Subgroups — #41004974

Order 26973.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582474	C₃	1	3	yes	maximal, complemented
2	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic
3	#15688511	—	3	3	no	—
4	C₃₇×#737	—	3	9	no	—
5	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
6	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
7	C₃×3₋¹⁺²×C₃₇	C₉	1	2	yes	—
8	C₃×H(3,𝔽₃)×C₃₇	C₉	1	1	yes	characteristic
9	C₃²×C₉×C₃₇	—	3	1	no	—
10	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
11	C₃³×C₃₇	—	3	1	no	—
12	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
13	3₋¹⁺²×C₃₇	—	3	9	no	—
14	3₋¹⁺²×C₃₇	—	3	9	no	—
15	3₋¹⁺²×C₃₇	—	3	6	no	—
16	H(3,𝔽₃)×C₃₇	—	3	3	no	—
17	C₃×C₉×C₃₇	—	3	9	no	—
18	C₃×C₉×C₃₇	—	3	2	no	—
19	C₃×C₉×C₃₇	—	3	3	no	—
20	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
22	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
23	#11758601	—	9	3	no	—
24	#10624511	—	37	1	no	maximal, Sylow
25	C₃²×C₃₇	—	3	1	no	—
26	C₃²×C₃₇	—	3	3	no	—
27	C₃²×C₃₇	—	9	1	no	—
28	C₃²×C₃₇	—	9	3	no	—
29	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
30	C₉×C₃₇	—	3	1	no	—
31	C₉×C₃₇	—	3	6	no	—
32	C₉×C₃₇	—	3	1	no	—
33	C₉×C₃₇	AΣL(1,𝔽₂₇)	1	3	yes	—
34	C₃×#737	C₃₇⋊C₃	1	1	yes	characteristic
35	#6869	—	37	3	no	—
36	C₃×C₃₇	—	3	1	no	—
37	C₃×C₃₇	—	9	1	no	—
38	C₃×C₃₇	—	9	3	no	—
39	C₃×C₃₇	#6866	1	1	yes	characteristic
40	#737	—	3	9	no	—
41	C₃×3₋¹⁺²	—	3	1	no	—
42	C₃×3₋¹⁺²	—	3	1	no	—
43	C₃×3₋¹⁺²	#66524	1	2	yes	—
44	C₃×H(3,𝔽₃)	#66524	1	1	yes	characteristic
45	C₃²×C₉	—	3	1	no	—
46	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
47	C₂₇⋊C₃	—	111	3	no	—
48	C₃₇	#10624511	1	1	yes	characteristic, complemented, Sylow
49	C₃³	—	3	1	no	—
50	C₃³	C₃×#66524	1	1	yes	characteristic
51	3₋¹⁺²	—	3	9	no	—
52	3₋¹⁺²	—	3	9	no	—
53	3₋¹⁺²	—	3	6	no	—
54	H(3,𝔽₃)	—	3	3	no	—
55	C₃×C₉	—	3	9	no	—
56	C₃×C₉	—	3	2	no	—
57	C₃×C₉	—	3	3	no	—
58	C₃×C₉	#11758608	1	1	yes	characteristic
59	C₃×C₉	#11758608	1	1	yes	characteristic
60	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
61	C₂₇	—	333	3	no	—
62	C₃²	—	3	1	no	—
63	C₃²	—	3	3	no	—
64	C₃²	—	9	1	no	—
65	C₃²	—	9	3	no	—
66	C₃²	C₃²⋊#66524	1	1	yes	characteristic
67	C₉	—	3	1	no	—
68	C₉	—	3	6	no	—
69	C₉	—	3	1	no	—
70	C₉	#15688513	1	3	yes	—
71	C₃	—	3	1	no	—
72	C₃	—	9	3	no	—
73	C₃	—	9	1	no	—
74	C₃	#22582468	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582468
commutator	C₃³×C₃₇	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃×C₃₇×#737	C₃
socle	C₃×C₃₇	#6866
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₉
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	C₃×#737	C₃₇⋊C₃
hypocenter	C₃₇	#10624511
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃×#737

Complements

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

N	H ≅ G/N	type
1. #22582474	C₃	semidirect
48. C₃₇	#10624511	semidirect

Maximal subgroups

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

subgroup	index
#22582474	3
C₃×C₃₇×#737	3
#10624511	37