Subgroups — #36709927

Order 22784.

482 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030765	C₂	1	1	yes	maximal, characteristic
2	#26030763	C₂	1	1	yes	maximal, characteristic
3	#26022539	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022538	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022537	C₂	1	2	yes	maximal, complemented
6	#26022536	C₂	1	2	yes	maximal, complemented
7	#26022535	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022035	C₂	1	2	yes	maximal
9	#26022033	C₂	1	2	yes	maximal
10	#346×D₈₉	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2255	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152154	—	2	1	no	—
13	#19152153	—	2	1	no	—
14	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152002	C₂²	1	1	yes	characteristic
17	#19152002	C₂²	1	1	yes	characteristic
18	#19152000	C₂²	1	1	yes	characteristic
19	#19151999	—	2	1	no	—
20	#19151999	C₂²	1	2	yes	—
21	#19151997	—	2	1	no	—
22	#19151995	—	2	1	no	—
23	#19151994	—	2	1	no	—
24	#19151635	—	2	1	no	—
25	#19151635	C₂²	1	1	yes	characteristic
26	#19151635	C₂²	1	2	yes	—
27	#19151635	C₂²	1	1	yes	characteristic
28	#19151633	—	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	#19151425	C₂²	1	2	yes	—
33	#19151424	C₂²	1	2	yes	—
34	#19151423	C₂²	1	2	yes	—
35	#19151423	C₂²	1	2	yes	—
36	#19151374	C₂²	1	1	yes	characteristic
37	#19151373	C₂²	1	2	yes	—
38	#500×C₈₉	C₂²	1	1	yes	characteristic
39	#499×C₈₉	C₂²	1	1	yes	characteristic
40	#367×C₈₉	C₂²	1	2	yes	—
41	#366×C₈₉	C₂²	1	2	yes	—
42	#346×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
44	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
45	#15668012	—	2	1	no	—
46	#15668011	—	2	1	no	—
47	#15668011	—	2	1	no	—
48	#15668004	—	2	1	no	—
49	#15668003	—	2	2	no	—
50	#15667992	—	4	1	no	—
51	#15667990	—	4	1	no	—
52	#15667990	—	4	1	no	—
53	#15667988	—	4	1	no	—
54	Q₈×Dic₈₉	—	4	1	no	—
55	#15667977	—	4	1	no	—
56	#15667977	—	4	1	no	—
57	#15667977	—	4	1	no	—
58	#15667970	—	2	1	no	—
59	#15667970	—	2	1	no	—
60	#15667970	—	2	2	no	—
61	C₄²×D₈₉	D₄	1	1	yes	characteristic
62	C₂×#13083549	C₂³	1	2	yes	—
63	C₁₆×D₈₉	Q₈	1	8	yes	complemented
64	#15667946	—	2	1	no	—
65	#15667946	C₂³	1	1	yes	characteristic
66	#15667946	C₂³	1	2	yes	—
67	#15667946	C₂³	1	1	yes	characteristic
68	#15667945	—	2	1	no	—
69	#15667945	—	2	1	no	—
70	#15667944	—	2	1	no	—
71	#15667943	—	2	2	no	—
72	#15667943	—	2	2	no	—
73	#15667943	—	2	1	no	—
74	#15667943	—	2	1	no	—
75	C₄×#10623494	D₄	1	1	yes	characteristic
76	C₈×Dic₈₉	D₄	1	1	yes	characteristic
77	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
78	C₄⋊Q₈×C₈₉	—	2	1	no	—
79	#125×C₈₉	—	2	1	no	—
80	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
81	#107×C₈₉	—	2	1	no	—
82	#107×C₈₉	C₂³	1	1	yes	characteristic
83	#107×C₈₉	C₂³	1	2	yes	—
84	#107×C₈₉	C₂³	1	1	yes	characteristic
85	#106×C₈₉	—	2	1	no	—
86	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
87	C₂×Dic₁₇₈	—	4	1	no	—
88	C₂×Dic₁₇₈	—	4	1	no	—
89	C₂×Dic₁₇₈	—	4	1	no	—
90	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
91	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
92	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
94	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
95	C₈×D₈₉	Q₁₆	1	4	yes	—
96	C₈×D₈₉	C₂×Q₈	1	4	yes	—
97	C₄⋊Dic₈₉	—	2	2	no	—
98	C₄⋊Dic₈₉	—	2	1	no	—
99	C₄⋊Dic₈₉	—	2	1	no	—
100	C₄⋊Dic₈₉	—	4	1	no	—
101	C₄⋊Dic₈₉	—	4	2	no	—
102	#13083553	—	4	1	no	—
103	#13083553	—	4	1	no	—
104	#13083553	—	4	2	no	—
105	#13083553	—	4	1	no	—
106	#13083553	—	4	1	no	—
107	#13083551	—	4	2	no	—
108	#13083551	—	4	1	no	—
109	#13083551	—	4	1	no	—
110	C₄×Dic₈₉	—	4	1	no	—
111	C₄×Dic₈₉	—	4	1	no	—
112	C₄×Dic₈₉	—	4	2	no	—
113	C₄×Dic₈₉	D₈	1	1	yes	characteristic
114	C₄×Dic₈₉	D₈	1	1	yes	characteristic
115	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
116	#13083549	C₂×Q₈	1	4	yes	—
117	C₂×Q₈×C₈₉	—	4	1	no	—
118	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₈₉	—	2	2	no	—
121	C₄⋊C₄×C₈₉	—	2	1	no	—
122	C₄⋊C₄×C₈₉	—	2	1	no	—
123	C₄⋊C₄×C₈₉	—	4	1	no	—
124	C₄⋊C₄×C₈₉	—	4	1	no	—
125	C₄⋊C₄×C₈₉	—	4	1	no	—
126	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₁₆×C₈₉	C₂×Q₈	1	4	yes	—
128	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
129	Q₈×C₈₉	—	8	2	no	—
130	C₂×C₄×C₈₉	—	4	1	no	—
131	C₂×C₄×C₈₉	—	4	1	no	—
132	C₂×C₄×C₈₉	—	4	2	no	—
133	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₈₉	—	4	1	no	—
137	C₂×Dic₈₉	—	4	1	no	—
138	C₂×Dic₈₉	—	4	2	no	—
139	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
140	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
141	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
142	C₄×D₈₉	—	4	1	no	—
143	C₄×D₈₉	—	4	1	no	—
144	C₄×D₈₉	C₄⋊Q₈	1	2	yes	—
145	C₄×D₈₉	C₂×Q₁₆	1	2	yes	—
146	Dic₁₇₈	—	8	2	no	—
147	Dic₁₇₈	—	8	2	no	—
148	Dic₁₇₈	—	8	2	no	—
149	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
150	C₈×C₈₉	C₂²×Q₈	1	2	yes	—
151	#10623494	C₄⋊Q₈	1	2	yes	—
152	#10623494	C₂×Q₁₆	1	2	yes	—
153	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
154	C₂×D₈₉	—	2	1	no	—
155	C₂×D₈₉	—	2	1	no	—
156	C₂×D₈₉	#500	1	2	yes	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	8	1	no	—
160	C₄×C₈₉	—	8	1	no	—
161	C₄×C₈₉	—	8	2	no	—
162	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
163	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
164	Dic₈₉	—	2	1	no	—
165	Dic₈₉	—	2	1	no	—
166	Dic₈₉	—	8	1	no	—
167	Dic₈₉	—	8	1	no	—
168	Dic₈₉	—	8	2	no	—
169	Dic₈₉	#500	1	2	yes	—
170	#33728	—	89	1	no	maximal, Sylow
171	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
172	C₂×C₈₉	#3414	1	1	yes	characteristic
173	C₂×C₈₉	#3414	1	1	yes	characteristic
174	D₈₉	—	2	2	no	—
175	#3160	—	89	1	no	—
176	#3160	—	89	1	no	—
177	#2255	—	89	1	no	—
178	#2255	—	89	1	no	—
179	#2255	—	89	2	no	—
180	#2255	—	89	2	no	—
181	#2255	—	89	1	no	—
182	#2255	D₈₉	1	1	yes	characteristic, complemented
183	#2158	—	89	2	no	—
184	#2158	—	89	2	no	—
185	C₂×#346	—	89	1	no	—
186	C₈₉	#33728	1	1	yes	characteristic, complemented, Sylow
187	#533	—	178	1	no	—
188	#533	—	178	1	no	—
189	C₂²×C₁₆	—	89	2	no	—
190	#500	—	89	1	no	—
191	#500	—	89	1	no	—
192	#500	—	89	1	no	—
193	#500	C₂×D₈₉	1	1	yes	characteristic
194	#499	—	89	2	no	—
195	#499	—	89	1	no	—
196	#499	—	178	1	no	—
197	#499	—	178	1	no	—
198	#499	C₂×D₈₉	1	1	yes	characteristic
199	#498	—	178	1	no	—
200	#498	—	178	1	no	—
201	#427	—	89	1	no	—
202	#427	—	89	2	no	—
203	#427	—	89	1	no	—
204	#427	—	178	1	no	—
205	#427	—	178	1	no	—
206	C₂×C₄×C₈	—	89	1	no	—
207	#367	—	89	2	no	—
208	#367	—	89	2	no	—
209	#367	—	89	2	no	—
210	#367	C₂×D₈₉	1	2	yes	—
211	#366	—	89	2	no	—
212	#366	—	89	2	no	—
213	#366	—	89	2	no	—
214	#366	C₂×D₈₉	1	2	yes	—
215	#346	—	89	2	no	—
216	#346	—	89	1	no	—
217	#346	C₂×D₈₉	1	1	yes	characteristic
218	C₂²×C₈	—	89	1	no	—
219	C₂²×C₈	—	89	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	178	1	no	—
222	C₄⋊Q₈	—	178	1	no	—
223	C₄⋊Q₈	—	178	1	no	—
224	#125	—	2	1	no	—
225	#125	—	178	2	no	—
226	#125	—	178	1	no	—
227	C₂²⋊Q₈	—	356	1	no	—
228	C₂²⋊Q₈	—	356	1	no	—
229	C₂²⋊Q₈	—	356	1	no	—
230	C₂²⋊Q₈	—	356	1	no	—
231	C₄×Q₈	—	356	1	no	—
232	C₄×Q₈	—	356	1	no	—
233	C₄×Q₈	—	356	1	no	—
234	C₄×Q₈	—	356	1	no	—
235	#117	—	178	1	no	—
236	#117	—	178	1	no	—
237	#117	—	178	2	no	—
238	C₂×C₄²	—	89	1	no	—
239	C₂×C₁₆	—	89	2	no	—
240	C₂×C₁₆	—	89	8	no	—
241	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
242	#107	—	2	1	no	—
243	#107	—	89	1	no	—
244	#107	—	89	2	no	—
245	#107	—	89	1	no	—
246	#107	—	178	1	no	—
247	#107	—	178	1	no	—
248	#107	—	178	1	no	—
249	#107	C₂²×D₈₉	1	1	yes	characteristic
250	#107	C₂²×D₈₉	1	2	yes	—
251	#107	C₂²×D₈₉	1	1	yes	characteristic
252	#106	—	2	1	no	—
253	#106	—	178	1	no	—
254	#106	—	178	1	no	—
255	#106	—	178	2	no	—
256	#106	—	178	2	no	—
257	#106	—	178	1	no	—
258	C₄×C₈	—	89	1	no	—
259	C₄×C₈	—	89	1	no	—
260	C₄×C₈	—	89	1	no	—
261	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
262	C₂×Q₈	—	4	1	no	—
263	C₂×Q₈	—	356	1	no	—
264	C₂×Q₈	—	356	1	no	—
265	C₂×Q₈	—	356	1	no	—
266	C₂²×C₄	—	89	1	no	—
267	C₂²×C₄	—	89	1	no	—
268	C₂²×C₄	—	89	1	no	—
269	C₂×C₈	—	89	1	no	—
270	C₂×C₈	—	89	1	no	—
271	C₂×C₈	—	89	4	no	—
272	C₂×C₈	—	89	4	no	—
273	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	356	2	no	—
282	C₄⋊C₄	—	356	2	no	—
283	C₄⋊C₄	—	356	1	no	—
284	C₄⋊C₄	—	356	1	no	—
285	C₄⋊C₄	—	356	1	no	—
286	C₄⋊C₄	—	356	1	no	—
287	C₄⋊C₄	—	356	1	no	—
288	C₄⋊C₄	—	178	2	no	—
289	C₄⋊C₄	—	178	1	no	—
290	C₄⋊C₄	—	178	1	no	—
291	C₂²⋊C₄	—	356	1	no	—
292	C₂²⋊C₄	—	356	1	no	—
293	C₂²⋊C₄	—	356	2	no	—
294	C₄²	—	89	1	no	—
295	C₄²	—	89	1	no	—
296	C₄²	—	89	1	no	—
297	C₄²	—	356	1	no	—
298	C₄²	—	356	1	no	—
299	C₄²	—	356	2	no	—
300	C₄²	D₄×D₈₉	1	1	yes	characteristic
301	C₁₆	—	89	4	no	—
302	C₁₆	Q₈×D₈₉	1	4	yes	—
303	C₂³	—	89	1	no	—
304	Q₈	—	8	2	no	—
305	Q₈	—	712	2	no	—
306	Q₈	—	712	2	no	—
307	Q₈	—	712	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	89	1	no	—
312	C₂×C₄	—	89	2	no	—
313	C₂×C₄	—	89	2	no	—
314	C₂×C₄	—	89	1	no	—
315	C₂×C₄	—	89	1	no	—
316	C₂×C₄	—	356	1	no	—
317	C₂×C₄	—	356	1	no	—
318	C₂×C₄	—	356	1	no	—
319	C₂×C₄	—	356	1	no	—
320	C₂×C₄	—	356	2	no	—
321	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
322	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
323	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
324	C₈	—	89	2	no	—
325	C₈	—	89	2	no	—
326	C₈	Q₁₆×D₈₉	1	2	yes	—
327	C₈	C₂×Q₈×D₈₉	1	2	yes	—
328	C₂²	—	89	2	no	—
329	C₂²	—	178	1	no	—
330	C₂²	—	178	1	no	—
331	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	1	no	—
337	C₄	—	89	2	no	—
338	C₄	—	178	1	no	—
339	C₄	—	178	1	no	—
340	C₄	—	712	1	no	—
341	C₄	—	712	2	no	—
342	C₄	—	712	1	no	—
343	C₄	C₄⋊Q₈×D₈₉	1	1	yes	characteristic
344	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
345	C₂	—	178	2	no	—
346	C₂	#500×D₈₉	1	1	yes	characteristic
347	C₂	#26033384	1	1	yes	characteristic
348	C₂	#26033387	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×#2255	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₈₉	C₂×D₈
hypercenter	#2255	D₈₉
hypocenter	C₈₉	#33728
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2255

Complements

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

N	H ≅ G/N	type
3. #26022539	C₂	semidirect
4. #26022538	C₂	semidirect
5. #26022537	C₂	semidirect
6. #26022536	C₂	semidirect
7. #26022535	C₂	semidirect
11. C₈₉×#2255	C₂	semidirect
63. C₁₆×D₈₉	Q₈	semidirect
182. #2255	D₈₉	semidirect
186. C₈₉	#33728	semidirect

Maximal subgroups

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

subgroup	index
#26030765	2
#26030763	2
#26022539	2
#26022538	2
#26022537	2
#26022536	2
#26022535	2
#26022035	2
#26022033	2
#346×D₈₉	2
C₈₉×#2255	2
#33728	89