Group #18073696 — subgroups

Subgroups — #18073696

Order 4864.

414 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 175 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	C₄²×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#14562381	C₂	1	2	yes	maximal
3	#14561919	C₂	1	2	yes	maximal
4	#14561468	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1276	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	2	yes	—
7	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	—
9	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
11	#11926296	C₄	1	4	yes	complemented
12	#11926294	C₄	1	4	yes	complemented
13	#11926266	—	2	2	no	—
14	C₈×C₁₉⋊C₈	—	2	2	no	—
15	C₁₉×#431	C₂²	1	2	yes	—
16	C₄³×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₁₉	—	2	2	no	—
18	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
20	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
23	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
24	C₁₉×C₂²⋊C₈	—	2	2	no	—
25	C₄×C₈×C₁₉	—	2	2	no	—
26	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
27	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
28	C₄×C₁₉⋊C₈	—	2	2	no	—
29	C₄×C₁₉⋊C₈	—	2	4	no	—
30	C₄×C₁₉⋊C₈	C₈	1	8	yes	complemented
31	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
32	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
33	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
34	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
35	C₄×C₁₉⋊C₈	Q₈	1	2	yes	—
36	C₂²×C₄×C₁₉	—	2	2	no	—
37	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	#41	1	2	yes	—
39	C₂×C₈×C₁₉	—	2	4	no	—
40	C₄²×C₁₉	—	2	8	no	—
41	C₄²×C₁₉	—	2	2	no	—
42	C₄²×C₁₉	C₂×C₈	1	4	yes	—
43	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
44	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₁₉⋊C₈	—	2	2	no	—
47	C₂×C₁₉⋊C₈	—	2	4	no	—
48	C₂×C₁₉⋊C₈	C₄²	1	2	yes	—
49	C₂×C₁₉⋊C₈	C₄²	1	1	yes	characteristic
50	C₂×C₁₉⋊C₈	C₄²	1	1	yes	characteristic
51	C₂×C₁₉⋊C₈	C₄⋊C₄	1	2	yes	—
52	C₂×C₁₉⋊C₈	C₄⋊C₄	1	2	yes	—
53	C₂×C₁₉⋊C₈	C₂×C₈	1	4	yes	—
54	C₂×C₁₉⋊C₈	C₂×C₈	1	4	yes	—
55	C₂×C₁₉⋊C₈	M₁₆	1	4	yes	—
56	#8266	—	19	1	no	maximal, Sylow
57	C₂³×C₁₉	#117	1	1	yes	characteristic
58	C₂×C₄×C₁₉	—	2	8	no	—
59	C₂×C₄×C₁₉	—	2	4	no	—
60	C₂×C₄×C₁₉	—	2	4	no	—
61	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
64	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
65	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
66	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
67	C₈×C₁₉	—	4	4	no	—
68	C₁₉⋊C₈	—	2	2	no	—
69	C₁₉⋊C₈	—	4	4	no	—
70	C₁₉⋊C₈	C₄×C₈	1	2	yes	—
71	C₁₉⋊C₈	C₄×C₈	1	2	yes	—
72	C₁₉⋊C₈	C₄⋊C₈	1	4	yes	—
73	C₁₉⋊C₈	C₄⋊C₈	1	4	yes	—
74	C₄²×C₈	—	19	1	no	—
75	#1564	—	19	2	no	—
76	#1450	—	19	2	no	—
77	#1276	—	19	1	no	—
78	#1276	D₁₉	1	1	yes	characteristic
79	C₂²×C₁₉	—	2	2	no	—
80	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₁₉	#431	1	2	yes	—
82	C₄×C₁₉	—	2	8	no	—
83	C₄×C₁₉	—	2	2	no	—
84	C₄×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
85	C₄×C₁₉	C₄×M₁₆	1	1	yes	characteristic
86	C₄×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
87	C₄×C₁₉	#443	1	4	yes	—
88	#431	—	19	2	no	—
89	#431	C₂×D₁₉	1	2	yes	—
90	C₂×C₄×C₈	—	19	2	no	—
91	C₂×C₄×C₈	—	19	1	no	—
92	C₂×C₄×C₈	—	19	1	no	—
93	C₂×C₄×C₈	—	19	2	no	—
94	C₄³	C₂×D₁₉	1	1	yes	characteristic
95	#335	—	19	4	no	—
96	#334	—	19	4	no	—
97	#322	—	38	2	no	—
98	C₈²	—	38	2	no	—
99	C₂×C₁₉	—	2	2	no	—
100	C₂×C₁₉	C₄×C₄⋊C₈	1	1	yes	characteristic
101	C₂×C₁₉	#1772	1	2	yes	—
102	C₂²×C₈	—	19	1	no	—
103	C₂²×C₈	—	19	2	no	—
104	C₂²×C₈	—	19	1	no	—
105	C₂×C₄²	—	2	2	no	—
106	C₂×C₄²	C₄×D₁₉	1	2	yes	—
107	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
108	C₄⋊C₈	—	19	4	no	—
109	C₄⋊C₈	C₄×D₁₉	1	4	yes	—
110	C₂²⋊C₈	—	2	2	no	—
111	C₂²⋊C₈	—	38	2	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	—	19	8	no	—
114	C₄×C₈	—	19	2	no	—
115	C₄×C₈	—	19	2	no	—
116	C₄×C₈	—	19	1	no	—
117	C₄×C₈	—	19	1	no	—
118	C₄×C₈	—	19	2	no	—
119	C₄×C₈	—	38	4	no	—
120	C₄×C₈	—	38	2	no	—
121	C₁₉	#8266	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
124	C₂²×C₄	#64543	1	2	yes	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	19	1	no	—
127	C₂×C₈	—	19	1	no	—
128	C₂×C₈	—	19	2	no	—
129	C₂×C₈	—	19	2	no	—
130	C₂×C₈	—	19	4	no	—
131	C₂×C₈	—	19	4	no	—
132	C₂×C₈	—	19	4	no	—
133	C₂×C₈	—	19	2	no	—
134	C₂×C₈	—	38	2	no	—
135	C₂×C₈	—	38	4	no	—
136	C₄²	—	2	8	no	—
137	C₄²	—	2	2	no	—
138	C₄²	C₈×D₁₉	1	4	yes	—
139	C₄²	C₂×C₄×D₁₉	1	2	yes	—
140	C₄²	D₄×D₁₉	1	1	yes	characteristic
141	C₄²	Q₈×D₁₉	1	1	yes	characteristic
142	C₂³	#10597323	1	1	yes	characteristic
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
147	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
148	C₂×C₄	#10597333	1	2	yes	—
149	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
150	C₂×C₄	C₂×C₈×D₁₉	1	2	yes	—
151	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
152	C₈	—	4	4	no	—
153	C₈	—	19	4	no	—
154	C₈	—	19	4	no	—
155	C₈	—	19	2	no	—
156	C₈	—	19	2	no	—
157	C₈	—	38	2	no	—
158	C₈	—	76	4	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#11926395	1	1	yes	characteristic
161	C₂²	#11926594	1	2	yes	—
162	C₄	—	2	8	no	—
163	C₄	—	2	2	no	—
164	C₄	C₄×C₈×D₁₉	1	1	yes	characteristic
165	C₄	#11926496	1	1	yes	characteristic
166	C₄	C₄⋊C₈×D₁₉	1	2	yes	—
167	C₄	#11926658	1	4	yes	—
168	C₂	—	2	2	no	—
169	C₂	#14563483	1	1	yes	characteristic
170	C₂	#14563513	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄²×C₁₉⋊C₈

C₂

yes

maximal, characteristic

#14562381

C₂

yes

maximal

#14561919

C₂

yes

maximal

#14561468

C₂

yes

maximal, characteristic

C₁₉×#1276

C₂

yes

maximal, characteristic

#11926597

C₂²

yes

—

C₂×C₄×C₁₉⋊C₈

C₄

yes

—

C₂×C₄×C₁₉⋊C₈

C₄

yes

—

C₂×C₄×C₁₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₁₉⋊C₈

C₂²

yes

characteristic

#11926296

C₄

yes

complemented

#11926294

C₄

yes

complemented

#11926266

—

C₈×C₁₉⋊C₈

—

C₁₉×#431

C₂²

yes

—

C₄³×C₁₉

C₂²

yes

characteristic

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

C₂×C₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂²×C₁₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₁₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₁₉⋊C₈

C₂×C₄

yes

—

C₁₉×C₄⋊C₈

C₂×C₄

yes

—

C₁₉×C₂²⋊C₈

—

C₄×C₈×C₁₉

—

C₂²⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

C₂×C₄

yes

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

C₈

yes

complemented

C₄×C₁₉⋊C₈

C₂×C₄

yes

—

C₄×C₁₉⋊C₈

C₂×C₄

yes

—

C₄×C₁₉⋊C₈

D₄

yes

characteristic

C₄×C₁₉⋊C₈

D₄

yes

characteristic

C₄×C₁₉⋊C₈

Q₈

yes

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂×C₈

yes

—

C₄²×C₁₉

C₂²×C₄

yes

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×Q₈

yes

characteristic

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

C₄²

yes

—

C₂×C₁₉⋊C₈

C₄²

yes

characteristic

C₂×C₁₉⋊C₈

C₄²

yes

characteristic

C₂×C₁₉⋊C₈

C₄⋊C₄

yes

—

C₂×C₁₉⋊C₈

C₄⋊C₄

yes

—

C₂×C₁₉⋊C₈

C₂×C₈

yes

—

C₂×C₁₉⋊C₈

C₂×C₈

yes

—

C₂×C₁₉⋊C₈

M₁₆

yes

—

—

C₂³×C₁₉

yes

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₄×D₄

yes

—

C₂×C₄×C₁₉

C₄×Q₈

yes

—

C₂×C₄×C₁₉

C₂²×C₈

yes

—

C₂×C₄×C₁₉

C₂×M₁₆

yes

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

C₄×C₈

yes

—

C₁₉⋊C₈

C₄×C₈

yes

—

C₁₉⋊C₈

C₄⋊C₈

yes

—

C₁₉⋊C₈

C₄⋊C₈

yes

—

C₄²×C₈

—

—

—

—

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₁₉

#431

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂×C₄×C₈

yes

characteristic

C₄×C₁₉

C₄×M₁₆

yes

characteristic

C₄×C₁₉

C₂×C₄⋊C₈

yes

—

C₄×C₁₉

#443

yes

—

#431

—

#431

C₂×D₁₉

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

C₂×D₁₉

yes

—

—

—

C₈²

—

C₂×C₁₉

—

100

C₂×C₁₉

C₄×C₄⋊C₈

yes

characteristic

101

C₂×C₁₉

#1772

yes

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂×C₄²

—

106

C₂×C₄²

C₄×D₁₉

yes

—

107

C₂×C₄²

C₂²×D₁₉

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

C₄×D₁₉

yes

—

110

C₂²⋊C₈

—

111

C₂²⋊C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₁₉

#8266

yes

characteristic, complemented, Sylow

122

C₂²×C₄

—

123

C₂²×C₄

C₂×C₄×D₁₉

yes

characteristic

124

C₂²×C₄

#64543

yes

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₄²

—

137

C₄²

—

138

C₄²

C₈×D₁₉

yes

—

139

C₄²

C₂×C₄×D₁₉

yes

—

140

C₄²

D₄×D₁₉

yes

characteristic

141

C₄²

Q₈×D₁₉

yes

characteristic

142

C₂³

#10597323

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₄²×D₁₉

yes

characteristic

147

C₂×C₄

(C₂²⋊D₁₉)⋊C₄

yes

—

148

C₂×C₄

#10597333

yes

—

149

C₂×C₄

C₄⋊C₄×D₁₉

yes

characteristic

150

C₂×C₄

C₂×C₈×D₁₉

yes

—

151

C₂×C₄

M₁₆×D₁₉

yes

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

156

C₈

—

157

C₈

—

158

C₈

—

159

C₂²

—

160

C₂²

#11926395

yes

characteristic

161

C₂²

#11926594

yes

—

162

C₄

—

163

C₄

—

164

C₄

C₄×C₈×D₁₉

yes

characteristic

165

C₄

#11926496

yes

characteristic

166

C₄

C₄⋊C₈×D₁₉

yes

—

167

C₄

#11926658

yes

—

168

C₂

—

169

C₂

#14563483

yes

characteristic

170

C₂

#14563513

yes

—

	subgroup	G/H
center	C₄²	D₄×D₁₉
commutator	C₄×C₁₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1276	C₂
socle	C₂²×C₁₉	C₄×C₄⋊C₄
prime radical	C₂³×C₁₉	#117
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1276	D₁₉
hypocenter	C₁₉	#8266
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₄²

D₄×D₁₉

commutator

C₄×C₁₉

C₂×C₄×C₈

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1276

C₂

socle

C₂²×C₁₉

C₄×C₄⋊C₄

prime radical

C₂³×C₁₉

#117

cosocle

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
11. #11926296	C₄	semidirect
12. #11926294	C₄	semidirect
30. C₄×C₁₉⋊C₈	C₈	semidirect
121. C₁₉	#8266	semidirect

H ≅ G/N

type

11. #11926296

C₄

semidirect

12. #11926294

C₄

semidirect

30. C₄×C₁₉⋊C₈

C₈

semidirect

121. C₁₉

#8266

semidirect

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

subgroup	index
C₄²×C₁₉⋊C₈	2
#14562381	2
#14561919	2
#14561468	2
C₁₉×#1276	2
#8266	19

subgroup

index

C₄²×C₁₉⋊C₈

C₁₉×#1276

Subgroups — #18073696

Conjugacy classes of subgroups

Complements

Maximal subgroups