Subgroups — C₅₃×#22206 #27229718

Order 13568.

774 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3050	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2930	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×#1810	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×#467	C₄	1	1	yes	characteristic, complemented
17	C₅₃×#467	C₄	1	1	yes	characteristic, complemented
18	C₅₃×#466	C₄	1	1	yes	characteristic, complemented
19	C₅₃×#466	C₄	1	1	yes	characteristic, complemented
20	C₅₃×#465	C₄	1	2	yes	complemented
21	C₅₃×#465	C₄	1	2	yes	complemented
22	C₅₃×#436	—	2	2	no	—
23	C₅₃×#436	—	2	2	no	—
24	C₅₃×#435	—	2	2	no	—
25	C₅₃×#435	—	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	D₄	1	1	yes	characteristic
54	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
55	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
57	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
58	C₂²×C₈×C₅₃	—	2	2	no	—
59	C₂²×C₈×C₅₃	—	4	1	no	—
60	C₂²×C₈×C₅₃	—	4	1	no	—
61	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₃	C₂×C₄	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	1	yes	characteristic
85	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
86	#106×C₅₃	C₂×C₄	1	2	yes	—
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	2	no	—
132	M₁₆×C₅₃	—	4	1	no	—
133	M₁₆×C₅₃	—	4	4	no	—
134	C₂×C₈×C₅₃	—	2	1	no	—
135	C₂×C₈×C₅₃	—	2	1	no	—
136	C₂×C₈×C₅₃	—	2	1	no	—
137	C₂×C₈×C₅₃	—	2	1	no	—
138	C₂×C₈×C₅₃	—	2	2	no	—
139	C₂×C₈×C₅₃	—	2	2	no	—
140	C₂×C₈×C₅₃	—	2	2	no	—
141	C₂×C₈×C₅₃	—	2	2	no	—
142	C₂×C₈×C₅₃	—	4	1	no	—
143	C₂×C₈×C₅₃	—	4	1	no	—
144	C₂×C₈×C₅₃	—	4	1	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	1	no	—
171	C₂³×C₅₃	—	4	1	no	—
172	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
173	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
174	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
175	Q₈×C₅₃	—	2	2	no	—
176	Q₈×C₅₃	—	2	2	no	—
177	D₄×C₅₃	—	2	2	no	—
178	D₄×C₅₃	—	2	2	no	—
179	D₄×C₅₃	—	4	2	no	—
180	D₄×C₅₃	—	4	2	no	—
181	D₄×C₅₃	—	8	2	no	—
182	D₄×C₅₃	—	8	2	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	1	no	—
193	C₂×C₄×C₅₃	—	2	2	no	—
194	C₂×C₄×C₅₃	—	2	2	no	—
195	C₂×C₄×C₅₃	—	2	2	no	—
196	C₂×C₄×C₅₃	—	4	2	no	—
197	C₂×C₄×C₅₃	—	4	2	no	—
198	C₂×C₄×C₅₃	—	4	2	no	—
199	C₂×C₄×C₅₃	—	4	2	no	—
200	C₂×C₄×C₅₃	—	4	2	no	—
201	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
202	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
203	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
204	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
206	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
207	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
208	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
209	C₈×C₅₃	—	2	1	no	—
210	C₈×C₅₃	—	2	1	no	—
211	C₈×C₅₃	—	2	1	no	—
212	C₈×C₅₃	—	2	1	no	—
213	C₈×C₅₃	—	2	2	no	—
214	C₈×C₅₃	—	2	2	no	—
215	C₈×C₅₃	—	4	2	no	—
216	C₈×C₅₃	—	4	2	no	—
217	#22206	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂²×C₅₃	—	2	1	no	—
219	C₂²×C₅₃	—	2	1	no	—
220	C₂²×C₅₃	—	2	1	no	—
221	C₂²×C₅₃	—	2	1	no	—
222	C₂²×C₅₃	—	2	1	no	—
223	C₂²×C₅₃	—	2	1	no	—
224	C₂²×C₅₃	—	2	1	no	—
225	C₂²×C₅₃	—	2	1	no	—
226	C₂²×C₅₃	—	4	2	no	—
227	C₂²×C₅₃	—	4	2	no	—
228	C₂²×C₅₃	—	4	2	no	—
229	C₂²×C₅₃	—	4	2	no	—
230	C₂²×C₅₃	—	4	2	no	—
231	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
232	C₂²×C₅₃	#538	1	1	yes	characteristic
233	C₂²×C₅₃	#539	1	1	yes	characteristic
234	C₄×C₅₃	—	2	1	no	—
235	C₄×C₅₃	—	2	1	no	—
236	C₄×C₅₃	—	4	2	no	—
237	C₄×C₅₃	—	4	2	no	—
238	C₄×C₅₃	—	4	2	no	—
239	C₄×C₅₃	—	8	2	no	—
240	C₄×C₅₃	#518	1	1	yes	characteristic
241	C₄×C₅₃	#520	1	1	yes	characteristic
242	C₄×C₅₃	#534	1	1	yes	characteristic
243	C₄×C₅₃	#537	1	1	yes	characteristic
244	#3050	C₂×C₅₃	1	1	yes	characteristic
245	#2930	C₂×C₅₃	1	2	yes	complemented
246	#1929	C₂×C₅₃	1	2	yes	complemented
247	#1922	C₂×C₅₃	1	2	yes	complemented
248	#1897	C₂×C₅₃	1	2	yes	complemented
249	#1876	C₂×C₅₃	1	2	yes	complemented
250	#1866	C₂×C₅₃	1	2	yes	complemented
251	#1810	C₂×C₅₃	1	1	yes	characteristic, complemented
252	#1729	C₂×C₅₃	1	1	yes	characteristic, complemented
253	C₂×C₅₃	—	2	1	no	—
254	C₂×C₅₃	—	2	1	no	—
255	C₂×C₅₃	—	4	1	no	—
256	C₂×C₅₃	—	4	1	no	—
257	C₂×C₅₃	—	8	2	no	—
258	C₂×C₅₃	#2325	1	1	yes	characteristic
259	C₂×C₅₃	#2957	1	1	yes	characteristic
260	C₂×C₅₃	#2957	1	1	yes	characteristic
261	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
262	C₂×#131	C₂²×C₅₃	1	2	yes	—
263	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
264	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
265	#525	C₂²×C₅₃	1	2	yes	—
266	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
267	#467	C₄×C₅₃	1	1	yes	characteristic, complemented
268	#467	C₄×C₅₃	1	1	yes	characteristic, complemented
269	#466	C₄×C₅₃	1	1	yes	characteristic, complemented
270	#466	C₄×C₅₃	1	1	yes	characteristic, complemented
271	#465	C₄×C₅₃	1	2	yes	complemented
272	#465	C₄×C₅₃	1	2	yes	complemented
273	#436	—	2	2	no	—
274	#436	—	2	2	no	—
275	#435	—	2	2	no	—
276	#435	—	2	2	no	—
277	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
278	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
279	#427	C₂²×C₅₃	1	1	yes	characteristic
280	#424	C₂²×C₅₃	1	2	yes	—
281	C₂×#104	C₂²×C₅₃	1	2	yes	—
282	C₂×#104	C₂²×C₅₃	1	2	yes	—
283	#416	C₂²×C₅₃	1	2	yes	—
284	#408	C₂²×C₅₃	1	2	yes	—
285	#407	C₂²×C₅₃	1	2	yes	—
286	#405	C₂²×C₅₃	1	2	yes	—
287	#341	C₂²×C₅₃	1	2	yes	—
288	#339	C₂²×C₅₃	1	2	yes	—
289	#339	C₂²×C₅₃	1	2	yes	—
290	#336	C₂²×C₅₃	1	2	yes	—
291	#336	C₂²×C₅₃	1	2	yes	—
292	C₅₃	#22206	1	1	yes	characteristic, direct product factor, Sylow
293	C₂×#41	C₂³×C₅₃	1	2	yes	—
294	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
295	#135	—	4	1	no	—
296	#135	—	4	1	no	—
297	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
298	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
299	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
300	#131	—	4	2	no	—
301	#131	—	4	2	no	—
302	C₂×M₁₆	—	2	2	no	—
303	C₂×M₁₆	—	4	1	no	—
304	C₂×M₁₆	—	4	1	no	—
305	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
306	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
307	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
308	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
309	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
310	C₂²×C₈	—	2	2	no	—
311	C₂²×C₈	—	4	1	no	—
312	C₂²×C₈	—	4	1	no	—
313	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
314	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
315	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
316	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
317	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
318	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
319	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
320	#123	—	2	2	no	—
321	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
322	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
323	#121	C₂×C₄×C₅₃	1	2	yes	—
324	#121	C₂×C₄×C₅₃	1	2	yes	—
325	#120	—	2	2	no	—
326	C₄×D₄	—	2	2	no	—
327	C₄×D₄	—	2	2	no	—
328	#117	—	4	1	no	—
329	#117	—	4	1	no	—
330	#117	C₂³×C₅₃	1	2	yes	—
331	C₂×C₄⋊C₄	—	2	1	no	—
332	C₂×C₂²⋊C₄	—	2	1	no	—
333	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
334	#108	—	4	1	no	—
335	#108	—	4	1	no	—
336	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
337	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
338	#106	C₂×C₄×C₅₃	1	2	yes	—
339	C₄⋊C₈	—	2	2	no	—
340	C₄⋊C₈	—	2	2	no	—
341	#104	—	4	2	no	—
342	#104	—	4	2	no	—
343	#103	C₂×C₄×C₅₃	1	2	yes	—
344	#103	C₂×C₄×C₅₃	1	2	yes	—
345	#102	C₂×C₄×C₅₃	1	2	yes	—
346	#102	C₂×C₄×C₅₃	1	2	yes	—
347	C₂²⋊C₈	—	2	2	no	—
348	C₂²⋊C₈	—	2	2	no	—
349	C₂²⋊C₈	—	2	2	no	—
350	C₂²⋊C₈	—	2	2	no	—
351	#97	—	2	2	no	—
352	C₄×C₈	—	2	2	no	—
353	C₂⁴	#41×C₅₃	1	1	yes	characteristic
354	#41	—	2	2	no	—
355	#41	—	2	2	no	—
356	#41	—	2	2	no	—
357	#41	—	2	2	no	—
358	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
370	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
373	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
374	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
375	Q₁₆	—	4	1	no	—
376	SD₁₆	—	4	2	no	—
377	D₈	—	4	1	no	—
378	M₁₆	—	2	2	no	—
379	M₁₆	—	2	2	no	—
380	M₁₆	—	2	2	no	—
381	M₁₆	—	2	2	no	—
382	M₁₆	—	4	1	no	—
383	M₁₆	—	4	2	no	—
384	M₁₆	—	4	1	no	—
385	M₁₆	—	4	4	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	4	1	no	—
395	C₂×C₈	—	4	1	no	—
396	C₂×C₈	—	4	1	no	—
397	C₂×C₈	—	4	1	no	—
398	C₂×C₈	—	4	4	no	—
399	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
400	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
401	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
402	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
403	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
404	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	4	2	no	—
408	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
409	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
410	C₂²⋊C₄	—	2	2	no	—
411	C₂²⋊C₄	—	2	2	no	—
412	C₂²⋊C₄	—	2	2	no	—
413	C₂²⋊C₄	—	2	2	no	—
414	C₂²⋊C₄	—	2	2	no	—
415	C₂²⋊C₄	—	4	2	no	—
416	C₄²	—	2	2	no	—
417	C₄²	—	2	2	no	—
418	C₄²	—	2	2	no	—
419	C₂³	—	2	1	no	—
420	C₂³	—	2	1	no	—
421	C₂³	—	2	2	no	—
422	C₂³	—	4	1	no	—
423	C₂³	—	4	1	no	—
424	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
425	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
426	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
427	Q₈	—	2	2	no	—
428	Q₈	—	2	2	no	—
429	D₄	—	2	2	no	—
430	D₄	—	2	2	no	—
431	D₄	—	4	2	no	—
432	D₄	—	4	2	no	—
433	D₄	—	8	2	no	—
434	D₄	—	8	2	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
454	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
455	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
456	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
457	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
458	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
459	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
460	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
461	C₈	—	2	1	no	—
462	C₈	—	2	1	no	—
463	C₈	—	2	1	no	—
464	C₈	—	2	1	no	—
465	C₈	—	2	2	no	—
466	C₈	—	2	2	no	—
467	C₈	—	4	2	no	—
468	C₈	—	4	2	no	—
469	C₂²	—	2	1	no	—
470	C₂²	—	2	1	no	—
471	C₂²	—	2	1	no	—
472	C₂²	—	2	1	no	—
473	C₂²	—	2	1	no	—
474	C₂²	—	2	1	no	—
475	C₂²	—	2	1	no	—
476	C₂²	—	2	1	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	2	no	—
480	C₂²	—	4	2	no	—
481	C₂²	—	4	2	no	—
482	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
483	C₂²	C₅₃×#538	1	1	yes	characteristic
484	C₂²	C₅₃×#539	1	1	yes	characteristic
485	C₄	—	2	1	no	—
486	C₄	—	2	1	no	—
487	C₄	—	4	2	no	—
488	C₄	—	4	2	no	—
489	C₄	—	4	2	no	—
490	C₄	—	8	2	no	—
491	C₄	C₅₃×#518	1	1	yes	characteristic
492	C₄	C₅₃×#520	1	1	yes	characteristic
493	C₄	C₅₃×#534	1	1	yes	characteristic
494	C₄	C₅₃×#537	1	1	yes	characteristic
495	C₂	—	2	1	no	—
496	C₂	—	2	1	no	—
497	C₂	—	4	1	no	—
498	C₂	—	4	1	no	—
499	C₂	—	8	2	no	—
500	C₂	C₅₃×#2325	1	1	yes	characteristic
501	C₂	C₅₃×#2957	1	1	yes	characteristic
502	C₂	C₅₃×#2957	1	1	yes	characteristic

	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×#3050	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×M₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2930	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×#1810	C₂	semidirect
9. C₅₃×#1729	C₂	semidirect
16. C₅₃×#467	C₄	semidirect
17. C₅₃×#467	C₄	semidirect
18. C₅₃×#466	C₄	semidirect
19. C₅₃×#466	C₄	semidirect
20. C₅₃×#465	C₄	semidirect
21. C₅₃×#465	C₄	semidirect
217. #22206	C₅₃	direct (N × H)
245. #2930	C₂×C₅₃	semidirect
246. #1929	C₂×C₅₃	semidirect
247. #1922	C₂×C₅₃	semidirect
248. #1897	C₂×C₅₃	semidirect
249. #1876	C₂×C₅₃	semidirect
250. #1866	C₂×C₅₃	semidirect
251. #1810	C₂×C₅₃	semidirect
252. #1729	C₂×C₅₃	semidirect
267. #467	C₄×C₅₃	semidirect
268. #467	C₄×C₅₃	semidirect
269. #466	C₄×C₅₃	semidirect
270. #466	C₄×C₅₃	semidirect
271. #465	C₄×C₅₃	semidirect
272. #465	C₄×C₅₃	semidirect
292. C₅₃	#22206	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×#3050	2
C₅₃×#2930	2
C₅₃×#1929	2
C₅₃×#1922	2
C₅₃×#1897	2
C₅₃×#1876	2
C₅₃×#1866	2
C₅₃×#1810	2
C₅₃×#1729	2
#22206	53

Subgroups — C53×#22206 #27229718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22206 #27229718