Subgroups — C₉×#5019 #13142737

Order 1728.

454 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#214	C₂	1	2	yes	maximal
3	C₉×#869	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#810	C₂	1	2	yes	maximal, complemented
5	C₉×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5019	C₃	1	1	yes	maximal, characteristic
7	C₉×#461	—	3	1	no	maximal
8	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	—	2	1	no	—
10	C₂×C₉×Dic₆	C₂²	1	2	yes	—
11	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₉×#214	—	2	4	no	—
15	C₉×C₄⋊Dic₃	—	2	1	no	—
16	C₉×C₄⋊Dic₃	—	2	1	no	—
17	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
18	C₃²×C₄×D₄	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×#214	C₂×C₃	1	2	yes	—
20	C₃×#869	C₂×C₃	1	1	yes	characteristic
21	C₃×#810	C₂×C₃	1	2	yes	—
22	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
23	C₂×C₉×SD₁₆	—	3	2	no	—
24	C₉×C₄⋊Q₈	—	3	1	no	—
25	C₄×D₄×C₉	—	2	1	no	—
26	C₄×D₄×C₉	D₃	1	1	yes	characteristic
27	C₉×C₄⋊C₈	—	3	1	no	—
28	C₉×#103	—	3	2	no	—
29	C₂³×C₃×C₉	—	2	1	no	—
30	C₃×D₄×C₉	—	2	1	no	—
31	C₃×D₄×C₉	D₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	—	2	1	no	—
33	C₂×C₃×C₄×C₉	—	4	2	no	—
34	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₂×C₉×Dic₃	—	2	2	no	—
38	C₉×Dic₆	—	2	2	no	—
39	C₉×Dic₆	—	2	2	no	—
40	C₉×C₃⋊C₈	—	2	2	no	—
41	C₃×#461	—	3	1	no	—
42	#5019	C₉	1	1	yes	characteristic, direct product factor
43	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
44	C₂²×C₃²×C₄	—	2	1	no	—
45	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
46	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
47	C₃²×C₂²⋊C₄	—	2	1	no	—
48	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
49	C₃×#214	—	2	4	no	—
50	C₃×C₄⋊Dic₃	—	2	1	no	—
51	C₃×C₄⋊Dic₃	—	2	1	no	—
52	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
53	C₂×Q₈×C₉	—	3	2	no	—
54	C₂×D₄×C₉	—	2	1	no	—
55	C₂×D₄×C₉	C₂×D₃	1	1	yes	characteristic
56	C₂²×C₄×C₉	—	2	2	no	—
57	C₂²×C₄×C₉	—	2	1	no	—
58	C₉×SD₁₆	—	6	4	no	—
59	C₂×C₈×C₉	—	3	2	no	—
60	C₉×C₄⋊C₄	—	2	1	no	—
61	C₉×C₄⋊C₄	—	6	1	no	—
62	C₉×C₄⋊C₄	—	6	1	no	—
63	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
64	C₉×C₂²⋊C₄	—	2	2	no	—
65	C₉×C₂²⋊C₄	—	2	1	no	—
66	C₄²×C₉	—	2	1	no	—
67	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
68	C₂²×C₃×C₉	—	2	2	no	—
69	C₂²×C₃×C₉	—	4	1	no	—
70	C₂²×C₃×C₉	—	4	1	no	—
71	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
72	C₃×C₄×C₉	—	2	1	no	—
73	C₃×C₄×C₉	—	4	1	no	—
74	C₃×C₄×C₉	SD₁₆	1	2	yes	complemented
75	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
76	C₃×C₄×C₉	#41	1	1	yes	characteristic
77	C₉×Dic₃	—	4	2	no	—
78	C₂×C₃×SD₁₆	—	3	2	no	—
79	C₃×C₄⋊Q₈	—	3	1	no	—
80	C₃×C₄×D₄	—	2	1	no	—
81	C₃×C₄×D₄	C₂×C₉	1	1	yes	characteristic
82	C₃×C₄×D₄	C₃×D₃	1	1	yes	characteristic
83	C₂×#214	C₂×C₉	1	2	yes	—
84	#869	C₂×C₉	1	1	yes	characteristic, complemented
85	C₃×C₄⋊C₈	—	3	1	no	—
86	C₃×#103	—	3	2	no	—
87	#810	C₂×C₉	1	2	yes	complemented
88	C₄⋊(C₃⋊C₈)	C₂×C₉	1	1	yes	characteristic, complemented
89	C₂³×C₃²	—	2	1	no	—
90	C₃²×D₄	—	2	1	no	—
91	C₃²×D₄	C₃×D₄	1	2	yes	—
92	C₂×C₃²×C₄	—	2	1	no	—
93	C₂×C₃²×C₄	—	4	2	no	—
94	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
97	C₂×C₃×Dic₃	—	2	2	no	—
98	C₃×Dic₆	—	2	2	no	—
99	C₃×Dic₆	—	2	2	no	—
100	C₂³×C₉	—	2	2	no	—
101	C₂³×C₉	—	2	1	no	—
102	C₃×C₃⋊C₈	—	2	2	no	—
103	Q₈×C₉	—	6	2	no	—
104	Q₈×C₉	—	6	2	no	—
105	D₄×C₉	—	2	2	no	—
106	D₄×C₉	—	2	1	no	—
107	D₄×C₉	—	2	2	no	—
108	D₄×C₉	D₁₂	1	2	yes	—
109	C₂×C₄×C₉	—	2	2	no	—
110	C₂×C₄×C₉	—	2	1	no	—
111	C₂×C₄×C₉	—	2	1	no	—
112	C₂×C₄×C₉	—	2	1	no	—
113	C₂×C₄×C₉	—	2	1	no	—
114	C₂×C₄×C₉	—	4	4	no	—
115	C₂×C₄×C₉	—	4	2	no	—
116	C₂×C₄×C₉	—	6	2	no	—
117	C₂×C₄×C₉	#67	1	1	yes	characteristic
118	C₂×C₄×C₉	#67	1	1	yes	characteristic
119	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
120	C₈×C₉	—	6	2	no	—
121	#461	—	3	1	no	Sylow
122	C₂×C₃×C₉	—	4	2	no	—
123	C₂×C₃×C₉	#121	1	1	yes	characteristic
124	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₃×C₉	#137	1	1	yes	characteristic
126	C₂×C₃×Q₈	—	3	2	no	—
127	C₂×C₃×D₄	—	2	1	no	—
128	C₂×C₃×D₄	C₂²×C₉	1	1	yes	characteristic
129	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
130	C₂²×C₃×C₄	—	2	2	no	—
131	C₂²×C₃×C₄	—	2	1	no	—
132	C₂²×C₃×C₄	—	2	1	no	—
133	C₂×Dic₆	C₂²×C₉	1	2	yes	—
134	C₃×SD₁₆	—	6	4	no	—
135	C₂×C₃×C₈	—	3	2	no	—
136	C₃×C₄⋊C₄	—	2	1	no	—
137	C₃×C₄⋊C₄	—	6	1	no	—
138	C₃×C₄⋊C₄	—	6	1	no	—
139	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
140	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
141	C₃×C₂²⋊C₄	—	2	2	no	—
142	C₃×C₂²⋊C₄	—	2	1	no	—
143	C₃×C₂²⋊C₄	—	2	1	no	—
144	C₃×C₄²	—	2	1	no	—
145	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
146	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
147	#214	—	2	4	no	—
148	C₄⋊Dic₃	—	2	1	no	—
149	C₄⋊Dic₃	—	2	1	no	—
150	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
151	C₂²×C₃²	—	2	2	no	—
152	C₂²×C₃²	—	4	1	no	—
153	C₂²×C₃²	—	4	1	no	—
154	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
155	C₃²×C₄	—	2	1	no	—
156	C₃²×C₄	—	4	1	no	—
157	C₃²×C₄	C₃×SD₁₆	1	2	yes	—
158	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
159	C₃²×C₄	C₃×#41	1	1	yes	characteristic
160	C₃×Dic₃	—	4	2	no	—
161	C₂²×C₉	—	2	4	no	—
162	C₂²×C₉	—	2	2	no	—
163	C₂²×C₉	—	2	1	no	—
164	C₂²×C₉	—	4	2	no	—
165	C₂²×C₉	—	4	2	no	—
166	C₂²×C₉	—	4	1	no	—
167	C₂²×C₉	—	4	1	no	—
168	C₂²×C₉	C₂×#67	1	1	yes	characteristic
169	C₄×C₉	—	2	1	no	—
170	C₄×C₉	—	2	2	no	—
171	C₄×C₉	—	2	1	no	—
172	C₄×C₉	—	2	1	no	—
173	C₄×C₉	—	4	1	no	—
174	C₄×C₉	—	4	2	no	—
175	C₄×C₉	—	4	1	no	—
176	C₄×C₉	—	12	2	no	—
177	C₄×C₉	#214	1	2	yes	complemented
178	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
179	C₄×C₉	#235	1	1	yes	characteristic
180	C₂×SD₁₆	—	3	2	no	—
181	C₄⋊Q₈	—	3	1	no	—
182	C₄×D₄	D₃×C₉	1	1	yes	characteristic
183	C₄⋊C₈	—	3	1	no	—
184	#103	—	3	2	no	—
185	C₃×C₉	#461	1	1	yes	characteristic, complemented, Sylow
186	C₂³×C₃	—	2	2	no	—
187	C₂³×C₃	—	2	1	no	—
188	C₂³×C₃	—	2	1	no	—
189	C₃×Q₈	—	6	2	no	—
190	C₃×Q₈	—	6	2	no	—
191	C₃×D₄	—	2	2	no	—
192	C₃×D₄	—	2	1	no	—
193	C₃×D₄	—	2	1	no	—
194	C₃×D₄	—	2	2	no	—
195	C₃×D₄	D₄×C₉	1	2	yes	—
196	C₃×D₄	C₃×D₁₂	1	2	yes	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	4	4	no	—
204	C₂×C₃×C₄	—	4	2	no	—
205	C₂×C₃×C₄	—	4	2	no	—
206	C₂×C₃×C₄	—	6	2	no	—
207	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
208	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
210	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
211	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
213	C₂×Dic₃	—	2	2	no	—
214	Dic₆	—	2	2	no	—
215	Dic₆	—	2	2	no	—
216	C₃×C₈	—	6	2	no	—
217	C₃⋊C₈	—	2	2	no	—
218	C₂×C₃²	—	4	2	no	—
219	C₂×C₃²	C₃×#121	1	1	yes	characteristic
220	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
221	C₂×C₃²	C₃×#137	1	1	yes	characteristic
222	C₂×C₉	—	2	1	no	—
223	C₂×C₉	—	2	1	no	—
224	C₂×C₉	—	2	1	no	—
225	C₂×C₉	—	4	4	no	—
226	C₂×C₉	—	4	2	no	—
227	C₂×C₉	#930	1	1	yes	characteristic
228	C₂×C₉	C₂×#214	1	1	yes	characteristic
229	C₂×C₉	#951	1	1	yes	characteristic
230	C₂×Q₈	—	3	2	no	—
231	C₂×D₄	C₂×D₃×C₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	SD₁₆	—	6	4	no	—
234	C₂×C₈	—	3	2	no	—
235	C₄⋊C₄	—	6	1	no	—
236	C₄⋊C₄	—	6	1	no	—
237	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
238	C₂²⋊C₄	—	2	1	no	—
239	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
240	C₂²×C₃	—	2	4	no	—
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	4	2	no	—
245	C₂²×C₃	—	4	2	no	—
246	C₂²×C₃	—	4	1	no	—
247	C₂²×C₃	—	4	1	no	—
248	C₂²×C₃	—	4	1	no	—
249	C₂²×C₃	—	4	1	no	—
250	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
251	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	4	1	no	—
258	C₃×C₄	—	4	2	no	—
259	C₃×C₄	—	4	1	no	—
260	C₃×C₄	—	4	1	no	—
261	C₃×C₄	—	12	2	no	—
262	C₃×C₄	C₉×SD₁₆	1	2	yes	complemented
263	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
264	C₃×C₄	C₉×#41	1	1	yes	characteristic
265	C₃×C₄	C₃×#214	1	2	yes	—
266	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
267	C₃×C₄	C₃×#235	1	1	yes	characteristic
268	Dic₃	—	4	2	no	—
269	C₃²	C₃×#461	1	1	yes	characteristic
270	C₉	—	2	1	no	—
271	C₉	#5019	1	1	yes	characteristic, direct product factor
272	C₂³	—	2	1	no	—
273	Q₈	—	6	2	no	—
274	Q₈	—	6	2	no	—
275	D₄	—	2	1	no	—
276	D₄	C₉×D₁₂	1	2	yes	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	6	2	no	—
280	C₂×C₄	C₉×#67	1	1	yes	characteristic
281	C₂×C₄	C₉×#67	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
283	C₈	—	6	2	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	4	4	no	—
288	C₂×C₃	—	4	2	no	—
289	C₂×C₃	—	4	2	no	—
290	C₂×C₃	C₉×#121	1	1	yes	characteristic
291	C₂×C₃	C₂×C₉×SD₁₆	1	1	yes	characteristic
292	C₂×C₃	C₉×#137	1	1	yes	characteristic
293	C₂×C₃	C₃×#930	1	1	yes	characteristic
294	C₂×C₃	C₂×C₃×#214	1	1	yes	characteristic
295	C₂×C₃	C₃×#951	1	1	yes	characteristic
296	C₂²	—	2	2	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	C₂×C₉×#67	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	12	2	no	—
303	C₄	C₉×#214	1	2	yes	complemented
304	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
305	C₄	C₉×#235	1	1	yes	characteristic
306	C₃	—	2	1	no	—
307	C₃	C₉×#461	1	1	yes	characteristic, complemented
308	C₃	C₃×#5019	1	1	yes	characteristic
309	C₂	—	4	2	no	—
310	C₂	C₉×#930	1	1	yes	characteristic
311	C₂	C₂×C₉×#214	1	1	yes	characteristic
312	C₂	C₉×#951	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×#67
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×D₄×C₉	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃²×D₄	C₂²×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₄×D₄×C₉	D₃
hypocenter	C₃	C₉×#461
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₄×D₄×C₉

Complements

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

N	H ≅ G/N	type
3. C₉×#869	C₂	semidirect
4. C₉×#810	C₂	semidirect
5. C₉×C₄⋊(C₃⋊C₈)	C₂	semidirect
42. #5019	C₉	direct (N × H)
74. C₃×C₄×C₉	SD₁₆	semidirect
84. #869	C₂×C₉	semidirect
87. #810	C₂×C₉	semidirect
88. C₄⋊(C₃⋊C₈)	C₂×C₉	semidirect
177. C₄×C₉	#214	semidirect
185. C₃×C₉	#461	semidirect
262. C₃×C₄	C₉×SD₁₆	semidirect
271. C₉	#5019	direct (N × H)
303. C₄	C₉×#214	semidirect
307. C₃	C₉×#461	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₄×C₉	2
C₂×C₉×#214	2
C₉×#869	2
C₉×#810	2
C₉×C₄⋊(C₃⋊C₈)	2
C₃×#5019	3
C₉×#461	3

Subgroups — C9×#5019 #13142737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5019 #13142737