Subgroups — #41006508

Order 26973.

596 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#15688512	C₃	1	9	yes	maximal, complemented
3	#22582554	C₃	1	2	yes	maximal, complemented
4	C₃₇×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₃₇⋊C₃	C₃²	1	3	yes	—
6	C₃×#11758608	C₃²	1	6	yes	—
7	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
8	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
9	#15688512	—	3	27	no	—
10	C₃²⋊#66524	—	3	6	no	—
11	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
12	C₃×H(3,𝔽₃)×C₃₇	—	3	3	no	—
13	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	3	yes	—
14	C₃³×C₃₇	—	3	3	no	—
15	C₃³×C₃₇	—	3	3	no	—
16	C₃³×C₃₇	—	9	3	no	—
17	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
18	C₃³×C₃₇	C₃³	1	1	yes	characteristic
19	C₃²×C₃₇⋊C₃	—	3	6	no	—
20	C₃²×C₃₇⋊C₃	—	3	3	no	—
21	C₃²×C₃₇⋊C₃	—	9	9	no	—
22	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	9	yes	complemented
23	H(3,𝔽₃)×C₃₇	—	3	9	no	—
24	H(3,𝔽₃)×C₃₇	—	9	9	no	—
25	C₃²⋊(C₃₇⋊C₃)	—	3	27	no	—
26	#11758608	—	3	18	no	—
27	C₃×#66524	—	3	6	no	—
28	#10624735	—	37	1	no	maximal, Sylow
29	C₃²×C₃₇	—	3	3	no	—
30	C₃²×C₃₇	—	3	3	no	—
31	C₃²×C₃₇	—	3	3	no	—
32	C₃²×C₃₇	—	3	6	no	—
33	C₃²×C₃₇	—	9	9	no	—
34	C₃²×C₃₇	—	9	3	no	—
35	C₃²×C₃₇	—	9	9	no	—
36	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
37	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
38	C₃×C₃₇⋊C₃	—	3	9	no	—
39	C₃×C₃₇⋊C₃	—	9	18	no	—
40	C₃×C₃₇⋊C₃	—	9	9	no	—
41	C₃×C₃₇⋊C₃	—	9	9	no	—
42	#66524	—	9	6	no	—
43	C₃²×H(3,𝔽₃)	—	37	1	no	—
44	C₃×AΣL(1,𝔽₂₇)	—	37	9	no	—
45	#6891	—	37	2	no	—
46	C₃²⋊H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic, complemented
47	C₃×C₃₇	—	3	3	no	—
48	C₃×C₃₇	—	9	3	no	—
49	C₃×C₃₇	—	27	3	no	—
50	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
51	C₃×C₃₇	#6909	1	3	yes	—
52	C₃₇⋊C₃	—	9	9	no	—
53	C₃⁴	—	37	3	no	—
54	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
55	C₃×3₋¹⁺²	—	37	6	no	—
56	C₃×H(3,𝔽₃)	—	3	3	no	—
57	C₃×H(3,𝔽₃)	—	111	9	no	—
58	C₃×H(3,𝔽₃)	—	111	3	no	—
59	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	3	yes	—
60	AΣL(1,𝔽₂₇)	—	111	27	no	—
61	C₃²⋊C₉	—	111	6	no	—
62	C₃₇	#10624735	1	1	yes	characteristic, complemented, Sylow
63	C₃³	—	3	3	no	—
64	C₃³	—	3	3	no	—
65	C₃³	—	9	3	no	—
66	C₃³	—	37	9	no	—
67	C₃³	—	333	9	no	—
68	C₃³	—	111	6	no	—
69	C₃³	—	111	3	no	—
70	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
71	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
72	3₋¹⁺²	—	111	18	no	—
73	H(3,𝔽₃)	—	3	9	no	—
74	H(3,𝔽₃)	—	9	9	no	—
75	H(3,𝔽₃)	—	111	27	no	—
76	C₃×C₉	—	111	6	no	—
77	C₃²	—	3	3	no	—
78	C₃²	—	3	3	no	—
79	C₃²	—	3	3	no	—
80	C₃²	—	3	6	no	—
81	C₃²	—	9	9	no	—
82	C₃²	—	9	3	no	—
83	C₃²	—	9	9	no	—
84	C₃²	—	333	9	no	—
85	C₃²	—	333	18	no	—
86	C₃²	—	333	9	no	—
87	C₃²	—	111	9	no	—
88	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
89	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
90	C₉	—	333	6	no	—
91	C₃	—	3	3	no	—
92	C₃	—	9	3	no	—
93	C₃	—	27	3	no	—
94	C₃	—	333	9	no	—
95	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic
96	C₃	#22582637	1	3	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×C₃²⋊H(3,𝔽₃)	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	C₃²⋊H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624735
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²⋊H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	semidirect
2. C₃×#15688512	C₃	semidirect
3. #22582554	C₃	semidirect
4. C₃₇×C₃²⋊H(3,𝔽₃)	C₃	semidirect
22. C₃²×C₃₇⋊C₃	H(3,𝔽₃)	semidirect
46. C₃²⋊H(3,𝔽₃)	C₃₇⋊C₃	semidirect
62. C₃₇	#10624735	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₃²⋊(C₃₇⋊C₃)	3
C₃×#15688512	3
#22582554	3
C₃₇×C₃²⋊H(3,𝔽₃)	3
#10624735	37