Subgroups — #41005024

Order 26973.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582633	C₃	1	1	yes	maximal, characteristic
2	#22582474	C₃	1	2	yes	maximal, complemented
3	C₃²×C₂₇×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
4	#15688531	—	3	1	no	—
5	#15688521	C₉	1	2	yes	—
6	#15688519	C₉	1	1	yes	characteristic
7	#15688516	—	3	1	no	—
8	#15688513	—	3	1	no	—
9	#15688511	—	3	2	no	—
10	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
11	C₃×C₂₇×C₃₇	—	3	1	no	—
12	C₃×C₂₇×C₃₇	—	3	3	no	—
13	C₃³×C₃₇	3₋¹⁺²	1	1	yes	characteristic
14	C₃×C₉×C₃₇	—	3	1	no	—
15	C₃×C₉×C₃₇	—	3	1	no	—
16	C₃×C₉×C₃₇	—	3	1	no	—
17	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
18	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
20	C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
21	#11758608	—	3	2	no	—
22	#11758606	—	9	1	no	—
23	#11758605	—	9	1	no	—
24	C₉×C₃₇⋊C₃	—	9	1	no	—
25	C₂₇×C₃₇	—	3	3	no	—
26	#11758601	—	9	2	no	—
27	#10624526	—	37	1	no	maximal, Sylow
28	C₃²×C₃₇	—	3	3	no	—
29	C₃²×C₃₇	—	3	1	no	—
30	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
31	C₃×C₃₇⋊C₃	—	9	1	no	—
32	C₉×C₃₇	—	3	1	no	—
33	C₉×C₃₇	—	3	1	no	—
34	C₉×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
35	C₉×C₃₇	#731	1	1	yes	characteristic
36	C₉×C₃₇	#732	1	1	yes	characteristic
37	#66524	—	9	2	no	—
38	#6908	—	37	1	no	—
39	C₃²×C₂₇	C₃₇⋊C₃	1	1	yes	characteristic, complemented
40	#6869	—	37	2	no	—
41	C₃×C₃₇	—	3	3	no	—
42	C₃×C₃₇	—	3	1	no	—
43	C₃×C₃₇	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
44	C₃₇⋊C₃	—	27	1	no	—
45	#737	—	111	1	no	—
46	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
47	#733	—	37	2	no	—
48	#732	—	37	1	no	—
49	#731	—	111	1	no	—
50	AΣL(1,𝔽₂₇)	—	111	1	no	—
51	C₂₇⋊C₃	—	111	2	no	—
52	C₃×C₂₇	—	3	1	no	—
53	C₃×C₂₇	—	3	3	no	—
54	C₃₇	#10624526	1	1	yes	characteristic, complemented, Sylow
55	C₃³	#11758605	1	1	yes	characteristic
56	3₋¹⁺²	—	333	1	no	—
57	3₋¹⁺²	—	333	1	no	—
58	3₋¹⁺²	—	111	2	no	—
59	H(3,𝔽₃)	—	111	1	no	—
60	C₃×C₉	—	3	1	no	—
61	C₃×C₉	—	3	1	no	—
62	C₃×C₉	—	3	1	no	—
63	C₃×C₉	—	333	1	no	—
64	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
65	C₃×C₉	#11758606	1	1	yes	characteristic
66	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
67	C₂₇	—	3	3	no	—
68	C₂₇	—	333	2	no	—
69	C₃²	—	3	3	no	—
70	C₃²	—	3	1	no	—
71	C₃²	—	333	1	no	—
72	C₃²	#15688502	1	1	yes	characteristic
73	C₉	—	3	1	no	—
74	C₉	—	3	1	no	—
75	C₉	—	333	2	no	—
76	C₉	#15688513	1	1	yes	characteristic
77	C₉	#15688516	1	1	yes	characteristic
78	C₉	#15688519	1	1	yes	characteristic
79	C₃	—	3	1	no	—
80	C₃	—	3	3	no	—
81	C₃	—	999	1	no	—
82	C₃	#22582477	1	1	yes	characteristic

	subgroup	G/H
center	C₉	#15688519
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃²×C₂₇×C₃₇	C₃
socle	C₃×C₃₇	H(3,𝔽₃)⋊C₉
prime radical	#22582633	C₃
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	C₃²×C₂₇	C₃₇⋊C₃
hypocenter	C₃₇	#10624526
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃²×C₂₇

Complements

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

N	H ≅ G/N	type
2. #22582474	C₃	semidirect
3. C₃²×C₂₇×C₃₇	C₃	semidirect
39. C₃²×C₂₇	C₃₇⋊C₃	semidirect
54. C₃₇	#10624526	semidirect

Maximal subgroups

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

subgroup	index
#22582633	3
#22582474	3
C₃²×C₂₇×C₃₇	3
#10624526	37