Subgroups — #1707×D₈₉ #36542073

Order 22784.

814 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₈₉	C₂	1	1	yes	maximal, characteristic
2	#498×D₈₉	C₂	1	2	yes	maximal
3	#494×D₈₉	C₂	1	1	yes	maximal, characteristic
4	#487×D₈₉	C₂	1	2	yes	maximal
5	#26019850	C₂	1	1	yes	maximal, characteristic, complemented
6	#26019849	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019848	C₂	1	2	yes	maximal, complemented
8	#26019847	C₂	1	2	yes	maximal, complemented
9	#26019846	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈²×D₈₉	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₈₉	—	2	2	no	—
13	C₄⋊Q₈×D₈₉	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₈₉	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152002	—	2	2	no	—
17	#19152001	C₂²	1	2	yes	—
18	#19151996	C₂²	1	2	yes	—
19	#19151995	—	2	4	no	—
20	#19151994	C₂²	1	2	yes	—
21	#19151994	C₂²	1	2	yes	—
22	#19151978	C₂²	1	1	yes	characteristic
23	#19151977	C₂²	1	2	yes	—
24	#19151976	—	2	2	no	—
25	#19151950	C₂²	1	2	yes	—
26	#19151949	C₂²	1	2	yes	—
27	#19151948	C₂²	1	2	yes	—
28	#19151947	—	2	4	no	—
29	#107×D₈₉	—	2	2	no	—
30	#106×D₈₉	—	2	4	no	—
31	#103×D₈₉	—	2	4	no	—
32	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
35	C₈×#10623494	C₂²	1	1	yes	characteristic
36	C₈×#10623494	C₂²	1	1	yes	characteristic
37	C₈×#10623494	C₂²	1	1	yes	characteristic
38	#500×C₈₉	C₂²	1	1	yes	characteristic
39	#498×C₈₉	C₂²	1	2	yes	—
40	#494×C₈₉	C₂²	1	1	yes	characteristic
41	#487×C₈₉	C₂²	1	2	yes	—
42	C₈²×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×Q₈×D₈₉	—	2	2	no	—
44	C₂×Q₈×D₈₉	—	2	2	no	—
45	C₂×Dic₃₅₆	—	2	2	no	—
46	C₂×#13083564	—	2	4	no	—
47	Q₁₆×D₈₉	—	4	8	no	—
48	C₂×C₈×D₈₉	D₄	1	2	yes	—
49	C₂×C₈×D₈₉	D₄	1	2	yes	—
50	C₂×C₈×D₈₉	D₄	1	2	yes	—
51	#15668012	C₂³	1	2	yes	—
52	#15668012	C₂³	1	2	yes	—
53	#15668011	—	2	2	no	—
54	#15668011	—	2	2	no	—
55	#15668011	—	2	2	no	—
56	#15668011	—	2	2	no	—
57	#15668010	—	2	2	no	—
58	#15668010	—	2	2	no	—
59	C₄⋊C₄×D₈₉	—	2	2	no	—
60	C₄⋊C₄×D₈₉	—	2	2	no	—
61	C₄⋊C₄×D₈₉	—	2	2	no	—
62	C₄⋊C₄×D₈₉	—	2	2	no	—
63	C₄²×D₈₉	C₂³	1	1	yes	characteristic
64	#15667946	—	2	2	no	—
65	#15667945	—	2	4	no	—
66	#15667944	—	2	4	no	—
67	#15667943	—	2	4	no	—
68	#15667943	—	2	4	no	—
69	#15667937	—	2	4	no	—
70	#15667936	—	2	4	no	—
71	#15667935	—	2	4	no	—
72	C₄×#10623494	C₂³	1	1	yes	characteristic
73	C₄×#10623494	C₂³	1	1	yes	characteristic
74	C₄×#10623494	C₂³	1	1	yes	characteristic
75	C₈×Dic₈₉	D₄	1	2	yes	—
76	C₈×Dic₈₉	D₄	1	2	yes	—
77	C₈×Dic₈₉	D₄	1	2	yes	—
78	C₂×Q₁₆×C₈₉	—	2	2	no	—
79	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
80	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
81	#107×C₈₉	—	2	2	no	—
82	#106×C₈₉	—	2	4	no	—
83	#103×C₈₉	—	2	4	no	—
84	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
87	C₂×Dic₁₇₈	—	2	2	no	—
88	C₂×Dic₁₇₈	—	2	2	no	—
89	C₂×Dic₁₇₈	—	4	2	no	—
90	C₂×Dic₁₇₈	—	4	2	no	—
91	Q₈×D₈₉	—	4	8	no	—
92	Q₈×D₈₉	—	4	8	no	—
93	C₂×C₄×D₈₉	—	4	2	no	—
94	C₂×C₄×D₈₉	—	4	2	no	—
95	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
98	Dic₃₅₆	—	4	4	no	—
99	#13083564	—	4	8	no	—
100	C₂×#10623494	C₂×D₄	1	2	yes	—
101	C₂×#10623494	C₂×D₄	1	2	yes	—
102	C₂×#10623494	C₂×D₄	1	2	yes	—
103	C₈×D₈₉	—	2	4	no	—
104	C₈×D₈₉	SD₁₆	1	8	yes	—
105	C₈×D₈₉	Q₁₆	1	8	yes	complemented
106	C₄⋊Dic₈₉	—	2	2	no	—
107	C₄⋊Dic₈₉	—	2	2	no	—
108	C₄⋊Dic₈₉	—	2	2	no	—
109	C₄⋊Dic₈₉	—	2	2	no	—
110	#13083553	—	4	2	no	—
111	#13083553	—	4	2	no	—
112	#13083553	—	4	2	no	—
113	#13083553	—	4	2	no	—
114	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
115	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
116	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×Q₈×C₈₉	—	2	2	no	—
118	C₂×Q₈×C₈₉	—	2	2	no	—
119	Q₁₆×C₈₉	—	4	4	no	—
120	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
123	C₄⋊C₄×C₈₉	—	2	2	no	—
124	C₄⋊C₄×C₈₉	—	2	2	no	—
125	C₄⋊C₄×C₈₉	—	2	2	no	—
126	C₄⋊C₄×C₈₉	—	2	2	no	—
127	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
128	C₂²×D₈₉	#127	1	1	yes	characteristic
129	Q₈×C₈₉	—	4	4	no	—
130	Q₈×C₈₉	—	4	4	no	—
131	C₂×C₄×C₈₉	—	4	2	no	—
132	C₂×C₄×C₈₉	—	4	2	no	—
133	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₈₉	—	4	2	no	—
137	C₂×Dic₈₉	—	4	2	no	—
138	C₂×Dic₈₉	#127	1	1	yes	characteristic
139	C₂×Dic₈₉	#127	1	1	yes	characteristic
140	C₂×Dic₈₉	#127	1	1	yes	characteristic
141	C₄×D₈₉	—	8	4	no	—
142	C₄×D₈₉	—	8	4	no	—
143	C₄×D₈₉	C₂×SD₁₆	1	4	yes	—
144	C₄×D₈₉	C₂×Q₁₆	1	4	yes	—
145	C₄×D₈₉	#135	1	4	yes	—
146	Dic₁₇₈	—	4	4	no	—
147	Dic₁₇₈	—	4	4	no	—
148	Dic₁₇₈	—	8	4	no	—
149	Dic₁₇₈	—	8	4	no	—
150	C₈×C₈₉	—	2	2	no	—
151	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
152	C₈×C₈₉	C₂×Q₁₆	1	4	yes	complemented
153	#10623494	—	2	2	no	—
154	#10623494	C₂×SD₁₆	1	4	yes	—
155	#10623494	C₂×Q₁₆	1	4	yes	complemented
156	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
157	C₂×D₈₉	#492	1	2	yes	—
158	C₂×D₈₉	#494	1	2	yes	—
159	C₂×D₈₉	#495	1	2	yes	—
160	C₄×C₈₉	—	8	2	no	—
161	C₄×C₈₉	—	8	2	no	—
162	C₄×C₈₉	C₂²×SD₁₆	1	2	yes	—
163	C₄×C₈₉	C₂²×Q₁₆	1	2	yes	—
164	C₄×C₈₉	C₂×#135	1	2	yes	—
165	Dic₈₉	—	8	2	no	—
166	Dic₈₉	—	8	2	no	—
167	Dic₈₉	#492	1	2	yes	—
168	Dic₈₉	#494	1	2	yes	—
169	Dic₈₉	#495	1	2	yes	—
170	C₂×#1707	—	89	1	no	maximal, Sylow
171	C₂×C₈₉	C₂×#492	1	1	yes	characteristic
172	C₂×C₈₉	C₂×#494	1	1	yes	characteristic
173	C₂×C₈₉	C₂×#495	1	1	yes	characteristic
174	D₈₉	#1707	1	4	yes	direct product factor
175	C₂×#500	—	89	1	no	—
176	C₂×#498	—	89	2	no	—
177	C₂×#494	—	89	1	no	—
178	C₂×#487	—	89	2	no	—
179	#1707	—	89	1	no	—
180	#1707	—	89	2	no	—
181	#1707	—	89	1	no	—
182	#1707	—	89	2	no	—
183	#1707	—	89	1	no	—
184	#1707	D₈₉	1	1	yes	characteristic, direct product factor
185	C₂×C₈²	—	89	1	no	—
186	C₈₉	C₂×#1707	1	1	yes	characteristic, complemented, Sylow
187	C₂²×Q₁₆	—	178	2	no	—
188	C₂×C₄⋊Q₈	—	89	2	no	—
189	C₂×C₄⋊Q₈	—	89	2	no	—
190	#500	—	89	2	no	—
191	#500	—	89	1	no	—
192	#500	—	178	2	no	—
193	#500	C₂×D₈₉	1	1	yes	characteristic
194	#498	—	89	2	no	—
195	#498	—	89	2	no	—
196	#498	—	89	2	no	—
197	#498	—	178	4	no	—
198	#498	C₂×D₈₉	1	2	yes	—
199	#494	—	89	2	no	—
200	#494	—	89	1	no	—
201	#494	—	178	2	no	—
202	#494	C₂×D₈₉	1	1	yes	characteristic
203	#487	—	89	2	no	—
204	#487	—	89	2	no	—
205	#487	—	89	2	no	—
206	#487	—	178	4	no	—
207	#487	C₂×D₈₉	1	2	yes	—
208	C₂×#107	—	178	2	no	—
209	C₂×#106	—	178	4	no	—
210	C₂×#103	—	178	4	no	—
211	C₂×C₄×C₈	—	89	1	no	—
212	C₂×C₄×C₈	—	89	1	no	—
213	C₂×C₄×C₈	—	89	1	no	—
214	C₈²	—	89	1	no	—
215	C₈²	—	89	1	no	—
216	C₈²	—	89	1	no	—
217	C₈²	C₂×D₈₉	1	1	yes	characteristic
218	C₂²×Q₈	—	178	2	no	—
219	C₂²×Q₈	—	178	2	no	—
220	C₂×Q₁₆	—	2	2	no	—
221	C₂×Q₁₆	—	356	8	no	—
222	C₂×Q₁₆	—	178	4	no	—
223	C₂×Q₁₆	—	178	2	no	—
224	C₂²×C₈	—	89	2	no	—
225	C₂²×C₈	—	89	2	no	—
226	C₂²×C₈	—	89	2	no	—
227	C₄⋊Q₈	—	89	2	no	—
228	C₄⋊Q₈	—	89	2	no	—
229	C₄⋊Q₈	—	178	2	no	—
230	C₄⋊Q₈	—	178	2	no	—
231	C₄⋊Q₈	—	178	2	no	—
232	C₄⋊Q₈	—	178	2	no	—
233	C₄⋊Q₈	—	178	2	no	—
234	C₄⋊Q₈	—	178	2	no	—
235	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
236	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
237	C₂×C₄⋊C₄	—	178	2	no	—
238	C₂×C₄⋊C₄	—	178	2	no	—
239	C₂×C₄⋊C₄	—	178	2	no	—
240	C₂×C₄⋊C₄	—	178	2	no	—
241	C₂×C₄²	—	89	1	no	—
242	#107	—	2	2	no	—
243	#107	—	178	4	no	—
244	#107	—	178	2	no	—
245	#106	—	2	4	no	—
246	#106	—	178	4	no	—
247	#106	—	178	4	no	—
248	#106	—	178	4	no	—
249	#103	—	2	4	no	—
250	#103	—	178	4	no	—
251	#103	—	178	4	no	—
252	#103	—	178	4	no	—
253	C₄×C₈	—	89	2	no	—
254	C₄×C₈	—	89	2	no	—
255	C₄×C₈	—	89	2	no	—
256	C₄×C₈	—	89	1	no	—
257	C₄×C₈	—	89	1	no	—
258	C₄×C₈	—	89	1	no	—
259	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	356	2	no	—
265	C₂×Q₈	—	356	2	no	—
266	C₂×Q₈	—	356	8	no	—
267	C₂×Q₈	—	356	8	no	—
268	C₂×Q₈	—	178	2	no	—
269	C₂×Q₈	—	178	2	no	—
270	C₂²×C₄	—	89	1	no	—
271	C₂²×C₄	—	89	1	no	—
272	C₂²×C₄	—	89	1	no	—
273	C₂²×C₄	—	356	2	no	—
274	C₂²×C₄	—	356	2	no	—
275	Q₁₆	—	4	4	no	—
276	Q₁₆	—	356	8	no	—
277	Q₁₆	—	356	4	no	—
278	C₂×C₈	—	89	2	no	—
279	C₂×C₈	—	89	2	no	—
280	C₂×C₈	—	89	8	no	—
281	C₂×C₈	—	89	8	no	—
282	C₂×C₈	—	89	2	no	—
283	C₂×C₈	—	178	4	no	—
284	C₂×C₈	D₄×D₈₉	1	2	yes	—
285	C₂×C₈	D₄×D₈₉	1	2	yes	—
286	C₂×C₈	D₄×D₈₉	1	2	yes	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	356	2	no	—
292	C₄⋊C₄	—	356	2	no	—
293	C₄⋊C₄	—	356	2	no	—
294	C₄⋊C₄	—	356	2	no	—
295	C₄⋊C₄	—	178	2	no	—
296	C₄⋊C₄	—	178	2	no	—
297	C₄⋊C₄	—	178	2	no	—
298	C₄⋊C₄	—	178	2	no	—
299	C₄²	—	89	1	no	—
300	C₄²	—	89	1	no	—
301	C₄²	—	89	1	no	—
302	C₄²	C₂³×D₈₉	1	1	yes	characteristic
303	C₂³	—	89	1	no	—
304	Q₈	—	4	4	no	—
305	Q₈	—	4	4	no	—
306	Q₈	—	356	4	no	—
307	Q₈	—	356	4	no	—
308	Q₈	—	712	4	no	—
309	Q₈	—	712	4	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	89	1	no	—
313	C₂×C₄	—	89	1	no	—
314	C₂×C₄	—	89	1	no	—
315	C₂×C₄	—	89	4	no	—
316	C₂×C₄	—	89	4	no	—
317	C₂×C₄	—	89	4	no	—
318	C₂×C₄	—	356	2	no	—
319	C₂×C₄	—	356	2	no	—
320	C₂×C₄	—	712	4	no	—
321	C₂×C₄	—	712	4	no	—
322	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
323	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
324	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	89	4	no	—
327	C₈	—	89	4	no	—
328	C₈	—	178	2	no	—
329	C₈	SD₁₆×D₈₉	1	4	yes	—
330	C₈	Q₁₆×D₈₉	1	4	yes	complemented
331	C₂²	—	89	2	no	—
332	C₂²	—	89	2	no	—
333	C₂²	—	89	2	no	—
334	C₂²	#127×D₈₉	1	1	yes	characteristic
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	89	2	no	—
338	C₄	—	89	2	no	—
339	C₄	—	89	2	no	—
340	C₄	—	712	2	no	—
341	C₄	—	712	2	no	—
342	C₄	C₂×SD₁₆×D₈₉	1	2	yes	—
343	C₄	C₂×Q₁₆×D₈₉	1	2	yes	—
344	C₄	#135×D₈₉	1	2	yes	—
345	C₂	—	89	4	no	—
346	C₂	#492×D₈₉	1	1	yes	characteristic
347	C₂	#494×D₈₉	1	1	yes	characteristic
348	C₂	#495×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₈₉
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1707	C₂
socle	C₂²×C₈₉	C₂×#127
prime radical	C₂²×D₈₉	#127
hypercenter	#1707	D₈₉
hypocenter	C₈₉	C₂×#1707
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
5. #26019850	C₂	semidirect
6. #26019849	C₂	semidirect
7. #26019848	C₂	semidirect
8. #26019847	C₂	semidirect
9. #26019846	C₂	semidirect
11. C₈₉×#1707	C₂	semidirect
105. C₈×D₈₉	Q₁₆	semidirect
152. C₈×C₈₉	C₂×Q₁₆	semidirect
155. #10623494	C₂×Q₁₆	semidirect
174. D₈₉	#1707	direct (N × H)
184. #1707	D₈₉	direct (N × H)
186. C₈₉	C₂×#1707	semidirect
330. C₈	Q₁₆×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#500×D₈₉	2
#498×D₈₉	2
#494×D₈₉	2
#487×D₈₉	2
#26019850	2
#26019849	2
#26019848	2
#26019847	2
#26019846	2
C₈²×D₈₉	2
C₈₉×#1707	2
C₂×#1707	89

Subgroups — #1707×D89 #36542073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1707×D₈₉ #36542073