Group #18076352 — subgroups

Subgroups — #18076352

Order 4864.

338 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 147 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	#14563806	C₂	1	2	yes	maximal
2	#14562394	C₂	1	1	yes	maximal, characteristic
3	#14562393	C₂	1	2	yes	maximal
4	C₂×#11926266	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1567	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597279	—	2	4	no	—
7	#11926568	C₂²	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
10	#11926297	—	2	2	no	—
11	#11926296	—	2	4	no	—
12	#11926266	C₄	1	4	yes	—
13	C₁₉×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#335	—	2	2	no	—
16	C₂×M₁₆×C₁₉	—	2	2	no	—
17	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
18	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
19	C₂×#64520	—	2	2	no	—
20	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
21	C₂²×C₁₉⋊C₈	Q₈	1	2	yes	—
22	C₁₉×C₄⋊C₈	—	2	2	no	—
23	C₁₉×C₄⋊C₈	—	2	2	no	—
24	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₉	D₄	1	2	yes	—
26	#10597279	—	4	8	no	—
27	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
28	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
29	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
30	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
31	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
32	C₄×C₁₉⋊C₈	Q₈	1	2	yes	—
33	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
36	M₁₆×C₁₉	—	4	4	no	—
37	C₂×C₈×C₁₉	—	4	2	no	—
38	C₂×C₈×C₁₉	—	4	2	no	—
39	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
40	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
44	#64520	—	4	4	no	—
45	C₂×C₁₉⋊C₈	—	4	2	no	—
46	C₂×C₁₉⋊C₈	—	4	2	no	—
47	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
48	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
49	C₂×C₁₉⋊C₈	D₈	1	4	yes	—
50	C₂×C₁₉⋊C₈	Q₁₆	1	4	yes	—
51	#8826	—	19	1	no	maximal, Sylow
52	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×C₄×C₁₉	—	2	2	no	—
54	C₂×C₄×C₁₉	—	2	2	no	—
55	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
59	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
62	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
63	C₈×C₁₉	—	2	4	no	—
64	C₈×C₁₉	—	8	2	no	—
65	C₁₉⋊C₈	—	2	4	no	—
66	C₁₉⋊C₈	—	8	2	no	—
67	C₁₉⋊C₈	#107	1	8	yes	—
68	#1848	—	19	2	no	—
69	#1567	—	19	2	no	—
70	#1567	—	19	1	no	—
71	#1567	D₁₉	1	1	yes	characteristic
72	C₂×#322	—	19	1	no	—
73	C₂²×C₁₉	—	2	1	no	—
74	C₂²×C₁₉	—	2	1	no	—
75	C₂²×C₁₉	#384	1	1	yes	characteristic
76	C₂²×C₁₉	#500	1	2	yes	—
77	C₄×C₁₉	—	2	1	no	—
78	C₄×C₁₉	—	2	1	no	—
79	C₄×C₁₉	C₂×#107	1	2	yes	—
80	C₄×C₁₉	#430	1	2	yes	—
81	C₄×C₁₉	#430	1	2	yes	—
82	C₄×C₁₉	#493	1	1	yes	characteristic
83	C₄×C₁₉	#494	1	1	yes	characteristic
84	C₂×#108	—	38	4	no	—
85	#423	—	19	2	no	—
86	#423	C₂×D₁₉	1	2	yes	—
87	C₂×C₄×C₈	—	19	1	no	—
88	C₂×C₄×C₈	—	19	1	no	—
89	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
90	#335	—	2	2	no	—
91	#335	—	38	4	no	—
92	#335	—	38	2	no	—
93	#322	—	19	4	no	—
94	C₂×C₁₉	—	2	2	no	—
95	C₂×C₁₉	#1845	1	1	yes	characteristic
96	C₂×C₁₉	#1849	1	1	yes	characteristic
97	C₂×C₁₉	#1849	1	1	yes	characteristic
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	38	2	no	—
100	C₂²×C₈	—	19	2	no	—
101	C₂²×C₈	—	19	2	no	—
102	C₂²×C₈	Dic₃₈	1	2	yes	—
103	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
104	#108	—	76	8	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	38	2	no	—
108	C₄⋊C₈	—	38	2	no	—
109	C₄×C₈	—	19	2	no	—
110	C₄×C₈	—	19	2	no	—
111	C₄×C₈	—	19	2	no	—
112	C₄×C₈	—	19	2	no	—
113	C₄×C₈	C₄×D₁₉	1	2	yes	—
114	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
115	C₁₉	#8826	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
117	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
118	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
119	M₁₆	—	4	4	no	—
120	M₁₆	—	76	4	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	19	4	no	—
124	C₂×C₈	—	19	4	no	—
125	C₂×C₈	—	19	4	no	—
126	C₂×C₈	—	19	4	no	—
127	C₂×C₈	—	76	2	no	—
128	C₂×C₈	—	76	2	no	—
129	C₂×C₈	Dic₁₉⋊C₄	1	4	yes	—
130	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
131	C₄²	D₄×D₁₉	1	1	yes	characteristic
132	C₄²	Q₈×D₁₉	1	1	yes	characteristic
133	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
134	C₂³	#10597334	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	#10597334	1	1	yes	characteristic
138	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
139	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
140	C₂×C₄	D₈×D₁₉	1	2	yes	—
141	C₂×C₄	Q₁₆×D₁₉	1	2	yes	—
142	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
143	C₂×C₄	#10597386	1	1	yes	characteristic
144	C₂×C₄	#10597390	1	1	yes	characteristic
145	C₈	—	2	4	no	—
146	C₈	—	8	2	no	—
147	C₈	—	19	8	no	—
148	C₈	—	38	4	no	—
149	C₈	—	152	2	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#11926413	1	1	yes	characteristic
153	C₂²	#11926948	1	2	yes	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#107×D₁₉	1	2	yes	—
157	C₄	#11926591	1	2	yes	—
158	C₄	#11926592	1	2	yes	—
159	C₄	#11926919	1	1	yes	characteristic
160	C₄	#11926922	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	#14563793	1	1	yes	characteristic
163	C₂	#14563808	1	1	yes	characteristic
164	C₂	#14563809	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14563806

C₂

yes

maximal

#14562394

C₂

yes

maximal, characteristic

#14562393

C₂

yes

maximal

C₂×#11926266

C₂

yes

maximal, characteristic

C₁₉×#1567

C₂

yes

maximal, characteristic

C₂×#10597279

—

#11926568

C₂²

yes

—

C₂×C₄×C₁₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₁₉⋊C₈

C₂²

yes

—

—

C₄

yes

—

C₁₉×#423

C₂²

yes

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×#335

—

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

Q₈

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×#64520

—

C₂²×C₁₉⋊C₈

D₄

yes

—

C₂²×C₁₉⋊C₈

Q₈

yes

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₄×C₈×C₁₉

C₂×C₄

yes

—

C₄×C₈×C₁₉

D₄

yes

—

#10597279

—

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

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₉

C₂×Q₈

yes

characteristic

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₄⋊C₄

yes

—

C₄²×C₁₉

C₂²×C₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×Q₈

yes

characteristic

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

C₄⋊C₄

yes

—

C₂×C₁₉⋊C₈

C₄⋊C₄

yes

—

C₂×C₁₉⋊C₈

D₈

yes

—

C₂×C₁₉⋊C₈

Q₁₆

yes

—

#8826

—

maximal, Sylow

C₂³×C₁₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉

#127

yes

characteristic

C₂×C₄×C₁₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

—

C₂×C₄×C₁₉

C₂×Q₁₆

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

#107

yes

—

—

—

—

D₁₉

yes

characteristic

C₂×#322

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#384

yes

characteristic

C₂²×C₁₉

#500

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂×#107

yes

—

C₄×C₁₉

#430

yes

—

C₄×C₁₉

#430

yes

—

C₄×C₁₉

yes

C₄×C₁₉

yes

—

—

C₂×D₁₉

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₉

yes

—

—

—

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

Dic₃₈

yes

—

103

C₂×C₄²

C₂²×D₁₉

yes

characteristic

104

#108

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

C₄×D₁₉

yes

—

114

C₄×C₈

C₂²⋊D₁₉

yes

—

115

C₁₉

#8826

yes

characteristic, complemented, Sylow

116

C₂²×C₄

C₂×Dic₃₈

yes

characteristic

117

C₂²×C₄

D₄×D₁₉

yes

characteristic

118

C₂²×C₄

Q₈×D₁₉

yes

characteristic

119

M₁₆

—

120

M₁₆

—

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

Dic₁₉⋊C₄

yes

—

130

C₄²

C₂×C₄×D₁₉

yes

characteristic

131

C₄²

D₄×D₁₉

yes

characteristic

132

C₄²

Q₈×D₁₉

yes

characteristic

133

C₄²

C₂×C₂²⋊D₁₉

yes

characteristic

134

C₂³

#10597334

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#10597334

yes

characteristic

138

C₂×C₄

C₄⋊C₄×D₁₉

yes

characteristic

139

C₂×C₄

C₄⋊C₄×D₁₉

yes

characteristic

140

C₂×C₄

D₈×D₁₉

yes

—

141

C₂×C₄

Q₁₆×D₁₉

yes

—

142

C₂×C₄

C₂×Dic₁₉⋊C₄

yes

characteristic

143

C₂×C₄

#10597386

yes

characteristic

144

C₂×C₄

#10597390

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

152

—

150

C₂²

—

151

C₂²

—

152

C₂²

#11926413

yes

characteristic

153

C₂²

#11926948

yes

—

154

C₄

—

155

C₄

—

156

C₄

#107×D₁₉

yes

—

157

C₄

#11926591

yes

—

158

C₄

#11926592

yes

—

159

C₄

#11926919

yes

characteristic

160

C₄

#11926922

yes

characteristic

161

C₂

—

162

C₂

#14563793

yes

characteristic

163

C₂

#14563808

yes

characteristic

164

C₂

#14563809

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597386
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1567	C₂
socle	C₂²×C₁₉	#384
prime radical	C₂³×C₁₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1567	D₁₉
hypocenter	C₁₉	#8826
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

subgroup

G/H

center

C₂×C₄

#10597386

commutator

C₄²×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1567

C₂

socle

C₂²×C₁₉

#384

prime radical

C₂³×C₁₉

C₄⋊Q₈

cosocle

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

N	H ≅ G/N	type
115. C₁₉	#8826	semidirect

H ≅ G/N

type

115. C₁₉

#8826

semidirect

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

subgroup	index
#14563806	2
#14562394	2
#14562393	2
C₂×#11926266	2
C₁₉×#1567	2
#8826	19

subgroup

index

C₁₉×#1567

Subgroups — #18076352

Conjugacy classes of subgroups

Complements

Maximal subgroups