Subgroups — C₅₃×#22205 #27229717

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3054	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2929	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1929	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1922	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1897	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1876	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1866	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1811	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1729	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#131×C₅₃	C₂²	1	2	yes	—
12	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#525	C₂²	1	2	yes	—
15	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#470	C₄	1	1	yes	characteristic, complemented
17	C₅₃×#470	C₄	1	1	yes	characteristic, complemented
18	C₅₃×#469	C₄	1	1	yes	characteristic, complemented
19	C₅₃×#469	C₄	1	1	yes	characteristic, complemented
20	C₅₃×#468	C₄	1	2	yes	complemented
21	C₅₃×#468	C₄	1	2	yes	complemented
22	C₅₃×#435	—	2	2	no	—
23	C₅₃×#435	—	2	2	no	—
24	C₈×D₄×C₅₃	—	2	2	no	—
25	C₈×D₄×C₅₃	—	2	2	no	—
26	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#427	C₂²	1	1	yes	characteristic
29	C₅₃×#424	C₂²	1	2	yes	—
30	C₂×#104×C₅₃	C₂²	1	2	yes	—
31	C₂×#104×C₅₃	C₂²	1	2	yes	—
32	C₅₃×#416	C₂²	1	2	yes	—
33	C₅₃×#408	C₂²	1	2	yes	—
34	C₅₃×#407	C₂²	1	2	yes	—
35	C₅₃×#405	C₂²	1	2	yes	—
36	C₅₃×#341	C₂²	1	2	yes	—
37	C₅₃×#339	C₂²	1	2	yes	—
38	C₅₃×#339	C₂²	1	2	yes	—
39	C₅₃×#336	C₂²	1	2	yes	—
40	C₅₃×#336	C₂²	1	2	yes	—
41	C₂×#41×C₅₃	C₂³	1	2	yes	—
42	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	#135×C₅₃	—	4	1	no	—
44	#135×C₅₃	—	4	1	no	—
45	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	#131×C₅₃	—	4	2	no	—
49	#131×C₅₃	—	4	2	no	—
50	C₂×M₁₆×C₅₃	—	2	2	no	—
51	C₂×M₁₆×C₅₃	—	4	1	no	—
52	C₂×M₁₆×C₅₃	—	4	1	no	—
53	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
58	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
59	C₂²×C₈×C₅₃	—	2	2	no	—
60	C₂²×C₈×C₅₃	—	4	1	no	—
61	C₂²×C₈×C₅₃	—	4	1	no	—
62	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	D₄	1	2	yes	—
64	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
65	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
68	#123×C₅₃	—	2	2	no	—
69	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
70	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
71	#121×C₅₃	C₂×C₄	1	2	yes	—
72	#121×C₅₃	C₂×C₄	1	2	yes	—
73	#120×C₅₃	—	2	2	no	—
74	C₄×D₄×C₅₃	—	2	2	no	—
75	C₄×D₄×C₅₃	—	2	2	no	—
76	#117×C₅₃	—	4	1	no	—
77	#117×C₅₃	—	4	1	no	—
78	#117×C₅₃	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
81	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
82	#108×C₅₃	—	4	1	no	—
83	#108×C₅₃	—	4	1	no	—
84	#107×C₅₃	C₂×C₄	1	2	yes	—
85	#106×C₅₃	C₂×C₄	1	1	yes	characteristic
86	#106×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₅₃	—	2	2	no	—
88	C₄⋊C₈×C₅₃	—	2	2	no	—
89	#104×C₅₃	—	4	2	no	—
90	#104×C₅₃	—	4	2	no	—
91	#103×C₅₃	C₂×C₄	1	2	yes	—
92	#103×C₅₃	C₂×C₄	1	2	yes	—
93	#102×C₅₃	C₂×C₄	1	2	yes	—
94	#102×C₅₃	C₂×C₄	1	2	yes	—
95	C₂²⋊C₈×C₅₃	—	2	2	no	—
96	C₂²⋊C₈×C₅₃	—	2	2	no	—
97	C₂²⋊C₈×C₅₃	—	2	2	no	—
98	C₂²⋊C₈×C₅₃	—	2	2	no	—
99	#97×C₅₃	—	2	2	no	—
100	C₄×C₈×C₅₃	—	2	2	no	—
101	C₂⁴×C₅₃	#41	1	1	yes	characteristic
102	#41×C₅₃	—	2	2	no	—
103	#41×C₅₃	—	2	2	no	—
104	#41×C₅₃	—	2	2	no	—
105	#41×C₅₃	—	2	2	no	—
106	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
107	C₂×D₄×C₅₃	—	2	2	no	—
108	C₂×D₄×C₅₃	—	2	2	no	—
109	C₂×D₄×C₅₃	—	2	2	no	—
110	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₅₃	—	2	1	no	—
112	C₂²×C₄×C₅₃	—	2	1	no	—
113	C₂²×C₄×C₅₃	—	2	1	no	—
114	C₂²×C₄×C₅₃	—	2	1	no	—
115	C₂²×C₄×C₅₃	—	2	2	no	—
116	C₂²×C₄×C₅₃	—	2	2	no	—
117	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
122	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
123	Q₁₆×C₅₃	—	4	1	no	—
124	SD₁₆×C₅₃	—	4	2	no	—
125	D₈×C₅₃	—	4	1	no	—
126	M₁₆×C₅₃	—	2	2	no	—
127	M₁₆×C₅₃	—	2	2	no	—
128	M₁₆×C₅₃	—	2	2	no	—
129	M₁₆×C₅₃	—	2	2	no	—
130	M₁₆×C₅₃	—	4	1	no	—
131	M₁₆×C₅₃	—	4	1	no	—
132	M₁₆×C₅₃	—	4	1	no	—
133	M₁₆×C₅₃	—	4	1	no	—
134	M₁₆×C₅₃	—	4	4	no	—
135	C₂×C₈×C₅₃	—	2	1	no	—
136	C₂×C₈×C₅₃	—	2	1	no	—
137	C₂×C₈×C₅₃	—	2	2	no	—
138	C₂×C₈×C₅₃	—	2	2	no	—
139	C₂×C₈×C₅₃	—	2	1	no	—
140	C₂×C₈×C₅₃	—	2	1	no	—
141	C₂×C₈×C₅₃	—	2	2	no	—
142	C₂×C₈×C₅₃	—	2	2	no	—
143	C₂×C₈×C₅₃	—	4	1	no	—
144	C₂×C₈×C₅₃	—	4	2	no	—
145	C₂×C₈×C₅₃	—	4	1	no	—
146	C₂×C₈×C₅₃	—	4	4	no	—
147	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
152	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
153	C₄⋊C₄×C₅₃	—	2	2	no	—
154	C₄⋊C₄×C₅₃	—	2	2	no	—
155	C₄⋊C₄×C₅₃	—	4	2	no	—
156	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
157	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
158	C₂²⋊C₄×C₅₃	—	2	2	no	—
159	C₂²⋊C₄×C₅₃	—	2	2	no	—
160	C₂²⋊C₄×C₅₃	—	2	2	no	—
161	C₂²⋊C₄×C₅₃	—	2	2	no	—
162	C₂²⋊C₄×C₅₃	—	2	2	no	—
163	C₂²⋊C₄×C₅₃	—	4	2	no	—
164	C₄²×C₅₃	—	2	2	no	—
165	C₄²×C₅₃	—	2	2	no	—
166	C₄²×C₅₃	—	2	2	no	—
167	C₂³×C₅₃	—	2	1	no	—
168	C₂³×C₅₃	—	2	1	no	—
169	C₂³×C₅₃	—	2	2	no	—
170	C₂³×C₅₃	—	4	2	no	—
171	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
172	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
173	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
174	Q₈×C₅₃	—	2	2	no	—
175	Q₈×C₅₃	—	2	2	no	—
176	D₄×C₅₃	—	2	2	no	—
177	D₄×C₅₃	—	2	2	no	—
178	D₄×C₅₃	—	4	4	no	—
179	D₄×C₅₃	—	4	4	no	—
180	D₄×C₅₃	—	4	2	no	—
181	D₄×C₅₃	—	4	2	no	—
182	C₂×C₄×C₅₃	—	2	1	no	—
183	C₂×C₄×C₅₃	—	2	1	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	1	no	—
186	C₂×C₄×C₅₃	—	2	1	no	—
187	C₂×C₄×C₅₃	—	2	1	no	—
188	C₂×C₄×C₅₃	—	2	1	no	—
189	C₂×C₄×C₅₃	—	2	1	no	—
190	C₂×C₄×C₅₃	—	2	1	no	—
191	C₂×C₄×C₅₃	—	2	1	no	—
192	C₂×C₄×C₅₃	—	2	2	no	—
193	C₂×C₄×C₅₃	—	2	2	no	—
194	C₂×C₄×C₅₃	—	2	2	no	—
195	C₂×C₄×C₅₃	—	4	2	no	—
196	C₂×C₄×C₅₃	—	4	2	no	—
197	C₂×C₄×C₅₃	—	4	4	no	—
198	C₂×C₄×C₅₃	—	4	2	no	—
199	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
200	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
201	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
202	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
204	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
205	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
206	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
207	C₈×C₅₃	—	2	1	no	—
208	C₈×C₅₃	—	2	1	no	—
209	C₈×C₅₃	—	2	1	no	—
210	C₈×C₅₃	—	2	1	no	—
211	C₈×C₅₃	—	2	2	no	—
212	C₈×C₅₃	—	2	2	no	—
213	C₈×C₅₃	—	4	2	no	—
214	C₈×C₅₃	—	4	2	no	—
215	#22205	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂²×C₅₃	—	2	2	no	—
217	C₂²×C₅₃	—	2	1	no	—
218	C₂²×C₅₃	—	2	1	no	—
219	C₂²×C₅₃	—	2	2	no	—
220	C₂²×C₅₃	—	2	2	no	—
221	C₂²×C₅₃	—	4	2	no	—
222	C₂²×C₅₃	—	4	2	no	—
223	C₂²×C₅₃	—	4	2	no	—
224	C₂²×C₅₃	—	4	2	no	—
225	C₂²×C₅₃	—	4	2	no	—
226	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
227	C₂²×C₅₃	#538	1	1	yes	characteristic
228	C₂²×C₅₃	#539	1	1	yes	characteristic
229	C₄×C₅₃	—	2	1	no	—
230	C₄×C₅₃	—	2	1	no	—
231	C₄×C₅₃	—	4	2	no	—
232	C₄×C₅₃	—	4	2	no	—
233	C₄×C₅₃	—	4	2	no	—
234	C₄×C₅₃	—	8	2	no	—
235	C₄×C₅₃	#518	1	1	yes	characteristic
236	C₄×C₅₃	#520	1	1	yes	characteristic
237	C₄×C₅₃	#534	1	1	yes	characteristic
238	C₄×C₅₃	#537	1	1	yes	characteristic
239	#3054	C₂×C₅₃	1	1	yes	characteristic
240	#2929	C₂×C₅₃	1	2	yes	complemented
241	#1929	C₂×C₅₃	1	2	yes	complemented
242	#1922	C₂×C₅₃	1	2	yes	complemented
243	#1897	C₂×C₅₃	1	2	yes	complemented
244	#1876	C₂×C₅₃	1	2	yes	complemented
245	#1866	C₂×C₅₃	1	2	yes	complemented
246	#1811	C₂×C₅₃	1	1	yes	characteristic, complemented
247	#1729	C₂×C₅₃	1	1	yes	characteristic, complemented
248	C₂×C₅₃	—	2	1	no	—
249	C₂×C₅₃	—	2	1	no	—
250	C₂×C₅₃	—	4	1	no	—
251	C₂×C₅₃	—	4	1	no	—
252	C₂×C₅₃	—	8	2	no	—
253	C₂×C₅₃	#2325	1	1	yes	characteristic
254	C₂×C₅₃	#2955	1	2	yes	—
255	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
256	C₂×#131	C₂²×C₅₃	1	2	yes	—
257	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
258	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
259	#525	C₂²×C₅₃	1	2	yes	—
260	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
261	#470	C₄×C₅₃	1	1	yes	characteristic, complemented
262	#470	C₄×C₅₃	1	1	yes	characteristic, complemented
263	#469	C₄×C₅₃	1	1	yes	characteristic, complemented
264	#469	C₄×C₅₃	1	1	yes	characteristic, complemented
265	#468	C₄×C₅₃	1	2	yes	complemented
266	#468	C₄×C₅₃	1	2	yes	complemented
267	#435	—	2	2	no	—
268	#435	—	2	2	no	—
269	C₈×D₄	—	2	2	no	—
270	C₈×D₄	—	2	2	no	—
271	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
272	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
273	#427	C₂²×C₅₃	1	1	yes	characteristic
274	#424	C₂²×C₅₃	1	2	yes	—
275	C₂×#104	C₂²×C₅₃	1	2	yes	—
276	C₂×#104	C₂²×C₅₃	1	2	yes	—
277	#416	C₂²×C₅₃	1	2	yes	—
278	#408	C₂²×C₅₃	1	2	yes	—
279	#407	C₂²×C₅₃	1	2	yes	—
280	#405	C₂²×C₅₃	1	2	yes	—
281	#341	C₂²×C₅₃	1	2	yes	—
282	#339	C₂²×C₅₃	1	2	yes	—
283	#339	C₂²×C₅₃	1	2	yes	—
284	#336	C₂²×C₅₃	1	2	yes	—
285	#336	C₂²×C₅₃	1	2	yes	—
286	C₅₃	#22205	1	1	yes	characteristic, direct product factor, Sylow
287	C₂×#41	C₂³×C₅₃	1	2	yes	—
288	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
289	#135	—	4	1	no	—
290	#135	—	4	1	no	—
291	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
292	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
293	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
294	#131	—	4	2	no	—
295	#131	—	4	2	no	—
296	C₂×M₁₆	—	2	2	no	—
297	C₂×M₁₆	—	4	1	no	—
298	C₂×M₁₆	—	4	1	no	—
299	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
300	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
301	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
302	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
303	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
304	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
305	C₂²×C₈	—	2	2	no	—
306	C₂²×C₈	—	4	1	no	—
307	C₂²×C₈	—	4	1	no	—
308	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
309	C₂²×C₈	D₄×C₅₃	1	2	yes	—
310	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
311	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
312	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
313	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
314	#123	—	2	2	no	—
315	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
316	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
317	#121	C₂×C₄×C₅₃	1	2	yes	—
318	#121	C₂×C₄×C₅₃	1	2	yes	—
319	#120	—	2	2	no	—
320	C₄×D₄	—	2	2	no	—
321	C₄×D₄	—	2	2	no	—
322	#117	—	4	1	no	—
323	#117	—	4	1	no	—
324	#117	C₂³×C₅₃	1	2	yes	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₂²⋊C₄	—	2	1	no	—
327	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
328	#108	—	4	1	no	—
329	#108	—	4	1	no	—
330	#107	C₂×C₄×C₅₃	1	2	yes	—
331	#106	C₂×C₄×C₅₃	1	1	yes	characteristic
332	#106	C₂×C₄×C₅₃	1	1	yes	characteristic
333	C₄⋊C₈	—	2	2	no	—
334	C₄⋊C₈	—	2	2	no	—
335	#104	—	4	2	no	—
336	#104	—	4	2	no	—
337	#103	C₂×C₄×C₅₃	1	2	yes	—
338	#103	C₂×C₄×C₅₃	1	2	yes	—
339	#102	C₂×C₄×C₅₃	1	2	yes	—
340	#102	C₂×C₄×C₅₃	1	2	yes	—
341	C₂²⋊C₈	—	2	2	no	—
342	C₂²⋊C₈	—	2	2	no	—
343	C₂²⋊C₈	—	2	2	no	—
344	C₂²⋊C₈	—	2	2	no	—
345	#97	—	2	2	no	—
346	C₄×C₈	—	2	2	no	—
347	C₂⁴	#41×C₅₃	1	1	yes	characteristic
348	#41	—	2	2	no	—
349	#41	—	2	2	no	—
350	#41	—	2	2	no	—
351	#41	—	2	2	no	—
352	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
364	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
365	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
366	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
367	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
368	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
369	Q₁₆	—	4	1	no	—
370	SD₁₆	—	4	2	no	—
371	D₈	—	4	1	no	—
372	M₁₆	—	2	2	no	—
373	M₁₆	—	2	2	no	—
374	M₁₆	—	2	2	no	—
375	M₁₆	—	2	2	no	—
376	M₁₆	—	4	1	no	—
377	M₁₆	—	4	1	no	—
378	M₁₆	—	4	1	no	—
379	M₁₆	—	4	1	no	—
380	M₁₆	—	4	4	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	4	1	no	—
390	C₂×C₈	—	4	2	no	—
391	C₂×C₈	—	4	1	no	—
392	C₂×C₈	—	4	4	no	—
393	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
394	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
395	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
396	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
397	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
398	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	4	2	no	—
402	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
403	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
404	C₂²⋊C₄	—	2	2	no	—
405	C₂²⋊C₄	—	2	2	no	—
406	C₂²⋊C₄	—	2	2	no	—
407	C₂²⋊C₄	—	2	2	no	—
408	C₂²⋊C₄	—	2	2	no	—
409	C₂²⋊C₄	—	4	2	no	—
410	C₄²	—	2	2	no	—
411	C₄²	—	2	2	no	—
412	C₄²	—	2	2	no	—
413	C₂³	—	2	1	no	—
414	C₂³	—	2	1	no	—
415	C₂³	—	2	2	no	—
416	C₂³	—	4	2	no	—
417	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
418	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
419	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
420	Q₈	—	2	2	no	—
421	Q₈	—	2	2	no	—
422	D₄	—	2	2	no	—
423	D₄	—	2	2	no	—
424	D₄	—	4	4	no	—
425	D₄	—	4	4	no	—
426	D₄	—	4	2	no	—
427	D₄	—	4	2	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	4	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
446	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
447	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
448	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
449	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
450	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
451	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
452	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
453	C₈	—	2	1	no	—
454	C₈	—	2	1	no	—
455	C₈	—	2	1	no	—
456	C₈	—	2	1	no	—
457	C₈	—	2	2	no	—
458	C₈	—	2	2	no	—
459	C₈	—	4	2	no	—
460	C₈	—	4	2	no	—
461	C₂²	—	2	2	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	2	2	no	—
464	C₂²	—	2	1	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	4	2	no	—
467	C₂²	—	4	2	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
472	C₂²	C₅₃×#538	1	1	yes	characteristic
473	C₂²	C₅₃×#539	1	1	yes	characteristic
474	C₄	—	2	1	no	—
475	C₄	—	2	1	no	—
476	C₄	—	4	2	no	—
477	C₄	—	4	2	no	—
478	C₄	—	4	2	no	—
479	C₄	—	8	2	no	—
480	C₄	C₅₃×#518	1	1	yes	characteristic
481	C₄	C₅₃×#520	1	1	yes	characteristic
482	C₄	C₅₃×#534	1	1	yes	characteristic
483	C₄	C₅₃×#537	1	1	yes	characteristic
484	C₂	—	2	1	no	—
485	C₂	—	2	1	no	—
486	C₂	—	4	1	no	—
487	C₂	—	4	1	no	—
488	C₂	—	8	2	no	—
489	C₂	C₅₃×#2325	1	1	yes	characteristic
490	C₂	C₅₃×#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#3054	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2929	C₂	semidirect
3. C₅₃×#1929	C₂	semidirect
4. C₅₃×#1922	C₂	semidirect
5. C₅₃×#1897	C₂	semidirect
6. C₅₃×#1876	C₂	semidirect
7. C₅₃×#1866	C₂	semidirect
8. C₅₃×#1811	C₂	semidirect
9. C₅₃×#1729	C₂	semidirect
16. C₅₃×#470	C₄	semidirect
17. C₅₃×#470	C₄	semidirect
18. C₅₃×#469	C₄	semidirect
19. C₅₃×#469	C₄	semidirect
20. C₅₃×#468	C₄	semidirect
21. C₅₃×#468	C₄	semidirect
215. #22205	C₅₃	direct (N × H)
240. #2929	C₂×C₅₃	semidirect
241. #1929	C₂×C₅₃	semidirect
242. #1922	C₂×C₅₃	semidirect
243. #1897	C₂×C₅₃	semidirect
244. #1876	C₂×C₅₃	semidirect
245. #1866	C₂×C₅₃	semidirect
246. #1811	C₂×C₅₃	semidirect
247. #1729	C₂×C₅₃	semidirect
261. #470	C₄×C₅₃	semidirect
262. #470	C₄×C₅₃	semidirect
263. #469	C₄×C₅₃	semidirect
264. #469	C₄×C₅₃	semidirect
265. #468	C₄×C₅₃	semidirect
266. #468	C₄×C₅₃	semidirect
286. C₅₃	#22205	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×#3054	2
C₅₃×#2929	2
C₅₃×#1929	2
C₅₃×#1922	2
C₅₃×#1897	2
C₅₃×#1876	2
C₅₃×#1866	2
C₅₃×#1811	2
C₅₃×#1729	2
#22205	53

Subgroups — C53×#22205 #27229717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22205 #27229717