Subgroups — C₃₇×3₋¹⁺²⋊C₂₇ #41004237

Order 26973.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₂₇×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₉×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃₇×C₃²⋊C₂₇	C₃	1	1	yes	maximal, characteristic
4	C₃₇×C₃²⋊C₂₇	C₃	1	1	yes	maximal, characteristic
5	C₃×3₋¹⁺²×C₃₇	C₉	1	1	yes	characteristic
6	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
7	C₃×C₂₇×C₃₇	—	3	1	no	—
8	C₃×C₂₇×C₃₇	—	3	3	no	—
9	C₃×C₂₇×C₃₇	—	3	1	no	—
10	C₃×C₂₇×C₃₇	—	3	1	no	—
11	C₃₇×C₉⋊C₉	—	3	2	no	—
12	C₃₇×C₃²⋊C₉	C₉	1	1	yes	characteristic
13	C₃₇×C₃²⋊C₉	C₉	1	1	yes	characteristic
14	C₉²×C₃₇	—	3	1	no	—
15	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
16	3₋¹⁺²×C₃₇	—	3	2	no	—
17	3₋¹⁺²×C₃₇	C₂₇	1	3	yes	complemented
18	C₃×C₉×C₃₇	—	3	1	no	—
19	C₃×C₉×C₃₇	—	3	1	no	—
20	C₃×C₉×C₃₇	—	3	1	no	—
21	C₃×C₉×C₃₇	—	3	1	no	—
22	C₃×C₉×C₃₇	—	3	1	no	—
23	C₃×C₉×C₃₇	—	3	1	no	—
24	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
26	C₂₇×C₃₇	—	3	3	no	—
27	C₂₇×C₃₇	—	9	1	no	—
28	C₂₇×C₃₇	—	9	1	no	—
29	3₋¹⁺²⋊C₂₇	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₃²×C₃₇	—	3	1	no	—
31	C₃²×C₃₇	—	3	1	no	—
32	C₃²×C₃₇	—	3	1	no	—
33	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
34	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
35	C₃²×C₃₇	C₂₇⋊C₃	1	2	yes	—
36	C₉×C₃₇	—	3	3	no	—
37	C₉×C₃₇	—	3	2	no	—
38	C₉×C₃₇	—	9	1	no	—
39	C₉×C₃₇	—	9	1	no	—
40	C₉×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
41	C₉×C₃₇	#731	1	1	yes	characteristic
42	C₉×C₃₇	#733	1	1	yes	characteristic
43	C₃²×C₂₇	C₃×C₃₇	1	1	yes	characteristic
44	C₉×3₋¹⁺²	C₃×C₃₇	1	1	yes	characteristic
45	C₃²⋊C₂₇	C₃×C₃₇	1	1	yes	characteristic
46	C₃²⋊C₂₇	C₃×C₃₇	1	1	yes	characteristic
47	C₃×C₃₇	—	3	1	no	—
48	C₃×C₃₇	—	3	2	no	—
49	C₃×C₃₇	C₃²⋊C₂₇	1	1	yes	characteristic
50	C₃×C₃₇	3₋¹⁺²⋊C₉	1	1	yes	characteristic
51	C₃×C₃₇	#6872	1	1	yes	characteristic
52	C₃×C₃₇	#6873	1	1	yes	characteristic
53	C₃×3₋¹⁺²	C₉×C₃₇	1	1	yes	characteristic
54	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
55	C₃×C₂₇	—	3	1	no	—
56	C₃×C₂₇	—	3	3	no	—
57	C₃×C₂₇	—	3	1	no	—
58	C₃×C₂₇	—	3	1	no	—
59	C₉⋊C₉	—	3	2	no	—
60	C₃²⋊C₉	C₉×C₃₇	1	1	yes	characteristic
61	C₃²⋊C₉	C₉×C₃₇	1	1	yes	characteristic
62	C₉²	—	3	1	no	—
63	C₃₇	3₋¹⁺²⋊C₂₇	1	1	yes	characteristic, direct product factor, Sylow
64	C₃³	C₃×C₉×C₃₇	1	1	yes	characteristic
65	3₋¹⁺²	—	3	2	no	—
66	3₋¹⁺²	C₂₇×C₃₇	1	3	yes	complemented
67	C₃×C₉	—	3	1	no	—
68	C₃×C₉	—	3	1	no	—
69	C₃×C₉	—	3	1	no	—
70	C₃×C₉	—	3	1	no	—
71	C₃×C₉	—	3	1	no	—
72	C₃×C₉	—	3	1	no	—
73	C₃×C₉	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
74	C₃×C₉	3₋¹⁺²×C₃₇	1	2	yes	—
75	C₂₇	—	3	3	no	—
76	C₂₇	—	9	1	no	—
77	C₂₇	—	9	1	no	—
78	C₃²	—	3	1	no	—
79	C₃²	—	3	1	no	—
80	C₃²	—	3	1	no	—
81	C₃²	C₃₇×C₃²⋊C₉	1	1	yes	characteristic
82	C₃²	C₃×C₂₇×C₃₇	1	1	yes	characteristic
83	C₃²	C₃₇×C₂₇⋊C₃	1	2	yes	—
84	C₉	—	3	3	no	—
85	C₉	—	3	2	no	—
86	C₉	—	9	1	no	—
87	C₉	—	9	1	no	—
88	C₉	C₃₇×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
89	C₉	C₃₇×#731	1	1	yes	characteristic
90	C₉	C₃₇×#733	1	1	yes	characteristic
91	C₃	—	3	1	no	—
92	C₃	—	3	2	no	—
93	C₃	C₃₇×C₃²⋊C₂₇	1	1	yes	characteristic
94	C₃	C₃₇×3₋¹⁺²⋊C₉	1	1	yes	characteristic
95	C₃	C₃₇×#6872	1	1	yes	characteristic
96	C₃	C₃₇×#6873	1	1	yes	characteristic

	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	C₃³×C₃₇	C₃×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
17. 3₋¹⁺²×C₃₇	C₂₇	semidirect
29. 3₋¹⁺²⋊C₂₇	C₃₇	direct (N × H)
63. C₃₇	3₋¹⁺²⋊C₂₇	direct (N × H)
66. 3₋¹⁺²	C₂₇×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₂₇×C₃₇	3
C₉×3₋¹⁺²×C₃₇	3
C₃₇×C₃²⋊C₂₇	3
3₋¹⁺²⋊C₂₇	37

Subgroups — C37×3−1+2⋊C27 #41004237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×3₋¹⁺²⋊C₂₇ #41004237