Group #18069231 — subgroups

Subgroups — #18069231

Order 4864.

204 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 45 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	#14564958	C₂	1	1	yes	maximal, characteristic
2	#14561505	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1322	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₇₆	C₄	1	1	yes	characteristic
5	#11926601	—	2	1	no	—
6	C₂×#10597273	—	2	1	no	—
7	#11926450	—	2	1	no	—
8	#11926307	C₄	1	1	yes	characteristic
9	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
11	C₂²×Dic₃₈	—	2	1	no	—
12	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₉	—	2	1	no	—
16	C₂×Dic₇₆	—	4	1	no	—
17	C₂×Dic₇₆	—	4	2	no	—
18	C₂×C₁₆×C₁₉	—	4	1	no	—
19	C₁₉×C₄⋊C₈	—	4	1	no	—
20	C₁₉⋊#95	—	4	1	no	—
21	C₁₉⋊#95	—	4	1	no	—
22	C₂²⋊(C₁₉⋊C₈)	—	4	1	no	—
23	#10597273	—	4	2	no	—
24	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
25	#10597260	—	2	2	no	—
26	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
28	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
29	C₂²×Dic₁₉	—	4	1	no	—
30	C₂²×Dic₁₉	—	4	1	no	—
31	C₂×Dic₃₈	—	2	2	no	—
32	C₂×Dic₃₈	—	4	1	no	—
33	C₂×Dic₃₈	—	4	1	no	—
34	C₂×Dic₃₈	—	4	1	no	—
35	C₂×Dic₃₈	—	4	1	no	—
36	C₂×Dic₃₈	—	4	1	no	—
37	C₂×C₈×C₁₉	—	2	1	no	—
38	C₂×C₈×C₁₉	—	2	1	no	—
39	C₂×C₈×C₁₉	—	4	1	no	—
40	C₂×C₈×C₁₉	—	4	1	no	—
41	C₄²×C₁₉	—	2	1	no	—
42	C₄²×C₁₉	—	2	1	no	—
43	C₄⋊Dic₁₉	—	4	1	no	—
44	C₄⋊Dic₁₉	—	4	1	no	—
45	C₂×C₁₉⋊C₈	—	4	2	no	—
46	Dic₇₆	—	4	2	no	—
47	Dic₇₆	—	4	2	no	—
48	C₁₆×C₁₉	—	8	1	no	—
49	#7082	—	19	1	no	maximal, Sylow
50	C₂³×C₁₉	#102	1	1	yes	characteristic
51	C₂×C₄×C₁₉	—	2	1	no	—
52	C₂×C₄×C₁₉	—	2	1	no	—
53	C₂×C₄×C₁₉	—	2	1	no	—
54	C₂×C₄×C₁₉	—	2	1	no	—
55	C₂×C₄×C₁₉	—	2	1	no	—
56	C₂×C₄×C₁₉	—	2	1	no	—
57	C₂×C₄×C₁₉	—	4	1	no	—
58	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
59	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
60	C₂×Dic₁₉	—	4	2	no	—
61	C₂×Dic₁₉	—	8	1	no	—
62	C₂×Dic₁₉	—	8	1	no	—
63	C₂×Dic₁₉	—	8	1	no	—
64	C₂×Dic₁₉	—	8	1	no	—
65	C₂×Dic₁₉	—	8	1	no	—
66	Dic₃₈	—	4	2	no	—
67	Dic₃₈	—	4	2	no	—
68	Dic₃₈	—	8	1	no	—
69	Dic₃₈	—	8	1	no	—
70	C₈×C₁₉	—	4	1	no	—
71	C₈×C₁₉	—	4	1	no	—
72	C₁₉⋊C₈	—	8	2	no	—
73	#2056	—	19	1	no	—
74	#1322	D₁₉	1	1	yes	characteristic
75	#1290	—	19	1	no	—
76	C₂²×C₁₉	—	2	1	no	—
77	C₂²×C₁₉	—	2	1	no	—
78	C₂²×C₁₉	#327	1	1	yes	characteristic
79	C₂²×C₁₉	#359	1	1	yes	characteristic
80	C₂²×C₁₉	#362	1	1	yes	characteristic
81	C₄×C₁₉	—	2	1	no	—
82	C₄×C₁₉	—	4	1	no	—
83	C₄×C₁₉	—	4	1	no	—
84	C₄×C₁₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
85	C₄×C₁₉	#355	1	1	yes	characteristic
86	Dic₁₉	—	8	2	no	—
87	Dic₁₉	—	16	1	no	—
88	C₂²×Q₁₆	—	19	1	no	—
89	#432	—	38	1	no	—
90	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
91	C₂×#103	—	38	1	no	—
92	#391	—	38	1	no	—
93	C₂²⋊C₁₆	—	2	1	no	—
94	#340	—	19	1	no	—
95	C₂×C₁₉	—	2	1	no	—
96	C₂×C₁₉	—	2	1	no	—
97	C₂×C₁₉	#1340	1	1	yes	characteristic
98	C₂×C₁₉	#1345	1	1	yes	characteristic
99	C₂×C₁₉	#1350	1	1	yes	characteristic
100	C₂²×Q₈	—	38	1	no	—
101	C₂×Q₁₆	—	76	1	no	—
102	C₂×Q₁₆	—	76	2	no	—
103	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
104	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	38	1	no	—
106	C₂×C₄²	D₇₆	1	1	yes	characteristic
107	C₂×C₁₆	—	4	1	no	—
108	C₄⋊C₈	—	4	1	no	—
109	C₄⋊C₈	—	76	1	no	—
110	#103	—	76	2	no	—
111	C₂²⋊C₈	—	76	1	no	—
112	#97	—	38	2	no	—
113	#95	—	76	1	no	—
114	#95	—	76	1	no	—
115	C₁₉	#7082	1	1	yes	characteristic, complemented, Sylow
116	C₂×Q₈	—	38	2	no	—
117	C₂×Q₈	—	76	1	no	—
118	C₂×Q₈	—	76	1	no	—
119	C₂×Q₈	—	76	1	no	—
120	C₂×Q₈	—	76	1	no	—
121	C₂×Q₈	—	76	1	no	—
122	C₂²×C₄	—	76	1	no	—
123	C₂²×C₄	—	76	1	no	—
124	C₂²×C₄	#64516	1	1	yes	characteristic
125	C₂²×C₄	D₁₅₂	1	1	yes	characteristic
126	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
127	Q₁₆	—	76	2	no	—
128	Q₁₆	—	76	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	76	2	no	—
134	C₄⋊C₄	—	76	1	no	—
135	C₄⋊C₄	—	76	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	2	1	no	—
138	C₁₆	—	8	1	no	—
139	C₂³	#10597278	1	1	yes	characteristic
140	Q₈	—	76	2	no	—
141	Q₈	—	76	2	no	—
142	Q₈	—	152	1	no	—
143	Q₈	—	152	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	76	2	no	—
152	C₂×C₄	—	152	1	no	—
153	C₂×C₄	—	152	1	no	—
154	C₂×C₄	—	152	1	no	—
155	C₂×C₄	—	152	1	no	—
156	C₂×C₄	—	152	1	no	—
157	C₂×C₄	#10597263	1	1	yes	characteristic
158	C₂×C₄	#10597282	1	1	yes	characteristic
159	C₈	—	4	1	no	—
160	C₈	—	4	1	no	—
161	C₈	—	152	2	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#11926279	1	1	yes	characteristic
165	C₂²	#11926352	1	1	yes	characteristic
166	C₂²	#11926361	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	304	1	no	—
171	C₄	—	152	2	no	—
172	C₄	#11926330	1	1	yes	characteristic
173	C₄	#11926340	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	#14561624	1	1	yes	characteristic
177	C₂	#14561639	1	1	yes	characteristic
178	C₂	#14561654	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564958

C₂

yes

maximal, characteristic

#14561505

C₂

yes

maximal, characteristic

C₁₉×#1322

C₂

yes

maximal, characteristic

C₂²×Dic₇₆

C₄

yes

—

—

—

C₄

yes

characteristic

C₂×C₁₉×C₄⋊C₈

C₂²

yes

characteristic

C₁₉×C₂²⋊C₁₆

—

C₂²×Dic₃₈

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂×Dic₇₆

—

C₂×Dic₇₆

—

C₂×C₁₆×C₁₉

—

C₁₉×C₄⋊C₈

—

C₁₉⋊#95

—

C₁₉⋊#95

—

C₂²⋊(C₁₉⋊C₈)

—

#10597273

—

C₄⋊(C₁₉⋊C₈)

—

#10597260

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

D₈

yes

characteristic

C₂²×C₄×C₁₉

SD₁₆

yes

characteristic

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₂×C₁₉⋊C₈

—

Dic₇₆

—

Dic₇₆

—

C₁₆×C₁₉

—

—

C₂³×C₁₉

yes

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#2056

—

#1322

D₁₉

yes

characteristic

#1290

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

AΣL(1,𝔽₁₆)

yes

characteristic

C₄×C₁₉

#355

yes

characteristic

Dic₁₉

—

Dic₁₉

—

C₂²×Q₁₆

—

#432

—

C₂×C₄⋊C₈

C₂×D₁₉

yes

characteristic

C₂×#103

—

#391

—

C₂²⋊C₁₆

—

#340

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

100

C₂²×Q₈

—

101

C₂×Q₁₆

—

102

C₂×Q₁₆

—

103

C₂²×C₈

C₄×D₁₉

yes

characteristic

104

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

D₇₆

yes

characteristic

107

C₂×C₁₆

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

#103

—

111

C₂²⋊C₈

—

112

#97

—

113

#95

—

114

#95

—

115

C₁₉

#7082

yes

characteristic, complemented, Sylow

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

#64516

yes

characteristic

125

C₂²×C₄

D₁₅₂

yes

characteristic

126

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

127

Q₁₆

—

128

Q₁₆

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₁₆

—

139

C₂³

#10597278

yes

characteristic

140

Q₈

—

141

Q₈

—

142

Q₈

—

152

—

143

Q₈

—

152

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

152

—

153

C₂×C₄

—

152

—

154

C₂×C₄

—

152

—

155

C₂×C₄

—

152

—

156

C₂×C₄

—

152

—

157

C₂×C₄

#10597263

yes

characteristic

158

C₂×C₄

#10597282

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

152

—

162

C₂²

—

163

C₂²

—

164

C₂²

#11926279

yes

characteristic

165

C₂²

#11926352

yes

characteristic

166

C₂²

#11926361

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

304

—

171

C₄

—

152

—

172

C₄

#11926330

yes

characteristic

173

C₄

#11926340

yes

characteristic

174

C₂

—

175

C₂

—

176

C₂

#14561624

yes

characteristic

177

C₂

#14561639

yes

characteristic

178

C₂

#14561654

yes

characteristic

	subgroup	G/H
center	C₂²	#11926279
commutator	C₂²×C₈×C₁₉	C₂×C₄
Frattini	C₂×C₄⋊C₈	C₂×D₁₉
Fitting	C₁₉×#1322	C₂
socle	C₂²×C₁₉	#327
prime radical	C₂³×C₁₉	#102
cosocle	C₂×C₁₉×C₄⋊C₈	C₂²
hypercenter	#1322	D₁₉
hypocenter	C₁₉	#7082
LCS	G ▷ C₂²×C₈×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1322

subgroup

G/H

center

C₂²

#11926279

commutator

C₂²×C₈×C₁₉

C₂×C₄

Frattini

C₂×C₄⋊C₈

C₂×D₁₉

Fitting

C₁₉×#1322

C₂

C₂²×C₁₉

C₂³×C₁₉

C₂×C₁₉×C₄⋊C₈

C₂²

D₁₉

C₁₉

G ▷ C₂²×C₈×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1322

N	H ≅ G/N	type
115. C₁₉	#7082	semidirect

H ≅ G/N

type

115. C₁₉

#7082

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14564958	2
#14561505	2
C₁₉×#1322	2
#7082	19

subgroup

index

#14564958

#14561505

C₁₉×#1322

#7082

Subgroups — #18069231

Conjugacy classes of subgroups

Complements

Maximal subgroups