Group #14516638 — subgroups

Subgroups — C₂²×#92162 #14516638

Order 1944.

439 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 62 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×#6857	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#92162	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃⋊#272	C₃	1	1	yes	maximal, characteristic
4	C₂²×(C₃⋊D₃)⋊C₉	—	3	1	no	maximal
5	C₂×#6857	C₂²	1	3	yes	complemented
6	#92162	C₂²	1	4	yes	direct product factor
7	C₂²×C₃×3₋¹⁺²	—	2	1	no	—
8	C₂²×C₃×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
9	C₂×C₃⋊#272	C₂×C₃	1	6	yes	—
10	C₂²×C₃²⋊C₉	D₃	1	1	yes	characteristic, complemented
11	C₂×(C₃⋊D₃)⋊C₉	—	3	6	no	—
12	#6857	C₂³	1	1	yes	characteristic, complemented, Sylow
13	C₂²×C₃×C₃⋊D₃	—	3	1	no	—
14	C₂²×C₃×C₃⋊D₃	—	9	1	no	—
15	C₂²×#272	—	9	3	no	—
16	C₂²×D₃×C₉	—	9	1	no	—
17	C₂×C₃×3₋¹⁺²	—	2	3	no	—
18	C₂×C₃×H(3,𝔽₃)	C₂²×C₃	1	3	yes	—
19	C₃⋊#272	C₂²×C₃	1	4	yes	—
20	C₂×C₃²⋊C₉	C₂×D₃	1	3	yes	complemented
21	(C₃⋊D₃)⋊C₉	—	3	4	no	—
22	C₂²×C₃³	—	3	1	no	—
23	C₂²×C₃³	C₃×D₃	1	1	yes	characteristic
24	C₂×C₃×C₃⋊D₃	—	3	6	no	—
25	C₂×C₃×C₃⋊D₃	—	9	6	no	—
26	C₂²×3₋¹⁺²	—	6	3	no	—
27	C₂²×H(3,𝔽₃)	—	3	3	no	—
28	C₂²×C₃×C₉	—	3	1	no	—
29	C₂²×C₃×C₉	—	6	1	no	—
30	C₂×#272	—	9	18	no	—
31	C₂×D₃×C₉	—	9	6	no	—
32	C₃×3₋¹⁺²	—	2	1	no	—
33	C₃×H(3,𝔽₃)	C₂³×C₃	1	1	yes	characteristic
34	C₃²⋊C₉	C₂²×D₃	1	1	yes	characteristic, complemented
35	C₂²×C₃⋊D₃	—	3	1	no	—
36	C₂²×C₃⋊D₃	—	9	1	no	—
37	C₂²×C₃×D₃	—	9	1	no	—
38	C₂²×C₃×D₃	—	27	1	no	—
39	C₂²×C₃×D₃	—	27	3	no	—
40	C₂³×C₉	—	27	1	no	—
41	C₂×C₃³	—	3	3	no	—
42	C₂×C₃³	C₂×C₃×D₃	1	3	yes	—
43	C₃×C₃⋊D₃	—	3	4	no	—
44	C₃×C₃⋊D₃	—	9	4	no	—
45	C₂×3₋¹⁺²	—	6	9	no	—
46	C₂×H(3,𝔽₃)	—	3	9	no	—
47	C₂×C₃×C₉	—	3	3	no	—
48	C₂×C₃×C₉	—	6	3	no	—
49	#272	—	9	12	no	—
50	D₃×C₉	—	9	4	no	—
51	C₂²×C₃²	—	3	1	no	—
52	C₂²×C₃²	—	3	3	no	—
53	C₂²×C₃²	—	3	1	no	—
54	C₂²×C₃²	—	6	1	no	—
55	C₂²×C₃²	—	9	1	no	—
56	C₂²×C₃²	—	18	1	no	—
57	C₂²×C₃²	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
58	C₂×C₃⋊D₃	—	3	6	no	—
59	C₂×C₃⋊D₃	—	9	6	no	—
60	C₂×C₃×D₃	—	9	6	no	—
61	C₂×C₃×D₃	—	27	6	no	—
62	C₂×C₃×D₃	—	27	18	no	—
63	C₂²×C₉	—	6	3	no	—
64	C₂²×C₉	—	9	1	no	—
65	C₂²×C₉	—	27	6	no	—
66	C₃³	—	3	1	no	—
67	C₃³	C₂²×C₃×D₃	1	1	yes	characteristic
68	3₋¹⁺²	—	6	3	no	—
69	H(3,𝔽₃)	—	3	3	no	—
70	C₃×C₉	—	3	1	no	—
71	C₃×C₉	—	6	1	no	—
72	C₂³×C₃	—	27	1	no	—
73	C₂²×D₃	—	9	1	no	—
74	C₂²×D₃	—	27	1	no	—
75	C₂²×D₃	—	27	3	no	—
76	C₂×C₃²	—	3	3	no	—
77	C₂×C₃²	—	3	9	no	—
78	C₂×C₃²	—	3	3	no	—
79	C₂×C₃²	—	6	3	no	—
80	C₂×C₃²	—	9	3	no	—
81	C₂×C₃²	—	18	3	no	—
82	C₂×C₃²	C₂×(C₃⋊D₃)⋊C₃	1	3	yes	—
83	C₃⋊D₃	—	3	4	no	—
84	C₃⋊D₃	—	9	4	no	—
85	C₃×D₃	—	9	4	no	—
86	C₃×D₃	—	27	4	no	—
87	C₃×D₃	—	27	12	no	—
88	C₂×C₉	—	6	9	no	—
89	C₂×C₉	—	9	3	no	—
90	C₂×C₉	—	27	4	no	—
91	C₂²×C₃	—	2	1	no	—
92	C₂²×C₃	—	9	1	no	—
93	C₂²×C₃	—	9	3	no	—
94	C₂²×C₃	—	27	6	no	—
95	C₂²×C₃	#4175	1	1	yes	characteristic
96	C₂²×C₃	#4176	1	1	yes	characteristic
97	C₂×D₃	—	9	6	no	—
98	C₂×D₃	—	27	6	no	—
99	C₂×D₃	—	27	18	no	—
100	C₃²	—	3	1	no	—
101	C₃²	—	3	3	no	—
102	C₃²	—	3	1	no	—
103	C₃²	—	6	1	no	—
104	C₃²	—	9	1	no	—
105	C₃²	—	18	1	no	—
106	C₃²	C₂²×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
107	C₉	—	6	3	no	—
108	C₉	—	9	1	no	—
109	C₂³	—	27	1	no	Sylow
110	C₂×C₃	—	2	3	no	—
111	C₂×C₃	—	9	3	no	—
112	C₂×C₃	—	9	9	no	—
113	C₂×C₃	—	27	4	no	—
114	C₂×C₃	C₂×#4175	1	3	yes	—
115	C₂×C₃	C₂×#4176	1	3	yes	—
116	D₃	—	9	4	no	—
117	D₃	—	27	4	no	—
118	D₃	—	27	12	no	—
119	C₂²	—	27	6	no	—
120	C₂²	#92162	1	1	yes	characteristic, direct product factor
121	C₃	—	2	1	no	—
122	C₃	—	9	1	no	—
123	C₃	—	9	3	no	—
124	C₃	C₂²×#4175	1	1	yes	characteristic
125	C₃	C₂²×#4176	1	1	yes	characteristic
126	C₂	—	27	4	no	—
127	C₂	C₂×#92162	1	3	yes	direct product factor

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#6857

C₂

yes

maximal, characteristic, complemented

C₂×#92162

C₂

yes

maximal, direct product factor

C₂²×C₃⋊#272

C₃

yes

maximal, characteristic

C₂²×(C₃⋊D₃)⋊C₉

—

maximal

C₂×#6857

C₂²

yes

complemented

#92162

C₂²

yes

direct product factor

C₂²×C₃×3₋¹⁺²

—

C₂²×C₃×H(3,𝔽₃)

C₂×C₃

yes

characteristic

C₂×C₃⋊#272

C₂×C₃

yes

—

C₂²×C₃²⋊C₉

D₃

yes

characteristic, complemented

C₂×(C₃⋊D₃)⋊C₉

—

#6857

C₂³

yes

characteristic, complemented, Sylow

C₂²×C₃×C₃⋊D₃

—

C₂²×C₃×C₃⋊D₃

—

C₂²×#272

—

C₂²×D₃×C₉

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃×H(3,𝔽₃)

C₂²×C₃

yes

—

C₃⋊#272

C₂²×C₃

yes

—

C₂×C₃²⋊C₉

C₂×D₃

yes

complemented

(C₃⋊D₃)⋊C₉

—

C₂²×C₃³

—

C₂²×C₃³

C₃×D₃

yes

characteristic

C₂×C₃×C₃⋊D₃

—

C₂×C₃×C₃⋊D₃

—

C₂²×3₋¹⁺²

—

C₂²×H(3,𝔽₃)

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂×#272

—

C₂×D₃×C₉

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

C₂³×C₃

yes

characteristic

C₃²⋊C₉

C₂²×D₃

yes

characteristic, complemented

C₂²×C₃⋊D₃

—

C₂²×C₃⋊D₃

—

C₂²×C₃×D₃

—

C₂²×C₃×D₃

—

C₂²×C₃×D₃

—

C₂³×C₉

—

C₂×C₃³

—

C₂×C₃³

C₂×C₃×D₃

yes

—

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₂×3₋¹⁺²

—

C₂×H(3,𝔽₃)

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

#272

—

D₃×C₉

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

(C₃⋊D₃)⋊C₃

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₃³

—

C₃³

C₂²×C₃×D₃

yes

characteristic

3₋¹⁺²

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₂³×C₃

—

C₂²×D₃

—

C₂²×D₃

—

C₂²×D₃

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂×(C₃⋊D₃)⋊C₃

yes

—

C₃⋊D₃

—

C₃⋊D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

yes

C₂²×C₃

yes

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

C₂²×(C₃⋊D₃)⋊C₃

yes

characteristic

107

C₉

—

108

C₉

—

109

C₂³

—

Sylow

110

C₂×C₃

—

111

C₂×C₃

—

112

C₂×C₃

—

113

C₂×C₃

—

114

C₂×C₃

C₂×#4175

yes

—

115

C₂×C₃

C₂×#4176

yes

—

116

D₃

—

117

D₃

—

118

D₃

—

119

C₂²

—

120

C₂²

#92162

yes

characteristic, direct product factor

121

C₃

—

122

C₃

—

123

C₃

—

124

C₃

C₂²×#4175

yes

characteristic

125

C₃

C₂²×#4176

yes

characteristic

126

C₂

—

127

C₂

C₂×#92162

yes

direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃	#4175
commutator, cosocle, hypocenter	C₃×H(3,𝔽₃)	C₂³×C₃
Frattini	C₃³	C₂²×C₃×D₃
Fitting	C₂²×#6857	C₂
socle	C₂²×C₃²	(C₃⋊D₃)⋊C₃
prime radical	C₂²×C₃⋊#272	C₃
LCS	G ▷ C₃×H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃

subgroup

G/H

center, hypercenter

C₂²×C₃

#4175

commutator, cosocle, hypocenter

C₃×H(3,𝔽₃)

C₂³×C₃

Frattini

C₃³

C₂²×C₃×D₃

Fitting

C₂²×#6857

C₂

socle

C₂²×C₃²

(C₃⋊D₃)⋊C₃

prime radical

C₂²×C₃⋊#272

C₃

LCS

G ▷ C₃×H(3,𝔽₃)

UCS

C₁ ◁ C₂²×C₃

N	H ≅ G/N	type
1. C₂²×#6857	C₂	semidirect
2. C₂×#92162	C₂	direct (N × H)
5. C₂×#6857	C₂²	semidirect
6. #92162	C₂²	direct (N × H)
10. C₂²×C₃²⋊C₉	D₃	semidirect
12. #6857	C₂³	semidirect
20. C₂×C₃²⋊C₉	C₂×D₃	semidirect
34. C₃²⋊C₉	C₂²×D₃	semidirect
120. C₂²	#92162	direct (N × H)
127. C₂	C₂×#92162	direct (N × H)

H ≅ G/N

type

1. C₂²×#6857

C₂

semidirect

2. C₂×#92162

C₂

direct (N × H)

5. C₂×#6857

C₂²

semidirect

6. #92162

C₂²

direct (N × H)

10. C₂²×C₃²⋊C₉

D₃

semidirect

12. #6857

C₂³

semidirect

20. C₂×C₃²⋊C₉

C₂×D₃

semidirect

34. C₃²⋊C₉

C₂²×D₃

semidirect

120. C₂²

#92162

direct (N × H)

127. C₂

C₂×#92162

direct (N × H)

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

subgroup	index
C₂²×#6857	2
C₂×#92162	2
C₂²×C₃⋊#272	3
C₂²×(C₃⋊D₃)⋊C₉	3

subgroup

index

C₂²×#6857

C₂×#92162

C₂²×C₃⋊#272

C₂²×(C₃⋊D₃)⋊C₉

Subgroups — C₂²×#92162 #14516638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×#92162 #14516638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#92162 #14516638