Group #19140362 — subgroups

Subgroups — #19140362

Order 5504.

528 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 157 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	C₂×#13063301	C₂	1	2	yes	maximal
2	C₂×#13063282	C₂	1	1	yes	maximal, characteristic
3	#14582830	C₂	1	4	yes	maximal, complemented
4	#14582825	C₂	1	4	yes	maximal
5	#14582806	C₂	1	2	yes	maximal, complemented
6	#14582765	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₄₃	C₂²	1	1	yes	characteristic
9	#13063301	—	2	8	no	—
10	#13063282	—	2	4	no	—
11	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
12	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
13	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	—
14	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
15	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	—
16	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
17	C₄₃⋊#95	C₂²	1	8	yes	—
18	C₄₃⋊#95	C₂²	1	4	yes	—
19	C₂²×Q₈×C₄₃	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₄₃	—	2	4	no	—
21	C₂×C₄⋊C₄×C₄₃	C₂²	1	4	yes	—
22	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
23	C₂³×D₄₃	C₂³	1	1	yes	characteristic
24	C₂×Q₈×C₄₃	—	2	4	no	—
25	C₂×Q₈×C₄₃	—	2	4	no	—
26	C₂×Q₈×C₄₃	—	2	4	no	—
27	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
28	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₄₃	C₂³	1	2	yes	—
31	C₂²×Dic₄₃	C₂³	1	4	yes	—
32	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
33	C₂×C₄×D₄₃	—	2	2	no	—
34	C₂×C₄×D₄₃	—	2	2	no	—
35	C₂×C₄×D₄₃	D₄	1	4	yes	—
36	C₄⋊C₄×C₄₃	—	2	8	no	—
37	C₄²×C₄₃	D₄	1	4	yes	—
38	(C₂×D₄₃)⋊C₄	—	2	4	no	—
39	(C₂×D₄₃)⋊C₄	—	2	8	no	—
40	C₄⋊Dic₄₃	—	2	4	no	—
41	Dic₄₃⋊C₄	—	2	4	no	—
42	Dic₄₃⋊C₄	—	2	8	no	—
43	C₄×Dic₄₃	—	2	2	no	—
44	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
45	C₂²×D₄₃	—	2	1	no	—
46	C₂²×D₄₃	—	2	1	no	—
47	C₂²×D₄₃	—	2	4	no	—
48	C₂²×D₄₃	C₂×D₄	1	2	yes	—
49	Q₈×C₄₃	—	4	8	no	—
50	C₂×C₄×C₄₃	—	2	4	no	—
51	C₂×C₄×C₄₃	—	2	4	no	—
52	C₂×C₄×C₄₃	—	2	4	no	—
53	C₂×C₄×C₄₃	—	2	4	no	—
54	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₄₃	C₂×D₄	1	4	yes	—
56	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₄₃	#41	1	2	yes	—
58	C₂×Dic₄₃	—	2	1	no	—
59	C₂×Dic₄₃	—	2	1	no	—
60	C₂×Dic₄₃	—	2	2	no	—
61	C₂×Dic₄₃	—	2	4	no	—
62	C₂×Dic₄₃	—	2	4	no	—
63	C₂×Dic₄₃	—	2	8	no	—
64	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
65	C₄×D₄₃	—	2	8	no	—
66	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
67	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
68	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
69	C₂²×C₄₃	2₋¹⁺⁴	1	4	yes	—
70	C₂×D₄₃	—	2	2	no	—
71	C₂×D₄₃	—	2	4	no	—
72	C₂×D₄₃	—	4	2	no	—
73	C₂×D₄₃	—	4	2	no	—
74	C₄×C₄₃	—	2	4	no	—
75	C₄×C₄₃	—	4	4	no	—
76	C₄×C₄₃	#121	1	4	yes	—
77	Dic₄₃	—	2	2	no	—
78	Dic₄₃	—	4	2	no	—
79	Dic₄₃	—	4	4	no	—
80	#2621	—	43	1	no	maximal, Sylow
81	C₂×C₄₃	C₂×#121	1	1	yes	characteristic
82	C₂×C₄₃	#536	1	2	yes	—
83	C₂×C₄₃	#536	1	2	yes	—
84	C₂×C₄₃	#541	1	2	yes	—
85	D₄₃	—	4	2	no	—
86	C₂×C₄⋊Q₈	D₄₃	1	1	yes	characteristic, complemented
87	C₂×C₂²⋊Q₈	—	43	2	no	—
88	C₂×#117	—	43	1	no	—
89	#398	—	43	4	no	—
90	#397	—	43	4	no	—
91	#391	—	43	2	no	—
92	#382	—	43	1	no	—
93	C₄₃	#2621	1	1	yes	characteristic, complemented, Sylow
94	C₂²×Q₈	C₂×D₄₃	1	2	yes	—
95	C₂³×C₄	—	43	1	no	—
96	C₄⋊Q₈	—	2	4	no	—
97	C₂²⋊Q₈	—	86	8	no	—
98	#117	—	86	4	no	—
99	C₂×C₄⋊C₄	—	43	2	no	—
100	C₂×C₄⋊C₄	—	43	4	no	—
101	C₂×C₄⋊C₄	—	43	2	no	—
102	C₂×C₄⋊C₄	C₂×D₄₃	1	4	yes	—
103	C₂×C₂²⋊C₄	—	43	2	no	—
104	C₂×C₂²⋊C₄	—	43	4	no	—
105	C₂×C₄²	—	43	1	no	—
106	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
107	#95	—	43	4	no	—
108	#95	—	43	8	no	—
109	C₂⁴	—	43	1	no	—
110	C₂×Q₈	—	2	4	no	—
111	C₂×Q₈	—	2	4	no	—
112	C₂×Q₈	—	2	4	no	—
113	C₂²×C₄	—	43	1	no	—
114	C₂²×C₄	—	43	4	no	—
115	C₂²×C₄	—	43	2	no	—
116	C₂²×C₄	—	43	4	no	—
117	C₂²×C₄	—	86	2	no	—
118	C₂²×C₄	—	86	2	no	—
119	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
120	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
121	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
122	C₄⋊C₄	—	2	8	no	—
123	C₄⋊C₄	—	86	8	no	—
124	C₄⋊C₄	—	86	4	no	—
125	C₄⋊C₄	—	86	4	no	—
126	C₂²⋊C₄	—	86	8	no	—
127	C₂²⋊C₄	—	86	4	no	—
128	C₄²	—	86	2	no	—
129	C₄²	C₂²⋊D₄₃	1	4	yes	—
130	C₂³	—	43	2	no	—
131	C₂³	—	86	1	no	—
132	C₂³	—	86	1	no	—
133	C₂³	—	86	4	no	—
134	C₂³	C₂³×D₄₃	1	1	yes	characteristic
135	Q₈	—	4	8	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	43	2	no	—
141	C₂×C₄	—	86	1	no	—
142	C₂×C₄	—	86	1	no	—
143	C₂×C₄	—	86	8	no	—
144	C₂×C₄	—	86	4	no	—
145	C₂×C₄	—	86	2	no	—
146	C₂×C₄	—	86	4	no	—
147	C₂×C₄	—	86	8	no	—
148	C₂×C₄	D₄×D₄₃	1	2	yes	—
149	C₂×C₄	#10621900	1	2	yes	—
150	C₂×C₄	C₂×C₂²⋊D₄₃	1	4	yes	—
151	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
152	C₂²	—	86	2	no	—
153	C₂²	—	86	4	no	—
154	C₂²	—	172	2	no	—
155	C₂²	—	172	2	no	—
156	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
157	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
158	C₂²	C₂×#10621900	1	1	yes	characteristic
159	C₂²	#13063373	1	4	yes	—
160	C₄	—	2	4	no	—
161	C₄	—	4	4	no	—
162	C₄	—	86	2	no	—
163	C₄	—	172	2	no	—
164	C₄	—	172	4	no	—
165	C₄	#13063296	1	4	yes	—
166	C₂	—	172	2	no	—
167	C₂	C₂×#13063296	1	1	yes	characteristic
168	C₂	#14583473	1	2	yes	—
169	C₂	#14583474	1	2	yes	—
170	C₂	#14583510	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063301

C₂

yes

maximal

C₂×#13063282

C₂

yes

maximal, characteristic

#14582830

C₂

yes

maximal, complemented

#14582825

C₂

yes

maximal

#14582806

C₂

yes

maximal, complemented

#14582765

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₄₃

C₂²

yes

characteristic

#13063301

—

#13063282

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

characteristic

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₃

C₂²

yes

—

C₄⋊Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂³×D₄₃

C₂³

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

characteristic

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

D₄

yes

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂³×C₄₃

C₂⁴

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

C₂×D₄

yes

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₄

yes

—

C₄×D₄₃

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂×#41

yes

characteristic

C₂²×C₄₃

2₋¹⁺⁴

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#121

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#536

yes

—

C₂×C₄₃

#541

yes

—

D₄₃

—

C₂×C₄⋊Q₈

D₄₃

yes

characteristic, complemented

C₂×C₂²⋊Q₈

—

—

—

—

—

—

C₄₃

#2621

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₃

yes

—

C₂³×C₄

—

C₄⋊Q₈

—

C₂²⋊Q₈

—

#117

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

103

C₂×C₂²⋊C₄

—

104

C₂×C₂²⋊C₄

—

105

C₂×C₄²

—

106

C₂×C₄²

C₂×D₄₃

yes

characteristic

107

#95

—

108

#95

—

109

C₂⁴

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂²×D₄₃

yes

—

120

C₂²×C₄

C₂²×D₄₃

yes

—

121

C₂²×C₄

C₂²×D₄₃

yes

characteristic

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₄²

C₂²⋊D₄₃

yes

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

C₂³×D₄₃

yes

characteristic

135

Q₈

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

D₄×D₄₃

yes

—

149

C₂×C₄

#10621900

yes

—

150

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

151

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

172

—

155

C₂²

—

172

—

156

C₂²

C₂×D₄×D₄₃

yes

characteristic

157

C₂²

C₂²×C₂²⋊D₄₃

yes

characteristic

158

C₂²

C₂×#10621900

yes

characteristic

159

C₂²

#13063373

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

172

—

164

C₄

—

172

—

165

C₄

#13063296

yes

—

166

C₂

—

172

—

167

C₂

C₂×#13063296

yes

characteristic

168

C₂

#14583473

yes

—

169

C₂

#14583474

yes

—

170

C₂

#14583510

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₄₃	C₂
hypercenter	C₂×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	#2621
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator, socle

C₂³×C₄₃

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₄₃

C₂

hypercenter

C₂×C₄⋊Q₈

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #14582830	C₂	semidirect
5. #14582806	C₂	semidirect
6. #14582765	C₂	semidirect
7. C₂×C₄⋊Q₈×C₄₃	C₂	semidirect
86. C₂×C₄⋊Q₈	D₄₃	semidirect
93. C₄₃	#2621	semidirect

H ≅ G/N

type

3. #14582830

C₂

semidirect

5. #14582806

C₂

semidirect

6. #14582765

C₂

semidirect

7. C₂×C₄⋊Q₈×C₄₃

C₂

semidirect

86. C₂×C₄⋊Q₈

D₄₃

semidirect

93. C₄₃

#2621

semidirect

Subgroups — #19140362

Conjugacy classes of subgroups

Complements