Subgroups — #36525107

Order 22784.

560 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030835	C₂	1	1	yes	maximal, characteristic
2	#26029348	C₂	1	2	yes	maximal
3	#26019159	C₂	1	2	yes	maximal, complemented
4	#26019150	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019068	C₂	1	2	yes	maximal, complemented
6	Q₁₆×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019058	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019014	C₂	1	1	yes	maximal, characteristic
9	#26018723	C₂	1	2	yes	maximal
10	#26018534	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152152	C₂²	1	2	yes	—
13	#19152002	C₄	1	1	yes	characteristic
14	#19151976	C₄	1	1	yes	characteristic
15	#19151947	C₄	1	2	yes	—
16	#19151840	C₄	1	1	yes	characteristic
17	#19151838	C₄	1	2	yes	—
18	#19151836	C₄	1	1	yes	characteristic
19	#19151728	C₂²	1	2	yes	—
20	#19151726	—	2	2	no	—
21	#19151725	—	2	2	no	—
22	Q₈×#10623494	C₂²	1	2	yes	—
23	#19151722	C₂²	1	2	yes	—
24	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
25	C₄×#13083564	C₂²	1	2	yes	—
26	#19151660	—	2	2	no	—
27	#19151658	—	2	2	no	—
28	#19151654	C₂²	1	2	yes	—
29	#19151649	C₂²	1	1	yes	characteristic
30	#19151649	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
32	#19151351	C₂²	1	1	yes	characteristic
33	#19151348	C₂²	1	2	yes	—
34	#19151330	C₂²	1	2	yes	—
35	Q₈⋊#10623494	C₂²	1	2	yes	—
36	#19151328	C₂²	1	2	yes	—
37	#19151320	C₂²	1	1	yes	characteristic
38	#19151320	C₂²	1	1	yes	characteristic
39	C₈×#10623494	C₂²	1	1	yes	characteristic
40	#446×C₈₉	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
42	#335×C₈₉	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₈₉	C₂²	1	2	yes	—
44	C₈²×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
46	C₂×#13083564	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
49	#15668011	C₂×C₄	1	2	yes	—
50	#15668010	C₂×C₄	1	2	yes	—
51	#15668002	—	2	2	no	—
52	#15667992	C₂×C₄	1	2	yes	—
53	#15667990	—	2	2	no	—
54	#15667988	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₇₈	—	2	2	no	—
56	C₄×Dic₁₇₈	C₂³	1	2	yes	—
57	#15667969	—	2	2	no	—
58	C₄²×D₈₉	C₂³	1	1	yes	characteristic
59	#15667946	C₂×C₄	1	1	yes	characteristic
60	#15667945	C₂×C₄	1	2	yes	—
61	C₄⋊#10623494	—	2	2	no	—
62	C₄⋊#10623494	C₂³	1	2	yes	—
63	#15667941	—	2	2	no	—
64	#15667941	—	2	1	no	—
65	#15667941	—	2	2	no	—
66	#15667941	D₄	1	1	yes	characteristic
67	#15667941	D₄	1	1	yes	characteristic
68	#15667937	C₂×C₄	1	2	yes	—
69	#15667936	C₂×C₄	1	2	yes	—
70	#15667935	C₂×C₄	1	2	yes	—
71	#15667924	—	2	2	no	—
72	#15667924	—	2	1	no	—
73	#15667924	D₄	1	1	yes	characteristic
74	#15667924	D₄	1	1	yes	characteristic
75	#15667922	—	2	2	no	—
76	#15667922	—	2	2	no	—
77	C₄×#10623494	—	2	2	no	—
78	C₄×#10623494	C₂³	1	1	yes	characteristic
79	C₄×#10623494	C₂³	1	1	yes	characteristic
80	C₄×#10623494	C₂³	1	1	yes	characteristic
81	C₈×Dic₈₉	—	2	2	no	—
82	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
84	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
86	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₈₉	—	2	2	no	—
88	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
89	#103×C₈₉	C₂×C₄	1	2	yes	—
90	#97×C₈₉	—	2	2	no	—
91	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₇₈	—	2	2	no	—
95	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₈₉	#41	1	1	yes	characteristic
99	Dic₃₅₆	—	2	2	no	—
100	#13083564	—	2	4	no	—
101	C₂×#10623494	—	2	2	no	—
102	C₂×#10623494	—	2	1	no	—
103	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
104	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
105	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
106	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
107	C₈×D₈₉	—	2	2	no	—
108	C₈×D₈₉	—	2	2	no	—
109	C₄⋊Dic₈₉	—	2	2	no	—
110	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
111	#13083553	—	2	2	no	—
112	#13083553	—	2	2	no	—
113	#13083553	—	2	2	no	—
114	#13083553	—	2	2	no	—
115	#13083553	—	2	2	no	—
116	#13083551	—	2	2	no	—
117	#13083551	—	2	2	no	—
118	C₄×Dic₈₉	—	2	2	no	—
119	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₈₉	#41	1	1	yes	characteristic
122	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
123	Q₁₆×C₈₉	—	2	2	no	—
124	C₂×C₈×C₈₉	—	2	2	no	—
125	C₂×C₈×C₈₉	—	2	1	no	—
126	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₄⋊C₄×C₈₉	—	2	2	no	—
131	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
132	C₄²×C₈₉	—	2	2	no	—
133	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
134	C₂²×D₈₉	C₄×D₄	1	1	yes	characteristic
135	Q₈×C₈₉	—	2	2	no	—
136	Q₈×C₈₉	—	2	2	no	—
137	C₂×C₄×C₈₉	—	2	2	no	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
142	C₂×Dic₈₉	—	2	2	no	—
143	C₂×Dic₈₉	—	2	2	no	—
144	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
146	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
147	C₄×D₈₉	—	2	1	no	—
148	C₄×D₈₉	—	2	1	no	—
149	C₄×D₈₉	—	2	1	no	—
150	C₄×D₈₉	—	2	1	no	—
151	C₄×D₈₉	—	4	1	no	—
152	Dic₁₇₈	—	2	2	no	—
153	Dic₁₇₈	—	2	2	no	—
154	Dic₁₇₈	—	8	2	no	—
155	C₈×C₈₉	—	2	1	no	—
156	C₈×C₈₉	—	2	1	no	—
157	C₈×C₈₉	—	2	1	no	—
158	C₈×C₈₉	—	4	1	no	—
159	C₈×C₈₉	—	4	2	no	—
160	C₈×C₈₉	#131	1	2	yes	—
161	#10623494	—	2	1	no	—
162	#10623494	—	2	1	no	—
163	#10623494	—	2	1	no	—
164	#10623494	—	4	1	no	—
165	#10623494	—	4	2	no	—
166	#10623494	#131	1	2	yes	—
167	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂×D₈₉	—	2	1	no	—
169	C₂×D₈₉	—	2	1	no	—
170	C₂×D₈₉	—	2	1	no	—
171	C₄×C₈₉	—	2	1	no	—
172	C₄×C₈₉	—	4	2	no	—
173	C₄×C₈₉	—	4	2	no	—
174	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
175	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
176	C₄×C₈₉	#578	1	1	yes	characteristic
177	C₄×C₈₉	#578	1	1	yes	characteristic
178	Dic₈₉	—	2	1	no	—
179	Dic₈₉	—	2	1	no	—
180	Dic₈₉	—	2	1	no	—
181	Dic₈₉	—	4	2	no	—
182	Dic₈₉	—	4	2	no	—
183	#17972	—	89	1	no	maximal, Sylow
184	C₂×C₈₉	#2931	1	1	yes	characteristic
185	C₂×C₈₉	#2950	1	1	yes	characteristic
186	C₂×C₈₉	#2957	1	1	yes	characteristic
187	D₈₉	—	4	1	no	—
188	#3170	—	89	1	no	—
189	#2985	—	89	2	no	—
190	#1593	—	89	2	no	—
191	#1591	—	89	1	no	—
192	#1591	D₈₉	1	1	yes	characteristic, complemented
193	#1579	—	89	2	no	—
194	C₈×Q₁₆	—	89	1	no	—
195	C₈×Q₁₆	—	89	1	no	—
196	#1568	—	89	1	no	—
197	#1497	—	89	2	no	—
198	#1450	—	89	1	no	—
199	C₈₉	#17972	1	1	yes	characteristic, complemented, Sylow
200	#533	—	89	2	no	—
201	#500	—	89	1	no	—
202	#494	—	89	1	no	—
203	#487	—	89	2	no	—
204	#467	—	89	1	no	—
205	#467	—	89	1	no	—
206	#467	—	89	2	no	—
207	#446	—	89	2	no	—
208	#446	—	178	2	no	—
209	#446	—	178	2	no	—
210	#446	C₂×D₈₉	1	2	yes	—
211	C₈×Q₈	—	89	2	no	—
212	C₈×Q₈	—	89	2	no	—
213	C₄×Q₁₆	—	89	2	no	—
214	C₄×Q₁₆	—	89	1	no	—
215	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
216	#433	—	178	2	no	—
217	#433	—	178	2	no	—
218	#432	—	89	2	no	—
219	#431	—	89	1	no	—
220	#431	—	89	1	no	—
221	C₂×C₄×C₈	—	89	1	no	—
222	#335	—	89	1	no	—
223	#335	C₂×D₈₉	1	1	yes	characteristic
224	#334	—	89	2	no	—
225	Q₈⋊C₈	—	89	2	no	—
226	Q₈⋊C₈	—	89	2	no	—
227	Q₈⋊C₈	—	89	2	no	—
228	Q₈⋊C₈	C₂×D₈₉	1	2	yes	—
229	#322	—	89	1	no	—
230	#322	—	89	1	no	—
231	C₈²	—	89	1	no	—
232	C₈²	C₂×D₈₉	1	1	yes	characteristic
233	C₂×Q₁₆	—	89	2	no	—
234	C₂×Q₁₆	—	89	1	no	—
235	C₂×Q₁₆	C₄×D₈₉	1	1	yes	characteristic
236	C₂²×C₈	—	89	1	no	—
237	C₂²×C₈	—	89	1	no	—
238	C₄⋊Q₈	—	89	2	no	—
239	C₄⋊Q₈	—	89	2	no	—
240	#125	—	178	2	no	—
241	C₂²⋊Q₈	—	89	2	no	—
242	C₂²⋊Q₈	—	89	2	no	—
243	C₂²⋊Q₈	—	178	2	no	—
244	C₄×Q₈	—	89	2	no	—
245	C₄×Q₈	—	178	2	no	—
246	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
247	#117	—	178	2	no	—
248	C₂×C₄²	—	89	1	no	—
249	#107	—	89	1	no	—
250	#107	—	89	2	no	—
251	#107	C₄×D₈₉	1	1	yes	characteristic
252	C₄⋊C₈	—	2	2	no	—
253	C₄⋊C₈	—	89	1	no	—
254	C₄⋊C₈	—	89	1	no	—
255	C₄⋊C₈	—	89	2	no	—
256	C₄⋊C₈	—	178	2	no	—
257	C₄⋊C₈	—	178	1	no	—
258	C₄⋊C₈	—	178	2	no	—
259	C₄⋊C₈	—	178	2	no	—
260	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
261	#103	—	89	2	no	—
262	#103	—	89	2	no	—
263	#103	—	89	2	no	—
264	#103	C₄×D₈₉	1	2	yes	—
265	C₂²⋊C₈	—	89	1	no	—
266	C₂²⋊C₈	—	89	1	no	—
267	C₂²⋊C₈	—	178	1	no	—
268	C₂²⋊C₈	—	178	2	no	—
269	#97	—	2	2	no	—
270	#97	—	178	2	no	—
271	#97	—	178	2	no	—
272	C₄×C₈	—	89	1	no	—
273	C₄×C₈	—	89	1	no	—
274	C₄×C₈	—	89	1	no	—
275	C₄×C₈	—	89	1	no	—
276	C₄×C₈	—	89	1	no	—
277	C₄×C₈	—	178	2	no	—
278	C₄×C₈	—	178	2	no	—
279	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
281	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
282	C₂×Q₈	—	89	2	no	—
283	C₂×Q₈	—	178	2	no	—
284	C₂×Q₈	C₂×C₄×D₈₉	1	2	yes	—
285	C₂²×C₄	—	89	1	no	—
286	C₂²×C₄	—	89	1	no	—
287	C₂²×C₄	—	89	1	no	—
288	Q₁₆	—	2	2	no	—
289	Q₁₆	—	178	4	no	—
290	Q₁₆	—	178	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	89	1	no	—
294	C₂×C₈	—	89	1	no	—
295	C₂×C₈	—	89	1	no	—
296	C₂×C₈	—	89	1	no	—
297	C₂×C₈	—	178	2	no	—
298	C₂×C₈	—	178	2	no	—
299	C₂×C₈	—	178	1	no	—
300	C₂×C₈	—	178	2	no	—
301	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
303	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
304	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	89	2	no	—
307	C₄⋊C₄	—	178	2	no	—
308	C₄⋊C₄	—	178	2	no	—
309	C₄⋊C₄	—	178	2	no	—
310	C₄⋊C₄	—	178	2	no	—
311	C₄⋊C₄	—	178	2	no	—
312	C₄⋊C₄	—	178	2	no	—
313	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
314	C₂²⋊C₄	—	178	2	no	—
315	C₂²⋊C₄	—	178	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	89	1	no	—
318	C₄²	—	89	1	no	—
319	C₄²	—	89	1	no	—
320	C₄²	—	178	2	no	—
321	C₄²	C₂³×D₈₉	1	1	yes	characteristic
322	C₂³	—	89	1	no	—
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	Q₈	—	178	2	no	—
326	Q₈	—	178	2	no	—
327	Q₈	—	712	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	89	1	no	—
331	C₂×C₄	—	89	1	no	—
332	C₂×C₄	—	89	1	no	—
333	C₂×C₄	—	356	1	no	—
334	C₂×C₄	—	178	2	no	—
335	C₂×C₄	—	178	2	no	—
336	C₂×C₄	—	178	1	no	—
337	C₂×C₄	—	178	1	no	—
338	C₂×C₄	—	178	1	no	—
339	C₂×C₄	—	178	1	no	—
340	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
342	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	4	1	no	—
347	C₈	—	4	2	no	—
348	C₈	—	89	2	no	—
349	C₈	—	356	1	no	—
350	C₈	—	356	2	no	—
351	C₈	—	178	1	no	—
352	C₈	—	178	1	no	—
353	C₈	—	178	1	no	—
354	C₈	#15668020	1	2	yes	—
355	C₂²	—	178	1	no	—
356	C₂²	—	178	1	no	—
357	C₂²	—	178	1	no	—
358	C₂²	C₄×D₄×D₈₉	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	2	no	—
362	C₄	—	356	2	no	—
363	C₄	—	356	2	no	—
364	C₄	—	178	1	no	—
365	C₄	—	178	1	no	—
366	C₄	—	178	1	no	—
367	C₄	#131×D₈₉	1	1	yes	characteristic
368	C₄	C₂×#15668020	1	1	yes	characteristic
369	C₄	#19152411	1	1	yes	characteristic
370	C₄	#19152412	1	1	yes	characteristic
371	C₂	—	356	1	no	—
372	C₂	#26028961	1	1	yes	characteristic
373	C₂	#26029115	1	1	yes	characteristic
374	C₂	#26029164	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1591	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂²×D₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1591	D₈₉
hypocenter	C₈₉	#17972
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
3. #26019159	C₂	semidirect
4. #26019150	C₂	semidirect
5. #26019068	C₂	semidirect
6. Q₁₆×#10623494	C₂	semidirect
7. #26019058	C₂	semidirect
11. C₈₉×#1591	C₂	semidirect
192. #1591	D₈₉	semidirect
199. C₈₉	#17972	semidirect

Maximal subgroups

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

subgroup	index
#26030835	2
#26029348	2
#26019159	2
#26019150	2
#26019068	2
Q₁₆×#10623494	2
#26019058	2
#26019014	2
#26018723	2
#26018534	2
C₈₉×#1591	2
#17972	89