Group #18076349 — subgroups

Subgroups — #18076349

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	#14562391	C₂	1	1	yes	maximal, characteristic
3	#14562390	C₂	1	2	yes	maximal
4	C₂×#11926266	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1566	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	#11926295	—	2	2	no	—
11	#11926294	—	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×#334	—	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	SD₁₆	1	8	yes	—
50	#8825	—	19	1	no	maximal, Sylow
51	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₉	—	2	2	no	—
53	C₂×C₄×C₁₉	—	2	2	no	—
54	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
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	#127	1	1	yes	characteristic
58	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₁₉	C₂×SD₁₆	1	4	yes	—
61	C₈×C₁₉	—	2	4	no	—
62	C₈×C₁₉	—	8	2	no	—
63	C₁₉⋊C₈	—	2	4	no	—
64	C₁₉⋊C₈	—	8	2	no	—
65	C₁₉⋊C₈	#106	1	8	yes	—
66	#1848	—	19	2	no	—
67	#1566	—	19	2	no	—
68	#1566	—	19	1	no	—
69	#1566	D₁₉	1	1	yes	characteristic
70	C₂×#322	—	19	1	no	—
71	C₂²×C₁₉	—	2	1	no	—
72	C₂²×C₁₉	—	2	1	no	—
73	C₂²×C₁₉	#384	1	1	yes	characteristic
74	C₂²×C₁₉	#498	1	2	yes	—
75	C₄×C₁₉	—	2	1	no	—
76	C₄×C₁₉	—	2	1	no	—
77	C₄×C₁₉	C₂×#106	1	2	yes	—
78	C₄×C₁₉	#430	1	2	yes	—
79	C₄×C₁₉	#430	1	2	yes	—
80	C₄×C₁₉	#492	1	2	yes	—
81	C₂×#108	—	38	4	no	—
82	#423	—	19	2	no	—
83	#423	C₂×D₁₉	1	2	yes	—
84	C₂×C₄×C₈	—	19	1	no	—
85	C₂×C₄×C₈	—	19	1	no	—
86	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
87	#334	—	2	2	no	—
88	#334	—	38	4	no	—
89	#334	—	38	2	no	—
90	#322	—	19	4	no	—
91	C₂×C₁₉	—	2	2	no	—
92	C₂×C₁₉	#1844	1	1	yes	characteristic
93	C₂×C₁₉	#1849	1	1	yes	characteristic
94	C₂×C₁₉	#1849	1	1	yes	characteristic
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	38	2	no	—
97	C₂²×C₈	—	19	2	no	—
98	C₂²×C₈	—	19	2	no	—
99	C₂²×C₈	Dic₃₈	1	2	yes	—
100	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
101	#108	—	76	8	no	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	38	2	no	—
105	C₄⋊C₈	—	38	2	no	—
106	C₄×C₈	—	19	2	no	—
107	C₄×C₈	—	19	2	no	—
108	C₄×C₈	—	19	2	no	—
109	C₄×C₈	—	19	2	no	—
110	C₄×C₈	C₄×D₁₉	1	2	yes	—
111	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
112	C₁₉	#8825	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
114	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
115	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
116	M₁₆	—	4	4	no	—
117	M₁₆	—	76	4	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	19	4	no	—
121	C₂×C₈	—	19	8	no	—
122	C₂×C₈	—	19	4	no	—
123	C₂×C₈	—	76	2	no	—
124	C₂×C₈	—	76	2	no	—
125	C₂×C₈	Dic₁₉⋊C₄	1	4	yes	—
126	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
127	C₄²	D₄×D₁₉	1	1	yes	characteristic
128	C₄²	Q₈×D₁₉	1	1	yes	characteristic
129	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
130	C₂³	#10597334	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	#10597334	1	1	yes	characteristic
134	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
135	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
136	C₂×C₄	SD₁₆×D₁₉	1	4	yes	—
137	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
138	C₂×C₄	#10597386	1	1	yes	characteristic
139	C₂×C₄	#10597390	1	1	yes	characteristic
140	C₈	—	2	4	no	—
141	C₈	—	8	2	no	—
142	C₈	—	19	8	no	—
143	C₈	—	38	4	no	—
144	C₈	—	152	2	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#11926413	1	1	yes	characteristic
148	C₂²	#11926941	1	2	yes	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	#106×D₁₉	1	2	yes	—
152	C₄	#11926591	1	2	yes	—
153	C₄	#11926592	1	2	yes	—
154	C₄	#11926915	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	#14563790	1	1	yes	characteristic
157	C₂	#14563808	1	1	yes	characteristic
158	C₂	#14563809	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14563806

C₂

yes

maximal

#14562391

C₂

yes

maximal, characteristic

#14562390

C₂

yes

maximal

C₂×#11926266

C₂

yes

maximal, characteristic

C₁₉×#1566

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×#334

—

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

SD₁₆

yes

—

#8825

—

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×SD₁₆

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

#106

yes

—

—

—

—

D₁₉

yes

characteristic

C₂×#322

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#384

yes

characteristic

C₂²×C₁₉

#498

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂×#106

yes

—

C₄×C₁₉

#430

yes

—

C₄×C₁₉

#430

yes

—

C₄×C₁₉

#492

yes

—

C₂×#108

—

#423

—

#423

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

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

Dic₃₈

yes

—

100

C₂×C₄²

C₂²×D₁₉

yes

characteristic

101

#108

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₄×D₁₉

yes

—

111

C₄×C₈

C₂²⋊D₁₉

yes

—

112

C₁₉

#8825

yes

characteristic, complemented, Sylow

113

C₂²×C₄

C₂×Dic₃₈

yes

characteristic

114

C₂²×C₄

D₄×D₁₉

yes

characteristic

115

C₂²×C₄

Q₈×D₁₉

yes

characteristic

116

M₁₆

—

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

Dic₁₉⋊C₄

yes

—

126

C₄²

C₂×C₄×D₁₉

yes

characteristic

127

C₄²

D₄×D₁₉

yes

characteristic

128

C₄²

Q₈×D₁₉

yes

characteristic

129

C₄²

C₂×C₂²⋊D₁₉

yes

characteristic

130

C₂³

#10597334

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#10597334

yes

characteristic

134

C₂×C₄

C₄⋊C₄×D₁₉

yes

characteristic

135

C₂×C₄

C₄⋊C₄×D₁₉

yes

characteristic

136

C₂×C₄

SD₁₆×D₁₉

yes

—

137

C₂×C₄

C₂×Dic₁₉⋊C₄

yes

characteristic

138

C₂×C₄

#10597386

yes

characteristic

139

C₂×C₄

#10597390

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

152

—

145

C₂²

—

146

C₂²

—

147

C₂²

#11926413

yes

characteristic

148

C₂²

#11926941

yes

—

149

C₄

—

150

C₄

—

151

C₄

#106×D₁₉

yes

—

152

C₄

#11926591

yes

—

153

C₄

#11926592

yes

—

154

C₄

#11926915

yes

—

155

C₂

—

156

C₂

#14563790

yes

characteristic

157

C₂

#14563808

yes

characteristic

158

C₂

#14563809

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597386
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1566	C₂
socle	C₂²×C₁₉	#384
prime radical	C₂³×C₁₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1566	D₁₉
hypocenter	C₁₉	#8825
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#10597386

commutator

C₄²×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1566

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 ◁ #1566

N	H ≅ G/N	type
112. C₁₉	#8825	semidirect

H ≅ G/N

type

112. C₁₉

#8825

semidirect

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

subgroup	index
#14563806	2
#14562391	2
#14562390	2
C₂×#11926266	2
C₁₉×#1566	2
#8825	19

subgroup

index

C₁₉×#1566

Subgroups — #18076349

Conjugacy classes of subgroups

Complements

Maximal subgroups