Subgroups — #24917569

Order 10624.

392 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128230	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#13050589	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128040	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128023	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128009	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128001	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127920	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127860	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127843	C₂	1	1	yes	maximal, characteristic
10	#19127813	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127805	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127802	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127798	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127757	C₂	1	1	yes	maximal, characteristic, complemented
15	#463×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050589	C₂²	1	1	yes	characteristic
17	C₂×#13050580	C₂²	1	1	yes	characteristic
18	C₂×#13050578	C₂²	1	1	yes	characteristic
19	#14581785	—	2	1	no	—
20	#14581784	C₂²	1	1	yes	characteristic
21	#14581774	—	2	1	no	—
22	#14581768	C₂²	1	1	yes	characteristic
23	#14581763	C₂²	1	1	yes	characteristic
24	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
26	#14581757	C₂²	1	1	yes	characteristic
27	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
28	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊D₈₃	—	2	1	no	—
30	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
31	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
32	#14581750	C₂²	1	1	yes	characteristic
33	#14581748	—	2	1	no	—
34	#14581748	—	2	1	no	—
35	C₂×C₄×Dic₈₃	—	2	1	no	—
36	C₄²×D₈₃	C₂²	1	1	yes	characteristic
37	#14581726	C₂²	1	1	yes	characteristic
38	#14581725	C₂²	1	1	yes	characteristic
39	#14581724	C₂²	1	1	yes	characteristic
40	#14581723	C₂²	1	1	yes	characteristic
41	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
42	#14581717	C₂²	1	1	yes	characteristic
43	#14581717	C₂²	1	1	yes	characteristic
44	#14581716	C₂²	1	1	yes	characteristic
45	#14581713	C₂²	1	1	yes	characteristic
46	#14581713	C₂²	1	1	yes	characteristic
47	#14581713	C₂²	1	1	yes	characteristic
48	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	#14581702	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₈₃	C₂²	1	1	yes	characteristic
54	#124×C₈₃	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	#103×C₈₃	C₂²	1	1	yes	characteristic
58	#102×C₈₃	C₂²	1	1	yes	characteristic
59	#13050589	—	2	2	no	—
60	#13050589	—	2	2	no	—
61	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₈₃	—	2	1	no	—
63	C₂×C₂²⋊D₈₃	—	2	1	no	—
64	C₂²×Dic₈₃	—	2	1	no	—
65	C₂²×Dic₈₃	—	2	1	no	—
66	C₂²×Dic₈₃	—	2	1	no	—
67	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
68	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
69	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
70	#13050580	—	2	2	no	—
71	#13050578	—	2	2	no	—
72	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₃	—	2	1	no	—
75	C₄⋊Dic₈₃	C₂³	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	Dic₈₃⋊C₄	—	2	1	no	—
81	Dic₈₃⋊C₄	—	2	1	no	—
82	Dic₈₃⋊C₄	—	2	1	no	—
83	C₂²⋊Dic₈₃	—	2	1	no	—
84	C₂²⋊Dic₈₃	—	2	1	no	—
85	C₂²⋊Dic₈₃	—	2	1	no	—
86	(C₂×D₈₃)⋊C₄	—	2	1	no	—
87	(C₂×D₈₃)⋊C₄	—	2	1	no	—
88	C₄×Dic₈₃	—	2	1	no	—
89	C₄×Dic₈₃	—	2	1	no	—
90	C₄×Dic₈₃	—	2	1	no	—
91	C₄×Dic₈₃	—	2	1	no	—
92	C₄×Dic₈₃	D₄	1	1	yes	characteristic
93	C₄×Dic₈₃	D₄	1	1	yes	characteristic
94	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₃	—	2	1	no	—
99	SD₁₆×C₈₃	—	2	2	no	—
100	D₈×C₈₃	—	2	2	no	—
101	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₃	—	2	1	no	—
105	C₂²⋊C₄×C₈₃	—	2	1	no	—
106	C₂²⋊C₄×C₈₃	—	2	1	no	—
107	C₄²×C₈₃	C₂³	1	1	yes	characteristic
108	C₂³×C₈₃	—	2	1	no	—
109	C₂³×C₈₃	—	2	1	no	—
110	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₈₃	—	2	1	no	—
112	Q₈×C₈₃	—	2	1	no	—
113	D₄×C₈₃	—	2	1	no	—
114	D₄×C₈₃	—	2	1	no	—
115	D₄×C₈₃	—	2	1	no	—
116	D₄×C₈₃	#41	1	2	yes	—
117	C₂×C₄×C₈₃	—	2	1	no	—
118	C₂×C₄×C₈₃	—	2	1	no	—
119	C₂×C₄×C₈₃	—	4	2	no	—
120	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₈₃	—	2	4	no	—
124	C₂²⋊D₈₃	—	4	2	no	—
125	C₂×Dic₈₃	—	2	4	no	—
126	C₂×Dic₈₃	—	2	1	no	—
127	C₂×Dic₈₃	—	2	1	no	—
128	C₂×Dic₈₃	—	2	1	no	—
129	C₂×Dic₈₃	—	2	1	no	—
130	C₂×Dic₈₃	—	4	2	no	—
131	C₂×Dic₈₃	—	4	2	no	—
132	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
135	C₄×D₈₃	—	2	1	no	—
136	C₄×D₈₃	—	2	1	no	—
137	C₄×D₈₃	—	4	1	no	—
138	C₄×D₈₃	—	4	1	no	—
139	Dic₁₆₆	—	2	1	no	—
140	Dic₁₆₆	#41	1	2	yes	—
141	C₈×C₈₃	—	2	1	no	—
142	C₈×C₈₃	—	2	1	no	—
143	C₈₃⋊C₈	—	2	1	no	—
144	C₈₃⋊C₈	—	2	1	no	—
145	C₂²×C₈₃	—	2	2	no	—
146	C₂²×C₈₃	—	4	1	no	—
147	C₂²×C₈₃	—	4	1	no	—
148	C₂²×C₈₃	—	4	1	no	—
149	C₂²×C₈₃	—	4	1	no	—
150	C₂²×C₈₃	—	4	1	no	—
151	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₈₃	—	2	1	no	—
153	C₂×D₈₃	—	2	1	no	—
154	C₂×D₈₃	—	2	1	no	—
155	C₄×C₈₃	—	2	1	no	—
156	C₄×C₈₃	—	2	1	no	—
157	C₄×C₈₃	—	4	1	no	—
158	C₄×C₈₃	—	4	1	no	—
159	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
160	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
161	Dic₈₃	—	2	2	no	—
162	Dic₈₃	—	2	1	no	—
163	Dic₈₃	—	2	1	no	—
164	Dic₈₃	—	2	1	no	—
165	Dic₈₃	—	4	1	no	—
166	Dic₈₃	—	4	1	no	—
167	Dic₈₃	—	4	1	no	—
168	C₂×C₈₃	—	4	2	no	—
169	C₂×C₈₃	—	8	1	no	—
170	C₂×C₈₃	#525	1	1	yes	characteristic
171	C₂×C₈₃	#575	1	1	yes	characteristic
172	C₂×C₈₃	#575	1	1	yes	characteristic
173	D₈₃	—	4	1	no	—
174	#3111	—	83	1	no	maximal, Sylow
175	C₈₃	#3111	1	1	yes	characteristic, complemented, Sylow
176	#529	—	83	1	no	—
177	C₄×#41	—	83	1	no	—
178	#485	—	83	1	no	—
179	#482	—	83	1	no	—
180	#479	—	83	1	no	—
181	#478	—	83	1	no	—
182	#464	—	83	1	no	—
183	#463	D₈₃	1	1	yes	characteristic, complemented
184	#452	—	83	1	no	—
185	#449	—	83	1	no	—
186	#442	—	83	1	no	—
187	#441	—	83	1	no	—
188	#440	—	83	1	no	—
189	#440	—	83	1	no	—
190	#432	—	83	1	no	—
191	C₂×#41	—	83	1	no	—
192	C₂×Q₁₆	—	83	1	no	—
193	C₂×SD₁₆	—	83	1	no	—
194	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
195	C₂×D₈	C₂×D₈₃	1	1	yes	characteristic
196	#126	—	166	1	no	—
197	#125	—	83	1	no	—
198	#124	C₂×D₈₃	1	1	yes	characteristic
199	#123	—	166	1	no	—
200	C₂²⋊Q₈	—	83	1	no	—
201	#121	—	83	1	no	—
202	C₄×Q₈	—	83	1	no	—
203	C₄×Q₈	—	83	1	no	—
204	C₄×Q₈	—	83	1	no	—
205	C₄×D₄	—	83	1	no	—
206	C₄×D₄	—	83	1	no	—
207	C₄×D₄	—	166	1	no	—
208	C₄×D₄	—	166	1	no	—
209	C₄×D₄	—	166	1	no	—
210	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
211	#117	—	83	1	no	—
212	#117	—	166	1	no	—
213	#117	—	166	1	no	—
214	C₂×C₄²	—	83	1	no	—
215	C₂×C₄²	—	166	1	no	—
216	#107	—	83	1	no	—
217	#107	—	83	1	no	—
218	#106	—	83	1	no	—
219	#106	—	83	1	no	—
220	C₄⋊C₈	—	83	1	no	—
221	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
222	#103	—	83	1	no	—
223	#103	—	83	1	no	—
224	#103	—	83	1	no	—
225	#103	C₂×D₈₃	1	1	yes	characteristic
226	#102	—	83	1	no	—
227	#102	—	83	1	no	—
228	#102	—	83	1	no	—
229	#102	C₂×D₈₃	1	1	yes	characteristic
230	C₂²⋊C₈	—	83	1	no	—
231	C₂²⋊C₈	—	83	1	no	—
232	#97	—	83	1	no	—
233	#97	—	83	1	no	—
234	#41	—	166	2	no	—
235	#41	—	166	2	no	—
236	C₂×Q₈	—	83	1	no	—
237	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
238	C₂×D₄	—	166	1	no	—
239	C₂×D₄	—	166	1	no	—
240	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
241	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	83	1	no	—
244	C₂²×C₄	—	83	1	no	—
245	C₂²×C₄	—	83	1	no	—
246	C₂²×C₄	—	166	1	no	—
247	C₂²×C₄	—	166	1	no	—
248	C₂²×C₄	—	166	1	no	—
249	Q₁₆	—	166	2	no	—
250	SD₁₆	—	2	2	no	—
251	SD₁₆	—	166	2	no	—
252	D₈	—	2	2	no	—
253	C₂×C₈	—	83	1	no	—
254	C₂×C₈	—	83	1	no	—
255	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
257	C₄⋊C₄	—	83	1	no	—
258	C₄⋊C₄	—	83	1	no	—
259	C₄⋊C₄	—	83	1	no	—
260	C₄⋊C₄	—	166	1	no	—
261	C₄⋊C₄	—	166	1	no	—
262	C₄⋊C₄	—	166	1	no	—
263	C₄⋊C₄	—	166	1	no	—
264	C₄⋊C₄	—	166	1	no	—
265	C₄⋊C₄	—	166	1	no	—
266	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	166	1	no	—
271	C₂²⋊C₄	—	166	1	no	—
272	C₂²⋊C₄	—	166	1	no	—
273	C₂²⋊C₄	—	166	1	no	—
274	C₂²⋊C₄	—	166	1	no	—
275	C₄²	—	83	1	no	—
276	C₄²	—	83	1	no	—
277	C₄²	—	83	1	no	—
278	C₄²	—	166	1	no	—
279	C₄²	—	166	1	no	—
280	C₄²	—	166	1	no	—
281	C₄²	—	166	1	no	—
282	C₄²	C₂²×D₈₃	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	83	1	no	—
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	83	2	no	—
289	Q₈	—	166	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	332	2	no	—
294	D₄	—	166	4	no	—
295	D₄	#13050589	1	2	yes	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	332	1	no	—
300	C₂×C₄	—	332	1	no	—
301	C₂×C₄	—	332	2	no	—
302	C₂×C₄	—	332	2	no	—
303	C₂×C₄	—	83	1	no	—
304	C₂×C₄	—	83	1	no	—
305	C₂×C₄	—	83	1	no	—
306	C₂×C₄	—	166	1	no	—
307	C₂×C₄	—	166	1	no	—
308	C₂×C₄	—	166	1	no	—
309	C₂×C₄	—	166	4	no	—
310	C₂×C₄	—	166	1	no	—
311	C₂×C₄	—	166	1	no	—
312	C₂×C₄	—	166	1	no	—
313	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
314	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	166	1	no	—
319	C₈	—	166	1	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	166	1	no	—
327	C₂²	—	166	1	no	—
328	C₂²	—	166	1	no	—
329	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	332	1	no	—
335	C₄	—	332	1	no	—
336	C₄	—	332	1	no	—
337	C₄	—	166	1	no	—
338	C₄	—	166	2	no	—
339	C₄	—	166	1	no	—
340	C₄	—	166	1	no	—
341	C₄	#41×D₈₃	1	1	yes	characteristic
342	C₄	C₂×#13050589	1	1	yes	characteristic
343	C₂	—	4	2	no	—
344	C₂	—	8	1	no	—
345	C₂	—	332	1	no	—
346	C₂	#19128211	1	1	yes	characteristic
347	C₂	#19128497	1	1	yes	characteristic
348	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	#463×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19127843	C₂
hypercenter	#463	D₈₃
hypocenter	C₈₃	#3111
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #19128230	C₂	semidirect
2. C₄×#13050589	C₂	semidirect
3. #19128040	C₂	semidirect
4. #19128023	C₂	semidirect
5. #19128009	C₂	semidirect
6. #19128001	C₂	semidirect
7. #19127920	C₂	semidirect
8. #19127860	C₂	semidirect
10. #19127813	C₂	semidirect
11. #19127805	C₂	semidirect
12. #19127802	C₂	semidirect
13. #19127798	C₂	semidirect
14. #19127757	C₂	semidirect
15. #463×C₈₃	C₂	semidirect
175. C₈₃	#3111	semidirect
183. #463	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128230	2
C₄×#13050589	2
#19128040	2
#19128023	2
#19128009	2
#19128001	2
#19127920	2
#19127860	2
#19127843	2
#19127813	2
#19127805	2
#19127802	2
#19127798	2
#19127757	2
#463×C₈₃	2
#3111	83