Group #18069931 — subgroups

Subgroups — #18069931

Order 4864.

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

normal subgroups only

This group has 176 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	#14564599	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×#10597316	C₄	1	1	yes	characteristic
5	C₂×C₄×C₁₉⋊C₈	C₄	1	1	yes	characteristic
6	#11926306	—	2	1	no	—
7	#11926306	—	2	1	no	—
8	#11926270	—	2	1	no	—
9	C₁₉×#382	C₂²	1	1	yes	characteristic
10	C₁₉×#324	—	2	1	no	—
11	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
13	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₉	—	2	1	no	—
15	C₂×Dic₁₉⋊C₄	—	4	1	no	—
16	C₂²×C₁₉⋊C₈	—	2	1	no	—
17	#10597316	—	2	1	no	—
18	#10597316	—	4	1	no	—
19	#10597316	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	C₄×C₁₉⋊C₈	—	4	1	no	—
24	C₂²×C₄×C₁₉	—	2	1	no	—
25	C₂²×C₄×C₁₉	—	2	1	no	—
26	C₂²×C₄×C₁₉	—	2	1	no	—
27	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×Dic₁₉	—	4	1	no	—
29	C₂×C₈×C₁₉	—	8	1	no	—
30	C₄⋊C₄×C₁₉	—	4	1	no	—
31	C₄⋊C₄×C₁₉	—	4	1	no	—
32	C₄²×C₁₉	—	2	1	no	—
33	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
34	C₄²×C₁₉	M₁₆	1	1	yes	characteristic
35	C₄⋊Dic₁₉	—	2	2	no	—
36	C₄⋊Dic₁₉	—	4	1	no	—
37	Dic₁₉⋊C₄	—	4	2	no	—
38	Dic₁₉⋊C₄	—	4	2	no	—
39	C₂×C₁₉⋊C₈	—	2	2	no	—
40	C₂×C₁₉⋊C₈	—	4	1	no	—
41	C₂×C₁₉⋊C₈	—	4	1	no	—
42	C₂×C₁₉⋊C₈	—	8	1	no	—
43	#7359	—	19	1	no	maximal, Sylow
44	C₂³×C₁₉	#99	1	1	yes	characteristic
45	C₂×C₄×C₁₉	—	2	1	no	—
46	C₂×C₄×C₁₉	—	2	1	no	—
47	C₂×C₄×C₁₉	—	2	1	no	—
48	C₂×C₄×C₁₉	—	2	1	no	—
49	C₂×C₄×C₁₉	—	4	1	no	—
50	C₂×C₄×C₁₉	—	4	1	no	—
51	C₂×C₄×C₁₉	—	4	1	no	—
52	C₂×C₄×C₁₉	—	4	1	no	—
53	C₂×C₄×C₁₉	—	8	1	no	—
54	C₂×C₄×C₁₉	—	8	1	no	—
55	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
57	C₂×Dic₁₉	—	4	2	no	—
58	C₂×Dic₁₉	—	8	1	no	—
59	C₂×Dic₁₉	—	8	1	no	—
60	C₈×C₁₉	—	16	1	no	—
61	C₁₉⋊C₈	—	4	2	no	—
62	C₁₉⋊C₈	—	16	1	no	—
63	#1991	—	19	1	no	—
64	#1324	—	19	1	no	—
65	#1324	D₁₉	1	1	yes	characteristic
66	C₂²×C₁₉	—	2	1	no	—
67	C₂²×C₁₉	—	2	1	no	—
68	C₂²×C₁₉	#323	1	1	yes	characteristic
69	C₂²×C₁₉	#354	1	1	yes	characteristic
70	C₂²×C₁₉	#356	1	1	yes	characteristic
71	C₄×C₁₉	—	2	1	no	—
72	C₄×C₁₉	—	2	1	no	—
73	C₄×C₁₉	—	4	1	no	—
74	C₄×C₁₉	—	4	1	no	—
75	C₄×C₁₉	—	8	1	no	—
76	C₄×C₁₉	—	8	1	no	—
77	Dic₁₉	—	8	2	no	—
78	C₂×#125	—	19	1	no	—
79	C₂×C₄×C₈	—	19	1	no	—
80	#382	C₂×D₁₉	1	1	yes	characteristic
81	#340	—	38	1	no	—
82	#340	—	38	1	no	—
83	#324	—	2	1	no	—
84	#324	—	38	1	no	—
85	C₂×C₁₉	—	2	1	no	—
86	C₂×C₁₉	—	2	1	no	—
87	C₂×C₁₉	#1319	1	1	yes	characteristic
88	C₂×C₁₉	#1414	1	1	yes	characteristic
89	C₂×C₁₉	#1414	1	1	yes	characteristic
90	C₂²×C₈	—	38	1	no	—
91	#125	—	19	2	no	—
92	#125	—	38	1	no	—
93	#125	—	76	1	no	—
94	C₂×C₄⋊C₄	—	38	1	no	—
95	C₂×C₄⋊C₄	—	76	1	no	—
96	C₂×C₄⋊C₄	D₇₆	1	1	yes	characteristic
97	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
98	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
99	C₂²⋊C₈	—	4	1	no	—
100	C₂²⋊C₈	—	76	1	no	—
101	C₄×C₈	—	76	1	no	—
102	#95	—	4	1	no	—
103	C₁₉	#7359	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	76	1	no	—
108	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
109	C₂×C₈	—	8	1	no	—
110	C₂×C₈	—	38	2	no	—
111	C₂×C₈	—	76	1	no	—
112	C₂×C₈	—	76	1	no	—
113	C₂×C₈	—	152	1	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₄⋊C₄	—	38	2	no	—
117	C₄⋊C₄	—	76	2	no	—
118	C₄⋊C₄	—	76	2	no	—
119	C₄⋊C₄	—	76	1	no	—
120	C₄²	—	2	1	no	—
121	C₄²	C₈×D₁₉	1	1	yes	characteristic
122	C₄²	#64515	1	1	yes	characteristic
123	C₂³	#10597263	1	1	yes	characteristic
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	8	1	no	—
134	C₂×C₄	—	76	2	no	—
135	C₂×C₄	—	152	1	no	—
136	C₂×C₄	—	152	1	no	—
137	C₂×C₄	(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
138	C₂×C₄	#10597280	1	1	yes	characteristic
139	C₈	—	16	1	no	—
140	C₈	—	304	1	no	—
141	C₈	—	76	2	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	#11926267	1	1	yes	characteristic
145	C₂²	#11926338	1	1	yes	characteristic
146	C₂²	#11926342	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	4	1	no	—
150	C₄	—	4	1	no	—
151	C₄	—	8	1	no	—
152	C₄	—	8	1	no	—
153	C₄	—	152	2	no	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#14561568	1	1	yes	characteristic
157	C₂	#14561809	1	1	yes	characteristic
158	C₂	#14561811	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564599

C₂

yes

maximal, characteristic

#14561579

C₂

yes

maximal, characteristic

C₁₉×#1324

C₂

yes

maximal, characteristic

C₂×#10597316

C₄

yes

characteristic

C₂×C₄×C₁₉⋊C₈

C₄

yes

—

—

—

C₁₉×#382

C₂²

yes

characteristic

C₁₉×#324

—

C₂×C₄⋊C₄×C₁₉

D₄

yes

characteristic

C₂×C₄⋊C₄×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂×Dic₁₉⋊C₄

—

C₂²×C₁₉⋊C₈

—

#10597316

—

#10597316

—

#10597316

C₈

yes

complemented

C₁₉×C₂²⋊C₈

—

C₁₉×#95

—

C₂²⋊(C₁₉⋊C₈)

—

C₄×C₁₉⋊C₈

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×Dic₁₉

—

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

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

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

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#1991

—

#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×#125

—

C₂×C₄×C₈

—

#382

C₂×D₁₉

yes

—

—

—

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂²×C₈

—

#125

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₇₆

yes

characteristic

C₂×C₄⋊C₄

C₂²⋊D₁₉

yes

characteristic

C₂×C₄²

C₄×D₁₉

yes

characteristic

C₂²⋊C₈

—

100

C₂²⋊C₈

—

101

C₄×C₈

—

102

#95

—

103

C₁₉

#7359

yes

characteristic, complemented, Sylow

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

152

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄²

—

121

C₄²

C₈×D₁₉

yes

characteristic

122

C₄²

#64515

yes

characteristic

123

C₂³

#10597263

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

152

—

136

C₂×C₄

—

152

—

137

C₂×C₄

(C₂×D₁₉)⋊C₈

yes

characteristic

138

C₂×C₄

#10597280

yes

characteristic

139

C₈

—

140

C₈

—

304

—

141

C₈

—

142

C₂²

—

143

C₂²

—

144

C₂²

#11926267

yes

characteristic

145

C₂²

#11926338

yes

characteristic

146

C₂²

#11926342

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

152

—

154

C₂

—

155

C₂

—

156

C₂

#14561568

yes

characteristic

157

C₂

#14561809

yes

characteristic

158

C₂

#14561811

yes

characteristic

	subgroup	G/H
center	C₂²	#11926338
commutator	C₄²×C₁₉	C₂×C₈
Frattini	#382	C₂×D₁₉
Fitting	C₁₉×#1324	C₂
socle	C₂²×C₁₉	#354
prime radical	C₂³×C₁₉	#99
cosocle	C₁₉×#382	C₂²
hypercenter	#1324	D₁₉
hypocenter	C₁₉	#7359
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

#11926338

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. #10597316	C₈	semidirect
103. C₁₉	#7359	semidirect

H ≅ G/N

type

19. #10597316

C₈

semidirect

103. C₁₉

#7359

semidirect

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

subgroup	index
#14564599	2
#14561579	2
C₁₉×#1324	2
#7359	19

subgroup

index

#14564599

#14561579

C₁₉×#1324

#7359

Subgroups — #18069931

Conjugacy classes of subgroups

Complements

Maximal subgroups