Subgroups — C₉×#1908 #11766629

Order 1152.

468 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#137	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#131	C₂	1	1	yes	maximal, characteristic
3	C₉×#417	C₂	1	1	yes	maximal, characteristic
4	C₂×C₉×#103	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₉×#101	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#343	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#341	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1908	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₉×#137	—	2	2	no	—
12	C₉×#137	C₄	1	4	yes	complemented
13	C₂×C₉×Q₁₆	—	2	1	no	—
14	C₂×C₉×SD₁₆	—	2	1	no	—
15	C₉×#131	—	2	2	no	—
16	C₉×#131	—	2	2	no	—
17	C₂×C₉×M₁₆	—	2	1	no	—
18	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₉	—	2	1	no	—
22	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
23	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₉×#103	—	2	1	no	—
25	C₉×#103	—	2	1	no	—
26	C₉×#103	—	2	1	no	—
27	C₉×#102	—	2	1	no	—
28	C₉×#101	—	2	2	no	—
29	C₂×C₃×#137	C₂×C₃	1	1	yes	characteristic
30	C₂×C₃×#131	C₂×C₃	1	1	yes	characteristic
31	C₃×#417	C₂×C₃	1	1	yes	characteristic
32	C₂×C₃×#103	C₂×C₃	1	1	yes	characteristic
33	C₂×C₃×#101	C₂×C₃	1	1	yes	characteristic
34	C₃×#343	C₂×C₃	1	1	yes	characteristic
35	C₃×#341	C₂×C₃	1	1	yes	characteristic
36	C₉×#41	—	2	1	no	—
37	C₉×#41	—	2	1	no	—
38	C₉×#41	C₂×C₄	1	2	yes	—
39	C₉×#41	D₄	1	2	yes	—
40	C₂×Q₈×C₉	—	2	1	no	—
41	C₂×Q₈×C₉	—	2	1	no	—
42	C₂×Q₈×C₉	—	2	1	no	—
43	C₂×Q₈×C₉	—	2	1	no	—
44	C₂×Q₈×C₉	—	2	1	no	—
45	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
46	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
47	C₂×D₄×C₉	—	2	1	no	—
48	C₂×D₄×C₉	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₉	—	2	1	no	—
50	C₂²×C₄×C₉	—	2	1	no	—
51	C₂²×C₄×C₉	—	2	1	no	—
52	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
53	C₉×Q₁₆	—	2	4	no	—
54	C₉×SD₁₆	—	2	4	no	—
55	C₉×M₁₆	—	2	4	no	—
56	C₉×M₁₆	—	2	1	no	—
57	C₉×M₁₆	—	2	1	no	—
58	C₉×M₁₆	—	2	1	no	—
59	C₉×M₁₆	—	2	1	no	—
60	C₉×M₁₆	C₂×C₄	1	2	yes	—
61	C₉×M₁₆	D₄	1	2	yes	—
62	C₂×C₈×C₉	—	2	4	no	—
63	C₂×C₈×C₉	—	2	1	no	—
64	C₂×C₈×C₉	—	2	1	no	—
65	C₂×C₈×C₉	—	2	1	no	—
66	C₂×C₈×C₉	—	2	1	no	—
67	C₂×C₈×C₉	—	2	1	no	—
68	C₂×C₈×C₉	D₄	1	1	yes	characteristic
69	C₂×C₈×C₉	D₄	1	1	yes	characteristic
70	C₉×C₄⋊C₄	—	2	1	no	—
71	C₉×C₄⋊C₄	—	2	1	no	—
72	#1908	C₉	1	1	yes	characteristic, direct product factor, Sylow
73	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
74	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
75	C₃×#137	—	2	2	no	—
76	C₃×#137	C₃×C₄	1	4	yes	—
77	C₂×C₃×Q₁₆	—	2	1	no	—
78	C₂×C₃×SD₁₆	—	2	1	no	—
79	C₃×#131	—	2	2	no	—
80	C₃×#131	—	2	2	no	—
81	C₂×C₃×M₁₆	—	2	1	no	—
82	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
85	C₂²×C₃×C₈	—	2	1	no	—
86	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
87	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
88	C₃×#103	—	2	1	no	—
89	C₃×#103	—	2	1	no	—
90	C₃×#103	—	2	1	no	—
91	C₃×#102	—	2	1	no	—
92	C₃×#101	—	2	2	no	—
93	C₂³×C₉	—	2	1	no	—
94	C₂³×C₉	#41	1	1	yes	characteristic
95	Q₈×C₉	—	2	1	no	—
96	Q₈×C₉	—	2	2	no	—
97	Q₈×C₉	—	2	2	no	—
98	Q₈×C₉	—	4	1	no	—
99	Q₈×C₉	—	4	1	no	—
100	Q₈×C₉	C₂²⋊C₄	1	2	yes	—
101	D₄×C₉	—	2	2	no	—
102	D₄×C₉	—	2	2	no	—
103	D₄×C₉	—	2	1	no	—
104	D₄×C₉	C₂²⋊C₄	1	2	yes	—
105	C₂×C₄×C₉	—	2	1	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	—	4	1	no	—
110	C₂×C₄×C₉	—	4	1	no	—
111	C₂×C₄×C₉	—	4	1	no	—
112	C₂×C₄×C₉	—	8	1	no	—
113	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₉	#41	1	1	yes	characteristic
119	C₈×C₉	—	2	2	no	—
120	C₈×C₉	—	2	1	no	—
121	C₈×C₉	—	2	2	no	—
122	C₈×C₉	—	2	1	no	—
123	C₈×C₉	—	4	1	no	—
124	C₂×#137	C₂×C₉	1	1	yes	characteristic
125	C₂×#131	C₂×C₉	1	1	yes	characteristic
126	#417	C₂×C₉	1	1	yes	characteristic
127	C₂×#103	C₂×C₉	1	1	yes	characteristic, complemented
128	C₂×#101	C₂×C₉	1	1	yes	characteristic, complemented
129	#343	C₂×C₉	1	1	yes	characteristic, complemented
130	#341	C₂×C₉	1	1	yes	characteristic, complemented
131	C₃×#41	—	2	1	no	—
132	C₃×#41	—	2	1	no	—
133	C₃×#41	C₂×C₃×C₄	1	2	yes	—
134	C₃×#41	C₃×D₄	1	2	yes	—
135	C₂×C₃×Q₈	—	2	1	no	—
136	C₂×C₃×Q₈	—	2	1	no	—
137	C₂×C₃×Q₈	—	2	1	no	—
138	C₂×C₃×Q₈	—	2	1	no	—
139	C₂×C₃×Q₈	—	2	1	no	—
140	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
141	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
142	C₂×C₃×D₄	—	2	1	no	—
143	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
144	C₂²×C₃×C₄	—	2	1	no	—
145	C₂²×C₃×C₄	—	2	1	no	—
146	C₂²×C₃×C₄	—	2	1	no	—
147	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
148	C₃×Q₁₆	—	2	4	no	—
149	C₃×SD₁₆	—	2	4	no	—
150	C₃×M₁₆	—	2	4	no	—
151	C₃×M₁₆	—	2	1	no	—
152	C₃×M₁₆	—	2	1	no	—
153	C₃×M₁₆	—	2	1	no	—
154	C₃×M₁₆	—	2	1	no	—
155	C₃×M₁₆	C₂×C₃×C₄	1	2	yes	—
156	C₃×M₁₆	C₃×D₄	1	2	yes	—
157	C₂×C₃×C₈	—	2	4	no	—
158	C₂×C₃×C₈	—	2	1	no	—
159	C₂×C₃×C₈	—	2	1	no	—
160	C₂×C₃×C₈	—	2	1	no	—
161	C₂×C₃×C₈	—	2	1	no	—
162	C₂×C₃×C₈	—	2	1	no	—
163	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
164	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
165	C₃×C₄⋊C₄	—	2	1	no	—
166	C₃×C₄⋊C₄	—	2	1	no	—
167	C₂²×C₉	—	2	1	no	—
168	C₂²×C₉	—	2	1	no	—
169	C₂²×C₉	—	2	2	no	—
170	C₂²×C₉	—	4	1	no	—
171	C₂²×C₉	—	4	1	no	—
172	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
173	C₂²×C₉	#121	1	1	yes	characteristic
174	C₂²×C₉	#123	1	1	yes	characteristic
175	C₄×C₉	—	2	2	no	—
176	C₄×C₉	—	4	2	no	—
177	C₄×C₉	—	8	1	no	—
178	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
179	C₄×C₉	C₄×D₄	1	1	yes	characteristic
180	C₄×C₉	#120	1	1	yes	characteristic
181	C₄×C₉	#121	1	1	yes	characteristic
182	C₂×#41	C₂²×C₉	1	1	yes	characteristic
183	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
184	#137	—	2	2	no	—
185	#137	C₄×C₉	1	4	yes	complemented
186	C₂×Q₁₆	—	2	1	no	—
187	C₂×SD₁₆	—	2	1	no	—
188	#131	—	2	2	no	—
189	#131	—	2	2	no	—
190	C₂×M₁₆	—	2	1	no	—
191	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
192	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
193	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
194	C₂²×C₈	—	2	1	no	—
195	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
196	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
197	#103	—	2	1	no	—
198	#103	—	2	1	no	—
199	#103	—	2	1	no	—
200	#102	—	2	1	no	—
201	#101	—	2	2	no	—
202	C₂³×C₃	—	2	1	no	—
203	C₂³×C₃	C₃×#41	1	1	yes	characteristic
204	C₃×Q₈	—	2	1	no	—
205	C₃×Q₈	—	2	2	no	—
206	C₃×Q₈	—	2	2	no	—
207	C₃×Q₈	—	4	1	no	—
208	C₃×Q₈	—	4	1	no	—
209	C₃×Q₈	C₃×C₂²⋊C₄	1	2	yes	—
210	C₃×D₄	—	2	2	no	—
211	C₃×D₄	—	2	2	no	—
212	C₃×D₄	—	2	1	no	—
213	C₃×D₄	C₃×C₂²⋊C₄	1	2	yes	—
214	C₂×C₃×C₄	—	2	1	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	4	1	no	—
219	C₂×C₃×C₄	—	4	1	no	—
220	C₂×C₃×C₄	—	4	1	no	—
221	C₂×C₃×C₄	—	8	1	no	—
222	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
227	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
228	C₃×C₈	—	2	2	no	—
229	C₃×C₈	—	2	1	no	—
230	C₃×C₈	—	2	2	no	—
231	C₃×C₈	—	2	1	no	—
232	C₃×C₈	—	4	1	no	—
233	C₂×C₉	—	2	1	no	—
234	C₂×C₉	—	2	1	no	—
235	C₂×C₉	—	4	2	no	—
236	C₂×C₉	#386	1	1	yes	characteristic
237	C₂×C₉	#471	1	1	yes	characteristic
238	C₂×C₉	#473	1	1	yes	characteristic
239	#41	—	2	1	no	—
240	#41	—	2	1	no	—
241	#41	C₂×C₄×C₉	1	2	yes	—
242	#41	D₄×C₉	1	2	yes	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	2	1	no	—
248	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
249	C₂×Q₈	D₄×C₉	1	1	yes	characteristic
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	D₄×C₉	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
256	Q₁₆	—	2	4	no	—
257	SD₁₆	—	2	4	no	—
258	M₁₆	—	2	4	no	—
259	M₁₆	—	2	1	no	—
260	M₁₆	—	2	1	no	—
261	M₁₆	—	2	1	no	—
262	M₁₆	—	2	1	no	—
263	M₁₆	C₂×C₄×C₉	1	2	yes	—
264	M₁₆	D₄×C₉	1	2	yes	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	D₄×C₉	1	1	yes	characteristic
272	C₂×C₈	D₄×C₉	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₂²×C₃	—	2	1	no	—
276	C₂²×C₃	—	2	1	no	—
277	C₂²×C₃	—	2	2	no	—
278	C₂²×C₃	—	4	1	no	—
279	C₂²×C₃	—	4	1	no	—
280	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
281	C₂²×C₃	C₃×#121	1	1	yes	characteristic
282	C₂²×C₃	C₃×#123	1	1	yes	characteristic
283	C₃×C₄	—	2	2	no	—
284	C₃×C₄	—	4	2	no	—
285	C₃×C₄	—	8	1	no	—
286	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
287	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
288	C₃×C₄	C₃×#120	1	1	yes	characteristic
289	C₃×C₄	C₃×#121	1	1	yes	characteristic
290	C₉	#1908	1	1	yes	characteristic, direct product factor, Sylow
291	C₂³	—	2	1	no	—
292	C₂³	C₉×#41	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	—	2	2	no	—
295	Q₈	—	2	2	no	—
296	Q₈	—	4	1	no	—
297	Q₈	—	4	1	no	—
298	Q₈	C₉×C₂²⋊C₄	1	2	yes	—
299	D₄	—	2	2	no	—
300	D₄	—	2	2	no	—
301	D₄	—	2	1	no	—
302	D₄	C₉×C₂²⋊C₄	1	2	yes	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	8	1	no	—
311	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
313	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
314	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
315	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
316	C₂×C₄	C₉×#41	1	1	yes	characteristic
317	C₈	—	2	2	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	2	no	—
320	C₈	—	2	1	no	—
321	C₈	—	4	1	no	—
322	C₂×C₃	—	2	1	no	—
323	C₂×C₃	—	2	1	no	—
324	C₂×C₃	—	4	2	no	—
325	C₂×C₃	C₃×#386	1	1	yes	characteristic
326	C₂×C₃	C₃×#471	1	1	yes	characteristic
327	C₂×C₃	C₃×#473	1	1	yes	characteristic
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	C₄×D₄×C₉	1	1	yes	characteristic
334	C₂²	C₉×#121	1	1	yes	characteristic
335	C₂²	C₉×#123	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	4	2	no	—
338	C₄	—	8	1	no	—
339	C₄	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
340	C₄	C₄×D₄×C₉	1	1	yes	characteristic
341	C₄	C₉×#120	1	1	yes	characteristic
342	C₄	C₉×#121	1	1	yes	characteristic
343	C₃	C₃×#1908	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	2	1	no	—
346	C₂	—	4	2	no	—
347	C₂	C₉×#386	1	1	yes	characteristic
348	C₂	C₉×#471	1	1	yes	characteristic
349	C₂	C₉×#473	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	#121
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₃×#121
prime radical	C₂×C₃×#41	C₂²×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₉×#103	C₂	semidirect
5. C₂×C₉×#101	C₂	semidirect
6. C₉×#343	C₂	semidirect
7. C₉×#341	C₂	semidirect
12. C₉×#137	C₄	semidirect
72. #1908	C₉	direct (N × H)
127. C₂×#103	C₂×C₉	semidirect
128. C₂×#101	C₂×C₉	semidirect
129. #343	C₂×C₉	semidirect
130. #341	C₂×C₉	semidirect
185. #137	C₄×C₉	semidirect
290. C₉	#1908	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×#137	2
C₂×C₉×#131	2
C₉×#417	2
C₂×C₉×#103	2
C₂×C₉×#101	2
C₉×#343	2
C₉×#341	2
C₃×#1908	3

Subgroups — C9×#1908 #11766629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1908 #11766629