Subgroups — #36470896

Order 22784.

420 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151418	C₂	1	1	yes	maximal, characteristic, complemented
3	#26021862	C₂	1	1	yes	maximal, characteristic
4	C₈×#13083551	C₂	1	1	yes	maximal, characteristic
5	#26018334	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018333	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#364×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
9	C₂×#15667952	C₂²	1	1	yes	characteristic
10	C₂×#15667952	C₂²	1	1	yes	characteristic
11	C₂×#15667951	—	2	1	no	—
12	M₃₂×D₈₉	—	2	2	no	—
13	M₃₂×D₈₉	—	2	2	no	—
14	C₂×C₁₆×D₈₉	—	2	1	no	—
15	C₂×#15667924	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
17	C₄×#13083551	C₄	1	1	yes	characteristic
18	#19151418	—	2	2	no	—
19	#19151377	—	2	1	no	—
20	#19151377	—	2	1	no	—
21	#19151353	C₄	1	1	yes	characteristic
22	#19151352	C₄	1	1	yes	characteristic
23	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	#364×C₈₉	—	2	2	no	—
27	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
29	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₈₉	—	2	1	no	—
31	C₂×C₈×D₈₉	—	2	1	no	—
32	C₂×C₈×D₈₉	—	2	1	no	—
33	C₂×C₈×D₈₉	—	2	1	no	—
34	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
35	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
36	C₂×C₈×D₈₉	Q₈	1	1	yes	characteristic
37	C₂×C₈×D₈₉	Q₈	1	1	yes	characteristic
38	C₂×#13083551	—	2	1	no	—
39	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
40	#15667952	—	2	1	no	—
41	#15667952	—	2	1	no	—
42	#15667952	—	2	1	no	—
43	#15667952	D₄	1	2	yes	—
44	#15667951	—	2	4	no	—
45	C₂×#13083549	—	2	1	no	—
46	C₂×#13083549	—	2	1	no	—
47	C₁₆×D₈₉	—	2	4	no	—
48	#15667924	C₈	1	2	yes	—
49	#15667924	C₈	1	2	yes	—
50	C₈×Dic₈₉	—	2	1	no	—
51	C₈×Dic₈₉	—	2	1	no	—
52	#15667919	—	2	1	no	—
53	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
56	M₃₂×C₈₉	—	2	1	no	—
57	M₃₂×C₈₉	—	2	1	no	—
58	M₃₂×C₈₉	—	2	1	no	—
59	M₃₂×C₈₉	D₄	1	2	yes	—
60	C₂×C₁₆×C₈₉	—	2	1	no	—
61	C₂×C₁₆×C₈₉	—	2	1	no	—
62	C₄×C₈×C₈₉	—	2	1	no	—
63	C₄×C₈×C₈₉	—	2	1	no	—
64	C₂³×D₈₉	M₁₆	1	1	yes	characteristic
65	C₂²×Dic₈₉	—	2	1	no	—
66	C₂²×Dic₈₉	M₁₆	1	1	yes	characteristic
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×D₈₉	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
74	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
75	C₂×#10623494	—	2	1	no	—
76	C₂×#10623494	—	2	1	no	—
77	C₂×#10623494	—	2	1	no	—
78	C₂×#10623494	—	2	1	no	—
79	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
80	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
81	C₂×#10623494	C₂×C₈	1	2	yes	—
82	C₈×D₈₉	—	2	1	no	—
83	C₈×D₈₉	—	2	1	no	—
84	C₈×D₈₉	—	2	1	no	—
85	C₈×D₈₉	—	2	1	no	—
86	C₈×D₈₉	—	2	1	no	—
87	C₈×D₈₉	—	2	1	no	—
88	C₈×D₈₉	—	2	1	no	—
89	C₈×D₈₉	—	2	1	no	—
90	#13083551	—	2	4	no	—
91	C₄×Dic₈₉	—	2	1	no	—
92	C₄×Dic₈₉	—	2	1	no	—
93	#13083549	—	2	2	no	—
94	#13083549	—	4	1	no	—
95	C₂²×C₄×C₈₉	—	2	1	no	—
96	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	—	2	1	no	—
98	C₂×C₈×C₈₉	—	2	1	no	—
99	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
100	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
104	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
105	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
106	C₄²×C₈₉	—	2	1	no	—
107	C₄²×C₈₉	—	2	1	no	—
108	C₁₆×C₈₉	—	2	2	no	—
109	C₁₆×C₈₉	—	4	1	no	—
110	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
111	C₂²×D₈₉	—	2	1	no	—
112	C₂²×D₈₉	—	2	1	no	—
113	C₂²×D₈₉	—	2	2	no	—
114	C₂²×D₈₉	—	4	1	no	—
115	C₂²×D₈₉	C₄⋊C₈	1	1	yes	characteristic
116	C₂²×D₈₉	C₄⋊C₈	1	1	yes	characteristic
117	C₂×C₄×C₈₉	—	2	2	no	—
118	C₂×C₄×C₈₉	—	4	1	no	—
119	C₂×C₄×C₈₉	—	4	1	no	—
120	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
125	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
126	C₂×Dic₈₉	—	2	1	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	2	2	no	—
129	C₂×Dic₈₉	—	4	1	no	—
130	C₂×Dic₈₉	—	4	1	no	—
131	C₂×Dic₈₉	C₄⋊C₈	1	1	yes	characteristic
132	C₂×Dic₈₉	C₄⋊C₈	1	1	yes	characteristic
133	C₄×D₈₉	—	2	1	no	—
134	C₄×D₈₉	—	2	1	no	—
135	C₄×D₈₉	—	2	1	no	—
136	C₄×D₈₉	—	2	1	no	—
137	C₄×D₈₉	—	2	1	no	—
138	C₄×D₈₉	—	2	1	no	—
139	C₄×D₈₉	—	2	1	no	—
140	C₄×D₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	2	no	—
142	C₈×C₈₉	#120	1	1	yes	characteristic
143	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
145	C₈×C₈₉	#123	1	1	yes	characteristic
146	#10623494	—	2	1	no	—
147	#10623494	—	2	1	no	—
148	#10623494	—	2	2	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
152	C₂²×C₈₉	#435	1	1	yes	characteristic
153	C₂²×C₈₉	#435	1	1	yes	characteristic
154	C₂×D₈₉	—	2	1	no	—
155	C₂×D₈₉	—	2	1	no	—
156	C₂×D₈₉	—	2	1	no	—
157	C₂×D₈₉	—	2	1	no	—
158	C₂×D₈₉	—	2	1	no	—
159	C₂×D₈₉	—	4	2	no	—
160	C₂×D₈₉	—	4	2	no	—
161	C₂×D₈₉	#364	1	2	yes	complemented
162	C₄×C₈₉	—	4	2	no	—
163	C₄×C₈₉	#385	1	1	yes	characteristic
164	C₄×C₈₉	#424	1	1	yes	characteristic
165	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
166	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
167	Dic₈₉	—	2	1	no	—
168	Dic₈₉	—	2	1	no	—
169	Dic₈₉	—	4	2	no	—
170	#12914	—	89	1	no	maximal, Sylow
171	C₂×C₈₉	—	2	1	no	—
172	C₂×C₈₉	—	2	1	no	—
173	C₂×C₈₉	#1850	1	1	yes	characteristic
174	C₂×C₈₉	C₂×#364	1	1	yes	characteristic
175	C₂×C₈₉	#2153	1	1	yes	characteristic
176	D₈₉	—	2	2	no	—
177	D₈₉	—	4	1	no	—
178	C₂²×M₃₂	—	89	1	no	—
179	C₂×#364	—	89	1	no	—
180	C₂×#364	D₈₉	1	1	yes	characteristic, complemented
181	#2112	—	89	1	no	—
182	C₈×C₂²⋊C₄	—	89	1	no	—
183	#1376	—	89	1	no	—
184	#1376	—	89	1	no	—
185	C₈₉	#12914	1	1	yes	characteristic, complemented, Sylow
186	C₂³×C₈	—	89	1	no	—
187	C₂×M₃₂	—	89	1	no	—
188	C₂×M₃₂	—	89	1	no	—
189	C₂×M₃₂	—	178	1	no	—
190	C₂×M₃₂	—	178	2	no	—
191	C₂×M₃₂	—	178	2	no	—
192	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
193	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
194	C₂²×C₁₆	—	178	1	no	—
195	C₂×C₂²⋊C₈	—	89	1	no	—
196	C₂×C₄×C₈	—	89	1	no	—
197	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
198	C₄×C₂²⋊C₄	—	89	1	no	—
199	#364	—	2	2	no	—
200	#364	—	178	2	no	—
201	C₂²⋊C₁₆	—	178	1	no	—
202	C₂²⋊C₁₆	—	178	1	no	—
203	#336	—	89	1	no	—
204	#336	—	89	1	no	—
205	C₂³×C₄	—	89	1	no	—
206	C₂²×C₈	—	89	1	no	—
207	C₂²×C₈	—	89	1	no	—
208	C₂²×C₈	—	89	1	no	—
209	C₂²×C₈	—	89	1	no	—
210	C₂²×C₈	—	89	1	no	—
211	C₂²×C₈	—	89	1	no	—
212	C₂²×C₈	—	178	1	no	—
213	C₂²×C₈	—	178	1	no	—
214	C₂²×C₈	—	178	1	no	—
215	C₂²×C₈	—	178	1	no	—
216	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
217	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
218	C₂×C₂²⋊C₄	—	178	1	no	—
219	C₂×C₄²	—	89	1	no	—
220	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
221	M₃₂	—	2	1	no	—
222	M₃₂	—	2	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	89	2	no	—
225	M₃₂	—	178	4	no	—
226	M₃₂	—	178	1	no	—
227	M₃₂	—	178	1	no	—
228	M₃₂	—	178	1	no	—
229	M₃₂	C₂²⋊D₈₉	1	2	yes	—
230	C₂×C₁₆	—	2	1	no	—
231	C₂×C₁₆	—	2	1	no	—
232	C₂×C₁₆	—	178	4	no	—
233	C₂×C₁₆	—	178	1	no	—
234	C₂×C₁₆	—	178	1	no	—
235	C₂²⋊C₈	—	89	2	no	—
236	C₂²⋊C₈	—	89	2	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	2	1	no	—
239	C₄×C₈	—	178	1	no	—
240	C₄×C₈	—	178	1	no	—
241	#95	—	178	1	no	—
242	C₂⁴	—	89	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	89	1	no	—
245	C₂²×C₄	—	89	1	no	—
246	C₂²×C₄	—	89	1	no	—
247	C₂²×C₄	—	89	1	no	—
248	C₂²×C₄	—	89	1	no	—
249	C₂²×C₄	—	178	1	no	—
250	C₂²×C₄	—	178	1	no	—
251	C₂²×C₄	—	178	1	no	—
252	C₂²×C₄	—	178	1	no	—
253	C₂²×C₄	—	178	1	no	—
254	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	89	2	no	—
258	C₂×C₈	—	89	1	no	—
259	C₂×C₈	—	89	1	no	—
260	C₂×C₈	—	178	1	no	—
261	C₂×C₈	—	178	1	no	—
262	C₂×C₈	—	178	1	no	—
263	C₂×C₈	—	178	1	no	—
264	C₂×C₈	—	178	1	no	—
265	C₂×C₈	—	178	1	no	—
266	C₂×C₈	—	178	1	no	—
267	C₂×C₈	—	178	1	no	—
268	C₂×C₈	—	178	1	no	—
269	C₂×C₈	—	178	1	no	—
270	C₂×C₈	—	178	1	no	—
271	C₂×C₈	—	178	1	no	—
272	C₂×C₈	C₈×D₈₉	1	2	yes	—
273	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
274	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
275	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
276	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
277	C₂×C₈	#13083574	1	1	yes	characteristic
278	C₂×C₈	#13083575	1	1	yes	characteristic
279	C₂²⋊C₄	—	178	4	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	178	1	no	—
283	C₄²	—	178	1	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	4	1	no	—
286	C₁₆	—	356	1	no	—
287	C₁₆	—	178	2	no	—
288	C₂³	—	89	1	no	—
289	C₂³	—	89	1	no	—
290	C₂³	—	356	1	no	—
291	C₂³	—	178	2	no	—
292	C₂³	—	178	1	no	—
293	C₂³	—	178	1	no	—
294	C₂³	M₁₆×D₈₉	1	1	yes	characteristic
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	89	1	no	—
299	C₂×C₄	—	89	1	no	—
300	C₂×C₄	—	356	1	no	—
301	C₂×C₄	—	356	1	no	—
302	C₂×C₄	—	178	2	no	—
303	C₂×C₄	—	178	1	no	—
304	C₂×C₄	—	178	1	no	—
305	C₂×C₄	—	178	1	no	—
306	C₂×C₄	—	178	1	no	—
307	C₂×C₄	—	178	1	no	—
308	C₂×C₄	—	178	1	no	—
309	C₂×C₄	—	178	1	no	—
310	C₂×C₄	—	178	1	no	—
311	C₂×C₄	—	178	1	no	—
312	C₂×C₄	—	178	1	no	—
313	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
314	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
315	C₂×C₄	#15667974	1	1	yes	characteristic
316	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
317	C₂×C₄	#15668018	1	1	yes	characteristic
318	C₂×C₄	#15668020	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	178	2	no	—
321	C₈	—	178	1	no	—
322	C₈	—	178	1	no	—
323	C₈	#15667980	1	1	yes	characteristic
324	C₈	#15667988	1	1	yes	characteristic
325	C₈	#15667990	1	1	yes	characteristic
326	C₈	#15667997	1	1	yes	characteristic
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	89	2	no	—
330	C₂²	—	356	2	no	—
331	C₂²	—	356	2	no	—
332	C₂²	—	178	1	no	—
333	C₂²	—	178	1	no	—
334	C₂²	—	178	1	no	—
335	C₂²	—	178	1	no	—
336	C₂²	—	178	1	no	—
337	C₂²	C₄⋊C₈×D₈₉	1	1	yes	characteristic
338	C₂²	#19151667	1	1	yes	characteristic
339	C₂²	#19151668	1	1	yes	characteristic
340	C₄	—	4	2	no	—
341	C₄	—	356	2	no	—
342	C₄	—	178	1	no	—
343	C₄	—	178	1	no	—
344	C₄	#19151469	1	1	yes	characteristic
345	C₄	#19151623	1	1	yes	characteristic
346	C₄	C₈×C₂²⋊D₈₉	1	1	yes	characteristic
347	C₄	#19151663	1	1	yes	characteristic
348	C₂	—	2	1	no	—
349	C₂	—	2	1	no	—
350	C₂	—	356	1	no	—
351	C₂	—	178	2	no	—
352	C₂	#26020428	1	1	yes	characteristic
353	C₂	#364×D₈₉	1	1	yes	characteristic
354	C₂	#26022016	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#364×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄⋊C₈
prime radical	C₂³×D₈₉	M₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#364	D₈₉
hypocenter	C₈₉	#12914
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. C₂×#19151418	C₂	semidirect
5. #26018334	C₂	semidirect
6. #26018333	C₂	semidirect
7. C₂×#364×C₈₉	C₂	semidirect
161. C₂×D₈₉	#364	semidirect
180. C₂×#364	D₈₉	semidirect
185. C₈₉	#12914	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₈₉	2
C₂×#19151418	2
#26021862	2
C₈×#13083551	2
#26018334	2
#26018333	2
C₂×#364×C₈₉	2
#12914	89