Subgroups — #41007810

Order 26973.

1082 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688532	C₃	1	2	yes	maximal, complemented
2	C₃×3₋¹⁺²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582565	C₃	1	2	yes	maximal, complemented
4	#22582564	C₃	1	2	yes	maximal, complemented
5	#22582562	C₃	1	2	yes	maximal, complemented
6	#22582559	C₃	1	3	yes	maximal, complemented
7	#22582558	C₃	1	3	yes	maximal, complemented
8	#22582555	C₃	1	6	yes	maximal, complemented
9	#22582555	C₃	1	6	yes	maximal, complemented
10	#22582555	C₃	1	6	yes	maximal, complemented
11	#22582551	C₃	1	1	yes	maximal, characteristic, complemented
12	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
13	C₉×C₃²⋊(C₃₇⋊C₃)	C₃	1	3	yes	maximal, complemented
14	C₃×#15688502	C₃	1	1	yes	maximal, characteristic, complemented
15	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic, complemented
16	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
17	#15688532	—	3	18	no	—
18	C₃×#11758608	C₃²	1	2	yes	—
19	C₃×#11758608	C₃²	1	2	yes	—
20	C₃×#11758605	C₃²	1	3	yes	—
21	C₃×#11758605	C₃²	1	6	yes	complemented
22	C₃×#11758606	C₃²	1	3	yes	—
23	C₃×#11758606	C₃²	1	6	yes	complemented
24	3₋¹⁺²×C₃₇⋊C₃	—	3	9	no	—
25	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
26	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	2	yes	—
27	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
28	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
29	C₃²×#66524	C₃²	1	2	yes	—
30	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
31	C₃×C₉×C₃₇⋊C₃	C₃²	1	6	yes	complemented
32	C₃²⋊#66524	C₃²	1	6	yes	complemented
33	C₃²⋊#66524	C₃²	1	6	yes	complemented
34	C₃²⋊#66524	C₃²	1	6	yes	complemented
35	#15688502	C₃²	1	1	yes	characteristic
36	#15688502	C₃²	1	2	yes	—
37	#15688502	C₃²	1	2	yes	—
38	#15688502	C₃²	1	2	yes	—
39	#15688502	C₃²	1	1	yes	characteristic
40	#15688502	C₃²	1	1	yes	characteristic
41	#15688502	C₃²	1	3	yes	—
42	#15688502	C₃²	1	3	yes	—
43	#15688502	C₃²	1	3	yes	—
44	#15688502	C₃²	1	6	yes	complemented
45	#15688502	C₃²	1	6	yes	complemented
46	#15688502	C₃²	1	12	yes	complemented
47	#15688502	C₃²	1	6	yes	complemented
48	#15688502	C₃²	1	6	yes	complemented
49	C₃₇×#737	—	3	9	no	—
50	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
51	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
52	C₃×3₋¹⁺²×C₃₇	C₃²	1	6	yes	complemented
53	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
54	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
55	C₃²×C₉×C₃₇	C₃²	1	3	yes	complemented
56	C₃³×C₃₇	C₃³	1	3	yes	—
57	C₃³×C₃₇	C₃³	1	1	yes	characteristic
58	C₃²×C₃₇⋊C₃	—	3	3	no	—
59	C₃²×C₃₇⋊C₃	—	3	3	no	—
60	C₃²×C₃₇⋊C₃	—	9	1	no	—
61	C₃²×C₃₇⋊C₃	—	9	1	no	—
62	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
63	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
64	C₃²×C₃₇⋊C₃	C₃³	1	6	yes	—
65	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
66	3₋¹⁺²×C₃₇	—	3	3	no	—
67	3₋¹⁺²×C₃₇	—	3	18	no	—
68	3₋¹⁺²×C₃₇	H(3,𝔽₃)	1	9	yes	complemented
69	H(3,𝔽₃)×C₃₇	—	3	3	no	—
70	C₃×C₉×C₃₇	—	3	3	no	—
71	C₃×C₉×C₃₇	—	3	9	no	—
72	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
73	C₃×C₉×C₃₇	C₃³	1	2	yes	—
74	C₃×C₉×C₃₇	C₃³	1	3	yes	—
75	C₃×C₉×C₃₇	C₃³	1	3	yes	—
76	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
77	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
78	C₃²⋊(C₃₇⋊C₃)	—	9	3	no	—
79	C₃²⋊(C₃₇⋊C₃)	—	9	3	no	—
80	#11758608	—	3	6	no	—
81	#11758608	—	3	6	no	—
82	C₃×#66524	—	3	6	no	—
83	C₃×#66524	C₃³	1	2	yes	—
84	C₃×#66524	C₃³	1	6	yes	—
85	C₃×#66524	C₃³	1	2	yes	—
86	C₃×#66524	C₃³	1	2	yes	—
87	C₃×#66524	C₃³	1	6	yes	—
88	#11758606	—	3	9	no	—
89	#11758606	—	3	18	no	—
90	#11758605	—	3	9	no	—
91	#11758605	—	3	18	no	—
92	C₉×C₃₇⋊C₃	—	3	9	no	—
93	C₉×C₃₇⋊C₃	—	3	18	no	—
94	#10624896	—	37	1	no	maximal, Sylow
95	C₃²×C₃₇	—	3	1	no	—
96	C₃²×C₃₇	—	3	1	no	—
97	C₃²×C₃₇	—	3	1	no	—
98	C₃²×C₃₇	—	3	3	no	—
99	C₃²×C₃₇	—	3	3	no	—
100	C₃²×C₃₇	—	3	3	no	—
101	C₃²×C₃₇	—	3	3	no	—
102	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
103	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
104	C₃×C₃₇⋊C₃	—	3	1	no	—
105	C₃×C₃₇⋊C₃	—	3	1	no	—
106	C₃×C₃₇⋊C₃	—	3	1	no	—
107	C₃×C₃₇⋊C₃	—	3	6	no	—
108	C₃×C₃₇⋊C₃	—	3	1	no	—
109	C₃×C₃₇⋊C₃	—	3	1	no	—
110	C₃×C₃₇⋊C₃	—	3	6	no	—
111	C₃×C₃₇⋊C₃	—	3	1	no	—
112	C₃×C₃₇⋊C₃	—	3	1	no	—
113	C₃×C₃₇⋊C₃	—	3	1	no	—
114	C₃×C₃₇⋊C₃	—	3	1	no	—
115	C₃×C₃₇⋊C₃	—	3	6	no	—
116	C₃×C₃₇⋊C₃	—	3	6	no	—
117	C₃×C₃₇⋊C₃	—	3	1	no	—
118	C₃×C₃₇⋊C₃	—	3	1	no	—
119	C₃×C₃₇⋊C₃	—	9	9	no	—
120	C₃×C₃₇⋊C₃	#737	1	3	yes	complemented
121	C₉×C₃₇	—	3	1	no	—
122	C₉×C₃₇	—	3	2	no	—
123	C₉×C₃₇	—	3	3	no	—
124	C₉×C₃₇	C₃×H(3,𝔽₃)	1	9	yes	—
125	#66524	—	3	2	no	—
126	#66524	—	3	6	no	—
127	#66524	—	3	6	no	—
128	#66524	—	3	2	no	—
129	#66524	—	3	2	no	—
130	C₃×#737	—	37	2	no	—
131	C₃×#737	C₃₇⋊C₃	1	1	yes	characteristic, complemented
132	C₃²×3₋¹⁺²	—	37	1	no	—
133	#6893	—	37	2	no	—
134	#6893	—	37	2	no	—
135	#6893	—	37	2	no	—
136	C₉⋊H(3,𝔽₃)	—	37	6	no	—
137	C₉⋊H(3,𝔽₃)	—	37	6	no	—
138	C₉⋊H(3,𝔽₃)	—	37	6	no	—
139	C₉⋊H(3,𝔽₃)	—	37	3	no	—
140	C₉⋊H(3,𝔽₃)	—	37	3	no	—
141	#6891	—	37	1	no	—
142	C₃²⋊H(3,𝔽₃)	—	37	1	no	—
143	C₉×H(3,𝔽₃)	—	37	3	no	—
144	C₃×C₃²⋊C₉	—	37	1	no	—
145	C₃×C₃₇	—	3	3	no	—
146	C₃×C₃₇	—	9	3	no	—
147	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
148	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
149	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
150	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
151	C₃₇⋊C₃	—	3	3	no	—
152	C₃₇⋊C₃	—	9	6	no	—
153	C₃₇⋊C₃	—	9	1	no	—
154	C₃₇⋊C₃	—	9	1	no	—
155	C₃⁴	—	37	1	no	—
156	#737	—	3	9	no	—
157	#737	—	111	18	no	—
158	C₃×3₋¹⁺²	—	37	2	no	—
159	C₃×3₋¹⁺²	—	37	2	no	—
160	C₃×3₋¹⁺²	—	37	3	no	—
161	C₃×3₋¹⁺²	—	37	3	no	—
162	C₃×3₋¹⁺²	—	37	6	no	—
163	C₃×3₋¹⁺²	—	37	6	no	—
164	C₃×3₋¹⁺²	—	111	9	no	—
165	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
166	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
167	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	6	yes	complemented
168	C₃×H(3,𝔽₃)	—	37	3	no	—
169	C₃×H(3,𝔽₃)	—	37	3	no	—
170	C₃×H(3,𝔽₃)	—	37	2	no	—
171	C₃×H(3,𝔽₃)	—	37	3	no	—
172	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
173	C₃²×C₉	—	37	2	no	—
174	C₃²×C₉	—	37	3	no	—
175	C₃²×C₉	—	37	6	no	—
176	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
177	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
178	C₃²⋊C₉	—	37	6	no	—
179	C₃²⋊C₉	—	37	6	no	—
180	C₃²⋊C₉	—	37	6	no	—
181	C₃²⋊C₉	—	37	1	no	—
182	C₃²⋊C₉	—	37	2	no	—
183	C₃²⋊C₉	—	37	2	no	—
184	C₃²⋊C₉	—	37	1	no	—
185	C₃²⋊C₉	—	37	1	no	—
186	C₃²⋊C₉	—	37	2	no	—
187	C₃²⋊C₉	—	37	3	no	—
188	C₃²⋊C₉	—	37	3	no	—
189	C₃²⋊C₉	—	37	3	no	—
190	C₃²⋊C₉	—	37	6	no	—
191	C₃²⋊C₉	—	37	6	no	—
192	C₃²⋊C₉	—	37	12	no	—
193	C₃²⋊C₉	—	37	6	no	—
194	C₃²⋊C₉	—	37	6	no	—
195	C₃₇	#10624896	1	1	yes	characteristic, complemented, Sylow
196	C₃³	—	37	1	no	—
197	C₃³	—	37	6	no	—
198	C₃³	—	37	1	no	—
199	C₃³	—	37	1	no	—
200	C₃³	—	333	1	no	—
201	C₃³	—	333	1	no	—
202	C₃³	—	111	3	no	—
203	C₃³	—	111	3	no	—
204	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
205	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
206	3₋¹⁺²	—	3	3	no	—
207	3₋¹⁺²	—	3	18	no	—
208	3₋¹⁺²	—	111	6	no	—
209	3₋¹⁺²	—	111	6	no	—
210	3₋¹⁺²	—	111	9	no	—
211	3₋¹⁺²	—	111	18	no	—
212	3₋¹⁺²	—	111	9	no	—
213	3₋¹⁺²	—	111	18	no	—
214	3₋¹⁺²	C₃²⋊(C₃₇⋊C₃)	1	9	yes	complemented
215	H(3,𝔽₃)	—	3	3	no	—
216	H(3,𝔽₃)	—	333	3	no	—
217	H(3,𝔽₃)	—	333	3	no	—
218	H(3,𝔽₃)	—	111	9	no	—
219	H(3,𝔽₃)	—	111	6	no	—
220	C₃×C₉	—	3	3	no	—
221	C₃×C₉	—	3	9	no	—
222	C₃×C₉	—	37	2	no	—
223	C₃×C₉	—	37	6	no	—
224	C₃×C₉	—	37	2	no	—
225	C₃×C₉	—	37	2	no	—
226	C₃×C₉	—	37	6	no	—
227	C₃×C₉	—	111	6	no	—
228	C₃×C₉	—	111	9	no	—
229	C₃×C₉	—	111	18	no	—
230	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
231	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
232	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
233	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
234	C₃²	—	3	1	no	—
235	C₃²	—	3	1	no	—
236	C₃²	—	3	1	no	—
237	C₃²	—	3	3	no	—
238	C₃²	—	3	3	no	—
239	C₃²	—	3	3	no	—
240	C₃²	—	3	3	no	—
241	C₃²	—	37	3	no	—
242	C₃²	—	333	9	no	—
243	C₃²	—	111	6	no	—
244	C₃²	—	111	1	no	—
245	C₃²	—	111	1	no	—
246	C₃²	—	111	6	no	—
247	C₃²	—	111	1	no	—
248	C₃²	—	111	1	no	—
249	C₃²	—	111	1	no	—
250	C₃²	—	111	6	no	—
251	C₃²	—	111	1	no	—
252	C₃²	—	111	1	no	—
253	C₃²	—	111	6	no	—
254	C₃²	—	111	1	no	—
255	C₃²	—	111	1	no	—
256	C₃²	—	111	1	no	—
257	C₃²	—	111	1	no	—
258	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
259	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
260	C₉	—	3	1	no	—
261	C₉	—	3	2	no	—
262	C₉	—	3	3	no	—
263	C₉	—	111	2	no	—
264	C₉	—	111	2	no	—
265	C₉	—	111	2	no	—
266	C₉	—	111	6	no	—
267	C₉	—	111	6	no	—
268	C₉	C₃×C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
269	C₃	—	3	3	no	—
270	C₃	—	9	3	no	—
271	C₃	—	333	1	no	—
272	C₃	—	333	1	no	—
273	C₃	—	333	6	no	—
274	C₃	—	111	3	no	—
275	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
276	C₃	#737×C₃₇⋊C₃	1	1	yes	characteristic
277	C₃	#22582677	1	1	yes	characteristic
278	C₃	#22582678	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃×C₃₇×#737	C₃
prime radical	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	C₃
hypercenter	C₃×#737	C₃₇⋊C₃
hypocenter	C₃₇	#10624896
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃×#737

Complements

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

N	H ≅ G/N	type
1. C₃×#15688532	C₃	semidirect
2. C₃×3₋¹⁺²×C₃₇⋊C₃	C₃	semidirect
3. #22582565	C₃	semidirect
4. #22582564	C₃	semidirect
5. #22582562	C₃	semidirect
6. #22582559	C₃	semidirect
7. #22582558	C₃	semidirect
8. #22582555	C₃	semidirect
9. #22582555	C₃	semidirect
10. #22582555	C₃	semidirect
11. #22582551	C₃	semidirect
13. C₉×C₃²⋊(C₃₇⋊C₃)	C₃	semidirect
14. C₃×#15688502	C₃	semidirect
15. C₃×C₃₇×#737	C₃	semidirect
21. C₃×#11758605	C₃²	semidirect
23. C₃×#11758606	C₃²	semidirect
31. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
32. C₃²⋊#66524	C₃²	semidirect
33. C₃²⋊#66524	C₃²	semidirect
34. C₃²⋊#66524	C₃²	semidirect
44. #15688502	C₃²	semidirect
45. #15688502	C₃²	semidirect
46. #15688502	C₃²	semidirect
47. #15688502	C₃²	semidirect
48. #15688502	C₃²	semidirect
52. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
55. C₃²×C₉×C₃₇	C₃²	semidirect
68. 3₋¹⁺²×C₃₇	H(3,𝔽₃)	semidirect
120. C₃×C₃₇⋊C₃	#737	semidirect
131. C₃×#737	C₃₇⋊C₃	semidirect
167. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
177. C₃²×C₉	C₃×C₃₇⋊C₃	semidirect
195. C₃₇	#10624896	semidirect
214. 3₋¹⁺²	C₃²⋊(C₃₇⋊C₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688532	3
C₃×3₋¹⁺²×C₃₇⋊C₃	3
#22582565	3
#22582564	3
#22582562	3
#22582559	3
#22582558	3
#22582555	3
#22582551	3
(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	3
C₉×C₃²⋊(C₃₇⋊C₃)	3
C₃×#15688502	3
C₃×C₃₇×#737	3
#10624896	37