Subgroups — #36445482

Order 22784.

436 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029228	C₂	1	1	yes	maximal, characteristic
2	#26019939	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018840	C₂	1	1	yes	maximal, characteristic
4	#26018542	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018152	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018145	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1500	C₂	1	1	yes	maximal, characteristic
8	#19151729	C₄	1	1	yes	characteristic, complemented
9	Q₈×#10623494	C₄	1	1	yes	characteristic, complemented
10	#19151676	C₄	1	1	yes	characteristic, complemented
11	#19151670	—	2	1	no	—
12	D₄×#10623494	—	2	1	no	—
13	D₄×#10623494	C₄	1	1	yes	characteristic, complemented
14	#19151659	—	2	1	no	—
15	#19151655	C₂²	1	1	yes	characteristic
16	#19151554	—	2	1	no	—
17	C₄×#13083558	C₂²	1	1	yes	characteristic
18	C₂×#15667923	—	2	1	no	—
19	C₂×#15667923	C₂²	1	1	yes	characteristic
20	C₂×C₄×#10623494	—	2	1	no	—
21	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
22	#19151340	C₄	1	2	yes	complemented
23	#19151338	C₄	1	2	yes	complemented
24	#19151320	—	2	1	no	—
25	#19151319	—	2	1	no	—
26	#529×C₈₉	C₂²	1	1	yes	characteristic
27	#431×C₈₉	C₂²	1	1	yes	characteristic
28	#376×C₈₉	C₂²	1	1	yes	characteristic
29	#328×C₈₉	C₄	1	2	yes	—
30	#327×C₈₉	C₄	1	2	yes	—
31	C₂×#13083558	—	2	1	no	—
32	C₂×#13083558	—	2	1	no	—
33	C₂²×#10623494	—	2	1	no	—
34	C₂²×#10623494	—	2	1	no	—
35	C₂²×#10623494	D₄	1	1	yes	characteristic
36	C₂²×#10623494	D₄	1	1	yes	characteristic
37	C₄⋊#10623494	—	2	1	no	—
38	C₄⋊#10623494	—	2	1	no	—
39	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
40	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
41	C₂²⋊#10623494	—	2	1	no	—
42	C₂²⋊#10623494	—	2	1	no	—
43	C₂²⋊#10623494	—	2	1	no	—
44	#15667923	—	2	4	no	—
45	#15667923	—	2	1	no	—
46	#15667923	—	2	1	no	—
47	#15667923	—	2	1	no	—
48	#15667923	C₂×C₄	1	2	yes	—
49	#15667923	C₂×C₄	1	2	yes	—
50	C₄×#10623494	—	2	4	no	—
51	C₄×#10623494	—	2	1	no	—
52	C₄×#10623494	—	2	1	no	—
53	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
54	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
55	C₄×#10623494	D₄	1	1	yes	characteristic
56	C₄×#10623494	D₄	1	1	yes	characteristic
57	#126×C₈₉	—	2	1	no	—
58	#125×C₈₉	C₂×C₄	1	1	yes	characteristic
59	#124×C₈₉	C₂×C₄	1	1	yes	characteristic
60	#123×C₈₉	—	2	1	no	—
61	C₂²⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
62	#121×C₈₉	C₂×C₄	1	1	yes	characteristic
63	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
64	C₄×D₄×C₈₉	—	2	1	no	—
65	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
66	#117×C₈₉	—	2	1	no	—
67	C₂×C₄²×C₈₉	—	2	1	no	—
68	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₈₉	—	2	1	no	—
70	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₈₉	—	2	1	no	—
72	#95×C₈₉	—	2	1	no	—
73	#95×C₈₉	C₂×C₄	1	2	yes	—
74	#13083558	—	4	2	no	—
75	#13083558	—	4	2	no	—
76	C₂×#10623494	—	2	2	no	—
77	C₂×#10623494	—	2	2	no	—
78	C₂×#10623494	—	2	2	no	—
79	C₂×#10623494	—	2	2	no	—
80	C₂×#10623494	—	2	2	no	—
81	C₂×#10623494	—	2	1	no	—
82	C₂×#10623494	—	2	1	no	—
83	C₂×#10623494	—	2	2	no	—
84	C₂×#10623494	—	4	2	no	—
85	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×Q₈×C₈₉	C₄²	1	1	yes	characteristic
90	C₂×D₄×C₈₉	—	2	1	no	—
91	C₂×D₄×C₈₉	C₄²	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	—	2	2	no	—
93	C₂²×C₄×C₈₉	—	2	1	no	—
94	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
97	C₂×C₈×C₈₉	—	2	2	no	—
98	C₄⋊C₄×C₈₉	—	2	1	no	—
99	C₄⋊C₄×C₈₉	—	2	1	no	—
100	C₄⋊C₄×C₈₉	—	2	1	no	—
101	C₄⋊C₄×C₈₉	—	2	1	no	—
102	C₄⋊C₄×C₈₉	C₄²	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	C₄²	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₉	—	2	1	no	—
105	C₂²⋊C₄×C₈₉	—	2	1	no	—
106	C₂²⋊C₄×C₈₉	—	2	1	no	—
107	C₂²⋊C₄×C₈₉	—	2	1	no	—
108	C₂²⋊C₄×C₈₉	—	2	1	no	—
109	C₄²×C₈₉	—	2	2	no	—
110	C₄²×C₈₉	—	2	2	no	—
111	C₄²×C₈₉	—	2	1	no	—
112	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₈₉	#41	1	1	yes	characteristic
116	C₂³×C₈₉	—	2	1	no	—
117	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
118	Q₈×C₈₉	—	4	1	no	—
119	D₄×C₈₉	—	4	1	no	—
120	D₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	—	2	2	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	4	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	4	2	no	—
131	C₂×C₄×C₈₉	—	4	1	no	—
132	C₂×C₄×C₈₉	—	4	1	no	—
133	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
139	C₈×C₈₉	—	4	2	no	—
140	#10623494	—	2	4	no	—
141	#10623494	—	2	1	no	—
142	#10623494	—	2	1	no	—
143	#10623494	—	4	2	no	—
144	#10623494	—	4	1	no	—
145	#10623494	—	4	1	no	—
146	#10623494	#104	1	2	yes	—
147	#10623494	#104	1	2	yes	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	—	4	1	no	—
151	C₂²×C₈₉	—	4	1	no	—
152	C₂²×C₈₉	—	4	1	no	—
153	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₈₉	#444	1	2	yes	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	—	4	4	no	—
161	C₄×C₈₉	—	4	1	no	—
162	C₄×C₈₉	—	4	1	no	—
163	C₄×C₈₉	—	4	1	no	—
164	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
165	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
166	#8249	—	89	1	no	maximal, Sylow
167	C₂×C₈₉	—	2	2	no	—
168	C₂×C₈₉	—	8	1	no	—
169	C₂×C₈₉	#1757	1	1	yes	characteristic
170	C₂×C₈₉	C₄×#104	1	1	yes	characteristic
171	C₂×C₈₉	#1765	1	1	yes	characteristic
172	#2966	—	89	1	no	—
173	#1726	—	89	1	no	—
174	#1524	—	89	1	no	—
175	#1500	D₈₉	1	1	yes	characteristic
176	#1451	—	89	1	no	—
177	#1302	—	89	1	no	—
178	#1300	—	89	1	no	—
179	C₈₉	#8249	1	1	yes	characteristic, complemented, Sylow
180	#529	C₂×D₈₉	1	1	yes	characteristic
181	#446	—	89	1	no	—
182	C₈×Q₈	—	89	1	no	—
183	#436	—	89	1	no	—
184	#435	—	178	1	no	—
185	C₈×D₄	—	89	1	no	—
186	C₈×D₄	—	178	1	no	—
187	#433	—	178	1	no	—
188	#432	—	89	1	no	—
189	#431	C₂×D₈₉	1	1	yes	characteristic
190	#405	—	178	1	no	—
191	C₄×M₁₆	—	89	1	no	—
192	C₂×#97	—	89	1	no	—
193	C₂×#97	—	178	1	no	—
194	C₂×C₄×C₈	—	89	1	no	—
195	C₂×C₄×C₈	—	178	1	no	—
196	#376	C₂×D₈₉	1	1	yes	characteristic
197	#330	—	89	2	no	—
198	#329	—	89	2	no	—
199	#328	Dic₈₉	1	2	yes	—
200	#327	Dic₈₉	1	2	yes	—
201	#322	—	178	1	no	—
202	#322	—	178	1	no	—
203	C₂×M₁₆	—	178	1	no	—
204	C₂×M₁₆	—	178	1	no	—
205	C₂²×C₈	—	89	1	no	—
206	C₂²×C₈	—	89	1	no	—
207	C₂²×C₈	—	178	1	no	—
208	C₂²×C₈	—	178	1	no	—
209	#126	—	2	1	no	—
210	#125	C₄×D₈₉	1	1	yes	characteristic
211	#124	C₄×D₈₉	1	1	yes	characteristic
212	#123	—	2	1	no	—
213	C₂²⋊Q₈	C₂×Dic₈₉	1	1	yes	characteristic
214	#121	C₂×Dic₈₉	1	1	yes	characteristic
215	C₄×Q₈	C₄×D₈₉	1	1	yes	characteristic
216	C₄×D₄	—	2	1	no	—
217	C₄×D₄	C₄×D₈₉	1	1	yes	characteristic
218	#117	—	2	1	no	—
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	89	1	no	—
223	C₄⋊C₈	—	89	1	no	—
224	C₄⋊C₈	—	178	1	no	—
225	C₄⋊C₈	—	178	1	no	—
226	C₂²⋊C₈	—	178	1	no	—
227	C₂²⋊C₈	—	178	1	no	—
228	C₂²⋊C₈	—	178	1	no	—
229	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
230	#97	—	89	2	no	—
231	#97	—	89	2	no	—
232	#97	—	178	1	no	—
233	#97	—	178	4	no	—
234	#97	—	178	1	no	—
235	#97	—	178	1	no	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	89	1	no	—
238	C₄×C₈	—	89	1	no	—
239	C₄×C₈	—	89	1	no	—
240	C₄×C₈	—	89	1	no	—
241	C₄×C₈	—	178	1	no	—
242	C₄×C₈	—	178	4	no	—
243	C₄×C₈	—	178	1	no	—
244	#95	—	2	1	no	—
245	#95	C₂×Dic₈₉	1	2	yes	—
246	C₂×Q₈	C₄×Dic₈₉	1	1	yes	characteristic
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₄×Dic₈₉	1	1	yes	characteristic
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
252	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
253	C₂²×C₄	#13083573	1	1	yes	characteristic
254	M₁₆	—	356	2	no	—
255	M₁₆	—	356	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	89	1	no	—
258	C₂×C₈	—	89	1	no	—
259	C₂×C₈	—	89	1	no	—
260	C₂×C₈	—	89	1	no	—
261	C₂×C₈	—	356	2	no	—
262	C₂×C₈	—	178	2	no	—
263	C₂×C₈	—	178	2	no	—
264	C₂×C₈	—	178	2	no	—
265	C₂×C₈	—	178	1	no	—
266	C₂×C₈	—	178	1	no	—
267	C₂×C₈	—	178	2	no	—
268	C₂×C₈	—	178	2	no	—
269	C₂×C₈	—	178	2	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	C₄×Dic₈₉	1	1	yes	characteristic
275	C₄⋊C₄	C₄×Dic₈₉	1	1	yes	characteristic
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	1	no	—
284	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
286	C₄²	D₄×D₈₉	1	1	yes	characteristic
287	C₄²	#13083573	1	1	yes	characteristic
288	C₂³	—	2	1	no	—
289	C₂³	D₄×Dic₈₉	1	1	yes	characteristic
290	Q₈	—	4	1	no	—
291	D₄	—	4	1	no	—
292	D₄	—	4	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
306	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
307	C₂×C₄	#15667968	1	1	yes	characteristic
308	C₂×C₄	#15667972	1	1	yes	characteristic
309	C₂×C₄	#15667972	1	1	yes	characteristic
310	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
311	C₈	—	4	2	no	—
312	C₈	—	89	2	no	—
313	C₈	—	89	2	no	—
314	C₈	—	356	2	no	—
315	C₈	—	356	1	no	—
316	C₈	—	356	1	no	—
317	C₈	—	178	4	no	—
318	C₈	—	178	1	no	—
319	C₈	—	178	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	C₂²⋊C₄×Dic₈₉	1	1	yes	characteristic
326	C₂²	#19151720	1	2	yes	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	4	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	#104×D₈₉	1	1	yes	characteristic
337	C₄	#104×D₈₉	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	8	1	no	—
340	C₂	#26020054	1	1	yes	characteristic
341	C₂	#104×Dic₈₉	1	1	yes	characteristic
342	C₂	#26020099	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1500	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	#121×C₈₉	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1500	D₈₉
hypocenter	C₈₉	#8249
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1500

Complements

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

N	H ≅ G/N	type
2. #26019939	C₂	semidirect
4. #26018542	C₂	semidirect
5. #26018152	C₂	semidirect
6. #26018145	C₂	semidirect
8. #19151729	C₄	semidirect
9. Q₈×#10623494	C₄	semidirect
10. #19151676	C₄	semidirect
13. D₄×#10623494	C₄	semidirect
22. #19151340	C₄	semidirect
23. #19151338	C₄	semidirect
179. C₈₉	#8249	semidirect

Maximal subgroups

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

subgroup	index
#26029228	2
#26019939	2
#26018840	2
#26018542	2
#26018152	2
#26018145	2
C₈₉×#1500	2
#8249	89