Subgroups — C₃₇⋊C₃×C₃²⋊H(3,𝔽₃) #41007530

Order 26973.

3824 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×H(3,𝔽₃)×C₃₇⋊C₃	C₃	1	12	yes	maximal, complemented
3	#22582550	C₃	1	2	yes	maximal, complemented
4	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	C₃	1	24	yes	maximal, complemented
5	C₃₇×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇⋊C₃	—	3	36	no	—
7	C₃³×C₃₇⋊C₃	C₃²	1	4	yes	—
8	C₃³×C₃₇⋊C₃	C₃²	1	8	yes	complemented
9	C₃³×C₃₇⋊C₃	C₃²	1	9	yes	complemented
10	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	72	yes	complemented
11	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	24	yes	complemented
12	H(3,𝔽₃)×C₃₇⋊C₃	—	3	108	no	—
13	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic, complemented
14	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	12	yes	complemented
15	C₃³×C₃₇	—	3	12	no	—
16	C₃³×C₃₇	C₃³	1	4	yes	—
17	C₃³×C₃₇	C₃³	1	9	yes	complemented
18	C₃²×C₃₇⋊C₃	—	3	243	no	—
19	C₃²×C₃₇⋊C₃	—	3	24	no	—
20	C₃²×C₃₇⋊C₃	—	3	36	no	—
21	C₃²×C₃₇⋊C₃	—	3	72	no	—
22	C₃²×C₃₇⋊C₃	—	3	108	no	—
23	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	36	yes	complemented
24	C₃²×C₃₇⋊C₃	C₃³	1	8	yes	—
25	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
26	C₃²×C₃₇⋊C₃	C₃³	1	18	yes	complemented
27	H(3,𝔽₃)×C₃₇	—	3	36	no	—
28	C₃²⋊(C₃₇⋊C₃)	—	3	72	no	—
29	C₃²⋊(C₃₇⋊C₃)	—	3	216	no	—
30	C₃×C₃²⋊H(3,𝔽₃)	—	37	1	no	maximal, Sylow
31	C₃²×C₃₇	—	3	27	no	—
32	C₃²×C₃₇	—	3	12	no	—
33	C₃²×C₃₇	—	3	36	no	—
34	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	12	yes	complemented
35	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
36	C₃×C₃₇⋊C₃	—	3	216	no	—
37	C₃×C₃₇⋊C₃	—	3	24	no	—
38	C₃×C₃₇⋊C₃	—	3	108	no	—
39	C₃×C₃₇⋊C₃	—	3	72	no	—
40	C₃×C₃₇⋊C₃	—	9	81	no	—
41	C₃×C₃₇⋊C₃	C₃×H(3,𝔽₃)	1	12	yes	—
42	C₃×C₃₇⋊C₃	C₃×H(3,𝔽₃)	1	24	yes	complemented
43	C₃⁵	—	37	1	no	—
44	C₃²×H(3,𝔽₃)	—	37	12	no	—
45	C₃²⋊H(3,𝔽₃)	—	37	2	no	—
46	C₃²⋊H(3,𝔽₃)	—	37	24	no	—
47	C₃²⋊H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic, direct product factor
48	C₃×C₃₇	—	3	12	no	—
49	C₃×C₃₇	—	9	9	no	—
50	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	4	yes	—
51	C₃₇⋊C₃	—	3	24	no	—
52	C₃₇⋊C₃	—	9	18	no	—
53	C₃₇⋊C₃	C₃²⋊H(3,𝔽₃)	1	9	yes	direct product factor
54	C₃⁴	—	37	4	no	—
55	C₃⁴	—	37	8	no	—
56	C₃⁴	—	37	9	no	—
57	C₃⁴	—	111	36	no	—
58	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented
59	C₃×H(3,𝔽₃)	—	37	24	no	—
60	C₃×H(3,𝔽₃)	—	37	72	no	—
61	C₃×H(3,𝔽₃)	—	111	108	no	—
62	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	12	yes	complemented
63	C₃₇	C₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
64	C₃³	—	3	12	no	—
65	C₃³	—	37	1	no	—
66	C₃³	—	37	36	no	—
67	C₃³	—	37	8	no	—
68	C₃³	—	37	18	no	—
69	C₃³	—	111	108	no	—
70	C₃³	—	111	36	no	—
71	C₃³	—	111	72	no	—
72	C₃³	—	111	243	no	—
73	C₃³	—	111	24	no	—
74	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
75	C₃³	C₃²×C₃₇⋊C₃	1	9	yes	complemented
76	H(3,𝔽₃)	—	3	36	no	—
77	H(3,𝔽₃)	—	111	216	no	—
78	H(3,𝔽₃)	—	111	72	no	—
79	C₃²	—	3	27	no	—
80	C₃²	—	3	12	no	—
81	C₃²	—	3	36	no	—
82	C₃²	—	37	12	no	—
83	C₃²	—	37	24	no	—
84	C₃²	—	333	81	no	—
85	C₃²	—	111	108	no	—
86	C₃²	—	111	216	no	—
87	C₃²	—	111	72	no	—
88	C₃²	—	111	24	no	—
89	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	12	yes	complemented
90	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
91	C₃	—	3	12	no	—
92	C₃	—	9	9	no	—
93	C₃	—	37	9	no	—
94	C₃	—	333	18	no	—
95	C₃	—	111	24	no	—
96	C₃	C₃×H(3,𝔽₃)×C₃₇⋊C₃	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃⁴×C₃₇⋊C₃	C₃	semidirect
2. C₃×H(3,𝔽₃)×C₃₇⋊C₃	C₃	semidirect
3. #22582550	C₃	semidirect
4. (C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	C₃	semidirect
5. C₃₇×C₃²⋊H(3,𝔽₃)	C₃	semidirect
8. C₃³×C₃₇⋊C₃	C₃²	semidirect
9. C₃³×C₃₇⋊C₃	C₃²	semidirect
10. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
11. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
13. C₃⁴×C₃₇	C₃²	semidirect
14. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
17. C₃³×C₃₇	C₃³	semidirect
23. C₃²×C₃₇⋊C₃	H(3,𝔽₃)	semidirect
26. C₃²×C₃₇⋊C₃	C₃³	semidirect
34. C₃²×C₃₇	C₃×H(3,𝔽₃)	semidirect
42. C₃×C₃₇⋊C₃	C₃×H(3,𝔽₃)	semidirect
47. C₃²⋊H(3,𝔽₃)	C₃₇⋊C₃	direct (N × H)
53. C₃₇⋊C₃	C₃²⋊H(3,𝔽₃)	direct (N × H)
58. C₃⁴	C₃×C₃₇⋊C₃	semidirect
62. C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	semidirect
63. C₃₇	C₃×C₃²⋊H(3,𝔽₃)	semidirect
75. C₃³	C₃²×C₃₇⋊C₃	semidirect
89. C₃²	H(3,𝔽₃)×C₃₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃⁴×C₃₇⋊C₃	3
C₃×H(3,𝔽₃)×C₃₇⋊C₃	3
#22582550	3
(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	3
C₃₇×C₃²⋊H(3,𝔽₃)	3
C₃×C₃²⋊H(3,𝔽₃)	37

Subgroups — C37⋊C3×C32⋊H(3,𝔽3) #41007530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇⋊C₃×C₃²⋊H(3,𝔽₃) #41007530