Subgroups — #1716×D₈₉ #36542151

Order 22784.

798 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#498×D₈₉	C₂	1	2	yes	maximal
2	#498×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#494×D₈₉	C₂	1	1	yes	maximal, characteristic
4	#487×D₈₉	C₂	1	2	yes	maximal
5	#26019887	C₂	1	2	yes	maximal, complemented
6	#26019886	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019885	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019884	C₂	1	2	yes	maximal, complemented
9	#26019883	C₂	1	1	yes	maximal, characteristic, complemented
10	#322×D₈₉	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1716	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₈₉	—	2	2	no	—
13	C₄⋊Q₈×D₈₉	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×D₈₉	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×D₈₉	C₂²	1	2	yes	—
16	#19151996	C₂²	1	1	yes	characteristic
17	#19151996	C₂²	1	2	yes	—
18	#19151995	—	2	4	no	—
19	#19151995	—	2	2	no	—
20	#19151994	C₂²	1	2	yes	—
21	#19151994	C₂²	1	2	yes	—
22	#19151994	C₂²	1	2	yes	—
23	#19151978	C₂²	1	1	yes	characteristic
24	#19151977	C₂²	1	1	yes	characteristic
25	#19151977	C₂²	1	1	yes	characteristic
26	#19151976	—	2	2	no	—
27	#19151950	C₂²	1	2	yes	—
28	#19151949	C₂²	1	2	yes	—
29	#19151948	C₂²	1	2	yes	—
30	#19151947	—	2	4	no	—
31	#106×D₈₉	—	2	4	no	—
32	#106×D₈₉	—	2	2	no	—
33	#103×D₈₉	—	2	4	no	—
34	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
35	C₄×C₈×D₈₉	C₂²	1	2	yes	—
36	#19151320	C₂²	1	2	yes	—
37	#19151319	C₂²	1	1	yes	characteristic
38	#498×C₈₉	C₂²	1	2	yes	—
39	#498×C₈₉	C₂²	1	1	yes	characteristic
40	#494×C₈₉	C₂²	1	1	yes	characteristic
41	#487×C₈₉	C₂²	1	2	yes	—
42	#322×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×Q₈×D₈₉	—	2	2	no	—
44	C₂×Q₈×D₈₉	—	2	1	no	—
45	C₂×Q₈×D₈₉	—	2	1	no	—
46	C₂×Dic₃₅₆	—	2	2	no	—
47	C₂×#13083564	—	2	2	no	—
48	C₂×#13083564	—	2	2	no	—
49	Q₁₆×D₈₉	—	4	8	no	—
50	C₂×C₈×D₈₉	D₄	1	4	yes	—
51	C₂×C₈×D₈₉	D₄	1	2	yes	—
52	#15668012	C₂³	1	1	yes	characteristic
53	#15668012	C₂³	1	2	yes	—
54	#15668012	C₂³	1	1	yes	characteristic
55	#15668011	—	2	2	no	—
56	#15668011	—	2	2	no	—
57	#15668011	—	2	2	no	—
58	#15668011	—	2	2	no	—
59	#15668010	—	2	2	no	—
60	#15668010	—	2	1	no	—
61	#15668010	—	2	1	no	—
62	C₄⋊C₄×D₈₉	—	2	2	no	—
63	C₄⋊C₄×D₈₉	—	2	2	no	—
64	C₄⋊C₄×D₈₉	—	2	2	no	—
65	C₄⋊C₄×D₈₉	—	2	2	no	—
66	C₄²×D₈₉	C₂³	1	1	yes	characteristic
67	#15667944	—	2	4	no	—
68	#15667944	—	2	2	no	—
69	#15667943	—	2	4	no	—
70	#15667943	—	2	4	no	—
71	#15667943	—	2	4	no	—
72	#15667937	—	2	4	no	—
73	#15667936	—	2	4	no	—
74	#15667935	—	2	4	no	—
75	C₄×#10623494	C₂³	1	2	yes	—
76	C₄×#10623494	C₂³	1	1	yes	characteristic
77	C₈×Dic₈₉	D₄	1	4	yes	—
78	C₈×Dic₈₉	D₄	1	2	yes	—
79	C₂×Q₁₆×C₈₉	—	2	2	no	—
80	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
81	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
83	#106×C₈₉	—	2	4	no	—
84	#106×C₈₉	—	2	2	no	—
85	#103×C₈₉	—	2	4	no	—
86	C₄×C₈×C₈₉	C₂³	1	2	yes	—
87	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
88	C₂×Dic₁₇₈	—	2	1	no	—
89	C₂×Dic₁₇₈	—	2	1	no	—
90	C₂×Dic₁₇₈	—	2	2	no	—
91	C₂×Dic₁₇₈	—	4	1	no	—
92	C₂×Dic₁₇₈	—	4	1	no	—
93	C₂×Dic₁₇₈	—	4	2	no	—
94	Q₈×D₈₉	—	4	4	no	—
95	Q₈×D₈₉	—	4	4	no	—
96	Q₈×D₈₉	—	4	8	no	—
97	C₂×C₄×D₈₉	—	4	1	no	—
98	C₂×C₄×D₈₉	—	4	1	no	—
99	C₂×C₄×D₈₉	—	4	2	no	—
100	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
102	Dic₃₅₆	—	4	4	no	—
103	#13083564	—	4	4	no	—
104	#13083564	—	4	4	no	—
105	C₂×#10623494	C₂×D₄	1	4	yes	—
106	C₂×#10623494	C₂×D₄	1	2	yes	—
107	C₈×D₈₉	—	2	8	no	—
108	C₈×D₈₉	SD₁₆	1	8	yes	—
109	C₄⋊Dic₈₉	—	2	2	no	—
110	C₄⋊Dic₈₉	—	2	2	no	—
111	C₄⋊Dic₈₉	—	2	2	no	—
112	C₄⋊Dic₈₉	—	2	2	no	—
113	#13083553	—	4	2	no	—
114	#13083553	—	4	2	no	—
115	#13083553	—	4	2	no	—
116	#13083553	—	4	2	no	—
117	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
118	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
119	C₂×Q₈×C₈₉	—	2	1	no	—
120	C₂×Q₈×C₈₉	—	2	1	no	—
121	C₂×Q₈×C₈₉	—	2	2	no	—
122	Q₁₆×C₈₉	—	4	4	no	—
123	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
124	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
125	C₄⋊C₄×C₈₉	—	2	2	no	—
126	C₄⋊C₄×C₈₉	—	2	2	no	—
127	C₄⋊C₄×C₈₉	—	2	2	no	—
128	C₄⋊C₄×C₈₉	—	2	2	no	—
129	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
130	C₂²×D₈₉	#127	1	1	yes	characteristic
131	Q₈×C₈₉	—	4	2	no	—
132	Q₈×C₈₉	—	4	2	no	—
133	Q₈×C₈₉	—	4	4	no	—
134	C₂×C₄×C₈₉	—	4	1	no	—
135	C₂×C₄×C₈₉	—	4	1	no	—
136	C₂×C₄×C₈₉	—	4	2	no	—
137	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
139	C₂×Dic₈₉	—	4	1	no	—
140	C₂×Dic₈₉	—	4	1	no	—
141	C₂×Dic₈₉	—	4	2	no	—
142	C₂×Dic₈₉	#127	1	1	yes	characteristic
143	C₂×Dic₈₉	#127	1	2	yes	—
144	C₄×D₈₉	—	8	2	no	—
145	C₄×D₈₉	—	8	2	no	—
146	C₄×D₈₉	—	8	4	no	—
147	C₄×D₈₉	C₂×SD₁₆	1	4	yes	—
148	C₄×D₈₉	#137	1	8	yes	—
149	Dic₁₇₈	—	4	2	no	—
150	Dic₁₇₈	—	4	2	no	—
151	Dic₁₇₈	—	4	4	no	—
152	Dic₁₇₈	—	8	2	no	—
153	Dic₁₇₈	—	8	2	no	—
154	Dic₁₇₈	—	8	4	no	—
155	C₈×C₈₉	—	2	4	no	—
156	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
157	#10623494	—	2	4	no	—
158	#10623494	C₂×SD₁₆	1	4	yes	—
159	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
160	C₂×D₈₉	#492	1	2	yes	—
161	C₂×D₈₉	#497	1	4	yes	—
162	C₄×C₈₉	—	8	1	no	—
163	C₄×C₈₉	—	8	1	no	—
164	C₄×C₈₉	—	8	2	no	—
165	C₄×C₈₉	C₂²×SD₁₆	1	2	yes	—
166	C₄×C₈₉	C₂×#137	1	4	yes	—
167	Dic₈₉	—	8	1	no	—
168	Dic₈₉	—	8	1	no	—
169	Dic₈₉	—	8	2	no	—
170	Dic₈₉	#492	1	2	yes	—
171	Dic₈₉	#497	1	4	yes	—
172	C₂×#1716	—	89	1	no	maximal, Sylow
173	C₂×C₈₉	C₂×#492	1	1	yes	characteristic
174	C₂×C₈₉	C₂×#497	1	2	yes	—
175	D₈₉	#1716	1	4	yes	direct product factor
176	C₂×#498	—	89	1	no	—
177	C₂×#498	—	89	2	no	—
178	C₂×#494	—	89	1	no	—
179	C₂×#487	—	89	2	no	—
180	#1716	—	89	2	no	—
181	#1716	—	89	1	no	—
182	#1716	—	89	1	no	—
183	#1716	—	89	2	no	—
184	#1716	—	89	1	no	—
185	#1716	D₈₉	1	1	yes	characteristic, direct product factor
186	C₂×#322	—	89	1	no	—
187	C₈₉	C₂×#1716	1	1	yes	characteristic, complemented, Sylow
188	C₂²×Q₁₆	—	178	2	no	—
189	C₂×C₄⋊Q₈	—	89	1	no	—
190	C₂×C₄⋊Q₈	—	89	2	no	—
191	C₂×C₄⋊Q₈	—	89	1	no	—
192	#498	—	89	2	no	—
193	#498	—	89	2	no	—
194	#498	—	89	2	no	—
195	#498	—	89	2	no	—
196	#498	—	89	1	no	—
197	#498	—	178	4	no	—
198	#498	—	178	2	no	—
199	#498	C₂×D₈₉	1	2	yes	—
200	#498	C₂×D₈₉	1	1	yes	characteristic
201	#494	—	89	1	no	—
202	#494	—	89	1	no	—
203	#494	—	89	1	no	—
204	#494	—	178	2	no	—
205	#494	C₂×D₈₉	1	1	yes	characteristic
206	#487	—	89	2	no	—
207	#487	—	89	2	no	—
208	#487	—	89	2	no	—
209	#487	—	178	4	no	—
210	#487	C₂×D₈₉	1	2	yes	—
211	C₂×#106	—	178	2	no	—
212	C₂×#106	—	178	4	no	—
213	C₂×#103	—	178	4	no	—
214	C₂×C₄×C₈	—	89	2	no	—
215	C₂×C₄×C₈	—	89	1	no	—
216	#322	—	89	1	no	—
217	#322	—	89	2	no	—
218	#322	C₂×D₈₉	1	1	yes	characteristic
219	C₂²×Q₈	—	178	1	no	—
220	C₂²×Q₈	—	178	1	no	—
221	C₂²×Q₈	—	178	2	no	—
222	C₂×Q₁₆	—	2	2	no	—
223	C₂×Q₁₆	—	356	8	no	—
224	C₂×Q₁₆	—	178	2	no	—
225	C₂×Q₁₆	—	178	2	no	—
226	C₂×Q₁₆	—	178	2	no	—
227	C₂²×C₈	—	89	2	no	—
228	C₂²×C₈	—	89	4	no	—
229	C₄⋊Q₈	—	89	1	no	—
230	C₄⋊Q₈	—	89	2	no	—
231	C₄⋊Q₈	—	89	1	no	—
232	C₄⋊Q₈	—	178	2	no	—
233	C₄⋊Q₈	—	178	2	no	—
234	C₄⋊Q₈	—	178	2	no	—
235	C₄⋊Q₈	—	178	2	no	—
236	C₄⋊Q₈	—	178	2	no	—
237	C₄⋊Q₈	—	178	1	no	—
238	C₄⋊Q₈	—	178	1	no	—
239	C₄⋊Q₈	C₂²×D₈₉	1	1	yes	characteristic
240	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
241	C₄⋊Q₈	C₂²×D₈₉	1	1	yes	characteristic
242	C₂×C₄⋊C₄	—	178	2	no	—
243	C₂×C₄⋊C₄	—	178	2	no	—
244	C₂×C₄⋊C₄	—	178	2	no	—
245	C₂×C₄⋊C₄	—	178	2	no	—
246	C₂×C₄²	—	89	1	no	—
247	#106	—	2	4	no	—
248	#106	—	2	2	no	—
249	#106	—	178	4	no	—
250	#106	—	178	4	no	—
251	#106	—	178	4	no	—
252	#106	—	178	4	no	—
253	#106	—	178	2	no	—
254	#103	—	2	4	no	—
255	#103	—	178	4	no	—
256	#103	—	178	4	no	—
257	#103	—	178	4	no	—
258	C₄×C₈	—	89	4	no	—
259	C₄×C₈	—	89	2	no	—
260	C₄×C₈	—	89	2	no	—
261	C₄×C₈	—	89	1	no	—
262	C₄×C₈	C₂²×D₈₉	1	2	yes	—
263	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	2	1	no	—
267	C₂×Q₈	—	356	2	no	—
268	C₂×Q₈	—	356	1	no	—
269	C₂×Q₈	—	356	1	no	—
270	C₂×Q₈	—	356	4	no	—
271	C₂×Q₈	—	356	4	no	—
272	C₂×Q₈	—	356	8	no	—
273	C₂×Q₈	—	178	2	no	—
274	C₂×Q₈	—	178	1	no	—
275	C₂×Q₈	—	178	1	no	—
276	C₂²×C₄	—	89	2	no	—
277	C₂²×C₄	—	89	1	no	—
278	C₂²×C₄	—	356	1	no	—
279	C₂²×C₄	—	356	1	no	—
280	C₂²×C₄	—	356	2	no	—
281	Q₁₆	—	4	4	no	—
282	Q₁₆	—	356	4	no	—
283	Q₁₆	—	356	4	no	—
284	Q₁₆	—	356	4	no	—
285	C₂×C₈	—	89	2	no	—
286	C₂×C₈	—	89	8	no	—
287	C₂×C₈	—	89	4	no	—
288	C₂×C₈	—	178	8	no	—
289	C₂×C₈	D₄×D₈₉	1	4	yes	—
290	C₂×C₈	D₄×D₈₉	1	2	yes	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	356	2	no	—
296	C₄⋊C₄	—	356	2	no	—
297	C₄⋊C₄	—	356	2	no	—
298	C₄⋊C₄	—	356	2	no	—
299	C₄⋊C₄	—	178	2	no	—
300	C₄⋊C₄	—	178	2	no	—
301	C₄⋊C₄	—	178	2	no	—
302	C₄⋊C₄	—	178	2	no	—
303	C₄²	—	89	1	no	—
304	C₄²	—	89	2	no	—
305	C₄²	C₂³×D₈₉	1	1	yes	characteristic
306	C₂³	—	89	1	no	—
307	Q₈	—	4	4	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	356	4	no	—
311	Q₈	—	356	2	no	—
312	Q₈	—	356	2	no	—
313	Q₈	—	712	2	no	—
314	Q₈	—	712	2	no	—
315	Q₈	—	712	4	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	89	2	no	—
320	C₂×C₄	—	89	1	no	—
321	C₂×C₄	—	89	8	no	—
322	C₂×C₄	—	89	4	no	—
323	C₂×C₄	—	356	1	no	—
324	C₂×C₄	—	356	1	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	712	2	no	—
327	C₂×C₄	—	712	2	no	—
328	C₂×C₄	—	712	4	no	—
329	C₂×C₄	C₂×D₄×D₈₉	1	2	yes	—
330	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
331	C₈	—	2	4	no	—
332	C₈	—	89	4	no	—
333	C₈	—	178	4	no	—
334	C₈	SD₁₆×D₈₉	1	4	yes	—
335	C₂²	—	89	2	no	—
336	C₂²	—	89	4	no	—
337	C₂²	#127×D₈₉	1	1	yes	characteristic
338	C₄	—	8	1	no	—
339	C₄	—	8	1	no	—
340	C₄	—	8	2	no	—
341	C₄	—	89	2	no	—
342	C₄	—	89	4	no	—
343	C₄	—	712	1	no	—
344	C₄	—	712	1	no	—
345	C₄	—	712	2	no	—
346	C₄	C₂×SD₁₆×D₈₉	1	2	yes	—
347	C₄	#137×D₈₉	1	4	yes	—
348	C₂	—	89	4	no	—
349	C₂	#492×D₈₉	1	1	yes	characteristic
350	C₂	#497×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂²	#127×D₈₉
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1716	C₂
socle	C₂²×C₈₉	C₂×#127
prime radical	C₂²×D₈₉	#127
hypercenter	#1716	D₈₉
hypocenter	C₈₉	C₂×#1716
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1716

Complements

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

N	H ≅ G/N	type
5. #26019887	C₂	semidirect
6. #26019886	C₂	semidirect
7. #26019885	C₂	semidirect
8. #26019884	C₂	semidirect
9. #26019883	C₂	semidirect
11. C₈₉×#1716	C₂	semidirect
175. D₈₉	#1716	direct (N × H)
185. #1716	D₈₉	direct (N × H)
187. C₈₉	C₂×#1716	semidirect

Maximal subgroups

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

subgroup	index
#498×D₈₉	2
#494×D₈₉	2
#487×D₈₉	2
#26019887	2
#26019886	2
#26019885	2
#26019884	2
#26019883	2
#322×D₈₉	2
C₈₉×#1716	2
C₂×#1716	89

Subgroups — #1716×D89 #36542151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1716×D₈₉ #36542151