Subgroups — C₅₃×#19885 #27227397

Order 13568.

830 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	C₅₃×#3391	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3274	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1712	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1687	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1668	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1621	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1619	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
13	C₂²×D₈×C₅₃	—	2	1	no	—
14	C₅₃×#550	C₂²	1	2	yes	complemented
15	D₄²×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#493	—	2	1	no	—
17	C₅₃×#493	C₂²	1	1	yes	characteristic
18	C₅₃×#493	C₂²	1	1	yes	characteristic, complemented
19	C₅₃×#493	C₂²	1	2	yes	complemented
20	C₅₃×#493	C₂²	1	2	yes	complemented
21	C₅₃×#469	—	2	2	no	—
22	C₅₃×#466	—	2	1	no	—
23	C₅₃×#460	C₂²	1	2	yes	complemented
24	C₅₃×#459	—	2	2	no	—
25	C₅₃×#459	C₂²	1	2	yes	—
26	C₅₃×#459	C₂²	1	2	yes	—
27	C₅₃×#459	C₂²	1	2	yes	complemented
28	C₅₃×#449	—	2	2	no	—
29	C₅₃×#447	—	2	2	no	—
30	C₅₃×#447	—	2	2	no	—
31	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
32	C₄×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
33	C₄×D₈×C₅₃	—	2	1	no	—
34	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#436	C₂²	1	2	yes	complemented
36	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#335	C₂²	1	1	yes	characteristic, complemented
38	C₅₃×#334	C₂²	1	1	yes	characteristic, complemented
39	C₅₃×#331	C₂²	1	2	yes	complemented
40	C₅₃×#331	C₂²	1	2	yes	complemented
41	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
44	C₈²×C₅₃	C₂²	1	1	yes	characteristic, complemented
45	C₂×#41×C₅₃	—	2	2	no	—
46	C₂²×D₄×C₅₃	—	2	2	no	—
47	C₂²×D₄×C₅₃	—	2	2	no	—
48	AGL(1,ℤ/8)×C₅₃	—	4	8	no	—
49	C₂×SD₁₆×C₅₃	—	2	2	no	—
50	C₂×SD₁₆×C₅₃	D₄	1	1	yes	characteristic
51	C₂×D₈×C₅₃	—	2	1	no	—
52	C₂×D₈×C₅₃	—	2	1	no	—
53	C₂×D₈×C₅₃	—	2	2	no	—
54	C₂×D₈×C₅₃	—	2	1	no	—
55	C₂×D₈×C₅₃	—	2	2	no	—
56	C₂×D₈×C₅₃	—	2	1	no	—
57	C₂×D₈×C₅₃	—	2	2	no	—
58	C₂×D₈×C₅₃	—	2	2	no	—
59	C₂×D₈×C₅₃	—	2	2	no	—
60	C₂×D₈×C₅₃	—	4	2	no	—
61	C₂×D₈×C₅₃	—	4	4	no	—
62	C₂×D₈×C₅₃	D₄	1	1	yes	characteristic
63	C₂×M₁₆×C₅₃	—	2	2	no	—
64	C₂²×C₈×C₅₃	—	2	1	no	—
65	#127×C₅₃	—	2	2	no	—
66	#127×C₅₃	C₂³	1	2	yes	—
67	#127×C₅₃	C₂³	1	2	yes	—
68	#121×C₅₃	—	2	2	no	—
69	#121×C₅₃	—	2	2	no	—
70	#121×C₅₃	—	2	2	no	—
71	#121×C₅₃	—	4	2	no	—
72	#120×C₅₃	—	4	2	no	—
73	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×D₄×C₅₃	—	2	2	no	—
75	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
76	C₄×D₄×C₅₃	C₂³	1	2	yes	—
77	#107×C₅₃	D₄	1	1	yes	characteristic
78	#106×C₅₃	D₄	1	1	yes	characteristic
79	C₄⋊C₈×C₅₃	—	2	1	no	—
80	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
82	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
83	#103×C₅₃	D₄	1	1	yes	characteristic
84	#102×C₅₃	—	2	2	no	—
85	#102×C₅₃	—	2	2	no	—
86	#102×C₅₃	—	2	2	no	—
87	#102×C₅₃	—	2	2	no	—
88	#102×C₅₃	D₄	1	2	yes	—
89	#102×C₅₃	D₄	1	1	yes	characteristic
90	C₂²⋊C₈×C₅₃	—	2	1	no	—
91	C₂²⋊C₈×C₅₃	—	2	2	no	—
92	C₄×C₈×C₅₃	—	2	1	no	—
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
96	C₂⁴×C₅₃	—	4	2	no	—
97	#41×C₅₃	—	4	4	no	—
98	#41×C₅₃	—	8	2	no	—
99	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂×D₄×C₅₃	—	2	2	no	—
101	C₂×D₄×C₅₃	—	2	2	no	—
102	C₂×D₄×C₅₃	—	2	2	no	—
103	C₂×D₄×C₅₃	—	2	2	no	—
104	C₂×D₄×C₅₃	—	2	2	no	—
105	C₂×D₄×C₅₃	—	2	2	no	—
106	C₂×D₄×C₅₃	—	4	2	no	—
107	C₂×D₄×C₅₃	—	4	2	no	—
108	C₂×D₄×C₅₃	—	4	2	no	—
109	C₂×D₄×C₅₃	—	4	2	no	—
110	C₂×D₄×C₅₃	—	4	4	no	—
111	C₂×D₄×C₅₃	—	4	4	no	—
112	C₂×D₄×C₅₃	—	8	2	no	—
113	C₂×D₄×C₅₃	—	8	2	no	—
114	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
115	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	—	2	1	no	—
117	C₂²×C₄×C₅₃	—	2	2	no	—
118	C₂²×C₄×C₅₃	—	4	2	no	—
119	SD₁₆×C₅₃	—	4	4	no	—
120	SD₁₆×C₅₃	D₈	1	4	yes	complemented
121	D₈×C₅₃	—	2	2	no	—
122	D₈×C₅₃	—	4	4	no	—
123	D₈×C₅₃	—	4	4	no	—
124	D₈×C₅₃	—	4	4	no	—
125	D₈×C₅₃	—	4	2	no	—
126	D₈×C₅₃	—	4	2	no	—
127	D₈×C₅₃	—	8	1	no	—
128	D₈×C₅₃	—	8	2	no	—
129	D₈×C₅₃	—	8	1	no	—
130	D₈×C₅₃	—	8	4	no	—
131	D₈×C₅₃	—	8	2	no	—
132	M₁₆×C₅₃	—	4	4	no	—
133	C₂×C₈×C₅₃	—	2	1	no	—
134	C₂×C₈×C₅₃	—	2	1	no	—
135	C₂×C₈×C₅₃	—	2	1	no	—
136	C₂×C₈×C₅₃	—	2	2	no	—
137	C₂×C₈×C₅₃	—	4	2	no	—
138	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
142	C₄⋊C₄×C₅₃	—	2	1	no	—
143	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
144	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
145	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
146	C₂²⋊C₄×C₅₃	—	2	1	no	—
147	C₂²⋊C₄×C₅₃	—	2	2	no	—
148	C₂²⋊C₄×C₅₃	—	4	2	no	—
149	C₂²⋊C₄×C₅₃	—	4	2	no	—
150	C₄²×C₅₃	—	2	1	no	—
151	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
152	C₂³×C₅₃	—	2	1	no	—
153	C₂³×C₅₃	—	2	2	no	—
154	C₂³×C₅₃	—	4	2	no	—
155	C₂³×C₅₃	—	4	2	no	—
156	C₂³×C₅₃	—	8	4	no	—
157	C₂³×C₅₃	—	8	4	no	—
158	C₂³×C₅₃	—	16	2	no	—
159	Q₈×C₅₃	—	2	1	no	—
160	Q₈×C₅₃	C₂×D₈	1	2	yes	complemented
161	D₄×C₅₃	—	2	2	no	—
162	D₄×C₅₃	—	2	1	no	—
163	D₄×C₅₃	—	2	2	no	—
164	D₄×C₅₃	—	4	2	no	—
165	D₄×C₅₃	—	4	2	no	—
166	D₄×C₅₃	—	4	2	no	—
167	D₄×C₅₃	—	4	2	no	—
168	D₄×C₅₃	—	4	2	no	—
169	D₄×C₅₃	—	4	2	no	—
170	D₄×C₅₃	—	4	2	no	—
171	D₄×C₅₃	—	4	2	no	—
172	D₄×C₅₃	—	8	4	no	—
173	D₄×C₅₃	—	8	2	no	—
174	D₄×C₅₃	—	8	2	no	—
175	D₄×C₅₃	—	8	4	no	—
176	D₄×C₅₃	—	8	4	no	—
177	D₄×C₅₃	—	16	2	no	—
178	D₄×C₅₃	C₂×D₈	1	2	yes	—
179	C₂×C₄×C₅₃	—	2	1	no	—
180	C₂×C₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	—	2	1	no	—
182	C₂×C₄×C₅₃	—	2	1	no	—
183	C₂×C₄×C₅₃	—	4	2	no	—
184	C₂×C₄×C₅₃	—	4	1	no	—
185	C₂×C₄×C₅₃	—	4	1	no	—
186	C₂×C₄×C₅₃	—	4	2	no	—
187	C₂×C₄×C₅₃	—	8	4	no	—
188	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
191	C₈×C₅₃	—	2	2	no	—
192	C₈×C₅₃	—	2	1	no	—
193	C₈×C₅₃	—	2	1	no	—
194	C₈×C₅₃	—	4	2	no	—
195	C₈×C₅₃	—	4	1	no	—
196	C₈×C₅₃	—	4	1	no	—
197	C₈×C₅₃	C₂×D₈	1	2	yes	complemented
198	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	complemented
199	#19885	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂²×C₅₃	—	2	2	no	—
201	C₂²×C₅₃	—	4	1	no	—
202	C₂²×C₅₃	—	4	1	no	—
203	C₂²×C₅₃	—	4	2	no	—
204	C₂²×C₅₃	—	4	2	no	—
205	C₂²×C₅₃	—	4	2	no	—
206	C₂²×C₅₃	—	8	2	no	—
207	C₂²×C₅₃	—	8	2	no	—
208	C₂²×C₅₃	—	8	2	no	—
209	C₂²×C₅₃	—	8	2	no	—
210	C₂²×C₅₃	—	8	2	no	—
211	C₂²×C₅₃	—	8	2	no	—
212	C₂²×C₅₃	—	16	8	no	—
213	C₂²×C₅₃	D₄²	1	1	yes	characteristic
214	C₄×C₅₃	—	2	2	no	—
215	C₄×C₅₃	—	2	1	no	—
216	C₄×C₅₃	—	4	1	no	—
217	C₄×C₅₃	—	4	2	no	—
218	C₄×C₅₃	—	4	1	no	—
219	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
220	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
221	C₄×C₅₃	#576	1	1	yes	characteristic
222	C₄×C₅₃	#576	1	1	yes	characteristic
223	#3391	C₂×C₅₃	1	1	yes	characteristic, complemented
224	D₄×D₈	C₂×C₅₃	1	1	yes	characteristic, complemented
225	#3274	C₂×C₅₃	1	2	yes	complemented
226	#1712	C₂×C₅₃	1	1	yes	characteristic, complemented
227	#1687	C₂×C₅₃	1	2	yes	complemented
228	#1668	C₂×C₅₃	1	1	yes	characteristic, complemented
229	#1667	C₂×C₅₃	1	1	yes	characteristic, complemented
230	#1621	C₂×C₅₃	1	2	yes	complemented
231	#1619	C₂×C₅₃	1	2	yes	complemented
232	#1589	C₂×C₅₃	1	1	yes	characteristic, complemented
233	C₈×SD₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
234	C₂×C₅₃	—	4	2	no	—
235	C₂×C₅₃	—	8	2	no	—
236	C₂×C₅₃	—	16	2	no	—
237	C₂×C₅₃	—	16	2	no	—
238	C₂×C₅₃	#3274	1	1	yes	characteristic
239	C₂×C₅₃	D₄×D₈	1	1	yes	characteristic
240	C₂×C₅₃	#3292	1	1	yes	characteristic
241	C₂×AGL(1,ℤ/8)	—	2	2	no	—
242	C₂²×D₈	—	2	1	no	—
243	#550	C₂²×C₅₃	1	2	yes	complemented
244	D₄²	C₂²×C₅₃	1	2	yes	—
245	#493	—	2	1	no	—
246	#493	C₂²×C₅₃	1	1	yes	characteristic
247	#493	C₂²×C₅₃	1	1	yes	characteristic, complemented
248	#493	C₂²×C₅₃	1	2	yes	complemented
249	#493	C₂²×C₅₃	1	2	yes	complemented
250	#469	—	2	2	no	—
251	#466	—	2	1	no	—
252	#460	C₂²×C₅₃	1	2	yes	complemented
253	#459	—	2	2	no	—
254	#459	C₂²×C₅₃	1	2	yes	—
255	#459	C₂²×C₅₃	1	2	yes	—
256	#459	C₂²×C₅₃	1	2	yes	complemented
257	#449	—	2	2	no	—
258	#447	—	2	2	no	—
259	#447	—	2	2	no	—
260	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
261	C₄×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
262	C₄×D₈	—	2	1	no	—
263	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
264	#436	C₂²×C₅₃	1	2	yes	complemented
265	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic
266	#335	C₂²×C₅₃	1	1	yes	characteristic, complemented
267	#334	C₂²×C₅₃	1	1	yes	characteristic, complemented
268	#331	C₂²×C₅₃	1	2	yes	complemented
269	#331	C₂²×C₅₃	1	2	yes	complemented
270	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
271	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
272	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
273	C₈²	C₂²×C₅₃	1	1	yes	characteristic, complemented
274	C₅₃	#19885	1	1	yes	characteristic, direct product factor, Sylow
275	C₂×#41	—	2	2	no	—
276	C₂²×D₄	—	2	2	no	—
277	C₂²×D₄	—	2	2	no	—
278	AGL(1,ℤ/8)	—	4	8	no	—
279	C₂×SD₁₆	—	2	2	no	—
280	C₂×SD₁₆	D₄×C₅₃	1	1	yes	characteristic
281	C₂×D₈	—	2	2	no	—
282	C₂×D₈	—	2	2	no	—
283	C₂×D₈	—	2	2	no	—
284	C₂×D₈	—	2	1	no	—
285	C₂×D₈	—	2	2	no	—
286	C₂×D₈	—	2	1	no	—
287	C₂×D₈	—	2	1	no	—
288	C₂×D₈	—	2	2	no	—
289	C₂×D₈	—	2	1	no	—
290	C₂×D₈	—	4	2	no	—
291	C₂×D₈	—	4	4	no	—
292	C₂×D₈	D₄×C₅₃	1	1	yes	characteristic
293	C₂×M₁₆	—	2	2	no	—
294	C₂²×C₈	—	2	1	no	—
295	#127	—	2	2	no	—
296	#127	C₂³×C₅₃	1	2	yes	—
297	#127	C₂³×C₅₃	1	2	yes	—
298	#121	—	2	2	no	—
299	#121	—	2	2	no	—
300	#121	—	2	2	no	—
301	#121	—	4	2	no	—
302	#120	—	4	2	no	—
303	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
304	C₄×D₄	—	2	2	no	—
305	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
306	C₄×D₄	C₂³×C₅₃	1	2	yes	—
307	#107	D₄×C₅₃	1	1	yes	characteristic
308	#106	D₄×C₅₃	1	1	yes	characteristic
309	C₄⋊C₈	—	2	1	no	—
310	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
311	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
312	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
313	#103	D₄×C₅₃	1	1	yes	characteristic
314	#102	—	2	2	no	—
315	#102	—	2	2	no	—
316	#102	—	2	2	no	—
317	#102	—	2	2	no	—
318	#102	D₄×C₅₃	1	2	yes	—
319	#102	D₄×C₅₃	1	1	yes	characteristic
320	C₂²⋊C₈	—	2	1	no	—
321	C₂²⋊C₈	—	2	2	no	—
322	C₄×C₈	—	2	1	no	—
323	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
324	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
325	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
326	C₂⁴	—	4	2	no	—
327	#41	—	4	4	no	—
328	#41	—	8	2	no	—
329	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	4	2	no	—
337	C₂×D₄	—	4	2	no	—
338	C₂×D₄	—	4	2	no	—
339	C₂×D₄	—	4	2	no	—
340	C₂×D₄	—	4	4	no	—
341	C₂×D₄	—	4	4	no	—
342	C₂×D₄	—	8	2	no	—
343	C₂×D₄	—	8	2	no	—
344	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
345	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	4	2	no	—
349	SD₁₆	—	4	4	no	—
350	SD₁₆	D₈×C₅₃	1	4	yes	complemented
351	D₈	—	2	2	no	—
352	D₈	—	4	4	no	—
353	D₈	—	4	4	no	—
354	D₈	—	4	4	no	—
355	D₈	—	4	2	no	—
356	D₈	—	4	2	no	—
357	D₈	—	8	1	no	—
358	D₈	—	8	1	no	—
359	D₈	—	8	2	no	—
360	D₈	—	8	4	no	—
361	D₈	—	8	2	no	—
362	M₁₆	—	4	4	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	4	2	no	—
368	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
369	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
370	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
371	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
374	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
375	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
376	C₂²⋊C₄	—	2	1	no	—
377	C₂²⋊C₄	—	2	2	no	—
378	C₂²⋊C₄	—	4	2	no	—
379	C₂²⋊C₄	—	4	2	no	—
380	C₄²	—	2	1	no	—
381	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
382	C₂³	—	2	1	no	—
383	C₂³	—	2	2	no	—
384	C₂³	—	4	2	no	—
385	C₂³	—	4	2	no	—
386	C₂³	—	8	4	no	—
387	C₂³	—	8	4	no	—
388	C₂³	—	16	2	no	—
389	Q₈	—	2	1	no	—
390	Q₈	C₂×D₈×C₅₃	1	2	yes	complemented
391	D₄	—	2	2	no	—
392	D₄	—	2	1	no	—
393	D₄	—	2	2	no	—
394	D₄	—	4	2	no	—
395	D₄	—	4	2	no	—
396	D₄	—	4	2	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	2	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	2	no	—
401	D₄	—	4	2	no	—
402	D₄	—	8	4	no	—
403	D₄	—	8	2	no	—
404	D₄	—	8	2	no	—
405	D₄	—	8	4	no	—
406	D₄	—	8	4	no	—
407	D₄	—	16	2	no	—
408	D₄	C₂×D₈×C₅₃	1	2	yes	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	8	4	no	—
418	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
419	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
420	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
421	C₈	—	2	2	no	—
422	C₈	—	2	1	no	—
423	C₈	—	2	1	no	—
424	C₈	—	4	2	no	—
425	C₈	—	4	1	no	—
426	C₈	—	4	1	no	—
427	C₈	C₂×D₈×C₅₃	1	2	yes	complemented
428	C₈	AGL(1,ℤ/8)×C₅₃	1	2	yes	complemented
429	C₂²	—	2	2	no	—
430	C₂²	—	4	1	no	—
431	C₂²	—	4	1	no	—
432	C₂²	—	4	2	no	—
433	C₂²	—	4	2	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	8	2	no	—
436	C₂²	—	8	2	no	—
437	C₂²	—	8	2	no	—
438	C₂²	—	8	2	no	—
439	C₂²	—	8	2	no	—
440	C₂²	—	8	2	no	—
441	C₂²	—	16	8	no	—
442	C₂²	D₄²×C₅₃	1	1	yes	characteristic
443	C₄	—	2	2	no	—
444	C₄	—	2	1	no	—
445	C₄	—	4	1	no	—
446	C₄	—	4	2	no	—
447	C₄	—	4	1	no	—
448	C₄	C₂²×D₈×C₅₃	1	1	yes	characteristic
449	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
450	C₄	C₅₃×#576	1	1	yes	characteristic
451	C₄	C₅₃×#576	1	1	yes	characteristic
452	C₂	—	4	2	no	—
453	C₂	—	8	2	no	—
454	C₂	—	16	2	no	—
455	C₂	—	16	2	no	—
456	C₂	C₅₃×#3274	1	1	yes	characteristic
457	C₂	D₄×D₈×C₅₃	1	1	yes	characteristic
458	C₂	C₅₃×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3391	C₂	semidirect
2. D₄×D₈×C₅₃	C₂	semidirect
3. C₅₃×#3274	C₂	semidirect
4. C₅₃×#1712	C₂	semidirect
5. C₅₃×#1687	C₂	semidirect
6. C₅₃×#1668	C₂	semidirect
7. C₅₃×#1667	C₂	semidirect
8. C₅₃×#1621	C₂	semidirect
9. C₅₃×#1619	C₂	semidirect
10. C₅₃×#1589	C₂	semidirect
11. C₈×SD₁₆×C₅₃	C₂	semidirect
14. C₅₃×#550	C₂²	semidirect
18. C₅₃×#493	C₂²	semidirect
19. C₅₃×#493	C₂²	semidirect
20. C₅₃×#493	C₂²	semidirect
23. C₅₃×#460	C₂²	semidirect
27. C₅₃×#459	C₂²	semidirect
31. C₈×Q₈×C₅₃	C₂²	semidirect
35. C₅₃×#436	C₂²	semidirect
37. C₅₃×#335	C₂²	semidirect
38. C₅₃×#334	C₂²	semidirect
39. C₅₃×#331	C₂²	semidirect
40. C₅₃×#331	C₂²	semidirect
41. C₅₃×Q₈⋊C₈	C₂²	semidirect
43. C₅₃×D₄⋊C₈	C₂²	semidirect
44. C₈²×C₅₃	C₂²	semidirect
120. SD₁₆×C₅₃	D₈	semidirect
160. Q₈×C₅₃	C₂×D₈	semidirect
197. C₈×C₅₃	C₂×D₈	semidirect
198. C₈×C₅₃	AGL(1,ℤ/8)	semidirect
199. #19885	C₅₃	direct (N × H)
223. #3391	C₂×C₅₃	semidirect
224. D₄×D₈	C₂×C₅₃	semidirect
225. #3274	C₂×C₅₃	semidirect
226. #1712	C₂×C₅₃	semidirect
227. #1687	C₂×C₅₃	semidirect
228. #1668	C₂×C₅₃	semidirect
229. #1667	C₂×C₅₃	semidirect
230. #1621	C₂×C₅₃	semidirect
231. #1619	C₂×C₅₃	semidirect
232. #1589	C₂×C₅₃	semidirect
233. C₈×SD₁₆	C₂×C₅₃	semidirect
243. #550	C₂²×C₅₃	semidirect
247. #493	C₂²×C₅₃	semidirect
248. #493	C₂²×C₅₃	semidirect
249. #493	C₂²×C₅₃	semidirect
252. #460	C₂²×C₅₃	semidirect
256. #459	C₂²×C₅₃	semidirect
260. C₈×Q₈	C₂²×C₅₃	semidirect
264. #436	C₂²×C₅₃	semidirect
266. #335	C₂²×C₅₃	semidirect
267. #334	C₂²×C₅₃	semidirect
268. #331	C₂²×C₅₃	semidirect
269. #331	C₂²×C₅₃	semidirect
270. Q₈⋊C₈	C₂²×C₅₃	semidirect
272. D₄⋊C₈	C₂²×C₅₃	semidirect
273. C₈²	C₂²×C₅₃	semidirect
274. C₅₃	#19885	direct (N × H)
350. SD₁₆	D₈×C₅₃	semidirect
390. Q₈	C₂×D₈×C₅₃	semidirect
427. C₈	C₂×D₈×C₅₃	semidirect
428. C₈	AGL(1,ℤ/8)×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3391	2
D₄×D₈×C₅₃	2
C₅₃×#3274	2
C₅₃×#1712	2
C₅₃×#1687	2
C₅₃×#1668	2
C₅₃×#1667	2
C₅₃×#1621	2
C₅₃×#1619	2
C₅₃×#1589	2
C₈×SD₁₆×C₅₃	2
#19885	53

Subgroups — C53×#19885 #27227397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19885 #27227397