Subgroups — C₉×#2500 #11767221

Order 1152.

852 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#121	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#394	C₂	1	4	yes	maximal, complemented
4	C₉×#390	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#390	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#388	C₂	1	4	yes	maximal, complemented
7	C₉×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₉×#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#2500	C₃	1	1	yes	maximal, characteristic
11	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₉	C₂²	1	2	yes	—
13	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
14	C₉×#121	—	2	4	no	—
15	C₉×#117	—	2	4	no	—
16	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
20	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
25	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₉	C₂²	1	2	yes	—
28	C₉×#95	C₂²	1	2	yes	—
29	C₉×#95	C₂²	1	4	yes	—
30	C₂×C₃×#121	C₂×C₃	1	1	yes	characteristic
31	C₂×C₃×#117	C₂×C₃	1	1	yes	characteristic
32	C₃×#394	C₂×C₃	1	4	yes	—
33	C₃×#390	C₂×C₃	1	1	yes	characteristic
34	C₃×#390	C₂×C₃	1	1	yes	characteristic
35	C₃×#388	C₂×C₃	1	4	yes	—
36	C₃×#384	C₂×C₃	1	1	yes	characteristic
37	C₃×#382	C₂×C₃	1	1	yes	characteristic
38	C₃×C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
39	C₂⁴×C₉	C₂³	1	2	yes	—
40	C₂⁴×C₉	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₉	—	2	4	no	—
42	C₂×D₄×C₉	—	2	2	no	—
43	C₂×D₄×C₉	—	2	2	no	—
44	C₂×D₄×C₉	—	2	2	no	—
45	C₂×D₄×C₉	—	4	4	no	—
46	C₂²×C₄×C₉	—	2	2	no	—
47	C₂²×C₄×C₉	—	2	2	no	—
48	C₂²×C₄×C₉	D₄	1	4	yes	—
49	C₂²×C₄×C₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₉	C₂³	1	2	yes	—
51	C₂²×C₄×C₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₉	C₂³	1	2	yes	—
53	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₉	C₂³	1	2	yes	—
55	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
56	C₉×C₄⋊C₄	—	2	2	no	—
57	C₉×C₄⋊C₄	—	2	2	no	—
58	C₉×C₄⋊C₄	—	2	2	no	—
59	C₉×C₄⋊C₄	—	4	4	no	—
60	C₉×C₂²⋊C₄	—	2	4	no	—
61	C₉×C₂²⋊C₄	—	2	4	no	—
62	C₉×C₂²⋊C₄	—	2	4	no	—
63	C₉×C₂²⋊C₄	—	2	8	no	—
64	C₉×C₂²⋊C₄	—	4	4	no	—
65	C₄²×C₉	—	2	4	no	—
66	C₄²×C₉	—	2	2	no	—
67	C₄²×C₉	—	2	2	no	—
68	#2500	C₉	1	1	yes	characteristic, direct product factor, Sylow
69	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
70	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
71	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
72	C₃×#121	—	2	4	no	—
73	C₃×#117	—	2	4	no	—
74	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
78	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
81	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
82	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
83	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
86	C₃×#95	C₂²×C₃	1	2	yes	—
87	C₃×#95	C₂²×C₃	1	4	yes	—
88	C₂³×C₉	—	2	2	no	—
89	C₂³×C₉	—	2	1	no	—
90	C₂³×C₉	—	2	1	no	—
91	C₂³×C₉	—	2	2	no	—
92	C₂³×C₉	—	2	2	no	—
93	C₂³×C₉	—	2	4	no	—
94	C₂³×C₉	—	2	2	no	—
95	C₂³×C₉	—	2	2	no	—
96	C₂³×C₉	—	2	4	no	—
97	C₂³×C₉	C₂×D₄	1	2	yes	—
98	C₂³×C₉	C₂⁴	1	1	yes	characteristic
99	D₄×C₉	—	4	8	no	—
100	D₄×C₉	—	4	4	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	2	1	no	—
103	C₂×C₄×C₉	—	2	1	no	—
104	C₂×C₄×C₉	—	2	2	no	—
105	C₂×C₄×C₉	—	2	8	no	—
106	C₂×C₄×C₉	—	2	2	no	—
107	C₂×C₄×C₉	—	2	2	no	—
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	—	2	2	no	—
110	C₂×C₄×C₉	—	2	4	no	—
111	C₂×C₄×C₉	—	2	2	no	—
112	C₂×C₄×C₉	—	2	2	no	—
113	C₂×C₄×C₉	—	2	4	no	—
114	C₂×C₄×C₉	—	2	4	no	—
115	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₉	#41	1	4	yes	—
118	C₂×C₄×C₉	#41	1	2	yes	—
119	C₂×C₄×C₉	#41	1	2	yes	—
120	C₂×#121	C₂×C₉	1	1	yes	characteristic
121	C₂×#117	C₂×C₉	1	1	yes	characteristic, complemented
122	#394	C₂×C₉	1	4	yes	complemented
123	#390	C₂×C₉	1	1	yes	characteristic, complemented
124	#390	C₂×C₉	1	1	yes	characteristic, complemented
125	#388	C₂×C₉	1	4	yes	complemented
126	#384	C₂×C₉	1	1	yes	characteristic, complemented
127	#382	C₂×C₉	1	1	yes	characteristic, complemented
128	C₄×C₂²⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
129	C₂⁴×C₃	C₂³×C₃	1	2	yes	—
130	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
131	C₂×C₃×D₄	—	2	4	no	—
132	C₂×C₃×D₄	—	2	2	no	—
133	C₂×C₃×D₄	—	2	2	no	—
134	C₂×C₃×D₄	—	2	2	no	—
135	C₂×C₃×D₄	—	4	4	no	—
136	C₂²×C₃×C₄	—	2	2	no	—
137	C₂²×C₃×C₄	—	2	2	no	—
138	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
139	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
140	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
141	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
142	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
143	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
144	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
145	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
146	C₃×C₄⋊C₄	—	2	2	no	—
147	C₃×C₄⋊C₄	—	2	2	no	—
148	C₃×C₄⋊C₄	—	2	2	no	—
149	C₃×C₄⋊C₄	—	4	4	no	—
150	C₃×C₂²⋊C₄	—	2	4	no	—
151	C₃×C₂²⋊C₄	—	2	4	no	—
152	C₃×C₂²⋊C₄	—	2	4	no	—
153	C₃×C₂²⋊C₄	—	2	8	no	—
154	C₃×C₂²⋊C₄	—	4	4	no	—
155	C₃×C₄²	—	2	4	no	—
156	C₃×C₄²	—	2	2	no	—
157	C₃×C₄²	—	2	2	no	—
158	C₂²×C₉	—	2	2	no	—
159	C₂²×C₉	—	2	2	no	—
160	C₂²×C₉	—	2	2	no	—
161	C₂²×C₉	—	4	2	no	—
162	C₂²×C₉	—	4	2	no	—
163	C₂²×C₉	—	4	2	no	—
164	C₂²×C₉	—	4	2	no	—
165	C₂²×C₉	—	4	2	no	—
166	C₂²×C₉	—	4	4	no	—
167	C₂²×C₉	—	4	4	no	—
168	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
169	C₂²×C₉	C₂×#41	1	1	yes	characteristic
170	C₂²×C₉	C₂×#41	1	1	yes	characteristic
171	C₂²×C₉	C₂×#41	1	2	yes	—
172	C₂²×C₉	H(5,𝔽₂)	1	2	yes	—
173	C₄×C₉	—	2	2	no	—
174	C₄×C₉	—	2	4	no	—
175	C₄×C₉	—	4	2	no	—
176	C₄×C₉	—	4	2	no	—
177	C₄×C₉	—	4	2	no	—
178	C₄×C₉	—	4	2	no	—
179	C₄×C₉	#123	1	4	yes	—
180	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
181	C₂²×D₄	C₂²×C₉	1	2	yes	—
182	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
183	#121	—	2	4	no	—
184	#117	—	2	4	no	—
185	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
186	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
187	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
188	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
189	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
190	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
191	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
192	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
193	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
194	C₂×C₄²	C₂²×C₉	1	2	yes	—
195	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
196	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
197	#95	C₂²×C₉	1	2	yes	—
198	#95	C₂²×C₉	1	4	yes	—
199	C₂³×C₃	—	2	2	no	—
200	C₂³×C₃	—	2	1	no	—
201	C₂³×C₃	—	2	1	no	—
202	C₂³×C₃	—	2	2	no	—
203	C₂³×C₃	—	2	2	no	—
204	C₂³×C₃	—	2	4	no	—
205	C₂³×C₃	—	2	2	no	—
206	C₂³×C₃	—	2	2	no	—
207	C₂³×C₃	—	2	4	no	—
208	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
209	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
210	C₃×D₄	—	4	8	no	—
211	C₃×D₄	—	4	4	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	2	1	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	8	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	—	2	4	no	—
222	C₂×C₃×C₄	—	2	2	no	—
223	C₂×C₃×C₄	—	2	2	no	—
224	C₂×C₃×C₄	—	2	4	no	—
225	C₂×C₃×C₄	—	2	4	no	—
226	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
227	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
228	C₂×C₃×C₄	C₃×#41	1	4	yes	—
229	C₂×C₃×C₄	C₃×#41	1	2	yes	—
230	C₂×C₃×C₄	C₃×#41	1	2	yes	—
231	C₂×C₉	—	4	2	no	—
232	C₂×C₉	—	8	2	no	—
233	C₂×C₉	C₂×#123	1	1	yes	characteristic
234	C₂×C₉	#532	1	1	yes	characteristic
235	C₂×C₉	#535	1	1	yes	characteristic
236	C₂×C₉	#553	1	2	yes	—
237	C₂×C₉	#555	1	2	yes	—
238	C₂⁴	C₂³×C₉	1	2	yes	—
239	C₂⁴	C₂³×C₉	1	1	yes	characteristic
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	4	4	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	D₄×C₉	1	4	yes	—
248	C₂²×C₄	C₂³×C₉	1	2	yes	—
249	C₂²×C₄	C₂³×C₉	1	2	yes	—
250	C₂²×C₄	C₂³×C₉	1	2	yes	—
251	C₂²×C₄	C₂³×C₉	1	2	yes	—
252	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
253	C₂²×C₄	C₂³×C₉	1	2	yes	—
254	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	4	4	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₂²⋊C₄	—	2	8	no	—
263	C₂²⋊C₄	—	4	4	no	—
264	C₄²	—	2	4	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₂²×C₃	—	2	2	no	—
268	C₂²×C₃	—	2	2	no	—
269	C₂²×C₃	—	2	2	no	—
270	C₂²×C₃	—	4	2	no	—
271	C₂²×C₃	—	4	2	no	—
272	C₂²×C₃	—	4	2	no	—
273	C₂²×C₃	—	4	2	no	—
274	C₂²×C₃	—	4	2	no	—
275	C₂²×C₃	—	4	4	no	—
276	C₂²×C₃	—	4	4	no	—
277	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
278	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
279	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
280	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
281	C₂²×C₃	C₃×H(5,𝔽₂)	1	2	yes	—
282	C₃×C₄	—	2	2	no	—
283	C₃×C₄	—	2	4	no	—
284	C₃×C₄	—	4	2	no	—
285	C₃×C₄	—	4	2	no	—
286	C₃×C₄	—	4	2	no	—
287	C₃×C₄	—	4	2	no	—
288	C₃×C₄	C₃×#123	1	4	yes	—
289	C₉	#2500	1	1	yes	characteristic, direct product factor, Sylow
290	C₂³	—	2	2	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	2	2	no	—
294	C₂³	—	2	2	no	—
295	C₂³	—	2	4	no	—
296	C₂³	—	2	2	no	—
297	C₂³	—	2	2	no	—
298	C₂³	—	2	4	no	—
299	C₂³	C₂×D₄×C₉	1	2	yes	—
300	C₂³	C₂⁴×C₉	1	1	yes	characteristic
301	D₄	—	4	8	no	—
302	D₄	—	4	4	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	8	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
318	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
319	C₂×C₄	C₉×#41	1	4	yes	—
320	C₂×C₄	C₉×#41	1	2	yes	—
321	C₂×C₄	C₉×#41	1	2	yes	—
322	C₂×C₃	—	4	2	no	—
323	C₂×C₃	—	8	2	no	—
324	C₂×C₃	C₂×C₃×#123	1	1	yes	characteristic
325	C₂×C₃	C₃×#532	1	1	yes	characteristic
326	C₂×C₃	C₃×#535	1	1	yes	characteristic
327	C₂×C₃	C₃×#553	1	2	yes	—
328	C₂×C₃	C₃×#555	1	2	yes	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	4	2	no	—
335	C₂²	—	4	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	4	4	no	—
338	C₂²	—	4	4	no	—
339	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
340	C₂²	C₂×C₉×#41	1	1	yes	characteristic
341	C₂²	C₂×C₉×#41	1	1	yes	characteristic
342	C₂²	C₂×C₉×#41	1	2	yes	—
343	C₂²	C₉×H(5,𝔽₂)	1	2	yes	—
344	C₄	—	2	2	no	—
345	C₄	—	2	4	no	—
346	C₄	—	4	2	no	—
347	C₄	—	4	2	no	—
348	C₄	—	4	2	no	—
349	C₄	—	4	2	no	—
350	C₄	C₉×#123	1	4	yes	—
351	C₃	C₃×#2500	1	1	yes	characteristic
352	C₂	—	4	2	no	—
353	C₂	—	8	2	no	—
354	C₂	C₂×C₉×#123	1	1	yes	characteristic
355	C₂	C₉×#532	1	1	yes	characteristic
356	C₂	C₉×#535	1	1	yes	characteristic
357	C₂	C₉×#553	1	2	yes	—
358	C₂	C₉×#555	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂³	C₂⁴×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂×C₃×#121	C₂×C₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#117	C₂	semidirect
3. C₉×#394	C₂	semidirect
4. C₉×#390	C₂	semidirect
5. C₉×#390	C₂	semidirect
6. C₉×#388	C₂	semidirect
7. C₉×#384	C₂	semidirect
8. C₉×#382	C₂	semidirect
9. C₄×C₉×C₂²⋊C₄	C₂	semidirect
68. #2500	C₉	direct (N × H)
121. C₂×#117	C₂×C₉	semidirect
122. #394	C₂×C₉	semidirect
123. #390	C₂×C₉	semidirect
124. #390	C₂×C₉	semidirect
125. #388	C₂×C₉	semidirect
126. #384	C₂×C₉	semidirect
127. #382	C₂×C₉	semidirect
128. C₄×C₂²⋊C₄	C₂×C₉	semidirect
289. C₉	#2500	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×#121	2
C₂×C₉×#117	2
C₉×#394	2
C₉×#390	2
C₉×#388	2
C₉×#384	2
C₉×#382	2
C₄×C₉×C₂²⋊C₄	2
C₃×#2500	3

Subgroups — C9×#2500 #11767221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2500 #11767221