Subgroups — #36537679

Order 22784.

582 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26031402	C₂	1	2	yes	maximal
2	#500×D₈₉	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	#26019777	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019655	C₂	1	2	yes	maximal, complemented
8	#26019654	C₂	1	2	yes	maximal, complemented
9	#26018997	C₂	1	1	yes	maximal, characteristic
10	#26018931	C₂	1	2	yes	maximal
11	C₈₉×#1696	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152152	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×D₈₉	C₂²	1	2	yes	—
14	#19152003	C₂²	1	1	yes	characteristic
15	#19152002	—	2	1	no	—
16	#19152002	—	2	1	no	—
17	#19152002	—	2	2	no	—
18	#19152001	C₂²	1	2	yes	—
19	#19151947	—	2	2	no	—
20	#19151935	—	2	2	no	—
21	#19151932	—	2	2	no	—
22	#19151894	C₂²	1	2	yes	—
23	#19151893	C₂²	1	2	yes	—
24	#19151890	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	#19151650	C₂²	1	2	yes	—
31	#107×D₈₉	—	2	1	no	—
32	#107×D₈₉	—	2	1	no	—
33	#19151623	—	2	1	no	—
34	#19151620	C₂²	1	1	yes	characteristic
35	#19151555	—	2	1	no	—
36	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
37	#19151350	C₂²	1	2	yes	—
38	#19151349	C₂²	1	1	yes	characteristic
39	#19151347	C₂²	1	2	yes	—
40	#19151346	C₂²	1	2	yes	—
41	#19151346	C₂²	1	2	yes	—
42	#500×C₈₉	C₂²	1	1	yes	characteristic
43	#477×C₈₉	C₂²	1	2	yes	—
44	#446×C₈₉	C₂²	1	1	yes	characteristic
45	#334×C₈₉	C₂²	1	1	yes	characteristic
46	#333×C₈₉	C₂²	1	2	yes	—
47	C₂×Q₈×D₈₉	—	2	2	no	—
48	C₂×#13083557	—	2	1	no	—
49	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
50	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
51	#15668012	C₂³	1	2	yes	—
52	#15668011	—	2	2	no	—
53	#15668011	—	2	2	no	—
54	#15668011	D₄	1	1	yes	characteristic
55	#15668010	—	2	2	no	—
56	#15668010	D₄	1	1	yes	characteristic
57	#15668002	—	2	1	no	—
58	#15667992	D₄	1	1	yes	characteristic
59	#15667990	—	2	1	no	—
60	#15667988	D₄	1	1	yes	characteristic
61	C₄×Dic₁₇₈	—	2	1	no	—
62	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	#15667969	—	2	1	no	—
64	C₄⋊C₄×D₈₉	—	2	2	no	—
65	C₄⋊C₄×D₈₉	—	2	2	no	—
66	C₄²×D₈₉	C₂³	1	1	yes	characteristic
67	#15667946	—	2	2	no	—
68	#15667946	—	2	1	no	—
69	#15667946	—	2	1	no	—
70	#15667945	—	2	2	no	—
71	#15667945	—	2	2	no	—
72	#15667945	—	2	2	no	—
73	#15667945	—	2	2	no	—
74	C₄⋊#10623494	—	2	1	no	—
75	C₄⋊#10623494	C₂³	1	1	yes	characteristic
76	C₄⋊#10623494	C₂³	1	2	yes	—
77	#15667941	—	2	1	no	—
78	#15667941	—	2	1	no	—
79	#15667941	—	2	1	no	—
80	#15667937	—	2	2	no	—
81	#15667936	—	2	2	no	—
82	#15667935	—	2	2	no	—
83	#15667924	—	2	2	no	—
84	#15667922	—	2	1	no	—
85	C₄×#10623494	—	2	1	no	—
86	C₄×#10623494	C₂³	1	1	yes	characteristic
87	C₈×Dic₈₉	—	2	1	no	—
88	C₈×Dic₈₉	—	2	2	no	—
89	C₈×Dic₈₉	D₄	1	1	yes	characteristic
90	C₈×Dic₈₉	D₄	1	1	yes	characteristic
91	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
92	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
93	#107×C₈₉	—	2	1	no	—
94	#107×C₈₉	—	2	1	no	—
95	#107×C₈₉	—	2	2	no	—
96	C₄⋊C₈×C₈₉	—	2	1	no	—
97	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
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	—	4	2	no	—
105	C₂×Dic₁₇₈	C₂×D₄	1	1	yes	characteristic
106	Q₈×D₈₉	—	8	4	no	—
107	C₂×C₄×D₈₉	—	4	2	no	—
108	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×D₈₉	#41	1	1	yes	characteristic
111	#13083557	—	4	2	no	—
112	C₂×#10623494	—	2	1	no	—
113	C₂×#10623494	—	2	2	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	Q₁₆	1	4	yes	—
118	C₄⋊Dic₈₉	—	2	1	no	—
119	C₄⋊Dic₈₉	—	2	2	no	—
120	C₄⋊Dic₈₉	—	2	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	#13083553	—	4	2	no	—
129	#13083551	—	2	1	no	—
130	#13083551	—	2	1	no	—
131	C₄×Dic₈₉	—	2	1	no	—
132	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
133	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
134	C₄×Dic₈₉	#41	1	1	yes	characteristic
135	C₂×Q₈×C₈₉	—	2	2	no	—
136	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₈₉	—	2	1	no	—
138	C₂×C₈×C₈₉	—	2	2	no	—
139	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
140	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
141	C₄⋊C₄×C₈₉	—	2	1	no	—
142	C₄⋊C₄×C₈₉	—	2	2	no	—
143	C₄⋊C₄×C₈₉	—	2	2	no	—
144	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
145	C₄²×C₈₉	—	2	1	no	—
146	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
147	C₂²×D₈₉	#121	1	1	yes	characteristic
148	Q₈×C₈₉	—	2	2	no	—
149	Q₈×C₈₉	—	8	2	no	—
150	C₂×C₄×C₈₉	—	2	1	no	—
151	C₂×C₄×C₈₉	—	2	1	no	—
152	C₂×C₄×C₈₉	—	4	2	no	—
153	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
156	C₂×Dic₈₉	—	2	1	no	—
157	C₂×Dic₈₉	—	2	1	no	—
158	C₂×Dic₈₉	—	4	2	no	—
159	C₂×Dic₈₉	#121	1	1	yes	characteristic
160	C₂×Dic₈₉	#121	1	1	yes	characteristic
161	C₂×Dic₈₉	#121	1	1	yes	characteristic
162	C₄×D₈₉	—	2	2	no	—
163	C₄×D₈₉	—	2	2	no	—
164	C₄×D₈₉	—	8	4	no	—
165	C₄×D₈₉	C₂×Q₁₆	1	2	yes	—
166	C₄×D₈₉	#137	1	2	yes	—
167	Dic₁₇₈	—	2	2	no	—
168	Dic₁₇₈	—	4	2	no	—
169	Dic₁₇₈	—	8	4	no	—
170	Dic₁₇₈	—	8	2	no	—
171	C₈×C₈₉	—	2	1	no	—
172	C₈×C₈₉	—	4	1	no	—
173	C₈×C₈₉	—	4	2	no	—
174	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
175	#10623494	—	2	1	no	—
176	#10623494	—	4	1	no	—
177	#10623494	—	4	2	no	—
178	#10623494	C₂×Q₁₆	1	2	yes	—
179	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
180	C₂×D₈₉	—	2	1	no	—
181	C₂×D₈₉	—	2	1	no	—
182	C₂×D₈₉	#462	1	2	yes	—
183	C₄×C₈₉	—	2	1	no	—
184	C₄×C₈₉	—	4	1	no	—
185	C₄×C₈₉	—	4	1	no	—
186	C₄×C₈₉	—	8	2	no	—
187	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
188	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
189	C₄×C₈₉	#578	1	2	yes	—
190	Dic₈₉	—	2	1	no	—
191	Dic₈₉	—	4	1	no	—
192	Dic₈₉	—	4	1	no	—
193	Dic₈₉	—	8	2	no	—
194	Dic₈₉	#462	1	2	yes	—
195	Dic₈₉	#472	1	2	yes	—
196	#19209	—	89	1	no	maximal, Sylow
197	C₂×C₈₉	C₂×#462	1	1	yes	characteristic
198	C₂×C₈₉	#3059	1	1	yes	characteristic
199	C₂×C₈₉	C₂×#472	1	1	yes	characteristic
200	D₈₉	—	2	2	no	—
201	#3230	—	89	2	no	—
202	C₂×#500	—	89	1	no	—
203	#2960	—	89	1	no	—
204	#1696	—	89	1	no	—
205	#1696	—	89	1	no	—
206	#1696	—	89	1	no	—
207	#1696	D₈₉	1	1	yes	characteristic, complemented
208	#1676	—	89	2	no	—
209	#1676	—	89	2	no	—
210	#1564	—	89	1	no	—
211	#1548	—	89	2	no	—
212	C₈₉	#19209	1	1	yes	characteristic, complemented, Sylow
213	#533	—	89	1	no	—
214	C₂×C₄⋊Q₈	—	89	2	no	—
215	#500	—	89	2	no	—
216	#500	—	89	1	no	—
217	#500	—	178	1	no	—
218	#500	—	178	1	no	—
219	#500	—	178	2	no	—
220	#500	C₂×D₈₉	1	1	yes	characteristic
221	#487	—	178	2	no	—
222	#484	—	178	2	no	—
223	#484	—	178	2	no	—
224	#477	—	89	2	no	—
225	#477	—	89	2	no	—
226	#477	—	89	2	no	—
227	#477	C₂×D₈₉	1	2	yes	—
228	#446	—	89	1	no	—
229	#446	—	178	1	no	—
230	#446	C₂×D₈₉	1	1	yes	characteristic
231	C₈×Q₈	—	89	1	no	—
232	C₈×Q₈	—	89	1	no	—
233	C₈×Q₈	—	178	1	no	—
234	#431	—	89	2	no	—
235	C₂×#107	—	178	1	no	—
236	C₂×#107	—	178	1	no	—
237	#424	—	178	1	no	—
238	#423	—	89	1	no	—
239	#405	—	178	1	no	—
240	C₂×C₄×C₈	—	89	1	no	—
241	#335	—	89	2	no	—
242	#334	—	89	1	no	—
243	#334	C₂×D₈₉	1	1	yes	characteristic
244	#333	—	89	2	no	—
245	#333	—	89	2	no	—
246	#333	—	89	2	no	—
247	#333	C₂×D₈₉	1	2	yes	—
248	C₂²×Q₈	—	178	2	no	—
249	C₂×M₁₆	—	178	1	no	—
250	C₂²×C₈	—	89	1	no	—
251	C₂²×C₈	—	89	1	no	—
252	C₄⋊Q₈	—	89	1	no	—
253	C₄⋊Q₈	—	89	1	no	—
254	C₄⋊Q₈	—	89	2	no	—
255	C₄⋊Q₈	—	178	2	no	—
256	C₄⋊Q₈	—	178	2	no	—
257	C₄⋊Q₈	—	178	2	no	—
258	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
259	#125	—	178	1	no	—
260	C₂²⋊Q₈	—	89	1	no	—
261	C₂²⋊Q₈	—	89	1	no	—
262	C₂²⋊Q₈	—	178	1	no	—
263	C₄×Q₈	—	89	1	no	—
264	C₄×Q₈	—	178	1	no	—
265	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
266	#117	—	178	1	no	—
267	C₂×C₄⋊C₄	—	178	2	no	—
268	C₂×C₄⋊C₄	—	178	2	no	—
269	C₂×C₄²	—	89	1	no	—
270	#107	—	2	1	no	—
271	#107	—	2	1	no	—
272	#107	—	2	2	no	—
273	#107	—	178	2	no	—
274	#107	—	178	2	no	—
275	#107	—	178	2	no	—
276	#107	—	178	2	no	—
277	#107	—	178	1	no	—
278	#107	—	178	1	no	—
279	#107	—	178	2	no	—
280	C₄⋊C₈	—	2	1	no	—
281	C₄⋊C₈	—	89	1	no	—
282	C₄⋊C₈	—	89	2	no	—
283	C₄⋊C₈	—	178	1	no	—
284	C₄⋊C₈	—	178	1	no	—
285	C₄⋊C₈	—	178	1	no	—
286	C₄⋊C₈	—	178	1	no	—
287	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
288	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
289	#103	—	2	2	no	—
290	#103	—	178	2	no	—
291	#103	—	178	2	no	—
292	#103	—	178	2	no	—
293	C₂²⋊C₈	—	178	2	no	—
294	#97	—	2	1	no	—
295	#97	—	178	1	no	—
296	C₄×C₈	—	89	1	no	—
297	C₄×C₈	—	89	1	no	—
298	C₄×C₈	—	89	1	no	—
299	C₄×C₈	—	178	1	no	—
300	C₄×C₈	—	178	2	no	—
301	C₄×C₈	—	178	1	no	—
302	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	—	89	1	no	—
305	C₂×Q₈	—	356	2	no	—
306	C₂×Q₈	—	178	1	no	—
307	C₂×Q₈	—	178	2	no	—
308	C₂×Q₈	—	712	4	no	—
309	C₂×Q₈	D₄×D₈₉	1	1	yes	characteristic
310	C₂²×C₄	—	89	1	no	—
311	C₂²×C₄	—	89	1	no	—
312	C₂²×C₄	—	89	1	no	—
313	C₂²×C₄	—	356	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	D₄×D₈₉	1	1	yes	characteristic
324	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	89	1	no	—
329	C₄⋊C₄	—	356	2	no	—
330	C₄⋊C₄	—	356	2	no	—
331	C₄⋊C₄	—	178	1	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	2	no	—
337	C₄⋊C₄	—	178	2	no	—
338	C₄⋊C₄	—	178	1	no	—
339	C₄⋊C₄	D₄×D₈₉	1	1	yes	characteristic
340	C₂²⋊C₄	—	178	1	no	—
341	C₂²⋊C₄	—	178	1	no	—
342	C₄²	—	2	1	no	—
343	C₄²	—	89	1	no	—
344	C₄²	—	89	1	no	—
345	C₄²	—	89	1	no	—
346	C₄²	—	178	1	no	—
347	C₄²	C₂³×D₈₉	1	1	yes	characteristic
348	C₂³	—	89	1	no	—
349	Q₈	—	2	2	no	—
350	Q₈	—	8	2	no	—
351	Q₈	—	356	2	no	—
352	Q₈	—	178	2	no	—
353	Q₈	—	712	2	no	—
354	Q₈	—	712	4	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	89	1	no	—
359	C₂×C₄	—	89	1	no	—
360	C₂×C₄	—	89	2	no	—
361	C₂×C₄	—	89	2	no	—
362	C₂×C₄	—	89	1	no	—
363	C₂×C₄	—	356	2	no	—
364	C₂×C₄	—	178	2	no	—
365	C₂×C₄	—	178	1	no	—
366	C₂×C₄	—	178	1	no	—
367	C₂×C₄	—	178	2	no	—
368	C₂×C₄	—	712	4	no	—
369	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
370	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
371	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
372	C₈	—	2	1	no	—
373	C₈	—	4	1	no	—
374	C₈	—	4	2	no	—
375	C₈	—	89	2	no	—
376	C₈	—	356	1	no	—
377	C₈	—	356	2	no	—
378	C₈	—	178	1	no	—
379	C₈	Q₁₆×D₈₉	1	2	yes	—
380	C₂²	—	89	2	no	—
381	C₂²	—	178	1	no	—
382	C₂²	—	178	1	no	—
383	C₂²	#121×D₈₉	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	4	1	no	—
386	C₄	—	4	1	no	—
387	C₄	—	8	2	no	—
388	C₄	—	89	2	no	—
389	C₄	—	89	2	no	—
390	C₄	—	356	1	no	—
391	C₄	—	356	1	no	—
392	C₄	—	178	1	no	—
393	C₄	—	712	2	no	—
394	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
395	C₄	#137×D₈₉	1	1	yes	characteristic
396	C₄	#19152414	1	2	yes	—
397	C₂	—	178	2	no	—
398	C₂	#462×D₈₉	1	1	yes	characteristic
399	C₂	#26029905	1	1	yes	characteristic
400	C₂	#472×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₈₉
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1696	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂²×D₈₉	#121
hypercenter	#1696	D₈₉
hypocenter	C₈₉	#19209
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1696

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. #26019777	C₂	semidirect
7. #26019655	C₂	semidirect
8. #26019654	C₂	semidirect
11. C₈₉×#1696	C₂	semidirect
207. #1696	D₈₉	semidirect
212. C₈₉	#19209	semidirect

Maximal subgroups

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

subgroup	index
#26031402	2
#500×D₈₉	2
#26029183	2
#26019781	2
#26019780	2
#26019777	2
#26019655	2
#26019654	2
#26018997	2
#26018931	2
C₈₉×#1696	2
#19209	89