Subgroups — C₉×#1905 #11766626

Order 1152.

465 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×C₉×#137	C₂	1	1	yes	maximal, characteristic
2	C₉×#519	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#421	C₂	1	1	yes	maximal, characteristic
4	C₉×#419	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#410	C₂	1	1	yes	maximal, characteristic
6	C₉×#344	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1905	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	—	2	2	no	—
13	C₂×C₉×Q₁₆	C₄	1	2	yes	complemented
14	C₂×C₉×SD₁₆	C₄	1	2	yes	complemented
15	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₉	—	2	2	no	—
18	C₄×Q₈×C₉	—	2	2	no	—
19	C₉×#117	—	2	2	no	—
20	C₉×#117	C₂²	1	1	yes	characteristic
21	C₉×#117	C₂²	1	1	yes	characteristic
22	C₉×#117	C₂²	1	1	yes	characteristic
23	C₉×#104	—	2	2	no	—
24	C₉×#103	—	2	2	no	—
25	C₉×#99	—	2	2	no	—
26	C₂×C₃×#137	C₂×C₃	1	1	yes	characteristic
27	C₃×#519	C₂×C₃	1	1	yes	characteristic
28	C₃×#421	C₂×C₃	1	1	yes	characteristic
29	C₃×#419	C₂×C₃	1	1	yes	characteristic
30	C₃×#410	C₂×C₃	1	1	yes	characteristic
31	C₃×#344	C₂×C₃	1	1	yes	characteristic
32	C₃×H(3,ℤ/4)	C₂×C₃	1	1	yes	characteristic
33	C₉×#41	—	2	1	no	—
34	C₉×#41	—	2	1	no	—
35	C₉×#41	—	2	1	no	—
36	C₉×#41	—	2	1	no	—
37	C₂×Q₈×C₉	—	2	1	no	—
38	C₂×Q₈×C₉	—	2	1	no	—
39	C₂×Q₈×C₉	—	2	1	no	—
40	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
41	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
42	C₂×Q₈×C₉	D₄	1	2	yes	—
43	C₂×Q₈×C₉	D₄	1	2	yes	—
44	C₂×D₄×C₉	—	2	1	no	—
45	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₉	—	2	1	no	—
47	C₂²×C₄×C₉	—	2	1	no	—
48	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
49	C₉×Q₁₆	—	2	4	no	—
50	C₉×SD₁₆	—	2	4	no	—
51	C₉×M₁₆	—	2	1	no	—
52	C₉×M₁₆	—	2	1	no	—
53	C₉×M₁₆	—	2	1	no	—
54	C₉×M₁₆	—	2	1	no	—
55	C₂×C₈×C₉	—	2	1	no	—
56	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
57	C₉×C₄⋊C₄	—	2	4	no	—
58	C₉×C₄⋊C₄	—	2	1	no	—
59	C₉×C₄⋊C₄	—	2	1	no	—
60	C₉×C₄⋊C₄	—	2	1	no	—
61	C₉×C₂²⋊C₄	—	2	1	no	—
62	C₉×C₂²⋊C₄	—	2	1	no	—
63	C₉×C₂²⋊C₄	D₄	1	2	yes	—
64	C₄²×C₉	—	2	4	no	—
65	C₄²×C₉	—	2	1	no	—
66	C₄²×C₉	—	2	1	no	—
67	C₄²×C₉	D₄	1	2	yes	—
68	#1905	C₉	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
70	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
71	C₃×#137	—	2	2	no	—
72	C₃×#137	—	2	2	no	—
73	C₂×C₃×Q₁₆	C₃×C₄	1	2	yes	—
74	C₂×C₃×SD₁₆	C₃×C₄	1	2	yes	—
75	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
77	C₃×C₄×Q₈	—	2	2	no	—
78	C₃×C₄×Q₈	—	2	2	no	—
79	C₃×#117	—	2	2	no	—
80	C₃×#117	C₂²×C₃	1	1	yes	characteristic
81	C₃×#117	C₂²×C₃	1	1	yes	characteristic
82	C₃×#117	C₂²×C₃	1	1	yes	characteristic
83	C₃×#104	—	2	2	no	—
84	C₃×#103	—	2	2	no	—
85	C₃×#99	—	2	2	no	—
86	C₂³×C₉	—	2	1	no	—
87	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
88	Q₈×C₉	—	2	2	no	—
89	Q₈×C₉	—	2	2	no	—
90	Q₈×C₉	—	2	2	no	—
91	Q₈×C₉	—	2	1	no	—
92	Q₈×C₉	—	2	1	no	—
93	Q₈×C₉	C₂²⋊C₄	1	4	yes	—
94	D₄×C₉	—	2	1	no	—
95	D₄×C₉	—	2	1	no	—
96	D₄×C₉	—	4	1	no	—
97	D₄×C₉	—	4	1	no	—
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	1	no	—
102	C₂×C₄×C₉	—	2	2	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	4	1	no	—
105	C₂×C₄×C₉	—	4	1	no	—
106	C₂×C₄×C₉	—	4	1	no	—
107	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₉	#41	1	1	yes	characteristic
112	C₂×C₄×C₉	#41	1	1	yes	characteristic
113	C₈×C₉	—	2	2	no	—
114	C₈×C₉	—	4	1	no	—
115	C₂×#137	C₂×C₉	1	1	yes	characteristic
116	#519	C₂×C₉	1	1	yes	characteristic, complemented
117	#421	C₂×C₉	1	1	yes	characteristic
118	#419	C₂×C₉	1	1	yes	characteristic, complemented
119	#410	C₂×C₉	1	1	yes	characteristic
120	#344	C₂×C₉	1	1	yes	characteristic, complemented
121	H(3,ℤ/4)	C₂×C₉	1	1	yes	characteristic, complemented
122	C₃×#41	—	2	1	no	—
123	C₃×#41	—	2	1	no	—
124	C₃×#41	—	2	1	no	—
125	C₃×#41	—	2	1	no	—
126	C₂×C₃×Q₈	—	2	1	no	—
127	C₂×C₃×Q₈	—	2	1	no	—
128	C₂×C₃×Q₈	—	2	1	no	—
129	C₂×C₃×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
130	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
131	C₂×C₃×Q₈	C₃×D₄	1	2	yes	—
132	C₂×C₃×Q₈	C₃×D₄	1	2	yes	—
133	C₂×C₃×D₄	—	2	1	no	—
134	C₂×C₃×D₄	C₂×C₃×C₄	1	1	yes	characteristic
135	C₂²×C₃×C₄	—	2	1	no	—
136	C₂²×C₃×C₄	—	2	1	no	—
137	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
138	C₃×Q₁₆	—	2	4	no	—
139	C₃×SD₁₆	—	2	4	no	—
140	C₃×M₁₆	—	2	1	no	—
141	C₃×M₁₆	—	2	1	no	—
142	C₃×M₁₆	—	2	1	no	—
143	C₃×M₁₆	—	2	1	no	—
144	C₂×C₃×C₈	—	2	1	no	—
145	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
146	C₃×C₄⋊C₄	—	2	4	no	—
147	C₃×C₄⋊C₄	—	2	1	no	—
148	C₃×C₄⋊C₄	—	2	1	no	—
149	C₃×C₄⋊C₄	—	2	1	no	—
150	C₃×C₂²⋊C₄	—	2	1	no	—
151	C₃×C₂²⋊C₄	—	2	1	no	—
152	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
153	C₃×C₄²	—	2	4	no	—
154	C₃×C₄²	—	2	1	no	—
155	C₃×C₄²	—	2	1	no	—
156	C₃×C₄²	C₃×D₄	1	2	yes	—
157	C₂²×C₉	—	4	1	no	—
158	C₂²×C₉	—	4	1	no	—
159	C₂²×C₉	—	8	1	no	—
160	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
161	C₂²×C₉	#120	1	1	yes	characteristic
162	C₂²×C₉	#121	1	1	yes	characteristic
163	C₄×C₉	—	2	4	no	—
164	C₄×C₉	—	4	2	no	—
165	C₄×C₉	—	4	2	no	—
166	C₄×C₉	—	4	1	no	—
167	C₄×C₉	—	8	1	no	—
168	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₄×C₉	C₄×D₄	1	1	yes	characteristic
170	C₄×C₉	#121	1	1	yes	characteristic
171	C₄×C₉	#123	1	1	yes	characteristic
172	C₂×#41	C₂²×C₉	1	1	yes	characteristic
173	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
174	#137	—	2	2	no	—
175	#137	—	2	2	no	—
176	C₂×Q₁₆	C₄×C₉	1	2	yes	complemented
177	C₂×SD₁₆	C₄×C₉	1	2	yes	complemented
178	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
179	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
180	C₄×Q₈	—	2	2	no	—
181	C₄×Q₈	—	2	2	no	—
182	#117	—	2	2	no	—
183	#117	C₂²×C₉	1	1	yes	characteristic
184	#117	C₂²×C₉	1	1	yes	characteristic
185	#117	C₂²×C₉	1	1	yes	characteristic
186	#104	—	2	2	no	—
187	#103	—	2	2	no	—
188	#99	—	2	2	no	—
189	C₂³×C₃	—	2	1	no	—
190	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
191	C₃×Q₈	—	2	2	no	—
192	C₃×Q₈	—	2	2	no	—
193	C₃×Q₈	—	2	2	no	—
194	C₃×Q₈	—	2	1	no	—
195	C₃×Q₈	—	2	1	no	—
196	C₃×Q₈	C₃×C₂²⋊C₄	1	4	yes	—
197	C₃×D₄	—	2	1	no	—
198	C₃×D₄	—	2	1	no	—
199	C₃×D₄	—	4	1	no	—
200	C₃×D₄	—	4	1	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	2	2	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	C₂²×C₃×C₄	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×C₃×D₄	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	C₃×C₈	—	2	2	no	—
217	C₃×C₈	—	4	1	no	—
218	C₂×C₉	—	2	1	no	—
219	C₂×C₉	—	2	1	no	—
220	C₂×C₉	—	2	1	no	—
221	C₂×C₉	—	8	1	no	—
222	C₂×C₉	#386	1	1	yes	characteristic
223	#41	—	2	1	no	—
224	#41	—	2	1	no	—
225	#41	—	2	1	no	—
226	#41	—	2	1	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	2	1	no	—
229	C₂×Q₈	—	2	1	no	—
230	C₂×Q₈	C₂×C₄×C₉	1	1	yes	characteristic
231	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
232	C₂×Q₈	D₄×C₉	1	2	yes	—
233	C₂×Q₈	D₄×C₉	1	2	yes	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	C₂×C₄×C₉	1	1	yes	characteristic
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
239	Q₁₆	—	2	4	no	—
240	SD₁₆	—	2	4	no	—
241	M₁₆	—	2	1	no	—
242	M₁₆	—	2	1	no	—
243	M₁₆	—	2	1	no	—
244	M₁₆	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	C₂×C₄×C₉	1	2	yes	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	D₄×C₉	1	2	yes	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	D₄×C₉	1	2	yes	—
258	C₂²×C₃	—	4	1	no	—
259	C₂²×C₃	—	4	1	no	—
260	C₂²×C₃	—	8	1	no	—
261	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
262	C₂²×C₃	C₃×#120	1	1	yes	characteristic
263	C₂²×C₃	C₃×#121	1	1	yes	characteristic
264	C₃×C₄	—	2	4	no	—
265	C₃×C₄	—	4	2	no	—
266	C₃×C₄	—	4	2	no	—
267	C₃×C₄	—	4	1	no	—
268	C₃×C₄	—	8	1	no	—
269	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
270	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
271	C₃×C₄	C₃×#121	1	1	yes	characteristic
272	C₃×C₄	C₃×#123	1	1	yes	characteristic
273	C₉	#1905	1	1	yes	characteristic, direct product factor, Sylow
274	C₂³	—	2	1	no	—
275	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
276	Q₈	—	2	2	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	2	2	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	C₉×C₂²⋊C₄	1	4	yes	—
282	D₄	—	2	1	no	—
283	D₄	—	2	1	no	—
284	D₄	—	4	1	no	—
285	D₄	—	4	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
299	C₂×C₄	C₉×#41	1	1	yes	characteristic
300	C₂×C₄	C₉×#41	1	1	yes	characteristic
301	C₈	—	2	2	no	—
302	C₈	—	4	1	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	—	2	1	no	—
305	C₂×C₃	—	2	1	no	—
306	C₂×C₃	—	8	1	no	—
307	C₂×C₃	C₃×#386	1	1	yes	characteristic
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	C₄×D₄×C₉	1	1	yes	characteristic
312	C₂²	C₉×#120	1	1	yes	characteristic
313	C₂²	C₉×#121	1	1	yes	characteristic
314	C₄	—	2	4	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	1	no	—
318	C₄	—	8	1	no	—
319	C₄	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
320	C₄	C₄×D₄×C₉	1	1	yes	characteristic
321	C₄	C₉×#121	1	1	yes	characteristic
322	C₄	C₉×#123	1	1	yes	characteristic
323	C₃	C₃×#1905	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	8	1	no	—
328	C₂	C₉×#386	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	#386
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#386
prime radical	C₂×C₃×#41	C₂²×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#519	C₂	semidirect
4. C₉×#419	C₂	semidirect
6. C₉×#344	C₂	semidirect
7. C₉×H(3,ℤ/4)	C₂	semidirect
13. C₂×C₉×Q₁₆	C₄	semidirect
14. C₂×C₉×SD₁₆	C₄	semidirect
68. #1905	C₉	direct (N × H)
116. #519	C₂×C₉	semidirect
118. #419	C₂×C₉	semidirect
120. #344	C₂×C₉	semidirect
121. H(3,ℤ/4)	C₂×C₉	semidirect
176. C₂×Q₁₆	C₄×C₉	semidirect
177. C₂×SD₁₆	C₄×C₉	semidirect
273. C₉	#1905	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
C₉×#519	2
C₉×#421	2
C₉×#419	2
C₉×#410	2
C₉×#344	2
C₉×H(3,ℤ/4)	2
C₃×#1905	3

Subgroups — C9×#1905 #11766626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1905 #11766626