Subgroups — #27274046

Order 13568.

568 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283347	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810679	C₂	1	1	yes	maximal, characteristic
3	#423×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20272590	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272589	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272589	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1292	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₅₃	—	2	1	no	—
9	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
10	#16810679	—	2	1	no	—
11	#16810679	—	2	1	no	—
12	#16810679	—	2	1	no	—
13	#16810679	—	2	1	no	—
14	#16810679	—	2	1	no	—
15	#16810679	C₄	1	2	yes	—
16	#16810675	—	2	1	no	—
17	#16810675	C₄	1	2	yes	—
18	#16810647	C₂²	1	1	yes	characteristic
19	#16810647	C₂²	1	1	yes	characteristic
20	#16810646	—	2	1	no	—
21	#16810645	—	2	1	no	—
22	#16810645	C₄	1	2	yes	—
23	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×D₅₃	—	2	1	no	—
25	C₄⋊C₈×D₅₃	C₄	1	2	yes	—
26	#16810586	—	2	1	no	—
27	C₂×#13133129	C₄	1	2	yes	—
28	M₁₆×Dic₅₃	—	2	1	no	—
29	C₂×#13133127	C₄	1	2	yes	—
30	C₅₃×#423	C₂²	1	1	yes	characteristic
31	C₅₃×#423	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
36	C₂×#11719917	—	2	1	no	—
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	C₂²×#88322	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅₃	—	2	2	no	—
43	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
46	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄²×D₅₃	D₄	1	1	yes	characteristic
51	C₄²×D₅₃	D₄	1	1	yes	characteristic
52	C₄²×D₅₃	D₄	1	1	yes	characteristic
53	C₄²×D₅₃	D₄	1	1	yes	characteristic
54	C₄⋊#88322	—	2	1	no	—
55	C₄⋊#88322	—	2	1	no	—
56	C₄⋊#88322	—	2	1	no	—
57	C₄⋊#88322	—	2	1	no	—
58	C₄⋊#88322	C₂×C₄	1	2	yes	—
59	C₄⋊#88322	C₂×C₄	1	2	yes	—
60	#13133146	—	2	2	no	—
61	#13133146	—	2	2	no	—
62	#13133129	—	2	2	no	—
63	#13133129	—	2	2	no	—
64	#13133129	—	2	2	no	—
65	#13133129	—	2	2	no	—
66	#13133127	—	2	2	no	—
67	#13133127	—	2	2	no	—
68	#13133127	—	2	2	no	—
69	C₈×Dic₅₃	—	2	2	no	—
70	C₂×M₁₆×C₅₃	—	2	1	no	—
71	C₂×M₁₆×C₅₃	—	2	1	no	—
72	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
73	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₅₃	—	2	1	no	—
75	C₄⋊C₈×C₅₃	—	2	1	no	—
76	C₄⋊C₈×C₅₃	—	2	1	no	—
77	C₄⋊C₈×C₅₃	—	2	1	no	—
78	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
79	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
80	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
84	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×D₅₃	—	2	1	no	—
88	C₂×C₄×D₅₃	—	2	1	no	—
89	C₂×C₄×D₅₃	—	2	1	no	—
90	C₂×C₄×D₅₃	—	2	1	no	—
91	C₂×C₄×D₅₃	—	4	1	no	—
92	C₂×C₄×D₅₃	—	4	1	no	—
93	C₂×C₄×D₅₃	—	4	2	no	—
94	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
95	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
96	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×D₅₃	C₂²⋊C₄	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	C₄⋊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	M₁₆×C₅₃	—	4	1	no	—
107	M₁₆×C₅₃	—	4	1	no	—
108	M₁₆×C₅₃	—	4	1	no	—
109	M₁₆×C₅₃	—	4	1	no	—
110	C₂×C₈×C₅₃	—	2	2	no	—
111	C₂×C₈×C₅₃	—	2	2	no	—
112	C₂×C₈×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	—	4	2	no	—
114	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
119	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
120	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
121	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
122	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
123	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
124	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
125	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
126	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
127	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
128	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
129	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
130	#11719917	—	4	1	no	—
131	#11719917	—	4	1	no	—
132	#11719917	—	4	1	no	—
133	#11719917	—	4	1	no	—
134	C₂×#88322	—	2	2	no	—
135	C₂×#88322	—	2	2	no	—
136	C₂×#88322	—	2	2	no	—
137	C₂×#88322	—	4	2	no	—
138	#11719912	—	4	4	no	—
139	C₈×D₅₃	—	4	4	no	—
140	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×D₅₃	—	2	1	no	—
142	C₂²×D₅₃	—	2	1	no	—
143	C₂²×D₅₃	—	2	2	no	—
144	C₂²×D₅₃	—	4	1	no	—
145	C₂²×D₅₃	#95	1	1	yes	characteristic
146	C₂²×D₅₃	#95	1	1	yes	characteristic
147	C₂×C₄×C₅₃	—	2	1	no	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	—	4	1	no	—
150	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
159	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
160	C₂×Dic₅₃	—	2	1	no	—
161	C₂×Dic₅₃	—	2	1	no	—
162	C₂×Dic₅₃	—	4	1	no	—
163	C₂×Dic₅₃	—	4	1	no	—
164	C₂×Dic₅₃	#95	1	1	yes	characteristic
165	C₂×Dic₅₃	#95	1	1	yes	characteristic
166	C₂×Dic₅₃	#95	1	1	yes	characteristic
167	C₂×Dic₅₃	#95	1	1	yes	characteristic
168	C₂×Dic₅₃	#95	1	1	yes	characteristic
169	C₂×Dic₅₃	#95	1	1	yes	characteristic
170	C₄×D₅₃	—	2	2	no	—
171	C₄×D₅₃	—	2	2	no	—
172	C₄×D₅₃	—	2	1	no	—
173	C₄×D₅₃	—	2	1	no	—
174	C₄×D₅₃	—	2	1	no	—
175	C₄×D₅₃	—	2	1	no	—
176	C₄×D₅₃	—	2	1	no	—
177	C₄×D₅₃	—	2	1	no	—
178	C₄×D₅₃	—	2	1	no	—
179	C₄×D₅₃	—	2	1	no	—
180	C₄×D₅₃	—	2	1	no	—
181	C₄×D₅₃	—	2	1	no	—
182	C₄×D₅₃	—	2	1	no	—
183	C₄×D₅₃	—	2	1	no	—
184	C₄×D₅₃	—	4	1	no	—
185	C₄×D₅₃	—	4	1	no	—
186	C₄×D₅₃	—	4	1	no	—
187	C₄×D₅₃	—	4	1	no	—
188	C₈×C₅₃	—	4	2	no	—
189	C₈×C₅₃	—	4	2	no	—
190	C₈×C₅₃	—	4	2	no	—
191	#88322	—	4	2	no	—
192	#88322	—	4	2	no	—
193	#88322	—	4	2	no	—
194	#8003	—	53	1	no	maximal, Sylow
195	C₂²×C₅₃	—	2	1	no	—
196	C₂²×C₅₃	—	2	1	no	—
197	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
198	C₂²×C₅₃	#428	1	1	yes	characteristic
199	C₂²×C₅₃	#428	1	1	yes	characteristic
200	C₂×D₅₃	—	2	1	no	—
201	C₂×D₅₃	—	2	1	no	—
202	C₂×D₅₃	—	2	1	no	—
203	C₂×D₅₃	—	2	1	no	—
204	C₂×D₅₃	—	2	1	no	—
205	C₂×D₅₃	—	4	2	no	—
206	C₂×D₅₃	—	4	2	no	—
207	C₂×D₅₃	#338	1	2	yes	—
208	C₄×C₅₃	—	2	1	no	—
209	C₄×C₅₃	—	2	1	no	—
210	C₄×C₅₃	—	2	1	no	—
211	C₄×C₅₃	#413	1	1	yes	characteristic
212	C₄×C₅₃	#413	1	1	yes	characteristic
213	C₄×C₅₃	#417	1	1	yes	characteristic
214	C₄×C₅₃	#417	1	1	yes	characteristic
215	C₄×C₅₃	#418	1	1	yes	characteristic
216	C₄×C₅₃	#419	1	1	yes	characteristic
217	Dic₅₃	—	2	1	no	—
218	Dic₅₃	—	2	1	no	—
219	Dic₅₃	—	2	1	no	—
220	Dic₅₃	—	2	1	no	—
221	Dic₅₃	—	2	1	no	—
222	Dic₅₃	—	2	1	no	—
223	#2920	—	53	1	no	—
224	C₂×#432	—	53	1	no	—
225	C₂×#423	—	53	1	no	—
226	#1292	—	53	1	no	—
227	#1292	—	53	1	no	—
228	#1292	—	53	1	no	—
229	#1292	D₅₃	1	1	yes	characteristic, complemented
230	C₂×C₅₃	—	2	1	no	—
231	C₂×C₅₃	—	2	1	no	—
232	C₂×C₅₃	C₂×#338	1	1	yes	characteristic
233	C₂×C₅₃	#1736	1	1	yes	characteristic
234	C₂×C₅₃	#1742	1	1	yes	characteristic
235	D₅₃	—	2	2	no	—
236	D₅₃	—	4	1	no	—
237	C₂²×M₁₆	—	106	1	no	—
238	C₂²×C₄²	—	53	1	no	—
239	#432	—	53	2	no	—
240	#432	—	106	1	no	—
241	#432	—	106	1	no	—
242	#432	—	106	1	no	—
243	#432	—	106	1	no	—
244	#432	—	106	1	no	—
245	#431	—	53	2	no	—
246	#431	—	106	1	no	—
247	#423	—	53	1	no	—
248	#423	—	53	1	no	—
249	#423	—	53	2	no	—
250	#423	—	106	1	no	—
251	#423	—	106	1	no	—
252	#423	C₂×D₅₃	1	1	yes	characteristic
253	#423	C₂×D₅₃	1	1	yes	characteristic
254	C₂×C₄⋊C₈	—	53	2	no	—
255	C₂×C₄⋊C₈	—	53	1	no	—
256	C₂×C₄⋊C₈	—	106	1	no	—
257	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
258	#407	—	106	1	no	—
259	C₂×C₂²⋊C₈	—	53	2	no	—
260	C₄×M₁₆	—	106	1	no	—
261	C₂×#97	—	53	2	no	—
262	C₅₃	#8003	1	1	yes	characteristic, complemented, Sylow
263	C₂³×C₄	—	53	1	no	—
264	C₂³×C₄	—	53	1	no	—
265	C₂³×C₄	—	53	1	no	—
266	C₂×M₁₆	—	2	1	no	—
267	C₂×M₁₆	—	2	1	no	—
268	C₂×M₁₆	—	106	2	no	—
269	C₂×M₁₆	—	106	1	no	—
270	C₂×M₁₆	—	106	1	no	—
271	C₂×M₁₆	—	212	2	no	—
272	C₂×M₁₆	—	212	2	no	—
273	C₂²×C₈	—	53	2	no	—
274	C₂²×C₈	—	106	2	no	—
275	C₂²×C₈	C₄×D₅₃	1	2	yes	—
276	C₂×C₄²	—	53	1	no	—
277	C₂×C₄²	—	53	1	no	—
278	C₂×C₄²	—	53	1	no	—
279	C₂×C₄²	—	53	1	no	—
280	C₂×C₄²	—	53	1	no	—
281	C₂×C₄²	—	53	1	no	—
282	C₂×C₄²	—	53	1	no	—
283	C₂×C₄²	—	53	1	no	—
284	C₂×C₄²	—	53	1	no	—
285	C₂×C₄²	—	53	1	no	—
286	C₂×C₄²	—	53	1	no	—
287	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
288	C₄⋊C₈	—	2	1	no	—
289	C₄⋊C₈	—	2	1	no	—
290	C₄⋊C₈	—	2	1	no	—
291	C₄⋊C₈	—	2	1	no	—
292	C₄⋊C₈	—	53	2	no	—
293	C₄⋊C₈	—	53	2	no	—
294	C₄⋊C₈	—	106	2	no	—
295	C₄⋊C₈	—	106	2	no	—
296	C₄⋊C₈	—	106	1	no	—
297	C₄⋊C₈	—	106	1	no	—
298	C₄⋊C₈	—	106	1	no	—
299	C₄⋊C₈	—	106	1	no	—
300	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
301	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
302	C₂²⋊C₈	—	106	2	no	—
303	C₂²⋊C₈	—	106	2	no	—
304	C₂²⋊C₈	—	106	2	no	—
305	C₂²⋊C₈	—	106	2	no	—
306	#97	—	106	2	no	—
307	#97	—	106	2	no	—
308	#97	—	106	2	no	—
309	C₄×C₈	—	106	2	no	—
310	C₂⁴	—	53	1	no	—
311	C₂²×C₄	—	53	1	no	—
312	C₂²×C₄	—	53	1	no	—
313	C₂²×C₄	—	53	1	no	—
314	C₂²×C₄	—	53	1	no	—
315	C₂²×C₄	—	53	1	no	—
316	C₂²×C₄	—	53	1	no	—
317	C₂²×C₄	—	53	1	no	—
318	C₂²×C₄	—	53	1	no	—
319	C₂²×C₄	—	53	1	no	—
320	C₂²×C₄	—	53	1	no	—
321	C₂²×C₄	—	53	1	no	—
322	C₂²×C₄	—	53	1	no	—
323	C₂²×C₄	—	53	1	no	—
324	C₂²×C₄	—	53	1	no	—
325	C₂²×C₄	—	53	1	no	—
326	C₂²×C₄	—	106	1	no	—
327	C₂²×C₄	—	106	1	no	—
328	C₂²×C₄	—	106	1	no	—
329	C₂²×C₄	—	106	1	no	—
330	C₂²×C₄	—	212	1	no	—
331	C₂²×C₄	—	212	1	no	—
332	C₂²×C₄	—	212	2	no	—
333	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
334	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
335	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
336	M₁₆	—	4	1	no	—
337	M₁₆	—	4	1	no	—
338	M₁₆	—	4	1	no	—
339	M₁₆	—	4	1	no	—
340	M₁₆	—	212	4	no	—
341	M₁₆	—	212	1	no	—
342	M₁₆	—	212	1	no	—
343	M₁₆	—	212	1	no	—
344	M₁₆	—	212	1	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	4	2	no	—
349	C₂×C₈	—	106	2	no	—
350	C₂×C₈	—	106	2	no	—
351	C₂×C₈	—	106	2	no	—
352	C₂×C₈	—	212	4	no	—
353	C₂×C₈	—	212	2	no	—
354	C₄²	—	53	1	no	—
355	C₄²	—	53	1	no	—
356	C₄²	—	53	1	no	—
357	C₄²	—	53	1	no	—
358	C₄²	—	53	1	no	—
359	C₄²	—	53	1	no	—
360	C₄²	—	53	1	no	—
361	C₄²	—	53	1	no	—
362	C₄²	—	53	1	no	—
363	C₄²	—	53	1	no	—
364	C₄²	—	53	1	no	—
365	C₄²	—	53	1	no	—
366	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
367	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
368	C₄²	D₄×D₅₃	1	1	yes	characteristic
369	C₄²	D₄×D₅₃	1	1	yes	characteristic
370	C₂³	—	53	1	no	—
371	C₂³	—	53	1	no	—
372	C₂³	—	106	2	no	—
373	C₂³	—	106	1	no	—
374	C₂³	—	106	1	no	—
375	C₂³	—	212	1	no	—
376	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	53	1	no	—
381	C₂×C₄	—	53	1	no	—
382	C₂×C₄	—	53	1	no	—
383	C₂×C₄	—	53	1	no	—
384	C₂×C₄	—	53	1	no	—
385	C₂×C₄	—	53	1	no	—
386	C₂×C₄	—	106	2	no	—
387	C₂×C₄	—	106	2	no	—
388	C₂×C₄	—	106	1	no	—
389	C₂×C₄	—	106	1	no	—
390	C₂×C₄	—	106	1	no	—
391	C₂×C₄	—	106	1	no	—
392	C₂×C₄	—	106	1	no	—
393	C₂×C₄	—	106	1	no	—
394	C₂×C₄	—	106	1	no	—
395	C₂×C₄	—	106	1	no	—
396	C₂×C₄	—	106	1	no	—
397	C₂×C₄	—	106	1	no	—
398	C₂×C₄	—	106	1	no	—
399	C₂×C₄	—	106	1	no	—
400	C₂×C₄	—	106	1	no	—
401	C₂×C₄	—	106	1	no	—
402	C₂×C₄	—	212	1	no	—
403	C₂×C₄	—	212	1	no	—
404	C₂×C₄	—	212	1	no	—
405	C₂×C₄	—	212	1	no	—
406	C₂×C₄	—	212	1	no	—
407	C₂×C₄	—	212	1	no	—
408	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
409	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
410	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
411	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
412	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
413	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
414	C₂×C₄	#13133241	1	1	yes	characteristic
415	C₂×C₄	#13133244	1	1	yes	characteristic
416	C₂×C₄	#13133246	1	1	yes	characteristic
417	C₂×C₄	#13133249	1	1	yes	characteristic
418	C₈	—	4	2	no	—
419	C₈	—	4	2	no	—
420	C₈	—	4	2	no	—
421	C₈	—	212	2	no	—
422	C₈	—	212	2	no	—
423	C₈	—	212	2	no	—
424	C₂²	—	2	1	no	—
425	C₂²	—	2	1	no	—
426	C₂²	—	53	2	no	—
427	C₂²	—	106	1	no	—
428	C₂²	—	106	1	no	—
429	C₂²	—	106	1	no	—
430	C₂²	—	106	1	no	—
431	C₂²	—	106	1	no	—
432	C₂²	—	212	2	no	—
433	C₂²	—	212	2	no	—
434	C₂²	#95×D₅₃	1	1	yes	characteristic
435	C₂²	#16810662	1	1	yes	characteristic
436	C₂²	#16810664	1	1	yes	characteristic
437	C₄	—	2	1	no	—
438	C₄	—	2	1	no	—
439	C₄	—	2	1	no	—
440	C₄	—	106	1	no	—
441	C₄	—	106	1	no	—
442	C₄	—	106	1	no	—
443	C₄	—	106	1	no	—
444	C₄	—	106	1	no	—
445	C₄	—	106	1	no	—
446	C₄	#16810604	1	1	yes	characteristic
447	C₄	#16810606	1	1	yes	characteristic
448	C₄	#16810621	1	1	yes	characteristic
449	C₄	#16810623	1	1	yes	characteristic
450	C₄	#16810626	1	1	yes	characteristic
451	C₄	#16810630	1	1	yes	characteristic
452	C₂	—	2	1	no	—
453	C₂	—	2	1	no	—
454	C₂	—	106	2	no	—
455	C₂	—	212	1	no	—
456	C₂	#338×D₅₃	1	1	yes	characteristic
457	C₂	#20274433	1	1	yes	characteristic
458	C₂	#20274452	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#95×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1292	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1292	D₅₃
hypocenter	C₅₃	#8003
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1292

Complements

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

N	H ≅ G/N	type
4. #20272590	C₂	semidirect
5. #20272589	C₂	semidirect
6. #20272589	C₂	semidirect
7. C₅₃×#1292	C₂	semidirect
229. #1292	D₅₃	semidirect
262. C₅₃	#8003	semidirect

Maximal subgroups

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

subgroup	index
#20283347	2
C₂×#16810679	2
#423×D₅₃	2
#20272590	2
#20272589	2
C₅₃×#1292	2
#8003	53