Subgroups — #44525030

Order 30861.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16812184	C₃	1	3	yes	maximal, complemented
2	C₃²×#11765917	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²⋊#11765917	C₃	1	6	yes	maximal, complemented
4	C₃²⋊#11765917	C₃	1	2	yes	maximal, complemented
5	C₃×H(3,𝔽₃)×C₁₂₇	C₃	1	1	yes	maximal, characteristic
6	#16812184	—	3	9	no	—
7	C₃×#11765917	—	3	3	no	—
8	C₃×#11765917	C₃²	1	2	yes	—
9	C₃×#11765917	C₃²	1	6	yes	—
10	C₃×#11765917	C₃²	1	1	yes	characteristic
11	C₃³×C₁₂₇	C₃²	1	3	yes	—
12	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
13	H(3,𝔽₃)×C₁₂₇	—	3	3	no	—
14	C₃²×C₁₂₇	—	3	1	no	—
15	C₃²×C₁₂₇	—	3	2	no	—
16	C₃²×C₁₂₇	—	3	3	no	—
17	C₃²×C₁₂₇	—	3	3	no	—
18	C₃²×C₁₂₇	—	3	6	no	—
19	C₃²×C₁₂₇	3₋¹⁺²	1	3	yes	complemented
20	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
21	#11765917	—	3	2	no	—
22	#11765917	—	3	6	no	—
23	#11765917	H(3,𝔽₃)	1	3	yes	complemented
24	C₃×C₁₂₇	—	3	3	no	—
25	C₃×C₁₂₇	—	9	3	no	—
26	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
27	C₃×C₁₂₇	C₃×3₋¹⁺²	1	1	yes	characteristic
28	C₃×C₁₂₇	#737	1	2	yes	—
29	C₉⋊H(3,𝔽₃)	—	127	1	no	maximal, Sylow
30	C₁₂₇	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
31	C₃×3₋¹⁺²	—	127	3	no	—
32	C₃×H(3,𝔽₃)	C₁₂₇⋊C₃	1	1	yes	characteristic
33	C₃²×C₉	—	127	1	no	—
34	C₃²⋊C₉	—	127	6	no	—
35	C₃²⋊C₉	—	127	2	no	—
36	C₃³	C₃×C₁₂₇⋊C₃	1	3	yes	—
37	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
38	3₋¹⁺²	—	381	9	no	—
39	H(3,𝔽₃)	—	3	3	no	—
40	C₃×C₉	—	381	3	no	—
41	C₃×C₉	—	127	2	no	—
42	C₃×C₉	—	127	6	no	—
43	C₃×C₉	—	127	1	no	—
44	C₃²	—	3	1	no	—
45	C₃²	—	3	2	no	—
46	C₃²	—	3	3	no	—
47	C₃²	—	3	3	no	—
48	C₃²	—	3	6	no	—
49	C₃²	#16812184	1	3	yes	complemented
50	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
51	C₉	—	381	2	no	—
52	C₉	—	381	6	no	—
53	C₉	—	127	3	no	—
54	C₃	—	3	3	no	—
55	C₃	—	9	3	no	—
56	C₃	H(3,𝔽₃)×C₁₂₇⋊C₃	1	1	yes	characteristic
57	C₃	C₃×#16812184	1	1	yes	characteristic
58	C₃	#23779859	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₁₂₇⋊C₃
commutator, socle, cosocle	C₃²×C₁₂₇	C₃³
Fitting, prime radical	C₃×H(3,𝔽₃)×C₁₂₇	C₃
hypercenter	C₃×H(3,𝔽₃)	C₁₂₇⋊C₃
hypocenter	C₁₂₇	C₉⋊H(3,𝔽₃)
LCS	G ▷ C₃²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃² ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃×#16812184	C₃	semidirect
2. C₃²×#11765917	C₃	semidirect
3. C₃²⋊#11765917	C₃	semidirect
4. C₃²⋊#11765917	C₃	semidirect
19. C₃²×C₁₂₇	3₋¹⁺²	semidirect
23. #11765917	H(3,𝔽₃)	semidirect
30. C₁₂₇	C₉⋊H(3,𝔽₃)	semidirect
49. C₃²	#16812184	semidirect

Maximal subgroups

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

subgroup	index
C₃×#16812184	3
C₃²×#11765917	3
C₃²⋊#11765917	3
C₃×H(3,𝔽₃)×C₁₂₇	3
C₉⋊H(3,𝔽₃)	127