Subgroups — #41004868

Order 26973.

130 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
2	#22582578	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582470	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#6895	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
6	#15688519	—	3	3	no	—
7	#15688504	—	3	3	no	—
8	C₃²⋊#66524	—	3	1	no	—
9	C₃²⋊#66524	—	3	1	no	—
10	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
11	C₃₇×C₉⋊C₉	—	3	3	no	—
12	C₃₇×C₃²⋊C₉	—	3	1	no	—
13	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
14	C₃²×C₃₇⋊C₃	—	9	1	no	—
15	C₃×C₉×C₃₇	—	3	3	no	—
16	C₃×C₉×C₃₇	—	3	1	no	—
17	C₃×C₉×C₃₇	—	3	1	no	—
18	C₃×C₉×C₃₇	—	3	1	no	—
19	C₃×C₉×C₃₇	—	3	1	no	—
20	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
21	C₃²⋊(C₃₇⋊C₃)	—	9	1	no	—
22	C₃²⋊(C₃₇⋊C₃)	—	9	1	no	—
23	C₃×#66524	—	3	3	no	—
24	C₃×#66524	—	9	1	no	—
25	#10624484	—	37	1	no	maximal, Sylow
26	C₃²×C₃₇	—	9	1	no	—
27	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
28	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
29	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
30	C₃²×C₃₇	#731	1	1	yes	characteristic
31	C₃×C₃₇⋊C₃	—	9	3	no	—
32	C₃×C₃₇⋊C₃	—	27	3	no	—
33	C₉×C₃₇	—	9	3	no	—
34	C₉×C₃₇	—	9	1	no	—
35	#66524	—	9	3	no	—
36	#66524	—	27	1	no	—
37	C₃×#732	—	37	1	no	—
38	#6895	—	37	1	no	—
39	#6895	C₃₇⋊C₃	1	1	yes	characteristic, complemented
40	#6867	—	37	1	no	—
41	C₃×C₃₇	—	3	1	no	—
42	C₃×C₃₇	—	3	1	no	—
43	C₃×C₃₇	—	3	1	no	—
44	C₃×C₃₇	—	3	1	no	—
45	C₃×C₃₇	#6857	1	1	yes	characteristic
46	C₃₇⋊C₃	—	27	3	no	—
47	C₃×H(3,𝔽₃)	—	111	1	no	—
48	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
49	#732	—	111	3	no	—
50	C₉⋊C₉	—	3	3	no	—
51	C₉⋊C₉	—	111	3	no	—
52	C₃²⋊C₉	—	3	1	no	—
53	C₃²⋊C₉	—	111	1	no	—
54	C₃²⋊C₉	—	111	1	no	—
55	C₃₇	#10624484	1	1	yes	characteristic, complemented, Sylow
56	C₃³	—	333	1	no	—
57	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
58	H(3,𝔽₃)	—	333	1	no	—
59	H(3,𝔽₃)	—	333	1	no	—
60	H(3,𝔽₃)	—	111	3	no	—
61	C₃×C₉	—	3	3	no	—
62	C₃×C₉	—	3	1	no	—
63	C₃×C₉	—	3	1	no	—
64	C₃×C₉	—	3	1	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	333	1	no	—
67	C₃×C₉	—	111	3	no	—
68	C₃²	—	9	1	no	—
69	C₃²	—	333	3	no	—
70	C₃²	—	999	3	no	—
71	C₃²	#15688513	1	1	yes	characteristic
72	C₃²	#15688514	1	1	yes	characteristic
73	C₃²	#15688514	1	1	yes	characteristic
74	C₃²	#15688517	1	1	yes	characteristic
75	C₉	—	9	3	no	—
76	C₉	—	9	1	no	—
77	C₉	—	333	3	no	—
78	C₉	—	999	1	no	—
79	C₃	—	3	1	no	—
80	C₃	—	3	1	no	—
81	C₃	—	3	1	no	—
82	C₃	—	3	1	no	—
83	C₃	—	999	3	no	—
84	C₃	#22582444	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582444
commutator, cosocle	C₃²×C₉×C₃₇	C₃²
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6895	C₃
socle	C₃×C₃₇	#6857
prime radical	C₃×#15688519	C₃
hypercenter	#6895	C₃₇⋊C₃
hypocenter	C₃₇	#10624484
LCS	G ▷ C₃²×C₉×C₃₇ ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ #6895

Complements

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

N	H ≅ G/N	type
2. #22582578	C₃	semidirect
3. #22582470	C₃	semidirect
4. C₃₇×#6895	C₃	semidirect
39. #6895	C₃₇⋊C₃	semidirect
55. C₃₇	#10624484	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
#22582578	3
#22582470	3
C₃₇×#6895	3
#10624484	37