Subgroups — #27353234

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 432 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	C₂×#16810645	C₂	1	1	yes	maximal, characteristic
3	#423×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20273487	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273486	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273485	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273484	C₂	1	4	yes	maximal, complemented
8	#20273469	C₂	1	4	yes	maximal, complemented
9	C₅₃×#1570	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#11719912	—	2	1	no	—
11	C₂×M₁₆×D₅₃	—	2	1	no	—
12	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
13	#16810676	C₂²	1	1	yes	characteristic
14	#16810675	C₄	1	2	yes	—
15	#16810674	C₂²	1	2	yes	—
16	#16810673	C₄	1	2	yes	—
17	#16810647	C₂²	1	1	yes	characteristic
18	#16810646	—	2	1	no	—
19	#16810645	—	2	1	no	—
20	#16810645	—	2	1	no	—
21	#16810645	—	2	1	no	—
22	#16810645	C₂²	1	2	yes	—
23	#16810645	C₂²	1	2	yes	—
24	#16810645	C₂²	1	2	yes	—
25	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
26	C₂×#13133146	—	2	1	no	—
27	C₄⋊C₈×D₅₃	—	2	1	no	—
28	C₄⋊C₈×D₅₃	C₄	1	2	yes	—
29	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
30	C₂²⋊C₈×D₅₃	C₄	1	2	yes	—
31	C₄×C₈×D₅₃	C₂²	1	2	yes	—
32	#16810374	C₂²	1	4	yes	complemented
33	#16810373	C₂²	1	4	yes	complemented
34	#16810373	C₂²	1	4	yes	complemented
35	C₅₃×#431	C₂²	1	1	yes	characteristic
36	C₅₃×#423	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#334	C₂²	1	4	yes	complemented
39	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
41	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
42	C₂×#11719917	—	2	1	no	—
43	C₂×#11719912	—	2	2	no	—
44	C₂×#11719912	—	2	2	no	—
45	C₂×#11719912	—	2	2	no	—
46	C₂×#11719912	—	4	4	no	—
47	M₁₆×D₅₃	—	4	4	no	—
48	C₂²×#88322	—	2	1	no	—
49	C₂×C₈×D₅₃	—	2	2	no	—
50	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
51	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄×Dic₅₃	Q₈	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	D₄	1	1	yes	characteristic
57	C₄²×D₅₃	D₄	1	1	yes	characteristic
58	C₄²×D₅₃	Q₈	1	1	yes	characteristic
59	C₄²×D₅₃	Q₈	1	1	yes	characteristic
60	C₄²×D₅₃	C₂³	1	1	yes	characteristic
61	C₄²×D₅₃	C₂³	1	1	yes	characteristic
62	C₄⋊#88322	—	2	1	no	—
63	C₄⋊#88322	—	2	1	no	—
64	C₄⋊#88322	C₂×C₄	1	2	yes	—
65	C₄⋊#88322	C₂³	1	2	yes	—
66	C₄⋊#88322	C₂³	1	2	yes	—
67	#13133146	—	2	2	no	—
68	#13133146	—	2	2	no	—
69	#13133146	—	2	2	no	—
70	#13133146	—	2	2	no	—
71	#13133146	C₂×C₄	1	4	yes	—
72	C₂²⋊#88322	C₂×C₄	1	2	yes	—
73	#13133129	C₂×C₄	1	4	yes	—
74	C₄×#88322	C₂³	1	2	yes	—
75	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
76	C₂×M₁₆×C₅₃	—	2	1	no	—
77	C₂²×C₈×C₅₃	—	2	1	no	—
78	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₅₃	—	2	1	no	—
80	C₄⋊C₈×C₅₃	—	2	1	no	—
81	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
82	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
83	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
84	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₅₃	C₂³	1	2	yes	—
86	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
90	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×D₅₃	—	4	2	no	—
94	C₂×C₄×D₅₃	—	4	2	no	—
95	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×D₅₃	SD₁₆	1	4	yes	—
102	C₂×C₄×D₅₃	SD₁₆	1	4	yes	—
103	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
108	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
109	M₁₆×C₅₃	—	4	2	no	—
110	C₂×C₈×C₅₃	—	2	2	no	—
111	C₂×C₈×C₅₃	—	2	2	no	—
112	C₂×C₈×C₅₃	—	4	2	no	—
113	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
114	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
117	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
118	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
119	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
120	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
121	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
122	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
123	C₄×Dic₅₃	C₄⋊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	C₂×D₄	1	1	yes	characteristic
127	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
128	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
129	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
130	#11719917	—	4	2	no	—
131	C₂×#88322	—	2	2	no	—
132	C₂×#88322	—	2	2	no	—
133	C₂×#88322	—	4	2	no	—
134	C₂×#88322	C₂²×C₄	1	4	yes	—
135	#11719912	—	4	4	no	—
136	#11719912	—	4	4	no	—
137	#11719912	—	4	4	no	—
138	C₈×D₅₃	—	4	4	no	—
139	C₈×D₅₃	—	4	4	no	—
140	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×D₅₃	—	4	1	no	—
142	C₂²×D₅₃	—	4	1	no	—
143	C₂²×D₅₃	#106	1	4	yes	—
144	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	4	1	no	—
148	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
152	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
153	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
156	C₂×Dic₅₃	—	2	2	no	—
157	C₂×Dic₅₃	—	4	1	no	—
158	C₂×Dic₅₃	#106	1	4	yes	—
159	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₄×D₅₃	—	2	2	no	—
166	C₄×D₅₃	—	2	2	no	—
167	C₄×D₅₃	—	2	2	no	—
168	C₄×D₅₃	—	2	2	no	—
169	C₄×D₅₃	—	4	1	no	—
170	C₄×D₅₃	—	4	1	no	—
171	C₄×D₅₃	—	4	1	no	—
172	C₄×D₅₃	—	4	1	no	—
173	C₄×D₅₃	#106	1	4	yes	—
174	C₄×D₅₃	#106	1	4	yes	—
175	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
176	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
177	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
178	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
179	C₈×C₅₃	—	2	4	no	—
180	C₈×C₅₃	—	4	2	no	—
181	C₈×C₅₃	—	4	2	no	—
182	#88322	—	2	4	no	—
183	#88322	—	4	2	no	—
184	#88322	—	4	2	no	—
185	#17813	—	53	1	no	maximal, Sylow
186	C₂²×C₅₃	—	2	1	no	—
187	C₂²×C₅₃	—	2	1	no	—
188	C₂²×C₅₃	C₂×#106	1	2	yes	—
189	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
190	C₂×D₅₃	—	2	1	no	—
191	C₂×D₅₃	—	2	1	no	—
192	C₂×D₅₃	—	2	1	no	—
193	C₂×D₅₃	—	2	1	no	—
194	C₂×D₅₃	—	4	4	no	—
195	C₂×D₅₃	C₂×#106	1	2	yes	—
196	C₂×D₅₃	C₂×#106	1	2	yes	—
197	C₄×C₅₃	—	2	1	no	—
198	C₄×C₅₃	—	2	1	no	—
199	C₄×C₅₃	—	2	1	no	—
200	C₄×C₅₃	C₂×#106	1	2	yes	—
201	C₄×C₅₃	#568	1	2	yes	—
202	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
203	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
204	Dic₅₃	—	2	1	no	—
205	Dic₅₃	—	2	1	no	—
206	Dic₅₃	—	2	1	no	—
207	Dic₅₃	C₂×#106	1	2	yes	—
208	Dic₅₃	C₂×#106	1	2	yes	—
209	Dic₅₃	#430	1	2	yes	—
210	C₂×#431	—	53	1	no	—
211	C₂×#423	—	53	1	no	—
212	C₂×#423	—	53	1	no	—
213	#1570	—	53	1	no	—
214	#1570	—	53	1	no	—
215	#1570	—	53	1	no	—
216	#1570	—	53	4	no	—
217	#1570	D₅₃	1	1	yes	characteristic, complemented
218	#1566	—	53	4	no	—
219	C₂×C₅₃	—	2	2	no	—
220	C₂×C₅₃	#2905	1	1	yes	characteristic
221	C₂×C₅₃	C₂²×#106	1	1	yes	characteristic
222	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
223	D₅₃	—	2	2	no	—
224	D₅₃	—	2	2	no	—
225	C₂²×M₁₆	—	106	1	no	—
226	C₂²×M₁₆	—	106	1	no	—
227	C₂²×C₄²	—	53	1	no	—
228	#431	—	53	1	no	—
229	#431	—	53	2	no	—
230	#431	—	53	2	no	—
231	#431	—	53	2	no	—
232	#431	C₂×D₅₃	1	1	yes	characteristic
233	#423	—	53	1	no	—
234	#423	—	53	2	no	—
235	#423	—	53	2	no	—
236	#423	—	53	2	no	—
237	#423	—	106	1	no	—
238	#423	—	106	1	no	—
239	#423	—	106	1	no	—
240	#423	—	106	1	no	—
241	#423	C₂×D₅₃	1	1	yes	characteristic
242	C₂×C₄⋊C₈	—	53	2	no	—
243	C₂×C₄⋊C₈	—	53	1	no	—
244	C₂×C₄⋊C₈	—	53	2	no	—
245	C₂×C₄⋊C₈	—	106	1	no	—
246	C₂×C₄⋊C₈	—	106	1	no	—
247	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
248	C₂×C₂²⋊C₈	—	53	2	no	—
249	C₂×C₄×C₈	—	53	2	no	—
250	#334	—	53	4	no	—
251	#334	—	53	4	no	—
252	#334	—	53	4	no	—
253	#334	C₂×D₅₃	1	4	yes	complemented
254	C₅₃	#17813	1	1	yes	characteristic, complemented, Sylow
255	C₂³×C₄	—	53	1	no	—
256	C₂³×C₄	—	53	1	no	—
257	C₂³×C₄	—	53	1	no	—
258	C₂×M₁₆	—	2	1	no	—
259	C₂×M₁₆	—	106	2	no	—
260	C₂×M₁₆	—	106	2	no	—
261	C₂×M₁₆	—	106	2	no	—
262	C₂×M₁₆	—	106	1	no	—
263	C₂×M₁₆	—	212	4	no	—
264	C₂×M₁₆	—	212	4	no	—
265	C₂²×C₈	—	2	1	no	—
266	C₂²×C₈	—	53	4	no	—
267	C₂²×C₈	—	106	2	no	—
268	C₂²×C₈	—	106	1	no	—
269	C₂×C₄²	—	53	1	no	—
270	C₂×C₄²	—	53	1	no	—
271	C₂×C₄²	—	53	1	no	—
272	C₂×C₄²	—	53	1	no	—
273	C₂×C₄²	—	53	1	no	—
274	C₂×C₄²	—	53	1	no	—
275	C₂×C₄²	—	53	1	no	—
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	C₂²×D₅₃	1	1	yes	characteristic
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	2	1	no	—
283	C₄⋊C₈	—	53	4	no	—
284	C₄⋊C₈	—	53	2	no	—
285	C₄⋊C₈	—	53	2	no	—
286	C₄⋊C₈	—	53	2	no	—
287	C₄⋊C₈	—	106	2	no	—
288	C₄⋊C₈	—	106	2	no	—
289	C₄⋊C₈	—	106	2	no	—
290	C₄⋊C₈	—	106	2	no	—
291	C₄⋊C₈	—	106	1	no	—
292	C₄⋊C₈	—	106	1	no	—
293	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
294	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
295	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
296	C₂²⋊C₈	—	53	2	no	—
297	C₂²⋊C₈	—	53	4	no	—
298	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
299	C₄×C₈	—	53	4	no	—
300	C₄×C₈	—	53	2	no	—
301	C₄×C₈	C₂²×D₅₃	1	2	yes	—
302	C₂⁴	—	53	1	no	—
303	C₂²×C₄	—	53	1	no	—
304	C₂²×C₄	—	53	1	no	—
305	C₂²×C₄	—	53	1	no	—
306	C₂²×C₄	—	53	4	no	—
307	C₂²×C₄	—	53	4	no	—
308	C₂²×C₄	—	53	1	no	—
309	C₂²×C₄	—	53	1	no	—
310	C₂²×C₄	—	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	—	212	2	no	—
321	C₂²×C₄	—	212	2	no	—
322	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
323	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
324	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
325	M₁₆	—	4	2	no	—
326	M₁₆	—	212	4	no	—
327	M₁₆	—	212	4	no	—
328	M₁₆	—	212	4	no	—
329	M₁₆	—	212	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	4	2	no	—
333	C₂×C₈	—	53	4	no	—
334	C₂×C₈	—	106	2	no	—
335	C₂×C₈	—	106	2	no	—
336	C₂×C₈	—	212	4	no	—
337	C₂×C₈	—	212	4	no	—
338	C₂×C₈	—	212	2	no	—
339	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
340	C₄²	—	53	1	no	—
341	C₄²	—	53	1	no	—
342	C₄²	—	53	1	no	—
343	C₄²	—	53	1	no	—
344	C₄²	—	53	1	no	—
345	C₄²	—	53	1	no	—
346	C₄²	—	53	1	no	—
347	C₄²	—	53	1	no	—
348	C₄²	—	53	1	no	—
349	C₄²	—	53	1	no	—
350	C₄²	—	53	1	no	—
351	C₄²	—	53	1	no	—
352	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
353	C₄²	D₄×D₅₃	1	1	yes	characteristic
354	C₄²	Q₈×D₅₃	1	1	yes	characteristic
355	C₄²	C₂³×D₅₃	1	1	yes	characteristic
356	C₂³	—	53	1	no	—
357	C₂³	—	53	1	no	—
358	C₂³	—	53	4	no	—
359	C₂³	—	212	1	no	—
360	C₂³	—	212	1	no	—
361	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	53	1	no	—
365	C₂×C₄	—	53	1	no	—
366	C₂×C₄	—	53	1	no	—
367	C₂×C₄	—	53	4	no	—
368	C₂×C₄	—	53	2	no	—
369	C₂×C₄	—	53	2	no	—
370	C₂×C₄	—	53	2	no	—
371	C₂×C₄	—	53	2	no	—
372	C₂×C₄	—	53	4	no	—
373	C₂×C₄	—	53	4	no	—
374	C₂×C₄	—	53	1	no	—
375	C₂×C₄	—	53	1	no	—
376	C₂×C₄	—	53	1	no	—
377	C₂×C₄	—	106	2	no	—
378	C₂×C₄	—	106	2	no	—
379	C₂×C₄	—	106	2	no	—
380	C₂×C₄	—	106	2	no	—
381	C₂×C₄	—	106	2	no	—
382	C₂×C₄	—	212	1	no	—
383	C₂×C₄	—	212	1	no	—
384	C₂×C₄	—	212	1	no	—
385	C₂×C₄	—	212	1	no	—
386	C₂×C₄	—	212	1	no	—
387	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
388	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
389	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
390	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
391	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
392	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
393	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
394	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
395	C₈	—	2	4	no	—
396	C₈	—	4	2	no	—
397	C₈	—	4	2	no	—
398	C₈	—	106	4	no	—
399	C₈	—	212	2	no	—
400	C₈	—	212	2	no	—
401	C₂²	—	2	1	no	—
402	C₂²	—	2	1	no	—
403	C₂²	—	53	2	no	—
404	C₂²	—	53	2	no	—
405	C₂²	—	106	1	no	—
406	C₂²	—	106	1	no	—
407	C₂²	—	106	1	no	—
408	C₂²	—	106	1	no	—
409	C₂²	—	212	4	no	—
410	C₂²	#106×D₅₃	1	2	yes	—
411	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	2	1	no	—
415	C₄	—	53	2	no	—
416	C₄	—	53	2	no	—
417	C₄	—	53	2	no	—
418	C₄	—	106	1	no	—
419	C₄	—	106	1	no	—
420	C₄	—	106	1	no	—
421	C₄	#106×D₅₃	1	2	yes	—
422	C₄	#16811388	1	2	yes	—
423	C₄	C₂×SD₁₆×D₅₃	1	1	yes	characteristic
424	C₄	C₂×SD₁₆×D₅₃	1	1	yes	characteristic
425	C₂	—	2	2	no	—
426	C₂	—	106	2	no	—
427	C₂	—	106	2	no	—
428	C₂	#20283276	1	1	yes	characteristic
429	C₂	C₂×#106×D₅₃	1	1	yes	characteristic
430	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×#1570	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1570	D₅₃
hypocenter	C₅₃	#17813
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1570

Complements

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

N	H ≅ G/N	type
4. #20273487	C₂	semidirect
5. #20273486	C₂	semidirect
6. #20273485	C₂	semidirect
7. #20273484	C₂	semidirect
8. #20273469	C₂	semidirect
9. C₅₃×#1570	C₂	semidirect
32. #16810374	C₂²	semidirect
33. #16810373	C₂²	semidirect
34. #16810373	C₂²	semidirect
38. C₅₃×#334	C₂²	semidirect
217. #1570	D₅₃	semidirect
253. #334	C₂×D₅₃	semidirect
254. C₅₃	#17813	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
C₂×#16810645	2
#423×D₅₃	2
#20273487	2
#20273486	2
#20273485	2
#20273484	2
#20273469	2
C₅₃×#1570	2
#17813	53