Subgroups — C₃₇×#10624709 #41004498

Order 26973.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃₇×C₂₇⋊C₃	C₃	1	2	yes	maximal
2	C₃×C₃₇×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic
4	C₃₇×#6875	C₃	1	9	yes	maximal, complemented
5	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
6	C₃₇×C₂₇⋊C₃	—	3	2	no	—
7	C₃₇×C₂₇⋊C₃	—	3	2	no	—
8	C₃₇×C₂₇⋊C₃	C₉	1	6	yes	complemented
9	C₃₇×C₂₇⋊C₃	C₃²	1	6	yes	—
10	C₃₇×C₂₇⋊C₃	C₃²	1	3	yes	—
11	C₃×C₂₇×C₃₇	—	3	2	no	—
12	C₃×C₂₇×C₃₇	C₉	1	3	yes	complemented
13	C₃₇×C₉⋊C₉	—	3	1	no	—
14	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
15	C₃₇×C₃²⋊C₉	—	3	1	no	—
16	C₉²×C₃₇	—	3	1	no	—
17	C₃³×C₃₇	3₋¹⁺²	1	1	yes	characteristic
18	C₃×C₉×C₃₇	—	3	3	no	—
19	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
20	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
21	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
22	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
23	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
24	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
25	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
26	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
27	C₂₇×C₃₇	—	3	6	no	—
28	C₂₇×C₃₇	C₃×C₉	1	9	yes	—
29	#10624709	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₃²×C₃₇	—	9	1	no	—
31	C₃²×C₃₇	C₉⋊C₉	1	2	yes	—
32	C₃²×C₃₇	C₉⋊C₉	1	1	yes	characteristic
33	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
34	C₉×C₃₇	—	9	3	no	—
35	C₉×C₃₇	C₉⋊C₉	1	2	yes	—
36	C₉×C₃₇	C₉⋊C₉	1	2	yes	—
37	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
38	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
39	C₉×C₃₇	C₃²×C₉	1	1	yes	characteristic
40	C₉×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
41	C₃×C₂₇⋊C₃	C₃×C₃₇	1	2	yes	—
42	C₃×C₂₇⋊C₃	C₃×C₃₇	1	1	yes	characteristic
43	#6887	C₃×C₃₇	1	1	yes	characteristic
44	#6875	C₃×C₃₇	1	9	yes	complemented
45	C₃×C₃₇	—	3	1	no	—
46	C₃×C₃₇	—	3	2	no	—
47	C₃×C₃₇	—	3	1	no	—
48	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
49	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
50	C₂₇⋊C₃	—	3	2	no	—
51	C₂₇⋊C₃	—	3	2	no	—
52	C₂₇⋊C₃	C₉×C₃₇	1	6	yes	complemented
53	C₂₇⋊C₃	C₃²×C₃₇	1	6	yes	—
54	C₂₇⋊C₃	C₃²×C₃₇	1	3	yes	—
55	C₃×C₂₇	—	3	2	no	—
56	C₃×C₂₇	C₉×C₃₇	1	3	yes	complemented
57	C₉⋊C₉	—	3	1	no	—
58	C₉⋊C₉	C₃²×C₃₇	1	3	yes	—
59	C₃²⋊C₉	—	3	1	no	—
60	C₉²	—	3	1	no	—
61	C₃₇	#10624709	1	1	yes	characteristic, direct product factor, Sylow
62	C₃³	3₋¹⁺²×C₃₇	1	1	yes	characteristic
63	C₃×C₉	—	3	3	no	—
64	C₃×C₉	C₃×C₉×C₃₇	1	1	yes	characteristic
65	C₃×C₉	C₃×C₉×C₃₇	1	2	yes	—
66	C₃×C₉	3₋¹⁺²×C₃₇	1	1	yes	characteristic
67	C₃×C₉	3₋¹⁺²×C₃₇	1	1	yes	characteristic
68	C₃×C₉	3₋¹⁺²×C₃₇	1	2	yes	—
69	C₃×C₉	3₋¹⁺²×C₃₇	1	2	yes	—
70	C₃×C₉	3₋¹⁺²×C₃₇	1	2	yes	—
71	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
72	C₂₇	—	3	6	no	—
73	C₂₇	C₃×C₉×C₃₇	1	9	yes	—
74	C₃²	—	9	1	no	—
75	C₃²	C₃₇×C₉⋊C₉	1	1	yes	characteristic
76	C₃²	C₃₇×C₉⋊C₉	1	2	yes	—
77	C₃²	C₃×3₋¹⁺²×C₃₇	1	1	yes	characteristic
78	C₉	—	9	3	no	—
79	C₉	C₃₇×C₉⋊C₉	1	1	yes	characteristic
80	C₉	C₃₇×C₉⋊C₉	1	2	yes	—
81	C₉	C₃₇×C₉⋊C₉	1	2	yes	—
82	C₉	C₃₇×C₉⋊C₉	1	1	yes	characteristic
83	C₉	C₃²×C₉×C₃₇	1	1	yes	characteristic
84	C₉	C₃×3₋¹⁺²×C₃₇	1	2	yes	—
85	C₃	—	3	1	no	—
86	C₃	—	3	1	no	—
87	C₃	—	3	2	no	—
88	C₃	C₃×C₃₇×C₉⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₉×C₃₇	C₉⋊C₉
commutator	C₉	C₃²×C₉×C₃₇
Frattini, cosocle	C₃×C₉	C₃³×C₃₇
socle	C₃×C₃₇	C₃×C₉⋊C₉
prime radical	C₃³×C₃₇	3₋¹⁺²
LCS	G ▷ C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₇×#6875	C₃	semidirect
8. C₃₇×C₂₇⋊C₃	C₉	semidirect
12. C₃×C₂₇×C₃₇	C₉	semidirect
29. #10624709	C₃₇	direct (N × H)
44. #6875	C₃×C₃₇	semidirect
52. C₂₇⋊C₃	C₉×C₃₇	semidirect
56. C₃×C₂₇	C₉×C₃₇	semidirect
61. C₃₇	#10624709	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₃₇×C₂₇⋊C₃	3
C₃₇×#6887	3
C₃₇×#6875	3
#10624709	37

Subgroups — C37×#10624709 #41004498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624709 #41004498