Subgroups — #36537640

Order 22784.

646 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	#26031395	C₂	1	2	yes	maximal
2	#494×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#26029183	C₂	1	1	yes	maximal, characteristic
4	#26019747	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019746	C₂	1	1	yes	maximal, characteristic, complemented
6	#26019743	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019621	C₂	1	2	yes	maximal, complemented
8	#26019620	C₂	1	2	yes	maximal, complemented
9	#26018997	C₂	1	1	yes	maximal, characteristic
10	#26018922	C₂	1	2	yes	maximal
11	C₈₉×#1690	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₈₉	—	2	1	no	—
13	C₂×Q₁₆×D₈₉	—	2	1	no	—
14	#19152152	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×D₈₉	C₂²	1	2	yes	—
16	#19151995	—	2	2	no	—
17	#19151978	C₂²	1	1	yes	characteristic
18	#19151977	C₂²	1	2	yes	—
19	#19151976	—	2	1	no	—
20	#19151976	—	2	1	no	—
21	#19151947	—	2	2	no	—
22	#19151930	—	2	2	no	—
23	#19151927	—	2	2	no	—
24	#19151884	C₂²	1	2	yes	—
25	#19151883	C₂²	1	2	yes	—
26	#19151880	C₂²	1	2	yes	—
27	#19151728	C₂²	1	1	yes	characteristic
28	#19151726	—	2	1	no	—
29	Q₈×#10623494	C₂²	1	1	yes	characteristic
30	#19151722	—	2	1	no	—
31	#19151722	C₂²	1	1	yes	characteristic
32	#19151650	C₂²	1	2	yes	—
33	#19151623	—	2	1	no	—
34	#19151620	C₂²	1	1	yes	characteristic
35	#19151555	—	2	1	no	—
36	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
37	#19151350	C₂²	1	2	yes	—
38	#19151349	C₂²	1	1	yes	characteristic
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	#446×C₈₉	C₂²	1	1	yes	characteristic
45	#334×C₈₉	C₂²	1	1	yes	characteristic
46	#332×C₈₉	C₂²	1	2	yes	—
47	C₂×Q₈×D₈₉	—	2	2	no	—
48	C₂×Dic₃₅₆	—	2	1	no	—
49	C₂×Dic₃₅₆	—	2	1	no	—
50	C₂×#13083564	—	2	2	no	—
51	C₂×#13083564	—	2	2	no	—
52	Q₁₆×D₈₉	—	4	4	no	—
53	Q₁₆×D₈₉	—	4	4	no	—
54	C₂×#13083557	—	2	1	no	—
55	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
56	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
57	#15668012	C₂³	1	2	yes	—
58	#15668011	—	2	2	no	—
59	#15668011	—	2	2	no	—
60	#15668011	D₄	1	1	yes	characteristic
61	#15668010	—	2	2	no	—
62	#15668010	D₄	1	1	yes	characteristic
63	#15668002	—	2	1	no	—
64	#15667992	D₄	1	1	yes	characteristic
65	#15667990	—	2	1	no	—
66	#15667988	D₄	1	1	yes	characteristic
67	C₄×Dic₁₇₈	—	2	1	no	—
68	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
69	#15667969	—	2	1	no	—
70	C₄⋊C₄×D₈₉	—	2	2	no	—
71	C₄⋊C₄×D₈₉	—	2	2	no	—
72	C₄²×D₈₉	C₂³	1	1	yes	characteristic
73	#15667944	—	2	2	no	—
74	#15667943	—	2	2	no	—
75	#15667943	—	2	2	no	—
76	C₄⋊#10623494	—	2	1	no	—
77	C₄⋊#10623494	C₂³	1	1	yes	characteristic
78	C₄⋊#10623494	C₂³	1	2	yes	—
79	#15667941	—	2	1	no	—
80	#15667941	—	2	1	no	—
81	#15667941	—	2	1	no	—
82	#15667937	—	2	2	no	—
83	#15667936	—	2	2	no	—
84	#15667935	—	2	2	no	—
85	#15667924	—	2	2	no	—
86	#15667922	—	2	1	no	—
87	C₄×#10623494	—	2	1	no	—
88	C₄×#10623494	C₂³	1	1	yes	characteristic
89	C₈×Dic₈₉	—	2	1	no	—
90	C₈×Dic₈₉	—	2	2	no	—
91	C₈×Dic₈₉	D₄	1	1	yes	characteristic
92	C₈×Dic₈₉	D₄	1	1	yes	characteristic
93	C₂×Q₁₆×C₈₉	—	2	1	no	—
94	C₂×Q₁₆×C₈₉	—	2	1	no	—
95	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
96	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
97	#106×C₈₉	—	2	2	no	—
98	C₄⋊C₈×C₈₉	—	2	1	no	—
99	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
101	#103×C₈₉	—	2	2	no	—
102	#97×C₈₉	—	2	1	no	—
103	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
104	C₂×Dic₁₇₈	—	2	1	no	—
105	C₂×Dic₁₇₈	—	2	2	no	—
106	C₂×Dic₁₇₈	—	4	2	no	—
107	C₂×Dic₁₇₈	C₂×D₄	1	1	yes	characteristic
108	Q₈×D₈₉	—	4	4	no	—
109	Q₈×D₈₉	—	4	4	no	—
110	C₂×C₄×D₈₉	—	4	2	no	—
111	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×D₈₉	#41	1	1	yes	characteristic
114	Dic₃₅₆	—	4	2	no	—
115	Dic₃₅₆	—	4	2	no	—
116	#13083564	—	4	4	no	—
117	#13083564	—	4	4	no	—
118	#13083557	—	4	2	no	—
119	C₂×#10623494	—	2	1	no	—
120	C₂×#10623494	—	2	2	no	—
121	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
122	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
123	C₈×D₈₉	—	2	2	no	—
124	C₈×D₈₉	SD₁₆	1	4	yes	—
125	C₄⋊Dic₈₉	—	2	1	no	—
126	C₄⋊Dic₈₉	—	2	2	no	—
127	C₄⋊Dic₈₉	—	2	2	no	—
128	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
129	#13083553	—	2	1	no	—
130	#13083553	—	2	1	no	—
131	#13083553	—	2	1	no	—
132	#13083553	—	2	1	no	—
133	#13083553	—	2	1	no	—
134	#13083553	—	4	2	no	—
135	#13083553	—	4	2	no	—
136	#13083551	—	2	1	no	—
137	#13083551	—	2	1	no	—
138	C₄×Dic₈₉	—	2	1	no	—
139	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
140	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
141	C₄×Dic₈₉	#41	1	1	yes	characteristic
142	C₂×Q₈×C₈₉	—	2	2	no	—
143	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
144	Q₁₆×C₈₉	—	4	2	no	—
145	Q₁₆×C₈₉	—	4	2	no	—
146	C₂×C₈×C₈₉	—	2	1	no	—
147	C₂×C₈×C₈₉	—	2	2	no	—
148	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
149	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
150	C₄⋊C₄×C₈₉	—	2	1	no	—
151	C₄⋊C₄×C₈₉	—	2	2	no	—
152	C₄⋊C₄×C₈₉	—	2	2	no	—
153	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₈₉	—	2	1	no	—
155	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
156	C₂²×D₈₉	#121	1	1	yes	characteristic
157	Q₈×C₈₉	—	2	2	no	—
158	Q₈×C₈₉	—	4	2	no	—
159	Q₈×C₈₉	—	4	2	no	—
160	C₂×C₄×C₈₉	—	2	1	no	—
161	C₂×C₄×C₈₉	—	2	1	no	—
162	C₂×C₄×C₈₉	—	4	2	no	—
163	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
166	C₂×Dic₈₉	—	2	1	no	—
167	C₂×Dic₈₉	—	2	1	no	—
168	C₂×Dic₈₉	—	4	2	no	—
169	C₂×Dic₈₉	#121	1	1	yes	characteristic
170	C₂×Dic₈₉	#121	1	1	yes	characteristic
171	C₂×Dic₈₉	#121	1	1	yes	characteristic
172	C₄×D₈₉	—	2	2	no	—
173	C₄×D₈₉	—	2	2	no	—
174	C₄×D₈₉	—	8	4	no	—
175	C₄×D₈₉	C₂×SD₁₆	1	2	yes	—
176	C₄×D₈₉	AGL(1,ℤ/8)	1	2	yes	—
177	Dic₁₇₈	—	2	2	no	—
178	Dic₁₇₈	—	4	2	no	—
179	Dic₁₇₈	—	4	2	no	—
180	Dic₁₇₈	—	4	2	no	—
181	Dic₁₇₈	—	8	4	no	—
182	C₈×C₈₉	—	2	1	no	—
183	C₈×C₈₉	—	4	1	no	—
184	C₈×C₈₉	—	4	2	no	—
185	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
186	#10623494	—	2	1	no	—
187	#10623494	—	4	1	no	—
188	#10623494	—	4	2	no	—
189	#10623494	C₂×SD₁₆	1	2	yes	—
190	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
191	C₂×D₈₉	—	2	1	no	—
192	C₂×D₈₉	—	2	1	no	—
193	C₂×D₈₉	#460	1	2	yes	—
194	C₄×C₈₉	—	2	1	no	—
195	C₄×C₈₉	—	4	1	no	—
196	C₄×C₈₉	—	4	1	no	—
197	C₄×C₈₉	—	8	2	no	—
198	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
199	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₄×C₈₉	#578	1	2	yes	—
201	Dic₈₉	—	2	1	no	—
202	Dic₈₉	—	4	1	no	—
203	Dic₈₉	—	4	1	no	—
204	Dic₈₉	—	8	2	no	—
205	Dic₈₉	#460	1	2	yes	—
206	Dic₈₉	#473	1	2	yes	—
207	#19204	—	89	1	no	maximal, Sylow
208	C₂×C₈₉	C₂×#460	1	1	yes	characteristic
209	C₂×C₈₉	#3059	1	1	yes	characteristic
210	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
211	D₈₉	—	2	2	no	—
212	#3229	—	89	2	no	—
213	C₂×#494	—	89	1	no	—
214	#2960	—	89	1	no	—
215	#1690	—	89	1	no	—
216	#1690	—	89	1	no	—
217	#1690	—	89	1	no	—
218	#1690	D₈₉	1	1	yes	characteristic, complemented
219	#1670	—	89	2	no	—
220	#1670	—	89	2	no	—
221	#1564	—	89	1	no	—
222	#1546	—	89	2	no	—
223	C₈₉	#19204	1	1	yes	characteristic, complemented, Sylow
224	C₂²×Q₁₆	—	178	1	no	—
225	C₂²×Q₁₆	—	178	1	no	—
226	#533	—	89	1	no	—
227	C₂×C₄⋊Q₈	—	89	2	no	—
228	#498	—	178	2	no	—
229	#494	—	89	2	no	—
230	#494	—	89	1	no	—
231	#494	—	178	1	no	—
232	#494	—	178	1	no	—
233	#494	C₂×D₈₉	1	1	yes	characteristic
234	#487	—	178	2	no	—
235	#483	—	178	2	no	—
236	#483	—	178	2	no	—
237	#475	—	89	2	no	—
238	#475	—	89	2	no	—
239	#475	—	89	2	no	—
240	#475	C₂×D₈₉	1	2	yes	—
241	#446	—	89	1	no	—
242	#446	—	178	1	no	—
243	#446	C₂×D₈₉	1	1	yes	characteristic
244	C₈×Q₈	—	89	1	no	—
245	C₈×Q₈	—	89	1	no	—
246	C₈×Q₈	—	178	1	no	—
247	#431	—	89	2	no	—
248	#424	—	178	1	no	—
249	#423	—	89	1	no	—
250	#405	—	178	1	no	—
251	C₂×C₄×C₈	—	89	1	no	—
252	#335	—	89	2	no	—
253	#334	—	89	1	no	—
254	#334	C₂×D₈₉	1	1	yes	characteristic
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	—	178	2	no	—
260	C₂×Q₁₆	—	2	1	no	—
261	C₂×Q₁₆	—	2	1	no	—
262	C₂×Q₁₆	—	356	4	no	—
263	C₂×Q₁₆	—	356	4	no	—
264	C₂×Q₁₆	—	178	2	no	—
265	C₂×Q₁₆	—	178	2	no	—
266	C₂×Q₁₆	—	178	1	no	—
267	C₂×Q₁₆	—	178	1	no	—
268	C₂×M₁₆	—	178	1	no	—
269	C₂²×C₈	—	89	1	no	—
270	C₂²×C₈	—	89	1	no	—
271	C₄⋊Q₈	—	89	1	no	—
272	C₄⋊Q₈	—	89	1	no	—
273	C₄⋊Q₈	—	89	2	no	—
274	C₄⋊Q₈	—	178	2	no	—
275	C₄⋊Q₈	—	178	2	no	—
276	C₄⋊Q₈	—	178	2	no	—
277	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
278	#125	—	178	1	no	—
279	C₂²⋊Q₈	—	89	1	no	—
280	C₂²⋊Q₈	—	89	1	no	—
281	C₂²⋊Q₈	—	178	1	no	—
282	C₄×Q₈	—	89	1	no	—
283	C₄×Q₈	—	178	1	no	—
284	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
285	#117	—	178	1	no	—
286	C₂×C₄⋊C₄	—	178	2	no	—
287	C₂×C₄⋊C₄	—	178	2	no	—
288	C₂×C₄²	—	89	1	no	—
289	#106	—	2	2	no	—
290	#106	—	178	2	no	—
291	#106	—	178	2	no	—
292	#106	—	178	2	no	—
293	C₄⋊C₈	—	2	1	no	—
294	C₄⋊C₈	—	89	1	no	—
295	C₄⋊C₈	—	89	2	no	—
296	C₄⋊C₈	—	178	1	no	—
297	C₄⋊C₈	—	178	1	no	—
298	C₄⋊C₈	—	178	1	no	—
299	C₄⋊C₈	—	178	1	no	—
300	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
301	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
302	#103	—	2	2	no	—
303	#103	—	178	2	no	—
304	#103	—	178	2	no	—
305	#103	—	178	2	no	—
306	C₂²⋊C₈	—	178	2	no	—
307	#97	—	2	1	no	—
308	#97	—	178	1	no	—
309	C₄×C₈	—	89	1	no	—
310	C₄×C₈	—	89	1	no	—
311	C₄×C₈	—	89	1	no	—
312	C₄×C₈	—	178	1	no	—
313	C₄×C₈	—	178	2	no	—
314	C₄×C₈	—	178	1	no	—
315	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
316	C₂×Q₈	—	2	2	no	—
317	C₂×Q₈	—	89	1	no	—
318	C₂×Q₈	—	356	2	no	—
319	C₂×Q₈	—	356	4	no	—
320	C₂×Q₈	—	356	4	no	—
321	C₂×Q₈	—	178	1	no	—
322	C₂×Q₈	—	178	2	no	—
323	C₂×Q₈	D₄×D₈₉	1	1	yes	characteristic
324	C₂²×C₄	—	89	1	no	—
325	C₂²×C₄	—	89	1	no	—
326	C₂²×C₄	—	89	1	no	—
327	C₂²×C₄	—	356	2	no	—
328	Q₁₆	—	4	2	no	—
329	Q₁₆	—	4	2	no	—
330	Q₁₆	—	356	2	no	—
331	Q₁₆	—	356	4	no	—
332	Q₁₆	—	356	4	no	—
333	Q₁₆	—	356	2	no	—
334	M₁₆	—	356	2	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	89	1	no	—
338	C₂×C₈	—	89	4	no	—
339	C₂×C₈	—	89	1	no	—
340	C₂×C₈	—	178	2	no	—
341	C₂×C₈	—	178	1	no	—
342	C₂×C₈	—	178	2	no	—
343	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
344	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	89	1	no	—
349	C₄⋊C₄	—	356	2	no	—
350	C₄⋊C₄	—	356	2	no	—
351	C₄⋊C₄	—	178	1	no	—
352	C₄⋊C₄	—	178	1	no	—
353	C₄⋊C₄	—	178	1	no	—
354	C₄⋊C₄	—	178	1	no	—
355	C₄⋊C₄	—	178	1	no	—
356	C₄⋊C₄	—	178	2	no	—
357	C₄⋊C₄	—	178	2	no	—
358	C₄⋊C₄	—	178	1	no	—
359	C₄⋊C₄	D₄×D₈₉	1	1	yes	characteristic
360	C₂²⋊C₄	—	178	1	no	—
361	C₂²⋊C₄	—	178	1	no	—
362	C₄²	—	2	1	no	—
363	C₄²	—	89	1	no	—
364	C₄²	—	89	1	no	—
365	C₄²	—	89	1	no	—
366	C₄²	—	178	1	no	—
367	C₄²	C₂³×D₈₉	1	1	yes	characteristic
368	C₂³	—	89	1	no	—
369	Q₈	—	2	2	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	4	2	no	—
372	Q₈	—	356	2	no	—
373	Q₈	—	356	2	no	—
374	Q₈	—	356	2	no	—
375	Q₈	—	178	2	no	—
376	Q₈	—	712	4	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	89	1	no	—
381	C₂×C₄	—	89	1	no	—
382	C₂×C₄	—	89	2	no	—
383	C₂×C₄	—	89	2	no	—
384	C₂×C₄	—	89	1	no	—
385	C₂×C₄	—	356	2	no	—
386	C₂×C₄	—	178	2	no	—
387	C₂×C₄	—	178	1	no	—
388	C₂×C₄	—	178	1	no	—
389	C₂×C₄	—	178	2	no	—
390	C₂×C₄	—	712	4	no	—
391	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
392	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
393	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
394	C₈	—	2	1	no	—
395	C₈	—	4	2	no	—
396	C₈	—	4	1	no	—
397	C₈	—	89	2	no	—
398	C₈	—	356	2	no	—
399	C₈	—	356	1	no	—
400	C₈	—	178	1	no	—
401	C₈	SD₁₆×D₈₉	1	2	yes	—
402	C₂²	—	89	2	no	—
403	C₂²	—	178	1	no	—
404	C₂²	—	178	1	no	—
405	C₂²	#121×D₈₉	1	1	yes	characteristic
406	C₄	—	2	1	no	—
407	C₄	—	4	1	no	—
408	C₄	—	4	1	no	—
409	C₄	—	8	2	no	—
410	C₄	—	89	2	no	—
411	C₄	—	89	2	no	—
412	C₄	—	356	1	no	—
413	C₄	—	356	1	no	—
414	C₄	—	178	1	no	—
415	C₄	—	712	2	no	—
416	C₄	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
417	C₄	AGL(1,ℤ/8)×D₈₉	1	1	yes	characteristic
418	C₄	#19152414	1	2	yes	—
419	C₂	—	178	2	no	—
420	C₂	#460×D₈₉	1	1	yes	characteristic
421	C₂	#26029905	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×#1690	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂²×D₈₉	#121
hypercenter	#1690	D₈₉
hypocenter	C₈₉	#19204
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1690

Complements

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

N	H ≅ G/N	type
4. #26019747	C₂	semidirect
5. #26019746	C₂	semidirect
6. #26019743	C₂	semidirect
7. #26019621	C₂	semidirect
8. #26019620	C₂	semidirect
11. C₈₉×#1690	C₂	semidirect
218. #1690	D₈₉	semidirect
223. C₈₉	#19204	semidirect

Maximal subgroups

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

subgroup	index
#26031395	2
#494×D₈₉	2
#26029183	2
#26019747	2
#26019746	2
#26019743	2
#26019621	2
#26019620	2
#26018997	2
#26018922	2
C₈₉×#1690	2
#19204	89