Group #14518934 — subgroups

Subgroups — #14518934

Order 1944.

669 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 669 conjugacy classes of subgroups — too many to draw. Only its 129 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×H(5,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#11758012	C₂	1	2	yes	maximal
3	#10619956	C₃	1	4	yes	maximal, complemented
4	C₃×#634⋊C₃	—	3	4	no	maximal
5	C₂×H(5,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₃×C₄×H(3,𝔽₃)	—	2	4	no	—
7	C₃×C₄×H(3,𝔽₃)	—	2	12	no	—
8	C₃×C₄×H(3,𝔽₃)	D₃	1	4	yes	—
9	C₃×C₄×H(3,𝔽₃)	C₂×C₃	1	4	yes	—
10	C₃⋊((C₃⋊Dic₃)⋊C₃)	C₂×C₃	1	8	yes	—
11	C₃×(C₃⋊Dic₃)⋊C₃	—	3	8	no	—
12	H(5,𝔽₃)	Q₈	1	1	yes	characteristic, complemented, Sylow
13	#6308	C₃²	1	1	yes	characteristic, complemented
14	C₃×#634	—	3	4	no	—
15	C₃²×Dic₆	—	9	1	no	—
16	#634⋊C₃	—	3	12	no	—
17	C₂×C₃×H(3,𝔽₃)	—	2	4	no	—
18	C₂×C₃×H(3,𝔽₃)	—	2	12	no	—
19	C₂×C₃×H(3,𝔽₃)	C₂×D₃	1	4	yes	—
20	C₂×C₃×H(3,𝔽₃)	C₂²×C₃	1	4	yes	—
21	C₃³×C₄	—	2	6	no	—
22	C₃³×C₄	—	2	3	no	—
23	C₃³×C₄	—	2	4	no	—
24	C₃³×C₄	—	2	4	no	—
25	C₃³×C₄	C₃×D₃	1	4	yes	—
26	C₃³×C₄	C₃⋊D₃	1	1	yes	characteristic
27	C₃³×C₄	C₂×C₃²	1	1	yes	characteristic
28	C₃⋊(C₃⋊Dic₃)	C₂×C₃²	1	2	yes	—
29	C₃×C₃⋊Dic₃	—	3	8	no	—
30	C₃²×Dic₃	—	9	2	no	—
31	C₄×H(3,𝔽₃)	—	2	1	no	—
32	C₄×H(3,𝔽₃)	—	2	8	no	—
33	C₄×H(3,𝔽₃)	—	2	6	no	—
34	C₄×H(3,𝔽₃)	—	2	12	no	—
35	C₄×H(3,𝔽₃)	—	2	12	no	—
36	C₄×H(3,𝔽₃)	C₃×D₃	1	12	yes	—
37	(C₃⋊Dic₃)⋊C₃	—	3	24	no	—
38	C₃×H(3,𝔽₃)	—	2	4	no	—
39	C₃×H(3,𝔽₃)	—	2	12	no	—
40	C₃×H(3,𝔽₃)	Dic₆	1	4	yes	complemented
41	C₃×H(3,𝔽₃)	C₃×Q₈	1	4	yes	complemented
42	C₃²×Q₈	—	27	1	no	—
43	#634	—	3	4	no	—
44	#634	—	27	1	no	—
45	C₃×Dic₆	—	9	4	no	—
46	C₃×Dic₆	—	27	4	no	—
47	C₂×C₃³	—	2	6	no	—
48	C₂×C₃³	—	2	3	no	—
49	C₂×C₃³	—	2	4	no	—
50	C₂×C₃³	—	2	4	no	—
51	C₂×C₃³	C₂×C₃×D₃	1	4	yes	—
52	C₂×C₃³	C₂×C₃⋊D₃	1	1	yes	characteristic
53	C₂×C₃³	C₂²×C₃²	1	1	yes	characteristic
54	C₂×H(3,𝔽₃)	—	2	1	no	—
55	C₂×H(3,𝔽₃)	—	2	8	no	—
56	C₂×H(3,𝔽₃)	—	2	6	no	—
57	C₂×H(3,𝔽₃)	—	2	12	no	—
58	C₂×H(3,𝔽₃)	—	2	12	no	—
59	C₂×H(3,𝔽₃)	C₂×C₃×D₃	1	12	yes	—
60	C₃²×C₄	—	2	4	no	—
61	C₃²×C₄	—	2	12	no	—
62	C₃²×C₄	—	9	1	no	—
63	C₃²×C₄	—	9	1	no	—
64	C₃²×C₄	—	9	4	no	—
65	C₃²×C₄	—	18	4	no	—
66	C₃²×C₄	—	18	4	no	—
67	C₃²×C₄	—	18	6	no	—
68	C₃²×C₄	—	18	3	no	—
69	C₃²×C₄	—	27	2	no	—
70	C₃²×C₄	C₃²×D₃	1	4	yes	—
71	C₃²×C₄	C₃×C₃⋊D₃	1	4	yes	—
72	C₃⋊Dic₃	—	3	8	no	—
73	C₃⋊Dic₃	—	27	2	no	—
74	C₃×Dic₃	—	9	8	no	—
75	C₃×Dic₃	—	27	8	no	—
76	C₃³	—	2	6	no	—
77	C₃³	—	2	3	no	—
78	C₃³	—	2	4	no	—
79	C₃³	—	2	4	no	—
80	C₃³	C₃×Dic₆	1	4	yes	complemented
81	C₃³	#634	1	1	yes	characteristic, complemented
82	C₃³	C₃²×Q₈	1	1	yes	characteristic, complemented
83	H(3,𝔽₃)	—	2	1	no	—
84	H(3,𝔽₃)	—	2	8	no	—
85	H(3,𝔽₃)	—	2	6	no	—
86	H(3,𝔽₃)	—	2	12	no	—
87	H(3,𝔽₃)	—	2	12	no	—
88	H(3,𝔽₃)	C₃×Dic₆	1	12	yes	complemented
89	C₃×Q₈	—	27	4	no	—
90	Dic₆	—	9	1	no	—
91	Dic₆	—	27	4	no	—
92	C₂×C₃²	—	2	4	no	—
93	C₂×C₃²	—	2	12	no	—
94	C₂×C₃²	—	9	1	no	—
95	C₂×C₃²	—	9	1	no	—
96	C₂×C₃²	—	9	4	no	—
97	C₂×C₃²	—	18	4	no	—
98	C₂×C₃²	—	18	4	no	—
99	C₂×C₃²	—	18	6	no	—
100	C₂×C₃²	—	18	3	no	—
101	C₂×C₃²	C₂×C₃²×D₃	1	4	yes	—
102	C₂×C₃²	C₂×C₃×C₃⋊D₃	1	4	yes	—
103	C₃×C₄	—	3	4	no	—
104	C₃×C₄	—	3	4	no	—
105	C₃×C₄	—	6	4	no	—
106	C₃×C₄	—	6	12	no	—
107	C₃×C₄	—	27	8	no	—
108	C₃×C₄	C₃²×C₃⋊D₃	1	1	yes	characteristic
109	Dic₃	—	9	2	no	—
110	Dic₃	—	27	8	no	—
111	C₃²	—	2	4	no	—
112	C₃²	—	2	12	no	—
113	C₃²	—	9	1	no	—
114	C₃²	—	9	1	no	—
115	C₃²	—	9	4	no	—
116	C₃²	—	18	4	no	—
117	C₃²	—	18	4	no	—
118	C₃²	—	18	6	no	—
119	C₃²	—	18	3	no	—
120	C₃²	C₃²×Dic₆	1	4	yes	—
121	C₃²	C₃×#634	1	4	yes	—
122	Q₈	—	27	1	no	Sylow
123	C₂×C₃	—	3	4	no	—
124	C₂×C₃	—	3	4	no	—
125	C₂×C₃	—	6	4	no	—
126	C₂×C₃	—	6	12	no	—
127	C₂×C₃	C₂×C₃²×C₃⋊D₃	1	1	yes	characteristic
128	C₄	—	27	2	no	—
129	C₄	#92391	1	1	yes	characteristic
130	C₃	—	3	4	no	—
131	C₃	—	3	4	no	—
132	C₃	—	6	4	no	—
133	C₃	—	6	12	no	—
134	C₃	C₃²×#634	1	1	yes	characteristic
135	C₂	C₂×#92391	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×H(5,𝔽₃)

C₂

yes

maximal, characteristic

#11758012

C₂

yes

maximal

#10619956

C₃

yes

maximal, complemented

C₃×#634⋊C₃

—

maximal

C₂×H(5,𝔽₃)

C₂²

yes

characteristic

C₃×C₄×H(3,𝔽₃)

—

C₃×C₄×H(3,𝔽₃)

—

C₃×C₄×H(3,𝔽₃)

D₃

yes

—

C₃×C₄×H(3,𝔽₃)

C₂×C₃

yes

—

C₃⋊((C₃⋊Dic₃)⋊C₃)

C₂×C₃

yes

—

C₃×(C₃⋊Dic₃)⋊C₃

—

H(5,𝔽₃)

Q₈

yes

characteristic, complemented, Sylow

#6308

C₃²

yes

characteristic, complemented

C₃×#634

—

C₃²×Dic₆

—

#634⋊C₃

—

C₂×C₃×H(3,𝔽₃)

—

C₂×C₃×H(3,𝔽₃)

—

C₂×C₃×H(3,𝔽₃)

C₂×D₃

yes

—

C₂×C₃×H(3,𝔽₃)

C₂²×C₃

yes

—

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

C₃×D₃

yes

—

C₃³×C₄

C₃⋊D₃

yes

characteristic

C₃³×C₄

C₂×C₃²

yes

characteristic

C₃⋊(C₃⋊Dic₃)

C₂×C₃²

yes

—

C₃×C₃⋊Dic₃

—

C₃²×Dic₃

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₃×D₃

yes

—

(C₃⋊Dic₃)⋊C₃

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

Dic₆

yes

complemented

C₃×H(3,𝔽₃)

C₃×Q₈

yes

complemented

C₃²×Q₈

—

#634

—

#634

—

C₃×Dic₆

—

C₃×Dic₆

—

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

C₂×C₃×D₃

yes

—

C₂×C₃³

C₂×C₃⋊D₃

yes

characteristic

C₂×C₃³

C₂²×C₃²

yes

characteristic

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

C₂×C₃×D₃

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₃²×D₃

yes

—

C₃²×C₄

C₃×C₃⋊D₃

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×Dic₆

yes

complemented

C₃³

#634

yes

characteristic, complemented

C₃³

C₃²×Q₈

yes

characteristic, complemented

—

—

—

—

—

C₃×Dic₆

yes

complemented

C₃×Q₈

—

Dic₆

—

Dic₆

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

100

C₂×C₃²

—

101

C₂×C₃²

C₂×C₃²×D₃

yes

—

102

C₂×C₃²

C₂×C₃×C₃⋊D₃

yes

—

103

C₃×C₄

—

104

C₃×C₄

—

105

C₃×C₄

—

106

C₃×C₄

—

107

C₃×C₄

—

108

C₃×C₄

C₃²×C₃⋊D₃

yes

characteristic

109

Dic₃

—

110

Dic₃

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

C₃²×Dic₆

yes

—

121

C₃²

C₃×#634

yes

—

122

Q₈

—

Sylow

123

C₂×C₃

—

124

C₂×C₃

—

125

C₂×C₃

—

126

C₂×C₃

—

127

C₂×C₃

C₂×C₃²×C₃⋊D₃

yes

characteristic

128

C₄

—

129

C₄

#92391

yes

characteristic

130

C₃

—

131

C₃

—

132

C₃

—

133

C₃

—

134

C₃

C₃²×#634

yes

characteristic

135

C₂

C₂×#92391

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#92391
commutator, cosocle	C₂×C₃³	C₂²×C₃²
Frattini, socle	C₂×C₃	C₂×C₃²×C₃⋊D₃
Fitting	C₄×H(5,𝔽₃)	C₂
prime radical	C₂×H(5,𝔽₃)	C₂²
hypercenter	C₄	#92391
hypocenter	C₃³	C₃²×Q₈
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₄

subgroup

G/H

center

C₂

C₂×#92391

commutator, cosocle

C₂×C₃³

C₂²×C₃²

Frattini, socle

C₂×C₃

C₂×C₃²×C₃⋊D₃

Fitting

C₄×H(5,𝔽₃)

C₂

prime radical

C₂×H(5,𝔽₃)

C₂²

hypercenter

C₄

#92391

hypocenter

C₃³

C₃²×Q₈

LCS

G ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂ ◁ C₄

N	H ≅ G/N	type
3. #10619956	C₃	semidirect
12. H(5,𝔽₃)	Q₈	semidirect
13. #6308	C₃²	semidirect
40. C₃×H(3,𝔽₃)	Dic₆	semidirect
41. C₃×H(3,𝔽₃)	C₃×Q₈	semidirect
80. C₃³	C₃×Dic₆	semidirect
81. C₃³	#634	semidirect
82. C₃³	C₃²×Q₈	semidirect
88. H(3,𝔽₃)	C₃×Dic₆	semidirect

H ≅ G/N

type

3. #10619956

C₃

semidirect

12. H(5,𝔽₃)

Q₈

semidirect

13. #6308

C₃²

semidirect

40. C₃×H(3,𝔽₃)

Dic₆

semidirect

41. C₃×H(3,𝔽₃)

C₃×Q₈

semidirect

80. C₃³

C₃×Dic₆

semidirect

81. C₃³

#634

semidirect

82. C₃³

C₃²×Q₈

semidirect

88. H(3,𝔽₃)

C₃×Dic₆

semidirect

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

subgroup	index
C₄×H(5,𝔽₃)	2
#11758012	2
#10619956	3
C₃×#634⋊C₃	3

subgroup

index

C₄×H(5,𝔽₃)

#11758012

#10619956

C₃×#634⋊C₃

Subgroups — #14518934

Conjugacy classes of subgroups

Complements

Maximal subgroups