Subgroups — C₉×#5493 #13152351

Order 1728.

1170 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	C₂²×C₉×#67	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#930	C₂	1	4	yes	maximal, complemented
4	C₉×#886	C₂	1	4	yes	maximal, complemented
5	C₉×#882	C₂	1	4	yes	maximal, complemented
6	C₃×#5493	C₃	1	1	yes	maximal, characteristic
7	C₉×#534	—	3	1	no	maximal
8	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
9	C₂³×D₃×C₉	C₂²	1	2	yes	complemented
10	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	complemented
11	C₂×C₉×#67	—	2	8	no	—
12	C₂×C₉×#67	C₂²	1	4	yes	—
13	C₂×C₉×#67	C₂²	1	4	yes	complemented
14	C₂²×C₉×Dic₃	C₂²	1	2	yes	complemented
15	C₂×C₉×D₁₂	C₂²	1	4	yes	complemented
16	C₃×C₉×C₂²⋊C₄	C₂²	1	4	yes	complemented
17	C₉×(C₂×D₃)⋊C₄	C₂²	1	8	yes	complemented
18	C₉×C₄⋊Dic₃	C₂²	1	4	yes	complemented
19	C₂²×C₃×#67	C₂×C₃	1	2	yes	—
20	C₂×C₃²×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
21	C₃×#930	C₂×C₃	1	4	yes	—
22	C₃×#886	C₂×C₃	1	4	yes	—
23	C₃×#882	C₂×C₃	1	4	yes	—
24	C₂²×D₄×C₉	—	3	2	no	—
25	C₉×#123	—	3	4	no	—
26	C₉×#121	—	3	4	no	—
27	C₉×#120	—	3	4	no	—
28	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
29	C₂×C₉×C₂²⋊C₄	D₃	1	1	yes	characteristic, complemented
30	C₂³×C₃×C₉	—	4	2	no	—
31	C₂³×C₃×C₉	D₄	1	4	yes	complemented
32	C₂³×C₃×C₉	C₂³	1	2	yes	—
33	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
34	C₂²×D₃×C₉	—	2	4	no	—
35	C₂²×D₃×C₉	—	4	4	no	—
36	C₂²×D₃×C₉	C₂³	1	4	yes	—
37	C₂×C₃×C₄×C₉	—	4	2	no	—
38	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
39	C₉×#67	—	2	8	no	—
40	C₉×#67	—	2	8	no	—
41	C₂×C₉×Dic₃	—	2	4	no	—
42	C₂×C₉×Dic₃	C₂³	1	4	yes	—
43	C₉×D₁₂	—	4	4	no	—
44	C₃×#534	—	3	1	no	—
45	#5493	C₉	1	1	yes	characteristic, direct product factor
46	C₂⁴×C₃²	C₂²×C₃	1	1	yes	characteristic
47	C₂³×C₃×D₃	C₂²×C₃	1	2	yes	—
48	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
49	C₂×C₃×#67	—	2	8	no	—
50	C₂×C₃×#67	C₂²×C₃	1	4	yes	—
51	C₂×C₃×#67	C₂²×C₃	1	4	yes	—
52	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
53	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
54	C₂⁴×C₉	—	2	1	no	—
55	C₂⁴×C₉	—	3	2	no	—
56	C₂⁴×C₉	C₂×D₃	1	1	yes	characteristic
57	C₃²×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
58	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	8	yes	—
59	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
60	C₂×D₄×C₉	—	3	4	no	—
61	C₂×D₄×C₉	—	3	4	no	—
62	C₂×D₄×C₉	—	3	4	no	—
63	C₂×D₄×C₉	—	6	8	no	—
64	C₂²×C₄×C₉	—	2	2	no	—
65	C₂²×C₄×C₉	—	3	2	no	—
66	C₂²×C₄×C₉	C₂×D₃	1	2	yes	complemented
67	C₉×C₄⋊C₄	—	3	4	no	—
68	C₉×C₂²⋊C₄	—	2	4	no	—
69	C₉×C₂²⋊C₄	—	3	8	no	—
70	C₉×C₂²⋊C₄	C₂×D₃	1	4	yes	complemented
71	C₂²×C₃×C₉	—	2	2	no	—
72	C₂²×C₃×C₉	—	2	4	no	—
73	C₂²×C₃×C₉	—	4	4	no	—
74	C₂²×C₃×C₉	C₂×D₄	1	4	yes	—
75	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
76	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
77	C₂×D₃×C₉	—	2	4	no	—
78	C₂×D₃×C₉	—	2	8	no	—
79	C₂×D₃×C₉	—	4	8	no	—
80	C₃×C₄×C₉	—	2	4	no	—
81	C₉×Dic₃	—	2	4	no	—
82	C₂²×C₃×D₄	—	3	2	no	—
83	C₂²×#67	C₂×C₉	1	2	yes	complemented
84	C₃×#123	—	3	4	no	—
85	C₃×#121	—	3	4	no	—
86	C₃×#120	—	3	4	no	—
87	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
88	C₂×C₃×C₂²⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
89	C₂×C₃×C₂²⋊C₄	C₃×D₃	1	1	yes	characteristic
90	#930	C₂×C₉	1	4	yes	complemented
91	#886	C₂×C₉	1	4	yes	complemented
92	#882	C₂×C₉	1	4	yes	complemented
93	C₂³×C₃²	—	4	2	no	—
94	C₂³×C₃²	C₃×D₄	1	4	yes	—
95	C₂³×C₃²	C₂³×C₃	1	2	yes	—
96	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
97	C₂²×C₃×D₃	—	2	4	no	—
98	C₂²×C₃×D₃	—	4	4	no	—
99	C₂²×C₃×D₃	C₂³×C₃	1	4	yes	—
100	C₂×C₃²×C₄	—	4	2	no	—
101	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
102	C₃×#67	—	2	8	no	—
103	C₃×#67	—	2	8	no	—
104	C₂×C₃×Dic₃	—	2	4	no	—
105	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
106	C₃×D₁₂	—	4	4	no	—
107	C₂³×C₉	—	2	4	no	—
108	C₂³×C₉	—	2	1	no	—
109	C₂³×C₉	—	2	2	no	—
110	C₂³×C₉	—	3	4	no	—
111	C₂³×C₉	—	4	2	no	—
112	C₂³×C₉	—	4	4	no	—
113	C₂³×C₉	—	6	4	no	—
114	C₂³×C₉	—	12	4	no	—
115	C₂³×C₉	D₁₂	1	4	yes	complemented
116	C₂³×C₉	C₂²×D₃	1	2	yes	—
117	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
118	D₄×C₉	—	6	8	no	—
119	D₄×C₉	—	6	8	no	—
120	D₄×C₉	—	12	4	no	—
121	C₂×C₄×C₉	—	2	4	no	—
122	C₂×C₄×C₉	—	3	4	no	—
123	C₂×C₄×C₉	—	4	2	no	—
124	C₂×C₄×C₉	—	4	4	no	—
125	C₂×C₄×C₉	—	6	4	no	—
126	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
127	#534	—	3	1	no	Sylow
128	C₂×C₃×C₉	—	2	2	no	—
129	C₂×C₃×C₉	—	2	4	no	—
130	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₃×C₉	H(5,𝔽₂)	1	2	yes	—
132	D₃×C₉	—	4	4	no	—
133	C₂⁴×C₃	—	2	1	no	—
134	C₂⁴×C₃	—	3	2	no	—
135	C₂⁴×C₃	C₂²×C₉	1	1	yes	characteristic
136	C₂⁴×C₃	C₂×C₃×D₃	1	1	yes	characteristic
137	C₂³×D₃	C₂²×C₉	1	2	yes	complemented
138	C₂×C₃×D₄	—	3	4	no	—
139	C₂×C₃×D₄	—	3	4	no	—
140	C₂×C₃×D₄	—	3	4	no	—
141	C₂×C₃×D₄	—	6	8	no	—
142	C₂²×C₃×C₄	—	2	2	no	—
143	C₂²×C₃×C₄	—	3	2	no	—
144	C₂²×C₃×C₄	C₂²×C₉	1	2	yes	complemented
145	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
146	C₂×#67	—	2	8	no	—
147	C₂×#67	C₂²×C₉	1	4	yes	—
148	C₂×#67	C₂²×C₉	1	4	yes	complemented
149	C₂²×Dic₃	C₂²×C₉	1	2	yes	complemented
150	C₂×D₁₂	C₂²×C₉	1	4	yes	complemented
151	C₃×C₄⋊C₄	—	3	4	no	—
152	C₃×C₂²⋊C₄	—	2	4	no	—
153	C₃×C₂²⋊C₄	—	3	8	no	—
154	C₃×C₂²⋊C₄	C₂²×C₉	1	4	yes	complemented
155	C₃×C₂²⋊C₄	C₂×C₃×D₃	1	4	yes	—
156	(C₂×D₃)⋊C₄	C₂²×C₉	1	8	yes	complemented
157	C₄⋊Dic₃	C₂²×C₉	1	4	yes	complemented
158	C₂²×C₃²	—	2	2	no	—
159	C₂²×C₃²	—	2	4	no	—
160	C₂²×C₃²	—	4	4	no	—
161	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
162	C₂²×C₃²	C₂×C₃×D₄	1	2	yes	—
163	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
164	C₂×C₃×D₃	—	2	4	no	—
165	C₂×C₃×D₃	—	2	8	no	—
166	C₂×C₃×D₃	—	4	8	no	—
167	C₃²×C₄	—	2	4	no	—
168	C₃×Dic₃	—	2	4	no	—
169	C₂²×C₉	—	2	4	no	—
170	C₂²×C₉	—	2	2	no	—
171	C₂²×C₉	—	2	4	no	—
172	C₂²×C₉	—	2	1	no	—
173	C₂²×C₉	—	2	2	no	—
174	C₂²×C₉	—	2	4	no	—
175	C₂²×C₉	—	4	4	no	—
176	C₂²×C₉	—	4	4	no	—
177	C₂²×C₉	—	4	8	no	—
178	C₂²×C₉	—	6	4	no	—
179	C₂²×C₉	—	6	8	no	—
180	C₂²×C₉	—	12	8	no	—
181	C₂²×C₉	C₂×D₁₂	1	4	yes	—
182	C₂²×C₉	C₂×D₁₂	1	2	yes	—
183	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
184	C₄×C₉	—	2	4	no	—
185	C₄×C₉	—	4	4	no	—
186	C₄×C₉	—	6	4	no	—
187	C₂²×D₄	—	3	2	no	—
188	#123	—	3	4	no	—
189	#121	—	3	4	no	—
190	#120	—	3	4	no	—
191	C₂×C₂²⋊C₄	D₃×C₉	1	1	yes	characteristic, complemented
192	C₃×C₉	#534	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₃	—	2	4	no	—
194	C₂³×C₃	—	2	1	no	—
195	C₂³×C₃	—	2	2	no	—
196	C₂³×C₃	—	3	4	no	—
197	C₂³×C₃	—	4	2	no	—
198	C₂³×C₃	—	4	2	no	—
199	C₂³×C₃	—	4	4	no	—
200	C₂³×C₃	—	6	4	no	—
201	C₂³×C₃	—	12	4	no	—
202	C₂³×C₃	D₄×C₉	1	4	yes	complemented
203	C₂³×C₃	C₂³×C₉	1	2	yes	—
204	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
205	C₂³×C₃	C₃×D₁₂	1	4	yes	—
206	C₂³×C₃	C₂²×C₃×D₃	1	2	yes	—
207	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
208	C₂²×D₃	—	2	4	no	—
209	C₂²×D₃	—	4	4	no	—
210	C₂²×D₃	C₂³×C₉	1	4	yes	—
211	C₃×D₄	—	6	8	no	—
212	C₃×D₄	—	6	8	no	—
213	C₃×D₄	—	12	4	no	—
214	C₂×C₃×C₄	—	2	4	no	—
215	C₂×C₃×C₄	—	3	4	no	—
216	C₂×C₃×C₄	—	4	2	no	—
217	C₂×C₃×C₄	—	4	2	no	—
218	C₂×C₃×C₄	—	4	4	no	—
219	C₂×C₃×C₄	—	6	4	no	—
220	C₂×C₃×C₄	C₂³×C₉	1	4	yes	—
221	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
222	#67	—	2	8	no	—
223	#67	—	2	8	no	—
224	C₂×Dic₃	—	2	4	no	—
225	C₂×Dic₃	C₂³×C₉	1	4	yes	—
226	D₁₂	—	4	4	no	—
227	C₂×C₃²	—	2	2	no	—
228	C₂×C₃²	—	2	4	no	—
229	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
230	C₂×C₃²	C₃×H(5,𝔽₂)	1	2	yes	—
231	C₃×D₃	—	4	4	no	—
232	C₂×C₉	—	2	4	no	—
233	C₂×C₉	—	2	4	no	—
234	C₂×C₉	—	2	2	no	—
235	C₂×C₉	—	2	1	no	—
236	C₂×C₉	—	2	2	no	—
237	C₂×C₉	—	4	4	no	—
238	C₂×C₉	—	12	4	no	—
239	C₂×C₉	C₂²×D₁₂	1	1	yes	characteristic
240	C₂×C₉	#1004	1	2	yes	—
241	C₂⁴	—	3	2	no	—
242	C₂⁴	C₂×D₃×C₉	1	1	yes	characteristic
243	C₂×D₄	—	3	4	no	—
244	C₂×D₄	—	3	4	no	—
245	C₂×D₄	—	3	4	no	—
246	C₂×D₄	—	6	8	no	—
247	C₂²×C₄	—	3	2	no	—
248	C₂²×C₄	C₂×D₃×C₉	1	2	yes	complemented
249	C₄⋊C₄	—	3	4	no	—
250	C₂²⋊C₄	—	3	8	no	—
251	C₂²⋊C₄	C₂×D₃×C₉	1	4	yes	complemented
252	C₂²×C₃	—	2	4	no	—
253	C₂²×C₃	—	2	2	no	—
254	C₂²×C₃	—	2	4	no	—
255	C₂²×C₃	—	2	2	no	—
256	C₂²×C₃	—	2	4	no	—
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	2	4	no	—
260	C₂²×C₃	—	4	4	no	—
261	C₂²×C₃	—	4	4	no	—
262	C₂²×C₃	—	4	4	no	—
263	C₂²×C₃	—	4	8	no	—
264	C₂²×C₃	—	6	4	no	—
265	C₂²×C₃	—	6	8	no	—
266	C₂²×C₃	—	12	8	no	—
267	C₂²×C₃	C₂×D₄×C₉	1	4	yes	—
268	C₂²×C₃	C₂×D₄×C₉	1	2	yes	—
269	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
270	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
271	C₂²×C₃	C₂×C₃×D₁₂	1	2	yes	—
272	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
273	C₂×D₃	—	2	4	no	—
274	C₂×D₃	—	2	8	no	—
275	C₂×D₃	—	4	8	no	—
276	C₃×C₄	—	2	4	no	—
277	C₃×C₄	—	2	4	no	—
278	C₃×C₄	—	4	4	no	—
279	C₃×C₄	—	6	4	no	—
280	Dic₃	—	2	4	no	—
281	C₃²	C₃×#534	1	1	yes	characteristic
282	C₉	—	2	1	no	—
283	C₉	#5493	1	1	yes	characteristic, direct product factor
284	C₂³	—	3	4	no	—
285	C₂³	—	4	2	no	—
286	C₂³	—	6	4	no	—
287	C₂³	—	12	4	no	—
288	C₂³	C₉×D₁₂	1	4	yes	complemented
289	C₂³	C₂²×D₃×C₉	1	2	yes	—
290	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
291	D₄	—	6	8	no	—
292	D₄	—	6	8	no	—
293	D₄	—	12	4	no	—
294	C₂×C₄	—	3	4	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	6	4	no	—
297	C₂×C₄	C₂²×D₃×C₉	1	4	yes	—
298	C₂×C₃	—	2	4	no	—
299	C₂×C₃	—	2	4	no	—
300	C₂×C₃	—	2	2	no	—
301	C₂×C₃	—	2	4	no	—
302	C₂×C₃	—	2	2	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	—	2	2	no	—
305	C₂×C₃	—	4	4	no	—
306	C₂×C₃	—	12	4	no	—
307	C₂×C₃	C₂²×D₄×C₉	1	1	yes	characteristic
308	C₂×C₃	C₉×H(5,𝔽₂)	1	2	yes	—
309	C₂×C₃	C₂²×C₃×D₁₂	1	1	yes	characteristic
310	C₂×C₃	C₃×#1004	1	2	yes	—
311	D₃	—	4	4	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	6	4	no	—
316	C₂²	—	6	8	no	—
317	C₂²	—	12	8	no	—
318	C₂²	C₂×C₉×D₁₂	1	4	yes	—
319	C₂²	C₂×C₉×D₁₂	1	2	yes	—
320	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	6	4	no	—
323	C₃	—	2	1	no	—
324	C₃	C₉×#534	1	1	yes	characteristic, complemented
325	C₃	C₃×#5493	1	1	yes	characteristic
326	C₂	—	2	4	no	—
327	C₂	—	2	2	no	—
328	C₂	—	12	4	no	—
329	C₂	C₂²×C₉×D₁₂	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	C₂×C₃×C₉×C₂²⋊C₄	C₂
socle, cosocle	C₂²×C₃²	C₂⁴×C₃
prime radical	C₃×#5493	C₃
hypercenter	C₂×C₉×C₂²⋊C₄	D₃
hypocenter	C₃	C₉×#534
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₉×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₉×#67	C₂	semidirect
2. C₂×C₃×C₉×C₂²⋊C₄	C₂	semidirect
3. C₉×#930	C₂	semidirect
4. C₉×#886	C₂	semidirect
5. C₉×#882	C₂	semidirect
9. C₂³×D₃×C₉	C₂²	semidirect
10. C₂²×C₃×C₄×C₉	C₂²	semidirect
13. C₂×C₉×#67	C₂²	semidirect
14. C₂²×C₉×Dic₃	C₂²	semidirect
15. C₂×C₉×D₁₂	C₂²	semidirect
16. C₃×C₉×C₂²⋊C₄	C₂²	semidirect
17. C₉×(C₂×D₃)⋊C₄	C₂²	semidirect
18. C₉×C₄⋊Dic₃	C₂²	semidirect
29. C₂×C₉×C₂²⋊C₄	D₃	semidirect
31. C₂³×C₃×C₉	D₄	semidirect
45. #5493	C₉	direct (N × H)
66. C₂²×C₄×C₉	C₂×D₃	semidirect
70. C₉×C₂²⋊C₄	C₂×D₃	semidirect
83. C₂²×#67	C₂×C₉	semidirect
88. C₂×C₃×C₂²⋊C₄	C₂×C₉	semidirect
90. #930	C₂×C₉	semidirect
91. #886	C₂×C₉	semidirect
92. #882	C₂×C₉	semidirect
115. C₂³×C₉	D₁₂	semidirect
137. C₂³×D₃	C₂²×C₉	semidirect
144. C₂²×C₃×C₄	C₂²×C₉	semidirect
148. C₂×#67	C₂²×C₉	semidirect
149. C₂²×Dic₃	C₂²×C₉	semidirect
150. C₂×D₁₂	C₂²×C₉	semidirect
154. C₃×C₂²⋊C₄	C₂²×C₉	semidirect
156. (C₂×D₃)⋊C₄	C₂²×C₉	semidirect
157. C₄⋊Dic₃	C₂²×C₉	semidirect
191. C₂×C₂²⋊C₄	D₃×C₉	semidirect
192. C₃×C₉	#534	semidirect
202. C₂³×C₃	D₄×C₉	semidirect
248. C₂²×C₄	C₂×D₃×C₉	semidirect
251. C₂²⋊C₄	C₂×D₃×C₉	semidirect
283. C₉	#5493	direct (N × H)
288. C₂³	C₉×D₁₂	semidirect
324. C₃	C₉×#534	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×#67	2
C₂×C₃×C₉×C₂²⋊C₄	2
C₉×#930	2
C₉×#886	2
C₉×#882	2
C₃×#5493	3
C₉×#534	3

Subgroups — C9×#5493 #13152351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5493 #13152351