Subgroups — #36460042

Order 22784.

918 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#10623494	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151353	C₂	1	2	yes	maximal
3	C₂×#19151353	C₂	1	2	yes	maximal
4	C₂×#19151353	C₂	1	1	yes	maximal, characteristic
5	C₄×#95×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×#10623494	C₂²	1	2	yes	—
7	C₂³×#10623494	C₂²	1	1	yes	characteristic
8	C₂³×#10623494	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10623494	—	2	4	no	—
10	C₂×C₄×#10623494	C₄	1	4	yes	complemented
11	#19151353	—	2	4	no	—
12	#19151353	—	2	4	no	—
13	#19151353	C₄	1	4	yes	complemented
14	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
15	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#95×C₈₉	C₄	1	2	yes	—
17	C₂×#95×C₈₉	C₄	1	2	yes	—
18	C₂²×#10623494	—	2	2	no	—
19	C₂²×#10623494	—	2	2	no	—
20	C₂²×#10623494	—	2	2	no	—
21	C₂²×#10623494	—	2	1	no	—
22	C₂²×#10623494	—	2	2	no	—
23	C₂²×#10623494	—	2	2	no	—
24	C₂²×#10623494	—	2	1	no	—
25	C₂²×#10623494	—	2	2	no	—
26	C₂²×#10623494	—	2	2	no	—
27	C₂²×#10623494	—	2	2	no	—
28	C₂²×#10623494	—	2	2	no	—
29	C₂²×#10623494	C₂×C₄	1	4	yes	—
30	C₂²×#10623494	C₂×C₄	1	2	yes	—
31	C₂²×#10623494	C₂×C₄	1	2	yes	—
32	C₂²×#10623494	D₄	1	4	yes	—
33	C₂²×#10623494	D₄	1	2	yes	—
34	C₂²×#10623494	Q₈	1	2	yes	—
35	C₄×#10623494	—	2	8	no	—
36	C₂³×C₄×C₈₉	C₂×C₄	1	4	yes	—
37	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	—	2	2	no	—
41	C₂×C₄²×C₈₉	—	2	2	no	—
42	C₂×C₄²×C₈₉	—	2	2	no	—
43	C₂×C₄²×C₈₉	—	2	2	no	—
44	C₂×C₄²×C₈₉	—	2	2	no	—
45	C₂×C₄²×C₈₉	—	2	2	no	—
46	C₂×C₄²×C₈₉	—	2	2	no	—
47	C₂×C₄²×C₈₉	—	2	2	no	—
48	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
50	#95×C₈₉	—	2	4	no	—
51	#95×C₈₉	C₈	1	8	yes	complemented
52	C₂×#10623494	—	2	2	no	—
53	C₂×#10623494	—	2	2	no	—
54	C₂×#10623494	—	2	2	no	—
55	C₂×#10623494	—	2	2	no	—
56	C₂×#10623494	—	2	4	no	—
57	C₂×#10623494	—	2	4	no	—
58	C₂×#10623494	—	2	4	no	—
59	C₂×#10623494	—	2	4	no	—
60	C₂×#10623494	—	2	4	no	—
61	C₂×#10623494	—	2	4	no	—
62	C₂×#10623494	—	2	4	no	—
63	C₂×#10623494	—	4	2	no	—
64	C₂×#10623494	—	4	2	no	—
65	C₂×#10623494	—	4	2	no	—
66	C₂×#10623494	—	4	2	no	—
67	C₂×#10623494	C₂²⋊C₄	1	4	yes	—
68	C₂×#10623494	C₄⋊C₄	1	4	yes	—
69	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	—	2	2	no	—
71	C₂²×C₄×C₈₉	—	2	2	no	—
72	C₂²×C₄×C₈₉	—	2	2	no	—
73	C₂²×C₄×C₈₉	—	2	2	no	—
74	C₂²×C₄×C₈₉	—	2	4	no	—
75	C₂²×C₄×C₈₉	—	2	4	no	—
76	C₂²×C₄×C₈₉	—	2	4	no	—
77	C₂²×C₄×C₈₉	—	2	4	no	—
78	C₂²×C₄×C₈₉	—	2	4	no	—
79	C₂²×C₄×C₈₉	—	2	4	no	—
80	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
81	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
82	C₂²×C₄×C₈₉	C₂×C₈	1	8	yes	—
83	C₂²×C₄×C₈₉	C₂×C₈	1	4	yes	—
84	C₂²×C₄×C₈₉	M₁₆	1	4	yes	—
85	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	#41	1	2	yes	—
91	C₂²×C₄×C₈₉	#41	1	2	yes	—
92	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	#41	1	2	yes	—
94	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
95	C₄²×C₈₉	—	2	2	no	—
96	C₄²×C₈₉	—	2	2	no	—
97	C₄²×C₈₉	—	2	4	no	—
98	C₄²×C₈₉	—	4	2	no	—
99	C₄²×C₈₉	—	4	2	no	—
100	C₄²×C₈₉	—	4	2	no	—
101	C₄²×C₈₉	—	4	2	no	—
102	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
103	C₂³×C₈₉	#117	1	2	yes	—
104	C₂³×C₈₉	C₄×D₄	1	2	yes	—
105	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
106	C₂³×C₈₉	C₄×Q₈	1	1	yes	characteristic
107	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
108	C₂³×C₈₉	C₂×M₁₆	1	2	yes	—
109	C₂³×C₈₉	#131	1	2	yes	—
110	C₂³×C₈₉	#131	1	2	yes	—
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	8	no	—
117	C₂×C₄×C₈₉	—	2	8	no	—
118	C₂×C₄×C₈₉	—	2	8	no	—
119	C₂×C₄×C₈₉	—	2	8	no	—
120	C₂×C₄×C₈₉	—	4	4	no	—
121	C₂×C₄×C₈₉	—	4	4	no	—
122	C₂×C₄×C₈₉	—	4	4	no	—
123	C₂×C₄×C₈₉	—	4	4	no	—
124	C₂×C₄×C₈₉	C₄×C₈	1	4	yes	—
125	C₂×C₄×C₈₉	#97	1	4	yes	—
126	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
129	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
130	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
131	C₂×C₄×C₈₉	#117	1	2	yes	—
132	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₈₉	#121	1	2	yes	—
137	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄×C₈₉	#123	1	2	yes	—
139	C₂×C₄×C₈₉	#123	1	2	yes	—
140	C₂×C₄×C₈₉	#124	1	2	yes	—
141	C₂×C₄×C₈₉	#125	1	2	yes	—
142	C₂×C₄×C₈₉	#126	1	2	yes	—
143	C₂×C₄×C₈₉	#126	1	2	yes	—
144	#10623494	—	2	4	no	—
145	#10623494	—	2	4	no	—
146	#10623494	—	4	4	no	—
147	C₂²×C₈₉	#376	1	1	yes	characteristic
148	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₈₉	#387	1	2	yes	—
151	C₂²×C₈₉	#388	1	2	yes	—
152	C₂²×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
153	C₂²×C₈₉	C₂×#97	1	1	yes	characteristic
154	C₂²×C₈₉	#405	1	1	yes	characteristic
155	C₂²×C₈₉	#405	1	1	yes	characteristic
156	C₂²×C₈₉	#431	1	4	yes	—
157	C₂²×C₈₉	#433	1	4	yes	—
158	C₂²×C₈₉	C₈×D₄	1	4	yes	—
159	C₂²×C₈₉	C₈×D₄	1	2	yes	—
160	C₂²×C₈₉	#435	1	4	yes	—
161	C₂²×C₈₉	#436	1	2	yes	—
162	C₂²×C₈₉	C₈×Q₈	1	2	yes	—
163	C₂²×C₈₉	#446	1	2	yes	—
164	C₄×C₈₉	—	2	4	no	—
165	C₄×C₈₉	—	2	4	no	—
166	C₄×C₈₉	—	4	8	no	—
167	C₄×C₈₉	#380	1	1	yes	characteristic
168	C₄×C₈₉	#381	1	1	yes	characteristic
169	C₄×C₈₉	#382	1	1	yes	characteristic
170	C₄×C₈₉	#383	1	1	yes	characteristic
171	C₄×C₈₉	#387	1	2	yes	—
172	C₄×C₈₉	#388	1	2	yes	—
173	#10797	—	89	1	no	maximal, Sylow
174	C₂×C₈₉	#1434	1	1	yes	characteristic
175	C₂×C₈₉	#1744	1	1	yes	characteristic
176	C₂×C₈₉	#1746	1	1	yes	characteristic
177	C₂×C₈₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
178	C₂×C₈₉	#1752	1	1	yes	characteristic
179	C₂×C₈₉	C₈×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₈₉	#1770	1	1	yes	characteristic
181	C₂×C₈₉	#1916	1	4	yes	—
182	C₂×C₈₉	#1923	1	4	yes	—
183	C₂²×C₄×C₈	—	89	1	no	—
184	C₂×#336	—	89	1	no	—
185	C₂×#336	—	89	2	no	—
186	C₂×#336	—	89	2	no	—
187	C₄×#95	D₈₉	1	1	yes	characteristic
188	C₈₉	#10797	1	1	yes	characteristic, complemented, Sylow
189	C₂³×C₈	—	89	1	no	—
190	C₂³×C₈	—	89	1	no	—
191	C₂³×C₈	—	89	2	no	—
192	C₂²×C₄²	C₂×D₈₉	1	2	yes	—
193	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
194	C₂×C₄×C₈	—	89	4	no	—
195	C₂×C₄×C₈	—	178	4	no	—
196	C₂×#95	Dic₈₉	1	2	yes	—
197	C₂×#95	Dic₈₉	1	2	yes	—
198	#336	—	89	4	no	—
199	#336	—	178	4	no	—
200	#336	—	178	4	no	—
201	C₂³×C₄	C₂×Dic₈₉	1	4	yes	—
202	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
203	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
204	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
205	C₂²×C₈	—	89	2	no	—
206	C₂²×C₈	—	89	2	no	—
207	C₂²×C₈	—	89	2	no	—
208	C₂²×C₈	—	89	4	no	—
209	C₂²×C₈	—	89	2	no	—
210	C₂²×C₈	—	89	4	no	—
211	C₂²×C₈	—	178	2	no	—
212	C₂²×C₈	—	178	2	no	—
213	C₂²×C₈	—	178	2	no	—
214	C₂²×C₈	—	178	2	no	—
215	C₂²×C₈	—	178	2	no	—
216	C₂²×C₈	—	178	2	no	—
217	C₂²×C₈	—	178	2	no	—
218	C₂²×C₈	—	178	1	no	—
219	C₂²×C₈	—	178	2	no	—
220	C₂²×C₈	—	178	2	no	—
221	C₂²×C₈	—	178	1	no	—
222	C₂×C₄²	—	2	2	no	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	—	2	2	no	—
225	C₂×C₄²	—	2	2	no	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	—	2	2	no	—
228	C₂×C₄²	—	2	2	no	—
229	C₂×C₄²	—	2	2	no	—
230	C₂×C₄²	C₄×D₈₉	1	2	yes	—
231	C₂×C₄²	C₄×D₈₉	1	2	yes	—
232	C₄×C₈	—	178	8	no	—
233	#95	—	2	4	no	—
234	#95	#10623494	1	8	yes	complemented
235	C₂⁴	C₂²×Dic₈₉	1	1	yes	characteristic
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
247	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
248	C₂²×C₄	C₂×#10623494	1	8	yes	—
249	C₂²×C₄	C₂×#10623494	1	4	yes	—
250	C₂²×C₄	#13083558	1	4	yes	—
251	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
253	C₂²×C₄	D₄×D₈₉	1	2	yes	—
254	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
255	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
256	C₂²×C₄	#13083573	1	2	yes	—
257	C₂²×C₄	#13083573	1	1	yes	characteristic
258	C₂²×C₄	#13083574	1	2	yes	—
259	C₂²×C₄	#13083574	1	2	yes	—
260	C₂²×C₄	#13083575	1	1	yes	characteristic
261	C₂×C₈	—	89	4	no	—
262	C₂×C₈	—	89	4	no	—
263	C₂×C₈	—	356	2	no	—
264	C₂×C₈	—	356	2	no	—
265	C₂×C₈	—	356	2	no	—
266	C₂×C₈	—	356	2	no	—
267	C₂×C₈	—	178	4	no	—
268	C₂×C₈	—	178	4	no	—
269	C₂×C₈	—	178	4	no	—
270	C₂×C₈	—	178	2	no	—
271	C₂×C₈	—	178	2	no	—
272	C₂×C₈	—	178	2	no	—
273	C₂×C₈	—	178	4	no	—
274	C₂×C₈	—	178	4	no	—
275	C₂×C₈	—	178	4	no	—
276	C₂×C₈	—	178	4	no	—
277	C₂×C₈	—	178	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	4	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	4	2	no	—
282	C₄²	—	4	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	—	4	2	no	—
285	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
286	C₂³	#15667971	1	2	yes	—
287	C₂³	D₄×Dic₈₉	1	2	yes	—
288	C₂³	D₄×Dic₈₉	1	1	yes	characteristic
289	C₂³	Q₈×Dic₈₉	1	1	yes	characteristic
290	C₂³	C₂²×#10623494	1	2	yes	—
291	C₂³	C₂×#13083558	1	2	yes	—
292	C₂³	#15668019	1	2	yes	—
293	C₂³	#15668019	1	2	yes	—
294	C₂×C₄	—	2	8	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	8	no	—
301	C₂×C₄	—	2	8	no	—
302	C₂×C₄	—	2	8	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	C₄×#10623494	1	4	yes	—
308	C₂×C₄	#15667923	1	4	yes	—
309	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
310	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
311	C₂×C₄	#15667968	1	1	yes	characteristic
312	C₂×C₄	#15667969	1	1	yes	characteristic
313	C₂×C₄	#15667969	1	1	yes	characteristic
314	C₂×C₄	#15667969	1	2	yes	—
315	C₂×C₄	#15667970	1	1	yes	characteristic
316	C₂×C₄	#15667972	1	2	yes	—
317	C₂×C₄	#15667974	1	1	yes	characteristic
318	C₂×C₄	#15667977	1	1	yes	characteristic
319	C₂×C₄	#15667985	1	2	yes	—
320	C₂×C₄	#15667990	1	2	yes	—
321	C₂×C₄	#15667994	1	2	yes	—
322	C₂×C₄	#15667997	1	2	yes	—
323	C₂×C₄	#15667998	1	2	yes	—
324	C₂×C₄	#15668002	1	2	yes	—
325	C₂×C₄	#15668005	1	2	yes	—
326	C₂×C₄	#15668007	1	2	yes	—
327	C₈	—	356	4	no	—
328	C₈	—	178	4	no	—
329	C₈	—	178	4	no	—
330	C₂²	#19151444	1	1	yes	characteristic
331	C₂²	C₂²⋊C₄×Dic₈₉	1	1	yes	characteristic
332	C₂²	C₄⋊C₄×Dic₈₉	1	1	yes	characteristic
333	C₂²	#19151484	1	2	yes	—
334	C₂²	#19151491	1	2	yes	—
335	C₂²	C₂×C₄×#10623494	1	1	yes	characteristic
336	C₂²	C₂×#15667923	1	1	yes	characteristic
337	C₂²	#19151554	1	1	yes	characteristic
338	C₂²	#19151554	1	1	yes	characteristic
339	C₂²	#19151651	1	4	yes	—
340	C₂²	#19151659	1	4	yes	—
341	C₂²	D₄×#10623494	1	2	yes	—
342	C₂²	D₄×#10623494	1	4	yes	—
343	C₂²	#19151670	1	4	yes	—
344	C₂²	#19151676	1	2	yes	—
345	C₂²	Q₈×#10623494	1	2	yes	—
346	C₂²	#19151729	1	2	yes	—
347	C₄	—	2	4	no	—
348	C₄	—	2	4	no	—
349	C₄	—	4	8	no	—
350	C₄	#19151457	1	1	yes	characteristic
351	C₄	#19151460	1	1	yes	characteristic
352	C₄	#19151462	1	1	yes	characteristic
353	C₄	#19151465	1	1	yes	characteristic
354	C₄	#19151480	1	2	yes	—
355	C₄	#19151487	1	2	yes	—
356	C₂	#26018477	1	1	yes	characteristic
357	C₂	#26019998	1	1	yes	characteristic
358	C₂	#26020004	1	1	yes	characteristic
359	C₂	C₂²⋊C₄×#10623494	1	1	yes	characteristic
360	C₂	#26020030	1	1	yes	characteristic
361	C₂	C₄⋊C₄×#10623494	1	1	yes	characteristic
362	C₂	#26020121	1	1	yes	characteristic
363	C₂	#26020811	1	4	yes	—
364	C₂	#26020860	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂×C₄×C₈
Fitting	C₄×#95×C₈₉	C₂
socle, prime radical	C₂⁴×C₈₉	C₂²×C₄
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₄×#95	D₈₉
hypocenter	C₈₉	#10797
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₄×#95

Complements

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

N	H ≅ G/N	type
10. C₂×C₄×#10623494	C₄	semidirect
13. #19151353	C₄	semidirect
51. #95×C₈₉	C₈	semidirect
188. C₈₉	#10797	semidirect
234. #95	#10623494	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×#10623494	2
C₂×#19151353	2
C₄×#95×C₈₉	2
#10797	89