Subgroups — C₈₉×#29185 #36407033

Order 22784.

590 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2647	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2330	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2088	C₂	1	2	yes	maximal
4	C₈₉×#2088	C₂	1	2	yes	maximal
5	C₈₉×#1992	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#1991	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1920	C₂	1	2	yes	maximal
8	C₈₉×#1919	C₂	1	2	yes	maximal
9	C₈₉×#1844	C₂	1	1	yes	maximal, characteristic
10	C₈₉×#1836	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1768	C₂	1	1	yes	maximal, characteristic
12	C₂×#125×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×#106×C₈₉	C₂²	1	2	yes	—
14	C₂×#106×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#400×C₈₉	—	2	2	no	—
20	#398×C₈₉	—	2	2	no	—
21	#389×C₈₉	—	2	2	no	—
22	#389×C₈₉	C₂²	1	2	yes	—
23	#389×C₈₉	C₂²	1	2	yes	—
24	#389×C₈₉	C₂²	1	2	yes	—
25	#387×C₈₉	—	2	2	no	—
26	#384×C₈₉	C₂²	1	1	yes	characteristic
27	#384×C₈₉	C₂²	1	1	yes	characteristic
28	#382×C₈₉	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
30	#376×C₈₉	C₂²	1	1	yes	characteristic
31	#340×C₈₉	C₂²	1	2	yes	—
32	#340×C₈₉	C₂²	1	2	yes	—
33	#340×C₈₉	C₂²	1	2	yes	—
34	#340×C₈₉	C₂²	1	2	yes	—
35	#340×C₈₉	C₂²	1	2	yes	—
36	#340×C₈₉	C₂²	1	2	yes	—
37	#336×C₈₉	C₂²	1	1	yes	characteristic
38	#336×C₈₉	C₂²	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
43	#125×C₈₉	—	2	2	no	—
44	#125×C₈₉	Q₈	1	4	yes	—
45	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
49	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
50	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
51	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
55	C₂×C₄²×C₈₉	—	2	2	no	—
56	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
59	#106×C₈₉	—	2	4	no	—
60	#106×C₈₉	—	2	4	no	—
61	C₄⋊C₈×C₈₉	—	2	2	no	—
62	C₄⋊C₈×C₈₉	—	2	2	no	—
63	#97×C₈₉	—	2	2	no	—
64	C₄×C₈×C₈₉	—	2	2	no	—
65	#95×C₈₉	—	2	2	no	—
66	#95×C₈₉	—	2	2	no	—
67	#95×C₈₉	—	2	2	no	—
68	#95×C₈₉	—	2	2	no	—
69	#95×C₈₉	—	2	2	no	—
70	#95×C₈₉	D₄	1	2	yes	—
71	#95×C₈₉	D₄	1	2	yes	—
72	C₂²×C₄×C₈₉	—	2	2	no	—
73	C₂²×C₄×C₈₉	—	2	2	no	—
74	C₂²×C₄×C₈₉	—	2	2	no	—
75	C₂²×C₄×C₈₉	—	2	2	no	—
76	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₈₉	—	2	1	no	—
82	C₂×C₈×C₈₉	—	2	1	no	—
83	C₂×C₈×C₈₉	—	2	1	no	—
84	C₂×C₈×C₈₉	—	2	1	no	—
85	C₂×C₈×C₈₉	—	2	1	no	—
86	C₂×C₈×C₈₉	—	2	1	no	—
87	C₂×C₈×C₈₉	—	2	1	no	—
88	C₂×C₈×C₈₉	—	2	1	no	—
89	C₂×C₈×C₈₉	—	2	1	no	—
90	C₂×C₈×C₈₉	—	2	1	no	—
91	C₂×C₈×C₈₉	#41	1	2	yes	—
92	C₂×C₈×C₈₉	#41	1	2	yes	—
93	C₄⋊C₄×C₈₉	—	2	8	no	—
94	C₄⋊C₄×C₈₉	—	2	2	no	—
95	C₄⋊C₄×C₈₉	—	2	2	no	—
96	C₄⋊C₄×C₈₉	—	2	2	no	—
97	C₄⋊C₄×C₈₉	—	2	2	no	—
98	C₄⋊C₄×C₈₉	—	2	2	no	—
99	C₄⋊C₄×C₈₉	—	2	2	no	—
100	C₄⋊C₄×C₈₉	—	2	2	no	—
101	C₄⋊C₄×C₈₉	—	8	2	no	—
102	C₄⋊C₄×C₈₉	—	8	2	no	—
103	C₄⋊C₄×C₈₉	—	8	2	no	—
104	C₄⋊C₄×C₈₉	—	8	2	no	—
105	C₄⋊C₄×C₈₉	—	8	2	no	—
106	C₄⋊C₄×C₈₉	C₂×Q₈	1	4	yes	—
107	C₄²×C₈₉	—	2	1	no	—
108	C₄²×C₈₉	—	2	1	no	—
109	C₄²×C₈₉	—	2	1	no	—
110	C₄²×C₈₉	—	2	1	no	—
111	C₄²×C₈₉	—	8	2	no	—
112	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
113	C₄²×C₈₉	#41	1	2	yes	—
114	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	—	2	2	no	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	2	4	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	2	no	—
121	C₂×C₄×C₈₉	—	2	2	no	—
122	C₂×C₄×C₈₉	—	2	2	no	—
123	C₂×C₄×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	4	2	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	4	2	no	—
131	C₂×C₄×C₈₉	—	4	2	no	—
132	C₂×C₄×C₈₉	—	4	2	no	—
133	C₂×C₄×C₈₉	—	4	2	no	—
134	C₂×C₄×C₈₉	—	4	2	no	—
135	C₂×C₄×C₈₉	—	4	2	no	—
136	C₂×C₄×C₈₉	C₂²⋊Q₈	1	4	yes	—
137	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
142	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
143	C₈×C₈₉	—	2	2	no	—
144	C₈×C₈₉	—	2	2	no	—
145	C₈×C₈₉	—	4	1	no	—
146	C₈×C₈₉	—	4	1	no	—
147	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
148	C₂²×C₈₉	#532	1	1	yes	characteristic
149	C₂²×C₈₉	#536	1	1	yes	characteristic
150	C₂²×C₈₉	#577	1	2	yes	—
151	C₂²×C₈₉	#577	1	2	yes	—
152	C₄×C₈₉	—	2	4	no	—
153	C₄×C₈₉	—	4	4	no	—
154	C₄×C₈₉	—	4	1	no	—
155	C₄×C₈₉	—	4	1	no	—
156	C₄×C₈₉	—	4	2	no	—
157	C₄×C₈₉	—	8	2	no	—
158	C₄×C₈₉	—	8	2	no	—
159	C₄×C₈₉	—	8	2	no	—
160	C₄×C₈₉	#552	1	1	yes	characteristic
161	C₄×C₈₉	#552	1	1	yes	characteristic
162	C₄×C₈₉	#557	1	1	yes	characteristic
163	C₄×C₈₉	#557	1	1	yes	characteristic
164	#29185	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂×C₈₉	#2565	1	1	yes	characteristic
166	C₂×C₈₉	#3084	1	2	yes	—
167	C₂×C₈₉	#3174	1	1	yes	characteristic
168	C₂×C₈₉	#3174	1	1	yes	characteristic
169	C₂×C₈₉	#3244	1	2	yes	—
170	#2647	C₂×C₈₉	1	1	yes	characteristic
171	#2330	C₂×C₈₉	1	1	yes	characteristic
172	#2088	C₂×C₈₉	1	2	yes	—
173	#2088	C₂×C₈₉	1	2	yes	—
174	#1992	C₂×C₈₉	1	1	yes	characteristic
175	#1991	C₂×C₈₉	1	1	yes	characteristic
176	#1920	C₂×C₈₉	1	2	yes	—
177	#1919	C₂×C₈₉	1	2	yes	—
178	#1844	C₂×C₈₉	1	1	yes	characteristic
179	#1836	C₂×C₈₉	1	1	yes	characteristic
180	#1768	C₂×C₈₉	1	1	yes	characteristic
181	C₈₉	#29185	1	1	yes	characteristic, direct product factor, Sylow
182	C₂×#125	C₂²×C₈₉	1	1	yes	characteristic
183	C₂×#106	C₂²×C₈₉	1	2	yes	—
184	C₂×#106	C₂²×C₈₉	1	2	yes	—
185	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
186	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
187	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
188	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
189	#400	—	2	2	no	—
190	#398	—	2	2	no	—
191	#389	—	2	2	no	—
192	#389	C₂²×C₈₉	1	2	yes	—
193	#389	C₂²×C₈₉	1	2	yes	—
194	#389	C₂²×C₈₉	1	2	yes	—
195	#387	—	2	2	no	—
196	#384	C₂²×C₈₉	1	1	yes	characteristic
197	#384	C₂²×C₈₉	1	1	yes	characteristic
198	#382	C₂²×C₈₉	1	1	yes	characteristic
199	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
200	#376	C₂²×C₈₉	1	1	yes	characteristic
201	#340	C₂²×C₈₉	1	2	yes	—
202	#340	C₂²×C₈₉	1	2	yes	—
203	#340	C₂²×C₈₉	1	2	yes	—
204	#340	C₂²×C₈₉	1	2	yes	—
205	#340	C₂²×C₈₉	1	2	yes	—
206	#340	C₂²×C₈₉	1	2	yes	—
207	#336	C₂²×C₈₉	1	1	yes	characteristic
208	#336	C₂²×C₈₉	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
212	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
213	#125	—	2	2	no	—
214	#125	Q₈×C₈₉	1	4	yes	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
222	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
223	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
224	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
225	C₂×C₄²	—	2	2	no	—
226	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
227	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
228	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
229	#106	—	2	4	no	—
230	#106	—	2	4	no	—
231	C₄⋊C₈	—	2	2	no	—
232	C₄⋊C₈	—	2	2	no	—
233	#97	—	2	2	no	—
234	C₄×C₈	—	2	2	no	—
235	#95	—	2	2	no	—
236	#95	—	2	2	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	D₄×C₈₉	1	2	yes	—
241	#95	D₄×C₈₉	1	2	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
250	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	#41×C₈₉	1	2	yes	—
262	C₂×C₈	#41×C₈₉	1	2	yes	—
263	C₄⋊C₄	—	2	8	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	8	2	no	—
272	C₄⋊C₄	—	8	2	no	—
273	C₄⋊C₄	—	8	2	no	—
274	C₄⋊C₄	—	8	2	no	—
275	C₄⋊C₄	—	8	2	no	—
276	C₄⋊C₄	C₂×Q₈×C₈₉	1	4	yes	—
277	C₄²	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	8	2	no	—
282	C₄²	C₂×Q₈×C₈₉	1	2	yes	—
283	C₄²	#41×C₈₉	1	2	yes	—
284	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	C₂²⋊Q₈×C₈₉	1	4	yes	—
307	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
308	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
309	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
310	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
311	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
312	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	2	2	no	—
315	C₈	—	4	1	no	—
316	C₈	—	4	1	no	—
317	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
318	C₂²	#532×C₈₉	1	1	yes	characteristic
319	C₂²	#536×C₈₉	1	1	yes	characteristic
320	C₂²	#577×C₈₉	1	2	yes	—
321	C₂²	#577×C₈₉	1	2	yes	—
322	C₄	—	2	4	no	—
323	C₄	—	4	4	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	2	no	—
327	C₄	—	8	2	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	2	no	—
330	C₄	#552×C₈₉	1	1	yes	characteristic
331	C₄	#552×C₈₉	1	1	yes	characteristic
332	C₄	#557×C₈₉	1	1	yes	characteristic
333	C₄	#557×C₈₉	1	1	yes	characteristic
334	C₂	C₈₉×#2565	1	1	yes	characteristic
335	C₂	C₈₉×#3084	1	2	yes	—
336	C₂	C₈₉×#3174	1	1	yes	characteristic
337	C₂	C₈₉×#3174	1	1	yes	characteristic
338	C₂	C₈₉×#3244	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ #376×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
164. #29185	C₈₉	direct (N × H)
181. C₈₉	#29185	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₉×#2647	2
C₈₉×#2330	2
C₈₉×#2088	2
C₈₉×#1992	2
C₈₉×#1991	2
C₈₉×#1920	2
C₈₉×#1919	2
C₈₉×#1844	2
C₈₉×#1836	2
C₈₉×#1768	2
#29185	89

Subgroups — C89×#29185 #36407033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29185 #36407033