Subgroups — #27367304

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285858	C₂	1	2	yes	maximal
2	#494×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20283646	C₂	1	1	yes	maximal, characteristic
4	#20274210	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274209	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274206	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274084	C₂	1	2	yes	maximal, complemented
8	#20274083	C₂	1	2	yes	maximal, complemented
9	#20273460	C₂	1	1	yes	maximal, characteristic
10	#20273385	C₂	1	2	yes	maximal
11	C₅₃×#1690	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₅₃	—	2	1	no	—
13	C₂×Q₁₆×D₅₃	—	2	1	no	—
14	#16811177	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×D₅₃	C₂²	1	2	yes	—
16	#16811020	—	2	2	no	—
17	#16811003	C₂²	1	1	yes	characteristic
18	#16811002	C₂²	1	2	yes	—
19	#16811001	—	2	1	no	—
20	#16811001	—	2	1	no	—
21	#16810972	—	2	2	no	—
22	#16810955	—	2	2	no	—
23	#16810952	—	2	2	no	—
24	#16810909	C₂²	1	2	yes	—
25	#16810908	C₂²	1	2	yes	—
26	#16810905	C₂²	1	2	yes	—
27	#16810753	C₂²	1	1	yes	characteristic
28	#16810751	—	2	1	no	—
29	Q₈×#88322	C₂²	1	1	yes	characteristic
30	#16810747	—	2	1	no	—
31	#16810747	C₂²	1	1	yes	characteristic
32	#16810675	C₂²	1	2	yes	—
33	#16810648	—	2	1	no	—
34	#16810645	C₂²	1	1	yes	characteristic
35	#16810580	—	2	1	no	—
36	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
37	#16810375	C₂²	1	2	yes	—
38	#16810374	C₂²	1	1	yes	characteristic
39	#16810370	C₂²	1	2	yes	—
40	#16810369	C₂²	1	2	yes	—
41	#16810368	C₂²	1	2	yes	—
42	C₅₃×#494	C₂²	1	1	yes	characteristic
43	C₅₃×#475	C₂²	1	2	yes	—
44	C₅₃×#446	C₂²	1	1	yes	characteristic
45	C₅₃×#334	C₂²	1	1	yes	characteristic
46	C₅₃×#332	C₂²	1	2	yes	—
47	C₂×Q₈×D₅₃	—	2	2	no	—
48	C₂×Dic₂₁₂	—	2	1	no	—
49	C₂×Dic₂₁₂	—	2	1	no	—
50	C₂×#11719925	—	2	2	no	—
51	C₂×#11719925	—	2	2	no	—
52	Q₁₆×D₅₃	—	4	4	no	—
53	Q₁₆×D₅₃	—	4	4	no	—
54	C₂×#11719912	—	2	1	no	—
55	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
56	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
57	#13133217	C₂³	1	2	yes	—
58	#13133216	—	2	2	no	—
59	#13133216	—	2	2	no	—
60	#13133216	D₄	1	1	yes	characteristic
61	#13133215	—	2	2	no	—
62	#13133215	D₄	1	1	yes	characteristic
63	#13133207	—	2	1	no	—
64	#13133197	D₄	1	1	yes	characteristic
65	#13133195	—	2	1	no	—
66	#13133193	D₄	1	1	yes	characteristic
67	C₄×Dic₁₀₆	—	2	1	no	—
68	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
69	#13133174	—	2	1	no	—
70	C₄⋊C₄×D₅₃	—	2	2	no	—
71	C₄⋊C₄×D₅₃	—	2	2	no	—
72	C₄²×D₅₃	C₂³	1	1	yes	characteristic
73	#13133149	—	2	2	no	—
74	#13133148	—	2	2	no	—
75	#13133148	—	2	2	no	—
76	C₄⋊#88322	—	2	1	no	—
77	C₄⋊#88322	C₂³	1	1	yes	characteristic
78	C₄⋊#88322	C₂³	1	2	yes	—
79	#13133146	—	2	1	no	—
80	#13133146	—	2	1	no	—
81	#13133146	—	2	1	no	—
82	#13133142	—	2	2	no	—
83	#13133141	—	2	2	no	—
84	#13133140	—	2	2	no	—
85	#13133129	—	2	2	no	—
86	#13133127	—	2	1	no	—
87	C₄×#88322	—	2	1	no	—
88	C₄×#88322	C₂³	1	1	yes	characteristic
89	C₈×Dic₅₃	—	2	2	no	—
90	C₈×Dic₅₃	—	2	1	no	—
91	C₈×Dic₅₃	D₄	1	1	yes	characteristic
92	C₈×Dic₅₃	D₄	1	1	yes	characteristic
93	C₂×Q₁₆×C₅₃	—	2	1	no	—
94	C₂×Q₁₆×C₅₃	—	2	1	no	—
95	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
96	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
97	#106×C₅₃	—	2	2	no	—
98	C₄⋊C₈×C₅₃	—	2	1	no	—
99	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
101	#103×C₅₃	—	2	2	no	—
102	#97×C₅₃	—	2	1	no	—
103	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
104	C₂×Q₈×C₅₃	—	2	2	no	—
105	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
106	Q₈×D₅₃	—	4	4	no	—
107	Q₈×D₅₃	—	4	4	no	—
108	C₂×C₄×D₅₃	—	4	2	no	—
109	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
112	C₂×Dic₁₀₆	—	2	1	no	—
113	C₂×Dic₁₀₆	—	2	2	no	—
114	C₂×Dic₁₀₆	—	4	2	no	—
115	C₂×Dic₁₀₆	C₂×D₄	1	1	yes	characteristic
116	Q₁₆×C₅₃	—	4	2	no	—
117	Q₁₆×C₅₃	—	4	2	no	—
118	C₂×C₈×C₅₃	—	2	1	no	—
119	C₂×C₈×C₅₃	—	2	2	no	—
120	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₄⋊C₄×C₅₃	—	2	1	no	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	2	no	—
125	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₅₃	—	2	1	no	—
127	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
128	#11719925	—	4	4	no	—
129	#11719925	—	4	4	no	—
130	#11719921	—	2	1	no	—
131	#11719921	—	2	1	no	—
132	C₄⋊Dic₅₃	—	2	1	no	—
133	C₄⋊Dic₅₃	—	2	2	no	—
134	C₄⋊Dic₅₃	—	2	2	no	—
135	C₄⋊Dic₅₃	C₂×D₄	1	1	yes	characteristic
136	#11719919	—	2	1	no	—
137	#11719919	—	2	1	no	—
138	#11719919	—	2	1	no	—
139	#11719919	—	2	1	no	—
140	#11719919	—	2	1	no	—
141	#11719919	—	4	2	no	—
142	#11719919	—	4	2	no	—
143	C₄×Dic₅₃	—	2	1	no	—
144	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
145	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
146	C₄×Dic₅₃	#41	1	1	yes	characteristic
147	C₂×#88322	—	2	2	no	—
148	C₂×#88322	—	2	1	no	—
149	C₂×#88322	C₂×D₄	1	1	yes	characteristic
150	C₂×#88322	C₂×D₄	1	1	yes	characteristic
151	Dic₂₁₂	—	4	2	no	—
152	Dic₂₁₂	—	4	2	no	—
153	#11719912	—	4	2	no	—
154	C₈×D₅₃	—	2	2	no	—
155	C₈×D₅₃	SD₁₆	1	4	yes	—
156	C₂²×D₅₃	#121	1	1	yes	characteristic
157	Q₈×C₅₃	—	2	2	no	—
158	Q₈×C₅₃	—	4	2	no	—
159	Q₈×C₅₃	—	4	2	no	—
160	C₂×C₄×C₅₃	—	2	1	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	4	2	no	—
163	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
166	C₂×Dic₅₃	—	2	1	no	—
167	C₂×Dic₅₃	—	2	1	no	—
168	C₂×Dic₅₃	—	4	2	no	—
169	C₂×Dic₅₃	#121	1	1	yes	characteristic
170	C₂×Dic₅₃	#121	1	1	yes	characteristic
171	C₂×Dic₅₃	#121	1	1	yes	characteristic
172	C₄×D₅₃	—	2	2	no	—
173	C₄×D₅₃	—	2	2	no	—
174	C₄×D₅₃	—	8	4	no	—
175	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
176	C₄×D₅₃	AGL(1,ℤ/8)	1	2	yes	—
177	Dic₁₀₆	—	2	2	no	—
178	Dic₁₀₆	—	4	2	no	—
179	Dic₁₀₆	—	4	2	no	—
180	Dic₁₀₆	—	4	2	no	—
181	Dic₁₀₆	—	8	4	no	—
182	C₈×C₅₃	—	2	1	no	—
183	C₈×C₅₃	—	4	1	no	—
184	C₈×C₅₃	—	4	2	no	—
185	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
186	#88322	—	2	1	no	—
187	#88322	—	4	1	no	—
188	#88322	—	4	2	no	—
189	#88322	C₂×SD₁₆	1	2	yes	—
190	#19204	—	53	1	no	maximal, Sylow
191	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
192	C₂×D₅₃	—	2	1	no	—
193	C₂×D₅₃	—	2	1	no	—
194	C₂×D₅₃	#460	1	2	yes	—
195	C₄×C₅₃	—	2	1	no	—
196	C₄×C₅₃	—	4	1	no	—
197	C₄×C₅₃	—	4	1	no	—
198	C₄×C₅₃	—	8	2	no	—
199	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
200	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
201	C₄×C₅₃	#578	1	2	yes	—
202	Dic₅₃	—	2	1	no	—
203	Dic₅₃	—	4	1	no	—
204	Dic₅₃	—	4	1	no	—
205	Dic₅₃	—	8	2	no	—
206	Dic₅₃	#460	1	2	yes	—
207	Dic₅₃	#473	1	2	yes	—
208	#3229	—	53	2	no	—
209	C₂×#494	—	53	1	no	—
210	#2960	—	53	1	no	—
211	#1690	—	53	1	no	—
212	#1690	—	53	1	no	—
213	#1690	—	53	1	no	—
214	#1690	D₅₃	1	1	yes	characteristic, complemented
215	#1670	—	53	2	no	—
216	#1670	—	53	2	no	—
217	#1564	—	53	1	no	—
218	#1546	—	53	2	no	—
219	C₂×C₅₃	C₂×#460	1	1	yes	characteristic
220	C₂×C₅₃	#3059	1	1	yes	characteristic
221	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
222	D₅₃	—	2	2	no	—
223	C₂²×Q₁₆	—	106	1	no	—
224	C₂²×Q₁₆	—	106	1	no	—
225	#533	—	53	1	no	—
226	C₂×C₄⋊Q₈	—	53	2	no	—
227	#498	—	106	2	no	—
228	#494	—	53	2	no	—
229	#494	—	53	1	no	—
230	#494	—	106	1	no	—
231	#494	—	106	1	no	—
232	#494	C₂×D₅₃	1	1	yes	characteristic
233	#487	—	106	2	no	—
234	#483	—	106	2	no	—
235	#483	—	106	2	no	—
236	#475	—	53	2	no	—
237	#475	—	53	2	no	—
238	#475	—	53	2	no	—
239	#475	C₂×D₅₃	1	2	yes	—
240	#446	—	53	1	no	—
241	#446	—	106	1	no	—
242	#446	C₂×D₅₃	1	1	yes	characteristic
243	C₈×Q₈	—	53	1	no	—
244	C₈×Q₈	—	53	1	no	—
245	C₈×Q₈	—	106	1	no	—
246	#431	—	53	2	no	—
247	#424	—	106	1	no	—
248	#423	—	53	1	no	—
249	#405	—	106	1	no	—
250	C₂×C₄×C₈	—	53	1	no	—
251	#335	—	53	2	no	—
252	#334	—	53	1	no	—
253	#334	C₂×D₅₃	1	1	yes	characteristic
254	#332	—	53	2	no	—
255	#332	—	53	2	no	—
256	#332	—	53	2	no	—
257	#332	C₂×D₅₃	1	2	yes	—
258	C₅₃	#19204	1	1	yes	characteristic, complemented, Sylow
259	C₂²×Q₈	—	106	2	no	—
260	C₂×Q₁₆	—	2	1	no	—
261	C₂×Q₁₆	—	2	1	no	—
262	C₂×Q₁₆	—	106	2	no	—
263	C₂×Q₁₆	—	106	2	no	—
264	C₂×Q₁₆	—	106	1	no	—
265	C₂×Q₁₆	—	106	1	no	—
266	C₂×Q₁₆	—	212	4	no	—
267	C₂×Q₁₆	—	212	4	no	—
268	C₂×M₁₆	—	106	1	no	—
269	C₂²×C₈	—	53	1	no	—
270	C₂²×C₈	—	53	1	no	—
271	C₄⋊Q₈	—	53	1	no	—
272	C₄⋊Q₈	—	53	1	no	—
273	C₄⋊Q₈	—	53	2	no	—
274	C₄⋊Q₈	—	106	2	no	—
275	C₄⋊Q₈	—	106	2	no	—
276	C₄⋊Q₈	—	106	2	no	—
277	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
278	#125	—	106	1	no	—
279	C₂²⋊Q₈	—	53	1	no	—
280	C₂²⋊Q₈	—	53	1	no	—
281	C₂²⋊Q₈	—	106	1	no	—
282	C₄×Q₈	—	53	1	no	—
283	C₄×Q₈	—	106	1	no	—
284	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
285	#117	—	106	1	no	—
286	C₂×C₄⋊C₄	—	106	2	no	—
287	C₂×C₄⋊C₄	—	106	2	no	—
288	C₂×C₄²	—	53	1	no	—
289	#106	—	2	2	no	—
290	#106	—	106	2	no	—
291	#106	—	106	2	no	—
292	#106	—	106	2	no	—
293	C₄⋊C₈	—	2	1	no	—
294	C₄⋊C₈	—	53	1	no	—
295	C₄⋊C₈	—	53	2	no	—
296	C₄⋊C₈	—	106	1	no	—
297	C₄⋊C₈	—	106	1	no	—
298	C₄⋊C₈	—	106	1	no	—
299	C₄⋊C₈	—	106	1	no	—
300	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
301	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
302	#103	—	2	2	no	—
303	#103	—	106	2	no	—
304	#103	—	106	2	no	—
305	#103	—	106	2	no	—
306	C₂²⋊C₈	—	106	2	no	—
307	#97	—	2	1	no	—
308	#97	—	106	1	no	—
309	C₄×C₈	—	53	1	no	—
310	C₄×C₈	—	53	1	no	—
311	C₄×C₈	—	53	1	no	—
312	C₄×C₈	—	106	1	no	—
313	C₄×C₈	—	106	2	no	—
314	C₄×C₈	—	106	1	no	—
315	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
316	C₂×Q₈	—	2	2	no	—
317	C₂×Q₈	—	53	1	no	—
318	C₂×Q₈	—	106	1	no	—
319	C₂×Q₈	—	106	2	no	—
320	C₂×Q₈	—	212	2	no	—
321	C₂×Q₈	—	212	4	no	—
322	C₂×Q₈	—	212	4	no	—
323	C₂×Q₈	D₄×D₅₃	1	1	yes	characteristic
324	C₂²×C₄	—	53	1	no	—
325	C₂²×C₄	—	53	1	no	—
326	C₂²×C₄	—	53	1	no	—
327	C₂²×C₄	—	212	2	no	—
328	Q₁₆	—	4	2	no	—
329	Q₁₆	—	4	2	no	—
330	Q₁₆	—	212	2	no	—
331	Q₁₆	—	212	4	no	—
332	Q₁₆	—	212	4	no	—
333	Q₁₆	—	212	2	no	—
334	M₁₆	—	212	2	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	53	1	no	—
338	C₂×C₈	—	53	4	no	—
339	C₂×C₈	—	53	1	no	—
340	C₂×C₈	—	106	2	no	—
341	C₂×C₈	—	106	1	no	—
342	C₂×C₈	—	106	2	no	—
343	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
344	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	53	1	no	—
349	C₄⋊C₄	—	106	1	no	—
350	C₄⋊C₄	—	106	1	no	—
351	C₄⋊C₄	—	106	1	no	—
352	C₄⋊C₄	—	106	1	no	—
353	C₄⋊C₄	—	106	1	no	—
354	C₄⋊C₄	—	106	1	no	—
355	C₄⋊C₄	—	106	2	no	—
356	C₄⋊C₄	—	106	2	no	—
357	C₄⋊C₄	—	212	2	no	—
358	C₄⋊C₄	—	212	2	no	—
359	C₄⋊C₄	D₄×D₅₃	1	1	yes	characteristic
360	C₂²⋊C₄	—	106	1	no	—
361	C₂²⋊C₄	—	106	1	no	—
362	C₄²	—	2	1	no	—
363	C₄²	—	53	1	no	—
364	C₄²	—	53	1	no	—
365	C₄²	—	53	1	no	—
366	C₄²	—	106	1	no	—
367	C₄²	C₂³×D₅₃	1	1	yes	characteristic
368	C₂³	—	53	1	no	—
369	Q₈	—	2	2	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	4	2	no	—
372	Q₈	—	106	2	no	—
373	Q₈	—	424	4	no	—
374	Q₈	—	212	2	no	—
375	Q₈	—	212	2	no	—
376	Q₈	—	212	2	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	53	1	no	—
381	C₂×C₄	—	53	1	no	—
382	C₂×C₄	—	53	2	no	—
383	C₂×C₄	—	53	2	no	—
384	C₂×C₄	—	53	1	no	—
385	C₂×C₄	—	106	2	no	—
386	C₂×C₄	—	106	2	no	—
387	C₂×C₄	—	106	1	no	—
388	C₂×C₄	—	106	1	no	—
389	C₂×C₄	—	424	4	no	—
390	C₂×C₄	—	212	2	no	—
391	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
392	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
393	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
394	C₈	—	2	1	no	—
395	C₈	—	4	1	no	—
396	C₈	—	4	2	no	—
397	C₈	—	53	2	no	—
398	C₈	—	106	1	no	—
399	C₈	—	212	1	no	—
400	C₈	—	212	2	no	—
401	C₈	SD₁₆×D₅₃	1	2	yes	—
402	C₂²	—	53	2	no	—
403	C₂²	—	106	1	no	—
404	C₂²	—	106	1	no	—
405	C₂²	#121×D₅₃	1	1	yes	characteristic
406	C₄	—	2	1	no	—
407	C₄	—	4	1	no	—
408	C₄	—	4	1	no	—
409	C₄	—	8	2	no	—
410	C₄	—	53	2	no	—
411	C₄	—	53	2	no	—
412	C₄	—	106	1	no	—
413	C₄	—	424	2	no	—
414	C₄	—	212	1	no	—
415	C₄	—	212	1	no	—
416	C₄	C₂×SD₁₆×D₅₃	1	1	yes	characteristic
417	C₄	AGL(1,ℤ/8)×D₅₃	1	1	yes	characteristic
418	C₄	#16811439	1	2	yes	—
419	C₂	—	106	2	no	—
420	C₂	#460×D₅₃	1	1	yes	characteristic
421	C₂	#20284368	1	1	yes	characteristic
422	C₂	#473×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1690	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂²×D₅₃	#121
hypercenter	#1690	D₅₃
hypocenter	C₅₃	#19204
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1690

Complements

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

N	H ≅ G/N	type
4. #20274210	C₂	semidirect
5. #20274209	C₂	semidirect
6. #20274206	C₂	semidirect
7. #20274084	C₂	semidirect
8. #20274083	C₂	semidirect
11. C₅₃×#1690	C₂	semidirect
214. #1690	D₅₃	semidirect
258. C₅₃	#19204	semidirect

Maximal subgroups

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

subgroup	index
#20285858	2
#494×D₅₃	2
#20283646	2
#20274210	2
#20274209	2
#20274206	2
#20274084	2
#20274083	2
#20273460	2
#20273385	2
C₅₃×#1690	2
#19204	53