Subgroups — C₅₃×#61293 #27268805

Order 13568.

1874 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3506	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3347	C₂	1	8	yes	maximal, complemented
3	C₅₃×#3345	C₂	1	8	yes	maximal, complemented
4	C₅₃×#3162	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3096	C₂	1	4	yes	maximal, complemented
6	C₅₃×#3077	C₂	1	2	yes	maximal, complemented
7	C₅₃×#3058	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₃×#442	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2932	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#2912	C₂	1	2	yes	maximal, complemented
11	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#131×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#559	—	2	4	no	—
14	C₅₃×#556	—	2	2	no	—
15	C₅₃×#555	C₂²	1	4	yes	complemented
16	C₅₃×#553	C₂²	1	8	yes	complemented
17	C₅₃×#550	—	2	2	no	—
18	C₅₃×#547	C₂²	1	4	yes	complemented
19	C₅₃×#538	—	2	4	no	—
20	C₅₃×#535	C₂²	1	4	yes	—
21	C₅₃×#533	C₂²	1	2	yes	—
22	C₅₃×#532	C₂²	1	1	yes	characteristic
23	C₅₃×#525	—	2	4	no	—
24	C₅₃×#520	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
26	C₅₃×#501	C₂²	1	4	yes	complemented
27	C₅₃×#501	C₂²	1	4	yes	complemented
28	C₅₃×#482	C₂²	1	8	yes	complemented
29	C₅₃×#482	C₂²	1	8	yes	complemented
30	C₅₃×#481	C₂²	1	8	yes	complemented
31	C₅₃×#480	C₂²	1	8	yes	complemented
32	C₅₃×#478	C₂²	1	8	yes	complemented
33	C₅₃×#476	C₂²	1	4	yes	complemented
34	C₅₃×#474	C₂²	1	4	yes	complemented
35	C₅₃×#469	C₂²	1	4	yes	—
36	C₅₃×#469	C₂²	1	4	yes	—
37	C₅₃×#466	C₂²	1	8	yes	—
38	C₅₃×#442	C₂²	1	8	yes	—
39	C₅₃×#435	C₂²	1	8	yes	complemented
40	C₅₃×#428	C₂²	1	4	yes	—
41	C₅₃×#428	C₂²	1	8	yes	complemented
42	C₅₃×#427	C₂²	1	2	yes	—
43	C₅₃×#427	C₂²	1	2	yes	—
44	C₅₃×#427	C₂²	1	4	yes	complemented
45	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×#106×C₅₃	C₂²	1	4	yes	complemented
47	C₅₃×#423	C₂²	1	1	yes	characteristic
48	C₅₃×#418	C₂²	1	4	yes	—
49	C₂×#102×C₅₃	C₂²	1	2	yes	—
50	C₅₃×#408	C₂²	1	2	yes	—
51	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
52	C₂×#41×C₅₃	—	2	4	no	—
53	C₂×#41×C₅₃	C₂³	1	2	yes	—
54	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
55	C₂³×C₄×C₅₃	—	2	2	no	—
56	C₂×D₈×C₅₃	—	2	4	no	—
57	C₂×D₈×C₅₃	C₂³	1	4	yes	—
58	#131×C₅₃	—	2	8	no	—
59	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
60	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
61	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
62	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
65	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
66	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
67	#127×C₅₃	—	2	4	no	—
68	#127×C₅₃	D₄	1	1	yes	characteristic
69	#126×C₅₃	—	2	8	no	—
70	#125×C₅₃	C₂³	1	4	yes	—
71	#124×C₅₃	—	2	8	no	—
72	#124×C₅₃	—	2	4	no	—
73	#124×C₅₃	D₄	1	2	yes	—
74	#123×C₅₃	—	2	8	no	—
75	#123×C₅₃	—	2	8	no	—
76	C₂²⋊Q₈×C₅₃	—	2	4	no	—
77	C₂²⋊Q₈×C₅₃	—	2	2	no	—
78	C₂²⋊Q₈×C₅₃	—	2	2	no	—
79	#121×C₅₃	—	2	4	no	—
80	#121×C₅₃	—	2	8	no	—
81	#121×C₅₃	D₄	1	4	yes	—
82	#121×C₅₃	C₂³	1	8	yes	—
83	#121×C₅₃	C₂³	1	8	yes	—
84	#120×C₅₃	—	2	8	no	—
85	C₄×Q₈×C₅₃	—	2	2	no	—
86	C₄×Q₈×C₅₃	—	2	2	no	—
87	C₄×Q₈×C₅₃	—	2	4	no	—
88	C₄×D₄×C₅₃	—	2	8	no	—
89	C₄×D₄×C₅₃	—	2	2	no	—
90	C₄×D₄×C₅₃	—	2	2	no	—
91	C₄×D₄×C₅₃	—	2	8	no	—
92	C₄×D₄×C₅₃	—	2	4	no	—
93	C₄×D₄×C₅₃	C₂³	1	4	yes	—
94	C₄×D₄×C₅₃	C₂³	1	4	yes	—
95	C₄×D₄×C₅₃	C₂³	1	4	yes	—
96	#117×C₅₃	—	2	4	no	—
97	#117×C₅₃	C₂³	1	4	yes	—
98	#117×C₅₃	C₂³	1	8	yes	—
99	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
101	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
102	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
103	#107×C₅₃	C₂³	1	4	yes	—
104	#107×C₅₃	C₂³	1	4	yes	—
105	#107×C₅₃	C₂³	1	8	yes	—
106	#106×C₅₃	C₂³	1	2	yes	—
107	#106×C₅₃	C₂³	1	2	yes	—
108	#106×C₅₃	C₂³	1	2	yes	—
109	#106×C₅₃	C₂³	1	2	yes	—
110	#106×C₅₃	C₂³	1	4	yes	—
111	#106×C₅₃	C₂³	1	8	yes	—
112	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
113	#102×C₅₃	C₂³	1	8	yes	—
114	#102×C₅₃	C₂³	1	8	yes	—
115	#102×C₅₃	C₂³	1	8	yes	—
116	C₂²⋊C₈×C₅₃	C₂³	1	8	yes	—
117	#97×C₅₃	C₂³	1	4	yes	—
118	C₂⁴×C₅₃	—	2	2	no	—
119	#41×C₅₃	—	2	4	no	—
120	#41×C₅₃	—	2	4	no	—
121	#41×C₅₃	—	8	8	no	—
122	C₂×Q₈×C₅₃	—	2	2	no	—
123	C₂×Q₈×C₅₃	—	2	2	no	—
124	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
125	C₂×D₄×C₅₃	—	2	8	no	—
126	C₂×D₄×C₅₃	—	2	8	no	—
127	C₂×D₄×C₅₃	—	2	2	no	—
128	C₂×D₄×C₅₃	—	2	4	no	—
129	C₂×D₄×C₅₃	—	2	4	no	—
130	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
131	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
132	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
133	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
134	C₂²×C₄×C₅₃	—	2	2	no	—
135	C₂²×C₄×C₅₃	—	2	8	no	—
136	C₂²×C₄×C₅₃	—	2	4	no	—
137	C₂²×C₄×C₅₃	—	2	4	no	—
138	C₂²×C₄×C₅₃	—	8	4	no	—
139	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
141	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
142	D₈×C₅₃	—	2	8	no	—
143	M₁₆×C₅₃	—	2	4	no	—
144	M₁₆×C₅₃	—	2	8	no	—
145	C₂×C₈×C₅₃	—	2	2	no	—
146	C₂×C₈×C₅₃	—	2	8	no	—
147	C₂×C₈×C₅₃	#41	1	4	yes	complemented
148	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
149	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
150	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
151	C₄⋊C₄×C₅₃	—	2	2	no	—
152	C₄⋊C₄×C₅₃	—	2	8	no	—
153	C₄⋊C₄×C₅₃	—	2	2	no	—
154	C₄⋊C₄×C₅₃	—	2	4	no	—
155	C₄⋊C₄×C₅₃	—	2	4	no	—
156	C₄⋊C₄×C₅₃	—	2	2	no	—
157	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
158	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
159	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
160	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
161	C₂²⋊C₄×C₅₃	—	2	4	no	—
162	C₂²⋊C₄×C₅₃	—	2	8	no	—
163	C₂²⋊C₄×C₅₃	—	2	4	no	—
164	C₂²⋊C₄×C₅₃	—	2	4	no	—
165	C₂²⋊C₄×C₅₃	—	2	8	no	—
166	C₂²⋊C₄×C₅₃	—	2	8	no	—
167	C₂²⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
168	C₄²×C₅₃	—	2	8	no	—
169	C₄²×C₅₃	—	2	4	no	—
170	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
171	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₅₃	C₂⁴	1	2	yes	—
173	C₂³×C₅₃	—	2	4	no	—
174	C₂³×C₅₃	—	4	4	no	—
175	C₂³×C₅₃	—	8	2	no	—
176	C₂³×C₅₃	—	8	2	no	—
177	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
178	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
179	Q₈×C₅₃	—	2	4	no	—
180	Q₈×C₅₃	—	4	4	no	—
181	Q₈×C₅₃	—	8	2	no	—
182	D₄×C₅₃	—	2	8	no	—
183	D₄×C₅₃	—	2	4	no	—
184	D₄×C₅₃	—	2	4	no	—
185	D₄×C₅₃	—	2	4	no	—
186	D₄×C₅₃	—	4	4	no	—
187	D₄×C₅₃	—	4	4	no	—
188	D₄×C₅₃	—	8	8	no	—
189	D₄×C₅₃	—	8	8	no	—
190	C₂×C₄×C₅₃	—	2	4	no	—
191	C₂×C₄×C₅₃	—	2	4	no	—
192	C₂×C₄×C₅₃	—	2	4	no	—
193	C₂×C₄×C₅₃	—	2	4	no	—
194	C₂×C₄×C₅₃	—	2	4	no	—
195	C₂×C₄×C₅₃	—	4	2	no	—
196	C₂×C₄×C₅₃	—	4	8	no	—
197	C₂×C₄×C₅₃	—	8	4	no	—
198	C₂×C₄×C₅₃	—	8	2	no	—
199	C₂×C₄×C₅₃	—	8	8	no	—
200	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
201	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
202	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
203	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
205	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
206	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
207	C₈×C₅₃	—	2	4	no	—
208	C₈×C₅₃	—	2	2	no	—
209	C₈×C₅₃	C₂×#41	1	4	yes	—
210	#61293	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂²×C₅₃	—	2	2	no	—
212	C₂²×C₅₃	—	2	8	no	—
213	C₂²×C₅₃	—	2	4	no	—
214	C₂²×C₅₃	—	4	4	no	—
215	C₂²×C₅₃	—	4	4	no	—
216	C₂²×C₅₃	—	4	4	no	—
217	C₂²×C₅₃	—	8	8	no	—
218	C₂²×C₅₃	#547	1	2	yes	—
219	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
220	C₄×C₅₃	—	2	2	no	—
221	C₄×C₅₃	—	2	2	no	—
222	C₄×C₅₃	—	2	4	no	—
223	C₄×C₅₃	—	4	4	no	—
224	C₄×C₅₃	—	4	4	no	—
225	C₄×C₅₃	—	4	4	no	—
226	C₄×C₅₃	#547	1	2	yes	—
227	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
228	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
229	#3506	C₂×C₅₃	1	2	yes	complemented
230	#3347	C₂×C₅₃	1	8	yes	complemented
231	#3345	C₂×C₅₃	1	8	yes	complemented
232	#3162	C₂×C₅₃	1	1	yes	characteristic, complemented
233	#3096	C₂×C₅₃	1	4	yes	complemented
234	#3077	C₂×C₅₃	1	2	yes	complemented
235	#3058	C₂×C₅₃	1	2	yes	complemented
236	C₂×#442	C₂×C₅₃	1	1	yes	characteristic, complemented
237	#2932	C₂×C₅₃	1	1	yes	characteristic, complemented
238	#2912	C₂×C₅₃	1	2	yes	complemented
239	C₂×C₅₃	—	2	2	no	—
240	C₂×C₅₃	—	4	4	no	—
241	C₂×C₅₃	—	8	4	no	—
242	C₂×C₅₃	C₂×#547	1	1	yes	characteristic
243	C₂×C₅₃	#3585	1	2	yes	—
244	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
245	C₂×#131	C₂²×C₅₃	1	2	yes	—
246	#559	—	2	4	no	—
247	#556	—	2	2	no	—
248	#555	C₂²×C₅₃	1	4	yes	complemented
249	#553	C₂²×C₅₃	1	8	yes	complemented
250	#550	—	2	2	no	—
251	#547	C₂²×C₅₃	1	4	yes	complemented
252	#538	—	2	4	no	—
253	#535	C₂²×C₅₃	1	4	yes	—
254	#533	C₂²×C₅₃	1	2	yes	—
255	#532	C₂²×C₅₃	1	1	yes	characteristic
256	#525	—	2	4	no	—
257	#520	C₂²×C₅₃	1	4	yes	—
258	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
259	#501	C₂²×C₅₃	1	4	yes	complemented
260	#501	C₂²×C₅₃	1	4	yes	complemented
261	#482	C₂²×C₅₃	1	8	yes	complemented
262	#482	C₂²×C₅₃	1	8	yes	complemented
263	#481	C₂²×C₅₃	1	8	yes	complemented
264	#480	C₂²×C₅₃	1	8	yes	complemented
265	#478	C₂²×C₅₃	1	8	yes	complemented
266	#476	C₂²×C₅₃	1	4	yes	complemented
267	#474	C₂²×C₅₃	1	4	yes	complemented
268	#469	C₂²×C₅₃	1	4	yes	—
269	#469	C₂²×C₅₃	1	4	yes	—
270	#466	C₂²×C₅₃	1	8	yes	—
271	#442	C₂²×C₅₃	1	8	yes	—
272	#435	C₂²×C₅₃	1	8	yes	complemented
273	#428	C₂²×C₅₃	1	4	yes	—
274	#428	C₂²×C₅₃	1	8	yes	complemented
275	#427	C₂²×C₅₃	1	2	yes	—
276	#427	C₂²×C₅₃	1	2	yes	—
277	#427	C₂²×C₅₃	1	4	yes	complemented
278	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
279	C₂×#106	C₂²×C₅₃	1	4	yes	complemented
280	#423	C₂²×C₅₃	1	1	yes	characteristic
281	#418	C₂²×C₅₃	1	4	yes	—
282	C₂×#102	C₂²×C₅₃	1	2	yes	—
283	#408	C₂²×C₅₃	1	2	yes	—
284	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
285	C₅₃	#61293	1	1	yes	characteristic, direct product factor, Sylow
286	C₂×#41	—	2	4	no	—
287	C₂×#41	C₂³×C₅₃	1	2	yes	—
288	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
289	C₂³×C₄	—	2	2	no	—
290	C₂×D₈	—	2	4	no	—
291	C₂×D₈	C₂³×C₅₃	1	4	yes	—
292	#131	—	2	8	no	—
293	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
294	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
295	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
296	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
297	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
298	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
299	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
300	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
301	#127	—	2	4	no	—
302	#127	D₄×C₅₃	1	1	yes	characteristic
303	#126	—	2	8	no	—
304	#125	C₂³×C₅₃	1	4	yes	—
305	#124	—	2	4	no	—
306	#124	—	2	8	no	—
307	#124	D₄×C₅₃	1	2	yes	—
308	#123	—	2	8	no	—
309	#123	—	2	8	no	—
310	C₂²⋊Q₈	—	2	2	no	—
311	C₂²⋊Q₈	—	2	2	no	—
312	C₂²⋊Q₈	—	2	4	no	—
313	#121	—	2	4	no	—
314	#121	—	2	8	no	—
315	#121	D₄×C₅₃	1	4	yes	—
316	#121	C₂³×C₅₃	1	8	yes	—
317	#121	C₂³×C₅₃	1	8	yes	—
318	#120	—	2	8	no	—
319	C₄×Q₈	—	2	4	no	—
320	C₄×Q₈	—	2	2	no	—
321	C₄×Q₈	—	2	2	no	—
322	C₄×D₄	—	2	8	no	—
323	C₄×D₄	—	2	4	no	—
324	C₄×D₄	—	2	8	no	—
325	C₄×D₄	—	2	2	no	—
326	C₄×D₄	—	2	2	no	—
327	C₄×D₄	C₂³×C₅₃	1	4	yes	—
328	C₄×D₄	C₂³×C₅₃	1	4	yes	—
329	C₄×D₄	C₂³×C₅₃	1	4	yes	—
330	#117	—	2	4	no	—
331	#117	C₂³×C₅₃	1	4	yes	—
332	#117	C₂³×C₅₃	1	8	yes	—
333	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
334	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
335	C₂×C₂²⋊C₄	—	2	2	no	—
336	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
337	#107	C₂³×C₅₃	1	4	yes	—
338	#107	C₂³×C₅₃	1	4	yes	—
339	#107	C₂³×C₅₃	1	8	yes	—
340	#106	C₂³×C₅₃	1	4	yes	—
341	#106	C₂³×C₅₃	1	2	yes	—
342	#106	C₂³×C₅₃	1	8	yes	—
343	#106	C₂³×C₅₃	1	2	yes	—
344	#106	C₂³×C₅₃	1	2	yes	—
345	#106	C₂³×C₅₃	1	2	yes	—
346	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
347	#102	C₂³×C₅₃	1	8	yes	—
348	#102	C₂³×C₅₃	1	8	yes	—
349	#102	C₂³×C₅₃	1	8	yes	—
350	C₂²⋊C₈	C₂³×C₅₃	1	8	yes	—
351	#97	C₂³×C₅₃	1	4	yes	—
352	C₂⁴	—	2	2	no	—
353	#41	—	2	4	no	—
354	#41	—	2	4	no	—
355	#41	—	8	8	no	—
356	C₂×Q₈	—	2	2	no	—
357	C₂×Q₈	—	2	2	no	—
358	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
359	C₂×D₄	—	2	8	no	—
360	C₂×D₄	—	2	8	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	—	2	4	no	—
363	C₂×D₄	—	2	4	no	—
364	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
365	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
366	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
367	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	2	8	no	—
370	C₂²×C₄	—	2	4	no	—
371	C₂²×C₄	—	2	4	no	—
372	C₂²×C₄	—	8	4	no	—
373	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
374	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
375	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
376	D₈	—	2	8	no	—
377	M₁₆	—	2	4	no	—
378	M₁₆	—	2	8	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	8	no	—
381	C₂×C₈	#41×C₅₃	1	4	yes	complemented
382	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
383	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
384	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	8	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	4	no	—
389	C₄⋊C₄	—	2	4	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
392	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
393	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
394	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
395	C₂²⋊C₄	—	2	4	no	—
396	C₂²⋊C₄	—	2	8	no	—
397	C₂²⋊C₄	—	2	4	no	—
398	C₂²⋊C₄	—	2	4	no	—
399	C₂²⋊C₄	—	2	8	no	—
400	C₂²⋊C₄	—	2	8	no	—
401	C₂²⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
402	C₄²	—	2	8	no	—
403	C₄²	—	2	4	no	—
404	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
405	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
406	C₄²	C₂⁴×C₅₃	1	2	yes	—
407	C₂³	—	2	4	no	—
408	C₂³	—	4	4	no	—
409	C₂³	—	8	2	no	—
410	C₂³	—	8	2	no	—
411	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
412	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
413	Q₈	—	2	4	no	—
414	Q₈	—	4	4	no	—
415	Q₈	—	8	2	no	—
416	D₄	—	2	8	no	—
417	D₄	—	2	4	no	—
418	D₄	—	2	4	no	—
419	D₄	—	2	4	no	—
420	D₄	—	4	4	no	—
421	D₄	—	4	4	no	—
422	D₄	—	8	8	no	—
423	D₄	—	8	8	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	8	no	—
431	C₂×C₄	—	8	4	no	—
432	C₂×C₄	—	8	2	no	—
433	C₂×C₄	—	8	8	no	—
434	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
435	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
436	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
437	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
438	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
439	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
440	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
441	C₈	—	2	4	no	—
442	C₈	—	2	2	no	—
443	C₈	C₂×#41×C₅₃	1	4	yes	—
444	C₂²	—	2	2	no	—
445	C₂²	—	2	4	no	—
446	C₂²	—	2	8	no	—
447	C₂²	—	4	4	no	—
448	C₂²	—	4	4	no	—
449	C₂²	—	4	4	no	—
450	C₂²	—	8	8	no	—
451	C₂²	C₅₃×#547	1	2	yes	—
452	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
453	C₄	—	2	4	no	—
454	C₄	—	2	2	no	—
455	C₄	—	2	2	no	—
456	C₄	—	4	4	no	—
457	C₄	—	4	4	no	—
458	C₄	—	4	4	no	—
459	C₄	C₅₃×#547	1	2	yes	—
460	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
461	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
462	C₂	—	2	2	no	—
463	C₂	—	4	4	no	—
464	C₂	—	8	4	no	—
465	C₂	C₂×C₅₃×#547	1	1	yes	characteristic
466	C₂	C₅₃×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₅₃×#532 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3506	C₂	semidirect
2. C₅₃×#3347	C₂	semidirect
3. C₅₃×#3345	C₂	semidirect
4. C₅₃×#3162	C₂	semidirect
5. C₅₃×#3096	C₂	semidirect
6. C₅₃×#3077	C₂	semidirect
7. C₅₃×#3058	C₂	semidirect
8. C₂×C₅₃×#442	C₂	semidirect
9. C₅₃×#2932	C₂	semidirect
10. C₅₃×#2912	C₂	semidirect
15. C₅₃×#555	C₂²	semidirect
16. C₅₃×#553	C₂²	semidirect
18. C₅₃×#547	C₂²	semidirect
26. C₅₃×#501	C₂²	semidirect
27. C₅₃×#501	C₂²	semidirect
28. C₅₃×#482	C₂²	semidirect
29. C₅₃×#482	C₂²	semidirect
30. C₅₃×#481	C₂²	semidirect
31. C₅₃×#480	C₂²	semidirect
32. C₅₃×#478	C₂²	semidirect
33. C₅₃×#476	C₂²	semidirect
34. C₅₃×#474	C₂²	semidirect
39. C₅₃×#435	C₂²	semidirect
41. C₅₃×#428	C₂²	semidirect
44. C₅₃×#427	C₂²	semidirect
46. C₂×#106×C₅₃	C₂²	semidirect
147. C₂×C₈×C₅₃	#41	semidirect
210. #61293	C₅₃	direct (N × H)
229. #3506	C₂×C₅₃	semidirect
230. #3347	C₂×C₅₃	semidirect
231. #3345	C₂×C₅₃	semidirect
232. #3162	C₂×C₅₃	semidirect
233. #3096	C₂×C₅₃	semidirect
234. #3077	C₂×C₅₃	semidirect
235. #3058	C₂×C₅₃	semidirect
236. C₂×#442	C₂×C₅₃	semidirect
237. #2932	C₂×C₅₃	semidirect
238. #2912	C₂×C₅₃	semidirect
248. #555	C₂²×C₅₃	semidirect
249. #553	C₂²×C₅₃	semidirect
251. #547	C₂²×C₅₃	semidirect
259. #501	C₂²×C₅₃	semidirect
260. #501	C₂²×C₅₃	semidirect
261. #482	C₂²×C₅₃	semidirect
262. #482	C₂²×C₅₃	semidirect
263. #481	C₂²×C₅₃	semidirect
264. #480	C₂²×C₅₃	semidirect
265. #478	C₂²×C₅₃	semidirect
266. #476	C₂²×C₅₃	semidirect
267. #474	C₂²×C₅₃	semidirect
272. #435	C₂²×C₅₃	semidirect
274. #428	C₂²×C₅₃	semidirect
277. #427	C₂²×C₅₃	semidirect
279. C₂×#106	C₂²×C₅₃	semidirect
285. C₅₃	#61293	direct (N × H)
381. C₂×C₈	#41×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3506	2
C₅₃×#3347	2
C₅₃×#3345	2
C₅₃×#3162	2
C₅₃×#3096	2
C₅₃×#3077	2
C₅₃×#3058	2
C₂×C₅₃×#442	2
C₅₃×#2932	2
C₅₃×#2912	2
#61293	53

Subgroups — C53×#61293 #27268805

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#61293 #27268805