Subgroups — #41004834

Order 26973.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688519	C₃	1	1	yes	maximal, characteristic, complemented
2	#22582470	C₃	1	2	yes	maximal, complemented
3	C₃×C₃₇×#732	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
5	#15688519	—	3	3	no	—
6	C₃²⋊#66524	—	3	2	no	—
7	C₃×H(3,𝔽₃)×C₃₇	—	3	1	no	—
8	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
9	C₃₇×#732	—	3	3	no	—
10	C₃³×C₃₇	—	9	1	no	—
11	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
12	C₃²×C₃₇⋊C₃	—	9	1	no	—
13	H(3,𝔽₃)×C₃₇	—	3	3	no	—
14	H(3,𝔽₃)×C₃₇	—	9	2	no	—
15	C₃×C₉×C₃₇	—	3	1	no	—
16	C₃×C₉×C₃₇	—	3	1	no	—
17	C₃×C₉×C₃₇	—	3	2	no	—
18	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
19	C₃²⋊(C₃₇⋊C₃)	—	9	2	no	—
20	C₃×#66524	—	9	2	no	—
21	#10624477	—	37	1	no	maximal, Sylow
22	C₃²×C₃₇	—	9	1	no	—
23	C₃²×C₃₇	—	9	3	no	—
24	C₃²×C₃₇	—	27	3	no	—
25	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
26	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
27	C₃²×C₃₇	#732	1	2	yes	—
28	C₃×C₃₇⋊C₃	—	9	3	no	—
29	C₃×C₃₇⋊C₃	—	27	3	no	—
30	C₉×C₃₇	—	9	1	no	—
31	#66524	—	27	2	no	—
32	C₃×#732	—	37	1	no	—
33	C₃×#732	C₃₇⋊C₃	1	1	yes	characteristic, complemented
34	#6867	—	37	2	no	—
35	C₃×C₃₇	—	3	2	no	—
36	C₃×C₃₇	—	3	1	no	—
37	C₃×C₃₇	—	3	1	no	—
38	C₃×C₃₇	—	27	3	no	—
39	C₃×C₃₇	#6856	1	1	yes	characteristic
40	C₃₇⋊C₃	—	27	3	no	—
41	C₃×H(3,𝔽₃)	—	3	1	no	—
42	C₃×H(3,𝔽₃)	—	111	1	no	—
43	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
44	#732	—	3	3	no	—
45	#732	—	111	3	no	—
46	C₃²⋊C₉	—	111	2	no	—
47	C₃₇	#10624477	1	1	yes	characteristic, complemented, Sylow
48	C₃³	—	9	1	no	—
49	C₃³	—	333	1	no	—
50	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
51	H(3,𝔽₃)	—	3	3	no	—
52	H(3,𝔽₃)	—	9	2	no	—
53	H(3,𝔽₃)	—	333	2	no	—
54	H(3,𝔽₃)	—	111	3	no	—
55	C₃×C₉	—	3	1	no	—
56	C₃×C₉	—	3	1	no	—
57	C₃×C₉	—	3	2	no	—
58	C₃×C₉	—	333	2	no	—
59	C₃²	—	9	1	no	—
60	C₃²	—	9	3	no	—
61	C₃²	—	27	3	no	—
62	C₃²	—	333	3	no	—
63	C₃²	—	999	3	no	—
64	C₃²	#15688512	1	1	yes	characteristic
65	C₃²	#15688513	1	1	yes	characteristic
66	C₃²	#15688518	1	2	yes	—
67	C₉	—	9	1	no	—
68	C₉	—	999	2	no	—
69	C₃	—	3	2	no	—
70	C₃	—	3	1	no	—
71	C₃	—	3	1	no	—
72	C₃	—	27	3	no	—
73	C₃	—	999	3	no	—
74	C₃	#22582442	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582442
commutator, cosocle	C₃²×C₉×C₃₇	C₃²
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃×C₃₇×#732	C₃
socle	C₃×C₃₇	#6856
hypercenter	C₃×#732	C₃₇⋊C₃
hypocenter	C₃₇	#10624477
LCS	G ▷ C₃²×C₉×C₃₇ ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ C₃×#732

Complements

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

N	H ≅ G/N	type
1. C₃×#15688519	C₃	semidirect
2. #22582470	C₃	semidirect
3. C₃×C₃₇×#732	C₃	semidirect
33. C₃×#732	C₃₇⋊C₃	semidirect
47. C₃₇	#10624477	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688519	3
#22582470	3
C₃×C₃₇×#732	3
#10624477	37