Group #18070100 — subgroups

Subgroups — #18070100

Order 4864.

150 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 47 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	C₂²⋊(C₁₉⋊C₃₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	#14561729	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2215	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597276	—	2	1	no	—
5	C₂×C₁₉⋊C₃₂	—	2	1	no	—
6	C₂×C₁₉×D₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
8	C₁₉×#466	—	2	1	no	—
9	C₁₉×#366	C₄	1	1	yes	characteristic
10	C₁₉×#357	—	2	1	no	—
11	C₂×D₈×C₁₉	—	2	1	no	—
12	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
13	C₁₉×#121	—	4	1	no	—
14	C₂×#64520	—	4	1	no	—
15	C₁₉×D₁₆	—	4	1	no	—
16	C₂×C₁₆×C₁₉	—	4	1	no	—
17	C₂×C₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
19	C₁₉×#107	—	2	1	no	—
20	C₁₉×#102	—	4	1	no	—
21	#10597276	—	2	2	no	—
22	#10597276	—	4	1	no	—
23	C₁₉⋊C₃₂	—	4	1	no	—
24	C₂×D₄×C₁₉	—	4	1	no	—
25	C₂×D₄×C₁₉	—	8	1	no	—
26	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
27	D₈×C₁₉	—	4	1	no	—
28	D₈×C₁₉	—	4	1	no	—
29	C₂×C₈×C₁₉	—	2	1	no	—
30	C₂×C₈×C₁₉	—	2	1	no	—
31	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
33	C₄⋊C₄×C₁₉	—	4	1	no	—
34	C₂²⋊C₄×C₁₉	—	8	1	no	—
35	#64520	—	4	2	no	—
36	#64520	—	8	1	no	—
37	C₂×C₁₉⋊C₈	—	8	1	no	—
38	C₁₆×C₁₉	—	2	1	no	—
39	C₁₆×C₁₉	—	2	1	no	—
40	#7420	—	19	1	no	maximal, Sylow
41	C₂³×C₁₉	—	8	1	no	—
42	C₂³×C₁₉	SD₃₂	1	1	yes	characteristic
43	D₄×C₁₉	—	8	1	no	—
44	D₄×C₁₉	—	8	1	no	—
45	D₄×C₁₉	—	16	2	no	—
46	C₂×C₄×C₁₉	—	2	1	no	—
47	C₂×C₄×C₁₉	—	2	1	no	—
48	C₂×C₄×C₁₉	—	8	1	no	—
49	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
50	C₂×C₄×C₁₉	D₁₆	1	1	yes	characteristic
51	C₈×C₁₉	—	2	1	no	—
52	C₈×C₁₉	#99	1	1	yes	characteristic
53	C₈×C₁₉	#104	1	1	yes	characteristic
54	C₁₉⋊C₈	—	8	2	no	—
55	#2215	D₁₉	1	1	yes	characteristic
56	C₂²⋊C₃₂	—	19	1	no	—
57	#1381	—	19	1	no	—
58	C₂²×C₁₉	—	2	1	no	—
59	C₂²×C₁₉	—	2	1	no	—
60	C₂²×C₁₉	—	2	1	no	—
61	C₂²×C₁₉	—	16	1	no	—
62	C₂²×C₁₉	—	16	1	no	—
63	C₂²×C₁₉	—	16	1	no	—
64	C₂²×C₁₉	#357	1	1	yes	characteristic
65	C₄×C₁₉	—	2	1	no	—
66	C₄×C₁₉	—	16	1	no	—
67	C₄×C₁₉	#327	1	1	yes	characteristic
68	C₄×C₁₉	#360	1	1	yes	characteristic
69	C₂×D₁₆	Dic₁₉	1	1	yes	characteristic
70	C₂²×C₁₆	C₂×D₁₉	1	1	yes	characteristic
71	#466	—	2	1	no	—
72	C₂×#108	—	38	1	no	—
73	C₂×C₃₂	—	38	1	no	—
74	#366	Dic₁₉	1	1	yes	characteristic
75	#357	—	2	1	no	—
76	C₂×C₁₉	—	4	1	no	—
77	C₂×C₁₉	—	32	1	no	—
78	C₂×C₁₉	#1347	1	1	yes	characteristic
79	C₂×C₁₉	#1417	1	1	yes	characteristic
80	C₂×C₁₉	#1419	1	1	yes	characteristic
81	C₂×D₈	—	2	1	no	—
82	C₂×M₁₆	—	76	1	no	—
83	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
84	#121	—	4	1	no	—
85	D₁₆	—	4	1	no	—
86	C₂×C₁₆	—	4	1	no	—
87	C₂×C₁₆	C₂×Dic₁₉	1	1	yes	characteristic
88	C₂×C₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
89	#108	—	38	2	no	—
90	#108	—	76	1	no	—
91	#107	—	2	1	no	—
92	#102	—	4	1	no	—
93	C₃₂	—	76	1	no	—
94	C₁₉	#7420	1	1	yes	characteristic, complemented, Sylow
95	C₂×D₄	—	4	1	no	—
96	C₂×D₄	—	8	1	no	—
97	C₂²×C₄	#64525	1	1	yes	characteristic
98	D₈	—	4	1	no	—
99	D₈	—	4	1	no	—
100	M₁₆	—	76	2	no	—
101	M₁₆	—	152	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	152	1	no	—
105	C₂×C₈	#64526	1	1	yes	characteristic
106	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
107	C₄⋊C₄	—	4	1	no	—
108	C₂²⋊C₄	—	8	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	2	1	no	—
111	C₂³	—	8	1	no	—
112	C₂³	#10597284	1	1	yes	characteristic
113	D₄	—	8	1	no	—
114	D₄	—	8	1	no	—
115	D₄	—	16	2	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	8	1	no	—
119	C₂×C₄	#10597283	1	1	yes	characteristic
120	C₂×C₄	#10597289	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	152	2	no	—
123	C₈	#10597291	1	1	yes	characteristic
124	C₈	#10597294	1	1	yes	characteristic
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	16	1	no	—
129	C₂²	—	16	1	no	—
130	C₂²	—	16	1	no	—
131	C₂²	#11926345	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	16	1	no	—
134	C₄	#11926278	1	1	yes	characteristic
135	C₄	#11926354	1	1	yes	characteristic
136	C₂	—	4	1	no	—
137	C₂	—	32	1	no	—
138	C₂	#14561644	1	1	yes	characteristic
139	C₂	#14561819	1	1	yes	characteristic
140	C₂	#14561825	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²⋊(C₁₉⋊C₃₂)

C₂

yes

maximal, characteristic, complemented

#14561729

C₂

yes

maximal, characteristic, complemented

C₁₉×#2215

C₂

yes

maximal, characteristic

C₂×#10597276

—

C₂×C₁₉⋊C₃₂

—

C₂×C₁₉×D₁₆

C₄

yes

characteristic

C₂²×C₁₆×C₁₉

C₂²

yes

characteristic

C₁₉×#466

—

C₁₉×#366

C₄

yes

characteristic

C₁₉×#357

—

C₂×D₈×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₁₉×#121

—

C₂×#64520

—

C₁₉×D₁₆

—

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₉

D₄

yes

characteristic

C₁₉×#107

—

C₁₉×#102

—

#10597276

—

#10597276

—

C₁₉⋊C₃₂

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

D₈

yes

characteristic

D₈×C₁₉

—

D₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

SD₁₆

yes

characteristic

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₁₆×C₁₉

—

C₁₆×C₁₉

—

#7420

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

SD₃₂

yes

characteristic

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#102

yes

characteristic

C₂×C₄×C₁₉

D₁₆

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

yes

C₈×C₁₉

yes

C₁₉⋊C₈

—

#2215

D₁₉

yes

characteristic

C₂²⋊C₃₂

—

#1381

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#357

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂×D₁₆

Dic₁₉

yes

characteristic

C₂²×C₁₆

C₂×D₁₉

yes

characteristic

#466

—

C₂×#108

—

C₂×C₃₂

—

#366

Dic₁₉

yes

characteristic

#357

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

#121

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₁₉

yes

characteristic

C₂×C₁₆

C₂²⋊D₁₉

yes

—

—

—

—

C₃₂

—

C₁₉

#7420

yes

characteristic, complemented, Sylow

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

#64525

yes

characteristic

D₈

—

D₈

—

100

M₁₆

—

101

M₁₆

—

152

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

152

—

105

C₂×C₈

#64526

yes

characteristic

106

C₂×C₈

C₂²⋊Dic₁₉

yes

characteristic

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₁₆

—

110

C₁₆

—

111

C₂³

—

112

C₂³

#10597284

yes

characteristic

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#10597283

yes

characteristic

120

C₂×C₄

#10597289

yes

characteristic

121

C₈

—

122

C₈

—

152

—

123

C₈

#10597291

yes

characteristic

124

C₈

#10597294

yes

characteristic

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

#11926345

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

#11926278

yes

characteristic

135

C₄

#11926354

yes

characteristic

136

C₂

—

137

C₂

—

138

C₂

#14561644

yes

characteristic

139

C₂

#14561819

yes

characteristic

140

C₂

#14561825

yes

characteristic

	subgroup	G/H
center	C₂²	#11926345
commutator	C₂×C₁₆×C₁₉	C₂×C₄
Frattini	C₂²×C₁₆	C₂×D₁₉
Fitting, prime radical	C₁₉×#2215	C₂
socle	C₂²×C₁₉	#357
cosocle	C₂²×C₁₆×C₁₉	C₂²
hypercenter	#2215	D₁₉
hypocenter	C₁₉	#7420
LCS	G ▷ C₂×C₁₆×C₁₉ ▷ C₂×C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ #2215

subgroup

G/H

center

C₂²

#11926345

commutator

C₂×C₁₆×C₁₉

C₂×C₄

Frattini

C₂²×C₁₆

C₂×D₁₉

Fitting, prime radical

C₁₉×#2215

C₂

socle

C₂²×C₁₉

#357

cosocle

C₂²×C₁₆×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂×C₁₆×C₁₉ ▷ C₂×C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ #2215

N	H ≅ G/N	type
1. C₂²⋊(C₁₉⋊C₃₂)	C₂	semidirect
2. #14561729	C₂	semidirect
94. C₁₉	#7420	semidirect

H ≅ G/N

type

1. C₂²⋊(C₁₉⋊C₃₂)

C₂

semidirect

2. #14561729

C₂

semidirect

94. C₁₉

#7420

semidirect

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

subgroup	index
C₂²⋊(C₁₉⋊C₃₂)	2
#14561729	2
C₁₉×#2215	2
#7420	19

subgroup

index

C₂²⋊(C₁₉⋊C₃₂)

#14561729

C₁₉×#2215

#7420

Subgroups — #18070100

Conjugacy classes of subgroups

Complements

Maximal subgroups