Subgroups — C₉×#1799 #11766520

Order 1152.

570 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#99	C₂	1	2	yes	maximal
3	C₉×#380	C₂	1	2	yes	maximal, complemented
4	C₉×#342	C₂	1	2	yes	maximal, complemented
5	C₃×#1799	C₃	1	1	yes	maximal, characteristic
6	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₉	C₄	1	1	yes	characteristic, complemented
8	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₄×D₄×C₉	—	2	2	no	—
10	C₄×D₄×C₉	—	2	2	no	—
11	C₂×C₉×C₄⋊C₄	—	2	2	no	—
12	C₂×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic, complemented
13	C₂×C₉×C₂²⋊C₄	C₄	1	1	yes	characteristic, complemented
14	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₂×C₉×C₂²⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₉	C₄	1	1	yes	characteristic, complemented
17	C₉×#99	—	2	4	no	—
18	C₉×#95	—	2	2	no	—
19	C₂×C₃×C₄×D₄	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×#99	C₂×C₃	1	2	yes	—
21	C₃×#380	C₂×C₃	1	2	yes	—
22	C₃×#342	C₂×C₃	1	2	yes	—
23	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂⁴×C₉	C₂³	1	1	yes	characteristic
25	C₂×D₄×C₉	—	2	2	no	—
26	C₂×D₄×C₉	—	2	2	no	—
27	C₂×D₄×C₉	D₄	1	2	yes	—
28	C₂×D₄×C₉	Q₈	1	2	yes	—
29	C₂²×C₄×C₉	—	2	1	no	—
30	C₂²×C₄×C₉	—	2	1	no	—
31	C₂²×C₄×C₉	—	2	4	no	—
32	C₂²×C₄×C₉	—	4	2	no	—
33	C₂²×C₄×C₉	—	4	2	no	—
34	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₉	D₄	1	2	yes	—
40	C₂²×C₄×C₉	D₄	1	2	yes	—
41	C₉×C₄⋊C₄	—	2	2	no	—
42	C₉×C₄⋊C₄	—	4	4	no	—
43	C₉×C₂²⋊C₄	—	2	2	no	—
44	C₉×C₂²⋊C₄	—	2	2	no	—
45	C₉×C₂²⋊C₄	—	2	4	no	—
46	C₉×C₂²⋊C₄	—	2	4	no	—
47	C₄²×C₉	—	2	1	no	—
48	C₄²×C₉	—	2	1	no	—
49	#1799	C₉	1	1	yes	characteristic, direct product factor, Sylow
50	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
51	C₂³×C₃×C₄	C₃×C₄	1	1	yes	characteristic
52	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
53	C₃×C₄×D₄	—	2	2	no	—
54	C₃×C₄×D₄	—	2	2	no	—
55	C₂×C₃×C₄⋊C₄	—	2	2	no	—
56	C₂×C₃×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
57	C₂×C₃×C₂²⋊C₄	C₃×C₄	1	1	yes	characteristic
58	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
59	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₃×C₄²	C₃×C₄	1	1	yes	characteristic
61	C₃×#99	—	2	4	no	—
62	C₃×#95	—	2	2	no	—
63	C₂³×C₉	—	2	1	no	—
64	C₂³×C₉	—	2	1	no	—
65	C₂³×C₉	—	4	1	no	—
66	C₂³×C₉	—	4	1	no	—
67	C₂³×C₉	—	4	1	no	—
68	C₂³×C₉	—	4	1	no	—
69	C₂³×C₉	C₄⋊C₄	1	2	yes	complemented
70	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
71	C₂³×C₉	C₂×D₄	1	2	yes	—
72	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
73	C₂³×C₉	C₂×Q₈	1	1	yes	characteristic
74	C₂³×C₉	#41	1	2	yes	—
75	D₄×C₉	—	2	4	no	—
76	D₄×C₉	—	2	4	no	—
77	C₂×C₄×C₉	—	2	1	no	—
78	C₂×C₄×C₉	—	2	1	no	—
79	C₂×C₄×C₉	—	2	4	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	2	1	no	—
82	C₂×C₄×C₉	—	2	1	no	—
83	C₂×C₄×C₉	—	4	1	no	—
84	C₂×C₄×C₉	—	4	2	no	—
85	C₂×C₄×C₉	—	4	2	no	—
86	C₂×C₄×C₉	—	4	2	no	—
87	C₂×C₄×C₉	—	4	1	no	—
88	C₂×C₄×C₉	—	4	1	no	—
89	C₂×C₄×C₉	—	4	4	no	—
90	C₂×C₄×C₉	—	4	4	no	—
91	C₂×C₄×C₉	—	4	4	no	—
92	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	complemented
96	C₂×C₄×D₄	C₂×C₉	1	1	yes	characteristic
97	C₂×#99	C₂×C₉	1	2	yes	—
98	#380	C₂×C₉	1	2	yes	complemented
99	#342	C₂×C₉	1	2	yes	complemented
100	C₂⁴×C₃	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
102	C₂×C₃×D₄	—	2	2	no	—
103	C₂×C₃×D₄	—	2	2	no	—
104	C₂×C₃×D₄	C₃×D₄	1	2	yes	—
105	C₂×C₃×D₄	C₃×Q₈	1	2	yes	—
106	C₂²×C₃×C₄	—	2	1	no	—
107	C₂²×C₃×C₄	—	2	1	no	—
108	C₂²×C₃×C₄	—	2	4	no	—
109	C₂²×C₃×C₄	—	4	2	no	—
110	C₂²×C₃×C₄	—	4	2	no	—
111	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
117	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
118	C₃×C₄⋊C₄	—	2	2	no	—
119	C₃×C₄⋊C₄	—	4	4	no	—
120	C₃×C₂²⋊C₄	—	2	2	no	—
121	C₃×C₂²⋊C₄	—	2	2	no	—
122	C₃×C₂²⋊C₄	—	2	4	no	—
123	C₃×C₂²⋊C₄	—	2	4	no	—
124	C₃×C₄²	—	2	1	no	—
125	C₃×C₄²	—	2	1	no	—
126	C₂²×C₉	—	2	2	no	—
127	C₂²×C₉	—	2	1	no	—
128	C₂²×C₉	—	2	2	no	—
129	C₂²×C₉	—	2	1	no	—
130	C₂²×C₉	—	2	2	no	—
131	C₂²×C₉	—	4	1	no	—
132	C₂²×C₉	—	4	1	no	—
133	C₂²×C₉	—	4	2	no	—
134	C₂²×C₉	—	4	2	no	—
135	C₂²×C₉	—	4	2	no	—
136	C₂²×C₉	—	4	1	no	—
137	C₂²×C₉	—	4	1	no	—
138	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₉	#117	1	1	yes	characteristic
141	C₂²×C₉	#121	1	2	yes	—
142	C₂²×C₉	C₂²⋊Q₈	1	2	yes	—
143	C₄×C₉	—	2	2	no	—
144	C₄×C₉	—	2	1	no	—
145	C₄×C₉	—	2	2	no	—
146	C₄×C₉	—	4	1	no	—
147	C₄×C₉	—	4	1	no	—
148	C₄×C₉	—	8	4	no	—
149	C₄×C₉	#99	1	2	yes	complemented
150	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
151	C₂³×C₄	C₄×C₉	1	1	yes	characteristic, complemented
152	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
153	C₄×D₄	—	2	2	no	—
154	C₄×D₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	C₄×C₉	1	1	yes	characteristic, complemented
157	C₂×C₂²⋊C₄	C₄×C₉	1	1	yes	characteristic, complemented
158	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
159	C₂×C₂²⋊C₄	C₂²×C₉	1	1	yes	characteristic
160	C₂×C₄²	C₄×C₉	1	1	yes	characteristic, complemented
161	#99	—	2	4	no	—
162	#95	—	2	2	no	—
163	C₂³×C₃	—	2	1	no	—
164	C₂³×C₃	—	2	1	no	—
165	C₂³×C₃	—	4	1	no	—
166	C₂³×C₃	—	4	1	no	—
167	C₂³×C₃	—	4	1	no	—
168	C₂³×C₃	—	4	1	no	—
169	C₂³×C₃	C₃×C₄⋊C₄	1	2	yes	—
170	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
171	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
172	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
173	C₂³×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
174	C₂³×C₃	C₃×#41	1	2	yes	—
175	C₃×D₄	—	2	4	no	—
176	C₃×D₄	—	2	4	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	2	4	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	4	1	no	—
184	C₂×C₃×C₄	—	4	2	no	—
185	C₂×C₃×C₄	—	4	2	no	—
186	C₂×C₃×C₄	—	4	2	no	—
187	C₂×C₃×C₄	—	4	1	no	—
188	C₂×C₃×C₄	—	4	1	no	—
189	C₂×C₃×C₄	—	4	4	no	—
190	C₂×C₃×C₄	—	4	4	no	—
191	C₂×C₃×C₄	—	4	4	no	—
192	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
193	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
194	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
195	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
196	C₂×C₉	—	2	2	no	—
197	C₂×C₉	—	2	1	no	—
198	C₂×C₉	—	2	1	no	—
199	C₂×C₉	—	2	2	no	—
200	C₂×C₉	—	4	2	no	—
201	C₂×C₉	#380	1	1	yes	characteristic
202	C₂×C₉	C₂×#99	1	1	yes	characteristic
203	C₂×C₉	#410	1	1	yes	characteristic
204	C₂⁴	C₂×C₄×C₉	1	1	yes	characteristic
205	C₂⁴	C₂³×C₉	1	1	yes	characteristic
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	D₄×C₉	1	2	yes	—
209	C₂×D₄	Q₈×C₉	1	2	yes	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
220	C₂²×C₄	D₄×C₉	1	2	yes	—
221	C₂²×C₄	D₄×C₉	1	2	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	2	1	no	—
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	4	1	no	—
236	C₂²×C₃	—	4	1	no	—
237	C₂²×C₃	—	4	2	no	—
238	C₂²×C₃	—	4	2	no	—
239	C₂²×C₃	—	4	2	no	—
240	C₂²×C₃	—	4	1	no	—
241	C₂²×C₃	—	4	1	no	—
242	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
243	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
244	C₂²×C₃	C₃×#117	1	1	yes	characteristic
245	C₂²×C₃	C₃×#121	1	2	yes	—
246	C₂²×C₃	C₃×C₂²⋊Q₈	1	2	yes	—
247	C₃×C₄	—	2	2	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	2	2	no	—
250	C₃×C₄	—	4	1	no	—
251	C₃×C₄	—	4	1	no	—
252	C₃×C₄	—	8	4	no	—
253	C₃×C₄	C₃×#99	1	2	yes	—
254	C₉	#1799	1	1	yes	characteristic, direct product factor, Sylow
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	—	4	1	no	—
258	C₂³	—	4	1	no	—
259	C₂³	—	4	1	no	—
260	C₂³	—	4	1	no	—
261	C₂³	C₉×C₄⋊C₄	1	2	yes	complemented
262	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
263	C₂³	C₂×D₄×C₉	1	2	yes	—
264	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
265	C₂³	C₂×Q₈×C₉	1	1	yes	characteristic
266	C₂³	C₉×#41	1	2	yes	—
267	D₄	—	2	4	no	—
268	D₄	—	2	4	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
285	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
286	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
287	C₂×C₄	C₉×C₄⋊C₄	1	2	yes	complemented
288	C₂×C₃	—	2	2	no	—
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	2	2	no	—
292	C₂×C₃	—	4	2	no	—
293	C₂×C₃	C₃×#380	1	1	yes	characteristic
294	C₂×C₃	C₂×C₃×#99	1	1	yes	characteristic
295	C₂×C₃	C₃×#410	1	1	yes	characteristic
296	C₂²	—	2	2	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
309	C₂²	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
310	C₂²	C₉×#117	1	1	yes	characteristic
311	C₂²	C₉×#121	1	2	yes	—
312	C₂²	C₉×C₂²⋊Q₈	1	2	yes	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	8	4	no	—
319	C₄	C₉×#99	1	2	yes	complemented
320	C₃	C₃×#1799	1	1	yes	characteristic
321	C₂	—	2	2	no	—
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	—	2	2	no	—
325	C₂	—	4	2	no	—
326	C₂	C₉×#380	1	1	yes	characteristic
327	C₂	C₂×C₉×#99	1	1	yes	characteristic
328	C₂	C₉×#410	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₉
Frattini, cosocle	C₂⁴×C₃	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×C₂²⋊C₄
prime radical	C₂²×C₃×D₄	C₂²×C₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂³×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₉×#380	C₂	semidirect
4. C₉×#342	C₂	semidirect
7. C₂³×C₄×C₉	C₄	semidirect
12. C₂×C₉×C₄⋊C₄	C₄	semidirect
13. C₂×C₉×C₂²⋊C₄	C₄	semidirect
16. C₂×C₄²×C₉	C₄	semidirect
49. #1799	C₉	direct (N × H)
69. C₂³×C₉	C₄⋊C₄	semidirect
95. C₂×C₄×C₉	C₄⋊C₄	semidirect
98. #380	C₂×C₉	semidirect
99. #342	C₂×C₉	semidirect
149. C₄×C₉	#99	semidirect
151. C₂³×C₄	C₄×C₉	semidirect
156. C₂×C₄⋊C₄	C₄×C₉	semidirect
157. C₂×C₂²⋊C₄	C₄×C₉	semidirect
160. C₂×C₄²	C₄×C₉	semidirect
254. C₉	#1799	direct (N × H)
261. C₂³	C₉×C₄⋊C₄	semidirect
287. C₂×C₄	C₉×C₄⋊C₄	semidirect
319. C₄	C₉×#99	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₉	2
C₂×C₉×#99	2
C₉×#380	2
C₉×#342	2
C₃×#1799	3

Subgroups — C9×#1799 #11766520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1799 #11766520