Subgroups — C₃₇×#10624694 #41004483

Order 26973.

312 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	3	yes	maximal, complemented
3	C₃₇×#6869	C₃	1	9	yes	maximal, complemented
4	C₃₇×#737	—	3	3	no	—
5	C₃₇×#737	—	3	6	no	—
6	C₃×3₋¹⁺²×C₃₇	C₉	1	2	yes	—
7	C₃×3₋¹⁺²×C₃₇	C₉	1	6	yes	—
8	C₃×H(3,𝔽₃)×C₃₇	C₉	1	1	yes	characteristic
9	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
10	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
11	C₃₇×C₂₇⋊C₃	—	3	6	no	—
12	C₃₇×C₂₇⋊C₃	C₃²	1	9	yes	—
13	C₃×C₂₇×C₃₇	—	3	3	no	—
14	C₃³×C₃₇	C₃×C₉	1	3	yes	—
15	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
16	3₋¹⁺²×C₃₇	—	3	6	no	—
17	3₋¹⁺²×C₃₇	—	3	18	no	—
18	H(3,𝔽₃)×C₃₇	—	3	3	no	—
19	C₃×C₉×C₃₇	—	3	3	no	—
20	C₃×C₉×C₃₇	—	3	6	no	—
21	C₃×C₉×C₃₇	C₃×C₉	1	6	yes	—
22	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
23	C₃×C₉×C₃₇	H(3,𝔽₃)	1	3	yes	—
24	C₃×C₉×C₃₇	3₋¹⁺²	1	6	yes	—
25	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
26	C₂₇×C₃₇	—	3	9	no	—
27	#10624694	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₃²×C₃₇	—	3	3	no	—
29	C₃²×C₃₇	—	9	1	no	—
30	C₃²×C₃₇	—	9	3	no	—
31	C₃²×C₃₇	C₃²⋊C₉	1	3	yes	—
32	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
33	C₉×C₃₇	—	3	6	no	—
34	C₉×C₃₇	C₃²⋊C₉	1	6	yes	—
35	C₉×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
36	C₉×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
37	C₃×#737	C₃×C₃₇	1	1	yes	characteristic
38	C₃×C₂₇⋊C₃	C₃×C₃₇	1	3	yes	complemented
39	#6869	C₃×C₃₇	1	9	yes	complemented
40	C₃×C₃₇	—	3	1	no	—
41	C₃×C₃₇	—	3	3	no	—
42	C₃×C₃₇	—	9	3	no	—
43	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
44	#737	—	3	3	no	—
45	#737	—	3	6	no	—
46	C₃×3₋¹⁺²	C₉×C₃₇	1	2	yes	—
47	C₃×3₋¹⁺²	C₉×C₃₇	1	6	yes	—
48	C₃×H(3,𝔽₃)	C₉×C₃₇	1	1	yes	characteristic
49	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
50	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
51	C₂₇⋊C₃	—	3	6	no	—
52	C₂₇⋊C₃	C₃²×C₃₇	1	9	yes	—
53	C₃×C₂₇	—	3	3	no	—
54	C₃₇	#10624694	1	1	yes	characteristic, direct product factor, Sylow
55	C₃³	C₃×C₉×C₃₇	1	3	yes	—
56	C₃³	C₃×C₉×C₃₇	1	1	yes	characteristic
57	3₋¹⁺²	—	3	6	no	—
58	3₋¹⁺²	—	3	18	no	—
59	H(3,𝔽₃)	—	3	3	no	—
60	C₃×C₉	—	3	3	no	—
61	C₃×C₉	—	3	6	no	—
62	C₃×C₉	C₃×C₉×C₃₇	1	6	yes	—
63	C₃×C₉	C₃×C₉×C₃₇	1	2	yes	—
64	C₃×C₉	H(3,𝔽₃)×C₃₇	1	3	yes	—
65	C₃×C₉	3₋¹⁺²×C₃₇	1	6	yes	—
66	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
67	C₂₇	—	3	9	no	—
68	C₃²	—	3	3	no	—
69	C₃²	—	9	1	no	—
70	C₃²	—	9	3	no	—
71	C₃²	C₃₇×C₃²⋊C₉	1	3	yes	—
72	C₃²	C₃²×C₉×C₃₇	1	1	yes	characteristic
73	C₉	—	3	6	no	—
74	C₉	C₃₇×C₃²⋊C₉	1	6	yes	—
75	C₉	C₃×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
76	C₉	C₃×3₋¹⁺²×C₃₇	1	2	yes	—
77	C₃	—	3	1	no	—
78	C₃	—	3	3	no	—
79	C₃	—	9	3	no	—
80	C₃	C₃×C₃₇×C₃²⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₉×C₃₇	C₃×H(3,𝔽₃)
commutator	C₃²	C₃²×C₉×C₃₇
Frattini, cosocle	C₃×C₉	C₃³×C₃₇
socle	C₃×C₃₇	C₃×C₃²⋊C₉
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₉
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉×C₃₇ ◁ C₃²×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. C₃₇×#6869	C₃	semidirect
27. #10624694	C₃₇	direct (N × H)
38. C₃×C₂₇⋊C₃	C₃×C₃₇	semidirect
39. #6869	C₃×C₃₇	semidirect
54. C₃₇	#10624694	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
C₃₇×#6869	3
#10624694	37

Subgroups — C37×#10624694 #41004483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624694 #41004483