Subgroups — #36537141

Order 22784.

596 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030836	C₂	1	1	yes	maximal, characteristic, complemented
2	#26030033	C₂	1	2	yes	maximal
3	#26029215	C₂	1	1	yes	maximal, characteristic
4	#26019833	C₂	1	2	yes	maximal, complemented
5	#26019825	C₂	1	2	yes	maximal, complemented
6	#26019781	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019775	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019743	C₂	1	1	yes	maximal, characteristic, complemented
9	#334×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#26018908	C₂	1	2	yes	maximal, complemented
11	C₈₉×#1689	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152153	C₂²	1	2	yes	—
13	C₄×#13083573	C₂²	1	1	yes	characteristic
14	#19152003	C₂²	1	1	yes	characteristic
15	#19151948	C₂²	1	2	yes	—
16	#19151904	—	2	2	no	—
17	#19151896	—	2	2	no	—
18	#19151894	C₂²	1	2	yes	—
19	#19151888	C₂²	1	2	yes	—
20	#19151880	C₂²	1	2	yes	—
21	#19151836	—	2	1	no	—
22	#19151836	—	2	1	no	—
23	#19151728	C₂²	1	1	yes	characteristic
24	#19151725	C₂²	1	1	yes	characteristic
25	Q₁₆×Dic₈₉	—	2	1	no	—
26	Q₁₆×Dic₈₉	—	2	1	no	—
27	#19151676	C₂²	1	1	yes	characteristic
28	#19151672	—	2	1	no	—
29	#19151667	—	2	1	no	—
30	#19151650	C₂²	1	1	yes	characteristic
31	#19151648	—	2	1	no	—
32	#19151635	—	2	2	no	—
33	C₄⋊C₈×D₈₉	C₂²	1	2	yes	—
34	#19151591	—	2	2	no	—
35	M₁₆×Dic₈₉	—	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	#494×C₈₉	C₂²	1	1	yes	characteristic
43	#474×C₈₉	C₂²	1	2	yes	—
44	#436×C₈₉	C₂²	1	1	yes	characteristic
45	#334×C₈₉	C₂²	1	1	yes	characteristic
46	#332×C₈₉	C₂²	1	2	yes	—
47	C₂×#13083573	D₄	1	1	yes	characteristic
48	C₂×#13083558	—	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	#15668003	—	2	2	no	—
54	#15667992	—	2	2	no	—
55	#15667988	—	2	2	no	—
56	Q₈×Dic₈₉	—	2	2	no	—
57	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
58	#15667977	—	2	2	no	—
59	#15667977	D₄	1	1	yes	characteristic
60	D₄×Dic₈₉	D₄	1	1	yes	characteristic
61	C₄×C₂²⋊D₈₉	—	2	1	no	—
62	#15667972	—	2	1	no	—
63	#15667970	—	2	2	no	—
64	#15667970	D₄	1	1	yes	characteristic
65	#15667968	—	2	1	no	—
66	C₂×C₄×Dic₈₉	—	2	1	no	—
67	C₄²×D₈₉	C₂³	1	1	yes	characteristic
68	#15667946	—	2	2	no	—
69	#15667946	—	2	1	no	—
70	#15667946	—	2	1	no	—
71	#15667943	—	2	2	no	—
72	#15667943	—	2	2	no	—
73	C₄⋊#10623494	C₂³	1	2	yes	—
74	C₄⋊#10623494	C₂³	1	1	yes	characteristic
75	#15667941	—	2	1	no	—
76	#15667941	—	2	2	no	—
77	#15667941	—	2	1	no	—
78	#15667937	—	2	2	no	—
79	#15667936	—	2	2	no	—
80	#15667936	—	2	2	no	—
81	#15667935	—	2	2	no	—
82	#15667933	—	2	2	no	—
83	C₂²⋊#10623494	—	2	1	no	—
84	#15667924	—	2	1	no	—
85	#15667922	—	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	1	no	—
92	C₂×Q₁₆×C₈₉	—	2	1	no	—
93	C₂×M₁₆×C₈₉	—	2	1	no	—
94	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
95	C₄×D₄×C₈₉	C₂³	1	1	yes	characteristic
96	#107×C₈₉	—	2	2	no	—
97	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
98	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
99	#102×C₈₉	—	2	2	no	—
100	C₂²⋊C₈×C₈₉	—	2	1	no	—
101	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	#13083573	—	2	4	no	—
103	C₂×Dic₁₇₈	—	2	2	no	—
104	C₂×Dic₁₇₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₂²⋊D₈₉	—	2	1	no	—
106	C₂²×Dic₈₉	—	2	1	no	—
107	C₂²×Dic₈₉	—	2	1	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	#13083558	—	4	2	no	—
112	C₂×#10623494	—	2	2	no	—
113	C₂×#10623494	—	2	1	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	—	4	1	no	—
118	C₈×D₈₉	—	4	4	no	—
119	C₄⋊Dic₈₉	—	2	2	no	—
120	C₄⋊Dic₈₉	—	2	2	no	—
121	C₄⋊Dic₈₉	—	4	2	no	—
122	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
123	#13083553	—	2	1	no	—
124	#13083553	—	2	1	no	—
125	#13083553	—	4	2	no	—
126	C₂²⋊Dic₈₉	—	2	1	no	—
127	#13083551	—	2	1	no	—
128	#13083551	—	4	2	no	—
129	C₄×Dic₈₉	—	2	1	no	—
130	C₄×Dic₈₉	—	2	1	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×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₈₉	—	2	1	no	—
139	Q₁₆×C₈₉	—	4	2	no	—
140	Q₁₆×C₈₉	—	4	2	no	—
141	M₁₆×C₈₉	—	4	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	2	no	—
147	C₄⋊C₄×C₈₉	—	2	2	no	—
148	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
149	C₂²⋊C₄×C₈₉	—	2	1	no	—
150	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
151	C₂³×C₈₉	—	2	1	no	—
152	C₂²×D₈₉	#121	1	1	yes	characteristic
153	Q₈×C₈₉	—	4	2	no	—
154	Q₈×C₈₉	—	4	2	no	—
155	D₄×C₈₉	—	2	2	no	—
156	C₂×C₄×C₈₉	—	2	1	no	—
157	C₂×C₄×C₈₉	—	4	1	no	—
158	C₂×C₄×C₈₉	—	4	1	no	—
159	C₂×C₄×C₈₉	—	4	2	no	—
160	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
163	C₂²⋊D₈₉	—	4	2	no	—
164	C₂×Dic₈₉	—	2	1	no	—
165	C₂×Dic₈₉	—	2	1	no	—
166	C₂×Dic₈₉	—	4	2	no	—
167	C₂×Dic₈₉	—	4	2	no	—
168	C₂×Dic₈₉	—	4	2	no	—
169	C₂×Dic₈₉	#121	1	1	yes	characteristic
170	C₂×Dic₈₉	#121	1	1	yes	characteristic
171	C₂×Dic₈₉	#121	1	1	yes	characteristic
172	C₄×D₈₉	—	2	1	no	—
173	C₄×D₈₉	—	2	1	no	—
174	C₄×D₈₉	—	4	1	no	—
175	C₄×D₈₉	#137	1	4	yes	—
176	Dic₁₇₈	—	2	2	no	—
177	Dic₁₇₈	—	8	2	no	—
178	C₈×C₈₉	—	2	1	no	—
179	C₈×C₈₉	—	4	1	no	—
180	C₈×C₈₉	—	4	2	no	—
181	C₈×C₈₉	#135	1	2	yes	—
182	#10623494	—	2	1	no	—
183	#10623494	—	2	1	no	—
184	#10623494	—	4	2	no	—
185	#10623494	—	4	1	no	—
186	C₂²×C₈₉	—	4	1	no	—
187	C₂²×C₈₉	—	4	1	no	—
188	C₂²×C₈₉	—	4	1	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	#470	1	2	yes	—
193	C₄×C₈₉	—	2	1	no	—
194	C₄×C₈₉	—	4	1	no	—
195	C₄×C₈₉	—	8	2	no	—
196	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
197	C₄×C₈₉	C₂×#137	1	2	yes	—
198	C₄×C₈₉	#575	1	1	yes	characteristic
199	Dic₈₉	—	2	1	no	—
200	Dic₈₉	—	2	1	no	—
201	Dic₈₉	—	4	1	no	—
202	Dic₈₉	—	4	1	no	—
203	Dic₈₉	—	8	2	no	—
204	Dic₈₉	#470	1	2	yes	—
205	#19158	—	89	1	no	maximal, Sylow
206	C₂×C₈₉	—	8	1	no	—
207	C₂×C₈₉	#3035	1	1	yes	characteristic
208	C₂×C₈₉	C₂×#470	1	1	yes	characteristic
209	C₂×C₈₉	#3067	1	1	yes	characteristic
210	D₈₉	—	2	2	no	—
211	#3170	—	89	1	no	—
212	#3076	—	89	2	no	—
213	#2965	—	89	1	no	—
214	#1704	—	89	2	no	—
215	#1703	—	89	2	no	—
216	#1696	—	89	1	no	—
217	#1695	—	89	1	no	—
218	#1690	—	89	1	no	—
219	#1689	D₈₉	1	1	yes	characteristic, complemented
220	C₂×#334	—	89	1	no	—
221	#1542	—	89	2	no	—
222	C₈₉	#19158	1	1	yes	characteristic, complemented, Sylow
223	#533	—	89	2	no	—
224	C₄×#41	—	89	1	no	—
225	#500	—	89	1	no	—
226	#494	C₂×D₈₉	1	1	yes	characteristic
227	#487	—	89	2	no	—
228	#479	—	178	2	no	—
229	#478	—	178	2	no	—
230	#477	—	89	2	no	—
231	#476	—	89	2	no	—
232	#475	—	89	2	no	—
233	#474	C₂×D₈₉	1	2	yes	—
234	#467	—	178	1	no	—
235	#467	—	178	1	no	—
236	#446	—	89	1	no	—
237	#446	—	89	1	no	—
238	C₄×Q₁₆	—	178	1	no	—
239	C₄×Q₁₆	—	178	1	no	—
240	#436	—	89	1	no	—
241	#436	C₂×D₈₉	1	1	yes	characteristic
242	#435	—	178	1	no	—
243	#435	—	178	1	no	—
244	#431	—	89	1	no	—
245	#431	—	178	1	no	—
246	#427	—	178	2	no	—
247	C₂×C₄⋊C₈	—	89	2	no	—
248	#416	—	178	2	no	—
249	C₄×M₁₆	—	178	1	no	—
250	C₂×C₄×C₈	—	89	1	no	—
251	#334	—	89	2	no	—
252	#334	—	89	1	no	—
253	#334	C₂×D₈₉	1	1	yes	characteristic
254	#333	—	89	2	no	—
255	#333	—	89	2	no	—
256	#332	—	89	2	no	—
257	#332	C₂×D₈₉	1	2	yes	—
258	C₂×#41	—	89	1	no	—
259	C₂×Q₁₆	—	2	1	no	—
260	C₂×Q₁₆	—	2	1	no	—
261	C₂×M₁₆	—	2	1	no	—
262	C₂×M₁₆	—	178	1	no	—
263	C₂²×C₈	—	89	1	no	—
264	C₂²×C₈	—	89	1	no	—
265	C₂²×C₈	—	178	2	no	—
266	C₄⋊Q₈	—	89	2	no	—
267	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
268	#125	—	178	2	no	—
269	C₂²⋊Q₈	—	178	2	no	—
270	C₂²⋊Q₈	—	178	2	no	—
271	C₄×Q₈	—	89	1	no	—
272	C₄×Q₈	—	89	1	no	—
273	C₄×Q₈	—	178	2	no	—
274	C₄×Q₈	—	178	2	no	—
275	C₄×D₄	—	89	1	no	—
276	C₄×D₄	—	178	1	no	—
277	C₄×D₄	—	178	1	no	—
278	C₄×D₄	C₂²×D₈₉	1	1	yes	characteristic
279	#117	—	89	1	no	—
280	#117	—	178	1	no	—
281	#117	—	178	2	no	—
282	C₂×C₄²	—	89	1	no	—
283	C₂×C₄²	—	178	1	no	—
284	#107	—	2	2	no	—
285	#107	—	178	2	no	—
286	#107	—	178	1	no	—
287	#107	—	178	1	no	—
288	#106	—	178	2	no	—
289	#106	—	178	2	no	—
290	C₄⋊C₈	—	89	1	no	—
291	C₄⋊C₈	—	89	2	no	—
292	C₄⋊C₈	—	178	2	no	—
293	C₄⋊C₈	—	178	1	no	—
294	C₄⋊C₈	—	178	1	no	—
295	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
296	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
297	#103	—	178	2	no	—
298	#103	—	178	2	no	—
299	#103	—	178	2	no	—
300	#103	—	178	2	no	—
301	#102	—	2	2	no	—
302	#102	—	178	2	no	—
303	C₂²⋊C₈	—	2	1	no	—
304	C₂²⋊C₈	—	178	1	no	—
305	C₂²⋊C₈	—	178	1	no	—
306	#97	—	178	1	no	—
307	#97	—	178	1	no	—
308	C₄×C₈	—	89	1	no	—
309	C₄×C₈	—	89	1	no	—
310	C₄×C₈	—	89	1	no	—
311	C₄×C₈	—	178	1	no	—
312	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
313	#41	—	178	4	no	—
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	89	1	no	—
316	C₂×Q₈	—	178	2	no	—
317	C₂×D₄	—	178	1	no	—
318	C₂×D₄	D₄×D₈₉	1	1	yes	characteristic
319	C₂²×C₄	—	2	1	no	—
320	C₂²×C₄	—	89	1	no	—
321	C₂²×C₄	—	89	1	no	—
322	C₂²×C₄	—	89	1	no	—
323	C₂²×C₄	—	178	1	no	—
324	C₂²×C₄	—	178	1	no	—
325	Q₁₆	—	4	2	no	—
326	Q₁₆	—	4	2	no	—
327	M₁₆	—	4	2	no	—
328	M₁₆	—	356	2	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	89	1	no	—
332	C₂×C₈	—	89	1	no	—
333	C₂×C₈	—	356	1	no	—
334	C₂×C₈	—	356	4	no	—
335	C₂×C₈	—	178	1	no	—
336	C₂×C₈	—	178	2	no	—
337	C₂×C₈	—	178	2	no	—
338	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
339	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	89	1	no	—
343	C₄⋊C₄	—	356	2	no	—
344	C₄⋊C₄	—	356	2	no	—
345	C₄⋊C₄	—	178	1	no	—
346	C₄⋊C₄	—	178	2	no	—
347	C₄⋊C₄	—	178	2	no	—
348	C₄⋊C₄	—	178	1	no	—
349	C₄⋊C₄	D₄×D₈₉	1	1	yes	characteristic
350	C₂²⋊C₄	—	2	1	no	—
351	C₂²⋊C₄	—	356	2	no	—
352	C₂²⋊C₄	—	178	1	no	—
353	C₂²⋊C₄	—	178	1	no	—
354	C₄²	—	89	1	no	—
355	C₄²	—	89	1	no	—
356	C₄²	—	89	1	no	—
357	C₄²	—	356	2	no	—
358	C₄²	—	178	1	no	—
359	C₄²	—	178	1	no	—
360	C₄²	—	178	1	no	—
361	C₄²	C₂³×D₈₉	1	1	yes	characteristic
362	C₂³	—	2	1	no	—
363	C₂³	—	89	1	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	4	2	no	—
366	Q₈	—	178	2	no	—
367	Q₈	—	712	2	no	—
368	D₄	—	2	2	no	—
369	D₄	—	356	2	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	89	1	no	—
375	C₂×C₄	—	89	4	no	—
376	C₂×C₄	—	89	1	no	—
377	C₂×C₄	—	89	1	no	—
378	C₂×C₄	—	356	1	no	—
379	C₂×C₄	—	356	2	no	—
380	C₂×C₄	—	356	2	no	—
381	C₂×C₄	—	356	2	no	—
382	C₂×C₄	—	178	1	no	—
383	C₂×C₄	—	178	1	no	—
384	C₂×C₄	—	178	1	no	—
385	C₂×C₄	—	178	1	no	—
386	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
387	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
388	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
389	C₈	—	2	1	no	—
390	C₈	—	4	2	no	—
391	C₈	—	4	1	no	—
392	C₈	—	356	2	no	—
393	C₈	—	356	1	no	—
394	C₈	—	178	1	no	—
395	C₈	—	178	1	no	—
396	C₈	#15668031	1	2	yes	—
397	C₂²	—	4	1	no	—
398	C₂²	—	4	1	no	—
399	C₂²	—	4	1	no	—
400	C₂²	—	89	2	no	—
401	C₂²	—	178	1	no	—
402	C₂²	—	178	1	no	—
403	C₂²	#121×D₈₉	1	1	yes	characteristic
404	C₄	—	2	1	no	—
405	C₄	—	4	1	no	—
406	C₄	—	8	2	no	—
407	C₄	—	89	2	no	—
408	C₄	—	356	1	no	—
409	C₄	—	356	1	no	—
410	C₄	—	178	1	no	—
411	C₄	—	178	1	no	—
412	C₄	—	712	2	no	—
413	C₄	C₂×#15668031	1	1	yes	characteristic
414	C₄	#137×D₈₉	1	2	yes	—
415	C₄	#19152394	1	1	yes	characteristic
416	C₂	—	8	1	no	—
417	C₂	—	178	2	no	—
418	C₂	#26029700	1	1	yes	characteristic
419	C₂	#470×D₈₉	1	1	yes	characteristic
420	C₂	#26029974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₈₉
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1689	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂×#13083573	D₄
hypercenter	#1689	D₈₉
hypocenter	C₈₉	#19158
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1689

Complements

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

N	H ≅ G/N	type
1. #26030836	C₂	semidirect
4. #26019833	C₂	semidirect
5. #26019825	C₂	semidirect
6. #26019781	C₂	semidirect
7. #26019775	C₂	semidirect
8. #26019743	C₂	semidirect
9. #334×D₈₉	C₂	semidirect
10. #26018908	C₂	semidirect
11. C₈₉×#1689	C₂	semidirect
219. #1689	D₈₉	semidirect
222. C₈₉	#19158	semidirect

Maximal subgroups

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

subgroup	index
#26030836	2
#26030033	2
#26029215	2
#26019833	2
#26019825	2
#26019781	2
#26019775	2
#26019743	2
#334×D₈₉	2
#26018908	2
C₈₉×#1689	2
#19158	89