Subgroups — #36526456

Order 22784.

564 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030834	C₂	1	1	yes	maximal, characteristic
2	#26029353	C₂	1	1	yes	maximal, characteristic
3	#26019159	C₂	1	2	yes	maximal, complemented
4	#26019102	C₂	1	2	yes	maximal, complemented
5	#26019100	C₂	1	2	yes	maximal, complemented
6	#26019069	C₂	1	2	yes	maximal, complemented
7	#26018998	C₂	1	1	yes	maximal, characteristic
8	#26018673	C₂	1	1	yes	maximal, characteristic
9	#26018672	C₂	1	1	yes	maximal, characteristic
10	#26018542	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2986	C₂	1	1	yes	maximal, characteristic
12	#19152155	C₂²	1	1	yes	characteristic
13	#19152001	C₄	1	1	yes	characteristic
14	#19151977	C₄	1	1	yes	characteristic
15	#19151950	C₄	1	1	yes	characteristic
16	#19151948	C₄	1	1	yes	characteristic
17	#19151837	C₄	1	2	yes	—
18	#19151837	C₄	1	2	yes	—
19	#19151728	C₂²	1	2	yes	—
20	#19151725	—	2	1	no	—
21	C₈×Dic₁₇₈	C₂²	1	2	yes	—
22	#19151722	—	2	1	no	—
23	C₄×#13083564	C₂²	1	2	yes	—
24	C₄×#13083564	C₂²	1	2	yes	—
25	#19151656	—	2	2	no	—
26	#19151651	C₂²	1	1	yes	characteristic
27	C₄×#13083558	C₂²	1	1	yes	characteristic
28	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
29	#19151350	C₂²	1	1	yes	characteristic
30	#19151350	C₂²	1	1	yes	characteristic
31	#19151348	C₂²	1	1	yes	characteristic
32	#19151348	C₂²	1	1	yes	characteristic
33	Q₈⋊#10623494	C₂²	1	2	yes	—
34	Q₈⋊#10623494	C₂²	1	2	yes	—
35	#19151328	C₂²	1	2	yes	—
36	#19151328	C₂²	1	2	yes	—
37	#19151320	C₂²	1	2	yes	—
38	#19151319	C₂²	1	2	yes	—
39	#533×C₈₉	C₂²	1	1	yes	characteristic
40	#446×C₈₉	—	2	1	no	—
41	#446×C₈₉	C₂²	1	2	yes	—
42	C₈×Q₈×C₈₉	—	2	1	no	—
43	C₈×Q₈×C₈₉	C₂²	1	2	yes	—
44	#433×C₈₉	—	2	2	no	—
45	#431×C₈₉	C₂²	1	1	yes	characteristic
46	#431×C₈₉	C₂²	1	1	yes	characteristic
47	C₂×#13083564	C₂×C₄	1	2	yes	—
48	C₂×#13083564	C₂×C₄	1	2	yes	—
49	C₂×#13083558	—	2	1	no	—
50	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
51	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
52	#15668012	C₂×C₄	1	1	yes	characteristic
53	#15668012	C₂×C₄	1	1	yes	characteristic
54	#15668004	—	2	1	no	—
55	#15667991	—	2	1	no	—
56	#15667991	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₇₈	—	2	1	no	—
58	C₄×Dic₁₇₈	C₂³	1	2	yes	—
59	#15667971	—	2	1	no	—
60	#15667945	C₂×C₄	1	2	yes	—
61	#15667945	C₂×C₄	1	2	yes	—
62	C₄⋊#10623494	C₂³	1	1	yes	characteristic
63	C₄⋊#10623494	C₂³	1	1	yes	characteristic
64	#15667941	—	2	2	no	—
65	#15667941	—	2	2	no	—
66	#15667936	C₂×C₄	1	2	yes	—
67	#15667936	C₂×C₄	1	2	yes	—
68	#15667935	C₂×C₄	1	2	yes	—
69	#15667935	C₂×C₄	1	2	yes	—
70	C₂²⋊#10623494	—	2	1	no	—
71	#15667923	—	2	1	no	—
72	#15667922	—	2	2	no	—
73	C₄×#10623494	—	2	1	no	—
74	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
75	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
76	C₄×#10623494	C₂³	1	2	yes	—
77	C₄×#10623494	C₂³	1	1	yes	characteristic
78	C₄×#10623494	C₂³	1	1	yes	characteristic
79	C₈×Dic₈₉	—	2	2	no	—
80	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
82	#125×C₈₉	—	2	1	no	—
83	C₂²⋊Q₈×C₈₉	—	2	1	no	—
84	C₂²⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
85	C₄×Q₈×C₈₉	—	2	1	no	—
86	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
87	#117×C₈₉	—	2	1	no	—
88	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
89	C₄⋊C₈×C₈₉	—	2	2	no	—
90	C₄⋊C₈×C₈₉	—	2	2	no	—
91	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
92	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
93	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
95	C₂²⋊C₈×C₈₉	—	2	1	no	—
96	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
97	#97×C₈₉	—	2	2	no	—
98	C₄×C₈×C₈₉	—	2	2	no	—
99	C₄×C₈×C₈₉	—	2	1	no	—
100	C₄×C₈×C₈₉	C₂³	1	2	yes	—
101	C₂×Dic₁₇₈	—	2	1	no	—
102	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
103	#13083564	—	2	4	no	—
104	#13083564	—	4	2	no	—
105	#13083558	—	4	2	no	—
106	C₂×#10623494	—	2	2	no	—
107	C₂×#10623494	—	2	1	no	—
108	C₂×#10623494	—	2	1	no	—
109	C₂×#10623494	—	2	2	no	—
110	C₂×#10623494	—	4	1	no	—
111	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
112	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
113	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
114	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
115	C₄⋊Dic₈₉	—	2	1	no	—
116	C₄⋊Dic₈₉	—	2	1	no	—
117	C₄⋊Dic₈₉	—	2	1	no	—
118	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
119	#13083553	—	2	2	no	—
120	#13083553	—	2	2	no	—
121	C₂²⋊Dic₈₉	—	2	2	no	—
122	C₄×Dic₈₉	—	2	2	no	—
123	C₂×Q₈×C₈₉	—	2	1	no	—
124	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
128	C₂×C₈×C₈₉	—	2	1	no	—
129	C₂×C₈×C₈₉	—	2	1	no	—
130	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
131	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
132	C₄⋊C₄×C₈₉	—	2	1	no	—
133	C₄⋊C₄×C₈₉	—	2	1	no	—
134	C₄⋊C₄×C₈₉	—	2	1	no	—
135	C₄⋊C₄×C₈₉	—	2	2	no	—
136	C₄⋊C₄×C₈₉	—	2	2	no	—
137	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
138	C₂²⋊C₄×C₈₉	—	2	2	no	—
139	C₄²×C₈₉	—	2	2	no	—
140	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
141	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
142	C₄²×C₈₉	#41	1	1	yes	characteristic
143	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
144	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
145	Q₈×C₈₉	—	2	2	no	—
146	Q₈×C₈₉	—	2	2	no	—
147	Q₈×C₈₉	—	4	2	no	—
148	C₂×C₄×C₈₉	—	2	1	no	—
149	C₂×C₄×C₈₉	—	2	1	no	—
150	C₂×C₄×C₈₉	—	2	1	no	—
151	C₂×C₄×C₈₉	—	2	1	no	—
152	C₂×C₄×C₈₉	—	2	2	no	—
153	C₂×C₄×C₈₉	—	2	2	no	—
154	C₂×C₄×C₈₉	—	2	2	no	—
155	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
161	C₂×Dic₈₉	—	2	2	no	—
162	C₂×Dic₈₉	—	2	2	no	—
163	Dic₁₇₈	—	2	2	no	—
164	Dic₁₇₈	—	2	2	no	—
165	Dic₁₇₈	—	4	2	no	—
166	C₈×C₈₉	—	2	2	no	—
167	C₈×C₈₉	—	2	2	no	—
168	C₈×C₈₉	—	4	1	no	—
169	C₈×C₈₉	—	4	1	no	—
170	#10623494	—	2	1	no	—
171	#10623494	—	2	1	no	—
172	#10623494	—	2	1	no	—
173	#10623494	—	4	2	no	—
174	#10623494	—	4	1	no	—
175	#10623494	—	4	1	no	—
176	#10623494	#131	1	2	yes	—
177	C₂²×C₈₉	—	2	1	no	—
178	C₂²×C₈₉	—	2	1	no	—
179	C₂²×C₈₉	—	2	1	no	—
180	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₈₉	—	2	1	no	—
182	C₄×C₈₉	—	2	1	no	—
183	C₄×C₈₉	—	2	1	no	—
184	C₄×C₈₉	—	4	2	no	—
185	C₄×C₈₉	—	4	2	no	—
186	C₄×C₈₉	#444	1	2	yes	—
187	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
188	C₄×C₈₉	#568	1	1	yes	characteristic
189	C₄×C₈₉	#578	1	1	yes	characteristic
190	C₄×C₈₉	#578	1	1	yes	characteristic
191	Dic₈₉	—	4	2	no	—
192	Dic₈₉	—	4	2	no	—
193	#18090	—	89	1	no	maximal, Sylow
194	C₂×C₈₉	—	4	1	no	—
195	C₂×C₈₉	#2935	1	1	yes	characteristic
196	C₂×C₈₉	#2950	1	1	yes	characteristic
197	C₂×C₈₉	C₂×#444	1	1	yes	characteristic
198	#3170	—	89	1	no	—
199	#2986	—	89	1	no	—
200	#2986	D₈₉	1	1	yes	characteristic
201	#1593	—	89	2	no	—
202	#1584	—	89	2	no	—
203	#1584	—	89	2	no	—
204	#1579	—	89	2	no	—
205	#1564	—	89	1	no	—
206	#1485	—	89	1	no	—
207	#1485	—	89	1	no	—
208	#1451	—	89	1	no	—
209	C₈₉	#18090	1	1	yes	characteristic, complemented, Sylow
210	#533	—	89	1	no	—
211	#533	C₂×D₈₉	1	1	yes	characteristic
212	#500	—	89	1	no	—
213	#494	—	89	1	no	—
214	#487	—	89	1	no	—
215	#487	—	89	1	no	—
216	#467	—	89	2	no	—
217	#467	—	89	2	no	—
218	#446	—	2	1	no	—
219	#446	—	89	2	no	—
220	#446	—	178	1	no	—
221	#446	C₂×D₈₉	1	2	yes	—
222	C₈×Q₈	—	2	1	no	—
223	C₈×Q₈	—	89	2	no	—
224	C₈×Q₈	—	178	1	no	—
225	C₈×Q₈	C₂×D₈₉	1	2	yes	—
226	C₄×Q₁₆	—	89	2	no	—
227	C₄×Q₁₆	—	89	2	no	—
228	#433	—	2	2	no	—
229	#433	—	178	2	no	—
230	#431	—	89	1	no	—
231	#431	C₂×D₈₉	1	1	yes	characteristic
232	#431	C₂×D₈₉	1	1	yes	characteristic
233	C₄×M₁₆	—	89	1	no	—
234	C₂×C₄×C₈	—	89	1	no	—
235	#335	—	89	1	no	—
236	#335	—	89	1	no	—
237	#334	—	89	1	no	—
238	#334	—	89	1	no	—
239	Q₈⋊C₈	—	89	2	no	—
240	Q₈⋊C₈	—	89	2	no	—
241	Q₈⋊C₈	—	89	2	no	—
242	Q₈⋊C₈	—	89	2	no	—
243	#322	—	89	2	no	—
244	#322	—	89	2	no	—
245	C₂×Q₁₆	—	89	2	no	—
246	C₂×Q₁₆	—	89	2	no	—
247	C₂×M₁₆	—	178	1	no	—
248	C₂²×C₈	—	89	1	no	—
249	C₂²×C₈	—	89	1	no	—
250	C₄⋊Q₈	—	89	1	no	—
251	C₄⋊Q₈	—	89	1	no	—
252	C₄⋊Q₈	C₄×D₈₉	1	1	yes	characteristic
253	C₄⋊Q₈	C₄×D₈₉	1	1	yes	characteristic
254	#125	—	2	1	no	—
255	#125	—	178	1	no	—
256	C₂²⋊Q₈	—	2	1	no	—
257	C₂²⋊Q₈	—	89	2	no	—
258	C₂²⋊Q₈	—	178	1	no	—
259	C₂²⋊Q₈	C₄×D₈₉	1	2	yes	—
260	C₄×Q₈	—	2	1	no	—
261	C₄×Q₈	—	89	2	no	—
262	C₄×Q₈	—	178	1	no	—
263	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
264	#117	—	2	1	no	—
265	#117	—	178	1	no	—
266	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
267	#107	—	89	2	no	—
268	#107	—	89	2	no	—
269	C₄⋊C₈	—	2	2	no	—
270	C₄⋊C₈	—	2	2	no	—
271	C₄⋊C₈	—	89	1	no	—
272	C₄⋊C₈	—	89	1	no	—
273	C₄⋊C₈	—	178	2	no	—
274	C₄⋊C₈	—	178	2	no	—
275	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
276	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
277	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
278	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
279	#103	—	89	2	no	—
280	#103	—	89	2	no	—
281	#103	—	89	2	no	—
282	#103	—	89	2	no	—
283	C₂²⋊C₈	—	2	1	no	—
284	C₂²⋊C₈	—	178	1	no	—
285	C₂²⋊C₈	C₂²⋊D₈₉	1	2	yes	—
286	#97	—	2	2	no	—
287	#97	—	178	2	no	—
288	#97	—	178	1	no	—
289	C₄×C₈	—	2	2	no	—
290	C₄×C₈	—	2	1	no	—
291	C₄×C₈	—	89	1	no	—
292	C₄×C₈	—	89	1	no	—
293	C₄×C₈	—	89	1	no	—
294	C₄×C₈	—	89	1	no	—
295	C₄×C₈	—	89	2	no	—
296	C₄×C₈	—	178	2	no	—
297	C₄×C₈	—	178	1	no	—
298	C₄×C₈	C₂²×D₈₉	1	2	yes	—
299	C₂×Q₈	—	2	1	no	—
300	C₂×Q₈	—	89	2	no	—
301	C₂×Q₈	—	178	1	no	—
302	C₂×Q₈	C₂×C₄×D₈₉	1	2	yes	—
303	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
304	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
305	C₂²×C₄	#13083574	1	1	yes	characteristic
306	Q₁₆	—	356	2	no	—
307	Q₁₆	—	178	4	no	—
308	M₁₆	—	356	2	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	89	1	no	—
312	C₂×C₈	—	89	1	no	—
313	C₂×C₈	—	89	1	no	—
314	C₂×C₈	—	89	1	no	—
315	C₂×C₈	—	356	1	no	—
316	C₂×C₈	—	178	2	no	—
317	C₂×C₈	—	178	1	no	—
318	C₂×C₈	—	178	1	no	—
319	C₂×C₈	—	178	2	no	—
320	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
321	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	89	2	no	—
328	C₄⋊C₄	—	178	1	no	—
329	C₄⋊C₄	—	178	1	no	—
330	C₄⋊C₄	—	178	2	no	—
331	C₄⋊C₄	—	178	1	no	—
332	C₄⋊C₄	—	178	2	no	—
333	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
334	C₂²⋊C₄	—	2	2	no	—
335	C₂²⋊C₄	—	178	2	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	178	2	no	—
338	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
339	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
340	C₄²	#13083574	1	1	yes	characteristic
341	C₄²	C₂³×D₈₉	1	1	yes	characteristic
342	C₂³	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
343	Q₈	—	2	2	no	—
344	Q₈	—	2	2	no	—
345	Q₈	—	4	2	no	—
346	Q₈	—	356	2	no	—
347	Q₈	—	178	2	no	—
348	Q₈	—	178	2	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	178	2	no	—
357	C₂×C₄	—	178	2	no	—
358	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
359	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
360	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
361	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
362	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
363	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
364	C₈	—	2	2	no	—
365	C₈	—	2	2	no	—
366	C₈	—	4	1	no	—
367	C₈	—	4	1	no	—
368	C₈	—	89	2	no	—
369	C₈	—	356	2	no	—
370	C₈	—	356	1	no	—
371	C₈	—	356	1	no	—
372	C₈	—	178	1	no	—
373	C₈	—	178	1	no	—
374	C₈	—	178	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	2	1	no	—
378	C₂²	C₂×C₄×C₂²⋊D₈₉	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	4	2	no	—
383	C₄	—	4	2	no	—
384	C₄	—	356	2	no	—
385	C₄	—	356	2	no	—
386	C₄	#19151718	1	2	yes	—
387	C₄	#131×D₈₉	1	1	yes	characteristic
388	C₄	#19152363	1	1	yes	characteristic
389	C₄	#19152413	1	1	yes	characteristic
390	C₄	#19152413	1	1	yes	characteristic
391	C₂	—	4	1	no	—
392	C₂	#26028994	1	1	yes	characteristic
393	C₂	#26029118	1	1	yes	characteristic
394	C₂	C₂×#19151718	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#2986	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#2986	D₈₉
hypocenter	C₈₉	#18090
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #2986

Complements

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

N	H ≅ G/N	type
3. #26019159	C₂	semidirect
4. #26019102	C₂	semidirect
5. #26019100	C₂	semidirect
6. #26019069	C₂	semidirect
209. C₈₉	#18090	semidirect

Maximal subgroups

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

subgroup	index
#26030834	2
#26029353	2
#26019159	2
#26019102	2
#26019100	2
#26019069	2
#26018998	2
#26018673	2
#26018672	2
#26018542	2
C₈₉×#2986	2
#18090	89