Subgroups — C₃₇×#10624900 #41004689

Order 26973.

730 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	C₃₇×#6895	C₃	1	8	yes	maximal, complemented
3	C₃₇×#6894	C₃	1	8	yes	maximal, complemented
4	C₃₇×#6894	C₃	1	16	yes	maximal, complemented
5	C₉×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
6	C₉×H(3,𝔽₃)×C₃₇	C₃	1	2	yes	maximal
7	C₃×C₃₇×C₉⋊C₉	C₃	1	4	yes	maximal, complemented
8	C₃₇×#737	—	3	9	no	—
9	C₃×3₋¹⁺²×C₃₇	C₃²	1	8	yes	—
10	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
11	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
12	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
13	C₃²×C₉×C₃₇	C₃²	1	8	yes	—
14	C₃₇×C₉⋊C₉	C₃²	1	1	yes	characteristic
15	C₃₇×C₉⋊C₉	C₃²	1	16	yes	—
16	C₃₇×C₉⋊C₉	C₃²	1	8	yes	—
17	C₃₇×C₉⋊C₉	C₃²	1	8	yes	—
18	C₃₇×C₉⋊C₉	C₃²	1	48	yes	—
19	C₃₇×C₃²⋊C₉	C₃²	1	8	yes	—
20	C₃₇×C₃²⋊C₉	C₃²	1	16	yes	—
21	C₃³×C₃₇	C₃³	1	4	yes	—
22	3₋¹⁺²×C₃₇	—	3	24	no	—
23	H(3,𝔽₃)×C₃₇	3₋¹⁺²	1	9	yes	—
24	C₃×C₉×C₃₇	—	3	12	no	—
25	C₃×C₉×C₃₇	—	3	12	no	—
26	C₃×C₉×C₃₇	—	9	8	no	—
27	C₃×C₉×C₃₇	C₃³	1	8	yes	—
28	C₃×C₉×C₃₇	C₃³	1	16	yes	—
29	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
30	C₃×C₉×C₃₇	C₃³	1	2	yes	—
31	C₃×C₉×C₃₇	C₃³	1	8	yes	—
32	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
33	#10624900	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₃²×C₃₇	—	3	4	no	—
35	C₃²×C₃₇	—	3	8	no	—
36	C₃²×C₃₇	C₃×3₋¹⁺²	1	12	yes	—
37	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
38	C₉×C₃₇	—	3	1	no	—
39	C₉×C₃₇	—	3	8	no	—
40	C₉×C₃₇	—	3	8	no	—
41	C₉×C₃₇	—	3	2	no	—
42	C₉×C₃₇	—	3	16	no	—
43	C₉×C₃₇	#737	1	3	yes	complemented
44	C₃×#737	C₃×C₃₇	1	1	yes	characteristic
45	#6895	C₃×C₃₇	1	8	yes	complemented
46	#6894	C₃×C₃₇	1	8	yes	complemented
47	#6894	C₃×C₃₇	1	16	yes	complemented
48	C₉×H(3,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic
49	C₉×H(3,𝔽₃)	C₃×C₃₇	1	2	yes	—
50	C₃×C₉⋊C₉	C₃×C₃₇	1	4	yes	complemented
51	C₃×C₃₇	—	3	12	no	—
52	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
53	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
54	C₃×C₃₇	3₋¹⁺⁴	1	2	yes	—
55	#737	—	3	9	no	—
56	C₃×3₋¹⁺²	C₃²×C₃₇	1	8	yes	—
57	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	1	yes	characteristic
58	C₃²×C₉	C₃²×C₃₇	1	4	yes	—
59	C₃²×C₉	C₃²×C₃₇	1	4	yes	—
60	C₃²×C₉	C₃²×C₃₇	1	8	yes	—
61	C₉⋊C₉	C₃²×C₃₇	1	1	yes	characteristic
62	C₉⋊C₉	C₃²×C₃₇	1	16	yes	—
63	C₉⋊C₉	C₃²×C₃₇	1	8	yes	—
64	C₉⋊C₉	C₃²×C₃₇	1	8	yes	—
65	C₉⋊C₉	C₃²×C₃₇	1	48	yes	—
66	C₃²⋊C₉	C₃²×C₃₇	1	8	yes	—
67	C₃²⋊C₉	C₃²×C₃₇	1	16	yes	—
68	C₃₇	#10624900	1	1	yes	characteristic, direct product factor, Sylow
69	C₃³	C₃³×C₃₇	1	4	yes	—
70	3₋¹⁺²	—	3	24	no	—
71	H(3,𝔽₃)	3₋¹⁺²×C₃₇	1	9	yes	—
72	C₃×C₉	—	3	12	no	—
73	C₃×C₉	—	3	12	no	—
74	C₃×C₉	—	9	8	no	—
75	C₃×C₉	C₃³×C₃₇	1	8	yes	—
76	C₃×C₉	C₃³×C₃₇	1	16	yes	—
77	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
78	C₃×C₉	C₃³×C₃₇	1	2	yes	—
79	C₃×C₉	C₃³×C₃₇	1	8	yes	—
80	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
81	C₃²	—	3	4	no	—
82	C₃²	—	3	8	no	—
83	C₃²	C₃×3₋¹⁺²×C₃₇	1	12	yes	—
84	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
85	C₉	—	3	1	no	—
86	C₉	—	3	8	no	—
87	C₉	—	3	8	no	—
88	C₉	—	3	2	no	—
89	C₉	—	3	16	no	—
90	C₉	C₃₇×#737	1	3	yes	complemented
91	C₃	—	3	12	no	—
92	C₃	C₃²×3₋¹⁺²×C₃₇	1	1	yes	characteristic
93	C₃	C₃×C₃₇×#737	1	1	yes	characteristic
94	C₃	C₃₇×3₋¹⁺⁴	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃₇×#6895	C₃	semidirect
3. C₃₇×#6894	C₃	semidirect
4. C₃₇×#6894	C₃	semidirect
7. C₃×C₃₇×C₉⋊C₉	C₃	semidirect
33. #10624900	C₃₇	direct (N × H)
43. C₉×C₃₇	#737	semidirect
45. #6895	C₃×C₃₇	semidirect
46. #6894	C₃×C₃₇	semidirect
47. #6894	C₃×C₃₇	semidirect
50. C₃×C₉⋊C₉	C₃×C₃₇	semidirect
68. C₃₇	#10624900	direct (N × H)
90. C₉	C₃₇×#737	semidirect

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
C₃₇×#6895	3
C₃₇×#6894	3
C₉×H(3,𝔽₃)×C₃₇	3
C₃×C₃₇×C₉⋊C₉	3
#10624900	37

Subgroups — C37×#10624900 #41004689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624900 #41004689