Subgroups — C₅₃×#23030 #27230542

Order 13568.

926 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#467	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#441	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2362	C₂	1	2	yes	maximal
5	C₅₃×#1971	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1943	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1928	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1873	C₂	1	2	yes	maximal
9	C₅₃×#1863	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1776	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1752	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
16	C₂×#117×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#467	C₄	1	8	yes	complemented
18	C₅₃×#441	—	2	4	no	—
19	C₅₃×#427	—	2	4	no	—
20	C₂×#107×C₅₃	C₂²	1	2	yes	—
21	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#103×C₅₃	C₂²	1	2	yes	—
25	C₂×#103×C₅₃	C₂²	1	2	yes	—
26	C₂×#103×C₅₃	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#391	C₂²	1	2	yes	—
31	C₅₃×#389	C₂²	1	2	yes	—
32	C₅₃×#388	—	2	2	no	—
33	C₅₃×#388	—	2	2	no	—
34	C₅₃×#387	—	2	2	no	—
35	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
37	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
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	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×Q₁₆×C₅₃	—	2	2	no	—
45	C₂×Q₁₆×C₅₃	—	2	2	no	—
46	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₅₃	—	2	2	no	—
48	C₂²×C₈×C₅₃	—	2	2	no	—
49	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²⋊Q₈×C₅₃	—	2	4	no	—
55	C₂²⋊Q₈×C₅₃	C₂×C₄	1	8	yes	—
56	C₄×Q₈×C₅₃	—	2	4	no	—
57	C₄×Q₈×C₅₃	—	4	2	no	—
58	#117×C₅₃	—	2	4	no	—
59	#117×C₅₃	—	4	2	no	—
60	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
69	C₂×C₄²×C₅₃	—	2	2	no	—
70	C₂×C₄²×C₅₃	—	2	2	no	—
71	C₂×C₄²×C₅₃	—	2	2	no	—
72	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	#107×C₅₃	—	2	4	no	—
75	#107×C₅₃	C₂×C₄	1	4	yes	—
76	#106×C₅₃	—	2	2	no	—
77	#106×C₅₃	D₄	1	4	yes	—
78	#103×C₅₃	—	2	4	no	—
79	#103×C₅₃	—	4	2	no	—
80	#103×C₅₃	C₂×C₄	1	8	yes	—
81	C₂²⋊C₈×C₅₃	—	2	2	no	—
82	#97×C₅₃	—	2	2	no	—
83	#97×C₅₃	—	2	2	no	—
84	#95×C₅₃	—	2	2	no	—
85	#95×C₅₃	—	2	2	no	—
86	#95×C₅₃	—	2	2	no	—
87	#95×C₅₃	—	2	2	no	—
88	#95×C₅₃	D₄	1	2	yes	—
89	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×Q₈×C₅₃	—	2	2	no	—
91	C₂×Q₈×C₅₃	—	2	2	no	—
92	C₂×Q₈×C₅₃	—	2	2	no	—
93	C₂×Q₈×C₅₃	—	2	2	no	—
94	C₂×Q₈×C₅₃	—	2	2	no	—
95	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
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	—	2	2	no	—
103	C₂²×C₄×C₅₃	—	2	2	no	—
104	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	#41	1	2	yes	—
109	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
110	Q₁₆×C₅₃	—	2	8	no	—
111	C₂×C₈×C₅₃	—	2	1	no	—
112	C₂×C₈×C₅₃	—	2	1	no	—
113	C₂×C₈×C₅₃	—	2	8	no	—
114	C₂×C₈×C₅₃	—	2	1	no	—
115	C₂×C₈×C₅₃	—	2	1	no	—
116	C₂×C₈×C₅₃	—	2	1	no	—
117	C₂×C₈×C₅₃	—	2	1	no	—
118	C₂×C₈×C₅₃	—	2	1	no	—
119	C₂×C₈×C₅₃	—	2	1	no	—
120	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
121	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
123	C₂×C₈×C₅₃	#41	1	2	yes	—
124	C₄⋊C₄×C₅₃	—	2	8	no	—
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	—	2	2	no	—
127	C₄⋊C₄×C₅₃	—	2	2	no	—
128	C₄⋊C₄×C₅₃	—	2	2	no	—
129	C₄⋊C₄×C₅₃	—	4	4	no	—
130	C₄⋊C₄×C₅₃	—	4	4	no	—
131	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
132	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
133	C₂²⋊C₄×C₅₃	—	2	8	no	—
134	C₂²⋊C₄×C₅₃	—	4	2	no	—
135	C₂²⋊C₄×C₅₃	—	4	4	no	—
136	C₄²×C₅₃	—	2	1	no	—
137	C₄²×C₅₃	—	2	1	no	—
138	C₄²×C₅₃	—	2	1	no	—
139	C₄²×C₅₃	—	2	1	no	—
140	C₄²×C₅₃	—	4	4	no	—
141	C₄²×C₅₃	—	4	4	no	—
142	C₄²×C₅₃	—	4	4	no	—
143	C₂³×C₅₃	—	2	1	no	—
144	C₂³×C₅₃	—	2	1	no	—
145	C₂³×C₅₃	—	2	2	no	—
146	C₂³×C₅₃	—	2	2	no	—
147	C₂³×C₅₃	C₄×D₄	1	2	yes	—
148	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
149	Q₈×C₅₃	—	2	4	no	—
150	Q₈×C₅₃	—	2	4	no	—
151	Q₈×C₅₃	—	4	2	no	—
152	Q₈×C₅₃	—	4	2	no	—
153	C₂×C₄×C₅₃	—	2	1	no	—
154	C₂×C₄×C₅₃	—	2	1	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	2	2	no	—
158	C₂×C₄×C₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	—	2	4	no	—
160	C₂×C₄×C₅₃	—	2	4	no	—
161	C₂×C₄×C₅₃	—	2	4	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	4	2	no	—
169	C₂×C₄×C₅₃	—	4	2	no	—
170	C₂×C₄×C₅₃	—	4	2	no	—
171	C₂×C₄×C₅₃	—	4	2	no	—
172	C₂×C₄×C₅₃	—	4	2	no	—
173	C₂×C₄×C₅₃	—	4	2	no	—
174	C₂×C₄×C₅₃	—	4	2	no	—
175	C₂×C₄×C₅₃	—	4	2	no	—
176	C₂×C₄×C₅₃	—	4	2	no	—
177	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
178	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
182	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
184	C₈×C₅₃	—	2	2	no	—
185	C₈×C₅₃	—	4	1	no	—
186	C₈×C₅₃	—	4	1	no	—
187	C₈×C₅₃	C₄×D₄	1	4	yes	—
188	#23030	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂²×C₅₃	—	2	2	no	—
190	C₂²×C₅₃	—	2	2	no	—
191	C₂²×C₅₃	—	2	2	no	—
192	C₂²×C₅₃	—	4	2	no	—
193	C₂²×C₅₃	—	4	2	no	—
194	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
195	C₂²×C₅₃	D₄²	1	1	yes	characteristic
196	C₂²×C₅₃	#547	1	1	yes	characteristic
197	C₂²×C₅₃	#575	1	2	yes	—
198	C₂²×C₅₃	#578	1	2	yes	—
199	C₄×C₅₃	—	2	2	no	—
200	C₄×C₅₃	—	4	2	no	—
201	C₄×C₅₃	—	4	4	no	—
202	C₄×C₅₃	—	4	4	no	—
203	C₄×C₅₃	—	4	2	no	—
204	C₄×C₅₃	—	4	4	no	—
205	C₄×C₅₃	—	8	2	no	—
206	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
207	C₄×C₅₃	#520	1	1	yes	characteristic
208	C₄×C₅₃	#548	1	1	yes	characteristic
209	C₄×C₅₃	#553	1	1	yes	characteristic
210	C₂×#467	C₂×C₅₃	1	1	yes	characteristic
211	C₂×#441	C₂×C₅₃	1	1	yes	characteristic, complemented
212	C₂×#427	C₂×C₅₃	1	1	yes	characteristic
213	#2362	C₂×C₅₃	1	2	yes	—
214	#1971	C₂×C₅₃	1	1	yes	characteristic, complemented
215	#1943	C₂×C₅₃	1	1	yes	characteristic, complemented
216	#1928	C₂×C₅₃	1	2	yes	complemented
217	#1873	C₂×C₅₃	1	2	yes	—
218	#1863	C₂×C₅₃	1	2	yes	complemented
219	#1776	C₂×C₅₃	1	1	yes	characteristic, complemented
220	#1752	C₂×C₅₃	1	1	yes	characteristic
221	C₂×C₅₃	—	4	2	no	—
222	C₂×C₅₃	#2361	1	1	yes	characteristic
223	C₂×C₅₃	#2943	1	1	yes	characteristic
224	C₂×C₅₃	#2950	1	1	yes	characteristic
225	C₂×C₅₃	#3269	1	2	yes	—
226	C₂×C₅₃	#3349	1	2	yes	—
227	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
228	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
229	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
230	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
231	C₂×#117	C₂²×C₅₃	1	2	yes	—
232	#467	C₄×C₅₃	1	8	yes	complemented
233	#441	—	2	4	no	—
234	#427	—	2	4	no	—
235	C₂×#107	C₂²×C₅₃	1	2	yes	—
236	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
237	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
238	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
239	C₂×#103	C₂²×C₅₃	1	2	yes	—
240	C₂×#103	C₂²×C₅₃	1	2	yes	—
241	C₂×#103	C₂²×C₅₃	1	2	yes	—
242	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
243	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
244	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
245	#391	C₂²×C₅₃	1	2	yes	—
246	#389	C₂²×C₅₃	1	2	yes	—
247	#388	—	2	2	no	—
248	#388	—	2	2	no	—
249	#387	—	2	2	no	—
250	C₄×C₄⋊C₄	—	2	2	no	—
251	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
252	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
253	#340	C₂²×C₅₃	1	2	yes	—
254	#340	C₂²×C₅₃	1	2	yes	—
255	#336	C₂²×C₅₃	1	1	yes	characteristic
256	#336	C₂²×C₅₃	1	1	yes	characteristic
257	C₅₃	#23030	1	1	yes	characteristic, direct product factor, Sylow
258	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
259	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
260	C₂×Q₁₆	—	2	2	no	—
261	C₂×Q₁₆	—	2	2	no	—
262	C₂×Q₁₆	C₂×C₄×C₅₃	1	4	yes	—
263	C₂²×C₈	—	2	2	no	—
264	C₂²×C₈	—	2	2	no	—
265	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
266	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
267	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
269	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
270	C₂²⋊Q₈	—	2	4	no	—
271	C₂²⋊Q₈	C₂×C₄×C₅₃	1	8	yes	—
272	C₄×Q₈	—	2	4	no	—
273	C₄×Q₈	—	4	2	no	—
274	#117	—	2	4	no	—
275	#117	—	4	2	no	—
276	C₂×C₄⋊C₄	—	2	2	no	—
277	C₂×C₄⋊C₄	—	2	2	no	—
278	C₂×C₄⋊C₄	—	2	2	no	—
279	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
280	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
281	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
282	C₂×C₂²⋊C₄	—	2	2	no	—
283	C₂×C₂²⋊C₄	—	2	2	no	—
284	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
285	C₂×C₄²	—	2	2	no	—
286	C₂×C₄²	—	2	2	no	—
287	C₂×C₄²	—	2	2	no	—
288	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
289	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
290	#107	—	2	4	no	—
291	#107	C₂×C₄×C₅₃	1	4	yes	—
292	#106	—	2	2	no	—
293	#106	D₄×C₅₃	1	4	yes	—
294	#103	—	2	4	no	—
295	#103	—	4	2	no	—
296	#103	C₂×C₄×C₅₃	1	8	yes	—
297	C₂²⋊C₈	—	2	2	no	—
298	#97	—	2	2	no	—
299	#97	—	2	2	no	—
300	#95	—	2	2	no	—
301	#95	—	2	2	no	—
302	#95	—	2	2	no	—
303	#95	—	2	2	no	—
304	#95	D₄×C₅₃	1	2	yes	—
305	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	—	2	2	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
312	C₂²×C₄	—	2	1	no	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
321	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
322	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
323	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
324	C₂²×C₄	#41×C₅₃	1	2	yes	—
325	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
326	Q₁₆	—	2	8	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	8	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
337	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
338	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
339	C₂×C₈	#41×C₅₃	1	2	yes	—
340	C₄⋊C₄	—	2	8	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	4	4	no	—
346	C₄⋊C₄	—	4	4	no	—
347	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
348	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
349	C₂²⋊C₄	—	2	8	no	—
350	C₂²⋊C₄	—	4	2	no	—
351	C₂²⋊C₄	—	4	4	no	—
352	C₄²	—	2	1	no	—
353	C₄²	—	2	1	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	2	1	no	—
356	C₄²	—	4	4	no	—
357	C₄²	—	4	4	no	—
358	C₄²	—	4	4	no	—
359	C₂³	—	2	1	no	—
360	C₂³	—	2	1	no	—
361	C₂³	—	2	2	no	—
362	C₂³	—	2	2	no	—
363	C₂³	C₄×D₄×C₅₃	1	2	yes	—
364	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
365	Q₈	—	2	4	no	—
366	Q₈	—	2	4	no	—
367	Q₈	—	4	2	no	—
368	Q₈	—	4	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	2	4	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	4	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
394	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
395	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
396	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
397	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
398	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
399	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
400	C₈	—	2	2	no	—
401	C₈	—	4	1	no	—
402	C₈	—	4	1	no	—
403	C₈	C₄×D₄×C₅₃	1	4	yes	—
404	C₂²	—	2	2	no	—
405	C₂²	—	2	2	no	—
406	C₂²	—	2	2	no	—
407	C₂²	—	4	2	no	—
408	C₂²	—	4	2	no	—
409	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
410	C₂²	D₄²×C₅₃	1	1	yes	characteristic
411	C₂²	C₅₃×#547	1	1	yes	characteristic
412	C₂²	C₅₃×#575	1	2	yes	—
413	C₂²	C₅₃×#578	1	2	yes	—
414	C₄	—	2	2	no	—
415	C₄	—	4	4	no	—
416	C₄	—	4	2	no	—
417	C₄	—	4	4	no	—
418	C₄	—	4	2	no	—
419	C₄	—	4	4	no	—
420	C₄	—	8	2	no	—
421	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
422	C₄	C₅₃×#520	1	1	yes	characteristic
423	C₄	C₅₃×#548	1	1	yes	characteristic
424	C₄	C₅₃×#553	1	1	yes	characteristic
425	C₂	—	4	2	no	—
426	C₂	C₅₃×#2361	1	1	yes	characteristic
427	C₂	C₅₃×#2943	1	1	yes	characteristic
428	C₂	C₅₃×#2950	1	1	yes	characteristic
429	C₂	C₅₃×#3269	1	2	yes	—
430	C₂	C₅₃×#3349	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#441	C₂	semidirect
5. C₅₃×#1971	C₂	semidirect
6. C₅₃×#1943	C₂	semidirect
7. C₅₃×#1928	C₂	semidirect
9. C₅₃×#1863	C₂	semidirect
10. C₅₃×#1776	C₂	semidirect
17. C₅₃×#467	C₄	semidirect
188. #23030	C₅₃	direct (N × H)
211. C₂×#441	C₂×C₅₃	semidirect
214. #1971	C₂×C₅₃	semidirect
215. #1943	C₂×C₅₃	semidirect
216. #1928	C₂×C₅₃	semidirect
218. #1863	C₂×C₅₃	semidirect
219. #1776	C₂×C₅₃	semidirect
232. #467	C₄×C₅₃	semidirect
257. C₅₃	#23030	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#467	2
C₂×C₅₃×#441	2
C₂×C₅₃×#427	2
C₅₃×#2362	2
C₅₃×#1971	2
C₅₃×#1943	2
C₅₃×#1928	2
C₅₃×#1873	2
C₅₃×#1863	2
C₅₃×#1776	2
C₅₃×#1752	2
#23030	53

Subgroups — C53×#23030 #27230542

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23030 #27230542