Subgroups — #36525003

Order 22784.

614 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030820	C₂	1	1	yes	maximal, characteristic
2	#26029224	C₂	1	2	yes	maximal, complemented
3	#26019166	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019160	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019151	C₂	1	2	yes	maximal, complemented
6	#26019072	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019064	C₂	1	2	yes	maximal, complemented
8	#26019012	C₂	1	1	yes	maximal, characteristic, complemented
9	#26018709	C₂	1	2	yes	maximal, complemented
10	#322×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1580	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152130	C₂²	1	2	yes	—
13	#19151999	C₄	1	1	yes	characteristic
14	#19151981	C₄	1	1	yes	characteristic
15	#19151951	C₄	1	2	yes	—
16	#19151840	C₄	1	1	yes	characteristic
17	#19151831	C₄	1	1	yes	characteristic
18	#19151826	C₄	1	2	yes	—
19	#19151725	C₂²	1	2	yes	—
20	C₈×Dic₁₇₈	C₂²	1	2	yes	—
21	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
22	C₄×#13083562	C₂²	1	2	yes	—
23	#19151676	C₂²	1	2	yes	—
24	#19151672	—	2	2	no	—
25	C₈×C₂²⋊D₈₉	—	2	2	no	—
26	#19151656	—	2	2	no	—
27	#19151654	C₂²	1	2	yes	—
28	#19151553	—	2	2	no	—
29	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
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	#19151328	C₂²	1	2	yes	—
36	D₄⋊#10623494	C₂²	1	2	yes	—
37	#19151320	C₂²	1	1	yes	characteristic
38	#19151320	C₂²	1	1	yes	characteristic
39	#19151319	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₈₉	C₂²	1	2	yes	—
42	#334×C₈₉	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₈₉	C₂²	1	2	yes	—
44	#322×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
46	C₂×#13083562	C₂×C₄	1	2	yes	—
47	C₂×#13083558	—	2	2	no	—
48	C₂×C₈×D₈₉	—	2	1	no	—
49	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₈₉	D₄	1	2	yes	—
52	#15668005	—	2	2	no	—
53	#15668003	C₂×C₄	1	2	yes	—
54	#15667999	C₂×C₄	1	2	yes	—
55	#15667994	—	2	2	no	—
56	#15667992	C₂×C₄	1	2	yes	—
57	#15667983	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₇₈	C₂³	1	2	yes	—
59	C₄×C₂²⋊D₈₉	—	2	2	no	—
60	#15667971	—	2	2	no	—
61	C₄²×D₈₉	C₂³	1	1	yes	characteristic
62	#15667946	C₂×C₄	1	1	yes	characteristic
63	#15667943	C₂×C₄	1	2	yes	—
64	C₄⋊#10623494	C₂³	1	2	yes	—
65	#15667941	—	2	2	no	—
66	#15667941	—	2	2	no	—
67	#15667937	C₂×C₄	1	2	yes	—
68	#15667935	C₂×C₄	1	2	yes	—
69	#15667933	C₂×C₄	1	2	yes	—
70	C₂²⋊#10623494	—	2	2	no	—
71	#15667924	—	2	2	no	—
72	#15667922	—	2	2	no	—
73	#15667922	—	2	2	no	—
74	C₄×#10623494	C₂³	1	1	yes	characteristic
75	C₄×#10623494	C₂³	1	1	yes	characteristic
76	C₄×#10623494	C₂³	1	1	yes	characteristic
77	C₈×Dic₈₉	—	2	1	no	—
78	C₈×Dic₈₉	—	2	2	no	—
79	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₈₉	D₄	1	2	yes	—
82	C₂×D₈×C₈₉	C₂×C₄	1	1	yes	characteristic
83	C₂²×C₈×C₈₉	—	2	2	no	—
84	C₄×D₄×C₈₉	C₂³	1	2	yes	—
85	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
86	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
87	#102×C₈₉	C₂×C₄	1	2	yes	—
88	C₂²⋊C₈×C₈₉	—	2	2	no	—
89	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
92	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
93	C₂×C₂²⋊D₈₉	—	2	2	no	—
94	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₈₉	#41	1	1	yes	characteristic
97	Dic₃₅₆	—	2	2	no	—
98	#13083562	—	2	4	no	—
99	#13083558	—	4	4	no	—
100	C₂×#10623494	—	2	2	no	—
101	C₂×#10623494	—	2	1	no	—
102	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
103	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
104	C₂×#10623494	C₂×D₄	1	2	yes	—
105	C₈×D₈₉	—	2	2	no	—
106	C₈×D₈₉	—	2	2	no	—
107	C₈×D₈₉	—	4	2	no	—
108	C₈×D₈₉	—	4	2	no	—
109	C₄⋊Dic₈₉	—	2	2	no	—
110	C₄⋊Dic₈₉	—	2	2	no	—
111	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
112	#13083553	—	2	2	no	—
113	#13083553	—	2	2	no	—
114	C₂²⋊Dic₈₉	—	2	2	no	—
115	C₂²⋊Dic₈₉	—	2	2	no	—
116	C₂²⋊Dic₈₉	—	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×D₄×C₈₉	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₈₉	—	2	2	no	—
124	D₈×C₈₉	—	2	2	no	—
125	C₂×C₈×C₈₉	—	2	2	no	—
126	C₂×C₈×C₈₉	—	2	1	no	—
127	C₂×C₈×C₈₉	—	4	4	no	—
128	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
131	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
132	C₂²⋊C₄×C₈₉	—	2	2	no	—
133	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
134	C₂³×C₈₉	—	2	2	no	—
135	C₂²×D₈₉	C₄×D₄	1	1	yes	characteristic
136	D₄×C₈₉	—	2	2	no	—
137	D₄×C₈₉	—	2	2	no	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	4	2	no	—
140	C₂×C₄×C₈₉	—	4	2	no	—
141	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
144	C₂²⋊D₈₉	—	4	4	no	—
145	C₂×Dic₈₉	—	2	2	no	—
146	C₂×Dic₈₉	—	2	2	no	—
147	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
148	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
149	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
150	C₄×D₈₉	—	2	2	no	—
151	C₄×D₈₉	—	2	1	no	—
152	C₄×D₈₉	—	2	1	no	—
153	C₄×D₈₉	—	2	2	no	—
154	Dic₁₇₈	—	2	2	no	—
155	Dic₁₇₈	—	2	2	no	—
156	C₈×C₈₉	—	2	1	no	—
157	C₈×C₈₉	—	2	2	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	2	no	—
163	#10623494	—	4	1	no	—
164	#10623494	—	4	2	no	—
165	#10623494	#131	1	2	yes	—
166	C₂²×C₈₉	—	4	2	no	—
167	C₂²×C₈₉	—	4	2	no	—
168	C₂²×C₈₉	—	4	2	no	—
169	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂×D₈₉	—	2	1	no	—
171	C₂×D₈₉	—	2	1	no	—
172	C₂×D₈₉	#444	1	2	yes	—
173	C₄×C₈₉	—	2	1	no	—
174	C₄×C₈₉	—	4	2	no	—
175	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
176	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
177	C₄×C₈₉	#575	1	2	yes	—
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	#17963	—	89	1	no	maximal, Sylow
184	C₂×C₈₉	—	8	2	no	—
185	C₂×C₈₉	#2931	1	1	yes	characteristic
186	C₂×C₈₉	#2943	1	1	yes	characteristic
187	C₂×C₈₉	C₂×#444	1	1	yes	characteristic
188	D₈₉	—	2	2	no	—
189	#3168	—	89	1	no	—
190	#2966	—	89	2	no	—
191	#1594	—	89	1	no	—
192	#1593	—	89	1	no	—
193	#1592	—	89	2	no	—
194	#1580	D₈₉	1	1	yes	characteristic, complemented
195	#1579	—	89	1	no	—
196	SD₁₆⋊C₈	—	89	2	no	—
197	#1568	—	89	1	no	—
198	#1494	—	89	2	no	—
199	C₂×#322	—	89	1	no	—
200	C₈₉	#17963	1	1	yes	characteristic, complemented, Sylow
201	#529	—	89	2	no	—
202	#499	—	89	1	no	—
203	#495	—	89	1	no	—
204	#488	—	89	2	no	—
205	#467	—	89	1	no	—
206	#466	—	89	1	no	—
207	#465	—	89	2	no	—
208	#446	—	89	2	no	—
209	C₈×Q₈	—	89	2	no	—
210	C₄×Q₁₆	—	89	1	no	—
211	C₄×SD₁₆	—	89	2	no	—
212	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
213	#436	—	89	2	no	—
214	#435	—	178	2	no	—
215	C₈×D₄	—	178	2	no	—
216	C₈×D₄	C₂×D₈₉	1	2	yes	—
217	#433	—	178	2	no	—
218	#432	—	89	2	no	—
219	#405	—	178	2	no	—
220	C₂×C₄×C₈	—	89	1	no	—
221	C₂×C₄×C₈	—	89	1	no	—
222	C₂×C₄×C₈	—	89	1	no	—
223	#335	—	89	2	no	—
224	#334	—	89	1	no	—
225	#334	C₂×D₈₉	1	1	yes	characteristic
226	Q₈⋊C₈	—	89	2	no	—
227	Q₈⋊C₈	—	89	2	no	—
228	D₄⋊C₈	—	89	2	no	—
229	D₄⋊C₈	C₂×D₈₉	1	2	yes	—
230	#322	—	89	1	no	—
231	#322	—	89	1	no	—
232	#322	—	89	1	no	—
233	#322	C₂×D₈₉	1	1	yes	characteristic
234	C₂×Q₁₆	—	89	1	no	—
235	C₂×SD₁₆	—	89	2	no	—
236	C₂×D₈	C₄×D₈₉	1	1	yes	characteristic
237	C₂×M₁₆	—	178	2	no	—
238	C₂²×C₈	—	2	2	no	—
239	C₂²×C₈	—	89	2	no	—
240	C₂²×C₈	—	89	1	no	—
241	C₂²×C₈	—	89	1	no	—
242	C₂²×C₈	—	178	1	no	—
243	#126	—	178	2	no	—
244	#125	—	89	2	no	—
245	#124	—	89	2	no	—
246	#123	—	178	2	no	—
247	C₂²⋊Q₈	—	89	2	no	—
248	#121	—	89	2	no	—
249	C₄×Q₈	—	89	2	no	—
250	C₄×D₄	—	178	2	no	—
251	C₄×D₄	C₂²×D₈₉	1	2	yes	—
252	#117	—	178	2	no	—
253	C₂×C₄²	—	89	1	no	—
254	#107	—	89	1	no	—
255	#107	C₄×D₈₉	1	1	yes	characteristic
256	#106	—	89	2	no	—
257	C₄⋊C₈	—	89	2	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	#102	—	89	2	no	—
264	#102	C₄×D₈₉	1	2	yes	—
265	C₂²⋊C₈	—	2	2	no	—
266	C₂²⋊C₈	—	178	2	no	—
267	C₂²⋊C₈	—	178	2	no	—
268	#97	—	178	2	no	—
269	#97	—	178	2	no	—
270	C₄×C₈	—	89	1	no	—
271	C₄×C₈	—	89	1	no	—
272	C₄×C₈	—	89	2	no	—
273	C₄×C₈	—	89	1	no	—
274	C₄×C₈	—	89	1	no	—
275	C₄×C₈	—	89	1	no	—
276	C₄×C₈	—	178	1	no	—
277	C₄×C₈	—	178	2	no	—
278	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
281	C₂×Q₈	—	89	2	no	—
282	C₂×D₄	—	178	2	no	—
283	C₂×D₄	C₂×C₄×D₈₉	1	2	yes	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	89	1	no	—
286	C₂²×C₄	—	89	1	no	—
287	C₂²×C₄	—	89	1	no	—
288	Q₁₆	—	178	2	no	—
289	SD₁₆	—	178	4	no	—
290	D₈	—	2	2	no	—
291	M₁₆	—	356	4	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	4	4	no	—
295	C₂×C₈	—	89	1	no	—
296	C₂×C₈	—	89	1	no	—
297	C₂×C₈	—	89	2	no	—
298	C₂×C₈	—	356	2	no	—
299	C₂×C₈	—	356	2	no	—
300	C₂×C₈	—	178	2	no	—
301	C₂×C₈	—	178	2	no	—
302	C₂×C₈	—	178	1	no	—
303	C₂×C₈	—	178	2	no	—
304	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
305	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
306	C₂×C₈	D₄×D₈₉	1	2	yes	—
307	C₄⋊C₄	—	89	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	C₂×C₄×D₈₉	1	2	yes	—
313	C₂²⋊C₄	—	2	2	no	—
314	C₂²⋊C₄	—	178	2	no	—
315	C₂²⋊C₄	—	178	2	no	—
316	C₂²⋊C₄	—	178	2	no	—
317	C₂²⋊C₄	—	178	2	no	—
318	C₄²	—	89	1	no	—
319	C₄²	—	89	1	no	—
320	C₄²	—	89	1	no	—
321	C₄²	—	178	2	no	—
322	C₄²	C₂³×D₈₉	1	1	yes	characteristic
323	C₂³	—	2	2	no	—
324	C₂³	—	89	1	no	—
325	Q₈	—	178	2	no	—
326	Q₈	—	178	2	no	—
327	D₄	—	2	2	no	—
328	D₄	—	2	2	no	—
329	D₄	—	356	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	89	1	no	—
334	C₂×C₄	—	89	1	no	—
335	C₂×C₄	—	89	1	no	—
336	C₂×C₄	—	178	2	no	—
337	C₂×C₄	—	178	2	no	—
338	C₂×C₄	—	178	2	no	—
339	C₂×C₄	—	178	2	no	—
340	C₂×C₄	—	178	1	no	—
341	C₂×C₄	—	178	1	no	—
342	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
344	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	—	2	1	no	—
347	C₈	—	4	1	no	—
348	C₈	—	4	2	no	—
349	C₈	—	89	2	no	—
350	C₈	—	356	1	no	—
351	C₈	—	356	2	no	—
352	C₈	—	178	2	no	—
353	C₈	—	178	1	no	—
354	C₈	#15668018	1	2	yes	—
355	C₂²	—	4	2	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	89	2	no	—
359	C₂²	—	178	1	no	—
360	C₂²	—	178	1	no	—
361	C₂²	C₄×D₄×D₈₉	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	4	2	no	—
364	C₄	—	356	2	no	—
365	C₄	—	356	2	no	—
366	C₄	—	178	1	no	—
367	C₄	—	178	1	no	—
368	C₄	—	178	1	no	—
369	C₄	#131×D₈₉	1	1	yes	characteristic
370	C₄	C₂×#15668018	1	1	yes	characteristic
371	C₄	#19152394	1	2	yes	—
372	C₂	—	8	2	no	—
373	C₂	—	178	2	no	—
374	C₂	#26028959	1	1	yes	characteristic
375	C₂	#26029049	1	1	yes	characteristic
376	C₂	#444×D₈₉	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×#1580	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#19151831	C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1580	D₈₉
hypocenter	C₈₉	#17963
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1580

Complements

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

N	H ≅ G/N	type
2. #26029224	C₂	semidirect
3. #26019166	C₂	semidirect
4. #26019160	C₂	semidirect
5. #26019151	C₂	semidirect
6. #26019072	C₂	semidirect
7. #26019064	C₂	semidirect
8. #26019012	C₂	semidirect
9. #26018709	C₂	semidirect
10. #322×D₈₉	C₂	semidirect
11. C₈₉×#1580	C₂	semidirect
194. #1580	D₈₉	semidirect
200. C₈₉	#17963	semidirect

Maximal subgroups

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

subgroup	index
#26030820	2
#26029224	2
#26019166	2
#26019160	2
#26019151	2
#26019072	2
#26019064	2
#26019012	2
#26018709	2
#322×D₈₉	2
C₈₉×#1580	2
#17963	89