Subgroups — #27537338

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287816	C₂	1	2	yes	maximal, complemented
2	#20283613	C₂	1	1	yes	maximal, characteristic
3	#20276558	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276554	C₂	1	2	yes	maximal, complemented
5	#20276552	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276551	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276550	C₂	1	2	yes	maximal, complemented
8	#20276479	C₂	1	1	yes	maximal, characteristic, complemented
9	#350×D₅₃	C₂	1	2	yes	maximal, complemented
10	#20276313	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2171	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811420	C₄	1	1	yes	characteristic
13	#131×D₅₃	C₂²	1	2	yes	—
14	#16811041	—	2	2	no	—
15	#16811041	C₂²	1	2	yes	complemented
16	#16811040	C₂²	1	2	yes	complemented
17	#16811039	—	2	2	no	—
18	#16811039	C₂²	1	2	yes	complemented
19	C₂×#13133156	—	2	2	no	—
20	C₂×#13133156	C₂²	1	1	yes	characteristic
21	M₃₂×D₅₃	—	2	2	no	—
22	M₃₂×D₅₃	C₂²	1	2	yes	—
23	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
24	#16810745	C₄	1	1	yes	characteristic
25	#16810744	C₄	1	1	yes	characteristic
26	#16810742	C₄	1	2	yes	—
27	#16810741	C₂²	1	1	yes	characteristic
28	#16810738	C₂²	1	2	yes	—
29	#16810670	C₄	1	1	yes	characteristic
30	#16810638	C₄	1	2	yes	—
31	#16810577	C₂²	1	1	yes	characteristic
32	#16810444	C₂²	1	1	yes	characteristic, complemented
33	#16810443	C₂²	1	2	yes	complemented
34	#16810408	C₂²	1	2	yes	complemented
35	#16810407	C₂²	1	2	yes	complemented
36	#16810406	C₂²	1	2	yes	complemented
37	#16810398	C₂²	1	1	yes	characteristic, complemented
38	C₄×#11719908	C₂²	1	1	yes	characteristic, complemented
39	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₅₃×#504	C₂²	1	2	yes	complemented
41	C₅₃×#443	C₂²	1	1	yes	characteristic
42	C₅₃×#364	C₂²	1	1	yes	characteristic, complemented
43	C₅₃×#350	C₂²	1	2	yes	complemented
44	C₅₃×#346	C₂²	1	1	yes	characteristic, complemented
45	#41×D₅₃	C₂×C₄	1	2	yes	—
46	#13133249	C₈	1	2	yes	—
47	#13133246	C₈	1	2	yes	—
48	#13133244	C₈	1	2	yes	—
49	#13133241	C₈	1	2	yes	—
50	#13133239	C₂×C₄	1	1	yes	characteristic
51	#13133237	C₂×C₄	1	2	yes	—
52	#13133225	—	2	2	no	—
53	#13133225	C₂×C₄	1	2	yes	—
54	#13133224	C₂×C₄	1	2	yes	—
55	#13133223	—	2	2	no	—
56	#13133223	C₂³	1	2	yes	—
57	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
58	M₁₆×D₅₃	—	2	2	no	—
59	M₁₆×D₅₃	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₅₃	—	2	2	no	—
61	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
62	#13133174	C₂×C₄	1	1	yes	characteristic
63	#13133157	—	2	2	no	—
64	#13133157	C₂³	1	2	yes	—
65	#13133156	—	2	2	no	—
66	#13133156	—	2	2	no	—
67	#13133156	—	2	2	no	—
68	#13133156	D₄	1	2	yes	—
69	#13133156	D₄	1	2	yes	—
70	C₂×#11719908	—	2	2	no	—
71	C₂×#11719908	C₂³	1	1	yes	characteristic
72	C₂×#11719908	C₂³	1	1	yes	characteristic
73	C₁₆×D₅₃	—	2	2	no	—
74	C₁₆×D₅₃	—	2	2	no	—
75	C₁₆×D₅₃	—	2	2	no	—
76	C₁₆×D₅₃	—	2	2	no	—
77	#13133153	C₂×C₄	1	1	yes	characteristic
78	#13133152	C₂×C₄	1	2	yes	—
79	#13133145	C₂×C₄	1	2	yes	—
80	#13133144	C₂×C₄	1	2	yes	—
81	#13133143	C₂×C₄	1	2	yes	—
82	#13133128	C₂×C₄	1	1	yes	characteristic
83	#13133127	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
85	#135×C₅₃	C₂×C₄	1	1	yes	characteristic
86	#131×C₅₃	C₂³	1	2	yes	—
87	M₃₂×C₅₃	—	2	2	no	—
88	M₃₂×C₅₃	C₂³	1	2	yes	—
89	C₂×C₁₆×C₅₃	—	2	2	no	—
90	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
91	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
92	#108×C₅₃	C₂×C₄	1	1	yes	characteristic
93	#104×C₅₃	C₂×C₄	1	2	yes	—
94	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
95	#41×C₅₃	C₂²×C₄	1	2	yes	—
96	#11719949	C₂×C₈	1	2	yes	—
97	Q₈×D₅₃	C₂×C₈	1	2	yes	—
98	#11719947	—	2	2	no	—
99	#11719947	C₂×C₈	1	2	yes	—
100	D₄×D₅₃	—	2	2	no	—
101	D₄×D₅₃	C₂×C₈	1	2	yes	—
102	#11719945	—	2	2	no	—
103	#11719945	C₂²×C₄	1	2	yes	—
104	C₂×C₄×D₅₃	—	2	2	no	—
105	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
106	Q₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
107	SD₁₆×C₅₃	C₂×C₈	1	2	yes	—
108	D₈×C₅₃	C₂×C₈	1	1	yes	characteristic
109	M₁₆×C₅₃	—	2	2	no	—
110	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₅₃	—	2	2	no	—
112	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
114	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	#11719925	C₂×C₈	1	2	yes	—
116	#11719924	C₂×C₈	1	2	yes	—
117	#11719923	C₂×C₈	1	2	yes	—
118	#11719922	C₂×C₈	1	2	yes	—
119	#11719921	—	2	1	no	—
120	#11719919	—	2	1	no	—
121	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
122	#11719917	—	2	2	no	—
123	#11719917	C₂²×C₄	1	2	yes	—
124	C₂×#88322	—	2	2	no	—
125	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
126	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
127	Dic₂₁₂	C₂×C₈	1	1	yes	characteristic
128	D₄₂₄	C₂×C₈	1	1	yes	characteristic
129	#11719913	C₂×C₈	1	2	yes	—
130	#11719912	—	2	2	no	—
131	#11719912	—	2	2	no	—
132	#11719912	—	2	2	no	—
133	#11719912	C₂×C₈	1	2	yes	—
134	#11719912	C₂×C₈	1	2	yes	—
135	C₈×D₅₃	—	2	2	no	—
136	C₈×D₅₃	—	2	2	no	—
137	C₈×D₅₃	—	2	2	no	—
138	C₈×D₅₃	C₂×D₄	1	2	yes	—
139	C₈×D₅₃	#41	1	2	yes	—
140	C₁₆×C₅₃	—	2	2	no	—
141	C₁₆×C₅₃	—	2	1	no	—
142	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
143	#11719908	—	2	2	no	—
144	#11719908	—	2	1	no	—
145	#11719908	C₂×D₄	1	1	yes	characteristic
146	#11719908	C₂×D₄	1	1	yes	characteristic
147	C₂²×D₅₃	—	2	2	no	—
148	C₂²×D₅₃	#131	1	1	yes	characteristic
149	Q₈×C₅₃	C₂²×C₈	1	2	yes	—
150	D₄×C₅₃	—	2	2	no	—
151	D₄×C₅₃	C₂²×C₈	1	2	yes	—
152	C₂×C₄×C₅₃	—	2	2	no	—
153	C₂×C₄×C₅₃	—	2	1	no	—
154	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
155	C₂²⋊D₅₃	—	2	2	no	—
156	C₂²⋊D₅₃	—	2	2	no	—
157	C₂²⋊D₅₃	—	2	2	no	—
158	C₂×Dic₅₃	—	2	2	no	—
159	C₂×Dic₅₃	—	2	1	no	—
160	C₂×Dic₅₃	#131	1	1	yes	characteristic
161	D₂₁₂	—	2	2	no	—
162	D₂₁₂	C₂²×C₈	1	2	yes	—
163	C₄×D₅₃	—	2	2	no	—
164	C₄×D₅₃	—	2	2	no	—
165	C₄×D₅₃	—	2	2	no	—
166	C₄×D₅₃	C₄×D₄	1	2	yes	—
167	C₄×D₅₃	C₂²×C₈	1	2	yes	—
168	Dic₁₀₆	—	2	2	no	—
169	Dic₁₀₆	C₂²×C₈	1	2	yes	—
170	C₈×C₅₃	—	2	2	no	—
171	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
172	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
173	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
174	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
175	#88322	—	2	2	no	—
176	#88322	C₄×D₄	1	2	yes	—
177	#88322	C₂²×C₈	1	2	yes	—
178	#33442	—	53	1	no	maximal, Sylow
179	C₂²×C₅₃	—	2	2	no	—
180	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
181	C₂×D₅₃	—	2	2	no	—
182	C₂×D₅₃	—	4	1	no	—
183	C₂×D₅₃	—	8	2	no	—
184	C₂×D₅₃	C₈×D₄	1	2	yes	—
185	C₄×C₅₃	—	2	2	no	—
186	C₄×C₅₃	—	4	1	no	—
187	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
188	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
189	Dic₅₃	—	2	2	no	—
190	Dic₅₃	—	4	1	no	—
191	Dic₅₃	C₈×D₄	1	2	yes	—
192	#3407	—	53	2	no	—
193	#2955	—	53	1	no	—
194	#2171	—	53	2	no	—
195	#2171	—	53	1	no	—
196	#2171	D₅₃	1	1	yes	characteristic, complemented
197	#2170	—	53	2	no	—
198	#2170	—	53	1	no	—
199	#2170	—	53	1	no	—
200	#2153	—	53	1	no	—
201	C₂×#350	—	53	2	no	—
202	#2108	—	53	1	no	—
203	C₂×C₅₃	—	2	1	no	—
204	C₂×C₅₃	—	4	2	no	—
205	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
206	D₅₃	—	2	2	no	—
207	D₅₃	—	4	2	no	—
208	#575	—	53	1	no	—
209	C₂×#131	—	53	2	no	—
210	#504	—	53	2	no	—
211	#504	—	53	2	no	—
212	#504	—	53	2	no	—
213	#504	—	106	2	no	—
214	#504	—	106	2	no	—
215	#504	C₂×D₅₃	1	2	yes	complemented
216	C₂×M₃₂	—	53	1	no	—
217	C₂×M₃₂	—	53	2	no	—
218	C₂×M₃₂	—	106	2	no	—
219	C₂×M₃₂	—	106	2	no	—
220	C₂²×C₁₆	—	53	1	no	—
221	#444	—	53	1	no	—
222	#444	—	53	1	no	—
223	#444	—	53	2	no	—
224	#443	—	53	2	no	—
225	#443	—	53	1	no	—
226	#443	C₂×D₅₃	1	1	yes	characteristic
227	#430	—	53	1	no	—
228	#421	—	53	2	no	—
229	#405	—	53	1	no	—
230	#364	—	53	2	no	—
231	#364	—	53	1	no	—
232	#364	C₂×D₅₃	1	1	yes	characteristic, complemented
233	#350	—	53	2	no	—
234	#350	—	53	2	no	—
235	#350	—	53	2	no	—
236	#350	C₂×D₅₃	1	2	yes	complemented
237	#346	—	53	1	no	—
238	#346	C₂×D₅₃	1	1	yes	characteristic, complemented
239	C₄×C₁₆	—	53	1	no	—
240	C₄×C₁₆	—	53	1	no	—
241	C₅₃	#33442	1	1	yes	characteristic, complemented, Sylow
242	C₂×#41	—	53	2	no	—
243	#137	—	53	2	no	—
244	#137	—	53	2	no	—
245	AGL(1,ℤ/8)	—	53	2	no	—
246	AGL(1,ℤ/8)	—	53	2	no	—
247	#135	—	53	2	no	—
248	#135	—	53	1	no	—
249	#135	C₄×D₅₃	1	1	yes	characteristic
250	#131	—	53	2	no	—
251	#131	—	53	2	no	—
252	#131	—	53	2	no	—
253	#131	—	106	2	no	—
254	#131	—	106	2	no	—
255	#131	C₂²×D₅₃	1	2	yes	—
256	C₂×M₁₆	—	53	1	no	—
257	C₂×M₁₆	—	53	2	no	—
258	C₂×M₁₆	—	106	2	no	—
259	C₂²×C₈	—	53	1	no	—
260	C₂²×C₈	—	106	2	no	—
261	#117	—	53	1	no	—
262	M₃₂	—	2	2	no	—
263	M₃₂	—	53	2	no	—
264	M₃₂	—	53	2	no	—
265	M₃₂	—	53	2	no	—
266	M₃₂	—	106	2	no	—
267	M₃₂	—	106	2	no	—
268	M₃₂	—	106	2	no	—
269	M₃₂	—	106	2	no	—
270	M₃₂	C₂²×D₅₃	1	2	yes	—
271	C₂×C₁₆	—	2	2	no	—
272	C₂×C₁₆	—	53	1	no	—
273	C₂×C₁₆	—	53	1	no	—
274	C₂×C₁₆	—	106	2	no	—
275	C₂×C₁₆	—	106	2	no	—
276	C₂×C₁₆	—	106	2	no	—
277	C₂×C₁₆	—	106	2	no	—
278	C₂×C₁₆	—	106	2	no	—
279	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
280	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
281	#108	—	53	2	no	—
282	#108	—	53	1	no	—
283	#108	C₄×D₅₃	1	1	yes	characteristic
284	#104	—	53	2	no	—
285	#104	—	53	2	no	—
286	#104	—	53	2	no	—
287	#104	C₄×D₅₃	1	2	yes	—
288	#97	—	53	1	no	—
289	#97	—	53	1	no	—
290	C₄×C₈	—	53	1	no	—
291	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
292	#41	—	53	2	no	—
293	#41	—	53	2	no	—
294	#41	—	53	2	no	—
295	#41	—	106	2	no	—
296	#41	—	106	2	no	—
297	#41	C₂×C₄×D₅₃	1	2	yes	—
298	C₂×Q₈	—	53	2	no	—
299	C₂×D₄	—	53	2	no	—
300	C₂×D₄	—	106	2	no	—
301	C₂²×C₄	—	53	1	no	—
302	C₂²×C₄	—	106	2	no	—
303	Q₁₆	—	53	2	no	—
304	Q₁₆	—	53	1	no	—
305	Q₁₆	C₈×D₅₃	1	1	yes	characteristic
306	SD₁₆	—	53	2	no	—
307	SD₁₆	—	53	2	no	—
308	SD₁₆	—	53	2	no	—
309	SD₁₆	C₈×D₅₃	1	2	yes	—
310	D₈	—	53	2	no	—
311	D₈	—	53	1	no	—
312	D₈	C₈×D₅₃	1	1	yes	characteristic
313	M₁₆	—	2	2	no	—
314	M₁₆	—	53	2	no	—
315	M₁₆	—	53	2	no	—
316	M₁₆	—	53	2	no	—
317	M₁₆	—	106	2	no	—
318	M₁₆	—	106	2	no	—
319	M₁₆	—	106	2	no	—
320	M₁₆	—	106	2	no	—
321	M₁₆	C₂×C₄×D₅₃	1	2	yes	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	53	1	no	—
324	C₂×C₈	—	53	1	no	—
325	C₂×C₈	—	53	2	no	—
326	C₂×C₈	—	53	2	no	—
327	C₂×C₈	—	106	2	no	—
328	C₂×C₈	—	106	2	no	—
329	C₂×C₈	—	106	2	no	—
330	C₂×C₈	—	106	2	no	—
331	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
332	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
333	C₄⋊C₄	—	106	1	no	—
334	C₂²⋊C₄	—	106	1	no	—
335	C₄²	—	53	1	no	—
336	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
337	C₁₆	—	2	2	no	—
338	C₁₆	—	2	1	no	—
339	C₁₆	—	53	1	no	—
340	C₁₆	—	53	1	no	—
341	C₁₆	—	106	2	no	—
342	C₁₆	—	106	1	no	—
343	C₁₆	D₄×D₅₃	1	2	yes	—
344	C₂³	—	53	1	no	—
345	C₂³	—	106	2	no	—
346	Q₈	—	53	2	no	—
347	Q₈	—	106	2	no	—
348	Q₈	C₂×C₈×D₅₃	1	2	yes	—
349	D₄	—	2	2	no	—
350	D₄	—	53	2	no	—
351	D₄	—	106	2	no	—
352	D₄	—	106	2	no	—
353	D₄	—	106	2	no	—
354	D₄	—	106	2	no	—
355	D₄	C₂×C₈×D₅₃	1	2	yes	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	53	1	no	—
359	C₂×C₄	—	53	2	no	—
360	C₂×C₄	—	53	2	no	—
361	C₂×C₄	—	106	2	no	—
362	C₂×C₄	—	106	2	no	—
363	C₂×C₄	—	106	2	no	—
364	C₂×C₄	—	106	2	no	—
365	C₂×C₄	—	106	1	no	—
366	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
367	C₈	—	2	2	no	—
368	C₈	—	53	2	no	—
369	C₈	—	53	2	no	—
370	C₈	—	106	2	no	—
371	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
372	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
373	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
374	C₈	#41×D₅₃	1	1	yes	characteristic
375	C₂²	—	2	2	no	—
376	C₂²	—	53	2	no	—
377	C₂²	—	106	2	no	—
378	C₂²	—	424	2	no	—
379	C₂²	—	212	1	no	—
380	C₂²	#131×D₅₃	1	1	yes	characteristic
381	C₄	—	2	2	no	—
382	C₄	—	4	1	no	—
383	C₄	—	53	2	no	—
384	C₄	—	106	2	no	—
385	C₄	—	212	1	no	—
386	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
387	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	4	2	no	—
390	C₂	—	106	2	no	—
391	C₂	—	212	2	no	—
392	C₂	C₈×D₄×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2171	C₂
socle	C₂×C₅₃	C₂×C₈×D₄
prime radical	#16811420	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2171	D₅₃
hypocenter	C₅₃	#33442
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #2171

Complements

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

N	H ≅ G/N	type
1. #20287816	C₂	semidirect
3. #20276558	C₂	semidirect
4. #20276554	C₂	semidirect
5. #20276552	C₂	semidirect
6. #20276551	C₂	semidirect
7. #20276550	C₂	semidirect
8. #20276479	C₂	semidirect
9. #350×D₅₃	C₂	semidirect
10. #20276313	C₂	semidirect
11. C₅₃×#2171	C₂	semidirect
15. #16811041	C₂²	semidirect
16. #16811040	C₂²	semidirect
18. #16811039	C₂²	semidirect
32. #16810444	C₂²	semidirect
33. #16810443	C₂²	semidirect
34. #16810408	C₂²	semidirect
35. #16810407	C₂²	semidirect
36. #16810406	C₂²	semidirect
37. #16810398	C₂²	semidirect
38. C₄×#11719908	C₂²	semidirect
39. C₁₆×Dic₅₃	C₂²	semidirect
40. C₅₃×#504	C₂²	semidirect
42. C₅₃×#364	C₂²	semidirect
43. C₅₃×#350	C₂²	semidirect
44. C₅₃×#346	C₂²	semidirect
196. #2171	D₅₃	semidirect
215. #504	C₂×D₅₃	semidirect
232. #364	C₂×D₅₃	semidirect
236. #350	C₂×D₅₃	semidirect
238. #346	C₂×D₅₃	semidirect
241. C₅₃	#33442	semidirect

Maximal subgroups

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

subgroup	index
#20287816	2
#20283613	2
#20276558	2
#20276554	2
#20276552	2
#20276551	2
#20276550	2
#20276479	2
#350×D₅₃	2
#20276313	2
C₅₃×#2171	2
#33442	53