Subgroups — C₉×#5003 #13142685

Order 1728.

394 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#930	C₂	1	1	yes	maximal, characteristic
3	C₉×#830	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#809	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#809	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#808	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#807	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5003	C₃	1	1	yes	maximal, characteristic
9	C₉×#482	—	3	1	no	maximal
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₉×#67	—	2	1	no	—
12	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
13	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
16	C₃×C₄²×C₉	—	2	1	no	—
17	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
18	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
19	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃²×#117	C₂×C₃	1	1	yes	characteristic
22	C₃×#930	C₂×C₃	1	1	yes	characteristic
23	C₃×#830	C₂×C₃	1	1	yes	characteristic
24	C₃×#809	C₂×C₃	1	1	yes	characteristic
25	C₃×#809	C₂×C₃	1	1	yes	characteristic
26	C₃×#808	C₂×C₃	1	1	yes	characteristic
27	C₃×#807	C₂×C₃	1	1	yes	characteristic
28	C₉×#121	—	3	1	no	—
29	C₉×#117	—	2	1	no	—
30	C₉×#117	D₃	1	1	yes	characteristic, complemented
31	C₉×#107	—	3	1	no	—
32	C₉×#106	—	3	1	no	—
33	C₉×#102	—	3	1	no	—
34	C₉×#102	—	3	1	no	—
35	C₉×C₂²⋊C₈	—	3	1	no	—
36	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
37	C₂²×D₃×C₉	—	2	1	no	—
38	C₂×C₃×C₄×C₉	—	2	1	no	—
39	C₂×C₃×C₄×C₉	—	2	1	no	—
40	C₂×C₃×C₄×C₉	—	2	1	no	—
41	C₂×C₃×C₄×C₉	—	2	1	no	—
42	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
43	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₉×#67	—	4	2	no	—
45	C₂×C₉×Dic₃	—	2	1	no	—
46	C₉×D₁₂	—	2	1	no	—
47	C₉×D₁₂	—	2	1	no	—
48	C₉×C₃⋊C₈	—	2	1	no	—
49	C₉×C₃⋊C₈	—	2	1	no	—
50	C₃×#482	—	3	1	no	—
51	#5003	C₉	1	1	yes	characteristic, direct product factor
52	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₃×#67	—	2	1	no	—
54	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
55	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
56	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
57	C₃²×C₂²⋊C₄	—	2	1	no	—
58	C₃²×C₄²	—	2	1	no	—
59	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
60	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
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	C₂×D₃	1	1	yes	characteristic
67	C₂×C₈×C₉	—	3	1	no	—
68	C₂×C₈×C₉	—	3	1	no	—
69	C₉×C₄⋊C₄	—	2	1	no	—
70	C₉×C₄⋊C₄	—	2	1	no	—
71	C₉×C₄⋊C₄	—	3	1	no	—
72	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
73	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
74	C₉×C₂²⋊C₄	—	2	2	no	—
75	C₉×C₂²⋊C₄	—	2	1	no	—
76	C₉×C₂²⋊C₄	—	6	1	no	—
77	C₄²×C₉	—	2	2	no	—
78	C₄²×C₉	—	2	1	no	—
79	C₂²×C₃×C₉	—	2	1	no	—
80	C₂²×C₃×C₉	—	2	1	no	—
81	C₂²×C₃×C₉	—	2	1	no	—
82	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
83	C₂×D₃×C₉	—	4	1	no	—
84	C₂×D₃×C₉	—	4	1	no	—
85	C₂×D₃×C₉	—	4	1	no	—
86	C₃×C₄×C₉	—	2	1	no	—
87	C₃×C₄×C₉	—	4	1	no	—
88	C₃×C₄×C₉	—	4	1	no	—
89	C₃×C₄×C₉	#41	1	1	yes	characteristic
90	C₃×C₄×C₉	#41	1	1	yes	characteristic
91	C₉×Dic₃	—	4	1	no	—
92	C₃×#121	—	3	1	no	—
93	C₃×#117	—	2	1	no	—
94	C₃×#117	C₂×C₉	1	1	yes	characteristic, complemented
95	C₃×#117	C₃×D₃	1	1	yes	characteristic
96	#930	C₂×C₉	1	1	yes	characteristic
97	C₃×#107	—	3	1	no	—
98	C₃×#106	—	3	1	no	—
99	C₃×#102	—	3	1	no	—
100	C₃×#102	—	3	1	no	—
101	C₃×C₂²⋊C₈	—	3	1	no	—
102	#830	C₂×C₉	1	1	yes	characteristic, complemented
103	#809	C₂×C₉	1	1	yes	characteristic, complemented
104	#809	C₂×C₉	1	1	yes	characteristic, complemented
105	#808	C₂×C₉	1	1	yes	characteristic, complemented
106	#807	C₂×C₉	1	1	yes	characteristic, complemented
107	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
108	C₂²×C₃×D₃	—	2	1	no	—
109	C₂×C₃²×C₄	—	2	1	no	—
110	C₂×C₃²×C₄	—	2	1	no	—
111	C₂×C₃²×C₄	—	2	1	no	—
112	C₂×C₃²×C₄	—	2	1	no	—
113	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
114	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
115	C₃×#67	—	4	2	no	—
116	C₂×C₃×Dic₃	—	2	1	no	—
117	C₃×D₁₂	—	2	1	no	—
118	C₃×D₁₂	—	2	1	no	—
119	C₂³×C₉	—	2	1	no	—
120	C₂³×C₉	—	6	1	no	—
121	C₂³×C₉	#67	1	1	yes	characteristic
122	C₃×C₃⋊C₈	—	2	1	no	—
123	C₃×C₃⋊C₈	—	2	1	no	—
124	D₄×C₉	—	6	1	no	—
125	D₄×C₉	—	6	1	no	—
126	D₄×C₉	—	12	2	no	—
127	C₂×C₄×C₉	—	2	2	no	—
128	C₂×C₄×C₉	—	2	2	no	—
129	C₂×C₄×C₉	—	2	1	no	—
130	C₂×C₄×C₉	—	2	1	no	—
131	C₂×C₄×C₉	—	2	2	no	—
132	C₂×C₄×C₉	—	2	2	no	—
133	C₂×C₄×C₉	—	2	1	no	—
134	C₂×C₄×C₉	—	2	1	no	—
135	C₂×C₄×C₉	—	2	1	no	—
136	C₂×C₄×C₉	—	2	1	no	—
137	C₂×C₄×C₉	—	6	1	no	—
138	C₂×C₄×C₉	#67	1	1	yes	characteristic
139	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
140	C₈×C₉	—	6	1	no	—
141	C₈×C₉	—	6	1	no	—
142	#482	—	3	1	no	Sylow
143	C₂×C₃×C₉	—	4	1	no	—
144	C₂×C₃×C₉	#123	1	1	yes	characteristic
145	C₂×C₃×C₉	#135	1	1	yes	characteristic
146	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
147	D₃×C₉	—	8	1	no	—
148	C₂×C₃×D₄	—	3	1	no	—
149	C₂×C₃×D₄	—	6	1	no	—
150	C₂²×C₃×C₄	—	2	1	no	—
151	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
152	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
153	C₂×#67	—	2	1	no	—
154	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
155	C₂×C₃×C₈	—	3	1	no	—
156	C₂×C₃×C₈	—	3	1	no	—
157	C₃×C₄⋊C₄	—	2	1	no	—
158	C₃×C₄⋊C₄	—	2	1	no	—
159	C₃×C₄⋊C₄	—	3	1	no	—
160	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
161	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
162	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
163	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
164	C₃×C₂²⋊C₄	—	2	2	no	—
165	C₃×C₂²⋊C₄	—	2	1	no	—
166	C₃×C₂²⋊C₄	—	2	1	no	—
167	C₃×C₂²⋊C₄	—	6	1	no	—
168	C₃×C₄²	—	2	2	no	—
169	C₃×C₄²	—	2	1	no	—
170	C₃×C₄²	—	2	1	no	—
171	(C₂×D₃)⋊C₄	—	2	1	no	—
172	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
173	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
174	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
175	C₂²×C₃²	—	2	1	no	—
176	C₂²×C₃²	—	2	1	no	—
177	C₂²×C₃²	—	2	1	no	—
178	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
179	C₂×C₃×D₃	—	4	1	no	—
180	C₂×C₃×D₃	—	4	1	no	—
181	C₂×C₃×D₃	—	4	1	no	—
182	C₃²×C₄	—	2	1	no	—
183	C₃²×C₄	—	4	1	no	—
184	C₃²×C₄	—	4	1	no	—
185	C₃²×C₄	C₃×#41	1	1	yes	characteristic
186	C₃²×C₄	C₃×#41	1	1	yes	characteristic
187	C₃×Dic₃	—	4	1	no	—
188	C₂²×C₉	—	2	2	no	—
189	C₂²×C₉	—	2	1	no	—
190	C₂²×C₉	—	2	1	no	—
191	C₂²×C₉	—	2	1	no	—
192	C₂²×C₉	—	2	1	no	—
193	C₂²×C₉	—	4	1	no	—
194	C₂²×C₉	—	4	1	no	—
195	C₂²×C₉	—	12	1	no	—
196	C₂²×C₉	—	12	1	no	—
197	C₂²×C₉	—	12	1	no	—
198	C₂²×C₉	C₂×#67	1	1	yes	characteristic
199	C₄×C₉	—	2	2	no	—
200	C₄×C₉	—	2	1	no	—
201	C₄×C₉	—	2	1	no	—
202	C₄×C₉	—	2	1	no	—
203	C₄×C₉	—	4	2	no	—
204	C₄×C₉	—	4	2	no	—
205	C₄×C₉	—	4	1	no	—
206	C₄×C₉	—	4	1	no	—
207	C₄×C₉	—	12	1	no	—
208	C₄×C₉	#235	1	1	yes	characteristic
209	C₄×C₉	#235	1	1	yes	characteristic
210	#121	—	3	1	no	—
211	#117	D₃×C₉	1	1	yes	characteristic, complemented
212	#107	—	3	1	no	—
213	#106	—	3	1	no	—
214	#102	—	3	1	no	—
215	#102	—	3	1	no	—
216	C₂²⋊C₈	—	3	1	no	—
217	C₃×C₉	#482	1	1	yes	characteristic, complemented, Sylow
218	C₂³×C₃	—	2	1	no	—
219	C₂³×C₃	—	6	1	no	—
220	C₂³×C₃	D₄×C₉	1	1	yes	characteristic
221	C₂³×C₃	C₃×#67	1	1	yes	characteristic
222	C₂²×D₃	—	2	1	no	—
223	C₃×D₄	—	6	1	no	—
224	C₃×D₄	—	6	1	no	—
225	C₃×D₄	—	12	2	no	—
226	C₂×C₃×C₄	—	2	2	no	—
227	C₂×C₃×C₄	—	2	2	no	—
228	C₂×C₃×C₄	—	2	1	no	—
229	C₂×C₃×C₄	—	2	1	no	—
230	C₂×C₃×C₄	—	2	1	no	—
231	C₂×C₃×C₄	—	2	1	no	—
232	C₂×C₃×C₄	—	2	2	no	—
233	C₂×C₃×C₄	—	2	2	no	—
234	C₂×C₃×C₄	—	2	1	no	—
235	C₂×C₃×C₄	—	2	1	no	—
236	C₂×C₃×C₄	—	2	1	no	—
237	C₂×C₃×C₄	—	2	1	no	—
238	C₂×C₃×C₄	—	2	1	no	—
239	C₂×C₃×C₄	—	2	1	no	—
240	C₂×C₃×C₄	—	6	1	no	—
241	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
242	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
243	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
245	#67	—	4	2	no	—
246	C₂×Dic₃	—	2	1	no	—
247	D₁₂	—	2	1	no	—
248	D₁₂	—	2	1	no	—
249	C₃×C₈	—	6	1	no	—
250	C₃×C₈	—	6	1	no	—
251	C₃⋊C₈	—	2	1	no	—
252	C₃⋊C₈	—	2	1	no	—
253	C₂×C₃²	—	4	1	no	—
254	C₂×C₃²	C₃×#123	1	1	yes	characteristic
255	C₂×C₃²	C₃×#135	1	1	yes	characteristic
256	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
257	C₃×D₃	—	8	1	no	—
258	C₂×C₉	—	2	1	no	—
259	C₂×C₉	—	2	1	no	—
260	C₂×C₉	—	2	1	no	—
261	C₂×C₉	—	4	2	no	—
262	C₂×C₉	—	4	1	no	—
263	C₂×C₉	—	24	1	no	—
264	C₂×C₉	#929	1	1	yes	characteristic
265	C₂×C₉	#949	1	1	yes	characteristic
266	C₂×C₉	#950	1	1	yes	characteristic
267	C₂×D₄	—	3	1	no	—
268	C₂×D₄	—	6	1	no	—
269	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
270	C₂×C₈	—	3	1	no	—
271	C₂×C₈	—	3	1	no	—
272	C₄⋊C₄	—	3	1	no	—
273	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
274	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	6	1	no	—
277	C₄²	—	2	1	no	—
278	C₂²×C₃	—	2	2	no	—
279	C₂²×C₃	—	2	1	no	—
280	C₂²×C₃	—	2	1	no	—
281	C₂²×C₃	—	2	1	no	—
282	C₂²×C₃	—	2	1	no	—
283	C₂²×C₃	—	2	1	no	—
284	C₂²×C₃	—	2	1	no	—
285	C₂²×C₃	—	2	1	no	—
286	C₂²×C₃	—	4	1	no	—
287	C₂²×C₃	—	4	1	no	—
288	C₂²×C₃	—	12	1	no	—
289	C₂²×C₃	—	12	1	no	—
290	C₂²×C₃	—	12	1	no	—
291	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
292	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
293	C₂×D₃	—	4	1	no	—
294	C₂×D₃	—	4	1	no	—
295	C₂×D₃	—	4	1	no	—
296	C₃×C₄	—	2	2	no	—
297	C₃×C₄	—	2	1	no	—
298	C₃×C₄	—	2	1	no	—
299	C₃×C₄	—	2	1	no	—
300	C₃×C₄	—	2	1	no	—
301	C₃×C₄	—	4	2	no	—
302	C₃×C₄	—	4	2	no	—
303	C₃×C₄	—	4	1	no	—
304	C₃×C₄	—	4	1	no	—
305	C₃×C₄	—	4	1	no	—
306	C₃×C₄	—	4	1	no	—
307	C₃×C₄	—	12	1	no	—
308	C₃×C₄	C₉×#41	1	1	yes	characteristic
309	C₃×C₄	C₉×#41	1	1	yes	characteristic
310	C₃×C₄	C₃×#235	1	1	yes	characteristic
311	C₃×C₄	C₃×#235	1	1	yes	characteristic
312	Dic₃	—	4	1	no	—
313	C₃²	C₃×#482	1	1	yes	characteristic
314	C₉	—	2	1	no	—
315	C₉	#5003	1	1	yes	characteristic, direct product factor
316	C₂³	—	6	1	no	—
317	C₂³	C₉×#67	1	1	yes	characteristic
318	D₄	—	6	1	no	—
319	D₄	—	6	1	no	—
320	D₄	—	12	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	6	1	no	—
326	C₂×C₄	C₉×#67	1	1	yes	characteristic
327	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
328	C₈	—	6	1	no	—
329	C₈	—	6	1	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	—	2	1	no	—
332	C₂×C₃	—	2	1	no	—
333	C₂×C₃	—	4	2	no	—
334	C₂×C₃	—	4	1	no	—
335	C₂×C₃	—	4	1	no	—
336	C₂×C₃	—	24	1	no	—
337	C₂×C₃	C₉×#123	1	1	yes	characteristic
338	C₂×C₃	C₉×#135	1	1	yes	characteristic
339	C₂×C₃	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
340	C₂×C₃	C₃×#929	1	1	yes	characteristic
341	C₂×C₃	C₃×#949	1	1	yes	characteristic
342	C₂×C₃	C₃×#950	1	1	yes	characteristic
343	D₃	—	8	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	12	1	no	—
348	C₂²	—	12	1	no	—
349	C₂²	—	12	1	no	—
350	C₂²	C₂×C₉×#67	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	12	1	no	—
355	C₄	C₉×#235	1	1	yes	characteristic
356	C₄	C₉×#235	1	1	yes	characteristic
357	C₃	—	2	1	no	—
358	C₃	C₉×#482	1	1	yes	characteristic, complemented
359	C₃	C₃×#5003	1	1	yes	characteristic
360	C₂	—	4	1	no	—
361	C₂	—	24	1	no	—
362	C₂	C₉×#929	1	1	yes	characteristic
363	C₂	C₉×#949	1	1	yes	characteristic
364	C₂	C₉×#950	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×#67
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#117	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃×#930	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#117	D₃
hypocenter	C₃	C₉×#482
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ C₉×#117

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×#117	C₂	semidirect
3. C₉×#830	C₂	semidirect
4. C₉×#809	C₂	semidirect
5. C₉×#809	C₂	semidirect
6. C₉×#808	C₂	semidirect
7. C₉×#807	C₂	semidirect
30. C₉×#117	D₃	semidirect
51. #5003	C₉	direct (N × H)
94. C₃×#117	C₂×C₉	semidirect
102. #830	C₂×C₉	semidirect
103. #809	C₂×C₉	semidirect
104. #809	C₂×C₉	semidirect
105. #808	C₂×C₉	semidirect
106. #807	C₂×C₉	semidirect
211. #117	D₃×C₉	semidirect
217. C₃×C₉	#482	semidirect
315. C₉	#5003	direct (N × H)
358. C₃	C₉×#482	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#117	2
C₉×#930	2
C₉×#830	2
C₉×#809	2
C₉×#808	2
C₉×#807	2
C₃×#5003	3
C₉×#482	3

Subgroups — C9×#5003 #13142685

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5003 #13142685