Subgroups — #41007205

Order 26973.

208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582578	C₃	1	2	yes	maximal, complemented
2	C₃×#15688505	C₃	1	1	yes	maximal, characteristic, complemented
3	3₋¹⁺²⋊#66524	C₃	1	6	yes	maximal, complemented
4	H(3,𝔽₃)⋊#66524	C₃	1	3	yes	maximal
5	C₃₇×#6893	C₃	1	1	yes	maximal, characteristic
6	C₃²×#66524	C₃²	1	3	yes	—
7	#15688505	—	3	3	no	—
8	#15688504	—	3	6	no	—
9	C₃²⋊#66524	C₃²	1	6	yes	—
10	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
11	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
12	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
13	C₃₇×C₃²⋊C₉	—	3	3	no	—
14	C₃³×C₃₇	—	3	1	no	—
15	C₃³×C₃₇	C₃³	1	1	yes	characteristic
16	3₋¹⁺²×C₃₇	—	3	6	no	—
17	H(3,𝔽₃)×C₃₇	—	3	3	no	—
18	C₃×C₉×C₃₇	—	3	3	no	—
19	C₃×C₉×C₃₇	—	3	2	no	—
20	C₃×C₉×C₃₇	H(3,𝔽₃)	1	3	yes	—
21	C₃×#66524	—	3	6	no	—
22	C₃×#66524	—	3	3	no	—
23	C₃×#66524	—	9	3	no	—
24	#10624817	—	37	1	no	maximal, Sylow
25	C₃²×C₃₇	—	3	3	no	—
26	C₃²×C₃₇	—	9	3	no	—
27	C₃²×C₃₇	—	9	1	no	—
28	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
29	C₃²×C₃₇	#737	1	3	yes	—
30	C₉×C₃₇	—	3	3	no	—
31	C₉×C₃₇	—	9	2	no	—
32	#66524	—	9	6	no	—
33	#66524	—	9	3	no	—
34	#6895	—	37	2	no	—
35	#6893	C₃₇⋊C₃	1	1	yes	characteristic
36	C₃×C₉⋊C₉	—	37	1	no	—
37	3₋¹⁺²⋊C₉	—	37	6	no	—
38	H(3,𝔽₃)⋊C₉	—	37	3	no	—
39	C₃×C₃₇	—	3	3	no	—
40	C₃×C₃₇	—	27	1	no	—
41	C₃×C₃₇	#6893	1	1	yes	characteristic
42	C₃×C₃₇	#6913	1	3	yes	—
43	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	—
44	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
45	C₃²×C₉	—	37	3	no	—
46	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
47	C₉⋊C₉	—	111	6	no	—
48	C₉⋊C₉	—	111	3	no	—
49	C₃²⋊C₉	—	3	3	no	—
50	C₃²⋊C₉	—	37	6	no	—
51	C₃₇	#10624817	1	1	yes	characteristic, complemented, Sylow
52	C₃³	—	3	1	no	—
53	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
54	3₋¹⁺²	—	3	6	no	—
55	H(3,𝔽₃)	—	3	3	no	—
56	C₃×C₉	—	3	3	no	—
57	C₃×C₉	—	3	2	no	—
58	C₃×C₉	—	333	3	no	—
59	C₃×C₉	—	111	6	no	—
60	C₃×C₉	—	111	3	no	—
61	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
62	C₃²	—	3	3	no	—
63	C₃²	—	9	1	no	—
64	C₃²	—	9	3	no	—
65	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
66	C₃²	#15688530	1	3	yes	—
67	C₉	—	3	3	no	—
68	C₉	—	9	2	no	—
69	C₉	—	333	3	no	—
70	C₉	—	333	6	no	—
71	C₃	—	3	3	no	—
72	C₃	—	27	1	no	—
73	C₃	#22582562	1	1	yes	characteristic
74	C₃	#22582655	1	3	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6893	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
hypercenter	#6893	C₃₇⋊C₃
hypocenter	C₃₇	#10624817
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ #6893

Complements

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

N	H ≅ G/N	type
1. #22582578	C₃	semidirect
2. C₃×#15688505	C₃	semidirect
3. 3₋¹⁺²⋊#66524	C₃	semidirect
51. C₃₇	#10624817	semidirect

Maximal subgroups

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

subgroup	index
#22582578	3
C₃×#15688505	3
3₋¹⁺²⋊#66524	3
H(3,𝔽₃)⋊#66524	3
C₃₇×#6893	3
#10624817	37