Subgroups — #41008441

Order 26973.

2150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊#66526	C₃	1	12	yes	maximal, complemented
2	C₃⁴×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₃²×#66526	C₃²	1	9	yes	complemented
4	C₃²⋊#66526	—	3	36	no	—
5	C₃³×C₃₇⋊C₃	—	3	12	no	—
6	C₃³×C₃₇⋊C₃	—	3	24	no	—
7	C₃³×C₃₇⋊C₃	C₉	1	8	yes	complemented
8	C₃³×C₃₇⋊C₃	C₃²	1	4	yes	—
9	C₃⁴×C₃₇	C₉	1	1	yes	characteristic, complemented
10	C₃³×C₃₇	—	3	12	no	—
11	C₃³×C₃₇	C₃×C₉	1	4	yes	—
12	C₃²×C₃₇⋊C₃	—	3	27	no	—
13	C₃²×C₃₇⋊C₃	—	3	216	no	—
14	C₃²×C₃₇⋊C₃	—	3	24	no	—
15	C₃²×C₃₇⋊C₃	—	3	72	no	—
16	C₃²×C₃₇⋊C₃	—	3	12	no	—
17	C₃²×C₃₇⋊C₃	—	3	24	no	—
18	C₃²×C₃₇⋊C₃	C₃×C₉	1	8	yes	—
19	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	12	yes	—
20	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	24	yes	—
21	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
22	C₃×#66526	—	3	36	no	—
23	#10624547	—	37	1	no	maximal, Sylow
24	C₃²×C₃₇	—	3	27	no	—
25	C₃²×C₃₇	—	3	12	no	—
26	C₃²×C₃₇	C₃²⋊C₉	1	12	yes	complemented
27	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
28	C₃×C₃₇⋊C₃	—	3	216	no	—
29	C₃×C₃₇⋊C₃	—	3	24	no	—
30	C₃×C₃₇⋊C₃	—	3	12	no	—
31	C₃×C₃₇⋊C₃	—	3	24	no	—
32	C₃×C₃₇⋊C₃	—	3	72	no	—
33	C₃×C₃₇⋊C₃	C₃²⋊C₉	1	24	yes	complemented
34	C₃×C₃₇⋊C₃	C₃×H(3,𝔽₃)	1	4	yes	—
35	C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	1	8	yes	—
36	#66526	—	9	9	no	—
37	C₃⁵	—	37	1	no	—
38	C₃×C₃²⋊C₉	—	37	12	no	—
39	C₃×C₃₇	—	3	12	no	—
40	C₃×C₃₇	C₃×C₃²⋊C₉	1	4	yes	—
41	C₃₇⋊C₃	—	3	24	no	—
42	C₃₇⋊C₃	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
43	C₃₇⋊C₃	#6891	1	8	yes	—
44	C₃⁴	—	37	4	no	—
45	C₃⁴	—	37	8	no	—
46	C₃⁴	—	111	12	no	—
47	C₃⁴	—	111	24	no	—
48	C₃⁴	#66526	1	1	yes	characteristic, complemented
49	C₃²×C₉	—	37	9	no	—
50	C₃²⋊C₉	—	111	36	no	—
51	C₃₇	#10624547	1	1	yes	characteristic, complemented, Sylow
52	C₃³	—	3	12	no	—
53	C₃³	—	37	1	no	—
54	C₃³	—	37	12	no	—
55	C₃³	—	37	24	no	—
56	C₃³	—	37	8	no	—
57	C₃³	—	111	12	no	—
58	C₃³	—	111	24	no	—
59	C₃³	—	111	72	no	—
60	C₃³	—	111	27	no	—
61	C₃³	—	111	216	no	—
62	C₃³	—	111	24	no	—
63	C₃³	C₃×#66526	1	4	yes	—
64	C₃×C₉	—	111	36	no	—
65	C₃²	—	3	27	no	—
66	C₃²	—	3	12	no	—
67	C₃²	—	37	4	no	—
68	C₃²	—	37	8	no	—
69	C₃²	—	37	24	no	—
70	C₃²	—	111	12	no	—
71	C₃²	—	111	24	no	—
72	C₃²	—	111	72	no	—
73	C₃²	—	111	216	no	—
74	C₃²	—	111	24	no	—
75	C₃²	C₃²⋊#66526	1	12	yes	complemented
76	C₃²	C₃²×#66526	1	1	yes	characteristic
77	C₉	—	333	9	no	—
78	C₃	—	3	12	no	—
79	C₃	—	37	1	no	—
80	C₃	—	37	8	no	—
81	C₃	—	111	24	no	—
82	C₃	C₃×C₃²⋊#66526	1	4	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×#66526
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting	C₃⁴×C₃₇	C₉
prime radical	C₃⁴×C₃₇⋊C₃	C₃
cosocle	C₃²×C₃₇⋊C₃	C₃³
hypercenter	C₃⁴	#66526
hypocenter	C₃₇	#10624547
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴

Complements

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

N	H ≅ G/N	type
1. C₃×C₃²⋊#66526	C₃	semidirect
3. C₃²×#66526	C₃²	semidirect
7. C₃³×C₃₇⋊C₃	C₉	semidirect
9. C₃⁴×C₃₇	C₉	semidirect
26. C₃²×C₃₇	C₃²⋊C₉	semidirect
33. C₃×C₃₇⋊C₃	C₃²⋊C₉	semidirect
48. C₃⁴	#66526	semidirect
51. C₃₇	#10624547	semidirect
75. C₃²	C₃²⋊#66526	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃²⋊#66526	3
C₃⁴×C₃₇⋊C₃	3
#10624547	37