Group #44535405 — subgroups

Subgroups — C₃×#23782923 #44535405

Order 31347.

5588 conjugacy classes of subgroups; only the 2884 normal subgroups are shown.

normal subgroups only

This group has 5588 conjugacy classes of subgroups — too many to draw. Only its 2884 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	#23782923	C₃	1	81	yes	maximal, direct product factor
2	C₃×#16812526	C₃	1	18	yes	maximal, complemented
3	C₃×#16812525	C₃	1	9	yes	maximal, complemented
4	C₃²×#11923958	C₃	1	2	yes	maximal, complemented
5	C₃×3₋¹⁺²×C₄₃⋊C₃	C₃	1	9	yes	maximal, complemented
6	C₃²×C₃²⋊(C₄₃⋊C₃)	C₃	1	1	yes	maximal, characteristic
7	C₃²×3₋¹⁺²×C₄₃	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₄₃⋊C₃	C₃²	1	3	yes	—
9	#16812526	C₃²	1	486	yes	complemented
10	#16812525	C₃²	1	243	yes	complemented
11	C₃×#11923958	C₃²	1	6	yes	—
12	C₃×#11923958	C₃²	1	6	yes	complemented
13	C₃×#11923958	C₃²	1	6	yes	complemented
14	C₃×#11923958	C₃²	1	54	yes	complemented
15	C₃×#11923957	C₃²	1	27	yes	complemented
16	C₃×#11923956	C₃²	1	27	yes	complemented
17	3₋¹⁺²×C₄₃⋊C₃	C₃²	1	243	yes	complemented
18	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	3	yes	—
19	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	6	yes	—
20	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	27	yes	—
21	C₃²×C₄₃⋊C₉	C₃²	1	6	yes	complemented
22	C₃×C₉×C₄₃⋊C₃	C₃²	1	27	yes	complemented
23	C₃⁴×C₄₃	C₃²	1	1	yes	characteristic
24	C₃×3₋¹⁺²×C₄₃	C₃²	1	3	yes	—
25	C₃×3₋¹⁺²×C₄₃	C₃²	1	6	yes	complemented
26	C₃×3₋¹⁺²×C₄₃	C₃²	1	27	yes	complemented
27	C₃²×C₉×C₄₃	C₃²	1	3	yes	complemented
28	C₃²×C₄₃⋊C₃	—	9	9	no	—
29	C₃²×C₄₃⋊C₃	C₃³	1	9	yes	—
30	C₃²×C₄₃⋊C₃	C₃³	1	27	yes	—
31	#11923958	C₃³	1	54	yes	—
32	#11923958	C₃³	1	54	yes	complemented
33	#11923958	C₃³	1	54	yes	complemented
34	#11923957	C₃³	1	243	yes	complemented
35	#11923956	C₃³	1	243	yes	complemented
36	C₃²⋊(C₄₃⋊C₃)	C₃³	1	27	yes	—
37	C₃²⋊(C₄₃⋊C₃)	C₃³	1	54	yes	—
38	C₃×C₄₃⋊C₉	C₃³	1	18	yes	—
39	C₃×C₄₃⋊C₉	C₃³	1	54	yes	complemented
40	C₉×C₄₃⋊C₃	C₃³	1	243	yes	complemented
41	C₃³×C₄₃	—	9	3	no	—
42	C₃³×C₄₃	C₃³	1	1	yes	characteristic
43	C₃³×C₄₃	C₃³	1	1	yes	characteristic
44	C₃³×C₄₃	C₃³	1	2	yes	—
45	C₃³×C₄₃	C₃³	1	9	yes	—
46	3₋¹⁺²×C₄₃	C₃³	1	27	yes	—
47	3₋¹⁺²×C₄₃	C₃³	1	54	yes	complemented
48	C₃×C₉×C₄₃	C₃³	1	9	yes	—
49	C₃×C₉×C₄₃	C₃³	1	27	yes	complemented
50	C₃×3₋¹⁺⁴	—	43	1	no	maximal, Sylow
51	C₃²×C₄₃	—	3	3	no	—
52	C₃²×C₄₃	—	3	6	no	—
53	C₃²×C₄₃	—	3	3	no	—
54	C₃²×C₄₃	—	9	9	no	—
55	C₃²×C₄₃	C₃⁴	1	3	yes	—
56	C₃²×C₄₃	C₃⁴	1	3	yes	—
57	C₃²×C₄₃	C₃⁴	1	6	yes	—
58	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
59	C₃×C₄₃⋊C₃	—	3	27	no	—
60	C₃×C₄₃⋊C₃	—	9	27	no	—
61	C₃×C₄₃⋊C₃	C₃⁴	1	27	yes	—
62	C₉×C₄₃	C₃⁴	1	27	yes	—
63	C₄₃⋊C₉	C₃⁴	1	54	yes	—
64	3₋¹⁺⁴	—	43	81	no	—
65	C₃×#737	—	43	18	no	—
66	C₃×#737	—	43	9	no	—
67	C₃²×3₋¹⁺²	—	43	2	no	—
68	C₃²×3₋¹⁺²	—	43	9	no	—
69	C₃²×3₋¹⁺²	C₄₃⋊C₃	1	1	yes	characteristic, complemented
70	C₃²×H(3,𝔽₃)	—	43	1	no	—
71	C₃×C₄₃	—	3	3	no	—
72	C₃×C₄₃	—	3	3	no	—
73	C₃×C₄₃	—	3	6	no	—
74	C₃×C₄₃	3₋¹⁺⁴	1	3	yes	complemented
75	C₃×C₄₃	C₃⁵	1	1	yes	characteristic
76	C₄₃⋊C₃	—	3	27	no	—
77	C₃⁴	—	43	3	no	—
78	C₃⁴	C₃×C₄₃⋊C₃	1	1	yes	characteristic
79	#737	—	43	486	no	—
80	#737	—	43	243	no	—
81	C₃×3₋¹⁺²	—	43	6	no	—
82	C₃×3₋¹⁺²	—	43	6	no	—
83	C₃×3₋¹⁺²	—	43	6	no	—
84	C₃×3₋¹⁺²	—	43	54	no	—
85	C₃×3₋¹⁺²	—	43	243	no	—
86	C₃×3₋¹⁺²	—	43	27	no	—
87	C₃×3₋¹⁺²	—	43	27	no	—
88	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	3	yes	—
89	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	6	yes	complemented
90	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	27	yes	complemented
91	C₃×H(3,𝔽₃)	—	43	6	no	—
92	C₃×H(3,𝔽₃)	—	43	3	no	—
93	C₃×H(3,𝔽₃)	—	43	27	no	—
94	C₃²×C₉	—	43	6	no	—
95	C₃²×C₉	—	43	27	no	—
96	C₃²×C₉	C₃×C₄₃⋊C₃	1	3	yes	complemented
97	C₄₃	C₃×3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
98	C₃³	—	9	3	no	—
99	C₃³	—	43	9	no	—
100	C₃³	—	43	27	no	—
101	C₃³	—	387	9	no	—
102	C₃³	C₃²×C₄₃⋊C₃	1	2	yes	—
103	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
104	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
105	C₃³	C₃²×C₄₃⋊C₃	1	9	yes	—
106	3₋¹⁺²	—	43	54	no	—
107	3₋¹⁺²	—	43	54	no	—
108	3₋¹⁺²	—	43	54	no	—
109	3₋¹⁺²	—	43	243	no	—
110	3₋¹⁺²	—	43	243	no	—
111	3₋¹⁺²	C₃²×C₄₃⋊C₃	1	27	yes	—
112	3₋¹⁺²	C₃²×C₄₃⋊C₃	1	54	yes	complemented
113	H(3,𝔽₃)	—	43	54	no	—
114	H(3,𝔽₃)	—	43	27	no	—
115	C₃×C₉	—	43	54	no	—
116	C₃×C₉	—	43	18	no	—
117	C₃×C₉	—	43	243	no	—
118	C₃×C₉	C₃²×C₄₃⋊C₃	1	9	yes	—
119	C₃×C₉	C₃²×C₄₃⋊C₃	1	27	yes	complemented
120	C₃²	—	3	3	no	—
121	C₃²	—	3	6	no	—
122	C₃²	—	3	3	no	—
123	C₃²	—	9	9	no	—
124	C₃²	—	43	27	no	—
125	C₃²	—	129	27	no	—
126	C₃²	—	387	27	no	—
127	C₃²	C₃³×C₄₃⋊C₃	1	6	yes	—
128	C₃²	C₃³×C₄₃⋊C₃	1	3	yes	—
129	C₃²	C₃³×C₄₃⋊C₃	1	3	yes	—
130	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
131	C₉	—	43	54	no	—
132	C₉	C₃³×C₄₃⋊C₃	1	27	yes	—
133	C₃	—	3	3	no	—
134	C₃	—	3	6	no	—
135	C₃	—	3	3	no	—
136	C₃	—	129	27	no	—
137	C₃	#23782923	1	3	yes	direct product factor
138	C₃	C₃⁴×C₄₃⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23782923

C₃

yes

maximal, direct product factor

C₃×#16812526

C₃

yes

maximal, complemented

C₃×#16812525

C₃

yes

maximal, complemented

C₃²×#11923958

C₃

yes

maximal, complemented

C₃×3₋¹⁺²×C₄₃⋊C₃

C₃

yes

maximal, complemented

C₃²×C₃²⋊(C₄₃⋊C₃)

C₃

yes

maximal, characteristic

C₃²×3₋¹⁺²×C₄₃

C₃

yes

maximal, characteristic, complemented

C₃³×C₄₃⋊C₃

C₃²

yes

—

#16812526

C₃²

486

yes

complemented

#16812525

C₃²

243

yes

complemented

C₃×#11923958

C₃²

yes

—

C₃×#11923958

C₃²

yes

complemented

C₃×#11923958

C₃²

yes

complemented

C₃×#11923958

C₃²

yes

complemented

C₃×#11923957

C₃²

yes

complemented

C₃×#11923956

C₃²

yes

complemented

3₋¹⁺²×C₄₃⋊C₃

C₃²

243

yes

complemented

C₃×C₃²⋊(C₄₃⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₄₃⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₄₃⋊C₃)

C₃²

yes

—

C₃²×C₄₃⋊C₉

C₃²

yes

complemented

C₃×C₉×C₄₃⋊C₃

C₃²

yes

complemented

C₃⁴×C₄₃

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₄₃

C₃²

yes

—

C₃×3₋¹⁺²×C₄₃

C₃²

yes

complemented

C₃×3₋¹⁺²×C₄₃

C₃²

yes

complemented

C₃²×C₉×C₄₃

C₃²

yes

complemented

C₃²×C₄₃⋊C₃

—

C₃²×C₄₃⋊C₃

C₃³

yes

—

C₃²×C₄₃⋊C₃

C₃³

yes

—

#11923958

C₃³

yes

—

#11923958

C₃³

yes

complemented

#11923958

C₃³

yes

complemented

#11923957

C₃³

243

yes

complemented

#11923956

C₃³

243

yes

complemented

C₃²⋊(C₄₃⋊C₃)

C₃³

yes

—

C₃²⋊(C₄₃⋊C₃)

C₃³

yes

—

C₃×C₄₃⋊C₉

C₃³

yes

—

C₃×C₄₃⋊C₉

C₃³

yes

complemented

C₉×C₄₃⋊C₃

C₃³

243

yes

complemented

C₃³×C₄₃

—

C₃³×C₄₃

C₃³

yes

characteristic

C₃³×C₄₃

C₃³

yes

characteristic

C₃³×C₄₃

C₃³

yes

—

C₃³×C₄₃

C₃³

yes

—

3₋¹⁺²×C₄₃

C₃³

yes

—

3₋¹⁺²×C₄₃

C₃³

yes

complemented

C₃×C₉×C₄₃

C₃³

yes

—

C₃×C₉×C₄₃

C₃³

yes

complemented

C₃×3₋¹⁺⁴

—

maximal, Sylow

C₃²×C₄₃

—

C₃²×C₄₃

—

C₃²×C₄₃

—

C₃²×C₄₃

—

C₃²×C₄₃

C₃⁴

yes

—

C₃²×C₄₃

C₃⁴

yes

—

C₃²×C₄₃

C₃⁴

yes

—

C₃²×C₄₃

C₃⁴

yes

characteristic

C₃×C₄₃⋊C₃

—

C₃×C₄₃⋊C₃

—

C₃×C₄₃⋊C₃

C₃⁴

yes

—

C₉×C₄₃

C₃⁴

yes

—

C₄₃⋊C₉

C₃⁴

yes

—

3₋¹⁺⁴

—

C₃×#737

—

C₃×#737

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₄₃⋊C₃

yes

characteristic, complemented

C₃²×H(3,𝔽₃)

—

C₃×C₄₃

—

C₃×C₄₃

—

C₃×C₄₃

—

C₃×C₄₃

3₋¹⁺⁴

yes

complemented

C₃×C₄₃

C₃⁵

yes

characteristic

C₄₃⋊C₃

—

C₃⁴

—

C₃⁴

C₃×C₄₃⋊C₃

yes

characteristic

#737

—

486

—

#737

—

243

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

243

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₄₃⋊C₃

yes

—

C₃×3₋¹⁺²

C₃×C₄₃⋊C₃

yes

complemented

C₃×3₋¹⁺²

C₃×C₄₃⋊C₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₄₃⋊C₃

yes

complemented

C₄₃

C₃×3₋¹⁺⁴

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

100

C₃³

—

101

C₃³

—

387

—

102

C₃³

C₃²×C₄₃⋊C₃

yes

—

103

C₃³

C₃²×C₄₃⋊C₃

yes

characteristic

104

C₃³

C₃²×C₄₃⋊C₃

yes

characteristic

105

C₃³

C₃²×C₄₃⋊C₃

yes

—

106

3₋¹⁺²

—

107

3₋¹⁺²

—

108

3₋¹⁺²

—

109

3₋¹⁺²

—

243

—

110

3₋¹⁺²

—

243

—

111

3₋¹⁺²

C₃²×C₄₃⋊C₃

yes

—

112

3₋¹⁺²

C₃²×C₄₃⋊C₃

yes

complemented

113

H(3,𝔽₃)

—

114

H(3,𝔽₃)

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

243

—

118

C₃×C₉

C₃²×C₄₃⋊C₃

yes

—

119

C₃×C₉

C₃²×C₄₃⋊C₃

yes

complemented

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

129

—

126

C₃²

—

387

—

127

C₃²

C₃³×C₄₃⋊C₃

yes

—

128

C₃²

C₃³×C₄₃⋊C₃

yes

—

129

C₃²

C₃³×C₄₃⋊C₃

yes

—

130

C₃²

C₃³×C₄₃⋊C₃

yes

characteristic

131

C₉

—

132

C₉

C₃³×C₄₃⋊C₃

yes

—

133

C₃

—

134

C₃

—

135

C₃

—

136

C₃

—

129

—

137

C₃

#23782923

yes

direct product factor

138

C₃

C₃⁴×C₄₃⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃²	C₃³×C₄₃⋊C₃
commutator, cosocle	C₃×C₄₃	C₃⁵
Frattini	C₃	C₃⁴×C₄₃⋊C₃
Fitting	C₃²×3₋¹⁺²×C₄₃	C₃
socle	C₃²×C₄₃	C₃⁴
prime radical	C₃²×C₃²⋊(C₄₃⋊C₃)	C₃
hypercenter	C₃²×3₋¹⁺²	C₄₃⋊C₃
hypocenter	C₄₃	C₃×3₋¹⁺⁴
LCS	G ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃² ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃²

C₃³×C₄₃⋊C₃

commutator, cosocle

C₃×C₄₃

C₃⁵

Frattini

C₃

C₃⁴×C₄₃⋊C₃

Fitting

C₃²×3₋¹⁺²×C₄₃

C₃

socle

C₃²×C₄₃

C₃⁴

prime radical

C₃²×C₃²⋊(C₄₃⋊C₃)

C₃

hypercenter

C₃²×3₋¹⁺²

C₄₃⋊C₃

hypocenter

C₄₃

C₃×3₋¹⁺⁴

LCS

G ▷ C₃×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₃² ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
1. #23782923	C₃	direct (N × H)
2. C₃×#16812526	C₃	semidirect
3. C₃×#16812525	C₃	semidirect
4. C₃²×#11923958	C₃	semidirect
5. C₃×3₋¹⁺²×C₄₃⋊C₃	C₃	semidirect
7. C₃²×3₋¹⁺²×C₄₃	C₃	semidirect
9. #16812526	C₃²	semidirect
10. #16812525	C₃²	semidirect
12. C₃×#11923958	C₃²	semidirect
13. C₃×#11923958	C₃²	semidirect
14. C₃×#11923958	C₃²	semidirect
15. C₃×#11923957	C₃²	semidirect
16. C₃×#11923956	C₃²	semidirect
17. 3₋¹⁺²×C₄₃⋊C₃	C₃²	semidirect
21. C₃²×C₄₃⋊C₉	C₃²	semidirect
22. C₃×C₉×C₄₃⋊C₃	C₃²	semidirect
25. C₃×3₋¹⁺²×C₄₃	C₃²	semidirect
26. C₃×3₋¹⁺²×C₄₃	C₃²	semidirect
27. C₃²×C₉×C₄₃	C₃²	semidirect
32. #11923958	C₃³	semidirect
33. #11923958	C₃³	semidirect
34. #11923957	C₃³	semidirect
35. #11923956	C₃³	semidirect
39. C₃×C₄₃⋊C₉	C₃³	semidirect
40. C₉×C₄₃⋊C₃	C₃³	semidirect
47. 3₋¹⁺²×C₄₃	C₃³	semidirect
49. C₃×C₉×C₄₃	C₃³	semidirect
69. C₃²×3₋¹⁺²	C₄₃⋊C₃	semidirect
74. C₃×C₄₃	3₋¹⁺⁴	semidirect
89. C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	semidirect
90. C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	semidirect
96. C₃²×C₉	C₃×C₄₃⋊C₃	semidirect
97. C₄₃	C₃×3₋¹⁺⁴	semidirect
112. 3₋¹⁺²	C₃²×C₄₃⋊C₃	semidirect
119. C₃×C₉	C₃²×C₄₃⋊C₃	semidirect
137. C₃	#23782923	direct (N × H)

H ≅ G/N

type

1. #23782923

C₃

direct (N × H)

2. C₃×#16812526

C₃

semidirect

3. C₃×#16812525

C₃

semidirect

4. C₃²×#11923958

C₃

semidirect

5. C₃×3₋¹⁺²×C₄₃⋊C₃

C₃

semidirect

7. C₃²×3₋¹⁺²×C₄₃

C₃

semidirect

9. #16812526

C₃²

semidirect

10. #16812525

C₃²

semidirect

12. C₃×#11923958

C₃²

semidirect

13. C₃×#11923958

C₃²

semidirect

14. C₃×#11923958

C₃²

semidirect

15. C₃×#11923957

C₃²

semidirect

16. C₃×#11923956

C₃²

semidirect

17. 3₋¹⁺²×C₄₃⋊C₃

C₃²

semidirect

21. C₃²×C₄₃⋊C₉

C₃²

semidirect

22. C₃×C₉×C₄₃⋊C₃

C₃²

semidirect

25. C₃×3₋¹⁺²×C₄₃

C₃²

semidirect

26. C₃×3₋¹⁺²×C₄₃

C₃²

semidirect

27. C₃²×C₉×C₄₃

C₃²

semidirect

32. #11923958

C₃³

semidirect

33. #11923958

C₃³

semidirect

34. #11923957

C₃³

semidirect

35. #11923956

C₃³

semidirect

39. C₃×C₄₃⋊C₉

C₃³

semidirect

40. C₉×C₄₃⋊C₃

C₃³

semidirect

47. 3₋¹⁺²×C₄₃

C₃³

semidirect

49. C₃×C₉×C₄₃

C₃³

semidirect

69. C₃²×3₋¹⁺²

C₄₃⋊C₃

semidirect

74. C₃×C₄₃

3₋¹⁺⁴

semidirect

89. C₃×3₋¹⁺²

C₃×C₄₃⋊C₃

semidirect

90. C₃×3₋¹⁺²

C₃×C₄₃⋊C₃

semidirect

96. C₃²×C₉

C₃×C₄₃⋊C₃

semidirect

97. C₄₃

C₃×3₋¹⁺⁴

semidirect

112. 3₋¹⁺²

C₃²×C₄₃⋊C₃

semidirect

119. C₃×C₉

C₃²×C₄₃⋊C₃

semidirect

137. C₃

#23782923

direct (N × H)

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

subgroup	index
#23782923	3
C₃×#16812526	3
C₃×#16812525	3
C₃²×#11923958	3
C₃×3₋¹⁺²×C₄₃⋊C₃	3
C₃²×C₃²⋊(C₄₃⋊C₃)	3
C₃²×3₋¹⁺²×C₄₃	3
C₃×3₋¹⁺⁴	43

subgroup

index

C₃×3₋¹⁺²×C₄₃⋊C₃

C₃²×C₃²⋊(C₄₃⋊C₃)

C₃²×3₋¹⁺²×C₄₃

C₃×3₋¹⁺⁴

Subgroups — C₃×#23782923 #44535405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#23782923 #44535405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#23782923 #44535405