Subgroups — #41004973

Order 26973.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688531	C₃	1	1	yes	maximal, characteristic
2	#22582474	C₃	1	2	yes	maximal, complemented
3	C₃₇×#6869	C₃	1	1	yes	maximal, characteristic, complemented
4	#15688531	—	3	3	no	—
5	#15688531	—	3	6	no	—
6	C₃×#11758608	C₉	1	2	yes	—
7	C₃×#11758605	—	3	1	no	—
8	C₃×#11758606	—	3	1	no	—
9	C₃×C₃²⋊(C₃₇⋊C₃)	C₉	1	1	yes	characteristic
10	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
11	#15688511	—	3	2	no	—
12	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
13	C₃₇×C₂₇⋊C₃	—	3	1	no	—
14	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
15	C₃²×C₃₇⋊C₃	—	3	1	no	—
16	C₃×C₉×C₃₇	—	3	1	no	—
17	C₃×C₉×C₃₇	—	3	2	no	—
18	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
20	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
21	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
22	#11758608	—	3	6	no	—
23	C₃×#66524	—	3	2	no	—
24	#11758606	—	3	3	no	—
25	#11758606	—	3	6	no	—
26	#11758605	—	3	9	no	—
27	C₉×C₃₇⋊C₃	—	3	9	no	—
28	C₂₇×C₃₇	—	9	1	no	—
29	#11758601	—	9	2	no	—
30	#10624511	—	37	1	no	maximal, Sylow
31	C₃²×C₃₇	—	3	1	no	—
32	C₃²×C₃₇	—	9	1	no	—
33	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
34	C₃×C₃₇⋊C₃	—	3	3	no	—
35	C₃×C₃₇⋊C₃	—	9	3	no	—
36	C₉×C₃₇	—	3	1	no	—
37	C₉×C₃₇	—	3	1	no	—
38	C₉×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
39	C₉×C₃₇	AΣL(1,𝔽₂₇)	1	2	yes	—
40	#66524	—	3	6	no	—
41	C₃×#737	—	37	1	no	—
42	#6869	—	37	2	no	—
43	#6869	C₃₇⋊C₃	1	1	yes	characteristic, complemented
44	C₃×C₃₇	—	3	1	no	—
45	C₃×C₃₇	—	9	1	no	—
46	C₃×C₃₇	#6866	1	1	yes	characteristic
47	C₃₇⋊C₃	—	9	3	no	—
48	#737	—	111	3	no	—
49	#737	—	111	6	no	—
50	C₃×3₋¹⁺²	—	37	2	no	—
51	C₃×3₋¹⁺²	—	111	1	no	—
52	C₃×3₋¹⁺²	—	111	1	no	—
53	C₃×H(3,𝔽₃)	—	37	1	no	—
54	C₃²×C₉	—	111	1	no	—
55	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
56	C₂₇⋊C₃	—	3	1	no	—
57	C₂₇⋊C₃	—	111	2	no	—
58	C₃₇	#10624511	1	1	yes	characteristic, complemented, Sylow
59	C₃³	—	111	1	no	—
60	C₃³	C₉×C₃₇⋊C₃	1	1	yes	characteristic
61	3₋¹⁺²	—	111	9	no	—
62	3₋¹⁺²	—	111	3	no	—
63	3₋¹⁺²	—	111	6	no	—
64	3₋¹⁺²	—	111	6	no	—
65	H(3,𝔽₃)	—	111	3	no	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	3	2	no	—
68	C₃×C₉	—	111	9	no	—
69	C₃×C₉	—	111	2	no	—
70	C₃×C₉	#11758605	1	1	yes	characteristic
71	C₃×C₉	#11758606	1	1	yes	characteristic
72	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
73	C₂₇	—	9	1	no	—
74	C₂₇	—	333	2	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	9	1	no	—
77	C₃²	—	333	3	no	—
78	C₃²	—	111	3	no	—
79	C₃²	#15688502	1	1	yes	characteristic
80	C₉	—	3	1	no	—
81	C₉	—	3	1	no	—
82	C₉	—	111	6	no	—
83	C₉	#15688512	1	1	yes	characteristic
84	C₉	#15688514	1	2	yes	—
85	C₃	—	3	1	no	—
86	C₃	—	9	1	no	—
87	C₃	—	333	3	no	—
88	C₃	#22582467	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582467
commutator	C₃³×C₃₇	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6869	C₃
socle	C₃×C₃₇	#6866
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₉
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	#6869	C₃₇⋊C₃
hypocenter	C₃₇	#10624511
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6869

Complements

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

N	H ≅ G/N	type
2. #22582474	C₃	semidirect
3. C₃₇×#6869	C₃	semidirect
43. #6869	C₃₇⋊C₃	semidirect
58. C₃₇	#10624511	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688531	3
#22582474	3
C₃₇×#6869	3
#10624511	37