Group #14517385 — subgroups

Subgroups — #14517385

Order 1944.

228 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 63 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	C₄×#6910	C₂	1	1	yes	maximal, characteristic
2	#11757587	C₂	1	2	yes	maximal
3	C₃×#6229	C₃	1	1	yes	maximal, characteristic
4	Dic₆×3₋¹⁺²	—	3	1	no	maximal
5	#10619613	C₃	1	3	yes	maximal, complemented
6	C₂×#6910	C₂²	1	1	yes	characteristic
7	C₃×C₄×3₋¹⁺²	—	2	1	no	—
8	C₃×C₄×3₋¹⁺²	D₃	1	1	yes	characteristic
9	C₃×C₄×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
10	C₃×#1084	C₂×C₃	1	2	yes	—
11	Dic₃×3₋¹⁺²	—	3	2	no	—
12	C₄×#731	—	2	3	no	—
13	C₄×#731	C₂×C₃	1	3	yes	—
14	#66323	C₂×C₃	1	6	yes	—
15	#6910	Q₈	1	1	yes	characteristic, complemented, Sylow
16	C₃²×Dic₆	—	3	1	no	—
17	C₃²×Dic₆	—	9	1	no	—
18	Q₈×3₋¹⁺²	—	9	1	no	—
19	#6229	C₃²	1	1	yes	characteristic
20	C₉×Dic₆	—	3	3	no	—
21	C₂×C₃×3₋¹⁺²	—	2	1	no	—
22	C₂×C₃×3₋¹⁺²	C₂×D₃	1	1	yes	characteristic
23	C₂×C₃×H(3,𝔽₃)	C₂²×C₃	1	1	yes	characteristic
24	C₂×#731	—	2	3	no	—
25	C₂×#731	C₂²×C₃	1	3	yes	—
26	C₃³×C₄	—	3	1	no	—
27	C₃³×C₄	C₃×D₃	1	1	yes	characteristic
28	C₃²×Dic₃	—	3	2	no	—
29	C₃²×Dic₃	—	9	2	no	—
30	C₄×3₋¹⁺²	—	2	3	no	—
31	C₄×3₋¹⁺²	—	3	1	no	—
32	C₄×3₋¹⁺²	—	6	1	no	—
33	C₄×3₋¹⁺²	—	6	1	no	—
34	C₄×3₋¹⁺²	—	6	1	no	—
35	C₄×3₋¹⁺²	—	9	2	no	—
36	C₄×H(3,𝔽₃)	—	2	1	no	—
37	C₄×H(3,𝔽₃)	—	6	1	no	—
38	C₄×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
39	C₃×C₄×C₉	—	6	1	no	—
40	C₃×C₄×C₉	C₃×D₃	1	3	yes	—
41	#1084	C₂×C₃²	1	2	yes	—
42	C₉×Dic₃	—	3	6	no	—
43	C₃×3₋¹⁺²	—	2	1	no	—
44	C₃×3₋¹⁺²	Dic₆	1	1	yes	characteristic, complemented
45	C₃×H(3,𝔽₃)	C₃×Q₈	1	1	yes	characteristic
46	#731	—	2	3	no	—
47	#731	C₃×Q₈	1	3	yes	complemented
48	C₃²×Q₈	—	9	1	no	—
49	C₃×Dic₆	—	3	1	no	—
50	C₃×Dic₆	—	9	1	no	—
51	C₃×Dic₆	—	9	1	no	—
52	C₃×Dic₆	—	9	3	no	—
53	Q₈×C₉	—	9	3	no	—
54	C₂×C₃³	—	3	1	no	—
55	C₂×C₃³	C₂×C₃×D₃	1	1	yes	characteristic
56	C₂×3₋¹⁺²	—	2	3	no	—
57	C₂×3₋¹⁺²	—	3	1	no	—
58	C₂×3₋¹⁺²	—	6	1	no	—
59	C₂×3₋¹⁺²	—	6	1	no	—
60	C₂×3₋¹⁺²	—	6	1	no	—
61	C₂×H(3,𝔽₃)	—	2	1	no	—
62	C₂×H(3,𝔽₃)	—	6	1	no	—
63	C₂×H(3,𝔽₃)	C₂²×C₃²	1	1	yes	characteristic
64	C₂×C₃×C₉	—	6	1	no	—
65	C₂×C₃×C₉	C₂×C₃×D₃	1	3	yes	—
66	C₃²×C₄	—	2	1	no	—
67	C₃²×C₄	—	3	1	no	—
68	C₃²×C₄	—	6	1	no	—
69	C₃²×C₄	—	9	1	no	—
70	C₃²×C₄	—	9	3	no	—
71	C₃²×C₄	—	9	2	no	—
72	C₃²×C₄	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
73	C₃²×C₄	C₃²×D₃	1	1	yes	characteristic
74	C₃×Dic₃	—	3	2	no	—
75	C₃×Dic₃	—	9	2	no	—
76	C₃×Dic₃	—	9	2	no	—
77	C₃×Dic₃	—	9	6	no	—
78	C₄×C₉	—	3	3	no	—
79	C₄×C₉	—	6	3	no	—
80	C₄×C₉	—	9	6	no	—
81	C₃³	—	3	1	no	—
82	C₃³	C₃×Dic₆	1	1	yes	characteristic
83	3₋¹⁺²	—	2	3	no	—
84	3₋¹⁺²	—	3	1	no	—
85	3₋¹⁺²	—	6	1	no	—
86	3₋¹⁺²	—	6	1	no	—
87	3₋¹⁺²	—	6	1	no	—
88	H(3,𝔽₃)	—	2	1	no	—
89	H(3,𝔽₃)	—	6	1	no	—
90	H(3,𝔽₃)	C₃²×Q₈	1	1	yes	characteristic
91	C₃×C₉	—	6	1	no	—
92	C₃×C₉	C₃×Dic₆	1	3	yes	complemented
93	C₃×Q₈	—	9	1	no	—
94	C₃×Q₈	—	27	1	no	—
95	Dic₆	—	3	1	no	—
96	Dic₆	—	9	1	no	—
97	C₂×C₃²	—	2	1	no	—
98	C₂×C₃²	—	3	1	no	—
99	C₂×C₃²	—	6	1	no	—
100	C₂×C₃²	—	9	1	no	—
101	C₂×C₃²	—	9	3	no	—
102	C₂×C₃²	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
103	C₂×C₃²	C₂×C₃²×D₃	1	1	yes	characteristic
104	C₂×C₉	—	3	3	no	—
105	C₂×C₉	—	6	3	no	—
106	C₃×C₄	—	3	1	no	—
107	C₃×C₄	—	3	1	no	—
108	C₃×C₄	—	6	1	no	—
109	C₃×C₄	—	9	2	no	—
110	C₃×C₄	—	9	1	no	—
111	C₃×C₄	—	18	1	no	—
112	C₃×C₄	—	27	2	no	—
113	C₃×C₄	C₃×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
114	Dic₃	—	3	2	no	—
115	Dic₃	—	9	2	no	—
116	C₃²	—	2	1	no	—
117	C₃²	—	3	1	no	—
118	C₃²	—	6	1	no	—
119	C₃²	—	9	1	no	—
120	C₃²	—	9	3	no	—
121	C₃²	#634⋊C₃	1	1	yes	characteristic
122	C₃²	C₃²×Dic₆	1	1	yes	characteristic
123	C₉	—	3	3	no	—
124	C₉	—	6	3	no	—
125	Q₈	—	9	1	no	Sylow
126	C₂×C₃	—	3	1	no	—
127	C₂×C₃	—	3	1	no	—
128	C₂×C₃	—	6	1	no	—
129	C₂×C₃	—	9	1	no	—
130	C₂×C₃	—	18	1	no	—
131	C₂×C₃	C₂×C₃×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
132	C₄	—	9	2	no	—
133	C₄	#92283	1	1	yes	characteristic
134	C₃	—	3	1	no	—
135	C₃	—	3	1	no	—
136	C₃	—	6	1	no	—
137	C₃	—	9	1	no	—
138	C₃	—	18	1	no	—
139	C₃	C₃×#634⋊C₃	1	1	yes	characteristic
140	C₂	C₂×#92283	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#6910

C₂

yes

maximal, characteristic

#11757587

C₂

yes

maximal

C₃×#6229

C₃

yes

maximal, characteristic

Dic₆×3₋¹⁺²

—

maximal

#10619613

C₃

yes

maximal, complemented

C₂×#6910

C₂²

yes

characteristic

C₃×C₄×3₋¹⁺²

—

C₃×C₄×3₋¹⁺²

D₃

yes

characteristic

C₃×C₄×H(3,𝔽₃)

C₂×C₃

yes

characteristic

C₃×#1084

C₂×C₃

yes

—

Dic₃×3₋¹⁺²

—

C₄×#731

—

C₄×#731

C₂×C₃

yes

—

#66323

C₂×C₃

yes

—

#6910

Q₈

yes

characteristic, complemented, Sylow

C₃²×Dic₆

—

C₃²×Dic₆

—

Q₈×3₋¹⁺²

—

#6229

C₃²

yes

characteristic

C₉×Dic₆

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

C₂×D₃

yes

characteristic

C₂×C₃×H(3,𝔽₃)

C₂²×C₃

yes

characteristic

C₂×#731

—

C₂×#731

C₂²×C₃

yes

—

C₃³×C₄

—

C₃³×C₄

C₃×D₃

yes

characteristic

C₃²×Dic₃

—

C₃²×Dic₃

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂×C₃²

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

C₃×D₃

yes

—

#1084

C₂×C₃²

yes

—

C₉×Dic₃

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

Dic₆

yes

characteristic, complemented

C₃×H(3,𝔽₃)

C₃×Q₈

yes

characteristic

#731

—

#731

C₃×Q₈

yes

complemented

C₃²×Q₈

—

C₃×Dic₆

—

C₃×Dic₆

—

C₃×Dic₆

—

C₃×Dic₆

—

Q₈×C₉

—

C₂×C₃³

—

C₂×C₃³

C₂×C₃×D₃

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

C₂²×C₃²

yes

characteristic

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂×C₃×D₃

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

(C₃⋊D₃)⋊C₃

yes

characteristic

C₃²×C₄

C₃²×D₃

yes

characteristic

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₃³

—

C₃³

C₃×Dic₆

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₃²×Q₈

yes

characteristic

C₃×C₉

—

C₃×C₉

C₃×Dic₆

yes

complemented

C₃×Q₈

—

C₃×Q₈

—

Dic₆

—

Dic₆

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

100

C₂×C₃²

—

101

C₂×C₃²

—

102

C₂×C₃²

C₂×(C₃⋊D₃)⋊C₃

yes

characteristic

103

C₂×C₃²

C₂×C₃²×D₃

yes

characteristic

104

C₂×C₉

—

105

C₂×C₉

—

106

C₃×C₄

—

107

C₃×C₄

—

108

C₃×C₄

—

109

C₃×C₄

—

110

C₃×C₄

—

111

C₃×C₄

—

112

C₃×C₄

—

113

C₃×C₄

C₃×(C₃⋊D₃)⋊C₃

yes

characteristic

114

Dic₃

—

115

Dic₃

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

#634⋊C₃

yes

characteristic

122

C₃²

C₃²×Dic₆

yes

characteristic

123

C₉

—

124

C₉

—

125

Q₈

—

Sylow

126

C₂×C₃

—

127

C₂×C₃

—

128

C₂×C₃

—

129

C₂×C₃

—

130

C₂×C₃

—

131

C₂×C₃

C₂×C₃×(C₃⋊D₃)⋊C₃

yes

characteristic

132

C₄

—

133

C₄

#92283

yes

characteristic

134

C₃

—

135

C₃

—

136

C₃

—

137

C₃

—

138

C₃

—

139

C₃

C₃×#634⋊C₃

yes

characteristic

140

C₂

C₂×#92283

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₂×C₃×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₂×H(3,𝔽₃)	C₂²×C₃²
Frattini	C₂×C₃²	C₂×C₃²×D₃
Fitting	C₄×#6910	C₂
prime radical	C₂×C₃×H(3,𝔽₃)	C₂²×C₃
hypercenter	C₃²×C₄	(C₃⋊D₃)⋊C₃
hypocenter	H(3,𝔽₃)	C₃²×Q₈
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃²×C₄

subgroup

G/H

center, socle

C₂×C₃

C₂×C₃×(C₃⋊D₃)⋊C₃

commutator, cosocle

C₂×H(3,𝔽₃)

C₂²×C₃²

Frattini

C₂×C₃²

C₂×C₃²×D₃

Fitting

C₄×#6910

C₂

prime radical

C₂×C₃×H(3,𝔽₃)

C₂²×C₃

hypercenter

C₃²×C₄

(C₃⋊D₃)⋊C₃

hypocenter

H(3,𝔽₃)

C₃²×Q₈

LCS

G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂×C₃ ◁ C₃²×C₄

N	H ≅ G/N	type
5. #10619613	C₃	semidirect
15. #6910	Q₈	semidirect
44. C₃×3₋¹⁺²	Dic₆	semidirect
47. #731	C₃×Q₈	semidirect
92. C₃×C₉	C₃×Dic₆	semidirect

H ≅ G/N

type

5. #10619613

C₃

semidirect

15. #6910

Q₈

semidirect

44. C₃×3₋¹⁺²

Dic₆

semidirect

47. #731

C₃×Q₈

semidirect

92. C₃×C₉

C₃×Dic₆

semidirect

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

subgroup	index
C₄×#6910	2
#11757587	2
C₃×#6229	3
Dic₆×3₋¹⁺²	3
#10619613	3

subgroup

index

Dic₆×3₋¹⁺²

Subgroups — #14517385

Conjugacy classes of subgroups

Complements

Maximal subgroups