Subgroups — #36538301

Order 22784.

606 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030836	C₂	1	1	yes	maximal, characteristic
2	#26030384	C₂	1	2	yes	maximal
3	C₈×Q₈×D₈₉	C₂	1	1	yes	maximal, characteristic
4	#26019681	C₂	1	2	yes	maximal, complemented
5	#26019680	C₂	1	2	yes	maximal, complemented
6	#26019647	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019646	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019619	C₂	1	1	yes	maximal, characteristic, complemented
9	#26019014	C₂	1	1	yes	maximal, characteristic
10	#26018951	C₂	1	2	yes	maximal
11	C₈₉×#1670	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152153	C₂²	1	2	yes	—
13	C₄×Q₈×D₈₉	C₂²	1	1	yes	characteristic
14	#19152003	C₂²	1	1	yes	characteristic
15	#19151948	C₂²	1	2	yes	—
16	#19151930	—	2	2	no	—
17	#19151880	C₂²	1	2	yes	—
18	#19151836	—	2	1	no	—
19	#19151836	—	2	1	no	—
20	#19151809	C₂²	1	2	yes	—
21	#19151808	C₂²	1	2	yes	—
22	#19151755	—	2	2	no	—
23	Q₈×#10623494	C₂²	1	1	yes	characteristic
24	C₈×Dic₁₇₈	—	2	1	no	—
25	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
26	#19151722	—	2	1	no	—
27	#19151722	C₂²	1	1	yes	characteristic
28	#19151704	—	2	2	no	—
29	#19151702	—	2	2	no	—
30	Q₁₆×Dic₈₉	—	2	1	no	—
31	Q₁₆×Dic₈₉	—	2	1	no	—
32	#19151654	C₂²	1	2	yes	—
33	C₄⋊C₈×D₈₉	—	2	1	no	—
34	C₄⋊C₈×D₈₉	C₂²	1	1	yes	characteristic
35	C₄×C₈×D₈₉	—	2	1	no	—
36	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
37	#19151351	C₂²	1	1	yes	characteristic
38	#19151348	C₂²	1	2	yes	—
39	#19151345	C₂²	1	2	yes	—
40	#19151344	C₂²	1	2	yes	—
41	#19151343	C₂²	1	2	yes	—
42	#494×C₈₉	C₂²	1	1	yes	characteristic
43	#475×C₈₉	C₂²	1	2	yes	—
44	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
45	#335×C₈₉	C₂²	1	1	yes	characteristic
46	#332×C₈₉	C₂²	1	2	yes	—
47	C₂×Q₈×D₈₉	D₄	1	1	yes	characteristic
48	C₂×Dic₃₅₆	—	2	2	no	—
49	C₂×#13083564	—	2	2	no	—
50	C₂×C₈×D₈₉	—	2	1	no	—
51	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
52	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
53	#15668012	C₂³	1	2	yes	—
54	#15668003	—	2	2	no	—
55	#15667992	—	2	2	no	—
56	#15667988	—	2	2	no	—
57	Q₈×Dic₈₉	—	2	2	no	—
58	Q₈×Dic₈₉	D₄	1	1	yes	characteristic
59	C₄×Dic₁₇₈	—	2	1	no	—
60	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
61	#15667977	—	2	1	no	—
62	#15667977	—	2	2	no	—
63	#15667977	D₄	1	1	yes	characteristic
64	#15667970	—	2	2	no	—
65	C₄⋊C₄×D₈₉	—	2	1	no	—
66	C₄⋊C₄×D₈₉	D₄	1	1	yes	characteristic
67	C₄²×D₈₉	—	2	1	no	—
68	C₄²×D₈₉	C₂³	1	1	yes	characteristic
69	#15667946	—	2	1	no	—
70	#15667946	—	2	1	no	—
71	#15667943	—	2	2	no	—
72	C₄⋊#10623494	—	2	1	no	—
73	C₄⋊#10623494	C₂³	1	1	yes	characteristic
74	C₄⋊#10623494	C₂³	1	2	yes	—
75	#15667941	—	2	1	no	—
76	#15667941	—	2	1	no	—
77	#15667941	—	2	1	no	—
78	#15667937	—	2	2	no	—
79	#15667936	—	2	2	no	—
80	#15667936	—	2	2	no	—
81	#15667935	—	2	2	no	—
82	#15667935	—	2	2	no	—
83	#15667924	—	2	2	no	—
84	#15667922	—	2	2	no	—
85	C₄×#10623494	—	2	1	no	—
86	C₄×#10623494	C₂³	1	1	yes	characteristic
87	C₈×Dic₈₉	—	2	1	no	—
88	C₈×Dic₈₉	—	2	1	no	—
89	C₈×Dic₈₉	D₄	1	1	yes	characteristic
90	C₈×Dic₈₉	D₄	1	1	yes	characteristic
91	C₂×Q₁₆×C₈₉	—	2	1	no	—
92	C₂×Q₁₆×C₈₉	—	2	1	no	—
93	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
94	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
95	#106×C₈₉	—	2	2	no	—
96	C₄⋊C₈×C₈₉	—	2	1	no	—
97	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
99	#103×C₈₉	—	2	2	no	—
100	C₄×C₈×C₈₉	—	2	1	no	—
101	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×Dic₁₇₈	—	2	1	no	—
103	C₂×Dic₁₇₈	—	2	2	no	—
104	C₂×Dic₁₇₈	C₂×D₄	1	1	yes	characteristic
105	Q₈×D₈₉	—	2	2	no	—
106	Q₈×D₈₉	—	2	2	no	—
107	C₂×C₄×D₈₉	—	2	1	no	—
108	C₂×C₄×D₈₉	—	2	1	no	—
109	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×D₈₉	#41	1	1	yes	characteristic
112	Dic₃₅₆	—	4	4	no	—
113	#13083564	—	4	4	no	—
114	C₂×#10623494	—	2	1	no	—
115	C₂×#10623494	—	2	2	no	—
116	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
117	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
118	C₈×D₈₉	—	2	2	no	—
119	C₈×D₈₉	—	4	1	no	—
120	C₈×D₈₉	—	4	2	no	—
121	C₄⋊Dic₈₉	—	2	1	no	—
122	C₄⋊Dic₈₉	—	2	2	no	—
123	C₄⋊Dic₈₉	—	2	2	no	—
124	C₄⋊Dic₈₉	—	4	2	no	—
125	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
126	#13083553	—	2	1	no	—
127	#13083553	—	2	1	no	—
128	#13083553	—	2	1	no	—
129	#13083553	—	4	2	no	—
130	#13083551	—	4	2	no	—
131	C₄×Dic₈₉	—	2	1	no	—
132	C₄×Dic₈₉	—	2	1	no	—
133	C₄×Dic₈₉	—	2	1	no	—
134	C₄×Dic₈₉	—	4	2	no	—
135	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
136	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
137	C₄×Dic₈₉	#41	1	1	yes	characteristic
138	C₂×Q₈×C₈₉	—	2	2	no	—
139	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
140	Q₁₆×C₈₉	—	4	2	no	—
141	Q₁₆×C₈₉	—	4	2	no	—
142	C₂×C₈×C₈₉	—	2	1	no	—
143	C₂×C₈×C₈₉	—	2	2	no	—
144	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
145	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
146	C₄⋊C₄×C₈₉	—	2	1	no	—
147	C₄⋊C₄×C₈₉	—	2	2	no	—
148	C₄⋊C₄×C₈₉	—	2	2	no	—
149	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₈₉	—	2	1	no	—
151	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
152	C₂²×D₈₉	#121	1	1	yes	characteristic
153	Q₈×C₈₉	—	2	1	no	—
154	Q₈×C₈₉	—	4	2	no	—
155	Q₈×C₈₉	—	4	2	no	—
156	Q₈×C₈₉	#135	1	2	yes	—
157	C₂×C₄×C₈₉	—	2	1	no	—
158	C₂×C₄×C₈₉	—	2	1	no	—
159	C₂×C₄×C₈₉	—	4	2	no	—
160	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
163	C₂×Dic₈₉	—	2	1	no	—
164	C₂×Dic₈₉	—	2	1	no	—
165	C₂×Dic₈₉	—	4	2	no	—
166	C₂×Dic₈₉	#121	1	1	yes	characteristic
167	C₂×Dic₈₉	#121	1	1	yes	characteristic
168	C₂×Dic₈₉	#121	1	1	yes	characteristic
169	C₄×D₈₉	—	2	4	no	—
170	C₄×D₈₉	—	2	2	no	—
171	C₄×D₈₉	—	2	1	no	—
172	C₄×D₈₉	—	2	1	no	—
173	C₄×D₈₉	—	2	1	no	—
174	C₄×D₈₉	—	2	1	no	—
175	C₄×D₈₉	—	4	2	no	—
176	Dic₁₇₈	—	2	4	no	—
177	Dic₁₇₈	—	2	1	no	—
178	Dic₁₇₈	—	4	2	no	—
179	Dic₁₇₈	—	4	2	no	—
180	Dic₁₇₈	#135	1	2	yes	—
181	C₈×C₈₉	—	2	1	no	—
182	C₈×C₈₉	—	4	2	no	—
183	C₈×C₈₉	—	4	1	no	—
184	C₈×C₈₉	#137	1	2	yes	—
185	#10623494	—	2	1	no	—
186	#10623494	—	2	1	no	—
187	#10623494	—	4	2	no	—
188	#10623494	—	4	1	no	—
189	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
190	C₂×D₈₉	—	2	1	no	—
191	C₂×D₈₉	—	2	1	no	—
192	C₂×D₈₉	#473	1	2	yes	—
193	C₄×C₈₉	—	2	2	no	—
194	C₄×C₈₉	—	2	1	no	—
195	C₄×C₈₉	—	4	1	no	—
196	C₄×C₈₉	—	8	2	no	—
197	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
198	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
199	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
200	C₄×C₈₉	#575	1	1	yes	characteristic
201	Dic₈₉	—	2	1	no	—
202	Dic₈₉	—	2	1	no	—
203	Dic₈₉	—	2	2	no	—
204	Dic₈₉	—	2	1	no	—
205	Dic₈₉	—	4	1	no	—
206	Dic₈₉	—	8	2	no	—
207	#19265	—	89	1	no	maximal, Sylow
208	C₂×C₈₉	#3037	1	1	yes	characteristic
209	C₂×C₈₉	#3050	1	1	yes	characteristic
210	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
211	D₈₉	—	2	2	no	—
212	#3170	—	89	1	no	—
213	#3113	—	89	2	no	—
214	C₂×C₈×Q₈	—	89	1	no	—
215	#1680	—	89	2	no	—
216	#1680	—	89	2	no	—
217	#1674	—	89	1	no	—
218	#1674	—	89	1	no	—
219	#1670	—	89	1	no	—
220	#1670	D₈₉	1	1	yes	characteristic, complemented
221	#1568	—	89	1	no	—
222	#1552	—	89	2	no	—
223	C₈₉	#19265	1	1	yes	characteristic, complemented, Sylow
224	#533	—	89	2	no	—
225	C₂×C₄×Q₈	—	89	1	no	—
226	#500	—	89	1	no	—
227	#494	C₂×D₈₉	1	1	yes	characteristic
228	#487	—	89	2	no	—
229	#483	—	178	2	no	—
230	#475	—	89	2	no	—
231	#475	C₂×D₈₉	1	2	yes	—
232	#467	—	178	1	no	—
233	#467	—	178	1	no	—
234	#462	—	89	2	no	—
235	#462	—	89	2	no	—
236	#451	—	178	2	no	—
237	C₈×Q₈	—	89	1	no	—
238	C₈×Q₈	—	89	1	no	—
239	C₈×Q₈	—	89	1	no	—
240	C₈×Q₈	—	178	1	no	—
241	C₈×Q₈	—	178	1	no	—
242	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
243	#441	—	178	2	no	—
244	#441	—	178	2	no	—
245	C₄×Q₁₆	—	178	1	no	—
246	C₄×Q₁₆	—	178	1	no	—
247	#432	—	89	2	no	—
248	C₂×C₄⋊C₈	—	89	1	no	—
249	C₂×C₄⋊C₈	—	178	1	no	—
250	C₂×C₄×C₈	—	89	1	no	—
251	C₂×C₄×C₈	—	178	1	no	—
252	#335	—	89	1	no	—
253	#335	C₂×D₈₉	1	1	yes	characteristic
254	#334	—	89	2	no	—
255	#332	—	89	2	no	—
256	#332	—	89	2	no	—
257	#332	—	89	2	no	—
258	#332	C₂×D₈₉	1	2	yes	—
259	C₂²×Q₈	—	89	1	no	—
260	C₂×Q₁₆	—	2	1	no	—
261	C₂×Q₁₆	—	2	1	no	—
262	C₂×Q₁₆	—	178	2	no	—
263	C₂×Q₁₆	—	178	2	no	—
264	C₂²×C₈	—	89	1	no	—
265	C₂²×C₈	—	89	1	no	—
266	C₂²×C₈	—	178	1	no	—
267	C₄⋊Q₈	—	89	2	no	—
268	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
269	#125	—	178	2	no	—
270	C₂²⋊Q₈	—	178	2	no	—
271	C₂²⋊Q₈	—	178	2	no	—
272	C₄×Q₈	—	89	1	no	—
273	C₄×Q₈	—	89	1	no	—
274	C₄×Q₈	—	89	1	no	—
275	C₄×Q₈	—	178	1	no	—
276	C₄×Q₈	—	178	2	no	—
277	C₄×Q₈	—	178	2	no	—
278	C₄×Q₈	—	178	1	no	—
279	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
280	#117	—	178	2	no	—
281	C₂×C₄⋊C₄	—	89	1	no	—
282	C₂×C₄⋊C₄	—	178	1	no	—
283	C₂×C₄²	—	89	1	no	—
284	C₂×C₄²	—	178	1	no	—
285	#107	—	178	1	no	—
286	#107	—	178	1	no	—
287	#106	—	2	2	no	—
288	#106	—	178	2	no	—
289	C₄⋊C₈	—	2	1	no	—
290	C₄⋊C₈	—	89	1	no	—
291	C₄⋊C₈	—	89	2	no	—
292	C₄⋊C₈	—	178	1	no	—
293	C₄⋊C₈	—	178	1	no	—
294	C₄⋊C₈	—	178	1	no	—
295	C₄⋊C₈	—	178	1	no	—
296	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
297	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
298	#103	—	2	2	no	—
299	#103	—	178	2	no	—
300	#103	—	178	2	no	—
301	#103	—	178	2	no	—
302	#103	—	178	2	no	—
303	#103	—	178	2	no	—
304	C₂²⋊C₈	—	178	2	no	—
305	#97	—	178	2	no	—
306	C₄×C₈	—	2	1	no	—
307	C₄×C₈	—	89	1	no	—
308	C₄×C₈	—	89	1	no	—
309	C₄×C₈	—	89	1	no	—
310	C₄×C₈	—	178	1	no	—
311	C₄×C₈	—	178	1	no	—
312	C₄×C₈	—	178	1	no	—
313	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	89	1	no	—
316	C₂×Q₈	—	178	1	no	—
317	C₂×Q₈	—	178	2	no	—
318	C₂×Q₈	—	178	2	no	—
319	C₂×Q₈	—	178	2	no	—
320	C₂×Q₈	D₄×D₈₉	1	1	yes	characteristic
321	C₂²×C₄	—	89	1	no	—
322	C₂²×C₄	—	89	1	no	—
323	C₂²×C₄	—	89	1	no	—
324	C₂²×C₄	—	178	1	no	—
325	C₂²×C₄	—	178	1	no	—
326	Q₁₆	—	4	2	no	—
327	Q₁₆	—	4	2	no	—
328	Q₁₆	—	356	4	no	—
329	Q₁₆	—	356	4	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	89	1	no	—
333	C₂×C₈	—	89	1	no	—
334	C₂×C₈	—	356	1	no	—
335	C₂×C₈	—	356	2	no	—
336	C₂×C₈	—	178	1	no	—
337	C₂×C₈	—	178	2	no	—
338	C₂×C₈	—	178	2	no	—
339	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
340	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
341	C₄⋊C₄	—	2	1	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	89	1	no	—
345	C₄⋊C₄	—	356	2	no	—
346	C₄⋊C₄	—	356	2	no	—
347	C₄⋊C₄	—	178	1	no	—
348	C₄⋊C₄	—	178	1	no	—
349	C₄⋊C₄	—	178	1	no	—
350	C₄⋊C₄	—	178	2	no	—
351	C₄⋊C₄	—	178	2	no	—
352	C₄⋊C₄	—	178	1	no	—
353	C₄⋊C₄	D₄×D₈₉	1	1	yes	characteristic
354	C₂²⋊C₄	—	356	2	no	—
355	C₄²	—	2	1	no	—
356	C₄²	—	89	1	no	—
357	C₄²	—	89	1	no	—
358	C₄²	—	89	1	no	—
359	C₄²	—	356	2	no	—
360	C₄²	—	178	1	no	—
361	C₄²	—	178	1	no	—
362	C₄²	—	178	1	no	—
363	C₄²	C₂³×D₈₉	1	1	yes	characteristic
364	C₂³	—	89	1	no	—
365	Q₈	—	2	1	no	—
366	Q₈	—	4	2	no	—
367	Q₈	—	4	2	no	—
368	Q₈	—	89	2	no	—
369	Q₈	—	356	2	no	—
370	Q₈	—	356	2	no	—
371	Q₈	—	178	4	no	—
372	Q₈	—	178	1	no	—
373	Q₈	#15668035	1	2	yes	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	89	1	no	—
378	C₂×C₄	—	89	1	no	—
379	C₂×C₄	—	89	1	no	—
380	C₂×C₄	—	356	2	no	—
381	C₂×C₄	—	356	2	no	—
382	C₂×C₄	—	178	2	no	—
383	C₂×C₄	—	178	4	no	—
384	C₂×C₄	—	178	1	no	—
385	C₂×C₄	—	178	1	no	—
386	C₂×C₄	—	178	1	no	—
387	C₂×C₄	—	178	1	no	—
388	C₂×C₄	—	178	1	no	—
389	C₂×C₄	—	178	1	no	—
390	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
391	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
392	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
393	C₈	—	2	1	no	—
394	C₈	—	4	1	no	—
395	C₈	—	4	2	no	—
396	C₈	—	356	1	no	—
397	C₈	—	356	2	no	—
398	C₈	—	178	1	no	—
399	C₈	—	178	1	no	—
400	C₈	#15668041	1	2	yes	—
401	C₂²	—	89	2	no	—
402	C₂²	—	178	1	no	—
403	C₂²	—	178	1	no	—
404	C₂²	#121×D₈₉	1	1	yes	characteristic
405	C₄	—	2	2	no	—
406	C₄	—	2	1	no	—
407	C₄	—	4	1	no	—
408	C₄	—	8	2	no	—
409	C₄	—	356	1	no	—
410	C₄	—	178	1	no	—
411	C₄	—	178	2	no	—
412	C₄	—	178	1	no	—
413	C₄	—	178	1	no	—
414	C₄	—	712	2	no	—
415	C₄	#135×D₈₉	1	1	yes	characteristic
416	C₄	C₂×#15668035	1	1	yes	characteristic
417	C₄	C₂×#15668041	1	1	yes	characteristic
418	C₄	#19152394	1	1	yes	characteristic
419	C₂	—	178	2	no	—
420	C₂	#26029719	1	1	yes	characteristic
421	C₂	#26029835	1	1	yes	characteristic
422	C₂	#473×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₈₉
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1670	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂²×D₈₉	#121
hypercenter	#1670	D₈₉
hypocenter	C₈₉	#19265
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1670

Complements

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

N	H ≅ G/N	type
4. #26019681	C₂	semidirect
5. #26019680	C₂	semidirect
6. #26019647	C₂	semidirect
7. #26019646	C₂	semidirect
8. #26019619	C₂	semidirect
11. C₈₉×#1670	C₂	semidirect
220. #1670	D₈₉	semidirect
223. C₈₉	#19265	semidirect

Maximal subgroups

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

subgroup	index
#26030836	2
#26030384	2
C₈×Q₈×D₈₉	2
#26019681	2
#26019680	2
#26019647	2
#26019646	2
#26019619	2
#26019014	2
#26018951	2
C₈₉×#1670	2
#19265	89