₈×D₄₃	—	2	4	no	—
49	C₂²×D₄₃	C₂×D₄	1	2	yes	—
50	Q₈×C₄₃	—	4	4	no	—
51	C₂×C₄×C₄₃	—	2	4	no	—
52	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₄₃	—	2	2	no	—
56	C₂×Dic₄₃	—	2	2	no	—
57	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
58	D₁₇₂	—	2	2	no	—
59	C₄×D₄₃	—	2	2	no	—
60	C₄×D₄₃	—	2	2	no	—
61	Dic₈₆	—	2	2	no	—
62	Dic₈₆	—	2	2	no	—
63	C₈×C₄₃	—	2	1	no	—
64	C₈×C₄₃	#41	1	2	yes	—
65	C₄₃⋊C₈	—	2	2	no	—
66	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
67	C₂×D₄₃	—	2	2	no	—
68	C₂×D₄₃	—	2	2	no	—
69	C₂×D₄₃	—	2	2	no	—
70	C₄×C₄₃	—	2	1	no	—
71	C₄×C₄₃	—	4	4	no	—
72	C₄×C₄₃	#135	1	2	yes	—
73	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
74	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
75	Dic₄₃	—	2	2	no	—
76	Dic₄₃	—	4	2	no	—
77	Dic₄₃	—	4	2	no	—
78	#3342	—	43	1	no	maximal, Sylow
79	C₂×C₄₃	#547	1	1	yes	characteristic
80	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
81	C₂×C₄₃	#578	1	1	yes	characteristic
82	D₄₃	—	4	2	no	—
83	#556	—	43	2	no	—
84	#500	D₄₃	1	1	yes	characteristic, complemented
85	#485	—	43	4	no	—
86	#475	—	43	2	no	—
87	#467	—	43	2	no	—
88	C₄×Q₁₆	—	43	1	no	—
89	C₈×D₄	—	43	1	no	—
90	#427	—	43	2	no	—
91	C₄₃	#3342	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	—	43	1	no	—
93	C₂²×C₈	—	43	2	no	—
94	C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
95	#126	—	86	4	no	—
96	#124	—	86	2	no	—
97	C₂²⋊Q₈	—	43	4	no	—
98	C₄×Q₈	—	43	2	no	—
99	C₄×Q₈	—	86	2	no	—
100	C₄×D₄	—	43	1	no	—
101	#117	—	43	4	no	—
102	#107	—	43	1	no	—
103	#107	C₂×D₄₃	1	2	yes	—
104	#107	C₂×D₄₃	1	2	yes	—
105	#106	—	43	4	no	—
106	C₄⋊C₈	—	43	1	no	—
107	#103	—	43	2	no	—
108	#103	—	43	4	no	—
109	C₂²⋊C₈	—	43	2	no	—
110	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
111	C₂×Q₈	—	2	2	no	—
112	C₂×Q₈	—	43	2	no	—
113	C₂×D₄	—	43	1	no	—
114	C₂²×C₄	—	43	2	no	—
115	Q₁₆	—	86	2	no	—
116	C₂×C₈	—	43	2	no	—
117	C₂×C₈	—	86	4	no	—
118	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
119	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	43	1	no	—
122	C₄⋊C₄	—	43	2	no	—
123	C₄⋊C₄	—	86	4	no	—
124	C₄⋊C₄	—	86	2	no	—
125	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
126	C₂²⋊C₄	—	43	2	no	—
127	C₂²⋊C₄	—	86	4	no	—
128	C₂²⋊C₄	—	86	4	no	—
129	C₄²	—	86	4	no	—
130	C₄²	—	86	2	no	—
131	C₄²	C₂²×D₄₃	1	1	yes	characteristic
132	C₂³	—	43	2	no	—
133	Q₈	—	4	4	no	—
134	Q₈	—	86	2	no	—
135	Q₈	—	86	2	no	—
136	D₄	—	86	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	43	2	no	—
139	C₂×C₄	—	86	2	no	—
140	C₂×C₄	—	86	2	no	—
141	C₂×C₄	—	86	2	no	—
142	C₂×C₄	—	86	2	no	—
143	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
144	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
145	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	86	2	no	—
148	C₈	#10621902	1	2	yes	—
149	C₂²	—	86	2	no	—
150	C₂²	—	86	2	no	—
151	C₂²	—	86	2	no	—
152	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	4	no	—
155	C₄	—	86	2	no	—
156	C₄	—	172	2	no	—
157	C₄	—	172	2	no	—
158	C₄	#13063344	1	2	yes	—
159	C₄	C₂×#10621902	1	1	yes	characteristic
160	C₄	#13063373	1	1	yes	characteristic
161	C₂	—	172	2	no	—
162	C₂	#14583552	1	1	yes	characteristic
163	C₂	C₂×#13063344	1	1	yes	characteristic
164	C₂	#14583712	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583623

C₂

yes

maximal

#14583242

C₂

yes

maximal, complemented

#14583181

C₂

yes

maximal, complemented

#14583141

C₂

yes

maximal, complemented

C₄×Dic₁₇₂

C₂

yes

maximal, characteristic, complemented

C₈×D₁₇₂

C₂

yes

maximal, characteristic

#14582936

C₂

yes

maximal, complemented

C₄₃×#500

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₇₂

C₂²

yes

characteristic

C₂×C₈×D₄₃

C₂²

yes

—

#13063320

—

#13063314

—

#13063305

C₂²

yes

—

Q₈×Dic₄₃

—

C₄×Dic₈₆

C₂²

yes

—

C₄×D₁₇₂

C₂²

yes

characteristic

#13063283

C₂²

yes

—

#13063259

C₂²

yes

characteristic

#13063256

C₂²

yes

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

#13063250

C₂²

yes

—

#13063248

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

D₄

yes

characteristic

C₂×C₄×D₄₃

C₂³

yes

—

C₂×Dic₈₆

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

D₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

D₄

yes

characteristic

C₄⋊Dic₄₃

C₂³

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

C₂³

yes

—

Dic₁₇₂

—

C₈×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×Dic₄₃

C₂×D₄

yes

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

#41

yes

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

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

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

#556

—

#500

D₄₃

yes

characteristic, complemented

#485

—

#475

—

#467

—

C₄×Q₁₆

—

C₈×D₄

—

#427

—

C₄₃

#3342

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₄₃

yes

—

#126

—

#124

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

100

C₄×D₄

—

101

#117

—

102

#107

—

103

#107

C₂×D₄₃

yes

—

104

#107

C₂×D₄₃

yes

—

105

#106

—

106

C₄⋊C₈

—

107

#103

—

108

#103

—

109

C₂²⋊C₈

—

110

C₄×C₈

C₂×D₄₃

yes

characteristic

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×D₄

—

114

C₂²×C₄

—

115

Q₁₆

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₂²×D₄₃

yes

characteristic

119

C₂×C₈

C₂²×D₄₃

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

C₂²×D₄₃

yes

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²×D₄₃

yes

characteristic

132

C₂³

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

D₄×D₄₃

yes

characteristic

144

C₂×C₄

D₄×D₄₃

yes

characteristic

145

C₂×C₄

C₂³×D₄₃

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

#10621902

yes

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×D₄×D₄₃

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

172

—

157

C₄

—

172

—

158

C₄

#13063344

yes

—

159

C₄

C₂×#10621902

yes

characteristic

160

C₄

#13063373

yes

characteristic

161

C₂

—

172

—

162

C₂

#14583552

yes

characteristic

163

C₂

C₂×#13063344

yes

characteristic

164

C₂

#14583712

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	#3342
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
2. #14583242	C₂	semidirect
3. #14583181	C₂	semidirect
4. #14583141	C₂	semidirect
5. C₄×Dic₁₇₂	C₂	semidirect
7. #14582936	C₂	semidirect
8. C₄₃×#500	C₂	semidirect
84. #500	D₄₃	semidirect
91. C₄₃	#3342	semidirect

H ≅ G/N

type

2. #14583242

C₂

semidirect

3. #14583181

C₂

semidirect

4. #14583141

C₂

semidirect

5. C₄×Dic₁₇₂

C₂

semidirect

7. #14582936

C₂

semidirect

8. C₄₃×#500

C₂

semidirect

84. #500

D₄₃

semidirect

91. C₄₃

#3342

semidirect

Subgroups — #19147025

Conjugacy classes of subgroups

Complements