Subgroups — C₃₇×#10624710 #41004499

Order 26973.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉²×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₃₇×C₉⋊C₂₇	C₃	1	6	yes	maximal, complemented
3	C₃₇×C₃²⋊C₂₇	C₃	1	3	yes	maximal
4	C₉×C₂₇×C₃₇	C₃	1	3	yes	maximal, complemented
5	C₃²×C₉×C₃₇	C₉	1	2	yes	—
6	C₃²×C₉×C₃₇	C₉	1	1	yes	characteristic
7	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
8	C₃×C₂₇×C₃₇	C₃²	1	9	yes	—
9	C₉²×C₃₇	C₉	1	2	yes	—
10	C₉²×C₃₇	C₉	1	4	yes	—
11	C₉²×C₃₇	C₃²	1	2	yes	—
12	C₉²×C₃₇	C₃²	1	1	yes	characteristic
13	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
14	C₃×C₉×C₃₇	—	3	6	no	—
15	C₃×C₉×C₃₇	—	3	3	no	—
16	C₃×C₉×C₃₇	C₂₇	1	9	yes	complemented
17	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
18	C₃×C₉×C₃₇	C₃×C₉	1	4	yes	—
19	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
20	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
21	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
22	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
23	C₂₇×C₃₇	—	3	9	no	—
24	#10624710	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₃²×C₃₇	—	3	1	no	—
26	C₃²×C₃₇	—	3	2	no	—
27	C₃²×C₃₇	C₃×C₂₇	1	3	yes	—
28	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
29	C₉×C₃₇	—	3	2	no	—
30	C₉×C₃₇	—	3	4	no	—
31	C₉×C₃₇	C₃×C₂₇	1	6	yes	—
32	C₉×C₃₇	C₃×C₂₇	1	3	yes	—
33	C₉×C₃₇	C₂₇⋊C₃	1	6	yes	—
34	C₉×C₃₇	#737	1	3	yes	—
35	C₃×C₉²	C₃×C₃₇	1	1	yes	characteristic
36	C₉⋊C₂₇	C₃×C₃₇	1	6	yes	complemented
37	C₃²⋊C₂₇	C₃×C₃₇	1	3	yes	—
38	C₉×C₂₇	C₃×C₃₇	1	3	yes	complemented
39	C₃×C₃₇	—	3	3	no	—
40	C₃×C₃₇	#6887	1	1	yes	characteristic
41	C₃×C₃₇	C₃²×C₂₇	1	1	yes	characteristic
42	C₃×C₃₇	C₃×C₂₇⋊C₃	1	2	yes	—
43	C₃²×C₉	C₉×C₃₇	1	2	yes	—
44	C₃²×C₉	C₉×C₃₇	1	1	yes	characteristic
45	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
46	C₃×C₂₇	C₃²×C₃₇	1	9	yes	—
47	C₉²	C₉×C₃₇	1	2	yes	—
48	C₉²	C₉×C₃₇	1	4	yes	—
49	C₉²	C₃²×C₃₇	1	2	yes	—
50	C₉²	C₃²×C₃₇	1	1	yes	characteristic
51	C₃₇	#10624710	1	1	yes	characteristic, direct product factor, Sylow
52	C₃³	C₃×C₉×C₃₇	1	1	yes	characteristic
53	C₃×C₉	—	3	6	no	—
54	C₃×C₉	—	3	3	no	—
55	C₃×C₉	C₂₇×C₃₇	1	9	yes	complemented
56	C₃×C₉	C₃×C₉×C₃₇	1	2	yes	—
57	C₃×C₉	C₃×C₉×C₃₇	1	4	yes	—
58	C₃×C₉	C₃×C₉×C₃₇	1	2	yes	—
59	C₃×C₉	C₃×C₉×C₃₇	1	1	yes	characteristic
60	C₃×C₉	C₃×C₉×C₃₇	1	2	yes	—
61	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
62	C₂₇	—	3	9	no	—
63	C₃²	—	3	1	no	—
64	C₃²	—	3	2	no	—
65	C₃²	C₃×C₂₇×C₃₇	1	3	yes	—
66	C₃²	C₃²×C₉×C₃₇	1	1	yes	characteristic
67	C₉	—	3	2	no	—
68	C₉	—	3	4	no	—
69	C₉	C₃×C₂₇×C₃₇	1	6	yes	—
70	C₉	C₃×C₂₇×C₃₇	1	3	yes	—
71	C₉	C₃₇×C₂₇⋊C₃	1	6	yes	—
72	C₉	C₃₇×#737	1	3	yes	—
73	C₃	—	3	3	no	—
74	C₃	C₃₇×#6887	1	1	yes	characteristic
75	C₃	C₃²×C₂₇×C₃₇	1	1	yes	characteristic
76	C₃	C₃×C₃₇×C₂₇⋊C₃	1	2	yes	—

	subgroup	G/H
center	C₉²×C₃₇	C₃²
commutator	C₃	C₃²×C₂₇×C₃₇
Frattini, cosocle	C₃×C₉	C₃³×C₃₇
socle	C₃²×C₃₇	C₃²×C₉
prime radical	C₃³×C₃₇	C₃×C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×C₉⋊C₂₇	C₃	semidirect
4. C₉×C₂₇×C₃₇	C₃	semidirect
16. C₃×C₉×C₃₇	C₂₇	semidirect
24. #10624710	C₃₇	direct (N × H)
36. C₉⋊C₂₇	C₃×C₃₇	semidirect
38. C₉×C₂₇	C₃×C₃₇	semidirect
51. C₃₇	#10624710	direct (N × H)
55. C₃×C₉	C₂₇×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉²×C₃₇	3
C₃₇×C₉⋊C₂₇	3
C₃₇×C₃²⋊C₂₇	3
C₉×C₂₇×C₃₇	3
#10624710	37

Subgroups — C37×#10624710 #41004499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624710 #41004499