Subgroups — C₁₀₁×H(3,𝔽₃)⋊C₉ #37589404

Order 24543.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(3,𝔽₃)×C₁₀₁	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₉×C₁₀₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²⋊C₉×C₁₀₁	C₃	1	2	yes	maximal, complemented
4	C₃³×C₁₀₁	—	3	1	no	—
5	C₃³×C₁₀₁	C₃²	1	1	yes	characteristic
6	H(3,𝔽₃)×C₁₀₁	—	3	2	no	—
7	H(3,𝔽₃)×C₁₀₁	C₉	1	3	yes	complemented
8	C₃×C₉×C₁₀₁	—	3	1	no	—
9	C₃×C₉×C₁₀₁	—	3	3	no	—
10	C₃×C₉×C₁₀₁	—	3	2	no	—
11	C₃²×C₁₀₁	—	3	1	no	—
12	C₃²×C₁₀₁	—	3	1	no	—
13	C₃²×C₁₀₁	—	3	1	no	—
14	C₃²×C₁₀₁	—	3	3	no	—
15	C₃²×C₁₀₁	—	9	1	no	—
16	C₃²×C₁₀₁	—	9	1	no	—
17	C₃²×C₁₀₁	—	9	1	no	—
18	C₃²×C₁₀₁	C₃×C₉	1	1	yes	characteristic
19	C₃²×C₁₀₁	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃²×C₁₀₁	3₋¹⁺²	1	2	yes	—
21	C₉×C₁₀₁	—	3	3	no	—
22	C₉×C₁₀₁	—	9	2	no	—
23	C₃×C₁₀₁	—	3	1	no	—
24	C₃×C₁₀₁	—	3	2	no	—
25	C₃×C₁₀₁	—	9	3	no	—
26	C₃×C₁₀₁	C₃²⋊C₉	1	1	yes	characteristic
27	C₃×C₁₀₁	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
28	C₃×C₁₀₁	#731	1	1	yes	characteristic
29	C₃×C₁₀₁	#732	1	1	yes	characteristic
30	H(3,𝔽₃)⋊C₉	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₁₀₁	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
32	C₃×H(3,𝔽₃)	C₃×C₁₀₁	1	1	yes	characteristic
33	C₃²×C₉	C₃×C₁₀₁	1	1	yes	characteristic, complemented
34	C₃²⋊C₉	C₃×C₁₀₁	1	2	yes	complemented
35	C₃³	—	3	1	no	—
36	C₃³	C₃²×C₁₀₁	1	1	yes	characteristic
37	H(3,𝔽₃)	—	3	2	no	—
38	H(3,𝔽₃)	C₉×C₁₀₁	1	3	yes	complemented
39	C₃×C₉	—	3	1	no	—
40	C₃×C₉	—	3	3	no	—
41	C₃×C₉	—	3	2	no	—
42	C₃²	—	3	1	no	—
43	C₃²	—	3	1	no	—
44	C₃²	—	3	1	no	—
45	C₃²	—	3	3	no	—
46	C₃²	—	9	1	no	—
47	C₃²	—	9	1	no	—
48	C₃²	—	9	1	no	—
49	C₃²	C₃×C₉×C₁₀₁	1	1	yes	characteristic
50	C₃²	H(3,𝔽₃)×C₁₀₁	1	1	yes	characteristic
51	C₃²	3₋¹⁺²×C₁₀₁	1	2	yes	—
52	C₉	—	3	3	no	—
53	C₉	—	9	2	no	—
54	C₃	—	3	1	no	—
55	C₃	—	3	2	no	—
56	C₃	—	9	3	no	—
57	C₃	C₃²⋊C₉×C₁₀₁	1	1	yes	characteristic
58	C₃	AΣL(1,𝔽₂₇)×C₁₀₁	1	1	yes	characteristic
59	C₃	#731×C₁₀₁	1	1	yes	characteristic
60	C₃	#732×C₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²×C₁₀₁	H(3,𝔽₃)
commutator	C₃²	C₃×C₉×C₁₀₁
Frattini, cosocle	C₃³	C₃²×C₁₀₁
prime radical	C₃×H(3,𝔽₃)×C₁₀₁	C₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₁₀₁ ◁ C₃³×C₁₀₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃²×C₉×C₁₀₁	C₃	semidirect
3. C₃²⋊C₉×C₁₀₁	C₃	semidirect
7. H(3,𝔽₃)×C₁₀₁	C₉	semidirect
30. H(3,𝔽₃)⋊C₉	C₁₀₁	direct (N × H)
31. C₁₀₁	H(3,𝔽₃)⋊C₉	direct (N × H)
33. C₃²×C₉	C₃×C₁₀₁	semidirect
34. C₃²⋊C₉	C₃×C₁₀₁	semidirect
38. H(3,𝔽₃)	C₉×C₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₃×H(3,𝔽₃)×C₁₀₁	3
C₃²×C₉×C₁₀₁	3
C₃²⋊C₉×C₁₀₁	3
H(3,𝔽₃)⋊C₉	101

Subgroups — C101×H(3,𝔽3)⋊C9 #37589404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×H(3,𝔽₃)⋊C₉ #37589404