Subgroups — #2264×D₈₉ #36712564

Order 22784.

462 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#508×D₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₃₂×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022572	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022571	C₂	1	2	yes	maximal, complemented
5	#26022570	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022569	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022568	C₂	1	2	yes	maximal, complemented
8	Q₆₄×D₈₉	C₂	1	2	yes	maximal, complemented
9	SD₆₄×D₈₉	C₂	1	2	yes	maximal, complemented
10	M₆₄×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2264	C₂	1	1	yes	maximal, characteristic, complemented
12	#135×D₈₉	—	2	1	no	—
13	C₂×Q₁₆×D₈₉	—	2	1	no	—
14	#19152031	C₂²	1	1	yes	characteristic
15	#19152030	C₂²	1	1	yes	characteristic
16	#19152029	—	2	1	no	—
17	#19152028	—	2	1	no	—
18	#19152027	C₂²	1	1	yes	characteristic
19	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic, complemented
20	C₂×#15667958	—	2	1	no	—
21	Q₃₂×D₈₉	—	2	1	no	—
22	Q₃₂×D₈₉	—	2	1	no	—
23	Q₃₂×D₈₉	C₂²	1	2	yes	—
24	Q₃₂×D₈₉	C₂²	1	1	yes	characteristic
25	Q₃₂×D₈₉	C₂²	1	2	yes	complemented
26	SD₃₂×D₈₉	—	2	1	no	—
27	D₁₆×D₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
29	Dic₁₄₂₄	C₂²	1	2	yes	complemented
30	#19151438	C₂²	1	2	yes	complemented
31	#19151438	C₂²	1	2	yes	complemented
32	#19151437	C₂²	1	2	yes	complemented
33	#19151436	C₂²	1	2	yes	complemented
34	#19151435	C₂²	1	2	yes	complemented
35	#19151432	C₂²	1	1	yes	characteristic, complemented
36	#19151431	C₂²	1	2	yes	complemented
37	C₃₂×D₈₉	C₂²	1	2	yes	—
38	#508×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic, complemented
40	Q₆₄×C₈₉	C₂²	1	2	yes	complemented
41	SD₆₄×C₈₉	C₂²	1	2	yes	complemented
42	M₆₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
43	#41×D₈₉	—	4	1	no	—
44	C₂×Q₈×D₈₉	—	4	1	no	—
45	#15668035	—	2	1	no	—
46	#15668034	—	2	1	no	—
47	#15668033	—	4	1	no	—
48	#15668032	—	4	1	no	—
49	#15668031	—	2	1	no	—
50	C₂×Dic₃₅₆	—	2	1	no	—
51	C₂×#13083564	—	4	1	no	—
52	Q₁₆×D₈₉	—	2	1	no	—
53	Q₁₆×D₈₉	—	2	2	no	—
54	Q₁₆×D₈₉	—	2	2	no	—
55	Q₁₆×D₈₉	—	4	1	no	—
56	Q₁₆×D₈₉	—	4	1	no	—
57	SD₁₆×D₈₉	—	4	1	no	—
58	D₈×D₈₉	—	2	1	no	—
59	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
60	Dic₇₁₂	—	2	1	no	—
61	Dic₇₁₂	C₂³	1	2	yes	—
62	Dic₇₁₂	C₂³	1	1	yes	characteristic
63	#15667958	—	2	2	no	—
64	#15667958	—	2	2	no	—
65	#15667958	—	2	1	no	—
66	#15667957	—	2	1	no	—
67	#15667956	—	2	1	no	—
68	#15667955	—	2	1	no	—
69	D₁₄₂₄	C₂³	1	1	yes	characteristic
70	#15667953	—	2	1	no	—
71	C₂×#13083549	D₄	1	1	yes	characteristic
72	C₁₆×D₈₉	D₄	1	1	yes	characteristic
73	C₁₆×D₈₉	D₄	1	1	yes	characteristic
74	C₁₆×D₈₉	C₂³	1	1	yes	characteristic
75	C₁₆×D₈₉	C₂³	1	1	yes	characteristic
76	#15667918	C₂³	1	2	yes	—
77	#135×C₈₉	—	2	1	no	—
78	C₂×Q₁₆×C₈₉	—	2	1	no	—
79	Q₃₂×C₈₉	—	2	1	no	—
80	Q₃₂×C₈₉	C₂³	1	2	yes	—
81	Q₃₂×C₈₉	C₂³	1	1	yes	characteristic
82	SD₃₂×C₈₉	—	2	1	no	—
83	D₁₆×C₈₉	C₂³	1	1	yes	characteristic
84	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
85	C₃₂×C₈₉	C₂³	1	2	yes	—
86	#13083575	—	4	1	no	—
87	#13083574	—	4	1	no	—
88	#13083574	—	8	1	no	—
89	#13083573	—	4	1	no	—
90	#13083573	—	8	1	no	—
91	C₂×Dic₁₇₈	—	4	1	no	—
92	C₂×Dic₁₇₈	—	8	1	no	—
93	Q₈×D₈₉	—	4	1	no	—
94	Q₈×D₈₉	—	4	2	no	—
95	Q₈×D₈₉	—	4	2	no	—
96	Q₈×D₈₉	—	8	1	no	—
97	Q₈×D₈₉	—	8	1	no	—
98	D₄×D₈₉	—	4	1	no	—
99	D₄×D₈₉	—	8	1	no	—
100	C₂×C₄×D₈₉	—	8	1	no	—
101	C₂×C₄×D₈₉	—	8	1	no	—
102	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
103	Dic₃₅₆	—	2	2	no	—
104	Dic₃₅₆	—	2	1	no	—
105	Dic₃₅₆	—	4	1	no	—
106	#13083564	—	4	2	no	—
107	#13083564	—	4	2	no	—
108	#13083564	—	4	1	no	—
109	#13083563	—	4	1	no	—
110	#13083562	—	4	1	no	—
111	#13083561	—	4	1	no	—
112	D₇₁₂	—	2	1	no	—
113	#13083559	—	4	1	no	—
114	C₂×#10623494	D₈	1	1	yes	characteristic
115	C₈×D₈₉	D₈	1	1	yes	characteristic
116	C₈×D₈₉	D₈	1	1	yes	characteristic
117	C₈×D₈₉	C₂×D₄	1	1	yes	characteristic
118	C₈×D₈₉	C₂×D₄	1	1	yes	characteristic
119	#13083549	C₂×D₄	1	1	yes	characteristic
120	#13083549	C₂×D₄	1	1	yes	characteristic
121	#41×C₈₉	—	4	1	no	—
122	C₂×Q₈×C₈₉	—	4	1	no	—
123	Q₁₆×C₈₉	—	2	2	no	—
124	Q₁₆×C₈₉	—	2	1	no	—
125	Q₁₆×C₈₉	—	4	1	no	—
126	SD₁₆×C₈₉	—	4	1	no	—
127	D₈×C₈₉	—	2	1	no	—
128	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₁₆×C₈₉	C₂⁴	1	1	yes	characteristic
131	C₂²×D₈₉	—	8	1	no	—
132	C₂²×D₈₉	D₁₆	1	1	yes	characteristic
133	Q₈×C₈₉	—	4	1	no	—
134	Q₈×C₈₉	—	4	2	no	—
135	Q₈×C₈₉	—	8	1	no	—
136	D₄×C₈₉	—	4	1	no	—
137	D₄×C₈₉	—	8	1	no	—
138	C₂×C₄×C₈₉	—	8	1	no	—
139	C₂×C₄×C₈₉	—	8	1	no	—
140	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
141	C₂²⋊D₈₉	—	8	1	no	—
142	C₂²⋊D₈₉	—	8	1	no	—
143	C₂²⋊D₈₉	—	8	1	no	—
144	C₂×Dic₈₉	—	8	1	no	—
145	C₂×Dic₈₉	—	8	1	no	—
146	C₂×Dic₈₉	D₁₆	1	1	yes	characteristic
147	D₃₅₆	—	4	1	no	—
148	D₃₅₆	—	8	1	no	—
149	C₄×D₈₉	—	8	1	no	—
150	C₄×D₈₉	—	8	2	no	—
151	C₄×D₈₉	—	8	2	no	—
152	C₄×D₈₉	—	8	1	no	—
153	C₄×D₈₉	—	8	1	no	—
154	C₄×D₈₉	D₁₆	1	1	yes	characteristic
155	C₄×D₈₉	D₁₆	1	1	yes	characteristic
156	C₄×D₈₉	C₂×D₈	1	1	yes	characteristic
157	C₄×D₈₉	C₂×D₈	1	1	yes	characteristic
158	Dic₁₇₈	—	4	1	no	—
159	Dic₁₇₈	—	4	2	no	—
160	Dic₁₇₈	—	8	1	no	—
161	Dic₁₇₈	—	8	2	no	—
162	Dic₁₇₈	—	8	2	no	—
163	Dic₁₇₈	—	8	1	no	—
164	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
165	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
166	#10623494	C₂×D₈	1	1	yes	characteristic
167	#10623494	C₂×D₈	1	1	yes	characteristic
168	C₂²×C₈₉	—	8	1	no	—
169	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
170	C₂×D₈₉	—	2	2	no	—
171	C₂×D₈₉	—	8	1	no	—
172	C₂×D₈₉	—	16	2	no	—
173	C₂×D₈₉	C₂×D₁₆	1	1	yes	characteristic
174	C₂×D₈₉	C₂×D₁₆	1	1	yes	characteristic
175	C₄×C₈₉	—	8	2	no	—
176	C₄×C₈₉	—	8	1	no	—
177	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
178	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
179	Dic₈₉	—	8	2	no	—
180	Dic₈₉	—	8	1	no	—
181	Dic₈₉	C₂×D₁₆	1	1	yes	characteristic
182	Dic₈₉	C₂×D₁₆	1	1	yes	characteristic
183	C₂×#2264	—	89	1	no	maximal, Sylow
184	C₂×C₈₉	—	2	1	no	—
185	C₂×C₈₉	—	16	1	no	—
186	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
187	D₈₉	—	2	1	no	—
188	D₈₉	—	16	1	no	—
189	D₈₉	#2264	1	2	yes	direct product factor
190	C₂×#508	—	89	1	no	—
191	C₂²×Q₃₂	—	89	1	no	—
192	#2264	—	89	1	no	—
193	#2264	—	89	1	no	—
194	#2264	—	89	2	no	—
195	#2264	—	89	2	no	—
196	#2264	—	89	1	no	—
197	#2264	D₈₉	1	1	yes	characteristic, direct product factor
198	C₂×Q₆₄	—	89	2	no	—
199	C₂×SD₆₄	—	89	2	no	—
200	C₂×M₆₄	—	89	1	no	—
201	C₈₉	C₂×#2264	1	1	yes	characteristic, complemented, Sylow
202	C₂×#135	—	178	1	no	—
203	C₂²×Q₁₆	—	178	1	no	—
204	#508	—	89	1	no	—
205	#508	—	89	1	no	—
206	#508	—	89	1	no	—
207	#508	—	178	1	no	—
208	#508	—	178	1	no	—
209	#508	C₂×D₈₉	1	1	yes	characteristic
210	C₂×Q₃₂	—	89	2	no	—
211	C₂×Q₃₂	—	89	2	no	—
212	C₂×Q₃₂	—	89	1	no	—
213	C₂×Q₃₂	—	89	1	no	—
214	C₂×Q₃₂	—	178	1	no	—
215	C₂×Q₃₂	—	178	1	no	—
216	C₂×Q₃₂	—	178	1	no	—
217	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic, complemented
218	C₂×SD₃₂	—	178	1	no	—
219	C₂×D₁₆	—	89	1	no	—
220	C₂²×C₁₆	—	89	1	no	—
221	Q₆₄	—	89	2	no	—
222	Q₆₄	—	89	2	no	—
223	Q₆₄	—	89	2	no	—
224	Q₆₄	C₂×D₈₉	1	2	yes	complemented
225	SD₆₄	—	89	2	no	—
226	SD₆₄	—	89	2	no	—
227	SD₆₄	—	89	2	no	—
228	SD₆₄	C₂×D₈₉	1	2	yes	complemented
229	M₆₄	—	89	1	no	—
230	M₆₄	—	89	2	no	—
231	M₆₄	C₂×D₈₉	1	1	yes	characteristic, complemented
232	C₂×C₃₂	—	89	2	no	—
233	C₂×#41	—	356	1	no	—
234	C₂²×Q₈	—	356	1	no	—
235	#135	—	2	1	no	—
236	#135	—	356	1	no	—
237	#135	—	356	1	no	—
238	#135	—	178	1	no	—
239	#135	—	178	1	no	—
240	#135	—	178	1	no	—
241	C₂×Q₁₆	—	2	1	no	—
242	C₂×Q₁₆	—	356	1	no	—
243	C₂×Q₁₆	—	356	1	no	—
244	C₂×Q₁₆	—	356	1	no	—
245	C₂×Q₁₆	—	178	2	no	—
246	C₂×Q₁₆	—	178	2	no	—
247	C₂×Q₁₆	—	178	1	no	—
248	C₂×Q₁₆	—	178	1	no	—
249	C₂×SD₁₆	—	356	1	no	—
250	C₂×D₈	—	178	1	no	—
251	C₂²×C₈	—	89	1	no	—
252	Q₃₂	—	2	1	no	—
253	Q₃₂	—	89	2	no	—
254	Q₃₂	—	89	1	no	—
255	Q₃₂	—	178	2	no	—
256	Q₃₂	—	178	2	no	—
257	Q₃₂	—	178	1	no	—
258	Q₃₂	—	178	1	no	—
259	Q₃₂	C₂²×D₈₉	1	2	yes	—
260	Q₃₂	C₂²×D₈₉	1	1	yes	characteristic
261	SD₃₂	—	2	1	no	—
262	SD₃₂	—	178	1	no	—
263	SD₃₂	—	178	1	no	—
264	SD₃₂	—	178	1	no	—
265	D₁₆	—	89	1	no	—
266	D₁₆	—	178	1	no	—
267	D₁₆	C₂²×D₈₉	1	1	yes	characteristic
268	C₂×C₁₆	—	89	1	no	—
269	C₂×C₁₆	—	89	1	no	—
270	C₂×C₁₆	—	89	1	no	—
271	C₂×C₁₆	—	89	1	no	—
272	C₂×C₁₆	—	89	1	no	—
273	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
274	C₃₂	—	89	2	no	—
275	C₃₂	C₂²×D₈₉	1	2	yes	—
276	#41	—	4	1	no	—
277	#41	—	356	1	no	—
278	#41	—	356	1	no	—
279	#41	—	356	1	no	—
280	#41	—	712	1	no	—
281	#41	—	712	1	no	—
282	C₂×Q₈	—	4	1	no	—
283	C₂×Q₈	—	356	1	no	—
284	C₂×Q₈	—	356	2	no	—
285	C₂×Q₈	—	356	2	no	—
286	C₂×Q₈	—	356	1	no	—
287	C₂×Q₈	—	712	1	no	—
288	C₂×Q₈	—	712	1	no	—
289	C₂×Q₈	—	712	1	no	—
290	C₂×D₄	—	356	1	no	—
291	C₂×D₄	—	712	1	no	—
292	C₂²×C₄	—	89	1	no	—
293	C₂²×C₄	—	712	1	no	—
294	C₂²×C₄	—	712	1	no	—
295	Q₁₆	—	2	1	no	—
296	Q₁₆	—	2	2	no	—
297	Q₁₆	—	4	1	no	—
298	Q₁₆	—	356	2	no	—
299	Q₁₆	—	356	2	no	—
300	Q₁₆	—	356	1	no	—
301	Q₁₆	—	356	1	no	—
302	Q₁₆	—	178	1	no	—
303	Q₁₆	—	178	2	no	—
304	SD₁₆	—	4	1	no	—
305	SD₁₆	—	356	1	no	—
306	SD₁₆	—	356	1	no	—
307	SD₁₆	—	356	1	no	—
308	D₈	—	2	1	no	—
309	D₈	—	356	1	no	—
310	D₈	—	178	1	no	—
311	C₂×C₈	—	89	1	no	—
312	C₂×C₈	—	89	1	no	—
313	C₂×C₈	—	89	1	no	—
314	C₂×C₈	—	89	1	no	—
315	C₂×C₈	—	89	1	no	—
316	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
317	C₁₆	—	89	1	no	—
318	C₁₆	—	89	1	no	—
319	C₁₆	D₄×D₈₉	1	1	yes	characteristic
320	C₁₆	C₂³×D₈₉	1	1	yes	characteristic
321	C₂³	—	89	1	no	—
322	C₂³	—	712	1	no	—
323	Q₈	—	4	1	no	—
324	Q₈	—	4	2	no	—
325	Q₈	—	8	1	no	—
326	Q₈	—	356	1	no	—
327	Q₈	—	356	2	no	—
328	Q₈	—	712	2	no	—
329	Q₈	—	712	2	no	—
330	Q₈	—	712	1	no	—
331	Q₈	—	712	1	no	—
332	D₄	—	4	1	no	—
333	D₄	—	8	1	no	—
334	D₄	—	356	1	no	—
335	D₄	—	712	1	no	—
336	D₄	—	712	1	no	—
337	D₄	—	712	1	no	—
338	D₄	—	712	1	no	—
339	C₂×C₄	—	8	1	no	—
340	C₂×C₄	—	8	1	no	—
341	C₂×C₄	—	89	1	no	—
342	C₂×C₄	—	89	1	no	—
343	C₂×C₄	—	89	1	no	—
344	C₂×C₄	—	89	1	no	—
345	C₂×C₄	—	89	1	no	—
346	C₂×C₄	—	712	1	no	—
347	C₂×C₄	—	712	1	no	—
348	C₂×C₄	—	712	2	no	—
349	C₂×C₄	—	712	2	no	—
350	C₂×C₄	—	712	1	no	—
351	C₂×C₄	—	712	1	no	—
352	C₂×C₄	—	712	1	no	—
353	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
354	C₈	—	89	1	no	—
355	C₈	—	89	1	no	—
356	C₈	D₈×D₈₉	1	1	yes	characteristic
357	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
358	C₂²	—	8	1	no	—
359	C₂²	—	89	1	no	—
360	C₂²	—	89	1	no	—
361	C₂²	—	1424	2	no	—
362	C₂²	—	178	2	no	—
363	C₂²	—	712	1	no	—
364	C₂²	D₁₆×D₈₉	1	1	yes	characteristic
365	C₄	—	8	1	no	—
366	C₄	—	8	2	no	—
367	C₄	—	89	1	no	—
368	C₄	—	89	1	no	—
369	C₄	—	712	1	no	—
370	C₄	—	712	2	no	—
371	C₄	D₁₆×D₈₉	1	1	yes	characteristic
372	C₄	C₂×D₈×D₈₉	1	1	yes	characteristic
373	C₂	—	2	1	no	—
374	C₂	—	16	1	no	—
375	C₂	—	89	2	no	—
376	C₂	—	1424	1	no	—
377	C₂	—	178	1	no	—
378	C₂	C₂×D₁₆×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₆×D₈₉
commutator, cosocle	C₁₆×C₈₉	C₂⁴
Frattini	C₁₆	C₂³×D₈₉
Fitting	C₈₉×#2264	C₂
socle	C₂×C₈₉	C₂²×D₁₆
prime radical	#508×D₈₉	C₂
hypercenter	#2264	D₈₉
hypocenter	C₈₉	C₂×#2264
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2264

Complements

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

N	H ≅ G/N	type
2. C₂×Q₃₂×D₈₉	C₂	semidirect
3. #26022572	C₂	semidirect
4. #26022571	C₂	semidirect
5. #26022570	C₂	semidirect
6. #26022569	C₂	semidirect
7. #26022568	C₂	semidirect
8. Q₆₄×D₈₉	C₂	semidirect
9. SD₆₄×D₈₉	C₂	semidirect
10. M₆₄×D₈₉	C₂	semidirect
11. C₈₉×#2264	C₂	semidirect
19. C₂×Dic₇₁₂	C₂²	semidirect
25. Q₃₂×D₈₉	C₂²	semidirect
29. Dic₁₄₂₄	C₂²	semidirect
30. #19151438	C₂²	semidirect
31. #19151438	C₂²	semidirect
32. #19151437	C₂²	semidirect
33. #19151436	C₂²	semidirect
34. #19151435	C₂²	semidirect
35. #19151432	C₂²	semidirect
36. #19151431	C₂²	semidirect
39. C₂×Q₃₂×C₈₉	C₂²	semidirect
40. Q₆₄×C₈₉	C₂²	semidirect
41. SD₆₄×C₈₉	C₂²	semidirect
42. M₆₄×C₈₉	C₂²	semidirect
189. D₈₉	#2264	direct (N × H)
197. #2264	D₈₉	direct (N × H)
201. C₈₉	C₂×#2264	semidirect
217. C₂×Q₃₂	C₂×D₈₉	semidirect
224. Q₆₄	C₂×D₈₉	semidirect
228. SD₆₄	C₂×D₈₉	semidirect
231. M₆₄	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#508×D₈₉	2
C₂×Q₃₂×D₈₉	2
#26022572	2
#26022571	2
#26022570	2
#26022569	2
#26022568	2
Q₆₄×D₈₉	2
SD₆₄×D₈₉	2
M₆₄×D₈₉	2
C₈₉×#2264	2
C₂×#2264	89

Subgroups — #2264×D89 #36712564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2264×D₈₉ #36712564