Subgroups — #27353270

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283346	C₂	1	1	yes	maximal, characteristic
2	#20283345	C₂	1	1	yes	maximal, characteristic
3	#423×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20273505	C₂	1	2	yes	maximal, complemented
5	#20273502	C₂	1	2	yes	maximal, complemented
6	#20273488	C₂	1	2	yes	maximal, complemented
7	#20273484	C₂	1	2	yes	maximal, complemented
8	#20273483	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273482	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273480	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1569	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×D₅₃	—	2	1	no	—
13	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
14	#16810680	C₂²	1	1	yes	characteristic
15	#16810678	—	2	1	no	—
16	#16810678	C₄	1	2	yes	—
17	#16810678	C₄	1	2	yes	—
18	#16810675	C₂²	1	1	yes	characteristic
19	#16810675	C₂²	1	1	yes	characteristic
20	#16810674	C₂²	1	2	yes	—
21	#16810674	C₂²	1	2	yes	—
22	#16810673	—	2	1	no	—
23	#16810647	C₂²	1	1	yes	characteristic
24	#16810646	—	2	1	no	—
25	#16810646	C₄	1	2	yes	—
26	#16810645	—	2	1	no	—
27	#16810645	C₂²	1	1	yes	characteristic
28	#16810645	C₂²	1	1	yes	characteristic
29	#16810645	C₂²	1	2	yes	—
30	C₄⋊C₈×D₅₃	—	2	1	no	—
31	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
32	#16810586	C₄	1	2	yes	—
33	#16810585	—	2	1	no	—
34	C₄×#11719917	C₂²	1	1	yes	characteristic
35	#16810575	—	2	1	no	—
36	#16810376	C₂²	1	2	yes	complemented
37	#16810375	C₂²	1	2	yes	complemented
38	#16810375	C₂²	1	2	yes	complemented
39	#16810374	C₂²	1	2	yes	complemented
40	#16810373	C₂²	1	2	yes	complemented
41	#16810373	C₂²	1	2	yes	complemented
42	C₅₃×#432	C₂²	1	1	yes	characteristic
43	C₅₃×#423	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#335	C₂²	1	2	yes	complemented
46	C₅₃×#334	C₂²	1	2	yes	complemented
47	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
49	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
50	C₂×#11719917	—	2	1	no	—
51	C₂×#11719917	C₂×C₄	1	2	yes	—
52	C₂×#11719912	—	2	1	no	—
53	C₂×#11719912	—	2	1	no	—
54	C₂×#11719912	—	2	2	no	—
55	M₁₆×D₅₃	—	4	4	no	—
56	C₂×C₈×D₅₃	—	2	2	no	—
57	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
60	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄²×D₅₃	D₄	1	1	yes	characteristic
63	C₄²×D₅₃	D₄	1	1	yes	characteristic
64	C₄²×D₅₃	Q₈	1	1	yes	characteristic
65	C₄²×D₅₃	Q₈	1	1	yes	characteristic
66	C₄²×D₅₃	C₂³	1	1	yes	characteristic
67	C₄²×D₅₃	C₂³	1	1	yes	characteristic
68	C₄⋊#88322	—	2	1	no	—
69	C₄⋊#88322	C₂³	1	1	yes	characteristic
70	C₄⋊#88322	C₂³	1	2	yes	—
71	C₄⋊#88322	C₂³	1	1	yes	characteristic
72	#13133146	—	2	2	no	—
73	#13133146	—	2	2	no	—
74	#13133146	—	2	1	no	—
75	#13133146	—	2	1	no	—
76	#13133146	C₂×C₄	1	4	yes	—
77	#13133146	C₂×C₄	1	4	yes	—
78	C₂²⋊#88322	—	2	1	no	—
79	#13133129	—	2	1	no	—
80	#13133129	—	2	1	no	—
81	#13133129	C₂×C₄	1	4	yes	—
82	#13133129	C₂×C₄	1	4	yes	—
83	#13133128	—	2	1	no	—
84	#13133128	C₂×C₄	1	2	yes	—
85	C₄×#88322	C₂³	1	2	yes	—
86	C₈×Dic₅₃	—	2	1	no	—
87	C₈×Dic₅₃	—	2	1	no	—
88	C₂×M₁₆×C₅₃	—	2	1	no	—
89	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
90	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
91	C₄⋊C₈×C₅₃	—	2	1	no	—
92	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
94	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₂²⋊C₈×C₅₃	—	2	1	no	—
96	#97×C₅₃	—	2	1	no	—
97	#97×C₅₃	C₂×C₄	1	2	yes	—
98	C₄×C₈×C₅₃	C₂³	1	2	yes	—
99	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
103	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×D₅₃	—	2	2	no	—
107	C₂×C₄×D₅₃	—	2	2	no	—
108	C₂×C₄×D₅₃	—	4	2	no	—
109	C₂×C₄×D₅₃	—	4	2	no	—
110	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
121	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
122	M₁₆×C₅₃	—	4	2	no	—
123	M₁₆×C₅₃	—	4	2	no	—
124	C₂×C₈×C₅₃	—	2	1	no	—
125	C₂×C₈×C₅₃	—	2	1	no	—
126	C₂×C₈×C₅₃	—	2	2	no	—
127	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
128	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
131	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
132	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
133	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
134	C₄×Dic₅₃	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
139	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
141	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
142	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
143	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
144	#11719917	—	4	2	no	—
145	#11719917	—	4	2	no	—
146	C₂×#88322	—	2	1	no	—
147	C₂×#88322	—	2	1	no	—
148	C₂×#88322	—	2	2	no	—
149	C₂×#88322	C₂²×C₄	1	4	yes	—
150	#11719912	—	4	4	no	—
151	#11719912	—	4	2	no	—
152	#11719912	—	4	2	no	—
153	C₈×D₅₃	—	4	4	no	—
154	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×D₅₃	—	2	2	no	—
156	C₂²×D₅₃	—	4	1	no	—
157	C₂²×D₅₃	—	4	1	no	—
158	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	1	no	—
163	C₂×C₄×C₅₃	—	4	1	no	—
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×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
170	C₂×Dic₅₃	—	2	2	no	—
171	C₂×Dic₅₃	—	2	2	no	—
172	C₂×Dic₅₃	—	4	1	no	—
173	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₄×D₅₃	—	2	2	no	—
180	C₄×D₅₃	—	2	2	no	—
181	C₄×D₅₃	—	2	2	no	—
182	C₄×D₅₃	—	2	1	no	—
183	C₄×D₅₃	—	2	1	no	—
184	C₄×D₅₃	—	2	2	no	—
185	C₄×D₅₃	—	2	2	no	—
186	C₄×D₅₃	—	2	2	no	—
187	C₄×D₅₃	—	4	1	no	—
188	C₄×D₅₃	—	4	1	no	—
189	C₄×D₅₃	—	4	1	no	—
190	C₄×D₅₃	—	4	1	no	—
191	C₄×D₅₃	AGL(1,ℤ/8)	1	2	yes	—
192	C₄×D₅₃	#137	1	2	yes	—
193	C₈×C₅₃	—	2	4	no	—
194	C₈×C₅₃	—	4	2	no	—
195	C₈×C₅₃	—	4	1	no	—
196	C₈×C₅₃	—	4	1	no	—
197	#88322	—	2	4	no	—
198	#88322	—	4	2	no	—
199	#88322	—	4	1	no	—
200	#88322	—	4	1	no	—
201	#17820	—	53	1	no	maximal, Sylow
202	C₂²×C₅₃	—	2	1	no	—
203	C₂²×C₅₃	—	2	1	no	—
204	C₂²×C₅₃	—	2	1	no	—
205	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
206	C₂×D₅₃	—	2	1	no	—
207	C₂×D₅₃	—	2	1	no	—
208	C₂×D₅₃	—	2	1	no	—
209	C₂×D₅₃	—	2	1	no	—
210	C₂×D₅₃	—	2	1	no	—
211	C₂×D₅₃	—	4	4	no	—
212	C₂×D₅₃	#428	1	2	yes	—
213	C₄×C₅₃	—	2	1	no	—
214	C₄×C₅₃	—	2	1	no	—
215	C₄×C₅₃	—	2	1	no	—
216	C₄×C₅₃	—	2	1	no	—
217	C₄×C₅₃	#568	1	2	yes	—
218	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
219	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
220	Dic₅₃	—	2	1	no	—
221	Dic₅₃	—	2	1	no	—
222	Dic₅₃	—	2	1	no	—
223	Dic₅₃	—	2	1	no	—
224	Dic₅₃	#428	1	2	yes	—
225	Dic₅₃	#430	1	2	yes	—
226	#2920	—	53	1	no	—
227	#2920	—	53	1	no	—
228	C₂×#423	—	53	1	no	—
229	#1574	—	53	2	no	—
230	#1574	—	53	2	no	—
231	#1571	—	53	2	no	—
232	#1570	—	53	2	no	—
233	#1569	—	53	1	no	—
234	#1569	—	53	1	no	—
235	#1569	—	53	1	no	—
236	#1569	D₅₃	1	1	yes	characteristic, complemented
237	C₂×C₅₃	—	4	1	no	—
238	C₂×C₅₃	#2905	1	1	yes	characteristic
239	C₂×C₅₃	C₂×#428	1	1	yes	characteristic
240	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
241	D₅₃	—	2	2	no	—
242	D₅₃	—	4	1	no	—
243	C₂²×M₁₆	—	106	1	no	—
244	C₂²×C₄²	—	53	1	no	—
245	#432	—	53	2	no	—
246	#432	—	53	2	no	—
247	#432	—	53	1	no	—
248	#432	—	106	1	no	—
249	#432	C₂×D₅₃	1	1	yes	characteristic
250	#431	—	53	2	no	—
251	#431	—	53	2	no	—
252	#431	—	53	1	no	—
253	#431	—	53	1	no	—
254	#431	—	106	1	no	—
255	#423	—	53	2	no	—
256	#423	—	53	1	no	—
257	#423	—	53	1	no	—
258	#423	—	53	1	no	—
259	#423	—	53	2	no	—
260	#423	—	106	1	no	—
261	#423	—	106	1	no	—
262	#423	C₂×D₅₃	1	1	yes	characteristic
263	C₂×C₄⋊C₈	—	53	2	no	—
264	C₂×C₄⋊C₈	—	106	1	no	—
265	#407	—	53	2	no	—
266	#407	—	106	1	no	—
267	C₄×M₁₆	—	53	1	no	—
268	C₄×M₁₆	—	106	1	no	—
269	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
270	#335	—	53	2	no	—
271	#335	—	53	2	no	—
272	#335	—	53	2	no	—
273	#335	C₂×D₅₃	1	2	yes	complemented
274	#334	—	53	2	no	—
275	#334	—	53	2	no	—
276	#334	—	53	2	no	—
277	#334	C₂×D₅₃	1	2	yes	complemented
278	C₅₃	#17820	1	1	yes	characteristic, complemented, Sylow
279	C₂³×C₄	—	53	1	no	—
280	C₂³×C₄	—	53	1	no	—
281	C₂³×C₄	—	53	1	no	—
282	C₂×M₁₆	—	2	1	no	—
283	C₂×M₁₆	—	53	2	no	—
284	C₂×M₁₆	—	106	2	no	—
285	C₂×M₁₆	—	106	1	no	—
286	C₂×M₁₆	—	106	1	no	—
287	C₂×M₁₆	—	106	1	no	—
288	C₂×M₁₆	—	212	4	no	—
289	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
290	C₂²×C₈	—	106	2	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	—	53	1	no	—
302	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
303	C₄⋊C₈	—	2	1	no	—
304	C₄⋊C₈	—	53	4	no	—
305	C₄⋊C₈	—	53	4	no	—
306	C₄⋊C₈	—	53	1	no	—
307	C₄⋊C₈	—	53	2	no	—
308	C₄⋊C₈	—	53	1	no	—
309	C₄⋊C₈	—	106	2	no	—
310	C₄⋊C₈	—	106	2	no	—
311	C₄⋊C₈	—	106	1	no	—
312	C₄⋊C₈	—	106	1	no	—
313	C₄⋊C₈	—	106	1	no	—
314	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
315	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
316	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
317	C₂²⋊C₈	—	2	1	no	—
318	C₂²⋊C₈	—	53	4	no	—
319	C₂²⋊C₈	—	53	4	no	—
320	C₂²⋊C₈	—	106	1	no	—
321	C₂²⋊C₈	—	106	1	no	—
322	C₂²⋊C₈	—	106	1	no	—
323	#97	—	2	1	no	—
324	#97	—	53	2	no	—
325	#97	—	106	1	no	—
326	#97	C₄×D₅₃	1	2	yes	—
327	C₄×C₈	—	53	2	no	—
328	C₄×C₈	—	106	1	no	—
329	C₄×C₈	—	106	1	no	—
330	C₄×C₈	C₂²×D₅₃	1	2	yes	—
331	C₂⁴	—	53	1	no	—
332	C₂²×C₄	—	53	1	no	—
333	C₂²×C₄	—	53	1	no	—
334	C₂²×C₄	—	53	1	no	—
335	C₂²×C₄	—	53	1	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	—	53	1	no	—
346	C₂²×C₄	—	53	1	no	—
347	C₂²×C₄	—	106	2	no	—
348	C₂²×C₄	—	106	2	no	—
349	C₂²×C₄	—	212	2	no	—
350	C₂²×C₄	—	212	2	no	—
351	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
352	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
353	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
354	M₁₆	—	4	2	no	—
355	M₁₆	—	4	2	no	—
356	M₁₆	—	212	2	no	—
357	M₁₆	—	212	2	no	—
358	M₁₆	—	212	4	no	—
359	M₁₆	—	212	2	no	—
360	M₁₆	—	212	2	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	53	4	no	—
365	C₂×C₈	—	106	1	no	—
366	C₂×C₈	—	106	1	no	—
367	C₂×C₈	—	106	2	no	—
368	C₂×C₈	—	212	4	no	—
369	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
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	C₂×C₄×D₅₃	1	1	yes	characteristic
383	C₄²	D₄×D₅₃	1	1	yes	characteristic
384	C₄²	Q₈×D₅₃	1	1	yes	characteristic
385	C₄²	C₂³×D₅₃	1	1	yes	characteristic
386	C₂³	—	53	1	no	—
387	C₂³	—	53	1	no	—
388	C₂³	—	106	2	no	—
389	C₂³	—	212	1	no	—
390	C₂³	—	212	1	no	—
391	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	53	1	no	—
397	C₂×C₄	—	53	1	no	—
398	C₂×C₄	—	53	2	no	—
399	C₂×C₄	—	53	2	no	—
400	C₂×C₄	—	53	1	no	—
401	C₂×C₄	—	53	1	no	—
402	C₂×C₄	—	53	1	no	—
403	C₂×C₄	—	53	1	no	—
404	C₂×C₄	—	106	2	no	—
405	C₂×C₄	—	106	2	no	—
406	C₂×C₄	—	106	2	no	—
407	C₂×C₄	—	106	2	no	—
408	C₂×C₄	—	106	2	no	—
409	C₂×C₄	—	106	2	no	—
410	C₂×C₄	—	106	2	no	—
411	C₂×C₄	—	106	2	no	—
412	C₂×C₄	—	106	1	no	—
413	C₂×C₄	—	106	1	no	—
414	C₂×C₄	—	212	1	no	—
415	C₂×C₄	—	212	1	no	—
416	C₂×C₄	—	212	1	no	—
417	C₂×C₄	—	212	1	no	—
418	C₂×C₄	—	212	1	no	—
419	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
420	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
421	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
422	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
423	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
424	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
425	C₈	—	2	4	no	—
426	C₈	—	4	2	no	—
427	C₈	—	4	1	no	—
428	C₈	—	4	1	no	—
429	C₈	—	106	4	no	—
430	C₈	—	212	2	no	—
431	C₈	—	212	1	no	—
432	C₈	—	212	1	no	—
433	C₂²	—	2	1	no	—
434	C₂²	—	2	1	no	—
435	C₂²	—	2	1	no	—
436	C₂²	—	53	2	no	—
437	C₂²	—	106	1	no	—
438	C₂²	—	106	1	no	—
439	C₂²	—	106	1	no	—
440	C₂²	—	106	1	no	—
441	C₂²	—	106	1	no	—
442	C₂²	—	212	4	no	—
443	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
444	C₄	—	2	1	no	—
445	C₄	—	2	1	no	—
446	C₄	—	2	1	no	—
447	C₄	—	2	1	no	—
448	C₄	—	53	2	no	—
449	C₄	—	53	2	no	—
450	C₄	—	106	1	no	—
451	C₄	—	106	1	no	—
452	C₄	—	106	1	no	—
453	C₄	—	106	1	no	—
454	C₄	#16811389	1	2	yes	—
455	C₄	AGL(1,ℤ/8)×D₅₃	1	1	yes	characteristic
456	C₄	#137×D₅₃	1	1	yes	characteristic
457	C₂	—	4	1	no	—
458	C₂	—	106	2	no	—
459	C₂	—	212	1	no	—
460	C₂	#20283277	1	1	yes	characteristic
461	C₂	#428×D₅₃	1	1	yes	characteristic
462	C₂	#430×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1569	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1569	D₅₃
hypocenter	C₅₃	#17820
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1569

Complements

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

N	H ≅ G/N	type
4. #20273505	C₂	semidirect
5. #20273502	C₂	semidirect
6. #20273488	C₂	semidirect
7. #20273484	C₂	semidirect
8. #20273483	C₂	semidirect
9. #20273482	C₂	semidirect
10. #20273480	C₂	semidirect
11. C₅₃×#1569	C₂	semidirect
36. #16810376	C₂²	semidirect
37. #16810375	C₂²	semidirect
38. #16810375	C₂²	semidirect
39. #16810374	C₂²	semidirect
40. #16810373	C₂²	semidirect
41. #16810373	C₂²	semidirect
45. C₅₃×#335	C₂²	semidirect
46. C₅₃×#334	C₂²	semidirect
236. #1569	D₅₃	semidirect
273. #335	C₂×D₅₃	semidirect
277. #334	C₂×D₅₃	semidirect
278. C₅₃	#17820	semidirect

Maximal subgroups

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

subgroup	index
#20283346	2
#20283345	2
#423×D₅₃	2
#20273505	2
#20273502	2
#20273488	2
#20273484	2
#20273483	2
#20273482	2
#20273480	2
C₅₃×#1569	2
#17820	53