Subgroups — #36470889

Order 22784.

462 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#13083549	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151416	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151377	C₂	1	1	yes	maximal, characteristic
5	C₈×#13083551	C₂	1	1	yes	maximal, characteristic
6	#26018329	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
9	C₂²×#13083549	C₂²	1	1	yes	characteristic
10	C₂²×#13083549	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×D₈₉	—	2	2	no	—
12	C₂×C₁₆×D₈₉	—	2	2	no	—
13	C₂×C₁₆×D₈₉	—	2	2	no	—
14	C₂×#15667924	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
16	C₄×#13083551	C₄	1	1	yes	characteristic
17	C₄⋊#13083549	—	2	2	no	—
18	#19151416	—	2	2	no	—
19	#19151377	—	2	2	no	—
20	#19151353	C₄	1	1	yes	characteristic
21	#19151352	C₄	1	1	yes	characteristic
22	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
27	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₈₉	—	2	1	no	—
29	C₂×C₈×D₈₉	—	2	1	no	—
30	C₂×C₈×D₈₉	—	2	1	no	—
31	C₂×C₈×D₈₉	—	2	1	no	—
32	C₂×C₈×D₈₉	D₄	1	2	yes	—
33	C₂×C₈×D₈₉	Q₈	1	2	yes	—
34	C₂×#13083551	C₈	1	1	yes	characteristic
35	C₂×#13083551	C₈	1	1	yes	characteristic
36	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂×#13083549	—	2	1	no	—
38	C₂×#13083549	—	2	1	no	—
39	C₂×#13083549	—	2	1	no	—
40	C₂×#13083549	—	2	1	no	—
41	C₂×#13083549	—	2	1	no	—
42	C₂×#13083549	D₄	1	2	yes	—
43	C₁₆×D₈₉	—	2	8	no	—
44	#15667924	—	2	2	no	—
45	C₈×Dic₈₉	—	2	1	no	—
46	C₈×Dic₈₉	—	2	1	no	—
47	#15667919	C₈	1	1	yes	characteristic
48	#15667919	C₈	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₁₆×C₈₉	—	2	1	no	—
53	C₂×C₁₆×C₈₉	—	2	1	no	—
54	C₂×C₁₆×C₈₉	—	2	1	no	—
55	C₂×C₁₆×C₈₉	—	2	1	no	—
56	C₂×C₁₆×C₈₉	—	2	1	no	—
57	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
58	C₄×C₈×C₈₉	—	2	1	no	—
59	C₄×C₈×C₈₉	—	2	1	no	—
60	C₂³×D₈₉	C₂×C₈	1	1	yes	characteristic
61	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
62	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
63	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
64	C₂×C₄×D₈₉	—	2	1	no	—
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×D₈₉	M₁₆	1	2	yes	—
70	C₂×#10623494	—	2	1	no	—
71	C₂×#10623494	—	2	1	no	—
72	C₂×#10623494	—	2	1	no	—
73	C₂×#10623494	—	2	1	no	—
74	C₂×#10623494	—	2	1	no	—
75	C₂×#10623494	C₄⋊C₄	1	2	yes	—
76	C₈×D₈₉	—	2	2	no	—
77	C₈×D₈₉	—	2	2	no	—
78	C₈×D₈₉	—	2	2	no	—
79	C₈×D₈₉	—	2	2	no	—
80	#13083551	—	2	2	no	—
81	#13083551	C₁₆	1	4	yes	complemented
82	C₄×Dic₈₉	—	2	1	no	—
83	C₄×Dic₈₉	—	2	1	no	—
84	#13083549	—	2	2	no	—
85	#13083549	—	4	1	no	—
86	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₈₉	—	2	1	no	—
90	C₂×C₈×C₈₉	—	2	1	no	—
91	C₂×C₈×C₈₉	—	2	1	no	—
92	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
97	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
98	C₄²×C₈₉	—	2	1	no	—
99	C₄²×C₈₉	—	2	1	no	—
100	C₁₆×C₈₉	—	2	2	no	—
101	C₁₆×C₈₉	—	4	1	no	—
102	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₈₉	—	2	1	no	—
104	C₂²×D₈₉	—	2	1	no	—
105	C₂²×D₈₉	—	2	1	no	—
106	C₂²×D₈₉	—	2	1	no	—
107	C₂²×D₈₉	C₄⋊C₈	1	2	yes	—
108	C₂²×D₈₉	C₂×C₁₆	1	2	yes	—
109	C₂²×D₈₉	M₃₂	1	2	yes	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	C₂×C₁₆	1	2	yes	complemented
116	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	—	2	1	no	—
124	C₂×Dic₈₉	—	2	1	no	—
125	C₂×Dic₈₉	—	2	1	no	—
126	C₂×Dic₈₉	—	2	1	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	C₂×Dic₈₉	C₄⋊C₈	1	2	yes	—
130	C₂×Dic₈₉	C₂×C₁₆	1	2	yes	complemented
131	C₄×D₈₉	—	2	2	no	—
132	C₄×D₈₉	—	2	2	no	—
133	C₄×D₈₉	—	2	2	no	—
134	C₄×D₈₉	—	2	2	no	—
135	C₈×C₈₉	—	4	1	no	—
136	C₈×C₈₉	#120	1	1	yes	characteristic
137	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
138	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
139	C₈×C₈₉	#123	1	1	yes	characteristic
140	#10623494	—	2	2	no	—
141	#10623494	—	4	1	no	—
142	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
143	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
144	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
145	C₂²×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
146	C₂²×C₈₉	C₂×M₃₂	1	1	yes	characteristic
147	C₂²×C₈₉	#504	1	1	yes	characteristic
148	C₂²×C₈₉	#504	1	1	yes	characteristic
149	C₂×D₈₉	—	2	2	no	—
150	C₂×D₈₉	—	2	2	no	—
151	C₂×D₈₉	—	2	2	no	—
152	C₂×D₈₉	—	2	2	no	—
153	C₂×D₈₉	—	2	2	no	—
154	C₂×D₈₉	—	2	2	no	—
155	C₂×D₈₉	C₄⋊C₁₆	1	4	yes	complemented
156	C₄×C₈₉	—	2	2	no	—
157	C₄×C₈₉	—	4	1	no	—
158	C₄×C₈₉	#385	1	1	yes	characteristic
159	C₄×C₈₉	#424	1	1	yes	characteristic
160	C₄×C₈₉	#435	1	1	yes	characteristic
161	C₄×C₈₉	#435	1	1	yes	characteristic
162	Dic₈₉	—	2	2	no	—
163	Dic₈₉	—	2	2	no	—
164	Dic₈₉	—	4	1	no	—
165	#12912	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	#1850	1	1	yes	characteristic
167	C₂×C₈₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
168	C₂×C₈₉	#2151	1	1	yes	characteristic
169	C₂×C₈₉	D₄×C₁₆	1	1	yes	characteristic
170	C₂×C₈₉	D₄×C₁₆	1	1	yes	characteristic
171	C₂×C₈₉	#2168	1	1	yes	characteristic
172	C₂×C₈₉	#2168	1	1	yes	characteristic
173	D₈₉	—	2	4	no	—
174	C₂³×C₁₆	—	89	1	no	—
175	C₂×C₄⋊C₁₆	—	89	1	no	—
176	C₂×C₄⋊C₁₆	—	89	1	no	—
177	C₂×C₂²⋊C₁₆	—	89	1	no	—
178	C₈×C₂²⋊C₄	—	89	1	no	—
179	#1374	—	89	1	no	—
180	#1374	D₈₉	1	1	yes	characteristic, complemented
181	C₈₉	#12912	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₈	—	89	1	no	—
183	C₂²×C₁₆	—	89	1	no	—
184	C₂²×C₁₆	—	89	1	no	—
185	C₂²×C₁₆	—	178	2	no	—
186	C₂²×C₁₆	—	178	2	no	—
187	C₂²×C₁₆	—	178	2	no	—
188	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
189	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
190	C₂×C₂²⋊C₈	—	89	1	no	—
191	C₂×C₄×C₈	—	89	1	no	—
192	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
193	C₄×C₂²⋊C₄	—	89	1	no	—
194	C₄⋊C₁₆	—	178	2	no	—
195	C₄⋊C₁₆	—	178	2	no	—
196	C₂²⋊C₁₆	—	178	2	no	—
197	#336	—	89	1	no	—
198	#336	—	89	1	no	—
199	C₂³×C₄	—	89	1	no	—
200	C₂²×C₈	—	89	1	no	—
201	C₂²×C₈	—	89	2	no	—
202	C₂²×C₈	—	89	2	no	—
203	C₂²×C₈	—	89	1	no	—
204	C₂²×C₈	—	178	1	no	—
205	C₂²×C₈	—	178	1	no	—
206	C₂²×C₈	—	178	1	no	—
207	C₂²×C₈	—	178	1	no	—
208	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	—	89	1	no	—
211	C₂×C₂²⋊C₄	—	89	1	no	—
212	C₂×C₄²	—	89	1	no	—
213	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	2	1	no	—
219	C₂×C₁₆	—	89	2	no	—
220	C₂×C₁₆	—	178	8	no	—
221	C₂×C₁₆	—	178	1	no	—
222	C₂×C₁₆	—	178	1	no	—
223	C₂×C₁₆	—	178	1	no	—
224	C₂×C₁₆	—	178	1	no	—
225	C₂×C₁₆	—	178	1	no	—
226	C₂×C₁₆	D₃₅₆	1	2	yes	—
227	C₂²⋊C₈	—	178	2	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	178	1	no	—
231	C₄×C₈	—	178	1	no	—
232	#95	—	89	1	no	—
233	#95	—	89	1	no	—
234	C₂⁴	—	89	1	no	—
235	C₂²×C₄	—	89	1	no	—
236	C₂²×C₄	—	89	1	no	—
237	C₂²×C₄	—	89	2	no	—
238	C₂²×C₄	—	89	2	no	—
239	C₂²×C₄	—	89	1	no	—
240	C₂²×C₄	—	178	1	no	—
241	C₂²×C₄	—	178	1	no	—
242	C₂²×C₄	—	178	1	no	—
243	C₂²×C₄	—	178	1	no	—
244	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
245	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	89	2	no	—
251	C₂×C₈	—	178	1	no	—
252	C₂×C₈	—	178	1	no	—
253	C₂×C₈	—	178	1	no	—
254	C₂×C₈	—	178	1	no	—
255	C₂×C₈	—	178	1	no	—
256	C₂×C₈	—	178	2	no	—
257	C₂×C₈	—	178	2	no	—
258	C₂×C₈	—	178	2	no	—
259	C₂×C₈	—	178	2	no	—
260	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
261	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
262	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
263	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
264	C₂×C₈	#13083573	1	1	yes	characteristic
265	C₂×C₈	#13083574	1	1	yes	characteristic
266	C₂²⋊C₄	—	89	4	no	—
267	C₂²⋊C₄	—	178	2	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	178	1	no	—
271	C₄²	—	178	1	no	—
272	C₁₆	—	2	2	no	—
273	C₁₆	—	4	1	no	—
274	C₁₆	—	356	1	no	—
275	C₁₆	—	178	2	no	—
276	C₂³	—	89	2	no	—
277	C₂³	—	89	2	no	—
278	C₂³	—	89	2	no	—
279	C₂³	—	178	1	no	—
280	C₂³	—	178	1	no	—
281	C₂³	—	178	1	no	—
282	C₂³	—	178	1	no	—
283	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	89	2	no	—
290	C₂×C₄	—	89	2	no	—
291	C₂×C₄	—	178	1	no	—
292	C₂×C₄	—	178	1	no	—
293	C₂×C₄	—	178	1	no	—
294	C₂×C₄	—	178	1	no	—
295	C₂×C₄	—	178	1	no	—
296	C₂×C₄	—	178	1	no	—
297	C₂×C₄	—	178	1	no	—
298	C₂×C₄	—	178	2	no	—
299	C₂×C₄	—	178	2	no	—
300	C₂×C₄	—	178	2	no	—
301	C₂×C₄	—	178	2	no	—
302	C₂×C₄	C₁₆×D₈₉	1	2	yes	complemented
303	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
304	C₂×C₄	#15667972	1	1	yes	characteristic
305	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
306	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
307	C₂×C₄	#15668018	1	1	yes	characteristic
308	C₂×C₄	#15668020	1	1	yes	characteristic
309	C₈	—	4	1	no	—
310	C₈	—	356	1	no	—
311	C₈	—	178	2	no	—
312	C₈	#15667982	1	1	yes	characteristic
313	C₈	#15667990	1	1	yes	characteristic
314	C₈	#15667992	1	1	yes	characteristic
315	C₈	#15667994	1	1	yes	characteristic
316	C₂²	—	89	4	no	—
317	C₂²	—	178	2	no	—
318	C₂²	—	178	2	no	—
319	C₂²	—	178	2	no	—
320	C₂²	—	178	2	no	—
321	C₂²	—	178	2	no	—
322	C₂²	—	178	2	no	—
323	C₂²	C₄⋊C₈×D₈₉	1	1	yes	characteristic
324	C₂²	#19151661	1	1	yes	characteristic
325	C₂²	C₈×D₃₅₆	1	1	yes	characteristic
326	C₂²	C₂×C₁₆×D₈₉	1	1	yes	characteristic
327	C₂²	M₃₂×D₈₉	1	1	yes	characteristic
328	C₂²	#19152014	1	1	yes	characteristic
329	C₂²	#19152016	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	4	1	no	—
332	C₄	—	356	1	no	—
333	C₄	—	178	2	no	—
334	C₄	—	178	2	no	—
335	C₄	#19151472	1	1	yes	characteristic
336	C₄	#19151623	1	1	yes	characteristic
337	C₄	#19151671	1	1	yes	characteristic
338	C₄	#19151672	1	1	yes	characteristic
339	C₂	—	178	4	no	—
340	C₂	#26020431	1	1	yes	characteristic
341	C₂	C₄⋊C₁₆×D₈₉	1	1	yes	characteristic
342	C₂	#26022009	1	1	yes	characteristic
343	C₂	#26022069	1	1	yes	characteristic
344	C₂	C₁₆×D₃₅₆	1	1	yes	characteristic
345	C₂	#26022079	1	1	yes	characteristic
346	C₂	#26022080	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂²×C₁₆
Fitting	C₈₉×#1374	C₂
socle	C₂³×C₈₉	C₂²×C₈
prime radical	C₂³×D₈₉	C₂×C₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1374	D₈₉
hypocenter	C₈₉	#12912
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊#13083549	C₂	semidirect
3. C₂×#19151416	C₂	semidirect
6. #26018329	C₂	semidirect
7. C₈₉×#1374	C₂	semidirect
81. #13083551	C₁₆	semidirect
115. C₂×C₄×C₈₉	C₂×C₁₆	semidirect
130. C₂×Dic₈₉	C₂×C₁₆	semidirect
155. C₂×D₈₉	C₄⋊C₁₆	semidirect
180. #1374	D₈₉	semidirect
181. C₈₉	#12912	semidirect
302. C₂×C₄	C₁₆×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₈₉	2
C₂×C₄⋊#13083549	2
C₂×#19151416	2
C₂×#19151377	2
C₈×#13083551	2
#26018329	2
C₈₉×#1374	2
#12912	89