Subgroups — C₃₇×#10624444 #41004233

Order 26973.

316 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₉×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₃₇×C₃²⋊C₂₇	C₃	1	3	yes	maximal, complemented
3	C₃⁴×C₃₇	C₉	1	1	yes	characteristic
4	C₃²×C₉×C₃₇	—	3	1	no	—
5	C₃²×C₉×C₃₇	—	3	2	no	—
6	C₃²×C₉×C₃₇	—	3	9	no	—
7	C₃²×C₉×C₃₇	C₉	1	2	yes	—
8	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
9	C₃×C₂₇×C₃₇	—	3	3	no	—
10	C₃³×C₃₇	—	3	3	no	—
11	C₃³×C₃₇	—	3	6	no	—
12	C₃³×C₃₇	—	3	1	no	—
13	C₃³×C₃₇	—	3	2	no	—
14	C₃³×C₃₇	C₂₇	1	3	yes	complemented
15	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
16	C₃×C₉×C₃₇	—	3	18	no	—
17	C₃×C₉×C₃₇	—	3	2	no	—
18	C₃×C₉×C₃₇	—	3	9	no	—
19	C₃×C₉×C₃₇	—	3	3	no	—
20	C₃×C₉×C₃₇	—	3	6	no	—
21	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
23	C₂₇×C₃₇	—	9	3	no	—
24	#10624444	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₃²×C₃₇	—	3	9	no	—
26	C₃²×C₃₇	—	3	18	no	—
27	C₃²×C₃₇	—	3	3	no	—
28	C₃²×C₃₇	—	3	3	no	—
29	C₃²×C₃₇	—	3	1	no	—
30	C₃²×C₃₇	—	3	6	no	—
31	C₃²×C₃₇	—	3	2	no	—
32	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
33	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
34	C₃²×C₃₇	C₂₇⋊C₃	1	2	yes	—
35	C₉×C₃₇	—	3	6	no	—
36	C₉×C₃₇	—	3	2	no	—
37	C₉×C₃₇	AΣL(1,𝔽₂₇)	1	3	yes	—
38	C₃³×C₉	C₃×C₃₇	1	1	yes	characteristic
39	C₃²⋊C₂₇	C₃×C₃₇	1	3	yes	complemented
40	C₃×C₃₇	—	3	9	no	—
41	C₃×C₃₇	—	3	1	no	—
42	C₃×C₃₇	—	3	2	no	—
43	C₃×C₃₇	C₃²⋊C₂₇	1	1	yes	characteristic
44	C₃×C₃₇	#6866	1	1	yes	characteristic
45	C₃×C₃₇	#6869	1	2	yes	—
46	C₃⁴	C₉×C₃₇	1	1	yes	characteristic
47	C₃²×C₉	—	3	1	no	—
48	C₃²×C₉	—	3	2	no	—
49	C₃²×C₉	—	3	9	no	—
50	C₃²×C₉	C₉×C₃₇	1	2	yes	—
51	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
52	C₃×C₂₇	—	3	3	no	—
53	C₃₇	#10624444	1	1	yes	characteristic, direct product factor, Sylow
54	C₃³	—	3	3	no	—
55	C₃³	—	3	6	no	—
56	C₃³	—	3	1	no	—
57	C₃³	—	3	2	no	—
58	C₃³	C₂₇×C₃₇	1	3	yes	complemented
59	C₃³	C₃×C₉×C₃₇	1	1	yes	characteristic
60	C₃×C₉	—	3	18	no	—
61	C₃×C₉	—	3	2	no	—
62	C₃×C₉	—	3	9	no	—
63	C₃×C₉	—	3	3	no	—
64	C₃×C₉	—	3	6	no	—
65	C₃×C₉	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
66	C₃×C₉	3₋¹⁺²×C₃₇	1	2	yes	—
67	C₂₇	—	9	3	no	—
68	C₃²	—	3	9	no	—
69	C₃²	—	3	18	no	—
70	C₃²	—	3	3	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	3	3	no	—
73	C₃²	—	3	2	no	—
74	C₃²	—	3	6	no	—
75	C₃²	C₃₇×C₃²⋊C₉	1	1	yes	characteristic
76	C₃²	C₃×C₂₇×C₃₇	1	1	yes	characteristic
77	C₃²	C₃₇×C₂₇⋊C₃	1	2	yes	—
78	C₉	—	3	2	no	—
79	C₉	—	3	6	no	—
80	C₉	C₃₇×AΣL(1,𝔽₂₇)	1	3	yes	—
81	C₃	—	3	1	no	—
82	C₃	—	3	2	no	—
83	C₃	—	3	9	no	—
84	C₃	C₃₇×C₃²⋊C₂₇	1	1	yes	characteristic
85	C₃	C₃₇×#6866	1	1	yes	characteristic
86	C₃	C₃₇×#6869	1	2	yes	—

	subgroup	G/H
center	C₃×C₉×C₃₇	H(3,𝔽₃)
commutator	C₃²	C₃×C₂₇×C₃₇
Frattini, cosocle	C₃²×C₉	C₃²×C₃₇
socle	C₃²×C₃₇	C₃²⋊C₉
prime radical	C₃⁴×C₃₇	C₉
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×C₃²⋊C₂₇	C₃	semidirect
14. C₃³×C₃₇	C₂₇	semidirect
24. #10624444	C₃₇	direct (N × H)
39. C₃²⋊C₂₇	C₃×C₃₇	semidirect
53. C₃₇	#10624444	direct (N × H)
58. C₃³	C₂₇×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₉×C₃₇	3
C₃₇×C₃²⋊C₂₇	3
#10624444	37

Subgroups — C37×#10624444 #41004233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624444 #41004233