Subgroups — C₉×#2957 #11767678

Order 1152.

678 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#135	C₂	1	1	yes	maximal, characteristic
2	C₉×#568	C₂	1	2	yes	maximal, complemented
3	C₉×#444	C₂	1	4	yes	maximal, complemented
4	C₉×#443	C₂	1	4	yes	maximal, complemented
5	C₂×C₉×#108	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#421	C₂	1	2	yes	maximal, complemented
7	C₉×#405	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2957	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	C₂²	1	2	yes	—
10	C₉×#135	C₄	1	2	yes	complemented
11	C₉×#135	C₄	1	2	yes	complemented
12	C₉×#135	C₂²	1	2	yes	—
13	C₉×#135	C₂²	1	2	yes	—
14	C₂×C₉×Q₁₆	C₄	1	1	yes	characteristic, complemented
15	C₂×C₉×SD₁₆	C₄	1	2	yes	complemented
16	C₂×C₉×D₈	C₄	1	1	yes	characteristic, complemented
17	C₉×#131	—	2	4	no	—
18	C₉×#131	C₂²	1	8	yes	complemented
19	C₂×C₉×M₁₆	—	2	4	no	—
20	C₂×C₉×M₁₆	C₂²	1	2	yes	—
21	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
23	C₉×#117	C₂²	1	1	yes	characteristic
24	C₉×#108	C₂²	1	2	yes	complemented
25	C₉×#108	C₂²	1	2	yes	complemented
26	C₉×#104	C₂²	1	8	yes	complemented
27	C₉×#97	C₂²	1	2	yes	complemented
28	C₄×C₈×C₉	C₂²	1	2	yes	complemented
29	C₂×C₃×#135	C₂×C₃	1	1	yes	characteristic
30	C₃×#568	C₂×C₃	1	2	yes	—
31	C₃×#444	C₂×C₃	1	4	yes	—
32	C₃×#443	C₂×C₃	1	4	yes	—
33	C₂×C₃×#108	C₂×C₃	1	1	yes	characteristic
34	C₃×#421	C₂×C₃	1	2	yes	—
35	C₃×#405	C₂×C₃	1	1	yes	characteristic
36	C₉×#41	—	2	2	no	—
37	C₉×#41	—	2	2	no	—
38	C₉×#41	C₂×C₄	1	4	yes	—
39	C₉×#41	C₂³	1	4	yes	—
40	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
41	C₂×D₄×C₉	—	2	2	no	—
42	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₉	—	2	2	no	—
44	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₉×Q₁₆	C₂×C₄	1	2	yes	—
46	C₉×Q₁₆	C₂×C₄	1	2	yes	—
47	C₉×SD₁₆	C₂×C₄	1	4	yes	—
48	C₉×SD₁₆	C₂×C₄	1	4	yes	—
49	C₉×D₈	C₂×C₄	1	2	yes	—
50	C₉×D₈	C₂×C₄	1	2	yes	—
51	C₉×M₁₆	—	2	2	no	—
52	C₉×M₁₆	—	2	8	no	—
53	C₉×M₁₆	D₄	1	2	yes	—
54	C₉×M₁₆	D₄	1	2	yes	—
55	C₉×M₁₆	C₂³	1	4	yes	—
56	C₂×C₈×C₉	—	2	2	no	—
57	C₂×C₈×C₉	—	2	8	no	—
58	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
60	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
61	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
62	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₉	C₂³	1	2	yes	—
65	C₉×C₄⋊C₄	—	2	1	no	—
66	C₉×C₂²⋊C₄	—	2	1	no	—
67	C₄²×C₉	C₂³	1	2	yes	—
68	#2957	C₉	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
70	C₃×#135	C₃×C₄	1	2	yes	—
71	C₃×#135	C₃×C₄	1	2	yes	—
72	C₃×#135	C₂²×C₃	1	2	yes	—
73	C₃×#135	C₂²×C₃	1	2	yes	—
74	C₂×C₃×Q₁₆	C₃×C₄	1	1	yes	characteristic
75	C₂×C₃×SD₁₆	C₃×C₄	1	2	yes	—
76	C₂×C₃×D₈	C₃×C₄	1	1	yes	characteristic
77	C₃×#131	—	2	4	no	—
78	C₃×#131	C₂²×C₃	1	8	yes	—
79	C₂×C₃×M₁₆	—	2	4	no	—
80	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
81	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
82	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
83	C₃×#117	C₂²×C₃	1	1	yes	characteristic
84	C₃×#108	C₂²×C₃	1	2	yes	—
85	C₃×#108	C₂²×C₃	1	2	yes	—
86	C₃×#104	C₂²×C₃	1	8	yes	—
87	C₃×#97	C₂²×C₃	1	2	yes	—
88	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
89	C₂³×C₉	—	2	2	no	—
90	C₂³×C₉	#41	1	1	yes	characteristic
91	Q₈×C₉	—	2	2	no	—
92	Q₈×C₉	C₂²×C₄	1	4	yes	—
93	D₄×C₉	—	2	4	no	—
94	D₄×C₉	—	2	4	no	—
95	D₄×C₉	—	2	2	no	—
96	D₄×C₉	C₂²×C₄	1	4	yes	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	4	no	—
99	C₂×C₄×C₉	—	2	4	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₉	#41	1	1	yes	characteristic
106	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
107	C₈×C₉	—	2	4	no	—
108	C₈×C₉	C₂²×C₄	1	1	yes	characteristic
109	C₈×C₉	C₂²×C₄	1	1	yes	characteristic
110	C₈×C₉	C₂²×C₄	1	1	yes	characteristic
111	C₈×C₉	C₂²×C₄	1	1	yes	characteristic
112	C₈×C₉	C₂×D₄	1	4	yes	—
113	C₂×#135	C₂×C₉	1	1	yes	characteristic
114	#568	C₂×C₉	1	2	yes	complemented
115	#444	C₂×C₉	1	4	yes	complemented
116	#443	C₂×C₉	1	4	yes	complemented
117	C₂×#108	C₂×C₉	1	1	yes	characteristic, complemented
118	#421	C₂×C₉	1	2	yes	complemented
119	#405	C₂×C₉	1	1	yes	characteristic, complemented
120	C₃×#41	—	2	2	no	—
121	C₃×#41	—	2	2	no	—
122	C₃×#41	C₂×C₃×C₄	1	4	yes	—
123	C₃×#41	C₂³×C₃	1	4	yes	—
124	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
125	C₂×C₃×D₄	—	2	2	no	—
126	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
127	C₂²×C₃×C₄	—	2	2	no	—
128	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
129	C₃×Q₁₆	C₂×C₃×C₄	1	2	yes	—
130	C₃×Q₁₆	C₂×C₃×C₄	1	2	yes	—
131	C₃×SD₁₆	C₂×C₃×C₄	1	4	yes	—
132	C₃×SD₁₆	C₂×C₃×C₄	1	4	yes	—
133	C₃×D₈	C₂×C₃×C₄	1	2	yes	—
134	C₃×D₈	C₂×C₃×C₄	1	2	yes	—
135	C₃×M₁₆	—	2	2	no	—
136	C₃×M₁₆	—	2	8	no	—
137	C₃×M₁₆	C₃×D₄	1	2	yes	—
138	C₃×M₁₆	C₃×D₄	1	2	yes	—
139	C₃×M₁₆	C₂³×C₃	1	4	yes	—
140	C₂×C₃×C₈	—	2	2	no	—
141	C₂×C₃×C₈	—	2	8	no	—
142	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
143	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
144	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
145	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
146	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
147	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
148	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
149	C₃×C₄⋊C₄	—	2	1	no	—
150	C₃×C₂²⋊C₄	—	2	1	no	—
151	C₃×C₄²	C₂³×C₃	1	2	yes	—
152	C₂²×C₉	—	2	4	no	—
153	C₂²×C₉	—	4	1	no	—
154	C₂²×C₉	—	8	2	no	—
155	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
156	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
157	C₂²×C₉	C₂×#41	1	1	yes	characteristic
158	C₄×C₉	—	2	4	no	—
159	C₄×C₉	—	4	2	no	—
160	C₄×C₉	C₄×D₄	1	1	yes	characteristic
161	C₄×C₉	C₄×D₄	1	1	yes	characteristic
162	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
163	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
164	C₂×#41	C₂²×C₉	1	2	yes	—
165	#135	C₄×C₉	1	2	yes	complemented
166	#135	C₄×C₉	1	2	yes	complemented
167	#135	C₂²×C₉	1	2	yes	—
168	#135	C₂²×C₉	1	2	yes	—
169	C₂×Q₁₆	C₄×C₉	1	1	yes	characteristic, complemented
170	C₂×SD₁₆	C₄×C₉	1	2	yes	complemented
171	C₂×D₈	C₄×C₉	1	1	yes	characteristic, complemented
172	#131	—	2	4	no	—
173	#131	C₂²×C₉	1	8	yes	complemented
174	C₂×M₁₆	—	2	4	no	—
175	C₂×M₁₆	C₂²×C₉	1	2	yes	—
176	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
177	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
178	#117	C₂²×C₉	1	1	yes	characteristic
179	#108	C₂²×C₉	1	2	yes	complemented
180	#108	C₂²×C₉	1	2	yes	complemented
181	#104	C₂²×C₉	1	8	yes	complemented
182	#97	C₂²×C₉	1	2	yes	complemented
183	C₄×C₈	C₂²×C₉	1	2	yes	complemented
184	C₂³×C₃	—	2	2	no	—
185	C₂³×C₃	C₃×#41	1	1	yes	characteristic
186	C₃×Q₈	—	2	2	no	—
187	C₃×Q₈	C₂²×C₃×C₄	1	4	yes	—
188	C₃×D₄	—	2	4	no	—
189	C₃×D₄	—	2	4	no	—
190	C₃×D₄	—	2	2	no	—
191	C₃×D₄	C₂²×C₃×C₄	1	4	yes	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	4	no	—
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
200	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
202	C₃×C₈	—	2	4	no	—
203	C₃×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
204	C₃×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
205	C₃×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
206	C₃×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
207	C₃×C₈	C₂×C₃×D₄	1	4	yes	—
208	C₂×C₉	—	2	1	no	—
209	C₂×C₉	—	2	1	no	—
210	C₂×C₉	—	2	1	no	—
211	C₂×C₉	—	4	4	no	—
212	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
213	#41	—	2	2	no	—
214	#41	—	2	2	no	—
215	#41	C₂×C₄×C₉	1	4	yes	—
216	#41	C₂³×C₉	1	4	yes	—
217	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
222	Q₁₆	C₂×C₄×C₉	1	2	yes	—
223	Q₁₆	C₂×C₄×C₉	1	2	yes	—
224	SD₁₆	C₂×C₄×C₉	1	4	yes	—
225	SD₁₆	C₂×C₄×C₉	1	4	yes	—
226	D₈	C₂×C₄×C₉	1	2	yes	—
227	D₈	C₂×C₄×C₉	1	2	yes	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	2	8	no	—
230	M₁₆	D₄×C₉	1	2	yes	—
231	M₁₆	D₄×C₉	1	2	yes	—
232	M₁₆	C₂³×C₉	1	4	yes	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	8	no	—
235	C₂×C₈	C₂×C₄×C₉	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×C₉	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×C₉	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×C₉	1	1	yes	characteristic
239	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
240	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
241	C₂×C₈	C₂³×C₉	1	2	yes	—
242	C₄⋊C₄	—	2	1	no	—
243	C₂²⋊C₄	—	2	1	no	—
244	C₄²	C₂³×C₉	1	2	yes	—
245	C₂²×C₃	—	2	4	no	—
246	C₂²×C₃	—	4	1	no	—
247	C₂²×C₃	—	8	2	no	—
248	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
249	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
250	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
251	C₃×C₄	—	2	4	no	—
252	C₃×C₄	—	4	2	no	—
253	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
254	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
255	C₃×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
256	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
257	C₉	#2957	1	1	yes	characteristic, direct product factor, Sylow
258	C₂³	—	2	2	no	—
259	C₂³	C₉×#41	1	1	yes	characteristic
260	Q₈	—	2	2	no	—
261	Q₈	C₂²×C₄×C₉	1	4	yes	—
262	D₄	—	2	4	no	—
263	D₄	—	2	4	no	—
264	D₄	—	2	2	no	—
265	D₄	C₂²×C₄×C₉	1	4	yes	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
272	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
273	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
274	C₂×C₄	C₉×#41	1	1	yes	characteristic
275	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	C₂²×C₄×C₉	1	1	yes	characteristic
278	C₈	C₂²×C₄×C₉	1	1	yes	characteristic
279	C₈	C₂²×C₄×C₉	1	1	yes	characteristic
280	C₈	C₂²×C₄×C₉	1	1	yes	characteristic
281	C₈	C₂×D₄×C₉	1	4	yes	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	4	4	no	—
286	C₂×C₃	C₂×C₃×C₄×D₄	1	1	yes	characteristic
287	C₂²	—	2	4	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	8	2	no	—
290	C₂²	C₄×D₄×C₉	1	1	yes	characteristic
291	C₂²	C₄×D₄×C₉	1	1	yes	characteristic
292	C₂²	C₂×C₉×#41	1	1	yes	characteristic
293	C₄	—	2	4	no	—
294	C₄	—	4	2	no	—
295	C₄	C₄×D₄×C₉	1	1	yes	characteristic
296	C₄	C₄×D₄×C₉	1	1	yes	characteristic
297	C₄	C₂³×C₄×C₉	1	1	yes	characteristic
298	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
299	C₃	C₃×#2957	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	4	4	no	—
304	C₂	C₂×C₄×D₄×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₉	C₂²×D₄
commutator	C₄	C₂³×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂×C₃	C₂×C₃×C₄×D₄
prime radical	C₂×C₃×#135	C₂×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ C₂×C₉×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#568	C₂	semidirect
3. C₉×#444	C₂	semidirect
4. C₉×#443	C₂	semidirect
5. C₂×C₉×#108	C₂	semidirect
6. C₉×#421	C₂	semidirect
7. C₉×#405	C₂	semidirect
10. C₉×#135	C₄	semidirect
11. C₉×#135	C₄	semidirect
14. C₂×C₉×Q₁₆	C₄	semidirect
15. C₂×C₉×SD₁₆	C₄	semidirect
16. C₂×C₉×D₈	C₄	semidirect
18. C₉×#131	C₂²	semidirect
24. C₉×#108	C₂²	semidirect
25. C₉×#108	C₂²	semidirect
26. C₉×#104	C₂²	semidirect
27. C₉×#97	C₂²	semidirect
28. C₄×C₈×C₉	C₂²	semidirect
68. #2957	C₉	direct (N × H)
114. #568	C₂×C₉	semidirect
115. #444	C₂×C₉	semidirect
116. #443	C₂×C₉	semidirect
117. C₂×#108	C₂×C₉	semidirect
118. #421	C₂×C₉	semidirect
119. #405	C₂×C₉	semidirect
165. #135	C₄×C₉	semidirect
166. #135	C₄×C₉	semidirect
169. C₂×Q₁₆	C₄×C₉	semidirect
170. C₂×SD₁₆	C₄×C₉	semidirect
171. C₂×D₈	C₄×C₉	semidirect
173. #131	C₂²×C₉	semidirect
179. #108	C₂²×C₉	semidirect
180. #108	C₂²×C₉	semidirect
181. #104	C₂²×C₉	semidirect
182. #97	C₂²×C₉	semidirect
183. C₄×C₈	C₂²×C₉	semidirect
257. C₉	#2957	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×#135	2
C₉×#568	2
C₉×#444	2
C₉×#443	2
C₂×C₉×#108	2
C₉×#421	2
C₉×#405	2
C₃×#2957	3

Subgroups — C9×#2957 #11767678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2957 #11767678