Subgroups — C₃₇×#10624849 #41004638

Order 26973.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#6908	C₃	1	3	yes	maximal, complemented
2	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#6883	C₃	1	3	yes	maximal, complemented
4	C₃₇×#6882	C₃	1	3	yes	maximal, complemented
5	C₃₇×#6881	C₃	1	3	yes	maximal, complemented
6	C₃₇×#737	—	3	3	no	—
7	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
8	C₃₇×#733	C₃²	1	3	yes	—
9	C₃₇×#733	C₃²	1	3	yes	—
10	C₃₇×#732	C₃²	1	3	yes	—
11	C₃₇×#731	—	3	3	no	—
12	C₃₇×AΣL(1,𝔽₂₇)	—	3	3	no	—
13	C₃₇×C₉⋊C₉	—	3	1	no	—
14	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
15	C₃₇×C₃²⋊C₉	—	3	1	no	—
16	C₉²×C₃₇	—	3	1	no	—
17	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
18	3₋¹⁺²×C₃₇	—	3	3	no	—
19	3₋¹⁺²×C₃₇	—	3	3	no	—
20	3₋¹⁺²×C₃₇	—	9	6	no	—
21	H(3,𝔽₃)×C₃₇	—	3	3	no	—
22	C₃×C₉×C₃₇	—	3	3	no	—
23	C₃×C₉×C₃₇	—	3	3	no	—
24	C₃×C₉×C₃₇	—	9	3	no	—
25	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
26	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
27	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
28	#10624849	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₃²×C₃₇	—	3	1	no	—
30	C₃²×C₃₇	—	9	1	no	—
31	C₃²×C₃₇	—	9	3	no	—
32	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
33	C₉×C₃₇	—	3	1	no	—
34	C₉×C₃₇	—	3	1	no	—
35	C₉×C₃₇	—	9	3	no	—
36	C₉×C₃₇	—	9	3	no	—
37	C₉×C₃₇	—	9	3	no	—
38	C₉×C₃₇	#732	1	3	yes	—
39	#6908	C₃×C₃₇	1	3	yes	complemented
40	#6887	C₃×C₃₇	1	1	yes	characteristic, complemented
41	#6883	C₃×C₃₇	1	3	yes	complemented
42	#6882	C₃×C₃₇	1	3	yes	complemented
43	#6881	C₃×C₃₇	1	3	yes	complemented
44	C₃×C₃₇	—	3	1	no	—
45	C₃×C₃₇	—	9	1	no	—
46	C₃×C₃₇	—	27	3	no	—
47	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
48	#737	—	3	3	no	—
49	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
50	#733	C₃²×C₃₇	1	3	yes	—
51	#733	C₃²×C₃₇	1	3	yes	—
52	#732	C₃²×C₃₇	1	3	yes	—
53	#731	—	3	3	no	—
54	AΣL(1,𝔽₂₇)	—	3	3	no	—
55	C₉⋊C₉	—	3	1	no	—
56	C₉⋊C₉	C₃²×C₃₇	1	3	yes	—
57	C₃²⋊C₉	—	3	1	no	—
58	C₉²	—	3	1	no	—
59	C₃₇	#10624849	1	1	yes	characteristic, direct product factor, Sylow
60	C₃³	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
61	3₋¹⁺²	—	3	3	no	—
62	3₋¹⁺²	—	3	3	no	—
63	3₋¹⁺²	—	9	6	no	—
64	H(3,𝔽₃)	—	3	3	no	—
65	C₃×C₉	—	3	3	no	—
66	C₃×C₉	—	3	3	no	—
67	C₃×C₉	—	9	3	no	—
68	C₃×C₉	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
69	C₃×C₉	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
70	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
71	C₃²	—	3	1	no	—
72	C₃²	—	9	1	no	—
73	C₃²	—	9	3	no	—
74	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
75	C₉	—	3	1	no	—
76	C₉	—	3	1	no	—
77	C₉	—	9	3	no	—
78	C₉	—	9	3	no	—
79	C₉	—	9	3	no	—
80	C₉	C₃₇×#732	1	3	yes	—
81	C₃	—	3	1	no	—
82	C₃	—	9	1	no	—
83	C₃	—	27	3	no	—
84	C₃	C₃×C₃₇×#732	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃×C₃₇	C₃×#732
commutator, Frattini, cosocle	C₃×C₉	C₃³×C₃₇
LCS	G ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₃₇ ◁ C₃×C₉×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×#6908	C₃	semidirect
2. C₃₇×#6887	C₃	semidirect
3. C₃₇×#6883	C₃	semidirect
4. C₃₇×#6882	C₃	semidirect
5. C₃₇×#6881	C₃	semidirect
28. #10624849	C₃₇	direct (N × H)
39. #6908	C₃×C₃₇	semidirect
40. #6887	C₃×C₃₇	semidirect
41. #6883	C₃×C₃₇	semidirect
42. #6882	C₃×C₃₇	semidirect
43. #6881	C₃×C₃₇	semidirect
59. C₃₇	#10624849	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×#6908	3
C₃₇×#6887	3
C₃₇×#6883	3
C₃₇×#6882	3
C₃₇×#6881	3
#10624849	37

Subgroups — C37×#10624849 #41004638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624849 #41004638