Subgroups — C₅₃×#21783 #27229295

Order 13568.

1230 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×#3499	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2889	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2884	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1912	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1909	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1886	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1884	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1827	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1796	C₂	1	2	yes	maximal, complemented
10	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#568	C₂²	1	2	yes	—
13	D₄×Q₈×C₅₃	—	2	2	no	—
14	C₅₃×#547	—	2	2	no	—
15	C₅₃×#547	—	2	4	no	—
16	C₅₃×#536	C₄	1	2	yes	complemented
17	C₅₃×#536	C₄	1	2	yes	complemented
18	C₅₃×#536	C₂²	1	1	yes	characteristic
19	C₅₃×#536	C₂²	1	1	yes	characteristic
20	C₅₃×#532	C₄	1	2	yes	complemented
21	C₅₃×#532	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Q₈×C₅₃	C₄	1	2	yes	complemented
23	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#519	C₂²	1	2	yes	—
25	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#421	C₂²	1	4	yes	—
27	C₅₃×#419	C₂²	1	4	yes	—
28	C₅₃×#413	—	2	2	no	—
29	C₅₃×#413	C₂²	1	2	yes	—
30	C₂×#101×C₅₃	—	2	2	no	—
31	C₂×#101×C₅₃	C₂²	1	2	yes	—
32	C₅₃×#410	—	2	4	no	—
33	C₅₃×#410	C₂²	1	2	yes	—
34	C₅₃×#410	C₂²	1	2	yes	—
35	C₅₃×#339	C₂²	1	4	yes	—
36	C₅₃×H(3,ℤ/4)	C₂²	1	4	yes	—
37	2₋¹⁺⁴×C₅₃	—	4	4	no	—
38	2₋¹⁺⁴×C₅₃	—	4	4	no	—
39	C₂×#41×C₅₃	—	2	1	no	—
40	C₂×#41×C₅₃	—	2	2	no	—
41	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×#41×C₅₃	D₄	1	2	yes	—
44	C₂×#41×C₅₃	D₄	1	4	yes	—
45	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×Q₈×C₅₃	—	2	1	no	—
49	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
50	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₂²×Q₈×C₅₃	D₄	1	2	yes	—
52	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
54	#131×C₅₃	—	4	4	no	—
55	C₂×M₁₆×C₅₃	—	2	4	no	—
56	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
57	C₄⋊Q₈×C₅₃	—	2	4	no	—
58	C₄⋊Q₈×C₅₃	—	2	2	no	—
59	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
62	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
63	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
64	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
65	#124×C₅₃	C₂×C₄	1	2	yes	—
66	#124×C₅₃	C₂×C₄	1	2	yes	—
67	#124×C₅₃	C₂×C₄	1	1	yes	characteristic
68	#124×C₅₃	C₂×C₄	1	1	yes	characteristic
69	#123×C₅₃	—	2	4	no	—
70	#123×C₅₃	—	2	2	no	—
71	#123×C₅₃	—	2	4	no	—
72	#123×C₅₃	—	2	2	no	—
73	C₂²⋊Q₈×C₅₃	—	2	4	no	—
74	C₂²⋊Q₈×C₅₃	—	2	2	no	—
75	C₂²⋊Q₈×C₅₃	—	2	4	no	—
76	C₂²⋊Q₈×C₅₃	—	2	2	no	—
77	#121×C₅₃	—	2	4	no	—
78	#121×C₅₃	—	2	2	no	—
79	C₄×Q₈×C₅₃	—	2	1	no	—
80	C₄×Q₈×C₅₃	—	4	4	no	—
81	C₄×D₄×C₅₃	—	2	1	no	—
82	C₄×D₄×C₅₃	—	2	2	no	—
83	C₄×D₄×C₅₃	—	2	4	no	—
84	C₄×D₄×C₅₃	—	2	2	no	—
85	#117×C₅₃	—	2	4	no	—
86	#117×C₅₃	C₂×C₄	1	2	yes	—
87	#117×C₅₃	C₂³	1	4	yes	—
88	#117×C₅₃	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
90	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
91	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
92	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
93	#104×C₅₃	—	4	4	no	—
94	#103×C₅₃	—	4	4	no	—
95	#101×C₅₃	—	2	2	no	—
96	#101×C₅₃	—	2	4	no	—
97	#100×C₅₃	—	2	2	no	—
98	#99×C₅₃	—	2	4	no	—
99	#99×C₅₃	—	2	4	no	—
100	#41×C₅₃	—	2	4	no	—
101	#41×C₅₃	—	2	4	no	—
102	#41×C₅₃	—	2	4	no	—
103	#41×C₅₃	—	4	8	no	—
104	#41×C₅₃	—	4	4	no	—
105	#41×C₅₃	—	4	4	no	—
106	#41×C₅₃	—	4	4	no	—
107	#41×C₅₃	—	4	8	no	—
108	#41×C₅₃	—	4	4	no	—
109	C₂×Q₈×C₅₃	—	2	2	no	—
110	C₂×Q₈×C₅₃	—	2	1	no	—
111	C₂×Q₈×C₅₃	—	2	4	no	—
112	C₂×Q₈×C₅₃	—	2	4	no	—
113	C₂×Q₈×C₅₃	—	4	4	no	—
114	C₂×Q₈×C₅₃	—	4	4	no	—
115	C₂×Q₈×C₅₃	—	4	4	no	—
116	C₂×Q₈×C₅₃	—	4	4	no	—
117	C₂×Q₈×C₅₃	C₂²⋊C₄	1	4	yes	—
118	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
120	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
122	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
124	C₂×D₄×C₅₃	—	2	2	no	—
125	C₂×D₄×C₅₃	—	2	2	no	—
126	C₂×D₄×C₅₃	—	2	1	no	—
127	C₂×D₄×C₅₃	C₂²⋊C₄	1	4	yes	—
128	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
131	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
132	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₅₃	—	2	2	no	—
134	C₂²×C₄×C₅₃	—	2	2	no	—
135	C₂²×C₄×C₅₃	—	2	1	no	—
136	C₂²×C₄×C₅₃	—	2	1	no	—
137	C₂²×C₄×C₅₃	—	2	2	no	—
138	C₂²×C₄×C₅₃	—	2	1	no	—
139	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
140	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
141	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
142	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
145	M₁₆×C₅₃	—	2	4	no	—
146	M₁₆×C₅₃	—	2	4	no	—
147	M₁₆×C₅₃	—	4	4	no	—
148	C₂×C₈×C₅₃	—	2	4	no	—
149	C₂×C₈×C₅₃	—	4	4	no	—
150	C₄⋊C₄×C₅₃	—	2	4	no	—
151	C₄⋊C₄×C₅₃	—	2	2	no	—
152	C₄⋊C₄×C₅₃	—	2	4	no	—
153	C₄⋊C₄×C₅₃	—	2	4	no	—
154	C₄⋊C₄×C₅₃	—	2	4	no	—
155	C₄⋊C₄×C₅₃	—	2	2	no	—
156	C₄⋊C₄×C₅₃	—	2	1	no	—
157	C₄⋊C₄×C₅₃	—	2	4	no	—
158	C₄⋊C₄×C₅₃	—	4	4	no	—
159	C₂²⋊C₄×C₅₃	—	2	2	no	—
160	C₂²⋊C₄×C₅₃	—	2	4	no	—
161	C₂²⋊C₄×C₅₃	—	2	2	no	—
162	C₂²⋊C₄×C₅₃	—	2	1	no	—
163	C₂²⋊C₄×C₅₃	—	2	4	no	—
164	C₂²⋊C₄×C₅₃	—	2	2	no	—
165	C₂²⋊C₄×C₅₃	—	2	4	no	—
166	C₄²×C₅₃	—	2	2	no	—
167	C₄²×C₅₃	—	2	1	no	—
168	C₄²×C₅₃	—	2	4	no	—
169	C₄²×C₅₃	—	4	4	no	—
170	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
171	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
172	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
173	C₂³×C₅₃	—	2	1	no	—
174	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
175	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
176	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
177	Q₈×C₅₃	—	2	4	no	—
178	Q₈×C₅₃	—	2	2	no	—
179	Q₈×C₅₃	—	2	4	no	—
180	Q₈×C₅₃	—	2	2	no	—
181	Q₈×C₅₃	—	2	4	no	—
182	Q₈×C₅₃	—	2	2	no	—
183	Q₈×C₅₃	—	4	4	no	—
184	Q₈×C₅₃	—	4	4	no	—
185	Q₈×C₅₃	—	4	2	no	—
186	D₄×C₅₃	—	2	4	no	—
187	D₄×C₅₃	—	2	4	no	—
188	D₄×C₅₃	—	2	2	no	—
189	D₄×C₅₃	—	2	2	no	—
190	D₄×C₅₃	—	2	2	no	—
191	D₄×C₅₃	—	4	4	no	—
192	D₄×C₅₃	—	4	4	no	—
193	D₄×C₅₃	—	4	4	no	—
194	D₄×C₅₃	—	4	2	no	—
195	C₂×C₄×C₅₃	—	2	1	no	—
196	C₂×C₄×C₅₃	—	2	4	no	—
197	C₂×C₄×C₅₃	—	2	4	no	—
198	C₂×C₄×C₅₃	—	2	4	no	—
199	C₂×C₄×C₅₃	—	2	1	no	—
200	C₂×C₄×C₅₃	—	2	2	no	—
201	C₂×C₄×C₅₃	—	2	2	no	—
202	C₂×C₄×C₅₃	—	2	1	no	—
203	C₂×C₄×C₅₃	—	4	4	no	—
204	C₂×C₄×C₅₃	—	4	4	no	—
205	C₂×C₄×C₅₃	—	4	2	no	—
206	C₂×C₄×C₅₃	—	4	4	no	—
207	C₂×C₄×C₅₃	—	4	4	no	—
208	C₂×C₄×C₅₃	—	4	4	no	—
209	C₂×C₄×C₅₃	—	4	2	no	—
210	C₂×C₄×C₅₃	—	4	2	no	—
211	C₂×C₄×C₅₃	—	4	4	no	—
212	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
213	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
214	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
215	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
216	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
218	C₈×C₅₃	—	4	4	no	—
219	#21783	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂²×C₅₃	—	2	2	no	—
221	C₂²×C₅₃	—	2	2	no	—
222	C₂²×C₅₃	—	4	1	no	—
223	C₂²×C₅₃	—	4	2	no	—
224	C₂²×C₅₃	—	4	2	no	—
225	C₂²×C₅₃	—	4	2	no	—
226	C₂²×C₅₃	—	4	1	no	—
227	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
228	C₂²×C₅₃	#534	1	2	yes	—
229	C₄×C₅₃	—	2	2	no	—
230	C₄×C₅₃	—	2	2	no	—
231	C₄×C₅₃	—	2	2	no	—
232	C₄×C₅₃	—	2	4	no	—
233	C₄×C₅₃	—	4	2	no	—
234	C₄×C₅₃	—	4	1	no	—
235	C₄×C₅₃	—	4	2	no	—
236	C₄×C₅₃	—	4	1	no	—
237	C₄×C₅₃	—	8	4	no	—
238	C₄×C₅₃	#518	1	2	yes	—
239	C₄×C₅₃	#534	1	2	yes	—
240	#3499	C₂×C₅₃	1	1	yes	characteristic
241	#2889	C₂×C₅₃	1	1	yes	characteristic, complemented
242	#2884	C₂×C₅₃	1	1	yes	characteristic, complemented
243	#1912	C₂×C₅₃	1	2	yes	complemented
244	#1909	C₂×C₅₃	1	2	yes	complemented
245	#1886	C₂×C₅₃	1	2	yes	complemented
246	#1884	C₂×C₅₃	1	2	yes	complemented
247	#1827	C₂×C₅₃	1	2	yes	complemented
248	#1796	C₂×C₅₃	1	2	yes	complemented
249	C₂×C₅₃	—	2	1	no	—
250	C₂×C₅₃	—	2	2	no	—
251	C₂×C₅₃	—	4	2	no	—
252	C₂×C₅₃	—	4	2	no	—
253	C₂×C₅₃	—	8	1	no	—
254	C₂×C₅₃	#2308	1	1	yes	characteristic
255	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	1	yes	characteristic
256	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	1	yes	characteristic
257	#568	C₂²×C₅₃	1	2	yes	—
258	D₄×Q₈	—	2	2	no	—
259	#547	—	2	4	no	—
260	#547	—	2	2	no	—
261	#536	C₄×C₅₃	1	2	yes	complemented
262	#536	C₄×C₅₃	1	2	yes	complemented
263	#536	C₂²×C₅₃	1	1	yes	characteristic
264	#536	C₂²×C₅₃	1	1	yes	characteristic
265	#532	C₄×C₅₃	1	2	yes	complemented
266	#532	C₂²×C₅₃	1	1	yes	characteristic
267	C₂×C₄⋊Q₈	C₄×C₅₃	1	2	yes	complemented
268	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
269	#519	C₂²×C₅₃	1	2	yes	—
270	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
271	#421	C₂²×C₅₃	1	4	yes	—
272	#419	C₂²×C₅₃	1	4	yes	—
273	#413	—	2	2	no	—
274	#413	C₂²×C₅₃	1	2	yes	—
275	C₂×#101	—	2	2	no	—
276	C₂×#101	C₂²×C₅₃	1	2	yes	—
277	#410	—	2	4	no	—
278	#410	C₂²×C₅₃	1	2	yes	—
279	#410	C₂²×C₅₃	1	2	yes	—
280	#339	C₂²×C₅₃	1	4	yes	—
281	H(3,ℤ/4)	C₂²×C₅₃	1	4	yes	—
282	C₅₃	#21783	1	1	yes	characteristic, direct product factor, Sylow
283	2₋¹⁺⁴	—	4	4	no	—
284	2₋¹⁺⁴	—	4	4	no	—
285	C₂×#41	—	2	1	no	—
286	C₂×#41	—	2	2	no	—
287	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
288	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
289	C₂×#41	D₄×C₅₃	1	4	yes	—
290	C₂×#41	D₄×C₅₃	1	2	yes	—
291	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
292	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
293	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
294	C₂²×Q₈	—	2	1	no	—
295	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
296	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
297	C₂²×Q₈	D₄×C₅₃	1	2	yes	—
298	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
299	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
300	#131	—	4	4	no	—
301	C₂×M₁₆	—	2	4	no	—
302	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
303	C₄⋊Q₈	—	2	2	no	—
304	C₄⋊Q₈	—	2	4	no	—
305	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
306	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
307	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
308	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
309	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
310	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
311	#124	C₂×C₄×C₅₃	1	2	yes	—
312	#124	C₂×C₄×C₅₃	1	2	yes	—
313	#124	C₂×C₄×C₅₃	1	1	yes	characteristic
314	#124	C₂×C₄×C₅₃	1	1	yes	characteristic
315	#123	—	2	2	no	—
316	#123	—	2	4	no	—
317	#123	—	2	2	no	—
318	#123	—	2	4	no	—
319	C₂²⋊Q₈	—	2	2	no	—
320	C₂²⋊Q₈	—	2	4	no	—
321	C₂²⋊Q₈	—	2	2	no	—
322	C₂²⋊Q₈	—	2	4	no	—
323	#121	—	2	2	no	—
324	#121	—	2	4	no	—
325	C₄×Q₈	—	2	1	no	—
326	C₄×Q₈	—	4	4	no	—
327	C₄×D₄	—	2	2	no	—
328	C₄×D₄	—	2	1	no	—
329	C₄×D₄	—	2	2	no	—
330	C₄×D₄	—	2	4	no	—
331	#117	—	2	4	no	—
332	#117	C₂×C₄×C₅₃	1	2	yes	—
333	#117	C₂³×C₅₃	1	4	yes	—
334	#117	C₂³×C₅₃	1	1	yes	characteristic
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	4	no	—
337	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
338	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
339	#104	—	4	4	no	—
340	#103	—	4	4	no	—
341	#101	—	2	2	no	—
342	#101	—	2	4	no	—
343	#100	—	2	2	no	—
344	#99	—	2	4	no	—
345	#99	—	2	4	no	—
346	#41	—	2	4	no	—
347	#41	—	2	4	no	—
348	#41	—	2	4	no	—
349	#41	—	4	8	no	—
350	#41	—	4	4	no	—
351	#41	—	4	4	no	—
352	#41	—	4	4	no	—
353	#41	—	4	8	no	—
354	#41	—	4	4	no	—
355	C₂×Q₈	—	2	2	no	—
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	4	no	—
358	C₂×Q₈	—	2	4	no	—
359	C₂×Q₈	—	4	4	no	—
360	C₂×Q₈	—	4	4	no	—
361	C₂×Q₈	—	4	4	no	—
362	C₂×Q₈	—	4	4	no	—
363	C₂×Q₈	C₂²⋊C₄×C₅₃	1	4	yes	—
364	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
365	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
366	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
367	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
368	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
369	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	C₂²⋊C₄×C₅₃	1	4	yes	—
374	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
375	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
376	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
377	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
378	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
386	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
387	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
388	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
390	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
391	M₁₆	—	2	4	no	—
392	M₁₆	—	2	4	no	—
393	M₁₆	—	4	4	no	—
394	C₂×C₈	—	2	4	no	—
395	C₂×C₈	—	4	4	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	4	no	—
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	4	4	no	—
405	C₂²⋊C₄	—	2	2	no	—
406	C₂²⋊C₄	—	2	2	no	—
407	C₂²⋊C₄	—	2	4	no	—
408	C₂²⋊C₄	—	2	1	no	—
409	C₂²⋊C₄	—	2	2	no	—
410	C₂²⋊C₄	—	2	4	no	—
411	C₂²⋊C₄	—	2	4	no	—
412	C₄²	—	2	2	no	—
413	C₄²	—	2	1	no	—
414	C₄²	—	2	4	no	—
415	C₄²	—	4	4	no	—
416	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
417	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
418	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
419	C₂³	—	2	1	no	—
420	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
421	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
422	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
423	Q₈	—	2	4	no	—
424	Q₈	—	2	2	no	—
425	Q₈	—	2	4	no	—
426	Q₈	—	2	2	no	—
427	Q₈	—	2	4	no	—
428	Q₈	—	2	2	no	—
429	Q₈	—	4	4	no	—
430	Q₈	—	4	4	no	—
431	Q₈	—	4	2	no	—
432	D₄	—	2	4	no	—
433	D₄	—	2	4	no	—
434	D₄	—	2	2	no	—
435	D₄	—	2	2	no	—
436	D₄	—	2	2	no	—
437	D₄	—	4	4	no	—
438	D₄	—	4	4	no	—
439	D₄	—	4	4	no	—
440	D₄	—	4	2	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	4	4	no	—
450	C₂×C₄	—	4	4	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	4	no	—
453	C₂×C₄	—	4	4	no	—
454	C₂×C₄	—	4	4	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	4	no	—
458	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
459	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
460	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
461	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
462	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
463	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
464	C₈	—	4	4	no	—
465	C₂²	—	2	2	no	—
466	C₂²	—	2	2	no	—
467	C₂²	—	4	1	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	1	no	—
472	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
473	C₂²	C₅₃×#534	1	2	yes	—
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	2	2	no	—
477	C₄	—	2	4	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	1	no	—
480	C₄	—	4	2	no	—
481	C₄	—	4	1	no	—
482	C₄	—	8	4	no	—
483	C₄	C₅₃×#518	1	2	yes	—
484	C₄	C₅₃×#534	1	2	yes	—
485	C₂	—	2	1	no	—
486	C₂	—	2	2	no	—
487	C₂	—	4	2	no	—
488	C₂	—	4	2	no	—
489	C₂	—	8	1	no	—
490	C₂	C₅₃×#2308	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#2308
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₅₃×#3499	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2889	C₂	semidirect
3. C₅₃×#2884	C₂	semidirect
4. C₅₃×#1912	C₂	semidirect
5. C₅₃×#1909	C₂	semidirect
6. C₅₃×#1886	C₂	semidirect
7. C₅₃×#1884	C₂	semidirect
8. C₅₃×#1827	C₂	semidirect
9. C₅₃×#1796	C₂	semidirect
16. C₅₃×#536	C₄	semidirect
17. C₅₃×#536	C₄	semidirect
20. C₅₃×#532	C₄	semidirect
22. C₂×C₄⋊Q₈×C₅₃	C₄	semidirect
219. #21783	C₅₃	direct (N × H)
241. #2889	C₂×C₅₃	semidirect
242. #2884	C₂×C₅₃	semidirect
243. #1912	C₂×C₅₃	semidirect
244. #1909	C₂×C₅₃	semidirect
245. #1886	C₂×C₅₃	semidirect
246. #1884	C₂×C₅₃	semidirect
247. #1827	C₂×C₅₃	semidirect
248. #1796	C₂×C₅₃	semidirect
261. #536	C₄×C₅₃	semidirect
262. #536	C₄×C₅₃	semidirect
265. #532	C₄×C₅₃	semidirect
267. C₂×C₄⋊Q₈	C₄×C₅₃	semidirect
282. C₅₃	#21783	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×#3499	2
C₅₃×#2889	2
C₅₃×#2884	2
C₅₃×#1912	2
C₅₃×#1909	2
C₅₃×#1886	2
C₅₃×#1884	2
C₅₃×#1827	2
C₅₃×#1796	2
#21783	53

Subgroups — C53×#21783 #27229295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21783 #27229295