Subgroups — #44525127

Order 30861.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23779861	C₃	1	2	yes	maximal, complemented
2	#23779859	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#16812184	C₃	1	1	yes	maximal, characteristic, complemented
4	#23779849	C₃	1	2	yes	maximal, complemented
5	#23779847	C₃	1	1	yes	maximal, characteristic, complemented
6	#23779846	C₃	1	2	yes	maximal, complemented
7	#23779844	C₃	1	1	yes	maximal, characteristic, complemented
8	#23779842	C₃	1	2	yes	maximal, complemented
9	AΣL(1,𝔽₂₇)×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
10	#16812184	—	3	2	no	—
11	#16812184	—	3	1	no	—
12	#16812184	—	3	2	no	—
13	#16812184	C₃²	1	2	yes	—
14	#16812184	C₃²	1	2	yes	—
15	#16812183	—	3	2	no	—
16	#16812182	—	3	2	no	—
17	C₃²⋊(C₁₂₇⋊C₃)	C₃²	1	2	yes	—
18	C₃×#11765917	—	3	1	no	—
19	C₃×#11765917	C₃²	1	1	yes	characteristic
20	C₃×#11765917	C₃²	1	1	yes	characteristic
21	C₃×#11765917	C₃²	1	1	yes	characteristic
22	C₉×C₁₂₇⋊C₃	—	3	2	no	—
23	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
24	3₋¹⁺²×C₁₂₇	C₃²	1	2	yes	—
25	H(3,𝔽₃)×C₁₂₇	C₃²	1	1	yes	characteristic
26	C₃²×C₁₂₇	—	3	1	no	—
27	C₃²×C₁₂₇	—	3	1	no	—
28	C₃²×C₁₂₇	—	9	1	no	—
29	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
30	C₃×C₁₂₇⋊C₃	—	3	2	no	—
31	C₉×C₁₂₇	—	3	2	no	—
32	#11765917	—	3	2	no	—
33	#11765917	—	3	2	no	—
34	#11765917	—	3	1	no	—
35	#11765917	—	3	1	no	—
36	#11765917	H(3,𝔽₃)	1	2	yes	—
37	#11765917	H(3,𝔽₃)	1	1	yes	characteristic
38	C₃×C₁₂₇	—	3	1	no	—
39	C₃×C₁₂₇	—	9	1	no	—
40	C₃×C₁₂₇	—	9	1	no	—
41	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	C₁₂₇⋊C₃	—	9	2	no	—
43	#6913	—	127	1	no	maximal, Sylow
44	C₁₂₇	#6913	1	1	yes	characteristic, complemented, Sylow
45	#737	—	127	2	no	—
46	#737	—	127	1	no	—
47	C₃×3₋¹⁺²	—	127	1	no	—
48	#733	—	127	2	no	—
49	#732	—	127	1	no	—
50	#731	—	127	2	no	—
51	#731	—	127	1	no	—
52	AΣL(1,𝔽₂₇)	—	127	2	no	—
53	AΣL(1,𝔽₂₇)	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
54	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
55	3₋¹⁺²	—	381	2	no	—
56	3₋¹⁺²	—	381	2	no	—
57	3₋¹⁺²	—	381	2	no	—
58	3₋¹⁺²	—	381	1	no	—
59	3₋¹⁺²	—	381	2	no	—
60	3₋¹⁺²	—	127	2	no	—
61	3₋¹⁺²	—	127	2	no	—
62	3₋¹⁺²	C₃×C₁₂₇⋊C₃	1	2	yes	—
63	H(3,𝔽₃)	—	127	2	no	—
64	H(3,𝔽₃)	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
65	C₃×C₉	—	381	2	no	—
66	C₃×C₉	—	381	1	no	—
67	C₃×C₉	—	127	1	no	—
68	C₃×C₉	—	127	1	no	—
69	C₃×C₉	—	127	1	no	—
70	C₃²	—	3	1	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	9	1	no	—
73	C₃²	—	381	2	no	—
74	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
75	C₉	—	3	2	no	—
76	C₉	—	381	2	no	—
77	C₉	—	381	2	no	—
78	C₉	—	381	1	no	—
79	C₉	—	381	1	no	—
80	C₉	—	127	2	no	—
81	C₉	—	127	1	no	—
82	C₃	—	3	1	no	—
83	C₃	—	9	1	no	—
84	C₃	—	9	1	no	—
85	C₃	—	1143	2	no	—
86	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	AΣL(1,𝔽₂₇)×C₁₂₇	C₃
socle	C₃×C₁₂₇	C₃×H(3,𝔽₃)
hypercenter	AΣL(1,𝔽₂₇)	C₁₂₇⋊C₃
hypocenter	C₁₂₇	#6913
LCS	G ▷ C₃²×C₁₂₇ ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ AΣL(1,𝔽₂₇)

Complements

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

N	H ≅ G/N	type
1. #23779861	C₃	semidirect
2. #23779859	C₃	semidirect
3. C₃×#16812184	C₃	semidirect
4. #23779849	C₃	semidirect
5. #23779847	C₃	semidirect
6. #23779846	C₃	semidirect
7. #23779844	C₃	semidirect
8. #23779842	C₃	semidirect
9. AΣL(1,𝔽₂₇)×C₁₂₇	C₃	semidirect
44. C₁₂₇	#6913	semidirect
53. AΣL(1,𝔽₂₇)	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
#23779859	3
C₃×#16812184	3
#23779849	3
#23779847	3
#23779846	3
#23779844	3
#23779842	3
AΣL(1,𝔽₂₇)×C₁₂₇	3
#6913	127