Group #18070096 — subgroups

Subgroups — #18070096

Order 4864.

152 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 49 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	#14565726	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₁₉⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1380	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₅₂	C₄	1	1	yes	characteristic, complemented
5	#11926785	—	2	1	no	—
6	C₂×C₁₉⋊C₃₂	—	2	1	no	—
7	#11926370	C₄	1	1	yes	characteristic, complemented
8	#11926349	—	2	1	no	—
9	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#107	—	2	1	no	—
11	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
13	#10597370	—	4	1	no	—
14	C₂×Dic₇₆	—	2	1	no	—
15	C₂×C₁₆×C₁₉	—	4	1	no	—
16	C₂×C₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
18	C₁₉×#107	—	2	2	no	—
19	C₁₉×#107	—	4	1	no	—
20	#10597275	—	2	1	no	—
21	#10597273	—	4	1	no	—
22	Dic₁₅₂	—	4	1	no	—
23	C₁₉⋊C₃₂	—	4	1	no	—
24	C₂²×C₄×C₁₉	—	8	1	no	—
25	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
26	C₂×Dic₃₈	—	4	1	no	—
27	C₂×C₈×C₁₉	—	2	1	no	—
28	C₂×C₈×C₁₉	—	2	1	no	—
29	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
31	C₄⋊C₄×C₁₉	—	4	2	no	—
32	C₄⋊C₄×C₁₉	—	8	1	no	—
33	C₂²⋊Dic₁₉	—	8	1	no	—
34	C₄⋊Dic₁₉	—	4	1	no	—
35	Dic₁₉⋊C₄	—	8	1	no	—
36	Dic₇₆	—	4	1	no	—
37	Dic₇₆	—	4	1	no	—
38	C₁₆×C₁₉	—	2	1	no	—
39	C₁₆×C₁₉	—	2	1	no	—
40	#7419	—	19	1	no	maximal, Sylow
41	C₂³×C₁₉	D₁₆	1	1	yes	characteristic
42	C₂×C₄×C₁₉	—	2	1	no	—
43	C₂×C₄×C₁₉	—	2	1	no	—
44	C₂×C₄×C₁₉	—	8	2	no	—
45	C₂×C₄×C₁₉	—	16	1	no	—
46	C₂×C₄×C₁₉	—	16	1	no	—
47	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
48	C₂×C₄×C₁₉	SD₃₂	1	1	yes	characteristic
49	C₂×Dic₁₉	—	8	1	no	—
50	C₂×Dic₁₉	—	8	1	no	—
51	Dic₃₈	—	8	1	no	—
52	Dic₃₈	—	8	1	no	—
53	C₈×C₁₉	—	2	1	no	—
54	C₈×C₁₉	#99	1	1	yes	characteristic
55	C₈×C₁₉	#104	1	1	yes	characteristic
56	#2216	—	19	1	no	—
57	C₂²⋊C₃₂	—	19	1	no	—
58	#1380	D₁₉	1	1	yes	characteristic
59	C₂²×C₁₉	—	2	1	no	—
60	C₂²×C₁₉	—	2	1	no	—
61	C₂²×C₁₉	#357	1	1	yes	characteristic
62	C₂²×C₁₉	SD₆₄	1	1	yes	characteristic
63	C₂²×C₁₉	Q₆₄	1	1	yes	characteristic
64	C₄×C₁₉	—	2	1	no	—
65	C₄×C₁₉	—	16	2	no	—
66	C₄×C₁₉	#327	1	1	yes	characteristic
67	C₄×C₁₉	#360	1	1	yes	characteristic
68	Dic₁₉	—	16	1	no	—
69	Dic₁₉	—	16	1	no	—
70	C₂×Q₃₂	—	19	1	no	—
71	C₂²×C₁₆	C₂×D₁₉	1	1	yes	characteristic
72	#467	—	38	1	no	—
73	C₂×#107	—	2	1	no	—
74	C₂×C₃₂	—	38	1	no	—
75	#366	—	19	1	no	—
76	#358	—	38	1	no	—
77	C₂×C₁₉	—	2	1	no	—
78	C₂×C₁₉	—	2	1	no	—
79	C₂×C₁₉	#1347	1	1	yes	characteristic
80	C₂×C₁₉	#1416	1	1	yes	characteristic
81	C₂×C₁₉	#1418	1	1	yes	characteristic
82	C₂×Q₁₆	—	38	1	no	—
83	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
84	C₂²⋊Q₈	—	76	1	no	—
85	C₂×C₄⋊C₄	—	4	1	no	—
86	Q₃₂	—	76	1	no	—
87	C₂×C₁₆	—	4	1	no	—
88	C₂×C₁₆	C₄×D₁₉	1	1	yes	characteristic
89	C₂×C₁₆	D₇₆	1	1	yes	characteristic
90	#107	—	2	2	no	—
91	#107	—	4	1	no	—
92	#107	—	38	1	no	—
93	#103	—	76	1	no	—
94	C₃₂	—	76	1	no	—
95	C₁₉	#7419	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₈	—	76	1	no	—
97	C₂²×C₄	—	8	1	no	—
98	C₂²×C₄	#64525	1	1	yes	characteristic
99	Q₁₆	—	76	1	no	—
100	Q₁₆	—	76	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
104	C₂×C₈	#64527	1	1	yes	characteristic
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	8	1	no	—
107	C₄⋊C₄	—	76	1	no	—
108	C₄⋊C₄	—	152	1	no	—
109	C₂²⋊C₄	—	152	1	no	—
110	C₁₆	—	2	1	no	—
111	C₁₆	—	2	1	no	—
112	C₂³	#10597283	1	1	yes	characteristic
113	Q₈	—	152	1	no	—
114	Q₈	—	152	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	8	2	no	—
118	C₂×C₄	—	16	1	no	—
119	C₂×C₄	—	16	1	no	—
120	C₂×C₄	—	152	1	no	—
121	C₂×C₄	—	152	1	no	—
122	C₂×C₄	#10597266	1	1	yes	characteristic
123	C₂×C₄	#10597285	1	1	yes	characteristic
124	C₈	—	2	1	no	—
125	C₈	#10597262	1	1	yes	characteristic
126	C₈	#10597263	1	1	yes	characteristic
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	#11926344	1	1	yes	characteristic
130	C₂²	#11926382	1	1	yes	characteristic
131	C₂²	#11926384	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	16	2	no	—
134	C₄	—	304	1	no	—
135	C₄	—	304	1	no	—
136	C₄	#11926277	1	1	yes	characteristic
137	C₄	#11926353	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	2	1	no	—
140	C₂	#14561643	1	1	yes	characteristic
141	C₂	#14561815	1	1	yes	characteristic
142	C₂	#14561821	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14565726

C₂

yes

maximal, characteristic

C₂²⋊(C₁₉⋊C₃₂)

C₂

yes

maximal, characteristic

C₁₉×#1380

C₂

yes

maximal, characteristic

C₂×Dic₁₅₂

C₄

yes

characteristic, complemented

#11926785

—

C₂×C₁₉⋊C₃₂

—

#11926370

C₄

yes

characteristic, complemented

#11926349

—

C₂²×C₁₆×C₁₉

C₂²

yes

characteristic

C₂×C₁₉×#107

—

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄⋊C₄×C₁₉

—

#10597370

—

C₂×Dic₇₆

—

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₉

D₄

yes

characteristic

C₁₉×#107

—

C₁₉×#107

—

#10597275

—

#10597273

—

Dic₁₅₂

—

C₁₉⋊C₃₂

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

D₈

yes

characteristic

C₂×Dic₃₈

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

SD₁₆

yes

characteristic

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

Dic₇₆

—

Dic₇₆

—

C₁₆×C₁₉

—

C₁₆×C₁₉

—

#7419

—

maximal, Sylow

C₂³×C₁₉

D₁₆

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#102

yes

characteristic

C₂×C₄×C₁₉

SD₃₂

yes

characteristic

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₈×C₁₉

yes

C₈×C₁₉

yes

—

C₂²⋊C₃₂

—

#1380

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#357

yes

characteristic

C₂²×C₁₉

SD₆₄

yes

characteristic

C₂²×C₁₉

Q₆₄

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

Dic₁₉

—

Dic₁₉

—

C₂×Q₃₂

—

C₂²×C₁₆

C₂×D₁₉

yes

characteristic

#467

—

C₂×#107

—

C₂×C₃₂

—

#366

—

#358

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×Q₁₆

—

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₁₉

yes

characteristic

C₂×C₁₆

D₇₆

yes

—

—

—

—

C₃₂

—

C₁₉

#7419

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#64525

yes

characteristic

Q₁₆

—

100

Q₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

characteristic

104

C₂×C₈

#64527

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

152

—

109

C₂²⋊C₄

—

152

—

110

C₁₆

—

111

C₁₆

—

112

C₂³

#10597283

yes

characteristic

113

Q₈

—

152

—

114

Q₈

—

152

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

152

—

121

C₂×C₄

—

152

—

122

C₂×C₄

#10597266

yes

characteristic

123

C₂×C₄

#10597285

yes

characteristic

124

C₈

—

125

C₈

#10597262

yes

characteristic

126

C₈

#10597263

yes

characteristic

127

C₂²

—

128

C₂²

—

129

C₂²

#11926344

yes

characteristic

130

C₂²

#11926382

yes

characteristic

131

C₂²

#11926384

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

304

—

135

C₄

—

304

—

136

C₄

#11926277

yes

characteristic

137

C₄

#11926353

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

#14561643

yes

characteristic

141

C₂

#14561815

yes

characteristic

142

C₂

#14561821

yes

characteristic

	subgroup	G/H
center	C₂²	#11926344
commutator	C₂×C₁₆×C₁₉	C₂×C₄
Frattini	C₂²×C₁₆	C₂×D₁₉
Fitting	C₁₉×#1380	C₂
socle	C₂²×C₁₉	#357
prime radical	C₂³×C₁₉	D₁₆
cosocle	C₂²×C₁₆×C₁₉	C₂²
hypercenter	#1380	D₁₉
hypocenter	C₁₉	#7419
LCS	G ▷ C₂×C₁₆×C₁₉ ▷ C₂×C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ #1380

subgroup

G/H

center

C₂²

#11926344

commutator

C₂×C₁₆×C₁₉

C₂×C₄

Frattini

C₂²×C₁₆

C₂×D₁₉

Fitting

C₁₉×#1380

C₂

socle

C₂²×C₁₉

#357

prime radical

C₂³×C₁₉

D₁₆

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 ◁ #1380

N	H ≅ G/N	type
4. C₂×Dic₁₅₂	C₄	semidirect
7. #11926370	C₄	semidirect
95. C₁₉	#7419	semidirect

H ≅ G/N

type

4. C₂×Dic₁₅₂

C₄

semidirect

7. #11926370

C₄

semidirect

95. C₁₉

#7419

semidirect

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

subgroup	index
#14565726	2
C₂²⋊(C₁₉⋊C₃₂)	2
C₁₉×#1380	2
#7419	19

subgroup

index

#14565726

C₂²⋊(C₁₉⋊C₃₂)

C₁₉×#1380

#7419

Subgroups — #18070096

Conjugacy classes of subgroups

Complements

Maximal subgroups