Group #18069919 — subgroups

Subgroups — #18069919

Order 4864.

204 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 43 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	#14564454	C₂	1	1	yes	maximal, characteristic
2	#14561579	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1324	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597315	C₄	1	1	yes	characteristic
5	C₂×#10597267	—	2	1	no	—
6	C₂×#10597267	—	2	1	no	—
7	C₂×#10597260	C₄	1	1	yes	characteristic
8	#11926270	—	2	1	no	—
9	C₁₉×#382	C₂²	1	1	yes	characteristic
10	C₁₉×#324	—	2	1	no	—
11	C₂²×Dic₃₈	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₉	—	4	1	no	—
16	C₂²×C₁₉⋊C₈	—	2	1	no	—
17	#10597315	—	2	1	no	—
18	#10597315	—	4	1	no	—
19	#10597315	C₈	1	2	yes	complemented
20	C₁₉×C₂²⋊C₈	—	4	1	no	—
21	C₁₉×#95	—	4	1	no	—
22	C₂²⋊(C₁₉⋊C₈)	—	4	1	no	—
23	#10597267	—	4	2	no	—
24	#10597267	—	4	2	no	—
25	#10597260	—	4	1	no	—
26	C₂²×C₄×C₁₉	—	2	1	no	—
27	C₂²×C₄×C₁₉	—	2	1	no	—
28	C₂²×C₄×C₁₉	—	2	1	no	—
29	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
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	—	8	1	no	—
38	C₄⋊C₄×C₁₉	—	4	1	no	—
39	C₄⋊C₄×C₁₉	—	4	1	no	—
40	C₄²×C₁₉	—	2	1	no	—
41	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₁₉	M₁₆	1	1	yes	characteristic
43	C₄⋊Dic₁₉	—	4	2	no	—
44	C₄⋊Dic₁₉	—	4	2	no	—
45	C₂×C₁₉⋊C₈	—	2	2	no	—
46	C₂×C₁₉⋊C₈	—	4	1	no	—
47	C₂×C₁₉⋊C₈	—	4	1	no	—
48	C₂×C₁₉⋊C₈	—	8	1	no	—
49	#7353	—	19	1	no	maximal, Sylow
50	C₂³×C₁₉	#99	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	—	4	1	no	—
56	C₂×C₄×C₁₉	—	4	1	no	—
57	C₂×C₄×C₁₉	—	4	1	no	—
58	C₂×C₄×C₁₉	—	4	1	no	—
59	C₂×C₄×C₁₉	—	8	1	no	—
60	C₂×C₄×C₁₉	—	8	1	no	—
61	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
63	C₂×Dic₁₉	—	4	2	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	—	16	1	no	—
71	C₁₉⋊C₈	—	4	2	no	—
72	C₁₉⋊C₈	—	16	1	no	—
73	#1960	—	19	1	no	—
74	#1324	—	19	1	no	—
75	#1324	D₁₉	1	1	yes	characteristic
76	C₂²×C₁₉	—	2	1	no	—
77	C₂²×C₁₉	—	2	1	no	—
78	C₂²×C₁₉	#323	1	1	yes	characteristic
79	C₂²×C₁₉	#353	1	1	yes	characteristic
80	C₂²×C₁₉	#355	1	1	yes	characteristic
81	C₄×C₁₉	—	2	1	no	—
82	C₄×C₁₉	—	2	1	no	—
83	C₄×C₁₉	—	4	1	no	—
84	C₄×C₁₉	—	4	1	no	—
85	C₄×C₁₉	—	8	1	no	—
86	C₄×C₁₉	—	8	1	no	—
87	Dic₁₉	—	8	2	no	—
88	C₂×C₄⋊Q₈	—	19	1	no	—
89	C₂×#103	—	38	1	no	—
90	C₂×#103	—	38	1	no	—
91	C₂×#97	—	19	1	no	—
92	#382	C₂×D₁₉	1	1	yes	characteristic
93	#324	—	2	1	no	—
94	#324	—	38	1	no	—
95	C₂×C₁₉	—	2	1	no	—
96	C₂×C₁₉	—	2	1	no	—
97	C₂×C₁₉	#1318	1	1	yes	characteristic
98	C₂×C₁₉	#1407	1	1	yes	characteristic
99	C₂×C₁₉	#1407	1	1	yes	characteristic
100	C₂²×Q₈	—	38	1	no	—
101	C₂²×C₈	—	38	1	no	—
102	C₄⋊Q₈	—	19	2	no	—
103	C₄⋊Q₈	—	38	1	no	—
104	C₄⋊Q₈	—	76	1	no	—
105	C₂×C₄⋊C₄	—	76	1	no	—
106	C₂×C₄⋊C₄	D₇₆	1	1	yes	characteristic
107	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
108	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
109	#103	—	76	2	no	—
110	#103	—	76	2	no	—
111	C₂²⋊C₈	—	4	1	no	—
112	C₂²⋊C₈	—	76	1	no	—
113	#97	—	76	1	no	—
114	#95	—	4	1	no	—
115	C₁₉	#7353	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	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	76	1	no	—
126	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
127	C₂×C₈	—	8	1	no	—
128	C₂×C₈	—	38	2	no	—
129	C₂×C₈	—	76	1	no	—
130	C₂×C₈	—	76	1	no	—
131	C₂×C₈	—	152	1	no	—
132	C₄⋊C₄	—	4	1	no	—
133	C₄⋊C₄	—	4	1	no	—
134	C₄⋊C₄	—	76	2	no	—
135	C₄⋊C₄	—	76	2	no	—
136	C₄²	—	2	1	no	—
137	C₄²	C₈×D₁₉	1	1	yes	characteristic
138	C₄²	#64515	1	1	yes	characteristic
139	C₂³	#10597263	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	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	8	1	no	—
154	C₂×C₄	—	76	2	no	—
155	C₂×C₄	—	152	1	no	—
156	C₂×C₄	—	152	1	no	—
157	C₂×C₄	(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
158	C₂×C₄	#10597280	1	1	yes	characteristic
159	C₈	—	16	1	no	—
160	C₈	—	304	1	no	—
161	C₈	—	76	2	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#11926267	1	1	yes	characteristic
165	C₂²	#11926336	1	1	yes	characteristic
166	C₂²	#11926340	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	4	1	no	—
171	C₄	—	8	1	no	—
172	C₄	—	8	1	no	—
173	C₄	—	152	2	no	—
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	#14561566	1	1	yes	characteristic
177	C₂	#14561794	1	1	yes	characteristic
178	C₂	#14561796	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564454

C₂

yes

maximal, characteristic

#14561579

C₂

yes

maximal, characteristic

C₁₉×#1324

C₂

yes

maximal, characteristic

C₂×#10597315

C₄

yes

—

—

C₄

yes

characteristic

#11926270

—

C₁₉×#382

C₂²

yes

characteristic

C₁₉×#324

—

C₂²×Dic₃₈

—

C₂×C₄⋊C₄×C₁₉

D₄

yes

characteristic

C₂×C₄⋊C₄×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂²×C₁₉⋊C₈

—

#10597315

—

#10597315

—

#10597315

C₈

yes

complemented

C₁₉×C₂²⋊C₈

—

C₁₉×#95

—

C₂²⋊(C₁₉⋊C₈)

—

#10597267

—

#10597267

—

#10597260

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×Dic₁₉

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂×C₈

yes

characteristic

C₄²×C₁₉

M₁₆

yes

characteristic

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

—

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

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₉

#100

yes

characteristic

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#1960

—

#1324

—

#1324

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

Dic₁₉

—

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×#103

—

C₂×#97

—

#382

C₂×D₁₉

yes

characteristic

#324

—

#324

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

100

C₂²×Q₈

—

101

C₂²×C₈

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

D₇₆

yes

characteristic

107

C₂×C₄⋊C₄

C₂²⋊D₁₉

yes

characteristic

108

C₂×C₄²

C₄×D₁₉

yes

characteristic

109

#103

—

110

#103

—

111

C₂²⋊C₈

—

112

C₂²⋊C₈

—

113

#97

—

114

#95

—

115

C₁₉

#7353

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

—

125

C₂²×C₄

—

126

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

152

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

—

137

C₄²

C₈×D₁₉

yes

characteristic

138

C₄²

#64515

yes

characteristic

139

C₂³

#10597263

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

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

152

—

156

C₂×C₄

—

152

—

157

C₂×C₄

(C₂×D₁₉)⋊C₈

yes

characteristic

158

C₂×C₄

#10597280

yes

characteristic

159

C₈

—

160

C₈

—

304

—

161

C₈

—

162

C₂²

—

163

C₂²

—

164

C₂²

#11926267

yes

characteristic

165

C₂²

#11926336

yes

characteristic

166

C₂²

#11926340

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

152

—

174

C₂

—

175

C₂

—

176

C₂

#14561566

yes

characteristic

177

C₂

#14561794

yes

characteristic

178

C₂

#14561796

yes

characteristic

	subgroup	G/H
center	C₂²	#11926336
commutator	C₄²×C₁₉	C₂×C₈
Frattini	#382	C₂×D₁₉
Fitting	C₁₉×#1324	C₂
socle	C₂²×C₁₉	#353
prime radical	C₂³×C₁₉	#99
cosocle	C₁₉×#382	C₂²
hypercenter	#1324	D₁₉
hypocenter	C₁₉	#7353
LCS	G ▷ C₄²×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #382 ◁ #1324

subgroup

G/H

center

C₂²

#11926336

commutator

C₄²×C₁₉

C₂×C₈

Frattini

#382

C₂×D₁₉

Fitting

C₁₉×#1324

C₂

C₂²×C₁₉

C₂³×C₁₉

C₁₉×#382

C₂²

D₁₉

C₁₉

G ▷ C₄²×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #382 ◁ #1324

N	H ≅ G/N	type
19. #10597315	C₈	semidirect
115. C₁₉	#7353	semidirect

H ≅ G/N

type

19. #10597315

C₈

semidirect

115. C₁₉

#7353

semidirect

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

subgroup	index
#14564454	2
#14561579	2
C₁₉×#1324	2
#7353	19

subgroup

index

#14564454

#14561579

C₁₉×#1324

#7353

Subgroups — #18069919

Conjugacy classes of subgroups

Complements

Maximal subgroups