Subgroups — #36707787

Order 22784.

718 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
2	#26029099	C₂	1	2	yes	maximal, complemented
3	#26022129	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022128	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022127	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#15667953	C₂	1	2	yes	maximal, complemented
7	#26022096	C₂	1	2	yes	maximal, complemented
8	#26022035	C₂	1	1	yes	maximal, characteristic, complemented
9	#26021966	C₂	1	2	yes	maximal, complemented
10	#26021850	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2177	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₈₉	C₂²	1	2	yes	—
13	#19152070	—	2	2	no	—
14	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
15	C₂×#15667953	C₂²	1	2	yes	—
16	D₁₆×D₈₉	C₄	1	8	yes	complemented
17	C₂×#15667951	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
19	#19151708	—	2	2	no	—
20	#19151707	—	2	2	no	—
21	C₄×D₇₁₂	C₂²	1	2	yes	complemented
22	D₈×Dic₈₉	C₂²	1	2	yes	complemented
23	#19151680	C₂²	1	2	yes	complemented
24	#19151635	C₂²	1	2	yes	—
25	#19151601	—	2	2	no	—
26	#19151552	C₂²	1	1	yes	characteristic
27	#19151426	C₂²	1	1	yes	characteristic, complemented
28	#19151423	C₂²	1	2	yes	complemented
29	#19151400	C₂²	1	2	yes	complemented
30	#19151399	C₂²	1	2	yes	complemented
31	#19151398	C₂²	1	2	yes	complemented
32	#19151373	C₂²	1	1	yes	characteristic, complemented
33	C₄×#13083549	C₂²	1	1	yes	characteristic, complemented
34	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₈₉	C₂²	1	2	yes	complemented
37	#367×C₈₉	C₂²	1	1	yes	characteristic, complemented
38	#357×C₈₉	C₂²	1	2	yes	complemented
39	#346×C₈₉	C₂²	1	1	yes	characteristic, complemented
40	C₂×D₄×D₈₉	—	2	2	no	—
41	C₂×D₇₁₂	C₂³	1	2	yes	—
42	C₂×#13083559	—	2	2	no	—
43	D₈×D₈₉	—	2	4	no	—
44	D₈×D₈₉	C₂×C₄	1	8	yes	complemented
45	C₂×#13083557	D₄	1	1	yes	characteristic
46	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
47	C₄×D₃₅₆	—	2	2	no	—
48	D₄×Dic₈₉	—	2	2	no	—
49	#15667974	—	2	2	no	—
50	C₄×C₂²⋊D₈₉	—	4	2	no	—
51	#15667972	—	4	2	no	—
52	#15667970	—	2	2	no	—
53	#15667969	D₄	1	1	yes	characteristic
54	C₂×#13083551	—	4	2	no	—
55	C₂²⋊C₄×D₈₉	—	4	2	no	—
56	D₁₄₂₄	C₂×C₄	1	4	yes	complemented
57	#15667953	C₂×C₄	1	8	yes	complemented
58	#15667951	—	2	2	no	—
59	C₂×#13083549	C₂³	1	1	yes	characteristic
60	C₂×#13083549	C₂³	1	1	yes	characteristic
61	C₁₆×D₈₉	C₂×C₄	1	4	yes	complemented
62	#15667946	C₂³	1	2	yes	—
63	#15667943	—	2	2	no	—
64	#15667934	—	2	2	no	—
65	#15667933	—	2	2	no	—
66	#15667932	—	2	2	no	—
67	#15667923	D₄	1	1	yes	characteristic
68	#15667922	D₄	1	1	yes	characteristic
69	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
70	C₂×D₈×C₈₉	C₂³	1	2	yes	—
71	C₄×D₄×C₈₉	—	2	2	no	—
72	D₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
75	#107×C₈₉	C₂³	1	2	yes	—
76	#102×C₈₉	—	2	2	no	—
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₂³×D₈₉	—	4	2	no	—
79	C₂×D₃₅₆	—	2	2	no	—
80	C₂×C₂²⋊D₈₉	—	4	2	no	—
81	D₄×D₈₉	—	2	8	no	—
82	D₄×D₈₉	—	4	4	no	—
83	C₂²×Dic₈₉	—	4	2	no	—
84	C₂×C₄×D₈₉	—	4	2	no	—
85	C₂×C₄×D₈₉	—	4	2	no	—
86	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
87	D₇₁₂	—	2	2	no	—
88	D₇₁₂	C₂²×C₄	1	4	yes	—
89	#13083559	—	2	8	no	—
90	#13083557	—	2	2	no	—
91	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
92	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
93	C₈×D₈₉	C₂²×C₄	1	2	yes	—
94	C₈×D₈₉	#41	1	2	yes	—
95	C₄⋊Dic₈₉	—	2	2	no	—
96	#13083553	D₈	1	2	yes	—
97	C₂²⋊Dic₈₉	—	4	2	no	—
98	#13083551	—	4	2	no	—
99	#13083551	—	4	2	no	—
100	#13083551	—	4	2	no	—
101	#13083551	D₈	1	2	yes	—
102	C₄×Dic₈₉	—	4	2	no	—
103	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
104	#13083549	—	2	1	no	—
105	#13083549	C₂²×C₄	1	2	yes	—
106	C₂×D₄×C₈₉	—	2	2	no	—
107	C₂²×C₄×C₈₉	—	4	2	no	—
108	D₈×C₈₉	—	2	2	no	—
109	D₈×C₈₉	C₂²×C₄	1	4	yes	—
110	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₈₉	—	2	2	no	—
113	C₂²⋊C₄×C₈₉	—	4	2	no	—
114	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₈₉	—	2	1	no	—
116	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
117	C₂³×C₈₉	—	4	2	no	—
118	C₂²×D₈₉	—	4	2	no	—
119	C₂²×D₈₉	—	4	8	no	—
120	C₂²×D₈₉	—	16	2	no	—
121	C₂²×D₈₉	—	16	2	no	—
122	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
123	D₄×C₈₉	—	2	4	no	—
124	D₄×C₈₉	—	4	2	no	—
125	C₂×C₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	16	2	no	—
127	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₈₉	—	4	8	no	—
130	C₂×Dic₈₉	—	4	2	no	—
131	C₂×Dic₈₉	—	16	2	no	—
132	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
133	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
134	D₃₅₆	—	2	4	no	—
135	D₃₅₆	—	4	2	no	—
136	C₄×D₈₉	—	16	2	no	—
137	C₄×D₈₉	—	16	2	no	—
138	C₄×D₈₉	C₄×D₄	1	2	yes	—
139	C₄×D₈₉	#135	1	2	yes	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
142	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
143	#10623494	—	2	1	no	—
144	#10623494	C₄×D₄	1	2	yes	—
145	C₂²×C₈₉	—	4	4	no	—
146	C₂²×C₈₉	—	8	2	no	—
147	C₂²×C₈₉	—	8	2	no	—
148	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₈₉	—	2	2	no	—
150	C₂×D₈₉	—	4	4	no	—
151	C₂×D₈₉	—	8	2	no	—
152	C₂×D₈₉	—	8	2	no	—
153	C₂×D₈₉	—	16	8	no	—
154	C₂×D₈₉	C₄×D₈	1	2	yes	—
155	C₄×C₈₉	—	2	2	no	—
156	C₄×C₈₉	—	8	2	no	—
157	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
158	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
159	Dic₈₉	—	2	2	no	—
160	Dic₈₉	—	8	2	no	—
161	Dic₈₉	C₄×D₈	1	2	yes	—
162	#33456	—	89	1	no	maximal, Sylow
163	C₂×C₈₉	—	8	4	no	—
164	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
165	C₂×C₈₉	#3415	1	2	yes	—
166	D₈₉	—	2	2	no	—
167	D₈₉	—	8	4	no	—
168	C₂²×D₁₆	—	89	1	no	—
169	#2948	—	89	2	no	—
170	#2177	—	89	1	no	—
171	#2177	—	89	1	no	—
172	#2177	—	89	1	no	—
173	#2177	D₈₉	1	1	yes	characteristic, complemented
174	C₄×D₁₆	—	89	2	no	—
175	C₄×D₁₆	—	89	2	no	—
176	#2158	—	89	1	no	—
177	#2140	—	89	2	no	—
178	#2108	—	89	1	no	—
179	C₈₉	#33456	1	1	yes	characteristic, complemented, Sylow
180	C₂²×D₈	—	89	2	no	—
181	#518	—	178	2	no	—
182	C₂×D₁₆	—	89	2	no	—
183	C₂×D₁₆	—	89	8	no	—
184	C₂×D₁₆	—	89	1	no	—
185	C₂×D₁₆	C₂×D₈₉	1	1	yes	characteristic
186	C₂×M₃₂	—	89	1	no	—
187	C₂²×C₁₆	—	89	1	no	—
188	#442	—	178	2	no	—
189	#442	—	178	2	no	—
190	C₄×D₈	—	89	2	no	—
191	C₄×D₈	—	89	2	no	—
192	C₄×D₈	—	89	2	no	—
193	C₄×D₈	C₂×D₈₉	1	2	yes	complemented
194	#427	—	89	2	no	—
195	#418	—	178	2	no	—
196	#405	—	89	1	no	—
197	#367	—	89	1	no	—
198	#367	C₂×D₈₉	1	1	yes	characteristic, complemented
199	#366	—	89	2	no	—
200	#357	—	89	2	no	—
201	#357	—	89	2	no	—
202	#357	—	89	2	no	—
203	#357	C₂×D₈₉	1	2	yes	complemented
204	#346	—	89	1	no	—
205	#346	C₂×D₈₉	1	1	yes	characteristic, complemented
206	C₄×C₁₆	—	89	1	no	—
207	C₄×C₁₆	—	89	1	no	—
208	C₂²×D₄	—	178	2	no	—
209	C₂×D₈	—	89	8	no	—
210	C₂×D₈	—	89	2	no	—
211	C₂×D₈	—	178	4	no	—
212	C₂×D₈	—	178	2	no	—
213	C₂×D₈	C₂²×D₈₉	1	2	yes	—
214	C₂×M₁₆	—	89	1	no	—
215	C₂²×C₈	—	89	1	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	356	2	no	—
218	C₄×D₄	—	356	2	no	—
219	C₄×D₄	—	178	2	no	—
220	C₄×D₄	—	178	2	no	—
221	C₄×D₄	—	178	2	no	—
222	#117	—	89	1	no	—
223	#117	—	178	2	no	—
224	C₂×C₂²⋊C₄	—	356	2	no	—
225	C₂×C₂²⋊C₄	—	356	2	no	—
226	D₁₆	—	89	8	no	—
227	D₁₆	—	89	4	no	—
228	D₁₆	C₄×D₈₉	1	4	yes	complemented
229	M₃₂	—	178	2	no	—
230	C₂×C₁₆	—	89	1	no	—
231	C₂×C₁₆	—	89	4	no	—
232	C₂×C₁₆	—	89	1	no	—
233	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
234	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
235	#107	—	89	2	no	—
236	#107	C₂²×D₈₉	1	2	yes	—
237	#106	—	178	2	no	—
238	#102	—	2	2	no	—
239	#102	—	178	2	no	—
240	#102	—	178	2	no	—
241	#102	—	178	2	no	—
242	#97	—	89	1	no	—
243	#97	—	89	1	no	—
244	C₄×C₈	—	89	1	no	—
245	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₂⁴	—	356	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	356	4	no	—
249	C₂×D₄	—	356	2	no	—
250	C₂×D₄	—	178	8	no	—
251	C₂×D₄	—	178	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	89	1	no	—
254	C₂²×C₄	—	356	2	no	—
255	C₂²×C₄	—	356	2	no	—
256	C₂²×C₄	—	356	2	no	—
257	D₈	—	2	2	no	—
258	D₈	—	89	4	no	—
259	D₈	—	178	8	no	—
260	D₈	—	178	2	no	—
261	D₈	C₂×C₄×D₈₉	1	4	yes	—
262	M₁₆	—	178	2	no	—
263	C₂×C₈	—	89	1	no	—
264	C₂×C₈	—	89	2	no	—
265	C₂×C₈	—	89	2	no	—
266	C₂×C₈	—	89	1	no	—
267	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
268	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	89	2	no	—
271	C₄⋊C₄	—	178	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	89	2	no	—
274	C₂²⋊C₄	—	356	2	no	—
275	C₂²⋊C₄	—	356	2	no	—
276	C₂²⋊C₄	—	356	2	no	—
277	C₂²⋊C₄	—	356	2	no	—
278	C₄²	—	89	1	no	—
279	C₄²	—	356	2	no	—
280	C₄²	D₄×D₈₉	1	1	yes	characteristic
281	C₁₆	—	2	1	no	—
282	C₁₆	—	89	2	no	—
283	C₁₆	—	178	1	no	—
284	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
285	C₂³	—	4	2	no	—
286	C₂³	—	89	1	no	—
287	C₂³	—	356	8	no	—
288	C₂³	—	356	2	no	—
289	C₂³	—	1424	2	no	—
290	C₂³	—	1424	2	no	—
291	D₄	—	2	4	no	—
292	D₄	—	4	2	no	—
293	D₄	—	356	8	no	—
294	D₄	—	356	2	no	—
295	D₄	—	178	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	16	2	no	—
298	C₂×C₄	—	89	1	no	—
299	C₂×C₄	—	89	2	no	—
300	C₂×C₄	—	89	2	no	—
301	C₂×C₄	—	89	2	no	—
302	C₂×C₄	—	356	2	no	—
303	C₂×C₄	—	1424	2	no	—
304	C₂×C₄	—	1424	2	no	—
305	C₂×C₄	—	1424	2	no	—
306	C₂×C₄	D₈×D₈₉	1	2	yes	—
307	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	89	2	no	—
310	C₈	—	178	1	no	—
311	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
312	C₈	#41×D₈₉	1	1	yes	characteristic
313	C₂²	—	4	4	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	89	2	no	—
317	C₂²	—	356	4	no	—
318	C₂²	—	1424	8	no	—
319	C₂²	—	178	2	no	—
320	C₂²	—	712	2	no	—
321	C₂²	—	712	2	no	—
322	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	89	2	no	—
326	C₄	—	178	2	no	—
327	C₄	—	712	2	no	—
328	C₄	C₄×D₄×D₈₉	1	1	yes	characteristic
329	C₄	#135×D₈₉	1	1	yes	characteristic
330	C₂	—	8	4	no	—
331	C₂	—	178	2	no	—
332	C₂	—	712	4	no	—
333	C₂	C₄×D₈×D₈₉	1	1	yes	characteristic
334	C₂	#26033392	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2177	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₁₆×D₈₉	C₂
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2177	D₈₉
hypocenter	C₈₉	#33456
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

Complements

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

N	H ≅ G/N	type
2. #26029099	C₂	semidirect
3. #26022129	C₂	semidirect
4. #26022128	C₂	semidirect
5. #26022127	C₂	semidirect
6. C₄×#15667953	C₂	semidirect
7. #26022096	C₂	semidirect
8. #26022035	C₂	semidirect
9. #26021966	C₂	semidirect
10. #26021850	C₂	semidirect
11. C₈₉×#2177	C₂	semidirect
16. D₁₆×D₈₉	C₄	semidirect
21. C₄×D₇₁₂	C₂²	semidirect
22. D₈×Dic₈₉	C₂²	semidirect
23. #19151680	C₂²	semidirect
27. #19151426	C₂²	semidirect
28. #19151423	C₂²	semidirect
29. #19151400	C₂²	semidirect
30. #19151399	C₂²	semidirect
31. #19151398	C₂²	semidirect
32. #19151373	C₂²	semidirect
33. C₄×#13083549	C₂²	semidirect
34. C₁₆×Dic₈₉	C₂²	semidirect
36. C₄×D₈×C₈₉	C₂²	semidirect
37. #367×C₈₉	C₂²	semidirect
38. #357×C₈₉	C₂²	semidirect
39. #346×C₈₉	C₂²	semidirect
44. D₈×D₈₉	C₂×C₄	semidirect
56. D₁₄₂₄	C₂×C₄	semidirect
57. #15667953	C₂×C₄	semidirect
61. C₁₆×D₈₉	C₂×C₄	semidirect
72. D₁₆×C₈₉	C₂×C₄	semidirect
173. #2177	D₈₉	semidirect
179. C₈₉	#33456	semidirect
193. C₄×D₈	C₂×D₈₉	semidirect
198. #367	C₂×D₈₉	semidirect
203. #357	C₂×D₈₉	semidirect
205. #346	C₂×D₈₉	semidirect
228. D₁₆	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆×D₈₉	2
#26029099	2
#26022129	2
#26022128	2
#26022127	2
C₄×#15667953	2
#26022096	2
#26022035	2
#26021966	2
#26021850	2
C₈₉×#2177	2
#33456	89