Subgroups — #41007850

Order 26973.

740 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#11758605	C₃	1	1	yes	maximal, characteristic
2	#22582597	C₃	1	6	yes	maximal, complemented
3	#22582596	C₃	1	3	yes	maximal, complemented
4	#22582582	C₃	1	6	yes	maximal, complemented
5	#22582581	C₃	1	3	yes	maximal, complemented
6	#22582573	C₃	1	2	yes	maximal, complemented
7	#22582571	C₃	1	2	yes	maximal, complemented
8	#22582570	C₃	1	6	yes	maximal, complemented
9	#22582568	C₃	1	2	yes	maximal, complemented
10	#22582552	C₃	1	1	yes	maximal, characteristic
11	#22582551	C₃	1	1	yes	maximal, characteristic
12	#22582548	C₃	1	2	yes	maximal, complemented
13	C₉×#11758605	C₃	1	3	yes	maximal, complemented
14	C₃²⋊C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
15	C₉×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
16	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
17	C₃×#11758608	C₃²	1	2	yes	—
18	C₃×#11758605	—	3	9	no	—
19	C₃×#11758605	C₃²	1	2	yes	—
20	C₃×#11758605	C₃²	1	1	yes	characteristic
21	C₃×#11758605	C₃²	1	2	yes	—
22	C₃×#11758605	C₃²	1	1	yes	characteristic
23	C₃×#11758605	C₃²	1	2	yes	—
24	C₃×#11758605	C₃²	1	1	yes	characteristic
25	C₃×#11758606	C₃²	1	3	yes	—
26	C₃²×#66524	C₃²	1	2	yes	—
27	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
28	#15688506	C₃²	1	6	yes	complemented
29	#15688505	C₃²	1	6	yes	complemented
30	#15688504	C₃²	1	6	yes	complemented
31	#15688504	C₃²	1	6	yes	complemented
32	#15688504	C₃²	1	6	yes	complemented
33	#15688504	C₃²	1	6	yes	complemented
34	#15688504	C₃²	1	6	yes	complemented
35	#15688504	C₃²	1	6	yes	complemented
36	#15688504	C₃²	1	12	yes	complemented
37	#15688504	C₃²	1	6	yes	complemented
38	C₃²⋊#66524	C₃²	1	2	yes	—
39	C₃²⋊#66524	C₃²	1	2	yes	—
40	#15688502	C₃²	1	3	yes	—
41	#15688502	C₃²	1	3	yes	—
42	#15688502	C₃²	1	3	yes	—
43	#15688502	C₃²	1	3	yes	—
44	#15688502	C₃²	1	3	yes	—
45	#15688502	C₃²	1	3	yes	—
46	#15688502	C₃²	1	3	yes	—
47	C₉×#66524	C₃²	1	6	yes	complemented
48	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
49	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
50	C₃₇×C₉⋊C₉	C₃²	1	6	yes	complemented
51	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
52	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
53	C₉²×C₃₇	C₃²	1	3	yes	complemented
54	C₃³×C₃₇	C₃³	1	1	yes	characteristic
55	C₃²×C₃₇⋊C₃	—	3	3	no	—
56	C₃²×C₃₇⋊C₃	—	3	3	no	—
57	C₃²×C₃₇⋊C₃	—	9	1	no	—
58	C₃²×C₃₇⋊C₃	—	9	1	no	—
59	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
60	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
61	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
62	3₋¹⁺²×C₃₇	—	3	3	no	—
63	C₃×C₉×C₃₇	—	3	3	no	—
64	C₃×C₉×C₃₇	C₃³	1	3	yes	—
65	C₃×C₉×C₃₇	C₃³	1	3	yes	—
66	C₃×C₉×C₃₇	C₃³	1	3	yes	—
67	C₃×C₉×C₃₇	C₃³	1	2	yes	—
68	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
69	#11758608	—	3	6	no	—
70	C₃×#66524	—	3	6	no	—
71	C₃×#66524	C₃³	1	6	yes	—
72	C₃×#66524	C₃³	1	6	yes	—
73	C₃×#66524	C₃³	1	6	yes	—
74	C₃×#66524	C₃³	1	2	yes	—
75	C₃×#66524	C₃³	1	2	yes	—
76	C₃×#66524	C₃³	1	2	yes	—
77	#11758606	—	9	3	no	—
78	#11758605	—	3	6	no	—
79	#11758605	—	3	3	no	—
80	#11758605	—	3	3	no	—
81	#11758605	—	3	6	no	—
82	#11758605	—	3	6	no	—
83	#11758605	3₋¹⁺²	1	9	yes	complemented
84	C₉×C₃₇⋊C₃	—	9	3	no	—
85	#10624899	—	37	1	no	maximal, Sylow
86	C₃²×C₃₇	—	3	1	no	—
87	C₃²×C₃₇	—	3	1	no	—
88	C₃²×C₃₇	—	3	1	no	—
89	C₃²×C₃₇	#737	1	3	yes	—
90	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
91	C₃×C₃₇⋊C₃	—	3	1	no	—
92	C₃×C₃₇⋊C₃	—	3	1	no	—
93	C₃×C₃₇⋊C₃	—	3	1	no	—
94	C₃×C₃₇⋊C₃	—	3	1	no	—
95	C₃×C₃₇⋊C₃	—	3	1	no	—
96	C₃×C₃₇⋊C₃	—	3	1	no	—
97	C₃×C₃₇⋊C₃	—	3	1	no	—
98	C₃×C₃₇⋊C₃	—	3	1	no	—
99	C₃×C₃₇⋊C₃	—	3	1	no	—
100	C₃×C₃₇⋊C₃	—	3	1	no	—
101	C₃×C₃₇⋊C₃	—	3	1	no	—
102	C₃×C₃₇⋊C₃	—	9	9	no	—
103	C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
104	C₉×C₃₇	—	3	2	no	—
105	C₉×C₃₇	—	3	3	no	—
106	C₉×C₃₇	—	3	3	no	—
107	C₉×C₃₇	—	3	3	no	—
108	C₉×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
109	#66524	—	3	2	no	—
110	#66524	—	3	2	no	—
111	#66524	—	3	6	no	—
112	#66524	—	3	6	no	—
113	#66524	—	3	6	no	—
114	#66524	C₃×3₋¹⁺²	1	6	yes	—
115	C₃²×3₋¹⁺²	—	37	1	no	—
116	#6899	—	37	6	no	—
117	#6899	—	37	3	no	—
118	#6896	—	37	6	no	—
119	#6896	—	37	3	no	—
120	#6894	—	37	2	no	—
121	#6894	—	37	2	no	—
122	#6894	—	37	6	no	—
123	#6894	—	37	2	no	—
124	#6891	—	37	1	no	—
125	#6891	—	37	1	no	—
126	C₉×3₋¹⁺²	—	37	2	no	—
127	C₉×3₋¹⁺²	—	37	3	no	—
128	C₉×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic, complemented
129	C₃×C₃²⋊C₉	—	37	1	no	—
130	C₃×C₃₇	—	3	3	no	—
131	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
132	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
133	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
134	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
135	C₃₇⋊C₃	—	3	3	no	—
136	C₃₇⋊C₃	—	9	1	no	—
137	C₃₇⋊C₃	—	9	1	no	—
138	C₃⁴	—	37	1	no	—
139	C₃×3₋¹⁺²	—	37	2	no	—
140	C₃×3₋¹⁺²	—	37	1	no	—
141	C₃×3₋¹⁺²	—	37	3	no	—
142	C₃×3₋¹⁺²	—	37	2	no	—
143	C₃×3₋¹⁺²	—	37	2	no	—
144	C₃×3₋¹⁺²	—	37	1	no	—
145	C₃×3₋¹⁺²	—	37	2	no	—
146	C₃×3₋¹⁺²	—	37	1	no	—
147	C₃×3₋¹⁺²	—	111	9	no	—
148	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
149	C₃²×C₉	—	37	2	no	—
150	C₃²×C₉	—	37	3	no	—
151	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
152	C₉⋊C₉	—	37	6	no	—
153	C₉⋊C₉	—	37	6	no	—
154	C₉⋊C₉	—	37	6	no	—
155	C₉⋊C₉	—	37	6	no	—
156	C₉⋊C₉	—	37	12	no	—
157	C₉⋊C₉	—	37	6	no	—
158	C₉⋊C₉	—	37	6	no	—
159	C₉⋊C₉	—	37	6	no	—
160	C₉⋊C₉	—	37	6	no	—
161	C₉⋊C₉	—	37	6	no	—
162	C₉⋊C₉	C₃×C₃₇⋊C₃	1	6	yes	complemented
163	C₃²⋊C₉	—	37	2	no	—
164	C₃²⋊C₉	—	37	2	no	—
165	C₃²⋊C₉	—	37	3	no	—
166	C₃²⋊C₉	—	37	3	no	—
167	C₃²⋊C₉	—	37	3	no	—
168	C₃²⋊C₉	—	37	3	no	—
169	C₃²⋊C₉	—	37	3	no	—
170	C₃²⋊C₉	—	37	3	no	—
171	C₃²⋊C₉	—	37	3	no	—
172	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
173	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
174	C₉²	—	37	6	no	—
175	C₉²	C₃×C₃₇⋊C₃	1	3	yes	complemented
176	C₃₇	#10624899	1	1	yes	characteristic, complemented, Sylow
177	C₃³	—	37	1	no	—
178	C₃³	—	37	1	no	—
179	C₃³	—	37	1	no	—
180	C₃³	—	333	1	no	—
181	C₃³	—	333	1	no	—
182	C₃³	—	111	3	no	—
183	C₃³	—	111	3	no	—
184	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
185	3₋¹⁺²	—	3	3	no	—
186	3₋¹⁺²	—	37	9	no	—
187	3₋¹⁺²	—	333	3	no	—
188	3₋¹⁺²	—	111	6	no	—
189	3₋¹⁺²	—	111	6	no	—
190	3₋¹⁺²	—	111	6	no	—
191	3₋¹⁺²	—	111	6	no	—
192	3₋¹⁺²	—	111	3	no	—
193	3₋¹⁺²	—	111	3	no	—
194	C₃×C₉	—	3	3	no	—
195	C₃×C₉	—	37	2	no	—
196	C₃×C₉	—	37	6	no	—
197	C₃×C₉	—	37	6	no	—
198	C₃×C₉	—	37	6	no	—
199	C₃×C₉	—	37	2	no	—
200	C₃×C₉	—	37	2	no	—
201	C₃×C₉	—	333	3	no	—
202	C₃×C₉	—	111	6	no	—
203	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
204	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
205	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
206	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
207	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
208	C₃²	—	3	1	no	—
209	C₃²	—	3	1	no	—
210	C₃²	—	3	1	no	—
211	C₃²	—	37	3	no	—
212	C₃²	—	333	9	no	—
213	C₃²	—	111	1	no	—
214	C₃²	—	111	1	no	—
215	C₃²	—	111	1	no	—
216	C₃²	—	111	1	no	—
217	C₃²	—	111	1	no	—
218	C₃²	—	111	1	no	—
219	C₃²	—	111	1	no	—
220	C₃²	—	111	1	no	—
221	C₃²	—	111	1	no	—
222	C₃²	—	111	1	no	—
223	C₃²	—	111	1	no	—
224	C₃²	#15688531	1	3	yes	—
225	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
226	C₉	—	3	2	no	—
227	C₉	—	3	3	no	—
228	C₉	—	3	3	no	—
229	C₉	—	3	3	no	—
230	C₉	—	37	6	no	—
231	C₉	—	111	2	no	—
232	C₉	—	111	2	no	—
233	C₉	—	111	6	no	—
234	C₉	—	111	6	no	—
235	C₉	—	111	6	no	—
236	C₉	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
237	C₃	—	3	3	no	—
238	C₃	—	333	1	no	—
239	C₃	—	333	1	no	—
240	C₃	—	111	3	no	—
241	C₃	C₃×3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
242	C₃	C₃×#15688531	1	1	yes	characteristic
243	C₃	#22582678	1	1	yes	characteristic
244	C₃	#22582678	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₉×3₋¹⁺²×C₃₇	C₃
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	C₉×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624899
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₉×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. #22582597	C₃	semidirect
3. #22582596	C₃	semidirect
4. #22582582	C₃	semidirect
5. #22582581	C₃	semidirect
6. #22582573	C₃	semidirect
7. #22582571	C₃	semidirect
8. #22582570	C₃	semidirect
9. #22582568	C₃	semidirect
12. #22582548	C₃	semidirect
13. C₉×#11758605	C₃	semidirect
15. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
28. #15688506	C₃²	semidirect
29. #15688505	C₃²	semidirect
30. #15688504	C₃²	semidirect
31. #15688504	C₃²	semidirect
32. #15688504	C₃²	semidirect
33. #15688504	C₃²	semidirect
34. #15688504	C₃²	semidirect
35. #15688504	C₃²	semidirect
36. #15688504	C₃²	semidirect
37. #15688504	C₃²	semidirect
47. C₉×#66524	C₃²	semidirect
50. C₃₇×C₉⋊C₉	C₃²	semidirect
53. C₉²×C₃₇	C₃²	semidirect
83. #11758605	3₋¹⁺²	semidirect
128. C₉×3₋¹⁺²	C₃₇⋊C₃	semidirect
162. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
175. C₉²	C₃×C₃₇⋊C₃	semidirect
176. C₃₇	#10624899	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃²×#11758605	3
#22582597	3
#22582596	3
#22582582	3
#22582581	3
#22582573	3
#22582571	3
#22582570	3
#22582568	3
#22582552	3
#22582551	3
#22582548	3
C₉×#11758605	3
C₃²⋊C₉×C₃₇⋊C₃	3
C₉×3₋¹⁺²×C₃₇	3
#10624899	37