Subgroups — #36523897

Order 22784.

678 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029331	C₂	1	2	yes	maximal
2	C₄×Q₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#26019069	C₂	1	2	yes	maximal, complemented
4	#26019068	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019058	C₂	1	1	yes	maximal, characteristic, complemented
8	#26018999	C₂	1	1	yes	maximal, characteristic
9	#26018645	C₂	1	2	yes	maximal
10	#26018534	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₈₉	C₄	1	1	yes	characteristic
13	C₄×Q₈×D₈₉	C₂²	1	2	yes	—
14	Q₈×#10623494	C₂²	1	2	yes	—
15	C₈×Dic₁₇₈	—	2	2	no	—
16	C₈×Dic₁₇₈	C₂²	1	2	yes	—
17	#19151722	—	2	2	no	—
18	#19151722	C₂²	1	2	yes	—
19	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
20	Q₁₆×Dic₈₉	C₄	1	1	yes	characteristic
21	#19151692	C₄	1	1	yes	characteristic
22	C₄×#13083564	C₂²	1	2	yes	—
23	#19151690	C₄	1	2	yes	—
24	#19151650	—	2	2	no	—
25	#19151650	C₂²	1	2	yes	—
26	#19151649	—	2	2	no	—
27	#19151649	C₂²	1	1	yes	characteristic
28	#19151649	C₂²	1	1	yes	characteristic
29	#107×D₈₉	C₄	1	1	yes	characteristic
30	#103×D₈₉	C₄	1	2	yes	—
31	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
32	#19151351	C₂²	1	1	yes	characteristic
33	#19151348	C₂²	1	2	yes	—
34	#19151330	C₂²	1	2	yes	—
35	Q₈⋊#10623494	C₂²	1	2	yes	—
36	#19151328	C₂²	1	2	yes	—
37	#19151320	C₂²	1	1	yes	characteristic
38	#19151320	C₂²	1	1	yes	characteristic
39	C₈×#10623494	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₈₉	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
42	#335×C₈₉	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₈₉	C₂²	1	2	yes	—
44	C₈²×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₈₉	C₂×C₄	1	2	yes	—
46	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
47	C₂×#13083564	C₂×C₄	1	2	yes	—
48	Q₁₆×D₈₉	C₈	1	8	yes	complemented
49	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
51	Q₈×Dic₈₉	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₇₈	—	2	2	no	—
53	C₄×Dic₁₇₈	C₂³	1	2	yes	—
54	#15667977	—	2	2	no	—
55	#15667977	C₂×C₄	1	2	yes	—
56	C₄⋊C₄×D₈₉	—	2	2	no	—
57	C₄⋊C₄×D₈₉	C₂×C₄	1	2	yes	—
58	C₄²×D₈₉	—	2	2	no	—
59	C₄²×D₈₉	C₂³	1	1	yes	characteristic
60	#15667946	C₂×C₄	1	1	yes	characteristic
61	#15667945	C₂×C₄	1	2	yes	—
62	C₄⋊#10623494	—	2	2	no	—
63	C₄⋊#10623494	C₂³	1	2	yes	—
64	#15667941	—	2	2	no	—
65	#15667941	—	2	1	no	—
66	#15667941	—	2	2	no	—
67	#15667941	D₄	1	1	yes	characteristic
68	#15667941	D₄	1	1	yes	characteristic
69	#15667937	C₂×C₄	1	2	yes	—
70	#15667936	C₂×C₄	1	2	yes	—
71	#15667935	C₂×C₄	1	2	yes	—
72	#15667924	—	2	2	no	—
73	#15667924	—	2	1	no	—
74	#15667924	D₄	1	1	yes	characteristic
75	#15667924	D₄	1	1	yes	characteristic
76	C₄×#10623494	—	2	2	no	—
77	C₄×#10623494	C₂³	1	1	yes	characteristic
78	C₄×#10623494	C₂³	1	1	yes	characteristic
79	C₄×#10623494	C₂³	1	1	yes	characteristic
80	C₈×Dic₈₉	—	2	2	no	—
81	C₈×Dic₈₉	—	2	2	no	—
82	C₈×Dic₈₉	—	2	2	no	—
83	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
85	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
87	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
88	C₄⋊C₈×C₈₉	—	2	2	no	—
89	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
90	#103×C₈₉	C₂×C₄	1	2	yes	—
91	C₄×C₈×C₈₉	—	2	2	no	—
92	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
95	C₂×Dic₁₇₈	—	2	2	no	—
96	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
97	Q₈×D₈₉	—	2	4	no	—
98	Q₈×D₈₉	C₂×C₈	1	8	yes	—
99	C₂×C₄×D₈₉	—	2	2	no	—
100	C₂×C₄×D₈₉	—	2	2	no	—
101	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×D₈₉	#41	1	1	yes	characteristic
104	Dic₃₅₆	C₂×C₈	1	4	yes	—
105	#13083564	C₂×C₈	1	8	yes	—
106	C₂×#10623494	—	2	2	no	—
107	C₂×#10623494	—	2	1	no	—
108	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
109	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
110	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
111	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
112	C₈×D₈₉	—	2	2	no	—
113	C₈×D₈₉	C₂×C₈	1	4	yes	—
114	C₄⋊Dic₈₉	—	2	2	no	—
115	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
116	#13083553	—	2	2	no	—
117	#13083553	—	2	2	no	—
118	#13083553	—	2	2	no	—
119	C₄×Dic₈₉	—	2	2	no	—
120	C₄×Dic₈₉	—	2	2	no	—
121	C₄×Dic₈₉	—	2	2	no	—
122	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₈₉	#41	1	1	yes	characteristic
125	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
126	Q₁₆×C₈₉	C₂×C₈	1	4	yes	—
127	C₂×C₈×C₈₉	—	2	2	no	—
128	C₂×C₈×C₈₉	—	2	1	no	—
129	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₄⋊C₄×C₈₉	—	2	2	no	—
134	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
135	C₄²×C₈₉	—	2	2	no	—
136	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
137	C₂²×D₈₉	C₄×D₄	1	1	yes	characteristic
138	Q₈×C₈₉	—	2	2	no	—
139	Q₈×C₈₉	C₂²×C₈	1	4	yes	—
140	C₂×C₄×C₈₉	—	2	2	no	—
141	C₂×C₄×C₈₉	—	2	2	no	—
142	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
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	8	no	—
151	C₄×D₈₉	—	2	1	no	—
152	C₄×D₈₉	—	2	1	no	—
153	C₄×D₈₉	—	4	1	no	—
154	C₄×D₈₉	—	8	2	no	—
155	C₄×D₈₉	C₂²×C₈	1	2	yes	—
156	C₄×D₈₉	#131	1	2	yes	—
157	Dic₁₇₈	—	2	2	no	—
158	Dic₁₇₈	—	2	8	no	—
159	Dic₁₇₈	C₂²×C₈	1	4	yes	—
160	C₈×C₈₉	—	2	1	no	—
161	C₈×C₈₉	—	2	1	no	—
162	C₈×C₈₉	—	2	1	no	—
163	C₈×C₈₉	—	4	1	no	—
164	C₈×C₈₉	—	4	2	no	—
165	C₈×C₈₉	C₂²×C₈	1	2	yes	—
166	#10623494	—	2	1	no	—
167	#10623494	—	2	1	no	—
168	#10623494	—	2	1	no	—
169	#10623494	—	4	1	no	—
170	#10623494	—	4	2	no	—
171	#10623494	C₂²×C₈	1	2	yes	—
172	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂×D₈₉	—	2	1	no	—
174	C₂×D₈₉	—	2	1	no	—
175	C₂×D₈₉	C₈×D₄	1	2	yes	—
176	C₄×C₈₉	—	2	4	no	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	—	4	2	no	—
179	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
180	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
181	C₄×C₈₉	#578	1	1	yes	characteristic
182	C₄×C₈₉	#578	1	1	yes	characteristic
183	Dic₈₉	—	2	4	no	—
184	Dic₈₉	—	2	1	no	—
185	Dic₈₉	—	2	1	no	—
186	Dic₈₉	—	4	2	no	—
187	Dic₈₉	C₈×D₄	1	2	yes	—
188	#17856	—	89	1	no	maximal, Sylow
189	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
190	C₂×C₈₉	#2950	1	1	yes	characteristic
191	C₂×C₈₉	#2957	1	1	yes	characteristic
192	D₈₉	—	2	2	no	—
193	#2982	—	89	2	no	—
194	C₂×C₄×Q₁₆	—	89	1	no	—
195	#1579	—	89	2	no	—
196	#1579	—	89	2	no	—
197	C₈×Q₁₆	—	89	1	no	—
198	C₈×Q₁₆	—	89	1	no	—
199	C₈×Q₁₆	—	89	1	no	—
200	C₈×Q₁₆	D₈₉	1	1	yes	characteristic, complemented
201	#1564	—	89	1	no	—
202	#1479	—	89	2	no	—
203	#1450	—	89	1	no	—
204	C₈₉	#17856	1	1	yes	characteristic, complemented, Sylow
205	C₂²×Q₁₆	—	89	1	no	—
206	C₂×C₄×Q₈	—	89	2	no	—
207	C₈×Q₈	—	89	2	no	—
208	C₈×Q₈	—	89	2	no	—
209	C₈×Q₈	—	89	2	no	—
210	C₈×Q₈	—	178	2	no	—
211	C₈×Q₈	—	178	2	no	—
212	C₈×Q₈	C₂×D₈₉	1	2	yes	—
213	C₄×Q₁₆	—	89	1	no	—
214	C₄×Q₁₆	—	89	2	no	—
215	C₄×Q₁₆	—	89	1	no	—
216	C₄×Q₁₆	—	89	2	no	—
217	C₄×Q₁₆	—	89	1	no	—
218	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
219	#431	—	89	1	no	—
220	#431	—	89	1	no	—
221	#431	—	89	2	no	—
222	#431	—	178	2	no	—
223	#431	—	178	2	no	—
224	C₂×#107	—	89	1	no	—
225	C₂×#103	—	89	2	no	—
226	C₂×C₄×C₈	—	89	1	no	—
227	#335	—	89	1	no	—
228	#335	C₂×D₈₉	1	1	yes	characteristic
229	#334	—	89	2	no	—
230	Q₈⋊C₈	—	89	2	no	—
231	Q₈⋊C₈	—	89	2	no	—
232	Q₈⋊C₈	—	89	2	no	—
233	Q₈⋊C₈	C₂×D₈₉	1	2	yes	—
234	#322	—	89	1	no	—
235	#322	—	89	1	no	—
236	C₈²	—	89	1	no	—
237	C₈²	C₂×D₈₉	1	1	yes	characteristic
238	C₂²×Q₈	—	89	2	no	—
239	C₂×Q₁₆	—	89	2	no	—
240	C₂×Q₁₆	—	89	8	no	—
241	C₂×Q₁₆	—	89	1	no	—
242	C₂×Q₁₆	C₄×D₈₉	1	1	yes	characteristic
243	C₂²×C₈	—	89	1	no	—
244	C₂²×C₈	—	89	1	no	—
245	C₄×Q₈	—	89	2	no	—
246	C₄×Q₈	—	89	2	no	—
247	C₄×Q₈	—	89	2	no	—
248	C₄×Q₈	—	178	2	no	—
249	C₄×Q₈	—	178	2	no	—
250	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
251	C₂×C₄⋊C₄	—	89	2	no	—
252	C₂×C₄⋊C₄	—	178	2	no	—
253	C₂×C₄²	—	89	1	no	—
254	C₂×C₄²	—	178	2	no	—
255	#107	—	89	1	no	—
256	#107	—	89	2	no	—
257	#107	C₄×D₈₉	1	1	yes	characteristic
258	C₄⋊C₈	—	2	2	no	—
259	C₄⋊C₈	—	89	1	no	—
260	C₄⋊C₈	—	89	1	no	—
261	C₄⋊C₈	—	89	2	no	—
262	C₄⋊C₈	—	178	2	no	—
263	C₄⋊C₈	—	178	1	no	—
264	C₄⋊C₈	—	178	2	no	—
265	C₄⋊C₈	—	178	2	no	—
266	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
267	#103	—	89	2	no	—
268	#103	—	89	2	no	—
269	#103	—	89	2	no	—
270	#103	C₄×D₈₉	1	2	yes	—
271	C₂²⋊C₈	—	89	1	no	—
272	C₂²⋊C₈	—	89	1	no	—
273	C₂²⋊C₈	—	178	2	no	—
274	C₂²⋊C₈	—	178	1	no	—
275	C₄×C₈	—	2	2	no	—
276	C₄×C₈	—	89	1	no	—
277	C₄×C₈	—	89	1	no	—
278	C₄×C₈	—	89	1	no	—
279	C₄×C₈	—	89	1	no	—
280	C₄×C₈	—	89	1	no	—
281	C₄×C₈	—	178	2	no	—
282	C₄×C₈	—	178	2	no	—
283	C₄×C₈	—	178	2	no	—
284	C₄×C₈	—	178	2	no	—
285	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
288	C₂×Q₈	—	89	8	no	—
289	C₂×Q₈	—	89	2	no	—
290	C₂×Q₈	—	178	2	no	—
291	C₂×Q₈	—	178	4	no	—
292	C₂×Q₈	C₂×C₄×D₈₉	1	2	yes	—
293	C₂²×C₄	—	89	1	no	—
294	C₂²×C₄	—	89	1	no	—
295	C₂²×C₄	—	89	1	no	—
296	C₂²×C₄	—	178	2	no	—
297	C₂²×C₄	—	178	2	no	—
298	Q₁₆	—	89	8	no	—
299	Q₁₆	—	89	4	no	—
300	Q₁₆	C₈×D₈₉	1	4	yes	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	89	1	no	—
304	C₂×C₈	—	89	4	no	—
305	C₂×C₈	—	89	1	no	—
306	C₂×C₈	—	89	1	no	—
307	C₂×C₈	—	89	1	no	—
308	C₂×C₈	—	178	2	no	—
309	C₂×C₈	—	178	2	no	—
310	C₂×C₈	—	178	1	no	—
311	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
312	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
313	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
314	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	89	2	no	—
317	C₄⋊C₄	—	178	2	no	—
318	C₄⋊C₄	—	178	2	no	—
319	C₄⋊C₄	—	178	2	no	—
320	C₄⋊C₄	—	178	2	no	—
321	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
322	C₄²	—	2	2	no	—
323	C₄²	—	89	1	no	—
324	C₄²	—	89	1	no	—
325	C₄²	—	89	1	no	—
326	C₄²	—	178	2	no	—
327	C₄²	—	178	2	no	—
328	C₄²	—	178	2	no	—
329	C₄²	C₂³×D₈₉	1	1	yes	characteristic
330	C₂³	—	89	1	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	89	4	no	—
333	Q₈	—	178	8	no	—
334	Q₈	—	178	2	no	—
335	Q₈	C₂×C₈×D₈₉	1	4	yes	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	89	1	no	—
339	C₂×C₄	—	89	2	no	—
340	C₂×C₄	—	89	2	no	—
341	C₂×C₄	—	89	1	no	—
342	C₂×C₄	—	89	1	no	—
343	C₂×C₄	—	356	1	no	—
344	C₂×C₄	—	178	8	no	—
345	C₂×C₄	—	178	2	no	—
346	C₂×C₄	—	178	2	no	—
347	C₂×C₄	—	178	1	no	—
348	C₂×C₄	—	178	1	no	—
349	C₂×C₄	—	712	2	no	—
350	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
351	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
352	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	4	1	no	—
357	C₈	—	4	2	no	—
358	C₈	—	89	2	no	—
359	C₈	—	356	1	no	—
360	C₈	—	356	2	no	—
361	C₈	—	178	1	no	—
362	C₈	—	178	1	no	—
363	C₈	—	178	1	no	—
364	C₈	C₂×C₈×D₈₉	1	2	yes	—
365	C₂²	—	89	2	no	—
366	C₂²	—	178	1	no	—
367	C₂²	—	178	1	no	—
368	C₂²	C₄×D₄×D₈₉	1	1	yes	characteristic
369	C₄	—	2	1	no	—
370	C₄	—	2	4	no	—
371	C₄	—	4	2	no	—
372	C₄	—	89	2	no	—
373	C₄	—	356	2	no	—
374	C₄	—	178	1	no	—
375	C₄	—	178	1	no	—
376	C₄	—	178	4	no	—
377	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
378	C₄	#131×D₈₉	1	1	yes	characteristic
379	C₄	#19152411	1	1	yes	characteristic
380	C₄	#19152412	1	1	yes	characteristic
381	C₂	—	178	2	no	—
382	C₂	C₈×D₄×D₈₉	1	1	yes	characteristic
383	C₂	#26029115	1	1	yes	characteristic
384	C₂	#26029164	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈×Q₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂²×D₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₈×Q₁₆	D₈₉
hypocenter	C₈₉	#17856
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #26019069	C₂	semidirect
4. #26019068	C₂	semidirect
5. C₈×Dic₃₅₆	C₂	semidirect
6. Q₁₆×#10623494	C₂	semidirect
7. #26019058	C₂	semidirect
11. C₈×Q₁₆×C₈₉	C₂	semidirect
48. Q₁₆×D₈₉	C₈	semidirect
200. C₈×Q₁₆	D₈₉	semidirect
204. C₈₉	#17856	semidirect

Maximal subgroups

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

subgroup	index
#26029331	2
C₄×Q₁₆×D₈₉	2
#26019069	2
#26019068	2
C₈×Dic₃₅₆	2
Q₁₆×#10623494	2
#26019058	2
#26018999	2
#26018645	2
#26018534	2
C₈×Q₁₆×C₈₉	2
#17856	89