Subgroups — 3₋¹⁺²×#11765917 #44525021

Order 30861.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#11765917	C₃	1	1	yes	maximal, characteristic
2	#23779835	C₃	1	3	yes	maximal, complemented
3	#23779834	C₃	1	3	yes	maximal, complemented
4	C₃²⋊#11765917	C₃	1	2	yes	maximal
5	C₉×#11765917	C₃	1	3	yes	maximal, complemented
6	C₃×3₋¹⁺²×C₁₂₇	C₃	1	1	yes	maximal, characteristic
7	C₃×#11765917	—	3	3	no	—
8	C₃×#11765917	C₃²	1	1	yes	characteristic
9	C₃×#11765917	C₃²	1	1	yes	characteristic
10	C₃×#11765917	C₃²	1	6	yes	—
11	C₃×#11765917	C₃²	1	1	yes	characteristic
12	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
13	3₋¹⁺²×C₁₂₇	C₉	1	9	yes	complemented
14	C₃×C₉×C₁₂₇	C₃²	1	3	yes	—
15	C₃²×C₁₂₇	—	3	1	no	—
16	C₃²×C₁₂₇	—	3	2	no	—
17	C₃²×C₁₂₇	C₃×C₉	1	3	yes	—
18	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
19	C₉×C₁₂₇	C₃×C₉	1	9	yes	complemented
20	#11765917	—	3	1	no	—
21	#11765917	—	3	1	no	—
22	#11765917	—	3	6	no	—
23	#11765917	3₋¹⁺²	1	3	yes	direct product factor
24	C₃×C₁₂₇	—	3	3	no	—
25	C₃×C₁₂₇	C₃²×C₉	1	1	yes	characteristic
26	C₃×C₁₂₇	C₃×3₋¹⁺²	1	1	yes	characteristic
27	C₃×C₁₂₇	#737	1	2	yes	—
28	C₉×3₋¹⁺²	—	127	1	no	maximal, Sylow
29	C₁₂₇	C₉×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
30	C₃×3₋¹⁺²	C₁₂₇⋊C₃	1	1	yes	characteristic
31	C₃²×C₉	—	127	1	no	—
32	C₉⋊C₉	—	127	3	no	—
33	C₉⋊C₉	—	127	3	no	—
34	C₃²⋊C₉	—	127	2	no	—
35	C₉²	—	127	3	no	—
36	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
37	3₋¹⁺²	#11765917	1	9	yes	direct product factor
38	C₃×C₉	—	381	3	no	—
39	C₃×C₉	—	127	6	no	—
40	C₃×C₉	—	127	1	no	—
41	C₃×C₉	—	127	1	no	—
42	C₃×C₉	—	127	1	no	—
43	C₃×C₉	C₃×C₁₂₇⋊C₃	1	3	yes	—
44	C₃²	—	3	1	no	—
45	C₃²	—	3	2	no	—
46	C₃²	C₃×#11765917	1	3	yes	—
47	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
48	C₉	—	381	1	no	—
49	C₉	—	381	1	no	—
50	C₉	—	381	6	no	—
51	C₉	—	127	3	no	—
52	C₉	C₃×#11765917	1	9	yes	complemented
53	C₃	—	3	3	no	—
54	C₃	C₃²×#11765917	1	1	yes	characteristic
55	C₃	3₋¹⁺²×C₁₂₇⋊C₃	1	1	yes	characteristic
56	C₃	#23779861	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₁₂₇⋊C₃
commutator	C₃×C₁₂₇	C₃²×C₉
Fitting	C₃×3₋¹⁺²×C₁₂₇	C₃
socle, cosocle	C₃²×C₁₂₇	C₃³
prime radical	C₃³×C₁₂₇	C₃²
hypercenter	C₃×3₋¹⁺²	C₁₂₇⋊C₃
hypocenter	C₁₂₇	C₉×3₋¹⁺²
LCS	G ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. #23779835	C₃	semidirect
3. #23779834	C₃	semidirect
5. C₉×#11765917	C₃	semidirect
13. 3₋¹⁺²×C₁₂₇	C₉	semidirect
19. C₉×C₁₂₇	C₃×C₉	semidirect
23. #11765917	3₋¹⁺²	direct (N × H)
29. C₁₂₇	C₉×3₋¹⁺²	semidirect
37. 3₋¹⁺²	#11765917	direct (N × H)
52. C₉	C₃×#11765917	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#11765917	3
#23779835	3
#23779834	3
C₃²⋊#11765917	3
C₉×#11765917	3
C₃×3₋¹⁺²×C₁₂₇	3
C₉×3₋¹⁺²	127

Subgroups — 3−1+2×#11765917 #44525021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#11765917 #44525021