Subgroups — C₁₁₃×#6858 #42117203

Order 27459.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×3₋¹⁺²×C₁₁₃	C₃	1	1	yes	maximal, characteristic
2	C₃²⋊C₉×C₁₁₃	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊C₉×C₁₁₃	C₃	1	2	yes	maximal
4	C₃³×C₁₁₃	C₃²	1	1	yes	characteristic
5	3₋¹⁺²×C₁₁₃	—	3	3	no	—
6	C₃×C₉×C₁₁₃	—	3	1	no	—
7	C₃×C₉×C₁₁₃	—	3	1	no	—
8	C₃×C₉×C₁₁₃	—	3	2	no	—
9	C₃²×C₁₁₃	—	3	1	no	—
10	C₃²×C₁₁₃	—	3	2	no	—
11	C₃²×C₁₁₃	—	3	1	no	—
12	C₃²×C₁₁₃	H(3,𝔽₃)	1	1	yes	characteristic
13	C₉×C₁₁₃	—	3	3	no	—
14	C₉×C₁₁₃	—	9	1	no	—
15	C₉×C₁₁₃	—	9	2	no	—
16	C₃×C₁₁₃	—	9	1	no	—
17	C₃×C₁₁₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
18	C₃×C₁₁₃	#731	1	2	yes	—
19	C₃×C₁₁₃	#733	1	1	yes	characteristic
20	#6858	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₁₁₃	#6858	1	1	yes	characteristic, direct product factor, Sylow
22	C₃×3₋¹⁺²	C₃×C₁₁₃	1	1	yes	characteristic
23	C₃²⋊C₉	C₃×C₁₁₃	1	1	yes	characteristic
24	C₃²⋊C₉	C₃×C₁₁₃	1	2	yes	—
25	C₃³	C₃²×C₁₁₃	1	1	yes	characteristic
26	3₋¹⁺²	—	3	3	no	—
27	C₃×C₉	—	3	1	no	—
28	C₃×C₉	—	3	1	no	—
29	C₃×C₉	—	3	2	no	—
30	C₃²	—	3	1	no	—
31	C₃²	—	3	2	no	—
32	C₃²	—	3	1	no	—
33	C₃²	H(3,𝔽₃)×C₁₁₃	1	1	yes	characteristic
34	C₉	—	3	3	no	—
35	C₉	—	9	1	no	—
36	C₉	—	9	2	no	—
37	C₃	—	9	1	no	—
38	C₃	AΣL(1,𝔽₂₇)×C₁₁₃	1	1	yes	characteristic
39	C₃	#731×C₁₁₃	1	2	yes	—
40	C₃	#733×C₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²×C₁₁₃	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃²×C₁₁₃
prime radical	C₃³×C₁₁₃	C₃²
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₁₁₃ ◁ C₃³×C₁₁₃ ◁ G

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

N	H ≅ G/N	type
20. #6858	C₁₁₃	direct (N × H)
21. C₁₁₃	#6858	direct (N × H)

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