Subgroups — #36524581

Order 22784.

586 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030291	C₂	1	1	yes	maximal, characteristic
2	#26029346	C₂	1	1	yes	maximal, characteristic
3	#26019123	C₂	1	2	yes	maximal, complemented
4	#26019102	C₂	1	2	yes	maximal, complemented
5	#26019069	C₂	1	2	yes	maximal, complemented
6	C₈×#13083564	C₂	1	2	yes	maximal, complemented
7	#26019030	C₂	1	1	yes	maximal, characteristic
8	#26018699	C₂	1	1	yes	maximal, characteristic
9	#26018672	C₂	1	1	yes	maximal, characteristic
10	#26018547	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2982	C₂	1	1	yes	maximal, characteristic
12	#19152155	C₂²	1	1	yes	characteristic
13	#19151933	C₄	1	2	yes	—
14	#19151893	C₄	1	2	yes	—
15	#19151808	C₄	1	2	yes	—
16	#19151754	C₄	1	2	yes	—
17	#19151728	C₂²	1	2	yes	—
18	#19151727	—	2	2	no	—
19	C₈×Dic₁₇₈	C₂²	1	2	yes	—
20	C₄×#13083564	C₂²	1	2	yes	—
21	C₄×#13083564	C₂²	1	2	yes	—
22	#19151656	—	2	2	no	—
23	#19151655	C₂²	1	1	yes	characteristic
24	#19151651	C₂²	1	1	yes	characteristic
25	C₄×#13083558	C₂²	1	1	yes	characteristic
26	#19151350	C₂²	1	2	yes	—
27	#19151348	C₂²	1	2	yes	—
28	Q₈⋊#10623494	C₂²	1	2	yes	—
29	Q₈⋊#10623494	C₂²	1	2	yes	—
30	#19151328	C₂²	1	2	yes	—
31	#19151328	C₂²	1	2	yes	—
32	#19151320	C₂²	1	1	yes	characteristic
33	#19151320	C₂²	1	1	yes	characteristic
34	#19151319	C₂²	1	1	yes	characteristic
35	C₈×#10623494	C₂²	1	1	yes	characteristic
36	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₈×Q₈×C₈₉	—	2	2	no	—
38	C₈×Q₈×C₈₉	C₂²	1	2	yes	—
39	C₈×Q₈×C₈₉	C₂²	1	2	yes	—
40	#431×C₈₉	—	2	2	no	—
41	#431×C₈₉	C₂²	1	1	yes	characteristic
42	#431×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×#13083564	C₂×C₄	1	2	yes	—
44	C₂×#13083564	C₂×C₄	1	2	yes	—
45	C₂×#13083558	—	2	1	no	—
46	#15668012	C₂×C₄	1	1	yes	characteristic
47	#15668012	C₂×C₄	1	1	yes	characteristic
48	#15668004	—	2	1	no	—
49	#15667991	—	2	1	no	—
50	#15667991	C₂×C₄	1	2	yes	—
51	C₄×Dic₁₇₈	—	2	1	no	—
52	C₄×Dic₁₇₈	C₂³	1	2	yes	—
53	#15667971	—	2	1	no	—
54	#15667945	C₂×C₄	1	2	yes	—
55	#15667945	C₂×C₄	1	2	yes	—
56	C₄⋊#10623494	C₂×C₄	1	2	yes	—
57	C₄⋊#10623494	C₂³	1	2	yes	—
58	#15667941	—	2	2	no	—
59	#15667941	—	2	2	no	—
60	#15667936	C₂×C₄	1	2	yes	—
61	#15667936	C₂×C₄	1	2	yes	—
62	#15667935	C₂×C₄	1	2	yes	—
63	#15667935	C₂×C₄	1	2	yes	—
64	C₂²⋊#10623494	—	2	1	no	—
65	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
66	#15667923	—	2	1	no	—
67	#15667923	—	2	1	no	—
68	#15667922	—	2	2	no	—
69	C₄×#10623494	C₂³	1	1	yes	characteristic
70	C₄×#10623494	C₂³	1	1	yes	characteristic
71	C₄×#10623494	C₂³	1	1	yes	characteristic
72	C₄×#10623494	C₂³	1	1	yes	characteristic
73	C₈×Dic₈₉	—	2	2	no	—
74	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
75	C₄×Q₈×C₈₉	—	2	1	no	—
76	C₄×Q₈×C₈₉	—	2	1	no	—
77	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
78	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
80	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄²×C₈₉	—	2	1	no	—
82	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₈₉	—	2	2	no	—
84	C₄⋊C₈×C₈₉	—	2	2	no	—
85	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
86	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
87	C₂²⋊C₈×C₈₉	—	2	1	no	—
88	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
89	C₄×C₈×C₈₉	—	2	2	no	—
90	C₄×C₈×C₈₉	—	2	2	no	—
91	C₄×C₈×C₈₉	—	2	1	no	—
92	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₇₈	—	2	1	no	—
95	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
96	#13083564	—	2	4	no	—
97	#13083564	—	2	4	no	—
98	#13083558	—	4	2	no	—
99	C₂×#10623494	—	2	1	no	—
100	C₂×#10623494	—	2	2	no	—
101	C₂×#10623494	—	2	1	no	—
102	C₂×#10623494	C₂²×C₄	1	2	yes	—
103	C₂×#10623494	C₂²×C₄	1	2	yes	—
104	C₄⋊Dic₈₉	—	2	1	no	—
105	C₄⋊Dic₈₉	—	2	1	no	—
106	C₄⋊Dic₈₉	—	2	1	no	—
107	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
108	#13083553	—	2	2	no	—
109	#13083553	—	2	2	no	—
110	C₂²⋊Dic₈₉	—	2	2	no	—
111	C₄×Dic₈₉	—	2	2	no	—
112	C₂×Q₈×C₈₉	—	2	4	no	—
113	C₂×Q₈×C₈₉	—	2	1	no	—
114	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₈₉	—	2	1	no	—
116	C₂²×C₄×C₈₉	—	2	1	no	—
117	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
120	C₂×C₈×C₈₉	—	2	2	no	—
121	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
123	C₄⋊C₄×C₈₉	—	2	2	no	—
124	C₄⋊C₄×C₈₉	—	2	1	no	—
125	C₄⋊C₄×C₈₉	—	2	2	no	—
126	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
127	C₄²×C₈₉	—	2	2	no	—
128	C₄²×C₈₉	—	2	2	no	—
129	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₈₉	#41	1	1	yes	characteristic
132	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
133	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
134	Q₈×C₈₉	—	2	4	no	—
135	Q₈×C₈₉	—	2	2	no	—
136	Q₈×C₈₉	#131	1	4	yes	—
137	C₂×C₄×C₈₉	—	2	4	no	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	2	2	no	—
140	C₂×C₄×C₈₉	—	2	2	no	—
141	C₂×C₄×C₈₉	—	2	2	no	—
142	C₂×C₄×C₈₉	—	4	1	no	—
143	C₂×C₄×C₈₉	—	8	1	no	—
144	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
150	C₂×Dic₈₉	—	2	2	no	—
151	C₂×Dic₈₉	—	2	2	no	—
152	Dic₁₇₈	—	2	2	no	—
153	Dic₁₇₈	—	2	2	no	—
154	Dic₁₇₈	—	8	1	no	—
155	C₈×C₈₉	—	2	2	no	—
156	C₈×C₈₉	—	2	2	no	—
157	C₈×C₈₉	—	4	2	no	—
158	#10623494	—	2	2	no	—
159	#10623494	—	2	2	no	—
160	#10623494	—	2	2	no	—
161	#10623494	—	4	1	no	—
162	#10623494	—	4	2	no	—
163	C₂²×C₈₉	—	2	1	no	—
164	C₂²×C₈₉	—	2	1	no	—
165	C₂²×C₈₉	—	2	1	no	—
166	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
167	C₄×C₈₉	—	2	4	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	—	2	1	no	—
171	C₄×C₈₉	—	2	1	no	—
172	C₄×C₈₉	—	4	2	no	—
173	C₄×C₈₉	C₂×#131	1	2	yes	—
174	C₄×C₈₉	#578	1	2	yes	—
175	Dic₈₉	—	4	2	no	—
176	Dic₈₉	—	4	2	no	—
177	#17925	—	89	1	no	maximal, Sylow
178	C₂×C₈₉	—	4	1	no	—
179	C₂×C₈₉	#2929	1	1	yes	characteristic
180	C₂×C₈₉	#2950	1	1	yes	characteristic
181	C₂×C₈₉	#2956	1	1	yes	characteristic
182	#3104	—	89	1	no	—
183	#2985	—	89	1	no	—
184	#2982	D₈₉	1	1	yes	characteristic
185	#1587	—	89	2	no	—
186	#1584	—	89	2	no	—
187	#1579	—	89	2	no	—
188	C₈×Q₁₆	—	89	2	no	—
189	#1572	—	89	1	no	—
190	#1491	—	89	1	no	—
191	#1485	—	89	1	no	—
192	#1453	—	89	1	no	—
193	C₈₉	#17925	1	1	yes	characteristic, complemented, Sylow
194	#533	—	89	1	no	—
195	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
196	#484	—	89	2	no	—
197	#477	—	89	2	no	—
198	#462	—	89	2	no	—
199	#451	—	89	2	no	—
200	#446	—	89	2	no	—
201	#446	—	178	2	no	—
202	C₈×Q₈	—	2	2	no	—
203	C₈×Q₈	—	89	2	no	—
204	C₈×Q₈	C₂×D₈₉	1	2	yes	—
205	C₈×Q₈	C₂×D₈₉	1	2	yes	—
206	C₄×Q₁₆	—	89	2	no	—
207	C₄×Q₁₆	—	89	2	no	—
208	#433	—	178	2	no	—
209	#432	—	89	1	no	—
210	#431	—	2	2	no	—
211	#431	—	89	1	no	—
212	#431	C₂×D₈₉	1	1	yes	characteristic
213	#431	C₂×D₈₉	1	1	yes	characteristic
214	C₄×M₁₆	—	89	1	no	—
215	#335	—	89	2	no	—
216	#334	—	89	2	no	—
217	Q₈⋊C₈	—	89	2	no	—
218	Q₈⋊C₈	—	89	2	no	—
219	Q₈⋊C₈	—	89	2	no	—
220	Q₈⋊C₈	—	89	2	no	—
221	#322	—	89	1	no	—
222	#322	—	89	1	no	—
223	#322	—	89	1	no	—
224	C₈²	—	89	1	no	—
225	C₂²×Q₈	C₄×D₈₉	1	1	yes	characteristic
226	C₂×Q₁₆	—	89	2	no	—
227	C₂×Q₁₆	—	89	2	no	—
228	C₂×M₁₆	—	178	1	no	—
229	C₄⋊Q₈	—	89	1	no	—
230	C₄⋊Q₈	—	89	1	no	—
231	#125	—	178	1	no	—
232	C₂²⋊Q₈	—	89	2	no	—
233	C₂²⋊Q₈	—	178	1	no	—
234	C₄×Q₈	—	2	1	no	—
235	C₄×Q₈	—	2	1	no	—
236	C₄×Q₈	—	89	2	no	—
237	C₄×Q₈	—	178	1	no	—
238	C₄×Q₈	C₄×D₈₉	1	2	yes	—
239	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
240	#117	—	178	1	no	—
241	C₂×C₄⋊C₄	—	2	1	no	—
242	C₂×C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
243	C₂×C₄²	—	2	1	no	—
244	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
245	#107	—	89	2	no	—
246	#107	—	89	2	no	—
247	C₄⋊C₈	—	2	2	no	—
248	C₄⋊C₈	—	2	2	no	—
249	C₄⋊C₈	—	89	2	no	—
250	C₄⋊C₈	—	89	2	no	—
251	C₄⋊C₈	—	178	2	no	—
252	C₄⋊C₈	—	178	2	no	—
253	C₄⋊C₈	C₂²⋊D₈₉	1	2	yes	—
254	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
255	#103	—	89	2	no	—
256	#103	—	89	2	no	—
257	#103	—	89	2	no	—
258	#103	—	89	2	no	—
259	C₂²⋊C₈	—	2	1	no	—
260	C₂²⋊C₈	—	89	2	no	—
261	C₂²⋊C₈	—	178	1	no	—
262	C₂²⋊C₈	C₂²⋊D₈₉	1	2	yes	—
263	#97	—	178	2	no	—
264	#97	—	178	1	no	—
265	#97	—	178	1	no	—
266	C₄×C₈	—	2	2	no	—
267	C₄×C₈	—	2	2	no	—
268	C₄×C₈	—	2	1	no	—
269	C₄×C₈	—	89	1	no	—
270	C₄×C₈	—	89	1	no	—
271	C₄×C₈	—	89	1	no	—
272	C₄×C₈	—	89	1	no	—
273	C₄×C₈	—	178	2	no	—
274	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
276	C₂×Q₈	—	2	4	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	89	2	no	—
279	C₂×Q₈	—	178	1	no	—
280	C₂×Q₈	C₂×C₄×D₈₉	1	2	yes	—
281	C₂²×C₄	—	2	1	no	—
282	C₂²×C₄	—	2	1	no	—
283	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
284	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
285	C₂²×C₄	#13083574	1	1	yes	characteristic
286	Q₁₆	—	178	4	no	—
287	Q₁₆	—	178	4	no	—
288	M₁₆	—	356	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	89	2	no	—
291	C₂×C₈	—	89	2	no	—
292	C₂×C₈	—	178	2	no	—
293	C₂×C₈	—	178	1	no	—
294	C₂×C₈	—	178	1	no	—
295	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
296	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	89	2	no	—
301	C₄⋊C₄	—	178	1	no	—
302	C₄⋊C₄	—	178	1	no	—
303	C₄⋊C₄	—	178	2	no	—
304	C₄⋊C₄	—	178	1	no	—
305	C₄⋊C₄	—	178	2	no	—
306	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
307	C₂²⋊C₄	—	178	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	2	2	no	—
310	C₄²	—	178	2	no	—
311	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
312	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
313	C₄²	#13083574	1	1	yes	characteristic
314	C₄²	C₂³×D₈₉	1	1	yes	characteristic
315	C₂³	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
316	Q₈	—	2	4	no	—
317	Q₈	—	2	2	no	—
318	Q₈	—	178	2	no	—
319	Q₈	—	178	2	no	—
320	Q₈	—	712	1	no	—
321	Q₈	#15668018	1	4	yes	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	8	1	no	—
329	C₂×C₄	—	178	2	no	—
330	C₂×C₄	—	178	2	no	—
331	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
332	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
333	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
335	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
336	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
337	C₈	—	2	2	no	—
338	C₈	—	2	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	356	1	no	—
341	C₈	—	356	2	no	—
342	C₈	—	178	2	no	—
343	C₈	—	178	2	no	—
344	C₈	—	178	2	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	C₂×C₄×C₂²⋊D₈₉	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	4	no	—
353	C₄	—	2	1	no	—
354	C₄	—	4	2	no	—
355	C₄	—	356	2	no	—
356	C₄	—	356	2	no	—
357	C₄	C₂×#15668018	1	2	yes	—
358	C₄	#19152413	1	2	yes	—
359	C₂	—	4	1	no	—
360	C₂	#26028946	1	1	yes	characteristic
361	C₂	#26029118	1	1	yes	characteristic
362	C₂	#26029159	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×#2982	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#2982	D₈₉
hypocenter	C₈₉	#17925
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #2982

Complements

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

N	H ≅ G/N	type
3. #26019123	C₂	semidirect
4. #26019102	C₂	semidirect
5. #26019069	C₂	semidirect
6. C₈×#13083564	C₂	semidirect
193. C₈₉	#17925	semidirect

Maximal subgroups

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

subgroup	index
#26030291	2
#26029346	2
#26019123	2
#26019102	2
#26019069	2
C₈×#13083564	2
#26019030	2
#26018699	2
#26018672	2
#26018547	2
C₈₉×#2982	2
#17925	89