Subgroups — #36537676

Order 22784.

582 conjugacy classes of subgroups.

The subgroup lattice has 410 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	#26029181	C₂	1	1	yes	maximal, characteristic
4	#26019779	C₂	1	2	yes	maximal, complemented
5	#26019778	C₂	1	2	yes	maximal, complemented
6	#26019657	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019656	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019653	C₂	1	1	yes	maximal, characteristic, complemented
9	#26018999	C₂	1	1	yes	maximal, characteristic
10	#26018931	C₂	1	2	yes	maximal
11	C₈₉×#1676	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	#19151729	C₂²	1	1	yes	characteristic
26	#19151727	—	2	1	no	—
27	#19151725	C₂²	1	1	yes	characteristic
28	C₈×Dic₁₇₈	—	2	1	no	—
29	C₈×Dic₁₇₈	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	#19151552	—	2	1	no	—
36	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
37	#19151351	C₂²	1	1	yes	characteristic
38	#19151348	C₂²	1	2	yes	—
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	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
45	#335×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	#15667923	—	2	1	no	—
85	#15667922	—	2	1	no	—
86	C₄×#10623494	C₂³	1	1	yes	characteristic
87	C₈×Dic₈₉	—	2	2	no	—
88	C₈×Dic₈₉	—	2	1	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	C₄×C₈×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	1	no	—
149	Q₈×C₈₉	—	2	1	no	—
150	Q₈×C₈₉	—	8	2	no	—
151	C₂×C₄×C₈₉	—	2	1	no	—
152	C₂×C₄×C₈₉	—	2	1	no	—
153	C₂×C₄×C₈₉	—	4	2	no	—
154	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
157	C₂×Dic₈₉	—	2	1	no	—
158	C₂×Dic₈₉	—	2	1	no	—
159	C₂×Dic₈₉	—	4	2	no	—
160	C₂×Dic₈₉	#121	1	1	yes	characteristic
161	C₂×Dic₈₉	#121	1	1	yes	characteristic
162	C₂×Dic₈₉	#121	1	1	yes	characteristic
163	C₄×D₈₉	—	2	1	no	—
164	C₄×D₈₉	—	2	1	no	—
165	C₄×D₈₉	—	2	2	no	—
166	C₄×D₈₉	—	8	4	no	—
167	C₄×D₈₉	C₂×Q₁₆	1	2	yes	—
168	C₄×D₈₉	#137	1	2	yes	—
169	Dic₁₇₈	—	2	1	no	—
170	Dic₁₇₈	—	2	1	no	—
171	Dic₁₇₈	—	4	2	no	—
172	Dic₁₇₈	—	8	4	no	—
173	Dic₁₇₈	—	8	2	no	—
174	C₈×C₈₉	—	2	1	no	—
175	C₈×C₈₉	—	4	1	no	—
176	C₈×C₈₉	—	4	2	no	—
177	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
178	#10623494	—	2	1	no	—
179	#10623494	—	4	1	no	—
180	#10623494	—	4	2	no	—
181	#10623494	C₂×Q₁₆	1	2	yes	—
182	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
183	C₂×D₈₉	—	2	1	no	—
184	C₂×D₈₉	—	2	1	no	—
185	C₂×D₈₉	#462	1	2	yes	—
186	C₄×C₈₉	—	2	1	no	—
187	C₄×C₈₉	—	4	1	no	—
188	C₄×C₈₉	—	4	1	no	—
189	C₄×C₈₉	—	8	2	no	—
190	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
191	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
192	C₄×C₈₉	#578	1	1	yes	characteristic
193	C₄×C₈₉	#578	1	1	yes	characteristic
194	Dic₈₉	—	2	1	no	—
195	Dic₈₉	—	4	1	no	—
196	Dic₈₉	—	4	1	no	—
197	Dic₈₉	—	8	2	no	—
198	Dic₈₉	#462	1	2	yes	—
199	Dic₈₉	#472	1	2	yes	—
200	#19209	—	89	1	no	maximal, Sylow
201	C₂×C₈₉	C₂×#462	1	1	yes	characteristic
202	C₂×C₈₉	#3059	1	1	yes	characteristic
203	C₂×C₈₉	C₂×#472	1	1	yes	characteristic
204	D₈₉	—	2	2	no	—
205	#3230	—	89	2	no	—
206	C₂×#500	—	89	1	no	—
207	#2960	—	89	1	no	—
208	#1696	—	89	2	no	—
209	#1696	—	89	2	no	—
210	#1676	—	89	1	no	—
211	#1676	—	89	1	no	—
212	#1676	—	89	1	no	—
213	#1676	D₈₉	1	1	yes	characteristic, complemented
214	#1564	—	89	1	no	—
215	#1548	—	89	2	no	—
216	C₈₉	#19209	1	1	yes	characteristic, complemented, Sylow
217	#533	—	89	1	no	—
218	C₂×C₄⋊Q₈	—	89	2	no	—
219	#500	—	89	2	no	—
220	#500	—	89	1	no	—
221	#500	—	178	1	no	—
222	#500	—	178	1	no	—
223	#500	—	178	2	no	—
224	#500	C₂×D₈₉	1	1	yes	characteristic
225	#487	—	178	2	no	—
226	#484	—	178	2	no	—
227	#484	—	178	2	no	—
228	#477	—	89	2	no	—
229	#477	—	89	2	no	—
230	#477	—	89	2	no	—
231	#477	C₂×D₈₉	1	2	yes	—
232	#446	—	89	1	no	—
233	#446	—	89	1	no	—
234	#446	—	178	1	no	—
235	C₈×Q₈	—	89	1	no	—
236	C₈×Q₈	—	178	1	no	—
237	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
238	#431	—	89	2	no	—
239	C₂×#107	—	178	1	no	—
240	C₂×#107	—	178	1	no	—
241	#424	—	178	1	no	—
242	#423	—	89	1	no	—
243	#405	—	178	1	no	—
244	C₂×C₄×C₈	—	89	1	no	—
245	#335	—	89	1	no	—
246	#335	C₂×D₈₉	1	1	yes	characteristic
247	#334	—	89	2	no	—
248	#333	—	89	2	no	—
249	#333	—	89	2	no	—
250	#333	—	89	2	no	—
251	#333	C₂×D₈₉	1	2	yes	—
252	C₂²×Q₈	—	178	2	no	—
253	C₂×M₁₆	—	178	1	no	—
254	C₂²×C₈	—	89	1	no	—
255	C₂²×C₈	—	89	1	no	—
256	C₄⋊Q₈	—	89	1	no	—
257	C₄⋊Q₈	—	89	1	no	—
258	C₄⋊Q₈	—	89	2	no	—
259	C₄⋊Q₈	—	178	2	no	—
260	C₄⋊Q₈	—	178	2	no	—
261	C₄⋊Q₈	—	178	2	no	—
262	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
263	#125	—	178	1	no	—
264	C₂²⋊Q₈	—	89	1	no	—
265	C₂²⋊Q₈	—	89	1	no	—
266	C₂²⋊Q₈	—	178	1	no	—
267	C₄×Q₈	—	89	1	no	—
268	C₄×Q₈	—	178	1	no	—
269	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
270	#117	—	178	1	no	—
271	C₂×C₄⋊C₄	—	178	2	no	—
272	C₂×C₄⋊C₄	—	178	2	no	—
273	C₂×C₄²	—	89	1	no	—
274	#107	—	2	1	no	—
275	#107	—	2	1	no	—
276	#107	—	2	2	no	—
277	#107	—	178	2	no	—
278	#107	—	178	2	no	—
279	#107	—	178	2	no	—
280	#107	—	178	2	no	—
281	#107	—	178	1	no	—
282	#107	—	178	1	no	—
283	#107	—	178	2	no	—
284	C₄⋊C₈	—	2	1	no	—
285	C₄⋊C₈	—	89	1	no	—
286	C₄⋊C₈	—	89	2	no	—
287	C₄⋊C₈	—	178	1	no	—
288	C₄⋊C₈	—	178	1	no	—
289	C₄⋊C₈	—	178	1	no	—
290	C₄⋊C₈	—	178	1	no	—
291	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
292	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
293	#103	—	2	2	no	—
294	#103	—	178	2	no	—
295	#103	—	178	2	no	—
296	#103	—	178	2	no	—
297	C₂²⋊C₈	—	178	2	no	—
298	#97	—	178	1	no	—
299	#97	—	178	1	no	—
300	C₄×C₈	—	2	1	no	—
301	C₄×C₈	—	89	1	no	—
302	C₄×C₈	—	89	1	no	—
303	C₄×C₈	—	89	1	no	—
304	C₄×C₈	—	178	2	no	—
305	C₄×C₈	—	178	1	no	—
306	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
307	C₂×Q₈	—	2	2	no	—
308	C₂×Q₈	—	89	1	no	—
309	C₂×Q₈	—	356	2	no	—
310	C₂×Q₈	—	178	1	no	—
311	C₂×Q₈	—	178	2	no	—
312	C₂×Q₈	—	712	4	no	—
313	C₂×Q₈	D₄×D₈₉	1	1	yes	characteristic
314	C₂²×C₄	—	89	1	no	—
315	C₂²×C₄	—	89	1	no	—
316	C₂²×C₄	—	89	1	no	—
317	C₂²×C₄	—	356	2	no	—
318	M₁₆	—	356	2	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	89	1	no	—
322	C₂×C₈	—	89	4	no	—
323	C₂×C₈	—	89	1	no	—
324	C₂×C₈	—	178	2	no	—
325	C₂×C₈	—	178	1	no	—
326	C₂×C₈	—	178	2	no	—
327	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
328	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	89	1	no	—
333	C₄⋊C₄	—	356	2	no	—
334	C₄⋊C₄	—	356	2	no	—
335	C₄⋊C₄	—	178	1	no	—
336	C₄⋊C₄	—	178	1	no	—
337	C₄⋊C₄	—	178	1	no	—
338	C₄⋊C₄	—	178	1	no	—
339	C₄⋊C₄	—	178	1	no	—
340	C₄⋊C₄	—	178	2	no	—
341	C₄⋊C₄	—	178	2	no	—
342	C₄⋊C₄	—	178	1	no	—
343	C₄⋊C₄	D₄×D₈₉	1	1	yes	characteristic
344	C₂²⋊C₄	—	178	1	no	—
345	C₂²⋊C₄	—	178	1	no	—
346	C₄²	—	2	1	no	—
347	C₄²	—	89	1	no	—
348	C₄²	—	89	1	no	—
349	C₄²	—	89	1	no	—
350	C₄²	—	178	1	no	—
351	C₄²	C₂³×D₈₉	1	1	yes	characteristic
352	C₂³	—	89	1	no	—
353	Q₈	—	2	1	no	—
354	Q₈	—	2	1	no	—
355	Q₈	—	8	2	no	—
356	Q₈	—	356	2	no	—
357	Q₈	—	178	1	no	—
358	Q₈	—	178	1	no	—
359	Q₈	—	712	2	no	—
360	Q₈	—	712	4	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	89	1	no	—
365	C₂×C₄	—	89	1	no	—
366	C₂×C₄	—	89	2	no	—
367	C₂×C₄	—	89	2	no	—
368	C₂×C₄	—	89	1	no	—
369	C₂×C₄	—	356	2	no	—
370	C₂×C₄	—	178	2	no	—
371	C₂×C₄	—	178	1	no	—
372	C₂×C₄	—	178	1	no	—
373	C₂×C₄	—	178	1	no	—
374	C₂×C₄	—	178	1	no	—
375	C₂×C₄	—	712	4	no	—
376	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
377	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
378	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
379	C₈	—	2	1	no	—
380	C₈	—	4	1	no	—
381	C₈	—	4	2	no	—
382	C₈	—	89	2	no	—
383	C₈	—	356	1	no	—
384	C₈	—	356	2	no	—
385	C₈	—	178	1	no	—
386	C₈	Q₁₆×D₈₉	1	2	yes	—
387	C₂²	—	89	2	no	—
388	C₂²	—	178	1	no	—
389	C₂²	—	178	1	no	—
390	C₂²	#121×D₈₉	1	1	yes	characteristic
391	C₄	—	2	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	4	1	no	—
394	C₄	—	8	2	no	—
395	C₄	—	89	2	no	—
396	C₄	—	89	2	no	—
397	C₄	—	356	1	no	—
398	C₄	—	356	1	no	—
399	C₄	—	178	1	no	—
400	C₄	—	712	2	no	—
401	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
402	C₄	#137×D₈₉	1	1	yes	characteristic
403	C₄	#19152411	1	1	yes	characteristic
404	C₄	#19152412	1	1	yes	characteristic
405	C₂	—	178	2	no	—
406	C₂	#462×D₈₉	1	1	yes	characteristic
407	C₂	#26029902	1	1	yes	characteristic
408	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×#1676	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂²×D₈₉	#121
hypercenter	#1676	D₈₉
hypocenter	C₈₉	#19209
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1676

Complements

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

N	H ≅ G/N	type
4. #26019779	C₂	semidirect
5. #26019778	C₂	semidirect
6. #26019657	C₂	semidirect
7. #26019656	C₂	semidirect
8. #26019653	C₂	semidirect
11. C₈₉×#1676	C₂	semidirect
213. #1676	D₈₉	semidirect
216. 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
#26029181	2
#26019779	2
#26019778	2
#26019657	2
#26019656	2
#26019653	2
#26018999	2
#26018931	2
C₈₉×#1676	2
#19209	89