Subgroups — #36537885

Order 22784.

584 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26031592	C₂	1	2	yes	maximal
2	#26030836	C₂	1	1	yes	maximal, characteristic, complemented
3	#26029226	C₂	1	1	yes	maximal, characteristic
4	#26019770	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019766	C₂	1	1	yes	maximal, characteristic, complemented
6	#26019738	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019680	C₂	1	2	yes	maximal, complemented
8	#26019674	C₂	1	2	yes	maximal, complemented
9	#334×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#26018949	C₂	1	2	yes	maximal, complemented
11	C₈₉×#1690	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152153	C₂²	1	2	yes	—
13	#19152135	C₂²	1	1	yes	characteristic
14	#19152003	C₂²	1	1	yes	characteristic
15	#19151948	C₂²	1	2	yes	—
16	#19151908	—	2	2	no	—
17	#19151875	C₂²	1	2	yes	—
18	#19151838	—	2	2	no	—
19	#19151836	—	2	1	no	—
20	#19151836	—	2	1	no	—
21	#19151828	—	2	2	no	—
22	#19151823	—	2	2	no	—
23	#19151808	C₂²	1	2	yes	—
24	#19151804	C₂²	1	2	yes	—
25	Q₈×#10623494	C₂²	1	1	yes	characteristic
26	Q₁₆×Dic₈₉	—	2	1	no	—
27	Q₁₆×Dic₈₉	—	2	1	no	—
28	#19151677	C₂²	1	1	yes	characteristic
29	#19151671	—	2	1	no	—
30	#19151663	—	2	1	no	—
31	#19151663	C₂²	1	1	yes	characteristic
32	#19151658	—	2	1	no	—
33	#19151654	C₂²	1	1	yes	characteristic
34	C₄⋊C₈×D₈₉	C₂²	1	2	yes	—
35	#19151555	—	2	1	no	—
36	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
37	#19151349	C₂²	1	1	yes	characteristic
38	#19151348	C₂²	1	2	yes	—
39	#19151345	C₂²	1	2	yes	—
40	#19151344	C₂²	1	2	yes	—
41	#19151343	C₂²	1	2	yes	—
42	#494×C₈₉	C₂²	1	1	yes	characteristic
43	#475×C₈₉	C₂²	1	2	yes	—
44	#446×C₈₉	C₂²	1	1	yes	characteristic
45	#334×C₈₉	C₂²	1	1	yes	characteristic
46	#332×C₈₉	C₂²	1	2	yes	—
47	C₂×#13083564	—	2	2	no	—
48	C₂×#13083563	—	2	2	no	—
49	C₂×#13083557	—	2	1	no	—
50	C₂×C₈×D₈₉	—	2	1	no	—
51	C₂×C₈×D₈₉	—	2	2	no	—
52	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
53	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
54	#15668012	C₂³	1	2	yes	—
55	#15668007	—	2	1	no	—
56	#15668003	—	2	2	no	—
57	#15668003	D₄	1	1	yes	characteristic
58	#15668001	D₄	1	1	yes	characteristic
59	#15667997	—	2	1	no	—
60	#15667992	—	2	2	no	—
61	#15667988	—	2	2	no	—
62	#15667988	D₄	1	1	yes	characteristic
63	#15667987	D₄	1	1	yes	characteristic
64	Q₈×Dic₈₉	—	2	2	no	—
65	#15667977	—	2	2	no	—
66	C₄×D₃₅₆	—	2	1	no	—
67	C₄×D₃₅₆	C₂³	1	1	yes	characteristic
68	#15667970	—	2	2	no	—
69	#15667969	—	2	1	no	—
70	C₄²×D₈₉	C₂³	1	1	yes	characteristic
71	#15667946	—	2	1	no	—
72	#15667946	—	2	1	no	—
73	#15667945	—	2	2	no	—
74	C₄⋊#10623494	—	2	1	no	—
75	C₄⋊#10623494	C₂³	1	2	yes	—
76	C₄⋊#10623494	C₂³	1	1	yes	characteristic
77	#15667941	—	2	2	no	—
78	#15667937	—	2	2	no	—
79	#15667936	—	2	2	no	—
80	#15667936	—	2	2	no	—
81	#15667935	—	2	2	no	—
82	#15667932	—	2	2	no	—
83	#15667924	—	2	1	no	—
84	#15667924	—	2	1	no	—
85	#15667924	—	2	1	no	—
86	#15667922	—	2	1	no	—
87	C₄×#10623494	—	2	1	no	—
88	C₄×#10623494	C₂³	1	1	yes	characteristic
89	C₈×Dic₈₉	D₄	1	1	yes	characteristic
90	C₈×Dic₈₉	D₄	1	1	yes	characteristic
91	C₂×Q₁₆×C₈₉	—	2	1	no	—
92	C₂×Q₁₆×C₈₉	—	2	1	no	—
93	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
94	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
95	#106×C₈₉	—	2	2	no	—
96	C₄⋊C₈×C₈₉	—	2	1	no	—
97	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
98	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
99	#103×C₈₉	—	2	2	no	—
100	#97×C₈₉	—	2	1	no	—
101	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×Dic₁₇₈	—	2	2	no	—
103	C₂×D₃₅₆	—	2	1	no	—
104	C₂×D₃₅₆	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×D₈₉	—	2	1	no	—
106	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×D₈₉	#41	1	1	yes	characteristic
109	#13083564	—	8	2	no	—
110	#13083563	—	8	2	no	—
111	#13083557	—	4	2	no	—
112	C₂×#10623494	—	2	2	no	—
113	C₂×#10623494	—	2	1	no	—
114	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
115	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
116	C₈×D₈₉	—	2	2	no	—
117	C₈×D₈₉	—	4	1	no	—
118	C₈×D₈₉	—	4	2	no	—
119	C₈×D₈₉	—	8	2	no	—
120	C₄⋊Dic₈₉	—	2	1	no	—
121	C₄⋊Dic₈₉	—	2	2	no	—
122	C₄⋊Dic₈₉	—	2	2	no	—
123	C₄⋊Dic₈₉	—	4	2	no	—
124	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
125	#13083553	—	2	1	no	—
126	#13083553	—	2	1	no	—
127	#13083553	—	4	2	no	—
128	#13083551	—	2	1	no	—
129	#13083551	—	2	1	no	—
130	#13083551	—	2	1	no	—
131	#13083551	—	2	1	no	—
132	#13083551	—	2	1	no	—
133	#13083551	—	4	2	no	—
134	C₄×Dic₈₉	—	4	2	no	—
135	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
136	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
137	C₄×Dic₈₉	#41	1	1	yes	characteristic
138	C₂×Q₈×C₈₉	—	2	2	no	—
139	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
140	Q₁₆×C₈₉	—	4	2	no	—
141	Q₁₆×C₈₉	—	4	2	no	—
142	C₂×C₈×C₈₉	—	2	2	no	—
143	C₂×C₈×C₈₉	—	2	1	no	—
144	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
145	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
146	C₄⋊C₄×C₈₉	—	2	1	no	—
147	C₄⋊C₄×C₈₉	—	2	2	no	—
148	C₄⋊C₄×C₈₉	—	2	2	no	—
149	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₈₉	—	2	1	no	—
151	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
152	C₂²×D₈₉	—	2	1	no	—
153	C₂²×D₈₉	#121	1	1	yes	characteristic
154	Q₈×C₈₉	—	2	2	no	—
155	Q₈×C₈₉	—	4	2	no	—
156	Q₈×C₈₉	—	4	2	no	—
157	C₂×C₄×C₈₉	—	2	1	no	—
158	C₂×C₄×C₈₉	—	2	1	no	—
159	C₂×C₄×C₈₉	—	4	2	no	—
160	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
163	C₂×Dic₈₉	—	2	1	no	—
164	C₂×Dic₈₉	—	4	2	no	—
165	C₂×Dic₈₉	#121	1	1	yes	characteristic
166	C₂×Dic₈₉	#121	1	1	yes	characteristic
167	C₂×Dic₈₉	#121	1	1	yes	characteristic
168	D₃₅₆	—	2	2	no	—
169	D₃₅₆	—	4	2	no	—
170	C₄×D₈₉	—	2	2	no	—
171	C₄×D₈₉	—	2	1	no	—
172	C₄×D₈₉	—	2	1	no	—
173	C₄×D₈₉	—	2	2	no	—
174	C₄×D₈₉	—	4	1	no	—
175	C₄×D₈₉	—	4	1	no	—
176	Dic₁₇₈	—	4	2	no	—
177	Dic₁₇₈	—	4	2	no	—
178	C₈×C₈₉	—	2	1	no	—
179	C₈×C₈₉	—	4	1	no	—
180	C₈×C₈₉	—	4	2	no	—
181	C₈×C₈₉	#135	1	2	yes	—
182	#10623494	—	2	1	no	—
183	#10623494	—	2	1	no	—
184	#10623494	—	4	1	no	—
185	#10623494	—	4	2	no	—
186	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
187	C₂×D₈₉	—	2	1	no	—
188	C₂×D₈₉	—	2	1	no	—
189	C₂×D₈₉	—	4	1	no	—
190	C₂×D₈₉	—	4	1	no	—
191	C₂×D₈₉	—	4	1	no	—
192	C₂×D₈₉	#473	1	2	yes	—
193	C₄×C₈₉	—	2	1	no	—
194	C₄×C₈₉	—	4	1	no	—
195	C₄×C₈₉	—	4	1	no	—
196	C₄×C₈₉	—	8	2	no	—
197	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
198	C₄×C₈₉	#575	1	1	yes	characteristic
199	C₄×C₈₉	#577	1	2	yes	—
200	Dic₈₉	—	2	1	no	—
201	Dic₈₉	—	2	1	no	—
202	Dic₈₉	—	2	1	no	—
203	Dic₈₉	—	4	1	no	—
204	Dic₈₉	—	8	2	no	—
205	#19228	—	89	1	no	maximal, Sylow
206	C₂×C₈₉	#3035	1	1	yes	characteristic
207	C₂×C₈₉	#3060	1	1	yes	characteristic
208	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
209	D₈₉	—	2	2	no	—
210	D₈₉	—	8	1	no	—
211	#3248	—	89	2	no	—
212	#3170	—	89	1	no	—
213	#2966	—	89	1	no	—
214	#1694	—	89	1	no	—
215	#1693	—	89	1	no	—
216	#1690	D₈₉	1	1	yes	characteristic, complemented
217	#1689	—	89	1	no	—
218	#1680	—	89	2	no	—
219	#1679	—	89	2	no	—
220	C₂×#334	—	89	1	no	—
221	#1552	—	89	2	no	—
222	C₈₉	#19228	1	1	yes	characteristic, complemented, Sylow
223	#533	—	89	2	no	—
224	#529	—	89	1	no	—
225	#500	—	89	1	no	—
226	#494	C₂×D₈₉	1	1	yes	characteristic
227	#487	—	89	2	no	—
228	#479	—	178	2	no	—
229	#475	C₂×D₈₉	1	2	yes	—
230	#474	—	89	2	no	—
231	#467	—	178	2	no	—
232	#467	—	178	1	no	—
233	#467	—	178	1	no	—
234	#465	—	178	2	no	—
235	#464	—	178	2	no	—
236	#462	—	89	2	no	—
237	#461	—	89	2	no	—
238	#446	C₂×D₈₉	1	1	yes	characteristic
239	C₈×Q₈	—	89	1	no	—
240	C₄×Q₁₆	—	178	1	no	—
241	C₄×Q₁₆	—	178	1	no	—
242	#436	—	89	1	no	—
243	#435	—	178	1	no	—
244	C₈×D₄	—	89	1	no	—
245	C₈×D₄	—	178	1	no	—
246	#433	—	178	1	no	—
247	#432	—	89	1	no	—
248	C₂×C₄⋊C₈	—	89	2	no	—
249	#405	—	178	1	no	—
250	C₂×C₄×C₈	—	89	1	no	—
251	#334	—	89	2	no	—
252	#334	—	89	1	no	—
253	#334	C₂×D₈₉	1	1	yes	characteristic
254	#332	—	89	2	no	—
255	#332	—	89	2	no	—
256	#332	—	89	2	no	—
257	#332	C₂×D₈₉	1	2	yes	—
258	C₂×Q₁₆	—	2	1	no	—
259	C₂×Q₁₆	—	2	1	no	—
260	C₂×Q₁₆	—	178	2	no	—
261	C₂×SD₁₆	—	178	2	no	—
262	C₂×M₁₆	—	178	1	no	—
263	C₂²×C₈	—	89	1	no	—
264	C₂²×C₈	—	89	1	no	—
265	C₂²×C₈	—	178	1	no	—
266	C₂²×C₈	—	178	2	no	—
267	C₄⋊Q₈	—	89	2	no	—
268	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
269	#126	—	178	1	no	—
270	#125	—	89	1	no	—
271	#125	—	178	2	no	—
272	#124	—	89	1	no	—
273	#123	—	178	1	no	—
274	C₂²⋊Q₈	—	89	1	no	—
275	C₂²⋊Q₈	—	178	2	no	—
276	C₂²⋊Q₈	—	178	2	no	—
277	#121	—	89	1	no	—
278	C₄×Q₈	—	178	2	no	—
279	C₄×Q₈	—	178	2	no	—
280	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
281	C₄×D₄	—	89	1	no	—
282	C₄×D₄	—	178	1	no	—
283	#117	—	178	1	no	—
284	#117	—	178	2	no	—
285	C₂×C₄²	—	89	1	no	—
286	#107	—	178	2	no	—
287	#107	—	178	1	no	—
288	#107	—	178	1	no	—
289	#106	—	2	2	no	—
290	C₄⋊C₈	—	2	1	no	—
291	C₄⋊C₈	—	89	1	no	—
292	C₄⋊C₈	—	89	2	no	—
293	C₄⋊C₈	—	178	1	no	—
294	C₄⋊C₈	—	178	2	no	—
295	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
296	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
297	#103	—	2	2	no	—
298	#103	—	178	2	no	—
299	#103	—	178	2	no	—
300	#103	—	178	2	no	—
301	#103	—	178	2	no	—
302	#102	—	178	2	no	—
303	C₂²⋊C₈	—	178	1	no	—
304	C₂²⋊C₈	—	178	1	no	—
305	C₂²⋊C₈	—	178	1	no	—
306	#97	—	2	1	no	—
307	#97	—	178	1	no	—
308	C₄×C₈	—	89	1	no	—
309	C₄×C₈	—	89	1	no	—
310	C₄×C₈	—	89	1	no	—
311	C₄×C₈	—	178	1	no	—
312	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	178	2	no	—
315	C₂×Q₈	D₄×D₈₉	1	1	yes	characteristic
316	C₂×D₄	—	89	1	no	—
317	C₂×D₄	—	178	1	no	—
318	C₂²×C₄	—	89	1	no	—
319	C₂²×C₄	—	89	1	no	—
320	C₂²×C₄	—	89	1	no	—
321	C₂²×C₄	—	178	1	no	—
322	Q₁₆	—	4	2	no	—
323	Q₁₆	—	4	2	no	—
324	Q₁₆	—	712	2	no	—
325	SD₁₆	—	712	2	no	—
326	M₁₆	—	356	2	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	89	1	no	—
330	C₂×C₈	—	89	1	no	—
331	C₂×C₈	—	356	1	no	—
332	C₂×C₈	—	356	2	no	—
333	C₂×C₈	—	178	1	no	—
334	C₂×C₈	—	178	2	no	—
335	C₂×C₈	—	178	2	no	—
336	C₂×C₈	—	712	2	no	—
337	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
338	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	89	1	no	—
343	C₄⋊C₄	—	356	2	no	—
344	C₄⋊C₄	—	356	2	no	—
345	C₄⋊C₄	—	178	1	no	—
346	C₄⋊C₄	—	178	1	no	—
347	C₄⋊C₄	—	178	1	no	—
348	C₄⋊C₄	—	178	2	no	—
349	C₄⋊C₄	—	178	2	no	—
350	C₄⋊C₄	D₄×D₈₉	1	1	yes	characteristic
351	C₂²⋊C₄	—	356	2	no	—
352	C₂²⋊C₄	—	178	1	no	—
353	C₂²⋊C₄	—	178	1	no	—
354	C₂²⋊C₄	—	178	1	no	—
355	C₂²⋊C₄	—	178	1	no	—
356	C₂²⋊C₄	—	178	1	no	—
357	C₄²	—	2	1	no	—
358	C₄²	—	89	1	no	—
359	C₄²	—	89	1	no	—
360	C₄²	—	89	1	no	—
361	C₄²	—	356	2	no	—
362	C₄²	C₂³×D₈₉	1	1	yes	characteristic
363	C₂³	—	89	1	no	—
364	C₂³	—	178	1	no	—
365	Q₈	—	2	2	no	—
366	Q₈	—	4	2	no	—
367	Q₈	—	4	2	no	—
368	Q₈	—	356	2	no	—
369	Q₈	—	356	2	no	—
370	D₄	—	356	2	no	—
371	D₄	—	178	2	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	89	1	no	—
376	C₂×C₄	—	89	1	no	—
377	C₂×C₄	—	89	1	no	—
378	C₂×C₄	—	356	1	no	—
379	C₂×C₄	—	356	1	no	—
380	C₂×C₄	—	356	2	no	—
381	C₂×C₄	—	178	2	no	—
382	C₂×C₄	—	178	1	no	—
383	C₂×C₄	—	178	1	no	—
384	C₂×C₄	—	178	2	no	—
385	C₂×C₄	—	178	1	no	—
386	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
387	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
388	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
389	C₈	—	2	1	no	—
390	C₈	—	4	1	no	—
391	C₈	—	4	2	no	—
392	C₈	—	356	2	no	—
393	C₈	—	356	1	no	—
394	C₈	—	178	1	no	—
395	C₈	—	178	1	no	—
396	C₈	#15668033	1	2	yes	—
397	C₂²	—	89	2	no	—
398	C₂²	—	356	1	no	—
399	C₂²	—	356	1	no	—
400	C₂²	—	356	1	no	—
401	C₂²	—	178	1	no	—
402	C₂²	—	178	1	no	—
403	C₂²	#121×D₈₉	1	1	yes	characteristic
404	C₄	—	2	1	no	—
405	C₄	—	4	1	no	—
406	C₄	—	4	1	no	—
407	C₄	—	8	2	no	—
408	C₄	—	356	1	no	—
409	C₄	—	178	1	no	—
410	C₄	—	178	1	no	—
411	C₄	—	178	1	no	—
412	C₄	—	712	2	no	—
413	C₄	C₂×#15668033	1	1	yes	characteristic
414	C₄	#19152394	1	1	yes	characteristic
415	C₄	#19152407	1	2	yes	—
416	C₂	—	178	2	no	—
417	C₂	—	712	1	no	—
418	C₂	#26029702	1	1	yes	characteristic
419	C₂	#26029916	1	1	yes	characteristic
420	C₂	#473×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₈₉
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1690	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	#15667987	D₄
hypercenter	#1690	D₈₉
hypocenter	C₈₉	#19228
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1690

Complements

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

N	H ≅ G/N	type
2. #26030836	C₂	semidirect
4. #26019770	C₂	semidirect
5. #26019766	C₂	semidirect
6. #26019738	C₂	semidirect
7. #26019680	C₂	semidirect
8. #26019674	C₂	semidirect
9. #334×D₈₉	C₂	semidirect
10. #26018949	C₂	semidirect
11. C₈₉×#1690	C₂	semidirect
216. #1690	D₈₉	semidirect
222. C₈₉	#19228	semidirect

Maximal subgroups

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

subgroup	index
#26031592	2
#26030836	2
#26029226	2
#26019770	2
#26019766	2
#26019738	2
#26019680	2
#26019674	2
#334×D₈₉	2
#26018949	2
C₈₉×#1690	2
#19228	89