Group #18073700 — subgroups

Subgroups — #18073700

Order 4864.

390 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 151 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	#14561928	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	#11926265	—	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	2	yes	—
22	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
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	4	no	—
29	C₄×C₁₉⋊C₈	—	2	2	no	—
30	C₄×C₁₉⋊C₈	—	2	4	no	—
31	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
32	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
33	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
34	C₄×C₁₉⋊C₈	Q₈	1	2	yes	—
35	C₂²×C₄×C₁₉	—	2	2	no	—
36	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	#41	1	2	yes	—
38	C₂×C₈×C₁₉	—	2	4	no	—
39	C₄²×C₁₉	—	2	8	no	—
40	C₄²×C₁₉	—	2	2	no	—
41	C₄²×C₁₉	M₁₆	1	4	yes	—
42	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
43	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₁₉⋊C₈	—	2	2	no	—
46	C₂×C₁₉⋊C₈	—	2	2	no	—
47	C₂×C₁₉⋊C₈	—	2	2	no	—
48	C₂×C₁₉⋊C₈	—	2	4	no	—
49	C₂×C₁₉⋊C₈	C₄²	1	2	yes	—
50	C₂×C₁₉⋊C₈	C₄²	1	1	yes	characteristic
51	C₂×C₁₉⋊C₈	C₄²	1	1	yes	characteristic
52	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
53	C₂×C₁₉⋊C₈	M₁₆	1	4	yes	—
54	#8267	—	19	1	no	maximal, Sylow
55	C₂³×C₁₉	#117	1	1	yes	characteristic
56	C₂×C₄×C₁₉	—	2	8	no	—
57	C₂×C₄×C₁₉	—	2	4	no	—
58	C₂×C₄×C₁₉	—	2	4	no	—
59	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
64	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
65	C₈×C₁₉	—	4	4	no	—
66	C₁₉⋊C₈	—	2	2	no	—
67	C₁₉⋊C₈	—	2	4	no	—
68	C₁₉⋊C₈	—	4	4	no	—
69	C₁₉⋊C₈	#97	1	2	yes	—
70	C₁₉⋊C₈	#97	1	2	yes	—
71	C₄²×C₈	—	19	1	no	—
72	#1564	—	19	2	no	—
73	#1452	—	19	2	no	—
74	#1276	—	19	1	no	—
75	#1276	D₁₉	1	1	yes	characteristic
76	C₂²×C₁₉	—	2	2	no	—
77	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₁₉	#432	1	2	yes	—
79	C₄×C₁₉	—	2	8	no	—
80	C₄×C₁₉	—	2	2	no	—
81	C₄×C₁₉	C₂×#97	1	1	yes	characteristic
82	C₄×C₁₉	C₄×M₁₆	1	1	yes	characteristic
83	C₄×C₁₉	#423	1	2	yes	—
84	C₄×C₁₉	#443	1	4	yes	—
85	#431	—	19	2	no	—
86	#431	C₂×D₁₉	1	2	yes	—
87	C₂×C₄×C₈	—	19	1	no	—
88	C₂×C₄×C₈	—	19	1	no	—
89	C₂×C₄×C₈	—	19	2	no	—
90	C₂×C₄×C₈	—	19	2	no	—
91	C₄³	C₂×D₁₉	1	1	yes	characteristic
92	#335	—	19	4	no	—
93	#334	—	19	4	no	—
94	#322	—	38	2	no	—
95	#322	—	38	2	no	—
96	C₂×C₁₉	—	2	2	no	—
97	C₂×C₁₉	#1767	1	1	yes	characteristic
98	C₂×C₁₉	#1772	1	2	yes	—
99	C₂²×C₈	—	19	1	no	—
100	C₂²×C₈	—	19	1	no	—
101	C₂²×C₈	—	19	2	no	—
102	C₂×C₄²	—	2	2	no	—
103	C₂×C₄²	C₄×D₁₉	1	2	yes	—
104	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
105	C₄⋊C₈	—	19	4	no	—
106	C₄⋊C₈	C₄×D₁₉	1	4	yes	—
107	C₂²⋊C₈	—	2	2	no	—
108	C₂²⋊C₈	—	38	2	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	—	19	2	no	—
111	C₄×C₈	—	19	2	no	—
112	C₄×C₈	—	19	2	no	—
113	C₄×C₈	—	19	2	no	—
114	C₄×C₈	—	38	4	no	—
115	C₄×C₈	—	38	4	no	—
116	C₄×C₈	—	38	2	no	—
117	C₁₉	#8267	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
120	C₂²×C₄	#64543	1	2	yes	—
121	C₂×C₈	—	2	4	no	—
122	C₂×C₈	—	19	1	no	—
123	C₂×C₈	—	19	1	no	—
124	C₂×C₈	—	19	4	no	—
125	C₂×C₈	—	19	2	no	—
126	C₂×C₈	—	19	4	no	—
127	C₂×C₈	—	38	2	no	—
128	C₂×C₈	—	38	2	no	—
129	C₂×C₈	—	38	2	no	—
130	C₂×C₈	—	38	4	no	—
131	C₄²	—	2	8	no	—
132	C₄²	—	2	2	no	—
133	C₄²	#64515	1	4	yes	—
134	C₄²	C₂×C₄×D₁₉	1	2	yes	—
135	C₄²	D₄×D₁₉	1	1	yes	characteristic
136	C₄²	Q₈×D₁₉	1	1	yes	characteristic
137	C₂³	#10597323	1	1	yes	characteristic
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
142	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
143	C₂×C₄	#10597333	1	2	yes	—
144	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
145	C₂×C₄	C₂×#64515	1	2	yes	—
146	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
147	C₈	—	4	4	no	—
148	C₈	—	19	2	no	—
149	C₈	—	19	2	no	—
150	C₈	—	38	4	no	—
151	C₈	—	38	2	no	—
152	C₈	—	76	4	no	—
153	C₂²	—	2	2	no	—
154	C₂²	#11926395	1	1	yes	characteristic
155	C₂²	#11926598	1	2	yes	—
156	C₄	—	2	8	no	—
157	C₄	—	2	2	no	—
158	C₄	#97×D₁₉	1	1	yes	characteristic
159	C₄	C₄×#64515	1	1	yes	characteristic
160	C₄	#11926566	1	2	yes	—
161	C₄	#11926658	1	4	yes	—
162	C₂	—	2	2	no	—
163	C₂	#14563487	1	1	yes	characteristic
164	C₂	#14563513	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄²×C₁₉⋊C₈

C₂

yes

maximal, characteristic

#14562381

C₂

yes

maximal

#14561928

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

—

#11926265

—

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

—

C₂²×C₁₉⋊C₈

C₂×C₄

yes

characteristic

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

—

C₄×C₁₉⋊C₈

C₂×C₄

yes

—

C₄×C₁₉⋊C₈

C₂×C₄

yes

—

C₄×C₁₉⋊C₈

D₄

yes

—

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

M₁₆

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

—

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

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

C₂×M₁₆

yes

—

C₂×C₄×C₁₉

C₂×M₁₆

yes

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

#97

yes

—

C₁₉⋊C₈

#97

yes

—

C₄²×C₈

—

—

—

—

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₁₉

#432

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂×#97

yes

characteristic

C₄×C₁₉

C₄×M₁₆

yes

characteristic

C₄×C₁₉

#423

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

—

—

—

—

C₂×C₁₉

—

C₂×C₁₉

#1767

yes

characteristic

C₂×C₁₉

#1772

yes

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂×C₄²

—

103

C₂×C₄²

C₄×D₁₉

yes

—

104

C₂×C₄²

C₂²×D₁₉

yes

characteristic

105

C₄⋊C₈

—

106

C₄⋊C₈

C₄×D₁₉

yes

—

107

C₂²⋊C₈

—

108

C₂²⋊C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₁₉

#8267

yes

characteristic, complemented, Sylow

118

C₂²×C₄

—

119

C₂²×C₄

C₂×C₄×D₁₉

yes

characteristic

120

C₂²×C₄

#64543

yes

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₄²

—

132

C₄²

—

133

C₄²

#64515

yes

—

134

C₄²

C₂×C₄×D₁₉

yes

—

135

C₄²

D₄×D₁₉

yes

characteristic

136

C₄²

Q₈×D₁₉

yes

characteristic

137

C₂³

#10597323

yes

characteristic

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

C₄²×D₁₉

yes

characteristic

142

C₂×C₄

(C₂²⋊D₁₉)⋊C₄

yes

—

143

C₂×C₄

#10597333

yes

—

144

C₂×C₄

C₄⋊C₄×D₁₉

yes

characteristic

145

C₂×C₄

C₂×#64515

yes

—

146

C₂×C₄

M₁₆×D₁₉

yes

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₂²

—

154

C₂²

#11926395

yes

characteristic

155

C₂²

#11926598

yes

—

156

C₄

—

157

C₄

—

158

C₄

#97×D₁₉

yes

characteristic

159

C₄

C₄×#64515

yes

characteristic

160

C₄

#11926566

yes

—

161

C₄

#11926658

yes

—

162

C₂

—

163

C₂

#14563487

yes

characteristic

164

C₂

#14563513

yes

—

	subgroup	G/H
center	C₄²	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
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	#8267
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₄²

D₄×D₁₉

commutator

C₂×C₄×C₁₉

C₂×C₄²

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 ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
11. #11926296	C₄	semidirect
12. #11926294	C₄	semidirect
117. C₁₉	#8267	semidirect

H ≅ G/N

type

11. #11926296

C₄

semidirect

12. #11926294

C₄

semidirect

117. C₁₉

#8267

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
#14561928	2
#14561468	2
C₁₉×#1276	2
#8267	19

subgroup

index

C₄²×C₁₉⋊C₈

C₁₉×#1276

Subgroups — #18073700

Conjugacy classes of subgroups

Complements

Maximal subgroups