Subgroups — C₃₇×#10624901 #41004690

Order 26973.

718 conjugacy classes of subgroups.

The subgroup lattice has 210 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	6	yes	maximal, complemented
5	C₃₇×#6896	C₃	1	3	yes	maximal, complemented
6	C₃₇×#6895	C₃	1	2	yes	maximal, complemented
7	C₃₇×#6894	C₃	1	2	yes	maximal, complemented
8	C₃₇×#6894	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃₇×#6894	C₃	1	6	yes	maximal, complemented
10	C₃₇×C₉⋊H(3,𝔽₃)	C₃	1	2	yes	maximal
11	C₃₇×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
12	C₉×3₋¹⁺²×C₃₇	C₃	1	2	yes	maximal, complemented
13	C₉×3₋¹⁺²×C₃₇	C₃	1	3	yes	maximal, complemented
14	C₉×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
15	C₃×C₃₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
16	C₃₇×#737	—	3	9	no	—
17	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
18	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
19	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
20	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
21	C₃×3₋¹⁺²×C₃₇	C₃²	1	6	yes	—
22	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
23	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
24	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
25	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
26	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
27	C₃²×C₉×C₃₇	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	2	yes	—
31	C₃₇×C₉⋊C₉	C₃²	1	1	yes	characteristic
32	C₃₇×C₉⋊C₉	C₃²	1	2	yes	—
33	C₃₇×C₉⋊C₉	C₃²	1	1	yes	characteristic
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	3	yes	—
40	C₃₇×C₉⋊C₉	C₃²	1	6	yes	—
41	C₃₇×C₉⋊C₉	C₃²	1	6	yes	—
42	C₃₇×C₉⋊C₉	C₃²	1	6	yes	—
43	C₃₇×C₉⋊C₉	C₃²	1	6	yes	—
44	C₃₇×C₉⋊C₉	C₃²	1	6	yes	—
45	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
46	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
47	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
48	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
49	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
50	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
51	C₉²×C₃₇	C₃²	1	6	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	9	no	—
57	3₋¹⁺²×C₃₇	—	3	9	no	—
58	3₋¹⁺²×C₃₇	—	3	9	no	—
59	3₋¹⁺²×C₃₇	—	9	6	no	—
60	3₋¹⁺²×C₃₇	3₋¹⁺²	1	9	yes	complemented
61	H(3,𝔽₃)×C₃₇	—	3	3	no	—
62	C₃×C₉×C₃₇	—	3	3	no	—
63	C₃×C₉×C₃₇	—	3	9	no	—
64	C₃×C₉×C₃₇	—	3	3	no	—
65	C₃×C₉×C₃₇	—	9	2	no	—
66	C₃×C₉×C₃₇	C₃³	1	2	yes	—
67	C₃×C₉×C₃₇	C₃³	1	6	yes	—
68	C₃×C₉×C₃₇	C₃³	1	6	yes	—
69	C₃×C₉×C₃₇	C₃³	1	6	yes	—
70	C₃×C₉×C₃₇	C₃³	1	2	yes	—
71	C₃×C₉×C₃₇	C₃³	1	2	yes	—
72	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
73	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
74	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
75	C₃×C₉×C₃₇	C₃³	1	3	yes	—
76	C₃×C₉×C₃₇	C₃³	1	2	yes	—
77	C₃×C₉×C₃₇	C₃³	1	3	yes	—
78	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
79	#10624901	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₃²×C₃₇	—	3	1	no	—
81	C₃²×C₃₇	—	3	2	no	—
82	C₃²×C₃₇	—	3	3	no	—
83	C₃²×C₃₇	—	3	3	no	—
84	C₃²×C₃₇	—	3	6	no	—
85	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
86	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
87	C₉×C₃₇	—	3	1	no	—
88	C₉×C₃₇	—	3	1	no	—
89	C₉×C₃₇	—	3	1	no	—
90	C₉×C₃₇	—	3	3	no	—
91	C₉×C₃₇	—	3	2	no	—
92	C₉×C₃₇	—	3	6	no	—
93	C₉×C₃₇	—	3	2	no	—
94	C₉×C₃₇	—	3	2	no	—
95	C₉×C₃₇	—	3	2	no	—
96	C₉×C₃₇	—	3	6	no	—
97	C₉×C₃₇	—	3	6	no	—
98	C₉×C₃₇	C₃×3₋¹⁺²	1	9	yes	—
99	C₉×C₃₇	#737	1	3	yes	complemented
100	C₃×#737	C₃×C₃₇	1	1	yes	characteristic
101	#6900	C₃×C₃₇	1	3	yes	complemented
102	#6899	C₃×C₃₇	1	6	yes	complemented
103	#6896	C₃×C₃₇	1	6	yes	complemented
104	#6896	C₃×C₃₇	1	3	yes	complemented
105	#6895	C₃×C₃₇	1	2	yes	complemented
106	#6894	C₃×C₃₇	1	2	yes	complemented
107	#6894	C₃×C₃₇	1	1	yes	characteristic, complemented
108	#6894	C₃×C₃₇	1	6	yes	complemented
109	C₉⋊H(3,𝔽₃)	C₃×C₃₇	1	2	yes	—
110	C₉⋊H(3,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic
111	C₉×3₋¹⁺²	C₃×C₃₇	1	2	yes	complemented
112	C₉×3₋¹⁺²	C₃×C₃₇	1	3	yes	complemented
113	C₉×3₋¹⁺²	C₃×C₃₇	1	1	yes	characteristic, complemented
114	C₃×C₉⋊C₉	C₃×C₃₇	1	1	yes	characteristic, complemented
115	C₃×C₃₇	—	3	3	no	—
116	C₃×C₃₇	—	9	3	no	—
117	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
118	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
119	C₃×C₃₇	3₋¹⁺⁴	1	2	yes	—
120	#737	—	3	9	no	—
121	C₃×3₋¹⁺²	C₃²×C₃₇	1	1	yes	characteristic
122	C₃×3₋¹⁺²	C₃²×C₃₇	1	3	yes	—
123	C₃×3₋¹⁺²	C₃²×C₃₇	1	3	yes	—
124	C₃×3₋¹⁺²	C₃²×C₃₇	1	3	yes	—
125	C₃×3₋¹⁺²	C₃²×C₃₇	1	6	yes	—
126	C₃×3₋¹⁺²	C₃²×C₃₇	1	1	yes	characteristic
127	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	1	yes	characteristic
128	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
129	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
130	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
131	C₃²×C₉	C₃²×C₃₇	1	2	yes	—
132	C₉⋊C₉	C₃²×C₃₇	1	1	yes	characteristic
133	C₉⋊C₉	C₃²×C₃₇	1	2	yes	—
134	C₉⋊C₉	C₃²×C₃₇	1	2	yes	—
135	C₉⋊C₉	C₃²×C₃₇	1	1	yes	characteristic
136	C₉⋊C₉	C₃²×C₃₇	1	2	yes	—
137	C₉⋊C₉	C₃²×C₃₇	1	1	yes	characteristic
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	6	yes	—
142	C₉⋊C₉	C₃²×C₃₇	1	6	yes	—
143	C₉⋊C₉	C₃²×C₃₇	1	3	yes	—
144	C₉⋊C₉	C₃²×C₃₇	1	6	yes	—
145	C₉⋊C₉	C₃²×C₃₇	1	6	yes	—
146	C₉⋊C₉	C₃²×C₃₇	1	6	yes	—
147	C₉⋊C₉	C₃²×C₃₇	1	6	yes	—
148	C₉⋊C₉	C₃²×C₃₇	1	6	yes	—
149	C₃²⋊C₉	C₃²×C₃₇	1	2	yes	—
150	C₃²⋊C₉	C₃²×C₃₇	1	2	yes	—
151	C₃²⋊C₉	C₃²×C₃₇	1	2	yes	—
152	C₃²⋊C₉	C₃²×C₃₇	1	6	yes	—
153	C₃²⋊C₉	C₃²×C₃₇	1	6	yes	—
154	C₃²⋊C₉	C₃²×C₃₇	1	6	yes	—
155	C₉²	C₃²×C₃₇	1	6	yes	—
156	C₉²	C₃²×C₃₇	1	3	yes	—
157	C₃₇	#10624901	1	1	yes	characteristic, direct product factor, Sylow
158	C₃³	C₃³×C₃₇	1	3	yes	—
159	C₃³	C₃³×C₃₇	1	1	yes	characteristic
160	3₋¹⁺²	—	3	3	no	—
161	3₋¹⁺²	—	3	9	no	—
162	3₋¹⁺²	—	3	9	no	—
163	3₋¹⁺²	—	3	9	no	—
164	3₋¹⁺²	—	9	6	no	—
165	3₋¹⁺²	3₋¹⁺²×C₃₇	1	9	yes	complemented
166	H(3,𝔽₃)	—	3	3	no	—
167	C₃×C₉	—	3	3	no	—
168	C₃×C₉	—	3	9	no	—
169	C₃×C₉	—	3	3	no	—
170	C₃×C₉	—	9	2	no	—
171	C₃×C₉	C₃³×C₃₇	1	2	yes	—
172	C₃×C₉	C₃³×C₃₇	1	6	yes	—
173	C₃×C₉	C₃³×C₃₇	1	6	yes	—
174	C₃×C₉	C₃³×C₃₇	1	6	yes	—
175	C₃×C₉	C₃³×C₃₇	1	2	yes	—
176	C₃×C₉	C₃³×C₃₇	1	2	yes	—
177	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
178	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
179	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
180	C₃×C₉	C₃³×C₃₇	1	3	yes	—
181	C₃×C₉	C₃³×C₃₇	1	2	yes	—
182	C₃×C₉	C₃³×C₃₇	1	3	yes	—
183	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
184	C₃²	—	3	1	no	—
185	C₃²	—	3	2	no	—
186	C₃²	—	3	3	no	—
187	C₃²	—	3	3	no	—
188	C₃²	—	3	6	no	—
189	C₃²	C₃×3₋¹⁺²×C₃₇	1	3	yes	—
190	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
191	C₉	—	3	1	no	—
192	C₉	—	3	1	no	—
193	C₉	—	3	1	no	—
194	C₉	—	3	3	no	—
195	C₉	—	3	2	no	—
196	C₉	—	3	6	no	—
197	C₉	—	3	2	no	—
198	C₉	—	3	2	no	—
199	C₉	—	3	2	no	—
200	C₉	—	3	6	no	—
201	C₉	—	3	6	no	—
202	C₉	C₃×3₋¹⁺²×C₃₇	1	9	yes	—
203	C₉	C₃₇×#737	1	3	yes	complemented
204	C₃	—	3	3	no	—
205	C₃	—	9	3	no	—
206	C₃	C₃²×3₋¹⁺²×C₃₇	1	1	yes	characteristic
207	C₃	C₃×C₃₇×#737	1	1	yes	characteristic
208	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×#6900	C₃	semidirect
3. C₃₇×#6899	C₃	semidirect
4. C₃₇×#6896	C₃	semidirect
5. C₃₇×#6896	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
14. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
15. C₃×C₃₇×C₉⋊C₉	C₃	semidirect
60. 3₋¹⁺²×C₃₇	3₋¹⁺²	semidirect
79. #10624901	C₃₇	direct (N × H)
99. C₉×C₃₇	#737	semidirect
101. #6900	C₃×C₃₇	semidirect
102. #6899	C₃×C₃₇	semidirect
103. #6896	C₃×C₃₇	semidirect
104. #6896	C₃×C₃₇	semidirect
105. #6895	C₃×C₃₇	semidirect
106. #6894	C₃×C₃₇	semidirect
107. #6894	C₃×C₃₇	semidirect
108. #6894	C₃×C₃₇	semidirect
111. C₉×3₋¹⁺²	C₃×C₃₇	semidirect
112. C₉×3₋¹⁺²	C₃×C₃₇	semidirect
113. C₉×3₋¹⁺²	C₃×C₃₇	semidirect
114. C₃×C₉⋊C₉	C₃×C₃₇	semidirect
157. C₃₇	#10624901	direct (N × H)
165. 3₋¹⁺²	3₋¹⁺²×C₃₇	semidirect
203. 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×#6900	3
C₃₇×#6899	3
C₃₇×#6896	3
C₃₇×#6895	3
C₃₇×#6894	3
C₃₇×C₉⋊H(3,𝔽₃)	3
C₉×3₋¹⁺²×C₃₇	3
C₃×C₃₇×C₉⋊C₉	3
#10624901	37

Subgroups — C37×#10624901 #41004690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624901 #41004690