Subgroups — #44525124

Order 30861.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16812182	C₃	1	1	yes	maximal, characteristic
2	3₋¹⁺²×C₁₂₇⋊C₃	C₃	1	3	yes	maximal
3	#23779850	C₃	1	2	yes	maximal, complemented
4	#23779849	C₃	1	6	yes	maximal, complemented
5	#733×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₁₂₇⋊C₃	C₃²	1	1	yes	characteristic
7	#16812184	C₃²	1	6	yes	—
8	#16812183	—	3	3	no	—
9	#16812182	—	3	3	no	—
10	#16812182	—	3	3	no	—
11	C₃×#11765917	C₃²	1	2	yes	—
12	C₉×C₁₂₇⋊C₃	—	3	3	no	—
13	3₋¹⁺²×C₁₂₇	C₃²	1	3	yes	—
14	C₃×C₉×C₁₂₇	C₃²	1	1	yes	characteristic
15	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
16	C₃×C₁₂₇⋊C₃	—	9	1	no	—
17	C₃×C₁₂₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
18	C₉×C₁₂₇	—	3	1	no	—
19	C₉×C₁₂₇	—	3	3	no	—
20	#11765917	—	3	6	no	—
21	#11765917	—	3	2	no	—
22	C₃×C₁₂₇	—	3	1	no	—
23	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
24	C₁₂₇⋊C₃	—	3	3	no	—
25	#6912	—	127	1	no	maximal, Sylow
26	C₁₂₇	#6912	1	1	yes	characteristic, complemented, Sylow
27	C₃×3₋¹⁺²	—	127	3	no	—
28	C₃×3₋¹⁺²	—	127	1	no	—
29	#733	—	127	6	no	—
30	#733	—	127	2	no	—
31	#733	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
32	C₃³	—	127	1	no	—
33	3₋¹⁺²	—	381	3	no	—
34	3₋¹⁺²	—	381	3	no	—
35	3₋¹⁺²	—	381	3	no	—
36	3₋¹⁺²	—	127	6	no	—
37	3₋¹⁺²	C₃×C₁₂₇⋊C₃	1	3	yes	—
38	C₃×C₉	—	381	3	no	—
39	C₃×C₉	—	127	2	no	—
40	C₃×C₉	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
41	C₃²	—	1143	1	no	—
42	C₃²	—	127	3	no	—
43	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
44	C₉	—	3	3	no	—
45	C₉	—	3	1	no	—
46	C₉	—	381	6	no	—
47	C₉	—	381	2	no	—
48	C₃	—	3	1	no	—
49	C₃	—	381	3	no	—
50	C₃	H(3,𝔽₃)×C₁₂₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×C₁₂₇⋊C₃
commutator, cosocle	C₃²×C₁₂₇	C₃³
Frattini	C₃²	C₃²×C₁₂₇⋊C₃
Fitting	#733×C₁₂₇	C₃
socle	C₃×C₁₂₇	C₃×H(3,𝔽₃)
prime radical	C₃²×C₁₂₇⋊C₃	C₃²
hypercenter	#733	C₁₂₇⋊C₃
hypocenter	C₁₂₇	#6912
LCS	G ▷ C₃²×C₁₂₇ ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #733

Complements

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

N	H ≅ G/N	type
3. #23779850	C₃	semidirect
4. #23779849	C₃	semidirect
5. #733×C₁₂₇	C₃	semidirect
26. C₁₂₇	#6912	semidirect
31. #733	C₁₂₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×#16812182	3
3₋¹⁺²×C₁₂₇⋊C₃	3
#23779850	3
#23779849	3
#733×C₁₂₇	3
#6912	127