Subgroups — #36470650

Order 22784.

440 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#405×D₈₉	C₂	1	1	yes	maximal, characteristic
2	#26021862	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151377	C₂	1	1	yes	maximal, characteristic
4	#26018342	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018341	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018340	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₈₉	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
10	#117×D₈₉	C₄	1	1	yes	characteristic
11	C₂×#15667952	C₂²	1	1	yes	characteristic
12	C₂²×#13083549	C₂²	1	1	yes	characteristic
13	#19151556	C₄	1	1	yes	characteristic
14	#19151555	—	2	1	no	—
15	#19151554	C₄	1	1	yes	characteristic
16	#19151553	C₂²	1	1	yes	characteristic
17	#19151552	—	2	1	no	—
18	#97×D₈₉	—	2	1	no	—
19	C₄×C₈×D₈₉	—	2	1	no	—
20	#19151377	—	2	1	no	—
21	#19151377	—	2	1	no	—
22	#19151377	—	2	1	no	—
23	#19151377	—	2	1	no	—
24	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	#405×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
29	C₂×#13083557	—	2	1	no	—
30	M₁₆×D₈₉	—	2	2	no	—
31	M₁₆×D₈₉	—	2	2	no	—
32	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₈₉	—	2	1	no	—
34	C₂×C₈×D₈₉	—	2	1	no	—
35	C₂×C₈×D₈₉	—	2	1	no	—
36	C₂×C₈×D₈₉	—	2	1	no	—
37	C₂×C₈×D₈₉	—	2	1	no	—
38	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
39	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
40	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
41	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
42	#15667971	C₂×C₄	1	1	yes	characteristic
43	#15667970	C₈	1	1	yes	characteristic
44	#15667970	C₈	1	1	yes	characteristic
45	#15667969	—	2	1	no	—
46	#15667968	—	2	1	no	—
47	C₄⋊C₄×D₈₉	C₈	1	1	yes	characteristic
48	C₄⋊C₄×D₈₉	C₈	1	1	yes	characteristic
49	C₂²⋊C₄×D₈₉	—	2	1	no	—
50	C₄²×D₈₉	—	2	1	no	—
51	#15667952	—	2	1	no	—
52	#15667952	—	2	1	no	—
53	C₂×#13083549	—	2	1	no	—
54	C₂×#13083549	—	2	1	no	—
55	C₂×#13083549	—	2	1	no	—
56	C₂×#13083549	—	2	1	no	—
57	#15667923	—	2	1	no	—
58	#15667922	—	2	1	no	—
59	#15667922	—	2	1	no	—
60	C₄×#10623494	—	2	1	no	—
61	C₈×Dic₈₉	—	2	1	no	—
62	C₈×Dic₈₉	—	2	1	no	—
63	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
66	M₃₂×C₈₉	—	2	1	no	—
67	M₃₂×C₈₉	—	2	1	no	—
68	C₂×C₁₆×C₈₉	—	2	1	no	—
69	C₂×C₁₆×C₈₉	—	2	1	no	—
70	C₂×C₁₆×C₈₉	—	2	1	no	—
71	C₂×C₁₆×C₈₉	—	2	1	no	—
72	#97×C₈₉	—	2	1	no	—
73	C₄×C₈×C₈₉	—	2	1	no	—
74	C₂³×D₈₉	M₁₆	1	1	yes	characteristic
75	C₂²×Dic₈₉	M₁₆	1	1	yes	characteristic
76	C₂×C₄×D₈₉	—	2	1	no	—
77	C₂×C₄×D₈₉	—	2	1	no	—
78	C₂×C₄×D₈₉	—	2	1	no	—
79	C₂×C₄×D₈₉	—	2	1	no	—
80	C₂×C₄×D₈₉	—	2	1	no	—
81	C₂×C₄×D₈₉	—	2	1	no	—
82	C₂×C₄×D₈₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×D₈₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
86	#13083558	—	2	1	no	—
87	#13083558	M₁₆	1	2	yes	—
88	#13083557	—	2	4	no	—
89	C₂×#10623494	—	2	1	no	—
90	C₂×#10623494	—	2	1	no	—
91	C₂×#10623494	—	2	1	no	—
92	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
94	C₈×D₈₉	—	2	4	no	—
95	C₈×D₈₉	—	2	1	no	—
96	C₈×D₈₉	—	2	1	no	—
97	C₈×D₈₉	—	2	1	no	—
98	C₈×D₈₉	—	2	1	no	—
99	C₈×D₈₉	—	2	1	no	—
100	C₈×D₈₉	—	2	1	no	—
101	C₈×D₈₉	—	2	1	no	—
102	C₈×D₈₉	—	2	1	no	—
103	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
104	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
105	#13083553	—	2	1	no	—
106	#13083553	—	2	1	no	—
107	C₂²⋊Dic₈₉	—	2	1	no	—
108	#13083551	—	2	1	no	—
109	#13083551	—	2	1	no	—
110	C₄×Dic₈₉	—	2	1	no	—
111	C₄×Dic₈₉	—	2	1	no	—
112	C₄×Dic₈₉	—	2	1	no	—
113	#13083549	—	4	1	no	—
114	#13083549	—	4	1	no	—
115	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
116	M₁₆×C₈₉	—	2	1	no	—
117	M₁₆×C₈₉	M₁₆	1	2	yes	—
118	C₂×C₈×C₈₉	—	2	1	no	—
119	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
122	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
123	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
124	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
125	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
126	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
127	C₂²⋊C₄×C₈₉	—	2	1	no	—
128	C₄²×C₈₉	—	2	1	no	—
129	C₁₆×C₈₉	—	4	1	no	—
130	C₁₆×C₈₉	—	4	1	no	—
131	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
132	C₂²×D₈₉	—	2	1	no	—
133	C₂²×D₈₉	—	2	1	no	—
134	C₂²×D₈₉	—	2	2	no	—
135	C₂²×D₈₉	—	4	1	no	—
136	C₂²×D₈₉	C₂²⋊C₈	1	1	yes	characteristic
137	C₂²×D₈₉	C₂²⋊C₈	1	1	yes	characteristic
138	C₂×C₄×C₈₉	—	2	1	no	—
139	C₂×C₄×C₈₉	—	2	1	no	—
140	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
142	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
146	C₂×Dic₈₉	—	2	1	no	—
147	C₂×Dic₈₉	—	2	1	no	—
148	C₂×Dic₈₉	—	2	1	no	—
149	C₂×Dic₈₉	—	2	1	no	—
150	C₂×Dic₈₉	C₂²⋊C₈	1	1	yes	characteristic
151	C₂×Dic₈₉	C₂²⋊C₈	1	1	yes	characteristic
152	C₄×D₈₉	—	2	1	no	—
153	C₄×D₈₉	—	2	1	no	—
154	C₄×D₈₉	—	2	1	no	—
155	C₄×D₈₉	—	2	1	no	—
156	C₄×D₈₉	—	2	1	no	—
157	C₄×D₈₉	—	2	1	no	—
158	C₄×D₈₉	—	2	1	no	—
159	C₄×D₈₉	—	2	1	no	—
160	C₄×D₈₉	—	4	2	no	—
161	C₄×D₈₉	—	4	2	no	—
162	C₈×C₈₉	—	2	2	no	—
163	C₈×C₈₉	#123	1	1	yes	characteristic
164	C₈×C₈₉	#123	1	1	yes	characteristic
165	C₈×C₈₉	#123	1	1	yes	characteristic
166	C₈×C₈₉	#123	1	1	yes	characteristic
167	#10623494	—	2	1	no	—
168	#10623494	—	2	1	no	—
169	#10623494	—	2	2	no	—
170	C₂²×C₈₉	—	2	1	no	—
171	C₂²×C₈₉	—	2	1	no	—
172	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
173	C₂²×C₈₉	#432	1	1	yes	characteristic
174	C₂²×C₈₉	#432	1	1	yes	characteristic
175	C₂×D₈₉	—	2	1	no	—
176	C₂×D₈₉	—	2	1	no	—
177	C₂×D₈₉	—	2	1	no	—
178	C₂×D₈₉	—	2	1	no	—
179	C₂×D₈₉	—	2	1	no	—
180	C₂×D₈₉	—	4	2	no	—
181	C₂×D₈₉	—	4	2	no	—
182	C₂×D₈₉	#350	1	2	yes	—
183	C₄×C₈₉	—	4	1	no	—
184	C₄×C₈₉	—	4	1	no	—
185	C₄×C₈₉	#381	1	1	yes	characteristic
186	C₄×C₈₉	#407	1	1	yes	characteristic
187	C₄×C₈₉	#431	1	1	yes	characteristic
188	C₄×C₈₉	#431	1	1	yes	characteristic
189	Dic₈₉	—	2	1	no	—
190	Dic₈₉	—	2	1	no	—
191	Dic₈₉	—	4	1	no	—
192	Dic₈₉	—	4	1	no	—
193	#12848	—	89	1	no	maximal, Sylow
194	C₂×C₈₉	—	2	1	no	—
195	C₂×C₈₉	—	2	1	no	—
196	C₂×C₈₉	#1817	1	1	yes	characteristic
197	C₂×C₈₉	C₂×#350	1	1	yes	characteristic
198	C₂×C₈₉	#2117	1	1	yes	characteristic
199	D₈₉	—	2	2	no	—
200	D₈₉	—	4	1	no	—
201	C₂×#405	—	89	1	no	—
202	#2112	—	89	1	no	—
203	C₂×C₂²⋊C₁₆	—	89	1	no	—
204	#1379	—	89	1	no	—
205	#1379	—	89	1	no	—
206	#1379	—	89	1	no	—
207	#1379	D₈₉	1	1	yes	characteristic, complemented
208	C₈₉	#12848	1	1	yes	characteristic, complemented, Sylow
209	C₂²×M₁₆	—	89	1	no	—
210	C₂³×C₈	—	89	1	no	—
211	C₂×#117	—	89	1	no	—
212	C₂×M₃₂	—	89	1	no	—
213	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
214	C₂²×C₁₆	—	89	1	no	—
215	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
216	#405	—	89	1	no	—
217	#405	—	89	1	no	—
218	#405	—	89	1	no	—
219	#405	—	178	1	no	—
220	#405	—	178	1	no	—
221	#405	C₂×D₈₉	1	1	yes	characteristic
222	C₂×#97	—	178	1	no	—
223	C₂×C₄×C₈	—	178	1	no	—
224	C₂²⋊C₁₆	—	178	1	no	—
225	C₂²⋊C₁₆	—	178	1	no	—
226	C₂²⋊C₁₆	—	178	1	no	—
227	C₂²⋊C₁₆	—	178	1	no	—
228	C₂³×C₄	—	89	1	no	—
229	C₂×M₁₆	—	89	1	no	—
230	C₂×M₁₆	—	178	1	no	—
231	C₂×M₁₆	—	178	2	no	—
232	C₂×M₁₆	—	178	2	no	—
233	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
234	C₂²×C₈	—	89	1	no	—
235	C₂²×C₈	—	89	1	no	—
236	C₂²×C₈	—	89	1	no	—
237	C₂²×C₈	—	89	1	no	—
238	C₂²×C₈	—	89	1	no	—
239	C₂²×C₈	—	178	1	no	—
240	C₂²×C₈	—	178	1	no	—
241	C₂²×C₈	—	178	1	no	—
242	C₂²×C₈	—	178	1	no	—
243	C₂²×C₈	—	178	1	no	—
244	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
245	#117	—	89	1	no	—
246	#117	—	89	1	no	—
247	#117	—	89	1	no	—
248	#117	—	178	1	no	—
249	#117	—	178	1	no	—
250	#117	C₄×D₈₉	1	1	yes	characteristic
251	C₂×C₄⋊C₄	—	89	1	no	—
252	C₂×C₄⋊C₄	—	89	1	no	—
253	C₂×C₂²⋊C₄	—	178	1	no	—
254	C₂×C₄²	—	178	1	no	—
255	M₃₂	—	2	1	no	—
256	M₃₂	—	2	1	no	—
257	M₃₂	—	178	1	no	—
258	M₃₂	—	178	1	no	—
259	C₂×C₁₆	—	2	1	no	—
260	C₂×C₁₆	—	2	1	no	—
261	C₂×C₁₆	—	2	1	no	—
262	C₂×C₁₆	—	2	1	no	—
263	C₂×C₁₆	—	178	1	no	—
264	C₂×C₁₆	—	178	1	no	—
265	C₂×C₁₆	—	178	1	no	—
266	C₂×C₁₆	—	178	1	no	—
267	#97	—	2	1	no	—
268	#97	—	178	1	no	—
269	#97	—	178	1	no	—
270	#97	—	178	1	no	—
271	C₄×C₈	—	2	1	no	—
272	C₄×C₈	—	178	1	no	—
273	C₄×C₈	—	178	1	no	—
274	C₄×C₈	—	178	1	no	—
275	C₂⁴	—	89	1	no	—
276	C₂²×C₄	—	89	1	no	—
277	C₂²×C₄	—	89	1	no	—
278	C₂²×C₄	—	89	1	no	—
279	C₂²×C₄	—	89	1	no	—
280	C₂²×C₄	—	89	1	no	—
281	C₂²×C₄	—	178	1	no	—
282	C₂²×C₄	—	178	1	no	—
283	C₂²×C₄	—	178	1	no	—
284	C₂²×C₄	—	178	1	no	—
285	C₂²×C₄	—	178	1	no	—
286	C₂²×C₄	—	178	1	no	—
287	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
288	M₁₆	—	2	1	no	—
289	M₁₆	—	89	2	no	—
290	M₁₆	—	178	4	no	—
291	M₁₆	—	178	1	no	—
292	M₁₆	#13083557	1	2	yes	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	89	1	no	—
295	C₂×C₈	—	89	1	no	—
296	C₂×C₈	—	178	4	no	—
297	C₂×C₈	—	178	1	no	—
298	C₂×C₈	—	178	1	no	—
299	C₂×C₈	—	178	1	no	—
300	C₂×C₈	—	178	1	no	—
301	C₂×C₈	—	178	1	no	—
302	C₂×C₈	—	178	1	no	—
303	C₂×C₈	—	178	1	no	—
304	C₂×C₈	—	178	1	no	—
305	C₂×C₈	—	178	1	no	—
306	C₂×C₈	—	178	1	no	—
307	C₂×C₈	—	178	1	no	—
308	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
309	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
310	C₂×C₈	#13083573	1	1	yes	characteristic
311	C₂×C₈	#13083574	1	1	yes	characteristic
312	C₂×C₈	#13083574	1	1	yes	characteristic
313	C₂×C₈	#13083575	1	1	yes	characteristic
314	C₄⋊C₄	—	89	1	no	—
315	C₄⋊C₄	—	89	1	no	—
316	C₄⋊C₄	—	178	1	no	—
317	C₄⋊C₄	—	178	1	no	—
318	C₄⋊C₄	C₈×D₈₉	1	1	yes	characteristic
319	C₄⋊C₄	C₈×D₈₉	1	1	yes	characteristic
320	C₂²⋊C₄	—	2	1	no	—
321	C₂²⋊C₄	—	178	1	no	—
322	C₂²⋊C₄	—	178	1	no	—
323	C₂²⋊C₄	—	178	1	no	—
324	C₄²	—	2	1	no	—
325	C₄²	—	178	1	no	—
326	C₄²	—	178	1	no	—
327	C₄²	—	178	1	no	—
328	C₁₆	—	4	1	no	—
329	C₁₆	—	4	1	no	—
330	C₁₆	—	356	1	no	—
331	C₁₆	—	356	1	no	—
332	C₂³	—	89	1	no	—
333	C₂³	—	89	1	no	—
334	C₂³	—	356	1	no	—
335	C₂³	—	178	2	no	—
336	C₂³	—	178	1	no	—
337	C₂³	—	178	1	no	—
338	C₂³	M₁₆×D₈₉	1	1	yes	characteristic
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	89	1	no	—
342	C₂×C₄	—	89	1	no	—
343	C₂×C₄	—	356	2	no	—
344	C₂×C₄	—	356	2	no	—
345	C₂×C₄	—	178	1	no	—
346	C₂×C₄	—	178	1	no	—
347	C₂×C₄	—	178	1	no	—
348	C₂×C₄	—	178	1	no	—
349	C₂×C₄	—	178	1	no	—
350	C₂×C₄	—	178	1	no	—
351	C₂×C₄	—	178	1	no	—
352	C₂×C₄	—	178	1	no	—
353	C₂×C₄	—	178	1	no	—
354	C₂×C₄	—	178	1	no	—
355	C₂×C₄	—	178	1	no	—
356	C₂×C₄	—	178	1	no	—
357	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
358	C₂×C₄	#15667969	1	1	yes	characteristic
359	C₂×C₄	#15667970	1	1	yes	characteristic
360	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
361	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
362	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
363	C₈	—	2	2	no	—
364	C₈	—	178	1	no	—
365	C₈	—	178	1	no	—
366	C₈	—	178	2	no	—
367	C₈	#15667994	1	1	yes	characteristic
368	C₈	#15667994	1	1	yes	characteristic
369	C₈	#15667997	1	1	yes	characteristic
370	C₈	#15667997	1	1	yes	characteristic
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	89	2	no	—
374	C₂²	—	356	2	no	—
375	C₂²	—	356	2	no	—
376	C₂²	—	178	1	no	—
377	C₂²	—	178	1	no	—
378	C₂²	—	178	1	no	—
379	C₂²	—	178	1	no	—
380	C₂²	—	178	1	no	—
381	C₂²	C₂²⋊C₈×D₈₉	1	1	yes	characteristic
382	C₂²	#19151653	1	1	yes	characteristic
383	C₂²	#19151654	1	1	yes	characteristic
384	C₄	—	4	1	no	—
385	C₄	—	4	1	no	—
386	C₄	—	356	1	no	—
387	C₄	—	356	1	no	—
388	C₄	—	178	1	no	—
389	C₄	—	178	1	no	—
390	C₄	#19151460	1	1	yes	characteristic
391	C₄	#19151560	1	1	yes	characteristic
392	C₄	#19151649	1	1	yes	characteristic
393	C₄	#19151650	1	1	yes	characteristic
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	356	1	no	—
397	C₂	—	178	2	no	—
398	C₂	#26020319	1	1	yes	characteristic
399	C₂	#350×D₈₉	1	1	yes	characteristic
400	C₂	#26021878	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1379	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂³×D₈₉	M₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1379	D₈₉
hypocenter	C₈₉	#12848
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
4. #26018342	C₂	semidirect
5. #26018341	C₂	semidirect
6. #26018340	C₂	semidirect
7. C₈₉×#1379	C₂	semidirect
207. #1379	D₈₉	semidirect
208. C₈₉	#12848	semidirect

Maximal subgroups

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

subgroup	index
#405×D₈₉	2
#26021862	2
C₂×#19151377	2
#26018342	2
#26018341	2
#26018340	2
C₈₉×#1379	2
#12848	89