Subgroups — C₃₇×#10624684 #41004473

Order 26973.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic
2	C₃×C₃₇×C₂₇⋊C₃	C₃	1	2	yes	maximal, complemented
3	C₃²×C₂₇×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃₇×C₃²⋊C₂₇	C₃	1	9	yes	maximal, complemented
5	C₃₇×#737	—	3	3	no	—
6	C₃₇×#737	—	3	6	no	—
7	C₃×3₋¹⁺²×C₃₇	C₉	1	2	yes	—
8	C₃×3₋¹⁺²×C₃₇	C₉	1	6	yes	—
9	C₃×H(3,𝔽₃)×C₃₇	C₉	1	1	yes	characteristic
10	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
11	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
12	C₃₇×C₂₇⋊C₃	—	3	6	no	—
13	C₃×C₂₇×C₃₇	—	3	3	no	—
14	C₃×C₂₇×C₃₇	C₃²	1	3	yes	—
15	C₃×C₂₇×C₃₇	C₃²	1	6	yes	—
16	C₃³×C₃₇	C₃×C₉	1	3	yes	—
17	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
18	3₋¹⁺²×C₃₇	—	3	6	no	—
19	3₋¹⁺²×C₃₇	—	3	18	no	—
20	H(3,𝔽₃)×C₃₇	—	3	3	no	—
21	C₃×C₉×C₃₇	—	3	3	no	—
22	C₃×C₉×C₃₇	—	3	6	no	—
23	C₃×C₉×C₃₇	C₃×C₉	1	6	yes	—
24	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
25	C₃×C₉×C₃₇	H(3,𝔽₃)	1	3	yes	—
26	C₃×C₉×C₃₇	3₋¹⁺²	1	6	yes	—
27	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
28	C₂₇×C₃₇	—	3	3	no	—
29	C₂₇×C₃₇	—	3	6	no	—
30	#10624684	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₃²×C₃₇	—	3	3	no	—
32	C₃²×C₃₇	—	3	3	no	—
33	C₃²×C₃₇	—	3	9	no	—
34	C₃²×C₃₇	C₃²⋊C₉	1	3	yes	—
35	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
36	C₉×C₃₇	—	3	6	no	—
37	C₉×C₃₇	C₃²⋊C₉	1	6	yes	—
38	C₉×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
39	C₉×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
40	C₃×#737	C₃×C₃₇	1	1	yes	characteristic
41	C₃×C₂₇⋊C₃	C₃×C₃₇	1	2	yes	complemented
42	C₃²×C₂₇	C₃×C₃₇	1	1	yes	characteristic, complemented
43	C₃²⋊C₂₇	C₃×C₃₇	1	9	yes	complemented
44	C₃×C₃₇	—	3	3	no	—
45	C₃×C₃₇	—	9	3	no	—
46	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
47	C₃×C₃₇	#6903	1	3	yes	—
48	#737	—	3	3	no	—
49	#737	—	3	6	no	—
50	C₃×3₋¹⁺²	C₉×C₃₇	1	2	yes	—
51	C₃×3₋¹⁺²	C₉×C₃₇	1	6	yes	—
52	C₃×H(3,𝔽₃)	C₉×C₃₇	1	1	yes	characteristic
53	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
54	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
55	C₂₇⋊C₃	—	3	6	no	—
56	C₃×C₂₇	—	3	3	no	—
57	C₃×C₂₇	C₃²×C₃₇	1	3	yes	—
58	C₃×C₂₇	C₃²×C₃₇	1	6	yes	—
59	C₃₇	#10624684	1	1	yes	characteristic, direct product factor, Sylow
60	C₃³	C₃×C₉×C₃₇	1	3	yes	—
61	C₃³	C₃×C₉×C₃₇	1	1	yes	characteristic
62	3₋¹⁺²	—	3	6	no	—
63	3₋¹⁺²	—	3	18	no	—
64	H(3,𝔽₃)	—	3	3	no	—
65	C₃×C₉	—	3	3	no	—
66	C₃×C₉	—	3	6	no	—
67	C₃×C₉	C₃×C₉×C₃₇	1	6	yes	—
68	C₃×C₉	C₃×C₉×C₃₇	1	2	yes	—
69	C₃×C₉	H(3,𝔽₃)×C₃₇	1	3	yes	—
70	C₃×C₉	3₋¹⁺²×C₃₇	1	6	yes	—
71	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
72	C₂₇	—	3	3	no	—
73	C₂₇	—	3	6	no	—
74	C₃²	—	3	3	no	—
75	C₃²	—	3	3	no	—
76	C₃²	—	3	9	no	—
77	C₃²	C₃₇×C₃²⋊C₉	1	3	yes	—
78	C₃²	C₃²×C₉×C₃₇	1	1	yes	characteristic
79	C₉	—	3	6	no	—
80	C₉	C₃₇×C₃²⋊C₉	1	6	yes	—
81	C₉	C₃×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
82	C₉	C₃×3₋¹⁺²×C₃₇	1	2	yes	—
83	C₃	—	3	3	no	—
84	C₃	—	9	3	no	—
85	C₃	C₃×C₃₇×C₃²⋊C₉	1	1	yes	characteristic
86	C₃	C₃₇×#6903	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃×C₃₇×C₂₇⋊C₃	C₃	semidirect
3. C₃²×C₂₇×C₃₇	C₃	semidirect
4. C₃₇×C₃²⋊C₂₇	C₃	semidirect
30. #10624684	C₃₇	direct (N × H)
41. C₃×C₂₇⋊C₃	C₃×C₃₇	semidirect
42. C₃²×C₂₇	C₃×C₃₇	semidirect
43. C₃²⋊C₂₇	C₃×C₃₇	semidirect
59. C₃₇	#10624684	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×#737	3
C₃×C₃₇×C₂₇⋊C₃	3
C₃²×C₂₇×C₃₇	3
C₃₇×C₃²⋊C₂₇	3
#10624684	37

Subgroups — C37×#10624684 #41004473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624684 #41004473