Subgroups — C₃₇×#10624894 #41004683

Order 26973.

1148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃₇×#737	C₃	1	3	yes	maximal, complemented
2	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃₇×#6893	C₃	1	24	yes	maximal, complemented
4	C₃₇×#6891	C₃	1	8	yes	maximal, complemented
5	C₃×C₃₇×C₃²⋊C₉	C₃	1	4	yes	maximal, complemented
6	C₃⁴×C₃₇	C₃²	1	4	yes	—
7	C₃₇×#737	—	3	27	no	—
8	C₃×3₋¹⁺²×C₃₇	C₃²	1	24	yes	complemented
9	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
10	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	8	yes	—
11	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
12	C₃²×C₉×C₃₇	C₃²	1	12	yes	complemented
13	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
14	C₃₇×C₃²⋊C₉	C₃²	1	8	yes	—
15	C₃₇×C₃²⋊C₉	C₃²	1	24	yes	complemented
16	C₃₇×C₃²⋊C₉	C₃²	1	48	yes	complemented
17	C₃³×C₃₇	—	3	12	no	—
18	C₃³×C₃₇	—	3	12	no	—
19	C₃³×C₃₇	—	9	8	no	—
20	C₃³×C₃₇	C₃³	1	8	yes	—
21	C₃³×C₃₇	C₃³	1	4	yes	—
22	C₃³×C₃₇	C₃³	1	1	yes	characteristic
23	3₋¹⁺²×C₃₇	—	3	72	no	—
24	H(3,𝔽₃)×C₃₇	—	3	24	no	—
25	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	9	yes	—
26	C₃×C₉×C₃₇	—	3	36	no	—
27	C₃×C₉×C₃₇	C₃³	1	3	yes	—
28	C₃×C₉×C₃₇	C₃³	1	24	yes	—
29	#10624894	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₃²×C₃₇	—	3	1	no	—
31	C₃²×C₃₇	—	3	4	no	—
32	C₃²×C₃₇	—	3	2	no	—
33	C₃²×C₃₇	—	3	8	no	—
34	C₃²×C₃₇	—	3	8	no	—
35	C₃²×C₃₇	—	3	8	no	—
36	C₃²×C₃₇	—	3	16	no	—
37	C₃²×C₃₇	—	9	36	no	—
38	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	12	yes	—
39	C₃²×C₃₇	#737	1	3	yes	complemented
40	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
41	C₉×C₃₇	—	3	3	no	—
42	C₉×C₃₇	—	3	24	no	—
43	C₃×#737	C₃×C₃₇	1	3	yes	complemented
44	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic
45	#6893	C₃×C₃₇	1	24	yes	complemented
46	#6891	C₃×C₃₇	1	8	yes	complemented
47	C₃×C₃²⋊C₉	C₃×C₃₇	1	4	yes	complemented
48	C₃×C₃₇	—	3	3	no	—
49	C₃×C₃₇	—	3	12	no	—
50	C₃×C₃₇	—	9	8	no	—
51	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
52	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
53	C₃×C₃₇	3₋¹⁺⁴	1	2	yes	—
54	C₃⁴	C₃²×C₃₇	1	4	yes	—
55	#737	—	3	27	no	—
56	C₃×3₋¹⁺²	C₃²×C₃₇	1	24	yes	complemented
57	C₃×H(3,𝔽₃)	—	3	9	no	—
58	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	8	yes	—
59	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	1	yes	characteristic
60	C₃²×C₉	C₃²×C₃₇	1	12	yes	complemented
61	C₃²⋊C₉	C₃²×C₃₇	1	1	yes	characteristic
62	C₃²⋊C₉	C₃²×C₃₇	1	8	yes	—
63	C₃²⋊C₉	C₃²×C₃₇	1	24	yes	complemented
64	C₃²⋊C₉	C₃²×C₃₇	1	48	yes	complemented
65	C₃₇	#10624894	1	1	yes	characteristic, direct product factor, Sylow
66	C₃³	—	3	12	no	—
67	C₃³	—	3	12	no	—
68	C₃³	—	9	8	no	—
69	C₃³	C₃³×C₃₇	1	8	yes	—
70	C₃³	C₃³×C₃₇	1	4	yes	—
71	C₃³	C₃³×C₃₇	1	1	yes	characteristic
72	3₋¹⁺²	—	3	72	no	—
73	H(3,𝔽₃)	—	3	24	no	—
74	H(3,𝔽₃)	H(3,𝔽₃)×C₃₇	1	9	yes	—
75	C₃×C₉	—	3	36	no	—
76	C₃×C₉	C₃³×C₃₇	1	3	yes	—
77	C₃×C₉	C₃³×C₃₇	1	24	yes	—
78	C₃²	—	3	1	no	—
79	C₃²	—	3	4	no	—
80	C₃²	—	3	2	no	—
81	C₃²	—	3	8	no	—
82	C₃²	—	3	8	no	—
83	C₃²	—	3	8	no	—
84	C₃²	—	3	16	no	—
85	C₃²	—	9	36	no	—
86	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	12	yes	—
87	C₃²	C₃₇×#737	1	3	yes	complemented
88	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
89	C₉	—	3	3	no	—
90	C₉	—	3	24	no	—
91	C₃	—	3	3	no	—
92	C₃	—	3	12	no	—
93	C₃	—	9	8	no	—
94	C₃	C₃²×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
95	C₃	C₃×C₃₇×#737	1	1	yes	characteristic
96	C₃	C₃₇×3₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₃₇×#737	C₃	semidirect
3. C₃₇×#6893	C₃	semidirect
4. C₃₇×#6891	C₃	semidirect
5. C₃×C₃₇×C₃²⋊C₉	C₃	semidirect
8. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
12. C₃²×C₉×C₃₇	C₃²	semidirect
15. C₃₇×C₃²⋊C₉	C₃²	semidirect
16. C₃₇×C₃²⋊C₉	C₃²	semidirect
29. #10624894	C₃₇	direct (N × H)
39. C₃²×C₃₇	#737	semidirect
43. C₃×#737	C₃×C₃₇	semidirect
45. #6893	C₃×C₃₇	semidirect
46. #6891	C₃×C₃₇	semidirect
47. C₃×C₃²⋊C₉	C₃×C₃₇	semidirect
56. C₃×3₋¹⁺²	C₃²×C₃₇	semidirect
60. C₃²×C₉	C₃²×C₃₇	semidirect
63. C₃²⋊C₉	C₃²×C₃₇	semidirect
64. C₃²⋊C₉	C₃²×C₃₇	semidirect
65. C₃₇	#10624894	direct (N × H)
87. C₃²	C₃₇×#737	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃₇×#737	3
C₃²×H(3,𝔽₃)×C₃₇	3
C₃₇×#6893	3
C₃₇×#6891	3
C₃×C₃₇×C₃²⋊C₉	3
#10624894	37

Subgroups — C37×#10624894 #41004683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624894 #41004683