Group #18069176 — subgroups

Subgroups — #18069176

Order 4864.

200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 59 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	#14561575	C₂	1	2	yes	maximal
2	C₁₉×#1842	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₁₉⋊C₁₆)	—	2	2	no	—
4	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
5	C₂×C₁₉×C₄⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
7	C₄³×C₁₉	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₁₉	—	2	1	no	—
9	C₂²×C₈×C₁₉	D₄	1	2	yes	—
10	C₂×C₄²×C₁₉	—	2	1	no	—
11	C₂×C₄²×C₁₉	—	2	1	no	—
12	C₂×C₄²×C₁₉	C₈	1	1	yes	characteristic
13	C₂×C₄²×C₁₉	C₈	1	1	yes	characteristic
14	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₁₉×C₄⋊C₈	—	4	1	no	—
16	C₁₉×C₄⋊C₈	—	4	4	no	—
17	C₁₉×C₄⋊C₈	—	4	1	no	—
18	C₂×C₁₉⋊C₁₆	—	4	2	no	—
19	C₂²×C₄×C₁₉	—	2	1	no	—
20	C₂²×C₄×C₁₉	—	2	1	no	—
21	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
23	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
24	C₂×C₈×C₁₉	—	2	2	no	—
25	C₂×C₈×C₁₉	—	4	2	no	—
26	C₂×C₈×C₁₉	—	4	2	no	—
27	C₂×C₈×C₁₉	—	4	1	no	—
28	C₄²×C₁₉	—	2	2	no	—
29	C₄²×C₁₉	—	2	2	no	—
30	C₄²×C₁₉	—	2	1	no	—
31	C₄²×C₁₉	—	2	1	no	—
32	C₄²×C₁₉	—	2	1	no	—
33	C₄²×C₁₉	—	4	1	no	—
34	C₄²×C₁₉	—	4	2	no	—
35	C₄²×C₁₉	C₁₆	1	2	yes	complemented
36	C₁₉⋊C₁₆	—	8	2	no	—
37	#7063	—	19	1	no	maximal, Sylow
38	C₂³×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
39	C₂×C₄×C₁₉	—	2	2	no	—
40	C₂×C₄×C₁₉	—	2	2	no	—
41	C₂×C₄×C₁₉	—	2	1	no	—
42	C₂×C₄×C₁₉	—	2	2	no	—
43	C₂×C₄×C₁₉	—	2	1	no	—
44	C₂×C₄×C₁₉	—	2	1	no	—
45	C₂×C₄×C₁₉	—	2	1	no	—
46	C₂×C₄×C₁₉	—	2	2	no	—
47	C₂×C₄×C₁₉	—	2	1	no	—
48	C₂×C₄×C₁₉	—	4	1	no	—
49	C₂×C₄×C₁₉	—	4	1	no	—
50	C₂×C₄×C₁₉	—	4	1	no	—
51	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
52	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
53	C₂×C₄×C₁₉	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₁₉	M₃₂	1	1	yes	characteristic
55	C₈×C₁₉	—	4	2	no	—
56	C₈×C₁₉	—	8	1	no	—
57	#1842	D₁₉	1	1	yes	characteristic
58	#1322	—	19	2	no	—
59	C₂²×C₁₉	—	2	1	no	—
60	C₂²×C₁₉	—	2	1	no	—
61	C₂²×C₁₉	#324	1	1	yes	characteristic
62	C₂²×C₁₉	C₂²⋊C₁₆	1	1	yes	characteristic
63	C₂²×C₁₉	#349	1	1	yes	characteristic
64	C₄×C₁₉	—	2	4	no	—
65	C₄×C₁₉	—	2	1	no	—
66	C₄×C₁₉	—	2	1	no	—
67	C₄×C₁₉	—	2	1	no	—
68	C₄×C₁₉	—	4	2	no	—
69	C₄×C₁₉	—	4	1	no	—
70	C₄×C₁₉	#353	1	1	yes	characteristic
71	C₄×C₁₉	#356	1	1	yes	characteristic
72	C₂×C₄⋊C₈	—	2	2	no	—
73	C₂×C₄⋊C₈	Dic₁₉	1	1	yes	characteristic
74	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
75	C₄³	Dic₁₉	1	1	yes	characteristic
76	C₂²⋊C₁₆	—	38	2	no	—
77	C₂×C₁₉	—	2	1	no	—
78	C₂×C₁₉	—	2	1	no	—
79	C₂×C₁₉	#1322	1	1	yes	characteristic
80	C₂×C₁₉	#1325	1	1	yes	characteristic
81	C₂×C₁₉	#1327	1	1	yes	characteristic
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	—	2	1	no	—
86	C₂×C₄²	C₁₉⋊C₈	1	1	yes	characteristic
87	C₂×C₄²	C₁₉⋊C₈	1	1	yes	characteristic
88	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
89	C₂×C₁₆	—	76	2	no	—
90	C₄⋊C₈	—	4	1	no	—
91	C₄⋊C₈	—	4	1	no	—
92	C₄⋊C₈	—	4	4	no	—
93	C₁₉	#7063	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	C₂×C₁₉⋊C₈	1	1	yes	characteristic
97	C₂²×C₄	#64520	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	4	2	no	—
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	4	1	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	2	2	no	—
105	C₄²	—	2	1	no	—
106	C₄²	—	2	1	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	4	1	no	—
109	C₄²	—	4	2	no	—
110	C₄²	C₁₉⋊C₁₆	1	2	yes	complemented
111	C₁₆	—	152	2	no	—
112	C₂³	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	C₂×C₁₉⋊C₁₆	1	1	yes	characteristic
126	C₂×C₄	#10597269	1	1	yes	characteristic
127	C₂×C₄	#10597291	1	1	yes	characteristic
128	C₂×C₄	#10597293	1	1	yes	characteristic
129	C₈	—	4	2	no	—
130	C₈	—	8	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	#11926270	1	1	yes	characteristic
134	C₂²	C₂²⋊(C₁₉⋊C₁₆)	1	1	yes	characteristic
135	C₂²	#11926326	1	1	yes	characteristic
136	C₄	—	2	4	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	2	no	—
141	C₄	—	4	1	no	—
142	C₄	#11926337	1	1	yes	characteristic
143	C₄	#11926343	1	1	yes	characteristic
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#14561575	1	1	yes	characteristic
147	C₂	#14561581	1	1	yes	characteristic
148	C₂	#14561586	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14561575

C₂

yes

maximal

C₁₉×#1842

C₂

yes

maximal, characteristic

C₂²⋊(C₁₉⋊C₁₆)

—

C₂×C₁₉×C₄⋊C₈

—

C₂×C₁₉×C₄⋊C₈

C₄

yes

characteristic

C₂×C₁₉×C₄⋊C₈

C₂²

yes

characteristic

C₄³×C₁₉

C₄

yes

characteristic

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

C₈

yes

characteristic

C₂×C₄²×C₁₉

C₈

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₂×C₁₉⋊C₁₆

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₉

M₁₆

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₁₆

yes

complemented

C₁₉⋊C₁₆

—

#7063

—

maximal, Sylow

C₂³×C₁₉

C₂²⋊C₈

yes

characteristic

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

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₉

M₃₂

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

#1842

D₁₉

yes

characteristic

#1322

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#324

yes

characteristic

C₂²×C₁₉

C₂²⋊C₁₆

yes

characteristic

C₂²×C₁₉

#349

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

Dic₁₉

yes

characteristic

C₂×C₄⋊C₈

C₂×D₁₉

yes

characteristic

C₄³

Dic₁₉

yes

characteristic

C₂²⋊C₁₆

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₁₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₁₉⋊C₈

yes

characteristic

C₂×C₄²

C₁₉⋊C₈

yes

characteristic

C₂×C₄²

C₂×Dic₁₉

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₁₉

#7063

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₁₉⋊C₈

yes

characteristic

C₂²×C₄

#64520

yes

characteristic

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

C₁₉⋊C₁₆

yes

complemented

111

C₁₆

—

152

—

112

C₂³

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×C₁₉⋊C₁₆

yes

characteristic

126

C₂×C₄

#10597269

yes

characteristic

127

C₂×C₄

#10597291

yes

characteristic

128

C₂×C₄

#10597293

yes

characteristic

129

C₈

—

130

C₈

—

131

C₂²

—

132

C₂²

—

133

C₂²

#11926270

yes

characteristic

134

C₂²

C₂²⋊(C₁₉⋊C₁₆)

yes

characteristic

135

C₂²

#11926326

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#11926337

yes

characteristic

143

C₄

#11926343

yes

characteristic

144

C₂

—

145

C₂

—

146

C₂

#14561575

yes

characteristic

147

C₂

#14561581

yes

characteristic

148

C₂

#14561586

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597291
commutator	C₂×C₄×C₁₉	C₂×C₁₆
Frattini	C₂×C₄⋊C₈	C₂×D₁₉
Fitting	C₁₉×#1842	C₂
socle	C₂²×C₁₉	#324
prime radical	C₂³×C₁₉	C₂²⋊C₈
cosocle	C₂×C₁₉×C₄⋊C₈	C₂²
hypercenter	#1842	D₁₉
hypocenter	C₁₉	#7063
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1842

subgroup

G/H

center

C₂×C₄

#10597291

commutator

C₂×C₄×C₁₉

C₂×C₁₆

Frattini

C₂×C₄⋊C₈

C₂×D₁₉

Fitting

C₁₉×#1842

C₂

socle

C₂²×C₁₉

#324

prime radical

C₂³×C₁₉

C₂²⋊C₈

cosocle

C₂×C₁₉×C₄⋊C₈

C₂²

D₁₉

C₁₉

G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1842

N	H ≅ G/N	type
35. C₄²×C₁₉	C₁₆	semidirect
93. C₁₉	#7063	semidirect
110. C₄²	C₁₉⋊C₁₆	semidirect

H ≅ G/N

type

35. C₄²×C₁₉

C₁₆

semidirect

93. C₁₉

#7063

semidirect

110. C₄²

C₁₉⋊C₁₆

semidirect

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

subgroup	index
#14561575	2
C₁₉×#1842	2
#7063	19

subgroup

index

#14561575

C₁₉×#1842

#7063

Subgroups — #18069176

Conjugacy classes of subgroups

Complements

Maximal subgroups