Subgroups — #36552676

Order 22784.

990 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151655	C₂	1	1	yes	maximal, characteristic
2	#26022650	C₂	1	1	yes	maximal, characteristic
3	#26020119	C₂	1	2	yes	maximal, complemented
4	C₈×C₄⋊Dic₈₉	C₂	1	2	yes	maximal, complemented
5	#26020028	C₂	1	2	yes	maximal
6	C₈×C₂²⋊Dic₈₉	C₂	1	2	yes	maximal
7	#26020003	C₂	1	2	yes	maximal, complemented
8	#26019938	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
10	#19151655	C₄	1	8	yes	complemented
11	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
12	C₂×C₂²⋊#10623494	C₂²	1	2	yes	—
13	C₂×#15667923	C₂²	1	2	yes	—
14	C₂×#15667922	C₂²	1	4	yes	—
15	C₂×C₈×Dic₈₉	C₂²	1	4	yes	—
16	#19151466	C₄	1	2	yes	—
17	#19151464	C₄	1	2	yes	—
18	#19151461	C₄	1	2	yes	—
19	#19151458	C₄	1	2	yes	—
20	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
21	C₄×C₂²⋊Dic₈₉	C₂²	1	2	yes	—
22	#19151444	C₂²	1	2	yes	—
23	#19151352	C₂²	1	4	yes	—
24	#19151352	C₂²	1	4	yes	—
25	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₈₉	C₄	1	8	yes	complemented
29	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
33	C₂²×#10623494	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊Dic₈₉	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₈₉	C₂³	1	4	yes	—
37	C₄⋊#10623494	C₂×C₄	1	8	yes	—
38	C₂²⋊#10623494	C₂×C₄	1	8	yes	—
39	#15667923	C₂×C₄	1	8	yes	—
40	#15667922	—	2	8	no	—
41	C₈×Dic₈₉	—	2	8	no	—
42	#15667919	C₂×C₄	1	4	yes	—
43	#15667919	C₂×C₄	1	4	yes	—
44	#15667919	C₂×C₄	1	4	yes	—
45	#15667919	C₂×C₄	1	4	yes	—
46	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	—	2	4	no	—
50	C₂²×C₈×C₈₉	—	2	4	no	—
51	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
60	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
61	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
64	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
65	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
66	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
67	C₂²×Dic₈₉	C₂²×C₄	1	8	yes	—
68	C₂×#10623494	—	2	4	no	—
69	C₂×#10623494	—	2	4	no	—
70	C₂×#10623494	C₂²×C₄	1	8	yes	—
71	C₄⋊Dic₈₉	—	2	8	no	—
72	C₂²⋊Dic₈₉	—	2	8	no	—
73	C₄×Dic₈₉	—	2	4	no	—
74	C₄×Dic₈₉	—	2	4	no	—
75	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	—	2	4	no	—
77	C₂²×C₄×C₈₉	—	2	4	no	—
78	C₂²×C₄×C₈₉	—	2	2	no	—
79	C₂²×C₄×C₈₉	—	2	2	no	—
80	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
81	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
82	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₈₉	—	2	2	no	—
91	C₂×C₈×C₈₉	—	2	2	no	—
92	C₂×C₈×C₈₉	—	2	8	no	—
93	C₂×C₈×C₈₉	—	2	8	no	—
94	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₈₉	#41	1	4	yes	—
99	C₂×C₈×C₈₉	#41	1	4	yes	—
100	C₄²×C₈₉	C₄²	1	4	yes	—
101	C₄²×C₈₉	C₄²	1	4	yes	—
102	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
103	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
104	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
105	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
106	C₂³×C₈₉	—	2	2	no	—
107	C₂³×C₈₉	—	2	2	no	—
108	C₂³×C₈₉	C₂×C₄²	1	2	yes	—
109	C₂³×C₈₉	#131	1	4	yes	—
110	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₈₉	—	2	2	no	—
112	C₂×C₄×C₈₉	—	2	2	no	—
113	C₂×C₄×C₈₉	—	2	2	no	—
114	C₂×C₄×C₈₉	—	2	2	no	—
115	C₂×C₄×C₈₉	—	2	8	no	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	2	2	no	—
118	C₂×C₄×C₈₉	—	2	4	no	—
119	C₂×C₄×C₈₉	—	2	8	no	—
120	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₈₉	#131	1	4	yes	—
126	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
129	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
130	C₂×Dic₈₉	—	2	8	no	—
131	C₂×Dic₈₉	—	2	8	no	—
132	C₂×Dic₈₉	—	2	8	no	—
133	C₈×C₈₉	—	2	2	no	—
134	C₈×C₈₉	—	2	2	no	—
135	C₈×C₈₉	#117	1	4	yes	—
136	C₈×C₈₉	#117	1	4	yes	—
137	#10623494	—	2	8	no	—
138	C₂²×C₈₉	—	2	2	no	—
139	C₂²×C₈₉	—	2	8	no	—
140	C₂²×C₈₉	—	2	2	no	—
141	C₂²×C₈₉	#405	1	4	yes	—
142	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
143	C₂²×C₈₉	C₄×#41	1	2	yes	—
144	C₂²×C₈₉	C₂×#131	1	2	yes	—
145	C₂²×C₈₉	C₂×#131	1	2	yes	—
146	C₄×C₈₉	—	2	2	no	—
147	C₄×C₈₉	—	2	2	no	—
148	C₄×C₈₉	—	2	2	no	—
149	C₄×C₈₉	—	2	2	no	—
150	C₄×C₈₉	#405	1	4	yes	—
151	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
152	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
153	C₄×C₈₉	C₄×#41	1	2	yes	—
154	Dic₈₉	—	4	8	no	—
155	#20476	—	89	1	no	maximal, Sylow
156	C₂×C₈₉	—	2	4	no	—
157	C₂×C₈₉	C₄×#117	1	1	yes	characteristic
158	C₂×C₈₉	C₂×#405	1	1	yes	characteristic
159	C₂×C₈₉	C₂×#405	1	1	yes	characteristic
160	C₂×C₈₉	#2966	1	4	yes	—
161	C₂×#432	—	89	1	no	—
162	C₂²×C₄×C₈	D₈₉	1	1	yes	characteristic
163	#2283	—	89	1	no	—
164	#1770	—	89	2	no	—
165	C₈×C₄⋊C₄	—	89	2	no	—
166	#1752	—	89	2	no	—
167	C₈×C₂²⋊C₄	—	89	2	no	—
168	#1746	—	89	2	no	—
169	#1726	—	89	2	no	—
170	C₈₉	#20476	1	1	yes	characteristic, complemented, Sylow
171	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
172	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
173	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
174	#432	—	89	8	no	—
175	C₂×C₄⋊C₈	—	89	2	no	—
176	C₂×C₂²⋊C₈	—	89	2	no	—
177	C₂×#97	—	89	2	no	—
178	C₂×#97	—	89	4	no	—
179	C₂×C₄×C₈	—	89	4	no	—
180	C₂×C₄×C₈	Dic₈₉	1	8	yes	complemented
181	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
182	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
185	#383	—	89	2	no	—
186	#382	—	89	2	no	—
187	#381	—	89	2	no	—
188	#380	—	89	2	no	—
189	C₄×C₄⋊C₄	—	89	2	no	—
190	C₄×C₂²⋊C₄	—	89	2	no	—
191	#376	—	89	2	no	—
192	#336	—	89	4	no	—
193	#336	—	89	4	no	—
194	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
195	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
196	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
197	C₂²×C₈	—	2	4	no	—
198	C₂²×C₈	—	2	4	no	—
199	C₂²×C₈	—	89	4	no	—
200	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
201	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
202	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
205	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
206	C₂×C₄⋊C₄	—	89	4	no	—
207	C₂×C₂²⋊C₄	—	89	4	no	—
208	C₂×C₄²	—	89	4	no	—
209	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
211	C₂×C₄²	C₄×D₈₉	1	4	yes	—
212	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
213	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
214	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
215	C₄⋊C₈	—	89	8	no	—
216	C₂²⋊C₈	—	89	8	no	—
217	#97	—	89	8	no	—
218	#97	—	178	8	no	—
219	C₄×C₈	—	178	8	no	—
220	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
221	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
222	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
223	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
224	#95	—	89	4	no	—
225	#95	—	89	4	no	—
226	#95	—	89	4	no	—
227	#95	—	89	4	no	—
228	C₂⁴	C₂×C₄×D₈₉	1	1	yes	characteristic
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	89	8	no	—
234	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
235	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
236	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
237	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
242	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
243	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	8	no	—
247	C₂×C₈	—	2	8	no	—
248	C₂×C₈	—	89	8	no	—
249	C₂×C₈	—	178	4	no	—
250	C₂×C₈	—	178	4	no	—
251	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
252	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
253	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
254	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
255	C₂×C₈	#13083574	1	4	yes	—
256	C₂×C₈	#13083574	1	4	yes	—
257	C₄⋊C₄	—	178	8	no	—
258	C₂²⋊C₄	—	178	8	no	—
259	C₄²	—	178	4	no	—
260	C₄²	—	178	4	no	—
261	C₄²	C₄×Dic₈₉	1	4	yes	—
262	C₄²	C₄×Dic₈₉	1	4	yes	—
263	C₄²	C₂×C₄×D₈₉	1	2	yes	—
264	C₄²	C₂×C₄×D₈₉	1	2	yes	—
265	C₄²	C₂²×Dic₈₉	1	2	yes	—
266	C₄²	C₂²×Dic₈₉	1	2	yes	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	C₂×C₄×Dic₈₉	1	2	yes	—
270	C₂³	#15668018	1	4	yes	—
271	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	178	8	no	—
282	C₂×C₄	—	178	8	no	—
283	C₂×C₄	—	178	8	no	—
284	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
285	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
286	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
287	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
288	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
289	C₂×C₄	#15668018	1	4	yes	—
290	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
291	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
292	C₂×C₄	C₂×#13083574	1	2	yes	—
293	C₂×C₄	C₂×#13083574	1	2	yes	—
294	C₈	—	2	2	no	—
295	C₈	—	2	2	no	—
296	C₈	—	178	8	no	—
297	C₈	#15667971	1	4	yes	—
298	C₈	#15667971	1	4	yes	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	8	no	—
301	C₂²	—	2	2	no	—
302	C₂²	#19151553	1	4	yes	—
303	C₂²	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
304	C₂²	C₄×#13083574	1	2	yes	—
305	C₂²	C₂×#15668018	1	2	yes	—
306	C₂²	C₂×#15668018	1	2	yes	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	356	8	no	—
312	C₄	#19151553	1	4	yes	—
313	C₄	C₂×#15667971	1	1	yes	characteristic
314	C₄	C₂×#15667971	1	1	yes	characteristic
315	C₄	C₄×#13083574	1	2	yes	—
316	C₂	—	2	4	no	—
317	C₂	C₄×#15667971	1	1	yes	characteristic
318	C₂	C₂×#19151553	1	1	yes	characteristic
319	C₂	C₂×#19151553	1	1	yes	characteristic
320	C₂	#26029227	1	4	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂²×C₄×C₈×C₈₉	C₂
socle	C₂³×C₈₉	C₂³×C₄
prime radical	C₂⁴×C₈₉	C₂²×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂²×C₄×C₈	D₈₉
hypocenter	C₈₉	#20476
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #26020119	C₂	semidirect
4. C₈×C₄⋊Dic₈₉	C₂	semidirect
7. #26020003	C₂	semidirect
8. #26019938	C₂	semidirect
10. #19151655	C₄	semidirect
28. C₂×C₄×C₈×C₈₉	C₄	semidirect
170. C₈₉	#20476	semidirect
180. C₂×C₄×C₈	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151655	2
#26022650	2
#26020119	2
C₈×C₄⋊Dic₈₉	2
#26020028	2
C₈×C₂²⋊Dic₈₉	2
#26020003	2
#26019938	2
C₂²×C₄×C₈×C₈₉	2
#20476	89