Subgroups — C₃₇×#10624723 #41004512

Order 26973.

250 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃₇×#733	C₃	1	1	yes	maximal, characteristic
2	C₉×3₋¹⁺²×C₃₇	C₃	1	3	yes	maximal
3	C₃×C₃₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
4	C₃₇×3₋¹⁺²⋊C₉	C₃	1	6	yes	maximal
5	C₃₇×#6868	C₃	1	2	yes	maximal
6	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
7	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
8	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
9	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
10	C₃₇×#733	C₉	1	9	yes	complemented
11	C₃₇×C₉⋊C₉	—	3	3	no	—
12	C₃₇×C₉⋊C₉	—	3	6	no	—
13	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
14	C₉²×C₃₇	—	3	3	no	—
15	C₃³×C₃₇	C₃³	1	1	yes	characteristic
16	3₋¹⁺²×C₃₇	—	3	6	no	—
17	3₋¹⁺²×C₃₇	C₃×C₉	1	9	yes	—
18	C₃×C₉×C₃₇	—	3	3	no	—
19	C₃×C₉×C₃₇	—	3	1	no	—
20	C₃×C₉×C₃₇	—	3	2	no	—
21	C₃×C₉×C₃₇	—	3	6	no	—
22	C₃×C₉×C₃₇	—	3	2	no	—
23	C₃×C₉×C₃₇	—	9	1	no	—
24	C₃×C₉×C₃₇	—	9	2	no	—
25	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
26	C₃×C₉×C₃₇	H(3,𝔽₃)	1	3	yes	—
27	#10624723	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₃²×C₃₇	—	3	1	no	—
29	C₃²×C₃₇	—	3	2	no	—
30	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
31	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
32	C₃²×C₃₇	#737	1	2	yes	—
33	C₉×C₃₇	—	3	3	no	—
34	C₉×C₃₇	—	3	3	no	—
35	C₉×C₃₇	—	3	9	no	—
36	C₉×C₃₇	—	9	6	no	—
37	C₉×C₃₇	—	9	2	no	—
38	C₃×#733	C₃×C₃₇	1	1	yes	characteristic
39	C₉×3₋¹⁺²	C₃×C₃₇	1	3	yes	—
40	C₃×C₉⋊C₉	C₃×C₃₇	1	1	yes	characteristic
41	3₋¹⁺²⋊C₉	C₃×C₃₇	1	6	yes	—
42	#6868	C₃×C₃₇	1	2	yes	—
43	C₃×C₃₇	—	3	1	no	—
44	C₃×C₃₇	—	3	2	no	—
45	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
46	C₃×C₃₇	#6912	1	1	yes	characteristic
47	C₃×C₃₇	#6913	1	2	yes	—
48	C₃×3₋¹⁺²	C₃²×C₃₇	1	3	yes	—
49	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
50	C₃²×C₉	C₃²×C₃₇	1	2	yes	—
51	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
52	#733	C₉×C₃₇	1	9	yes	complemented
53	C₉⋊C₉	—	3	3	no	—
54	C₉⋊C₉	—	3	6	no	—
55	C₃²⋊C₉	C₃²×C₃₇	1	6	yes	—
56	C₉²	—	3	3	no	—
57	C₃₇	#10624723	1	1	yes	characteristic, direct product factor, Sylow
58	C₃³	C₃³×C₃₇	1	1	yes	characteristic
59	3₋¹⁺²	—	3	6	no	—
60	3₋¹⁺²	C₃×C₉×C₃₇	1	9	yes	—
61	C₃×C₉	—	3	3	no	—
62	C₃×C₉	—	3	1	no	—
63	C₃×C₉	—	3	2	no	—
64	C₃×C₉	—	3	6	no	—
65	C₃×C₉	—	3	2	no	—
66	C₃×C₉	—	9	1	no	—
67	C₃×C₉	—	9	2	no	—
68	C₃×C₉	C₃×C₉×C₃₇	1	3	yes	—
69	C₃×C₉	H(3,𝔽₃)×C₃₇	1	3	yes	—
70	C₃²	—	3	1	no	—
71	C₃²	—	3	2	no	—
72	C₃²	C₃²×C₉×C₃₇	1	1	yes	characteristic
73	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
74	C₃²	C₃₇×#737	1	2	yes	—
75	C₉	—	3	3	no	—
76	C₉	—	3	9	no	—
77	C₉	—	3	3	no	—
78	C₉	—	9	2	no	—
79	C₉	—	9	6	no	—
80	C₃	—	3	1	no	—
81	C₃	—	3	2	no	—
82	C₃	C₉×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
83	C₃	C₃₇×#6912	1	1	yes	characteristic
84	C₃	C₃₇×#6913	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
commutator	C₃²	C₃²×C₉×C₃₇
Frattini, cosocle	C₃³	C₃³×C₃₇
prime radical	C₃³×C₃₇	C₃³
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₃₇×#733	C₉	semidirect
27. #10624723	C₃₇	direct (N × H)
52. #733	C₉×C₃₇	semidirect
57. C₃₇	#10624723	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×#733	3
C₉×3₋¹⁺²×C₃₇	3
C₃×C₃₇×C₉⋊C₉	3
C₃₇×3₋¹⁺²⋊C₉	3
C₃₇×#6868	3
#10624723	37

Subgroups — C37×#10624723 #41004512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624723 #41004512