Subgroups — C₉×#4953 #13142475

Order 1728.

642 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#217	C₂	1	2	yes	maximal
3	C₂×C₉×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×C₃⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×C₃⋊#95	C₂	1	2	yes	maximal, complemented
6	C₃×#4953	C₃	1	1	yes	maximal, characteristic
7	C₉×#397	—	3	1	no	maximal
8	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
12	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
13	C₃×C₉×C₂²⋊C₄	—	2	2	no	—
14	C₉×#217	—	2	4	no	—
15	C₉×C₄⋊Dic₃	—	2	2	no	—
16	C₂×C₃²×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
17	C₂×C₃×#217	C₂×C₃	1	2	yes	—
18	C₂×C₃×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic
19	C₃×C₃⋊#95	C₂×C₃	1	2	yes	—
20	C₃×C₃⋊#95	C₂×C₃	1	1	yes	characteristic
21	C₂×C₉×C₄⋊C₄	—	3	1	no	—
22	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
23	C₂×C₉×C₂²⋊C₄	—	3	2	no	—
24	C₂×C₉×C₂²⋊C₄	D₃	1	1	yes	characteristic
25	C₉×#95	—	3	2	no	—
26	C₉×#95	—	3	1	no	—
27	C₂³×C₃×C₉	—	2	1	no	—
28	C₂³×C₃×C₉	—	2	1	no	—
29	C₂³×C₃×C₉	—	2	2	no	—
30	C₂³×C₃×C₉	—	2	2	no	—
31	C₂³×C₃×C₉	D₄	1	2	yes	—
32	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	—	2	2	no	—
34	C₂×C₃×C₄×C₉	—	2	1	no	—
35	C₂×C₃×C₄×C₉	—	2	2	no	—
36	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
37	C₂×C₉×Dic₃	—	2	2	no	—
38	C₂×C₉×Dic₃	—	2	2	no	—
39	C₂×C₉×Dic₃	—	2	2	no	—
40	C₂×C₉×Dic₃	—	2	2	no	—
41	C₂×C₉×Dic₃	—	2	2	no	—
42	C₂×C₉×Dic₃	—	2	2	no	—
43	C₃×#397	—	3	1	no	—
44	#4953	C₉	1	1	yes	characteristic, direct product factor
45	C₂⁴×C₃²	C₂²×C₃	1	1	yes	characteristic
46	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
47	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
49	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
50	C₂⁴×C₉	—	2	1	no	—
51	C₂⁴×C₉	C₂×D₃	1	1	yes	characteristic
52	C₃²×C₂²⋊C₄	—	2	2	no	—
53	C₃×#217	—	2	4	no	—
54	C₃×C₄⋊Dic₃	—	2	2	no	—
55	C₂²×C₄×C₉	—	2	1	no	—
56	C₂²×C₄×C₉	—	2	1	no	—
57	C₂²×C₄×C₉	—	3	2	no	—
58	C₂²×C₄×C₉	—	3	2	no	—
59	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
60	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
61	C₉×C₄⋊C₄	—	6	2	no	—
62	C₉×C₂²⋊C₄	—	2	4	no	—
63	C₉×C₂²⋊C₄	—	2	2	no	—
64	C₉×C₂²⋊C₄	—	6	4	no	—
65	C₂²×C₃×C₉	—	2	2	no	—
66	C₂²×C₃×C₉	—	2	4	no	—
67	C₂²×C₃×C₉	—	4	1	no	—
68	C₂²×C₃×C₉	—	4	1	no	—
69	C₂²×C₃×C₉	—	4	2	no	—
70	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₃×C₉	#41	1	1	yes	characteristic
73	C₂²×C₃×C₉	#41	1	2	yes	—
74	C₂²×C₃×C₉	#41	1	2	yes	—
75	C₃×C₄×C₉	—	2	2	no	—
76	C₃×C₄×C₉	—	4	1	no	—
77	C₉×Dic₃	—	4	2	no	—
78	C₉×Dic₃	—	4	2	no	—
79	C₂×C₃×C₄⋊C₄	—	3	1	no	—
80	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
81	C₂×C₃×C₂²⋊C₄	—	3	2	no	—
82	C₂×C₃×C₂²⋊C₄	C₂×C₉	1	1	yes	characteristic
83	C₂×C₃×C₂²⋊C₄	C₃×D₃	1	1	yes	characteristic
84	C₂×#217	C₂×C₉	1	2	yes	—
85	C₂×C₄⋊Dic₃	C₂×C₉	1	1	yes	characteristic, complemented
86	C₃×#95	—	3	1	no	—
87	C₃×#95	—	3	2	no	—
88	C₃⋊#95	C₂×C₉	1	1	yes	characteristic, complemented
89	C₃⋊#95	C₂×C₉	1	2	yes	complemented
90	C₂³×C₃²	—	2	1	no	—
91	C₂³×C₃²	—	2	1	no	—
92	C₂³×C₃²	—	2	2	no	—
93	C₂³×C₃²	—	2	2	no	—
94	C₂³×C₃²	C₃×D₄	1	2	yes	—
95	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
96	C₂×C₃²×C₄	—	2	1	no	—
97	C₂×C₃²×C₄	—	2	2	no	—
98	C₂×C₃²×C₄	—	2	2	no	—
99	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
100	C₂×C₃×Dic₃	—	2	2	no	—
101	C₂×C₃×Dic₃	—	2	2	no	—
102	C₂×C₃×Dic₃	—	2	2	no	—
103	C₂×C₃×Dic₃	—	2	2	no	—
104	C₂×C₃×Dic₃	—	2	2	no	—
105	C₂×C₃×Dic₃	—	2	2	no	—
106	C₂³×C₉	—	2	2	no	—
107	C₂³×C₉	—	2	2	no	—
108	C₂³×C₉	—	2	1	no	—
109	C₂³×C₉	—	2	2	no	—
110	C₂³×C₉	—	2	2	no	—
111	C₂³×C₉	—	2	1	no	—
112	C₂³×C₉	—	2	2	no	—
113	C₂³×C₉	—	2	1	no	—
114	C₂³×C₉	—	4	2	no	—
115	C₂³×C₉	—	4	2	no	—
116	C₂³×C₉	D₁₂	1	2	yes	—
117	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
118	C₂×C₄×C₉	—	2	2	no	—
119	C₂×C₄×C₉	—	2	2	no	—
120	C₂×C₄×C₉	—	2	2	no	—
121	C₂×C₄×C₉	—	2	1	no	—
122	C₂×C₄×C₉	—	2	2	no	—
123	C₂×C₄×C₉	—	4	2	no	—
124	C₂×C₄×C₉	—	4	2	no	—
125	C₂×C₄×C₉	—	6	2	no	—
126	C₂×C₄×C₉	—	6	2	no	—
127	C₂×C₄×C₉	—	6	2	no	—
128	C₂×C₄×C₉	—	6	2	no	—
129	C₂×C₄×C₉	—	6	2	no	—
130	C₂×C₄×C₉	—	6	2	no	—
131	C₂×C₄×C₉	#67	1	2	yes	—
132	#397	—	3	1	no	Sylow
133	C₂×C₃×C₉	—	4	2	no	—
134	C₂×C₃×C₉	#121	1	1	yes	characteristic
135	C₂×C₃×C₉	#123	1	2	yes	—
136	C₂×C₃×C₉	#123	1	1	yes	characteristic
137	C₂×C₃×C₉	#124	1	1	yes	characteristic
138	C₂×C₃×C₉	#126	1	2	yes	—
139	C₂⁴×C₃	—	2	1	no	—
140	C₂⁴×C₃	C₂²×C₉	1	1	yes	characteristic
141	C₂⁴×C₃	C₂×C₃×D₃	1	1	yes	characteristic
142	C₂²×C₃×C₄	—	2	1	no	—
143	C₂²×C₃×C₄	—	2	1	no	—
144	C₂²×C₃×C₄	—	3	2	no	—
145	C₂²×C₃×C₄	—	3	2	no	—
146	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
147	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
148	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
149	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
150	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
151	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
152	C₃×C₄⋊C₄	—	6	2	no	—
153	C₃×C₂²⋊C₄	—	2	4	no	—
154	C₃×C₂²⋊C₄	—	2	2	no	—
155	C₃×C₂²⋊C₄	—	2	2	no	—
156	C₃×C₂²⋊C₄	—	6	4	no	—
157	#217	—	2	4	no	—
158	C₄⋊Dic₃	—	2	2	no	—
159	C₂²×C₃²	—	2	2	no	—
160	C₂²×C₃²	—	2	4	no	—
161	C₂²×C₃²	—	4	1	no	—
162	C₂²×C₃²	—	4	1	no	—
163	C₂²×C₃²	—	4	2	no	—
164	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
166	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
167	C₂²×C₃²	C₃×#41	1	2	yes	—
168	C₂²×C₃²	C₃×#41	1	2	yes	—
169	C₃²×C₄	—	2	2	no	—
170	C₃²×C₄	—	4	1	no	—
171	C₃×Dic₃	—	4	2	no	—
172	C₃×Dic₃	—	4	2	no	—
173	C₂²×C₉	—	2	4	no	—
174	C₂²×C₉	—	2	4	no	—
175	C₂²×C₉	—	2	2	no	—
176	C₂²×C₉	—	2	1	no	—
177	C₂²×C₉	—	2	1	no	—
178	C₂²×C₉	—	2	2	no	—
179	C₂²×C₉	—	2	2	no	—
180	C₂²×C₉	—	2	1	no	—
181	C₂²×C₉	—	4	4	no	—
182	C₂²×C₉	—	4	4	no	—
183	C₂²×C₉	—	4	2	no	—
184	C₂²×C₉	—	4	2	no	—
185	C₂²×C₉	—	4	2	no	—
186	C₂²×C₉	—	4	1	no	—
187	C₂²×C₉	—	4	1	no	—
188	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
189	C₂²×C₉	#235	1	1	yes	characteristic
190	C₂²×C₉	#237	1	2	yes	—
191	C₂²×C₉	#237	1	2	yes	—
192	C₂²×C₉	C₂×#67	1	1	yes	characteristic
193	C₄×C₉	—	2	2	no	—
194	C₄×C₉	—	4	2	no	—
195	C₄×C₉	—	4	2	no	—
196	C₄×C₉	—	4	1	no	—
197	C₄×C₉	—	12	2	no	—
198	C₄×C₉	—	12	2	no	—
199	C₂×C₄⋊C₄	—	3	1	no	—
200	C₂×C₂²⋊C₄	—	3	2	no	—
201	C₂×C₂²⋊C₄	D₃×C₉	1	1	yes	characteristic
202	#95	—	3	1	no	—
203	#95	—	3	2	no	—
204	C₃×C₉	#397	1	1	yes	characteristic, complemented, Sylow
205	C₂³×C₃	—	2	2	no	—
206	C₂³×C₃	—	2	2	no	—
207	C₂³×C₃	—	2	1	no	—
208	C₂³×C₃	—	2	1	no	—
209	C₂³×C₃	—	2	2	no	—
210	C₂³×C₃	—	2	2	no	—
211	C₂³×C₃	—	2	1	no	—
212	C₂³×C₃	—	2	2	no	—
213	C₂³×C₃	—	2	2	no	—
214	C₂³×C₃	—	2	1	no	—
215	C₂³×C₃	—	2	2	no	—
216	C₂³×C₃	—	2	1	no	—
217	C₂³×C₃	—	4	2	no	—
218	C₂³×C₃	—	4	2	no	—
219	C₂³×C₃	D₄×C₉	1	2	yes	—
220	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
221	C₂³×C₃	C₃×D₁₂	1	2	yes	—
222	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
223	C₂×C₃×C₄	—	2	2	no	—
224	C₂×C₃×C₄	—	2	2	no	—
225	C₂×C₃×C₄	—	2	2	no	—
226	C₂×C₃×C₄	—	2	1	no	—
227	C₂×C₃×C₄	—	2	2	no	—
228	C₂×C₃×C₄	—	2	2	no	—
229	C₂×C₃×C₄	—	2	1	no	—
230	C₂×C₃×C₄	—	2	2	no	—
231	C₂×C₃×C₄	—	4	2	no	—
232	C₂×C₃×C₄	—	4	2	no	—
233	C₂×C₃×C₄	—	6	2	no	—
234	C₂×C₃×C₄	—	6	2	no	—
235	C₂×C₃×C₄	—	6	2	no	—
236	C₂×C₃×C₄	—	6	2	no	—
237	C₂×C₃×C₄	—	6	2	no	—
238	C₂×C₃×C₄	—	6	2	no	—
239	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
240	C₂×C₃×C₄	C₃×#67	1	2	yes	—
241	C₂×Dic₃	—	2	2	no	—
242	C₂×Dic₃	—	2	2	no	—
243	C₂×Dic₃	—	2	2	no	—
244	C₂×Dic₃	—	2	2	no	—
245	C₂×Dic₃	—	2	2	no	—
246	C₂×Dic₃	—	2	2	no	—
247	C₂×C₃²	—	4	2	no	—
248	C₂×C₃²	C₃×#121	1	1	yes	characteristic
249	C₂×C₃²	C₃×#123	1	2	yes	—
250	C₂×C₃²	C₃×#123	1	1	yes	characteristic
251	C₂×C₃²	C₃×#124	1	1	yes	characteristic
252	C₂×C₃²	C₃×#126	1	2	yes	—
253	C₂×C₉	—	2	2	no	—
254	C₂×C₉	—	2	2	no	—
255	C₂×C₉	—	2	1	no	—
256	C₂×C₉	—	2	1	no	—
257	C₂×C₉	—	2	1	no	—
258	C₂×C₉	—	4	4	no	—
259	C₂×C₉	—	4	2	no	—
260	C₂×C₉	#879	1	2	yes	—
261	C₂×C₉	#886	1	2	yes	—
262	C₂×C₉	#930	1	1	yes	characteristic
263	C₂×C₉	#935	1	1	yes	characteristic
264	C₂×C₉	#936	1	1	yes	characteristic
265	C₂⁴	C₂×D₃×C₉	1	1	yes	characteristic
266	C₂²×C₄	—	3	2	no	—
267	C₂²×C₄	—	3	2	no	—
268	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
270	C₄⋊C₄	—	6	2	no	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	6	4	no	—
273	C₂²×C₃	—	2	4	no	—
274	C₂²×C₃	—	2	4	no	—
275	C₂²×C₃	—	2	2	no	—
276	C₂²×C₃	—	2	4	no	—
277	C₂²×C₃	—	2	2	no	—
278	C₂²×C₃	—	2	1	no	—
279	C₂²×C₃	—	2	1	no	—
280	C₂²×C₃	—	2	2	no	—
281	C₂²×C₃	—	2	2	no	—
282	C₂²×C₃	—	2	1	no	—
283	C₂²×C₃	—	4	4	no	—
284	C₂²×C₃	—	4	2	no	—
285	C₂²×C₃	—	4	2	no	—
286	C₂²×C₃	—	4	4	no	—
287	C₂²×C₃	—	4	2	no	—
288	C₂²×C₃	—	4	1	no	—
289	C₂²×C₃	—	4	1	no	—
290	C₂²×C₃	—	4	2	no	—
291	C₂²×C₃	—	4	1	no	—
292	C₂²×C₃	—	4	1	no	—
293	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
294	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
295	C₂²×C₃	C₉×#41	1	1	yes	characteristic
296	C₂²×C₃	C₉×#41	1	2	yes	—
297	C₂²×C₃	C₉×#41	1	2	yes	—
298	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
299	C₂²×C₃	C₃×#235	1	1	yes	characteristic
300	C₂²×C₃	C₃×#237	1	2	yes	—
301	C₂²×C₃	C₃×#237	1	2	yes	—
302	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
303	C₃×C₄	—	2	2	no	—
304	C₃×C₄	—	2	2	no	—
305	C₃×C₄	—	4	2	no	—
306	C₃×C₄	—	4	2	no	—
307	C₃×C₄	—	4	1	no	—
308	C₃×C₄	—	4	1	no	—
309	C₃×C₄	—	12	2	no	—
310	C₃×C₄	—	12	2	no	—
311	Dic₃	—	4	2	no	—
312	Dic₃	—	4	2	no	—
313	C₃²	C₃×#397	1	1	yes	characteristic
314	C₉	—	2	1	no	—
315	C₉	#4953	1	1	yes	characteristic, direct product factor
316	C₂³	—	2	1	no	—
317	C₂³	—	2	2	no	—
318	C₂³	—	2	2	no	—
319	C₂³	—	2	1	no	—
320	C₂³	C₉×D₁₂	1	2	yes	—
321	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	6	2	no	—
326	C₂×C₄	—	6	2	no	—
327	C₂×C₄	—	6	2	no	—
328	C₂×C₄	—	6	2	no	—
329	C₂×C₄	—	6	2	no	—
330	C₂×C₄	—	6	2	no	—
331	C₂×C₄	C₉×#67	1	2	yes	—
332	C₂×C₃	—	2	1	no	—
333	C₂×C₃	—	2	1	no	—
334	C₂×C₃	—	2	1	no	—
335	C₂×C₃	—	2	2	no	—
336	C₂×C₃	—	2	2	no	—
337	C₂×C₃	—	4	4	no	—
338	C₂×C₃	—	4	2	no	—
339	C₂×C₃	—	4	2	no	—
340	C₂×C₃	C₉×#121	1	1	yes	characteristic
341	C₂×C₃	C₉×#123	1	1	yes	characteristic
342	C₂×C₃	C₉×#123	1	2	yes	—
343	C₂×C₃	C₉×#124	1	1	yes	characteristic
344	C₂×C₃	C₉×#126	1	2	yes	—
345	C₂×C₃	C₃×#879	1	2	yes	—
346	C₂×C₃	C₃×#886	1	2	yes	—
347	C₂×C₃	C₃×#930	1	1	yes	characteristic
348	C₂×C₃	C₃×#935	1	1	yes	characteristic
349	C₂×C₃	C₃×#936	1	1	yes	characteristic
350	C₂²	—	2	2	no	—
351	C₂²	—	2	4	no	—
352	C₂²	—	4	1	no	—
353	C₂²	—	4	1	no	—
354	C₂²	—	4	2	no	—
355	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
356	C₂²	C₉×#235	1	1	yes	characteristic
357	C₂²	C₉×#237	1	2	yes	—
358	C₂²	C₉×#237	1	2	yes	—
359	C₂²	C₂×C₉×#67	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	4	1	no	—
362	C₄	—	12	2	no	—
363	C₄	—	12	2	no	—
364	C₃	—	2	1	no	—
365	C₃	C₉×#397	1	1	yes	characteristic, complemented
366	C₃	C₃×#4953	1	1	yes	characteristic
367	C₂	—	4	2	no	—
368	C₂	C₉×#879	1	2	yes	—
369	C₂	C₉×#886	1	2	yes	—
370	C₂	C₉×#930	1	1	yes	characteristic
371	C₂	C₉×#935	1	1	yes	characteristic
372	C₂	C₉×#936	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂²×D₃
commutator	C₂³×C₃	C₂³×C₉
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₃×C₉×C₂²⋊C₄	C₂
socle, cosocle	C₂³×C₃²	C₂³×C₃
prime radical	C₂⁴×C₃²	C₂²×C₃
hypercenter	C₂×C₉×C₂²⋊C₄	D₃
hypocenter	C₃	C₉×#397
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₉×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₉×C₄⋊Dic₃	C₂	semidirect
4. C₉×C₃⋊#95	C₂	semidirect
5. C₉×C₃⋊#95	C₂	semidirect
44. #4953	C₉	direct (N × H)
85. C₂×C₄⋊Dic₃	C₂×C₉	semidirect
88. C₃⋊#95	C₂×C₉	semidirect
89. C₃⋊#95	C₂×C₉	semidirect
204. C₃×C₉	#397	semidirect
315. C₉	#4953	direct (N × H)
365. C₃	C₉×#397	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×C₂²⋊C₄	2
C₂×C₉×#217	2
C₂×C₉×C₄⋊Dic₃	2
C₉×C₃⋊#95	2
C₃×#4953	3
C₉×#397	3

Subgroups — C9×#4953 #13142475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4953 #13142475