Subgroups — C₉×#5535 #13152506

Order 1728.

796 conjugacy classes of subgroups.

The subgroup lattice has 340 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	2	yes	maximal, complemented
3	C₉×#894	C₂	1	4	yes	maximal, complemented
4	C₉×#884	C₂	1	4	yes	maximal, complemented
5	C₉×#874	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₉×D₁₂	C₂	1	2	yes	maximal, complemented
7	C₉×#870	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5535	C₃	1	1	yes	maximal, characteristic
9	C₉×#540	—	3	1	no	maximal
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₉×#67	C₂²	1	4	yes	—
12	C₂×C₉×D₁₂	C₂²	1	2	yes	—
13	C₂×C₉×D₁₂	C₂²	1	4	yes	—
14	C₂×C₄×D₃×C₉	C₂²	1	4	yes	—
15	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
18	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
19	C₉×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
20	C₉×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
21	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
22	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
23	C₃²×#117	C₂×C₃	1	1	yes	characteristic
24	C₃×#930	C₂×C₃	1	2	yes	—
25	C₃×#894	C₂×C₃	1	4	yes	—
26	C₃×#884	C₂×C₃	1	4	yes	—
27	C₃×#874	C₂×C₃	1	1	yes	characteristic
28	C₃×C₄×D₁₂	C₂×C₃	1	2	yes	—
29	C₃×#870	C₂×C₃	1	1	yes	characteristic
30	C₉×#127	—	3	1	no	—
31	C₉×#125	—	3	1	no	—
32	C₉×#123	—	3	4	no	—
33	C₉×#121	—	3	2	no	—
34	C₉×#121	—	3	4	no	—
35	C₄×D₄×C₉	—	3	2	no	—
36	C₉×#117	—	2	1	no	—
37	C₉×#117	D₃	1	1	yes	characteristic, complemented
38	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
39	C₂²×D₃×C₉	C₂³	1	4	yes	—
40	C₂×C₃×C₄×C₉	—	2	2	no	—
41	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
42	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
43	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₉×#67	—	4	4	no	—
46	C₂×C₉×Dic₃	C₂³	1	4	yes	—
47	C₉×D₁₂	—	2	4	no	—
48	C₉×D₁₂	—	4	4	no	—
49	C₄×D₃×C₉	—	4	4	no	—
50	C₃×#540	—	3	1	no	—
51	#5535	C₉	1	1	yes	characteristic, direct product factor
52	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₃×#67	C₂²×C₃	1	4	yes	—
54	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
55	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
56	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
57	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
58	C₃²×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
59	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
60	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
61	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
62	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
63	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
64	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
65	C₂×D₄×C₉	—	3	4	no	—
66	C₂×D₄×C₉	—	3	2	no	—
67	C₂×D₄×C₉	—	3	4	no	—
68	C₂²×C₄×C₉	—	2	1	no	—
69	C₂²×C₄×C₉	—	3	4	no	—
70	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
71	C₉×C₄⋊C₄	—	2	2	no	—
72	C₉×C₄⋊C₄	—	3	2	no	—
73	C₉×C₄⋊C₄	—	3	4	no	—
74	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
75	C₉×C₂²⋊C₄	—	2	2	no	—
76	C₉×C₂²⋊C₄	—	3	4	no	—
77	C₉×C₂²⋊C₄	—	3	4	no	—
78	C₉×C₂²⋊C₄	C₂×D₃	1	2	yes	—
79	C₄²×C₉	—	2	1	no	—
80	C₄²×C₉	—	2	1	no	—
81	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
82	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
83	C₂²×C₃×C₉	—	2	1	no	—
84	C₂²×C₃×C₉	—	2	2	no	—
85	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
86	C₂×D₃×C₉	—	2	4	no	—
87	C₂×D₃×C₉	—	2	4	no	—
88	C₂×D₃×C₉	—	2	4	no	—
89	C₃×C₄×C₉	—	2	1	no	—
90	C₃×C₄×C₉	—	2	2	no	—
91	C₃×C₄×C₉	—	2	2	no	—
92	C₃×C₄×C₉	#41	1	2	yes	—
93	C₉×Dic₃	—	2	4	no	—
94	C₃×#127	—	3	1	no	—
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	—	2	1	no	—
101	C₃×#117	C₂×C₉	1	1	yes	characteristic, complemented
102	C₃×#117	C₃×D₃	1	1	yes	characteristic
103	#930	C₂×C₉	1	2	yes	complemented
104	#894	C₂×C₉	1	4	yes	complemented
105	#884	C₂×C₉	1	4	yes	complemented
106	#874	C₂×C₉	1	1	yes	characteristic, complemented
107	C₄×D₁₂	C₂×C₉	1	2	yes	complemented
108	#870	C₂×C₉	1	1	yes	characteristic, complemented
109	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
110	C₂²×C₃×D₃	C₂³×C₃	1	4	yes	—
111	C₂×C₃²×C₄	—	2	2	no	—
112	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
113	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
114	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
115	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
116	C₃×#67	—	4	4	no	—
117	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
118	C₃×D₁₂	—	2	4	no	—
119	C₃×D₁₂	—	4	4	no	—
120	C₃×C₄×D₃	—	4	4	no	—
121	C₂³×C₉	—	2	1	no	—
122	C₂³×C₉	—	3	4	no	—
123	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
124	D₄×C₉	—	6	4	no	—
125	D₄×C₉	—	12	4	no	—
126	D₄×C₉	—	12	4	no	—
127	C₂×C₄×C₉	—	2	4	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	—	3	4	no	—
134	C₂×C₄×C₉	—	12	4	no	—
135	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
136	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
137	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
138	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
139	#540	—	3	1	no	Sylow
140	C₂×C₃×C₉	—	4	1	no	—
141	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
142	C₂×C₃×C₉	H(5,𝔽₂)	1	2	yes	—
143	D₃×C₉	—	4	4	no	—
144	C₂×C₃×D₄	—	3	4	no	—
145	C₂×C₃×D₄	—	3	2	no	—
146	C₂×C₃×D₄	—	3	4	no	—
147	C₂²×C₃×C₄	—	2	1	no	—
148	C₂²×C₃×C₄	—	3	4	no	—
149	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
150	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
151	C₂×#67	C₂²×C₉	1	4	yes	—
152	C₂×D₁₂	C₂²×C₉	1	2	yes	—
153	C₂×D₁₂	C₂²×C₉	1	4	yes	—
154	C₂×C₄×D₃	C₂²×C₉	1	4	yes	—
155	C₃×C₄⋊C₄	—	2	2	no	—
156	C₃×C₄⋊C₄	—	3	4	no	—
157	C₃×C₄⋊C₄	—	3	2	no	—
158	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
159	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
160	C₃×C₂²⋊C₄	—	2	2	no	—
161	C₃×C₂²⋊C₄	—	3	4	no	—
162	C₃×C₂²⋊C₄	—	3	4	no	—
163	C₃×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
164	C₃×C₂²⋊C₄	C₂×C₃×D₃	1	2	yes	—
165	C₃×C₄²	—	2	1	no	—
166	C₃×C₄²	—	2	1	no	—
167	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
168	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
169	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
170	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
171	(C₂×D₃)⋊C₄	C₂²×C₉	1	4	yes	—
172	(C₂×D₃)⋊C₄	C₂²×C₉	1	4	yes	—
173	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
174	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
175	C₂²×C₃²	—	2	1	no	—
176	C₂²×C₃²	—	2	2	no	—
177	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
178	C₂×C₃×D₃	—	2	4	no	—
179	C₂×C₃×D₃	—	2	4	no	—
180	C₂×C₃×D₃	—	2	4	no	—
181	C₃²×C₄	—	2	1	no	—
182	C₃²×C₄	—	2	2	no	—
183	C₃²×C₄	—	2	2	no	—
184	C₃²×C₄	C₃×#41	1	2	yes	—
185	C₃×Dic₃	—	2	4	no	—
186	C₂²×C₉	—	2	2	no	—
187	C₂²×C₉	—	2	1	no	—
188	C₂²×C₉	—	2	2	no	—
189	C₂²×C₉	—	2	1	no	—
190	C₂²×C₉	—	4	2	no	—
191	C₂²×C₉	—	6	4	no	—
192	C₂²×C₉	—	6	4	no	—
193	C₂²×C₉	—	6	4	no	—
194	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
195	C₄×C₉	—	2	2	no	—
196	C₄×C₉	—	2	1	no	—
197	C₄×C₉	—	2	2	no	—
198	C₄×C₉	—	2	2	no	—
199	C₄×C₉	—	2	2	no	—
200	C₄×C₉	—	4	2	no	—
201	C₄×C₉	—	4	2	no	—
202	C₄×C₉	—	6	4	no	—
203	C₄×C₉	#235	1	2	yes	—
204	#127	—	3	1	no	—
205	#125	—	3	1	no	—
206	#123	—	3	4	no	—
207	#121	—	3	2	no	—
208	#121	—	3	4	no	—
209	C₄×D₄	—	3	2	no	—
210	#117	D₃×C₉	1	1	yes	characteristic, complemented
211	C₃×C₉	#540	1	1	yes	characteristic, complemented, Sylow
212	C₂³×C₃	—	2	1	no	—
213	C₂³×C₃	—	3	4	no	—
214	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
215	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
216	C₂²×D₃	C₂³×C₉	1	4	yes	—
217	C₃×D₄	—	6	4	no	—
218	C₃×D₄	—	12	4	no	—
219	C₃×D₄	—	12	4	no	—
220	C₂×C₃×C₄	—	2	4	no	—
221	C₂×C₃×C₄	—	2	2	no	—
222	C₂×C₃×C₄	—	2	2	no	—
223	C₂×C₃×C₄	—	2	1	no	—
224	C₂×C₃×C₄	—	2	1	no	—
225	C₂×C₃×C₄	—	2	2	no	—
226	C₂×C₃×C₄	—	2	2	no	—
227	C₂×C₃×C₄	—	3	4	no	—
228	C₂×C₃×C₄	—	12	4	no	—
229	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
230	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
232	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
234	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
235	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
237	#67	—	4	4	no	—
238	C₂×Dic₃	C₂³×C₉	1	4	yes	—
239	D₁₂	—	2	4	no	—
240	D₁₂	—	4	4	no	—
241	C₄×D₃	—	4	4	no	—
242	C₂×C₃²	—	4	1	no	—
243	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
244	C₂×C₃²	C₃×H(5,𝔽₂)	1	2	yes	—
245	C₃×D₃	—	4	4	no	—
246	C₂×C₉	—	2	1	no	—
247	C₂×C₉	—	2	2	no	—
248	C₂×C₉	—	4	2	no	—
249	C₂×C₉	—	4	1	no	—
250	C₂×C₉	—	12	4	no	—
251	C₂×C₉	C₂×#235	1	1	yes	characteristic
252	C₂×C₉	#1009	1	2	yes	—
253	C₂×D₄	—	3	4	no	—
254	C₂×D₄	—	3	2	no	—
255	C₂×D₄	—	3	4	no	—
256	C₂²×C₄	—	3	4	no	—
257	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
258	C₄⋊C₄	—	3	2	no	—
259	C₄⋊C₄	—	3	4	no	—
260	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
261	C₂²⋊C₄	—	3	4	no	—
262	C₂²⋊C₄	—	3	4	no	—
263	C₂²⋊C₄	C₂×D₃×C₉	1	2	yes	—
264	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
265	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
266	C₂²×C₃	—	2	2	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	2	2	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	2	2	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	—	4	2	no	—
273	C₂²×C₃	—	6	4	no	—
274	C₂²×C₃	—	6	4	no	—
275	C₂²×C₃	—	6	4	no	—
276	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
277	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
278	C₂×D₃	—	2	4	no	—
279	C₂×D₃	—	2	4	no	—
280	C₂×D₃	—	2	4	no	—
281	C₃×C₄	—	2	2	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	2	2	no	—
284	C₃×C₄	—	2	2	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	2	2	no	—
287	C₃×C₄	—	2	2	no	—
288	C₃×C₄	—	2	2	no	—
289	C₃×C₄	—	4	2	no	—
290	C₃×C₄	—	4	2	no	—
291	C₃×C₄	—	6	4	no	—
292	C₃×C₄	C₉×#41	1	2	yes	—
293	C₃×C₄	C₃×#235	1	2	yes	—
294	Dic₃	—	2	4	no	—
295	C₃²	C₃×#540	1	1	yes	characteristic
296	C₉	—	2	1	no	—
297	C₉	#5535	1	1	yes	characteristic, direct product factor
298	C₂³	—	3	4	no	—
299	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
300	D₄	—	6	4	no	—
301	D₄	—	12	4	no	—
302	D₄	—	12	4	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	3	4	no	—
305	C₂×C₄	—	12	4	no	—
306	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
307	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
308	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
310	C₂×C₃	—	2	1	no	—
311	C₂×C₃	—	2	2	no	—
312	C₂×C₃	—	4	2	no	—
313	C₂×C₃	—	4	1	no	—
314	C₂×C₃	—	4	1	no	—
315	C₂×C₃	—	12	4	no	—
316	C₂×C₃	C₂×C₉×#41	1	1	yes	characteristic
317	C₂×C₃	C₉×H(5,𝔽₂)	1	2	yes	—
318	C₂×C₃	C₂×C₃×#235	1	1	yes	characteristic
319	C₂×C₃	C₃×#1009	1	2	yes	—
320	D₃	—	4	4	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	6	4	no	—
324	C₂²	—	6	4	no	—
325	C₂²	—	6	4	no	—
326	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
327	C₄	—	2	1	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	6	4	no	—
331	C₄	C₉×#235	1	2	yes	—
332	C₃	—	2	1	no	—
333	C₃	C₉×#540	1	1	yes	characteristic, complemented
334	C₃	C₃×#5535	1	1	yes	characteristic
335	C₂	—	4	1	no	—
336	C₂	—	12	4	no	—
337	C₂	C₂×C₉×#235	1	1	yes	characteristic
338	C₂	C₉×#1009	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×#117	C₂
socle, cosocle	C₂²×C₃²	C₂⁴×C₃
prime radical	C₃×#5535	C₃
hypercenter	C₉×#117	D₃
hypocenter	C₃	C₉×#540
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×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
2. C₉×#930	C₂	semidirect
3. C₉×#894	C₂	semidirect
4. C₉×#884	C₂	semidirect
5. C₉×#874	C₂	semidirect
6. C₄×C₉×D₁₂	C₂	semidirect
7. C₉×#870	C₂	semidirect
37. C₉×#117	D₃	semidirect
51. #5535	C₉	direct (N × H)
101. C₃×#117	C₂×C₉	semidirect
103. #930	C₂×C₉	semidirect
104. #894	C₂×C₉	semidirect
105. #884	C₂×C₉	semidirect
106. #874	C₂×C₉	semidirect
107. C₄×D₁₂	C₂×C₉	semidirect
108. #870	C₂×C₉	semidirect
210. #117	D₃×C₉	semidirect
211. C₃×C₉	#540	semidirect
297. C₉	#5535	direct (N × H)
333. 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×#117	2
C₉×#930	2
C₉×#894	2
C₉×#884	2
C₉×#874	2
C₄×C₉×D₁₂	2
C₉×#870	2
C₃×#5535	3
C₉×#540	3

Subgroups — C9×#5535 #13152506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5535 #13152506