Subgroups — #2227×D₈₉ #36709053

Order 22784.

574 conjugacy classes of subgroups.

The subgroup lattice has 388 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	C₄×Q₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#26022405	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022404	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022403	C₂	1	2	yes	maximal, complemented
6	#26022402	C₂	1	2	yes	maximal, complemented
7	#26022401	C₂	1	1	yes	maximal, characteristic, complemented
8	#366×D₈₉	C₂	1	2	yes	maximal
9	C₄⋊C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
10	#358×D₈₉	C₂	1	2	yes	maximal
11	C₈₉×#2227	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₈₉	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×D₈₉	—	2	1	no	—
14	C₄×Q₈×D₈₉	—	2	1	no	—
15	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
16	#19152003	C₂²	1	1	yes	characteristic
17	#19152002	—	2	1	no	—
18	#19152002	C₂²	1	2	yes	—
19	#19152001	—	2	1	no	—
20	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
21	Q₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
22	#19151692	C₂²	1	1	yes	characteristic
23	C₄×#13083564	—	2	1	no	—
24	#19151690	—	2	1	no	—
25	#107×D₈₉	—	2	1	no	—
26	#107×D₈₉	C₂²	1	2	yes	—
27	#107×D₈₉	C₂²	1	1	yes	characteristic
28	#103×D₈₉	—	2	1	no	—
29	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
30	#19151424	C₂²	1	2	yes	—
31	#19151423	C₂²	1	2	yes	—
32	#19151423	C₂²	1	2	yes	—
33	C₄⋊#13083549	C₂²	1	1	yes	characteristic
34	#19151416	C₂²	1	2	yes	—
35	#19151403	C₂²	1	2	yes	—
36	#19151402	C₂²	1	2	yes	—
37	#19151401	C₂²	1	2	yes	—
38	#500×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	#366×C₈₉	C₂²	1	2	yes	—
41	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
42	#358×C₈₉	C₂²	1	2	yes	—
43	C₂×Q₈×D₈₉	—	2	1	no	—
44	C₂×Q₈×D₈₉	—	4	1	no	—
45	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
46	C₂×#13083564	—	2	1	no	—
47	Q₁₆×D₈₉	—	2	2	no	—
48	Q₁₆×D₈₉	Q₈	1	4	yes	complemented
49	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
50	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
51	#15668012	—	2	1	no	—
52	#15668011	—	2	2	no	—
53	#15668011	—	4	1	no	—
54	#15668010	—	4	1	no	—
55	Q₈×Dic₈₉	—	2	1	no	—
56	C₄×Dic₁₇₈	—	2	1	no	—
57	C₄×Dic₁₇₈	—	4	1	no	—
58	#15667977	—	2	1	no	—
59	#15667977	—	4	1	no	—
60	C₄⋊C₄×D₈₉	—	2	2	no	—
61	C₄⋊C₄×D₈₉	—	2	1	no	—
62	C₄⋊C₄×D₈₉	—	4	1	no	—
63	C₄⋊C₄×D₈₉	—	4	1	no	—
64	C₄²×D₈₉	—	4	1	no	—
65	C₄²×D₈₉	D₄	1	1	yes	characteristic
66	C₂×#13083549	C₂³	1	2	yes	—
67	C₁₆×D₈₉	—	2	4	no	—
68	#15667946	—	2	1	no	—
69	#15667946	C₂³	1	2	yes	—
70	#15667946	C₂³	1	1	yes	characteristic
71	#15667945	—	2	2	no	—
72	#15667945	—	2	1	no	—
73	#15667945	—	2	2	no	—
74	#15667937	—	2	1	no	—
75	#15667936	—	2	1	no	—
76	#15667935	—	2	1	no	—
77	C₄×#10623494	D₄	1	1	yes	characteristic
78	C₈×Dic₈₉	D₄	1	1	yes	characteristic
79	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
80	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
81	C₄⋊Q₈×C₈₉	—	2	1	no	—
82	C₄×Q₈×C₈₉	—	2	1	no	—
83	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
84	#107×C₈₉	—	2	1	no	—
85	#107×C₈₉	C₂³	1	2	yes	—
86	#107×C₈₉	C₂³	1	1	yes	characteristic
87	#103×C₈₉	—	2	1	no	—
88	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
89	C₂×Dic₁₇₈	—	2	1	no	—
90	C₂×Dic₁₇₈	—	4	1	no	—
91	C₂×Dic₁₇₈	—	4	1	no	—
92	C₂×Dic₁₇₈	—	4	2	no	—
93	Q₈×D₈₉	—	2	4	no	—
94	Q₈×D₈₉	—	4	2	no	—
95	Q₈×D₈₉	—	8	4	no	—
96	C₂×C₄×D₈₉	—	4	1	no	—
97	C₂×C₄×D₈₉	—	4	2	no	—
98	C₂×C₄×D₈₉	—	4	1	no	—
99	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
100	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
101	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
102	Dic₃₅₆	—	2	1	no	—
103	Dic₃₅₆	C₂×Q₈	1	2	yes	complemented
104	#13083564	—	2	4	no	—
105	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
106	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
107	C₈×D₈₉	—	2	2	no	—
108	C₈×D₈₉	C₂×Q₈	1	2	yes	—
109	C₈×D₈₉	#41	1	2	yes	—
110	C₄⋊Dic₈₉	—	2	1	no	—
111	C₄⋊Dic₈₉	—	2	2	no	—
112	C₄⋊Dic₈₉	—	4	1	no	—
113	C₄⋊Dic₈₉	—	4	1	no	—
114	#13083553	—	4	1	no	—
115	#13083553	—	4	1	no	—
116	#13083553	—	4	1	no	—
117	#13083553	—	4	2	no	—
118	#13083553	—	4	2	no	—
119	C₄×Dic₈₉	—	4	1	no	—
120	C₄×Dic₈₉	—	4	1	no	—
121	C₄×Dic₈₉	—	4	1	no	—
122	C₄×Dic₈₉	D₈	1	1	yes	characteristic
123	C₄×Dic₈₉	D₈	1	1	yes	characteristic
124	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
125	#13083549	—	2	2	no	—
126	C₂×Q₈×C₈₉	—	2	1	no	—
127	C₂×Q₈×C₈₉	—	4	1	no	—
128	Q₁₆×C₈₉	—	2	1	no	—
129	Q₁₆×C₈₉	C₂×Q₈	1	2	yes	complemented
130	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
132	C₄⋊C₄×C₈₉	—	2	1	no	—
133	C₄⋊C₄×C₈₉	—	2	2	no	—
134	C₄⋊C₄×C₈₉	—	4	1	no	—
135	C₄⋊C₄×C₈₉	—	4	1	no	—
136	C₄²×C₈₉	—	4	1	no	—
137	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
138	C₁₆×C₈₉	—	2	2	no	—
139	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
140	Q₈×C₈₉	—	2	2	no	—
141	Q₈×C₈₉	—	4	1	no	—
142	Q₈×C₈₉	—	8	2	no	—
143	C₂×C₄×C₈₉	—	4	1	no	—
144	C₂×C₄×C₈₉	—	4	1	no	—
145	C₂×C₄×C₈₉	—	4	2	no	—
146	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×Dic₈₉	—	4	1	no	—
150	C₂×Dic₈₉	—	4	1	no	—
151	C₂×Dic₈₉	—	4	2	no	—
152	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
153	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
154	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
155	C₄×D₈₉	—	2	2	no	—
156	C₄×D₈₉	—	4	4	no	—
157	C₄×D₈₉	—	8	2	no	—
158	C₄×D₈₉	—	8	4	no	—
159	C₄×D₈₉	Q₃₂	1	4	yes	—
160	C₄×D₈₉	C₂²⋊Q₈	1	2	yes	—
161	C₄×D₈₉	#137	1	2	yes	—
162	Dic₁₇₈	—	2	2	no	—
163	Dic₁₇₈	—	4	1	no	—
164	Dic₁₇₈	—	4	4	no	—
165	Dic₁₇₈	—	8	4	no	—
166	Dic₁₇₈	—	8	2	no	—
167	C₈×C₈₉	—	2	1	no	—
168	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
169	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
170	#10623494	—	2	1	no	—
171	#10623494	C₂²⋊Q₈	1	2	yes	—
172	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
173	C₂×D₈₉	#474	1	2	yes	—
174	C₂×D₈₉	C₂×Q₃₂	1	2	yes	—
175	C₂×D₈₉	#509	1	2	yes	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	—	4	2	no	—
178	C₄×C₈₉	—	8	2	no	—
179	C₄×C₈₉	—	8	1	no	—
180	C₄×C₈₉	C₂×Q₃₂	1	2	yes	—
181	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
182	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
183	Dic₈₉	—	2	1	no	—
184	Dic₈₉	—	4	2	no	—
185	Dic₈₉	—	8	2	no	—
186	Dic₈₉	—	8	1	no	—
187	Dic₈₉	#474	1	2	yes	—
188	Dic₈₉	C₂×Q₃₂	1	2	yes	—
189	C₂×#2227	—	89	1	no	maximal, Sylow
190	C₂×C₈₉	C₂×#474	1	1	yes	characteristic
191	C₂×C₈₉	C₂²×Q₃₂	1	1	yes	characteristic
192	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
193	D₈₉	#2227	1	4	yes	direct product factor
194	C₂×#500	—	89	1	no	—
195	C₂×C₄×Q₁₆	—	89	1	no	—
196	#2227	—	89	1	no	—
197	#2227	—	89	1	no	—
198	#2227	—	89	2	no	—
199	#2227	—	89	1	no	—
200	#2227	—	89	2	no	—
201	#2227	D₈₉	1	1	yes	characteristic, direct product factor
202	C₂×#366	—	89	2	no	—
203	C₂×C₄⋊C₁₆	—	89	1	no	—
204	C₂×#358	—	89	2	no	—
205	C₈₉	C₂×#2227	1	1	yes	characteristic, complemented, Sylow
206	C₂²×Q₁₆	—	89	1	no	—
207	C₂×C₄⋊Q₈	—	178	1	no	—
208	C₂×C₄×Q₈	—	178	1	no	—
209	C₂²×C₁₆	—	89	2	no	—
210	#500	—	89	2	no	—
211	#500	—	89	1	no	—
212	#500	—	178	1	no	—
213	#500	—	178	1	no	—
214	#500	C₂×D₈₉	1	1	yes	characteristic
215	C₄×Q₁₆	—	89	1	no	—
216	C₄×Q₁₆	—	89	1	no	—
217	C₄×Q₁₆	—	89	1	no	—
218	C₄×Q₁₆	—	178	1	no	—
219	C₄×Q₁₆	—	178	1	no	—
220	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
221	C₂×#107	—	89	2	no	—
222	C₂×#107	—	89	1	no	—
223	C₂×#107	—	178	1	no	—
224	C₂×#103	—	178	1	no	—
225	C₂×C₄×C₈	—	89	1	no	—
226	#366	—	89	2	no	—
227	#366	—	89	2	no	—
228	#366	—	89	2	no	—
229	#366	C₂×D₈₉	1	2	yes	—
230	C₄⋊C₁₆	—	89	2	no	—
231	C₄⋊C₁₆	—	89	1	no	—
232	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
233	#358	—	89	2	no	—
234	#358	—	89	2	no	—
235	#358	—	89	2	no	—
236	#358	C₂×D₈₉	1	2	yes	—
237	C₂²×Q₈	—	356	1	no	—
238	C₂²×Q₈	—	178	1	no	—
239	C₂×Q₁₆	—	89	4	no	—
240	C₂×Q₁₆	—	89	1	no	—
241	C₂×Q₁₆	—	178	2	no	—
242	C₂×Q₁₆	—	178	1	no	—
243	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
244	C₂²×C₈	—	89	1	no	—
245	C₂²×C₈	—	89	1	no	—
246	C₄⋊Q₈	—	2	1	no	—
247	C₄⋊Q₈	—	356	1	no	—
248	C₄⋊Q₈	—	356	1	no	—
249	C₄⋊Q₈	—	178	2	no	—
250	C₄⋊Q₈	—	178	1	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	—	356	1	no	—
253	C₄×Q₈	—	356	1	no	—
254	C₄×Q₈	—	178	1	no	—
255	C₄×Q₈	—	178	1	no	—
256	C₄×Q₈	—	178	1	no	—
257	C₂×C₄⋊C₄	—	356	1	no	—
258	C₂×C₄⋊C₄	—	356	1	no	—
259	C₂×C₄⋊C₄	—	178	1	no	—
260	C₂×C₄⋊C₄	—	178	2	no	—
261	C₂×C₄²	—	89	1	no	—
262	C₂×C₄²	—	356	1	no	—
263	C₂×C₁₆	—	89	2	no	—
264	C₂×C₁₆	—	178	4	no	—
265	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
266	#107	—	2	1	no	—
267	#107	—	89	2	no	—
268	#107	—	89	1	no	—
269	#107	—	178	1	no	—
270	#107	—	178	2	no	—
271	#107	—	178	2	no	—
272	#107	—	178	1	no	—
273	#107	C₂²×D₈₉	1	2	yes	—
274	#107	C₂²×D₈₉	1	1	yes	characteristic
275	#103	—	2	1	no	—
276	#103	—	178	1	no	—
277	#103	—	178	1	no	—
278	#103	—	178	1	no	—
279	C₄×C₈	—	89	1	no	—
280	C₄×C₈	—	89	1	no	—
281	C₄×C₈	—	89	1	no	—
282	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
283	C₂×Q₈	—	2	1	no	—
284	C₂×Q₈	—	4	1	no	—
285	C₂×Q₈	—	356	1	no	—
286	C₂×Q₈	—	356	2	no	—
287	C₂×Q₈	—	356	2	no	—
288	C₂×Q₈	—	356	1	no	—
289	C₂×Q₈	—	178	4	no	—
290	C₂×Q₈	—	178	1	no	—
291	C₂×Q₈	—	712	4	no	—
292	C₂²×C₄	—	89	1	no	—
293	C₂²×C₄	—	89	1	no	—
294	C₂²×C₄	—	89	1	no	—
295	C₂²×C₄	—	356	1	no	—
296	C₂²×C₄	—	356	1	no	—
297	C₂²×C₄	—	356	2	no	—
298	Q₁₆	—	2	1	no	—
299	Q₁₆	—	89	2	no	—
300	Q₁₆	—	178	4	no	—
301	Q₁₆	—	178	1	no	—
302	Q₁₆	Q₈×D₈₉	1	2	yes	complemented
303	C₂×C₈	—	89	1	no	—
304	C₂×C₈	—	89	2	no	—
305	C₂×C₈	—	89	2	no	—
306	C₂×C₈	—	89	1	no	—
307	C₂×C₈	—	178	2	no	—
308	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
309	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	356	1	no	—
315	C₄⋊C₄	—	356	1	no	—
316	C₄⋊C₄	—	356	2	no	—
317	C₄⋊C₄	—	356	1	no	—
318	C₄⋊C₄	—	356	2	no	—
319	C₄⋊C₄	—	356	1	no	—
320	C₄⋊C₄	—	356	1	no	—
321	C₄⋊C₄	—	178	1	no	—
322	C₄⋊C₄	—	178	2	no	—
323	C₄²	—	4	1	no	—
324	C₄²	—	89	1	no	—
325	C₄²	—	89	1	no	—
326	C₄²	—	89	1	no	—
327	C₄²	—	356	1	no	—
328	C₄²	—	356	1	no	—
329	C₄²	—	356	1	no	—
330	C₄²	D₄×D₈₉	1	1	yes	characteristic
331	C₁₆	—	2	2	no	—
332	C₁₆	—	178	2	no	—
333	C₂³	—	89	1	no	—
334	Q₈	—	2	2	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	8	2	no	—
337	Q₈	—	356	4	no	—
338	Q₈	—	356	1	no	—
339	Q₈	—	178	2	no	—
340	Q₈	—	712	2	no	—
341	Q₈	—	712	4	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	89	1	no	—
346	C₂×C₄	—	89	1	no	—
347	C₂×C₄	—	89	2	no	—
348	C₂×C₄	—	89	4	no	—
349	C₂×C₄	—	89	2	no	—
350	C₂×C₄	—	89	1	no	—
351	C₂×C₄	—	356	4	no	—
352	C₂×C₄	—	356	1	no	—
353	C₂×C₄	—	356	1	no	—
354	C₂×C₄	—	356	2	no	—
355	C₂×C₄	—	178	2	no	—
356	C₂×C₄	—	712	2	no	—
357	C₂×C₄	—	712	4	no	—
358	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
359	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
360	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
361	C₈	—	2	1	no	—
362	C₈	—	89	2	no	—
363	C₈	—	178	1	no	—
364	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
365	C₈	#41×D₈₉	1	1	yes	characteristic
366	C₂²	—	89	2	no	—
367	C₂²	—	89	2	no	—
368	C₂²	—	89	2	no	—
369	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	4	2	no	—
372	C₄	—	8	2	no	—
373	C₄	—	8	1	no	—
374	C₄	—	89	2	no	—
375	C₄	—	89	2	no	—
376	C₄	—	356	2	no	—
377	C₄	—	178	1	no	—
378	C₄	—	712	2	no	—
379	C₄	—	712	1	no	—
380	C₄	Q₃₂×D₈₉	1	2	yes	—
381	C₄	C₂²⋊Q₈×D₈₉	1	1	yes	characteristic
382	C₄	#137×D₈₉	1	1	yes	characteristic
383	C₂	—	89	4	no	—
384	C₂	#474×D₈₉	1	1	yes	characteristic
385	C₂	C₂×Q₃₂×D₈₉	1	1	yes	characteristic
386	C₂	#509×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2227	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₈₉	C₂×D₈
hypercenter	#2227	D₈₉
hypocenter	C₈₉	C₂×#2227
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2227

Complements

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

N	H ≅ G/N	type
3. #26022405	C₂	semidirect
4. #26022404	C₂	semidirect
5. #26022403	C₂	semidirect
6. #26022402	C₂	semidirect
7. #26022401	C₂	semidirect
11. C₈₉×#2227	C₂	semidirect
48. Q₁₆×D₈₉	Q₈	semidirect
103. Dic₃₅₆	C₂×Q₈	semidirect
129. Q₁₆×C₈₉	C₂×Q₈	semidirect
193. D₈₉	#2227	direct (N × H)
201. #2227	D₈₉	direct (N × H)
205. C₈₉	C₂×#2227	semidirect
302. Q₁₆	Q₈×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
C₄×Q₁₆×D₈₉	2
#26022405	2
#26022404	2
#26022403	2
#26022402	2
#26022401	2
#366×D₈₉	2
C₄⋊C₁₆×D₈₉	2
#358×D₈₉	2
C₈₉×#2227	2
C₂×#2227	89

Subgroups — #2227×D89 #36709053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2227×D₈₉ #36709053