Group #19134611 — subgroups

Subgroups — #19134611

Order 5504.

292 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063250	C₂	1	2	yes	maximal
3	#14582765	C₂	1	1	yes	maximal, characteristic
4	#14582662	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
7	C₄×Dic₈₆	—	2	2	no	—
8	C₄×Dic₈₆	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₄₃⋊C₄	—	2	1	no	—
12	C₂×C₄×Dic₄₃	—	2	1	no	—
13	#13063250	—	2	4	no	—
14	C₄₃⋊#95	—	2	1	no	—
15	C₄₃⋊#95	—	2	1	no	—
16	C₂²×C₈×C₄₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₄₃	—	2	2	no	—
19	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₄₃	—	2	1	no	—
23	C₂²×Dic₄₃	—	2	1	no	—
24	C₂²×Dic₄₃	—	2	2	no	—
25	C₂×Dic₈₆	—	2	1	no	—
26	C₂×Dic₈₆	—	2	1	no	—
27	C₂×Dic₈₆	—	2	1	no	—
28	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
29	C₂×Dic₈₆	D₄	1	2	yes	—
30	C₂×Dic₈₆	Q₈	1	2	yes	—
31	C₂×C₈×C₄₃	—	2	2	no	—
32	C₂×C₈×C₄₃	—	2	2	no	—
33	C₂×C₈×C₄₃	—	2	2	no	—
34	C₄²×C₄₃	C₂×C₄	1	2	yes	—
35	C₄²×C₄₃	D₄	1	2	yes	—
36	C₄⋊Dic₄₃	—	2	1	no	—
37	C₄⋊Dic₄₃	—	2	2	no	—
38	C₄⋊Dic₄₃	—	2	2	no	—
39	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
40	Dic₄₃⋊C₄	—	2	4	no	—
41	C₄×Dic₄₃	—	2	4	no	—
42	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₄₃	—	2	2	no	—
44	C₂×C₄×C₄₃	—	2	2	no	—
45	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
52	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
53	C₂×Dic₄₃	—	2	2	no	—
54	C₂×Dic₄₃	—	2	2	no	—
55	C₂×Dic₄₃	—	2	2	no	—
56	C₂×Dic₄₃	—	2	2	no	—
57	C₂×Dic₄₃	—	4	2	no	—
58	C₂×Dic₄₃	—	4	2	no	—
59	C₂×Dic₄₃	—	4	2	no	—
60	C₂×Dic₄₃	—	4	2	no	—
61	Dic₈₆	—	2	2	no	—
62	Dic₈₆	—	2	2	no	—
63	Dic₈₆	—	2	2	no	—
64	Dic₈₆	C₄⋊C₄	1	4	yes	—
65	C₈×C₄₃	—	4	2	no	—
66	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄₃	#121	1	1	yes	characteristic
68	C₂²×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂²×C₄₃	#135	1	2	yes	—
70	C₂²×C₄₃	#137	1	2	yes	—
71	C₄×C₄₃	—	2	2	no	—
72	C₄×C₄₃	—	2	2	no	—
73	C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₄×C₄₃	#117	1	1	yes	characteristic
75	C₄×C₄₃	#121	1	1	yes	characteristic
76	C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
77	Dic₄₃	—	2	4	no	—
78	Dic₄₃	—	4	2	no	—
79	Dic₄₃	—	8	2	no	—
80	#1805	—	43	1	no	maximal, Sylow
81	C₂×C₄₃	#380	1	1	yes	characteristic
82	C₂×C₄₃	#416	1	1	yes	characteristic
83	C₂×C₄₃	#419	1	1	yes	characteristic
84	C₂×C₄₃	#464	1	2	yes	—
85	C₂×C₄₃	#479	1	2	yes	—
86	C₂×C₄×Q₈	—	43	1	no	—
87	C₂×C₄⋊C₈	D₄₃	1	1	yes	characteristic
88	C₂×#103	—	43	2	no	—
89	#382	—	43	1	no	—
90	#340	—	43	2	no	—
91	C₄₃	#1805	1	1	yes	characteristic, complemented, Sylow
92	C₂²×Q₈	—	43	1	no	—
93	C₂²×C₈	C₂×D₄₃	1	2	yes	—
94	C₄×Q₈	—	43	4	no	—
95	C₄×Q₈	—	86	2	no	—
96	C₂×C₄⋊C₄	—	43	1	no	—
97	C₂×C₄⋊C₄	—	43	2	no	—
98	C₂×C₄⋊C₄	—	86	1	no	—
99	C₂×C₄²	—	86	1	no	—
100	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
101	C₄⋊C₈	—	2	2	no	—
102	#103	—	86	4	no	—
103	#95	—	86	1	no	—
104	#95	—	86	1	no	—
105	C₂×Q₈	—	43	2	no	—
106	C₂×Q₈	—	43	2	no	—
107	C₂×Q₈	—	43	2	no	—
108	C₂×Q₈	—	86	1	no	—
109	C₂×Q₈	—	86	1	no	—
110	C₂×Q₈	—	86	1	no	—
111	C₂²×C₄	—	86	1	no	—
112	C₂²×C₄	—	86	1	no	—
113	C₂²×C₄	—	86	2	no	—
114	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
115	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₄⋊C₄	—	43	2	no	—
121	C₄⋊C₄	—	86	4	no	—
122	C₄⋊C₄	—	86	1	no	—
123	C₄⋊C₄	—	86	2	no	—
124	C₄⋊C₄	—	86	2	no	—
125	C₄²	—	86	4	no	—
126	C₄²	C₄×D₄₃	1	2	yes	—
127	C₄²	C₂²⋊D₄₃	1	2	yes	—
128	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
129	Q₈	—	43	4	no	—
130	Q₈	—	86	2	no	—
131	Q₈	—	86	2	no	—
132	Q₈	—	86	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	86	2	no	—
136	C₂×C₄	—	86	2	no	—
137	C₂×C₄	—	86	2	no	—
138	C₂×C₄	—	86	2	no	—
139	C₂×C₄	—	172	2	no	—
140	C₂×C₄	—	172	2	no	—
141	C₂×C₄	—	172	2	no	—
142	C₂×C₄	—	172	2	no	—
143	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
144	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
145	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
146	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
147	C₂×C₄	#10621900	1	1	yes	characteristic
148	C₂×C₄	Q₈×D₄₃	1	1	yes	characteristic
149	C₂×C₄	#10621902	1	1	yes	characteristic
150	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
151	C₈	—	4	2	no	—
152	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
153	C₂²	#13063300	1	1	yes	characteristic
154	C₂²	#13063305	1	1	yes	characteristic
155	C₂²	#13063347	1	2	yes	—
156	C₂²	#13063358	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	86	4	no	—
160	C₄	—	344	2	no	—
161	C₄	—	172	2	no	—
162	C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
163	C₄	#13063283	1	1	yes	characteristic
164	C₄	#13063296	1	1	yes	characteristic
165	C₄	#13063301	1	1	yes	characteristic
166	C₂	#14582756	1	1	yes	characteristic
167	C₂	#14582896	1	1	yes	characteristic
168	C₂	#14582909	1	1	yes	characteristic
169	C₂	#14583122	1	2	yes	—
170	C₂	#14583207	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₈₆

C₂

yes

maximal, characteristic

C₂×#13063250

C₂

yes

maximal

#14582765

C₂

yes

maximal, characteristic

#14582662

C₂

yes

maximal

C₂×C₄⋊C₈×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

characteristic

C₄×Dic₈₆

—

C₄×Dic₈₆

C₄

yes

complemented

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×Dic₄₃⋊C₄

—

C₂×C₄×Dic₄₃

—

#13063250

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₂×Dic₈₆

D₄

yes

—

C₂×Dic₈₆

Q₈

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

D₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

C₄⋊C₄

yes

—

C₈×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

#121

yes

characteristic

C₂²×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₃

#135

yes

—

C₂²×C₄₃

#137

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×C₄⋊C₄

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#479

yes

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

D₄₃

yes

—

—

—

C₄₃

#1805

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

C₂×D₄₃

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

100

C₂×C₄²

C₂×D₄₃

yes

characteristic

101

C₄⋊C₈

—

102

#103

—

103

#95

—

104

#95

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

D₁₇₂

yes

characteristic

115

C₂²×C₄

D₁₇₂

yes

characteristic

116

C₂²×C₄

C₂²×D₄₃

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄²

—

126

C₄²

C₄×D₄₃

yes

—

127

C₄²

C₂²⋊D₄₃

yes

—

128

C₂³

C₂×D₁₇₂

yes

characteristic

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

172

—

140

C₂×C₄

—

172

—

141

C₂×C₄

—

172

—

142

C₂×C₄

—

172

—

143

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

144

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

145

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

146

C₂×C₄

D₄×D₄₃

yes

characteristic

147

C₂×C₄

#10621900

yes

characteristic

148

C₂×C₄

Q₈×D₄₃

yes

characteristic

149

C₂×C₄

#10621902

yes

characteristic

150

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

151

C₈

—

152

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

153

C₂²

#13063300

yes

characteristic

154

C₂²

#13063305

yes

characteristic

155

C₂²

#13063347

yes

—

156

C₂²

#13063358

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

344

—

161

C₄

—

172

—

162

C₄

C₄⋊C₄×D₄₃

yes

characteristic

163

C₄

#13063283

yes

characteristic

164

C₄

#13063296

yes

characteristic

165

C₄

#13063301

yes

characteristic

166

C₂

#14582756

yes

characteristic

167

C₂

#14582896

yes

characteristic

168

C₂

#14582909

yes

characteristic

169

C₂

#14583122

yes

—

170

C₂

#14583207

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×C₄⋊C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#1805
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×C₄⋊C₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₄₃

hypocenter

C₄₃

#1805

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
8. C₄×Dic₈₆	C₄	semidirect
91. C₄₃	#1805	semidirect

H ≅ G/N

type

8. C₄×Dic₈₆

C₄

semidirect

91. C₄₃

#1805

semidirect

Subgroups — #19134611

Conjugacy classes of subgroups

Complements