Subgroups — #36537830

Order 22784.

584 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26031587	C₂	1	2	yes	maximal
2	#26030839	C₂	1	1	yes	maximal, characteristic, complemented
3	#26029229	C₂	1	1	yes	maximal, characteristic
4	#26019808	C₂	1	2	yes	maximal, complemented
5	#26019802	C₂	1	2	yes	maximal, complemented
6	#26019638	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019622	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019618	C₂	1	1	yes	maximal, characteristic, complemented
9	#335×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#26018951	C₂	1	2	yes	maximal, complemented
11	C₈₉×#1674	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152153	C₂²	1	2	yes	—
13	#19152135	C₂²	1	1	yes	characteristic
14	#19151978	C₂²	1	1	yes	characteristic
15	#19151950	C₂²	1	2	yes	—
16	#19151907	—	2	2	no	—
17	#19151883	C₂²	1	2	yes	—
18	#19151879	C₂²	1	2	yes	—
19	#19151840	—	2	1	no	—
20	#19151840	—	2	1	no	—
21	#19151836	—	2	2	no	—
22	#19151830	—	2	2	no	—
23	#19151822	—	2	2	no	—
24	#19151800	C₂²	1	2	yes	—
25	#19151729	C₂²	1	1	yes	characteristic
26	#19151692	—	2	1	no	—
27	#19151692	—	2	1	no	—
28	#19151675	C₂²	1	1	yes	characteristic
29	#19151668	—	2	1	no	—
30	C₈×D₃₅₆	—	2	1	no	—
31	C₈×D₃₅₆	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	#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	#19151345	C₂²	1	2	yes	—
40	#19151344	C₂²	1	2	yes	—
41	#19151343	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×Dic₃₅₆	—	2	1	no	—
48	C₂×Dic₃₅₆	—	2	1	no	—
49	C₂×#13083561	—	2	2	no	—
50	C₂×#13083557	—	2	1	no	—
51	C₂×C₈×D₈₉	—	2	2	no	—
52	C₂×C₈×D₈₉	—	2	1	no	—
53	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
54	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
55	#15668012	C₂³	1	2	yes	—
56	#15668007	—	2	1	no	—
57	#15668003	—	2	2	no	—
58	#15668003	D₄	1	1	yes	characteristic
59	#15668001	D₄	1	1	yes	characteristic
60	#15667997	—	2	1	no	—
61	#15667992	—	2	2	no	—
62	#15667988	—	2	2	no	—
63	#15667988	D₄	1	1	yes	characteristic
64	#15667987	D₄	1	1	yes	characteristic
65	Q₈×Dic₈₉	—	2	2	no	—
66	#15667977	—	2	2	no	—
67	C₄×D₃₅₆	—	2	1	no	—
68	C₄×D₃₅₆	C₂³	1	1	yes	characteristic
69	#15667970	—	2	2	no	—
70	#15667969	—	2	1	no	—
71	C₄²×D₈₉	C₂³	1	1	yes	characteristic
72	#15667946	—	2	2	no	—
73	#15667943	—	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	#15667936	—	2	2	no	—
79	#15667936	—	2	2	no	—
80	#15667935	—	2	2	no	—
81	#15667935	—	2	2	no	—
82	#15667934	—	2	2	no	—
83	#15667924	—	2	1	no	—
84	#15667924	—	2	1	no	—
85	#15667924	—	2	1	no	—
86	#15667923	—	2	1	no	—
87	#15667922	—	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	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	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	Dic₃₅₆	—	4	2	no	—
110	Dic₃₅₆	—	4	2	no	—
111	#13083561	—	8	2	no	—
112	#13083557	—	4	2	no	—
113	C₂×#10623494	—	2	2	no	—
114	C₂×#10623494	—	2	1	no	—
115	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
116	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
117	C₈×D₈₉	—	2	2	no	—
118	C₈×D₈₉	—	4	1	no	—
119	C₈×D₈₉	—	4	2	no	—
120	C₈×D₈₉	—	8	2	no	—
121	C₄⋊Dic₈₉	—	2	1	no	—
122	C₄⋊Dic₈₉	—	2	2	no	—
123	C₄⋊Dic₈₉	—	2	2	no	—
124	C₄⋊Dic₈₉	—	4	2	no	—
125	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
126	#13083553	—	2	1	no	—
127	#13083553	—	2	1	no	—
128	#13083553	—	4	2	no	—
129	#13083551	—	2	1	no	—
130	#13083551	—	2	1	no	—
131	#13083551	—	2	1	no	—
132	#13083551	—	2	1	no	—
133	#13083551	—	2	1	no	—
134	#13083551	—	4	2	no	—
135	C₄×Dic₈₉	—	4	2	no	—
136	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
137	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
138	C₄×Dic₈₉	#41	1	1	yes	characteristic
139	C₂×Q₈×C₈₉	—	2	2	no	—
140	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
141	Q₁₆×C₈₉	—	8	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	1	no	—
155	Q₈×C₈₉	—	2	1	no	—
156	Q₈×C₈₉	—	4	2	no	—
157	Q₈×C₈₉	—	4	2	no	—
158	C₂×C₄×C₈₉	—	2	1	no	—
159	C₂×C₄×C₈₉	—	2	1	no	—
160	C₂×C₄×C₈₉	—	4	2	no	—
161	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
164	C₂×Dic₈₉	—	2	1	no	—
165	C₂×Dic₈₉	—	4	2	no	—
166	C₂×Dic₈₉	#121	1	1	yes	characteristic
167	C₂×Dic₈₉	#121	1	1	yes	characteristic
168	C₂×Dic₈₉	#121	1	1	yes	characteristic
169	D₃₅₆	—	2	1	no	—
170	D₃₅₆	—	2	1	no	—
171	D₃₅₆	—	4	2	no	—
172	C₄×D₈₉	—	2	2	no	—
173	C₄×D₈₉	—	2	1	no	—
174	C₄×D₈₉	—	2	1	no	—
175	C₄×D₈₉	—	2	1	no	—
176	C₄×D₈₉	—	2	1	no	—
177	C₄×D₈₉	—	4	1	no	—
178	C₄×D₈₉	—	4	1	no	—
179	Dic₁₇₈	—	4	2	no	—
180	Dic₁₇₈	—	4	2	no	—
181	C₈×C₈₉	—	2	1	no	—
182	C₈×C₈₉	—	4	1	no	—
183	C₈×C₈₉	—	4	2	no	—
184	C₈×C₈₉	#135	1	2	yes	—
185	#10623494	—	2	1	no	—
186	#10623494	—	2	1	no	—
187	#10623494	—	4	1	no	—
188	#10623494	—	4	2	no	—
189	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
190	C₂×D₈₉	—	2	1	no	—
191	C₂×D₈₉	—	2	1	no	—
192	C₂×D₈₉	—	4	1	no	—
193	C₂×D₈₉	—	4	1	no	—
194	C₂×D₈₉	—	4	1	no	—
195	C₂×D₈₉	#473	1	2	yes	—
196	C₄×C₈₉	—	2	1	no	—
197	C₄×C₈₉	—	4	1	no	—
198	C₄×C₈₉	—	4	1	no	—
199	C₄×C₈₉	—	8	2	no	—
200	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
201	C₄×C₈₉	#575	1	1	yes	characteristic
202	C₄×C₈₉	#576	1	1	yes	characteristic
203	C₄×C₈₉	#578	1	1	yes	characteristic
204	Dic₈₉	—	2	1	no	—
205	Dic₈₉	—	2	1	no	—
206	Dic₈₉	—	2	1	no	—
207	Dic₈₉	—	4	1	no	—
208	Dic₈₉	—	8	2	no	—
209	#19223	—	89	1	no	maximal, Sylow
210	C₂×C₈₉	#3035	1	1	yes	characteristic
211	C₂×C₈₉	#3061	1	1	yes	characteristic
212	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
213	D₈₉	—	2	2	no	—
214	D₈₉	—	8	1	no	—
215	#3248	—	89	2	no	—
216	#3170	—	89	1	no	—
217	#2966	—	89	1	no	—
218	#1700	—	89	2	no	—
219	#1699	—	89	2	no	—
220	#1674	D₈₉	1	1	yes	characteristic, complemented
221	#1673	—	89	1	no	—
222	#1670	—	89	1	no	—
223	#1669	—	89	1	no	—
224	C₂×#335	—	89	1	no	—
225	#1552	—	89	2	no	—
226	C₈₉	#19223	1	1	yes	characteristic, complemented, Sylow
227	#533	—	89	2	no	—
228	#529	—	89	1	no	—
229	#500	C₂×D₈₉	1	1	yes	characteristic
230	#494	—	89	1	no	—
231	#487	—	89	2	no	—
232	#479	—	178	2	no	—
233	#475	—	89	2	no	—
234	#474	—	89	2	no	—
235	#467	—	178	1	no	—
236	#467	—	178	1	no	—
237	#467	—	178	2	no	—
238	#465	—	178	2	no	—
239	#464	—	178	2	no	—
240	#462	C₂×D₈₉	1	2	yes	—
241	#461	—	89	2	no	—
242	#446	—	89	1	no	—
243	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
244	C₄×Q₁₆	—	178	1	no	—
245	C₄×Q₁₆	—	178	1	no	—
246	#436	—	89	1	no	—
247	#435	—	178	1	no	—
248	C₈×D₄	—	89	1	no	—
249	C₈×D₄	—	178	1	no	—
250	#433	—	178	1	no	—
251	#432	—	89	1	no	—
252	C₂×C₄⋊C₈	—	89	2	no	—
253	#405	—	178	1	no	—
254	C₂×C₄×C₈	—	89	1	no	—
255	#335	—	89	2	no	—
256	#335	—	89	1	no	—
257	#335	C₂×D₈₉	1	1	yes	characteristic
258	#332	—	89	2	no	—
259	#332	—	89	2	no	—
260	#332	—	89	2	no	—
261	#332	C₂×D₈₉	1	2	yes	—
262	C₂×Q₁₆	—	2	2	no	—
263	C₂×Q₁₆	—	178	1	no	—
264	C₂×Q₁₆	—	178	1	no	—
265	C₂×SD₁₆	—	178	2	no	—
266	C₂×M₁₆	—	178	1	no	—
267	C₂²×C₈	—	89	1	no	—
268	C₂²×C₈	—	89	1	no	—
269	C₂²×C₈	—	178	2	no	—
270	C₂²×C₈	—	178	1	no	—
271	C₄⋊Q₈	—	89	2	no	—
272	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
273	#126	—	178	1	no	—
274	#125	—	89	1	no	—
275	#125	—	178	2	no	—
276	#124	—	89	1	no	—
277	#123	—	178	1	no	—
278	C₂²⋊Q₈	—	89	1	no	—
279	C₂²⋊Q₈	—	178	2	no	—
280	C₂²⋊Q₈	—	178	2	no	—
281	#121	—	89	1	no	—
282	C₄×Q₈	—	178	2	no	—
283	C₄×Q₈	—	178	2	no	—
284	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
285	C₄×D₄	—	89	1	no	—
286	C₄×D₄	—	178	1	no	—
287	#117	—	178	1	no	—
288	#117	—	178	2	no	—
289	C₂×C₄²	—	89	1	no	—
290	#107	—	2	1	no	—
291	#107	—	2	1	no	—
292	#107	—	178	2	no	—
293	#106	—	178	2	no	—
294	C₄⋊C₈	—	2	1	no	—
295	C₄⋊C₈	—	89	1	no	—
296	C₄⋊C₈	—	89	2	no	—
297	C₄⋊C₈	—	178	1	no	—
298	C₄⋊C₈	—	178	2	no	—
299	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
300	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
301	#103	—	2	2	no	—
302	#103	—	178	2	no	—
303	#103	—	178	2	no	—
304	#103	—	178	2	no	—
305	#103	—	178	2	no	—
306	#102	—	178	2	no	—
307	C₂²⋊C₈	—	178	1	no	—
308	C₂²⋊C₈	—	178	1	no	—
309	C₂²⋊C₈	—	178	1	no	—
310	#97	—	178	1	no	—
311	#97	—	178	1	no	—
312	C₄×C₈	—	2	1	no	—
313	C₄×C₈	—	89	1	no	—
314	C₄×C₈	—	89	1	no	—
315	C₄×C₈	—	89	1	no	—
316	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
317	C₂×Q₈	—	2	2	no	—
318	C₂×Q₈	—	178	2	no	—
319	C₂×Q₈	D₄×D₈₉	1	1	yes	characteristic
320	C₂×D₄	—	89	1	no	—
321	C₂×D₄	—	178	1	no	—
322	C₂²×C₄	—	89	1	no	—
323	C₂²×C₄	—	89	1	no	—
324	C₂²×C₄	—	89	1	no	—
325	C₂²×C₄	—	178	1	no	—
326	Q₁₆	—	8	2	no	—
327	Q₁₆	—	356	2	no	—
328	Q₁₆	—	356	2	no	—
329	SD₁₆	—	712	2	no	—
330	M₁₆	—	356	2	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	89	1	no	—
334	C₂×C₈	—	89	1	no	—
335	C₂×C₈	—	356	1	no	—
336	C₂×C₈	—	356	2	no	—
337	C₂×C₈	—	178	1	no	—
338	C₂×C₈	—	178	2	no	—
339	C₂×C₈	—	178	2	no	—
340	C₂×C₈	—	712	2	no	—
341	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
342	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	89	1	no	—
347	C₄⋊C₄	—	356	2	no	—
348	C₄⋊C₄	—	356	2	no	—
349	C₄⋊C₄	—	178	1	no	—
350	C₄⋊C₄	—	178	1	no	—
351	C₄⋊C₄	—	178	1	no	—
352	C₄⋊C₄	—	178	2	no	—
353	C₄⋊C₄	—	178	2	no	—
354	C₄⋊C₄	D₄×D₈₉	1	1	yes	characteristic
355	C₂²⋊C₄	—	356	2	no	—
356	C₂²⋊C₄	—	178	1	no	—
357	C₂²⋊C₄	—	178	1	no	—
358	C₂²⋊C₄	—	178	1	no	—
359	C₂²⋊C₄	—	178	1	no	—
360	C₂²⋊C₄	—	178	1	no	—
361	C₄²	—	2	1	no	—
362	C₄²	—	89	1	no	—
363	C₄²	—	89	1	no	—
364	C₄²	—	89	1	no	—
365	C₄²	—	356	2	no	—
366	C₄²	C₂³×D₈₉	1	1	yes	characteristic
367	C₂³	—	89	1	no	—
368	C₂³	—	178	1	no	—
369	Q₈	—	2	1	no	—
370	Q₈	—	2	1	no	—
371	Q₈	—	4	2	no	—
372	Q₈	—	4	2	no	—
373	Q₈	—	356	2	no	—
374	Q₈	—	356	2	no	—
375	D₄	—	356	2	no	—
376	D₄	—	178	1	no	—
377	D₄	—	178	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	89	1	no	—
382	C₂×C₄	—	89	1	no	—
383	C₂×C₄	—	89	1	no	—
384	C₂×C₄	—	356	1	no	—
385	C₂×C₄	—	356	1	no	—
386	C₂×C₄	—	356	2	no	—
387	C₂×C₄	—	178	2	no	—
388	C₂×C₄	—	178	1	no	—
389	C₂×C₄	—	178	1	no	—
390	C₂×C₄	—	178	1	no	—
391	C₂×C₄	—	178	1	no	—
392	C₂×C₄	—	178	1	no	—
393	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
394	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
395	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
396	C₈	—	2	1	no	—
397	C₈	—	4	1	no	—
398	C₈	—	4	2	no	—
399	C₈	—	356	2	no	—
400	C₈	—	356	1	no	—
401	C₈	—	178	1	no	—
402	C₈	—	178	1	no	—
403	C₈	#15668035	1	2	yes	—
404	C₂²	—	89	2	no	—
405	C₂²	—	356	1	no	—
406	C₂²	—	356	1	no	—
407	C₂²	—	356	1	no	—
408	C₂²	—	178	1	no	—
409	C₂²	—	178	1	no	—
410	C₂²	#121×D₈₉	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	4	1	no	—
413	C₄	—	4	1	no	—
414	C₄	—	8	2	no	—
415	C₄	—	356	1	no	—
416	C₄	—	178	1	no	—
417	C₄	—	178	1	no	—
418	C₄	—	178	1	no	—
419	C₄	—	712	2	no	—
420	C₄	C₂×#15668035	1	1	yes	characteristic
421	C₄	#19152397	1	1	yes	characteristic
422	C₄	#19152399	1	1	yes	characteristic
423	C₄	#19152412	1	1	yes	characteristic
424	C₂	—	178	2	no	—
425	C₂	—	712	1	no	—
426	C₂	#26029710	1	1	yes	characteristic
427	C₂	#26029917	1	1	yes	characteristic
428	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×#1674	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	#15667987	D₄
hypercenter	#1674	D₈₉
hypocenter	C₈₉	#19223
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
2. #26030839	C₂	semidirect
4. #26019808	C₂	semidirect
5. #26019802	C₂	semidirect
6. #26019638	C₂	semidirect
7. #26019622	C₂	semidirect
8. #26019618	C₂	semidirect
9. #335×D₈₉	C₂	semidirect
10. #26018951	C₂	semidirect
11. C₈₉×#1674	C₂	semidirect
220. #1674	D₈₉	semidirect
226. C₈₉	#19223	semidirect

Maximal subgroups

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

subgroup	index
#26031587	2
#26030839	2
#26029229	2
#26019808	2
#26019802	2
#26019638	2
#26019622	2
#26019618	2
#335×D₈₉	2
#26018951	2
C₈₉×#1674	2
#19223	89