Subgroups — #41008367

Order 26973.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582691	C₃	1	3	yes	maximal, complemented
2	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#66526	—	3	3	no	—
4	C₃³×C₃₇⋊C₃	C₉	1	2	yes	complemented
5	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
6	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
7	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
8	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
9	C₃⁴×C₃₇	C₉	1	1	yes	characteristic, complemented
10	C₃³×C₃₇	—	3	3	no	—
11	C₃³×C₃₇	—	3	1	no	—
12	C₃³×C₃₇	—	9	3	no	—
13	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
14	C₃²×C₃₇⋊C₃	—	3	2	no	—
15	C₃²×C₃₇⋊C₃	—	3	3	no	—
16	C₃²×C₃₇⋊C₃	—	3	6	no	—
17	C₃²×C₃₇⋊C₃	—	9	6	no	—
18	C₃²×C₃₇⋊C₃	—	9	3	no	—
19	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	2	yes	—
21	C₃×#66526	—	9	3	no	—
22	C₃²⋊(C₃₇⋊C₃)	—	3	18	no	—
23	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
24	#10624505	—	37	1	no	maximal, Sylow
25	C₃²×C₃₇	—	3	3	no	—
26	C₃²×C₃₇	—	3	1	no	—
27	C₃²×C₃₇	—	3	3	no	—
28	C₃²×C₃₇	—	9	9	no	—
29	C₃²×C₃₇	—	9	3	no	—
30	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
31	C₃×C₃₇⋊C₃	—	3	6	no	—
32	C₃×C₃₇⋊C₃	—	3	2	no	—
33	C₃×C₃₇⋊C₃	—	9	1	no	—
34	C₃×C₃₇⋊C₃	—	9	9	no	—
35	C₃×C₃₇⋊C₃	—	9	6	no	—
36	C₃×C₃₇⋊C₃	—	9	18	no	—
37	C₃×C₃₇⋊C₃	—	9	2	no	—
38	C₃×C₃₇⋊C₃	AΣL(1,𝔽₂₇)	1	3	yes	—
39	#66526	—	27	3	no	—
40	C₃²×H(3,𝔽₃)	—	37	1	no	—
41	#6866	—	37	3	no	—
42	C₃×C₃₇	—	3	3	no	—
43	C₃×C₃₇	—	3	1	no	—
44	C₃×C₃₇	—	9	3	no	—
45	C₃×C₃₇	#6866	1	1	yes	characteristic
46	C₃₇⋊C₃	—	3	3	no	—
47	C₃₇⋊C₃	—	9	6	no	—
48	C₃₇⋊C₃	—	9	2	no	—
49	C₃⁴	—	37	1	no	—
50	C₃⁴	—	37	2	no	—
51	C₃⁴	#66526	1	1	yes	characteristic, complemented
52	C₃×H(3,𝔽₃)	—	111	1	no	—
53	C₃×H(3,𝔽₃)	—	111	2	no	—
54	C₃×H(3,𝔽₃)	—	111	9	no	—
55	C₃²⋊C₉	—	111	3	no	—
56	C₃₇	#10624505	1	1	yes	characteristic, complemented, Sylow
57	C₃³	—	3	3	no	—
58	C₃³	—	3	1	no	—
59	C₃³	—	9	3	no	—
60	C₃³	—	37	1	no	—
61	C₃³	—	37	2	no	—
62	C₃³	—	333	3	no	—
63	C₃³	—	333	6	no	—
64	C₃³	—	111	3	no	—
65	C₃³	—	111	6	no	—
66	C₃³	—	111	2	no	—
67	C₃³	C₃×#66526	1	1	yes	characteristic
68	H(3,𝔽₃)	—	111	9	no	—
69	H(3,𝔽₃)	—	111	18	no	—
70	C₃×C₉	—	333	3	no	—
71	C₃²	—	3	3	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	3	3	no	—
74	C₃²	—	9	9	no	—
75	C₃²	—	9	3	no	—
76	C₃²	—	37	3	no	—
77	C₃²	—	333	2	no	—
78	C₃²	—	333	18	no	—
79	C₃²	—	333	6	no	—
80	C₃²	—	333	9	no	—
81	C₃²	—	333	1	no	—
82	C₃²	—	111	2	no	—
83	C₃²	—	111	6	no	—
84	C₃²	C₃²⋊#66526	1	1	yes	characteristic
85	C₉	—	999	3	no	—
86	C₃	—	3	1	no	—
87	C₃	—	3	3	no	—
88	C₃	—	9	3	no	—
89	C₃	—	333	2	no	—
90	C₃	—	333	6	no	—
91	C₃	—	111	3	no	—
92	C₃	#22582691	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582691
commutator	C₃³×C₃₇	C₃×C₉
Frattini	C₃³	C₃×#66526
Fitting	C₃⁴×C₃₇	C₉
socle	C₃×C₃₇	#6866
prime radical	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃
cosocle	C₃³×C₃₇⋊C₃	C₃²
hypercenter	C₃⁴	#66526
hypocenter	C₃₇	#10624505
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃³ ◁ C₃⁴

Complements

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

N	H ≅ G/N	type
1. #22582691	C₃	semidirect
4. C₃³×C₃₇⋊C₃	C₉	semidirect
9. C₃⁴×C₃₇	C₉	semidirect
51. C₃⁴	#66526	semidirect
56. C₃₇	#10624505	semidirect

Maximal subgroups

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

subgroup	index
#22582691	3
C₃²×C₃²⋊(C₃₇⋊C₃)	3
#10624505	37