Subgroups — #35375083

Order 21248.

560 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24918046	C₂	1	1	yes	maximal, characteristic
2	#24916559	C₂	1	2	yes	maximal
3	#24906370	C₂	1	2	yes	maximal, complemented
4	#24906361	C₂	1	1	yes	maximal, characteristic, complemented
5	#24906279	C₂	1	2	yes	maximal, complemented
6	Q₁₆×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#24906269	C₂	1	1	yes	maximal, characteristic, complemented
8	#24906225	C₂	1	1	yes	maximal, characteristic
9	#24905934	C₂	1	2	yes	maximal
10	#24905745	C₂	1	1	yes	maximal, characteristic
11	C₈₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	#19128255	C₂²	1	2	yes	—
13	#19128105	C₄	1	1	yes	characteristic
14	#19128079	C₄	1	1	yes	characteristic
15	#19128050	C₄	1	2	yes	—
16	#19127943	C₄	1	1	yes	characteristic
17	#19127941	C₄	1	2	yes	—
18	#19127939	C₄	1	1	yes	characteristic
19	#19127831	C₂²	1	2	yes	—
20	#19127829	—	2	2	no	—
21	#19127828	—	2	2	no	—
22	Q₈×C₈₃⋊C₈	C₂²	1	2	yes	—
23	#19127825	C₂²	1	2	yes	—
24	C₄×Dic₃₃₂	C₂²	1	1	yes	characteristic
25	C₄×#13050580	C₂²	1	2	yes	—
26	#19127763	—	2	2	no	—
27	#19127761	—	2	2	no	—
28	#19127757	C₂²	1	2	yes	—
29	#19127752	C₂²	1	1	yes	characteristic
30	#19127752	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
32	#19127454	C₂²	1	1	yes	characteristic
33	#19127451	C₂²	1	2	yes	—
34	#19127433	C₂²	1	2	yes	—
35	Q₈⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
36	#19127431	C₂²	1	2	yes	—
37	#19127423	C₂²	1	1	yes	characteristic
38	#19127423	C₂²	1	1	yes	characteristic
39	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
40	#446×C₈₃	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
42	#335×C₈₃	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₈₃	C₂²	1	2	yes	—
44	C₈²×C₈₃	C₂²	1	1	yes	characteristic
45	C₂×Dic₃₃₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#13050580	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
49	#14581791	C₂×C₄	1	2	yes	—
50	#14581790	C₂×C₄	1	2	yes	—
51	#14581782	—	2	2	no	—
52	#14581772	C₂×C₄	1	2	yes	—
53	#14581770	—	2	2	no	—
54	#14581768	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₆₆	—	2	2	no	—
56	C₄×Dic₁₆₆	C₂³	1	2	yes	—
57	#14581749	—	2	2	no	—
58	C₄²×D₈₃	C₂³	1	1	yes	characteristic
59	#14581726	C₂×C₄	1	1	yes	characteristic
60	#14581725	C₂×C₄	1	2	yes	—
61	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
62	C₄⋊(C₈₃⋊C₈)	C₂³	1	2	yes	—
63	Dic₈₃⋊C₈	—	2	2	no	—
64	Dic₈₃⋊C₈	—	2	1	no	—
65	Dic₈₃⋊C₈	—	2	2	no	—
66	Dic₈₃⋊C₈	D₄	1	1	yes	characteristic
67	Dic₈₃⋊C₈	D₄	1	1	yes	characteristic
68	#14581717	C₂×C₄	1	2	yes	—
69	#14581716	C₂×C₄	1	2	yes	—
70	#14581715	C₂×C₄	1	2	yes	—
71	(C₂×D₈₃)⋊C₈	—	2	2	no	—
72	(C₂×D₈₃)⋊C₈	—	2	1	no	—
73	(C₂×D₈₃)⋊C₈	D₄	1	1	yes	characteristic
74	(C₂×D₈₃)⋊C₈	D₄	1	1	yes	characteristic
75	#14581702	—	2	2	no	—
76	#14581702	—	2	2	no	—
77	C₄×C₈₃⋊C₈	—	2	2	no	—
78	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₈×Dic₈₃	—	2	2	no	—
82	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
84	C₂×Q₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₈₃	C₂³	1	2	yes	—
86	#107×C₈₃	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₈₃	—	2	2	no	—
88	C₄⋊C₈×C₈₃	C₂³	1	2	yes	—
89	#103×C₈₃	C₂×C₄	1	2	yes	—
90	#97×C₈₃	—	2	2	no	—
91	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₆₆	—	2	2	no	—
95	C₂×Dic₁₆₆	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₈₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₈₃	#41	1	1	yes	characteristic
99	Dic₃₃₂	—	2	2	no	—
100	#13050580	—	2	4	no	—
101	C₂×C₈₃⋊C₈	—	2	2	no	—
102	C₂×C₈₃⋊C₈	—	2	1	no	—
103	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₈×D₈₃	—	2	2	no	—
108	C₈×D₈₃	—	2	2	no	—
109	C₄⋊Dic₈₃	—	2	2	no	—
110	C₄⋊Dic₈₃	C₂²×C₄	1	2	yes	—
111	Dic₈₃⋊C₄	—	2	2	no	—
112	Dic₈₃⋊C₄	—	2	2	no	—
113	Dic₈₃⋊C₄	—	2	2	no	—
114	Dic₈₃⋊C₄	—	2	2	no	—
115	Dic₈₃⋊C₄	—	2	2	no	—
116	(C₂×D₈₃)⋊C₄	—	2	2	no	—
117	(C₂×D₈₃)⋊C₄	—	2	2	no	—
118	C₄×Dic₈₃	—	2	2	no	—
119	C₄×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₈₃	#41	1	1	yes	characteristic
122	C₂×Q₈×C₈₃	C₂²×C₄	1	2	yes	—
123	Q₁₆×C₈₃	—	2	2	no	—
124	C₂×C₈×C₈₃	—	2	2	no	—
125	C₂×C₈×C₈₃	—	2	1	no	—
126	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
130	C₄⋊C₄×C₈₃	—	2	2	no	—
131	C₄⋊C₄×C₈₃	C₂²×C₄	1	2	yes	—
132	C₄²×C₈₃	—	2	2	no	—
133	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
134	C₂²×D₈₃	C₄×D₄	1	1	yes	characteristic
135	Q₈×C₈₃	—	2	2	no	—
136	Q₈×C₈₃	—	2	2	no	—
137	C₂×C₄×C₈₃	—	2	2	no	—
138	C₂×C₄×C₈₃	—	2	2	no	—
139	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
142	C₂×Dic₈₃	—	2	2	no	—
143	C₂×Dic₈₃	—	2	2	no	—
144	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
146	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
147	C₄×D₈₃	—	2	1	no	—
148	C₄×D₈₃	—	2	1	no	—
149	C₄×D₈₃	—	2	1	no	—
150	C₄×D₈₃	—	2	1	no	—
151	C₄×D₈₃	—	4	1	no	—
152	Dic₁₆₆	—	2	2	no	—
153	Dic₁₆₆	—	2	2	no	—
154	Dic₁₆₆	—	8	2	no	—
155	C₈×C₈₃	—	2	1	no	—
156	C₈×C₈₃	—	2	1	no	—
157	C₈×C₈₃	—	2	1	no	—
158	C₈×C₈₃	—	4	1	no	—
159	C₈×C₈₃	—	4	2	no	—
160	C₈×C₈₃	#131	1	2	yes	—
161	C₈₃⋊C₈	—	2	1	no	—
162	C₈₃⋊C₈	—	2	1	no	—
163	C₈₃⋊C₈	—	2	1	no	—
164	C₈₃⋊C₈	—	4	1	no	—
165	C₈₃⋊C₈	—	4	2	no	—
166	C₈₃⋊C₈	#131	1	2	yes	—
167	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂×D₈₃	—	2	1	no	—
169	C₂×D₈₃	—	2	1	no	—
170	C₂×D₈₃	—	2	1	no	—
171	C₄×C₈₃	—	2	1	no	—
172	C₄×C₈₃	—	4	2	no	—
173	C₄×C₈₃	—	4	2	no	—
174	C₄×C₈₃	C₂×#131	1	1	yes	characteristic
175	C₄×C₈₃	C₂×#131	1	1	yes	characteristic
176	C₄×C₈₃	#578	1	1	yes	characteristic
177	C₄×C₈₃	#578	1	1	yes	characteristic
178	Dic₈₃	—	2	1	no	—
179	Dic₈₃	—	2	1	no	—
180	Dic₈₃	—	2	1	no	—
181	Dic₈₃	—	4	2	no	—
182	Dic₈₃	—	4	2	no	—
183	#17972	—	83	1	no	maximal, Sylow
184	C₂×C₈₃	#2931	1	1	yes	characteristic
185	C₂×C₈₃	#2950	1	1	yes	characteristic
186	C₂×C₈₃	#2957	1	1	yes	characteristic
187	D₈₃	—	4	1	no	—
188	#3170	—	83	1	no	—
189	#2985	—	83	2	no	—
190	#1593	—	83	2	no	—
191	#1591	—	83	1	no	—
192	#1591	D₈₃	1	1	yes	characteristic, complemented
193	#1579	—	83	2	no	—
194	C₈×Q₁₆	—	83	1	no	—
195	C₈×Q₁₆	—	83	1	no	—
196	#1568	—	83	1	no	—
197	#1497	—	83	2	no	—
198	#1450	—	83	1	no	—
199	C₈₃	#17972	1	1	yes	characteristic, complemented, Sylow
200	#533	—	83	2	no	—
201	#500	—	83	1	no	—
202	#494	—	83	1	no	—
203	#487	—	83	2	no	—
204	#467	—	83	1	no	—
205	#467	—	83	1	no	—
206	#467	—	83	2	no	—
207	#446	—	83	2	no	—
208	#446	—	166	2	no	—
209	#446	—	166	2	no	—
210	#446	C₂×D₈₃	1	2	yes	—
211	C₈×Q₈	—	83	2	no	—
212	C₈×Q₈	—	83	2	no	—
213	C₄×Q₁₆	—	83	2	no	—
214	C₄×Q₁₆	—	83	1	no	—
215	C₄×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
216	#433	—	166	2	no	—
217	#433	—	166	2	no	—
218	#432	—	83	2	no	—
219	#431	—	83	1	no	—
220	#431	—	83	1	no	—
221	C₂×C₄×C₈	—	83	1	no	—
222	#335	—	83	1	no	—
223	#335	C₂×D₈₃	1	1	yes	characteristic
224	#334	—	83	2	no	—
225	Q₈⋊C₈	—	83	2	no	—
226	Q₈⋊C₈	—	83	2	no	—
227	Q₈⋊C₈	—	83	2	no	—
228	Q₈⋊C₈	C₂×D₈₃	1	2	yes	—
229	#322	—	83	1	no	—
230	#322	—	83	1	no	—
231	C₈²	—	83	1	no	—
232	C₈²	C₂×D₈₃	1	1	yes	characteristic
233	C₂×Q₁₆	—	83	2	no	—
234	C₂×Q₁₆	—	83	1	no	—
235	C₂×Q₁₆	C₄×D₈₃	1	1	yes	characteristic
236	C₂²×C₈	—	83	1	no	—
237	C₂²×C₈	—	83	1	no	—
238	C₄⋊Q₈	—	83	2	no	—
239	C₄⋊Q₈	—	83	2	no	—
240	#125	—	166	2	no	—
241	C₂²⋊Q₈	—	83	2	no	—
242	C₂²⋊Q₈	—	83	2	no	—
243	C₂²⋊Q₈	—	166	2	no	—
244	C₄×Q₈	—	83	2	no	—
245	C₄×Q₈	—	166	2	no	—
246	C₄×Q₈	C₂²×D₈₃	1	2	yes	—
247	#117	—	166	2	no	—
248	C₂×C₄²	—	83	1	no	—
249	#107	—	83	1	no	—
250	#107	—	83	2	no	—
251	#107	C₄×D₈₃	1	1	yes	characteristic
252	C₄⋊C₈	—	2	2	no	—
253	C₄⋊C₈	—	83	1	no	—
254	C₄⋊C₈	—	83	1	no	—
255	C₄⋊C₈	—	83	2	no	—
256	C₄⋊C₈	—	166	2	no	—
257	C₄⋊C₈	—	166	1	no	—
258	C₄⋊C₈	—	166	2	no	—
259	C₄⋊C₈	—	166	2	no	—
260	C₄⋊C₈	C₂²×D₈₃	1	2	yes	—
261	#103	—	83	2	no	—
262	#103	—	83	2	no	—
263	#103	—	83	2	no	—
264	#103	C₄×D₈₃	1	2	yes	—
265	C₂²⋊C₈	—	83	1	no	—
266	C₂²⋊C₈	—	83	1	no	—
267	C₂²⋊C₈	—	166	1	no	—
268	C₂²⋊C₈	—	166	2	no	—
269	#97	—	2	2	no	—
270	#97	—	166	2	no	—
271	#97	—	166	2	no	—
272	C₄×C₈	—	83	1	no	—
273	C₄×C₈	—	83	1	no	—
274	C₄×C₈	—	83	1	no	—
275	C₄×C₈	—	83	1	no	—
276	C₄×C₈	—	83	1	no	—
277	C₄×C₈	—	166	2	no	—
278	C₄×C₈	—	166	2	no	—
279	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
281	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
282	C₂×Q₈	—	83	2	no	—
283	C₂×Q₈	—	166	2	no	—
284	C₂×Q₈	C₂×C₄×D₈₃	1	2	yes	—
285	C₂²×C₄	—	83	1	no	—
286	C₂²×C₄	—	83	1	no	—
287	C₂²×C₄	—	83	1	no	—
288	Q₁₆	—	2	2	no	—
289	Q₁₆	—	166	4	no	—
290	Q₁₆	—	166	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	83	1	no	—
294	C₂×C₈	—	83	1	no	—
295	C₂×C₈	—	83	1	no	—
296	C₂×C₈	—	83	1	no	—
297	C₂×C₈	—	166	2	no	—
298	C₂×C₈	—	166	2	no	—
299	C₂×C₈	—	166	1	no	—
300	C₂×C₈	—	166	2	no	—
301	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
303	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
304	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	83	2	no	—
307	C₄⋊C₄	—	166	2	no	—
308	C₄⋊C₄	—	166	2	no	—
309	C₄⋊C₄	—	166	2	no	—
310	C₄⋊C₄	—	166	2	no	—
311	C₄⋊C₄	—	166	2	no	—
312	C₄⋊C₄	—	166	2	no	—
313	C₄⋊C₄	C₂×C₄×D₈₃	1	2	yes	—
314	C₂²⋊C₄	—	166	2	no	—
315	C₂²⋊C₄	—	166	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	83	1	no	—
318	C₄²	—	83	1	no	—
319	C₄²	—	83	1	no	—
320	C₄²	—	166	2	no	—
321	C₄²	C₂³×D₈₃	1	1	yes	characteristic
322	C₂³	—	83	1	no	—
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	Q₈	—	664	2	no	—
326	Q₈	—	166	2	no	—
327	Q₈	—	166	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	332	1	no	—
331	C₂×C₄	—	83	1	no	—
332	C₂×C₄	—	83	1	no	—
333	C₂×C₄	—	83	1	no	—
334	C₂×C₄	—	166	2	no	—
335	C₂×C₄	—	166	2	no	—
336	C₂×C₄	—	166	1	no	—
337	C₂×C₄	—	166	1	no	—
338	C₂×C₄	—	166	1	no	—
339	C₂×C₄	—	166	1	no	—
340	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
342	C₂×C₄	#41×D₈₃	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	4	1	no	—
347	C₈	—	4	2	no	—
348	C₈	—	332	1	no	—
349	C₈	—	332	2	no	—
350	C₈	—	83	2	no	—
351	C₈	—	166	1	no	—
352	C₈	—	166	1	no	—
353	C₈	—	166	1	no	—
354	C₈	#14581800	1	2	yes	—
355	C₂²	—	166	1	no	—
356	C₂²	—	166	1	no	—
357	C₂²	—	166	1	no	—
358	C₂²	C₄×D₄×D₈₃	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	2	no	—
362	C₄	—	332	2	no	—
363	C₄	—	332	2	no	—
364	C₄	—	166	1	no	—
365	C₄	—	166	1	no	—
366	C₄	—	166	1	no	—
367	C₄	#131×D₈₃	1	1	yes	characteristic
368	C₄	C₂×#14581800	1	1	yes	characteristic
369	C₄	#19128514	1	1	yes	characteristic
370	C₄	#19128515	1	1	yes	characteristic
371	C₂	—	332	1	no	—
372	C₂	#24916172	1	1	yes	characteristic
373	C₂	#24916326	1	1	yes	characteristic
374	C₂	#24916375	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₄²	C₂³×D₈₃
Fitting	C₈₃×#1591	C₂
socle	C₂²×C₈₃	C₂×C₄×D₄
prime radical	C₂²×D₈₃	C₄×D₄
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	#1591	D₈₃
hypocenter	C₈₃	#17972
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
3. #24906370	C₂	semidirect
4. #24906361	C₂	semidirect
5. #24906279	C₂	semidirect
6. Q₁₆×C₈₃⋊C₈	C₂	semidirect
7. #24906269	C₂	semidirect
11. C₈₃×#1591	C₂	semidirect
192. #1591	D₈₃	semidirect
199. C₈₃	#17972	semidirect

Maximal subgroups

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

subgroup	index
#24918046	2
#24916559	2
#24906370	2
#24906361	2
#24906279	2
Q₁₆×C₈₃⋊C₈	2
#24906269	2
#24906225	2
#24905934	2
#24905745	2
C₈₃×#1591	2
#17972	83