Subgroups — C₅₃×#61286 #27268798

Order 13568.

1754 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3507	C₂	1	2	yes	maximal
2	C₅₃×#3356	C₂	1	4	yes	maximal, complemented
3	C₅₃×#3352	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3342	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3338	C₂	1	2	yes	maximal, complemented
6	C₅₃×#3336	C₂	1	4	yes	maximal, complemented
7	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#3097	C₂	1	4	yes	maximal, complemented
9	C₅₃×#3078	C₂	1	2	yes	maximal, complemented
10	C₅₃×#3059	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
13	C₅₃×#2913	C₂	1	2	yes	maximal, complemented
14	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#131×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#559	—	2	2	no	—
17	C₅₃×#557	—	2	2	no	—
18	C₅₃×#556	—	2	4	no	—
19	C₅₃×#556	C₂²	1	4	yes	—
20	C₅₃×#554	C₂²	1	4	yes	—
21	C₅₃×#552	C₂²	1	8	yes	—
22	C₅₃×#539	—	2	4	no	—
23	C₅₃×#536	C₂²	1	4	yes	—
24	C₅₃×#533	C₂²	1	2	yes	—
25	C₅₃×#532	C₂²	1	1	yes	characteristic
26	C₅₃×#529	—	2	4	no	—
27	C₅₃×#520	C₂²	1	4	yes	—
28	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
29	C₅₃×#500	C₂²	1	1	yes	characteristic
30	C₅₃×#500	C₂²	1	1	yes	characteristic
31	C₅₃×#499	C₂²	1	2	yes	—
32	C₅₃×#499	C₂²	1	2	yes	—
33	C₅₃×#498	C₂²	1	2	yes	—
34	C₅₃×#485	C₂²	1	8	yes	—
35	C₅₃×#485	C₂²	1	8	yes	—
36	C₅₃×#484	C₂²	1	8	yes	—
37	C₅₃×#483	C₂²	1	8	yes	—
38	C₅₃×#479	C₂²	1	8	yes	—
39	C₅₃×#477	C₂²	1	4	yes	—
40	C₅₃×#475	C₂²	1	4	yes	—
41	C₅₃×#470	C₂²	1	8	yes	—
42	C₅₃×#467	C₂²	1	4	yes	—
43	C₅₃×#467	C₂²	1	4	yes	—
44	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
45	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
46	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
47	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
48	C₅₃×#436	C₂²	1	2	yes	—
49	C₅₃×#436	C₂²	1	2	yes	—
50	C₈×D₄×C₅₃	C₂²	1	2	yes	—
51	C₈×D₄×C₅₃	C₂²	1	2	yes	—
52	C₅₃×#428	C₂²	1	4	yes	—
53	C₅₃×#428	C₂²	1	8	yes	—
54	C₅₃×#427	C₂²	1	2	yes	—
55	C₅₃×#427	C₂²	1	4	yes	—
56	C₅₃×#427	C₂²	1	2	yes	—
57	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
58	C₂×#107×C₅₃	C₂²	1	4	yes	—
59	C₅₃×#423	C₂²	1	1	yes	characteristic
60	C₅₃×#419	C₂²	1	4	yes	—
61	C₂×#103×C₅₃	C₂²	1	2	yes	—
62	C₅₃×#408	C₂²	1	2	yes	—
63	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
64	C₂×#41×C₅₃	C₂³	1	2	yes	—
65	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
66	C₂×Q₁₆×C₅₃	—	2	4	no	—
67	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
68	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
69	#131×C₅₃	—	2	8	no	—
70	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
71	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
72	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
73	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄⋊Q₈×C₅₃	—	2	4	no	—
76	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
77	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
78	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
79	#127×C₅₃	D₄	1	1	yes	characteristic
80	#126×C₅₃	—	2	8	no	—
81	#126×C₅₃	—	2	8	no	—
82	#125×C₅₃	—	2	4	no	—
83	#125×C₅₃	—	2	8	no	—
84	#125×C₅₃	C₂³	1	4	yes	—
85	#124×C₅₃	—	2	4	no	—
86	#124×C₅₃	—	2	4	no	—
87	#124×C₅₃	D₄	1	2	yes	—
88	#123×C₅₃	—	2	8	no	—
89	#123×C₅₃	—	2	8	no	—
90	C₂²⋊Q₈×C₅₃	—	2	8	no	—
91	C₂²⋊Q₈×C₅₃	—	2	2	no	—
92	C₂²⋊Q₈×C₅₃	—	2	2	no	—
93	C₂²⋊Q₈×C₅₃	C₂³	1	8	yes	—
94	C₂²⋊Q₈×C₅₃	C₂³	1	8	yes	—
95	#121×C₅₃	D₄	1	2	yes	—
96	#121×C₅₃	D₄	1	2	yes	—
97	C₄×Q₈×C₅₃	—	2	4	no	—
98	C₄×Q₈×C₅₃	—	2	2	no	—
99	C₄×Q₈×C₅₃	—	2	2	no	—
100	C₄×Q₈×C₅₃	—	2	2	no	—
101	C₄×Q₈×C₅₃	—	2	2	no	—
102	C₄×Q₈×C₅₃	—	2	4	no	—
103	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
104	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
105	C₄×D₄×C₅₃	—	2	8	no	—
106	C₄×D₄×C₅₃	—	2	4	no	—
107	C₄×D₄×C₅₃	—	2	4	no	—
108	C₄×D₄×C₅₃	C₂³	1	2	yes	—
109	C₄×D₄×C₅₃	C₂³	1	2	yes	—
110	#117×C₅₃	—	2	8	no	—
111	#117×C₅₃	C₂³	1	8	yes	—
112	#117×C₅₃	C₂³	1	4	yes	—
113	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
114	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
116	C₂×C₄²×C₅₃	—	2	2	no	—
117	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
118	#107×C₅₃	C₂³	1	2	yes	—
119	#107×C₅₃	C₂³	1	8	yes	—
120	#107×C₅₃	C₂³	1	2	yes	—
121	#107×C₅₃	C₂³	1	2	yes	—
122	#107×C₅₃	C₂³	1	2	yes	—
123	#107×C₅₃	C₂³	1	2	yes	—
124	#107×C₅₃	C₂³	1	2	yes	—
125	#106×C₅₃	C₂³	1	4	yes	—
126	#106×C₅₃	C₂³	1	8	yes	—
127	#106×C₅₃	C₂³	1	4	yes	—
128	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
129	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
130	#103×C₅₃	C₂³	1	4	yes	—
131	#103×C₅₃	C₂³	1	4	yes	—
132	#103×C₅₃	C₂³	1	8	yes	—
133	#103×C₅₃	C₂³	1	8	yes	—
134	C₂²⋊C₈×C₅₃	C₂³	1	4	yes	—
135	C₂²⋊C₈×C₅₃	C₂³	1	4	yes	—
136	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
137	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
140	#41×C₅₃	—	2	4	no	—
141	#41×C₅₃	—	2	4	no	—
142	C₂×Q₈×C₅₃	—	2	2	no	—
143	C₂×Q₈×C₅₃	—	2	2	no	—
144	C₂×Q₈×C₅₃	—	2	2	no	—
145	C₂×Q₈×C₅₃	—	2	4	no	—
146	C₂×Q₈×C₅₃	—	2	4	no	—
147	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
148	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
149	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
150	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
151	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
152	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₅₃	—	2	2	no	—
154	C₂²×C₄×C₅₃	—	2	2	no	—
155	C₂²×C₄×C₅₃	—	2	8	no	—
156	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
160	Q₁₆×C₅₃	—	2	4	no	—
161	Q₁₆×C₅₃	—	2	4	no	—
162	M₁₆×C₅₃	—	2	4	no	—
163	M₁₆×C₅₃	—	2	8	no	—
164	C₂×C₈×C₅₃	—	2	2	no	—
165	C₂×C₈×C₅₃	—	2	8	no	—
166	C₂×C₈×C₅₃	#41	1	4	yes	—
167	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
168	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
169	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
170	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
171	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
172	C₄⋊C₄×C₅₃	—	2	4	no	—
173	C₄⋊C₄×C₅₃	—	2	4	no	—
174	C₄⋊C₄×C₅₃	—	2	2	no	—
175	C₄⋊C₄×C₅₃	—	2	8	no	—
176	C₄⋊C₄×C₅₃	—	2	4	no	—
177	C₄⋊C₄×C₅₃	—	2	8	no	—
178	C₄⋊C₄×C₅₃	—	2	2	no	—
179	C₄⋊C₄×C₅₃	—	2	4	no	—
180	C₄⋊C₄×C₅₃	—	2	2	no	—
181	C₄⋊C₄×C₅₃	—	2	8	no	—
182	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
183	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
184	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
185	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
186	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
187	C₂²⋊C₄×C₅₃	—	2	8	no	—
188	C₂²⋊C₄×C₅₃	—	2	4	no	—
189	C₂²⋊C₄×C₅₃	—	2	8	no	—
190	C₂²⋊C₄×C₅₃	—	2	8	no	—
191	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
192	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
193	C₄²×C₅₃	—	2	4	no	—
194	C₄²×C₅₃	—	2	4	no	—
195	C₄²×C₅₃	—	2	8	no	—
196	C₄²×C₅₃	—	2	4	no	—
197	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
199	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
200	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
201	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
202	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
203	Q₈×C₅₃	—	2	4	no	—
204	Q₈×C₅₃	—	2	4	no	—
205	Q₈×C₅₃	—	2	4	no	—
206	Q₈×C₅₃	—	4	4	no	—
207	Q₈×C₅₃	—	4	4	no	—
208	Q₈×C₅₃	—	8	2	no	—
209	D₄×C₅₃	—	2	8	no	—
210	D₄×C₅₃	—	2	4	no	—
211	D₄×C₅₃	—	2	4	no	—
212	D₄×C₅₃	—	4	2	no	—
213	C₂×C₄×C₅₃	—	2	2	no	—
214	C₂×C₄×C₅₃	—	2	4	no	—
215	C₂×C₄×C₅₃	—	2	4	no	—
216	C₂×C₄×C₅₃	—	2	4	no	—
217	C₂×C₄×C₅₃	—	2	4	no	—
218	C₂×C₄×C₅₃	—	2	4	no	—
219	C₂×C₄×C₅₃	—	2	4	no	—
220	C₂×C₄×C₅₃	—	4	1	no	—
221	C₂×C₄×C₅₃	—	4	4	no	—
222	C₂×C₄×C₅₃	—	8	2	no	—
223	C₂×C₄×C₅₃	—	8	8	no	—
224	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
228	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
231	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
232	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
233	C₈×C₅₃	—	2	4	no	—
234	C₈×C₅₃	—	2	1	no	—
235	C₈×C₅₃	—	2	1	no	—
236	C₈×C₅₃	C₂×#41	1	2	yes	—
237	C₈×C₅₃	C₂×#41	1	2	yes	—
238	#61286	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
239	C₂²×C₅₃	—	2	1	no	—
240	C₂²×C₅₃	—	2	1	no	—
241	C₂²×C₅₃	—	2	4	no	—
242	C₂²×C₅₃	—	2	4	no	—
243	C₂²×C₅₃	—	2	4	no	—
244	C₂²×C₅₃	#547	1	2	yes	—
245	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
246	C₄×C₅₃	—	2	1	no	—
247	C₄×C₅₃	—	2	1	no	—
248	C₄×C₅₃	—	2	4	no	—
249	C₄×C₅₃	—	2	1	no	—
250	C₄×C₅₃	—	4	4	no	—
251	C₄×C₅₃	—	4	4	no	—
252	C₄×C₅₃	—	4	4	no	—
253	C₄×C₅₃	—	4	4	no	—
254	C₄×C₅₃	#547	1	2	yes	—
255	C₄×C₅₃	#578	1	2	yes	—
256	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
257	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
258	#3507	C₂×C₅₃	1	2	yes	—
259	#3356	C₂×C₅₃	1	4	yes	complemented
260	#3352	C₂×C₅₃	1	4	yes	complemented
261	#3342	C₂×C₅₃	1	2	yes	complemented
262	#3338	C₂×C₅₃	1	2	yes	complemented
263	#3336	C₂×C₅₃	1	4	yes	complemented
264	#3160	C₂×C₅₃	1	1	yes	characteristic, complemented
265	#3097	C₂×C₅₃	1	4	yes	complemented
266	#3078	C₂×C₅₃	1	2	yes	complemented
267	#3059	C₂×C₅₃	1	2	yes	complemented
268	C₂×C₄×Q₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
269	#2931	C₂×C₅₃	1	1	yes	characteristic
270	#2913	C₂×C₅₃	1	2	yes	complemented
271	C₂×C₅₃	—	2	2	no	—
272	C₂×C₅₃	—	4	4	no	—
273	C₂×C₅₃	C₂×#547	1	1	yes	characteristic
274	C₂×C₅₃	C₂×#578	1	1	yes	characteristic
275	C₂×C₅₃	#3584	1	1	yes	characteristic
276	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
277	C₂×#131	C₂²×C₅₃	1	2	yes	—
278	#559	—	2	2	no	—
279	#557	—	2	2	no	—
280	#556	—	2	4	no	—
281	#556	C₂²×C₅₃	1	4	yes	—
282	#554	C₂²×C₅₃	1	4	yes	—
283	#552	C₂²×C₅₃	1	8	yes	—
284	#539	—	2	4	no	—
285	#536	C₂²×C₅₃	1	4	yes	—
286	#533	C₂²×C₅₃	1	2	yes	—
287	#532	C₂²×C₅₃	1	1	yes	characteristic
288	#529	—	2	4	no	—
289	#520	C₂²×C₅₃	1	4	yes	—
290	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
291	#500	C₂²×C₅₃	1	1	yes	characteristic
292	#500	C₂²×C₅₃	1	1	yes	characteristic
293	#499	C₂²×C₅₃	1	2	yes	—
294	#499	C₂²×C₅₃	1	2	yes	—
295	#498	C₂²×C₅₃	1	2	yes	—
296	#485	C₂²×C₅₃	1	8	yes	—
297	#485	C₂²×C₅₃	1	8	yes	—
298	#484	C₂²×C₅₃	1	8	yes	—
299	#483	C₂²×C₅₃	1	8	yes	—
300	#479	C₂²×C₅₃	1	8	yes	—
301	#477	C₂²×C₅₃	1	4	yes	—
302	#475	C₂²×C₅₃	1	4	yes	—
303	#470	C₂²×C₅₃	1	8	yes	—
304	#467	C₂²×C₅₃	1	4	yes	—
305	#467	C₂²×C₅₃	1	4	yes	—
306	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
307	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
308	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
309	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
310	#436	C₂²×C₅₃	1	2	yes	—
311	#436	C₂²×C₅₃	1	2	yes	—
312	C₈×D₄	C₂²×C₅₃	1	2	yes	—
313	C₈×D₄	C₂²×C₅₃	1	2	yes	—
314	#428	C₂²×C₅₃	1	4	yes	—
315	#428	C₂²×C₅₃	1	8	yes	—
316	#427	C₂²×C₅₃	1	4	yes	—
317	#427	C₂²×C₅₃	1	2	yes	—
318	#427	C₂²×C₅₃	1	2	yes	—
319	C₂×#107	C₂²×C₅₃	1	4	yes	—
320	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
321	#423	C₂²×C₅₃	1	1	yes	characteristic
322	#419	C₂²×C₅₃	1	4	yes	—
323	C₂×#103	C₂²×C₅₃	1	2	yes	—
324	#408	C₂²×C₅₃	1	2	yes	—
325	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
326	C₅₃	#61286	1	1	yes	characteristic, direct product factor, Sylow
327	C₂×#41	C₂³×C₅₃	1	2	yes	—
328	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
329	C₂×Q₁₆	—	2	4	no	—
330	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
331	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
332	#131	—	2	8	no	—
333	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
334	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
335	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
336	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
337	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
338	C₄⋊Q₈	—	2	4	no	—
339	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
340	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
341	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
342	#127	D₄×C₅₃	1	1	yes	characteristic
343	#126	—	2	8	no	—
344	#126	—	2	8	no	—
345	#125	—	2	4	no	—
346	#125	—	2	8	no	—
347	#125	C₂³×C₅₃	1	4	yes	—
348	#124	—	2	4	no	—
349	#124	—	2	4	no	—
350	#124	D₄×C₅₃	1	2	yes	—
351	#123	—	2	8	no	—
352	#123	—	2	8	no	—
353	C₂²⋊Q₈	—	2	2	no	—
354	C₂²⋊Q₈	—	2	2	no	—
355	C₂²⋊Q₈	—	2	8	no	—
356	C₂²⋊Q₈	C₂³×C₅₃	1	8	yes	—
357	C₂²⋊Q₈	C₂³×C₅₃	1	8	yes	—
358	#121	D₄×C₅₃	1	2	yes	—
359	#121	D₄×C₅₃	1	2	yes	—
360	C₄×Q₈	—	2	4	no	—
361	C₄×Q₈	—	2	4	no	—
362	C₄×Q₈	—	2	2	no	—
363	C₄×Q₈	—	2	2	no	—
364	C₄×Q₈	—	2	2	no	—
365	C₄×Q₈	—	2	2	no	—
366	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
367	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
368	C₄×D₄	—	2	8	no	—
369	C₄×D₄	—	2	4	no	—
370	C₄×D₄	—	2	4	no	—
371	C₄×D₄	C₂³×C₅₃	1	2	yes	—
372	C₄×D₄	C₂³×C₅₃	1	2	yes	—
373	#117	—	2	8	no	—
374	#117	C₂³×C₅₃	1	4	yes	—
375	#117	C₂³×C₅₃	1	8	yes	—
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
378	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
379	C₂×C₄²	—	2	2	no	—
380	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
381	#107	C₂³×C₅₃	1	2	yes	—
382	#107	C₂³×C₅₃	1	2	yes	—
383	#107	C₂³×C₅₃	1	2	yes	—
384	#107	C₂³×C₅₃	1	8	yes	—
385	#107	C₂³×C₅₃	1	2	yes	—
386	#107	C₂³×C₅₃	1	2	yes	—
387	#107	C₂³×C₅₃	1	2	yes	—
388	#106	C₂³×C₅₃	1	4	yes	—
389	#106	C₂³×C₅₃	1	8	yes	—
390	#106	C₂³×C₅₃	1	4	yes	—
391	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
392	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
393	#103	C₂³×C₅₃	1	4	yes	—
394	#103	C₂³×C₅₃	1	4	yes	—
395	#103	C₂³×C₅₃	1	8	yes	—
396	#103	C₂³×C₅₃	1	8	yes	—
397	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
398	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
399	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
400	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
401	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
402	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
403	#41	—	2	4	no	—
404	#41	—	2	4	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	—	2	2	no	—
408	C₂×Q₈	—	2	4	no	—
409	C₂×Q₈	—	2	4	no	—
410	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
411	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
412	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
413	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
414	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
415	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	—	2	8	no	—
419	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
420	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
421	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
422	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
423	Q₁₆	—	2	4	no	—
424	Q₁₆	—	2	4	no	—
425	M₁₆	—	2	4	no	—
426	M₁₆	—	2	8	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	8	no	—
429	C₂×C₈	#41×C₅₃	1	4	yes	—
430	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
431	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
432	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
433	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
434	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
435	C₄⋊C₄	—	2	4	no	—
436	C₄⋊C₄	—	2	4	no	—
437	C₄⋊C₄	—	2	2	no	—
438	C₄⋊C₄	—	2	8	no	—
439	C₄⋊C₄	—	2	4	no	—
440	C₄⋊C₄	—	2	8	no	—
441	C₄⋊C₄	—	2	2	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	2	8	no	—
445	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
446	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
447	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
448	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
449	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
450	C₂²⋊C₄	—	2	8	no	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₂²⋊C₄	—	2	8	no	—
453	C₂²⋊C₄	—	2	8	no	—
454	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
455	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
456	C₄²	—	2	4	no	—
457	C₄²	—	2	4	no	—
458	C₄²	—	2	8	no	—
459	C₄²	—	2	4	no	—
460	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
461	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
462	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
463	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
464	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
465	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
466	Q₈	—	2	4	no	—
467	Q₈	—	2	4	no	—
468	Q₈	—	2	4	no	—
469	Q₈	—	4	4	no	—
470	Q₈	—	4	4	no	—
471	Q₈	—	8	2	no	—
472	D₄	—	2	8	no	—
473	D₄	—	2	4	no	—
474	D₄	—	2	4	no	—
475	D₄	—	4	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	4	1	no	—
484	C₂×C₄	—	4	4	no	—
485	C₂×C₄	—	8	2	no	—
486	C₂×C₄	—	8	8	no	—
487	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
488	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
489	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
490	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
491	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
492	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
493	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
494	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
495	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
496	C₈	—	2	4	no	—
497	C₈	—	2	1	no	—
498	C₈	—	2	1	no	—
499	C₈	C₂×#41×C₅₃	1	2	yes	—
500	C₈	C₂×#41×C₅₃	1	2	yes	—
501	C₂²	—	2	1	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	2	4	no	—
504	C₂²	—	2	4	no	—
505	C₂²	—	2	4	no	—
506	C₂²	C₅₃×#547	1	2	yes	—
507	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
508	C₄	—	2	4	no	—
509	C₄	—	2	1	no	—
510	C₄	—	2	1	no	—
511	C₄	—	2	1	no	—
512	C₄	—	4	4	no	—
513	C₄	—	4	4	no	—
514	C₄	—	4	4	no	—
515	C₄	—	4	4	no	—
516	C₄	C₅₃×#547	1	2	yes	—
517	C₄	C₅₃×#578	1	2	yes	—
518	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
519	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
520	C₂	—	2	2	no	—
521	C₂	—	4	4	no	—
522	C₂	C₂×C₅₃×#547	1	1	yes	characteristic
523	C₂	C₂×C₅₃×#578	1	1	yes	characteristic
524	C₂	C₅₃×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₅₃×#532	C₂²
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
2. C₅₃×#3356	C₂	semidirect
3. C₅₃×#3352	C₂	semidirect
4. C₅₃×#3342	C₂	semidirect
5. C₅₃×#3338	C₂	semidirect
6. C₅₃×#3336	C₂	semidirect
7. C₅₃×#3160	C₂	semidirect
8. C₅₃×#3097	C₂	semidirect
9. C₅₃×#3078	C₂	semidirect
10. C₅₃×#3059	C₂	semidirect
11. C₂×C₄×Q₁₆×C₅₃	C₂	semidirect
13. C₅₃×#2913	C₂	semidirect
238. #61286	C₅₃	direct (N × H)
259. #3356	C₂×C₅₃	semidirect
260. #3352	C₂×C₅₃	semidirect
261. #3342	C₂×C₅₃	semidirect
262. #3338	C₂×C₅₃	semidirect
263. #3336	C₂×C₅₃	semidirect
264. #3160	C₂×C₅₃	semidirect
265. #3097	C₂×C₅₃	semidirect
266. #3078	C₂×C₅₃	semidirect
267. #3059	C₂×C₅₃	semidirect
268. C₂×C₄×Q₁₆	C₂×C₅₃	semidirect
270. #2913	C₂×C₅₃	semidirect
326. C₅₃	#61286	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×#3507	2
C₅₃×#3356	2
C₅₃×#3352	2
C₅₃×#3342	2
C₅₃×#3338	2
C₅₃×#3336	2
C₅₃×#3160	2
C₅₃×#3097	2
C₅₃×#3078	2
C₅₃×#3059	2
C₂×C₄×Q₁₆×C₅₃	2
C₅₃×#2931	2
C₅₃×#2913	2
#61286	53

Subgroups — C53×#61286 #27268798

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#61286 #27268798