Subgroups — C₃×#10588331 #13141275

Order 1728.

490 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
2	C₃×C₉×#95	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉⋊#95	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉⋊#95	C₂	1	2	yes	maximal
5	C₃×C₉⋊#95	C₂	1	2	yes	maximal
6	C₃×#4651	—	3	1	no	maximal
7	#10588331	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
9	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
11	C₂²×C₃×Dic₉	C₂²	1	1	yes	characteristic
12	C₂²×C₃×Dic₉	C₂²	1	1	yes	characteristic
13	C₃×C₄×Dic₉	C₄	1	4	yes	complemented
14	C₂×C₃×C₄×Dic₃	—	3	1	no	—
15	C₃²×#95	D₃	1	1	yes	characteristic
16	C₃×C₃⋊#95	—	3	2	no	—
17	C₃×C₃⋊#95	—	3	2	no	—
18	C₃×C₃⋊#95	—	3	1	no	—
19	C₂×C₄×Dic₉	C₂×C₃	1	1	yes	characteristic
20	C₉×#95	—	2	1	no	—
21	C₉×#95	C₂×C₃	1	1	yes	characteristic
22	C₉⋊#95	C₂×C₃	1	1	yes	characteristic
23	C₉⋊#95	C₂×C₃	1	2	yes	—
24	C₉⋊#95	C₂×C₃	1	2	yes	—
25	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	—	2	2	no	—
27	C₂×C₃×C₄×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	—	2	2	no	—
29	C₂×C₃×C₄×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂×C₃×Dic₉	—	2	2	no	—
32	C₂×C₃×Dic₉	—	2	1	no	—
33	C₂×C₃×Dic₉	—	2	1	no	—
34	C₂×C₃×Dic₉	—	2	2	no	—
35	C₂×C₃×Dic₉	—	2	2	no	—
36	C₂×C₃×Dic₉	—	2	2	no	—
37	C₂×C₃×Dic₉	C₂×C₄	1	2	yes	—
38	C₂×C₃×Dic₉	C₂×C₄	1	2	yes	—
39	C₃×#383	—	9	1	no	—
40	#4651	—	3	1	no	—
41	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
42	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃×Dic₃	—	3	1	no	—
44	C₂²×C₃×Dic₃	—	3	1	no	—
45	C₂²×C₃×Dic₃	—	3	2	no	—
46	C₃×C₄×Dic₃	—	3	4	no	—
47	C₂²×C₄×C₉	—	2	2	no	—
48	C₂²×C₄×C₉	—	2	1	no	—
49	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
50	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
51	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
52	C₂²×Dic₉	C₂²×C₃	1	1	yes	characteristic
53	C₂²×Dic₉	C₂²×C₃	1	1	yes	characteristic
54	C₄×Dic₉	C₃×C₄	1	4	yes	complemented
55	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₃×C₉	#41	1	2	yes	—
57	C₂²×C₃×C₉	#41	1	2	yes	—
58	C₂²×C₃×C₉	#41	1	1	yes	characteristic
59	C₂²×C₃×C₉	#41	1	1	yes	characteristic
60	C₃×C₄×C₉	—	2	2	no	—
61	C₃×C₄×C₉	—	4	2	no	—
62	C₃×Dic₉	—	2	2	no	—
63	C₃×Dic₉	—	2	2	no	—
64	C₃×Dic₉	—	4	2	no	—
65	C₂×C₃×C₄²	—	9	1	no	—
66	C₂×C₄×Dic₃	—	3	1	no	—
67	C₃×#95	—	2	1	no	—
68	C₃×#95	—	9	2	no	—
69	C₃×#95	—	9	2	no	—
70	C₃×#95	—	9	1	no	—
71	C₃×#95	D₉	1	1	yes	characteristic
72	C₃×#95	C₃×D₃	1	1	yes	characteristic
73	C₃⋊#95	—	3	2	no	—
74	C₃⋊#95	—	3	2	no	—
75	C₃⋊#95	—	3	1	no	—
76	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
77	C₂×C₃²×C₄	—	2	2	no	—
78	C₂×C₃²×C₄	—	2	2	no	—
79	C₂×C₃²×C₄	—	2	2	no	—
80	C₂×C₃²×C₄	—	2	2	no	—
81	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
82	C₂×C₃×Dic₃	—	3	2	no	—
83	C₂×C₃×Dic₃	—	3	2	no	—
84	C₂×C₃×Dic₃	—	6	2	no	—
85	C₂×C₃×Dic₃	—	6	1	no	—
86	C₂×C₃×Dic₃	—	6	1	no	—
87	C₂×C₃×Dic₃	—	6	2	no	—
88	C₂×C₃×Dic₃	—	6	2	no	—
89	C₂×C₃×Dic₃	—	6	2	no	—
90	C₂³×C₉	—	2	1	no	—
91	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	4	no	—
94	C₂×C₄×C₉	—	2	2	no	—
95	C₂×C₄×C₉	—	2	2	no	—
96	C₂×C₄×C₉	—	2	2	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	4	2	no	—
99	C₂×C₄×C₉	—	4	2	no	—
100	C₂×C₄×C₉	—	4	2	no	—
101	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
102	C₂×Dic₉	—	2	2	no	—
103	C₂×Dic₉	—	2	1	no	—
104	C₂×Dic₉	—	2	1	no	—
105	C₂×Dic₉	—	2	2	no	—
106	C₂×Dic₉	—	2	2	no	—
107	C₂×Dic₉	—	2	2	no	—
108	C₂×Dic₉	C₂×C₃×C₄	1	2	yes	—
109	C₂×Dic₉	C₂×C₃×C₄	1	2	yes	—
110	#383	—	9	1	no	Sylow
111	C₂×C₃×C₉	#117	1	1	yes	characteristic
112	C₂×C₃×C₉	#117	1	1	yes	characteristic
113	C₂×C₃×C₉	#117	1	1	yes	characteristic
114	C₂×C₃×C₉	#126	1	2	yes	—
115	C₂×C₃×C₉	#126	1	2	yes	—
116	C₂²×C₃×C₄	—	2	2	no	—
117	C₂²×C₃×C₄	—	2	1	no	—
118	C₂²×C₃×C₄	—	9	1	no	—
119	C₂²×C₃×C₄	—	9	1	no	—
120	C₂²×C₃×C₄	—	9	2	no	—
121	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
122	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
124	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
125	C₂²×Dic₃	—	3	1	no	—
126	C₂²×Dic₃	—	3	1	no	—
127	C₂²×Dic₃	—	3	2	no	—
128	C₃×C₄²	—	9	4	no	—
129	C₄×Dic₃	—	3	4	no	—
130	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
131	C₂²×C₃²	#235	1	2	yes	—
132	C₂²×C₃²	#237	1	2	yes	—
133	C₂²×C₃²	#237	1	1	yes	characteristic
134	C₂²×C₃²	#239	1	1	yes	characteristic
135	C₃²×C₄	—	2	2	no	—
136	C₃²×C₄	—	4	2	no	—
137	C₃×Dic₃	—	6	2	no	—
138	C₃×Dic₃	—	6	2	no	—
139	C₃×Dic₃	—	12	2	no	—
140	C₂²×C₉	—	2	1	no	—
141	C₂²×C₉	—	2	2	no	—
142	C₂²×C₉	—	2	2	no	—
143	C₂²×C₉	—	2	1	no	—
144	C₂²×C₉	—	2	1	no	—
145	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
146	C₂²×C₉	C₃×#41	1	2	yes	—
147	C₂²×C₉	C₃×#41	1	2	yes	—
148	C₂²×C₉	C₃×#41	1	1	yes	characteristic
149	C₂²×C₉	C₃×#41	1	1	yes	characteristic
150	C₄×C₉	—	2	2	no	—
151	C₄×C₉	—	4	2	no	—
152	C₄×C₉	—	4	4	no	—
153	C₄×C₉	—	4	2	no	—
154	Dic₉	—	2	2	no	—
155	Dic₉	—	2	2	no	—
156	Dic₉	—	4	2	no	—
157	C₂×C₄²	—	9	1	no	—
158	#95	—	9	2	no	—
159	#95	—	9	2	no	—
160	#95	—	9	1	no	—
161	#95	C₃×D₉	1	1	yes	characteristic
162	C₃×C₉	#383	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₃	—	2	1	no	—
164	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
165	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	4	2	no	—
177	C₂×C₃×C₄	—	4	2	no	—
178	C₂×C₃×C₄	—	4	2	no	—
179	C₂×C₃×C₄	—	9	2	no	—
180	C₂×C₃×C₄	—	9	2	no	—
181	C₂×C₃×C₄	—	18	2	no	—
182	C₂×C₃×C₄	—	18	1	no	—
183	C₂×C₃×C₄	—	18	1	no	—
184	C₂×C₃×C₄	—	18	2	no	—
185	C₂×C₃×C₄	—	18	2	no	—
186	C₂×C₃×C₄	—	18	2	no	—
187	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
188	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
189	C₂×Dic₃	—	3	2	no	—
190	C₂×Dic₃	—	3	2	no	—
191	C₂×Dic₃	—	6	2	no	—
192	C₂×Dic₃	—	6	1	no	—
193	C₂×Dic₃	—	6	1	no	—
194	C₂×Dic₃	—	6	2	no	—
195	C₂×Dic₃	—	6	2	no	—
196	C₂×Dic₃	—	6	2	no	—
197	C₂×C₃²	#872	1	1	yes	characteristic
198	C₂×C₃²	#877	1	1	yes	characteristic
199	C₂×C₃²	#879	1	2	yes	—
200	C₂×C₃²	#892	1	1	yes	characteristic
201	C₂×C₃²	#898	1	2	yes	—
202	C₂×C₉	—	2	1	no	—
203	C₂×C₉	—	2	2	no	—
204	C₂×C₉	—	2	1	no	—
205	C₂×C₉	—	2	2	no	—
206	C₂×C₉	—	2	1	no	—
207	C₂×C₉	C₃×#117	1	1	yes	characteristic
208	C₂×C₉	C₃×#117	1	1	yes	characteristic
209	C₂×C₉	C₃×#117	1	1	yes	characteristic
210	C₂×C₉	C₃×#126	1	2	yes	—
211	C₂×C₉	C₃×#126	1	2	yes	—
212	C₂²×C₄	—	9	1	no	—
213	C₂²×C₄	—	9	1	no	—
214	C₂²×C₄	—	9	2	no	—
215	C₂²×C₄	C₂×C₃×D₉	1	2	yes	—
216	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
217	C₄²	—	9	4	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	—	2	2	no	—
220	C₂²×C₃	—	2	2	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	—	2	1	no	—
223	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
224	C₂²×C₃	#3697	1	2	yes	—
225	C₂²×C₃	#3699	1	2	yes	—
226	C₂²×C₃	#3699	1	1	yes	characteristic
227	C₂²×C₃	#3701	1	1	yes	characteristic
228	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
229	C₂²×C₃	C₃×#235	1	2	yes	—
230	C₂²×C₃	C₃×#237	1	2	yes	—
231	C₂²×C₃	C₃×#237	1	1	yes	characteristic
232	C₂²×C₃	C₃×#239	1	1	yes	characteristic
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	2	2	no	—
235	C₃×C₄	—	4	2	no	—
236	C₃×C₄	—	4	4	no	—
237	C₃×C₄	—	4	2	no	—
238	C₃×C₄	—	4	2	no	—
239	C₃×C₄	—	18	2	no	—
240	C₃×C₄	—	18	2	no	—
241	C₃×C₄	—	36	2	no	—
242	Dic₃	—	6	2	no	—
243	Dic₃	—	6	2	no	—
244	Dic₃	—	12	2	no	—
245	C₃²	#4651	1	1	yes	characteristic
246	C₉	—	2	1	no	—
247	C₉	C₃×#383	1	1	yes	characteristic, complemented
248	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	9	2	no	—
254	C₂×C₄	—	9	2	no	—
255	C₂×C₄	—	18	2	no	—
256	C₂×C₄	—	18	1	no	—
257	C₂×C₄	—	18	1	no	—
258	C₂×C₄	—	18	2	no	—
259	C₂×C₄	—	18	2	no	—
260	C₂×C₄	—	18	2	no	—
261	C₂×C₄	C₃×C₄×D₉	1	2	yes	—
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	—	2	1	no	—
264	C₂×C₃	—	2	2	no	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	2	no	—
267	C₂×C₃	#63434	1	1	yes	characteristic
268	C₂×C₃	#63439	1	1	yes	characteristic
269	C₂×C₃	#63441	1	2	yes	—
270	C₂×C₃	#63454	1	1	yes	characteristic
271	C₂×C₃	#63460	1	2	yes	—
272	C₂×C₃	C₃×#872	1	1	yes	characteristic
273	C₂×C₃	C₃×#877	1	1	yes	characteristic
274	C₂×C₃	C₃×#879	1	2	yes	—
275	C₂×C₃	C₃×#892	1	1	yes	characteristic
276	C₂×C₃	C₃×#898	1	2	yes	—
277	C₂²	C₂×C₃×C₄×D₉	1	1	yes	characteristic
278	C₂²	C₃×#3697	1	2	yes	—
279	C₂²	C₃×#3699	1	2	yes	—
280	C₂²	C₃×#3699	1	1	yes	characteristic
281	C₂²	C₃×#3701	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	4	2	no	—
284	C₄	—	18	2	no	—
285	C₄	—	18	2	no	—
286	C₄	—	36	2	no	—
287	C₃	—	2	1	no	—
288	C₃	#10588331	1	1	yes	characteristic, direct product factor
289	C₃	C₃×#4651	1	1	yes	characteristic
290	C₂	C₃×#63434	1	1	yes	characteristic
291	C₂	C₃×#63439	1	1	yes	characteristic
292	C₂	C₃×#63441	1	2	yes	—
293	C₂	C₃×#63454	1	1	yes	characteristic
294	C₂	C₃×#63460	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₃×C₄
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₃×C₉×#95	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₉	C₂³×C₃
hypercenter	C₃×#95	D₉
hypocenter	C₉	C₃×#383
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#95

Complements

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

N	H ≅ G/N	type
7. #10588331	C₃	direct (N × H)
13. C₃×C₄×Dic₉	C₄	semidirect
54. C₄×Dic₉	C₃×C₄	semidirect
162. C₃×C₉	#383	semidirect
247. C₉	C₃×#383	semidirect
288. C₃	#10588331	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×Dic₉	2
C₃×C₉×#95	2
C₃×C₉⋊#95	2
C₃×#4651	3
#10588331	3

Subgroups — C3×#10588331 #13141275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588331 #13141275