Subgroups — #36523899

Order 22784.

678 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029331	C₂	1	2	yes	maximal
2	C₄×Q₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#26019072	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019071	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019070	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#13083564	C₂	1	2	yes	maximal, complemented
7	#26019056	C₂	1	2	yes	maximal, complemented
8	#26018997	C₂	1	1	yes	maximal, characteristic
9	#26018645	C₂	1	2	yes	maximal
10	#26018537	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1579	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₈₉	C₄	1	1	yes	characteristic
13	C₄×Q₈×D₈₉	C₂²	1	2	yes	—
14	Q₈×#10623494	C₂²	1	2	yes	—
15	C₈×Dic₁₇₈	—	2	2	no	—
16	C₈×Dic₁₇₈	C₂²	1	2	yes	—
17	#19151722	—	2	2	no	—
18	#19151722	C₂²	1	2	yes	—
19	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
20	Q₁₆×Dic₈₉	C₄	1	1	yes	characteristic
21	#19151692	C₄	1	1	yes	characteristic
22	C₄×#13083564	C₂²	1	2	yes	—
23	#19151690	C₄	1	2	yes	—
24	#19151650	—	2	2	no	—
25	#19151650	C₂²	1	2	yes	—
26	#19151649	—	2	2	no	—
27	#19151649	C₂²	1	1	yes	characteristic
28	#19151649	C₂²	1	1	yes	characteristic
29	#107×D₈₉	C₄	1	1	yes	characteristic
30	#103×D₈₉	C₄	1	2	yes	—
31	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
32	#19151350	C₂²	1	2	yes	—
33	#19151349	C₂²	1	1	yes	characteristic
34	#19151330	C₂²	1	2	yes	—
35	Q₈⋊#10623494	C₂²	1	2	yes	—
36	#19151328	C₂²	1	2	yes	—
37	#19151319	C₂²	1	1	yes	characteristic
38	C₈×#10623494	C₂²	1	1	yes	characteristic
39	C₈×#10623494	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₈₉	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
42	#334×C₈₉	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₈₉	C₂²	1	2	yes	—
44	#322×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₈₉	C₂×C₄	1	2	yes	—
46	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
47	C₂×#13083564	C₂×C₄	1	2	yes	—
48	Q₁₆×D₈₉	C₈	1	8	yes	complemented
49	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
51	Q₈×Dic₈₉	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₇₈	—	2	2	no	—
53	C₄×Dic₁₇₈	C₂³	1	2	yes	—
54	#15667977	—	2	2	no	—
55	#15667977	C₂×C₄	1	2	yes	—
56	C₄⋊C₄×D₈₉	—	2	2	no	—
57	C₄⋊C₄×D₈₉	C₂×C₄	1	2	yes	—
58	C₄²×D₈₉	—	2	2	no	—
59	C₄²×D₈₉	C₂³	1	1	yes	characteristic
60	#15667946	C₂×C₄	1	1	yes	characteristic
61	#15667945	C₂×C₄	1	2	yes	—
62	C₄⋊#10623494	—	2	2	no	—
63	C₄⋊#10623494	C₂³	1	2	yes	—
64	#15667941	—	2	2	no	—
65	#15667941	—	2	1	no	—
66	#15667941	—	2	2	no	—
67	#15667941	D₄	1	2	yes	—
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	2	yes	—
74	C₄×#10623494	—	2	2	no	—
75	C₄×#10623494	C₂³	1	1	yes	characteristic
76	C₄×#10623494	C₂³	1	1	yes	characteristic
77	C₄×#10623494	C₂³	1	1	yes	characteristic
78	C₈×Dic₈₉	—	2	2	no	—
79	C₈×Dic₈₉	—	2	2	no	—
80	C₈×Dic₈₉	—	2	2	no	—
81	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
83	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
84	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
85	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
86	C₄⋊C₈×C₈₉	—	2	2	no	—
87	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
88	#103×C₈₉	C₂×C₄	1	2	yes	—
89	C₄×C₈×C₈₉	—	2	2	no	—
90	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₂×Dic₁₇₈	—	2	2	no	—
94	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
95	Q₈×D₈₉	—	2	4	no	—
96	Q₈×D₈₉	C₂×C₈	1	8	yes	—
97	C₂×C₄×D₈₉	—	2	2	no	—
98	C₂×C₄×D₈₉	—	2	2	no	—
99	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×D₈₉	#41	1	1	yes	characteristic
102	Dic₃₅₆	C₂×C₈	1	4	yes	—
103	#13083564	C₂×C₈	1	8	yes	—
104	C₂×#10623494	—	2	2	no	—
105	C₂×#10623494	—	2	1	no	—
106	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
107	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
108	C₂×#10623494	C₂×D₄	1	2	yes	—
109	C₈×D₈₉	—	2	2	no	—
110	C₈×D₈₉	C₂×C₈	1	4	yes	—
111	C₄⋊Dic₈₉	—	2	2	no	—
112	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
113	#13083553	—	2	2	no	—
114	#13083553	—	2	2	no	—
115	#13083553	—	2	2	no	—
116	C₄×Dic₈₉	—	2	2	no	—
117	C₄×Dic₈₉	—	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	C₂×C₈	1	4	yes	—
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	2	yes	—
129	C₄⋊C₄×C₈₉	—	2	2	no	—
130	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
131	C₄²×C₈₉	—	2	2	no	—
132	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
133	C₂²×D₈₉	C₄×D₄	1	1	yes	characteristic
134	Q₈×C₈₉	—	2	2	no	—
135	Q₈×C₈₉	C₂²×C₈	1	4	yes	—
136	C₂×C₄×C₈₉	—	2	2	no	—
137	C₂×C₄×C₈₉	—	2	2	no	—
138	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
141	C₂×Dic₈₉	—	2	2	no	—
142	C₂×Dic₈₉	—	2	2	no	—
143	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
144	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
146	C₄×D₈₉	—	2	8	no	—
147	C₄×D₈₉	—	2	2	no	—
148	C₄×D₈₉	—	4	1	no	—
149	C₄×D₈₉	—	8	2	no	—
150	C₄×D₈₉	C₂²×C₈	1	2	yes	—
151	C₄×D₈₉	#131	1	2	yes	—
152	Dic₁₇₈	—	2	2	no	—
153	Dic₁₇₈	—	2	8	no	—
154	Dic₁₇₈	C₂²×C₈	1	4	yes	—
155	C₈×C₈₉	—	2	1	no	—
156	C₈×C₈₉	—	2	2	no	—
157	C₈×C₈₉	—	4	1	no	—
158	C₈×C₈₉	—	4	2	no	—
159	C₈×C₈₉	C₂²×C₈	1	2	yes	—
160	#10623494	—	2	1	no	—
161	#10623494	—	2	2	no	—
162	#10623494	—	4	1	no	—
163	#10623494	—	4	2	no	—
164	#10623494	C₂²×C₈	1	2	yes	—
165	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂×D₈₉	—	2	1	no	—
167	C₂×D₈₉	—	2	1	no	—
168	C₂×D₈₉	C₈×D₄	1	2	yes	—
169	C₄×C₈₉	—	2	4	no	—
170	C₄×C₈₉	—	2	1	no	—
171	C₄×C₈₉	—	4	2	no	—
172	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
173	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
174	C₄×C₈₉	#578	1	2	yes	—
175	Dic₈₉	—	2	4	no	—
176	Dic₈₉	—	2	1	no	—
177	Dic₈₉	—	2	1	no	—
178	Dic₈₉	—	4	2	no	—
179	Dic₈₉	C₈×D₄	1	2	yes	—
180	#17856	—	89	1	no	maximal, Sylow
181	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
182	C₂×C₈₉	#2950	1	1	yes	characteristic
183	C₂×C₈₉	#2957	1	1	yes	characteristic
184	D₈₉	—	2	2	no	—
185	#2982	—	89	2	no	—
186	C₂×C₄×Q₁₆	—	89	1	no	—
187	#1579	—	89	1	no	—
188	#1579	—	89	1	no	—
189	#1579	—	89	1	no	—
190	#1579	D₈₉	1	1	yes	characteristic, complemented
191	C₈×Q₁₆	—	89	2	no	—
192	C₈×Q₁₆	—	89	2	no	—
193	#1564	—	89	1	no	—
194	#1479	—	89	2	no	—
195	#1450	—	89	1	no	—
196	C₈₉	#17856	1	1	yes	characteristic, complemented, Sylow
197	C₂²×Q₁₆	—	89	1	no	—
198	C₂×C₄×Q₈	—	89	2	no	—
199	C₈×Q₈	—	89	2	no	—
200	C₈×Q₈	—	89	2	no	—
201	C₈×Q₈	—	89	2	no	—
202	C₈×Q₈	—	178	2	no	—
203	C₈×Q₈	—	178	2	no	—
204	C₈×Q₈	C₂×D₈₉	1	2	yes	—
205	C₄×Q₁₆	—	89	1	no	—
206	C₄×Q₁₆	—	89	2	no	—
207	C₄×Q₁₆	—	89	1	no	—
208	C₄×Q₁₆	—	89	2	no	—
209	C₄×Q₁₆	—	89	1	no	—
210	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
211	#431	—	89	1	no	—
212	#431	—	89	1	no	—
213	#431	—	89	2	no	—
214	#431	—	178	2	no	—
215	#431	—	178	2	no	—
216	C₂×#107	—	89	1	no	—
217	C₂×#103	—	89	2	no	—
218	C₂×C₄×C₈	—	89	1	no	—
219	#335	—	89	2	no	—
220	#334	—	89	1	no	—
221	#334	C₂×D₈₉	1	1	yes	characteristic
222	Q₈⋊C₈	—	89	2	no	—
223	Q₈⋊C₈	—	89	2	no	—
224	Q₈⋊C₈	—	89	2	no	—
225	Q₈⋊C₈	C₂×D₈₉	1	2	yes	—
226	#322	—	89	1	no	—
227	#322	C₂×D₈₉	1	1	yes	characteristic
228	C₈²	—	89	1	no	—
229	C₈²	—	89	1	no	—
230	C₂²×Q₈	—	89	2	no	—
231	C₂×Q₁₆	—	89	2	no	—
232	C₂×Q₁₆	—	89	8	no	—
233	C₂×Q₁₆	—	89	1	no	—
234	C₂×Q₁₆	C₄×D₈₉	1	1	yes	characteristic
235	C₂²×C₈	—	89	1	no	—
236	C₂²×C₈	—	89	1	no	—
237	C₄×Q₈	—	89	2	no	—
238	C₄×Q₈	—	89	2	no	—
239	C₄×Q₈	—	89	2	no	—
240	C₄×Q₈	—	178	2	no	—
241	C₄×Q₈	—	178	2	no	—
242	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
243	C₂×C₄⋊C₄	—	89	2	no	—
244	C₂×C₄⋊C₄	—	178	2	no	—
245	C₂×C₄²	—	89	1	no	—
246	C₂×C₄²	—	178	2	no	—
247	#107	—	89	1	no	—
248	#107	—	89	2	no	—
249	#107	C₄×D₈₉	1	1	yes	characteristic
250	C₄⋊C₈	—	2	2	no	—
251	C₄⋊C₈	—	89	2	no	—
252	C₄⋊C₈	—	89	2	no	—
253	C₄⋊C₈	—	178	2	no	—
254	C₄⋊C₈	—	178	1	no	—
255	C₄⋊C₈	—	178	2	no	—
256	C₄⋊C₈	—	178	2	no	—
257	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
258	#103	—	89	2	no	—
259	#103	—	89	2	no	—
260	#103	—	89	2	no	—
261	#103	C₄×D₈₉	1	2	yes	—
262	C₂²⋊C₈	—	89	2	no	—
263	C₂²⋊C₈	—	178	2	no	—
264	C₂²⋊C₈	—	178	1	no	—
265	C₄×C₈	—	2	2	no	—
266	C₄×C₈	—	89	1	no	—
267	C₄×C₈	—	89	1	no	—
268	C₄×C₈	—	89	1	no	—
269	C₄×C₈	—	89	1	no	—
270	C₄×C₈	—	89	1	no	—
271	C₄×C₈	—	178	2	no	—
272	C₄×C₈	—	178	2	no	—
273	C₄×C₈	—	178	2	no	—
274	C₄×C₈	—	178	2	no	—
275	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
278	C₂×Q₈	—	89	8	no	—
279	C₂×Q₈	—	89	2	no	—
280	C₂×Q₈	—	178	2	no	—
281	C₂×Q₈	—	178	4	no	—
282	C₂×Q₈	C₂×C₄×D₈₉	1	2	yes	—
283	C₂²×C₄	—	89	1	no	—
284	C₂²×C₄	—	89	1	no	—
285	C₂²×C₄	—	89	1	no	—
286	C₂²×C₄	—	178	2	no	—
287	C₂²×C₄	—	178	2	no	—
288	Q₁₆	—	89	8	no	—
289	Q₁₆	—	89	4	no	—
290	Q₁₆	C₈×D₈₉	1	4	yes	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	89	1	no	—
294	C₂×C₈	—	89	4	no	—
295	C₂×C₈	—	89	1	no	—
296	C₂×C₈	—	89	2	no	—
297	C₂×C₈	—	178	2	no	—
298	C₂×C₈	—	178	2	no	—
299	C₂×C₈	—	178	1	no	—
300	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
301	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
302	C₂×C₈	D₄×D₈₉	1	2	yes	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	89	2	no	—
305	C₄⋊C₄	—	178	2	no	—
306	C₄⋊C₄	—	178	2	no	—
307	C₄⋊C₄	—	178	2	no	—
308	C₄⋊C₄	—	178	2	no	—
309	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
310	C₄²	—	2	2	no	—
311	C₄²	—	89	1	no	—
312	C₄²	—	89	1	no	—
313	C₄²	—	89	1	no	—
314	C₄²	—	178	2	no	—
315	C₄²	—	178	2	no	—
316	C₄²	—	178	2	no	—
317	C₄²	C₂³×D₈₉	1	1	yes	characteristic
318	C₂³	—	89	1	no	—
319	Q₈	—	2	2	no	—
320	Q₈	—	89	4	no	—
321	Q₈	—	178	8	no	—
322	Q₈	—	178	2	no	—
323	Q₈	C₂×C₈×D₈₉	1	4	yes	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	89	1	no	—
327	C₂×C₄	—	89	2	no	—
328	C₂×C₄	—	89	2	no	—
329	C₂×C₄	—	89	1	no	—
330	C₂×C₄	—	89	1	no	—
331	C₂×C₄	—	356	1	no	—
332	C₂×C₄	—	178	8	no	—
333	C₂×C₄	—	178	2	no	—
334	C₂×C₄	—	178	2	no	—
335	C₂×C₄	—	178	2	no	—
336	C₂×C₄	—	712	2	no	—
337	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
338	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
339	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	2	1	no	—
342	C₈	—	4	1	no	—
343	C₈	—	4	2	no	—
344	C₈	—	89	2	no	—
345	C₈	—	356	1	no	—
346	C₈	—	356	2	no	—
347	C₈	—	178	2	no	—
348	C₈	—	178	1	no	—
349	C₈	C₂×C₈×D₈₉	1	2	yes	—
350	C₂²	—	89	2	no	—
351	C₂²	—	178	1	no	—
352	C₂²	—	178	1	no	—
353	C₂²	C₄×D₄×D₈₉	1	1	yes	characteristic
354	C₄	—	2	1	no	—
355	C₄	—	2	4	no	—
356	C₄	—	4	2	no	—
357	C₄	—	89	2	no	—
358	C₄	—	356	2	no	—
359	C₄	—	178	1	no	—
360	C₄	—	178	1	no	—
361	C₄	—	178	4	no	—
362	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
363	C₄	#131×D₈₉	1	1	yes	characteristic
364	C₄	#19152414	1	2	yes	—
365	C₂	—	178	2	no	—
366	C₂	C₈×D₄×D₈₉	1	1	yes	characteristic
367	C₂	#26029117	1	1	yes	characteristic
368	C₂	#26029166	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1579	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂²×D₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1579	D₈₉
hypocenter	C₈₉	#17856
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

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

N	H ≅ G/N	type
3. #26019072	C₂	semidirect
4. #26019071	C₂	semidirect
5. #26019070	C₂	semidirect
6. C₈×#13083564	C₂	semidirect
7. #26019056	C₂	semidirect
11. C₈₉×#1579	C₂	semidirect
48. Q₁₆×D₈₉	C₈	semidirect
190. #1579	D₈₉	semidirect
196. C₈₉	#17856	semidirect

Maximal subgroups

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

subgroup	index
#26029331	2
C₄×Q₁₆×D₈₉	2
#26019072	2
#26019071	2
#26019070	2
C₈×#13083564	2
#26019056	2
#26018997	2
#26018645	2
#26018537	2
C₈₉×#1579	2
#17856	89