Subgroups — C₅₃×#61890 #27269402

Order 13568.

2042 conjugacy classes of subgroups.

The subgroup lattice has 676 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	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3399	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3397	C₂	1	4	yes	maximal, complemented
5	C₅₃×#3368	C₂	1	4	yes	maximal, complemented
6	C₅₃×#3366	C₂	1	4	yes	maximal, complemented
7	C₅₃×#3217	C₂	1	4	yes	maximal, complemented
8	C₅₃×#3208	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#3205	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#3114	C₂	1	4	yes	maximal, complemented
11	C₅₃×#3101	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#3100	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#2983	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₅₃	—	2	1	no	—
15	C₅₃×#559	C₂²	1	2	yes	complemented
16	C₅₃×#557	C₂²	1	2	yes	complemented
17	C₅₃×#556	—	2	2	no	—
18	C₅₃×#556	C₂²	1	2	yes	complemented
19	C₅₃×#554	C₂²	1	2	yes	complemented
20	C₅₃×#552	—	2	2	no	—
21	C₅₃×#550	—	2	2	no	—
22	C₅₃×#550	—	2	2	no	—
23	C₅₃×#550	C₂²	1	4	yes	—
24	C₅₃×#550	C₂²	1	4	yes	—
25	C₅₃×#539	—	2	2	no	—
26	C₅₃×#536	—	2	1	no	—
27	C₅₃×#533	C₂²	1	1	yes	characteristic
28	C₅₃×#532	C₂²	1	1	yes	characteristic
29	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#127×C₅₃	C₂²	1	2	yes	—
31	C₅₃×#529	C₂²	1	4	yes	—
32	C₂×#121×C₅₃	—	2	1	no	—
33	C₂×#121×C₅₃	—	2	2	no	—
34	C₅₃×#520	—	2	1	no	—
35	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
38	C₅₃×#496	C₂²	1	8	yes	complemented
39	C₅₃×#493	C₂²	1	2	yes	complemented
40	C₅₃×#492	C₂²	1	2	yes	complemented
41	C₅₃×#490	C₂²	1	8	yes	complemented
42	C₅₃×#486	C₂²	1	2	yes	complemented
43	C₅₃×#486	C₂²	1	2	yes	complemented
44	C₅₃×#483	C₂²	1	2	yes	—
45	C₅₃×#480	C₂²	1	2	yes	—
46	C₅₃×#478	C₂²	1	8	yes	complemented
47	C₅₃×#476	C₂²	1	2	yes	complemented
48	C₅₃×#474	C₂²	1	2	yes	complemented
49	C₅₃×#463	C₂²	1	8	yes	complemented
50	C₅₃×#461	C₂²	1	2	yes	complemented
51	C₅₃×#460	C₂²	1	4	yes	complemented
52	C₅₃×#460	C₂²	1	8	yes	complemented
53	C₅₃×#459	C₂²	1	4	yes	complemented
54	C₅₃×#459	C₂²	1	8	yes	complemented
55	C₅₃×#459	C₂²	1	2	yes	complemented
56	C₅₃×#450	C₂²	1	4	yes	—
57	C₅₃×#450	C₂²	1	2	yes	—
58	C₅₃×#447	C₂²	1	4	yes	—
59	C₅₃×#447	C₂²	1	2	yes	—
60	C₅₃×#446	C₂²	1	4	yes	complemented
61	C₅₃×#446	C₂²	1	4	yes	complemented
62	C₅₃×#442	C₂²	1	8	yes	complemented
63	C₅₃×#440	C₂²	1	8	yes	complemented
64	C₄×SD₁₆×C₅₃	C₂²	1	4	yes	complemented
65	C₄×D₈×C₅₃	C₂²	1	4	yes	complemented
66	C₅₃×#432	C₂²	1	4	yes	—
67	C₅₃×#431	C₂²	1	1	yes	characteristic
68	C₅₃×#431	C₂²	1	1	yes	characteristic
69	C₂×#41×C₅₃	—	2	4	no	—
70	C₂×#41×C₅₃	—	2	4	no	—
71	C₂×#41×C₅₃	—	2	1	no	—
72	C₂×#41×C₅₃	—	4	8	no	—
73	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
74	C₂²×D₄×C₅₃	—	2	2	no	—
75	C₂²×D₄×C₅₃	—	2	1	no	—
76	C₂²×D₄×C₅₃	—	2	2	no	—
77	C₂²×D₄×C₅₃	—	2	1	no	—
78	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
79	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
80	C₂³×C₄×C₅₃	—	2	2	no	—
81	C₂³×C₄×C₅₃	—	2	1	no	—
82	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	—
83	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
84	C₂×D₈×C₅₃	C₂³	1	8	yes	—
85	C₂×D₈×C₅₃	C₂³	1	4	yes	—
86	C₄⋊Q₈×C₅₃	—	2	2	no	—
87	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
90	#127×C₅₃	—	2	4	no	—
91	#127×C₅₃	—	2	1	no	—
92	#127×C₅₃	—	2	1	no	—
93	#127×C₅₃	C₂³	1	2	yes	—
94	#127×C₅₃	C₂³	1	2	yes	—
95	#127×C₅₃	C₂³	1	8	yes	—
96	#127×C₅₃	C₂³	1	1	yes	characteristic
97	#126×C₅₃	—	2	4	no	—
98	#126×C₅₃	—	2	4	no	—
99	#125×C₅₃	—	2	4	no	—
100	#125×C₅₃	—	2	1	no	—
101	#125×C₅₃	C₂³	1	4	yes	—
102	#124×C₅₃	—	2	1	no	—
103	#124×C₅₃	—	2	2	no	—
104	#124×C₅₃	C₂³	1	4	yes	—
105	#123×C₅₃	—	2	4	no	—
106	#123×C₅₃	—	2	4	no	—
107	C₂²⋊Q₈×C₅₃	—	2	4	no	—
108	C₂²⋊Q₈×C₅₃	—	2	2	no	—
109	C₂²⋊Q₈×C₅₃	—	2	2	no	—
110	C₂²⋊Q₈×C₅₃	—	2	1	no	—
111	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
112	#121×C₅₃	—	2	4	no	—
113	#121×C₅₃	—	2	4	no	—
114	#121×C₅₃	—	2	8	no	—
115	#121×C₅₃	—	2	8	no	—
116	#121×C₅₃	—	2	1	no	—
117	#121×C₅₃	C₂³	1	2	yes	—
118	C₄×Q₈×C₅₃	—	2	2	no	—
119	C₄×Q₈×C₅₃	—	2	1	no	—
120	C₄×Q₈×C₅₃	D₄	1	2	yes	—
121	C₄×Q₈×C₅₃	D₄	1	2	yes	—
122	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
123	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
124	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
125	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
126	C₄×D₄×C₅₃	—	2	2	no	—
127	C₄×D₄×C₅₃	—	2	4	no	—
128	C₄×D₄×C₅₃	—	2	1	no	—
129	C₄×D₄×C₅₃	—	2	1	no	—
130	C₄×D₄×C₅₃	—	2	1	no	—
131	C₄×D₄×C₅₃	—	2	4	no	—
132	C₄×D₄×C₅₃	C₂³	1	2	yes	—
133	C₄×D₄×C₅₃	C₂³	1	2	yes	—
134	C₄×D₄×C₅₃	C₂³	1	8	yes	—
135	C₄×D₄×C₅₃	C₂³	1	4	yes	—
136	C₄×D₄×C₅₃	C₂³	1	2	yes	—
137	#117×C₅₃	—	2	2	no	—
138	#117×C₅₃	—	2	4	no	—
139	#117×C₅₃	—	2	1	no	—
140	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
141	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
142	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
143	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
144	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
145	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
146	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
147	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
148	#107×C₅₃	C₂³	1	4	yes	—
149	#106×C₅₃	C₂³	1	4	yes	—
150	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
151	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
152	C₄⋊C₈×C₅₃	C₂³	1	8	yes	—
153	#103×C₅₃	C₂³	1	4	yes	—
154	#102×C₅₃	C₂³	1	8	yes	—
155	#102×C₅₃	C₂³	1	8	yes	—
156	#102×C₅₃	C₂³	1	4	yes	—
157	#102×C₅₃	C₂³	1	4	yes	—
158	#102×C₅₃	C₂³	1	4	yes	—
159	C₂²⋊C₈×C₅₃	C₂³	1	2	yes	—
160	C₂²⋊C₈×C₅₃	C₂³	1	2	yes	—
161	#97×C₅₃	C₂³	1	8	yes	—
162	C₄×C₈×C₅₃	C₂³	1	2	yes	—
163	C₄×C₈×C₅₃	C₂³	1	2	yes	—
164	C₂⁴×C₅₃	—	2	2	no	—
165	C₂⁴×C₅₃	—	2	1	no	—
166	#41×C₅₃	—	4	16	no	—
167	#41×C₅₃	—	4	16	no	—
168	#41×C₅₃	—	4	4	no	—
169	C₂×Q₈×C₅₃	—	2	4	no	—
170	C₂×Q₈×C₅₃	—	2	1	no	—
171	C₂×Q₈×C₅₃	—	2	1	no	—
172	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
173	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
174	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
175	C₂×D₄×C₅₃	—	2	2	no	—
176	C₂×D₄×C₅₃	—	2	1	no	—
177	C₂×D₄×C₅₃	—	2	2	no	—
178	C₂×D₄×C₅₃	—	2	1	no	—
179	C₂×D₄×C₅₃	—	2	8	no	—
180	C₂×D₄×C₅₃	—	2	2	no	—
181	C₂×D₄×C₅₃	—	2	2	no	—
182	C₂×D₄×C₅₃	—	2	1	no	—
183	C₂×D₄×C₅₃	—	2	8	no	—
184	C₂×D₄×C₅₃	—	2	2	no	—
185	C₂×D₄×C₅₃	—	2	2	no	—
186	C₂×D₄×C₅₃	—	4	2	no	—
187	C₂×D₄×C₅₃	—	4	4	no	—
188	C₂×D₄×C₅₃	—	4	8	no	—
189	C₂×D₄×C₅₃	—	4	4	no	—
190	C₂×D₄×C₅₃	—	8	4	no	—
191	C₂×D₄×C₅₃	—	8	2	no	—
192	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
193	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
194	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
195	C₂²×C₄×C₅₃	—	2	1	no	—
196	C₂²×C₄×C₅₃	—	2	2	no	—
197	C₂²×C₄×C₅₃	—	2	8	no	—
198	C₂²×C₄×C₅₃	—	2	2	no	—
199	C₂²×C₄×C₅₃	—	2	2	no	—
200	C₂²×C₄×C₅₃	—	2	4	no	—
201	C₂²×C₄×C₅₃	—	4	8	no	—
202	C₂²×C₄×C₅₃	—	4	4	no	—
203	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
204	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
205	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
206	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
208	SD₁₆×C₅₃	—	4	4	no	—
209	SD₁₆×C₅₃	—	4	8	no	—
210	D₈×C₅₃	—	4	8	no	—
211	D₈×C₅₃	—	4	4	no	—
212	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
213	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
214	C₄⋊C₄×C₅₃	—	2	1	no	—
215	C₄⋊C₄×C₅₃	—	2	4	no	—
216	C₄⋊C₄×C₅₃	—	2	1	no	—
217	C₄⋊C₄×C₅₃	—	2	1	no	—
218	C₄⋊C₄×C₅₃	—	2	2	no	—
219	C₄⋊C₄×C₅₃	—	2	4	no	—
220	C₄⋊C₄×C₅₃	—	2	1	no	—
221	C₄⋊C₄×C₅₃	—	2	1	no	—
222	C₄⋊C₄×C₅₃	—	2	2	no	—
223	C₄⋊C₄×C₅₃	—	2	4	no	—
224	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
225	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
226	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
227	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
228	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
229	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
230	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
231	C₂²⋊C₄×C₅₃	—	2	2	no	—
232	C₂²⋊C₄×C₅₃	—	2	8	no	—
233	C₂²⋊C₄×C₅₃	—	2	2	no	—
234	C₂²⋊C₄×C₅₃	—	2	2	no	—
235	C₂²⋊C₄×C₅₃	—	2	2	no	—
236	C₂²⋊C₄×C₅₃	—	2	4	no	—
237	C₂²⋊C₄×C₅₃	—	2	4	no	—
238	C₂²⋊C₄×C₅₃	—	2	2	no	—
239	C₂²⋊C₄×C₅₃	—	2	4	no	—
240	C₄²×C₅₃	—	2	4	no	—
241	C₄²×C₅₃	—	2	2	no	—
242	C₄²×C₅₃	C₂×D₄	1	4	yes	—
243	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
244	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
245	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
246	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
247	C₄²×C₅₃	C₂⁴	1	4	yes	—
248	C₂³×C₅₃	—	2	2	no	—
249	C₂³×C₅₃	—	2	2	no	—
250	C₂³×C₅₃	—	2	4	no	—
251	C₂³×C₅₃	—	4	2	no	—
252	C₂³×C₅₃	—	4	4	no	—
253	C₂³×C₅₃	—	4	2	no	—
254	C₂³×C₅₃	—	4	4	no	—
255	C₂³×C₅₃	—	4	4	no	—
256	C₂³×C₅₃	—	4	2	no	—
257	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
258	Q₈×C₅₃	—	2	4	no	—
259	Q₈×C₅₃	—	2	4	no	—
260	Q₈×C₅₃	—	2	4	no	—
261	Q₈×C₅₃	—	4	2	no	—
262	Q₈×C₅₃	—	8	1	no	—
263	D₄×C₅₃	—	2	2	no	—
264	D₄×C₅₃	—	2	8	no	—
265	D₄×C₅₃	—	2	2	no	—
266	D₄×C₅₃	—	2	4	no	—
267	D₄×C₅₃	—	4	2	no	—
268	D₄×C₅₃	—	4	4	no	—
269	D₄×C₅₃	—	4	16	no	—
270	D₄×C₅₃	—	4	4	no	—
271	D₄×C₅₃	—	4	4	no	—
272	D₄×C₅₃	—	4	4	no	—
273	D₄×C₅₃	—	4	2	no	—
274	D₄×C₅₃	—	8	1	no	—
275	D₄×C₅₃	—	8	8	no	—
276	D₄×C₅₃	—	8	2	no	—
277	D₄×C₅₃	—	8	2	no	—
278	C₂×C₄×C₅₃	—	2	4	no	—
279	C₂×C₄×C₅₃	—	2	2	no	—
280	C₂×C₄×C₅₃	—	2	2	no	—
281	C₂×C₄×C₅₃	—	2	2	no	—
282	C₂×C₄×C₅₃	—	2	2	no	—
283	C₂×C₄×C₅₃	—	2	2	no	—
284	C₂×C₄×C₅₃	—	4	2	no	—
285	C₂×C₄×C₅₃	—	4	1	no	—
286	C₂×C₄×C₅₃	—	4	16	no	—
287	C₂×C₄×C₅₃	—	4	2	no	—
288	C₂×C₄×C₅₃	—	4	4	no	—
289	C₂×C₄×C₅₃	—	4	4	no	—
290	C₂×C₄×C₅₃	—	4	4	no	—
291	C₂×C₄×C₅₃	—	8	4	no	—
292	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
293	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
294	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
295	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
296	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
297	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
298	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
300	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
301	C₈×C₅₃	—	2	4	no	—
302	C₈×C₅₃	—	2	4	no	—
303	#61890	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
304	C₂²×C₅₃	—	2	1	no	—
305	C₂²×C₅₃	—	2	1	no	—
306	C₂²×C₅₃	—	4	2	no	—
307	C₂²×C₅₃	—	4	2	no	—
308	C₂²×C₅₃	—	4	4	no	—
309	C₂²×C₅₃	—	4	2	no	—
310	C₂²×C₅₃	—	4	4	no	—
311	C₂²×C₅₃	—	4	2	no	—
312	C₂²×C₅₃	—	4	4	no	—
313	C₂²×C₅₃	—	4	2	no	—
314	C₂²×C₅₃	—	4	2	no	—
315	C₂²×C₅₃	—	8	8	no	—
316	C₂²×C₅₃	—	8	4	no	—
317	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
318	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
319	C₄×C₅₃	—	2	4	no	—
320	C₄×C₅₃	—	2	1	no	—
321	C₄×C₅₃	—	2	1	no	—
322	C₄×C₅₃	—	2	1	no	—
323	C₄×C₅₃	—	2	4	no	—
324	C₄×C₅₃	—	4	2	no	—
325	C₄×C₅₃	—	4	2	no	—
326	C₄×C₅₃	—	4	2	no	—
327	C₄×C₅₃	—	4	2	no	—
328	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
329	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
330	C₄×C₅₃	#585	1	2	yes	—
331	#3507	C₂×C₅₃	1	1	yes	characteristic, complemented
332	C₂×#550	C₂×C₅₃	1	1	yes	characteristic, complemented
333	#3399	C₂×C₅₃	1	4	yes	complemented
334	#3397	C₂×C₅₃	1	4	yes	complemented
335	#3368	C₂×C₅₃	1	4	yes	complemented
336	#3366	C₂×C₅₃	1	4	yes	complemented
337	#3217	C₂×C₅₃	1	4	yes	complemented
338	#3208	C₂×C₅₃	1	1	yes	characteristic, complemented
339	#3205	C₂×C₅₃	1	1	yes	characteristic, complemented
340	#3114	C₂×C₅₃	1	4	yes	complemented
341	#3101	C₂×C₅₃	1	1	yes	characteristic, complemented
342	#3100	C₂×C₅₃	1	1	yes	characteristic, complemented
343	#2983	C₂×C₅₃	1	1	yes	characteristic, complemented
344	C₂×C₅₃	—	2	2	no	—
345	C₂×C₅₃	—	8	2	no	—
346	C₂×C₅₃	—	8	2	no	—
347	C₂×C₅₃	—	8	4	no	—
348	C₂×C₅₃	#3487	1	1	yes	characteristic
349	C₂×C₅₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
350	C₂×C₅₃	#3585	1	1	yes	characteristic
351	C₂²×#41	—	2	1	no	—
352	#559	C₂²×C₅₃	1	2	yes	complemented
353	#557	C₂²×C₅₃	1	2	yes	complemented
354	#556	—	2	2	no	—
355	#556	C₂²×C₅₃	1	2	yes	complemented
356	#554	C₂²×C₅₃	1	2	yes	complemented
357	#552	—	2	2	no	—
358	#550	—	2	2	no	—
359	#550	—	2	2	no	—
360	#550	C₂²×C₅₃	1	4	yes	—
361	#550	C₂²×C₅₃	1	4	yes	—
362	#539	—	2	2	no	—
363	#536	—	2	1	no	—
364	#533	C₂²×C₅₃	1	1	yes	characteristic
365	#532	C₂²×C₅₃	1	1	yes	characteristic
366	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
367	C₂×#127	C₂²×C₅₃	1	2	yes	—
368	#529	C₂²×C₅₃	1	4	yes	—
369	C₂×#121	—	2	1	no	—
370	C₂×#121	—	2	2	no	—
371	#520	—	2	1	no	—
372	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
373	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
374	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
375	#496	C₂²×C₅₃	1	8	yes	complemented
376	#493	C₂²×C₅₃	1	2	yes	complemented
377	#492	C₂²×C₅₃	1	2	yes	complemented
378	#490	C₂²×C₅₃	1	8	yes	complemented
379	#486	C₂²×C₅₃	1	2	yes	complemented
380	#486	C₂²×C₅₃	1	2	yes	complemented
381	#483	C₂²×C₅₃	1	2	yes	—
382	#480	C₂²×C₅₃	1	2	yes	—
383	#478	C₂²×C₅₃	1	8	yes	complemented
384	#476	C₂²×C₅₃	1	2	yes	complemented
385	#474	C₂²×C₅₃	1	2	yes	complemented
386	#463	C₂²×C₅₃	1	8	yes	complemented
387	#461	C₂²×C₅₃	1	2	yes	complemented
388	#460	C₂²×C₅₃	1	8	yes	complemented
389	#460	C₂²×C₅₃	1	4	yes	complemented
390	#459	C₂²×C₅₃	1	2	yes	complemented
391	#459	C₂²×C₅₃	1	4	yes	complemented
392	#459	C₂²×C₅₃	1	8	yes	complemented
393	#450	C₂²×C₅₃	1	2	yes	—
394	#450	C₂²×C₅₃	1	4	yes	—
395	#447	C₂²×C₅₃	1	2	yes	—
396	#447	C₂²×C₅₃	1	4	yes	—
397	#446	C₂²×C₅₃	1	4	yes	complemented
398	#446	C₂²×C₅₃	1	4	yes	complemented
399	#442	C₂²×C₅₃	1	8	yes	complemented
400	#440	C₂²×C₅₃	1	8	yes	complemented
401	C₄×SD₁₆	C₂²×C₅₃	1	4	yes	complemented
402	C₄×D₈	C₂²×C₅₃	1	4	yes	complemented
403	#432	C₂²×C₅₃	1	4	yes	—
404	#431	C₂²×C₅₃	1	1	yes	characteristic
405	#431	C₂²×C₅₃	1	1	yes	characteristic
406	C₅₃	#61890	1	1	yes	characteristic, direct product factor, Sylow
407	C₂×#41	—	2	4	no	—
408	C₂×#41	—	2	4	no	—
409	C₂×#41	—	2	1	no	—
410	C₂×#41	—	4	8	no	—
411	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
412	C₂²×D₄	—	2	2	no	—
413	C₂²×D₄	—	2	1	no	—
414	C₂²×D₄	—	2	2	no	—
415	C₂²×D₄	—	2	1	no	—
416	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
417	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
418	C₂³×C₄	—	2	2	no	—
419	C₂³×C₄	—	2	1	no	—
420	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
421	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	—
422	C₂×D₈	C₂³×C₅₃	1	4	yes	—
423	C₂×D₈	C₂³×C₅₃	1	8	yes	—
424	C₄⋊Q₈	—	2	2	no	—
425	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
426	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
427	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
428	#127	—	2	4	no	—
429	#127	—	2	1	no	—
430	#127	—	2	1	no	—
431	#127	C₂³×C₅₃	1	2	yes	—
432	#127	C₂³×C₅₃	1	8	yes	—
433	#127	C₂³×C₅₃	1	2	yes	—
434	#127	C₂³×C₅₃	1	1	yes	characteristic
435	#126	—	2	4	no	—
436	#126	—	2	4	no	—
437	#125	—	2	4	no	—
438	#125	—	2	1	no	—
439	#125	C₂³×C₅₃	1	4	yes	—
440	#124	—	2	2	no	—
441	#124	—	2	1	no	—
442	#124	C₂³×C₅₃	1	4	yes	—
443	#123	—	2	4	no	—
444	#123	—	2	4	no	—
445	C₂²⋊Q₈	—	2	2	no	—
446	C₂²⋊Q₈	—	2	2	no	—
447	C₂²⋊Q₈	—	2	1	no	—
448	C₂²⋊Q₈	—	2	4	no	—
449	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
450	#121	—	2	1	no	—
451	#121	—	2	4	no	—
452	#121	—	2	4	no	—
453	#121	—	2	8	no	—
454	#121	—	2	8	no	—
455	#121	C₂³×C₅₃	1	2	yes	—
456	C₄×Q₈	—	2	2	no	—
457	C₄×Q₈	—	2	1	no	—
458	C₄×Q₈	D₄×C₅₃	1	2	yes	—
459	C₄×Q₈	D₄×C₅₃	1	2	yes	—
460	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
461	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
462	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
463	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
464	C₄×D₄	—	2	4	no	—
465	C₄×D₄	—	2	2	no	—
466	C₄×D₄	—	2	4	no	—
467	C₄×D₄	—	2	1	no	—
468	C₄×D₄	—	2	1	no	—
469	C₄×D₄	—	2	1	no	—
470	C₄×D₄	C₂³×C₅₃	1	2	yes	—
471	C₄×D₄	C₂³×C₅₃	1	2	yes	—
472	C₄×D₄	C₂³×C₅₃	1	8	yes	—
473	C₄×D₄	C₂³×C₅₃	1	4	yes	—
474	C₄×D₄	C₂³×C₅₃	1	2	yes	—
475	#117	—	2	1	no	—
476	#117	—	2	2	no	—
477	#117	—	2	4	no	—
478	C₂×C₄⋊C₄	—	2	2	no	—
479	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
480	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
481	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
482	C₂×C₂²⋊C₄	—	2	2	no	—
483	C₂×C₂²⋊C₄	—	2	1	no	—
484	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
485	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
486	#107	C₂³×C₅₃	1	4	yes	—
487	#106	C₂³×C₅₃	1	4	yes	—
488	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
489	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
490	C₄⋊C₈	C₂³×C₅₃	1	8	yes	—
491	#103	C₂³×C₅₃	1	4	yes	—
492	#102	C₂³×C₅₃	1	4	yes	—
493	#102	C₂³×C₅₃	1	4	yes	—
494	#102	C₂³×C₅₃	1	4	yes	—
495	#102	C₂³×C₅₃	1	8	yes	—
496	#102	C₂³×C₅₃	1	8	yes	—
497	C₂²⋊C₈	C₂³×C₅₃	1	2	yes	—
498	C₂²⋊C₈	C₂³×C₅₃	1	2	yes	—
499	#97	C₂³×C₅₃	1	8	yes	—
500	C₄×C₈	C₂³×C₅₃	1	2	yes	—
501	C₄×C₈	C₂³×C₅₃	1	2	yes	—
502	C₂⁴	—	2	2	no	—
503	C₂⁴	—	2	1	no	—
504	#41	—	4	16	no	—
505	#41	—	4	16	no	—
506	#41	—	4	4	no	—
507	C₂×Q₈	—	2	4	no	—
508	C₂×Q₈	—	2	1	no	—
509	C₂×Q₈	—	2	1	no	—
510	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
511	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
512	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
513	C₂×D₄	—	2	2	no	—
514	C₂×D₄	—	2	1	no	—
515	C₂×D₄	—	2	2	no	—
516	C₂×D₄	—	2	1	no	—
517	C₂×D₄	—	2	8	no	—
518	C₂×D₄	—	2	2	no	—
519	C₂×D₄	—	2	2	no	—
520	C₂×D₄	—	2	1	no	—
521	C₂×D₄	—	2	8	no	—
522	C₂×D₄	—	2	2	no	—
523	C₂×D₄	—	2	2	no	—
524	C₂×D₄	—	4	2	no	—
525	C₂×D₄	—	4	4	no	—
526	C₂×D₄	—	4	8	no	—
527	C₂×D₄	—	4	4	no	—
528	C₂×D₄	—	8	4	no	—
529	C₂×D₄	—	8	2	no	—
530	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
531	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
532	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
533	C₂²×C₄	—	2	1	no	—
534	C₂²×C₄	—	2	2	no	—
535	C₂²×C₄	—	2	8	no	—
536	C₂²×C₄	—	2	2	no	—
537	C₂²×C₄	—	2	2	no	—
538	C₂²×C₄	—	2	4	no	—
539	C₂²×C₄	—	4	8	no	—
540	C₂²×C₄	—	4	4	no	—
541	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
542	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
543	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
544	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
545	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
546	SD₁₆	—	4	4	no	—
547	SD₁₆	—	4	8	no	—
548	D₈	—	4	8	no	—
549	D₈	—	4	4	no	—
550	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
551	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
552	C₄⋊C₄	—	2	1	no	—
553	C₄⋊C₄	—	2	4	no	—
554	C₄⋊C₄	—	2	1	no	—
555	C₄⋊C₄	—	2	1	no	—
556	C₄⋊C₄	—	2	2	no	—
557	C₄⋊C₄	—	2	4	no	—
558	C₄⋊C₄	—	2	1	no	—
559	C₄⋊C₄	—	2	1	no	—
560	C₄⋊C₄	—	2	2	no	—
561	C₄⋊C₄	—	2	4	no	—
562	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
563	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
564	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
565	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
566	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
567	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
568	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
569	C₂²⋊C₄	—	2	2	no	—
570	C₂²⋊C₄	—	2	8	no	—
571	C₂²⋊C₄	—	2	2	no	—
572	C₂²⋊C₄	—	2	2	no	—
573	C₂²⋊C₄	—	2	2	no	—
574	C₂²⋊C₄	—	2	4	no	—
575	C₂²⋊C₄	—	2	4	no	—
576	C₂²⋊C₄	—	2	2	no	—
577	C₂²⋊C₄	—	2	4	no	—
578	C₄²	—	2	4	no	—
579	C₄²	—	2	2	no	—
580	C₄²	C₂×D₄×C₅₃	1	4	yes	—
581	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
582	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
583	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
584	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
585	C₄²	C₂⁴×C₅₃	1	4	yes	—
586	C₂³	—	2	2	no	—
587	C₂³	—	2	2	no	—
588	C₂³	—	2	4	no	—
589	C₂³	—	4	2	no	—
590	C₂³	—	4	4	no	—
591	C₂³	—	4	2	no	—
592	C₂³	—	4	4	no	—
593	C₂³	—	4	4	no	—
594	C₂³	—	4	2	no	—
595	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
596	Q₈	—	2	4	no	—
597	Q₈	—	2	4	no	—
598	Q₈	—	2	4	no	—
599	Q₈	—	4	2	no	—
600	Q₈	—	8	1	no	—
601	D₄	—	2	4	no	—
602	D₄	—	2	2	no	—
603	D₄	—	2	8	no	—
604	D₄	—	2	2	no	—
605	D₄	—	4	2	no	—
606	D₄	—	4	4	no	—
607	D₄	—	4	16	no	—
608	D₄	—	4	4	no	—
609	D₄	—	4	4	no	—
610	D₄	—	4	4	no	—
611	D₄	—	4	2	no	—
612	D₄	—	8	1	no	—
613	D₄	—	8	8	no	—
614	D₄	—	8	2	no	—
615	D₄	—	8	2	no	—
616	C₂×C₄	—	2	4	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	4	2	no	—
623	C₂×C₄	—	4	1	no	—
624	C₂×C₄	—	4	16	no	—
625	C₂×C₄	—	4	2	no	—
626	C₂×C₄	—	4	4	no	—
627	C₂×C₄	—	4	4	no	—
628	C₂×C₄	—	4	4	no	—
629	C₂×C₄	—	8	4	no	—
630	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
631	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
632	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
633	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
634	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
635	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
636	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
637	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
638	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
639	C₈	—	2	4	no	—
640	C₈	—	2	4	no	—
641	C₂²	—	2	1	no	—
642	C₂²	—	2	1	no	—
643	C₂²	—	4	2	no	—
644	C₂²	—	4	4	no	—
645	C₂²	—	4	2	no	—
646	C₂²	—	4	4	no	—
647	C₂²	—	4	2	no	—
648	C₂²	—	4	2	no	—
649	C₂²	—	4	2	no	—
650	C₂²	—	4	2	no	—
651	C₂²	—	4	4	no	—
652	C₂²	—	8	8	no	—
653	C₂²	—	8	4	no	—
654	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
655	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
656	C₄	—	2	1	no	—
657	C₄	—	2	4	no	—
658	C₄	—	2	4	no	—
659	C₄	—	2	1	no	—
660	C₄	—	2	1	no	—
661	C₄	—	4	2	no	—
662	C₄	—	4	2	no	—
663	C₄	—	4	2	no	—
664	C₄	—	4	2	no	—
665	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
666	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
667	C₄	C₅₃×#585	1	2	yes	—
668	C₂	—	2	2	no	—
669	C₂	—	8	2	no	—
670	C₂	—	8	4	no	—
671	C₂	—	8	2	no	—
672	C₂	C₅₃×#3487	1	1	yes	characteristic
673	C₂	C₂²×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
674	C₂	C₅₃×#3585	1	1	yes	characteristic

	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 ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3507	C₂	semidirect
2. C₂×C₅₃×#550	C₂	semidirect
3. C₅₃×#3399	C₂	semidirect
4. C₅₃×#3397	C₂	semidirect
5. C₅₃×#3368	C₂	semidirect
6. C₅₃×#3366	C₂	semidirect
7. C₅₃×#3217	C₂	semidirect
8. C₅₃×#3208	C₂	semidirect
9. C₅₃×#3205	C₂	semidirect
10. C₅₃×#3114	C₂	semidirect
11. C₅₃×#3101	C₂	semidirect
12. C₅₃×#3100	C₂	semidirect
13. C₅₃×#2983	C₂	semidirect
15. C₅₃×#559	C₂²	semidirect
16. C₅₃×#557	C₂²	semidirect
18. C₅₃×#556	C₂²	semidirect
19. C₅₃×#554	C₂²	semidirect
38. C₅₃×#496	C₂²	semidirect
39. C₅₃×#493	C₂²	semidirect
40. C₅₃×#492	C₂²	semidirect
41. C₅₃×#490	C₂²	semidirect
42. C₅₃×#486	C₂²	semidirect
43. C₅₃×#486	C₂²	semidirect
46. C₅₃×#478	C₂²	semidirect
47. C₅₃×#476	C₂²	semidirect
48. C₅₃×#474	C₂²	semidirect
49. C₅₃×#463	C₂²	semidirect
50. C₅₃×#461	C₂²	semidirect
51. C₅₃×#460	C₂²	semidirect
52. C₅₃×#460	C₂²	semidirect
53. C₅₃×#459	C₂²	semidirect
54. C₅₃×#459	C₂²	semidirect
55. C₅₃×#459	C₂²	semidirect
60. C₅₃×#446	C₂²	semidirect
61. C₅₃×#446	C₂²	semidirect
62. C₅₃×#442	C₂²	semidirect
63. C₅₃×#440	C₂²	semidirect
64. C₄×SD₁₆×C₅₃	C₂²	semidirect
65. C₄×D₈×C₅₃	C₂²	semidirect
303. #61890	C₅₃	direct (N × H)
331. #3507	C₂×C₅₃	semidirect
332. C₂×#550	C₂×C₅₃	semidirect
333. #3399	C₂×C₅₃	semidirect
334. #3397	C₂×C₅₃	semidirect
335. #3368	C₂×C₅₃	semidirect
336. #3366	C₂×C₅₃	semidirect
337. #3217	C₂×C₅₃	semidirect
338. #3208	C₂×C₅₃	semidirect
339. #3205	C₂×C₅₃	semidirect
340. #3114	C₂×C₅₃	semidirect
341. #3101	C₂×C₅₃	semidirect
342. #3100	C₂×C₅₃	semidirect
343. #2983	C₂×C₅₃	semidirect
352. #559	C₂²×C₅₃	semidirect
353. #557	C₂²×C₅₃	semidirect
355. #556	C₂²×C₅₃	semidirect
356. #554	C₂²×C₅₃	semidirect
375. #496	C₂²×C₅₃	semidirect
376. #493	C₂²×C₅₃	semidirect
377. #492	C₂²×C₅₃	semidirect
378. #490	C₂²×C₅₃	semidirect
379. #486	C₂²×C₅₃	semidirect
380. #486	C₂²×C₅₃	semidirect
383. #478	C₂²×C₅₃	semidirect
384. #476	C₂²×C₅₃	semidirect
385. #474	C₂²×C₅₃	semidirect
386. #463	C₂²×C₅₃	semidirect
387. #461	C₂²×C₅₃	semidirect
388. #460	C₂²×C₅₃	semidirect
389. #460	C₂²×C₅₃	semidirect
390. #459	C₂²×C₅₃	semidirect
391. #459	C₂²×C₅₃	semidirect
392. #459	C₂²×C₅₃	semidirect
397. #446	C₂²×C₅₃	semidirect
398. #446	C₂²×C₅₃	semidirect
399. #442	C₂²×C₅₃	semidirect
400. #440	C₂²×C₅₃	semidirect
401. C₄×SD₁₆	C₂²×C₅₃	semidirect
402. C₄×D₈	C₂²×C₅₃	semidirect
406. C₅₃	#61890	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
C₂×C₅₃×#550	2
C₅₃×#3399	2
C₅₃×#3397	2
C₅₃×#3368	2
C₅₃×#3366	2
C₅₃×#3217	2
C₅₃×#3208	2
C₅₃×#3205	2
C₅₃×#3114	2
C₅₃×#3101	2
C₅₃×#3100	2
C₅₃×#2983	2
#61890	53

Subgroups — C53×#61890 #27269402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#61890 #27269402