Group #36382189 — subgroups

Subgroups — #36382189

Order 22599.

188 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 43 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	#21419420	C₃	1	2	yes	maximal, complemented
2	#21419407	C₃	1	1	yes	maximal, characteristic, complemented
3	#21419363	C₃	1	1	yes	maximal, characteristic, complemented
4	#21419281	C₃	1	2	yes	maximal, complemented
5	#21419277	C₃	1	2	yes	maximal, complemented
6	#21419272	C₃	1	1	yes	maximal, characteristic, complemented
7	#21419267	C₃	1	1	yes	maximal, characteristic, complemented
8	#21419263	C₃	1	2	yes	maximal, complemented
9	C₃₁×#6882	C₃	1	1	yes	maximal, characteristic, complemented
10	#14578913	—	3	2	no	—
11	C₃×#11719686	—	3	1	no	—
12	C₃×#11719686	C₃²	1	1	yes	characteristic
13	3₋¹⁺²×C₃₁⋊C₃	—	3	1	no	—
14	#14578902	C₃²	1	2	yes	—
15	#14578902	C₃²	1	2	yes	—
16	#14578901	—	3	2	no	—
17	#14578900	C₃²	1	2	yes	—
18	#14578898	—	3	2	no	—
19	#14578895	—	3	2	no	—
20	#14578887	C₃²	1	1	yes	characteristic
21	(C₃₁⋊C₉)⋊C₉	—	3	2	no	—
22	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
23	C₃₁×#733	C₃²	1	1	yes	characteristic
24	C₃₁×#733	C₃²	1	2	yes	—
25	C₃₁×C₉⋊C₉	C₃²	1	1	yes	characteristic
26	C₃²×C₃₁⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
27	3₋¹⁺²×C₃₁	—	3	2	no	—
28	3₋¹⁺²×C₃₁	—	3	1	no	—
29	C₃×C₉×C₃₁	—	3	1	no	—
30	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
31	C₃²⋊(C₃₁⋊C₃)	—	3	2	no	—
32	#11719689	—	3	2	no	—
33	#11719689	—	3	2	no	—
34	C₃×C₃₁⋊C₉	—	3	1	no	—
35	C₃×C₃₁⋊C₉	—	3	1	no	—
36	C₃×C₃₁⋊C₉	H(3,𝔽₃)	1	2	yes	—
37	#11719687	—	9	2	no	—
38	#11719687	—	9	1	no	—
39	#11719686	—	9	1	no	—
40	#11719686	—	9	3	no	—
41	#11719686	—	9	2	no	—
42	#11719686	—	9	1	no	—
43	C₉×C₃₁⋊C₃	—	9	2	no	—
44	C₉×C₃₁⋊C₃	—	9	1	no	—
45	#10624852	—	31	1	no	maximal, Sylow
46	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
47	C₃×C₃₁⋊C₃	—	3	1	no	—
48	C₃×C₃₁⋊C₃	—	9	1	no	—
49	C₃×C₃₁⋊C₃	—	9	2	no	—
50	C₉×C₃₁	—	3	1	no	—
51	C₉×C₃₁	—	9	1	no	—
52	C₉×C₃₁	—	9	2	no	—
53	C₉×C₃₁	—	9	1	no	—
54	C₃₁⋊C₉	—	3	3	no	—
55	C₃₁⋊C₉	—	3	2	no	—
56	C₃₁⋊C₉	—	9	1	no	—
57	C₃₁⋊C₉	—	9	2	no	—
58	C₃₁⋊C₉	—	9	2	no	—
59	#6913	—	31	2	no	—
60	#6912	—	31	1	no	—
61	#6900	—	31	1	no	—
62	#6883	—	31	2	no	—
63	#6882	—	31	2	no	—
64	#6882	C₃₁⋊C₃	1	1	yes	characteristic, complemented
65	#6881	—	31	1	no	—
66	#6880	—	31	1	no	—
67	#6878	—	31	2	no	—
68	C₃×C₃₁	—	3	1	no	—
69	C₃×C₃₁	#6911	1	1	yes	characteristic
70	C₃₁⋊C₃	—	9	1	no	—
71	C₃₁⋊C₃	—	27	2	no	—
72	#737	—	93	2	no	—
73	C₃×3₋¹⁺²	—	31	1	no	—
74	C₃×3₋¹⁺²	—	93	1	no	—
75	C₃×3₋¹⁺²	—	93	1	no	—
76	#733	—	31	2	no	—
77	#733	—	31	2	no	—
78	#733	—	93	2	no	—
79	#733	C₃×C₃₁⋊C₃	1	1	yes	characteristic
80	#733	C₃×C₃₁⋊C₃	1	2	yes	—
81	#732	—	31	2	no	—
82	#731	—	93	2	no	—
83	AΣL(1,𝔽₂₇)	—	93	2	no	—
84	C₉⋊C₉	—	31	1	no	—
85	C₉⋊C₉	—	93	2	no	—
86	C₉⋊C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
87	C₉²	—	31	1	no	—
88	C₃₁	#10624852	1	1	yes	characteristic, complemented, Sylow
89	C₃³	—	31	1	no	—
90	3₋¹⁺²	—	3	2	no	—
91	3₋¹⁺²	—	3	1	no	—
92	3₋¹⁺²	—	279	1	no	—
93	3₋¹⁺²	—	279	1	no	—
94	3₋¹⁺²	—	279	1	no	—
95	3₋¹⁺²	—	279	3	no	—
96	3₋¹⁺²	—	279	2	no	—
97	3₋¹⁺²	—	279	2	no	—
98	3₋¹⁺²	—	93	2	no	—
99	3₋¹⁺²	—	93	2	no	—
100	H(3,𝔽₃)	—	93	2	no	—
101	C₃×C₉	—	3	1	no	—
102	C₃×C₉	—	279	1	no	—
103	C₃×C₉	—	279	2	no	—
104	C₃×C₉	—	31	2	no	—
105	C₃×C₉	—	93	1	no	—
106	C₃×C₉	—	93	1	no	—
107	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
108	C₃²	—	279	1	no	—
109	C₃²	—	279	2	no	—
110	C₃²	—	93	1	no	—
111	C₃²	H(3,𝔽₃)×C₃₁⋊C₃	1	1	yes	characteristic
112	C₉	—	3	1	no	—
113	C₉	—	9	1	no	—
114	C₉	—	9	1	no	—
115	C₉	—	9	2	no	—
116	C₉	—	279	1	no	—
117	C₉	—	279	2	no	—
118	C₉	—	279	2	no	—
119	C₉	—	93	3	no	—
120	C₉	—	93	2	no	—
121	C₃	—	3	1	no	—
122	C₃	—	279	1	no	—
123	C₃	—	837	2	no	—
124	C₃	#21419403	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21419420

C₃

yes

maximal, complemented

#21419407

C₃

yes

maximal, characteristic, complemented

#21419363

C₃

yes

maximal, characteristic, complemented

#21419281

C₃

yes

maximal, complemented

#21419277

C₃

yes

maximal, complemented

#21419272

C₃

yes

maximal, characteristic, complemented

#21419267

C₃

yes

maximal, characteristic, complemented

#21419263

C₃

yes

maximal, complemented

C₃₁×#6882

C₃

yes

maximal, characteristic, complemented

#14578913

—

C₃×#11719686

—

C₃×#11719686

C₃²

yes

characteristic

3₋¹⁺²×C₃₁⋊C₃

—

#14578902

C₃²

yes

—

#14578902

C₃²

yes

—

#14578901

—

#14578900

C₃²

yes

—

#14578898

—

#14578895

—

#14578887

C₃²

yes

characteristic

(C₃₁⋊C₉)⋊C₉

—

C₉×C₃₁⋊C₉

C₃²

yes

characteristic

C₃₁×#733

C₃²

yes

characteristic

C₃₁×#733

C₃²

yes

—

C₃₁×C₉⋊C₉

C₃²

yes

characteristic

C₃²×C₃₁⋊C₃

H(3,𝔽₃)

yes

characteristic

3₋¹⁺²×C₃₁

—

3₋¹⁺²×C₃₁

—

C₃×C₉×C₃₁

—

C₃×C₉×C₃₁

C₃³

yes

characteristic

C₃²⋊(C₃₁⋊C₃)

—

#11719689

—

#11719689

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

H(3,𝔽₃)

yes

—

—

—

—

—

—

—

C₉×C₃₁⋊C₃

—

C₉×C₃₁⋊C₃

—

#10624852

—

maximal, Sylow

C₃²×C₃₁

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₁⋊C₃

—

C₃×C₃₁⋊C₃

—

C₃×C₃₁⋊C₃

—

C₉×C₃₁

—

C₉×C₃₁

—

C₉×C₃₁

—

C₉×C₃₁

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

—

—

—

—

—

C₃₁⋊C₃

yes

characteristic, complemented

#6881

—

#6880

—

#6878

—

C₃×C₃₁

—

C₃×C₃₁

#6911

yes

characteristic

C₃₁⋊C₃

—

C₃₁⋊C₃

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

—

—

—

C₃×C₃₁⋊C₃

yes

characteristic

#733

C₃×C₃₁⋊C₃

yes

—

#732

—

#731

—

AΣL(1,𝔽₂₇)

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

C₃×C₃₁⋊C₃

yes

characteristic

C₉²

—

C₃₁

#10624852

yes

characteristic, complemented, Sylow

C₃³

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

279

—

3₋¹⁺²

—

279

—

3₋¹⁺²

—

279

—

3₋¹⁺²

—

279

—

3₋¹⁺²

—

279

—

3₋¹⁺²

—

279

—

3₋¹⁺²

—

3₋¹⁺²

—

100

H(3,𝔽₃)

—

101

C₃×C₉

—

102

C₃×C₉

—

279

—

103

C₃×C₉

—

279

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

C₃²×C₃₁⋊C₃

yes

characteristic

108

C₃²

—

279

—

109

C₃²

—

279

—

110

C₃²

—

111

C₃²

H(3,𝔽₃)×C₃₁⋊C₃

yes

characteristic

112

C₉

—

113

C₉

—

114

C₉

—

115

C₉

—

116

C₉

—

279

—

117

C₉

—

279

—

118

C₉

—

279

—

119

C₉

—

120

C₉

—

121

C₃

—

122

C₃

—

279

—

123

C₃

—

837

—

124

C₃

#21419403

yes

characteristic

	subgroup	G/H
center	C₃	#21419403
commutator, cosocle	C₃×C₉×C₃₁	C₃³
Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
Fitting	C₃₁×#6882	C₃
socle	C₃×C₃₁	#6911
hypercenter	#6882	C₃₁⋊C₃
hypocenter	C₃₁	#10624852
LCS	G ▷ C₃×C₉×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6882

subgroup

G/H

center

C₃

#21419403

commutator, cosocle

C₃×C₉×C₃₁

C₃³

Frattini

C₃×C₉

C₃²×C₃₁⋊C₃

Fitting

C₃₁×#6882

C₃

C₃×C₃₁

C₃₁⋊C₃

C₃₁

G ▷ C₃×C₉×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6882

N	H ≅ G/N	type
1. #21419420	C₃	semidirect
2. #21419407	C₃	semidirect
3. #21419363	C₃	semidirect
4. #21419281	C₃	semidirect
5. #21419277	C₃	semidirect
6. #21419272	C₃	semidirect
7. #21419267	C₃	semidirect
8. #21419263	C₃	semidirect
9. C₃₁×#6882	C₃	semidirect
64. #6882	C₃₁⋊C₃	semidirect
88. C₃₁	#10624852	semidirect

H ≅ G/N

type

1. #21419420

C₃

semidirect

2. #21419407

C₃

semidirect

3. #21419363

C₃

semidirect

4. #21419281

C₃

semidirect

5. #21419277

C₃

semidirect

6. #21419272

C₃

semidirect

7. #21419267

C₃

semidirect

8. #21419263

C₃

semidirect

9. C₃₁×#6882

C₃

semidirect

64. #6882

C₃₁⋊C₃

semidirect

88. C₃₁

#10624852

semidirect

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

subgroup	index
#21419420	3
#21419407	3
#21419363	3
#21419281	3
#21419277	3
#21419272	3
#21419267	3
#21419263	3
C₃₁×#6882	3
#10624852	31

subgroup

index

C₃₁×#6882

Subgroups — #36382189

Conjugacy classes of subgroups

Complements

Maximal subgroups