Subgroups — #27354696

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	#20285311	C₂	1	1	yes	maximal, characteristic
2	#20283428	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273605	C₂	1	2	yes	maximal, complemented
4	#20273598	C₂	1	2	yes	maximal, complemented
5	#20273537	C₂	1	2	yes	maximal, complemented
6	#20273525	C₂	1	2	yes	maximal, complemented
7	#20273483	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273183	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273180	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2966	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133223	—	2	1	no	—
13	#16811173	C₂²	1	1	yes	characteristic
14	#16811033	C₄	1	1	yes	characteristic
15	#16811013	C₄	1	1	yes	characteristic
16	#16810989	C₄	1	1	yes	characteristic
17	#16810982	C₄	1	1	yes	characteristic
18	#16810874	C₄	1	2	yes	—
19	#16810871	C₄	1	2	yes	—
20	C₄×#11719924	C₂²	1	2	yes	—
21	C₄×#11719922	C₂²	1	2	yes	—
22	#16810702	—	2	1	no	—
23	#16810702	C₂²	1	2	yes	—
24	C₈×D₂₁₂	—	2	1	no	—
25	C₈×D₂₁₂	C₂²	1	2	yes	—
26	#16810676	C₂²	1	1	yes	characteristic
27	#16810647	C₂²	1	1	yes	characteristic
28	#16810589	—	2	1	no	—
29	C₄×#11719917	C₂²	1	1	yes	characteristic
30	#16810375	C₂²	1	2	yes	—
31	#16810373	C₂²	1	2	yes	—
32	Q₈⋊#88322	C₂²	1	2	yes	—
33	D₄⋊#88322	C₂²	1	2	yes	—
34	#16810350	C₂²	1	2	yes	—
35	#16810350	C₂²	1	2	yes	—
36	#16810345	C₂²	1	2	yes	—
37	C₈×#88322	C₂²	1	2	yes	—
38	C₅₃×#529	C₂²	1	1	yes	characteristic
39	C₅₃×#446	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#436	C₂²	1	1	yes	characteristic
42	C₅₃×#435	—	2	1	no	—
43	C₈×D₄×C₅₃	—	2	1	no	—
44	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#433	—	2	1	no	—
46	C₅₃×#432	C₂²	1	1	yes	characteristic
47	C₅₃×#405	—	2	1	no	—
48	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
49	C₂×#11719945	—	2	1	no	—
50	C₂×#11719924	C₂×C₄	1	2	yes	—
51	C₂×#11719922	C₂×C₄	1	2	yes	—
52	#13133223	—	4	4	no	—
53	C₂×#11719917	—	2	1	no	—
54	C₂×#11719917	C₂×C₄	1	2	yes	—
55	C₂×#11719912	—	2	2	no	—
56	C₂×C₈×D₅₃	—	2	2	no	—
57	#13133217	C₂×C₄	1	1	yes	characteristic
58	#13133214	C₂×C₄	1	1	yes	characteristic
59	#13133205	—	2	1	no	—
60	#13133191	—	2	1	no	—
61	#13133191	C₂×C₄	1	2	yes	—
62	C₄×D₂₁₂	—	2	1	no	—
63	C₄×D₂₁₂	C₂³	1	2	yes	—
64	#13133150	C₂×C₄	1	2	yes	—
65	#13133148	C₂×C₄	1	2	yes	—
66	C₄⋊#88322	—	2	1	no	—
67	C₄⋊#88322	—	2	1	no	—
68	C₄⋊#88322	C₂³	1	2	yes	—
69	#13133141	C₂×C₄	1	2	yes	—
70	#13133138	C₂×C₄	1	2	yes	—
71	#13133137	C₂×C₄	1	2	yes	—
72	#13133137	C₂×C₄	1	2	yes	—
73	C₂²⋊#88322	—	2	1	no	—
74	#13133129	—	2	2	no	—
75	#13133129	—	2	2	no	—
76	#13133128	C₂×C₄	1	1	yes	characteristic
77	#13133128	C₂×C₄	1	1	yes	characteristic
78	C₄×#88322	C₂³	1	2	yes	—
79	C₄×#88322	C₂³	1	2	yes	—
80	C₂×M₁₆×C₅₃	—	2	1	no	—
81	C₂²×C₈×C₅₃	—	2	1	no	—
82	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
83	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
84	#126×C₅₃	—	2	1	no	—
85	#125×C₅₃	C₂×C₄	1	1	yes	characteristic
86	#124×C₅₃	C₂×C₄	1	1	yes	characteristic
87	#123×C₅₃	—	2	1	no	—
88	C₂²⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
89	#121×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄×D₄×C₅₃	—	2	1	no	—
92	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
93	#117×C₅₃	—	2	1	no	—
94	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
95	C₄⋊C₈×C₅₃	—	2	1	no	—
96	C₄⋊C₈×C₅₃	—	2	1	no	—
97	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
99	C₂²⋊C₈×C₅₃	—	2	1	no	—
100	C₂²⋊C₈×C₅₃	—	2	1	no	—
101	C₂²⋊C₈×C₅₃	—	2	1	no	—
102	#97×C₅₃	—	2	1	no	—
103	#97×C₅₃	—	2	1	no	—
104	C₄×C₈×C₅₃	—	2	1	no	—
105	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
106	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
107	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
108	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
109	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×D₄×C₅₃	—	2	1	no	—
111	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	—	2	1	no	—
113	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
116	C₂×C₂²⋊D₅₃	—	2	2	no	—
117	#11719945	—	4	2	no	—
118	#11719945	—	4	2	no	—
119	C₂×D₂₁₂	—	2	1	no	—
120	C₂×D₂₁₂	—	2	1	no	—
121	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
122	C₂×C₄×D₅₃	—	2	2	no	—
123	C₂×Dic₁₀₆	—	2	1	no	—
124	M₁₆×C₅₃	—	4	2	no	—
125	C₂×C₈×C₅₃	—	2	2	no	—
126	C₂×C₈×C₅₃	—	2	1	no	—
127	C₂×C₈×C₅₃	—	2	1	no	—
128	C₂×C₈×C₅₃	—	4	1	no	—
129	C₂×C₈×C₅₃	—	4	2	no	—
130	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
134	C₄⋊C₄×C₅₃	—	2	1	no	—
135	C₄⋊C₄×C₅₃	—	2	1	no	—
136	C₄⋊C₄×C₅₃	—	2	1	no	—
137	C₄⋊C₄×C₅₃	—	2	1	no	—
138	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₂²⋊C₄×C₅₃	—	2	1	no	—
141	C₂²⋊C₄×C₅₃	—	2	1	no	—
142	C₂²⋊C₄×C₅₃	—	2	1	no	—
143	C₂²⋊C₄×C₅₃	—	2	1	no	—
144	C₂²⋊C₄×C₅₃	—	2	1	no	—
145	C₄²×C₅₃	—	2	1	no	—
146	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₅₃	#41	1	1	yes	characteristic
149	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
150	#11719924	—	2	4	no	—
151	#11719922	—	2	4	no	—
152	#11719921	—	2	2	no	—
153	#11719921	—	2	2	no	—
154	C₄⋊Dic₅₃	—	2	1	no	—
155	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
156	#11719917	—	2	4	no	—
157	#11719917	—	4	2	no	—
158	C₂×#88322	—	2	2	no	—
159	C₂×#88322	—	2	2	no	—
160	C₂×#88322	C₂²×C₄	1	2	yes	—
161	C₂×#88322	C₂²×C₄	1	2	yes	—
162	#11719912	—	4	4	no	—
163	C₈×D₅₃	—	4	4	no	—
164	C₂³×C₅₃	—	2	1	no	—
165	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
166	C₂²×D₅₃	—	2	2	no	—
167	Q₈×C₅₃	—	2	2	no	—
168	D₄×C₅₃	—	2	2	no	—
169	D₄×C₅₃	—	4	2	no	—
170	C₂×C₄×C₅₃	—	2	2	no	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	4	1	no	—
177	C₂×C₄×C₅₃	—	4	1	no	—
178	C₂×C₄×C₅₃	—	4	1	no	—
179	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
185	C₂²⋊D₅₃	—	4	4	no	—
186	C₂×Dic₅₃	—	2	2	no	—
187	D₂₁₂	—	2	2	no	—
188	D₂₁₂	—	2	2	no	—
189	D₂₁₂	—	4	2	no	—
190	D₂₁₂	—	4	2	no	—
191	C₄×D₅₃	—	4	2	no	—
192	C₄×D₅₃	—	4	2	no	—
193	Dic₁₀₆	—	4	2	no	—
194	C₈×C₅₃	—	2	1	no	—
195	C₈×C₅₃	—	2	1	no	—
196	C₈×C₅₃	—	2	1	no	—
197	C₈×C₅₃	—	4	1	no	—
198	C₈×C₅₃	—	4	1	no	—
199	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
200	#88322	—	2	2	no	—
201	#88322	—	4	2	no	—
202	#88322	—	4	2	no	—
203	#88322	#131	1	4	yes	complemented
204	#17966	—	53	1	no	maximal, Sylow
205	C₂²×C₅₃	—	2	1	no	—
206	C₂²×C₅₃	—	2	1	no	—
207	C₂²×C₅₃	—	2	1	no	—
208	C₂²×C₅₃	—	4	1	no	—
209	C₂²×C₅₃	—	4	1	no	—
210	C₂²×C₅₃	—	4	1	no	—
211	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
212	C₂×D₅₃	—	4	2	no	—
213	C₂×D₅₃	—	4	2	no	—
214	C₂×D₅₃	—	4	2	no	—
215	C₄×C₅₃	—	2	1	no	—
216	C₄×C₅₃	—	2	1	no	—
217	C₄×C₅₃	—	2	1	no	—
218	C₄×C₅₃	—	2	1	no	—
219	C₄×C₅₃	—	4	1	no	—
220	C₄×C₅₃	—	4	1	no	—
221	C₄×C₅₃	—	4	1	no	—
222	C₄×C₅₃	C₂×#131	1	2	yes	—
223	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
224	C₄×C₅₃	#575	1	1	yes	characteristic
225	Dic₅₃	—	4	2	no	—
226	#3171	—	53	1	no	—
227	#2966	D₅₃	1	1	yes	characteristic, complemented
228	#2931	—	53	1	no	—
229	#1590	—	53	2	no	—
230	#1589	—	53	2	no	—
231	#1580	—	53	2	no	—
232	SD₁₆⋊C₈	—	53	2	no	—
233	#1569	—	53	1	no	—
234	#1496	—	53	1	no	—
235	#1495	—	53	1	no	—
236	C₈×M₁₆	—	53	1	no	—
237	C₂×C₅₃	—	4	1	no	—
238	C₂×C₅₃	—	8	1	no	—
239	C₂×C₅₃	#2931	1	1	yes	characteristic
240	C₂×C₅₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
241	C₂×C₅₃	#2955	1	1	yes	characteristic
242	D₅₃	—	8	2	no	—
243	C₂×#131	—	106	1	no	—
244	#532	—	53	1	no	—
245	#529	C₂×D₅₃	1	1	yes	characteristic
246	#501	—	53	1	no	—
247	#496	—	53	1	no	—
248	#490	—	53	1	no	—
249	#489	—	53	1	no	—
250	#469	—	53	2	no	—
251	#468	—	53	2	no	—
252	#446	C₂×D₅₃	1	1	yes	characteristic
253	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
254	C₄×SD₁₆	—	53	2	no	—
255	C₄×D₈	—	53	2	no	—
256	#436	—	53	2	no	—
257	#436	—	106	1	no	—
258	#436	C₂×D₅₃	1	1	yes	characteristic
259	#435	—	2	1	no	—
260	C₈×D₄	—	2	1	no	—
261	C₈×D₄	—	53	2	no	—
262	C₈×D₄	—	106	1	no	—
263	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
264	#433	—	2	1	no	—
265	#432	C₂×D₅₃	1	1	yes	characteristic
266	#431	—	53	1	no	—
267	#423	—	53	1	no	—
268	#408	—	106	1	no	—
269	#405	—	2	1	no	—
270	C₄×M₁₆	—	53	1	no	—
271	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
272	#335	—	53	2	no	—
273	#334	—	53	2	no	—
274	Q₈⋊C₈	—	53	2	no	—
275	D₄⋊C₈	—	53	2	no	—
276	D₄⋊C₈	—	53	2	no	—
277	D₄⋊C₈	—	53	2	no	—
278	#322	—	53	2	no	—
279	C₈²	—	53	2	no	—
280	C₅₃	#17966	1	1	yes	characteristic, complemented, Sylow
281	C₂×#41	—	106	1	no	—
282	C₂×SD₁₆	—	53	2	no	—
283	C₂×D₈	—	53	2	no	—
284	#131	—	212	4	no	—
285	C₂×M₁₆	—	2	1	no	—
286	C₂×M₁₆	—	53	2	no	—
287	C₂×M₁₆	—	106	1	no	—
288	C₂×M₁₆	—	106	2	no	—
289	C₂²×C₈	—	2	1	no	—
290	C₂²×C₈	—	106	2	no	—
291	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
292	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
293	C₄⋊Q₈	—	53	1	no	—
294	#127	—	53	1	no	—
295	#126	—	2	1	no	—
296	#125	C₄×D₅₃	1	1	yes	characteristic
297	#124	—	106	1	no	—
298	#124	C₄×D₅₃	1	1	yes	characteristic
299	#123	—	2	1	no	—
300	C₂²⋊Q₈	C₄×D₅₃	1	1	yes	characteristic
301	#121	—	53	2	no	—
302	#121	—	106	1	no	—
303	#121	C₄×D₅₃	1	1	yes	characteristic
304	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
305	C₄×D₄	—	2	1	no	—
306	C₄×D₄	—	53	2	no	—
307	C₄×D₄	—	106	1	no	—
308	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
309	#117	—	2	1	no	—
310	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
311	#107	—	53	2	no	—
312	#106	—	53	2	no	—
313	C₄⋊C₈	—	2	1	no	—
314	C₄⋊C₈	—	2	1	no	—
315	C₄⋊C₈	—	53	2	no	—
316	C₄⋊C₈	—	106	1	no	—
317	C₄⋊C₈	—	106	1	no	—
318	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
319	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
320	#103	—	53	2	no	—
321	#102	—	53	2	no	—
322	#102	—	53	2	no	—
323	#102	—	53	2	no	—
324	C₂²⋊C₈	—	2	1	no	—
325	C₂²⋊C₈	—	2	1	no	—
326	C₂²⋊C₈	—	2	1	no	—
327	C₂²⋊C₈	—	106	1	no	—
328	C₂²⋊C₈	—	106	2	no	—
329	C₂²⋊C₈	—	106	2	no	—
330	#97	—	2	1	no	—
331	#97	—	2	1	no	—
332	#97	—	53	1	no	—
333	#97	—	53	1	no	—
334	C₄×C₈	—	2	1	no	—
335	C₄×C₈	—	53	2	no	—
336	C₄×C₈	—	53	2	no	—
337	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
338	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
339	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
340	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
341	#41	—	212	2	no	—
342	#41	—	212	2	no	—
343	C₂×Q₈	—	106	1	no	—
344	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	53	2	no	—
347	C₂×D₄	—	106	2	no	—
348	C₂×D₄	—	106	1	no	—
349	C₂×D₄	—	106	1	no	—
350	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	106	2	no	—
353	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
354	C₂²×C₄	#11719945	1	1	yes	characteristic
355	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
356	SD₁₆	—	106	4	no	—
357	D₈	—	106	4	no	—
358	M₁₆	—	4	2	no	—
359	M₁₆	—	106	4	no	—
360	M₁₆	—	212	2	no	—
361	M₁₆	—	212	4	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	4	1	no	—
366	C₂×C₈	—	4	2	no	—
367	C₂×C₈	—	53	2	no	—
368	C₂×C₈	—	53	2	no	—
369	C₂×C₈	—	106	2	no	—
370	C₂×C₈	—	106	2	no	—
371	C₂×C₈	—	212	4	no	—
372	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
373	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
374	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
375	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	53	2	no	—
381	C₄⋊C₄	—	106	1	no	—
382	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
383	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
384	C₂²⋊C₄	—	2	1	no	—
385	C₂²⋊C₄	—	2	1	no	—
386	C₂²⋊C₄	—	2	1	no	—
387	C₂²⋊C₄	—	2	1	no	—
388	C₂²⋊C₄	—	2	1	no	—
389	C₂²⋊C₄	—	106	2	no	—
390	C₂²⋊C₄	—	106	2	no	—
391	C₄²	—	2	1	no	—
392	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
393	C₄²	#11719945	1	1	yes	characteristic
394	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
395	C₄²	C₂³×D₅₃	1	1	yes	characteristic
396	C₂³	—	2	1	no	—
397	C₂³	—	106	2	no	—
398	C₂³	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
399	Q₈	—	2	2	no	—
400	Q₈	—	212	2	no	—
401	D₄	—	2	2	no	—
402	D₄	—	4	2	no	—
403	D₄	—	106	2	no	—
404	D₄	—	106	2	no	—
405	D₄	—	212	2	no	—
406	D₄	—	212	2	no	—
407	D₄	—	212	4	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	106	2	no	—
418	C₂×C₄	—	212	2	no	—
419	C₂×C₄	—	212	2	no	—
420	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
421	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
422	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
423	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
424	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
425	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
426	C₈	—	2	1	no	—
427	C₈	—	2	1	no	—
428	C₈	—	2	1	no	—
429	C₈	—	4	1	no	—
430	C₈	—	4	1	no	—
431	C₈	—	53	4	no	—
432	C₈	—	106	2	no	—
433	C₈	—	212	2	no	—
434	C₈	—	212	2	no	—
435	C₈	#13133242	1	2	yes	—
436	C₂²	—	2	1	no	—
437	C₂²	—	2	1	no	—
438	C₂²	—	2	1	no	—
439	C₂²	—	4	1	no	—
440	C₂²	—	4	1	no	—
441	C₂²	—	4	1	no	—
442	C₂²	—	212	2	no	—
443	C₂²	—	212	2	no	—
444	C₂²	—	212	2	no	—
445	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
446	C₄	—	2	1	no	—
447	C₄	—	2	1	no	—
448	C₄	—	2	1	no	—
449	C₄	—	2	1	no	—
450	C₄	—	4	1	no	—
451	C₄	—	4	1	no	—
452	C₄	—	4	1	no	—
453	C₄	—	212	2	no	—
454	C₄	#131×D₅₃	1	2	yes	—
455	C₄	C₂×#13133242	1	1	yes	characteristic
456	C₄	#16811421	1	1	yes	characteristic
457	C₂	—	4	1	no	—
458	C₂	—	8	1	no	—
459	C₂	—	424	2	no	—
460	C₂	#20283425	1	1	yes	characteristic
461	C₂	C₄×#13133242	1	1	yes	characteristic
462	C₂	#20283616	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#2966	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#20285311	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2966	D₅₃
hypocenter	C₅₃	#17966
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2966

Complements

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

N	H ≅ G/N	type
2. #20283428	C₂	semidirect
3. #20273605	C₂	semidirect
4. #20273598	C₂	semidirect
5. #20273537	C₂	semidirect
6. #20273525	C₂	semidirect
7. #20273483	C₂	semidirect
8. #20273183	C₂	semidirect
9. #20273180	C₂	semidirect
10. C₈×#11719917	C₂	semidirect
11. C₅₃×#2966	C₂	semidirect
203. #88322	#131	semidirect
227. #2966	D₅₃	semidirect
280. C₅₃	#17966	semidirect

Maximal subgroups

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

subgroup	index
#20285311	2
#20283428	2
#20273605	2
#20273598	2
#20273537	2
#20273525	2
#20273483	2
#20273183	2
#20273180	2
C₈×#11719917	2
C₅₃×#2966	2
#17966	53