Subgroups — C₃₇×#10624887 #41004676

Order 26973.

698 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic
2	C₃₇×#6900	C₃	1	3	yes	maximal, complemented
3	C₃₇×#6899	C₃	1	6	yes	maximal, complemented
4	C₃₇×#6896	C₃	1	9	yes	maximal, complemented
5	C₃₇×#6894	C₃	1	6	yes	maximal, complemented
6	C₃₇×#6894	C₃	1	9	yes	maximal, complemented
7	C₃₇×C₉⋊H(3,𝔽₃)	C₃	1	3	yes	maximal
8	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃×C₃₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
11	C₃₇×#737	—	3	9	no	—
12	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
13	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
14	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
15	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
16	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
17	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
18	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
19	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
20	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
21	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
22	C₃₇×C₉⋊C₉	C₃²	1	2	yes	—
23	C₃₇×C₉⋊C₉	C₃²	1	1	yes	characteristic
24	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
25	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
26	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
27	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
28	C₃₇×C₉⋊C₉	C₃²	1	36	yes	—
29	C₃₇×C₉⋊C₉	C₃²	1	18	yes	—
30	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
31	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
32	C₃₇×C₃²⋊C₉	C₃²	1	18	yes	—
33	C₉²×C₃₇	C₃²	1	3	yes	—
34	C₉²×C₃₇	C₃²	1	3	yes	—
35	C₃³×C₃₇	C₃³	1	3	yes	—
36	C₃³×C₃₇	C₃³	1	1	yes	characteristic
37	3₋¹⁺²×C₃₇	—	3	6	no	—
38	3₋¹⁺²×C₃₇	—	3	9	no	—
39	3₋¹⁺²×C₃₇	—	3	9	no	—
40	3₋¹⁺²×C₃₇	—	9	9	no	—
41	H(3,𝔽₃)×C₃₇	—	3	3	no	—
42	C₃×C₉×C₃₇	—	3	9	no	—
43	C₃×C₉×C₃₇	—	9	3	no	—
44	C₃×C₉×C₃₇	3₋¹⁺²	1	9	yes	complemented
45	C₃×C₉×C₃₇	C₃³	1	9	yes	—
46	C₃×C₉×C₃₇	C₃³	1	9	yes	—
47	C₃×C₉×C₃₇	C₃³	1	6	yes	—
48	C₃×C₉×C₃₇	C₃³	1	6	yes	—
49	C₃×C₉×C₃₇	C₃³	1	3	yes	—
50	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
51	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
52	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
53	#10624887	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₃²×C₃₇	—	3	3	no	—
55	C₃²×C₃₇	—	3	3	no	—
56	C₃²×C₃₇	—	3	9	no	—
57	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
58	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
59	C₉×C₃₇	—	3	6	no	—
60	C₉×C₃₇	—	3	6	no	—
61	C₉×C₃₇	—	3	3	no	—
62	C₉×C₃₇	—	3	9	no	—
63	C₉×C₃₇	—	3	9	no	—
64	C₉×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
65	C₉×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
66	C₉×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
67	C₃×#737	C₃×C₃₇	1	1	yes	characteristic
68	#6900	C₃×C₃₇	1	3	yes	complemented
69	#6899	C₃×C₃₇	1	6	yes	complemented
70	#6896	C₃×C₃₇	1	9	yes	complemented
71	#6894	C₃×C₃₇	1	6	yes	complemented
72	#6894	C₃×C₃₇	1	9	yes	complemented
73	C₉⋊H(3,𝔽₃)	C₃×C₃₇	1	3	yes	—
74	#6887	C₃×C₃₇	1	1	yes	characteristic, complemented
75	#6887	C₃×C₃₇	1	1	yes	characteristic, complemented
76	C₃×C₉⋊C₉	C₃×C₃₇	1	1	yes	characteristic, complemented
77	C₃×C₃₇	—	3	3	no	—
78	C₃×C₃₇	—	9	3	no	—
79	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
80	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
81	#737	—	3	9	no	—
82	C₃×3₋¹⁺²	C₃²×C₃₇	1	2	yes	—
83	C₃×3₋¹⁺²	C₃²×C₃₇	1	3	yes	—
84	C₃×3₋¹⁺²	C₃²×C₃₇	1	3	yes	—
85	C₃×3₋¹⁺²	C₃²×C₃₇	1	9	yes	—
86	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	1	yes	characteristic
87	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
88	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
89	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
90	C₉⋊C₉	C₃²×C₃₇	1	3	yes	—
91	C₉⋊C₉	C₃²×C₃₇	1	3	yes	—
92	C₉⋊C₉	C₃²×C₃₇	1	2	yes	—
93	C₉⋊C₉	C₃²×C₃₇	1	1	yes	characteristic
94	C₉⋊C₉	C₃²×C₃₇	1	3	yes	—
95	C₉⋊C₉	C₃²×C₃₇	1	3	yes	—
96	C₉⋊C₉	C₃²×C₃₇	1	3	yes	—
97	C₉⋊C₉	C₃²×C₃₇	1	3	yes	—
98	C₉⋊C₉	C₃²×C₃₇	1	36	yes	—
99	C₉⋊C₉	C₃²×C₃₇	1	18	yes	—
100	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
101	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
102	C₃²⋊C₉	C₃²×C₃₇	1	18	yes	—
103	C₉²	C₃²×C₃₇	1	3	yes	—
104	C₉²	C₃²×C₃₇	1	3	yes	—
105	C₃₇	#10624887	1	1	yes	characteristic, direct product factor, Sylow
106	C₃³	C₃³×C₃₇	1	3	yes	—
107	C₃³	C₃³×C₃₇	1	1	yes	characteristic
108	3₋¹⁺²	—	3	6	no	—
109	3₋¹⁺²	—	3	9	no	—
110	3₋¹⁺²	—	3	9	no	—
111	3₋¹⁺²	—	9	9	no	—
112	H(3,𝔽₃)	—	3	3	no	—
113	C₃×C₉	—	3	9	no	—
114	C₃×C₉	—	9	3	no	—
115	C₃×C₉	3₋¹⁺²×C₃₇	1	9	yes	complemented
116	C₃×C₉	C₃³×C₃₇	1	9	yes	—
117	C₃×C₉	C₃³×C₃₇	1	9	yes	—
118	C₃×C₉	C₃³×C₃₇	1	6	yes	—
119	C₃×C₉	C₃³×C₃₇	1	6	yes	—
120	C₃×C₉	C₃³×C₃₇	1	3	yes	—
121	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
122	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
123	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
124	C₃²	—	3	3	no	—
125	C₃²	—	3	3	no	—
126	C₃²	—	3	9	no	—
127	C₃²	C₃×3₋¹⁺²×C₃₇	1	3	yes	—
128	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
129	C₉	—	3	6	no	—
130	C₉	—	3	3	no	—
131	C₉	—	3	6	no	—
132	C₉	—	3	9	no	—
133	C₉	—	3	9	no	—
134	C₉	C₃×3₋¹⁺²×C₃₇	1	3	yes	—
135	C₉	C₃×3₋¹⁺²×C₃₇	1	3	yes	—
136	C₉	C₃×3₋¹⁺²×C₃₇	1	3	yes	—
137	C₃	—	3	3	no	—
138	C₃	—	9	3	no	—
139	C₃	C₃²×3₋¹⁺²×C₃₇	1	1	yes	characteristic
140	C₃	C₃₇×3₋¹⁺⁴	1	3	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×#6900	C₃	semidirect
3. C₃₇×#6899	C₃	semidirect
4. C₃₇×#6896	C₃	semidirect
5. C₃₇×#6894	C₃	semidirect
6. C₃₇×#6894	C₃	semidirect
8. C₃₇×#6887	C₃	semidirect
9. C₃₇×#6887	C₃	semidirect
10. C₃×C₃₇×C₉⋊C₉	C₃	semidirect
44. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
53. #10624887	C₃₇	direct (N × H)
68. #6900	C₃×C₃₇	semidirect
69. #6899	C₃×C₃₇	semidirect
70. #6896	C₃×C₃₇	semidirect
71. #6894	C₃×C₃₇	semidirect
72. #6894	C₃×C₃₇	semidirect
74. #6887	C₃×C₃₇	semidirect
75. #6887	C₃×C₃₇	semidirect
76. C₃×C₉⋊C₉	C₃×C₃₇	semidirect
105. C₃₇	#10624887	direct (N × H)
115. C₃×C₉	3₋¹⁺²×C₃₇	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×#6900	3
C₃₇×#6899	3
C₃₇×#6896	3
C₃₇×#6894	3
C₃₇×C₉⋊H(3,𝔽₃)	3
C₃₇×#6887	3
C₃×C₃₇×C₉⋊C₉	3
#10624887	37

Subgroups — C37×#10624887 #41004676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624887 #41004676