Group #18070102 — subgroups

Subgroups — #18070102

Order 4864.

140 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 45 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×#1381	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₅₂	C₄	1	1	yes	characteristic
5	#11926785	—	2	1	no	—
6	C₂×C₁₉⋊C₃₂	—	2	1	no	—
7	#11926370	C₄	1	1	yes	characteristic
8	#11926349	—	2	1	no	—
9	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#108	—	2	1	no	—
11	C₂×M₁₆×C₁₉	—	4	1	no	—
12	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
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×#108	—	2	2	no	—
19	C₁₉×#108	—	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	D₈	1	1	yes	characteristic
25	C₂×Dic₃₈	—	4	1	no	—
26	M₁₆×C₁₉	—	4	2	no	—
27	M₁₆×C₁₉	—	8	1	no	—
28	C₂×C₈×C₁₉	—	2	1	no	—
29	C₂×C₈×C₁₉	—	2	1	no	—
30	C₂×C₈×C₁₉	—	8	1	no	—
31	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
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	#7421	—	19	1	no	maximal, Sylow
41	C₂³×C₁₉	SD₃₂	1	1	yes	characteristic
42	C₂×C₄×C₁₉	—	2	1	no	—
43	C₂×C₄×C₁₉	—	2	1	no	—
44	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
45	C₂×C₄×C₁₉	D₁₆	1	1	yes	characteristic
46	C₂×Dic₁₉	—	8	1	no	—
47	C₂×Dic₁₉	—	8	1	no	—
48	Dic₃₈	—	8	1	no	—
49	Dic₃₈	—	8	1	no	—
50	C₈×C₁₉	—	2	1	no	—
51	C₈×C₁₉	—	8	2	no	—
52	C₈×C₁₉	#99	1	1	yes	characteristic
53	C₈×C₁₉	#104	1	1	yes	characteristic
54	#2216	—	19	1	no	—
55	C₂²⋊C₃₂	—	19	1	no	—
56	#1381	D₁₉	1	1	yes	characteristic
57	C₂²×C₁₉	—	2	1	no	—
58	C₂²×C₁₉	—	2	1	no	—
59	C₂²×C₁₉	—	2	1	no	—
60	C₂²×C₁₉	#357	1	1	yes	characteristic
61	C₄×C₁₉	—	2	1	no	—
62	C₄×C₁₉	#327	1	1	yes	characteristic
63	C₄×C₁₉	#360	1	1	yes	characteristic
64	Dic₁₉	—	16	1	no	—
65	Dic₁₉	—	16	1	no	—
66	C₂×Q₃₂	—	19	1	no	—
67	C₂²×C₁₆	C₂×D₁₉	1	1	yes	characteristic
68	#467	—	38	1	no	—
69	C₂×#108	—	2	1	no	—
70	C₂×C₃₂	—	38	1	no	—
71	#366	—	19	1	no	—
72	#358	—	38	1	no	—
73	C₂×C₁₉	—	4	1	no	—
74	C₂×C₁₉	#1347	1	1	yes	characteristic
75	C₂×C₁₉	#1417	1	1	yes	characteristic
76	C₂×C₁₉	#1420	1	1	yes	characteristic
77	C₂×Q₁₆	—	38	1	no	—
78	C₂×M₁₆	—	4	1	no	—
79	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
80	C₂²⋊Q₈	—	76	1	no	—
81	Q₃₂	—	76	1	no	—
82	C₂×C₁₆	—	4	1	no	—
83	C₂×C₁₆	C₄×D₁₉	1	1	yes	characteristic
84	C₂×C₁₆	D₇₆	1	1	yes	characteristic
85	#108	—	2	2	no	—
86	#108	—	4	1	no	—
87	#107	—	38	1	no	—
88	#103	—	76	1	no	—
89	C₃₂	—	76	1	no	—
90	C₁₉	#7421	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	—	76	1	no	—
92	C₂²×C₄	#64525	1	1	yes	characteristic
93	Q₁₆	—	76	1	no	—
94	Q₁₆	—	76	1	no	—
95	M₁₆	—	4	2	no	—
96	M₁₆	—	8	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	8	1	no	—
100	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
101	C₂×C₈	#64527	1	1	yes	characteristic
102	C₄⋊C₄	—	76	1	no	—
103	C₄⋊C₄	—	152	1	no	—
104	C₂²⋊C₄	—	152	1	no	—
105	C₁₆	—	2	1	no	—
106	C₁₆	—	2	1	no	—
107	C₂³	#10597285	1	1	yes	characteristic
108	Q₈	—	152	1	no	—
109	Q₈	—	152	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	152	1	no	—
113	C₂×C₄	—	152	1	no	—
114	C₂×C₄	#10597266	1	1	yes	characteristic
115	C₂×C₄	#10597283	1	1	yes	characteristic
116	C₈	—	2	1	no	—
117	C₈	—	8	2	no	—
118	C₈	#10597262	1	1	yes	characteristic
119	C₈	#10597263	1	1	yes	characteristic
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	#11926344	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	304	1	no	—
126	C₄	—	304	1	no	—
127	C₄	#11926277	1	1	yes	characteristic
128	C₄	#11926353	1	1	yes	characteristic
129	C₂	—	4	1	no	—
130	C₂	#14561643	1	1	yes	characteristic
131	C₂	#14561818	1	1	yes	characteristic
132	C₂	#14561827	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×#1381

C₂

yes

maximal, characteristic

C₂×Dic₁₅₂

C₄

yes

characteristic

#11926785

—

C₂×C₁₉⋊C₃₂

—

#11926370

C₄

yes

characteristic

#11926349

—

C₂²×C₁₆×C₁₉

C₂²

yes

characteristic

C₂×C₁₉×#108

—

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

characteristic

#10597370

—

C₂×Dic₇₆

—

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₉

D₄

yes

characteristic

C₁₉×#108

—

C₁₉×#108

—

#10597275

—

#10597273

—

Dic₁₅₂

—

C₁₉⋊C₃₂

—

C₂²×C₄×C₁₉

D₈

yes

characteristic

C₂×Dic₃₈

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

SD₁₆

yes

characteristic

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

Dic₇₆

—

Dic₇₆

—

C₁₆×C₁₉

—

C₁₆×C₁₉

—

#7421

—

maximal, Sylow

C₂³×C₁₉

SD₃₂

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#102

yes

characteristic

C₂×C₄×C₁₉

D₁₆

yes

characteristic

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

yes

C₈×C₁₉

yes

—

C₂²⋊C₃₂

—

#1381

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#357

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

Dic₁₉

—

Dic₁₉

—

C₂×Q₃₂

—

C₂²×C₁₆

C₂×D₁₉

yes

characteristic

#467

—

C₂×#108

—

C₂×C₃₂

—

#366

—

#358

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

C₂²⋊Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₁₉

yes

characteristic

C₂×C₁₆

D₇₆

yes

—

—

—

—

C₃₂

—

C₁₉

#7421

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

#64525

yes

characteristic

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

characteristic

101

C₂×C₈

#64527

yes

characteristic

102

C₄⋊C₄

—

103

C₄⋊C₄

—

152

—

104

C₂²⋊C₄

—

152

—

105

C₁₆

—

106

C₁₆

—

107

C₂³

#10597285

yes

characteristic

108

Q₈

—

152

—

109

Q₈

—

152

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

152

—

113

C₂×C₄

—

152

—

114

C₂×C₄

#10597266

yes

characteristic

115

C₂×C₄

#10597283

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

#10597262

yes

characteristic

119

C₈

#10597263

yes

characteristic

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

#11926344

yes

characteristic

124

C₄

—

125

C₄

—

304

—

126

C₄

—

304

—

127

C₄

#11926277

yes

characteristic

128

C₄

#11926353

yes

characteristic

129

C₂

—

130

C₂

#14561643

yes

characteristic

131

C₂

#14561818

yes

characteristic

132

C₂

#14561827

yes

characteristic

	subgroup	G/H
center	C₂²	#11926344
commutator	C₂×C₁₆×C₁₉	C₂×C₄
Frattini	C₂²×C₁₆	C₂×D₁₉
Fitting	C₁₉×#1381	C₂
socle	C₂²×C₁₉	#357
prime radical	C₂³×C₁₉	SD₃₂
cosocle	C₂²×C₁₆×C₁₉	C₂²
hypercenter	#1381	D₁₉
hypocenter	C₁₉	#7421
LCS	G ▷ C₂×C₁₆×C₁₉ ▷ C₂×C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ #1381

subgroup

G/H

center

C₂²

#11926344

commutator

C₂×C₁₆×C₁₉

C₂×C₄

Frattini

C₂²×C₁₆

C₂×D₁₉

Fitting

C₁₉×#1381

C₂

socle

C₂²×C₁₉

#357

prime radical

C₂³×C₁₉

SD₃₂

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 ◁ #1381

N	H ≅ G/N	type
90. C₁₉	#7421	semidirect

H ≅ G/N

type

90. C₁₉

#7421

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×#1381	2
#7421	19

subgroup

index

#14565726

C₂²⋊(C₁₉⋊C₃₂)

C₁₉×#1381

#7421

Subgroups — #18070102

Conjugacy classes of subgroups

Complements

Maximal subgroups