Subgroups — #27537500

Order 13568.

652 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287848	C₂	1	1	yes	maximal, characteristic
2	#443×D₅₃	C₂	1	2	yes	maximal, complemented
3	#20276601	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276600	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276598	C₂	1	2	yes	maximal, complemented
6	#20276597	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276594	C₂	1	2	yes	maximal, complemented
8	#20276508	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276441	C₂	1	2	yes	maximal, complemented
10	C₄×#13133156	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2178	C₂	1	1	yes	maximal, characteristic, complemented
12	#135×D₅₃	C₂²	1	2	yes	—
13	#131×D₅₃	—	2	2	no	—
14	#16811065	C₄	1	2	yes	complemented
15	#16811062	C₄	1	2	yes	complemented
16	#16811060	C₄	1	2	yes	complemented
17	#16811057	C₄	1	2	yes	complemented
18	#16811055	C₂²	1	1	yes	characteristic
19	#16811053	C₂²	1	2	yes	—
20	C₂×#13133156	C₂²	1	1	yes	characteristic
21	C₂×#13133156	C₂²	1	1	yes	characteristic
22	#16810741	C₂²	1	2	yes	complemented
23	#16810740	C₂²	1	2	yes	complemented
24	#16810739	C₂²	1	2	yes	complemented
25	#16810738	—	2	2	no	—
26	#16810737	—	2	2	no	—
27	#108×D₅₃	C₂²	1	2	yes	—
28	#104×D₅₃	—	2	2	no	—
29	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
30	#16810453	C₂²	1	1	yes	characteristic, complemented
31	#16810452	C₂²	1	2	yes	complemented
32	#16810431	C₂²	1	2	yes	complemented
33	#16810430	C₂²	1	2	yes	complemented
34	#16810429	C₂²	1	2	yes	complemented
35	#16810398	C₂²	1	1	yes	characteristic, complemented
36	#16810398	C₂²	1	1	yes	characteristic, complemented
37	C₄×#11719908	C₂²	1	1	yes	characteristic, complemented
38	C₅₃×#508	C₂²	1	1	yes	characteristic
39	C₅₃×#443	C₂²	1	2	yes	complemented
40	C₅₃×#368	C₂²	1	1	yes	characteristic, complemented
41	C₅₃×#359	C₂²	1	2	yes	complemented
42	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	#41×D₅₃	—	2	2	no	—
44	#13133240	C₂×C₄	1	2	yes	—
45	#13133239	C₂³	1	2	yes	—
46	#13133238	—	2	2	no	—
47	#13133237	—	2	2	no	—
48	#13133236	C₂×C₄	1	2	yes	—
49	Q₁₆×D₅₃	C₂×C₄	1	2	yes	—
50	SD₁₆×D₅₃	—	2	2	no	—
51	D₈×D₅₃	C₂×C₄	1	2	yes	—
52	#13133225	—	2	2	no	—
53	#13133225	—	4	2	no	—
54	#13133224	—	2	2	no	—
55	#13133223	—	2	2	no	—
56	#13133223	—	4	2	no	—
57	M₁₆×D₅₃	—	2	2	no	—
58	M₁₆×D₅₃	—	4	2	no	—
59	C₂×C₈×D₅₃	—	4	2	no	—
60	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
61	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
62	C₄²×D₅₃	D₄	1	1	yes	characteristic
63	Dic₄₂₄	C₂×C₄	1	1	yes	characteristic, complemented
64	#13133163	C₂×C₄	1	2	yes	complemented
65	#13133162	C₂×C₄	1	2	yes	complemented
66	#13133161	C₂×C₄	1	2	yes	complemented
67	#13133160	C₂×C₄	1	2	yes	complemented
68	D₈₄₈	C₂×C₄	1	1	yes	characteristic, complemented
69	#13133158	C₂×C₄	1	2	yes	complemented
70	#13133156	—	2	2	no	—
71	#13133156	C₂×C₄	1	2	yes	—
72	#13133156	C₂×C₄	1	2	yes	—
73	C₂×#11719908	C₂³	1	1	yes	characteristic
74	C₂×#11719908	C₂³	1	1	yes	characteristic
75	#13133153	C₂³	1	2	yes	—
76	#13133152	—	2	2	no	—
77	#13133145	—	2	2	no	—
78	#13133144	—	2	2	no	—
79	#13133143	—	2	2	no	—
80	C₄×#88322	D₄	1	1	yes	characteristic
81	C₈×Dic₅₃	D₄	1	1	yes	characteristic
82	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
83	#135×C₅₃	C₂³	1	2	yes	—
84	#131×C₅₃	—	2	2	no	—
85	Q₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
86	SD₃₂×C₅₃	C₂×C₄	1	2	yes	complemented
87	D₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
88	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
89	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
90	#108×C₅₃	C₂³	1	2	yes	—
91	#104×C₅₃	—	2	2	no	—
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	#41×C₅₃	—	2	2	no	—
94	#11719949	—	2	2	no	—
95	Q₈×D₅₃	—	2	2	no	—
96	#11719947	—	2	2	no	—
97	#11719947	—	4	2	no	—
98	D₄×D₅₃	—	2	2	no	—
99	D₄×D₅₃	—	4	2	no	—
100	#11719945	—	2	2	no	—
101	#11719945	—	4	2	no	—
102	C₂×C₄×D₅₃	—	2	1	no	—
103	C₂×C₄×D₅₃	—	4	2	no	—
104	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
105	Q₁₆×C₅₃	C₂²×C₄	1	2	yes	—
106	SD₁₆×C₅₃	—	2	2	no	—
107	D₈×C₅₃	C₂²×C₄	1	2	yes	—
108	M₁₆×C₅₃	—	2	2	no	—
109	M₁₆×C₅₃	—	4	2	no	—
110	C₂×C₈×C₅₃	—	4	2	no	—
111	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
113	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
114	#11719925	—	2	2	no	—
115	#11719924	—	2	2	no	—
116	#11719923	—	2	2	no	—
117	#11719922	—	2	2	no	—
118	C₄×Dic₅₃	—	2	1	no	—
119	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
120	#11719917	—	2	2	no	—
121	#11719917	—	4	2	no	—
122	C₂×#88322	—	4	2	no	—
123	C₂×#88322	C₂×D₄	1	1	yes	characteristic
124	C₂×#88322	C₂×D₄	1	1	yes	characteristic
125	Dic₂₁₂	C₂²×C₄	1	2	yes	—
126	D₄₂₄	C₂²×C₄	1	2	yes	—
127	#11719913	—	2	2	no	—
128	#11719912	—	4	2	no	—
129	#11719912	—	4	2	no	—
130	#11719912	—	4	2	no	—
131	C₈×D₅₃	—	4	2	no	—
132	C₈×D₅₃	—	4	2	no	—
133	C₈×D₅₃	—	4	2	no	—
134	C₈×D₅₃	D₈	1	2	yes	—
135	C₈×D₅₃	D₈	1	2	yes	—
136	C₈×D₅₃	C₂²×C₄	1	2	yes	—
137	C₈×D₅₃	#41	1	2	yes	—
138	C₁₆×C₅₃	—	2	1	no	—
139	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
141	#11719908	—	2	1	no	—
142	#11719908	C₂²×C₄	1	2	yes	—
143	C₂²×D₅₃	—	4	2	no	—
144	C₂²×D₅₃	#135	1	1	yes	characteristic
145	Q₈×C₅₃	—	2	2	no	—
146	D₄×C₅₃	—	2	2	no	—
147	D₄×C₅₃	—	4	2	no	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	—	4	2	no	—
150	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
151	C₂²⋊D₅₃	—	4	2	no	—
152	C₂²⋊D₅₃	—	4	2	no	—
153	C₂²⋊D₅₃	—	4	2	no	—
154	C₂×Dic₅₃	—	2	1	no	—
155	C₂×Dic₅₃	—	4	2	no	—
156	C₂×Dic₅₃	#135	1	1	yes	characteristic
157	D₂₁₂	—	2	2	no	—
158	D₂₁₂	—	4	2	no	—
159	C₄×D₅₃	—	4	2	no	—
160	C₄×D₅₃	—	4	2	no	—
161	C₄×D₅₃	—	4	2	no	—
162	C₄×D₅₃	—	4	2	no	—
163	C₄×D₅₃	C₄×D₄	1	2	yes	—
164	C₄×D₅₃	C₂×D₈	1	2	yes	—
165	Dic₁₀₆	—	2	2	no	—
166	Dic₁₀₆	—	4	2	no	—
167	C₈×C₅₃	—	4	2	no	—
168	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
169	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
170	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
171	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
172	#88322	—	4	2	no	—
173	#88322	C₄×D₄	1	2	yes	—
174	#88322	C₂×D₈	1	2	yes	—
175	#33463	—	53	1	no	maximal, Sylow
176	C₂²×C₅₃	—	4	2	no	—
177	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
178	C₂×D₅₃	—	4	1	no	—
179	C₂×D₅₃	—	4	2	no	—
180	C₂×D₅₃	—	8	4	no	—
181	C₂×D₅₃	C₄×D₈	1	2	yes	—
182	C₄×C₅₃	—	2	2	no	—
183	C₄×C₅₃	—	4	2	no	—
184	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
185	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
186	Dic₅₃	—	4	1	no	—
187	Dic₅₃	—	4	2	no	—
188	Dic₅₃	C₄×D₈	1	2	yes	—
189	#3414	—	53	1	no	—
190	C₂×#443	—	53	2	no	—
191	#2179	—	53	1	no	—
192	#2179	—	53	1	no	—
193	#2179	—	53	2	no	—
194	#2178	—	53	2	no	—
195	#2178	—	53	1	no	—
196	#2178	D₅₃	1	1	yes	characteristic, complemented
197	#2161	—	53	1	no	—
198	#2143	—	53	2	no	—
199	C₄×M₃₂	—	53	1	no	—
200	C₂×C₅₃	—	2	1	no	—
201	C₂×C₅₃	—	8	2	no	—
202	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
203	D₅₃	—	2	2	no	—
204	D₅₃	—	8	2	no	—
205	C₂×#135	—	53	2	no	—
206	C₂×#131	—	106	2	no	—
207	#510	—	53	2	no	—
208	#510	—	53	2	no	—
209	#509	—	53	2	no	—
210	#509	—	53	2	no	—
211	#508	—	53	2	no	—
212	#508	—	53	1	no	—
213	#508	C₂×D₅₃	1	1	yes	characteristic
214	C₂×M₃₂	—	53	1	no	—
215	C₂×M₃₂	—	53	1	no	—
216	#443	—	53	2	no	—
217	#443	—	53	2	no	—
218	#443	—	53	2	no	—
219	#443	—	106	2	no	—
220	#443	—	106	2	no	—
221	#443	C₂×D₅₃	1	2	yes	complemented
222	C₂×#108	—	53	2	no	—
223	C₂×#104	—	106	2	no	—
224	C₂×C₄×C₈	—	53	1	no	—
225	#368	—	53	1	no	—
226	#368	—	53	2	no	—
227	#368	C₂×D₅₃	1	1	yes	characteristic, complemented
228	#359	—	53	2	no	—
229	#359	—	53	2	no	—
230	#359	—	53	2	no	—
231	#359	C₂×D₅₃	1	2	yes	complemented
232	#346	—	53	1	no	—
233	#346	—	53	1	no	—
234	C₄×C₁₆	—	53	1	no	—
235	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
236	C₅₃	#33463	1	1	yes	characteristic, complemented, Sylow
237	C₂×#41	—	106	2	no	—
238	#135	—	53	2	no	—
239	#135	—	53	2	no	—
240	#135	—	53	2	no	—
241	#135	—	106	2	no	—
242	#135	—	106	2	no	—
243	#135	C₂²×D₅₃	1	2	yes	—
244	C₂×Q₁₆	—	53	2	no	—
245	C₂×SD₁₆	—	106	2	no	—
246	C₂×D₈	—	53	2	no	—
247	#131	—	2	2	no	—
248	#131	—	106	2	no	—
249	#131	—	106	2	no	—
250	#131	—	106	2	no	—
251	#131	—	212	2	no	—
252	#131	—	212	2	no	—
253	C₂×M₁₆	—	106	2	no	—
254	C₂×M₁₆	—	212	2	no	—
255	C₂²×C₈	—	53	1	no	—
256	C₂²×C₈	—	53	1	no	—
257	C₂²×C₈	—	212	2	no	—
258	C₂×C₄²	—	53	1	no	—
259	Q₃₂	—	53	2	no	—
260	Q₃₂	—	53	1	no	—
261	Q₃₂	C₄×D₅₃	1	1	yes	characteristic, complemented
262	SD₃₂	—	53	2	no	—
263	SD₃₂	—	53	2	no	—
264	SD₃₂	—	53	2	no	—
265	SD₃₂	C₄×D₅₃	1	2	yes	complemented
266	D₁₆	—	53	2	no	—
267	D₁₆	—	53	1	no	—
268	D₁₆	C₄×D₅₃	1	1	yes	characteristic, complemented
269	M₃₂	—	53	2	no	—
270	M₃₂	—	53	2	no	—
271	M₃₂	—	106	2	no	—
272	C₂×C₁₆	—	53	1	no	—
273	C₂×C₁₆	—	53	1	no	—
274	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
275	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
276	#108	—	53	2	no	—
277	#108	—	106	2	no	—
278	#108	C₂²×D₅₃	1	2	yes	—
279	#104	—	2	2	no	—
280	#104	—	106	2	no	—
281	#104	—	106	2	no	—
282	#104	—	106	2	no	—
283	C₄×C₈	—	53	1	no	—
284	C₄×C₈	—	53	1	no	—
285	C₄×C₈	—	53	1	no	—
286	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
287	#41	—	2	2	no	—
288	#41	—	106	2	no	—
289	#41	—	106	2	no	—
290	#41	—	106	2	no	—
291	#41	—	212	2	no	—
292	#41	—	212	2	no	—
293	C₂×Q₈	—	106	2	no	—
294	C₂×D₄	—	106	2	no	—
295	C₂×D₄	—	212	2	no	—
296	C₂²×C₄	—	53	1	no	—
297	C₂²×C₄	—	106	1	no	—
298	C₂²×C₄	—	212	2	no	—
299	Q₁₆	—	53	2	no	—
300	Q₁₆	—	106	2	no	—
301	Q₁₆	C₂×C₄×D₅₃	1	2	yes	—
302	SD₁₆	—	2	2	no	—
303	SD₁₆	—	106	2	no	—
304	SD₁₆	—	106	2	no	—
305	SD₁₆	—	106	2	no	—
306	D₈	—	53	2	no	—
307	D₈	—	106	2	no	—
308	D₈	C₂×C₄×D₅₃	1	2	yes	—
309	M₁₆	—	2	2	no	—
310	M₁₆	—	4	2	no	—
311	M₁₆	—	106	2	no	—
312	M₁₆	—	212	2	no	—
313	M₁₆	—	212	2	no	—
314	M₁₆	—	212	2	no	—
315	M₁₆	—	212	2	no	—
316	C₂×C₈	—	4	2	no	—
317	C₂×C₈	—	53	1	no	—
318	C₂×C₈	—	53	1	no	—
319	C₂×C₈	—	53	2	no	—
320	C₂×C₈	—	53	2	no	—
321	C₂×C₈	—	53	2	no	—
322	C₂×C₈	—	53	2	no	—
323	C₂×C₈	—	212	2	no	—
324	C₂×C₈	—	212	2	no	—
325	C₂×C₈	—	212	2	no	—
326	C₂×C₈	—	212	2	no	—
327	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
328	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
329	C₄²	—	53	1	no	—
330	C₄²	—	106	1	no	—
331	C₄²	D₄×D₅₃	1	1	yes	characteristic
332	C₁₆	—	2	1	no	—
333	C₁₆	—	53	2	no	—
334	C₁₆	—	106	1	no	—
335	C₁₆	C₂×C₄×D₅₃	1	1	yes	characteristic
336	C₁₆	C₂×C₄×D₅₃	1	1	yes	characteristic
337	C₂³	—	53	1	no	—
338	C₂³	—	212	2	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	106	2	no	—
341	Q₈	—	212	2	no	—
342	D₄	—	2	2	no	—
343	D₄	—	4	2	no	—
344	D₄	—	106	2	no	—
345	D₄	—	212	2	no	—
346	D₄	—	212	2	no	—
347	D₄	—	212	2	no	—
348	D₄	—	212	2	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	53	1	no	—
352	C₂×C₄	—	53	2	no	—
353	C₂×C₄	—	53	2	no	—
354	C₂×C₄	—	106	1	no	—
355	C₂×C₄	—	212	2	no	—
356	C₂×C₄	—	212	2	no	—
357	C₂×C₄	—	212	2	no	—
358	C₂×C₄	—	212	2	no	—
359	C₂×C₄	—	212	2	no	—
360	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
361	C₈	—	4	2	no	—
362	C₈	—	53	2	no	—
363	C₈	—	53	2	no	—
364	C₈	—	212	2	no	—
365	C₈	D₈×D₅₃	1	1	yes	characteristic
366	C₈	D₈×D₅₃	1	1	yes	characteristic
367	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
368	C₈	#41×D₅₃	1	1	yes	characteristic
369	C₂²	—	4	2	no	—
370	C₂²	—	53	2	no	—
371	C₂²	—	424	4	no	—
372	C₂²	—	212	1	no	—
373	C₂²	—	212	2	no	—
374	C₂²	#135×D₅₃	1	1	yes	characteristic
375	C₄	—	2	2	no	—
376	C₄	—	4	2	no	—
377	C₄	—	53	2	no	—
378	C₄	—	212	1	no	—
379	C₄	—	212	2	no	—
380	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
381	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
382	C₂	—	2	1	no	—
383	C₂	—	8	2	no	—
384	C₂	—	106	2	no	—
385	C₂	—	424	2	no	—
386	C₂	C₄×D₈×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2178	C₂
socle	C₂×C₅₃	C₂×C₄×D₈
prime radical	#20287848	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2178	D₅₃
hypocenter	C₅₃	#33463
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2178

Complements

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

N	H ≅ G/N	type
2. #443×D₅₃	C₂	semidirect
3. #20276601	C₂	semidirect
4. #20276600	C₂	semidirect
5. #20276598	C₂	semidirect
6. #20276597	C₂	semidirect
7. #20276594	C₂	semidirect
8. #20276508	C₂	semidirect
9. #20276441	C₂	semidirect
10. C₄×#13133156	C₂	semidirect
11. C₅₃×#2178	C₂	semidirect
14. #16811065	C₄	semidirect
15. #16811062	C₄	semidirect
16. #16811060	C₄	semidirect
17. #16811057	C₄	semidirect
22. #16810741	C₂²	semidirect
23. #16810740	C₂²	semidirect
24. #16810739	C₂²	semidirect
30. #16810453	C₂²	semidirect
31. #16810452	C₂²	semidirect
32. #16810431	C₂²	semidirect
33. #16810430	C₂²	semidirect
34. #16810429	C₂²	semidirect
35. #16810398	C₂²	semidirect
36. #16810398	C₂²	semidirect
37. C₄×#11719908	C₂²	semidirect
39. C₅₃×#443	C₂²	semidirect
40. C₅₃×#368	C₂²	semidirect
41. C₅₃×#359	C₂²	semidirect
42. C₄×C₁₆×C₅₃	C₂²	semidirect
63. Dic₄₂₄	C₂×C₄	semidirect
64. #13133163	C₂×C₄	semidirect
65. #13133162	C₂×C₄	semidirect
66. #13133161	C₂×C₄	semidirect
67. #13133160	C₂×C₄	semidirect
68. D₈₄₈	C₂×C₄	semidirect
69. #13133158	C₂×C₄	semidirect
85. Q₃₂×C₅₃	C₂×C₄	semidirect
86. SD₃₂×C₅₃	C₂×C₄	semidirect
87. D₁₆×C₅₃	C₂×C₄	semidirect
196. #2178	D₅₃	semidirect
221. #443	C₂×D₅₃	semidirect
227. #368	C₂×D₅₃	semidirect
231. #359	C₂×D₅₃	semidirect
235. C₄×C₁₆	C₂×D₅₃	semidirect
236. C₅₃	#33463	semidirect
261. Q₃₂	C₄×D₅₃	semidirect
265. SD₃₂	C₄×D₅₃	semidirect
268. D₁₆	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20287848	2
#443×D₅₃	2
#20276601	2
#20276600	2
#20276598	2
#20276597	2
#20276594	2
#20276508	2
#20276441	2
C₄×#13133156	2
C₅₃×#2178	2
#33463	53