Subgroups — #36471319

Order 22784.

508 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#26028944	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151382	C₂	1	1	yes	maximal, characteristic, complemented
4	#26021862	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018342	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018340	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#350×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
9	#19152105	C₄	1	1	yes	characteristic
10	C₂×#15667952	C₂²	1	1	yes	characteristic
11	C₂²×#13083549	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₈₉	—	2	2	no	—
13	C₂×C₁₆×D₈₉	—	2	2	no	—
14	C₂×C₁₆×D₈₉	—	2	2	no	—
15	#19151667	—	2	2	no	—
16	C₈×C₂²⋊D₈₉	—	2	2	no	—
17	#19151626	C₄	1	1	yes	characteristic
18	#19151565	C₄	1	1	yes	characteristic
19	#19151561	C₄	1	1	yes	characteristic
20	#19151553	C₂²	1	1	yes	characteristic
21	#19151382	—	2	2	no	—
22	#19151377	—	2	2	no	—
23	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	#350×C₈₉	—	2	2	no	—
27	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
29	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₈₉	—	2	1	no	—
31	C₂×C₈×D₈₉	—	2	1	no	—
32	C₂×C₈×D₈₉	—	2	1	no	—
33	C₂×C₈×D₈₉	—	2	1	no	—
34	C₂×C₈×D₈₉	D₄	1	2	yes	—
35	C₂×C₈×D₈₉	D₄	1	2	yes	—
36	#15667993	—	2	1	no	—
37	#15667988	C₈	1	2	yes	—
38	#15667983	C₈	1	2	yes	—
39	#15667980	—	2	1	no	—
40	C₄×C₂²⋊D₈₉	—	2	2	no	—
41	#15667971	C₂×C₄	1	1	yes	characteristic
42	#15667952	—	2	1	no	—
43	#15667952	—	2	1	no	—
44	C₂×#13083549	—	2	1	no	—
45	C₂×#13083549	—	2	1	no	—
46	C₂×#13083549	—	2	1	no	—
47	C₂×#13083549	D₄	1	2	yes	complemented
48	C₁₆×D₈₉	—	2	8	no	—
49	#15667941	—	2	2	no	—
50	C₂²⋊#10623494	—	2	2	no	—
51	#15667924	—	2	2	no	—
52	#15667922	—	2	1	no	—
53	C₈×Dic₈₉	—	2	1	no	—
54	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
55	#131×C₈₉	—	2	1	no	—
56	#131×C₈₉	—	2	1	no	—
57	#131×C₈₉	—	2	1	no	—
58	#131×C₈₉	—	2	1	no	—
59	C₂×M₁₆×C₈₉	—	2	1	no	—
60	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₈₉	—	2	1	no	—
62	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
63	M₃₂×C₈₉	—	2	1	no	—
64	M₃₂×C₈₉	—	2	1	no	—
65	C₂×C₁₆×C₈₉	—	2	1	no	—
66	C₂×C₁₆×C₈₉	—	2	1	no	—
67	C₂×C₁₆×C₈₉	—	2	1	no	—
68	C₂×C₁₆×C₈₉	D₄	1	2	yes	complemented
69	C₂³×D₈₉	M₁₆	1	1	yes	characteristic
70	C₂×C₂²⋊D₈₉	—	2	2	no	—
71	C₂²×Dic₈₉	M₁₆	1	1	yes	characteristic
72	C₂×C₄×D₈₉	—	2	1	no	—
73	C₂×C₄×D₈₉	—	2	1	no	—
74	C₂×C₄×D₈₉	—	2	1	no	—
75	C₂×C₄×D₈₉	—	2	1	no	—
76	C₂×C₄×D₈₉	C₂²⋊C₄	1	2	yes	—
77	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
78	#13083558	—	2	1	no	—
79	#13083558	—	2	1	no	—
80	C₂×#10623494	—	2	1	no	—
81	C₂×#10623494	—	2	1	no	—
82	C₂×#10623494	—	2	1	no	—
83	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
84	C₈×D₈₉	—	2	2	no	—
85	C₈×D₈₉	—	2	2	no	—
86	C₈×D₈₉	—	2	2	no	—
87	C₈×D₈₉	—	2	2	no	—
88	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
89	#13083553	—	2	2	no	—
90	C₂²⋊Dic₈₉	—	2	1	no	—
91	C₂²⋊Dic₈₉	—	2	2	no	—
92	#13083551	—	2	2	no	—
93	C₄×Dic₈₉	—	2	1	no	—
94	#13083549	—	2	2	no	—
95	#13083549	—	4	1	no	—
96	#41×C₈₉	—	2	1	no	—
97	#41×C₈₉	—	2	1	no	—
98	#41×C₈₉	—	2	1	no	—
99	#41×C₈₉	—	2	1	no	—
100	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
101	C₂×D₄×C₈₉	—	2	1	no	—
102	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	—	2	1	no	—
104	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
105	M₁₆×C₈₉	—	2	1	no	—
106	M₁₆×C₈₉	—	2	1	no	—
107	M₁₆×C₈₉	—	4	1	no	—
108	M₁₆×C₈₉	—	4	1	no	—
109	C₂×C₈×C₈₉	—	2	1	no	—
110	C₂×C₈×C₈₉	—	4	1	no	—
111	C₂×C₈×C₈₉	—	4	1	no	—
112	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
117	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
118	C₁₆×C₈₉	—	2	2	no	—
119	C₁₆×C₈₉	—	4	1	no	—
120	C₂³×C₈₉	—	2	1	no	—
121	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
122	C₂²×D₈₉	—	2	1	no	—
123	C₂²×D₈₉	—	2	1	no	—
124	C₂²×D₈₉	—	2	4	no	—
125	C₂²×D₈₉	C₂²⋊C₈	1	2	yes	—
126	Q₈×C₈₉	—	2	1	no	—
127	Q₈×C₈₉	—	2	1	no	—
128	D₄×C₈₉	—	2	1	no	—
129	D₄×C₈₉	—	2	1	no	—
130	D₄×C₈₉	—	4	1	no	—
131	D₄×C₈₉	—	4	1	no	—
132	C₂×C₄×C₈₉	—	2	1	no	—
133	C₂×C₄×C₈₉	—	4	1	no	—
134	C₂×C₄×C₈₉	—	4	1	no	—
135	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
140	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
141	C₂²⋊D₈₉	—	4	4	no	—
142	C₂×Dic₈₉	—	2	1	no	—
143	C₂×Dic₈₉	—	2	1	no	—
144	C₂×Dic₈₉	—	2	2	no	—
145	C₂×Dic₈₉	C₂²⋊C₈	1	2	yes	—
146	C₄×D₈₉	—	2	2	no	—
147	C₄×D₈₉	—	2	2	no	—
148	C₄×D₈₉	—	2	2	no	—
149	C₄×D₈₉	—	2	2	no	—
150	C₈×C₈₉	—	4	1	no	—
151	C₈×C₈₉	#120	1	1	yes	characteristic
152	C₈×C₈₉	#121	1	1	yes	characteristic
153	C₈×C₈₉	#121	1	1	yes	characteristic
154	C₈×C₈₉	#123	1	1	yes	characteristic
155	#10623494	—	2	2	no	—
156	#10623494	—	4	1	no	—
157	C₂²×C₈₉	—	2	2	no	—
158	C₂²×C₈₉	—	4	2	no	—
159	C₂²×C₈₉	—	4	1	no	—
160	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
161	C₂²×C₈₉	#435	1	1	yes	characteristic
162	C₂²×C₈₉	#435	1	1	yes	characteristic
163	C₂×D₈₉	—	2	4	no	—
164	C₂×D₈₉	—	2	4	no	—
165	C₂×D₈₉	—	2	2	no	—
166	C₂×D₈₉	—	2	2	no	—
167	C₂×D₈₉	#350	1	4	yes	complemented
168	C₄×C₈₉	—	4	1	no	—
169	C₄×C₈₉	#386	1	1	yes	characteristic
170	C₄×C₈₉	#408	1	1	yes	characteristic
171	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
172	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
173	Dic₈₉	—	2	2	no	—
174	Dic₈₉	—	4	2	no	—
175	#12983	—	89	1	no	maximal, Sylow
176	C₂×C₈₉	—	2	1	no	—
177	C₂×C₈₉	—	2	1	no	—
178	C₂×C₈₉	—	8	1	no	—
179	C₂×C₈₉	#1869	1	1	yes	characteristic
180	C₂×C₈₉	C₂×#350	1	2	yes	—
181	D₈₉	—	2	4	no	—
182	C₂³×C₁₆	—	89	1	no	—
183	#2929	—	89	1	no	—
184	C₂×#350	—	89	1	no	—
185	C₂×#350	D₈₉	1	1	yes	characteristic, complemented
186	#2112	—	89	1	no	—
187	#1379	—	89	1	no	—
188	#1379	—	89	1	no	—
189	C₈₉	#12983	1	1	yes	characteristic, complemented, Sylow
190	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
191	C₂³×C₈	—	89	1	no	—
192	#525	—	89	1	no	—
193	C₂×M₃₂	—	89	1	no	—
194	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
195	C₂²×C₁₆	—	89	1	no	—
196	C₂²×C₁₆	—	178	2	no	—
197	C₂²×C₁₆	—	178	2	no	—
198	C₂²×C₁₆	—	178	2	no	—
199	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
200	#435	—	178	2	no	—
201	C₈×D₄	—	178	2	no	—
202	#424	—	89	1	no	—
203	#408	—	89	1	no	—
204	#407	—	89	1	no	—
205	#405	—	89	1	no	—
206	#350	—	2	2	no	—
207	#350	—	178	2	no	—
208	C₂²⋊C₁₆	—	178	2	no	—
209	C₂×#41	C₄×D₈₉	1	1	yes	characteristic
210	C₂³×C₄	—	89	1	no	—
211	#131	—	2	1	no	—
212	#131	—	2	1	no	—
213	#131	—	2	1	no	—
214	#131	—	2	1	no	—
215	C₂×M₁₆	—	2	1	no	—
216	C₂×M₁₆	—	89	1	no	—
217	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
218	C₂²×C₈	—	2	1	no	—
219	C₂²×C₈	—	89	1	no	—
220	C₂²×C₈	—	89	2	no	—
221	C₂²×C₈	—	89	2	no	—
222	C₂²×C₈	—	178	1	no	—
223	C₂²×C₈	—	178	1	no	—
224	C₂²×C₈	—	178	1	no	—
225	C₂²×C₈	—	178	1	no	—
226	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
227	#123	—	178	1	no	—
228	C₂²⋊Q₈	—	89	2	no	—
229	#121	—	89	2	no	—
230	#120	—	178	1	no	—
231	C₄×D₄	—	178	2	no	—
232	#117	—	89	1	no	—
233	M₃₂	—	2	1	no	—
234	M₃₂	—	2	1	no	—
235	M₃₂	—	178	1	no	—
236	M₃₂	—	178	1	no	—
237	C₂×C₁₆	—	2	1	no	—
238	C₂×C₁₆	—	2	1	no	—
239	C₂×C₁₆	—	2	1	no	—
240	C₂×C₁₆	—	89	2	no	—
241	C₂×C₁₆	—	178	8	no	—
242	C₂×C₁₆	—	178	1	no	—
243	C₂×C₁₆	—	178	1	no	—
244	C₂×C₁₆	—	178	1	no	—
245	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	complemented
246	C₄⋊C₈	—	178	2	no	—
247	C₂²⋊C₈	—	178	2	no	—
248	C₂²⋊C₈	—	178	2	no	—
249	#97	—	178	1	no	—
250	C₄×C₈	—	178	1	no	—
251	C₂⁴	—	89	1	no	—
252	#41	—	2	1	no	—
253	#41	—	2	1	no	—
254	#41	—	2	1	no	—
255	#41	—	2	1	no	—
256	C₂×Q₈	C₈×D₈₉	1	1	yes	characteristic
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	178	2	no	—
259	C₂×D₄	C₈×D₈₉	1	1	yes	characteristic
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	89	1	no	—
262	C₂²×C₄	—	89	2	no	—
263	C₂²×C₄	—	89	2	no	—
264	C₂²×C₄	—	178	1	no	—
265	C₂²×C₄	—	178	1	no	—
266	C₂²×C₄	—	178	1	no	—
267	C₂²×C₄	—	178	1	no	—
268	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
269	M₁₆	—	2	1	no	—
270	M₁₆	—	2	1	no	—
271	M₁₆	—	4	1	no	—
272	M₁₆	—	4	1	no	—
273	M₁₆	—	178	1	no	—
274	M₁₆	—	178	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	89	2	no	—
279	C₂×C₈	—	178	1	no	—
280	C₂×C₈	—	178	1	no	—
281	C₂×C₈	—	178	1	no	—
282	C₂×C₈	—	178	2	no	—
283	C₂×C₈	—	178	2	no	—
284	C₂×C₈	—	178	2	no	—
285	C₂×C₈	—	178	2	no	—
286	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
287	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
288	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
289	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
290	C₂×C₈	#13083573	1	1	yes	characteristic
291	C₂×C₈	#13083574	1	1	yes	characteristic
292	C₄⋊C₄	—	89	2	no	—
293	C₄⋊C₄	—	178	2	no	—
294	C₂²⋊C₄	—	178	2	no	—
295	C₂²⋊C₄	—	178	2	no	—
296	C₂²⋊C₄	—	178	1	no	—
297	C₄²	—	178	1	no	—
298	C₁₆	—	2	2	no	—
299	C₁₆	—	4	1	no	—
300	C₁₆	—	356	1	no	—
301	C₁₆	—	178	2	no	—
302	C₂³	—	2	1	no	—
303	C₂³	—	89	2	no	—
304	C₂³	—	178	4	no	—
305	C₂³	—	178	1	no	—
306	C₂³	—	178	1	no	—
307	C₂³	M₁₆×D₈₉	1	1	yes	characteristic
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	D₄	—	2	1	no	—
311	D₄	—	2	1	no	—
312	D₄	—	4	1	no	—
313	D₄	—	4	1	no	—
314	D₄	—	356	4	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	89	2	no	—
319	C₂×C₄	—	178	2	no	—
320	C₂×C₄	—	178	1	no	—
321	C₂×C₄	—	178	1	no	—
322	C₂×C₄	—	178	2	no	—
323	C₂×C₄	—	178	2	no	—
324	C₂×C₄	—	178	2	no	—
325	C₂×C₄	—	178	2	no	—
326	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
327	C₂×C₄	#15667972	1	1	yes	characteristic
328	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
329	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
330	C₂×C₄	#15668018	1	1	yes	characteristic
331	C₂×C₄	#15668020	1	1	yes	characteristic
332	C₈	—	4	1	no	—
333	C₈	—	356	1	no	—
334	C₈	—	178	2	no	—
335	C₈	#15667980	1	1	yes	characteristic
336	C₈	#15667983	1	1	yes	characteristic
337	C₈	#15667985	1	1	yes	characteristic
338	C₈	#15667994	1	1	yes	characteristic
339	C₂²	—	2	2	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	89	4	no	—
343	C₂²	—	178	4	no	—
344	C₂²	—	178	4	no	—
345	C₂²	—	178	2	no	—
346	C₂²	—	178	2	no	—
347	C₂²	C₂²⋊C₈×D₈₉	1	1	yes	characteristic
348	C₂²	#19151667	1	1	yes	characteristic
349	C₂²	#19151672	1	1	yes	characteristic
350	C₄	—	4	1	no	—
351	C₄	—	356	2	no	—
352	C₄	—	178	2	no	—
353	C₄	#19151474	1	1	yes	characteristic
354	C₄	#19151563	1	1	yes	characteristic
355	C₄	#19151661	1	1	yes	characteristic
356	C₄	C₈×C₂²⋊D₈₉	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	8	1	no	—
360	C₂	—	178	4	no	—
361	C₂	#26020541	1	1	yes	characteristic
362	C₂	#350×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#350×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	#19152105	C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#350	D₈₉
hypocenter	C₈₉	#12983
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#350

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₆×D₈₉	C₂	semidirect
3. C₂×#19151382	C₂	semidirect
4. #26021862	C₂	semidirect
5. #26018342	C₂	semidirect
6. #26018340	C₂	semidirect
7. C₂×#350×C₈₉	C₂	semidirect
47. C₂×#13083549	D₄	semidirect
68. C₂×C₁₆×C₈₉	D₄	semidirect
167. C₂×D₈₉	#350	semidirect
185. C₂×#350	D₈₉	semidirect
189. C₈₉	#12983	semidirect
245. C₂×C₁₆	C₂²⋊D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₈₉	2
#26028944	2
C₂×#19151382	2
#26021862	2
#26018342	2
#26018340	2
C₂×#350×C₈₉	2
#12983	89