Subgroups — #36523449

Order 22784.

710 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028880	C₂	1	2	yes	maximal
2	C₂×C₄×#13083557	C₂	1	1	yes	maximal, characteristic
3	#26019018	C₂	1	4	yes	maximal, complemented
4	#26019016	C₂	1	4	yes	maximal, complemented
5	#26019000	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018998	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#13083557	C₄	1	1	yes	characteristic
9	C₂²×#13083557	C₄	1	1	yes	characteristic
10	C₂×C₄²×D₈₉	C₂²	1	1	yes	characteristic
11	#19151654	C₂²	1	4	yes	—
12	#19151652	—	2	2	no	—
13	#19151651	C₂²	1	2	yes	—
14	#19151649	—	2	2	no	—
15	#19151620	C₂²	1	4	yes	—
16	#19151560	—	2	2	no	—
17	C₄×#13083557	—	2	2	no	—
18	C₄×#13083557	C₄	1	2	yes	—
19	C₄×#13083557	C₄	1	2	yes	—
20	C₄×#13083557	C₂²	1	2	yes	—
21	C₄×#13083557	C₂²	1	2	yes	—
22	C₂×#15667922	C₄	1	1	yes	characteristic
23	C₂×#15667922	C₄	1	1	yes	characteristic
24	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
25	#19151351	C₂²	1	2	yes	complemented
26	#19151350	C₂²	1	4	yes	complemented
27	#19151349	C₂²	1	2	yes	complemented
28	#19151348	C₂²	1	4	yes	complemented
29	#431×C₈₉	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	#335×C₈₉	C₂²	1	2	yes	complemented
32	#334×C₈₉	C₂²	1	2	yes	complemented
33	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×D₈₉	Q₈	1	1	yes	characteristic
36	C₂×#13083557	—	2	4	no	—
37	C₂×#13083557	—	2	4	no	—
38	C₂×#13083557	—	2	4	no	—
39	C₂×#13083557	C₂×C₄	1	2	yes	—
40	C₂×#13083557	C₂×C₄	1	2	yes	—
41	C₂×#13083557	C₂×C₄	1	2	yes	—
42	C₂×#13083557	C₂×C₄	1	2	yes	—
43	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
44	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
47	C₂×C₄×Dic₈₉	Q₈	1	1	yes	characteristic
48	C₄²×D₈₉	C₂×C₄	1	2	yes	—
49	C₄²×D₈₉	D₄	1	2	yes	—
50	C₄²×D₈₉	Q₈	1	2	yes	—
51	C₄²×D₈₉	C₂³	1	2	yes	—
52	C₄⋊#10623494	C₂³	1	4	yes	—
53	#15667941	—	2	4	no	—
54	C₂²⋊#10623494	—	2	2	no	—
55	#15667924	—	2	4	no	—
56	#15667922	—	2	4	no	—
57	#15667922	C₂×C₄	1	2	yes	—
58	#15667922	C₂×C₄	1	2	yes	—
59	#15667922	C₂×C₄	1	2	yes	—
60	#15667922	C₂×C₄	1	2	yes	—
61	C₄×#10623494	—	2	2	no	—
62	C₄×#10623494	C₂×C₄	1	2	yes	—
63	C₄×#10623494	C₂³	1	2	yes	—
64	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
68	C₂²⋊C₈×C₈₉	—	2	2	no	—
69	C₄×C₈×C₈₉	—	2	2	no	—
70	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	C₂³×D₈₉	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×D₈₉	—	2	2	no	—
79	C₂×C₄×D₈₉	—	2	2	no	—
80	C₂×C₄×D₈₉	—	4	2	no	—
81	C₂×C₄×D₈₉	—	4	2	no	—
82	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
83	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
84	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
85	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
86	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
87	C₂×C₄×D₈₉	C₂×Q₈	1	2	yes	—
88	#13083557	—	2	4	no	—
89	#13083557	—	2	4	no	—
90	#13083557	—	2	4	no	—
91	#13083557	—	2	4	no	—
92	#13083557	—	4	8	no	—
93	C₂×#10623494	—	2	2	no	—
94	C₂×#10623494	—	2	4	no	—
95	C₂×#10623494	M₁₆	1	4	yes	complemented
96	C₂×#10623494	C₂²×C₄	1	2	yes	—
97	C₂×#10623494	C₂²×C₄	1	2	yes	—
98	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
99	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
100	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
101	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
102	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
103	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
104	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
107	C₂×C₈×C₈₉	—	2	2	no	—
108	C₂×C₈×C₈₉	—	2	4	no	—
109	C₂×C₈×C₈₉	M₁₆	1	4	yes	complemented
110	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
117	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
118	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×D₈₉	—	2	2	no	—
120	C₂²×D₈₉	—	4	1	no	—
121	C₂²×D₈₉	—	4	1	no	—
122	C₂²×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
133	C₂×Dic₈₉	—	2	2	no	—
134	C₂×Dic₈₉	—	4	1	no	—
135	C₂×Dic₈₉	—	4	1	no	—
136	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
139	C₄×D₈₉	—	2	2	no	—
140	C₄×D₈₉	—	2	2	no	—
141	C₄×D₈₉	—	2	4	no	—
142	C₄×D₈₉	—	2	4	no	—
143	C₄×D₈₉	—	2	4	no	—
144	C₄×D₈₉	—	4	2	no	—
145	C₄×D₈₉	—	4	2	no	—
146	C₈×C₈₉	—	2	1	no	—
147	C₈×C₈₉	—	2	1	no	—
148	C₈×C₈₉	—	4	4	no	—
149	C₈×C₈₉	C₂×M₁₆	1	2	yes	—
150	C₈×C₈₉	C₂×M₁₆	1	2	yes	—
151	#10623494	—	2	2	no	—
152	#10623494	—	4	4	no	—
153	#10623494	C₂×M₁₆	1	4	yes	—
154	C₂²×C₈₉	—	2	1	no	—
155	C₂²×C₈₉	—	2	1	no	—
156	C₂²×C₈₉	#423	1	2	yes	—
157	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₂×D₈₉	—	2	2	no	—
159	C₂×D₈₉	—	2	2	no	—
160	C₂×D₈₉	—	2	2	no	—
161	C₂×D₈₉	—	4	4	no	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	—	2	1	no	—
164	C₄×C₈₉	#423	1	2	yes	—
165	C₄×C₈₉	#430	1	2	yes	—
166	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
167	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
168	C₄×C₈₉	#575	1	2	yes	—
169	Dic₈₉	—	2	2	no	—
170	Dic₈₉	—	2	2	no	—
171	Dic₈₉	—	2	2	no	—
172	#17795	—	89	1	no	maximal, Sylow
173	C₂×C₈₉	—	2	2	no	—
174	C₂×C₈₉	C₂×#423	1	1	yes	characteristic
175	C₂×C₈₉	#2911	1	1	yes	characteristic
176	C₂×C₈₉	C₂×#430	1	1	yes	characteristic
177	D₈₉	—	4	2	no	—
178	#2920	—	89	2	no	—
179	C₂×C₄×M₁₆	—	89	1	no	—
180	#1569	—	89	4	no	—
181	#1569	—	89	4	no	—
182	#1564	—	89	2	no	—
183	#1564	—	89	1	no	—
184	#1564	D₈₉	1	1	yes	characteristic, complemented
185	C₈₉	#17795	1	1	yes	characteristic, complemented, Sylow
186	C₂²×M₁₆	—	89	1	no	—
187	C₂²×M₁₆	—	89	1	no	—
188	C₂²×C₄²	—	89	1	no	—
189	#432	—	89	4	no	—
190	#432	—	178	2	no	—
191	#431	—	89	2	no	—
192	#431	—	178	2	no	—
193	#431	C₂×D₈₉	1	2	yes	—
194	#423	—	89	4	no	—
195	#407	—	178	2	no	—
196	C₄×M₁₆	—	89	2	no	—
197	C₄×M₁₆	—	89	2	no	—
198	C₄×M₁₆	—	89	2	no	—
199	C₄×M₁₆	—	89	2	no	—
200	C₄×M₁₆	—	178	2	no	—
201	C₂×#97	—	89	1	no	—
202	C₂×#97	—	89	1	no	—
203	C₂×C₄×C₈	—	89	1	no	—
204	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
205	#335	—	89	4	no	—
206	#335	—	89	2	no	—
207	#335	C₂×D₈₉	1	2	yes	complemented
208	#334	—	89	4	no	—
209	#334	—	89	2	no	—
210	#334	C₂×D₈₉	1	2	yes	complemented
211	C₂³×C₄	—	89	1	no	—
212	C₂³×C₄	—	89	1	no	—
213	C₂³×C₄	—	89	1	no	—
214	C₂×M₁₆	—	89	2	no	—
215	C₂×M₁₆	—	89	2	no	—
216	C₂×M₁₆	—	89	2	no	—
217	C₂×M₁₆	—	89	2	no	—
218	C₂×M₁₆	—	178	4	no	—
219	C₂×M₁₆	—	178	4	no	—
220	C₂×M₁₆	—	178	4	no	—
221	C₂²×C₈	—	89	1	no	—
222	C₂²×C₈	—	89	1	no	—
223	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
224	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
225	C₂×C₄²	—	89	1	no	—
226	C₂×C₄²	—	89	1	no	—
227	C₂×C₄²	—	89	1	no	—
228	C₂×C₄²	—	89	2	no	—
229	C₂×C₄²	—	89	2	no	—
230	C₂×C₄²	—	89	2	no	—
231	C₂×C₄²	—	89	2	no	—
232	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
233	C₄⋊C₈	—	89	4	no	—
234	C₄⋊C₈	—	178	4	no	—
235	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
236	C₂²⋊C₈	—	2	2	no	—
237	C₂²⋊C₈	—	178	4	no	—
238	C₂²⋊C₈	—	178	2	no	—
239	#97	—	89	2	no	—
240	#97	—	89	2	no	—
241	#97	—	89	2	no	—
242	#97	—	89	2	no	—
243	#97	—	178	4	no	—
244	C₄×C₈	—	2	2	no	—
245	C₄×C₈	—	89	2	no	—
246	C₄×C₈	—	89	2	no	—
247	C₄×C₈	—	178	2	no	—
248	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
249	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
251	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
252	C₂⁴	—	89	1	no	—
253	C₂²×C₄	—	89	1	no	—
254	C₂²×C₄	—	89	1	no	—
255	C₂²×C₄	—	89	1	no	—
256	C₂²×C₄	—	89	2	no	—
257	C₂²×C₄	—	89	2	no	—
258	C₂²×C₄	—	89	2	no	—
259	C₂²×C₄	—	89	2	no	—
260	C₂²×C₄	—	89	2	no	—
261	C₂²×C₄	—	89	2	no	—
262	C₂²×C₄	—	356	2	no	—
263	C₂²×C₄	—	356	2	no	—
264	C₂²×C₄	—	178	2	no	—
265	C₂²×C₄	—	178	2	no	—
266	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
267	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
268	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
269	M₁₆	—	356	8	no	—
270	M₁₆	—	178	4	no	—
271	M₁₆	—	178	4	no	—
272	M₁₆	—	178	4	no	—
273	M₁₆	—	178	4	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	—	89	4	no	—
277	C₂×C₈	—	89	2	no	—
278	C₂×C₈	—	89	2	no	—
279	C₂×C₈	—	178	4	no	—
280	C₂×C₈	—	178	2	no	—
281	C₂×C₈	#13083557	1	4	yes	complemented
282	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
283	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
284	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
285	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
286	C₄²	—	89	2	no	—
287	C₄²	—	89	2	no	—
288	C₄²	—	89	2	no	—
289	C₄²	—	89	2	no	—
290	C₄²	—	89	2	no	—
291	C₄²	—	89	2	no	—
292	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
293	C₄²	D₄×D₈₉	1	1	yes	characteristic
294	C₄²	Q₈×D₈₉	1	1	yes	characteristic
295	C₄²	C₂³×D₈₉	1	1	yes	characteristic
296	C₂³	—	89	2	no	—
297	C₂³	—	356	1	no	—
298	C₂³	—	356	1	no	—
299	C₂³	—	178	2	no	—
300	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	89	2	no	—
304	C₂×C₄	—	89	2	no	—
305	C₂×C₄	—	89	2	no	—
306	C₂×C₄	—	356	2	no	—
307	C₂×C₄	—	356	2	no	—
308	C₂×C₄	—	356	1	no	—
309	C₂×C₄	—	356	1	no	—
310	C₂×C₄	—	178	2	no	—
311	C₂×C₄	—	178	4	no	—
312	C₂×C₄	—	178	4	no	—
313	C₂×C₄	—	178	4	no	—
314	C₂×C₄	—	178	2	no	—
315	C₂×C₄	—	178	2	no	—
316	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
317	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
318	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
319	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
320	C₂×C₄	C₂×#13083557	1	2	yes	—
321	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
322	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
323	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	4	4	no	—
327	C₈	—	89	4	no	—
328	C₈	—	356	4	no	—
329	C₈	—	178	2	no	—
330	C₈	C₂×#13083557	1	2	yes	—
331	C₈	C₂×#13083557	1	2	yes	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	356	4	no	—
335	C₂²	—	178	2	no	—
336	C₂²	—	178	2	no	—
337	C₂²	—	178	2	no	—
338	C₂²	#19151620	1	2	yes	—
339	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	178	2	no	—
343	C₄	—	178	2	no	—
344	C₄	—	178	2	no	—
345	C₄	#19151620	1	2	yes	—
346	C₄	#19151645	1	2	yes	—
347	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
348	C₄	C₂²×#13083557	1	1	yes	characteristic
349	C₄	#19152395	1	2	yes	—
350	C₂	—	2	2	no	—
351	C₂	—	356	2	no	—
352	C₂	C₂×#19151620	1	1	yes	characteristic
353	C₂	#26028840	1	1	yes	characteristic
354	C₂	C₂×#19151645	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂³×D₈₉	C₄⋊C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#17795
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #26019018	C₂	semidirect
4. #26019016	C₂	semidirect
5. #26019000	C₂	semidirect
6. #26018998	C₂	semidirect
7. C₈₉×#1564	C₂	semidirect
25. #19151351	C₂²	semidirect
26. #19151350	C₂²	semidirect
27. #19151349	C₂²	semidirect
28. #19151348	C₂²	semidirect
31. #335×C₈₉	C₂²	semidirect
32. #334×C₈₉	C₂²	semidirect
95. C₂×#10623494	M₁₆	semidirect
109. C₂×C₈×C₈₉	M₁₆	semidirect
184. #1564	D₈₉	semidirect
185. C₈₉	#17795	semidirect
207. #335	C₂×D₈₉	semidirect
210. #334	C₂×D₈₉	semidirect
281. C₂×C₈	#13083557	semidirect

Maximal subgroups

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

subgroup	index
#26028880	2
C₂×C₄×#13083557	2
#26019018	2
#26019016	2
#26019000	2
#26018998	2
C₈₉×#1564	2
#17795	89