Subgroups — C₅₃×#32787 #27240299

Order 13568.

672 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2857	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2851	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2101	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2100	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2100	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#2077	C₂	1	2	yes	maximal
7	C₅₃×#2075	C₂	1	2	yes	maximal
8	C₅₃×#2059	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2058	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#2057	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1817	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#107×C₅₃	C₂²	1	2	yes	—
17	C₂×#106×C₅₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₅₃×#391	—	2	1	no	—
20	C₅₃×#389	—	2	1	no	—
21	C₅₃×#389	C₂²	1	1	yes	characteristic
22	C₅₃×#389	C₂²	1	1	yes	characteristic
23	C₅₃×#389	C₂²	1	2	yes	—
24	C₅₃×#387	—	2	1	no	—
25	C₅₃×#387	—	2	1	no	—
26	C₅₃×#385	—	2	1	no	—
27	C₅₃×#385	—	2	1	no	—
28	C₅₃×#384	C₂²	1	1	yes	characteristic
29	C₅₃×#382	C₂²	1	1	yes	characteristic
30	C₅₃×#380	—	2	1	no	—
31	C₅₃×#380	C₂²	1	1	yes	characteristic
32	C₅₃×#380	C₂²	1	2	yes	—
33	C₅₃×#380	C₂²	1	1	yes	characteristic
34	C₅₃×#340	C₂²	1	2	yes	—
35	C₅₃×#340	C₂²	1	2	yes	—
36	C₅₃×#340	C₂²	1	2	yes	—
37	C₅₃×#340	C₂²	1	2	yes	—
38	C₅₃×#340	C₂²	1	2	yes	—
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₅₃×#336	C₂²	1	1	yes	characteristic
41	C₅₃×#336	C₂²	1	1	yes	characteristic
42	C₅₃×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₅₃	—	2	1	no	—
44	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₅₃	—	4	2	no	—
50	#125×C₅₃	—	4	2	no	—
51	C₂²⋊Q₈×C₅₃	—	4	4	no	—
52	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
69	C₂×C₄²×C₅₃	—	4	2	no	—
70	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	#107×C₅₃	—	4	2	no	—
75	#106×C₅₃	—	4	2	no	—
76	C₂²⋊C₈×C₅₃	—	4	2	no	—
77	#95×C₅₃	—	2	2	no	—
78	#95×C₅₃	—	2	1	no	—
79	#95×C₅₃	—	2	1	no	—
80	#95×C₅₃	—	2	2	no	—
81	#95×C₅₃	—	2	1	no	—
82	#95×C₅₃	—	2	1	no	—
83	#95×C₅₃	—	2	1	no	—
84	#95×C₅₃	—	2	1	no	—
85	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×Q₈×C₅₃	—	4	2	no	—
87	C₂×Q₈×C₅₃	—	4	2	no	—
88	C₂×Q₈×C₅₃	—	4	2	no	—
89	C₂²×C₄×C₅₃	—	2	2	no	—
90	C₂²×C₄×C₅₃	—	2	2	no	—
91	C₂²×C₄×C₅₃	—	2	2	no	—
92	C₂²×C₄×C₅₃	—	2	2	no	—
93	C₂²×C₄×C₅₃	—	2	2	no	—
94	C₂²×C₄×C₅₃	—	2	1	no	—
95	C₂²×C₄×C₅₃	—	2	1	no	—
96	C₂²×C₄×C₅₃	—	2	1	no	—
97	C₂²×C₄×C₅₃	—	2	1	no	—
98	C₂²×C₄×C₅₃	—	2	1	no	—
99	C₂²×C₄×C₅₃	—	2	1	no	—
100	C₂²×C₄×C₅₃	—	2	2	no	—
101	C₂²×C₄×C₅₃	—	2	2	no	—
102	C₂²×C₄×C₅₃	—	4	1	no	—
103	C₂²×C₄×C₅₃	—	4	1	no	—
104	C₂²×C₄×C₅₃	—	4	1	no	—
105	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
111	C₂×C₈×C₅₃	—	2	2	no	—
112	C₂×C₈×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	2	no	—
117	C₄⋊C₄×C₅₃	—	2	2	no	—
118	C₄⋊C₄×C₅₃	—	2	2	no	—
119	C₄⋊C₄×C₅₃	—	2	2	no	—
120	C₄⋊C₄×C₅₃	—	2	2	no	—
121	C₄⋊C₄×C₅₃	—	2	1	no	—
122	C₄⋊C₄×C₅₃	—	2	1	no	—
123	C₄⋊C₄×C₅₃	—	2	1	no	—
124	C₄⋊C₄×C₅₃	—	2	1	no	—
125	C₄⋊C₄×C₅₃	—	2	4	no	—
126	C₄⋊C₄×C₅₃	—	4	2	no	—
127	C₄⋊C₄×C₅₃	—	4	2	no	—
128	C₄⋊C₄×C₅₃	—	4	4	no	—
129	C₄⋊C₄×C₅₃	—	4	2	no	—
130	C₄⋊C₄×C₅₃	—	4	2	no	—
131	C₄⋊C₄×C₅₃	—	4	2	no	—
132	C₄⋊C₄×C₅₃	—	4	2	no	—
133	C₄⋊C₄×C₅₃	—	4	2	no	—
134	C₂²⋊C₄×C₅₃	—	4	2	no	—
135	C₂²⋊C₄×C₅₃	—	4	2	no	—
136	C₂²⋊C₄×C₅₃	—	4	4	no	—
137	C₄²×C₅₃	—	2	2	no	—
138	C₄²×C₅₃	—	2	2	no	—
139	C₄²×C₅₃	—	4	1	no	—
140	C₄²×C₅₃	—	4	1	no	—
141	C₂³×C₅₃	—	2	1	no	—
142	C₂³×C₅₃	—	2	1	no	—
143	C₂³×C₅₃	—	2	1	no	—
144	C₂³×C₅₃	—	2	1	no	—
145	C₂³×C₅₃	—	2	1	no	—
146	C₂³×C₅₃	—	2	1	no	—
147	C₂³×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
148	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
149	Q₈×C₅₃	—	8	4	no	—
150	C₂×C₄×C₅₃	—	2	1	no	—
151	C₂×C₄×C₅₃	—	2	1	no	—
152	C₂×C₄×C₅₃	—	2	1	no	—
153	C₂×C₄×C₅₃	—	2	1	no	—
154	C₂×C₄×C₅₃	—	2	1	no	—
155	C₂×C₄×C₅₃	—	2	1	no	—
156	C₂×C₄×C₅₃	—	2	1	no	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	2	1	no	—
160	C₂×C₄×C₅₃	—	2	1	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	1	no	—
163	C₂×C₄×C₅₃	—	4	4	no	—
164	C₂×C₄×C₅₃	—	4	4	no	—
165	C₂×C₄×C₅₃	—	4	2	no	—
166	C₂×C₄×C₅₃	—	4	2	no	—
167	C₂×C₄×C₅₃	—	4	2	no	—
168	C₂×C₄×C₅₃	—	4	1	no	—
169	C₂×C₄×C₅₃	—	4	1	no	—
170	C₂×C₄×C₅₃	—	4	1	no	—
171	C₂×C₄×C₅₃	—	4	2	no	—
172	C₂×C₄×C₅₃	—	4	1	no	—
173	C₂×C₄×C₅₃	—	4	2	no	—
174	C₂×C₄×C₅₃	—	4	1	no	—
175	C₂×C₄×C₅₃	—	4	1	no	—
176	C₂×C₄×C₅₃	—	4	2	no	—
177	C₂×C₄×C₅₃	—	4	1	no	—
178	C₂×C₄×C₅₃	—	4	1	no	—
179	C₂×C₄×C₅₃	—	4	2	no	—
180	C₂×C₄×C₅₃	—	4	1	no	—
181	C₂×C₄×C₅₃	—	4	1	no	—
182	C₂×C₄×C₅₃	—	4	1	no	—
183	C₂×C₄×C₅₃	—	4	1	no	—
184	C₂×C₄×C₅₃	—	4	2	no	—
185	C₂×C₄×C₅₃	#135	1	2	yes	—
186	C₂×C₄×C₅₃	#137	1	2	yes	—
187	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
188	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
189	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
190	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
191	C₈×C₅₃	—	4	2	no	—
192	C₈×C₅₃	—	4	2	no	—
193	#32787	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂²×C₅₃	—	2	2	no	—
195	C₂²×C₅₃	—	2	2	no	—
196	C₂²×C₅₃	—	2	2	no	—
197	C₂²×C₅₃	—	4	1	no	—
198	C₂²×C₅₃	—	4	1	no	—
199	C₂²×C₅₃	—	4	1	no	—
200	C₂²×C₅₃	—	4	1	no	—
201	C₂²×C₅₃	#534	1	1	yes	characteristic
202	C₂²×C₅₃	#536	1	1	yes	characteristic
203	C₂²×C₅₃	#536	1	1	yes	characteristic
204	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
205	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₂²×C₅₃	C₂×#137	1	1	yes	characteristic
207	C₂²×C₅₃	#575	1	1	yes	characteristic
208	C₄×C₅₃	—	2	2	no	—
209	C₄×C₅₃	—	2	2	no	—
210	C₄×C₅₃	—	4	1	no	—
211	C₄×C₅₃	—	4	1	no	—
212	C₄×C₅₃	—	4	1	no	—
213	C₄×C₅₃	—	8	1	no	—
214	C₄×C₅₃	—	8	2	no	—
215	C₄×C₅₃	—	8	1	no	—
216	C₄×C₅₃	—	8	1	no	—
217	C₄×C₅₃	—	8	2	no	—
218	C₄×C₅₃	—	8	1	no	—
219	C₄×C₅₃	#563	1	2	yes	—
220	C₄×C₅₃	#563	1	2	yes	—
221	#2857	C₂×C₅₃	1	1	yes	characteristic
222	#2851	C₂×C₅₃	1	1	yes	characteristic
223	#2101	C₂×C₅₃	1	2	yes	complemented
224	#2100	C₂×C₅₃	1	1	yes	characteristic, complemented
225	#2100	C₂×C₅₃	1	1	yes	characteristic, complemented
226	#2077	C₂×C₅₃	1	2	yes	—
227	#2075	C₂×C₅₃	1	2	yes	—
228	#2059	C₂×C₅₃	1	2	yes	complemented
229	#2058	C₂×C₅₃	1	1	yes	characteristic, complemented
230	#2057	C₂×C₅₃	1	1	yes	characteristic, complemented
231	#1817	C₂×C₅₃	1	1	yes	characteristic
232	C₂×C₅₃	—	4	2	no	—
233	C₂×C₅₃	#2817	1	1	yes	characteristic
234	C₂×C₅₃	#3188	1	1	yes	characteristic
235	C₂×C₅₃	#3196	1	1	yes	characteristic
236	C₂×C₅₃	#3231	1	1	yes	characteristic
237	C₂×C₅₃	#3233	1	1	yes	characteristic
238	C₂×C₅₃	#3234	1	1	yes	characteristic
239	C₂×C₅₃	#3238	1	1	yes	characteristic
240	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
241	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic
242	C₂×C₂²⋊Q₈	—	2	1	no	—
243	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
244	C₂×#107	C₂²×C₅₃	1	2	yes	—
245	C₂×#106	C₂²×C₅₃	1	2	yes	—
246	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
247	#391	—	2	1	no	—
248	#389	—	2	1	no	—
249	#389	C₂²×C₅₃	1	1	yes	characteristic
250	#389	C₂²×C₅₃	1	1	yes	characteristic
251	#389	C₂²×C₅₃	1	2	yes	—
252	#387	—	2	1	no	—
253	#387	—	2	1	no	—
254	#385	—	2	1	no	—
255	#385	—	2	1	no	—
256	#384	C₂²×C₅₃	1	1	yes	characteristic
257	#382	C₂²×C₅₃	1	1	yes	characteristic
258	#380	—	2	1	no	—
259	#380	C₂²×C₅₃	1	1	yes	characteristic
260	#380	C₂²×C₅₃	1	2	yes	—
261	#380	C₂²×C₅₃	1	1	yes	characteristic
262	#340	C₂²×C₅₃	1	2	yes	—
263	#340	C₂²×C₅₃	1	2	yes	—
264	#340	C₂²×C₅₃	1	2	yes	—
265	#340	C₂²×C₅₃	1	2	yes	—
266	#340	C₂²×C₅₃	1	2	yes	—
267	#340	C₂²×C₅₃	1	2	yes	—
268	#336	C₂²×C₅₃	1	1	yes	characteristic
269	#336	C₂²×C₅₃	1	1	yes	characteristic
270	#336	C₂²×C₅₃	1	2	yes	—
271	C₅₃	#32787	1	1	yes	characteristic, direct product factor, Sylow
272	C₂²×Q₈	—	2	1	no	—
273	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
274	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
275	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
276	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
277	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
278	C₄⋊Q₈	—	4	2	no	—
279	#125	—	4	2	no	—
280	C₂²⋊Q₈	—	4	4	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	—	2	2	no	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	—	2	1	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
291	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
292	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
293	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
294	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
295	C₂×C₂²⋊C₄	—	2	2	no	—
296	C₂×C₂²⋊C₄	—	2	1	no	—
297	C₂×C₂²⋊C₄	—	2	1	no	—
298	C₂×C₄²	—	4	2	no	—
299	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
300	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
301	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
302	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
303	#107	—	4	2	no	—
304	#106	—	4	2	no	—
305	C₂²⋊C₈	—	4	2	no	—
306	#95	—	2	2	no	—
307	#95	—	2	1	no	—
308	#95	—	2	1	no	—
309	#95	—	2	2	no	—
310	#95	—	2	1	no	—
311	#95	—	2	1	no	—
312	#95	—	2	1	no	—
313	#95	—	2	1	no	—
314	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
315	C₂×Q₈	—	4	2	no	—
316	C₂×Q₈	—	4	2	no	—
317	C₂×Q₈	—	4	2	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	4	1	no	—
332	C₂²×C₄	—	4	1	no	—
333	C₂²×C₄	—	4	1	no	—
334	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
335	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
336	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
337	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
338	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
339	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	4	no	—
355	C₄⋊C₄	—	4	2	no	—
356	C₄⋊C₄	—	4	4	no	—
357	C₄⋊C₄	—	4	2	no	—
358	C₄⋊C₄	—	4	2	no	—
359	C₄⋊C₄	—	4	2	no	—
360	C₄⋊C₄	—	4	2	no	—
361	C₄⋊C₄	—	4	2	no	—
362	C₄⋊C₄	—	4	2	no	—
363	C₂²⋊C₄	—	4	2	no	—
364	C₂²⋊C₄	—	4	4	no	—
365	C₂²⋊C₄	—	4	2	no	—
366	C₄²	—	2	2	no	—
367	C₄²	—	2	2	no	—
368	C₄²	—	4	1	no	—
369	C₄²	—	4	1	no	—
370	C₂³	—	2	1	no	—
371	C₂³	—	2	1	no	—
372	C₂³	—	2	1	no	—
373	C₂³	—	2	1	no	—
374	C₂³	—	2	1	no	—
375	C₂³	—	2	1	no	—
376	C₂³	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
377	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
378	Q₈	—	8	4	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	4	4	no	—
393	C₂×C₄	—	4	4	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	#135×C₅₃	1	2	yes	—
415	C₂×C₄	#137×C₅₃	1	2	yes	—
416	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
417	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
418	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
419	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
420	C₈	—	4	2	no	—
421	C₈	—	4	2	no	—
422	C₂²	—	2	2	no	—
423	C₂²	—	2	2	no	—
424	C₂²	—	2	2	no	—
425	C₂²	—	4	1	no	—
426	C₂²	—	4	1	no	—
427	C₂²	—	4	1	no	—
428	C₂²	—	4	1	no	—
429	C₂²	C₅₃×#534	1	1	yes	characteristic
430	C₂²	C₅₃×#536	1	1	yes	characteristic
431	C₂²	C₅₃×#536	1	1	yes	characteristic
432	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
433	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
434	C₂²	C₂×#137×C₅₃	1	1	yes	characteristic
435	C₂²	C₅₃×#575	1	1	yes	characteristic
436	C₄	—	2	2	no	—
437	C₄	—	2	2	no	—
438	C₄	—	4	1	no	—
439	C₄	—	4	1	no	—
440	C₄	—	4	1	no	—
441	C₄	—	8	1	no	—
442	C₄	—	8	1	no	—
443	C₄	—	8	2	no	—
444	C₄	—	8	1	no	—
445	C₄	—	8	1	no	—
446	C₄	—	8	2	no	—
447	C₄	C₅₃×#563	1	2	yes	—
448	C₄	C₅₃×#563	1	2	yes	—
449	C₂	—	4	2	no	—
450	C₂	C₅₃×#2817	1	1	yes	characteristic
451	C₂	C₅₃×#3188	1	1	yes	characteristic
452	C₂	C₅₃×#3196	1	1	yes	characteristic
453	C₂	C₅₃×#3231	1	1	yes	characteristic
454	C₂	C₅₃×#3233	1	1	yes	characteristic
455	C₂	C₅₃×#3234	1	1	yes	characteristic
456	C₂	C₅₃×#3238	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2101	C₂	semidirect
4. C₅₃×#2100	C₂	semidirect
5. C₅₃×#2100	C₂	semidirect
8. C₅₃×#2059	C₂	semidirect
9. C₅₃×#2058	C₂	semidirect
10. C₅₃×#2057	C₂	semidirect
193. #32787	C₅₃	direct (N × H)
223. #2101	C₂×C₅₃	semidirect
224. #2100	C₂×C₅₃	semidirect
225. #2100	C₂×C₅₃	semidirect
228. #2059	C₂×C₅₃	semidirect
229. #2058	C₂×C₅₃	semidirect
230. #2057	C₂×C₅₃	semidirect
271. C₅₃	#32787	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2857	2
C₅₃×#2851	2
C₅₃×#2101	2
C₅₃×#2100	2
C₅₃×#2077	2
C₅₃×#2075	2
C₅₃×#2059	2
C₅₃×#2058	2
C₅₃×#2057	2
C₅₃×#1817	2
#32787	53

Subgroups — C53×#32787 #27240299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#32787 #27240299