Subgroups — #36709100

Order 22784.

574 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#15668035	C₂	1	1	yes	maximal, characteristic
3	#26022404	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022400	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022396	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022392	C₂	1	2	yes	maximal, complemented
7	#26022388	C₂	1	2	yes	maximal, complemented
8	#26022037	C₂	1	2	yes	maximal, complemented
9	#26022006	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021963	C₂	1	2	yes	maximal
11	C₈₉×#2227	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668035	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×D₈₉	—	2	1	no	—
14	C₄×#13083575	—	2	1	no	—
15	C₂×#15667951	C₂²	1	2	yes	—
16	#19152003	C₂²	1	1	yes	characteristic
17	#19152002	—	2	1	no	—
18	#19152002	C₂²	1	2	yes	—
19	#19152001	—	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	C₂²	1	1	yes	characteristic
26	#107×D₈₉	—	2	1	no	—
27	#107×D₈₉	C₂²	1	2	yes	—
28	#19151593	—	2	1	no	—
29	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
30	#19151426	C₂²	1	2	yes	—
31	#19151425	C₂²	1	2	yes	—
32	#19151423	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	C₂×Q₈×D₈₉	—	4	1	no	—
45	#15668035	—	2	2	no	—
46	#15668035	Q₈	1	4	yes	complemented
47	C₂×#13083561	—	2	1	no	—
48	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
49	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
50	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
51	#15668012	—	2	1	no	—
52	#15668011	—	2	2	no	—
53	#15668011	—	4	1	no	—
54	#15668010	—	4	1	no	—
55	Q₈×Dic₈₉	—	2	1	no	—
56	C₄×D₃₅₆	—	2	1	no	—
57	C₄×D₃₅₆	—	4	1	no	—
58	#15667974	—	2	1	no	—
59	#15667974	—	4	1	no	—
60	#15667970	—	2	1	no	—
61	#15667970	—	4	1	no	—
62	C₄⋊C₄×D₈₉	—	2	2	no	—
63	C₄⋊C₄×D₈₉	—	4	1	no	—
64	C₄²×D₈₉	—	4	1	no	—
65	C₄²×D₈₉	D₄	1	1	yes	characteristic
66	#15667951	—	2	4	no	—
67	C₂×#13083549	C₂³	1	2	yes	—
68	#15667946	—	2	1	no	—
69	#15667946	C₂³	1	2	yes	—
70	#15667946	C₂³	1	1	yes	characteristic
71	#15667945	—	2	2	no	—
72	#15667945	—	2	2	no	—
73	#15667943	—	2	1	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×Dic₁₇₈	—	4	2	no	—
93	Q₈×D₈₉	—	8	4	no	—
94	C₂×D₃₅₆	—	2	1	no	—
95	C₂×D₃₅₆	—	4	1	no	—
96	C₂×C₄×D₈₉	—	4	1	no	—
97	C₂×C₄×D₈₉	—	4	2	no	—
98	C₂×C₄×D₈₉	—	4	1	no	—
99	C₂×C₄×D₈₉	—	4	1	no	—
100	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
101	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
102	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
103	#13083561	—	2	4	no	—
104	D₇₁₂	—	2	1	no	—
105	D₇₁₂	C₂×Q₈	1	2	yes	complemented
106	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
107	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
108	C₈×D₈₉	—	2	2	no	—
109	C₈×D₈₉	C₂×Q₈	1	2	yes	—
110	C₈×D₈₉	#41	1	2	yes	—
111	C₄⋊Dic₈₉	—	2	1	no	—
112	C₄⋊Dic₈₉	—	2	2	no	—
113	C₄⋊Dic₈₉	—	4	1	no	—
114	C₄⋊Dic₈₉	—	4	1	no	—
115	#13083553	—	4	2	no	—
116	#13083553	—	4	2	no	—
117	#13083551	—	4	1	no	—
118	#13083551	—	4	1	no	—
119	#13083551	—	4	1	no	—
120	C₄×Dic₈₉	—	4	1	no	—
121	C₄×Dic₈₉	—	4	1	no	—
122	C₄×Dic₈₉	D₈	1	1	yes	characteristic
123	C₄×Dic₈₉	D₈	1	1	yes	characteristic
124	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
125	#13083549	—	2	2	no	—
126	C₂×Q₈×C₈₉	—	2	1	no	—
127	C₂×Q₈×C₈₉	—	4	1	no	—
128	Q₁₆×C₈₉	—	2	1	no	—
129	Q₁₆×C₈₉	C₂×Q₈	1	2	yes	complemented
130	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
132	C₄⋊C₄×C₈₉	—	2	1	no	—
133	C₄⋊C₄×C₈₉	—	2	2	no	—
134	C₄⋊C₄×C₈₉	—	4	1	no	—
135	C₄⋊C₄×C₈₉	—	4	1	no	—
136	C₄²×C₈₉	—	4	1	no	—
137	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
138	C₁₆×C₈₉	—	2	2	no	—
139	C₂²×D₈₉	—	4	1	no	—
140	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
141	Q₈×C₈₉	—	2	2	no	—
142	Q₈×C₈₉	—	4	1	no	—
143	Q₈×C₈₉	—	8	2	no	—
144	C₂×C₄×C₈₉	—	4	1	no	—
145	C₂×C₄×C₈₉	—	4	1	no	—
146	C₂×C₄×C₈₉	—	4	2	no	—
147	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₈₉	—	4	1	no	—
151	C₂×Dic₈₉	—	4	2	no	—
152	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
153	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
154	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
155	D₃₅₆	—	2	2	no	—
156	D₃₅₆	—	4	4	no	—
157	D₃₅₆	—	4	1	no	—
158	C₄×D₈₉	—	2	2	no	—
159	C₄×D₈₉	—	2	2	no	—
160	C₄×D₈₉	—	4	4	no	—
161	C₄×D₈₉	—	8	4	no	—
162	C₄×D₈₉	—	8	2	no	—
163	C₄×D₈₉	—	8	2	no	—
164	C₄×D₈₉	C₂²⋊Q₈	1	2	yes	—
165	C₄×D₈₉	#137	1	2	yes	—
166	Dic₁₇₈	—	8	2	no	—
167	Dic₁₇₈	—	8	4	no	—
168	C₈×C₈₉	—	2	1	no	—
169	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
170	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
171	#10623494	—	2	1	no	—
172	#10623494	C₂²⋊Q₈	1	2	yes	—
173	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
174	C₂×D₈₉	—	2	1	no	—
175	C₂×D₈₉	—	2	1	no	—
176	C₂×D₈₉	—	4	2	no	—
177	C₂×D₈₉	—	8	1	no	—
178	C₂×D₈₉	—	8	1	no	—
179	C₂×D₈₉	#474	1	2	yes	—
180	C₄×C₈₉	—	2	1	no	—
181	C₄×C₈₉	—	4	2	no	—
182	C₄×C₈₉	—	8	2	no	—
183	C₄×C₈₉	—	8	1	no	—
184	C₄×C₈₉	#510	1	2	yes	—
185	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
186	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
187	Dic₈₉	—	2	1	no	—
188	Dic₈₉	—	8	2	no	—
189	Dic₈₉	—	8	1	no	—
190	Dic₈₉	#474	1	2	yes	—
191	Dic₈₉	#509	1	2	yes	—
192	#33629	—	89	1	no	maximal, Sylow
193	C₂×C₈₉	C₂×#474	1	1	yes	characteristic
194	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
195	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
196	D₈₉	—	2	2	no	—
197	D₈₉	—	8	2	no	—
198	C₂×#500	—	89	1	no	—
199	C₄×#135	—	89	1	no	—
200	#2227	—	89	1	no	—
201	#2227	D₈₉	1	1	yes	characteristic, complemented
202	#2226	—	89	1	no	—
203	#2226	—	89	1	no	—
204	#2225	—	89	2	no	—
205	#2224	—	89	2	no	—
206	#2159	—	89	2	no	—
207	#2150	—	89	1	no	—
208	#2139	—	89	2	no	—
209	C₈₉	#33629	1	1	yes	characteristic, complemented, Sylow
210	C₂×#135	—	89	1	no	—
211	C₂×C₄⋊Q₈	—	178	1	no	—
212	C₄×#41	—	178	1	no	—
213	C₂×M₃₂	—	89	2	no	—
214	#500	—	89	2	no	—
215	#500	—	89	1	no	—
216	#500	—	178	1	no	—
217	#500	—	178	1	no	—
218	#500	C₂×D₈₉	1	1	yes	characteristic
219	C₄×Q₁₆	—	89	1	no	—
220	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
221	C₄×SD₁₆	—	178	1	no	—
222	C₄×SD₁₆	—	178	1	no	—
223	C₄×D₈	—	89	1	no	—
224	C₄×D₈	—	89	1	no	—
225	#427	—	89	1	no	—
226	C₂×#107	—	89	2	no	—
227	C₂×#107	—	178	1	no	—
228	#416	—	178	1	no	—
229	C₂×C₄×C₈	—	89	1	no	—
230	#367	—	89	2	no	—
231	#367	—	89	2	no	—
232	#366	—	89	2	no	—
233	#366	C₂×D₈₉	1	2	yes	—
234	C₄⋊C₁₆	—	89	2	no	—
235	C₄⋊C₁₆	—	89	1	no	—
236	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
237	#358	—	89	2	no	—
238	#358	C₂×D₈₉	1	2	yes	—
239	#357	—	89	2	no	—
240	#357	—	89	2	no	—
241	C₂×#41	—	178	1	no	—
242	C₂²×Q₈	—	356	1	no	—
243	#135	—	89	4	no	—
244	#135	—	178	2	no	—
245	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
246	C₂×SD₁₆	—	178	1	no	—
247	C₂×D₈	—	89	1	no	—
248	C₂²×C₈	—	89	1	no	—
249	C₂²×C₈	—	89	1	no	—
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	—	356	1	no	—
252	C₄⋊Q₈	—	356	1	no	—
253	C₄⋊Q₈	—	178	2	no	—
254	C₄⋊Q₈	—	178	1	no	—
255	C₄×Q₈	—	2	1	no	—
256	C₄×Q₈	—	178	1	no	—
257	C₄×D₄	—	356	1	no	—
258	C₄×D₄	—	356	1	no	—
259	C₄×D₄	—	178	1	no	—
260	C₄×D₄	—	178	1	no	—
261	#117	—	356	1	no	—
262	#117	—	178	1	no	—
263	C₂×C₄⋊C₄	—	356	1	no	—
264	C₂×C₄⋊C₄	—	178	2	no	—
265	C₂×C₄²	—	89	1	no	—
266	C₂×C₄²	—	356	1	no	—
267	M₃₂	—	178	4	no	—
268	C₂×C₁₆	—	89	2	no	—
269	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
270	#107	—	2	1	no	—
271	#107	—	89	2	no	—
272	#107	—	89	1	no	—
273	#107	—	178	2	no	—
274	#107	—	178	2	no	—
275	#107	—	178	1	no	—
276	#107	C₂²×D₈₉	1	2	yes	—
277	#107	C₂²×D₈₉	1	1	yes	characteristic
278	#106	—	178	1	no	—
279	#103	—	2	1	no	—
280	#103	—	178	1	no	—
281	#102	—	178	1	no	—
282	#102	—	178	1	no	—
283	C₄×C₈	—	89	1	no	—
284	C₄×C₈	—	89	1	no	—
285	C₄×C₈	—	89	1	no	—
286	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
287	#41	—	356	2	no	—
288	#41	—	178	4	no	—
289	C₂×Q₈	—	2	1	no	—
290	C₂×Q₈	—	4	1	no	—
291	C₂×Q₈	—	356	2	no	—
292	C₂×Q₈	—	356	1	no	—
293	C₂×Q₈	—	712	4	no	—
294	C₂×D₄	—	356	1	no	—
295	C₂×D₄	—	178	1	no	—
296	C₂²×C₄	—	89	1	no	—
297	C₂²×C₄	—	89	1	no	—
298	C₂²×C₄	—	89	1	no	—
299	C₂²×C₄	—	356	1	no	—
300	C₂²×C₄	—	356	1	no	—
301	C₂²×C₄	—	356	2	no	—
302	C₂²×C₄	—	356	1	no	—
303	Q₁₆	—	2	1	no	—
304	Q₁₆	Q₈×D₈₉	1	2	yes	complemented
305	SD₁₆	—	178	4	no	—
306	D₈	—	89	2	no	—
307	D₈	—	178	1	no	—
308	C₂×C₈	—	89	1	no	—
309	C₂×C₈	—	89	2	no	—
310	C₂×C₈	—	89	2	no	—
311	C₂×C₈	—	89	1	no	—
312	C₂×C₈	—	178	2	no	—
313	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
314	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
315	C₄⋊C₄	—	2	1	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	4	1	no	—
318	C₄⋊C₄	—	4	1	no	—
319	C₄⋊C₄	—	356	2	no	—
320	C₄⋊C₄	—	356	2	no	—
321	C₄⋊C₄	—	356	1	no	—
322	C₄⋊C₄	—	356	1	no	—
323	C₄⋊C₄	—	178	1	no	—
324	C₄⋊C₄	—	178	2	no	—
325	C₂²⋊C₄	—	356	1	no	—
326	C₂²⋊C₄	—	356	1	no	—
327	C₂²⋊C₄	—	356	1	no	—
328	C₄²	—	4	1	no	—
329	C₄²	—	89	1	no	—
330	C₄²	—	89	1	no	—
331	C₄²	—	89	1	no	—
332	C₄²	—	356	1	no	—
333	C₄²	—	356	1	no	—
334	C₄²	D₄×D₈₉	1	1	yes	characteristic
335	C₁₆	—	2	2	no	—
336	C₁₆	—	178	2	no	—
337	C₂³	—	89	1	no	—
338	C₂³	—	356	1	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	4	1	no	—
341	Q₈	—	8	2	no	—
342	Q₈	—	712	2	no	—
343	Q₈	—	712	4	no	—
344	D₄	—	356	4	no	—
345	D₄	—	356	1	no	—
346	D₄	—	178	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	89	1	no	—
351	C₂×C₄	—	89	1	no	—
352	C₂×C₄	—	89	2	no	—
353	C₂×C₄	—	89	2	no	—
354	C₂×C₄	—	89	1	no	—
355	C₂×C₄	—	356	4	no	—
356	C₂×C₄	—	356	1	no	—
357	C₂×C₄	—	356	2	no	—
358	C₂×C₄	—	178	2	no	—
359	C₂×C₄	—	178	2	no	—
360	C₂×C₄	—	712	2	no	—
361	C₂×C₄	—	712	4	no	—
362	C₂×C₄	—	712	2	no	—
363	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
364	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
365	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	89	2	no	—
368	C₈	—	178	1	no	—
369	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
370	C₈	#41×D₈₉	1	1	yes	characteristic
371	C₂²	—	89	2	no	—
372	C₂²	—	356	2	no	—
373	C₂²	—	178	1	no	—
374	C₂²	—	178	1	no	—
375	C₂²	—	712	1	no	—
376	C₂²	—	712	1	no	—
377	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
378	C₄	—	2	1	no	—
379	C₄	—	4	2	no	—
380	C₄	—	8	2	no	—
381	C₄	—	8	1	no	—
382	C₄	—	89	2	no	—
383	C₄	—	89	2	no	—
384	C₄	—	178	1	no	—
385	C₄	—	712	2	no	—
386	C₄	—	712	1	no	—
387	C₄	#19152040	1	2	yes	—
388	C₄	C₂²⋊Q₈×D₈₉	1	1	yes	characteristic
389	C₄	#137×D₈₉	1	1	yes	characteristic
390	C₂	—	178	2	no	—
391	C₂	—	712	2	no	—
392	C₂	#474×D₈₉	1	1	yes	characteristic
393	C₂	#509×D₈₉	1	1	yes	characteristic
394	C₂	C₂×#19152040	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	#33629
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. #500×D₈₉	C₂	semidirect
3. #26022404	C₂	semidirect
4. #26022400	C₂	semidirect
5. #26022396	C₂	semidirect
6. #26022392	C₂	semidirect
7. #26022388	C₂	semidirect
8. #26022037	C₂	semidirect
9. #26022006	C₂	semidirect
11. C₈₉×#2227	C₂	semidirect
46. #15668035	Q₈	semidirect
105. D₇₁₂	C₂×Q₈	semidirect
129. Q₁₆×C₈₉	C₂×Q₈	semidirect
201. #2227	D₈₉	semidirect
209. C₈₉	#33629	semidirect
304. Q₁₆	Q₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#500×D₈₉	2
C₄×#15668035	2
#26022404	2
#26022400	2
#26022396	2
#26022392	2
#26022388	2
#26022037	2
#26022006	2
#26021963	2
C₈₉×#2227	2
#33629	89