Subgroups — #2249×D₈₉ #36709823

Order 22784.

854 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×SD₃₂×D₈₉	C₂	1	2	yes	maximal, complemented
3	C₂×D₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
4	#495×D₈₉	C₂	1	2	yes	maximal, complemented
5	#26022514	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022513	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022512	C₂	1	2	yes	maximal, complemented
8	#26022511	C₂	1	2	yes	maximal, complemented
9	#26022510	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2249	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₈₉	C₂²	1	2	yes	—
13	C₂×SD₁₆×D₈₉	—	2	2	no	—
14	C₂×D₈×D₈₉	C₂²	1	2	yes	—
15	#124×D₈₉	—	2	2	no	—
16	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic, complemented
17	C₂×#15667958	C₂²	1	2	yes	complemented
18	Q₃₂×D₈₉	—	2	4	no	—
19	C₂×#15667957	C₂²	1	2	yes	complemented
20	C₂×#15667956	C₂²	1	2	yes	complemented
21	C₂×#15667955	C₂²	1	2	yes	complemented
22	SD₃₂×D₈₉	—	2	8	no	—
23	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
24	C₂×#15667953	C₂²	1	2	yes	—
25	D₁₆×D₈₉	—	2	4	no	—
26	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
28	#19151983	C₂²	1	2	yes	complemented
29	#19151982	C₂²	1	2	yes	complemented
30	#19151981	C₂²	1	2	yes	complemented
31	#19151980	—	2	2	no	—
32	#19151979	—	2	2	no	—
33	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
34	C₄×#13083549	C₂²	1	1	yes	characteristic, complemented
35	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
36	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
37	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic, complemented
38	C₂×SD₃₂×C₈₉	C₂²	1	2	yes	complemented
39	C₂×D₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	#495×C₈₉	C₂²	1	2	yes	complemented
41	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
42	C₂×Q₈×D₈₉	—	2	2	no	—
43	C₂×D₄×D₈₉	—	2	2	no	—
44	C₂×Dic₃₅₆	C₂³	1	2	yes	—
45	C₂×#13083564	—	2	2	no	—
46	Q₁₆×D₈₉	—	2	4	no	—
47	Q₁₆×D₈₉	—	2	4	no	—
48	C₂×#13083563	—	2	2	no	—
49	C₂×#13083562	—	2	2	no	—
50	C₂×#13083561	—	2	2	no	—
51	SD₁₆×D₈₉	—	4	8	no	—
52	C₂×D₇₁₂	C₂³	1	2	yes	—
53	C₂×#13083559	—	2	2	no	—
54	D₈×D₈₉	—	2	4	no	—
55	D₈×D₈₉	—	2	4	no	—
56	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
57	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
58	#15668001	—	2	2	no	—
59	#15668000	—	2	2	no	—
60	#15667999	—	2	2	no	—
61	#15667998	—	4	4	no	—
62	C₂²⋊C₄×D₈₉	—	4	4	no	—
63	C₄²×D₈₉	D₄	1	1	yes	characteristic
64	Dic₇₁₂	—	2	2	no	—
65	#15667958	—	2	4	no	—
66	#15667957	—	2	4	no	—
67	#15667956	—	2	4	no	—
68	#15667955	—	2	4	no	—
69	D₁₄₂₄	—	2	2	no	—
70	#15667953	—	2	4	no	—
71	C₂×#13083549	C₂³	1	1	yes	characteristic
72	C₂×#13083549	C₂³	1	1	yes	characteristic
73	C₁₆×D₈₉	D₄	1	4	yes	—
74	C₁₆×D₈₉	D₄	1	4	yes	—
75	C₄×#10623494	D₄	1	1	yes	characteristic
76	C₈×Dic₈₉	D₄	1	1	yes	characteristic
77	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
78	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
79	C₂×SD₁₆×C₈₉	—	2	2	no	—
80	C₂×D₈×C₈₉	C₂³	1	2	yes	—
81	#124×C₈₉	—	2	2	no	—
82	Q₃₂×C₈₉	—	2	2	no	—
83	SD₃₂×C₈₉	—	2	4	no	—
84	D₁₆×C₈₉	—	2	2	no	—
85	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
86	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
88	C₂³×D₈₉	—	4	2	no	—
89	C₂×Dic₁₇₈	—	2	2	no	—
90	C₂×Dic₁₇₈	—	4	2	no	—
91	Q₈×D₈₉	—	4	4	no	—
92	Q₈×D₈₉	—	4	4	no	—
93	C₂×D₃₅₆	—	2	2	no	—
94	C₂×C₂²⋊D₈₉	—	4	2	no	—
95	D₄×D₈₉	—	4	4	no	—
96	D₄×D₈₉	—	4	4	no	—
97	C₂×C₄×D₈₉	—	4	2	no	—
98	C₂×C₄×D₈₉	D₈	1	2	yes	—
99	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
100	Dic₃₅₆	—	2	2	no	—
101	Dic₃₅₆	—	2	2	no	—
102	#13083564	—	4	4	no	—
103	#13083563	—	4	4	no	—
104	#13083562	—	4	4	no	—
105	#13083561	—	4	4	no	—
106	D₇₁₂	—	2	2	no	—
107	D₇₁₂	—	2	2	no	—
108	#13083559	—	4	4	no	—
109	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
110	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
111	C₈×D₈₉	D₈	1	4	yes	—
112	C₈×D₈₉	C₂×D₄	1	2	yes	—
113	C₈×D₈₉	C₂×D₄	1	2	yes	—
114	C₂²⋊Dic₈₉	—	4	4	no	—
115	#13083551	—	4	4	no	—
116	#13083551	—	4	4	no	—
117	C₄×Dic₈₉	D₈	1	2	yes	—
118	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
119	#13083549	C₂×D₄	1	2	yes	—
120	#13083549	C₂×D₄	1	2	yes	—
121	C₂×Q₈×C₈₉	—	2	2	no	—
122	C₂×D₄×C₈₉	—	2	2	no	—
123	Q₁₆×C₈₉	—	2	2	no	—
124	Q₁₆×C₈₉	—	2	2	no	—
125	SD₁₆×C₈₉	—	4	4	no	—
126	D₈×C₈₉	—	2	2	no	—
127	D₈×C₈₉	—	2	2	no	—
128	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
130	C₂²⋊C₄×C₈₉	—	4	4	no	—
131	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
133	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
134	C₂³×C₈₉	—	4	2	no	—
135	C₂²×D₈₉	—	4	2	no	—
136	C₂²×D₈₉	—	8	4	no	—
137	C₂²×D₈₉	—	8	8	no	—
138	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
139	Q₈×C₈₉	—	4	2	no	—
140	Q₈×C₈₉	—	4	2	no	—
141	D₄×C₈₉	—	4	2	no	—
142	D₄×C₈₉	—	4	2	no	—
143	C₂×C₄×C₈₉	—	4	2	no	—
144	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₈₉	—	8	4	no	—
147	C₂×Dic₈₉	—	4	2	no	—
148	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
149	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
150	D₃₅₆	—	4	2	no	—
151	D₃₅₆	—	4	2	no	—
152	C₄×D₈₉	—	2	4	no	—
153	C₄×D₈₉	—	8	4	no	—
154	C₄×D₈₉	#127	1	2	yes	—
155	C₄×D₈₉	C₂×D₈	1	2	yes	—
156	Dic₁₇₈	—	4	2	no	—
157	Dic₁₇₈	—	4	2	no	—
158	Dic₁₇₈	—	8	4	no	—
159	C₈×C₈₉	C₂×D₈	1	2	yes	—
160	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
161	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
162	#10623494	#127	1	2	yes	—
163	#10623494	C₂×D₈	1	2	yes	—
164	C₂²×C₈₉	—	8	2	no	—
165	C₂²×C₈₉	—	8	4	no	—
166	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₈₉	—	8	2	no	—
168	C₂×D₈₉	—	8	4	no	—
169	C₂×D₈₉	—	16	8	no	—
170	C₂×D₈₉	#493	1	2	yes	—
171	C₂×D₈₉	#508	1	4	yes	—
172	C₄×C₈₉	—	2	2	no	—
173	C₄×C₈₉	—	8	2	no	—
174	C₄×C₈₉	C₂×#127	1	1	yes	characteristic
175	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
176	Dic₈₉	—	2	2	no	—
177	Dic₈₉	—	8	2	no	—
178	Dic₈₉	#493	1	2	yes	—
179	C₂×#2249	—	89	1	no	maximal, Sylow
180	C₂×C₈₉	—	16	2	no	—
181	C₂×C₈₉	C₂×#493	1	1	yes	characteristic
182	C₂×C₈₉	C₂×#508	1	2	yes	—
183	D₈₉	—	16	2	no	—
184	D₈₉	#2249	1	4	yes	direct product factor
185	C₂²×Q₃₂	—	89	1	no	—
186	C₂²×SD₃₂	—	89	2	no	—
187	C₂²×D₁₆	—	89	1	no	—
188	C₂×#495	—	89	2	no	—
189	#2249	—	89	1	no	—
190	#2249	—	89	2	no	—
191	#2249	—	89	1	no	—
192	#2249	—	89	2	no	—
193	#2249	—	89	1	no	—
194	#2249	D₈₉	1	1	yes	characteristic, direct product factor
195	C₂×C₄×C₁₆	—	89	1	no	—
196	C₈₉	C₂×#2249	1	1	yes	characteristic, complemented, Sylow
197	C₂²×Q₁₆	—	89	2	no	—
198	C₂²×SD₁₆	—	178	2	no	—
199	C₂²×D₈	—	89	2	no	—
200	C₂×#124	—	178	2	no	—
201	C₂×Q₃₂	—	89	2	no	—
202	C₂×Q₃₂	—	89	1	no	—
203	C₂×Q₃₂	—	178	4	no	—
204	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic, complemented
205	C₂×SD₃₂	—	89	2	no	—
206	C₂×SD₃₂	—	89	2	no	—
207	C₂×SD₃₂	—	89	2	no	—
208	C₂×SD₃₂	—	178	8	no	—
209	C₂×SD₃₂	C₂×D₈₉	1	2	yes	complemented
210	C₂×D₁₆	—	89	2	no	—
211	C₂×D₁₆	—	89	1	no	—
212	C₂×D₁₆	—	178	4	no	—
213	C₂×D₁₆	C₂×D₈₉	1	1	yes	characteristic
214	C₂²×C₁₆	—	89	1	no	—
215	C₂²×C₁₆	—	89	1	no	—
216	#495	—	89	2	no	—
217	#495	—	89	2	no	—
218	#495	—	89	2	no	—
219	#495	—	178	2	no	—
220	#495	—	178	2	no	—
221	#495	C₂×D₈₉	1	2	yes	complemented
222	C₂×C₄×C₈	—	89	1	no	—
223	C₄×C₁₆	—	89	1	no	—
224	C₄×C₁₆	—	89	1	no	—
225	C₄×C₁₆	—	89	1	no	—
226	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
227	C₂²×Q₈	—	178	2	no	—
228	C₂²×D₄	—	178	2	no	—
229	C₂×Q₁₆	—	89	2	no	—
230	C₂×Q₁₆	—	178	4	no	—
231	C₂×Q₁₆	—	178	4	no	—
232	C₂×Q₁₆	—	178	2	no	—
233	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
234	C₂×SD₁₆	—	2	2	no	—
235	C₂×SD₁₆	—	356	8	no	—
236	C₂×SD₁₆	—	178	2	no	—
237	C₂×SD₁₆	—	178	2	no	—
238	C₂×SD₁₆	—	178	2	no	—
239	C₂×D₈	—	89	2	no	—
240	C₂×D₈	—	178	4	no	—
241	C₂×D₈	—	178	4	no	—
242	C₂×D₈	—	178	2	no	—
243	C₂×D₈	C₂²×D₈₉	1	2	yes	—
244	C₂²×C₈	—	89	1	no	—
245	C₂²×C₈	—	89	1	no	—
246	#124	—	2	2	no	—
247	#124	—	356	4	no	—
248	#124	—	178	2	no	—
249	#124	—	178	2	no	—
250	#124	—	178	2	no	—
251	C₂×C₂²⋊C₄	—	356	4	no	—
252	C₂×C₄²	—	89	1	no	—
253	Q₃₂	—	2	2	no	—
254	Q₃₂	—	178	4	no	—
255	Q₃₂	—	178	2	no	—
256	SD₃₂	—	2	4	no	—
257	SD₃₂	—	178	4	no	—
258	SD₃₂	—	178	4	no	—
259	SD₃₂	—	178	4	no	—
260	D₁₆	—	2	2	no	—
261	D₁₆	—	178	4	no	—
262	D₁₆	—	178	2	no	—
263	C₂×C₁₆	—	89	1	no	—
264	C₂×C₁₆	—	89	1	no	—
265	C₂×C₁₆	—	89	4	no	—
266	C₂×C₁₆	—	89	4	no	—
267	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
268	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
269	C₄×C₈	—	89	1	no	—
270	C₄×C₈	—	89	1	no	—
271	C₄×C₈	—	89	1	no	—
272	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
273	C₂⁴	—	356	2	no	—
274	C₂×Q₈	—	2	2	no	—
275	C₂×Q₈	—	356	2	no	—
276	C₂×Q₈	—	356	4	no	—
277	C₂×Q₈	—	356	4	no	—
278	C₂×Q₈	—	178	2	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	—	356	4	no	—
281	C₂×D₄	—	356	4	no	—
282	C₂×D₄	—	356	2	no	—
283	C₂×D₄	—	178	2	no	—
284	C₂²×C₄	—	89	1	no	—
285	C₂²×C₄	—	89	2	no	—
286	C₂²×C₄	—	356	2	no	—
287	Q₁₆	—	2	2	no	—
288	Q₁₆	—	2	2	no	—
289	Q₁₆	—	356	4	no	—
290	Q₁₆	—	178	2	no	—
291	Q₁₆	—	178	2	no	—
292	SD₁₆	—	4	4	no	—
293	SD₁₆	—	356	4	no	—
294	SD₁₆	—	356	4	no	—
295	SD₁₆	—	356	4	no	—
296	D₈	—	2	2	no	—
297	D₈	—	2	2	no	—
298	D₈	—	356	4	no	—
299	D₈	—	178	2	no	—
300	D₈	—	178	2	no	—
301	C₂×C₈	—	89	1	no	—
302	C₂×C₈	—	89	1	no	—
303	C₂×C₈	—	89	4	no	—
304	C₂×C₈	—	89	2	no	—
305	C₂×C₈	—	89	2	no	—
306	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
307	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
308	C₂²⋊C₄	—	4	4	no	—
309	C₂²⋊C₄	—	356	4	no	—
310	C₂²⋊C₄	—	356	4	no	—
311	C₂²⋊C₄	—	356	4	no	—
312	C₄²	—	89	2	no	—
313	C₄²	—	89	1	no	—
314	C₄²	D₄×D₈₉	1	1	yes	characteristic
315	C₁₆	—	89	2	no	—
316	C₁₆	—	89	2	no	—
317	C₁₆	D₄×D₈₉	1	2	yes	—
318	C₁₆	D₄×D₈₉	1	2	yes	—
319	C₂³	—	4	2	no	—
320	C₂³	—	89	1	no	—
321	C₂³	—	356	2	no	—
322	C₂³	—	712	4	no	—
323	C₂³	—	712	8	no	—
324	Q₈	—	4	2	no	—
325	Q₈	—	4	2	no	—
326	Q₈	—	356	2	no	—
327	Q₈	—	356	2	no	—
328	Q₈	—	712	4	no	—
329	D₄	—	4	2	no	—
330	D₄	—	4	2	no	—
331	D₄	—	356	2	no	—
332	D₄	—	356	2	no	—
333	D₄	—	712	4	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	89	1	no	—
336	C₂×C₄	—	89	2	no	—
337	C₂×C₄	—	89	2	no	—
338	C₂×C₄	—	89	2	no	—
339	C₂×C₄	—	356	2	no	—
340	C₂×C₄	—	178	4	no	—
341	C₂×C₄	—	712	4	no	—
342	C₂×C₄	D₈×D₈₉	1	2	yes	—
343	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
344	C₈	—	89	2	no	—
345	C₈	—	89	2	no	—
346	C₈	D₈×D₈₉	1	2	yes	—
347	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
348	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
349	C₂²	—	8	2	no	—
350	C₂²	—	8	4	no	—
351	C₂²	—	89	2	no	—
352	C₂²	—	89	4	no	—
353	C₂²	—	1424	8	no	—
354	C₂²	—	712	2	no	—
355	C₂²	—	712	4	no	—
356	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
357	C₄	—	2	2	no	—
358	C₄	—	8	2	no	—
359	C₄	—	89	2	no	—
360	C₄	—	178	2	no	—
361	C₄	—	712	2	no	—
362	C₄	#127×D₈₉	1	1	yes	characteristic
363	C₄	C₂×D₈×D₈₉	1	1	yes	characteristic
364	C₂	—	16	2	no	—
365	C₂	—	89	4	no	—
366	C₂	—	1424	2	no	—
367	C₂	#493×D₈₉	1	1	yes	characteristic
368	C₂	#508×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2249	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₁₆×D₈₉	C₂
hypercenter	#2249	D₈₉
hypocenter	C₈₉	C₂×#2249
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2249

Complements

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

N	H ≅ G/N	type
1. C₂×Q₃₂×D₈₉	C₂	semidirect
2. C₂×SD₃₂×D₈₉	C₂	semidirect
4. #495×D₈₉	C₂	semidirect
5. #26022514	C₂	semidirect
6. #26022513	C₂	semidirect
7. #26022512	C₂	semidirect
8. #26022511	C₂	semidirect
9. #26022510	C₂	semidirect
10. C₄×C₁₆×D₈₉	C₂	semidirect
11. C₈₉×#2249	C₂	semidirect
16. C₂×Dic₇₁₂	C₂²	semidirect
17. C₂×#15667958	C₂²	semidirect
19. C₂×#15667957	C₂²	semidirect
20. C₂×#15667956	C₂²	semidirect
21. C₂×#15667955	C₂²	semidirect
28. #19151983	C₂²	semidirect
29. #19151982	C₂²	semidirect
30. #19151981	C₂²	semidirect
34. C₄×#13083549	C₂²	semidirect
35. C₁₆×Dic₈₉	C₂²	semidirect
36. C₁₆×Dic₈₉	C₂²	semidirect
37. C₂×Q₃₂×C₈₉	C₂²	semidirect
38. C₂×SD₃₂×C₈₉	C₂²	semidirect
40. #495×C₈₉	C₂²	semidirect
41. C₄×C₁₆×C₈₉	C₂²	semidirect
184. D₈₉	#2249	direct (N × H)
194. #2249	D₈₉	direct (N × H)
196. C₈₉	C₂×#2249	semidirect
204. C₂×Q₃₂	C₂×D₈₉	semidirect
209. C₂×SD₃₂	C₂×D₈₉	semidirect
221. #495	C₂×D₈₉	semidirect
226. C₄×C₁₆	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₈₉	2
C₂×SD₃₂×D₈₉	2
C₂×D₁₆×D₈₉	2
#495×D₈₉	2
#26022514	2
#26022513	2
#26022512	2
#26022511	2
#26022510	2
C₄×C₁₆×D₈₉	2
C₈₉×#2249	2
C₂×#2249	89

Subgroups — #2249×D89 #36709823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2249×D₈₉ #36709823