Subgroups — C₉×#5207 #13143511

Order 1728.

806 conjugacy classes of subgroups.

The subgroup lattice has 300 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×#208	C₂	1	2	yes	maximal, complemented
3	C₉×#830	C₂	1	4	yes	maximal, complemented
4	C₃×#5207	C₃	1	1	yes	maximal, characteristic
5	C₉×#407	—	3	1	no	maximal
6	C₂⁴×C₃×C₉	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₉	—	2	4	no	—
8	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
10	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₉×#208	—	2	4	no	—
13	C₂×C₉×C₃⋊C₈	C₂²	1	4	yes	complemented
14	C₂³×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
15	C₂×C₃×#208	C₂×C₃	1	2	yes	—
16	C₂³×C₄×C₉	—	2	1	no	—
17	C₂³×C₄×C₉	D₃	1	1	yes	characteristic
18	C₃×#830	C₂×C₃	1	4	yes	—
19	C₂×C₉×M₁₆	—	3	2	no	—
20	C₉×C₂²⋊C₈	—	3	4	no	—
21	C₂³×C₃×C₉	—	2	2	no	—
22	C₂³×C₃×C₉	—	2	4	no	—
23	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄×C₉	—	2	2	no	—
26	C₂×C₃×C₄×C₉	—	2	4	no	—
27	C₂×C₃×C₄×C₉	—	2	4	no	—
28	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	D₄	1	4	yes	—
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₉×C₃⋊C₈	—	2	4	no	—
33	C₃×#407	—	3	1	no	—
34	#5207	C₉	1	1	yes	characteristic, direct product factor
35	C₂⁴×C₃²	C₃×C₄	1	1	yes	characteristic
36	C₂²×C₃²×C₄	—	2	4	no	—
37	C₂²×C₃²×C₄	C₃×C₄	1	1	yes	characteristic
38	C₂²×C₃²×C₄	C₃×C₄	1	2	yes	—
39	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
40	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
41	C₂⁴×C₉	—	2	1	no	—
42	C₂⁴×C₉	Dic₃	1	1	yes	characteristic
43	C₃×#208	—	2	4	no	—
44	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
45	C₂²×C₄×C₉	—	2	8	no	—
46	C₂²×C₄×C₉	—	2	1	no	—
47	C₂²×C₄×C₉	—	2	2	no	—
48	C₂²×C₄×C₉	—	2	2	no	—
49	C₂²×C₄×C₉	—	2	1	no	—
50	C₂²×C₄×C₉	—	2	4	no	—
51	C₂²×C₄×C₉	Dic₃	1	1	yes	characteristic
52	C₂²×C₄×C₉	Dic₃	1	2	yes	—
53	C₂²×C₄×C₉	C₂×D₃	1	2	yes	—
54	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
55	C₉×M₁₆	—	6	4	no	—
56	C₂×C₈×C₉	—	3	4	no	—
57	C₂²×C₃×C₉	—	2	8	no	—
58	C₂²×C₃×C₉	—	2	2	no	—
59	C₂²×C₃×C₉	—	2	2	no	—
60	C₂²×C₃×C₉	—	2	2	no	—
61	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₃×C₉	M₁₆	1	4	yes	complemented
63	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₃×C₄×C₉	—	2	2	no	—
65	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
66	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
67	C₂³×C₃×C₄	—	2	1	no	—
68	C₂³×C₃×C₄	C₂×C₉	1	1	yes	characteristic
69	C₂³×C₃×C₄	C₃×D₃	1	1	yes	characteristic
70	C₂×C₃×M₁₆	—	3	2	no	—
71	C₂×#208	C₂×C₉	1	2	yes	complemented
72	C₃×C₂²⋊C₈	—	3	4	no	—
73	#830	C₂×C₉	1	4	yes	complemented
74	C₂³×C₃²	—	2	2	no	—
75	C₂³×C₃²	—	2	4	no	—
76	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
77	C₂³×C₃²	C₂×C₃×C₄	1	2	yes	—
78	C₂×C₃²×C₄	—	2	2	no	—
79	C₂×C₃²×C₄	—	2	4	no	—
80	C₂×C₃²×C₄	—	2	4	no	—
81	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
82	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
84	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
85	C₂³×C₉	—	2	4	no	—
86	C₂³×C₉	—	2	8	no	—
87	C₂³×C₉	—	2	4	no	—
88	C₂³×C₉	—	2	2	no	—
89	C₂³×C₉	—	2	2	no	—
90	C₂³×C₉	—	2	1	no	—
91	C₂³×C₉	C₂×Dic₃	1	2	yes	—
92	C₂³×C₉	C₂×Dic₃	1	1	yes	characteristic
93	C₃×C₃⋊C₈	—	2	4	no	—
94	C₂×C₄×C₉	—	2	4	no	—
95	C₂×C₄×C₉	—	2	8	no	—
96	C₂×C₄×C₉	—	2	8	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	4	no	—
99	C₂×C₄×C₉	—	2	4	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	1	no	—
102	C₂×C₄×C₉	—	2	4	no	—
103	C₂×C₄×C₉	—	2	1	no	—
104	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
105	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
106	C₂×C₄×C₉	#67	1	4	yes	—
107	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
108	C₈×C₉	—	6	4	no	—
109	#407	—	3	1	no	Sylow
110	C₂×C₃×C₉	—	2	2	no	—
111	C₂×C₃×C₉	—	2	4	no	—
112	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₃×C₉	C₂×M₁₆	1	2	yes	—
114	C₂⁴×C₃	—	2	1	no	—
115	C₂⁴×C₃	C₄×C₉	1	1	yes	characteristic
116	C₂⁴×C₃	C₃×Dic₃	1	1	yes	characteristic
117	C₂²×C₃×C₄	—	2	8	no	—
118	C₂²×C₃×C₄	—	2	4	no	—
119	C₂²×C₃×C₄	—	2	4	no	—
120	C₂²×C₃×C₄	—	2	1	no	—
121	C₂²×C₃×C₄	—	2	2	no	—
122	C₂²×C₃×C₄	—	2	2	no	—
123	C₂²×C₃×C₄	—	2	1	no	—
124	C₂²×C₃×C₄	C₄×C₉	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₄×C₉	1	2	yes	—
126	C₂²×C₃×C₄	C₂²×C₉	1	2	yes	—
127	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₃×Dic₃	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₃×Dic₃	1	2	yes	—
130	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
131	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
132	C₃×M₁₆	—	6	4	no	—
133	C₂×C₃×C₈	—	3	4	no	—
134	#208	—	2	4	no	—
135	C₂×C₃⋊C₈	C₂²×C₉	1	4	yes	complemented
136	C₂²×C₃²	—	2	8	no	—
137	C₂²×C₃²	—	2	2	no	—
138	C₂²×C₃²	—	2	2	no	—
139	C₂²×C₃²	—	2	2	no	—
140	C₂²×C₃²	C₃×C₂²⋊C₄	1	2	yes	—
141	C₂²×C₃²	C₃×M₁₆	1	4	yes	—
142	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
143	C₃²×C₄	—	2	2	no	—
144	C₃²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
145	C₃²×C₄	C₂×C₃×D₄	1	2	yes	—
146	C₂²×C₉	—	2	4	no	—
147	C₂²×C₉	—	2	16	no	—
148	C₂²×C₉	—	2	8	no	—
149	C₂²×C₉	—	2	4	no	—
150	C₂²×C₉	—	2	4	no	—
151	C₂²×C₉	—	2	2	no	—
152	C₂²×C₉	—	2	2	no	—
153	C₂²×C₉	—	2	2	no	—
154	C₂²×C₉	—	2	4	no	—
155	C₂²×C₉	—	2	2	no	—
156	C₂²×C₉	—	2	1	no	—
157	C₂²×C₉	#208	1	4	yes	complemented
158	C₂²×C₉	#217	1	2	yes	—
159	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
160	C₄×C₉	—	2	4	no	—
161	C₄×C₉	—	2	2	no	—
162	C₄×C₉	—	2	2	no	—
163	C₄×C₉	—	2	2	no	—
164	C₄×C₉	#217	1	2	yes	—
165	C₄×C₉	C₂×#67	1	2	yes	—
166	C₂³×C₄	D₃×C₉	1	1	yes	characteristic
167	C₂×M₁₆	—	3	2	no	—
168	C₂²⋊C₈	—	3	4	no	—
169	C₃×C₉	#407	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₃	—	2	4	no	—
171	C₂³×C₃	—	2	8	no	—
172	C₂³×C₃	—	2	2	no	—
173	C₂³×C₃	—	2	4	no	—
174	C₂³×C₃	—	2	4	no	—
175	C₂³×C₃	—	2	2	no	—
176	C₂³×C₃	—	2	2	no	—
177	C₂³×C₃	—	2	1	no	—
178	C₂³×C₃	C₂×C₄×C₉	1	1	yes	characteristic
179	C₂³×C₃	C₂×C₄×C₉	1	2	yes	—
180	C₂³×C₃	C₂×C₃×Dic₃	1	2	yes	—
181	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
182	C₂×C₃×C₄	—	2	4	no	—
183	C₂×C₃×C₄	—	2	8	no	—
184	C₂×C₃×C₄	—	2	8	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	4	no	—
189	C₂×C₃×C₄	—	2	4	no	—
190	C₂×C₃×C₄	—	2	4	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	4	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
196	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
197	C₂×C₃×C₄	D₄×C₉	1	4	yes	—
198	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
200	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
201	C₂×C₃×C₄	C₃×#67	1	4	yes	—
202	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
203	C₃×C₈	—	6	4	no	—
204	C₃⋊C₈	—	2	4	no	—
205	C₂×C₃²	—	2	2	no	—
206	C₂×C₃²	—	2	4	no	—
207	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₃²	C₂×C₃×M₁₆	1	2	yes	—
209	C₂×C₉	—	2	8	no	—
210	C₂×C₉	—	2	4	no	—
211	C₂×C₉	—	2	4	no	—
212	C₂×C₉	—	2	2	no	—
213	C₂×C₉	—	2	2	no	—
214	C₂×C₉	—	2	1	no	—
215	C₂×C₉	C₂×#208	1	2	yes	—
216	C₂×C₉	C₂×#217	1	1	yes	characteristic
217	C₂⁴	C₉×Dic₃	1	1	yes	characteristic
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	C₉×Dic₃	1	1	yes	characteristic
220	C₂²×C₄	C₉×Dic₃	1	2	yes	—
221	C₂²×C₄	C₂×D₃×C₉	1	2	yes	—
222	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
223	M₁₆	—	6	4	no	—
224	C₂×C₈	—	3	4	no	—
225	C₂²×C₃	—	2	4	no	—
226	C₂²×C₃	—	2	8	no	—
227	C₂²×C₃	—	2	16	no	—
228	C₂²×C₃	—	2	8	no	—
229	C₂²×C₃	—	2	4	no	—
230	C₂²×C₃	—	2	4	no	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	4	no	—
239	C₂²×C₃	—	2	2	no	—
240	C₂²×C₃	C₉×C₂²⋊C₄	1	2	yes	—
241	C₂²×C₃	C₉×M₁₆	1	4	yes	complemented
242	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
243	C₂²×C₃	C₃×#208	1	4	yes	—
244	C₂²×C₃	C₃×#217	1	2	yes	—
245	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
246	C₃×C₄	—	2	4	no	—
247	C₃×C₄	—	2	2	no	—
248	C₃×C₄	—	2	2	no	—
249	C₃×C₄	—	2	2	no	—
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	C₉×C₂²⋊C₄	1	2	yes	—
252	C₃×C₄	C₂×D₄×C₉	1	2	yes	—
253	C₃×C₄	C₃×#217	1	2	yes	—
254	C₃×C₄	C₂×C₃×#67	1	2	yes	—
255	C₃²	C₃×#407	1	1	yes	characteristic
256	C₉	—	2	1	no	—
257	C₉	#5207	1	1	yes	characteristic, direct product factor
258	C₂³	—	2	4	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×C₉×Dic₃	1	2	yes	—
261	C₂³	C₂×C₉×Dic₃	1	1	yes	characteristic
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
266	C₂×C₄	C₂×C₉×Dic₃	1	1	yes	characteristic
267	C₂×C₄	C₉×#67	1	4	yes	—
268	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
269	C₈	—	6	4	no	—
270	C₂×C₃	—	2	8	no	—
271	C₂×C₃	—	2	4	no	—
272	C₂×C₃	—	2	2	no	—
273	C₂×C₃	—	2	4	no	—
274	C₂×C₃	—	2	4	no	—
275	C₂×C₃	—	2	2	no	—
276	C₂×C₃	—	2	2	no	—
277	C₂×C₃	—	2	1	no	—
278	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₃	C₂×C₉×M₁₆	1	2	yes	—
280	C₂×C₃	C₂×C₃×#208	1	2	yes	—
281	C₂×C₃	C₂×C₃×#217	1	1	yes	characteristic
282	C₂²	—	2	8	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	C₉×#208	1	4	yes	complemented
287	C₂²	C₉×#217	1	2	yes	—
288	C₂²	C₂²×C₉×Dic₃	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	C₉×#217	1	2	yes	—
291	C₄	C₂×C₉×#67	1	2	yes	—
292	C₃	—	2	1	no	—
293	C₃	C₉×#407	1	1	yes	characteristic, complemented
294	C₃	C₃×#5207	1	1	yes	characteristic
295	C₂	—	2	4	no	—
296	C₂	—	2	2	no	—
297	C₂	C₂×C₉×#208	1	2	yes	—
298	C₂	C₂×C₉×#217	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂³×C₃×C₄×C₉	C₂
socle	C₂²×C₃²	C₂²×C₃×C₄
prime radical	C₂⁴×C₃²	C₃×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂³×C₄×C₉	D₃
hypocenter	C₃	C₉×#407
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂³×C₄×C₉

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#208	C₂	semidirect
3. C₉×#830	C₂	semidirect
13. C₂×C₉×C₃⋊C₈	C₂²	semidirect
34. #5207	C₉	direct (N × H)
62. C₂²×C₃×C₉	M₁₆	semidirect
71. C₂×#208	C₂×C₉	semidirect
73. #830	C₂×C₉	semidirect
135. C₂×C₃⋊C₈	C₂²×C₉	semidirect
157. C₂²×C₉	#208	semidirect
169. C₃×C₉	#407	semidirect
241. C₂²×C₃	C₉×M₁₆	semidirect
257. C₉	#5207	direct (N × H)
286. C₂²	C₉×#208	semidirect
293. C₃	C₉×#407	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×#208	2
C₉×#830	2
C₃×#5207	3
C₉×#407	3

Subgroups — C9×#5207 #13143511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5207 #13143511