Subgroups — C₉×#5719 #13153269

Order 1728.

828 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	C₃×C₉×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#939	C₂	1	4	yes	maximal, complemented
3	C₉×#938	C₂	1	2	yes	maximal, complemented
4	C₉×#935	C₂	1	4	yes	maximal, complemented
5	D₄×C₉×Dic₃	C₂	1	2	yes	maximal, complemented
6	C₉×#872	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#870	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5719	C₃	1	1	yes	maximal, characteristic
9	C₉×#540	—	3	1	no	maximal
10	C₂×C₃×D₄×C₉	C₂²	1	2	yes	—
11	C₂×C₃×D₄×C₉	C₂²	1	4	yes	—
12	C₂×C₉×#67	C₂²	1	4	yes	—
13	C₂²×C₉×Dic₃	C₂²	1	4	yes	—
14	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
15	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₉×#217	C₂²	1	4	yes	—
17	C₉×#217	C₂²	1	4	yes	—
18	C₉×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
19	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
20	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
21	C₉×Dic₃⋊C₄	C₂²	1	2	yes	—
22	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
23	C₃²×#127	C₂×C₃	1	1	yes	characteristic
24	C₃×#939	C₂×C₃	1	4	yes	—
25	C₃×#938	C₂×C₃	1	2	yes	—
26	C₃×#935	C₂×C₃	1	4	yes	—
27	C₃×D₄×Dic₃	C₂×C₃	1	2	yes	—
28	C₃×#872	C₂×C₃	1	1	yes	characteristic
29	C₃×#870	C₂×C₃	1	1	yes	characteristic
30	C₉×#127	—	2	1	no	—
31	C₉×#127	D₃	1	1	yes	characteristic, complemented
32	C₉×#125	—	3	1	no	—
33	C₉×#123	—	3	4	no	—
34	C₉×#121	—	3	2	no	—
35	C₉×#121	—	3	4	no	—
36	C₄×D₄×C₉	—	3	2	no	—
37	C₉×#117	—	3	1	no	—
38	C₂³×C₃×C₉	C₂³	1	4	yes	—
39	C₂²×D₃×C₉	C₂³	1	1	yes	characteristic
40	C₃×D₄×C₉	—	2	4	no	—
41	C₃×D₄×C₉	—	4	4	no	—
42	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
43	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₉×#67	—	4	4	no	—
45	C₂×C₉×Dic₃	—	4	4	no	—
46	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
47	C₂×C₉×Dic₃	C₂³	1	2	yes	—
48	C₂×C₉×Dic₃	C₂³	1	4	yes	—
49	C₄×D₃×C₉	—	2	2	no	—
50	C₃×#540	—	3	1	no	—
51	#5719	C₉	1	1	yes	characteristic, direct product factor
52	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
53	C₂×C₃²×D₄	C₂²×C₃	1	4	yes	—
54	C₂×C₃×#67	C₂²×C₃	1	4	yes	—
55	C₂²×C₃×Dic₃	C₂²×C₃	1	4	yes	—
56	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
57	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
58	C₃×#217	C₂²×C₃	1	4	yes	—
59	C₃×#217	C₂²×C₃	1	4	yes	—
60	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
61	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
62	C₃×Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
63	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
64	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
65	C₂×D₄×C₉	—	2	2	no	—
66	C₂×D₄×C₉	—	2	4	no	—
67	C₂×D₄×C₉	—	3	4	no	—
68	C₂×D₄×C₉	C₂×D₃	1	2	yes	—
69	C₂×D₄×C₉	C₂×D₃	1	4	yes	—
70	C₂²×C₄×C₉	—	3	4	no	—
71	C₂²×C₄×C₉	—	3	1	no	—
72	C₉×C₄⋊C₄	—	3	2	no	—
73	C₉×C₄⋊C₄	—	3	2	no	—
74	C₉×C₄⋊C₄	—	3	4	no	—
75	C₉×C₂²⋊C₄	—	3	4	no	—
76	C₉×C₂²⋊C₄	—	3	4	no	—
77	C₉×C₂²⋊C₄	—	3	2	no	—
78	C₄²×C₉	—	2	1	no	—
79	C₄²×C₉	—	3	1	no	—
80	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
81	C₂²×C₃×C₉	—	2	4	no	—
82	C₂²×C₃×C₉	—	2	4	no	—
83	C₂²×C₃×C₉	—	2	4	no	—
84	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
85	C₂×D₃×C₉	—	2	1	no	—
86	C₂×D₃×C₉	—	2	2	no	—
87	C₃×C₄×C₉	—	2	2	no	—
88	C₃×C₄×C₉	#41	1	2	yes	—
89	C₉×Dic₃	—	2	1	no	—
90	C₉×Dic₃	—	2	2	no	—
91	C₉×Dic₃	—	2	4	no	—
92	C₃×#127	—	2	1	no	—
93	C₃×#127	C₂×C₉	1	1	yes	characteristic, complemented
94	C₃×#127	C₃×D₃	1	1	yes	characteristic
95	C₃×#125	—	3	1	no	—
96	C₃×#123	—	3	4	no	—
97	C₃×#121	—	3	2	no	—
98	C₃×#121	—	3	4	no	—
99	C₃×C₄×D₄	—	3	2	no	—
100	C₃×#117	—	3	1	no	—
101	#939	C₂×C₉	1	4	yes	complemented
102	#938	C₂×C₉	1	2	yes	complemented
103	#935	C₂×C₉	1	4	yes	complemented
104	D₄×Dic₃	C₂×C₉	1	2	yes	complemented
105	#872	C₂×C₉	1	1	yes	characteristic, complemented
106	#870	C₂×C₉	1	1	yes	characteristic, complemented
107	C₂³×C₃²	C₂³×C₃	1	4	yes	—
108	C₂²×C₃×D₃	C₂³×C₃	1	1	yes	characteristic
109	C₃²×D₄	—	2	4	no	—
110	C₃²×D₄	—	4	4	no	—
111	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
112	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
113	C₃×#67	—	4	4	no	—
114	C₂×C₃×Dic₃	—	4	4	no	—
115	C₂×C₃×Dic₃	C₂³×C₃	1	2	yes	—
116	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
117	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
118	C₃×C₄×D₃	—	2	2	no	—
119	C₂³×C₉	—	2	4	no	—
120	C₂³×C₉	—	3	1	no	—
121	C₂³×C₉	C₂²×D₃	1	4	yes	—
122	D₄×C₉	—	2	4	no	—
123	D₄×C₉	—	4	4	no	—
124	D₄×C₉	—	4	4	no	—
125	D₄×C₉	—	4	8	no	—
126	D₄×C₉	—	12	4	no	—
127	C₂×C₄×C₉	—	2	2	no	—
128	C₂×C₄×C₉	—	2	1	no	—
129	C₂×C₄×C₉	—	3	2	no	—
130	C₂×C₄×C₉	—	3	1	no	—
131	C₂×C₄×C₉	—	3	4	no	—
132	C₂×C₄×C₉	—	6	2	no	—
133	C₂×C₄×C₉	—	12	4	no	—
134	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
135	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
136	#540	—	3	1	no	Sylow
137	C₂×C₃×C₉	—	4	4	no	—
138	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
139	C₂×C₃×C₉	H(5,𝔽₂)	1	2	yes	—
140	D₃×C₉	—	4	1	no	—
141	C₂×C₃×D₄	—	2	2	no	—
142	C₂×C₃×D₄	—	2	4	no	—
143	C₂×C₃×D₄	—	3	4	no	—
144	C₂×C₃×D₄	C₂²×C₉	1	2	yes	—
145	C₂×C₃×D₄	C₂²×C₉	1	4	yes	—
146	C₂×C₃×D₄	C₂×C₃×D₃	1	2	yes	—
147	C₂×C₃×D₄	C₂×C₃×D₃	1	4	yes	—
148	C₂²×C₃×C₄	—	3	4	no	—
149	C₂²×C₃×C₄	—	3	1	no	—
150	C₂×#67	C₂²×C₉	1	4	yes	—
151	C₂²×Dic₃	C₂²×C₉	1	4	yes	—
152	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
153	C₃×C₄⋊C₄	—	3	2	no	—
154	C₃×C₄⋊C₄	—	3	4	no	—
155	C₃×C₄⋊C₄	—	3	2	no	—
156	C₃×C₂²⋊C₄	—	3	4	no	—
157	C₃×C₂²⋊C₄	—	3	4	no	—
158	C₃×C₂²⋊C₄	—	3	2	no	—
159	C₃×C₄²	—	2	1	no	—
160	C₃×C₄²	—	3	1	no	—
161	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
162	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
163	#217	C₂²×C₉	1	4	yes	—
164	#217	C₂²×C₉	1	4	yes	—
165	(C₂×D₃)⋊C₄	C₂²×C₉	1	2	yes	—
166	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
167	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
168	Dic₃⋊C₄	C₂²×C₉	1	2	yes	—
169	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
170	C₂²×C₃²	—	2	4	no	—
171	C₂²×C₃²	—	2	4	no	—
172	C₂²×C₃²	—	2	4	no	—
173	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
174	C₂×C₃×D₃	—	2	1	no	—
175	C₂×C₃×D₃	—	2	2	no	—
176	C₃²×C₄	—	2	2	no	—
177	C₃²×C₄	C₃×#41	1	2	yes	—
178	C₃×Dic₃	—	2	1	no	—
179	C₃×Dic₃	—	2	2	no	—
180	C₃×Dic₃	—	2	4	no	—
181	C₂²×C₉	—	2	4	no	—
182	C₂²×C₉	—	2	4	no	—
183	C₂²×C₉	—	2	4	no	—
184	C₂²×C₉	—	2	1	no	—
185	C₂²×C₉	—	4	4	no	—
186	C₂²×C₉	—	4	4	no	—
187	C₂²×C₉	—	4	4	no	—
188	C₂²×C₉	—	6	1	no	—
189	C₂²×C₉	—	6	2	no	—
190	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
191	C₄×C₉	—	2	4	no	—
192	C₄×C₉	—	2	2	no	—
193	C₄×C₉	—	2	2	no	—
194	C₄×C₉	—	6	1	no	—
195	C₄×C₉	—	6	2	no	—
196	C₄×C₉	—	6	4	no	—
197	C₄×C₉	#237	1	2	yes	—
198	#127	D₃×C₉	1	1	yes	characteristic, complemented
199	#125	—	3	1	no	—
200	#123	—	3	4	no	—
201	#121	—	3	2	no	—
202	#121	—	3	4	no	—
203	C₄×D₄	—	3	2	no	—
204	#117	—	3	1	no	—
205	C₃×C₉	#540	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₃	—	2	4	no	—
207	C₂³×C₃	—	3	1	no	—
208	C₂³×C₃	C₂³×C₉	1	4	yes	—
209	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
210	C₂²×D₃	C₂³×C₉	1	1	yes	characteristic
211	C₃×D₄	—	2	4	no	—
212	C₃×D₄	—	2	4	no	—
213	C₃×D₄	—	4	4	no	—
214	C₃×D₄	—	4	4	no	—
215	C₃×D₄	—	4	4	no	—
216	C₃×D₄	—	4	8	no	—
217	C₃×D₄	—	12	4	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	2	1	no	—
220	C₂×C₃×C₄	—	3	2	no	—
221	C₂×C₃×C₄	—	3	1	no	—
222	C₂×C₃×C₄	—	3	4	no	—
223	C₂×C₃×C₄	—	6	2	no	—
224	C₂×C₃×C₄	—	12	4	no	—
225	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
226	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
228	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
229	#67	—	4	4	no	—
230	C₂×Dic₃	—	4	4	no	—
231	C₂×Dic₃	C₂³×C₉	1	2	yes	—
232	C₂×Dic₃	C₂³×C₉	1	4	yes	—
233	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
234	C₄×D₃	—	2	2	no	—
235	C₂×C₃²	—	4	4	no	—
236	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
237	C₂×C₃²	C₃×H(5,𝔽₂)	1	2	yes	—
238	C₃×D₃	—	4	1	no	—
239	C₂×C₉	—	2	1	no	—
240	C₂×C₉	—	2	2	no	—
241	C₂×C₉	—	4	4	no	—
242	C₂×C₉	—	8	4	no	—
243	C₂×C₉	—	12	1	no	—
244	C₂×C₉	C₂×#237	1	1	yes	characteristic
245	C₂×C₉	#1004	1	2	yes	—
246	C₂×D₄	—	3	4	no	—
247	C₂×D₄	C₂×D₃×C₉	1	2	yes	—
248	C₂×D₄	C₂×D₃×C₉	1	4	yes	—
249	C₂²×C₄	—	3	4	no	—
250	C₂²×C₄	—	3	1	no	—
251	C₄⋊C₄	—	3	2	no	—
252	C₄⋊C₄	—	3	2	no	—
253	C₄⋊C₄	—	3	4	no	—
254	C₂²⋊C₄	—	3	4	no	—
255	C₂²⋊C₄	—	3	4	no	—
256	C₂²⋊C₄	—	3	2	no	—
257	C₄²	—	3	1	no	—
258	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
259	C₂²×C₃	—	2	4	no	—
260	C₂²×C₃	—	2	4	no	—
261	C₂²×C₃	—	2	4	no	—
262	C₂²×C₃	—	2	4	no	—
263	C₂²×C₃	—	2	4	no	—
264	C₂²×C₃	—	2	4	no	—
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	—	4	4	no	—
267	C₂²×C₃	—	4	4	no	—
268	C₂²×C₃	—	4	4	no	—
269	C₂²×C₃	—	6	1	no	—
270	C₂²×C₃	—	6	2	no	—
271	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
272	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
273	C₂×D₃	—	2	1	no	—
274	C₂×D₃	—	2	2	no	—
275	C₃×C₄	—	2	4	no	—
276	C₃×C₄	—	2	2	no	—
277	C₃×C₄	—	2	2	no	—
278	C₃×C₄	—	2	2	no	—
279	C₃×C₄	—	6	1	no	—
280	C₃×C₄	—	6	2	no	—
281	C₃×C₄	—	6	4	no	—
282	C₃×C₄	C₉×#41	1	2	yes	—
283	C₃×C₄	C₃×#237	1	2	yes	—
284	Dic₃	—	2	1	no	—
285	Dic₃	—	2	2	no	—
286	Dic₃	—	2	4	no	—
287	C₃²	C₃×#540	1	1	yes	characteristic
288	C₉	—	2	1	no	—
289	C₉	#5719	1	1	yes	characteristic, direct product factor
290	C₂³	—	3	1	no	—
291	C₂³	C₂²×D₃×C₉	1	4	yes	—
292	D₄	—	2	4	no	—
293	D₄	—	4	4	no	—
294	D₄	—	12	4	no	—
295	C₂×C₄	—	3	2	no	—
296	C₂×C₄	—	3	1	no	—
297	C₂×C₄	—	3	4	no	—
298	C₂×C₄	—	6	2	no	—
299	C₂×C₄	—	12	4	no	—
300	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
301	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
302	C₂×C₃	—	2	1	no	—
303	C₂×C₃	—	2	2	no	—
304	C₂×C₃	—	4	4	no	—
305	C₂×C₃	—	4	4	no	—
306	C₂×C₃	—	8	4	no	—
307	C₂×C₃	—	12	1	no	—
308	C₂×C₃	C₂×C₉×#41	1	1	yes	characteristic
309	C₂×C₃	C₉×H(5,𝔽₂)	1	2	yes	—
310	C₂×C₃	C₂×C₃×#237	1	1	yes	characteristic
311	C₂×C₃	C₃×#1004	1	2	yes	—
312	D₃	—	4	1	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	2	4	no	—
316	C₂²	—	6	1	no	—
317	C₂²	—	6	2	no	—
318	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
319	C₄	—	2	2	no	—
320	C₄	—	6	1	no	—
321	C₄	—	6	2	no	—
322	C₄	—	6	4	no	—
323	C₄	C₉×#237	1	2	yes	—
324	C₃	—	2	1	no	—
325	C₃	C₉×#540	1	1	yes	characteristic, complemented
326	C₃	C₃×#5719	1	1	yes	characteristic
327	C₂	—	4	4	no	—
328	C₂	—	12	1	no	—
329	C₂	C₂×C₉×#237	1	1	yes	characteristic
330	C₂	C₉×#1004	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂³×D₃
commutator	C₂²×C₃	C₂⁴×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₉×#127	C₂
socle, cosocle	C₂²×C₃²	C₂⁴×C₃
prime radical	C₃×#5719	C₃
hypercenter	C₉×#127	D₃
hypocenter	C₃	C₉×#540
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#127

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×#127	C₂	semidirect
2. C₉×#939	C₂	semidirect
3. C₉×#938	C₂	semidirect
4. C₉×#935	C₂	semidirect
5. D₄×C₉×Dic₃	C₂	semidirect
6. C₉×#872	C₂	semidirect
7. C₉×#870	C₂	semidirect
31. C₉×#127	D₃	semidirect
51. #5719	C₉	direct (N × H)
93. C₃×#127	C₂×C₉	semidirect
101. #939	C₂×C₉	semidirect
102. #938	C₂×C₉	semidirect
103. #935	C₂×C₉	semidirect
104. D₄×Dic₃	C₂×C₉	semidirect
105. #872	C₂×C₉	semidirect
106. #870	C₂×C₉	semidirect
198. #127	D₃×C₉	semidirect
205. C₃×C₉	#540	semidirect
289. C₉	#5719	direct (N × H)
325. C₃	C₉×#540	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#127	2
C₉×#939	2
C₉×#938	2
C₉×#935	2
D₄×C₉×Dic₃	2
C₉×#872	2
C₉×#870	2
C₃×#5719	3
C₉×#540	3

Subgroups — C9×#5719 #13153269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5719 #13153269