Subgroups — C₃₇×#10624891 #41004680

Order 26973.

602 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#6899	C₃	1	6	yes	maximal, complemented
2	C₃₇×#6898	C₃	1	3	yes	maximal, complemented
3	C₃₇×#6897	C₃	1	6	yes	maximal, complemented
4	C₃₇×#6896	C₃	1	3	yes	maximal, complemented
5	C₃₇×#6895	C₃	1	2	yes	maximal, complemented
6	C₃₇×#6895	C₃	1	3	yes	maximal, complemented
7	C₃₇×#6894	C₃	1	2	yes	maximal, complemented
8	C₃₇×#6894	C₃	1	2	yes	maximal, complemented
9	C₃₇×#6894	C₃	1	6	yes	maximal, complemented
10	C₃₇×#6893	C₃	1	1	yes	maximal, characteristic
11	C₃₇×C₉⋊H(3,𝔽₃)	C₃	1	2	yes	maximal
12	C₉×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
13	C₉×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
14	C₉×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
15	C₃×C₃₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
16	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
17	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
18	C₃×3₋¹⁺²×C₃₇	C₃²	1	6	yes	—
19	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
20	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
21	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
22	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
23	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
24	C₃₇×C₉⋊C₉	C₃²	1	2	yes	—
25	C₃₇×C₉⋊C₉	C₃²	1	1	yes	characteristic
26	C₃₇×C₉⋊C₉	C₃²	1	1	yes	characteristic
27	C₃₇×C₉⋊C₉	C₃²	1	2	yes	—
28	C₃₇×C₉⋊C₉	C₃²	1	1	yes	characteristic
29	C₃₇×C₉⋊C₉	C₃²	1	2	yes	—
30	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
31	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
32	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
33	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
34	C₃₇×C₉⋊C₉	C₃²	1	6	yes	—
35	C₃₇×C₉⋊C₉	C₃²	1	6	yes	—
36	C₃₇×C₉⋊C₉	C₃²	1	6	yes	—
37	C₃₇×C₉⋊C₉	C₃²	1	6	yes	—
38	C₃₇×C₉⋊C₉	C₃²	1	6	yes	—
39	C₃₇×C₉⋊C₉	C₃²	1	6	yes	—
40	C₃₇×C₉⋊C₉	C₃²	1	12	yes	—
41	C₃₇×C₉⋊C₉	C₃²	1	6	yes	—
42	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
43	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
44	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
45	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
46	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
47	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
48	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
49	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
50	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
51	C₉²×C₃₇	C₃²	1	3	yes	—
52	C₉²×C₃₇	C₃²	1	3	yes	—
53	C₃³×C₃₇	C₃³	1	3	yes	—
54	C₃³×C₃₇	C₃³	1	1	yes	characteristic
55	3₋¹⁺²×C₃₇	—	3	3	no	—
56	3₋¹⁺²×C₃₇	—	3	3	no	—
57	3₋¹⁺²×C₃₇	—	9	6	no	—
58	H(3,𝔽₃)×C₃₇	—	3	3	no	—
59	C₃×C₉×C₃₇	—	3	3	no	—
60	C₃×C₉×C₃₇	—	9	1	no	—
61	C₃×C₉×C₃₇	—	9	3	no	—
62	C₃×C₉×C₃₇	—	9	2	no	—
63	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
64	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
65	C₃×C₉×C₃₇	C₃³	1	3	yes	—
66	C₃×C₉×C₃₇	C₃³	1	3	yes	—
67	C₃×C₉×C₃₇	C₃³	1	6	yes	—
68	C₃×C₉×C₃₇	C₃³	1	6	yes	—
69	C₃×C₉×C₃₇	C₃³	1	2	yes	—
70	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
71	C₃×C₉×C₃₇	C₃³	1	6	yes	—
72	C₃×C₉×C₃₇	C₃³	1	2	yes	—
73	C₃×C₉×C₃₇	C₃³	1	2	yes	—
74	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
75	C₃×C₉×C₃₇	C₃³	1	2	yes	—
76	#10624891	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₃²×C₃₇	—	3	1	no	—
78	C₃²×C₃₇	—	3	2	no	—
79	C₃²×C₃₇	—	3	3	no	—
80	C₃²×C₃₇	—	3	3	no	—
81	C₃²×C₃₇	—	3	6	no	—
82	C₃²×C₃₇	#737	1	3	yes	—
83	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
84	C₉×C₃₇	—	3	2	no	—
85	C₉×C₃₇	—	3	1	no	—
86	C₉×C₃₇	—	3	3	no	—
87	C₉×C₃₇	—	3	3	no	—
88	C₉×C₃₇	—	3	1	no	—
89	C₉×C₃₇	—	3	1	no	—
90	C₉×C₃₇	—	3	1	no	—
91	C₉×C₃₇	—	3	6	no	—
92	C₉×C₃₇	—	3	2	no	—
93	C₉×C₃₇	—	3	2	no	—
94	C₉×C₃₇	—	3	2	no	—
95	C₉×C₃₇	—	3	6	no	—
96	C₉×C₃₇	—	3	6	no	—
97	#6899	C₃×C₃₇	1	6	yes	complemented
98	#6898	C₃×C₃₇	1	3	yes	complemented
99	#6897	C₃×C₃₇	1	6	yes	complemented
100	#6896	C₃×C₃₇	1	3	yes	complemented
101	#6895	C₃×C₃₇	1	2	yes	complemented
102	#6895	C₃×C₃₇	1	3	yes	complemented
103	#6894	C₃×C₃₇	1	2	yes	complemented
104	#6894	C₃×C₃₇	1	2	yes	complemented
105	#6894	C₃×C₃₇	1	6	yes	complemented
106	#6893	C₃×C₃₇	1	1	yes	characteristic
107	C₉⋊H(3,𝔽₃)	C₃×C₃₇	1	2	yes	—
108	C₉×3₋¹⁺²	C₃×C₃₇	1	1	yes	characteristic, complemented
109	C₉×3₋¹⁺²	C₃×C₃₇	1	1	yes	characteristic, complemented
110	C₉×H(3,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic
111	C₃×C₉⋊C₉	C₃×C₃₇	1	1	yes	characteristic, complemented
112	C₃×C₃₇	—	3	3	no	—
113	C₃×C₃₇	—	9	3	no	—
114	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
115	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
116	C₃×C₃₇	3₋¹⁺⁴	1	2	yes	—
117	C₃×3₋¹⁺²	C₃²×C₃₇	1	1	yes	characteristic
118	C₃×3₋¹⁺²	C₃²×C₃₇	1	1	yes	characteristic
119	C₃×3₋¹⁺²	C₃²×C₃₇	1	6	yes	—
120	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	1	yes	characteristic
121	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
122	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
123	C₃²×C₉	C₃²×C₃₇	1	2	yes	—
124	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
125	C₉⋊C₉	C₃²×C₃₇	1	2	yes	—
126	C₉⋊C₉	C₃²×C₃₇	1	1	yes	characteristic
127	C₉⋊C₉	C₃²×C₃₇	1	1	yes	characteristic
128	C₉⋊C₉	C₃²×C₃₇	1	2	yes	—
129	C₉⋊C₉	C₃²×C₃₇	1	1	yes	characteristic
130	C₉⋊C₉	C₃²×C₃₇	1	2	yes	—
131	C₉⋊C₉	C₃²×C₃₇	1	3	yes	—
132	C₉⋊C₉	C₃²×C₃₇	1	3	yes	—
133	C₉⋊C₉	C₃²×C₃₇	1	3	yes	—
134	C₉⋊C₉	C₃²×C₃₇	1	3	yes	—
135	C₉⋊C₉	C₃²×C₃₇	1	6	yes	—
136	C₉⋊C₉	C₃²×C₃₇	1	6	yes	—
137	C₉⋊C₉	C₃²×C₃₇	1	6	yes	—
138	C₉⋊C₉	C₃²×C₃₇	1	6	yes	—
139	C₉⋊C₉	C₃²×C₃₇	1	6	yes	—
140	C₉⋊C₉	C₃²×C₃₇	1	6	yes	—
141	C₉⋊C₉	C₃²×C₃₇	1	12	yes	—
142	C₉⋊C₉	C₃²×C₃₇	1	6	yes	—
143	C₃²⋊C₉	C₃²×C₃₇	1	2	yes	—
144	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
145	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
146	C₃²⋊C₉	C₃²×C₃₇	1	2	yes	—
147	C₃²⋊C₉	C₃²×C₃₇	1	2	yes	—
148	C₃²⋊C₉	C₃²×C₃₇	1	6	yes	—
149	C₃²⋊C₉	C₃²×C₃₇	1	6	yes	—
150	C₃²⋊C₉	C₃²×C₃₇	1	6	yes	—
151	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
152	C₉²	C₃²×C₃₇	1	3	yes	—
153	C₉²	C₃²×C₃₇	1	3	yes	—
154	C₃₇	#10624891	1	1	yes	characteristic, direct product factor, Sylow
155	C₃³	C₃³×C₃₇	1	3	yes	—
156	C₃³	C₃³×C₃₇	1	1	yes	characteristic
157	3₋¹⁺²	—	3	3	no	—
158	3₋¹⁺²	—	3	3	no	—
159	3₋¹⁺²	—	9	6	no	—
160	H(3,𝔽₃)	—	3	3	no	—
161	C₃×C₉	—	3	3	no	—
162	C₃×C₉	—	9	1	no	—
163	C₃×C₉	—	9	3	no	—
164	C₃×C₉	—	9	2	no	—
165	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
166	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
167	C₃×C₉	C₃³×C₃₇	1	3	yes	—
168	C₃×C₉	C₃³×C₃₇	1	3	yes	—
169	C₃×C₉	C₃³×C₃₇	1	6	yes	—
170	C₃×C₉	C₃³×C₃₇	1	6	yes	—
171	C₃×C₉	C₃³×C₃₇	1	2	yes	—
172	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
173	C₃×C₉	C₃³×C₃₇	1	6	yes	—
174	C₃×C₉	C₃³×C₃₇	1	2	yes	—
175	C₃×C₉	C₃³×C₃₇	1	2	yes	—
176	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
177	C₃×C₉	C₃³×C₃₇	1	2	yes	—
178	C₃²	—	3	1	no	—
179	C₃²	—	3	2	no	—
180	C₃²	—	3	3	no	—
181	C₃²	—	3	3	no	—
182	C₃²	—	3	6	no	—
183	C₃²	C₃₇×#737	1	3	yes	—
184	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
185	C₉	—	3	2	no	—
186	C₉	—	3	1	no	—
187	C₉	—	3	3	no	—
188	C₉	—	3	3	no	—
189	C₉	—	3	1	no	—
190	C₉	—	3	2	no	—
191	C₉	—	3	1	no	—
192	C₉	—	3	1	no	—
193	C₉	—	3	6	no	—
194	C₉	—	3	2	no	—
195	C₉	—	3	2	no	—
196	C₉	—	3	6	no	—
197	C₉	—	3	6	no	—
198	C₃	—	3	3	no	—
199	C₃	—	9	3	no	—
200	C₃	C₃×C₃₇×#737	1	1	yes	characteristic
201	C₃	C₃₇×3₋¹⁺⁴	1	1	yes	characteristic
202	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
1. C₃₇×#6899	C₃	semidirect
2. C₃₇×#6898	C₃	semidirect
3. C₃₇×#6897	C₃	semidirect
4. C₃₇×#6896	C₃	semidirect
5. C₃₇×#6895	C₃	semidirect
6. C₃₇×#6895	C₃	semidirect
7. C₃₇×#6894	C₃	semidirect
8. C₃₇×#6894	C₃	semidirect
9. C₃₇×#6894	C₃	semidirect
12. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
13. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
15. C₃×C₃₇×C₉⋊C₉	C₃	semidirect
76. #10624891	C₃₇	direct (N × H)
97. #6899	C₃×C₃₇	semidirect
98. #6898	C₃×C₃₇	semidirect
99. #6897	C₃×C₃₇	semidirect
100. #6896	C₃×C₃₇	semidirect
101. #6895	C₃×C₃₇	semidirect
102. #6895	C₃×C₃₇	semidirect
103. #6894	C₃×C₃₇	semidirect
104. #6894	C₃×C₃₇	semidirect
105. #6894	C₃×C₃₇	semidirect
108. C₉×3₋¹⁺²	C₃×C₃₇	semidirect
109. C₉×3₋¹⁺²	C₃×C₃₇	semidirect
111. C₃×C₉⋊C₉	C₃×C₃₇	semidirect
154. C₃₇	#10624891	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×#6899	3
C₃₇×#6898	3
C₃₇×#6897	3
C₃₇×#6896	3
C₃₇×#6895	3
C₃₇×#6894	3
C₃₇×#6893	3
C₃₇×C₉⋊H(3,𝔽₃)	3
C₉×3₋¹⁺²×C₃₇	3
C₉×H(3,𝔽₃)×C₃₇	3
C₃×C₃₇×C₉⋊C₉	3
#10624891	37

Subgroups — C37×#10624891 #41004680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624891 #41004680