Subgroups — D₅₃×#1303 #27274378

Order 13568.

810 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#431×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#423×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₄×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20272618	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272617	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272616	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1303	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₅₃	—	2	1	no	—
9	C₂×C₄⋊C₄×D₅₃	—	2	1	no	—
10	C₂×C₄²×D₅₃	—	2	1	no	—
11	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
12	#16810676	C₂²	1	1	yes	characteristic
13	#16810675	C₄	1	2	yes	complemented
14	#16810674	—	2	1	no	—
15	#16810673	—	2	1	no	—
16	#16810647	C₂²	1	1	yes	characteristic
17	#16810646	—	2	1	no	—
18	#16810645	—	2	1	no	—
19	#16810645	C₄	1	2	yes	complemented
20	C₄⋊C₈×D₅₃	—	2	1	no	—
21	C₄⋊C₈×D₅₃	C₄	1	2	yes	complemented
22	C₄⋊C₈×D₅₃	C₄	1	2	yes	complemented
23	C₂²⋊C₈×D₅₃	—	2	1	no	—
24	C₄×C₈×D₅₃	—	2	1	no	—
25	C₄⋊C₄×Dic₅₃	—	2	1	no	—
26	C₄×#11719919	—	2	1	no	—
27	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
28	#16810474	C₄	1	2	yes	—
29	#95×D₅₃	C₄	1	2	yes	—
30	C₅₃×#431	C₂²	1	1	yes	characteristic
31	C₅₃×#423	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₄×D₅₃	—	2	2	no	—
34	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
37	C₂×#11719917	—	2	1	no	—
38	C₂×#11719912	—	2	2	no	—
39	M₁₆×D₅₃	—	4	2	no	—
40	M₁₆×D₅₃	—	4	2	no	—
41	C₂×C₈×D₅₃	—	2	2	no	—
42	C₂×C₈×D₅₃	—	2	2	no	—
43	C₂×C₄⋊Dic₅₃	—	2	1	no	—
44	C₂×#11719919	—	2	2	no	—
45	C₄⋊C₄×D₅₃	—	2	8	no	—
46	C₂×C₄×Dic₅₃	—	2	1	no	—
47	C₂×C₄×Dic₅₃	—	2	2	no	—
48	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
51	C₄²×D₅₃	—	4	2	no	—
52	C₄²×D₅₃	—	4	2	no	—
53	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
54	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄²×D₅₃	D₄	1	1	yes	characteristic
58	C₄²×D₅₃	D₄	1	1	yes	characteristic
59	C₄²×D₅₃	Q₈	1	1	yes	characteristic
60	C₄²×D₅₃	Q₈	1	1	yes	characteristic
61	C₄⋊#88322	—	2	1	no	—
62	C₄⋊#88322	C₂×C₄	1	2	yes	complemented
63	C₄⋊#88322	C₂×C₄	1	2	yes	complemented
64	#13133146	—	2	2	no	—
65	#13133146	—	2	2	no	—
66	#13133146	—	2	2	no	—
67	C₂²⋊#88322	—	2	1	no	—
68	#13133129	—	2	2	no	—
69	C₄×#88322	—	2	1	no	—
70	C₈×Dic₅₃	—	2	2	no	—
71	#13133124	—	2	2	no	—
72	#13133124	C₂×C₄	1	2	yes	—
73	C₂×M₁₆×C₅₃	—	2	1	no	—
74	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
75	C₂×C₄²×C₅₃	—	2	1	no	—
76	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₅₃	—	2	1	no	—
78	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	complemented
79	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	complemented
80	C₂²⋊C₈×C₅₃	—	2	1	no	—
81	C₄×C₈×C₅₃	—	2	1	no	—
82	#95×C₅₃	C₂×C₄	1	2	yes	—
83	C₂³×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	—	2	2	no	—
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×Dic₅₃	—	2	2	no	—
89	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×D₅₃	—	2	4	no	—
93	C₂×C₄×D₅₃	—	2	8	no	—
94	C₂×C₄×D₅₃	—	4	2	no	—
95	C₂×C₄×D₅₃	—	4	4	no	—
96	C₂×C₄×D₅₃	—	4	4	no	—
97	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
98	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
99	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×D₅₃	D₈	1	2	yes	—
110	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
111	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
112	C₂×C₄×D₅₃	Q₁₆	1	2	yes	—
113	M₁₆×C₅₃	—	4	1	no	—
114	M₁₆×C₅₃	—	4	1	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	2	no	—
117	C₄⋊C₄×C₅₃	—	2	4	no	—
118	C₄²×C₅₃	—	4	1	no	—
119	C₄²×C₅₃	—	4	1	no	—
120	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
124	C₄⋊Dic₅₃	—	2	4	no	—
125	#11719919	—	2	8	no	—
126	C₄×Dic₅₃	—	4	1	no	—
127	C₄×Dic₅₃	—	4	1	no	—
128	C₄×Dic₅₃	—	4	4	no	—
129	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
130	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
131	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
132	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
133	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
134	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
135	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
136	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
137	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
138	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
139	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
140	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
141	#11719917	—	4	1	no	—
142	#11719917	—	4	1	no	—
143	C₂×#88322	—	2	2	no	—
144	C₂×#88322	—	2	2	no	—
145	#11719912	—	4	4	no	—
146	C₈×D₅₃	—	4	4	no	—
147	C₈×D₅₃	—	4	4	no	—
148	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×D₅₃	—	2	2	no	—
150	C₂²×D₅₃	—	2	2	no	—
151	C₂²×D₅₃	#95	1	1	yes	characteristic
152	C₂²×D₅₃	#95	1	1	yes	characteristic
153	C₂²×D₅₃	#102	1	2	yes	—
154	C₂²×D₅₃	#103	1	2	yes	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	4	no	—
157	C₂×C₄×C₅₃	—	4	1	no	—
158	C₂×C₄×C₅₃	—	4	2	no	—
159	C₂×C₄×C₅₃	—	4	2	no	—
160	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
170	C₂×Dic₅₃	—	2	2	no	—
171	C₂×Dic₅₃	—	2	4	no	—
172	C₂×Dic₅₃	—	4	1	no	—
173	C₂×Dic₅₃	—	4	2	no	—
174	C₂×Dic₅₃	—	4	2	no	—
175	C₂×Dic₅₃	#95	1	1	yes	characteristic
176	C₂×Dic₅₃	#95	1	1	yes	characteristic
177	C₂×Dic₅₃	#95	1	1	yes	characteristic
178	C₂×Dic₅₃	#95	1	1	yes	characteristic
179	C₂×Dic₅₃	#95	1	1	yes	characteristic
180	C₂×Dic₅₃	#95	1	1	yes	characteristic
181	C₂×Dic₅₃	#102	1	2	yes	—
182	C₂×Dic₅₃	#103	1	2	yes	—
183	C₄×D₅₃	—	2	2	no	—
184	C₄×D₅₃	—	2	2	no	—
185	C₄×D₅₃	—	2	2	no	—
186	C₄×D₅₃	—	2	2	no	—
187	C₄×D₅₃	—	2	2	no	—
188	C₄×D₅₃	—	4	1	no	—
189	C₄×D₅₃	—	4	1	no	—
190	C₄×D₅₃	—	4	16	no	—
191	C₄×D₅₃	#102	1	2	yes	—
192	C₄×D₅₃	#102	1	2	yes	—
193	C₄×D₅₃	#103	1	2	yes	—
194	C₄×D₅₃	#103	1	2	yes	—
195	C₄×D₅₃	#104	1	4	yes	—
196	C₄×D₅₃	#106	1	2	yes	—
197	C₄×D₅₃	#106	1	2	yes	—
198	C₄×D₅₃	#107	1	2	yes	—
199	C₄×D₅₃	#107	1	2	yes	—
200	C₈×C₅₃	—	4	2	no	—
201	C₈×C₅₃	—	4	2	no	—
202	#88322	—	4	2	no	—
203	#88322	—	4	2	no	—
204	C₂×#1303	—	53	1	no	maximal, Sylow
205	C₂²×C₅₃	—	2	1	no	—
206	C₂²×C₅₃	—	2	1	no	—
207	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
208	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
209	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
210	C₂×D₅₃	—	2	1	no	—
211	C₂×D₅₃	—	2	1	no	—
212	C₂×D₅₃	—	2	4	no	—
213	C₂×D₅₃	—	2	4	no	—
214	C₂×D₅₃	#339	1	2	yes	—
215	C₂×D₅₃	#340	1	2	yes	—
216	C₂×D₅₃	#340	1	2	yes	—
217	C₂×D₅₃	#344	1	2	yes	—
218	C₄×C₅₃	—	2	1	no	—
219	C₄×C₅₃	—	2	1	no	—
220	C₄×C₅₃	—	2	1	no	—
221	C₄×C₅₃	—	4	4	no	—
222	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
223	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
224	C₄×C₅₃	C₂×#104	1	2	yes	—
225	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
226	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
227	Dic₅₃	—	2	1	no	—
228	Dic₅₃	—	2	1	no	—
229	Dic₅₃	—	2	1	no	—
230	Dic₅₃	—	4	4	no	—
231	Dic₅₃	#339	1	2	yes	—
232	Dic₅₃	#340	1	2	yes	—
233	Dic₅₃	#340	1	2	yes	—
234	C₂×#431	—	53	1	no	—
235	C₂×#423	—	53	1	no	—
236	C₂×C₄×C₄⋊C₄	—	53	1	no	—
237	#1303	—	53	1	no	—
238	#1303	—	53	1	no	—
239	#1303	—	53	1	no	—
240	#1303	D₅₃	1	1	yes	characteristic, direct product factor
241	C₂×C₅₃	—	2	1	no	—
242	C₂×C₅₃	—	2	1	no	—
243	C₂×C₅₃	C₂×#339	1	1	yes	characteristic
244	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
245	C₂×C₅₃	C₂×#344	1	1	yes	characteristic
246	D₅₃	—	2	2	no	—
247	D₅₃	#1303	1	4	yes	direct product factor
248	C₂²×M₁₆	—	106	1	no	—
249	C₂²×C₄⋊C₄	—	106	1	no	—
250	C₂²×C₄²	—	53	1	no	—
251	C₂²×C₄²	—	106	1	no	—
252	#431	—	53	1	no	—
253	#431	—	53	2	no	—
254	#431	—	106	1	no	—
255	#431	—	106	1	no	—
256	#431	C₂×D₅₃	1	1	yes	characteristic
257	#423	—	53	1	no	—
258	#423	—	53	2	no	—
259	#423	—	106	1	no	—
260	#423	—	106	1	no	—
261	#423	C₂×D₅₃	1	1	yes	characteristic
262	C₂×C₄⋊C₈	—	53	2	no	—
263	C₂×C₄⋊C₈	—	53	2	no	—
264	C₂×C₄⋊C₈	—	106	1	no	—
265	C₂×C₂²⋊C₈	—	106	1	no	—
266	C₂×C₄×C₈	—	106	1	no	—
267	C₄×C₄⋊C₄	—	53	1	no	—
268	C₄×C₄⋊C₄	—	53	2	no	—
269	C₄×C₄⋊C₄	—	106	1	no	—
270	C₄×C₄⋊C₄	—	106	1	no	—
271	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
272	C₂×#95	—	53	2	no	—
273	C₅₃	C₂×#1303	1	1	yes	characteristic, complemented, Sylow
274	C₂³×C₄	—	53	1	no	—
275	C₂³×C₄	—	53	1	no	—
276	C₂³×C₄	—	53	1	no	—
277	C₂³×C₄	—	106	2	no	—
278	C₂×M₁₆	—	2	1	no	—
279	C₂×M₁₆	—	106	2	no	—
280	C₂×M₁₆	—	106	1	no	—
281	C₂×M₁₆	—	212	2	no	—
282	C₂×M₁₆	—	212	2	no	—
283	C₂²×C₈	—	106	2	no	—
284	C₂²×C₈	—	106	2	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	106	8	no	—
287	C₂×C₄⋊C₄	—	106	2	no	—
288	C₂×C₄⋊C₄	—	106	1	no	—
289	C₂×C₄²	—	2	1	no	—
290	C₂×C₄²	—	53	1	no	—
291	C₂×C₄²	—	53	1	no	—
292	C₂×C₄²	—	53	1	no	—
293	C₂×C₄²	—	53	1	no	—
294	C₂×C₄²	—	53	1	no	—
295	C₂×C₄²	—	53	1	no	—
296	C₂×C₄²	—	53	1	no	—
297	C₂×C₄²	—	53	1	no	—
298	C₂×C₄²	—	53	1	no	—
299	C₂×C₄²	—	53	1	no	—
300	C₂×C₄²	—	53	1	no	—
301	C₂×C₄²	—	106	2	no	—
302	C₂×C₄²	—	106	1	no	—
303	C₂×C₄²	—	212	2	no	—
304	C₂×C₄²	—	212	2	no	—
305	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
306	C₄⋊C₈	—	2	1	no	—
307	C₄⋊C₈	—	53	2	no	—
308	C₄⋊C₈	—	53	2	no	—
309	C₄⋊C₈	—	106	2	no	—
310	C₄⋊C₈	—	106	2	no	—
311	C₄⋊C₈	—	106	2	no	—
312	C₄⋊C₈	—	106	1	no	—
313	C₄⋊C₈	C₄×D₅₃	1	2	yes	complemented
314	C₄⋊C₈	C₄×D₅₃	1	2	yes	complemented
315	C₂²⋊C₈	—	2	1	no	—
316	C₂²⋊C₈	—	106	2	no	—
317	C₂²⋊C₈	—	106	1	no	—
318	C₄×C₈	—	2	1	no	—
319	C₄×C₈	—	106	2	no	—
320	C₄×C₈	—	106	1	no	—
321	#95	—	53	2	no	—
322	#95	—	106	2	no	—
323	#95	C₄×D₅₃	1	2	yes	—
324	C₂⁴	—	53	1	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	53	1	no	—
327	C₂²×C₄	—	53	1	no	—
328	C₂²×C₄	—	53	1	no	—
329	C₂²×C₄	—	53	1	no	—
330	C₂²×C₄	—	53	1	no	—
331	C₂²×C₄	—	53	1	no	—
332	C₂²×C₄	—	53	2	no	—
333	C₂²×C₄	—	53	2	no	—
334	C₂²×C₄	—	53	2	no	—
335	C₂²×C₄	—	53	2	no	—
336	C₂²×C₄	—	53	1	no	—
337	C₂²×C₄	—	53	1	no	—
338	C₂²×C₄	—	53	1	no	—
339	C₂²×C₄	—	53	1	no	—
340	C₂²×C₄	—	53	1	no	—
341	C₂²×C₄	—	53	1	no	—
342	C₂²×C₄	—	53	1	no	—
343	C₂²×C₄	—	53	1	no	—
344	C₂²×C₄	—	53	1	no	—
345	C₂²×C₄	—	106	4	no	—
346	C₂²×C₄	—	106	8	no	—
347	C₂²×C₄	—	106	2	no	—
348	C₂²×C₄	—	212	2	no	—
349	C₂²×C₄	—	212	4	no	—
350	C₂²×C₄	—	212	4	no	—
351	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
352	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
353	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
354	M₁₆	—	4	1	no	—
355	M₁₆	—	4	1	no	—
356	M₁₆	—	212	4	no	—
357	M₁₆	—	212	1	no	—
358	M₁₆	—	212	1	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	106	2	no	—
362	C₂×C₈	—	106	2	no	—
363	C₂×C₈	—	212	4	no	—
364	C₂×C₈	—	212	4	no	—
365	C₄⋊C₄	—	2	4	no	—
366	C₄⋊C₄	—	106	8	no	—
367	C₄⋊C₄	—	106	4	no	—
368	C₄²	—	4	1	no	—
369	C₄²	—	4	1	no	—
370	C₄²	—	53	1	no	—
371	C₄²	—	53	1	no	—
372	C₄²	—	53	1	no	—
373	C₄²	—	53	1	no	—
374	C₄²	—	53	1	no	—
375	C₄²	—	53	1	no	—
376	C₄²	—	53	1	no	—
377	C₄²	—	53	1	no	—
378	C₄²	—	53	1	no	—
379	C₄²	—	53	1	no	—
380	C₄²	—	53	1	no	—
381	C₄²	—	53	1	no	—
382	C₄²	—	212	4	no	—
383	C₄²	—	212	1	no	—
384	C₄²	—	212	1	no	—
385	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
386	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
387	C₄²	D₄×D₅₃	1	1	yes	characteristic
388	C₄²	Q₈×D₅₃	1	1	yes	characteristic
389	C₂³	—	53	1	no	—
390	C₂³	—	53	2	no	—
391	C₂³	—	53	2	no	—
392	C₂³	—	53	1	no	—
393	C₂³	—	106	2	no	—
394	C₂³	—	106	2	no	—
395	C₂³	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	53	2	no	—
402	C₂×C₄	—	53	2	no	—
403	C₂×C₄	—	53	1	no	—
404	C₂×C₄	—	53	1	no	—
405	C₂×C₄	—	53	1	no	—
406	C₂×C₄	—	53	2	no	—
407	C₂×C₄	—	53	2	no	—
408	C₂×C₄	—	53	2	no	—
409	C₂×C₄	—	53	2	no	—
410	C₂×C₄	—	53	4	no	—
411	C₂×C₄	—	53	2	no	—
412	C₂×C₄	—	53	2	no	—
413	C₂×C₄	—	53	2	no	—
414	C₂×C₄	—	53	2	no	—
415	C₂×C₄	—	53	1	no	—
416	C₂×C₄	—	53	1	no	—
417	C₂×C₄	—	53	1	no	—
418	C₂×C₄	—	106	2	no	—
419	C₂×C₄	—	106	2	no	—
420	C₂×C₄	—	106	2	no	—
421	C₂×C₄	—	106	2	no	—
422	C₂×C₄	—	106	2	no	—
423	C₂×C₄	—	106	2	no	—
424	C₂×C₄	—	106	4	no	—
425	C₂×C₄	—	212	1	no	—
426	C₂×C₄	—	212	1	no	—
427	C₂×C₄	—	212	16	no	—
428	C₂×C₄	—	212	1	no	—
429	C₂×C₄	—	212	2	no	—
430	C₂×C₄	—	212	2	no	—
431	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
432	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
433	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
434	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
435	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
436	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
437	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
438	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
439	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
440	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
441	C₈	—	4	2	no	—
442	C₈	—	4	2	no	—
443	C₈	—	212	2	no	—
444	C₈	—	212	2	no	—
445	C₂²	—	2	1	no	—
446	C₂²	—	2	1	no	—
447	C₂²	—	53	2	no	—
448	C₂²	—	53	2	no	—
449	C₂²	—	53	2	no	—
450	C₂²	—	53	2	no	—
451	C₂²	—	106	4	no	—
452	C₂²	—	106	4	no	—
453	C₂²	—	106	1	no	—
454	C₂²	—	106	1	no	—
455	C₂²	#95×D₅₃	1	1	yes	characteristic
456	C₂²	#102×D₅₃	1	1	yes	characteristic
457	C₂²	#103×D₅₃	1	1	yes	characteristic
458	C₄	—	2	1	no	—
459	C₄	—	2	1	no	—
460	C₄	—	2	1	no	—
461	C₄	—	4	4	no	—
462	C₄	—	53	2	no	—
463	C₄	—	53	2	no	—
464	C₄	—	53	2	no	—
465	C₄	—	106	1	no	—
466	C₄	—	106	1	no	—
467	C₄	—	106	1	no	—
468	C₄	—	212	4	no	—
469	C₄	#102×D₅₃	1	1	yes	characteristic
470	C₄	#103×D₅₃	1	1	yes	characteristic
471	C₄	#104×D₅₃	1	2	yes	—
472	C₄	#106×D₅₃	1	1	yes	characteristic
473	C₄	#107×D₅₃	1	1	yes	characteristic
474	C₂	—	2	1	no	—
475	C₂	—	2	1	no	—
476	C₂	—	53	4	no	—
477	C₂	—	106	2	no	—
478	C₂	#339×D₅₃	1	1	yes	characteristic
479	C₂	#340×D₅₃	1	1	yes	characteristic
480	C₂	#344×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1303	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂³×D₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1303	D₅₃
hypocenter	C₅₃	C₂×#1303
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1303

Complements

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

N	H ≅ G/N	type
4. #20272618	C₂	semidirect
5. #20272617	C₂	semidirect
6. #20272616	C₂	semidirect
7. C₅₃×#1303	C₂	semidirect
13. #16810675	C₄	semidirect
19. #16810645	C₄	semidirect
21. C₄⋊C₈×D₅₃	C₄	semidirect
22. C₄⋊C₈×D₅₃	C₄	semidirect
62. C₄⋊#88322	C₂×C₄	semidirect
63. C₄⋊#88322	C₂×C₄	semidirect
78. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
79. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
240. #1303	D₅₃	direct (N × H)
247. D₅₃	#1303	direct (N × H)
273. C₅₃	C₂×#1303	semidirect
313. C₄⋊C₈	C₄×D₅₃	semidirect
314. C₄⋊C₈	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#431×D₅₃	2
#423×D₅₃	2
C₄×C₄⋊C₄×D₅₃	2
#20272618	2
#20272617	2
#20272616	2
C₅₃×#1303	2
C₂×#1303	53

Subgroups — D53×#1303 #27274378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1303 #27274378