Subgroups — #35387047

Order 21248.

586 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24917974	C₂	1	1	yes	maximal, characteristic
2	#24917504	C₂	1	2	yes	maximal
3	#446×D₈₃	C₂	1	1	yes	maximal, characteristic
4	#24906988	C₂	1	1	yes	maximal, characteristic, complemented
5	#24906982	C₂	1	1	yes	maximal, characteristic, complemented
6	#24906981	C₂	1	1	yes	maximal, characteristic, complemented
7	#24906972	C₂	1	2	yes	maximal, complemented
8	#24906971	C₂	1	2	yes	maximal, complemented
9	#24906208	C₂	1	1	yes	maximal, characteristic
10	#24906117	C₂	1	2	yes	maximal
11	C₈₃×#1696	C₂	1	1	yes	maximal, characteristic, complemented
12	#19128256	C₂²	1	2	yes	—
13	C₄×Q₈×D₈₃	C₂²	1	1	yes	characteristic
14	#19128106	C₂²	1	1	yes	characteristic
15	#19128102	—	2	1	no	—
16	#19128102	—	2	1	no	—
17	#19128097	C₂²	1	2	yes	—
18	#19128038	—	2	2	no	—
19	#19127993	C₂²	1	2	yes	—
20	#19127912	C₂²	1	2	yes	—
21	#19127911	C₂²	1	2	yes	—
22	#19127858	—	2	2	no	—
23	#19127832	C₂²	1	1	yes	characteristic
24	#19127831	C₂²	1	1	yes	characteristic
25	#19127830	—	2	1	no	—
26	#19127829	—	2	1	no	—
27	#19127828	C₂²	1	1	yes	characteristic
28	#19127795	—	2	2	no	—
29	#19127793	—	2	2	no	—
30	#19127753	C₂²	1	2	yes	—
31	#19127738	—	2	1	no	—
32	#19127738	—	2	1	no	—
33	C₄⋊C₈×D₈₃	—	2	1	no	—
34	C₄⋊C₈×D₈₃	C₂²	1	1	yes	characteristic
35	#97×D₈₃	—	2	1	no	—
36	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
37	#19127453	C₂²	1	2	yes	—
38	#19127452	C₂²	1	1	yes	characteristic
39	#19127450	C₂²	1	2	yes	—
40	#19127448	C₂²	1	2	yes	—
41	#19127446	C₂²	1	2	yes	—
42	#500×C₈₃	C₂²	1	1	yes	characteristic
43	#477×C₈₃	C₂²	1	2	yes	—
44	#446×C₈₃	C₂²	1	1	yes	characteristic
45	#334×C₈₃	C₂²	1	1	yes	characteristic
46	#333×C₈₃	C₂²	1	2	yes	—
47	C₂×Q₈×D₈₃	D₄	1	1	yes	characteristic
48	C₂×Dic₃₃₂	—	2	2	no	—
49	C₂×#13050580	—	2	2	no	—
50	C₂×C₈×D₈₃	—	2	1	no	—
51	C₂×C₈×D₈₃	D₄	1	1	yes	characteristic
52	C₂×C₈×D₈₃	D₄	1	1	yes	characteristic
53	#14581792	C₂³	1	2	yes	—
54	#14581783	—	2	2	no	—
55	#14581772	—	2	2	no	—
56	#14581768	—	2	2	no	—
57	Q₈×Dic₈₃	—	2	2	no	—
58	Q₈×Dic₈₃	D₄	1	1	yes	characteristic
59	C₄×Dic₁₆₆	—	2	1	no	—
60	C₄×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	#14581757	—	2	1	no	—
62	#14581757	—	2	2	no	—
63	#14581757	D₄	1	1	yes	characteristic
64	#14581750	—	2	2	no	—
65	C₄⋊C₄×D₈₃	—	2	1	no	—
66	C₄⋊C₄×D₈₃	D₄	1	1	yes	characteristic
67	C₄²×D₈₃	—	2	1	no	—
68	C₄²×D₈₃	C₂³	1	1	yes	characteristic
69	#14581726	—	2	1	no	—
70	#14581726	—	2	1	no	—
71	#14581725	—	2	2	no	—
72	#14581723	—	2	2	no	—
73	#14581723	—	2	2	no	—
74	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
75	C₄⋊(C₈₃⋊C₈)	C₂³	1	1	yes	characteristic
76	C₄⋊(C₈₃⋊C₈)	C₂³	1	2	yes	—
77	Dic₈₃⋊C₈	—	2	1	no	—
78	Dic₈₃⋊C₈	—	2	1	no	—
79	Dic₈₃⋊C₈	—	2	1	no	—
80	#14581717	—	2	2	no	—
81	#14581715	—	2	2	no	—
82	#14581715	—	2	2	no	—
83	(C₂×D₈₃)⋊C₈	—	2	2	no	—
84	#14581703	—	2	1	no	—
85	#14581702	—	2	1	no	—
86	#14581702	—	2	1	no	—
87	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
88	C₈×Dic₈₃	—	2	2	no	—
89	C₈×Dic₈₃	D₄	1	1	yes	characteristic
90	C₈×Dic₈₃	D₄	1	1	yes	characteristic
91	C₄⋊Q₈×C₈₃	C₂³	1	2	yes	—
92	C₄×Q₈×C₈₃	C₂³	1	1	yes	characteristic
93	#107×C₈₃	—	2	1	no	—
94	#107×C₈₃	—	2	1	no	—
95	#107×C₈₃	—	2	2	no	—
96	C₄⋊C₈×C₈₃	—	2	1	no	—
97	C₄⋊C₈×C₈₃	C₂³	1	2	yes	—
98	C₄⋊C₈×C₈₃	C₂³	1	1	yes	characteristic
99	#103×C₈₃	—	2	2	no	—
100	#97×C₈₃	—	2	1	no	—
101	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₂×Dic₁₆₆	—	2	1	no	—
103	C₂×Dic₁₆₆	—	2	2	no	—
104	C₂×Dic₁₆₆	C₂×D₄	1	1	yes	characteristic
105	Q₈×D₈₃	—	4	2	no	—
106	C₂×C₄×D₈₃	—	2	1	no	—
107	C₂×C₄×D₈₃	—	2	1	no	—
108	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×D₈₃	#41	1	1	yes	characteristic
111	Dic₃₃₂	—	4	4	no	—
112	#13050580	—	4	4	no	—
113	C₂×C₈₃⋊C₈	—	2	1	no	—
114	C₂×C₈₃⋊C₈	—	2	2	no	—
115	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
117	C₈×D₈₃	—	2	2	no	—
118	C₈×D₈₃	—	4	2	no	—
119	C₈×D₈₃	Q₁₆	1	4	yes	complemented
120	C₄⋊Dic₈₃	—	2	1	no	—
121	C₄⋊Dic₈₃	—	2	2	no	—
122	C₄⋊Dic₈₃	—	2	2	no	—
123	C₄⋊Dic₈₃	—	4	2	no	—
124	C₄⋊Dic₈₃	C₂×D₄	1	1	yes	characteristic
125	Dic₈₃⋊C₄	—	2	1	no	—
126	Dic₈₃⋊C₄	—	2	1	no	—
127	Dic₈₃⋊C₄	—	2	1	no	—
128	Dic₈₃⋊C₄	—	4	2	no	—
129	(C₂×D₈₃)⋊C₄	—	4	2	no	—
130	C₄×Dic₈₃	—	2	1	no	—
131	C₄×Dic₈₃	—	2	1	no	—
132	C₄×Dic₈₃	—	2	1	no	—
133	C₄×Dic₈₃	—	4	2	no	—
134	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
135	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
136	C₄×Dic₈₃	#41	1	1	yes	characteristic
137	C₂×Q₈×C₈₃	—	2	2	no	—
138	C₂×Q₈×C₈₃	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈×C₈₃	—	2	1	no	—
140	C₂×C₈×C₈₃	—	2	2	no	—
141	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
143	C₄⋊C₄×C₈₃	—	2	1	no	—
144	C₄⋊C₄×C₈₃	—	2	2	no	—
145	C₄⋊C₄×C₈₃	—	2	2	no	—
146	C₄⋊C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₈₃	—	2	1	no	—
148	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
149	C₂²×D₈₃	#121	1	1	yes	characteristic
150	Q₈×C₈₃	—	2	2	no	—
151	Q₈×C₈₃	—	8	2	no	—
152	C₂×C₄×C₈₃	—	2	1	no	—
153	C₂×C₄×C₈₃	—	2	1	no	—
154	C₂×C₄×C₈₃	—	4	2	no	—
155	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
158	C₂×Dic₈₃	—	2	1	no	—
159	C₂×Dic₈₃	—	2	1	no	—
160	C₂×Dic₈₃	—	4	2	no	—
161	C₂×Dic₈₃	#121	1	1	yes	characteristic
162	C₂×Dic₈₃	#121	1	1	yes	characteristic
163	C₂×Dic₈₃	#121	1	1	yes	characteristic
164	C₄×D₈₃	—	2	2	no	—
165	C₄×D₈₃	—	4	1	no	—
166	C₄×D₈₃	—	4	2	no	—
167	C₄×D₈₃	—	4	2	no	—
168	C₄×D₈₃	C₂×Q₁₆	1	2	yes	—
169	C₄×D₈₃	#137	1	2	yes	—
170	Dic₁₆₆	—	2	2	no	—
171	Dic₁₆₆	—	4	2	no	—
172	Dic₁₆₆	—	4	2	no	—
173	Dic₁₆₆	—	4	2	no	—
174	C₈×C₈₃	—	2	1	no	—
175	C₈×C₈₃	—	4	1	no	—
176	C₈×C₈₃	—	4	2	no	—
177	C₈×C₈₃	C₂×Q₁₆	1	2	yes	—
178	C₈₃⋊C₈	—	2	1	no	—
179	C₈₃⋊C₈	—	4	1	no	—
180	C₈₃⋊C₈	—	4	2	no	—
181	C₈₃⋊C₈	C₂×Q₁₆	1	2	yes	—
182	C₂²×C₈₃	C₂×#121	1	1	yes	characteristic
183	C₂×D₈₃	—	2	1	no	—
184	C₂×D₈₃	—	2	1	no	—
185	C₂×D₈₃	#462	1	2	yes	—
186	C₄×C₈₃	—	2	1	no	—
187	C₄×C₈₃	—	4	1	no	—
188	C₄×C₈₃	—	4	1	no	—
189	C₄×C₈₃	—	8	2	no	—
190	C₄×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
191	C₄×C₈₃	C₂×#137	1	1	yes	characteristic
192	C₄×C₈₃	#575	1	2	yes	—
193	Dic₈₃	—	2	1	no	—
194	Dic₈₃	—	2	1	no	—
195	Dic₈₃	—	4	1	no	—
196	Dic₈₃	—	4	1	no	—
197	Dic₈₃	—	8	2	no	—
198	Dic₈₃	#462	1	2	yes	—
199	#19150	—	83	1	no	maximal, Sylow
200	C₂×C₈₃	C₂×#462	1	1	yes	characteristic
201	C₂×C₈₃	#3054	1	1	yes	characteristic
202	C₂×C₈₃	#3067	1	1	yes	characteristic
203	D₈₃	—	2	2	no	—
204	#3160	—	83	1	no	—
205	#3104	—	83	2	no	—
206	C₂×#446	—	83	1	no	—
207	#1696	—	83	1	no	—
208	#1696	D₈₃	1	1	yes	characteristic, complemented
209	#1694	—	83	1	no	—
210	#1694	—	83	1	no	—
211	#1692	—	83	2	no	—
212	#1692	—	83	2	no	—
213	#1564	—	83	1	no	—
214	#1542	—	83	2	no	—
215	C₈₃	#19150	1	1	yes	characteristic, complemented, Sylow
216	#533	—	83	2	no	—
217	C₂×C₄×Q₈	—	83	1	no	—
218	#500	—	83	1	no	—
219	#500	C₂×D₈₃	1	1	yes	characteristic
220	#499	—	166	1	no	—
221	#499	—	166	1	no	—
222	#498	—	83	2	no	—
223	#484	—	166	2	no	—
224	#477	—	83	2	no	—
225	#477	C₂×D₈₃	1	2	yes	—
226	#462	—	83	2	no	—
227	#462	—	83	2	no	—
228	#451	—	166	2	no	—
229	#446	—	83	1	no	—
230	#446	—	83	1	no	—
231	#446	—	83	1	no	—
232	#446	—	166	1	no	—
233	#446	—	166	1	no	—
234	#446	C₂×D₈₃	1	1	yes	characteristic
235	C₄×Q₁₆	—	166	2	no	—
236	C₄×Q₁₆	—	166	2	no	—
237	#431	—	83	2	no	—
238	#427	—	166	1	no	—
239	#427	—	166	1	no	—
240	C₂×C₄⋊C₈	—	83	1	no	—
241	C₂×C₄⋊C₈	—	166	1	no	—
242	C₂×#97	—	166	1	no	—
243	C₂×C₄×C₈	—	83	1	no	—
244	#335	—	83	2	no	—
245	#334	—	83	1	no	—
246	#334	C₂×D₈₃	1	1	yes	characteristic
247	#333	—	83	2	no	—
248	#333	C₂×D₈₃	1	2	yes	—
249	#332	—	83	2	no	—
250	#332	—	83	2	no	—
251	C₂²×Q₈	—	83	1	no	—
252	C₂×Q₁₆	—	166	2	no	—
253	C₂×Q₁₆	—	166	2	no	—
254	C₂²×C₈	—	83	1	no	—
255	C₂²×C₈	—	83	1	no	—
256	C₂²×C₈	—	166	1	no	—
257	C₄⋊Q₈	—	83	2	no	—
258	C₄⋊Q₈	C₂²×D₈₃	1	2	yes	—
259	#125	—	166	2	no	—
260	C₂²⋊Q₈	—	166	2	no	—
261	C₂²⋊Q₈	—	166	2	no	—
262	C₄×Q₈	—	83	1	no	—
263	C₄×Q₈	—	83	1	no	—
264	C₄×Q₈	—	83	1	no	—
265	C₄×Q₈	—	166	1	no	—
266	C₄×Q₈	—	166	2	no	—
267	C₄×Q₈	—	166	2	no	—
268	C₄×Q₈	—	166	1	no	—
269	C₄×Q₈	C₂²×D₈₃	1	1	yes	characteristic
270	#117	—	166	2	no	—
271	C₂×C₄⋊C₄	—	83	1	no	—
272	C₂×C₄⋊C₄	—	166	1	no	—
273	C₂×C₄²	—	83	1	no	—
274	C₂×C₄²	—	166	1	no	—
275	#107	—	2	1	no	—
276	#107	—	2	1	no	—
277	#107	—	2	2	no	—
278	#107	—	166	2	no	—
279	#107	—	166	1	no	—
280	#107	—	166	1	no	—
281	#106	—	166	2	no	—
282	#106	—	166	2	no	—
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	—	83	1	no	—
285	C₄⋊C₈	—	83	2	no	—
286	C₄⋊C₈	—	166	1	no	—
287	C₄⋊C₈	—	166	1	no	—
288	C₄⋊C₈	—	166	1	no	—
289	C₄⋊C₈	—	166	1	no	—
290	C₄⋊C₈	C₂²×D₈₃	1	1	yes	characteristic
291	C₄⋊C₈	C₂²×D₈₃	1	2	yes	—
292	#103	—	2	2	no	—
293	#103	—	166	2	no	—
294	#103	—	166	2	no	—
295	#103	—	166	2	no	—
296	C₂²⋊C₈	—	166	2	no	—
297	#97	—	2	1	no	—
298	#97	—	166	1	no	—
299	#97	—	166	1	no	—
300	#97	—	166	1	no	—
301	C₄×C₈	—	83	1	no	—
302	C₄×C₈	—	83	1	no	—
303	C₄×C₈	—	83	1	no	—
304	C₄×C₈	—	166	2	no	—
305	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	—	332	2	no	—
308	C₂×Q₈	—	83	1	no	—
309	C₂×Q₈	—	166	1	no	—
310	C₂×Q₈	—	166	2	no	—
311	C₂×Q₈	D₄×D₈₃	1	1	yes	characteristic
312	C₂²×C₄	—	83	1	no	—
313	C₂²×C₄	—	83	1	no	—
314	C₂²×C₄	—	83	1	no	—
315	C₂²×C₄	—	166	1	no	—
316	C₂²×C₄	—	166	1	no	—
317	Q₁₆	—	332	4	no	—
318	Q₁₆	—	332	4	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	332	2	no	—
322	C₂×C₈	—	83	1	no	—
323	C₂×C₈	—	83	4	no	—
324	C₂×C₈	—	83	1	no	—
325	C₂×C₈	—	166	2	no	—
326	C₂×C₈	—	166	2	no	—
327	C₂×C₈	—	166	1	no	—
328	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
329	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	332	2	no	—
334	C₄⋊C₄	—	332	2	no	—
335	C₄⋊C₄	—	83	1	no	—
336	C₄⋊C₄	—	166	1	no	—
337	C₄⋊C₄	—	166	1	no	—
338	C₄⋊C₄	—	166	1	no	—
339	C₄⋊C₄	—	166	2	no	—
340	C₄⋊C₄	—	166	2	no	—
341	C₄⋊C₄	—	166	1	no	—
342	C₄⋊C₄	D₄×D₈₃	1	1	yes	characteristic
343	C₂²⋊C₄	—	332	2	no	—
344	C₄²	—	2	1	no	—
345	C₄²	—	332	2	no	—
346	C₄²	—	83	1	no	—
347	C₄²	—	83	1	no	—
348	C₄²	—	83	1	no	—
349	C₄²	—	166	1	no	—
350	C₄²	—	166	1	no	—
351	C₄²	—	166	1	no	—
352	C₄²	C₂³×D₈₃	1	1	yes	characteristic
353	C₂³	—	83	1	no	—
354	Q₈	—	2	2	no	—
355	Q₈	—	8	2	no	—
356	Q₈	—	332	2	no	—
357	Q₈	—	332	2	no	—
358	Q₈	—	332	2	no	—
359	Q₈	—	166	2	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	332	1	no	—
364	C₂×C₄	—	332	2	no	—
365	C₂×C₄	—	332	2	no	—
366	C₂×C₄	—	332	2	no	—
367	C₂×C₄	—	83	1	no	—
368	C₂×C₄	—	83	2	no	—
369	C₂×C₄	—	83	2	no	—
370	C₂×C₄	—	83	1	no	—
371	C₂×C₄	—	83	1	no	—
372	C₂×C₄	—	166	1	no	—
373	C₂×C₄	—	166	1	no	—
374	C₂×C₄	—	166	2	no	—
375	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
376	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
377	C₂×C₄	#41×D₈₃	1	1	yes	characteristic
378	C₈	—	2	1	no	—
379	C₈	—	4	1	no	—
380	C₈	—	4	2	no	—
381	C₈	—	332	1	no	—
382	C₈	—	332	2	no	—
383	C₈	—	83	2	no	—
384	C₈	—	166	1	no	—
385	C₈	Q₁₆×D₈₃	1	2	yes	—
386	C₂²	—	83	2	no	—
387	C₂²	—	166	1	no	—
388	C₂²	—	166	1	no	—
389	C₂²	#121×D₈₃	1	1	yes	characteristic
390	C₄	—	2	1	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	8	2	no	—
394	C₄	—	332	1	no	—
395	C₄	—	332	1	no	—
396	C₄	—	83	2	no	—
397	C₄	—	664	2	no	—
398	C₄	—	166	1	no	—
399	C₄	—	166	1	no	—
400	C₄	C₂×Q₁₆×D₈₃	1	1	yes	characteristic
401	C₄	#137×D₈₃	1	1	yes	characteristic
402	C₄	#19128500	1	2	yes	—
403	C₂	—	166	2	no	—
404	C₂	#462×D₈₃	1	1	yes	characteristic
405	C₂	#24917074	1	1	yes	characteristic
406	C₂	#24917181	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₈₃
commutator, cosocle	C₄²×C₈₃	C₂⁴
Frattini	C₄²	C₂³×D₈₃
Fitting	C₈₃×#1696	C₂
socle	C₂²×C₈₃	C₂×#121
prime radical	C₂²×D₈₃	#121
hypercenter	#1696	D₈₃
hypocenter	C₈₃	#19150
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1696

Complements

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

N	H ≅ G/N	type
4. #24906988	C₂	semidirect
5. #24906982	C₂	semidirect
6. #24906981	C₂	semidirect
7. #24906972	C₂	semidirect
8. #24906971	C₂	semidirect
11. C₈₃×#1696	C₂	semidirect
119. C₈×D₈₃	Q₁₆	semidirect
208. #1696	D₈₃	semidirect
215. C₈₃	#19150	semidirect

Maximal subgroups

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

subgroup	index
#24917974	2
#24917504	2
#446×D₈₃	2
#24906988	2
#24906982	2
#24906981	2
#24906972	2
#24906971	2
#24906208	2
#24906117	2
C₈₃×#1696	2
#19150	83