Subgroups — C₄×C₉×#217 #13143518

Order 1728.

1010 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	C₂³×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#217	C₂	1	2	yes	maximal
3	C₂×C₄×C₉×Dic₃	C₂	1	2	yes	maximal, complemented
4	C₉×C₃⋊#95	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×#217	C₃	1	1	yes	maximal, characteristic
6	C₄×C₉×C₂²⋊C₄	—	3	1	no	maximal
7	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₉	—	2	4	no	—
9	C₂²×C₃×C₄×C₉	C₄	1	4	yes	complemented
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₉×Dic₃	C₂²	1	4	yes	—
13	C₉×#217	C₄	1	8	yes	direct product factor
14	C₄×C₉×Dic₃	—	2	4	no	—
15	C₂³×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
16	C₂×C₃×#217	C₂×C₃	1	2	yes	—
17	C₂×C₃×C₄×Dic₃	C₂×C₃	1	2	yes	—
18	C₂³×C₄×C₉	—	2	1	no	—
19	C₂³×C₄×C₉	D₃	1	1	yes	characteristic
20	C₃×C₃⋊#95	C₂×C₃	1	2	yes	—
21	C₂×C₉×C₂²⋊C₄	—	3	2	no	—
22	C₂×C₄²×C₉	—	3	2	no	—
23	C₉×#95	—	3	2	no	—
24	C₂³×C₃×C₉	—	2	2	no	—
25	C₂³×C₃×C₉	—	2	2	no	—
26	C₂³×C₃×C₉	C₂×C₄	1	4	yes	—
27	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
28	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	—	2	8	no	—
31	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×C₉	D₄	1	4	yes	—
34	C₂×C₉×Dic₃	—	2	4	no	—
35	C₂×C₉×Dic₃	—	2	4	no	—
36	C₂×C₉×Dic₃	C₂×C₄	1	8	yes	complemented
37	C₃×C₄×C₂²⋊C₄	—	3	1	no	—
38	C₄×#217	C₉	1	1	yes	characteristic, direct product factor
39	C₂⁴×C₃²	C₂²×C₃	1	1	yes	characteristic
40	C₂²×C₃²×C₄	—	2	4	no	—
41	C₂²×C₃²×C₄	C₃×C₄	1	4	yes	—
42	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
43	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
44	C₂²×C₃×Dic₃	C₂²×C₃	1	4	yes	—
45	C₂⁴×C₉	—	2	1	no	—
46	C₂⁴×C₉	C₂×D₃	1	1	yes	characteristic
47	C₃×#217	C₃×C₄	1	8	yes	—
48	C₃×C₄×Dic₃	—	2	4	no	—
49	C₂²×C₄×C₉	—	2	8	no	—
50	C₂²×C₄×C₉	—	2	4	no	—
51	C₂²×C₄×C₉	—	2	1	no	—
52	C₂²×C₄×C₉	—	2	1	no	—
53	C₂²×C₄×C₉	—	2	4	no	—
54	C₂²×C₄×C₉	—	3	4	no	—
55	C₂²×C₄×C₉	Dic₃	1	4	yes	complemented
56	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
57	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
58	C₉×C₂²⋊C₄	—	3	8	no	—
59	C₄²×C₉	—	6	4	no	—
60	C₂²×C₃×C₉	—	2	2	no	—
61	C₂²×C₃×C₉	—	2	2	no	—
62	C₂²×C₃×C₉	—	2	8	no	—
63	C₂²×C₃×C₉	C₄²	1	4	yes	complemented
64	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₃×C₉	C₂²×C₄	1	2	yes	—
66	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
67	C₂²×C₃×C₉	#41	1	2	yes	—
68	C₃×C₄×C₉	—	2	2	no	—
69	C₃×C₄×C₉	C₂²⋊C₄	1	4	yes	complemented
70	C₉×Dic₃	—	2	8	no	—
71	C₂³×C₃×C₄	—	2	1	no	—
72	C₂³×C₃×C₄	C₂×C₉	1	1	yes	characteristic
73	C₂³×C₃×C₄	C₃×D₃	1	1	yes	characteristic
74	C₂×C₃×C₂²⋊C₄	—	3	2	no	—
75	C₂×C₃×C₄²	—	3	2	no	—
76	C₂×#217	C₂×C₉	1	2	yes	—
77	C₂×C₄×Dic₃	C₂×C₉	1	2	yes	complemented
78	C₃×#95	—	3	2	no	—
79	C₃⋊#95	C₂×C₉	1	2	yes	complemented
80	C₂³×C₃²	—	2	2	no	—
81	C₂³×C₃²	—	2	2	no	—
82	C₂³×C₃²	C₂×C₃×C₄	1	4	yes	—
83	C₂³×C₃²	C₂×C₃×C₄	1	2	yes	—
84	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
85	C₂×C₃²×C₄	—	2	2	no	—
86	C₂×C₃²×C₄	—	2	8	no	—
87	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
88	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
89	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
90	C₂×C₃×Dic₃	—	2	4	no	—
91	C₂×C₃×Dic₃	—	2	4	no	—
92	C₂×C₃×Dic₃	C₂×C₃×C₄	1	8	yes	—
93	C₂³×C₉	—	2	4	no	—
94	C₂³×C₉	—	2	4	no	—
95	C₂³×C₉	—	2	2	no	—
96	C₂³×C₉	—	2	2	no	—
97	C₂³×C₉	—	2	4	no	—
98	C₂³×C₉	—	2	2	no	—
99	C₂³×C₉	—	2	1	no	—
100	C₂³×C₉	C₄×D₃	1	4	yes	—
101	C₂³×C₉	C₂×Dic₃	1	2	yes	—
102	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
103	C₂×C₄×C₉	—	2	4	no	—
104	C₂×C₄×C₉	—	2	16	no	—
105	C₂×C₄×C₉	—	2	2	no	—
106	C₂×C₄×C₉	—	2	8	no	—
107	C₂×C₄×C₉	—	2	2	no	—
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	—	2	4	no	—
110	C₂×C₄×C₉	—	3	8	no	—
111	C₂×C₄×C₉	—	6	4	no	—
112	C₂×C₄×C₉	—	6	4	no	—
113	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
114	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
115	C₂×C₄×C₉	#67	1	4	yes	—
116	C₄×C₂²⋊C₄	—	3	1	no	Sylow
117	C₂×C₃×C₉	—	2	4	no	—
118	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₃×C₉	#117	1	1	yes	characteristic
121	C₂×C₃×C₉	C₄×D₄	1	4	yes	—
122	C₂⁴×C₃	—	2	1	no	—
123	C₂⁴×C₃	C₂²×C₉	1	1	yes	characteristic
124	C₂⁴×C₃	C₂×C₃×D₃	1	1	yes	characteristic
125	C₂²×C₃×C₄	—	2	8	no	—
126	C₂²×C₃×C₄	—	2	4	no	—
127	C₂²×C₃×C₄	—	2	4	no	—
128	C₂²×C₃×C₄	—	2	4	no	—
129	C₂²×C₃×C₄	—	2	1	no	—
130	C₂²×C₃×C₄	—	2	1	no	—
131	C₂²×C₃×C₄	—	3	4	no	—
132	C₂²×C₃×C₄	C₄×C₉	1	4	yes	complemented
133	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
134	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
135	C₂²×C₃×C₄	C₃×Dic₃	1	4	yes	—
136	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
137	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
138	C₂²×Dic₃	C₂²×C₉	1	4	yes	—
139	C₃×C₂²⋊C₄	—	3	8	no	—
140	C₃×C₄²	—	6	4	no	—
141	#217	C₄×C₉	1	8	yes	direct product factor
142	C₄×Dic₃	—	2	4	no	—
143	C₂²×C₃²	—	2	2	no	—
144	C₂²×C₃²	—	2	2	no	—
145	C₂²×C₃²	—	2	8	no	—
146	C₂²×C₃²	C₃×C₄²	1	4	yes	—
147	C₂²×C₃²	C₂²×C₃×C₄	1	2	yes	—
148	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
149	C₂²×C₃²	C₂×C₃×D₄	1	2	yes	—
150	C₂²×C₃²	C₃×#41	1	2	yes	—
151	C₃²×C₄	—	2	2	no	—
152	C₃²×C₄	C₃×C₂²⋊C₄	1	4	yes	—
153	C₃×Dic₃	—	2	8	no	—
154	C₂²×C₉	—	2	4	no	—
155	C₂²×C₉	—	2	4	no	—
156	C₂²×C₉	—	2	16	no	—
157	C₂²×C₉	—	2	2	no	—
158	C₂²×C₉	—	2	8	no	—
159	C₂²×C₉	—	2	2	no	—
160	C₂²×C₉	—	2	4	no	—
161	C₂²×C₉	—	2	2	no	—
162	C₂²×C₉	—	2	2	no	—
163	C₂²×C₉	—	2	1	no	—
164	C₂²×C₉	—	2	2	no	—
165	C₂²×C₉	C₄×Dic₃	1	4	yes	complemented
166	C₂²×C₉	C₂×C₄×D₃	1	2	yes	—
167	C₂²×C₉	#235	1	2	yes	—
168	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
169	C₂²×C₉	C₂×#67	1	2	yes	—
170	C₄×C₉	—	2	4	no	—
171	C₄×C₉	—	2	2	no	—
172	C₄×C₉	—	2	4	no	—
173	C₄×C₉	—	6	8	no	—
174	C₄×C₉	#217	1	4	yes	direct product factor
175	C₂³×C₄	D₃×C₉	1	1	yes	characteristic
176	C₂×C₂²⋊C₄	—	3	2	no	—
177	C₂×C₄²	—	3	2	no	—
178	#95	—	3	2	no	—
179	C₃×C₉	C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₃	—	2	4	no	—
181	C₂³×C₃	—	2	4	no	—
182	C₂³×C₃	—	2	2	no	—
183	C₂³×C₃	—	2	2	no	—
184	C₂³×C₃	—	2	2	no	—
185	C₂³×C₃	—	2	2	no	—
186	C₂³×C₃	—	2	4	no	—
187	C₂³×C₃	—	2	2	no	—
188	C₂³×C₃	—	2	1	no	—
189	C₂³×C₃	C₂×C₄×C₉	1	4	yes	—
190	C₂³×C₃	C₂×C₄×C₉	1	2	yes	—
191	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
192	C₂³×C₃	C₃×C₄×D₃	1	4	yes	—
193	C₂³×C₃	C₂×C₃×Dic₃	1	2	yes	—
194	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
195	C₂×C₃×C₄	—	2	4	no	—
196	C₂×C₃×C₄	—	2	16	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	2	8	no	—
200	C₂×C₃×C₄	—	2	8	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	4	no	—
204	C₂×C₃×C₄	—	3	8	no	—
205	C₂×C₃×C₄	—	6	4	no	—
206	C₂×C₃×C₄	—	6	4	no	—
207	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
208	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
209	C₂×C₃×C₄	D₄×C₉	1	4	yes	—
210	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
211	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
212	C₂×C₃×C₄	C₃×#67	1	4	yes	—
213	C₂×Dic₃	—	2	4	no	—
214	C₂×Dic₃	—	2	4	no	—
215	C₂×Dic₃	C₂×C₄×C₉	1	8	yes	complemented
216	C₂×C₃²	—	2	4	no	—
217	C₂×C₃²	C₂×C₃×C₄²	1	1	yes	characteristic
218	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₃²	C₃×#117	1	1	yes	characteristic
220	C₂×C₃²	C₃×C₄×D₄	1	4	yes	—
221	C₂×C₉	—	2	8	no	—
222	C₂×C₉	—	2	4	no	—
223	C₂×C₉	—	2	1	no	—
224	C₂×C₉	—	2	1	no	—
225	C₂×C₉	—	2	4	no	—
226	C₂×C₉	—	2	1	no	—
227	C₂×C₉	C₂×C₄×Dic₃	1	1	yes	characteristic
228	C₂×C₉	#926	1	1	yes	characteristic
229	C₂×C₉	C₄×#67	1	4	yes	—
230	C₂×C₉	C₂×#217	1	1	yes	characteristic
231	C₂⁴	C₂×D₃×C₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	3	4	no	—
234	C₂²×C₄	C₉×Dic₃	1	4	yes	complemented
235	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
237	C₂²⋊C₄	—	3	8	no	—
238	C₄²	—	6	4	no	—
239	C₂²×C₃	—	2	4	no	—
240	C₂²×C₃	—	2	4	no	—
241	C₂²×C₃	—	2	16	no	—
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	2	8	no	—
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	—	2	8	no	—
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	2	2	no	—
249	C₂²×C₃	—	2	2	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	2	2	no	—
252	C₂²×C₃	—	2	4	no	—
253	C₂²×C₃	C₄²×C₉	1	4	yes	complemented
254	C₂²×C₃	C₂²×C₄×C₉	1	2	yes	—
255	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
256	C₂²×C₃	C₂×D₄×C₉	1	2	yes	—
257	C₂²×C₃	C₉×#41	1	2	yes	—
258	C₂²×C₃	C₃×C₄×Dic₃	1	4	yes	—
259	C₂²×C₃	C₂×C₃×C₄×D₃	1	2	yes	—
260	C₂²×C₃	C₃×#235	1	2	yes	—
261	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
262	C₂²×C₃	C₂×C₃×#67	1	2	yes	—
263	C₃×C₄	—	2	4	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	2	2	no	—
266	C₃×C₄	—	2	4	no	—
267	C₃×C₄	—	6	8	no	—
268	C₃×C₄	C₉×C₂²⋊C₄	1	4	yes	complemented
269	C₃×C₄	C₃×#217	1	4	yes	—
270	Dic₃	—	2	8	no	—
271	C₃²	C₃×C₄×C₂²⋊C₄	1	1	yes	characteristic
272	C₉	—	2	1	no	—
273	C₉	C₄×#217	1	1	yes	characteristic, direct product factor
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	C₄×D₃×C₉	1	4	yes	—
277	C₂³	C₂×C₉×Dic₃	1	2	yes	—
278	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	3	8	no	—
282	C₂×C₄	—	6	4	no	—
283	C₂×C₄	—	6	4	no	—
284	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
285	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
286	C₂×C₄	C₉×#67	1	4	yes	—
287	C₂×C₃	—	2	8	no	—
288	C₂×C₃	—	2	4	no	—
289	C₂×C₃	—	2	4	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	2	4	no	—
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	C₂×C₄²×C₉	1	1	yes	characteristic
295	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
296	C₂×C₃	C₉×#117	1	1	yes	characteristic
297	C₂×C₃	C₄×D₄×C₉	1	4	yes	—
298	C₂×C₃	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
299	C₂×C₃	C₃×#926	1	1	yes	characteristic
300	C₂×C₃	C₃×C₄×#67	1	4	yes	—
301	C₂×C₃	C₂×C₃×#217	1	1	yes	characteristic
302	C₂²	—	2	2	no	—
303	C₂²	—	2	8	no	—
304	C₂²	—	2	2	no	—
305	C₂²	C₄×C₉×Dic₃	1	4	yes	complemented
306	C₂²	C₂×C₄×D₃×C₉	1	2	yes	—
307	C₂²	C₉×#235	1	2	yes	—
308	C₂²	C₂²×C₉×Dic₃	1	1	yes	characteristic
309	C₂²	C₂×C₉×#67	1	2	yes	—
310	C₄	—	2	2	no	—
311	C₄	—	6	8	no	—
312	C₄	C₉×#217	1	4	yes	direct product factor
313	C₃	—	2	1	no	—
314	C₃	C₄×C₉×C₂²⋊C₄	1	1	yes	characteristic, complemented
315	C₃	C₃×C₄×#217	1	1	yes	characteristic
316	C₂	—	2	4	no	—
317	C₂	C₂×C₄×C₉×Dic₃	1	1	yes	characteristic
318	C₂	C₉×#926	1	1	yes	characteristic
319	C₂	C₄×C₉×#67	1	4	yes	—
320	C₂	C₂×C₉×#217	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₉	C₂×D₃
commutator	C₂×C₃	C₂×C₄²×C₉
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂³×C₃×C₄×C₉	C₂
socle, cosocle	C₂³×C₃²	C₂³×C₃
prime radical	C₂⁴×C₃²	C₂²×C₃
hypercenter	C₂³×C₄×C₉	D₃
hypocenter	C₃	C₄×C₉×C₂²⋊C₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ C₂³×C₄×C₉

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₉×Dic₃	C₂	semidirect
4. C₉×C₃⋊#95	C₂	semidirect
9. C₂²×C₃×C₄×C₉	C₄	semidirect
13. C₉×#217	C₄	direct (N × H)
36. C₂×C₉×Dic₃	C₂×C₄	semidirect
38. C₄×#217	C₉	direct (N × H)
55. C₂²×C₄×C₉	Dic₃	semidirect
63. C₂²×C₃×C₉	C₄²	semidirect
69. C₃×C₄×C₉	C₂²⋊C₄	semidirect
77. C₂×C₄×Dic₃	C₂×C₉	semidirect
79. C₃⋊#95	C₂×C₉	semidirect
132. C₂²×C₃×C₄	C₄×C₉	semidirect
141. #217	C₄×C₉	direct (N × H)
165. C₂²×C₉	C₄×Dic₃	semidirect
174. C₄×C₉	#217	direct (N × H)
179. C₃×C₉	C₄×C₂²⋊C₄	semidirect
215. C₂×Dic₃	C₂×C₄×C₉	semidirect
234. C₂²×C₄	C₉×Dic₃	semidirect
253. C₂²×C₃	C₄²×C₉	semidirect
268. C₃×C₄	C₉×C₂²⋊C₄	semidirect
273. C₉	C₄×#217	direct (N × H)
305. C₂²	C₄×C₉×Dic₃	semidirect
312. C₄	C₉×#217	direct (N × H)
314. C₃	C₄×C₉×C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₉	2
C₂×C₉×#217	2
C₂×C₄×C₉×Dic₃	2
C₉×C₃⋊#95	2
C₃×C₄×#217	3
C₄×C₉×C₂²⋊C₄	3

Subgroups — C4×C9×#217 #13143518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₉×#217 #13143518