Subgroups — #36537686

Order 22784.

570 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26031595	C₂	1	2	yes	maximal
2	#26030763	C₂	1	1	yes	maximal, characteristic
3	#26029183	C₂	1	1	yes	maximal, characteristic
4	#26019781	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019780	C₂	1	1	yes	maximal, characteristic, complemented
6	#26019767	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019633	C₂	1	2	yes	maximal, complemented
8	#26019632	C₂	1	2	yes	maximal, complemented
9	#334×D₈₉	C₂	1	1	yes	maximal, characteristic
10	#26018941	C₂	1	2	yes	maximal
11	C₈₉×#1694	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152153	C₂²	1	2	yes	—
13	#19152152	C₂²	1	1	yes	characteristic
14	#19152003	C₂²	1	1	yes	characteristic
15	#19151999	—	2	1	no	—
16	#19151999	—	2	1	no	—
17	#19151994	C₂²	1	2	yes	—
18	#19151894	C₂²	1	2	yes	—
19	#19151893	C₂²	1	2	yes	—
20	#19151891	—	2	2	no	—
21	#19151838	—	2	2	no	—
22	#19151836	—	2	2	no	—
23	#19151806	—	2	2	no	—
24	#19151805	C₂²	1	2	yes	—
25	#19151728	C₂²	1	1	yes	characteristic
26	#19151726	—	2	1	no	—
27	Q₈×#10623494	C₂²	1	1	yes	characteristic
28	#19151722	—	2	1	no	—
29	#19151722	C₂²	1	1	yes	characteristic
30	#19151635	—	2	1	no	—
31	#19151635	—	2	1	no	—
32	#19151623	—	2	1	no	—
33	#19151620	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	#19151346	C₂²	1	2	yes	—
40	#19151346	C₂²	1	2	yes	—
41	#19151344	C₂²	1	2	yes	—
42	#500×C₈₉	C₂²	1	1	yes	characteristic
43	#462×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×#13083557	—	2	1	no	—
49	C₂×C₈×D₈₉	—	2	2	no	—
50	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
51	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
52	#15668012	C₂³	1	2	yes	—
53	#15668011	D₄	1	1	yes	characteristic
54	#15668010	D₄	1	1	yes	characteristic
55	#15668003	—	2	2	no	—
56	#15668002	—	2	1	no	—
57	#15667992	—	2	2	no	—
58	#15667992	D₄	1	1	yes	characteristic
59	#15667990	—	2	1	no	—
60	#15667988	—	2	2	no	—
61	#15667988	D₄	1	1	yes	characteristic
62	Q₈×Dic₈₉	—	2	2	no	—
63	C₄×Dic₁₇₈	—	2	1	no	—
64	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
65	#15667977	—	2	2	no	—
66	#15667970	—	2	2	no	—
67	#15667969	—	2	1	no	—
68	C₄²×D₈₉	C₂³	1	1	yes	characteristic
69	#15667946	—	2	2	no	—
70	#15667946	—	2	1	no	—
71	#15667946	—	2	1	no	—
72	#15667945	—	2	2	no	—
73	#15667943	—	2	2	no	—
74	#15667943	—	2	2	no	—
75	C₄⋊#10623494	—	2	1	no	—
76	C₄⋊#10623494	C₂³	1	2	yes	—
77	C₄⋊#10623494	C₂³	1	1	yes	characteristic
78	#15667941	—	2	1	no	—
79	#15667941	—	2	2	no	—
80	#15667941	—	2	1	no	—
81	#15667941	—	2	1	no	—
82	#15667937	—	2	2	no	—
83	#15667936	—	2	2	no	—
84	#15667936	—	2	2	no	—
85	#15667922	—	2	1	no	—
86	C₄×#10623494	—	2	1	no	—
87	C₄×#10623494	C₂³	1	1	yes	characteristic
88	C₈×Dic₈₉	—	2	1	no	—
89	C₈×Dic₈₉	D₄	1	1	yes	characteristic
90	C₈×Dic₈₉	D₄	1	1	yes	characteristic
91	C₂×Q₁₆×C₈₉	—	2	2	no	—
92	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
93	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
94	#107×C₈₉	—	2	1	no	—
95	#107×C₈₉	—	2	1	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	1	no	—
103	C₂×Dic₁₇₈	—	2	2	no	—
104	C₂×Dic₁₇₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×D₈₉	#41	1	1	yes	characteristic
108	#13083564	—	8	2	no	—
109	#13083557	—	4	2	no	—
110	C₂×#10623494	—	2	2	no	—
111	C₂×#10623494	—	2	1	no	—
112	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
113	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
114	C₈×D₈₉	—	2	2	no	—
115	C₈×D₈₉	—	8	2	no	—
116	C₈×D₈₉	Q₁₆	1	4	yes	complemented
117	C₄⋊Dic₈₉	—	2	1	no	—
118	C₄⋊Dic₈₉	—	2	2	no	—
119	C₄⋊Dic₈₉	—	2	2	no	—
120	C₄⋊Dic₈₉	—	4	2	no	—
121	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
122	#13083553	—	2	1	no	—
123	#13083553	—	2	1	no	—
124	#13083553	—	2	1	no	—
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	—	4	2	no	—
131	C₄×Dic₈₉	—	2	1	no	—
132	C₄×Dic₈₉	—	4	2	no	—
133	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
134	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
135	C₄×Dic₈₉	#41	1	1	yes	characteristic
136	C₂×Q₈×C₈₉	—	2	2	no	—
137	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
138	Q₁₆×C₈₉	—	8	2	no	—
139	C₂×C₈×C₈₉	—	2	2	no	—
140	C₂×C₈×C₈₉	—	2	1	no	—
141	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
143	C₄⋊C₄×C₈₉	—	2	1	no	—
144	C₄⋊C₄×C₈₉	—	2	2	no	—
145	C₄⋊C₄×C₈₉	—	2	2	no	—
146	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₈₉	—	2	1	no	—
148	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
149	C₂²×D₈₉	#121	1	1	yes	characteristic
150	Q₈×C₈₉	—	2	2	no	—
151	Q₈×C₈₉	—	4	2	no	—
152	Q₈×C₈₉	—	4	2	no	—
153	C₂×C₄×C₈₉	—	2	1	no	—
154	C₂×C₄×C₈₉	—	2	1	no	—
155	C₂×C₄×C₈₉	—	4	2	no	—
156	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
159	C₂×Dic₈₉	—	2	1	no	—
160	C₂×Dic₈₉	—	2	1	no	—
161	C₂×Dic₈₉	—	4	2	no	—
162	C₂×Dic₈₉	#121	1	1	yes	characteristic
163	C₂×Dic₈₉	#121	1	1	yes	characteristic
164	C₂×Dic₈₉	#121	1	1	yes	characteristic
165	C₄×D₈₉	—	2	2	no	—
166	C₄×D₈₉	—	4	1	no	—
167	C₄×D₈₉	C₂×Q₁₆	1	2	yes	—
168	C₄×D₈₉	#137	1	2	yes	—
169	Dic₁₇₈	—	2	2	no	—
170	Dic₁₇₈	—	4	2	no	—
171	Dic₁₇₈	—	8	2	no	—
172	C₈×C₈₉	—	2	1	no	—
173	C₈×C₈₉	—	4	1	no	—
174	C₈×C₈₉	—	4	2	no	—
175	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
176	#10623494	—	2	1	no	—
177	#10623494	—	4	1	no	—
178	#10623494	—	4	2	no	—
179	#10623494	C₂×Q₁₆	1	2	yes	—
180	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
181	C₂×D₈₉	—	2	1	no	—
182	C₂×D₈₉	—	2	1	no	—
183	C₂×D₈₉	#462	1	2	yes	—
184	C₄×C₈₉	—	2	1	no	—
185	C₄×C₈₉	—	4	1	no	—
186	C₄×C₈₉	—	4	1	no	—
187	C₄×C₈₉	—	8	2	no	—
188	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
189	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
190	C₄×C₈₉	#577	1	2	yes	—
191	Dic₈₉	—	2	1	no	—
192	Dic₈₉	—	2	1	no	—
193	Dic₈₉	—	4	1	no	—
194	Dic₈₉	—	4	1	no	—
195	Dic₈₉	—	8	2	no	—
196	Dic₈₉	#462	1	2	yes	—
197	#19210	—	89	1	no	maximal, Sylow
198	C₂×C₈₉	C₂×#462	1	1	yes	characteristic
199	C₂×C₈₉	#3060	1	1	yes	characteristic
200	C₂×C₈₉	#3067	1	1	yes	characteristic
201	D₈₉	—	2	2	no	—
202	#3249	—	89	2	no	—
203	#3160	—	89	1	no	—
204	#2960	—	89	1	no	—
205	#1696	—	89	1	no	—
206	#1696	—	89	1	no	—
207	#1694	—	89	1	no	—
208	#1694	D₈₉	1	1	yes	characteristic, complemented
209	#1672	—	89	2	no	—
210	#1672	—	89	2	no	—
211	C₂×#334	—	89	1	no	—
212	#1550	—	89	2	no	—
213	C₈₉	#19210	1	1	yes	characteristic, complemented, Sylow
214	#533	—	89	1	no	—
215	#533	—	89	2	no	—
216	#500	—	89	1	no	—
217	#500	C₂×D₈₉	1	1	yes	characteristic
218	#499	—	178	1	no	—
219	#499	—	178	1	no	—
220	#498	—	89	2	no	—
221	#477	—	89	2	no	—
222	#477	—	89	2	no	—
223	#477	—	178	2	no	—
224	#467	—	178	2	no	—
225	#467	—	178	2	no	—
226	#462	—	89	2	no	—
227	#462	—	178	2	no	—
228	#462	C₂×D₈₉	1	2	yes	—
229	#446	—	89	1	no	—
230	#446	—	178	1	no	—
231	#446	C₂×D₈₉	1	1	yes	characteristic
232	C₈×Q₈	—	89	1	no	—
233	C₈×Q₈	—	89	1	no	—
234	C₈×Q₈	—	178	1	no	—
235	#427	—	178	1	no	—
236	#427	—	178	1	no	—
237	#424	—	178	1	no	—
238	#423	—	89	1	no	—
239	C₂×C₄⋊C₈	—	89	2	no	—
240	#405	—	178	1	no	—
241	C₂×C₄×C₈	—	89	1	no	—
242	#334	—	89	2	no	—
243	#334	—	89	1	no	—
244	#334	C₂×D₈₉	1	1	yes	characteristic
245	#333	—	89	2	no	—
246	#333	—	89	2	no	—
247	#332	—	89	2	no	—
248	#332	C₂×D₈₉	1	2	yes	—
249	C₂×Q₁₆	—	2	2	no	—
250	C₂×Q₁₆	—	178	2	no	—
251	C₂×M₁₆	—	178	1	no	—
252	C₂²×C₈	—	89	1	no	—
253	C₂²×C₈	—	89	1	no	—
254	C₂²×C₈	—	178	2	no	—
255	C₄⋊Q₈	—	89	1	no	—
256	C₄⋊Q₈	—	89	1	no	—
257	C₄⋊Q₈	—	89	2	no	—
258	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
259	#125	—	178	2	no	—
260	#125	—	178	1	no	—
261	C₂²⋊Q₈	—	89	1	no	—
262	C₂²⋊Q₈	—	89	1	no	—
263	C₂²⋊Q₈	—	178	2	no	—
264	C₂²⋊Q₈	—	178	2	no	—
265	C₂²⋊Q₈	—	178	1	no	—
266	C₄×Q₈	—	89	1	no	—
267	C₄×Q₈	—	178	1	no	—
268	C₄×Q₈	—	178	2	no	—
269	C₄×Q₈	—	178	2	no	—
270	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
271	#117	—	178	1	no	—
272	#117	—	178	2	no	—
273	C₂×C₄²	—	89	1	no	—
274	#107	—	2	1	no	—
275	#107	—	2	1	no	—
276	#107	—	178	2	no	—
277	#107	—	178	1	no	—
278	#107	—	178	1	no	—
279	#107	—	178	2	no	—
280	#106	—	178	2	no	—
281	#106	—	178	2	no	—
282	C₄⋊C₈	—	2	1	no	—
283	C₄⋊C₈	—	89	1	no	—
284	C₄⋊C₈	—	89	2	no	—
285	C₄⋊C₈	—	178	1	no	—
286	C₄⋊C₈	—	178	2	no	—
287	C₄⋊C₈	—	178	1	no	—
288	C₄⋊C₈	—	178	1	no	—
289	C₄⋊C₈	—	178	1	no	—
290	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
291	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
292	#103	—	2	2	no	—
293	#103	—	178	2	no	—
294	#103	—	178	2	no	—
295	#103	—	178	2	no	—
296	#97	—	2	1	no	—
297	#97	—	178	1	no	—
298	C₄×C₈	—	89	1	no	—
299	C₄×C₈	—	89	1	no	—
300	C₄×C₈	—	89	1	no	—
301	C₄×C₈	—	178	1	no	—
302	C₄×C₈	—	178	1	no	—
303	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	—	89	1	no	—
306	C₂×Q₈	—	178	1	no	—
307	C₂×Q₈	—	178	2	no	—
308	C₂×Q₈	D₄×D₈₉	1	1	yes	characteristic
309	C₂²×C₄	—	89	1	no	—
310	C₂²×C₄	—	89	1	no	—
311	C₂²×C₄	—	89	1	no	—
312	Q₁₆	—	8	2	no	—
313	Q₁₆	—	712	2	no	—
314	M₁₆	—	356	2	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	89	1	no	—
318	C₂×C₈	—	89	4	no	—
319	C₂×C₈	—	89	1	no	—
320	C₂×C₈	—	178	2	no	—
321	C₂×C₈	—	178	1	no	—
322	C₂×C₈	—	178	2	no	—
323	C₂×C₈	—	712	2	no	—
324	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
325	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	89	1	no	—
330	C₄⋊C₄	—	356	2	no	—
331	C₄⋊C₄	—	356	2	no	—
332	C₄⋊C₄	—	178	1	no	—
333	C₄⋊C₄	—	178	1	no	—
334	C₄⋊C₄	—	178	1	no	—
335	C₄⋊C₄	—	178	1	no	—
336	C₄⋊C₄	—	178	1	no	—
337	C₄⋊C₄	—	178	2	no	—
338	C₄⋊C₄	—	178	2	no	—
339	C₄⋊C₄	—	178	1	no	—
340	C₄⋊C₄	D₄×D₈₉	1	1	yes	characteristic
341	C₂²⋊C₄	—	356	2	no	—
342	C₂²⋊C₄	—	178	1	no	—
343	C₂²⋊C₄	—	178	1	no	—
344	C₄²	—	2	1	no	—
345	C₄²	—	89	1	no	—
346	C₄²	—	89	1	no	—
347	C₄²	—	89	1	no	—
348	C₄²	—	356	2	no	—
349	C₄²	—	178	1	no	—
350	C₄²	C₂³×D₈₉	1	1	yes	characteristic
351	C₂³	—	89	1	no	—
352	Q₈	—	2	2	no	—
353	Q₈	—	4	2	no	—
354	Q₈	—	4	2	no	—
355	Q₈	—	356	2	no	—
356	Q₈	—	178	2	no	—
357	Q₈	—	712	2	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	89	1	no	—
362	C₂×C₄	—	89	2	no	—
363	C₂×C₄	—	89	2	no	—
364	C₂×C₄	—	89	1	no	—
365	C₂×C₄	—	89	1	no	—
366	C₂×C₄	—	356	1	no	—
367	C₂×C₄	—	356	2	no	—
368	C₂×C₄	—	178	1	no	—
369	C₂×C₄	—	178	1	no	—
370	C₂×C₄	—	178	2	no	—
371	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
372	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
373	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
374	C₈	—	2	1	no	—
375	C₈	—	4	1	no	—
376	C₈	—	4	2	no	—
377	C₈	—	89	2	no	—
378	C₈	—	356	1	no	—
379	C₈	—	356	2	no	—
380	C₈	—	178	1	no	—
381	C₈	Q₁₆×D₈₉	1	2	yes	—
382	C₂²	—	89	2	no	—
383	C₂²	—	178	1	no	—
384	C₂²	—	178	1	no	—
385	C₂²	#121×D₈₉	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	4	1	no	—
388	C₄	—	4	1	no	—
389	C₄	—	8	2	no	—
390	C₄	—	89	2	no	—
391	C₄	—	356	1	no	—
392	C₄	—	356	1	no	—
393	C₄	—	178	1	no	—
394	C₄	—	178	1	no	—
395	C₄	—	712	2	no	—
396	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
397	C₄	#137×D₈₉	1	1	yes	characteristic
398	C₄	#19152408	1	2	yes	—
399	C₂	—	178	2	no	—
400	C₂	#462×D₈₉	1	1	yes	characteristic
401	C₂	#26029912	1	1	yes	characteristic
402	C₂	#26029974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₈₉
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1694	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂²×D₈₉	#121
hypercenter	#1694	D₈₉
hypocenter	C₈₉	#19210
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1694

Complements

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

N	H ≅ G/N	type
4. #26019781	C₂	semidirect
5. #26019780	C₂	semidirect
6. #26019767	C₂	semidirect
7. #26019633	C₂	semidirect
8. #26019632	C₂	semidirect
11. C₈₉×#1694	C₂	semidirect
116. C₈×D₈₉	Q₁₆	semidirect
208. #1694	D₈₉	semidirect
213. C₈₉	#19210	semidirect

Maximal subgroups

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

subgroup	index
#26031595	2
#26030763	2
#26029183	2
#26019781	2
#26019780	2
#26019767	2
#26019633	2
#26019632	2
#334×D₈₉	2
#26018941	2
C₈₉×#1694	2
#19210	89