Subgroups — #24919977

Order 10624.

348 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	#19128417	C₂	1	1	yes	maximal, characteristic
2	#19128389	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128045	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128044	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128028	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128021	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127998	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127983	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127948	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127931	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127803	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127774	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127740	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127738	C₂	1	1	yes	maximal, characteristic, complemented
15	#501×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050579	C₂²	1	1	yes	characteristic
17	C₂×#13050573	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
19	#14581787	—	2	1	no	—
20	#14581783	—	2	1	no	—
21	#14581781	—	2	1	no	—
22	#14581781	—	2	1	no	—
23	#14581777	—	2	1	no	—
24	#14581772	C₂²	1	1	yes	characteristic
25	#14581772	C₂²	1	1	yes	characteristic
26	#14581767	—	2	1	no	—
27	#14581767	C₂²	1	1	yes	characteristic
28	#14581767	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
30	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
31	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
32	#14581750	—	2	1	no	—
33	#14581750	—	2	1	no	—
34	#14581750	C₂²	1	1	yes	characteristic
35	#14581750	C₂²	1	1	yes	characteristic
36	#14581750	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
38	#14581726	C₂²	1	1	yes	characteristic
39	#14581725	C₂²	1	1	yes	characteristic
40	#14581723	C₂²	1	1	yes	characteristic
41	#14581723	C₂²	1	1	yes	characteristic
42	#14581723	C₂²	1	1	yes	characteristic
43	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
44	#14581717	C₂²	1	1	yes	characteristic
45	#14581715	C₂²	1	1	yes	characteristic
46	#14581715	C₂²	1	1	yes	characteristic
47	#14581714	C₂²	1	1	yes	characteristic
48	#14581712	C₂²	1	1	yes	characteristic
49	#14581712	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×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	#97×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
60	C₂×D₃₃₂	—	2	1	no	—
61	C₂×D₃₃₂	—	2	1	no	—
62	C₂×D₃₃₂	D₄	1	1	yes	characteristic
63	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₈₃	—	2	1	no	—
65	C₂×C₄×D₈₃	—	2	1	no	—
66	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
68	#13050579	—	2	2	no	—
69	#13050573	—	2	2	no	—
70	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
72	C₈×D₈₃	—	2	2	no	—
73	C₄⋊Dic₈₃	—	2	1	no	—
74	C₄⋊Dic₈₃	—	2	1	no	—
75	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
76	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
78	Dic₈₃⋊C₄	—	2	1	no	—
79	Dic₈₃⋊C₄	—	2	1	no	—
80	(C₂×D₈₃)⋊C₄	—	2	1	no	—
81	(C₂×D₈₃)⋊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	(C₂×D₈₃)⋊C₄	—	2	1	no	—
88	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
89	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
90	C₄×Dic₈₃	—	2	1	no	—
91	C₄×Dic₈₃	—	2	1	no	—
92	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₂×Q₈×C₈₃	—	2	1	no	—
95	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₈₃	—	2	1	no	—
98	C₄⋊C₄×C₈₃	—	2	1	no	—
99	C₄⋊C₄×C₈₃	—	2	1	no	—
100	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₄²×C₈₃	C₂³	1	1	yes	characteristic
105	C₂²×D₈₃	—	2	1	no	—
106	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
108	Q₈×C₈₃	—	8	1	no	—
109	C₂×C₄×C₈₃	—	2	1	no	—
110	C₂×C₄×C₈₃	—	2	1	no	—
111	C₂×C₄×C₈₃	—	2	1	no	—
112	C₂×C₄×C₈₃	—	2	1	no	—
113	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₈₃	—	2	1	no	—
117	C₂×Dic₈₃	—	2	1	no	—
118	C₂×Dic₈₃	—	2	1	no	—
119	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
121	D₃₃₂	—	2	1	no	—
122	D₃₃₂	—	2	1	no	—
123	D₃₃₂	—	4	1	no	—
124	D₃₃₂	—	8	1	no	—
125	D₃₃₂	—	8	1	no	—
126	C₄×D₈₃	—	2	1	no	—
127	C₄×D₈₃	—	2	1	no	—
128	C₄×D₈₃	—	2	1	no	—
129	C₄×D₈₃	—	2	1	no	—
130	C₄×D₈₃	—	4	2	no	—
131	C₄×D₈₃	—	4	2	no	—
132	Dic₁₆₆	—	2	1	no	—
133	Dic₁₆₆	—	2	1	no	—
134	C₈×C₈₃	—	2	1	no	—
135	C₈×C₈₃	#41	1	2	yes	—
136	C₈₃⋊C₈	—	2	1	no	—
137	C₈₃⋊C₈	—	2	1	no	—
138	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₈₃	—	2	1	no	—
140	C₂×D₈₃	—	2	1	no	—
141	C₂×D₈₃	—	2	1	no	—
142	C₂×D₈₃	—	2	1	no	—
143	C₂×D₈₃	—	2	1	no	—
144	C₂×D₈₃	—	2	1	no	—
145	C₂×D₈₃	—	4	1	no	—
146	C₂×D₈₃	—	4	1	no	—
147	C₂×D₈₃	—	4	1	no	—
148	C₄×C₈₃	—	2	1	no	—
149	C₄×C₈₃	—	2	1	no	—
150	C₄×C₈₃	—	4	1	no	—
151	C₄×C₈₃	—	4	1	no	—
152	C₄×C₈₃	—	4	1	no	—
153	C₄×C₈₃	—	4	1	no	—
154	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
155	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
156	Dic₈₃	—	2	1	no	—
157	Dic₈₃	—	2	1	no	—
158	Dic₈₃	—	4	1	no	—
159	Dic₈₃	—	4	1	no	—
160	Dic₈₃	—	4	1	no	—
161	C₂×C₈₃	#547	1	1	yes	characteristic
162	C₂×C₈₃	#575	1	1	yes	characteristic
163	C₂×C₈₃	#577	1	1	yes	characteristic
164	D₈₃	—	4	1	no	—
165	D₈₃	—	4	1	no	—
166	D₈₃	—	8	1	no	—
167	#3348	—	83	1	no	maximal, Sylow
168	C₈₃	#3348	1	1	yes	characteristic, complemented, Sylow
169	#555	—	83	1	no	—
170	#552	—	83	1	no	—
171	#501	D₈₃	1	1	yes	characteristic, complemented
172	#485	—	83	1	no	—
173	#485	—	83	1	no	—
174	#482	—	83	1	no	—
175	#481	—	83	1	no	—
176	#478	—	83	1	no	—
177	#475	—	83	1	no	—
178	#468	—	83	1	no	—
179	#465	—	83	1	no	—
180	#440	—	83	1	no	—
181	#435	—	83	1	no	—
182	#428	—	83	1	no	—
183	#427	—	83	1	no	—
184	C₂×SD₁₆	—	83	1	no	—
185	C₂×M₁₆	—	83	1	no	—
186	C₂²×C₈	—	83	1	no	—
187	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
188	#126	—	166	1	no	—
189	#125	—	166	1	no	—
190	#125	C₂×D₈₃	1	1	yes	characteristic
191	#124	—	166	1	no	—
192	#124	—	166	1	no	—
193	#123	—	166	1	no	—
194	C₂²⋊Q₈	—	83	1	no	—
195	C₂²⋊Q₈	—	83	1	no	—
196	#121	—	83	1	no	—
197	#121	—	83	1	no	—
198	#121	—	166	1	no	—
199	C₄×Q₈	—	83	1	no	—
200	C₄×D₄	—	83	1	no	—
201	C₄×D₄	—	83	1	no	—
202	#117	—	83	1	no	—
203	#117	—	83	1	no	—
204	#117	—	83	1	no	—
205	#117	—	166	1	no	—
206	#117	—	166	1	no	—
207	C₂×C₄⋊C₄	—	83	1	no	—
208	#107	—	83	1	no	—
209	#107	—	83	1	no	—
210	#107	C₂×D₈₃	1	1	yes	characteristic
211	#107	C₂×D₈₃	1	1	yes	characteristic
212	#106	—	83	1	no	—
213	#106	—	83	1	no	—
214	#106	—	83	1	no	—
215	#106	C₂×D₈₃	1	1	yes	characteristic
216	#106	C₂×D₈₃	1	1	yes	characteristic
217	C₄⋊C₈	—	83	1	no	—
218	#103	—	83	1	no	—
219	#103	—	83	1	no	—
220	#103	—	83	1	no	—
221	#102	—	83	1	no	—
222	#102	—	83	1	no	—
223	#102	—	83	1	no	—
224	C₂²⋊C₈	—	83	1	no	—
225	C₂²⋊C₈	—	83	1	no	—
226	#97	C₂×D₈₃	1	1	yes	characteristic
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	83	1	no	—
229	C₂×D₄	—	83	1	no	—
230	C₂×D₄	—	83	1	no	—
231	C₂×D₄	—	166	1	no	—
232	C₂×D₄	—	166	1	no	—
233	C₂²×C₄	—	83	1	no	—
234	C₂²×C₄	—	83	1	no	—
235	C₂²×C₄	—	166	1	no	—
236	C₂²×C₄	—	166	1	no	—
237	SD₁₆	—	166	2	no	—
238	M₁₆	—	166	2	no	—
239	C₂×C₈	—	83	1	no	—
240	C₂×C₈	—	83	1	no	—
241	C₂×C₈	—	166	2	no	—
242	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	83	1	no	—
248	C₄⋊C₄	—	83	1	no	—
249	C₄⋊C₄	—	83	1	no	—
250	C₄⋊C₄	—	166	1	no	—
251	C₄⋊C₄	—	166	1	no	—
252	C₄⋊C₄	—	166	1	no	—
253	C₄⋊C₄	—	166	1	no	—
254	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
258	C₂²⋊C₄	—	83	1	no	—
259	C₂²⋊C₄	—	83	1	no	—
260	C₂²⋊C₄	—	166	1	no	—
261	C₂²⋊C₄	—	166	1	no	—
262	C₂²⋊C₄	—	166	1	no	—
263	C₂²⋊C₄	—	166	1	no	—
264	C₂²⋊C₄	—	166	1	no	—
265	C₂²⋊C₄	—	166	1	no	—
266	C₂²⋊C₄	—	166	1	no	—
267	C₂²⋊C₄	—	166	1	no	—
268	C₄²	—	166	1	no	—
269	C₄²	—	166	1	no	—
270	C₄²	—	166	1	no	—
271	C₄²	—	166	1	no	—
272	C₄²	C₂²×D₈₃	1	1	yes	characteristic
273	C₂³	—	83	1	no	—
274	C₂³	—	83	1	no	—
275	C₂³	—	166	1	no	—
276	Q₈	—	8	1	no	—
277	Q₈	—	166	1	no	—
278	Q₈	—	166	1	no	—
279	D₄	—	332	1	no	—
280	D₄	—	664	1	no	—
281	D₄	—	664	1	no	—
282	D₄	—	166	1	no	—
283	D₄	—	166	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	332	2	no	—
289	C₂×C₄	—	332	2	no	—
290	C₂×C₄	—	83	1	no	—
291	C₂×C₄	—	83	1	no	—
292	C₂×C₄	—	166	1	no	—
293	C₂×C₄	—	166	1	no	—
294	C₂×C₄	—	166	1	no	—
295	C₂×C₄	—	166	1	no	—
296	C₂×C₄	—	166	1	no	—
297	C₂×C₄	—	166	1	no	—
298	C₂×C₄	—	166	1	no	—
299	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
300	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
301	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	166	1	no	—
304	C₈	—	166	1	no	—
305	C₈	#13050591	1	2	yes	—
306	C₂²	—	332	1	no	—
307	C₂²	—	332	1	no	—
308	C₂²	—	332	1	no	—
309	C₂²	—	166	1	no	—
310	C₂²	—	166	1	no	—
311	C₂²	—	166	1	no	—
312	C₂²	—	166	1	no	—
313	C₂²	—	166	1	no	—
314	C₂²	—	166	1	no	—
315	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	332	1	no	—
323	C₄	—	332	1	no	—
324	C₄	—	332	1	no	—
325	C₄	—	166	1	no	—
326	C₄	—	166	1	no	—
327	C₄	C₂×#13050591	1	1	yes	characteristic
328	C₄	#14581840	1	1	yes	characteristic
329	C₂	—	332	1	no	—
330	C₂	—	332	1	no	—
331	C₂	—	664	1	no	—
332	C₂	#19128354	1	1	yes	characteristic
333	C₂	#19128500	1	1	yes	characteristic
334	C₂	#19128511	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#501×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128417	C₂
hypercenter	#501	D₈₃
hypocenter	C₈₃	#3348
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
2. #19128389	C₂	semidirect
3. #19128045	C₂	semidirect
4. #19128044	C₂	semidirect
5. #19128028	C₂	semidirect
6. #19128021	C₂	semidirect
7. #19127998	C₂	semidirect
8. #19127983	C₂	semidirect
9. #19127948	C₂	semidirect
10. #19127931	C₂	semidirect
11. #19127803	C₂	semidirect
12. #19127774	C₂	semidirect
13. #19127740	C₂	semidirect
14. #19127738	C₂	semidirect
15. #501×C₈₃	C₂	semidirect
168. C₈₃	#3348	semidirect
171. #501	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128417	2
#19128389	2
#19128045	2
#19128044	2
#19128028	2
#19128021	2
#19127998	2
#19127983	2
#19127948	2
#19127931	2
#19127803	2
#19127774	2
#19127740	2
#19127738	2
#501×C₈₃	2
#3348	83