Subgroups — D₅₃×#1950 #27387608

Order 13568.

938 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108×D₅₃	C₂	1	2	yes	maximal
2	#423×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
5	#20275491	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275490	C₂	1	1	yes	maximal, characteristic, complemented
7	#20275489	C₂	1	2	yes	maximal, complemented
8	#20275488	C₂	1	1	yes	maximal, characteristic, complemented
9	#20275487	C₂	1	2	yes	maximal, complemented
10	#341×D₅₃	C₂	1	2	yes	maximal
11	C₅₃×#1950	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×D₅₃	C₂²	1	2	yes	—
13	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
16	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
17	C₂×#13133153	C₂²	1	2	yes	—
18	C₂×#13133152	C₂²	1	2	yes	—
19	C₂×#13133152	C₂²	1	2	yes	—
20	#108×D₅₃	—	2	8	no	—
21	#16810647	C₂²	1	1	yes	characteristic
22	#16810646	C₂²	1	2	yes	—
23	#16810645	—	2	1	no	—
24	#16810645	—	2	1	no	—
25	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
26	C₂×#13133146	C₂²	1	2	yes	—
27	C₄⋊C₈×D₅₃	—	2	1	no	—
28	C₄⋊C₈×D₅₃	—	2	1	no	—
29	C₄⋊C₈×D₅₃	C₄	1	2	yes	—
30	C₄⋊C₈×D₅₃	C₄	1	2	yes	—
31	C₄⋊C₈×D₅₃	C₄	1	2	yes	—
32	C₄⋊C₈×D₅₃	C₄	1	2	yes	—
33	C₄×#11719917	C₂²	1	1	yes	characteristic
34	#16810575	—	2	1	no	—
35	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
36	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
37	C₄×#11719912	—	2	1	no	—
38	#97×D₅₃	—	2	1	no	—
39	C₄×C₈×D₅₃	—	2	1	no	—
40	#16810388	C₂²	1	2	yes	—
41	#16810387	C₂²	1	2	yes	—
42	#16810387	C₂²	1	2	yes	—
43	C₂×#108×C₅₃	C₂²	1	2	yes	—
44	C₅₃×#423	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
46	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×#341	C₂²	1	2	yes	—
48	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
49	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
50	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
51	C₂×#11719917	C₂³	1	2	yes	—
52	C₂×#11719917	C₂³	1	1	yes	characteristic
53	C₂×#11719917	C₂³	1	1	yes	characteristic
54	C₂×#11719912	—	2	1	no	—
55	C₂×#11719912	—	2	1	no	—
56	C₂×#11719912	—	2	2	no	—
57	M₁₆×D₅₃	—	2	4	no	—
58	M₁₆×D₅₃	—	2	2	no	—
59	M₁₆×D₅₃	—	2	2	no	—
60	M₁₆×D₅₃	—	2	4	no	—
61	M₁₆×D₅₃	D₄	1	4	yes	—
62	M₁₆×D₅₃	Q₈	1	4	yes	—
63	C₂²×#88322	C₂³	1	2	yes	—
64	C₂×C₈×D₅₃	—	2	1	no	—
65	C₂×C₈×D₅₃	—	2	1	no	—
66	C₂×C₈×D₅₃	—	2	4	no	—
67	C₂×C₈×D₅₃	—	2	2	no	—
68	C₂×C₈×D₅₃	—	2	4	no	—
69	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
70	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
71	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
72	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
73	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
74	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
75	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
76	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
77	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
78	C₄²×D₅₃	D₄	1	1	yes	characteristic
79	C₄²×D₅₃	D₄	1	1	yes	characteristic
80	C₄²×D₅₃	D₄	1	1	yes	characteristic
81	C₄²×D₅₃	D₄	1	1	yes	characteristic
82	#13133153	—	2	4	no	—
83	#13133152	—	2	4	no	—
84	#13133152	—	2	4	no	—
85	C₄⋊#88322	—	2	1	no	—
86	C₄⋊#88322	—	2	1	no	—
87	C₄⋊#88322	C₂×C₄	1	2	yes	—
88	C₄⋊#88322	C₂×C₄	1	2	yes	—
89	#13133146	—	2	2	no	—
90	#13133146	—	2	2	no	—
91	#13133146	C₂×C₄	1	4	yes	—
92	#13133146	C₂×C₄	1	4	yes	—
93	#13133128	—	2	1	no	—
94	#13133127	—	2	1	no	—
95	#13133127	—	2	1	no	—
96	C₄×#88322	—	2	1	no	—
97	C₈×Dic₅₃	—	2	1	no	—
98	C₈×Dic₅₃	—	2	1	no	—
99	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
100	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
101	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
102	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
103	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
104	#108×C₅₃	—	2	4	no	—
105	C₄⋊C₈×C₅₃	—	2	1	no	—
106	C₄⋊C₈×C₅₃	—	2	1	no	—
107	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
108	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
109	#97×C₅₃	—	2	1	no	—
110	C₄×C₈×C₅₃	—	2	1	no	—
111	C₂³×D₅₃	C₂×Q₈	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
115	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
116	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
117	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
118	C₂×C₄×D₅₃	—	2	4	no	—
119	C₂×C₄×D₅₃	—	4	2	no	—
120	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
131	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
134	C₂×C₄×D₅₃	#41	1	2	yes	—
135	C₂×C₄×D₅₃	#41	1	2	yes	—
136	M₁₆×C₅₃	—	2	1	no	—
137	M₁₆×C₅₃	—	2	1	no	—
138	M₁₆×C₅₃	—	2	2	no	—
139	M₁₆×C₅₃	—	2	2	no	—
140	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
141	M₁₆×C₅₃	C₂×Q₈	1	2	yes	—
142	C₂×C₈×C₅₃	—	2	1	no	—
143	C₂×C₈×C₅₃	—	2	1	no	—
144	C₂×C₈×C₅₃	—	2	2	no	—
145	C₂×C₈×C₅₃	—	2	2	no	—
146	C₂×C₈×C₅₃	—	2	2	no	—
147	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
148	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
152	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
153	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
154	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
155	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
156	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
157	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
158	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
159	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
160	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
162	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
163	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
164	#11719917	—	2	1	no	—
165	#11719917	—	2	1	no	—
166	#11719917	—	2	2	no	—
167	#11719917	—	2	2	no	—
168	#11719917	C₂×D₄	1	2	yes	—
169	#11719917	C₂×Q₈	1	2	yes	—
170	C₂×#88322	—	2	1	no	—
171	C₂×#88322	—	2	1	no	—
172	C₂×#88322	—	2	2	no	—
173	C₂×#88322	—	2	2	no	—
174	C₂×#88322	—	2	2	no	—
175	C₂×#88322	C₂²×C₄	1	4	yes	—
176	#11719912	—	2	4	no	—
177	#11719912	—	2	4	no	—
178	#11719912	—	4	4	no	—
179	C₈×D₅₃	—	2	8	no	—
180	C₈×D₅₃	—	2	8	no	—
181	C₈×D₅₃	—	2	4	no	—
182	C₈×D₅₃	—	2	4	no	—
183	C₈×D₅₃	—	4	4	no	—
184	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
185	C₂²×D₅₃	—	2	2	no	—
186	C₂²×D₅₃	—	2	2	no	—
187	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂²×D₅₃	#125	1	2	yes	—
190	C₂²×D₅₃	C₄⋊Q₈	1	2	yes	—
191	C₂×C₄×C₅₃	—	2	2	no	—
192	C₂×C₄×C₅₃	—	4	1	no	—
193	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
198	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
201	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
202	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
203	C₂×Dic₅₃	—	2	2	no	—
204	C₂×Dic₅₃	—	4	1	no	—
205	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
206	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
207	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
208	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
210	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
211	C₂×Dic₅₃	#125	1	2	yes	—
212	C₂×Dic₅₃	C₄⋊Q₈	1	2	yes	—
213	C₄×D₅₃	—	2	2	no	—
214	C₄×D₅₃	—	2	2	no	—
215	C₄×D₅₃	—	2	2	no	—
216	C₄×D₅₃	—	2	2	no	—
217	C₄×D₅₃	—	2	2	no	—
218	C₄×D₅₃	—	4	1	no	—
219	C₄×D₅₃	—	4	1	no	—
220	C₄×D₅₃	#108	1	4	yes	complemented
221	C₄×D₅₃	C₄×D₄	1	2	yes	—
222	C₄×D₅₃	C₄×D₄	1	2	yes	—
223	C₄×D₅₃	C₄×Q₈	1	2	yes	—
224	C₄×D₅₃	C₄×Q₈	1	2	yes	—
225	C₄×D₅₃	#121	1	2	yes	—
226	C₄×D₅₃	#121	1	2	yes	—
227	C₄×D₅₃	C₂²⋊Q₈	1	2	yes	—
228	C₄×D₅₃	C₂²⋊Q₈	1	2	yes	—
229	C₈×C₅₃	—	2	4	no	—
230	C₈×C₅₃	—	2	2	no	—
231	C₈×C₅₃	—	2	2	no	—
232	C₈×C₅₃	—	4	2	no	—
233	#88322	—	2	4	no	—
234	#88322	—	2	2	no	—
235	#88322	—	2	2	no	—
236	#88322	—	4	2	no	—
237	C₂×#1950	—	53	1	no	maximal, Sylow
238	C₂²×C₅₃	—	2	1	no	—
239	C₂²×C₅₃	—	2	1	no	—
240	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
241	C₂²×C₅₃	C₂×#125	1	1	yes	characteristic
242	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
243	C₂×D₅₃	—	2	1	no	—
244	C₂×D₅₃	—	2	1	no	—
245	C₂×D₅₃	—	2	4	no	—
246	C₂×D₅₃	—	2	4	no	—
247	C₂×D₅₃	#389	1	2	yes	—
248	C₂×D₅₃	#389	1	2	yes	—
249	C₂×D₅₃	C₂×#108	1	2	yes	—
250	C₂×D₅₃	#430	1	2	yes	—
251	C₄×C₅₃	—	2	1	no	—
252	C₄×C₅₃	—	2	1	no	—
253	C₄×C₅₃	—	2	1	no	—
254	C₄×C₅₃	C₂×#108	1	2	yes	complemented
255	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
256	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
257	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
258	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
259	Dic₅₃	—	2	1	no	—
260	Dic₅₃	—	2	1	no	—
261	Dic₅₃	—	2	1	no	—
262	Dic₅₃	#389	1	2	yes	—
263	Dic₅₃	#389	1	2	yes	—
264	Dic₅₃	C₂×#108	1	2	yes	complemented
265	C₂²×#108	—	53	2	no	—
266	C₂×#423	—	53	1	no	—
267	C₂²×C₄⋊C₈	—	53	1	no	—
268	C₂×C₄×M₁₆	—	53	1	no	—
269	#1950	—	53	2	no	—
270	#1950	—	53	1	no	—
271	#1950	—	53	1	no	—
272	#1950	—	53	1	no	—
273	#1950	—	53	2	no	—
274	#1950	D₅₃	1	1	yes	characteristic, direct product factor
275	C₂×#341	—	53	2	no	—
276	C₂×C₅₃	—	2	1	no	—
277	C₂×C₅₃	—	2	1	no	—
278	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
279	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
280	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
281	D₅₃	—	2	2	no	—
282	D₅₃	#1950	1	4	yes	direct product factor
283	C₂²×M₁₆	—	53	1	no	—
284	C₂²×M₁₆	—	53	1	no	—
285	C₂²×M₁₆	—	53	2	no	—
286	C₂³×C₈	—	53	2	no	—
287	C₂²×C₄²	—	53	1	no	—
288	C₂×#108	—	53	2	no	—
289	C₂×#108	—	53	2	no	—
290	C₂×#108	—	53	2	no	—
291	C₂×#108	—	106	8	no	—
292	C₂×#108	C₂×D₅₃	1	2	yes	—
293	#423	—	53	2	no	—
294	#423	—	53	1	no	—
295	#423	—	106	1	no	—
296	#423	—	106	1	no	—
297	#423	C₂×D₅₃	1	1	yes	characteristic
298	C₂×C₄⋊C₈	—	53	2	no	—
299	C₂×C₄⋊C₈	—	53	1	no	—
300	C₂×C₄⋊C₈	—	53	2	no	—
301	C₂×C₄⋊C₈	—	53	2	no	—
302	C₂×C₄⋊C₈	—	53	2	no	—
303	C₂×C₄⋊C₈	—	53	2	no	—
304	C₂×C₄⋊C₈	—	106	1	no	—
305	C₂×C₄⋊C₈	—	106	1	no	—
306	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
307	C₄×M₁₆	—	53	1	no	—
308	C₄×M₁₆	—	53	1	no	—
309	C₄×M₁₆	—	53	1	no	—
310	C₄×M₁₆	—	106	1	no	—
311	C₄×M₁₆	—	106	1	no	—
312	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
313	C₂×#97	—	106	1	no	—
314	C₂×C₄×C₈	—	106	1	no	—
315	#341	—	53	2	no	—
316	#341	—	53	2	no	—
317	#341	—	53	2	no	—
318	#341	C₂×D₅₃	1	2	yes	—
319	C₅₃	C₂×#1950	1	1	yes	characteristic, complemented, Sylow
320	C₂³×C₄	—	53	1	no	—
321	C₂³×C₄	—	53	1	no	—
322	C₂³×C₄	—	53	1	no	—
323	C₂×M₁₆	—	53	1	no	—
324	C₂×M₁₆	—	53	1	no	—
325	C₂×M₁₆	—	53	4	no	—
326	C₂×M₁₆	—	53	4	no	—
327	C₂×M₁₆	—	53	2	no	—
328	C₂×M₁₆	—	106	1	no	—
329	C₂×M₁₆	—	106	1	no	—
330	C₂×M₁₆	—	106	2	no	—
331	C₂×M₁₆	—	106	2	no	—
332	C₂×M₁₆	—	106	2	no	—
333	C₂×M₁₆	—	106	4	no	—
334	C₂×M₁₆	—	106	4	no	—
335	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
336	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
337	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
338	C₂²×C₈	—	53	2	no	—
339	C₂²×C₈	—	53	4	no	—
340	C₂²×C₈	—	53	4	no	—
341	C₂²×C₈	—	106	4	no	—
342	C₂²×C₈	—	106	4	no	—
343	C₂²×C₈	—	106	1	no	—
344	C₂²×C₈	—	106	1	no	—
345	C₂²×C₈	—	106	2	no	—
346	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
347	C₂×C₄²	—	53	1	no	—
348	C₂×C₄²	—	53	1	no	—
349	C₂×C₄²	—	53	1	no	—
350	C₂×C₄²	—	53	1	no	—
351	C₂×C₄²	—	53	1	no	—
352	C₂×C₄²	—	53	1	no	—
353	C₂×C₄²	—	53	1	no	—
354	C₂×C₄²	—	53	1	no	—
355	C₂×C₄²	—	53	1	no	—
356	C₂×C₄²	—	53	1	no	—
357	C₂×C₄²	—	53	1	no	—
358	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
359	#108	—	2	4	no	—
360	#108	—	106	4	no	—
361	#108	—	106	4	no	—
362	#108	—	106	4	no	—
363	C₄⋊C₈	—	2	1	no	—
364	C₄⋊C₈	—	2	1	no	—
365	C₄⋊C₈	—	53	4	no	—
366	C₄⋊C₈	—	53	4	no	—
367	C₄⋊C₈	—	53	2	no	—
368	C₄⋊C₈	—	53	2	no	—
369	C₄⋊C₈	—	106	2	no	—
370	C₄⋊C₈	—	106	2	no	—
371	C₄⋊C₈	—	106	1	no	—
372	C₄⋊C₈	—	106	1	no	—
373	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
374	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
375	#97	—	2	1	no	—
376	#97	—	106	1	no	—
377	#97	—	106	1	no	—
378	#97	—	106	1	no	—
379	C₄×C₈	—	2	1	no	—
380	C₄×C₈	—	106	1	no	—
381	C₄×C₈	—	106	1	no	—
382	C₄×C₈	—	106	1	no	—
383	C₂⁴	—	53	1	no	—
384	C₂²×C₄	—	53	1	no	—
385	C₂²×C₄	—	53	1	no	—
386	C₂²×C₄	—	53	1	no	—
387	C₂²×C₄	—	53	1	no	—
388	C₂²×C₄	—	53	2	no	—
389	C₂²×C₄	—	53	1	no	—
390	C₂²×C₄	—	53	1	no	—
391	C₂²×C₄	—	53	2	no	—
392	C₂²×C₄	—	53	2	no	—
393	C₂²×C₄	—	53	2	no	—
394	C₂²×C₄	—	53	1	no	—
395	C₂²×C₄	—	53	1	no	—
396	C₂²×C₄	—	53	1	no	—
397	C₂²×C₄	—	53	1	no	—
398	C₂²×C₄	—	53	1	no	—
399	C₂²×C₄	—	53	1	no	—
400	C₂²×C₄	—	53	1	no	—
401	C₂²×C₄	—	53	1	no	—
402	C₂²×C₄	—	53	1	no	—
403	C₂²×C₄	—	106	4	no	—
404	C₂²×C₄	—	212	2	no	—
405	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
406	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
407	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
408	M₁₆	—	2	1	no	—
409	M₁₆	—	2	1	no	—
410	M₁₆	—	2	2	no	—
411	M₁₆	—	2	2	no	—
412	M₁₆	—	53	2	no	—
413	M₁₆	—	53	2	no	—
414	M₁₆	—	106	4	no	—
415	M₁₆	—	106	4	no	—
416	M₁₆	—	106	1	no	—
417	M₁₆	—	106	1	no	—
418	M₁₆	—	106	2	no	—
419	M₁₆	—	106	2	no	—
420	M₁₆	—	212	4	no	—
421	M₁₆	D₄×D₅₃	1	2	yes	—
422	M₁₆	Q₈×D₅₃	1	2	yes	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	53	4	no	—
429	C₂×C₈	—	106	8	no	—
430	C₂×C₈	—	106	8	no	—
431	C₂×C₈	—	106	4	no	—
432	C₂×C₈	—	106	4	no	—
433	C₂×C₈	—	106	2	no	—
434	C₂×C₈	—	106	1	no	—
435	C₂×C₈	—	106	1	no	—
436	C₂×C₈	—	106	2	no	—
437	C₂×C₈	—	106	2	no	—
438	C₂×C₈	—	212	4	no	—
439	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
440	C₄²	—	53	1	no	—
441	C₄²	—	53	1	no	—
442	C₄²	—	53	1	no	—
443	C₄²	—	53	1	no	—
444	C₄²	—	53	1	no	—
445	C₄²	—	53	1	no	—
446	C₄²	—	53	1	no	—
447	C₄²	—	53	1	no	—
448	C₄²	—	53	1	no	—
449	C₄²	—	53	1	no	—
450	C₄²	—	53	1	no	—
451	C₄²	—	53	1	no	—
452	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
453	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
454	C₄²	D₄×D₅₃	1	1	yes	characteristic
455	C₄²	D₄×D₅₃	1	1	yes	characteristic
456	C₂³	—	53	1	no	—
457	C₂³	—	53	1	no	—
458	C₂³	—	53	2	no	—
459	C₂³	—	53	2	no	—
460	C₂³	—	106	2	no	—
461	C₂³	—	106	2	no	—
462	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	53	1	no	—
466	C₂×C₄	—	53	1	no	—
467	C₂×C₄	—	53	1	no	—
468	C₂×C₄	—	53	2	no	—
469	C₂×C₄	—	53	2	no	—
470	C₂×C₄	—	53	2	no	—
471	C₂×C₄	—	53	2	no	—
472	C₂×C₄	—	53	2	no	—
473	C₂×C₄	—	53	2	no	—
474	C₂×C₄	—	53	2	no	—
475	C₂×C₄	—	53	2	no	—
476	C₂×C₄	—	53	2	no	—
477	C₂×C₄	—	53	2	no	—
478	C₂×C₄	—	53	4	no	—
479	C₂×C₄	—	53	1	no	—
480	C₂×C₄	—	53	1	no	—
481	C₂×C₄	—	53	1	no	—
482	C₂×C₄	—	106	2	no	—
483	C₂×C₄	—	106	2	no	—
484	C₂×C₄	—	106	2	no	—
485	C₂×C₄	—	106	2	no	—
486	C₂×C₄	—	106	2	no	—
487	C₂×C₄	—	106	2	no	—
488	C₂×C₄	—	212	1	no	—
489	C₂×C₄	—	212	1	no	—
490	C₂×C₄	—	212	1	no	—
491	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
492	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
493	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
494	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
495	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
496	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
497	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
498	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
499	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
500	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
501	C₈	—	2	4	no	—
502	C₈	—	2	2	no	—
503	C₈	—	2	2	no	—
504	C₈	—	4	2	no	—
505	C₈	—	106	4	no	—
506	C₈	—	106	2	no	—
507	C₈	—	106	2	no	—
508	C₈	—	212	2	no	—
509	C₂²	—	2	1	no	—
510	C₂²	—	2	1	no	—
511	C₂²	—	53	2	no	—
512	C₂²	—	53	2	no	—
513	C₂²	—	53	2	no	—
514	C₂²	—	53	2	no	—
515	C₂²	—	106	4	no	—
516	C₂²	—	106	4	no	—
517	C₂²	—	106	1	no	—
518	C₂²	—	106	1	no	—
519	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
520	C₂²	#125×D₅₃	1	1	yes	characteristic
521	C₂²	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
522	C₄	—	2	1	no	—
523	C₄	—	2	1	no	—
524	C₄	—	2	1	no	—
525	C₄	—	53	2	no	—
526	C₄	—	53	2	no	—
527	C₄	—	53	2	no	—
528	C₄	—	106	1	no	—
529	C₄	—	106	1	no	—
530	C₄	—	106	1	no	—
531	C₄	#108×D₅₃	1	2	yes	complemented
532	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
533	C₄	C₄×Q₈×D₅₃	1	1	yes	characteristic
534	C₄	#121×D₅₃	1	1	yes	characteristic
535	C₄	C₂²⋊Q₈×D₅₃	1	1	yes	characteristic
536	C₂	—	2	1	no	—
537	C₂	—	2	1	no	—
538	C₂	—	53	4	no	—
539	C₂	—	106	2	no	—
540	C₂	#389×D₅₃	1	1	yes	characteristic
541	C₂	C₂×#108×D₅₃	1	1	yes	characteristic
542	C₂	#430×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1950	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1950	D₅₃
hypocenter	C₅₃	C₂×#1950
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1950

Complements

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

N	H ≅ G/N	type
5. #20275491	C₂	semidirect
6. #20275490	C₂	semidirect
7. #20275489	C₂	semidirect
8. #20275488	C₂	semidirect
9. #20275487	C₂	semidirect
11. C₅₃×#1950	C₂	semidirect
220. C₄×D₅₃	#108	semidirect
254. C₄×C₅₃	C₂×#108	semidirect
264. Dic₅₃	C₂×#108	semidirect
274. #1950	D₅₃	direct (N × H)
282. D₅₃	#1950	direct (N × H)
319. C₅₃	C₂×#1950	semidirect
531. C₄	#108×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#108×D₅₃	2
#423×D₅₃	2
C₂×C₄⋊C₈×D₅₃	2
C₄×M₁₆×D₅₃	2
#20275491	2
#20275490	2
#20275489	2
#20275488	2
#20275487	2
#341×D₅₃	2
C₅₃×#1950	2
C₂×#1950	53

Subgroups — D53×#1950 #27387608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1950 #27387608