Subgroups — C₃×#10589799 #13153158

Order 1728.

776 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#63457	C₂	1	2	yes	maximal, complemented
3	C₃×#63457	C₂	1	4	yes	maximal, complemented
4	C₃×#63456	C₂	1	4	yes	maximal, complemented
5	C₃×#63455	C₂	1	2	yes	maximal, complemented
6	C₃×#63436	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#63434	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5697	—	3	1	no	maximal
9	#10589799	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×D₃₆	C₂²	1	2	yes	—
11	C₂×C₃×D₃₆	C₂²	1	4	yes	—
12	C₂×C₃×D₃₆	C₂²	1	4	yes	—
13	C₂×C₃×C₄×D₉	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×D₉	C₂²	1	4	yes	—
15	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₉×C₄⋊C₄	C₂²	1	4	yes	—
17	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
18	C₃×(C₂×D₉)⋊C₄	C₂²	1	2	yes	—
19	C₃×(C₂×D₉)⋊C₄	C₂²	1	4	yes	—
20	C₃×(C₂×D₉)⋊C₄	C₂²	1	4	yes	—
21	C₃×Dic₉⋊C₄	C₂²	1	2	yes	—
22	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
23	C₃²×#125	D₃	1	1	yes	characteristic
24	C₃×#895	—	3	4	no	—
25	C₃×#895	—	3	2	no	—
26	C₃×#894	—	3	4	no	—
27	C₃×#893	—	3	2	no	—
28	C₃×#874	—	3	1	no	—
29	C₃×#872	—	3	1	no	—
30	C₉×#125	—	2	1	no	—
31	C₉×#125	C₂×C₃	1	1	yes	characteristic, complemented
32	#63457	C₂×C₃	1	4	yes	complemented
33	#63457	C₂×C₃	1	2	yes	complemented
34	#63456	C₂×C₃	1	4	yes	complemented
35	#63455	C₂×C₃	1	2	yes	complemented
36	#63436	C₂×C₃	1	1	yes	characteristic, complemented
37	#63434	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂²×C₃×D₉	C₂³	1	1	yes	characteristic
39	C₂²×C₃×D₉	C₂³	1	4	yes	—
40	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
41	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
42	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₂×C₃×Dic₉	C₂³	1	2	yes	—
44	C₂×C₃×Dic₉	C₂³	1	1	yes	characteristic
45	C₃×D₃₆	—	2	4	no	—
46	C₃×D₃₆	—	4	4	no	—
47	C₃×D₃₆	—	4	4	no	—
48	C₃×C₄×D₉	—	2	2	no	—
49	C₃×C₄×D₉	—	4	4	no	—
50	C₃×#540	—	9	1	no	—
51	#5697	—	3	1	no	—
52	C₂×C₃×D₁₂	—	3	2	no	—
53	C₂×C₃×D₁₂	—	3	4	no	—
54	C₂×C₃×D₁₂	—	3	4	no	—
55	C₂×C₃×C₄×D₃	—	3	1	no	—
56	C₂×C₃×C₄×D₃	—	3	4	no	—
57	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
58	C₃²×C₄⋊C₄	C₂×D₃	1	4	yes	—
59	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
60	C₃×(C₂×D₃)⋊C₄	—	3	4	no	—
61	C₃×(C₂×D₃)⋊C₄	—	3	4	no	—
62	C₃×(C₂×D₃)⋊C₄	—	3	2	no	—
63	C₃×Dic₃⋊C₄	—	3	2	no	—
64	C₃×C₄×Dic₃	—	3	1	no	—
65	C₂×D₃₆	C₂²×C₃	1	2	yes	—
66	C₂×D₃₆	C₂²×C₃	1	4	yes	—
67	C₂×D₃₆	C₂²×C₃	1	4	yes	—
68	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₄×D₉	C₂²×C₃	1	4	yes	—
70	C₉×C₄⋊C₄	—	2	2	no	—
71	C₉×C₄⋊C₄	—	2	4	no	—
72	C₉×C₄⋊C₄	C₂²×C₃	1	2	yes	—
73	C₉×C₄⋊C₄	C₂²×C₃	1	4	yes	—
74	C₄²×C₉	—	2	1	no	—
75	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
76	(C₂×D₉)⋊C₄	C₂²×C₃	1	2	yes	—
77	(C₂×D₉)⋊C₄	C₂²×C₃	1	4	yes	—
78	(C₂×D₉)⋊C₄	C₂²×C₃	1	4	yes	—
79	Dic₉⋊C₄	C₂²×C₃	1	2	yes	—
80	C₄×Dic₉	C₂²×C₃	1	1	yes	characteristic
81	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
82	C₂×C₃×D₉	—	2	1	no	—
83	C₂×C₃×D₉	—	2	4	no	—
84	C₂×C₃×D₉	—	2	4	no	—
85	C₂×C₃×D₉	—	2	2	no	—
86	C₂×C₃×D₉	—	2	4	no	—
87	C₃×C₄×C₉	—	2	4	no	—
88	C₃×C₄×C₉	—	2	2	no	—
89	C₃×C₄×C₉	#41	1	2	yes	—
90	C₃×Dic₉	—	2	1	no	—
91	C₃×Dic₉	—	2	2	no	—
92	C₃×#127	—	9	1	no	—
93	C₃×#125	—	2	1	no	—
94	C₃×#125	D₉	1	1	yes	characteristic, complemented
95	C₃×#125	C₃×D₃	1	1	yes	characteristic
96	C₃×#123	—	9	4	no	—
97	C₃×#121	—	9	4	no	—
98	C₃×#121	—	9	2	no	—
99	C₃×C₄×D₄	—	9	2	no	—
100	C₃×#117	—	9	1	no	—
101	#895	—	3	4	no	—
102	#895	—	3	2	no	—
103	#894	—	3	4	no	—
104	#893	—	3	2	no	—
105	#874	—	3	1	no	—
106	#872	—	3	1	no	—
107	C₂²×C₃×D₃	—	3	1	no	—
108	C₂²×C₃×D₃	—	3	4	no	—
109	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
110	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
111	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
112	C₂×C₃×Dic₃	—	3	2	no	—
113	C₂×C₃×Dic₃	—	3	1	no	—
114	C₃×D₁₂	—	6	4	no	—
115	C₃×D₁₂	—	12	4	no	—
116	C₃×D₁₂	—	12	4	no	—
117	C₃×C₄×D₃	—	6	2	no	—
118	C₃×C₄×D₃	—	12	4	no	—
119	C₂²×D₉	C₂³×C₃	1	1	yes	characteristic
120	C₂²×D₉	C₂³×C₃	1	4	yes	—
121	C₂×C₄×C₉	—	2	1	no	—
122	C₂×C₄×C₉	—	2	2	no	—
123	C₂×C₄×C₉	—	2	4	no	—
124	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
125	C₂×C₄×C₉	C₂³×C₃	1	4	yes	—
126	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
127	C₂×Dic₉	C₂³×C₃	1	2	yes	—
128	C₂×Dic₉	C₂³×C₃	1	1	yes	characteristic
129	D₃₆	—	2	4	no	—
130	D₃₆	—	4	4	no	—
131	D₃₆	—	4	4	no	—
132	C₄×D₉	—	2	2	no	—
133	C₄×D₉	—	4	4	no	—
134	#540	—	9	1	no	Sylow
135	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
136	C₂×C₃×C₉	H(5,𝔽₂)	1	2	yes	—
137	C₃×D₉	—	4	1	no	—
138	C₃×D₉	—	4	4	no	—
139	C₂×C₃×D₄	—	9	2	no	—
140	C₂×C₃×D₄	—	9	4	no	—
141	C₂×C₃×D₄	—	9	4	no	—
142	C₂²×C₃×C₄	—	9	1	no	—
143	C₂²×C₃×C₄	—	9	4	no	—
144	C₂×D₁₂	—	3	2	no	—
145	C₂×D₁₂	—	3	4	no	—
146	C₂×D₁₂	—	3	4	no	—
147	C₂×C₄×D₃	—	3	1	no	—
148	C₂×C₄×D₃	—	3	4	no	—
149	C₃×C₄⋊C₄	—	2	2	no	—
150	C₃×C₄⋊C₄	—	2	4	no	—
151	C₃×C₄⋊C₄	—	9	2	no	—
152	C₃×C₄⋊C₄	C₂×D₉	1	2	yes	—
153	C₃×C₄⋊C₄	C₂×D₉	1	4	yes	—
154	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
155	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
156	C₃×C₂²⋊C₄	—	9	2	no	—
157	C₃×C₂²⋊C₄	—	9	4	no	—
158	C₃×C₂²⋊C₄	—	9	4	no	—
159	C₃×C₄²	—	2	1	no	—
160	C₃×C₄²	—	9	1	no	—
161	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
162	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
163	(C₂×D₃)⋊C₄	—	3	2	no	—
164	(C₂×D₃)⋊C₄	—	3	4	no	—
165	(C₂×D₃)⋊C₄	—	3	4	no	—
166	Dic₃⋊C₄	—	3	2	no	—
167	C₄×Dic₃	—	3	1	no	—
168	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
169	C₂×C₃×D₃	—	6	1	no	—
170	C₂×C₃×D₃	—	6	4	no	—
171	C₂×C₃×D₃	—	6	4	no	—
172	C₂×C₃×D₃	—	6	2	no	—
173	C₂×C₃×D₃	—	6	4	no	—
174	C₃²×C₄	—	2	4	no	—
175	C₃²×C₄	—	2	2	no	—
176	C₃²×C₄	#239	1	2	yes	—
177	C₃×Dic₃	—	6	1	no	—
178	C₃×Dic₃	—	6	2	no	—
179	C₂²×C₉	—	2	1	no	—
180	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
181	C₂×D₉	—	2	1	no	—
182	C₂×D₉	—	2	4	no	—
183	C₂×D₉	—	2	4	no	—
184	C₂×D₉	—	2	2	no	—
185	C₂×D₉	—	2	4	no	—
186	C₄×C₉	—	2	4	no	—
187	C₄×C₉	—	2	2	no	—
188	C₄×C₉	—	2	2	no	—
189	C₄×C₉	—	4	4	no	—
190	C₄×C₉	—	4	2	no	—
191	C₄×C₉	C₃×#41	1	2	yes	—
192	Dic₉	—	2	1	no	—
193	Dic₉	—	2	2	no	—
194	#127	—	9	1	no	—
195	#125	C₃×D₉	1	1	yes	characteristic, complemented
196	#123	—	9	4	no	—
197	#121	—	9	4	no	—
198	#121	—	9	2	no	—
199	C₄×D₄	—	9	2	no	—
200	#117	—	9	1	no	—
201	C₃×C₉	#540	1	1	yes	characteristic, complemented, Sylow
202	C₂³×C₃	—	9	1	no	—
203	C₂³×C₃	—	9	4	no	—
204	C₂²×D₃	—	3	1	no	—
205	C₂²×D₃	—	3	4	no	—
206	C₃×D₄	—	18	4	no	—
207	C₃×D₄	—	36	4	no	—
208	C₃×D₄	—	36	4	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	2	4	no	—
212	C₂×C₃×C₄	—	9	2	no	—
213	C₂×C₃×C₄	—	9	1	no	—
214	C₂×C₃×C₄	—	18	2	no	—
215	C₂×C₃×C₄	—	36	4	no	—
216	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
217	C₂×C₃×C₄	C₂²×D₉	1	4	yes	—
218	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
220	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
221	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
222	C₂×Dic₃	—	3	2	no	—
223	C₂×Dic₃	—	3	1	no	—
224	D₁₂	—	6	4	no	—
225	D₁₂	—	12	4	no	—
226	D₁₂	—	12	4	no	—
227	C₄×D₃	—	6	2	no	—
228	C₄×D₃	—	12	4	no	—
229	C₂×C₃²	C₂×#239	1	1	yes	characteristic
230	C₂×C₃²	#1009	1	2	yes	—
231	C₃×D₃	—	12	1	no	—
232	C₃×D₃	—	12	4	no	—
233	C₂×C₉	—	2	1	no	—
234	C₂×C₉	—	2	2	no	—
235	C₂×C₉	C₂×C₃×#41	1	1	yes	characteristic
236	C₂×C₉	C₃×H(5,𝔽₂)	1	2	yes	—
237	D₉	—	4	1	no	—
238	D₉	—	4	4	no	—
239	C₂×D₄	—	9	2	no	—
240	C₂×D₄	—	9	4	no	—
241	C₂×D₄	—	9	4	no	—
242	C₂²×C₄	—	9	1	no	—
243	C₂²×C₄	—	9	4	no	—
244	C₄⋊C₄	—	9	2	no	—
245	C₄⋊C₄	C₂×C₃×D₉	1	2	yes	—
246	C₄⋊C₄	C₂×C₃×D₉	1	4	yes	—
247	C₂²⋊C₄	—	9	2	no	—
248	C₂²⋊C₄	—	9	4	no	—
249	C₂²⋊C₄	—	9	4	no	—
250	C₄²	—	9	1	no	—
251	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	—	18	1	no	—
254	C₂²×C₃	—	18	4	no	—
255	C₂²×C₃	—	18	4	no	—
256	C₂²×C₃	—	18	2	no	—
257	C₂²×C₃	—	18	4	no	—
258	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
259	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
260	C₂×D₃	—	6	1	no	—
261	C₂×D₃	—	6	4	no	—
262	C₂×D₃	—	6	4	no	—
263	C₂×D₃	—	6	2	no	—
264	C₂×D₃	—	6	4	no	—
265	C₃×C₄	—	2	4	no	—
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	2	4	no	—
268	C₃×C₄	—	2	2	no	—
269	C₃×C₄	—	2	2	no	—
270	C₃×C₄	—	4	4	no	—
271	C₃×C₄	—	4	2	no	—
272	C₃×C₄	—	18	1	no	—
273	C₃×C₄	—	18	2	no	—
274	C₃×C₄	#3701	1	2	yes	—
275	C₃×C₄	C₃×#239	1	2	yes	—
276	Dic₃	—	6	1	no	—
277	Dic₃	—	6	2	no	—
278	C₃²	#5697	1	1	yes	characteristic
279	C₉	—	2	1	no	—
280	C₉	C₃×#540	1	1	yes	characteristic, complemented
281	C₂³	—	9	1	no	—
282	C₂³	—	9	4	no	—
283	D₄	—	18	4	no	—
284	D₄	—	36	4	no	—
285	D₄	—	36	4	no	—
286	C₂×C₄	—	9	2	no	—
287	C₂×C₄	—	9	1	no	—
288	C₂×C₄	—	18	2	no	—
289	C₂×C₄	—	36	4	no	—
290	C₂×C₄	C₂²×C₃×D₉	1	2	yes	—
291	C₂×C₄	C₂²×C₃×D₉	1	4	yes	—
292	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	—	2	2	no	—
295	C₂×C₃	—	36	1	no	—
296	C₂×C₃	—	36	4	no	—
297	C₂×C₃	C₂×#3701	1	1	yes	characteristic
298	C₂×C₃	#63715	1	2	yes	—
299	C₂×C₃	C₂×C₃×#239	1	1	yes	characteristic
300	C₂×C₃	C₃×#1009	1	2	yes	—
301	D₃	—	12	1	no	—
302	D₃	—	12	4	no	—
303	C₂²	—	18	1	no	—
304	C₂²	—	18	4	no	—
305	C₂²	—	18	4	no	—
306	C₂²	—	18	2	no	—
307	C₂²	—	18	4	no	—
308	C₂²	C₂³×C₃×D₉	1	1	yes	characteristic
309	C₄	—	2	4	no	—
310	C₄	—	2	2	no	—
311	C₄	—	18	1	no	—
312	C₄	—	18	2	no	—
313	C₄	C₃×#3701	1	2	yes	—
314	C₃	—	2	1	no	—
315	C₃	#10589799	1	1	yes	characteristic, direct product factor
316	C₃	C₃×#5697	1	1	yes	characteristic
317	C₂	—	36	1	no	—
318	C₂	—	36	4	no	—
319	C₂	C₂×C₃×#3701	1	1	yes	characteristic
320	C₂	C₃×#63715	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₉×#125	C₂
socle	C₂²×C₃²	C₂³×D₃
hypercenter	C₃×#125	D₉
hypocenter	C₉	C₃×#540
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#125

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×#125	C₂	semidirect
2. C₃×#63457	C₂	semidirect
3. C₃×#63457	C₂	semidirect
4. C₃×#63456	C₂	semidirect
5. C₃×#63455	C₂	semidirect
6. C₃×#63436	C₂	semidirect
7. C₃×#63434	C₂	semidirect
9. #10589799	C₃	direct (N × H)
31. C₉×#125	C₂×C₃	semidirect
32. #63457	C₂×C₃	semidirect
33. #63457	C₂×C₃	semidirect
34. #63456	C₂×C₃	semidirect
35. #63455	C₂×C₃	semidirect
36. #63436	C₂×C₃	semidirect
37. #63434	C₂×C₃	semidirect
94. C₃×#125	D₉	semidirect
195. #125	C₃×D₉	semidirect
201. C₃×C₉	#540	semidirect
280. C₉	C₃×#540	semidirect
315. C₃	#10589799	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₉×#125	2
C₃×#63457	2
C₃×#63456	2
C₃×#63455	2
C₃×#63436	2
C₃×#63434	2
C₃×#5697	3
#10589799	3

Subgroups — C3×#10589799 #13153158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10589799 #13153158