Subgroups — #35559076

Order 21248.

574 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#14581815	C₂	1	1	yes	maximal, characteristic
3	#24909615	C₂	1	1	yes	maximal, characteristic, complemented
4	#24909611	C₂	1	1	yes	maximal, characteristic, complemented
5	#24909607	C₂	1	1	yes	maximal, characteristic, complemented
6	#24909603	C₂	1	2	yes	maximal, complemented
7	#24909599	C₂	1	2	yes	maximal, complemented
8	#24909248	C₂	1	2	yes	maximal, complemented
9	#24909217	C₂	1	1	yes	maximal, characteristic, complemented
10	#24909174	C₂	1	2	yes	maximal
11	C₈₃×#2227	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14581815	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×D₈₃	—	2	1	no	—
14	C₄×#13050591	—	2	1	no	—
15	C₂×#14581731	C₂²	1	2	yes	—
16	#19128106	C₂²	1	1	yes	characteristic
17	#19128105	—	2	1	no	—
18	#19128105	C₂²	1	2	yes	—
19	#19128104	—	2	1	no	—
20	Q₁₆×Dic₈₃	C₂²	1	1	yes	characteristic
21	C₄×#13050577	—	2	1	no	—
22	#19127786	—	2	1	no	—
23	C₄×D₆₆₄	C₂²	1	1	yes	characteristic
24	#19127783	C₂²	1	1	yes	characteristic
25	#19127738	C₂²	1	1	yes	characteristic
26	#107×D₈₃	—	2	1	no	—
27	#107×D₈₃	C₂²	1	2	yes	—
28	#19127696	—	2	1	no	—
29	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
30	#19127529	C₂²	1	2	yes	—
31	#19127528	C₂²	1	2	yes	—
32	#19127526	C₂²	1	2	yes	—
33	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	1	yes	characteristic
34	Dic₈₃⋊C₁₆	C₂²	1	2	yes	—
35	#19127505	C₂²	1	2	yes	—
36	#19127503	C₂²	1	2	yes	—
37	#19127501	C₂²	1	2	yes	—
38	#500×C₈₃	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
40	#366×C₈₃	C₂²	1	2	yes	—
41	C₄⋊C₁₆×C₈₃	C₂²	1	1	yes	characteristic
42	#358×C₈₃	C₂²	1	2	yes	—
43	C₂×#13050591	—	2	1	no	—
44	C₂×Q₈×D₈₃	—	4	1	no	—
45	#14581815	—	2	2	no	—
46	#14581815	Q₈	1	4	yes	complemented
47	C₂×#13050577	—	2	1	no	—
48	C₂×D₆₆₄	C₂³	1	1	yes	characteristic
49	C₂×C₈×D₈₃	D₄	1	1	yes	characteristic
50	C₂×C₈×D₈₃	C₂³	1	1	yes	characteristic
51	#14581792	—	2	1	no	—
52	#14581791	—	2	2	no	—
53	#14581791	—	4	1	no	—
54	#14581790	—	4	1	no	—
55	Q₈×Dic₈₃	—	2	1	no	—
56	C₄×D₃₃₂	—	2	1	no	—
57	C₄×D₃₃₂	—	4	1	no	—
58	#14581754	—	2	1	no	—
59	#14581754	—	4	1	no	—
60	#14581750	—	2	1	no	—
61	#14581750	—	4	1	no	—
62	C₄⋊C₄×D₈₃	—	2	2	no	—
63	C₄⋊C₄×D₈₃	—	4	1	no	—
64	C₄²×D₈₃	—	4	1	no	—
65	C₄²×D₈₃	D₄	1	1	yes	characteristic
66	#14581731	—	2	4	no	—
67	C₂×C₈₃⋊C₁₆	C₂³	1	2	yes	—
68	#14581726	—	2	1	no	—
69	#14581726	C₂³	1	2	yes	—
70	#14581726	C₂³	1	1	yes	characteristic
71	#14581725	—	2	2	no	—
72	#14581725	—	2	2	no	—
73	#14581723	—	2	1	no	—
74	#14581716	—	2	1	no	—
75	#14581714	—	2	1	no	—
76	#14581712	—	2	1	no	—
77	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
78	C₈×Dic₈₃	D₄	1	1	yes	characteristic
79	C₈×Dic₈₃	C₂³	1	1	yes	characteristic
80	C₂×Q₁₆×C₈₃	C₂³	1	1	yes	characteristic
81	C₄⋊Q₈×C₈₃	—	2	1	no	—
82	C₄×Q₈×C₈₃	—	2	1	no	—
83	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
84	#107×C₈₃	—	2	1	no	—
85	#107×C₈₃	C₂³	1	2	yes	—
86	#107×C₈₃	C₂³	1	1	yes	characteristic
87	#103×C₈₃	—	2	1	no	—
88	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
89	#13050591	—	2	4	no	—
90	#13050591	—	4	2	no	—
91	C₂×Dic₁₆₆	—	4	1	no	—
92	C₂×Dic₁₆₆	—	4	2	no	—
93	Q₈×D₈₃	—	8	4	no	—
94	C₂×D₃₃₂	—	2	1	no	—
95	C₂×D₃₃₂	—	4	1	no	—
96	C₂×C₄×D₈₃	—	4	1	no	—
97	C₂×C₄×D₈₃	—	4	2	no	—
98	C₂×C₄×D₈₃	—	4	1	no	—
99	C₂×C₄×D₈₃	—	4	1	no	—
100	C₂×C₄×D₈₃	D₈	1	1	yes	characteristic
101	C₂×C₄×D₈₃	D₈	1	1	yes	characteristic
102	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
103	#13050577	—	2	4	no	—
104	D₆₆₄	—	2	1	no	—
105	D₆₆₄	C₂×Q₈	1	2	yes	complemented
106	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
108	C₈×D₈₃	—	2	2	no	—
109	C₈×D₈₃	C₂×Q₈	1	2	yes	—
110	C₈×D₈₃	#41	1	2	yes	—
111	C₄⋊Dic₈₃	—	2	2	no	—
112	C₄⋊Dic₈₃	—	2	1	no	—
113	C₄⋊Dic₈₃	—	4	1	no	—
114	C₄⋊Dic₈₃	—	4	1	no	—
115	Dic₈₃⋊C₄	—	4	2	no	—
116	Dic₈₃⋊C₄	—	4	2	no	—
117	(C₂×D₈₃)⋊C₄	—	4	1	no	—
118	(C₂×D₈₃)⋊C₄	—	4	1	no	—
119	(C₂×D₈₃)⋊C₄	—	4	1	no	—
120	C₄×Dic₈₃	—	4	1	no	—
121	C₄×Dic₈₃	—	4	1	no	—
122	C₄×Dic₈₃	D₈	1	1	yes	characteristic
123	C₄×Dic₈₃	D₈	1	1	yes	characteristic
124	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
125	C₈₃⋊C₁₆	—	2	2	no	—
126	C₂×Q₈×C₈₃	—	2	1	no	—
127	C₂×Q₈×C₈₃	—	4	1	no	—
128	Q₁₆×C₈₃	—	2	1	no	—
129	Q₁₆×C₈₃	C₂×Q₈	1	2	yes	complemented
130	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
132	C₄⋊C₄×C₈₃	—	2	2	no	—
133	C₄⋊C₄×C₈₃	—	2	1	no	—
134	C₄⋊C₄×C₈₃	—	4	1	no	—
135	C₄⋊C₄×C₈₃	—	4	1	no	—
136	C₄²×C₈₃	—	4	1	no	—
137	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
138	C₁₆×C₈₃	—	2	2	no	—
139	C₂²×D₈₃	—	4	1	no	—
140	C₂²×D₈₃	C₂×D₈	1	1	yes	characteristic
141	Q₈×C₈₃	—	2	2	no	—
142	Q₈×C₈₃	—	4	1	no	—
143	Q₈×C₈₃	—	8	2	no	—
144	C₂×C₄×C₈₃	—	4	1	no	—
145	C₂×C₄×C₈₃	—	4	1	no	—
146	C₂×C₄×C₈₃	—	4	2	no	—
147	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₈₃	—	4	1	no	—
151	C₂×Dic₈₃	—	4	2	no	—
152	C₂×Dic₈₃	C₂×D₈	1	1	yes	characteristic
153	C₂×Dic₈₃	C₂×D₈	1	1	yes	characteristic
154	C₂×Dic₈₃	C₂×D₈	1	1	yes	characteristic
155	D₃₃₂	—	2	2	no	—
156	D₃₃₂	—	4	4	no	—
157	D₃₃₂	—	4	1	no	—
158	C₄×D₈₃	—	2	2	no	—
159	C₄×D₈₃	—	2	2	no	—
160	C₄×D₈₃	—	4	4	no	—
161	C₄×D₈₃	—	8	4	no	—
162	C₄×D₈₃	—	8	2	no	—
163	C₄×D₈₃	—	8	2	no	—
164	C₄×D₈₃	C₂²⋊Q₈	1	2	yes	—
165	C₄×D₈₃	#137	1	2	yes	—
166	Dic₁₆₆	—	8	2	no	—
167	Dic₁₆₆	—	8	4	no	—
168	C₈×C₈₃	—	2	1	no	—
169	C₈×C₈₃	C₂²×Q₈	1	1	yes	characteristic
170	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
171	C₈₃⋊C₈	—	2	1	no	—
172	C₈₃⋊C₈	C₂²⋊Q₈	1	2	yes	—
173	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
174	C₂×D₈₃	—	2	1	no	—
175	C₂×D₈₃	—	2	1	no	—
176	C₂×D₈₃	—	4	2	no	—
177	C₂×D₈₃	—	8	1	no	—
178	C₂×D₈₃	—	8	1	no	—
179	C₂×D₈₃	#474	1	2	yes	—
180	C₄×C₈₃	—	2	1	no	—
181	C₄×C₈₃	—	4	2	no	—
182	C₄×C₈₃	—	8	2	no	—
183	C₄×C₈₃	—	8	1	no	—
184	C₄×C₈₃	#510	1	2	yes	—
185	C₄×C₈₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
186	C₄×C₈₃	C₂×#137	1	1	yes	characteristic
187	Dic₈₃	—	2	1	no	—
188	Dic₈₃	—	8	2	no	—
189	Dic₈₃	—	8	1	no	—
190	Dic₈₃	#474	1	2	yes	—
191	Dic₈₃	#509	1	2	yes	—
192	#33629	—	83	1	no	maximal, Sylow
193	C₂×C₈₃	C₂×#474	1	1	yes	characteristic
194	C₂×C₈₃	C₂×#509	1	1	yes	characteristic
195	C₂×C₈₃	C₂×#510	1	1	yes	characteristic
196	D₈₃	—	2	2	no	—
197	D₈₃	—	8	2	no	—
198	C₂×#500	—	83	1	no	—
199	C₄×#135	—	83	1	no	—
200	#2227	—	83	1	no	—
201	#2227	D₈₃	1	1	yes	characteristic, complemented
202	#2226	—	83	1	no	—
203	#2226	—	83	1	no	—
204	#2225	—	83	2	no	—
205	#2224	—	83	2	no	—
206	#2159	—	83	2	no	—
207	#2150	—	83	1	no	—
208	#2139	—	83	2	no	—
209	C₈₃	#33629	1	1	yes	characteristic, complemented, Sylow
210	C₂×#135	—	83	1	no	—
211	C₂×C₄⋊Q₈	—	166	1	no	—
212	C₄×#41	—	166	1	no	—
213	C₂×M₃₂	—	83	2	no	—
214	#500	—	83	2	no	—
215	#500	—	83	1	no	—
216	#500	—	166	1	no	—
217	#500	—	166	1	no	—
218	#500	C₂×D₈₃	1	1	yes	characteristic
219	C₄×Q₁₆	—	83	1	no	—
220	C₄×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
221	C₄×SD₁₆	—	166	1	no	—
222	C₄×SD₁₆	—	166	1	no	—
223	C₄×D₈	—	83	1	no	—
224	C₄×D₈	—	83	1	no	—
225	#427	—	83	1	no	—
226	C₂×#107	—	83	2	no	—
227	C₂×#107	—	166	1	no	—
228	#416	—	166	1	no	—
229	C₂×C₄×C₈	—	83	1	no	—
230	#367	—	83	2	no	—
231	#367	—	83	2	no	—
232	#366	—	83	2	no	—
233	#366	C₂×D₈₃	1	2	yes	—
234	C₄⋊C₁₆	—	83	2	no	—
235	C₄⋊C₁₆	—	83	1	no	—
236	C₄⋊C₁₆	C₂×D₈₃	1	1	yes	characteristic
237	#358	—	83	2	no	—
238	#358	C₂×D₈₃	1	2	yes	—
239	#357	—	83	2	no	—
240	#357	—	83	2	no	—
241	C₂×#41	—	166	1	no	—
242	C₂²×Q₈	—	332	1	no	—
243	#135	—	83	4	no	—
244	#135	—	166	2	no	—
245	C₂×Q₁₆	C₂²×D₈₃	1	1	yes	characteristic
246	C₂×SD₁₆	—	166	1	no	—
247	C₂×D₈	—	83	1	no	—
248	C₂²×C₈	—	83	1	no	—
249	C₂²×C₈	—	83	1	no	—
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	—	332	1	no	—
252	C₄⋊Q₈	—	332	1	no	—
253	C₄⋊Q₈	—	166	2	no	—
254	C₄⋊Q₈	—	166	1	no	—
255	C₄×Q₈	—	2	1	no	—
256	C₄×Q₈	—	166	1	no	—
257	C₄×D₄	—	332	1	no	—
258	C₄×D₄	—	332	1	no	—
259	C₄×D₄	—	166	1	no	—
260	C₄×D₄	—	166	1	no	—
261	#117	—	332	1	no	—
262	#117	—	166	1	no	—
263	C₂×C₄⋊C₄	—	332	1	no	—
264	C₂×C₄⋊C₄	—	166	2	no	—
265	C₂×C₄²	—	332	1	no	—
266	C₂×C₄²	—	83	1	no	—
267	M₃₂	—	166	4	no	—
268	C₂×C₁₆	—	83	2	no	—
269	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
270	#107	—	2	1	no	—
271	#107	—	83	2	no	—
272	#107	—	83	1	no	—
273	#107	—	166	2	no	—
274	#107	—	166	2	no	—
275	#107	—	166	1	no	—
276	#107	C₂²×D₈₃	1	2	yes	—
277	#107	C₂²×D₈₃	1	1	yes	characteristic
278	#106	—	166	1	no	—
279	#103	—	2	1	no	—
280	#103	—	166	1	no	—
281	#102	—	166	1	no	—
282	#102	—	166	1	no	—
283	C₄×C₈	—	83	1	no	—
284	C₄×C₈	—	83	1	no	—
285	C₄×C₈	—	83	1	no	—
286	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
287	#41	—	332	2	no	—
288	#41	—	166	4	no	—
289	C₂×Q₈	—	2	1	no	—
290	C₂×Q₈	—	4	1	no	—
291	C₂×Q₈	—	332	2	no	—
292	C₂×Q₈	—	332	1	no	—
293	C₂×Q₈	—	664	4	no	—
294	C₂×D₄	—	332	1	no	—
295	C₂×D₄	—	166	1	no	—
296	C₂²×C₄	—	332	1	no	—
297	C₂²×C₄	—	332	2	no	—
298	C₂²×C₄	—	332	1	no	—
299	C₂²×C₄	—	332	1	no	—
300	C₂²×C₄	—	83	1	no	—
301	C₂²×C₄	—	83	1	no	—
302	C₂²×C₄	—	83	1	no	—
303	Q₁₆	—	2	1	no	—
304	Q₁₆	Q₈×D₈₃	1	2	yes	complemented
305	SD₁₆	—	166	4	no	—
306	D₈	—	83	2	no	—
307	D₈	—	166	1	no	—
308	C₂×C₈	—	83	1	no	—
309	C₂×C₈	—	83	2	no	—
310	C₂×C₈	—	83	2	no	—
311	C₂×C₈	—	83	1	no	—
312	C₂×C₈	—	166	2	no	—
313	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
314	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
315	C₄⋊C₄	—	2	1	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	4	1	no	—
318	C₄⋊C₄	—	4	1	no	—
319	C₄⋊C₄	—	332	2	no	—
320	C₄⋊C₄	—	332	2	no	—
321	C₄⋊C₄	—	332	1	no	—
322	C₄⋊C₄	—	332	1	no	—
323	C₄⋊C₄	—	166	1	no	—
324	C₄⋊C₄	—	166	2	no	—
325	C₂²⋊C₄	—	332	1	no	—
326	C₂²⋊C₄	—	332	1	no	—
327	C₂²⋊C₄	—	332	1	no	—
328	C₄²	—	4	1	no	—
329	C₄²	—	332	1	no	—
330	C₄²	—	332	1	no	—
331	C₄²	—	83	1	no	—
332	C₄²	—	83	1	no	—
333	C₄²	—	83	1	no	—
334	C₄²	D₄×D₈₃	1	1	yes	characteristic
335	C₁₆	—	2	2	no	—
336	C₁₆	—	166	2	no	—
337	C₂³	—	332	1	no	—
338	C₂³	—	83	1	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	4	1	no	—
341	Q₈	—	8	2	no	—
342	Q₈	—	664	2	no	—
343	Q₈	—	664	4	no	—
344	D₄	—	332	4	no	—
345	D₄	—	332	1	no	—
346	D₄	—	166	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	332	4	no	—
351	C₂×C₄	—	332	1	no	—
352	C₂×C₄	—	332	2	no	—
353	C₂×C₄	—	83	1	no	—
354	C₂×C₄	—	83	1	no	—
355	C₂×C₄	—	83	2	no	—
356	C₂×C₄	—	83	2	no	—
357	C₂×C₄	—	83	1	no	—
358	C₂×C₄	—	664	2	no	—
359	C₂×C₄	—	664	4	no	—
360	C₂×C₄	—	664	2	no	—
361	C₂×C₄	—	166	2	no	—
362	C₂×C₄	—	166	2	no	—
363	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
364	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
365	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	83	2	no	—
368	C₈	—	166	1	no	—
369	C₈	C₂×Q₈×D₈₃	1	1	yes	characteristic
370	C₈	#41×D₈₃	1	1	yes	characteristic
371	C₂²	—	332	2	no	—
372	C₂²	—	83	2	no	—
373	C₂²	—	664	1	no	—
374	C₂²	—	664	1	no	—
375	C₂²	—	166	1	no	—
376	C₂²	—	166	1	no	—
377	C₂²	C₂×D₈×D₈₃	1	1	yes	characteristic
378	C₄	—	2	1	no	—
379	C₄	—	4	2	no	—
380	C₄	—	8	2	no	—
381	C₄	—	8	1	no	—
382	C₄	—	83	2	no	—
383	C₄	—	83	2	no	—
384	C₄	—	664	2	no	—
385	C₄	—	664	1	no	—
386	C₄	—	166	1	no	—
387	C₄	#19128143	1	2	yes	—
388	C₄	C₂²⋊Q₈×D₈₃	1	1	yes	characteristic
389	C₄	#137×D₈₃	1	1	yes	characteristic
390	C₂	—	664	2	no	—
391	C₂	—	166	2	no	—
392	C₂	#474×D₈₃	1	1	yes	characteristic
393	C₂	#509×D₈₃	1	1	yes	characteristic
394	C₂	C₂×#19128143	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₃
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#2227	C₂
socle	C₂²×C₈₃	C₂²×D₈
prime radical	C₂×#14581815	C₂²
hypercenter	#2227	D₈₃
hypocenter	C₈₃	#33629
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2227

Complements

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

N	H ≅ G/N	type
1. #500×D₈₃	C₂	semidirect
3. #24909615	C₂	semidirect
4. #24909611	C₂	semidirect
5. #24909607	C₂	semidirect
6. #24909603	C₂	semidirect
7. #24909599	C₂	semidirect
8. #24909248	C₂	semidirect
9. #24909217	C₂	semidirect
11. C₈₃×#2227	C₂	semidirect
46. #14581815	Q₈	semidirect
105. D₆₆₄	C₂×Q₈	semidirect
129. Q₁₆×C₈₃	C₂×Q₈	semidirect
201. #2227	D₈₃	semidirect
209. C₈₃	#33629	semidirect
304. Q₁₆	Q₈×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#500×D₈₃	2
C₄×#14581815	2
#24909615	2
#24909611	2
#24909607	2
#24909603	2
#24909599	2
#24909248	2
#24909217	2
#24909174	2
C₈₃×#2227	2
#33629	83