Subgroups — #36709077

Order 22784.

546 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030763	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#15668035	C₂	1	1	yes	maximal, characteristic
3	#26022418	C₂	1	2	yes	maximal, complemented
4	#26022412	C₂	1	2	yes	maximal, complemented
5	#26022396	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022394	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022390	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022033	C₂	1	2	yes	maximal, complemented
9	C₄⋊C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021958	C₂	1	2	yes	maximal
11	C₈₉×#2227	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668035	C₂²	1	1	yes	characteristic
13	#19152153	—	2	1	no	—
14	C₄×#13083575	—	2	1	no	—
15	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
16	#19152003	C₂²	1	1	yes	characteristic
17	#19151999	—	2	1	no	—
18	#19151999	C₂²	1	2	yes	—
19	#19151994	—	2	1	no	—
20	Q₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
21	C₄×#13083561	—	2	1	no	—
22	#19151683	—	2	1	no	—
23	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
24	#19151680	C₂²	1	1	yes	characteristic
25	#19151635	—	2	1	no	—
26	#19151635	C₂²	1	2	yes	—
27	#19151635	C₂²	1	1	yes	characteristic
28	#19151593	—	2	1	no	—
29	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
30	#19151425	C₂²	1	2	yes	—
31	#19151425	C₂²	1	2	yes	—
32	#19151424	C₂²	1	2	yes	—
33	C₄⋊#13083549	C₂²	1	1	yes	characteristic
34	#19151416	C₂²	1	2	yes	—
35	#19151402	C₂²	1	2	yes	—
36	#19151400	C₂²	1	2	yes	—
37	#19151398	C₂²	1	2	yes	—
38	#500×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	#366×C₈₉	C₂²	1	2	yes	—
41	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
42	#358×C₈₉	C₂²	1	2	yes	—
43	C₂×#13083575	—	2	1	no	—
44	#15668035	—	2	2	no	—
45	#15668035	Q₈	1	4	yes	complemented
46	C₂×#13083561	—	2	1	no	—
47	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
49	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
50	#15668012	—	2	1	no	—
51	#15668003	—	2	2	no	—
52	#15667992	—	4	1	no	—
53	#15667988	—	4	1	no	—
54	Q₈×Dic₈₉	—	2	1	no	—
55	Q₈×Dic₈₉	—	4	1	no	—
56	#15667977	—	4	1	no	—
57	C₄×D₃₅₆	—	2	1	no	—
58	C₄×D₃₅₆	—	4	1	no	—
59	#15667974	—	2	1	no	—
60	#15667974	—	4	1	no	—
61	#15667970	—	2	2	no	—
62	#15667970	—	2	1	no	—
63	#15667970	—	4	1	no	—
64	C₄²×D₈₉	—	4	1	no	—
65	C₄²×D₈₉	D₄	1	1	yes	characteristic
66	C₂×#13083549	C₂³	1	2	yes	—
67	C₁₆×D₈₉	—	2	4	no	—
68	#15667946	—	2	1	no	—
69	#15667946	C₂³	1	2	yes	—
70	#15667946	C₂³	1	1	yes	characteristic
71	#15667943	—	2	2	no	—
72	#15667943	—	2	1	no	—
73	#15667943	—	2	2	no	—
74	#15667936	—	2	1	no	—
75	#15667934	—	2	1	no	—
76	#15667932	—	2	1	no	—
77	C₄×#10623494	D₄	1	1	yes	characteristic
78	C₈×Dic₈₉	D₄	1	1	yes	characteristic
79	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
80	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
81	C₄⋊Q₈×C₈₉	—	2	1	no	—
82	C₄×Q₈×C₈₉	—	2	1	no	—
83	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
84	#107×C₈₉	—	2	1	no	—
85	#107×C₈₉	C₂³	1	2	yes	—
86	#107×C₈₉	C₂³	1	1	yes	characteristic
87	#103×C₈₉	—	2	1	no	—
88	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
89	#13083575	—	2	4	no	—
90	#13083575	—	4	2	no	—
91	C₂×Dic₁₇₈	—	4	1	no	—
92	C₂×D₃₅₆	—	2	1	no	—
93	C₂×D₃₅₆	—	4	1	no	—
94	C₂×C₄×D₈₉	—	4	1	no	—
95	C₂×C₄×D₈₉	—	4	1	no	—
96	C₂×C₄×D₈₉	—	4	1	no	—
97	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
98	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
99	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
100	#13083561	—	2	4	no	—
101	D₇₁₂	—	2	1	no	—
102	D₇₁₂	C₂×Q₈	1	2	yes	—
103	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
104	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
105	C₈×D₈₉	—	2	2	no	—
106	C₈×D₈₉	C₂×Q₈	1	2	yes	—
107	C₈×D₈₉	#41	1	2	yes	—
108	C₄⋊Dic₈₉	—	2	1	no	—
109	C₄⋊Dic₈₉	—	2	2	no	—
110	C₄⋊Dic₈₉	—	4	1	no	—
111	C₄⋊Dic₈₉	—	4	1	no	—
112	C₄⋊Dic₈₉	—	4	2	no	—
113	#13083553	—	4	2	no	—
114	#13083551	—	4	1	no	—
115	#13083551	—	4	2	no	—
116	#13083551	—	4	1	no	—
117	#13083551	—	4	1	no	—
118	C₄×Dic₈₉	—	4	1	no	—
119	C₄×Dic₈₉	—	4	2	no	—
120	C₄×Dic₈₉	—	4	1	no	—
121	C₄×Dic₈₉	D₈	1	1	yes	characteristic
122	C₄×Dic₈₉	D₈	1	1	yes	characteristic
123	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
124	#13083549	—	2	2	no	—
125	C₂×Q₈×C₈₉	—	2	1	no	—
126	C₂×Q₈×C₈₉	—	4	1	no	—
127	Q₁₆×C₈₉	—	2	1	no	—
128	Q₁₆×C₈₉	C₂×Q₈	1	2	yes	—
129	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
131	C₄⋊C₄×C₈₉	—	2	1	no	—
132	C₄⋊C₄×C₈₉	—	2	2	no	—
133	C₄⋊C₄×C₈₉	—	4	1	no	—
134	C₄⋊C₄×C₈₉	—	4	1	no	—
135	C₄²×C₈₉	—	4	1	no	—
136	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
137	C₁₆×C₈₉	—	2	2	no	—
138	C₂²×D₈₉	—	4	1	no	—
139	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
140	Q₈×C₈₉	—	2	2	no	—
141	Q₈×C₈₉	—	4	1	no	—
142	Q₈×C₈₉	—	8	2	no	—
143	C₂×C₄×C₈₉	—	4	1	no	—
144	C₂×C₄×C₈₉	—	4	1	no	—
145	C₂×C₄×C₈₉	—	4	2	no	—
146	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×Dic₈₉	—	4	1	no	—
150	C₂×Dic₈₉	—	4	2	no	—
151	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
152	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
153	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
154	D₃₅₆	—	2	2	no	—
155	D₃₅₆	—	4	4	no	—
156	D₃₅₆	—	4	1	no	—
157	C₄×D₈₉	—	2	2	no	—
158	C₄×D₈₉	—	4	1	no	—
159	C₄×D₈₉	—	4	4	no	—
160	C₄×D₈₉	—	8	2	no	—
161	C₄×D₈₉	—	8	2	no	—
162	C₄×D₈₉	C₂²⋊Q₈	1	2	yes	—
163	C₄×D₈₉	#137	1	2	yes	—
164	Dic₁₇₈	—	8	2	no	—
165	C₈×C₈₉	—	2	1	no	—
166	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
167	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
168	#10623494	—	2	1	no	—
169	#10623494	C₂²⋊Q₈	1	2	yes	—
170	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
171	C₂×D₈₉	—	2	1	no	—
172	C₂×D₈₉	—	2	1	no	—
173	C₂×D₈₉	—	4	2	no	—
174	C₂×D₈₉	—	8	1	no	—
175	C₂×D₈₉	—	8	1	no	—
176	C₂×D₈₉	#474	1	2	yes	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	—	4	2	no	—
179	C₄×C₈₉	—	8	1	no	—
180	C₄×C₈₉	—	8	2	no	—
181	C₄×C₈₉	#508	1	2	yes	—
182	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
183	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
184	Dic₈₉	—	2	1	no	—
185	Dic₈₉	—	2	1	no	—
186	Dic₈₉	—	8	2	no	—
187	Dic₈₉	—	8	1	no	—
188	Dic₈₉	#474	1	2	yes	—
189	#33626	—	89	1	no	maximal, Sylow
190	C₂×C₈₉	C₂×#474	1	1	yes	characteristic
191	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
192	C₂×C₈₉	#3414	1	1	yes	characteristic
193	D₈₉	—	2	2	no	—
194	D₈₉	—	8	2	no	—
195	#3160	—	89	1	no	—
196	C₄×#135	—	89	1	no	—
197	#2229	—	89	2	no	—
198	#2228	—	89	2	no	—
199	#2227	D₈₉	1	1	yes	characteristic, complemented
200	#2226	—	89	1	no	—
201	#2225	—	89	1	no	—
202	#2224	—	89	1	no	—
203	#2158	—	89	2	no	—
204	C₂×C₄⋊C₁₆	—	89	1	no	—
205	#2138	—	89	2	no	—
206	C₈₉	#33626	1	1	yes	characteristic, complemented, Sylow
207	C₂×#135	—	89	1	no	—
208	#533	—	178	1	no	—
209	C₄×#41	—	178	1	no	—
210	C₂²×C₁₆	—	89	2	no	—
211	#500	—	89	1	no	—
212	#500	C₂×D₈₉	1	1	yes	characteristic
213	#499	—	89	2	no	—
214	#499	—	178	1	no	—
215	#498	—	178	1	no	—
216	C₄×Q₁₆	—	89	1	no	—
217	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
218	C₄×SD₁₆	—	178	1	no	—
219	C₄×SD₁₆	—	178	1	no	—
220	C₄×D₈	—	89	1	no	—
221	C₄×D₈	—	89	1	no	—
222	#427	—	89	2	no	—
223	#427	—	89	1	no	—
224	#427	—	178	1	no	—
225	#416	—	178	1	no	—
226	C₂×C₄×C₈	—	89	1	no	—
227	#367	—	89	2	no	—
228	#367	—	89	2	no	—
229	#366	—	89	2	no	—
230	#366	C₂×D₈₉	1	2	yes	—
231	C₄⋊C₁₆	—	89	2	no	—
232	C₄⋊C₁₆	—	89	1	no	—
233	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
234	#358	—	89	2	no	—
235	#358	C₂×D₈₉	1	2	yes	—
236	#357	—	89	2	no	—
237	#357	—	89	2	no	—
238	C₂×#41	—	178	1	no	—
239	#135	—	89	4	no	—
240	#135	—	178	2	no	—
241	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
242	C₂×SD₁₆	—	178	1	no	—
243	C₂×D₈	—	89	1	no	—
244	C₂²×C₈	—	89	1	no	—
245	C₂²×C₈	—	89	1	no	—
246	C₄⋊Q₈	—	2	1	no	—
247	C₄⋊Q₈	—	178	1	no	—
248	#125	—	178	2	no	—
249	C₂²⋊Q₈	—	356	1	no	—
250	C₂²⋊Q₈	—	356	1	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	—	356	1	no	—
253	C₄×Q₈	—	356	1	no	—
254	C₄×Q₈	—	178	1	no	—
255	C₄×D₄	—	356	1	no	—
256	C₄×D₄	—	356	1	no	—
257	C₄×D₄	—	178	1	no	—
258	C₄×D₄	—	178	1	no	—
259	#117	—	356	1	no	—
260	#117	—	178	1	no	—
261	#117	—	178	2	no	—
262	C₂×C₄²	—	89	1	no	—
263	C₂×C₄²	—	356	1	no	—
264	C₂×C₁₆	—	89	2	no	—
265	C₂×C₁₆	—	178	4	no	—
266	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
267	#107	—	2	1	no	—
268	#107	—	89	2	no	—
269	#107	—	89	1	no	—
270	#107	—	178	1	no	—
271	#107	C₂²×D₈₉	1	2	yes	—
272	#107	C₂²×D₈₉	1	1	yes	characteristic
273	#106	—	178	1	no	—
274	#106	—	178	2	no	—
275	#106	—	178	2	no	—
276	#103	—	2	1	no	—
277	#103	—	178	1	no	—
278	#102	—	178	1	no	—
279	#102	—	178	1	no	—
280	C₄×C₈	—	89	1	no	—
281	C₄×C₈	—	89	1	no	—
282	C₄×C₈	—	89	1	no	—
283	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
284	#41	—	356	2	no	—
285	#41	—	178	4	no	—
286	C₂×Q₈	—	2	1	no	—
287	C₂×Q₈	—	4	1	no	—
288	C₂×Q₈	—	356	1	no	—
289	C₂×D₄	—	356	1	no	—
290	C₂×D₄	—	178	1	no	—
291	C₂²×C₄	—	89	1	no	—
292	C₂²×C₄	—	89	1	no	—
293	C₂²×C₄	—	89	1	no	—
294	C₂²×C₄	—	356	1	no	—
295	C₂²×C₄	—	356	1	no	—
296	C₂²×C₄	—	356	1	no	—
297	Q₁₆	—	2	1	no	—
298	Q₁₆	Q₈×D₈₉	1	2	yes	—
299	SD₁₆	—	178	4	no	—
300	D₈	—	89	2	no	—
301	D₈	—	178	1	no	—
302	C₂×C₈	—	89	1	no	—
303	C₂×C₈	—	89	2	no	—
304	C₂×C₈	—	89	2	no	—
305	C₂×C₈	—	89	1	no	—
306	C₂×C₈	—	178	2	no	—
307	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	356	1	no	—
314	C₄⋊C₄	—	356	2	no	—
315	C₄⋊C₄	—	356	2	no	—
316	C₄⋊C₄	—	356	1	no	—
317	C₄⋊C₄	—	178	1	no	—
318	C₄⋊C₄	—	178	2	no	—
319	C₂²⋊C₄	—	356	1	no	—
320	C₂²⋊C₄	—	356	1	no	—
321	C₂²⋊C₄	—	356	2	no	—
322	C₂²⋊C₄	—	356	1	no	—
323	C₄²	—	4	1	no	—
324	C₄²	—	89	1	no	—
325	C₄²	—	89	1	no	—
326	C₄²	—	89	1	no	—
327	C₄²	—	356	1	no	—
328	C₄²	—	356	1	no	—
329	C₄²	—	356	2	no	—
330	C₄²	D₄×D₈₉	1	1	yes	characteristic
331	C₁₆	—	2	2	no	—
332	C₁₆	—	178	2	no	—
333	C₂³	—	89	1	no	—
334	C₂³	—	356	1	no	—
335	Q₈	—	2	2	no	—
336	Q₈	—	4	1	no	—
337	Q₈	—	8	2	no	—
338	Q₈	—	712	2	no	—
339	D₄	—	356	4	no	—
340	D₄	—	356	1	no	—
341	D₄	—	178	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	89	1	no	—
346	C₂×C₄	—	89	2	no	—
347	C₂×C₄	—	89	2	no	—
348	C₂×C₄	—	89	1	no	—
349	C₂×C₄	—	89	1	no	—
350	C₂×C₄	—	356	1	no	—
351	C₂×C₄	—	356	4	no	—
352	C₂×C₄	—	356	1	no	—
353	C₂×C₄	—	356	2	no	—
354	C₂×C₄	—	178	2	no	—
355	C₂×C₄	—	712	2	no	—
356	C₂×C₄	—	712	2	no	—
357	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
358	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
359	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
360	C₈	—	2	1	no	—
361	C₈	—	89	2	no	—
362	C₈	—	178	1	no	—
363	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
364	C₈	#41×D₈₉	1	1	yes	characteristic
365	C₂²	—	89	2	no	—
366	C₂²	—	356	2	no	—
367	C₂²	—	178	1	no	—
368	C₂²	—	178	1	no	—
369	C₂²	—	712	1	no	—
370	C₂²	—	712	1	no	—
371	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	4	2	no	—
374	C₄	—	8	1	no	—
375	C₄	—	8	2	no	—
376	C₄	—	89	2	no	—
377	C₄	—	178	1	no	—
378	C₄	—	178	1	no	—
379	C₄	—	712	1	no	—
380	C₄	—	712	2	no	—
381	C₄	#19152031	1	2	yes	—
382	C₄	C₂²⋊Q₈×D₈₉	1	1	yes	characteristic
383	C₄	#137×D₈₉	1	1	yes	characteristic
384	C₂	—	178	2	no	—
385	C₂	—	712	2	no	—
386	C₂	#474×D₈₉	1	1	yes	characteristic
387	C₂	C₂×#19152031	1	1	yes	characteristic
388	C₂	#26033384	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2227	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#15668035	C₂²
hypercenter	#2227	D₈₉
hypocenter	C₈₉	#33626
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2227

Complements

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

N	H ≅ G/N	type
1. #26030763	C₂	semidirect
3. #26022418	C₂	semidirect
4. #26022412	C₂	semidirect
5. #26022396	C₂	semidirect
6. #26022394	C₂	semidirect
7. #26022390	C₂	semidirect
8. #26022033	C₂	semidirect
9. C₄⋊C₁₆×D₈₉	C₂	semidirect
11. C₈₉×#2227	C₂	semidirect
45. #15668035	Q₈	semidirect
199. #2227	D₈₉	semidirect
206. C₈₉	#33626	semidirect

Maximal subgroups

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

subgroup	index
#26030763	2
C₄×#15668035	2
#26022418	2
#26022412	2
#26022396	2
#26022394	2
#26022390	2
#26022033	2
C₄⋊C₁₆×D₈₉	2
#26021958	2
C₈₉×#2227	2
#33626	89