Subgroups — #44525118

Order 30861.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16812184	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#16812184	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
4	#23779846	C₃	1	9	yes	maximal, complemented
5	C₃×3₋¹⁺²×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₁₂₇⋊C₃	—	3	1	no	—
7	#16812184	—	3	1	no	—
8	#16812184	—	3	1	no	—
9	#16812184	—	3	1	no	—
10	#16812184	—	3	1	no	—
11	#16812184	—	3	1	no	—
12	#16812184	C₃²	1	3	yes	complemented
13	C₃²⋊(C₁₂₇⋊C₃)	—	3	1	no	—
14	C₃²⋊(C₁₂₇⋊C₃)	—	3	1	no	—
15	C₃²⋊(C₁₂₇⋊C₃)	C₃²	1	3	yes	—
16	C₃×#11765917	—	3	1	no	—
17	C₃×#11765917	C₃²	1	3	yes	complemented
18	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
19	3₋¹⁺²×C₁₂₇	—	3	1	no	—
20	3₋¹⁺²×C₁₂₇	—	3	1	no	—
21	3₋¹⁺²×C₁₂₇	C₃²	1	3	yes	complemented
22	C₃×C₉×C₁₂₇	—	3	1	no	—
23	C₃²×C₁₂₇	—	9	1	no	—
24	C₃²×C₁₂₇	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃²×C₁₂₇	H(3,𝔽₃)	1	1	yes	characteristic
26	C₃²×C₁₂₇	H(3,𝔽₃)	1	1	yes	characteristic
27	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
28	C₃×C₁₂₇⋊C₃	—	3	3	no	—
29	C₃×C₁₂₇⋊C₃	—	9	3	no	—
30	C₉×C₁₂₇	—	3	3	no	—
31	#11765917	—	3	3	no	—
32	#11765917	—	3	3	no	—
33	C₃×C₁₂₇	—	3	1	no	—
34	C₃×C₁₂₇	—	3	1	no	—
35	C₃×C₁₂₇	—	3	1	no	—
36	C₃×C₁₂₇	—	3	1	no	—
37	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
38	C₁₂₇⋊C₃	—	9	3	no	—
39	#6910	—	127	1	no	maximal, Sylow
40	C₁₂₇	#6910	1	1	yes	characteristic, complemented, Sylow
41	C₃×3₋¹⁺²	—	127	1	no	—
42	C₃×3₋¹⁺²	—	127	1	no	—
43	C₃×3₋¹⁺²	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
44	C₃×H(3,𝔽₃)	—	127	1	no	—
45	#731	—	127	9	no	—
46	C₃³	—	381	1	no	—
47	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
48	3₋¹⁺²	—	3	1	no	—
49	3₋¹⁺²	—	3	1	no	—
50	3₋¹⁺²	—	381	1	no	—
51	3₋¹⁺²	—	381	1	no	—
52	3₋¹⁺²	—	381	1	no	—
53	3₋¹⁺²	—	381	1	no	—
54	3₋¹⁺²	—	381	1	no	—
55	3₋¹⁺²	—	127	3	no	—
56	3₋¹⁺²	C₃×C₁₂₇⋊C₃	1	3	yes	complemented
57	H(3,𝔽₃)	—	381	1	no	—
58	H(3,𝔽₃)	—	381	1	no	—
59	H(3,𝔽₃)	—	127	3	no	—
60	C₃×C₉	—	3	1	no	—
61	C₃×C₉	—	381	1	no	—
62	C₃×C₉	—	127	3	no	—
63	C₃²	—	9	1	no	—
64	C₃²	—	1143	3	no	—
65	C₃²	—	381	3	no	—
66	C₃²	C₃²⋊(C₁₂₇⋊C₃)	1	1	yes	characteristic
67	C₃²	C₃²⋊(C₁₂₇⋊C₃)	1	1	yes	characteristic
68	C₃²	C₃²⋊(C₁₂₇⋊C₃)	1	1	yes	characteristic
69	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
70	C₉	—	3	3	no	—
71	C₉	—	381	3	no	—
72	C₉	—	381	3	no	—
73	C₃	—	3	1	no	—
74	C₃	—	3	1	no	—
75	C₃	—	3	1	no	—
76	C₃	—	3	1	no	—
77	C₃	—	1143	3	no	—
78	C₃	C₃×C₃²⋊(C₁₂₇⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×C₃²⋊(C₁₂₇⋊C₃)
commutator, cosocle	C₃²×C₁₂₇	C₃³
Frattini	C₃²	C₃²×C₁₂₇⋊C₃
Fitting	C₃×3₋¹⁺²×C₁₂₇	C₃
socle	C₃×C₁₂₇	C₃×H(3,𝔽₃)
prime radical	C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃
hypercenter	C₃×3₋¹⁺²	C₁₂₇⋊C₃
hypocenter	C₁₂₇	#6910
LCS	G ▷ C₃²×C₁₂₇ ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₃×#16812184	C₃	semidirect
2. C₃×#16812184	C₃	semidirect
4. #23779846	C₃	semidirect
5. C₃×3₋¹⁺²×C₁₂₇	C₃	semidirect
12. #16812184	C₃²	semidirect
17. C₃×#11765917	C₃²	semidirect
21. 3₋¹⁺²×C₁₂₇	C₃²	semidirect
40. C₁₂₇	#6910	semidirect
43. C₃×3₋¹⁺²	C₁₂₇⋊C₃	semidirect
56. 3₋¹⁺²	C₃×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×#16812184	3
C₃×C₃²⋊(C₁₂₇⋊C₃)	3
#23779846	3
C₃×3₋¹⁺²×C₁₂₇	3
#6910	127