Subgroups — C₈₉×#9280 #36387128

Order 22784.

382 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3231	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2985	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1506	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1506	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1497	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#1287	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1274	C₂	1	1	yes	maximal, characteristic
8	#533×C₈₉	C₂²	1	1	yes	characteristic
9	#533×C₈₉	C₂²	1	1	yes	characteristic
10	#500×C₈₉	C₄	1	1	yes	characteristic, complemented
11	#498×C₈₉	C₄	1	1	yes	characteristic, complemented
12	#487×C₈₉	C₄	1	1	yes	characteristic, complemented
13	#487×C₈₉	C₄	1	1	yes	characteristic, complemented
14	#485×C₈₉	—	2	2	no	—
15	#479×C₈₉	—	2	2	no	—
16	#446×C₈₉	—	2	2	no	—
17	#446×C₈₉	—	2	1	no	—
18	C₈×Q₈×C₈₉	—	2	1	no	—
19	#433×C₈₉	—	2	2	no	—
20	#432×C₈₉	C₂²	1	1	yes	characteristic
21	#432×C₈₉	C₂²	1	1	yes	characteristic
22	#431×C₈₉	C₂²	1	1	yes	characteristic
23	#431×C₈₉	C₂²	1	1	yes	characteristic
24	#376×C₈₉	C₂²	1	1	yes	characteristic
25	#328×C₈₉	—	2	2	no	—
26	#328×C₈₉	—	2	2	no	—
27	Q₈⋊C₈×C₈₉	—	2	1	no	—
28	Q₈⋊C₈×C₈₉	—	2	1	no	—
29	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	#125×C₈₉	—	2	1	no	—
34	#125×C₈₉	—	2	1	no	—
35	C₂²⋊Q₈×C₈₉	—	2	1	no	—
36	C₂²⋊Q₈×C₈₉	—	2	1	no	—
37	C₂²⋊Q₈×C₈₉	—	2	1	no	—
38	C₂²⋊Q₈×C₈₉	D₄	1	2	yes	—
39	C₄×Q₈×C₈₉	—	2	1	no	—
40	C₄×Q₈×C₈₉	—	2	1	no	—
41	C₄×Q₈×C₈₉	—	2	1	no	—
42	C₄×Q₈×C₈₉	D₄	1	2	yes	—
43	#117×C₈₉	—	2	1	no	—
44	#117×C₈₉	—	2	1	no	—
45	C₂×C₄²×C₈₉	—	2	1	no	—
46	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
47	#107×C₈₉	—	2	2	no	—
48	#106×C₈₉	—	2	2	no	—
49	C₄⋊C₈×C₈₉	—	2	2	no	—
50	C₄⋊C₈×C₈₉	—	2	2	no	—
51	C₄⋊C₈×C₈₉	—	2	1	no	—
52	C₄⋊C₈×C₈₉	—	2	1	no	—
53	C₄⋊C₈×C₈₉	—	2	1	no	—
54	C₄⋊C₈×C₈₉	—	2	1	no	—
55	#103×C₈₉	—	2	2	no	—
56	#103×C₈₉	—	2	2	no	—
57	C₂²⋊C₈×C₈₉	—	2	1	no	—
58	C₂²⋊C₈×C₈₉	—	2	1	no	—
59	C₂²⋊C₈×C₈₉	—	2	1	no	—
60	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
61	#97×C₈₉	—	2	2	no	—
62	#97×C₈₉	—	2	1	no	—
63	#97×C₈₉	—	2	1	no	—
64	C₄×C₈×C₈₉	—	2	2	no	—
65	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
68	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
69	#95×C₈₉	—	2	1	no	—
70	#95×C₈₉	—	2	1	no	—
71	C₂×Q₈×C₈₉	—	2	1	no	—
72	C₂×Q₈×C₈₉	—	2	1	no	—
73	C₂×Q₈×C₈₉	—	2	1	no	—
74	C₂×Q₈×C₈₉	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₈₉	—	4	1	no	—
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
79	C₂×C₈×C₈₉	—	2	1	no	—
80	C₂×C₈×C₈₉	—	2	1	no	—
81	C₂×C₈×C₈₉	—	2	1	no	—
82	C₂×C₈×C₈₉	—	2	2	no	—
83	C₂×C₈×C₈₉	—	2	1	no	—
84	C₂×C₈×C₈₉	—	4	1	no	—
85	C₄⋊C₄×C₈₉	—	2	1	no	—
86	C₄⋊C₄×C₈₉	—	2	1	no	—
87	C₄⋊C₄×C₈₉	—	2	1	no	—
88	C₄⋊C₄×C₈₉	—	2	2	no	—
89	C₄⋊C₄×C₈₉	—	2	2	no	—
90	C₄⋊C₄×C₈₉	—	2	1	no	—
91	C₄⋊C₄×C₈₉	—	2	1	no	—
92	C₄⋊C₄×C₈₉	—	2	1	no	—
93	C₄⋊C₄×C₈₉	—	2	1	no	—
94	C₄⋊C₄×C₈₉	—	4	1	no	—
95	C₄⋊C₄×C₈₉	—	4	1	no	—
96	C₄⋊C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
97	C₂²⋊C₄×C₈₉	—	2	2	no	—
98	C₂²⋊C₄×C₈₉	—	4	1	no	—
99	C₄²×C₈₉	—	2	2	no	—
100	C₄²×C₈₉	—	2	2	no	—
101	C₄²×C₈₉	—	4	1	no	—
102	C₄²×C₈₉	—	4	1	no	—
103	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₈₉	#41	1	1	yes	characteristic
107	C₂³×C₈₉	#121	1	1	yes	characteristic
108	Q₈×C₈₉	—	4	2	no	—
109	Q₈×C₈₉	—	4	1	no	—
110	Q₈×C₈₉	—	4	1	no	—
111	Q₈×C₈₉	—	4	1	no	—
112	Q₈×C₈₉	—	4	1	no	—
113	Q₈×C₈₉	—	8	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	2	2	no	—
118	C₂×C₄×C₈₉	—	4	1	no	—
119	C₂×C₄×C₈₉	—	4	1	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	—	4	1	no	—
122	C₂×C₄×C₈₉	—	4	1	no	—
123	C₂×C₄×C₈₉	—	8	1	no	—
124	C₂×C₄×C₈₉	—	8	1	no	—
125	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
129	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
130	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
131	C₈×C₈₉	—	4	1	no	—
132	C₈×C₈₉	—	4	2	no	—
133	C₈×C₈₉	—	4	1	no	—
134	C₈×C₈₉	—	4	1	no	—
135	C₈×C₈₉	—	4	1	no	—
136	C₈×C₈₉	—	8	1	no	—
137	C₂²×C₈₉	—	2	1	no	—
138	C₂²×C₈₉	—	2	1	no	—
139	C₂²×C₈₉	—	2	1	no	—
140	C₂²×C₈₉	#386	1	1	yes	characteristic
141	C₄×C₈₉	—	2	1	no	—
142	C₄×C₈₉	—	2	1	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	—	2	1	no	—
146	C₄×C₈₉	—	4	2	no	—
147	C₄×C₈₉	—	4	2	no	—
148	C₄×C₈₉	—	8	2	no	—
149	C₄×C₈₉	—	8	1	no	—
150	C₄×C₈₉	—	8	1	no	—
151	C₄×C₈₉	#443	1	1	yes	characteristic
152	C₄×C₈₉	#444	1	1	yes	characteristic
153	#9280	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂×C₈₉	—	4	1	no	—
155	C₂×C₈₉	#1876	1	1	yes	characteristic
156	C₂×C₈₉	#1905	1	1	yes	characteristic
157	C₂×C₈₉	#1909	1	1	yes	characteristic
158	#3231	C₂×C₈₉	1	1	yes	characteristic
159	#2985	C₂×C₈₉	1	1	yes	characteristic
160	#1506	C₂×C₈₉	1	1	yes	characteristic
161	#1506	C₂×C₈₉	1	1	yes	characteristic
162	#1497	C₂×C₈₉	1	1	yes	characteristic
163	#1287	C₂×C₈₉	1	1	yes	characteristic
164	#1274	C₂×C₈₉	1	1	yes	characteristic
165	C₈₉	#9280	1	1	yes	characteristic, direct product factor, Sylow
166	#533	C₂²×C₈₉	1	1	yes	characteristic
167	#533	C₂²×C₈₉	1	1	yes	characteristic
168	#500	C₄×C₈₉	1	1	yes	characteristic, complemented
169	#498	C₄×C₈₉	1	1	yes	characteristic, complemented
170	#487	C₄×C₈₉	1	1	yes	characteristic, complemented
171	#487	C₄×C₈₉	1	1	yes	characteristic, complemented
172	#485	—	2	2	no	—
173	#479	—	2	2	no	—
174	#446	—	2	2	no	—
175	#446	—	2	1	no	—
176	C₈×Q₈	—	2	1	no	—
177	#433	—	2	2	no	—
178	#432	C₂²×C₈₉	1	1	yes	characteristic
179	#432	C₂²×C₈₉	1	1	yes	characteristic
180	#431	C₂²×C₈₉	1	1	yes	characteristic
181	#431	C₂²×C₈₉	1	1	yes	characteristic
182	#376	C₂²×C₈₉	1	1	yes	characteristic
183	#328	—	2	2	no	—
184	#328	—	2	2	no	—
185	Q₈⋊C₈	—	2	1	no	—
186	Q₈⋊C₈	—	2	1	no	—
187	C₄⋊Q₈	C₂×C₄×C₈₉	1	1	yes	characteristic
188	C₄⋊Q₈	C₂×C₄×C₈₉	1	1	yes	characteristic
189	C₄⋊Q₈	C₂×C₄×C₈₉	1	1	yes	characteristic
190	C₄⋊Q₈	C₂×C₄×C₈₉	1	1	yes	characteristic
191	#125	—	2	1	no	—
192	#125	—	2	1	no	—
193	C₂²⋊Q₈	—	2	1	no	—
194	C₂²⋊Q₈	—	2	1	no	—
195	C₂²⋊Q₈	—	2	1	no	—
196	C₂²⋊Q₈	D₄×C₈₉	1	2	yes	—
197	C₄×Q₈	—	2	1	no	—
198	C₄×Q₈	—	2	1	no	—
199	C₄×Q₈	—	2	1	no	—
200	C₄×Q₈	D₄×C₈₉	1	2	yes	—
201	#117	—	2	1	no	—
202	#117	—	2	1	no	—
203	C₂×C₄²	—	2	1	no	—
204	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
205	#107	—	2	2	no	—
206	#106	—	2	2	no	—
207	C₄⋊C₈	—	2	2	no	—
208	C₄⋊C₈	—	2	2	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	#103	—	2	2	no	—
214	#103	—	2	2	no	—
215	C₂²⋊C₈	—	2	1	no	—
216	C₂²⋊C₈	—	2	1	no	—
217	C₂²⋊C₈	—	2	1	no	—
218	C₂²⋊C₈	D₄×C₈₉	1	2	yes	—
219	#97	—	2	2	no	—
220	#97	—	2	1	no	—
221	#97	—	2	1	no	—
222	C₄×C₈	—	2	2	no	—
223	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
224	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
225	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
226	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
227	#95	—	2	1	no	—
228	#95	—	2	1	no	—
229	C₂×Q₈	—	2	1	no	—
230	C₂×Q₈	—	2	1	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	C₂²⋊C₄×C₈₉	1	2	yes	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
236	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	C₂²⋊C₄×C₈₉	1	2	yes	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	4	1	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	4	1	no	—
260	C₄²	—	4	1	no	—
261	C₄²	C₂²×C₄×C₈₉	1	1	yes	characteristic
262	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
264	C₄²	#41×C₈₉	1	1	yes	characteristic
265	C₂³	#121×C₈₉	1	1	yes	characteristic
266	Q₈	—	4	2	no	—
267	Q₈	—	4	1	no	—
268	Q₈	—	4	1	no	—
269	Q₈	—	4	1	no	—
270	Q₈	—	4	1	no	—
271	Q₈	—	8	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	8	1	no	—
282	C₂×C₄	—	8	1	no	—
283	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
284	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
285	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
286	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
287	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
288	C₂×C₄	#123×C₈₉	1	1	yes	characteristic
289	C₈	—	4	1	no	—
290	C₈	—	4	2	no	—
291	C₈	—	4	1	no	—
292	C₈	—	4	1	no	—
293	C₈	—	4	1	no	—
294	C₈	—	8	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	#386×C₈₉	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	1	no	—
308	C₄	—	8	1	no	—
309	C₄	#443×C₈₉	1	1	yes	characteristic
310	C₄	#444×C₈₉	1	1	yes	characteristic
311	C₂	—	4	1	no	—
312	C₂	C₈₉×#1876	1	1	yes	characteristic
313	C₂	C₈₉×#1905	1	1	yes	characteristic
314	C₂	C₈₉×#1909	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	#386
commutator	C₄²	C₂²×C₄×C₈₉
Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
prime radical	C₂³×C₈₉	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
10. #500×C₈₉	C₄	semidirect
11. #498×C₈₉	C₄	semidirect
12. #487×C₈₉	C₄	semidirect
13. #487×C₈₉	C₄	semidirect
153. #9280	C₈₉	direct (N × H)
165. C₈₉	#9280	direct (N × H)
168. #500	C₄×C₈₉	semidirect
169. #498	C₄×C₈₉	semidirect
170. #487	C₄×C₈₉	semidirect
171. #487	C₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3231	2
C₈₉×#2985	2
C₈₉×#1506	2
C₈₉×#1497	2
C₈₉×#1287	2
C₈₉×#1274	2
#9280	89

Subgroups — C89×#9280 #36387128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#9280 #36387128