Group #18075154 — subgroups

Subgroups — #18075154

Order 4864.

910 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 139 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	#14563709	C₂	1	6	yes	maximal
2	C₂²×C₁₉×#126	C₂	1	1	yes	maximal, characteristic
3	#11926400	C₂²	1	4	yes	—
4	#11926311	—	2	12	no	—
5	C₂×C₁₉×#126	—	2	12	no	—
6	C₂²×C₄⋊C₄×C₁₉	C₄	1	3	yes	complemented
7	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	3	yes	—
8	C₂²×C₄²×C₁₉	C₄	1	1	yes	characteristic, complemented
9	C₂⁵×C₁₉	C₂³	1	1	yes	characteristic
10	C₂³×C₄×C₁₉	C₂×C₄	1	3	yes	—
11	C₂³×C₄×C₁₉	C₂×C₄	1	3	yes	—
12	C₁₉×#126	—	2	32	no	—
13	C₂×C₄⋊C₄×C₁₉	—	2	6	no	—
14	C₂×C₄⋊C₄×C₁₉	—	2	6	no	—
15	C₂×C₄⋊C₄×C₁₉	—	2	6	no	—
16	C₂×C₂²⋊C₄×C₁₉	—	2	6	no	—
17	C₂×C₂²⋊C₄×C₁₉	—	2	6	no	—
18	C₂×C₂²⋊C₄×C₁₉	D₄	1	12	yes	—
19	C₂×C₄²×C₁₉	—	2	6	no	—
20	C₂×C₂²⋊Dic₁₉	—	2	12	no	—
21	C₂⁴×C₁₉	—	4	6	no	—
22	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
23	C₂⁴×C₁₉	C₂×D₄	1	6	yes	—
24	C₂²×C₄×C₁₉	—	2	3	no	—
25	C₂²×C₄×C₁₉	—	2	3	no	—
26	C₂²×C₄×C₁₉	—	2	3	no	—
27	C₂²×C₄×C₁₉	—	2	3	no	—
28	C₂²×C₄×C₁₉	—	4	6	no	—
29	C₂²×C₄×C₁₉	—	4	6	no	—
30	C₂²×C₄×C₁₉	C₂²⋊C₄	1	6	yes	—
31	C₂²×C₄×C₁₉	C₂²⋊C₄	1	6	yes	—
32	C₂²×Dic₁₉	—	4	4	no	—
33	C₄⋊C₄×C₁₉	—	2	24	no	—
34	C₂²⋊C₄×C₁₉	—	2	24	no	—
35	C₄²×C₁₉	—	2	8	no	—
36	C₂²⋊Dic₁₉	—	4	24	no	—
37	#8545	—	19	1	no	maximal, Sylow
38	C₂³×C₁₉	—	2	6	no	—
39	C₂³×C₁₉	—	2	12	no	—
40	C₂³×C₁₉	—	4	6	no	—
41	C₂³×C₁₉	—	4	6	no	—
42	C₂³×C₁₉	—	8	8	no	—
43	C₂³×C₁₉	C₂×C₂²⋊C₄	1	3	yes	—
44	C₂³×C₁₉	#120	1	4	yes	—
45	C₂×C₄×C₁₉	—	2	6	no	—
46	C₂×C₄×C₁₉	—	2	6	no	—
47	C₂×C₄×C₁₉	—	4	6	no	—
48	C₂×C₄×C₁₉	—	4	6	no	—
49	C₂×C₄×C₁₉	—	4	6	no	—
50	C₂×C₄×C₁₉	—	4	6	no	—
51	C₂×C₄×C₁₉	—	4	6	no	—
52	C₂×C₄×C₁₉	—	4	6	no	—
53	C₂×Dic₁₉	—	8	12	no	—
54	C₂²×#126	D₁₉	1	1	yes	characteristic
55	#1819	—	19	6	no	—
56	C₂²×C₁₉	—	2	8	no	—
57	C₂²×C₁₉	—	2	3	no	—
58	C₂²×C₁₉	—	2	3	no	—
59	C₂²×C₁₉	—	2	12	no	—
60	C₂²×C₁₉	—	8	12	no	—
61	C₂²×C₁₉	#379	1	1	yes	characteristic
62	C₂²×C₁₉	#410	1	6	yes	—
63	C₄×C₁₉	—	4	6	no	—
64	C₄×C₁₉	—	4	6	no	—
65	Dic₁₉	—	16	4	no	—
66	C₂×#126	—	2	12	no	—
67	C₂²×C₄⋊C₄	Dic₁₉	1	3	yes	complemented
68	C₂²×C₂²⋊C₄	C₂×D₁₉	1	3	yes	—
69	C₂²×C₄²	Dic₁₉	1	1	yes	characteristic, complemented
70	#379	—	19	4	no	—
71	#342	—	38	12	no	—
72	C₂×C₁₉	—	2	6	no	—
73	C₂×C₁₉	—	4	4	no	—
74	C₂×C₁₉	#1782	1	3	yes	—
75	C₂⁵	C₂²×D₁₉	1	1	yes	characteristic
76	C₂³×C₄	C₂×Dic₁₉	1	3	yes	—
77	C₂³×C₄	C₂×Dic₁₉	1	3	yes	—
78	#126	—	2	32	no	—
79	C₂×C₄⋊C₄	—	2	6	no	—
80	C₂×C₄⋊C₄	—	2	6	no	—
81	C₂×C₄⋊C₄	—	2	6	no	—
82	C₂×C₂²⋊C₄	—	2	6	no	—
83	C₂×C₂²⋊C₄	—	2	6	no	—
84	C₂×C₂²⋊C₄	—	38	12	no	—
85	C₂×C₂²⋊C₄	C₂²⋊D₁₉	1	12	yes	—
86	C₂×C₄²	—	2	6	no	—
87	C₁₉	#8545	1	1	yes	characteristic, complemented, Sylow
88	C₂⁴	—	4	6	no	—
89	C₂⁴	C₂²×Dic₁₉	1	1	yes	characteristic
90	C₂⁴	C₂×C₂²⋊D₁₉	1	6	yes	—
91	C₂²×C₄	—	2	3	no	—
92	C₂²×C₄	—	2	3	no	—
93	C₂²×C₄	—	2	3	no	—
94	C₂²×C₄	—	2	3	no	—
95	C₂²×C₄	—	4	6	no	—
96	C₂²×C₄	—	4	6	no	—
97	C₂²×C₄	—	76	4	no	—
98	C₂²×C₄	C₂²⋊Dic₁₉	1	6	yes	—
99	C₂²×C₄	C₂²⋊Dic₁₉	1	6	yes	—
100	C₄⋊C₄	—	2	24	no	—
101	C₂²⋊C₄	—	2	24	no	—
102	C₂²⋊C₄	—	76	24	no	—
103	C₄²	—	2	8	no	—
104	C₂³	—	2	6	no	—
105	C₂³	—	2	12	no	—
106	C₂³	—	4	6	no	—
107	C₂³	—	4	6	no	—
108	C₂³	—	8	8	no	—
109	C₂³	C₂×C₂²⋊Dic₁₉	1	3	yes	—
110	C₂³	#10597399	1	4	yes	—
111	C₂×C₄	—	2	6	no	—
112	C₂×C₄	—	2	6	no	—
113	C₂×C₄	—	4	6	no	—
114	C₂×C₄	—	4	6	no	—
115	C₂×C₄	—	4	6	no	—
116	C₂×C₄	—	4	6	no	—
117	C₂×C₄	—	4	6	no	—
118	C₂×C₄	—	4	6	no	—
119	C₂×C₄	—	152	12	no	—
120	C₂²	—	2	8	no	—
121	C₂²	—	2	3	no	—
122	C₂²	—	2	3	no	—
123	C₂²	—	2	12	no	—
124	C₂²	—	8	12	no	—
125	C₂²	#11926400	1	1	yes	characteristic
126	C₂²	#11926518	1	6	yes	—
127	C₄	—	4	6	no	—
128	C₄	—	4	6	no	—
129	C₄	—	304	4	no	—
130	C₂	—	2	6	no	—
131	C₂	—	4	4	no	—
132	C₂	#14563553	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14563709

C₂

yes

maximal

C₂²×C₁₉×#126

C₂

yes

maximal, characteristic

#11926400

C₂²

yes

—

#11926311

—

C₂×C₁₉×#126

—

C₂²×C₄⋊C₄×C₁₉

C₄

yes

complemented

C₂²×C₂²⋊C₄×C₁₉

C₂²

yes

—

C₂²×C₄²×C₁₉

C₄

yes

characteristic, complemented

C₂⁵×C₁₉

C₂³

yes

characteristic

C₂³×C₄×C₁₉

C₂×C₄

yes

—

C₂³×C₄×C₁₉

C₂×C₄

yes

—

C₁₉×#126

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₂²⋊C₄×C₁₉

—

C₂×C₂²⋊C₄×C₁₉

—

C₂×C₂²⋊C₄×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

—

C₂×C₂²⋊Dic₁₉

—

C₂⁴×C₁₉

—

C₂⁴×C₁₉

C₂²×C₄

yes

characteristic

C₂⁴×C₁₉

C₂×D₄

yes

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×Dic₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₂²⋊Dic₁₉

—

#8545

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

C₂×C₂²⋊C₄

yes

—

C₂³×C₁₉

#120

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×Dic₁₉

—

C₂²×#126

D₁₉

yes

characteristic

#1819

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#379

yes

characteristic

C₂²×C₁₉

#410

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

Dic₁₉

—

C₂×#126

—

C₂²×C₄⋊C₄

Dic₁₉

yes

complemented

C₂²×C₂²⋊C₄

C₂×D₁₉

yes

—

C₂²×C₄²

Dic₁₉

yes

characteristic, complemented

#379

—

#342

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

#1782

yes

—

C₂⁵

C₂²×D₁₉

yes

characteristic

C₂³×C₄

C₂×Dic₁₉

yes

—

C₂³×C₄

C₂×Dic₁₉

yes

—

#126

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂²⋊D₁₉

yes

—

C₂×C₄²

—

C₁₉

#8545

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂⁴

C₂²×Dic₁₉

yes

characteristic

C₂⁴

C₂×C₂²⋊D₁₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₁₉

yes

—

C₂²×C₄

C₂²⋊Dic₁₉

yes

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

C₂×C₂²⋊Dic₁₉

yes

—

110

C₂³

#10597399

yes

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

152

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

#11926400

yes

characteristic

126

C₂²

#11926518

yes

—

127

C₄

—

128

C₄

—

129

C₄

—

304

—

130

C₂

—

131

C₂

—

132

C₂

#14563553

yes

—

	subgroup	G/H
center	C₂²	#11926400
commutator	C₂⁴×C₁₉	C₂²×C₄
Frattini	C₂⁵	C₂²×D₁₉
Fitting	C₂²×C₁₉×#126	C₂
socle	C₂²×C₁₉	#379
prime radical, cosocle	C₂⁵×C₁₉	C₂³
hypercenter	C₂²×#126	D₁₉
hypocenter	C₁₉	#8545
LCS	G ▷ C₂⁴×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂²×#126

subgroup

G/H

center

C₂²

#11926400

commutator

C₂⁴×C₁₉

C₂²×C₄

Frattini

C₂⁵

C₂²×D₁₉

Fitting

C₂²×C₁₉×#126

C₂

socle

C₂²×C₁₉

#379

prime radical, cosocle

C₂⁵×C₁₉

C₂³

hypercenter

C₂²×#126

D₁₉

hypocenter

C₁₉

#8545

LCS

G ▷ C₂⁴×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂²×#126

N	H ≅ G/N	type
6. C₂²×C₄⋊C₄×C₁₉	C₄	semidirect
8. C₂²×C₄²×C₁₉	C₄	semidirect
67. C₂²×C₄⋊C₄	Dic₁₉	semidirect
69. C₂²×C₄²	Dic₁₉	semidirect
87. C₁₉	#8545	semidirect

H ≅ G/N

type

6. C₂²×C₄⋊C₄×C₁₉

C₄

semidirect

8. C₂²×C₄²×C₁₉

C₄

semidirect

67. C₂²×C₄⋊C₄

Dic₁₉

semidirect

69. C₂²×C₄²

Dic₁₉

semidirect

87. C₁₉

#8545

semidirect

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

subgroup	index
#14563709	2
C₂²×C₁₉×#126	2
#8545	19

subgroup

index

#14563709

C₂²×C₁₉×#126

#8545

Subgroups — #18075154

Conjugacy classes of subgroups

Complements

Maximal subgroups