Subgroups — #36537691

Order 22784.

570 conjugacy classes of subgroups.

The subgroup lattice has 412 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	#26029181	C₂	1	1	yes	maximal, characteristic
4	#26019757	C₂	1	2	yes	maximal, complemented
5	#26019756	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	#26019643	C₂	1	1	yes	maximal, characteristic, complemented
9	#335×D₈₉	C₂	1	1	yes	maximal, characteristic
10	#26018941	C₂	1	2	yes	maximal
11	C₈₉×#1674	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	#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	#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	#19151552	—	2	1	no	—
36	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
37	#19151351	C₂²	1	1	yes	characteristic
38	#19151350	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	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
45	#335×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	#15667923	—	2	1	no	—
86	#15667922	—	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	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	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	1	no	—
151	Q₈×C₈₉	—	2	1	no	—
152	Q₈×C₈₉	—	4	2	no	—
153	Q₈×C₈₉	—	4	2	no	—
154	C₂×C₄×C₈₉	—	2	1	no	—
155	C₂×C₄×C₈₉	—	2	1	no	—
156	C₂×C₄×C₈₉	—	4	2	no	—
157	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
160	C₂×Dic₈₉	—	2	1	no	—
161	C₂×Dic₈₉	—	2	1	no	—
162	C₂×Dic₈₉	—	4	2	no	—
163	C₂×Dic₈₉	#121	1	1	yes	characteristic
164	C₂×Dic₈₉	#121	1	1	yes	characteristic
165	C₂×Dic₈₉	#121	1	1	yes	characteristic
166	C₄×D₈₉	—	2	1	no	—
167	C₄×D₈₉	—	2	1	no	—
168	C₄×D₈₉	—	4	1	no	—
169	C₄×D₈₉	C₂×Q₁₆	1	2	yes	—
170	C₄×D₈₉	#137	1	2	yes	—
171	Dic₁₇₈	—	2	1	no	—
172	Dic₁₇₈	—	2	1	no	—
173	Dic₁₇₈	—	4	2	no	—
174	Dic₁₇₈	—	8	2	no	—
175	C₈×C₈₉	—	2	1	no	—
176	C₈×C₈₉	—	4	1	no	—
177	C₈×C₈₉	—	4	2	no	—
178	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
179	#10623494	—	2	1	no	—
180	#10623494	—	4	1	no	—
181	#10623494	—	4	2	no	—
182	#10623494	C₂×Q₁₆	1	2	yes	—
183	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
184	C₂×D₈₉	—	2	1	no	—
185	C₂×D₈₉	—	2	1	no	—
186	C₂×D₈₉	#462	1	2	yes	—
187	C₄×C₈₉	—	2	1	no	—
188	C₄×C₈₉	—	4	1	no	—
189	C₄×C₈₉	—	4	1	no	—
190	C₄×C₈₉	—	8	2	no	—
191	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
192	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
193	C₄×C₈₉	#576	1	1	yes	characteristic
194	C₄×C₈₉	#578	1	1	yes	characteristic
195	Dic₈₉	—	2	1	no	—
196	Dic₈₉	—	2	1	no	—
197	Dic₈₉	—	4	1	no	—
198	Dic₈₉	—	4	1	no	—
199	Dic₈₉	—	8	2	no	—
200	Dic₈₉	#462	1	2	yes	—
201	#19211	—	89	1	no	maximal, Sylow
202	C₂×C₈₉	C₂×#462	1	1	yes	characteristic
203	C₂×C₈₉	#3061	1	1	yes	characteristic
204	C₂×C₈₉	#3067	1	1	yes	characteristic
205	D₈₉	—	2	2	no	—
206	#3249	—	89	2	no	—
207	#3160	—	89	1	no	—
208	#2960	—	89	1	no	—
209	#1692	—	89	2	no	—
210	#1692	—	89	2	no	—
211	#1676	—	89	1	no	—
212	#1676	—	89	1	no	—
213	#1674	—	89	1	no	—
214	#1674	D₈₉	1	1	yes	characteristic, complemented
215	C₂×#335	—	89	1	no	—
216	#1550	—	89	2	no	—
217	C₈₉	#19211	1	1	yes	characteristic, complemented, Sylow
218	#533	—	89	1	no	—
219	#533	—	89	2	no	—
220	#500	—	89	1	no	—
221	#500	C₂×D₈₉	1	1	yes	characteristic
222	#499	—	178	1	no	—
223	#499	—	178	1	no	—
224	#498	—	89	2	no	—
225	#477	—	89	2	no	—
226	#477	—	89	2	no	—
227	#477	—	178	2	no	—
228	#467	—	178	2	no	—
229	#467	—	178	2	no	—
230	#462	—	89	2	no	—
231	#462	—	178	2	no	—
232	#462	C₂×D₈₉	1	2	yes	—
233	#446	—	89	1	no	—
234	#446	—	89	1	no	—
235	#446	—	178	1	no	—
236	C₈×Q₈	—	89	1	no	—
237	C₈×Q₈	—	178	1	no	—
238	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
239	#427	—	178	1	no	—
240	#427	—	178	1	no	—
241	#424	—	178	1	no	—
242	#423	—	89	1	no	—
243	C₂×C₄⋊C₈	—	89	2	no	—
244	#405	—	178	1	no	—
245	C₂×C₄×C₈	—	89	1	no	—
246	#335	—	89	2	no	—
247	#335	—	89	1	no	—
248	#335	C₂×D₈₉	1	1	yes	characteristic
249	#333	—	89	2	no	—
250	#333	—	89	2	no	—
251	#332	—	89	2	no	—
252	#332	C₂×D₈₉	1	2	yes	—
253	C₂×Q₁₆	—	2	2	no	—
254	C₂×Q₁₆	—	178	2	no	—
255	C₂×M₁₆	—	178	1	no	—
256	C₂²×C₈	—	89	1	no	—
257	C₂²×C₈	—	89	1	no	—
258	C₂²×C₈	—	178	2	no	—
259	C₄⋊Q₈	—	89	1	no	—
260	C₄⋊Q₈	—	89	1	no	—
261	C₄⋊Q₈	—	89	2	no	—
262	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
263	#125	—	178	2	no	—
264	#125	—	178	1	no	—
265	C₂²⋊Q₈	—	89	1	no	—
266	C₂²⋊Q₈	—	89	1	no	—
267	C₂²⋊Q₈	—	178	2	no	—
268	C₂²⋊Q₈	—	178	2	no	—
269	C₂²⋊Q₈	—	178	1	no	—
270	C₄×Q₈	—	89	1	no	—
271	C₄×Q₈	—	178	1	no	—
272	C₄×Q₈	—	178	2	no	—
273	C₄×Q₈	—	178	2	no	—
274	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
275	#117	—	178	1	no	—
276	#117	—	178	2	no	—
277	C₂×C₄²	—	89	1	no	—
278	#107	—	2	1	no	—
279	#107	—	2	1	no	—
280	#107	—	178	2	no	—
281	#107	—	178	1	no	—
282	#107	—	178	1	no	—
283	#107	—	178	2	no	—
284	#106	—	178	2	no	—
285	#106	—	178	2	no	—
286	C₄⋊C₈	—	2	1	no	—
287	C₄⋊C₈	—	89	1	no	—
288	C₄⋊C₈	—	89	2	no	—
289	C₄⋊C₈	—	178	1	no	—
290	C₄⋊C₈	—	178	2	no	—
291	C₄⋊C₈	—	178	1	no	—
292	C₄⋊C₈	—	178	1	no	—
293	C₄⋊C₈	—	178	1	no	—
294	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
295	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
296	#103	—	2	2	no	—
297	#103	—	178	2	no	—
298	#103	—	178	2	no	—
299	#103	—	178	2	no	—
300	#97	—	178	1	no	—
301	#97	—	178	1	no	—
302	C₄×C₈	—	2	1	no	—
303	C₄×C₈	—	89	1	no	—
304	C₄×C₈	—	89	1	no	—
305	C₄×C₈	—	89	1	no	—
306	C₄×C₈	—	178	1	no	—
307	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	89	1	no	—
310	C₂×Q₈	—	178	1	no	—
311	C₂×Q₈	—	178	2	no	—
312	C₂×Q₈	D₄×D₈₉	1	1	yes	characteristic
313	C₂²×C₄	—	89	1	no	—
314	C₂²×C₄	—	89	1	no	—
315	C₂²×C₄	—	89	1	no	—
316	Q₁₆	—	8	2	no	—
317	Q₁₆	—	712	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	—	712	2	no	—
328	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
329	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	89	1	no	—
334	C₄⋊C₄	—	356	2	no	—
335	C₄⋊C₄	—	356	2	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	1	no	—
341	C₄⋊C₄	—	178	2	no	—
342	C₄⋊C₄	—	178	2	no	—
343	C₄⋊C₄	—	178	1	no	—
344	C₄⋊C₄	D₄×D₈₉	1	1	yes	characteristic
345	C₂²⋊C₄	—	356	2	no	—
346	C₂²⋊C₄	—	178	1	no	—
347	C₂²⋊C₄	—	178	1	no	—
348	C₄²	—	2	1	no	—
349	C₄²	—	89	1	no	—
350	C₄²	—	89	1	no	—
351	C₄²	—	89	1	no	—
352	C₄²	—	356	2	no	—
353	C₄²	—	178	1	no	—
354	C₄²	C₂³×D₈₉	1	1	yes	characteristic
355	C₂³	—	89	1	no	—
356	Q₈	—	2	1	no	—
357	Q₈	—	2	1	no	—
358	Q₈	—	4	2	no	—
359	Q₈	—	4	2	no	—
360	Q₈	—	356	2	no	—
361	Q₈	—	178	1	no	—
362	Q₈	—	178	1	no	—
363	Q₈	—	712	2	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	89	1	no	—
368	C₂×C₄	—	89	2	no	—
369	C₂×C₄	—	89	2	no	—
370	C₂×C₄	—	89	1	no	—
371	C₂×C₄	—	89	1	no	—
372	C₂×C₄	—	356	1	no	—
373	C₂×C₄	—	356	2	no	—
374	C₂×C₄	—	178	1	no	—
375	C₂×C₄	—	178	1	no	—
376	C₂×C₄	—	178	1	no	—
377	C₂×C₄	—	178	1	no	—
378	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
379	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
380	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
381	C₈	—	2	1	no	—
382	C₈	—	4	1	no	—
383	C₈	—	4	2	no	—
384	C₈	—	89	2	no	—
385	C₈	—	356	1	no	—
386	C₈	—	356	2	no	—
387	C₈	—	178	1	no	—
388	C₈	Q₁₆×D₈₉	1	2	yes	—
389	C₂²	—	89	2	no	—
390	C₂²	—	178	1	no	—
391	C₂²	—	178	1	no	—
392	C₂²	#121×D₈₉	1	1	yes	characteristic
393	C₄	—	2	1	no	—
394	C₄	—	4	1	no	—
395	C₄	—	4	1	no	—
396	C₄	—	8	2	no	—
397	C₄	—	89	2	no	—
398	C₄	—	356	1	no	—
399	C₄	—	356	1	no	—
400	C₄	—	178	1	no	—
401	C₄	—	178	1	no	—
402	C₄	—	712	2	no	—
403	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
404	C₄	#137×D₈₉	1	1	yes	characteristic
405	C₄	#19152399	1	1	yes	characteristic
406	C₄	#19152412	1	1	yes	characteristic
407	C₂	—	178	2	no	—
408	C₂	#462×D₈₉	1	1	yes	characteristic
409	C₂	#26029917	1	1	yes	characteristic
410	C₂	#26029974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₈₉
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1674	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂²×D₈₉	#121
hypercenter	#1674	D₈₉
hypocenter	C₈₉	#19211
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1674

Complements

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

N	H ≅ G/N	type
4. #26019757	C₂	semidirect
5. #26019756	C₂	semidirect
6. #26019657	C₂	semidirect
7. #26019656	C₂	semidirect
8. #26019643	C₂	semidirect
11. C₈₉×#1674	C₂	semidirect
116. C₈×D₈₉	Q₁₆	semidirect
214. #1674	D₈₉	semidirect
217. C₈₉	#19211	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
#26029181	2
#26019757	2
#26019756	2
#26019657	2
#26019656	2
#26019643	2
#335×D₈₉	2
#26018941	2
C₈₉×#1674	2
#19211	89