Subgroups — C₅₃×#60724 #27268236

Order 13568.

1874 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3512	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3510	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3183	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3179	C₂	1	4	yes	maximal, complemented
5	C₅₃×#3178	C₂	1	4	yes	maximal, complemented
6	C₅₃×#3168	C₂	1	4	yes	maximal, complemented
7	C₅₃×#3166	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#3153	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#3131	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅₃×#488	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3063	C₂	1	2	yes	maximal, complemented
12	C₅₃×#3062	C₂	1	2	yes	maximal, complemented
13	C₅₃×#2947	C₂	1	2	yes	maximal, complemented
14	C₅₃×#2946	C₂	1	2	yes	maximal, complemented
15	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#137×C₅₃	C₂²	1	2	yes	—
17	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	2	yes	—
18	C₂×#131×C₅₃	C₂²	1	2	yes	—
19	C₅₃×#559	—	2	2	no	—
20	C₅₃×#556	—	2	2	no	—
21	C₅₃×#553	—	2	2	no	—
22	C₅₃×#553	—	2	4	no	—
23	C₅₃×#552	—	2	2	no	—
24	C₅₃×#551	—	2	4	no	—
25	C₅₃×#542	C₂²	1	4	yes	—
26	C₅₃×#541	C₂²	1	2	yes	complemented
27	C₅₃×#540	C₂²	1	2	yes	complemented
28	C₅₃×#537	C₂²	1	2	yes	—
29	C₅₃×#536	C₂²	1	1	yes	characteristic
30	C₅₃×#535	C₂²	1	1	yes	characteristic
31	C₅₃×#533	C₂²	1	1	yes	characteristic
32	C₅₃×#532	C₂²	1	1	yes	characteristic
33	C₅₃×#529	C₂²	1	4	yes	—
34	C₅₃×#529	C₂²	1	4	yes	complemented
35	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×#124×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#520	C₂²	1	2	yes	—
38	C₅₃×#519	C₂²	1	1	yes	characteristic
39	C₅₃×#518	C₂²	1	1	yes	characteristic
40	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#501	C₂²	1	4	yes	complemented
42	C₅₃×#499	C₂²	1	4	yes	complemented
43	C₅₃×#497	C₂²	1	2	yes	—
44	C₅₃×#496	C₂²	1	2	yes	—
45	C₅₃×#495	C₂²	1	4	yes	—
46	C₅₃×#488	C₂²	1	2	yes	—
47	C₅₃×#488	C₂²	1	4	yes	—
48	C₅₃×#488	C₂²	1	2	yes	—
49	C₅₃×#488	C₂²	1	2	yes	complemented
50	C₅₃×#488	C₂²	1	2	yes	complemented
51	C₅₃×#487	C₂²	1	2	yes	complemented
52	C₅₃×#486	C₂²	1	2	yes	complemented
53	C₅₃×#470	C₂²	1	4	yes	complemented
54	C₅₃×#469	C₂²	1	4	yes	—
55	C₅₃×#468	C₂²	1	4	yes	—
56	C₅₃×#468	C₂²	1	4	yes	complemented
57	C₅₃×#467	C₂²	1	4	yes	complemented
58	C₅₃×#466	C₂²	1	4	yes	—
59	C₅₃×#465	C₂²	1	4	yes	—
60	C₅₃×#465	C₂²	1	4	yes	complemented
61	C₅₃×#442	C₂²	1	4	yes	—
62	C₅₃×#441	C₂²	1	4	yes	complemented
63	C₅₃×#440	C₂²	1	4	yes	—
64	C₅₃×#440	C₂²	1	4	yes	complemented
65	C₄×Q₁₆×C₅₃	C₂²	1	4	yes	complemented
66	C₄×SD₁₆×C₅₃	C₂²	1	4	yes	—
67	C₄×SD₁₆×C₅₃	C₂²	1	4	yes	complemented
68	C₄×D₈×C₅₃	C₂²	1	4	yes	—
69	C₅₃×#428	C₂²	1	2	yes	—
70	C₅₃×#428	C₂²	1	2	yes	—
71	C₅₃×#416	C₂²	1	2	yes	—
72	C₅₃×#416	C₂²	1	2	yes	—
73	C₂×#103×C₅₃	C₂²	1	2	yes	—
74	C₂×#102×C₅₃	C₂²	1	2	yes	—
75	C₅₃×#405	C₂²	1	2	yes	—
76	C₅₃×#405	C₂²	1	4	yes	complemented
77	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
78	C₄×M₁₆×C₅₃	C₂²	1	2	yes	complemented
79	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
80	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	complemented
81	C₂×#41×C₅₃	C₂³	1	2	yes	—
82	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
83	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
85	#137×C₅₃	—	2	8	no	—
86	AGL(1,ℤ/8)×C₅₃	—	2	8	no	—
87	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
88	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
89	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
90	C₂×D₈×C₅₃	C₂³	1	4	yes	—
91	#131×C₅₃	—	2	8	no	—
92	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
93	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
94	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
95	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
96	C₄⋊Q₈×C₅₃	—	2	1	no	—
97	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
98	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
99	#127×C₅₃	—	2	1	no	—
100	#127×C₅₃	C₂³	1	1	yes	characteristic
101	#126×C₅₃	—	2	4	no	—
102	#126×C₅₃	—	2	4	no	—
103	#126×C₅₃	—	2	4	no	—
104	#126×C₅₃	—	2	4	no	—
105	#125×C₅₃	—	2	4	no	—
106	#125×C₅₃	—	2	2	no	—
107	#125×C₅₃	C₂³	1	2	yes	—
108	#125×C₅₃	C₂³	1	2	yes	—
109	#125×C₅₃	C₂³	1	2	yes	—
110	#124×C₅₃	—	2	4	no	—
111	#124×C₅₃	—	2	2	no	—
112	#124×C₅₃	—	2	1	no	—
113	#124×C₅₃	—	2	1	no	—
114	#124×C₅₃	C₂³	1	2	yes	—
115	#124×C₅₃	C₂³	1	2	yes	—
116	#124×C₅₃	C₂³	1	2	yes	—
117	#123×C₅₃	—	2	4	no	—
118	#123×C₅₃	—	2	4	no	—
119	#123×C₅₃	—	2	2	no	—
120	#123×C₅₃	—	2	4	no	—
121	#123×C₅₃	—	2	4	no	—
122	C₂²⋊Q₈×C₅₃	—	2	4	no	—
123	C₂²⋊Q₈×C₅₃	—	2	2	no	—
124	C₂²⋊Q₈×C₅₃	—	2	2	no	—
125	C₂²⋊Q₈×C₅₃	C₂³	1	4	yes	—
126	C₂²⋊Q₈×C₅₃	C₂³	1	4	yes	—
127	#121×C₅₃	—	2	4	no	—
128	#121×C₅₃	—	2	4	no	—
129	#121×C₅₃	—	2	2	no	—
130	#121×C₅₃	—	2	2	no	—
131	#121×C₅₃	C₂³	1	4	yes	—
132	#121×C₅₃	C₂³	1	4	yes	—
133	#120×C₅₃	—	2	2	no	—
134	C₄×Q₈×C₅₃	—	2	2	no	—
135	C₄×Q₈×C₅₃	—	2	2	no	—
136	C₄×Q₈×C₅₃	D₄	1	2	yes	—
137	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
138	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
139	C₄×D₄×C₅₃	—	2	2	no	—
140	C₄×D₄×C₅₃	—	2	4	no	—
141	C₄×D₄×C₅₃	—	2	2	no	—
142	C₄×D₄×C₅₃	—	2	4	no	—
143	C₄×D₄×C₅₃	—	2	4	no	—
144	C₄×D₄×C₅₃	D₄	1	4	yes	—
145	C₄×D₄×C₅₃	D₄	1	2	yes	—
146	C₄×D₄×C₅₃	C₂³	1	4	yes	—
147	C₄×D₄×C₅₃	C₂³	1	4	yes	—
148	#117×C₅₃	—	2	2	no	—
149	#117×C₅₃	—	2	2	no	—
150	#117×C₅₃	—	2	4	no	—
151	#117×C₅₃	—	2	4	no	—
152	#117×C₅₃	C₂³	1	1	yes	characteristic
153	#117×C₅₃	C₂³	1	1	yes	characteristic
154	#117×C₅₃	C₂³	1	1	yes	characteristic
155	#117×C₅₃	C₂³	1	2	yes	—
156	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
157	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
158	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
159	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
160	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
161	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
162	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
163	#107×C₅₃	C₂³	1	4	yes	—
164	#107×C₅₃	C₂³	1	4	yes	—
165	#106×C₅₃	C₂³	1	4	yes	—
166	#106×C₅₃	C₂³	1	4	yes	—
167	#103×C₅₃	C₂³	1	2	yes	—
168	#103×C₅₃	C₂³	1	2	yes	—
169	#103×C₅₃	C₂³	1	2	yes	—
170	#103×C₅₃	C₂³	1	2	yes	—
171	#103×C₅₃	C₂³	1	4	yes	—
172	#103×C₅₃	C₂³	1	4	yes	—
173	#102×C₅₃	C₂³	1	2	yes	—
174	#102×C₅₃	C₂³	1	2	yes	—
175	#102×C₅₃	C₂³	1	2	yes	—
176	#102×C₅₃	C₂³	1	2	yes	—
177	#102×C₅₃	C₂³	1	4	yes	—
178	#102×C₅₃	C₂³	1	4	yes	—
179	#97×C₅₃	C₂³	1	2	yes	—
180	#97×C₅₃	C₂³	1	4	yes	—
181	C₄×C₈×C₅₃	C₂³	1	2	yes	—
182	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
183	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
184	C₄×C₈×C₅₃	C₂³	1	2	yes	—
185	C₄×C₈×C₅₃	C₂³	1	4	yes	—
186	C₂⁴×C₅₃	—	2	1	no	—
187	#41×C₅₃	—	2	4	no	—
188	#41×C₅₃	—	2	4	no	—
189	#41×C₅₃	—	2	4	no	—
190	C₂×Q₈×C₅₃	—	2	1	no	—
191	C₂×Q₈×C₅₃	—	2	2	no	—
192	C₂×Q₈×C₅₃	—	2	4	no	—
193	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
194	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
195	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
196	C₂×D₄×C₅₃	—	2	2	no	—
197	C₂×D₄×C₅₃	—	2	4	no	—
198	C₂×D₄×C₅₃	—	2	4	no	—
199	C₂×D₄×C₅₃	—	2	1	no	—
200	C₂×D₄×C₅₃	—	2	2	no	—
201	C₂×D₄×C₅₃	—	2	4	no	—
202	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
203	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
204	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
205	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
206	C₂²×C₄×C₅₃	—	2	1	no	—
207	C₂²×C₄×C₅₃	—	2	2	no	—
208	C₂²×C₄×C₅₃	—	2	2	no	—
209	C₂²×C₄×C₅₃	—	2	4	no	—
210	C₂²×C₄×C₅₃	—	2	4	no	—
211	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
212	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
213	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
214	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
215	Q₁₆×C₅₃	—	2	8	no	—
216	SD₁₆×C₅₃	—	2	8	no	—
217	SD₁₆×C₅₃	—	2	8	no	—
218	D₈×C₅₃	—	2	8	no	—
219	M₁₆×C₅₃	—	2	8	no	—
220	M₁₆×C₅₃	#41	1	8	yes	—
221	C₂×C₈×C₅₃	—	2	4	no	—
222	C₂×C₈×C₅₃	—	2	8	no	—
223	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
224	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
225	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
226	C₄⋊C₄×C₅₃	—	2	2	no	—
227	C₄⋊C₄×C₅₃	—	2	4	no	—
228	C₄⋊C₄×C₅₃	—	2	4	no	—
229	C₄⋊C₄×C₅₃	—	2	4	no	—
230	C₄⋊C₄×C₅₃	—	2	2	no	—
231	C₄⋊C₄×C₅₃	—	2	4	no	—
232	C₄⋊C₄×C₅₃	—	2	4	no	—
233	C₄⋊C₄×C₅₃	—	2	4	no	—
234	C₄⋊C₄×C₅₃	—	2	2	no	—
235	C₄⋊C₄×C₅₃	—	2	2	no	—
236	C₄⋊C₄×C₅₃	—	2	2	no	—
237	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
238	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
239	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
240	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
241	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
242	C₂²⋊C₄×C₅₃	—	2	1	no	—
243	C₂²⋊C₄×C₅₃	—	2	1	no	—
244	C₂²⋊C₄×C₅₃	—	2	4	no	—
245	C₂²⋊C₄×C₅₃	—	2	4	no	—
246	C₂²⋊C₄×C₅₃	—	2	4	no	—
247	C₂²⋊C₄×C₅₃	—	2	4	no	—
248	C₂²⋊C₄×C₅₃	—	2	4	no	—
249	C₂²⋊C₄×C₅₃	—	2	2	no	—
250	C₂²⋊C₄×C₅₃	—	2	4	no	—
251	C₂²⋊C₄×C₅₃	—	2	4	no	—
252	C₂²⋊C₄×C₅₃	—	2	4	no	—
253	C₂²⋊C₄×C₅₃	—	2	2	no	—
254	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
255	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
256	C₄²×C₅₃	—	2	4	no	—
257	C₄²×C₅₃	—	2	4	no	—
258	C₄²×C₅₃	—	2	1	no	—
259	C₄²×C₅₃	—	2	1	no	—
260	C₄²×C₅₃	C₂×D₄	1	4	yes	—
261	C₄²×C₅₃	C₂×D₄	1	2	yes	—
262	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
263	C₄²×C₅₃	C₂⁴	1	2	yes	—
264	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
265	C₂³×C₅₃	—	2	2	no	—
266	C₂³×C₅₃	—	2	2	no	—
267	C₂³×C₅₃	—	4	2	no	—
268	C₂³×C₅₃	—	8	2	no	—
269	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
270	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
271	Q₈×C₅₃	—	2	2	no	—
272	Q₈×C₅₃	—	2	4	no	—
273	Q₈×C₅₃	—	2	2	no	—
274	Q₈×C₅₃	—	2	2	no	—
275	Q₈×C₅₃	—	4	2	no	—
276	Q₈×C₅₃	—	8	2	no	—
277	D₄×C₅₃	—	2	4	no	—
278	D₄×C₅₃	—	2	2	no	—
279	D₄×C₅₃	—	2	4	no	—
280	D₄×C₅₃	—	2	2	no	—
281	D₄×C₅₃	—	2	2	no	—
282	D₄×C₅₃	—	4	4	no	—
283	D₄×C₅₃	—	4	2	no	—
284	D₄×C₅₃	—	8	2	no	—
285	D₄×C₅₃	—	8	4	no	—
286	D₄×C₅₃	—	8	4	no	—
287	C₂×C₄×C₅₃	—	2	4	no	—
288	C₂×C₄×C₅₃	—	2	4	no	—
289	C₂×C₄×C₅₃	—	2	2	no	—
290	C₂×C₄×C₅₃	—	2	2	no	—
291	C₂×C₄×C₅₃	—	2	2	no	—
292	C₂×C₄×C₅₃	—	2	2	no	—
293	C₂×C₄×C₅₃	—	4	4	no	—
294	C₂×C₄×C₅₃	—	4	2	no	—
295	C₂×C₄×C₅₃	—	4	2	no	—
296	C₂×C₄×C₅₃	—	4	4	no	—
297	C₂×C₄×C₅₃	—	8	4	no	—
298	C₂×C₄×C₅₃	—	8	4	no	—
299	C₂×C₄×C₅₃	—	8	2	no	—
300	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
301	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
302	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
303	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
304	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
305	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
306	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
307	C₈×C₅₃	—	2	2	no	—
308	C₈×C₅₃	—	2	2	no	—
309	C₈×C₅₃	C₂×#41	1	8	yes	—
310	#60724	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
311	C₂²×C₅₃	—	2	2	no	—
312	C₂²×C₅₃	—	2	4	no	—
313	C₂²×C₅₃	—	2	2	no	—
314	C₂²×C₅₃	—	4	2	no	—
315	C₂²×C₅₃	—	4	4	no	—
316	C₂²×C₅₃	—	4	2	no	—
317	C₂²×C₅₃	—	4	2	no	—
318	C₂²×C₅₃	—	4	2	no	—
319	C₂²×C₅₃	—	16	2	no	—
320	C₂²×C₅₃	#532	1	2	yes	—
321	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
322	C₄×C₅₃	—	2	2	no	—
323	C₄×C₅₃	—	2	4	no	—
324	C₄×C₅₃	—	2	2	no	—
325	C₄×C₅₃	—	2	2	no	—
326	C₄×C₅₃	—	4	4	no	—
327	C₄×C₅₃	—	4	2	no	—
328	C₄×C₅₃	—	4	2	no	—
329	C₄×C₅₃	—	4	2	no	—
330	C₄×C₅₃	—	4	2	no	—
331	C₄×C₅₃	#532	1	2	yes	—
332	C₄×C₅₃	C₂²×#41	1	2	yes	—
333	#3512	C₂×C₅₃	1	1	yes	characteristic, complemented
334	#3510	C₂×C₅₃	1	1	yes	characteristic, complemented
335	#3183	C₂×C₅₃	1	4	yes	complemented
336	#3179	C₂×C₅₃	1	4	yes	complemented
337	#3178	C₂×C₅₃	1	4	yes	complemented
338	#3168	C₂×C₅₃	1	4	yes	complemented
339	#3166	C₂×C₅₃	1	1	yes	characteristic, complemented
340	#3153	C₂×C₅₃	1	1	yes	characteristic, complemented
341	#3131	C₂×C₅₃	1	1	yes	characteristic, complemented
342	C₂×#488	C₂×C₅₃	1	1	yes	characteristic, complemented
343	#3063	C₂×C₅₃	1	2	yes	complemented
344	#3062	C₂×C₅₃	1	2	yes	complemented
345	#2947	C₂×C₅₃	1	2	yes	complemented
346	#2946	C₂×C₅₃	1	2	yes	complemented
347	C₄×#131	C₂×C₅₃	1	1	yes	characteristic, complemented
348	C₂×C₅₃	—	2	2	no	—
349	C₂×C₅₃	—	4	2	no	—
350	C₂×C₅₃	—	8	2	no	—
351	C₂×C₅₃	—	8	2	no	—
352	C₂×C₅₃	C₂×#532	1	1	yes	characteristic
353	C₂×C₅₃	#3584	1	2	yes	—
354	C₂×#137	C₂²×C₅₃	1	2	yes	—
355	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	2	yes	—
356	C₂×#131	C₂²×C₅₃	1	2	yes	—
357	#559	—	2	2	no	—
358	#556	—	2	2	no	—
359	#553	—	2	4	no	—
360	#553	—	2	2	no	—
361	#552	—	2	2	no	—
362	#551	—	2	4	no	—
363	#542	C₂²×C₅₃	1	4	yes	—
364	#541	C₂²×C₅₃	1	2	yes	complemented
365	#540	C₂²×C₅₃	1	2	yes	complemented
366	#537	C₂²×C₅₃	1	2	yes	—
367	#536	C₂²×C₅₃	1	1	yes	characteristic
368	#535	C₂²×C₅₃	1	1	yes	characteristic
369	#533	C₂²×C₅₃	1	1	yes	characteristic
370	#532	C₂²×C₅₃	1	1	yes	characteristic
371	#529	C₂²×C₅₃	1	4	yes	—
372	#529	C₂²×C₅₃	1	4	yes	complemented
373	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic
374	C₂×#124	C₂²×C₅₃	1	1	yes	characteristic
375	#520	C₂²×C₅₃	1	2	yes	—
376	#519	C₂²×C₅₃	1	1	yes	characteristic
377	#518	C₂²×C₅₃	1	1	yes	characteristic
378	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
379	#501	C₂²×C₅₃	1	4	yes	complemented
380	#499	C₂²×C₅₃	1	4	yes	complemented
381	#497	C₂²×C₅₃	1	2	yes	—
382	#496	C₂²×C₅₃	1	2	yes	—
383	#495	C₂²×C₅₃	1	4	yes	—
384	#488	C₂²×C₅₃	1	2	yes	—
385	#488	C₂²×C₅₃	1	4	yes	—
386	#488	C₂²×C₅₃	1	2	yes	—
387	#488	C₂²×C₅₃	1	2	yes	complemented
388	#488	C₂²×C₅₃	1	2	yes	complemented
389	#487	C₂²×C₅₃	1	2	yes	complemented
390	#486	C₂²×C₅₃	1	2	yes	complemented
391	#470	C₂²×C₅₃	1	4	yes	complemented
392	#469	C₂²×C₅₃	1	4	yes	—
393	#468	C₂²×C₅₃	1	4	yes	—
394	#468	C₂²×C₅₃	1	4	yes	complemented
395	#467	C₂²×C₅₃	1	4	yes	complemented
396	#466	C₂²×C₅₃	1	4	yes	—
397	#465	C₂²×C₅₃	1	4	yes	—
398	#465	C₂²×C₅₃	1	4	yes	complemented
399	#442	C₂²×C₅₃	1	4	yes	—
400	#441	C₂²×C₅₃	1	4	yes	complemented
401	#440	C₂²×C₅₃	1	4	yes	—
402	#440	C₂²×C₅₃	1	4	yes	complemented
403	C₄×Q₁₆	C₂²×C₅₃	1	4	yes	complemented
404	C₄×SD₁₆	C₂²×C₅₃	1	4	yes	—
405	C₄×SD₁₆	C₂²×C₅₃	1	4	yes	complemented
406	C₄×D₈	C₂²×C₅₃	1	4	yes	—
407	#428	C₂²×C₅₃	1	2	yes	—
408	#428	C₂²×C₅₃	1	2	yes	—
409	#416	C₂²×C₅₃	1	2	yes	—
410	#416	C₂²×C₅₃	1	2	yes	—
411	C₂×#103	C₂²×C₅₃	1	2	yes	—
412	C₂×#102	C₂²×C₅₃	1	2	yes	—
413	#405	C₂²×C₅₃	1	2	yes	—
414	#405	C₂²×C₅₃	1	4	yes	complemented
415	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
416	C₄×M₁₆	C₂²×C₅₃	1	2	yes	complemented
417	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
418	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	complemented
419	C₅₃	#60724	1	1	yes	characteristic, direct product factor, Sylow
420	C₂×#41	C₂³×C₅₃	1	2	yes	—
421	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
422	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
423	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
424	#137	—	2	8	no	—
425	AGL(1,ℤ/8)	—	2	8	no	—
426	C₂×Q₁₆	C₂³×C₅₃	1	4	yes	—
427	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
428	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
429	C₂×D₈	C₂³×C₅₃	1	4	yes	—
430	#131	—	2	8	no	—
431	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
432	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
433	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
434	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
435	C₄⋊Q₈	—	2	1	no	—
436	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
437	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
438	#127	—	2	1	no	—
439	#127	C₂³×C₅₃	1	1	yes	characteristic
440	#126	—	2	4	no	—
441	#126	—	2	4	no	—
442	#126	—	2	4	no	—
443	#126	—	2	4	no	—
444	#125	—	2	4	no	—
445	#125	—	2	2	no	—
446	#125	C₂³×C₅₃	1	2	yes	—
447	#125	C₂³×C₅₃	1	2	yes	—
448	#125	C₂³×C₅₃	1	2	yes	—
449	#124	—	2	1	no	—
450	#124	—	2	1	no	—
451	#124	—	2	4	no	—
452	#124	—	2	2	no	—
453	#124	C₂³×C₅₃	1	2	yes	—
454	#124	C₂³×C₅₃	1	2	yes	—
455	#124	C₂³×C₅₃	1	2	yes	—
456	#123	—	2	2	no	—
457	#123	—	2	4	no	—
458	#123	—	2	4	no	—
459	#123	—	2	4	no	—
460	#123	—	2	4	no	—
461	C₂²⋊Q₈	—	2	2	no	—
462	C₂²⋊Q₈	—	2	2	no	—
463	C₂²⋊Q₈	—	2	4	no	—
464	C₂²⋊Q₈	C₂³×C₅₃	1	4	yes	—
465	C₂²⋊Q₈	C₂³×C₅₃	1	4	yes	—
466	#121	—	2	2	no	—
467	#121	—	2	4	no	—
468	#121	—	2	2	no	—
469	#121	—	2	4	no	—
470	#121	C₂³×C₅₃	1	4	yes	—
471	#121	C₂³×C₅₃	1	4	yes	—
472	#120	—	2	2	no	—
473	C₄×Q₈	—	2	2	no	—
474	C₄×Q₈	—	2	2	no	—
475	C₄×Q₈	D₄×C₅₃	1	2	yes	—
476	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
477	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
478	C₄×D₄	—	2	4	no	—
479	C₄×D₄	—	2	4	no	—
480	C₄×D₄	—	2	2	no	—
481	C₄×D₄	—	2	4	no	—
482	C₄×D₄	—	2	2	no	—
483	C₄×D₄	D₄×C₅₃	1	4	yes	—
484	C₄×D₄	D₄×C₅₃	1	2	yes	—
485	C₄×D₄	C₂³×C₅₃	1	4	yes	—
486	C₄×D₄	C₂³×C₅₃	1	4	yes	—
487	#117	—	2	2	no	—
488	#117	—	2	2	no	—
489	#117	—	2	4	no	—
490	#117	—	2	4	no	—
491	#117	C₂³×C₅₃	1	1	yes	characteristic
492	#117	C₂³×C₅₃	1	1	yes	characteristic
493	#117	C₂³×C₅₃	1	1	yes	characteristic
494	#117	C₂³×C₅₃	1	2	yes	—
495	C₂×C₄⋊C₄	—	2	2	no	—
496	C₂×C₄⋊C₄	—	2	2	no	—
497	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
498	C₂×C₂²⋊C₄	—	2	2	no	—
499	C₂×C₂²⋊C₄	—	2	2	no	—
500	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
501	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
502	#107	C₂³×C₅₃	1	4	yes	—
503	#107	C₂³×C₅₃	1	4	yes	—
504	#106	C₂³×C₅₃	1	4	yes	—
505	#106	C₂³×C₅₃	1	4	yes	—
506	#103	C₂³×C₅₃	1	4	yes	—
507	#103	C₂³×C₅₃	1	4	yes	—
508	#103	C₂³×C₅₃	1	2	yes	—
509	#103	C₂³×C₅₃	1	2	yes	—
510	#103	C₂³×C₅₃	1	2	yes	—
511	#103	C₂³×C₅₃	1	2	yes	—
512	#102	C₂³×C₅₃	1	4	yes	—
513	#102	C₂³×C₅₃	1	4	yes	—
514	#102	C₂³×C₅₃	1	2	yes	—
515	#102	C₂³×C₅₃	1	2	yes	—
516	#102	C₂³×C₅₃	1	2	yes	—
517	#102	C₂³×C₅₃	1	2	yes	—
518	#97	C₂³×C₅₃	1	2	yes	—
519	#97	C₂³×C₅₃	1	4	yes	—
520	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
521	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
522	C₄×C₈	C₂³×C₅₃	1	2	yes	—
523	C₄×C₈	C₂³×C₅₃	1	2	yes	—
524	C₄×C₈	C₂³×C₅₃	1	4	yes	—
525	C₂⁴	—	2	1	no	—
526	#41	—	2	4	no	—
527	#41	—	2	4	no	—
528	#41	—	2	4	no	—
529	C₂×Q₈	—	2	1	no	—
530	C₂×Q₈	—	2	2	no	—
531	C₂×Q₈	—	2	4	no	—
532	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
533	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
534	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
535	C₂×D₄	—	2	2	no	—
536	C₂×D₄	—	2	4	no	—
537	C₂×D₄	—	2	4	no	—
538	C₂×D₄	—	2	1	no	—
539	C₂×D₄	—	2	2	no	—
540	C₂×D₄	—	2	4	no	—
541	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
542	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
543	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
544	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
545	C₂²×C₄	—	2	1	no	—
546	C₂²×C₄	—	2	2	no	—
547	C₂²×C₄	—	2	2	no	—
548	C₂²×C₄	—	2	4	no	—
549	C₂²×C₄	—	2	4	no	—
550	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
551	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
552	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
553	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
554	Q₁₆	—	2	8	no	—
555	SD₁₆	—	2	8	no	—
556	SD₁₆	—	2	8	no	—
557	D₈	—	2	8	no	—
558	M₁₆	—	2	8	no	—
559	M₁₆	#41×C₅₃	1	8	yes	—
560	C₂×C₈	—	2	4	no	—
561	C₂×C₈	—	2	8	no	—
562	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
563	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
564	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
565	C₄⋊C₄	—	2	2	no	—
566	C₄⋊C₄	—	2	4	no	—
567	C₄⋊C₄	—	2	4	no	—
568	C₄⋊C₄	—	2	4	no	—
569	C₄⋊C₄	—	2	2	no	—
570	C₄⋊C₄	—	2	4	no	—
571	C₄⋊C₄	—	2	4	no	—
572	C₄⋊C₄	—	2	4	no	—
573	C₄⋊C₄	—	2	2	no	—
574	C₄⋊C₄	—	2	2	no	—
575	C₄⋊C₄	—	2	2	no	—
576	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
577	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
578	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
579	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
580	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
581	C₂²⋊C₄	—	2	1	no	—
582	C₂²⋊C₄	—	2	1	no	—
583	C₂²⋊C₄	—	2	4	no	—
584	C₂²⋊C₄	—	2	4	no	—
585	C₂²⋊C₄	—	2	4	no	—
586	C₂²⋊C₄	—	2	4	no	—
587	C₂²⋊C₄	—	2	4	no	—
588	C₂²⋊C₄	—	2	2	no	—
589	C₂²⋊C₄	—	2	4	no	—
590	C₂²⋊C₄	—	2	4	no	—
591	C₂²⋊C₄	—	2	4	no	—
592	C₂²⋊C₄	—	2	2	no	—
593	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
594	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
595	C₄²	—	2	4	no	—
596	C₄²	—	2	4	no	—
597	C₄²	—	2	1	no	—
598	C₄²	—	2	1	no	—
599	C₄²	C₂×D₄×C₅₃	1	4	yes	—
600	C₄²	C₂×D₄×C₅₃	1	2	yes	—
601	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
602	C₄²	C₂⁴×C₅₃	1	2	yes	—
603	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
604	C₂³	—	2	2	no	—
605	C₂³	—	2	2	no	—
606	C₂³	—	4	2	no	—
607	C₂³	—	8	2	no	—
608	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
609	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
610	Q₈	—	2	2	no	—
611	Q₈	—	2	2	no	—
612	Q₈	—	2	4	no	—
613	Q₈	—	2	2	no	—
614	Q₈	—	4	2	no	—
615	Q₈	—	8	2	no	—
616	D₄	—	2	4	no	—
617	D₄	—	2	2	no	—
618	D₄	—	2	2	no	—
619	D₄	—	2	4	no	—
620	D₄	—	2	2	no	—
621	D₄	—	4	4	no	—
622	D₄	—	4	2	no	—
623	D₄	—	8	2	no	—
624	D₄	—	8	4	no	—
625	D₄	—	8	4	no	—
626	C₂×C₄	—	2	4	no	—
627	C₂×C₄	—	2	2	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	2	2	no	—
630	C₂×C₄	—	2	2	no	—
631	C₂×C₄	—	2	4	no	—
632	C₂×C₄	—	4	4	no	—
633	C₂×C₄	—	4	2	no	—
634	C₂×C₄	—	4	2	no	—
635	C₂×C₄	—	4	4	no	—
636	C₂×C₄	—	8	4	no	—
637	C₂×C₄	—	8	4	no	—
638	C₂×C₄	—	8	2	no	—
639	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
640	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
641	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
642	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
643	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
644	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
645	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
646	C₈	—	2	2	no	—
647	C₈	—	2	2	no	—
648	C₈	C₂×#41×C₅₃	1	8	yes	—
649	C₂²	—	2	2	no	—
650	C₂²	—	2	4	no	—
651	C₂²	—	2	2	no	—
652	C₂²	—	4	2	no	—
653	C₂²	—	4	4	no	—
654	C₂²	—	4	2	no	—
655	C₂²	—	4	2	no	—
656	C₂²	—	4	2	no	—
657	C₂²	—	16	2	no	—
658	C₂²	C₅₃×#532	1	2	yes	—
659	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
660	C₄	—	2	2	no	—
661	C₄	—	2	4	no	—
662	C₄	—	2	2	no	—
663	C₄	—	2	2	no	—
664	C₄	—	4	4	no	—
665	C₄	—	4	2	no	—
666	C₄	—	4	2	no	—
667	C₄	—	4	2	no	—
668	C₄	—	4	2	no	—
669	C₄	C₅₃×#532	1	2	yes	—
670	C₄	C₂²×#41×C₅₃	1	2	yes	—
671	C₂	—	2	2	no	—
672	C₂	—	4	2	no	—
673	C₂	—	8	2	no	—
674	C₂	—	8	2	no	—
675	C₂	C₂×C₅₃×#532	1	1	yes	characteristic
676	C₂	C₅₃×#3584	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 ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3512	C₂	semidirect
2. C₅₃×#3510	C₂	semidirect
3. C₅₃×#3183	C₂	semidirect
4. C₅₃×#3179	C₂	semidirect
5. C₅₃×#3178	C₂	semidirect
6. C₅₃×#3168	C₂	semidirect
7. C₅₃×#3166	C₂	semidirect
8. C₅₃×#3153	C₂	semidirect
9. C₅₃×#3131	C₂	semidirect
10. C₂×C₅₃×#488	C₂	semidirect
11. C₅₃×#3063	C₂	semidirect
12. C₅₃×#3062	C₂	semidirect
13. C₅₃×#2947	C₂	semidirect
14. C₅₃×#2946	C₂	semidirect
15. C₄×#131×C₅₃	C₂	semidirect
26. C₅₃×#541	C₂²	semidirect
27. C₅₃×#540	C₂²	semidirect
34. C₅₃×#529	C₂²	semidirect
41. C₅₃×#501	C₂²	semidirect
42. C₅₃×#499	C₂²	semidirect
49. C₅₃×#488	C₂²	semidirect
50. C₅₃×#488	C₂²	semidirect
51. C₅₃×#487	C₂²	semidirect
52. C₅₃×#486	C₂²	semidirect
53. C₅₃×#470	C₂²	semidirect
56. C₅₃×#468	C₂²	semidirect
57. C₅₃×#467	C₂²	semidirect
60. C₅₃×#465	C₂²	semidirect
62. C₅₃×#441	C₂²	semidirect
64. C₅₃×#440	C₂²	semidirect
65. C₄×Q₁₆×C₅₃	C₂²	semidirect
67. C₄×SD₁₆×C₅₃	C₂²	semidirect
76. C₅₃×#405	C₂²	semidirect
78. C₄×M₁₆×C₅₃	C₂²	semidirect
80. C₂×C₄×C₈×C₅₃	C₂²	semidirect
310. #60724	C₅₃	direct (N × H)
333. #3512	C₂×C₅₃	semidirect
334. #3510	C₂×C₅₃	semidirect
335. #3183	C₂×C₅₃	semidirect
336. #3179	C₂×C₅₃	semidirect
337. #3178	C₂×C₅₃	semidirect
338. #3168	C₂×C₅₃	semidirect
339. #3166	C₂×C₅₃	semidirect
340. #3153	C₂×C₅₃	semidirect
341. #3131	C₂×C₅₃	semidirect
342. C₂×#488	C₂×C₅₃	semidirect
343. #3063	C₂×C₅₃	semidirect
344. #3062	C₂×C₅₃	semidirect
345. #2947	C₂×C₅₃	semidirect
346. #2946	C₂×C₅₃	semidirect
347. C₄×#131	C₂×C₅₃	semidirect
364. #541	C₂²×C₅₃	semidirect
365. #540	C₂²×C₅₃	semidirect
372. #529	C₂²×C₅₃	semidirect
379. #501	C₂²×C₅₃	semidirect
380. #499	C₂²×C₅₃	semidirect
387. #488	C₂²×C₅₃	semidirect
388. #488	C₂²×C₅₃	semidirect
389. #487	C₂²×C₅₃	semidirect
390. #486	C₂²×C₅₃	semidirect
391. #470	C₂²×C₅₃	semidirect
394. #468	C₂²×C₅₃	semidirect
395. #467	C₂²×C₅₃	semidirect
398. #465	C₂²×C₅₃	semidirect
400. #441	C₂²×C₅₃	semidirect
402. #440	C₂²×C₅₃	semidirect
403. C₄×Q₁₆	C₂²×C₅₃	semidirect
405. C₄×SD₁₆	C₂²×C₅₃	semidirect
414. #405	C₂²×C₅₃	semidirect
416. C₄×M₁₆	C₂²×C₅₃	semidirect
418. C₂×C₄×C₈	C₂²×C₅₃	semidirect
419. C₅₃	#60724	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×#3512	2
C₅₃×#3510	2
C₅₃×#3183	2
C₅₃×#3179	2
C₅₃×#3178	2
C₅₃×#3168	2
C₅₃×#3166	2
C₅₃×#3153	2
C₅₃×#3131	2
C₂×C₅₃×#488	2
C₅₃×#3063	2
C₅₃×#3062	2
C₅₃×#2947	2
C₅₃×#2946	2
C₄×#131×C₅₃	2
#60724	53

Subgroups — C53×#60724 #27268236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60724 #27268236