Subgroups — #24918632

Order 10624.

334 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128236	C₂	1	1	yes	maximal, characteristic
2	#125×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128100	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128058	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128056	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128038	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128033	C₂	1	1	yes	maximal, characteristic, complemented
8	#19128021	C₂	1	1	yes	maximal, characteristic
9	#19128016	C₂	1	1	yes	maximal, characteristic
10	#19128011	C₂	1	1	yes	maximal, characteristic, complemented
11	#19128007	C₂	1	1	yes	maximal, characteristic, complemented
12	#19128004	C₂	1	1	yes	maximal, characteristic, complemented
13	#19128000	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127752	C₂	1	1	yes	maximal, characteristic, complemented
15	#499×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	#14581787	—	2	1	no	—
17	#14581784	C₂²	1	1	yes	characteristic
18	#14581783	C₂²	1	1	yes	characteristic
19	#14581783	C₂²	1	1	yes	characteristic
20	#14581782	—	2	1	no	—
21	#14581782	—	2	1	no	—
22	#14581780	C₂²	1	1	yes	characteristic
23	#14581777	—	2	1	no	—
24	#14581772	C₂²	1	1	yes	characteristic
25	#14581767	C₂²	1	1	yes	characteristic
26	#14581757	C₂²	1	1	yes	characteristic
27	#14581754	—	2	1	no	—
28	#14581754	C₂²	1	1	yes	characteristic
29	#14581750	—	2	1	no	—
30	C₄⋊C₄×D₈₃	—	2	1	no	—
31	C₄⋊C₄×D₈₃	—	2	1	no	—
32	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
34	C₄²×D₈₃	C₂²	1	1	yes	characteristic
35	#14581725	C₂²	1	1	yes	characteristic
36	#14581725	C₂²	1	1	yes	characteristic
37	#14581723	C₂²	1	1	yes	characteristic
38	#14581723	C₂²	1	1	yes	characteristic
39	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
41	#14581717	C₂²	1	1	yes	characteristic
42	#14581717	C₂²	1	1	yes	characteristic
43	#14581715	C₂²	1	1	yes	characteristic
44	#14581715	C₂²	1	1	yes	characteristic
45	#14581714	C₂²	1	1	yes	characteristic
46	#14581714	C₂²	1	1	yes	characteristic
47	#14581712	C₂²	1	1	yes	characteristic
48	#14581712	C₂²	1	1	yes	characteristic
49	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
52	#125×C₈₃	C₂²	1	1	yes	characteristic
53	#125×C₈₃	C₂²	1	1	yes	characteristic
54	#107×C₈₃	C₂²	1	1	yes	characteristic
55	#107×C₈₃	C₂²	1	1	yes	characteristic
56	#106×C₈₃	C₂²	1	1	yes	characteristic
57	#106×C₈₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
60	C₂×D₃₃₂	—	2	1	no	—
61	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₈₃	—	2	1	no	—
63	C₂×C₄×D₈₃	—	2	1	no	—
64	C₂×C₄×D₈₃	—	2	1	no	—
65	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
66	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
67	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
68	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₈₃	—	2	1	no	—
71	C₄⋊Dic₈₃	—	2	1	no	—
72	C₄⋊Dic₈₃	—	2	1	no	—
73	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
75	Dic₈₃⋊C₄	—	2	1	no	—
76	Dic₈₃⋊C₄	—	2	1	no	—
77	Dic₈₃⋊C₄	—	2	1	no	—
78	Dic₈₃⋊C₄	—	2	1	no	—
79	Dic₈₃⋊C₄	—	2	1	no	—
80	Dic₈₃⋊C₄	—	2	1	no	—
81	Dic₈₃⋊C₄	—	2	1	no	—
82	(C₂×D₈₃)⋊C₄	—	2	1	no	—
83	(C₂×D₈₃)⋊C₄	—	2	1	no	—
84	(C₂×D₈₃)⋊C₄	—	2	1	no	—
85	(C₂×D₈₃)⋊C₄	—	2	1	no	—
86	(C₂×D₈₃)⋊C₄	—	2	1	no	—
87	C₄×Dic₈₃	—	2	1	no	—
88	C₄×Dic₈₃	D₄	1	1	yes	characteristic
89	C₄×Dic₈₃	D₄	1	1	yes	characteristic
90	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
91	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₈₃	—	2	1	no	—
94	C₄⋊C₄×C₈₃	—	2	1	no	—
95	C₄⋊C₄×C₈₃	—	2	1	no	—
96	C₄⋊C₄×C₈₃	—	2	1	no	—
97	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
101	C₄²×C₈₃	C₂³	1	1	yes	characteristic
102	C₂²×D₈₃	—	2	1	no	—
103	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₈₃	—	2	1	no	—
105	C₂×C₄×C₈₃	—	2	1	no	—
106	C₂×C₄×C₈₃	—	2	1	no	—
107	C₂×C₄×C₈₃	—	2	1	no	—
108	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
111	C₂×Dic₈₃	—	2	1	no	—
112	C₂×Dic₈₃	—	2	1	no	—
113	C₂×Dic₈₃	—	2	1	no	—
114	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
117	D₃₃₂	—	2	1	no	—
118	D₃₃₂	—	2	1	no	—
119	D₃₃₂	—	4	2	no	—
120	C₄×D₈₃	—	2	2	no	—
121	C₄×D₈₃	—	2	1	no	—
122	C₄×D₈₃	—	2	1	no	—
123	C₄×D₈₃	—	4	1	no	—
124	C₄×D₈₃	—	4	2	no	—
125	C₄×D₈₃	—	4	2	no	—
126	C₄×D₈₃	—	4	2	no	—
127	Dic₁₆₆	—	2	1	no	—
128	Dic₁₆₆	—	2	1	no	—
129	C₈×C₈₃	—	2	1	no	—
130	C₈×C₈₃	—	2	1	no	—
131	C₈₃⋊C₈	—	2	1	no	—
132	C₈₃⋊C₈	—	2	1	no	—
133	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×D₈₃	—	2	1	no	—
135	C₂×D₈₃	—	2	1	no	—
136	C₂×D₈₃	—	4	1	no	—
137	C₂×D₈₃	—	4	1	no	—
138	C₂×D₈₃	—	4	1	no	—
139	C₂×D₈₃	#135	1	2	yes	—
140	C₄×C₈₃	—	2	1	no	—
141	C₄×C₈₃	—	2	1	no	—
142	C₄×C₈₃	—	4	1	no	—
143	C₄×C₈₃	—	4	1	no	—
144	C₄×C₈₃	—	4	1	no	—
145	C₄×C₈₃	—	4	1	no	—
146	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
147	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
148	Dic₈₃	—	2	1	no	—
149	Dic₈₃	—	2	1	no	—
150	Dic₈₃	—	2	1	no	—
151	Dic₈₃	—	4	1	no	—
152	Dic₈₃	—	4	1	no	—
153	Dic₈₃	—	4	1	no	—
154	C₂×C₈₃	#536	1	1	yes	characteristic
155	C₂×C₈₃	C₂×#135	1	1	yes	characteristic
156	C₂×C₈₃	#575	1	1	yes	characteristic
157	D₈₃	—	2	2	no	—
158	D₈₃	—	8	1	no	—
159	#3232	—	83	1	no	maximal, Sylow
160	C₈₃	#3232	1	1	yes	characteristic, complemented, Sylow
161	#529	—	83	1	no	—
162	C₂×#125	—	83	1	no	—
163	#499	—	83	1	no	—
164	#499	D₈₃	1	1	yes	characteristic, complemented
165	#488	—	83	1	no	—
166	#488	—	83	1	no	—
167	#484	—	83	1	no	—
168	#483	—	83	1	no	—
169	#481	—	83	1	no	—
170	#480	—	83	1	no	—
171	#479	—	83	1	no	—
172	#479	—	83	1	no	—
173	#478	—	83	1	no	—
174	#478	—	83	1	no	—
175	#431	—	83	1	no	—
176	#126	—	166	1	no	—
177	#125	—	83	1	no	—
178	#125	—	83	1	no	—
179	#125	—	83	1	no	—
180	#125	—	166	1	no	—
181	#125	—	166	1	no	—
182	#125	C₂×D₈₃	1	1	yes	characteristic
183	#125	C₂×D₈₃	1	1	yes	characteristic
184	#124	—	83	1	no	—
185	#123	—	166	1	no	—
186	C₂²⋊Q₈	—	83	1	no	—
187	#121	—	83	1	no	—
188	C₄×Q₈	—	83	1	no	—
189	C₄×D₄	—	83	1	no	—
190	C₄×D₄	—	166	1	no	—
191	#117	—	166	1	no	—
192	C₂×C₄⋊C₄	—	83	1	no	—
193	C₂×C₄⋊C₄	—	83	1	no	—
194	C₂×C₄⋊C₄	—	166	1	no	—
195	C₂×C₄⋊C₄	—	166	1	no	—
196	C₂×C₄²	—	83	1	no	—
197	#107	—	83	1	no	—
198	#107	—	83	1	no	—
199	#107	C₂×D₈₃	1	1	yes	characteristic
200	#107	C₂×D₈₃	1	1	yes	characteristic
201	#106	—	83	1	no	—
202	#106	—	83	1	no	—
203	#106	C₂×D₈₃	1	1	yes	characteristic
204	#106	C₂×D₈₃	1	1	yes	characteristic
205	C₄⋊C₈	—	83	1	no	—
206	C₄⋊C₈	—	83	1	no	—
207	#103	—	83	1	no	—
208	#103	—	83	1	no	—
209	#103	—	83	1	no	—
210	#103	—	83	1	no	—
211	#102	—	83	1	no	—
212	#102	—	83	1	no	—
213	#102	—	83	1	no	—
214	#102	—	83	1	no	—
215	C₂²⋊C₈	—	83	1	no	—
216	C₂²⋊C₈	—	83	1	no	—
217	C₄×C₈	—	83	1	no	—
218	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
219	C₂×Q₈	—	83	1	no	—
220	C₂×D₄	—	83	1	no	—
221	C₂×D₄	—	166	1	no	—
222	C₂²×C₄	—	83	1	no	—
223	C₂²×C₄	—	83	1	no	—
224	C₂²×C₄	—	83	1	no	—
225	C₂²×C₄	—	166	1	no	—
226	C₂²×C₄	—	166	1	no	—
227	C₂²×C₄	—	166	1	no	—
228	C₂×C₈	—	83	1	no	—
229	C₂×C₈	—	83	1	no	—
230	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	83	1	no	—
237	C₄⋊C₄	—	83	1	no	—
238	C₄⋊C₄	—	166	1	no	—
239	C₄⋊C₄	—	166	1	no	—
240	C₄⋊C₄	—	166	1	no	—
241	C₄⋊C₄	—	166	1	no	—
242	C₄⋊C₄	—	166	1	no	—
243	C₄⋊C₄	—	166	1	no	—
244	C₄⋊C₄	—	166	1	no	—
245	C₄⋊C₄	—	166	1	no	—
246	C₄⋊C₄	—	166	1	no	—
247	C₄⋊C₄	—	166	1	no	—
248	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
251	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
252	C₂²⋊C₄	—	166	1	no	—
253	C₂²⋊C₄	—	166	1	no	—
254	C₂²⋊C₄	—	166	1	no	—
255	C₂²⋊C₄	—	166	1	no	—
256	C₂²⋊C₄	—	166	1	no	—
257	C₄²	—	83	1	no	—
258	C₄²	—	83	1	no	—
259	C₄²	—	83	1	no	—
260	C₄²	—	166	1	no	—
261	C₄²	C₂²×D₈₃	1	1	yes	characteristic
262	C₂³	—	83	1	no	—
263	C₂³	—	166	1	no	—
264	Q₈	—	166	1	no	—
265	Q₈	—	166	1	no	—
266	D₄	—	332	2	no	—
267	D₄	—	166	1	no	—
268	D₄	—	166	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	332	1	no	—
274	C₂×C₄	—	332	2	no	—
275	C₂×C₄	—	332	2	no	—
276	C₂×C₄	—	332	2	no	—
277	C₂×C₄	—	83	1	no	—
278	C₂×C₄	—	83	1	no	—
279	C₂×C₄	—	83	1	no	—
280	C₂×C₄	—	166	2	no	—
281	C₂×C₄	—	166	1	no	—
282	C₂×C₄	—	166	1	no	—
283	C₂×C₄	—	166	1	no	—
284	C₂×C₄	—	166	1	no	—
285	C₂×C₄	—	166	1	no	—
286	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
287	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	166	1	no	—
292	C₈	—	166	1	no	—
293	C₂²	—	332	1	no	—
294	C₂²	—	332	1	no	—
295	C₂²	—	332	1	no	—
296	C₂²	—	83	2	no	—
297	C₂²	—	166	1	no	—
298	C₂²	—	166	1	no	—
299	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	332	1	no	—
307	C₄	—	332	1	no	—
308	C₄	—	332	1	no	—
309	C₄	—	166	1	no	—
310	C₄	—	166	1	no	—
311	C₄	—	166	1	no	—
312	C₄	#14581840	1	1	yes	characteristic
313	C₄	#14581840	1	1	yes	characteristic
314	C₂	—	664	1	no	—
315	C₂	—	166	2	no	—
316	C₂	#19128276	1	1	yes	characteristic
317	C₂	#135×D₈₃	1	1	yes	characteristic
318	C₂	#19128498	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#499×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#14581767	C₂²
hypercenter	#499	D₈₃
hypocenter	C₈₃	#3232
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

Complements

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

N	H ≅ G/N	type
2. #125×D₈₃	C₂	semidirect
3. #19128100	C₂	semidirect
4. #19128058	C₂	semidirect
5. #19128056	C₂	semidirect
6. #19128038	C₂	semidirect
7. #19128033	C₂	semidirect
10. #19128011	C₂	semidirect
11. #19128007	C₂	semidirect
12. #19128004	C₂	semidirect
13. #19128000	C₂	semidirect
14. #19127752	C₂	semidirect
15. #499×C₈₃	C₂	semidirect
160. C₈₃	#3232	semidirect
164. #499	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128236	2
#125×D₈₃	2
#19128100	2
#19128058	2
#19128056	2
#19128038	2
#19128033	2
#19128021	2
#19128016	2
#19128011	2
#19128007	2
#19128004	2
#19128000	2
#19127752	2
#499×C₈₃	2
#3232	83