Subgroups — C₉×#1925 #11766646

Order 1152.

435 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#107	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#102	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#102	C₂	1	1	yes	maximal, characteristic
4	C₉×#390	C₂	1	1	yes	maximal, characteristic
5	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#340	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1925	C₃	1	1	yes	maximal, characteristic
9	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
16	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
17	C₂×C₄²×C₉	—	2	1	no	—
18	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
19	C₉×#107	—	2	2	no	—
20	C₉×#102	—	2	2	no	—
21	C₉×#102	C₄	1	4	yes	complemented
22	C₉×#95	—	2	1	no	—
23	C₂×C₃×#107	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×#102	C₂×C₃	1	1	yes	characteristic
25	C₂×C₃×#102	C₂×C₃	1	1	yes	characteristic
26	C₃×#390	C₂×C₃	1	1	yes	characteristic
27	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
28	C₃×#340	C₂×C₃	1	1	yes	characteristic
29	C₃×#336	C₂×C₃	1	1	yes	characteristic
30	C₂⁴×C₉	—	2	1	no	—
31	C₂×D₄×C₉	—	2	1	no	—
32	C₂×D₄×C₉	—	2	1	no	—
33	C₂×D₄×C₉	—	2	1	no	—
34	C₂×D₄×C₉	—	2	1	no	—
35	C₂×D₄×C₉	—	2	1	no	—
36	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₉	—	2	1	no	—
38	C₂²×C₄×C₉	—	2	1	no	—
39	C₂²×C₄×C₉	—	2	1	no	—
40	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₉	—	2	1	no	—
44	C₂×C₈×C₉	—	2	1	no	—
45	C₂×C₈×C₉	—	2	1	no	—
46	C₂×C₈×C₉	—	2	1	no	—
47	C₂×C₈×C₉	—	2	1	no	—
48	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
49	C₉×C₄⋊C₄	—	2	1	no	—
50	C₉×C₄⋊C₄	—	2	1	no	—
51	C₉×C₄⋊C₄	—	2	1	no	—
52	C₉×C₄⋊C₄	—	2	1	no	—
53	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
54	C₉×C₄⋊C₄	D₄	1	2	yes	—
55	C₉×C₂²⋊C₄	—	4	2	no	—
56	C₉×C₂²⋊C₄	—	4	2	no	—
57	C₄²×C₉	—	2	1	no	—
58	C₄²×C₉	—	2	1	no	—
59	C₄²×C₉	—	4	2	no	—
60	#1925	C₉	1	1	yes	characteristic, direct product factor, Sylow
61	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
68	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
69	C₂×C₃×C₄²	—	2	1	no	—
70	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
71	C₃×#107	—	2	2	no	—
72	C₃×#102	—	2	2	no	—
73	C₃×#102	C₃×C₄	1	4	yes	—
74	C₃×#95	—	2	1	no	—
75	C₂³×C₉	—	2	2	no	—
76	C₂³×C₉	—	4	1	no	—
77	C₂³×C₉	—	4	1	no	—
78	C₂³×C₉	—	4	1	no	—
79	C₂³×C₉	—	4	1	no	—
80	C₂³×C₉	—	4	1	no	—
81	C₂³×C₉	—	4	1	no	—
82	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
83	D₄×C₉	—	2	2	no	—
84	D₄×C₉	—	2	2	no	—
85	D₄×C₉	—	4	1	no	—
86	D₄×C₉	—	4	1	no	—
87	C₂×C₄×C₉	—	2	1	no	—
88	C₂×C₄×C₉	—	2	1	no	—
89	C₂×C₄×C₉	—	2	2	no	—
90	C₂×C₄×C₉	—	2	2	no	—
91	C₂×C₄×C₉	—	2	1	no	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	—	2	1	no	—
94	C₂×C₄×C₉	—	4	1	no	—
95	C₂×C₄×C₉	—	4	1	no	—
96	C₂×C₄×C₉	—	4	1	no	—
97	C₂×C₄×C₉	—	4	1	no	—
98	C₂×C₄×C₉	—	4	1	no	—
99	C₂×C₄×C₉	—	4	1	no	—
100	C₂×C₄×C₉	—	4	1	no	—
101	C₂×C₄×C₉	D₈	1	2	yes	—
102	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₉	#41	1	1	yes	characteristic
105	C₂×C₄×C₉	#41	1	1	yes	characteristic
106	C₂×C₄×C₉	#41	1	1	yes	characteristic
107	C₂×C₄×C₉	#41	1	1	yes	characteristic
108	C₈×C₉	—	2	2	no	—
109	C₈×C₉	—	4	1	no	—
110	C₂×#107	C₂×C₉	1	1	yes	characteristic, complemented
111	C₂×#102	C₂×C₉	1	1	yes	characteristic
112	C₂×#102	C₂×C₉	1	1	yes	characteristic
113	#390	C₂×C₉	1	1	yes	characteristic
114	C₄×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
115	#340	C₂×C₉	1	1	yes	characteristic, complemented
116	#336	C₂×C₉	1	1	yes	characteristic, complemented
117	C₂⁴×C₃	—	2	1	no	—
118	C₂×C₃×D₄	—	2	1	no	—
119	C₂×C₃×D₄	—	2	1	no	—
120	C₂×C₃×D₄	—	2	1	no	—
121	C₂×C₃×D₄	—	2	1	no	—
122	C₂×C₃×D₄	—	2	1	no	—
123	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
124	C₂²×C₃×C₄	—	2	1	no	—
125	C₂²×C₃×C₄	—	2	1	no	—
126	C₂²×C₃×C₄	—	2	1	no	—
127	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₂×C₃×C₈	—	2	1	no	—
133	C₂×C₃×C₈	—	2	1	no	—
134	C₂×C₃×C₈	—	2	1	no	—
135	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
136	C₃×C₄⋊C₄	—	2	1	no	—
137	C₃×C₄⋊C₄	—	2	1	no	—
138	C₃×C₄⋊C₄	—	2	1	no	—
139	C₃×C₄⋊C₄	—	2	1	no	—
140	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
141	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
142	C₃×C₂²⋊C₄	—	4	2	no	—
143	C₃×C₂²⋊C₄	—	4	2	no	—
144	C₃×C₄²	—	2	1	no	—
145	C₃×C₄²	—	2	1	no	—
146	C₃×C₄²	—	4	2	no	—
147	C₂²×C₉	—	4	2	no	—
148	C₂²×C₉	—	4	2	no	—
149	C₂²×C₉	—	4	2	no	—
150	C₂²×C₉	—	8	1	no	—
151	C₂²×C₉	—	8	1	no	—
152	C₂²×C₉	—	8	1	no	—
153	C₂²×C₉	—	8	1	no	—
154	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
155	C₂²×C₉	#121	1	1	yes	characteristic
156	C₂²×C₉	#123	1	1	yes	characteristic
157	C₂²×C₉	C₂×D₈	1	1	yes	characteristic
158	C₂²×C₉	#135	1	1	yes	characteristic
159	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₂²×C₉	#137	1	1	yes	characteristic
161	C₄×C₉	—	2	2	no	—
162	C₄×C₉	—	4	2	no	—
163	C₄×C₉	—	4	2	no	—
164	C₄×C₉	—	4	1	no	—
165	C₄×C₉	—	8	1	no	—
166	C₄×C₉	#117	1	1	yes	characteristic
167	C₄×C₉	C₄×D₄	1	1	yes	characteristic
168	C₄×C₉	#124	1	1	yes	characteristic
169	C₄×C₉	#126	1	1	yes	characteristic
170	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
171	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
172	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
174	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
176	C₂×C₂²⋊C₄	—	2	1	no	—
177	C₂×C₂²⋊C₄	—	2	1	no	—
178	C₂×C₄²	—	2	1	no	—
179	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
180	#107	—	2	2	no	—
181	#102	—	2	2	no	—
182	#102	C₄×C₉	1	4	yes	complemented
183	#95	—	2	1	no	—
184	C₂³×C₃	—	2	2	no	—
185	C₂³×C₃	—	4	1	no	—
186	C₂³×C₃	—	4	1	no	—
187	C₂³×C₃	—	4	1	no	—
188	C₂³×C₃	—	4	1	no	—
189	C₂³×C₃	—	4	1	no	—
190	C₂³×C₃	—	4	1	no	—
191	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
192	C₃×D₄	—	2	2	no	—
193	C₃×D₄	—	2	2	no	—
194	C₃×D₄	—	4	1	no	—
195	C₃×D₄	—	4	1	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	2	2	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	1	no	—
204	C₂×C₃×C₄	—	4	1	no	—
205	C₂×C₃×C₄	—	4	1	no	—
206	C₂×C₃×C₄	—	4	1	no	—
207	C₂×C₃×C₄	—	4	1	no	—
208	C₂×C₃×C₄	—	4	1	no	—
209	C₂×C₃×C₄	—	4	1	no	—
210	C₂×C₃×C₄	C₃×D₈	1	2	yes	—
211	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
214	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
215	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
216	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
217	C₃×C₈	—	2	2	no	—
218	C₃×C₈	—	4	1	no	—
219	C₂×C₉	—	8	2	no	—
220	C₂×C₉	#388	1	1	yes	characteristic
221	C₂×C₉	C₄×D₈	1	1	yes	characteristic
222	C₂×C₉	#440	1	1	yes	characteristic
223	C₂×C₉	#459	1	1	yes	characteristic
224	C₂×C₉	#464	1	1	yes	characteristic
225	C₂×C₉	#480	1	1	yes	characteristic
226	C₂×C₉	#482	1	1	yes	characteristic
227	C₂⁴	—	2	1	no	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
238	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
239	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	C₂×C₄×C₉	1	2	yes	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
251	C₄⋊C₄	D₄×C₉	1	2	yes	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	4	2	no	—
257	C₂²×C₃	—	4	2	no	—
258	C₂²×C₃	—	4	2	no	—
259	C₂²×C₃	—	4	2	no	—
260	C₂²×C₃	—	8	1	no	—
261	C₂²×C₃	—	8	1	no	—
262	C₂²×C₃	—	8	1	no	—
263	C₂²×C₃	—	8	1	no	—
264	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
265	C₂²×C₃	C₃×#121	1	1	yes	characteristic
266	C₂²×C₃	C₃×#123	1	1	yes	characteristic
267	C₂²×C₃	C₂×C₃×D₈	1	1	yes	characteristic
268	C₂²×C₃	C₃×#135	1	1	yes	characteristic
269	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
270	C₂²×C₃	C₃×#137	1	1	yes	characteristic
271	C₃×C₄	—	2	2	no	—
272	C₃×C₄	—	4	2	no	—
273	C₃×C₄	—	4	2	no	—
274	C₃×C₄	—	4	1	no	—
275	C₃×C₄	—	8	1	no	—
276	C₃×C₄	C₃×#117	1	1	yes	characteristic
277	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
278	C₃×C₄	C₃×#124	1	1	yes	characteristic
279	C₃×C₄	C₃×#126	1	1	yes	characteristic
280	C₉	#1925	1	1	yes	characteristic, direct product factor, Sylow
281	C₂³	—	2	2	no	—
282	C₂³	—	4	1	no	—
283	C₂³	—	4	1	no	—
284	C₂³	—	4	1	no	—
285	C₂³	—	4	1	no	—
286	C₂³	—	4	1	no	—
287	C₂³	—	4	1	no	—
288	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
289	D₄	—	2	2	no	—
290	D₄	—	2	2	no	—
291	D₄	—	4	1	no	—
292	D₄	—	4	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	C₉×D₈	1	2	yes	—
308	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
310	C₂×C₄	C₉×#41	1	1	yes	characteristic
311	C₂×C₄	C₉×#41	1	1	yes	characteristic
312	C₂×C₄	C₉×#41	1	1	yes	characteristic
313	C₂×C₄	C₉×#41	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	4	1	no	—
316	C₂×C₃	—	8	2	no	—
317	C₂×C₃	C₃×#388	1	1	yes	characteristic
318	C₂×C₃	C₃×C₄×D₈	1	1	yes	characteristic
319	C₂×C₃	C₃×#440	1	1	yes	characteristic
320	C₂×C₃	C₃×#459	1	1	yes	characteristic
321	C₂×C₃	C₃×#464	1	1	yes	characteristic
322	C₂×C₃	C₃×#480	1	1	yes	characteristic
323	C₂×C₃	C₃×#482	1	1	yes	characteristic
324	C₂²	—	4	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	—	8	1	no	—
331	C₂²	C₄×D₄×C₉	1	1	yes	characteristic
332	C₂²	C₉×#121	1	1	yes	characteristic
333	C₂²	C₉×#123	1	1	yes	characteristic
334	C₂²	C₂×C₉×D₈	1	1	yes	characteristic
335	C₂²	C₉×#135	1	1	yes	characteristic
336	C₂²	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
337	C₂²	C₉×#137	1	1	yes	characteristic
338	C₄	—	2	2	no	—
339	C₄	—	4	2	no	—
340	C₄	—	4	2	no	—
341	C₄	—	4	1	no	—
342	C₄	—	8	1	no	—
343	C₄	C₉×#117	1	1	yes	characteristic
344	C₄	C₄×D₄×C₉	1	1	yes	characteristic
345	C₄	C₉×#124	1	1	yes	characteristic
346	C₄	C₉×#126	1	1	yes	characteristic
347	C₃	C₃×#1925	1	1	yes	characteristic
348	C₂	—	8	2	no	—
349	C₂	C₉×#388	1	1	yes	characteristic
350	C₂	C₄×C₉×D₈	1	1	yes	characteristic
351	C₂	C₉×#440	1	1	yes	characteristic
352	C₂	C₉×#459	1	1	yes	characteristic
353	C₂	C₉×#464	1	1	yes	characteristic
354	C₂	C₉×#480	1	1	yes	characteristic
355	C₂	C₉×#482	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂²×C₃×D₄	C₂²×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×#107	C₂	semidirect
5. C₄×C₉×C₄⋊C₄	C₂	semidirect
6. C₉×#340	C₂	semidirect
7. C₉×#336	C₂	semidirect
21. C₉×#102	C₄	semidirect
60. #1925	C₉	direct (N × H)
110. C₂×#107	C₂×C₉	semidirect
114. C₄×C₄⋊C₄	C₂×C₉	semidirect
115. #340	C₂×C₉	semidirect
116. #336	C₂×C₉	semidirect
182. #102	C₄×C₉	semidirect
280. C₉	#1925	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×#107	2
C₂×C₉×#102	2
C₉×#390	2
C₄×C₉×C₄⋊C₄	2
C₉×#340	2
C₉×#336	2
C₃×#1925	3

Subgroups — C9×#1925 #11766646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1925 #11766646