Subgroups — C₉×#5002 #13142683

Order 1728.

550 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#235	C₂	1	1	yes	maximal, characteristic
2	C₃×C₉×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#810	C₂	1	2	yes	maximal, complemented
5	C₉×#809	C₂	1	2	yes	maximal, complemented
6	C₃×#5002	C₃	1	1	yes	maximal, characteristic
7	C₉×#416	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×#67	—	2	1	no	—
10	C₉×#235	—	2	2	no	—
11	C₉×#235	C₄	1	4	yes	complemented
12	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₃×C₉	—	2	1	no	—
14	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
15	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
17	C₃×C₄²×C₉	—	2	1	no	—
18	C₂×C₉×C₃⋊C₈	—	2	2	no	—
19	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃×#235	C₂×C₃	1	1	yes	characteristic
21	C₃²×#117	C₂×C₃	1	1	yes	characteristic
22	C₂²×C₃×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
23	C₂×C₉×#41	—	3	1	no	—
24	C₃×#810	C₂×C₃	1	2	yes	—
25	C₃×#809	C₂×C₃	1	2	yes	—
26	C₂²×C₈×C₉	—	3	1	no	—
27	C₉×#117	—	2	1	no	—
28	C₉×#117	D₃	1	1	yes	characteristic, complemented
29	C₉×#103	—	3	2	no	—
30	C₉×#102	—	3	2	no	—
31	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
32	C₂²×D₃×C₉	—	2	1	no	—
33	C₂×C₃×C₄×C₉	—	2	2	no	—
34	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
37	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₉×#67	—	2	4	no	—
39	C₂×C₉×Dic₃	—	2	1	no	—
40	C₉×D₁₂	—	2	1	no	—
41	C₉×D₁₂	C₂×C₄	1	2	yes	—
42	C₄×D₃×C₉	—	2	4	no	—
43	C₉×Dic₆	—	2	1	no	—
44	C₉×Dic₆	C₂×C₄	1	2	yes	—
45	C₉×C₃⋊C₈	—	2	2	no	—
46	C₃×#416	—	3	1	no	—
47	#5002	C₉	1	1	yes	characteristic, direct product factor
48	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₃×#67	—	2	1	no	—
50	C₃×#235	—	2	2	no	—
51	C₃×#235	C₃×C₄	1	4	yes	—
52	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₃×C₄×D₃	—	2	1	no	—
54	C₂×C₃×Dic₆	C₂²×C₃	1	1	yes	characteristic
55	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
56	C₃²×C₂²⋊C₄	—	2	1	no	—
57	C₃²×C₄²	—	2	1	no	—
58	C₂×C₃×C₃⋊C₈	—	2	2	no	—
59	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
60	C₉×#41	—	3	4	no	—
61	C₉×#41	—	6	2	no	—
62	C₂×Q₈×C₉	—	3	1	no	—
63	C₂×D₄×C₉	—	3	1	no	—
64	C₂×D₄×C₉	—	6	1	no	—
65	C₂²×C₄×C₉	—	2	1	no	—
66	C₂²×C₄×C₉	—	6	1	no	—
67	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
68	C₂×C₈×C₉	—	3	2	no	—
69	C₂×C₈×C₉	—	6	2	no	—
70	C₉×C₄⋊C₄	—	2	2	no	—
71	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
72	C₉×C₂²⋊C₄	—	2	2	no	—
73	C₉×C₂²⋊C₄	—	2	1	no	—
74	C₄²×C₉	—	2	2	no	—
75	C₄²×C₉	—	2	1	no	—
76	C₂²×C₃×C₉	—	2	2	no	—
77	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
79	C₂×D₃×C₉	—	2	2	no	—
80	C₂×D₃×C₉	—	4	2	no	—
81	C₃×C₄×C₉	—	4	2	no	—
82	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
83	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
84	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
85	C₉×Dic₃	—	2	2	no	—
86	C₂×C₃×#41	—	3	1	no	—
87	C₂×#235	C₂×C₉	1	1	yes	characteristic
88	C₂²×C₃×C₈	—	3	1	no	—
89	C₃×#117	—	2	1	no	—
90	C₃×#117	C₂×C₉	1	1	yes	characteristic, complemented
91	C₃×#117	C₃×D₃	1	1	yes	characteristic
92	C₂²×C₃⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
93	C₃×#103	—	3	2	no	—
94	C₃×#102	—	3	2	no	—
95	#810	C₂×C₉	1	2	yes	complemented
96	#809	C₂×C₉	1	2	yes	complemented
97	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
98	C₂²×C₃×D₃	—	2	1	no	—
99	C₂×C₃²×C₄	—	2	2	no	—
100	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
101	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
102	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
103	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
104	C₃×#67	—	2	4	no	—
105	C₂×C₃×Dic₃	—	2	1	no	—
106	C₃×D₁₂	—	2	1	no	—
107	C₃×D₁₂	C₂×C₃×C₄	1	2	yes	—
108	C₃×C₄×D₃	—	2	4	no	—
109	C₃×Dic₆	—	2	1	no	—
110	C₃×Dic₆	C₂×C₃×C₄	1	2	yes	—
111	C₂³×C₉	—	2	1	no	—
112	C₂³×C₉	—	6	1	no	—
113	C₂³×C₉	#67	1	1	yes	characteristic
114	C₃×C₃⋊C₈	—	2	2	no	—
115	Q₈×C₉	—	3	2	no	—
116	Q₈×C₉	—	6	1	no	—
117	D₄×C₉	—	3	2	no	—
118	D₄×C₉	—	6	4	no	—
119	D₄×C₉	—	6	1	no	—
120	C₂×C₄×C₉	—	2	4	no	—
121	C₂×C₄×C₉	—	2	2	no	—
122	C₂×C₄×C₉	—	2	2	no	—
123	C₂×C₄×C₉	—	2	2	no	—
124	C₂×C₄×C₉	—	2	1	no	—
125	C₂×C₄×C₉	—	2	1	no	—
126	C₂×C₄×C₉	—	6	1	no	—
127	C₂×C₄×C₉	—	6	4	no	—
128	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
129	C₂×C₄×C₉	D₁₂	1	2	yes	—
130	C₂×C₄×C₉	#67	1	1	yes	characteristic
131	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
132	C₈×C₉	—	6	2	no	—
133	#416	—	3	1	no	Sylow
134	C₂×C₃×C₉	—	2	2	no	—
135	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₃×C₉	#135	1	2	yes	—
137	D₃×C₉	—	4	2	no	—
138	C₃×#41	—	3	4	no	—
139	C₃×#41	—	6	2	no	—
140	C₂×C₃×Q₈	—	3	1	no	—
141	C₂×C₃×D₄	—	3	1	no	—
142	C₂×C₃×D₄	—	6	1	no	—
143	C₂²×C₃×C₄	—	2	1	no	—
144	C₂²×C₃×C₄	—	6	1	no	—
145	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
146	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
147	C₂×#67	—	2	1	no	—
148	#235	—	2	2	no	—
149	#235	C₄×C₉	1	4	yes	complemented
150	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
151	C₂×C₄×D₃	—	2	1	no	—
152	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic
153	C₂×C₃×C₈	—	3	2	no	—
154	C₂×C₃×C₈	—	6	2	no	—
155	C₃×C₄⋊C₄	—	2	2	no	—
156	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
157	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
158	C₃×C₂²⋊C₄	—	2	2	no	—
159	C₃×C₂²⋊C₄	—	2	1	no	—
160	C₃×C₂²⋊C₄	—	2	1	no	—
161	C₃×C₄²	—	2	2	no	—
162	C₃×C₄²	—	2	1	no	—
163	C₃×C₄²	—	2	1	no	—
164	C₂×C₃⋊C₈	—	2	2	no	—
165	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
166	C₂²×C₃²	—	2	2	no	—
167	C₂²×C₃²	C₃×C₂²⋊C₄	1	2	yes	—
168	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
169	C₂×C₃×D₃	—	2	2	no	—
170	C₂×C₃×D₃	—	4	2	no	—
171	C₃²×C₄	—	4	2	no	—
172	C₃²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
173	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
174	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
175	C₃×Dic₃	—	2	2	no	—
176	C₂²×C₉	—	2	2	no	—
177	C₂²×C₉	—	2	2	no	—
178	C₂²×C₉	—	2	1	no	—
179	C₂²×C₉	—	4	2	no	—
180	C₂²×C₉	—	6	2	no	—
181	C₂²×C₉	—	12	2	no	—
182	C₂²×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
183	C₂²×C₉	C₂×#67	1	1	yes	characteristic
184	C₄×C₉	—	2	2	no	—
185	C₄×C₉	—	2	1	no	—
186	C₄×C₉	—	2	1	no	—
187	C₄×C₉	—	4	4	no	—
188	C₄×C₉	—	4	2	no	—
189	C₄×C₉	—	6	2	no	—
190	C₄×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
191	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
192	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
193	C₂×#41	—	3	1	no	—
194	C₂²×C₈	—	3	1	no	—
195	#117	D₃×C₉	1	1	yes	characteristic, complemented
196	#103	—	3	2	no	—
197	#102	—	3	2	no	—
198	C₃×C₉	#416	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₃	—	2	1	no	—
200	C₂³×C₃	—	6	1	no	—
201	C₂³×C₃	D₄×C₉	1	1	yes	characteristic
202	C₂³×C₃	C₃×#67	1	1	yes	characteristic
203	C₂²×D₃	—	2	1	no	—
204	C₃×Q₈	—	3	2	no	—
205	C₃×Q₈	—	6	1	no	—
206	C₃×D₄	—	3	2	no	—
207	C₃×D₄	—	6	4	no	—
208	C₃×D₄	—	6	1	no	—
209	C₂×C₃×C₄	—	2	4	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	2	1	no	—
215	C₂×C₃×C₄	—	2	1	no	—
216	C₂×C₃×C₄	—	6	1	no	—
217	C₂×C₃×C₄	—	6	4	no	—
218	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
219	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
220	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
222	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
223	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
224	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
226	#67	—	2	4	no	—
227	C₂×Dic₃	—	2	1	no	—
228	D₁₂	—	2	1	no	—
229	D₁₂	C₂×C₄×C₉	1	2	yes	—
230	C₄×D₃	—	2	4	no	—
231	Dic₆	—	2	1	no	—
232	Dic₆	C₂×C₄×C₉	1	2	yes	—
233	C₃×C₈	—	6	2	no	—
234	C₃⋊C₈	—	2	2	no	—
235	C₂×C₃²	—	2	2	no	—
236	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₃²	C₃×#135	1	2	yes	—
238	C₃×D₃	—	4	2	no	—
239	C₂×C₉	—	2	2	no	—
240	C₂×C₉	—	2	1	no	—
241	C₂×C₉	—	2	2	no	—
242	C₂×C₉	—	4	2	no	—
243	C₂×C₉	—	12	2	no	—
244	C₂×C₉	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
245	C₂×C₉	#950	1	2	yes	—
246	#41	—	3	4	no	—
247	#41	—	6	2	no	—
248	C₂×Q₈	—	3	1	no	—
249	C₂×D₄	—	3	1	no	—
250	C₂×D₄	—	6	1	no	—
251	C₂²×C₄	—	6	1	no	—
252	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
253	C₂×C₈	—	3	2	no	—
254	C₂×C₈	—	6	2	no	—
255	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
256	C₂²⋊C₄	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	2	2	no	—
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	2	2	no	—
262	C₂²×C₃	—	4	2	no	—
263	C₂²×C₃	—	6	2	no	—
264	C₂²×C₃	—	12	2	no	—
265	C₂²×C₃	C₉×C₂²⋊C₄	1	2	yes	—
266	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
267	C₂²×C₃	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
268	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
269	C₂×D₃	—	2	2	no	—
270	C₂×D₃	—	4	2	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	2	2	no	—
274	C₃×C₄	—	4	4	no	—
275	C₃×C₄	—	4	2	no	—
276	C₃×C₄	—	4	2	no	—
277	C₃×C₄	—	6	2	no	—
278	C₃×C₄	C₉×C₂²⋊C₄	1	2	yes	—
279	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
280	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
281	C₃×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
282	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
283	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
284	Dic₃	—	2	2	no	—
285	C₃²	C₃×#416	1	1	yes	characteristic
286	C₉	—	2	1	no	—
287	C₉	#5002	1	1	yes	characteristic, direct product factor
288	C₂³	—	6	1	no	—
289	C₂³	C₉×#67	1	1	yes	characteristic
290	Q₈	—	3	2	no	—
291	Q₈	—	6	1	no	—
292	D₄	—	3	2	no	—
293	D₄	—	6	4	no	—
294	D₄	—	6	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	6	1	no	—
297	C₂×C₄	—	6	4	no	—
298	C₂×C₄	C₄×D₃×C₉	1	2	yes	—
299	C₂×C₄	C₉×D₁₂	1	2	yes	—
300	C₂×C₄	C₉×#67	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
302	C₈	—	6	2	no	—
303	C₂×C₃	—	2	2	no	—
304	C₂×C₃	—	2	2	no	—
305	C₂×C₃	—	2	1	no	—
306	C₂×C₃	—	2	2	no	—
307	C₂×C₃	—	4	2	no	—
308	C₂×C₃	—	12	2	no	—
309	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
310	C₂×C₃	C₉×#135	1	2	yes	—
311	C₂×C₃	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
312	C₂×C₃	C₃×#950	1	2	yes	—
313	D₃	—	4	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	6	2	no	—
316	C₂²	—	12	2	no	—
317	C₂²	C₉×(C₂×D₃)⋊C₄	1	2	yes	—
318	C₂²	C₂×C₉×#67	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	6	2	no	—
321	C₄	C₉×(C₂×D₃)⋊C₄	1	2	yes	—
322	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
323	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
324	C₃	—	2	1	no	—
325	C₃	C₉×#416	1	1	yes	characteristic, complemented
326	C₃	C₃×#5002	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	—	12	2	no	—
329	C₂	C₂×C₉×(C₂×D₃)⋊C₄	1	1	yes	characteristic
330	C₂	C₉×#950	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₉	#67
commutator	C₃×C₄	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#117	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃×#235	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#117	D₃
hypocenter	C₃	C₉×#416
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂²×C₄×C₉ ◁ C₉×#117

Complements

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

N	H ≅ G/N	type
2. C₃×C₉×#117	C₂	semidirect
3. C₂²×C₉×C₃⋊C₈	C₂	semidirect
4. C₉×#810	C₂	semidirect
5. C₉×#809	C₂	semidirect
11. C₉×#235	C₄	semidirect
28. C₉×#117	D₃	semidirect
47. #5002	C₉	direct (N × H)
90. C₃×#117	C₂×C₉	semidirect
92. C₂²×C₃⋊C₈	C₂×C₉	semidirect
95. #810	C₂×C₉	semidirect
96. #809	C₂×C₉	semidirect
149. #235	C₄×C₉	semidirect
195. #117	D₃×C₉	semidirect
198. C₃×C₉	#416	semidirect
287. C₉	#5002	direct (N × H)
325. C₃	C₉×#416	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#235	2
C₃×C₉×#117	2
C₂²×C₉×C₃⋊C₈	2
C₉×#810	2
C₉×#809	2
C₃×#5002	3
C₉×#416	3

Subgroups — C9×#5002 #13142683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5002 #13142683