Subgroups — #44525108

Order 30861.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23779861	C₃	1	3	yes	maximal, complemented
2	C₃×C₉×C₁₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#23779850	C₃	1	1	yes	maximal, characteristic, complemented
4	#23779848	C₃	1	1	yes	maximal, characteristic, complemented
5	#23779846	C₃	1	3	yes	maximal, complemented
6	#23779843	C₃	1	3	yes	maximal, complemented
7	#733×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₁₂₇⋊C₃	C₃²	1	1	yes	characteristic
9	#16812184	C₃²	1	3	yes	—
10	#16812183	—	3	3	no	—
11	#16812182	—	3	3	no	—
12	C₃²⋊(C₁₂₇⋊C₃)	C₃²	1	3	yes	—
13	C₃×#11765917	C₃²	1	1	yes	characteristic
14	C₃×#11765917	C₃²	1	1	yes	characteristic
15	C₉×C₁₂₇⋊C₃	—	3	1	no	—
16	C₉×C₁₂₇⋊C₃	—	3	1	no	—
17	C₉×C₁₂₇⋊C₃	—	3	1	no	—
18	C₉×C₁₂₇⋊C₃	—	3	3	no	—
19	3₋¹⁺²×C₁₂₇	C₃²	1	3	yes	—
20	C₃×C₉×C₁₂₇	C₃²	1	1	yes	characteristic
21	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
22	C₃×C₁₂₇⋊C₃	—	3	1	no	—
23	C₃×C₁₂₇⋊C₃	—	3	3	no	—
24	C₃×C₁₂₇⋊C₃	—	3	3	no	—
25	C₉×C₁₂₇	—	3	1	no	—
26	C₉×C₁₂₇	—	3	3	no	—
27	#11765917	—	3	1	no	—
28	#11765917	—	3	3	no	—
29	#11765917	H(3,𝔽₃)	1	1	yes	characteristic
30	#11765917	H(3,𝔽₃)	1	1	yes	characteristic
31	#11765917	H(3,𝔽₃)	1	1	yes	characteristic
32	C₃×C₁₂₇	—	3	1	no	—
33	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
34	C₁₂₇⋊C₃	—	3	3	no	—
35	C₁₂₇⋊C₃	—	9	3	no	—
36	#6908	—	127	1	no	maximal, Sylow
37	C₁₂₇	#6908	1	1	yes	characteristic, complemented, Sylow
38	#737	—	127	3	no	—
39	C₃²×C₉	—	127	1	no	—
40	#733	—	127	1	no	—
41	#733	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
42	#732	—	127	1	no	—
43	#731	—	127	3	no	—
44	AΣL(1,𝔽₂₇)	—	127	3	no	—
45	C₃³	—	127	1	no	—
46	3₋¹⁺²	—	381	3	no	—
47	3₋¹⁺²	—	381	3	no	—
48	3₋¹⁺²	—	127	3	no	—
49	3₋¹⁺²	C₃×C₁₂₇⋊C₃	1	3	yes	—
50	H(3,𝔽₃)	—	127	3	no	—
51	C₃×C₉	—	381	1	no	—
52	C₃×C₉	—	381	1	no	—
53	C₃×C₉	—	381	1	no	—
54	C₃×C₉	—	381	3	no	—
55	C₃×C₉	—	127	1	no	—
56	C₃×C₉	—	127	1	no	—
57	C₃×C₉	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
58	C₃²	—	381	3	no	—
59	C₃²	—	381	3	no	—
60	C₃²	—	381	1	no	—
61	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
62	C₉	—	3	1	no	—
63	C₉	—	3	3	no	—
64	C₉	—	381	1	no	—
65	C₉	—	381	3	no	—
66	C₉	—	127	1	no	—
67	C₉	—	127	1	no	—
68	C₉	—	127	1	no	—
69	C₃	—	3	1	no	—
70	C₃	—	1143	3	no	—
71	C₃	—	381	3	no	—
72	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)
hypercenter	#733	C₁₂₇⋊C₃
hypocenter	C₁₂₇	#6908
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
1. #23779861	C₃	semidirect
2. C₃×C₉×C₁₂₇⋊C₃	C₃	semidirect
3. #23779850	C₃	semidirect
4. #23779848	C₃	semidirect
5. #23779846	C₃	semidirect
6. #23779843	C₃	semidirect
7. #733×C₁₂₇	C₃	semidirect
37. C₁₂₇	#6908	semidirect
41. #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
#23779861	3
C₃×C₉×C₁₂₇⋊C₃	3
#23779850	3
#23779848	3
#23779846	3
#23779843	3
#733×C₁₂₇	3
#6908	127