Subgroups — D₅₃×#1674 #27366444

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#462×D₅₃	C₂	1	2	yes	maximal
3	C₈×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20274110	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274109	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274108	C₂	1	2	yes	maximal, complemented
7	#20274107	C₂	1	2	yes	maximal, complemented
8	#20274106	C₂	1	1	yes	maximal, characteristic, complemented
9	#335×D₅₃	C₂	1	1	yes	maximal, characteristic
10	#332×D₅₃	C₂	1	2	yes	maximal
11	C₅₃×#1674	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₅₃	—	2	2	no	—
13	C₄⋊Q₈×D₅₃	C₂²	1	2	yes	—
14	C₄×Q₈×D₅₃	C₂²	1	1	yes	characteristic
15	#16811028	C₂²	1	1	yes	characteristic
16	#16811027	—	2	1	no	—
17	#16811027	—	2	1	no	—
18	#16811026	C₂²	1	2	yes	—
19	#16810834	C₂²	1	2	yes	—
20	#16810833	C₂²	1	2	yes	—
21	#16810832	—	2	2	no	—
22	#16810831	—	2	2	no	—
23	#16810830	C₂²	1	2	yes	—
24	Q₈×#88322	C₂²	1	1	yes	characteristic
25	C₈×Dic₁₀₆	—	2	1	no	—
26	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
27	#16810747	—	2	1	no	—
28	#16810747	C₂²	1	1	yes	characteristic
29	#107×D₅₃	—	2	1	no	—
30	#107×D₅₃	—	2	1	no	—
31	C₄⋊C₈×D₅₃	—	2	1	no	—
32	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
33	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
34	#103×D₅₃	—	2	2	no	—
35	C₄×C₈×D₅₃	—	2	1	no	—
36	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
37	#16810376	C₂²	1	1	yes	characteristic
38	#16810375	C₂²	1	2	yes	—
39	#16810370	C₂²	1	2	yes	—
40	#16810369	C₂²	1	2	yes	—
41	#16810368	C₂²	1	2	yes	—
42	C₅₃×#500	C₂²	1	1	yes	characteristic
43	C₅₃×#462	C₂²	1	2	yes	—
44	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#335	C₂²	1	1	yes	characteristic
46	C₅₃×#332	C₂²	1	2	yes	—
47	C₂×Q₈×D₅₃	—	2	2	no	—
48	C₂×Q₈×D₅₃	D₄	1	1	yes	characteristic
49	C₂×Dic₂₁₂	—	2	2	no	—
50	C₂×#11719925	—	2	2	no	—
51	C₂×#11719925	—	2	2	no	—
52	Q₁₆×D₅₃	—	4	8	no	—
53	C₂×C₈×D₅₃	—	2	1	no	—
54	C₂×C₈×D₅₃	—	2	2	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	#13133215	—	2	2	no	—
61	Q₈×Dic₅₃	D₄	1	1	yes	characteristic
62	C₄×Dic₁₀₆	—	2	1	no	—
63	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
64	#13133182	—	2	1	no	—
65	#13133182	D₄	1	1	yes	characteristic
66	C₄⋊C₄×D₅₃	—	2	1	no	—
67	C₄⋊C₄×D₅₃	—	2	2	no	—
68	C₄⋊C₄×D₅₃	—	2	2	no	—
69	C₄⋊C₄×D₅₃	D₄	1	1	yes	characteristic
70	C₄²×D₅₃	—	2	1	no	—
71	C₄²×D₅₃	C₂³	1	1	yes	characteristic
72	#13133151	—	2	1	no	—
73	#13133151	—	2	1	no	—
74	#13133150	—	2	2	no	—
75	#13133150	—	2	2	no	—
76	C₄⋊#88322	—	2	1	no	—
77	C₄⋊#88322	C₂³	1	2	yes	—
78	C₄⋊#88322	C₂³	1	1	yes	characteristic
79	#13133146	—	2	1	no	—
80	#13133146	—	2	2	no	—
81	#13133146	—	2	1	no	—
82	#13133146	—	2	1	no	—
83	#13133142	—	2	2	no	—
84	#13133141	—	2	2	no	—
85	#13133140	—	2	2	no	—
86	C₄×#88322	—	2	1	no	—
87	C₄×#88322	C₂³	1	1	yes	characteristic
88	C₈×Dic₅₃	—	2	1	no	—
89	C₈×Dic₅₃	—	2	1	no	—
90	C₈×Dic₅₃	D₄	1	1	yes	characteristic
91	C₈×Dic₅₃	D₄	1	1	yes	characteristic
92	C₂×Q₁₆×C₅₃	—	2	2	no	—
93	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
94	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
95	#107×C₅₃	—	2	1	no	—
96	#107×C₅₃	—	2	1	no	—
97	C₄⋊C₈×C₅₃	—	2	1	no	—
98	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
99	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
100	#103×C₅₃	—	2	2	no	—
101	C₄×C₈×C₅₃	—	2	1	no	—
102	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
103	C₂×Q₈×C₅₃	—	2	2	no	—
104	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
105	Q₈×D₅₃	—	2	2	no	—
106	Q₈×D₅₃	—	4	4	no	—
107	Q₈×D₅₃	—	4	4	no	—
108	Q₈×D₅₃	D₈	1	4	yes	—
109	C₂×C₄×D₅₃	—	2	1	no	—
110	C₂×C₄×D₅₃	—	2	1	no	—
111	C₂×C₄×D₅₃	—	4	2	no	—
112	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
115	C₂×Dic₁₀₆	—	2	1	no	—
116	C₂×Dic₁₀₆	—	2	2	no	—
117	C₂×Dic₁₀₆	—	4	2	no	—
118	C₂×Dic₁₀₆	C₂×D₄	1	1	yes	characteristic
119	Q₁₆×C₅₃	—	4	4	no	—
120	C₂×C₈×C₅₃	—	2	1	no	—
121	C₂×C₈×C₅₃	—	2	2	no	—
122	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
124	C₄⋊C₄×C₅₃	—	2	1	no	—
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	—	2	2	no	—
127	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₅₃	—	2	1	no	—
129	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
130	#11719925	—	4	4	no	—
131	#11719925	—	4	4	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	—	4	2	no	—
140	#11719919	—	4	2	no	—
141	C₄×Dic₅₃	—	2	1	no	—
142	C₄×Dic₅₃	—	2	1	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	1	no	—
148	C₂×#88322	—	2	2	no	—
149	C₂×#88322	C₂×D₄	1	1	yes	characteristic
150	C₂×#88322	C₂×D₄	1	1	yes	characteristic
151	Dic₂₁₂	—	4	4	no	—
152	C₈×D₅₃	—	2	2	no	—
153	C₈×D₅₃	—	4	4	no	—
154	C₈×D₅₃	—	4	2	no	—
155	C₈×D₅₃	Q₁₆	1	4	yes	complemented
156	C₂²×D₅₃	#121	1	1	yes	characteristic
157	Q₈×C₅₃	—	2	1	no	—
158	Q₈×C₅₃	—	4	2	no	—
159	Q₈×C₅₃	—	4	2	no	—
160	Q₈×C₅₃	C₂×D₈	1	2	yes	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	1	no	—
163	C₂×C₄×C₅₃	—	4	2	no	—
164	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
167	C₂×Dic₅₃	—	2	1	no	—
168	C₂×Dic₅₃	—	2	1	no	—
169	C₂×Dic₅₃	—	4	2	no	—
170	C₂×Dic₅₃	#121	1	1	yes	characteristic
171	C₂×Dic₅₃	#121	1	1	yes	characteristic
172	C₂×Dic₅₃	#121	1	1	yes	characteristic
173	C₄×D₅₃	—	2	4	no	—
174	C₄×D₅₃	—	2	2	no	—
175	C₄×D₅₃	—	4	2	no	—
176	C₄×D₅₃	—	8	4	no	—
177	C₄×D₅₃	C₂×D₈	1	2	yes	—
178	C₄×D₅₃	C₂×Q₁₆	1	2	yes	—
179	C₄×D₅₃	#135	1	2	yes	—
180	C₄×D₅₃	#137	1	2	yes	—
181	Dic₁₀₆	—	2	1	no	—
182	Dic₁₀₆	—	2	4	no	—
183	Dic₁₀₆	—	4	2	no	—
184	Dic₁₀₆	—	4	2	no	—
185	Dic₁₀₆	—	8	4	no	—
186	Dic₁₀₆	C₂×D₈	1	2	yes	—
187	C₈×C₅₃	—	2	1	no	—
188	C₈×C₅₃	—	4	2	no	—
189	C₈×C₅₃	—	4	1	no	—
190	C₈×C₅₃	C₂×Q₁₆	1	2	yes	complemented
191	#88322	—	2	1	no	—
192	#88322	—	4	2	no	—
193	#88322	—	4	1	no	—
194	#88322	C₂×Q₁₆	1	2	yes	complemented
195	C₂×#1674	—	53	1	no	maximal, Sylow
196	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
197	C₂×D₅₃	#462	1	2	yes	—
198	C₂×D₅₃	#466	1	2	yes	—
199	C₂×D₅₃	#473	1	2	yes	—
200	C₄×C₅₃	—	2	2	no	—
201	C₄×C₅₃	—	2	1	no	—
202	C₄×C₅₃	—	4	1	no	—
203	C₄×C₅₃	—	8	2	no	—
204	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
205	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
206	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
207	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
208	Dic₅₃	—	2	2	no	—
209	Dic₅₃	—	2	1	no	—
210	Dic₅₃	—	4	1	no	—
211	Dic₅₃	—	8	2	no	—
212	Dic₅₃	#462	1	2	yes	—
213	Dic₅₃	#466	1	2	yes	—
214	C₂×#500	—	53	1	no	—
215	C₂×#462	—	53	2	no	—
216	C₂×C₈×Q₈	—	53	1	no	—
217	#1674	—	53	1	no	—
218	#1674	—	53	2	no	—
219	#1674	—	53	1	no	—
220	#1674	—	53	1	no	—
221	#1674	—	53	2	no	—
222	#1674	D₅₃	1	1	yes	characteristic, direct product factor
223	C₂×#335	—	53	1	no	—
224	C₂×#332	—	53	2	no	—
225	C₂×C₅₃	C₂×#462	1	1	yes	characteristic
226	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
227	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
228	D₅₃	#1674	1	4	yes	direct product factor
229	C₂²×Q₁₆	—	106	2	no	—
230	C₂×C₄⋊Q₈	—	53	2	no	—
231	C₂×C₄×Q₈	—	53	1	no	—
232	#500	—	53	2	no	—
233	#500	—	53	1	no	—
234	#500	—	106	1	no	—
235	#500	—	106	1	no	—
236	#500	C₂×D₅₃	1	1	yes	characteristic
237	#462	—	53	2	no	—
238	#462	—	53	2	no	—
239	#462	—	53	2	no	—
240	#462	—	106	2	no	—
241	#462	—	106	2	no	—
242	#462	C₂×D₅₃	1	2	yes	—
243	C₈×Q₈	—	53	1	no	—
244	C₈×Q₈	—	53	1	no	—
245	C₈×Q₈	—	53	1	no	—
246	C₈×Q₈	—	106	1	no	—
247	C₈×Q₈	—	106	1	no	—
248	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
249	C₂×#107	—	106	1	no	—
250	C₂×#107	—	106	1	no	—
251	C₂×C₄⋊C₈	—	53	1	no	—
252	C₂×C₄⋊C₈	—	53	2	no	—
253	C₂×C₄⋊C₈	—	106	1	no	—
254	C₂×#103	—	106	2	no	—
255	C₂×C₄×C₈	—	53	1	no	—
256	C₂×C₄×C₈	—	106	1	no	—
257	#335	—	53	2	no	—
258	#335	—	53	1	no	—
259	#335	C₂×D₅₃	1	1	yes	characteristic
260	#332	—	53	2	no	—
261	#332	—	53	2	no	—
262	#332	—	53	2	no	—
263	#332	C₂×D₅₃	1	2	yes	—
264	C₅₃	C₂×#1674	1	1	yes	characteristic, complemented, Sylow
265	C₂²×Q₈	—	53	1	no	—
266	C₂²×Q₈	—	106	2	no	—
267	C₂×Q₁₆	—	2	2	no	—
268	C₂×Q₁₆	—	106	2	no	—
269	C₂×Q₁₆	—	106	2	no	—
270	C₂×Q₁₆	—	106	2	no	—
271	C₂×Q₁₆	—	212	8	no	—
272	C₂²×C₈	—	53	1	no	—
273	C₂²×C₈	—	53	1	no	—
274	C₂²×C₈	—	106	2	no	—
275	C₂²×C₈	—	106	1	no	—
276	C₄⋊Q₈	—	53	2	no	—
277	C₄⋊Q₈	—	106	2	no	—
278	C₄⋊Q₈	—	106	2	no	—
279	C₄⋊Q₈	—	106	2	no	—
280	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
281	C₄×Q₈	—	53	1	no	—
282	C₄×Q₈	—	53	1	no	—
283	C₄×Q₈	—	53	1	no	—
284	C₄×Q₈	—	106	1	no	—
285	C₄×Q₈	—	106	1	no	—
286	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
287	C₂×C₄⋊C₄	—	53	1	no	—
288	C₂×C₄⋊C₄	—	106	1	no	—
289	C₂×C₄⋊C₄	—	106	2	no	—
290	C₂×C₄⋊C₄	—	106	2	no	—
291	C₂×C₄²	—	53	1	no	—
292	C₂×C₄²	—	106	1	no	—
293	#107	—	2	1	no	—
294	#107	—	2	1	no	—
295	#107	—	106	2	no	—
296	#107	—	106	2	no	—
297	#107	—	106	1	no	—
298	#107	—	106	1	no	—
299	C₄⋊C₈	—	2	1	no	—
300	C₄⋊C₈	—	53	1	no	—
301	C₄⋊C₈	—	53	2	no	—
302	C₄⋊C₈	—	106	1	no	—
303	C₄⋊C₈	—	106	1	no	—
304	C₄⋊C₈	—	106	2	no	—
305	C₄⋊C₈	—	106	1	no	—
306	C₄⋊C₈	—	106	1	no	—
307	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
308	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
309	#103	—	2	2	no	—
310	#103	—	106	2	no	—
311	#103	—	106	2	no	—
312	#103	—	106	2	no	—
313	C₄×C₈	—	2	1	no	—
314	C₄×C₈	—	53	1	no	—
315	C₄×C₈	—	53	1	no	—
316	C₄×C₈	—	53	1	no	—
317	C₄×C₈	—	106	1	no	—
318	C₄×C₈	—	106	1	no	—
319	C₄×C₈	—	106	1	no	—
320	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
321	C₂×Q₈	—	2	2	no	—
322	C₂×Q₈	—	53	4	no	—
323	C₂×Q₈	—	53	1	no	—
324	C₂×Q₈	—	106	1	no	—
325	C₂×Q₈	—	106	2	no	—
326	C₂×Q₈	—	106	2	no	—
327	C₂×Q₈	—	212	2	no	—
328	C₂×Q₈	—	212	4	no	—
329	C₂×Q₈	—	212	4	no	—
330	C₂×Q₈	D₄×D₅₃	1	1	yes	characteristic
331	C₂²×C₄	—	53	1	no	—
332	C₂²×C₄	—	53	1	no	—
333	C₂²×C₄	—	53	1	no	—
334	C₂²×C₄	—	106	1	no	—
335	C₂²×C₄	—	106	1	no	—
336	C₂²×C₄	—	212	2	no	—
337	Q₁₆	—	4	4	no	—
338	Q₁₆	—	212	4	no	—
339	Q₁₆	—	212	4	no	—
340	Q₁₆	—	212	4	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	53	1	no	—
344	C₂×C₈	—	53	4	no	—
345	C₂×C₈	—	53	1	no	—
346	C₂×C₈	—	106	2	no	—
347	C₂×C₈	—	106	1	no	—
348	C₂×C₈	—	106	2	no	—
349	C₂×C₈	—	212	4	no	—
350	C₂×C₈	—	212	2	no	—
351	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
352	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	53	1	no	—
357	C₄⋊C₄	—	106	1	no	—
358	C₄⋊C₄	—	106	1	no	—
359	C₄⋊C₄	—	106	1	no	—
360	C₄⋊C₄	—	106	1	no	—
361	C₄⋊C₄	—	106	2	no	—
362	C₄⋊C₄	—	106	2	no	—
363	C₄⋊C₄	—	212	2	no	—
364	C₄⋊C₄	—	212	2	no	—
365	C₄⋊C₄	D₄×D₅₃	1	1	yes	characteristic
366	C₄²	—	2	1	no	—
367	C₄²	—	53	1	no	—
368	C₄²	—	53	1	no	—
369	C₄²	—	53	1	no	—
370	C₄²	—	106	1	no	—
371	C₄²	—	106	1	no	—
372	C₄²	—	106	1	no	—
373	C₄²	C₂³×D₅₃	1	1	yes	characteristic
374	C₂³	—	53	1	no	—
375	Q₈	—	2	1	no	—
376	Q₈	—	4	2	no	—
377	Q₈	—	4	2	no	—
378	Q₈	—	53	2	no	—
379	Q₈	—	106	4	no	—
380	Q₈	—	106	1	no	—
381	Q₈	—	424	4	no	—
382	Q₈	—	212	2	no	—
383	Q₈	—	212	2	no	—
384	Q₈	D₈×D₅₃	1	2	yes	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	53	1	no	—
389	C₂×C₄	—	53	1	no	—
390	C₂×C₄	—	53	2	no	—
391	C₂×C₄	—	53	2	no	—
392	C₂×C₄	—	53	2	no	—
393	C₂×C₄	—	53	2	no	—
394	C₂×C₄	—	53	1	no	—
395	C₂×C₄	—	106	2	no	—
396	C₂×C₄	—	106	4	no	—
397	C₂×C₄	—	106	1	no	—
398	C₂×C₄	—	106	1	no	—
399	C₂×C₄	—	424	4	no	—
400	C₂×C₄	—	212	2	no	—
401	C₂×C₄	—	212	2	no	—
402	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
403	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
404	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
405	C₈	—	2	1	no	—
406	C₈	—	4	2	no	—
407	C₈	—	4	1	no	—
408	C₈	—	53	2	no	—
409	C₈	—	106	1	no	—
410	C₈	—	212	2	no	—
411	C₈	—	212	1	no	—
412	C₈	Q₁₆×D₅₃	1	2	yes	complemented
413	C₂²	—	53	2	no	—
414	C₂²	—	53	2	no	—
415	C₂²	—	53	2	no	—
416	C₂²	#121×D₅₃	1	1	yes	characteristic
417	C₄	—	2	1	no	—
418	C₄	—	2	2	no	—
419	C₄	—	4	1	no	—
420	C₄	—	8	2	no	—
421	C₄	—	53	2	no	—
422	C₄	—	53	2	no	—
423	C₄	—	106	1	no	—
424	C₄	—	106	2	no	—
425	C₄	—	424	2	no	—
426	C₄	—	212	1	no	—
427	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
428	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
429	C₄	#135×D₅₃	1	1	yes	characteristic
430	C₄	#137×D₅₃	1	1	yes	characteristic
431	C₂	—	53	4	no	—
432	C₂	#462×D₅₃	1	1	yes	characteristic
433	C₂	#466×D₅₃	1	1	yes	characteristic
434	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×#1674	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂²×D₅₃	#121
hypercenter	#1674	D₅₃
hypocenter	C₅₃	C₂×#1674
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1674

Complements

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

N	H ≅ G/N	type
4. #20274110	C₂	semidirect
5. #20274109	C₂	semidirect
6. #20274108	C₂	semidirect
7. #20274107	C₂	semidirect
8. #20274106	C₂	semidirect
11. C₅₃×#1674	C₂	semidirect
155. C₈×D₅₃	Q₁₆	semidirect
190. C₈×C₅₃	C₂×Q₁₆	semidirect
194. #88322	C₂×Q₁₆	semidirect
222. #1674	D₅₃	direct (N × H)
228. D₅₃	#1674	direct (N × H)
264. C₅₃	C₂×#1674	semidirect
412. C₈	Q₁₆×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#500×D₅₃	2
#462×D₅₃	2
C₈×Q₈×D₅₃	2
#20274110	2
#20274109	2
#20274108	2
#20274107	2
#20274106	2
#335×D₅₃	2
#332×D₅₃	2
C₅₃×#1674	2
C₂×#1674	53

Subgroups — D53×#1674 #27366444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1674 #27366444