Subgroups — C₅₃×#61337 #27268849

Order 13568.

1852 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3371	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3370	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×#3136	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₃×#490	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#3081	C₂	1	2	yes	maximal, complemented
10	C₅₃×#3080	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3045	C₂	1	2	yes	maximal, complemented
12	C₅₃×#3038	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#2961	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#2946	C₂	1	2	yes	maximal, complemented
15	C₅₃×#2943	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	2	yes	—
17	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#559	C₂²	1	4	yes	—
19	C₅₃×#557	C₂²	1	4	yes	complemented
20	C₅₃×#554	—	2	4	no	—
21	C₅₃×#554	C₂²	1	4	yes	complemented
22	C₅₃×#550	C₂²	1	4	yes	—
23	C₅₃×#546	—	2	4	no	—
24	C₅₃×#544	—	2	2	no	—
25	C₅₃×#544	C₂²	1	2	yes	—
26	C₅₃×#540	—	2	1	no	—
27	C₅₃×#540	—	2	1	no	—
28	C₅₃×#535	C₂²	1	2	yes	—
29	C₅₃×#533	—	2	1	no	—
30	C₅₃×#533	C₂²	1	1	yes	characteristic
31	C₅₃×#532	—	2	2	no	—
32	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×#125×C₅₃	—	2	1	no	—
34	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#520	C₂²	1	2	yes	—
36	C₅₃×#518	C₂²	1	2	yes	—
37	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
38	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#490	C₂²	1	4	yes	—
40	C₅₃×#490	C₂²	1	4	yes	—
41	C₅₃×#490	C₂²	1	4	yes	complemented
42	C₅₃×#489	C₂²	1	4	yes	complemented
43	C₅₃×#488	C₂²	1	4	yes	complemented
44	C₅₃×#486	C₂²	1	4	yes	complemented
45	C₅₃×#479	C₂²	1	4	yes	complemented
46	C₅₃×#478	C₂²	1	4	yes	complemented
47	C₅₃×#478	C₂²	1	4	yes	complemented
48	C₅₃×#478	C₂²	1	4	yes	complemented
49	C₅₃×#476	C₂²	1	4	yes	complemented
50	C₅₃×#474	C₂²	1	4	yes	complemented
51	C₅₃×#464	C₂²	1	4	yes	complemented
52	C₅₃×#463	C₂²	1	4	yes	complemented
53	C₅₃×#460	C₂²	1	4	yes	—
54	C₅₃×#460	C₂²	1	4	yes	complemented
55	C₅₃×#459	C₂²	1	4	yes	—
56	C₅₃×#459	C₂²	1	4	yes	complemented
57	C₅₃×#446	C₂²	1	4	yes	complemented
58	C₅₃×#446	C₂²	1	4	yes	complemented
59	C₅₃×#442	C₂²	1	2	yes	complemented
60	C₅₃×#442	C₂²	1	4	yes	complemented
61	C₅₃×#441	C₂²	1	2	yes	complemented
62	C₅₃×#440	C₂²	1	2	yes	complemented
63	C₅₃×#440	C₂²	1	2	yes	complemented
64	C₅₃×#440	C₂²	1	4	yes	complemented
65	C₄×SD₁₆×C₅₃	C₂²	1	4	yes	complemented
66	C₄×D₈×C₅₃	C₂²	1	4	yes	complemented
67	C₅₃×#428	C₂²	1	2	yes	—
68	C₅₃×#427	C₂²	1	1	yes	characteristic
69	C₅₃×#424	C₂²	1	2	yes	—
70	C₅₃×#423	C₂²	1	1	yes	characteristic
71	C₅₃×#418	C₂²	1	2	yes	—
72	C₅₃×#417	C₂²	1	2	yes	—
73	C₅₃×#416	C₂²	1	1	yes	characteristic
74	C₅₃×#416	C₂²	1	1	yes	characteristic
75	C₂×#102×C₅₃	C₂²	1	2	yes	—
76	C₂×#102×C₅₃	C₂²	1	2	yes	—
77	C₅₃×#405	C₂²	1	2	yes	—
78	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
79	C₂×#41×C₅₃	—	2	4	no	—
80	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
81	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
82	C₂²×D₄×C₅₃	—	2	2	no	—
83	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
84	AGL(1,ℤ/8)×C₅₃	—	2	8	no	—
85	#135×C₅₃	—	2	4	no	—
86	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
87	C₂×SD₁₆×C₅₃	C₂³	1	1	yes	characteristic
88	C₂×SD₁₆×C₅₃	C₂³	1	1	yes	characteristic
89	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
90	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₂×D₈×C₅₃	C₂³	1	4	yes	—
92	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
93	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₄⋊Q₈×C₅₃	—	2	2	no	—
96	C₄⋊Q₈×C₅₃	—	2	4	no	—
97	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
98	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
99	#127×C₅₃	—	2	1	no	—
100	#127×C₅₃	—	2	1	no	—
101	#127×C₅₃	C₂³	1	4	yes	—
102	#127×C₅₃	C₂³	1	4	yes	—
103	#125×C₅₃	—	2	2	no	—
104	#125×C₅₃	—	2	4	no	—
105	#125×C₅₃	—	2	4	no	—
106	#125×C₅₃	—	2	1	no	—
107	#125×C₅₃	—	2	1	no	—
108	#125×C₅₃	D₄	1	1	yes	characteristic
109	#125×C₅₃	D₄	1	1	yes	characteristic
110	#125×C₅₃	C₂³	1	4	yes	—
111	#125×C₅₃	C₂³	1	4	yes	—
112	#124×C₅₃	—	2	4	no	—
113	#124×C₅₃	C₂³	1	4	yes	—
114	#123×C₅₃	—	2	4	no	—
115	#123×C₅₃	—	2	4	no	—
116	C₂²⋊Q₈×C₅₃	—	2	4	no	—
117	C₂²⋊Q₈×C₅₃	—	2	4	no	—
118	C₂²⋊Q₈×C₅₃	—	2	4	no	—
119	C₂²⋊Q₈×C₅₃	—	2	4	no	—
120	C₂²⋊Q₈×C₅₃	D₄	1	4	yes	—
121	#121×C₅₃	—	2	2	no	—
122	#121×C₅₃	—	2	2	no	—
123	#121×C₅₃	—	2	4	no	—
124	#121×C₅₃	—	2	4	no	—
125	#121×C₅₃	—	2	4	no	—
126	#120×C₅₃	—	2	4	no	—
127	C₄×Q₈×C₅₃	—	2	4	no	—
128	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
129	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
130	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
131	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
132	C₄×D₄×C₅₃	—	2	2	no	—
133	C₄×D₄×C₅₃	—	2	2	no	—
134	C₄×D₄×C₅₃	—	2	4	no	—
135	C₄×D₄×C₅₃	—	2	4	no	—
136	C₄×D₄×C₅₃	—	2	2	no	—
137	C₄×D₄×C₅₃	—	2	2	no	—
138	C₄×D₄×C₅₃	C₂³	1	2	yes	—
139	C₄×D₄×C₅₃	C₂³	1	2	yes	—
140	C₄×D₄×C₅₃	C₂³	1	4	yes	—
141	C₄×D₄×C₅₃	C₂³	1	4	yes	—
142	C₄×D₄×C₅₃	C₂³	1	4	yes	—
143	#117×C₅₃	—	2	2	no	—
144	#117×C₅₃	—	2	1	no	—
145	#117×C₅₃	—	2	1	no	—
146	#117×C₅₃	C₂³	1	1	yes	characteristic
147	#117×C₅₃	C₂³	1	1	yes	characteristic
148	#117×C₅₃	C₂³	1	2	yes	—
149	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
150	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
151	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
152	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
153	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
154	C₂×C₄²×C₅₃	—	2	1	no	—
155	C₂×C₄²×C₅₃	—	2	1	no	—
156	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
157	#107×C₅₃	C₂³	1	2	yes	—
158	#107×C₅₃	C₂³	1	4	yes	—
159	#106×C₅₃	C₂³	1	2	yes	—
160	#106×C₅₃	C₂³	1	4	yes	—
161	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
162	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
163	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
164	#103×C₅₃	C₂³	1	2	yes	—
165	#103×C₅₃	C₂³	1	4	yes	—
166	#103×C₅₃	C₂³	1	2	yes	—
167	#102×C₅₃	C₂³	1	2	yes	—
168	#102×C₅₃	C₂³	1	4	yes	—
169	#102×C₅₃	C₂³	1	4	yes	—
170	#102×C₅₃	C₂³	1	4	yes	—
171	#102×C₅₃	C₂³	1	4	yes	—
172	#102×C₅₃	C₂³	1	4	yes	—
173	#102×C₅₃	C₂³	1	4	yes	—
174	#102×C₅₃	C₂³	1	4	yes	—
175	#102×C₅₃	C₂³	1	2	yes	—
176	#97×C₅₃	C₂³	1	4	yes	—
177	#97×C₅₃	C₂³	1	2	yes	—
178	#97×C₅₃	C₂³	1	2	yes	—
179	C₄×C₈×C₅₃	C₂³	1	4	yes	—
180	C₂⁴×C₅₃	—	2	2	no	—
181	#41×C₅₃	—	2	2	no	—
182	#41×C₅₃	—	2	2	no	—
183	#41×C₅₃	—	2	2	no	—
184	#41×C₅₃	—	8	8	no	—
185	#41×C₅₃	#41	1	4	yes	complemented
186	C₂×Q₈×C₅₃	—	2	4	no	—
187	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
188	C₂×Q₈×C₅₃	C₂⁴	1	1	yes	characteristic
189	C₂×Q₈×C₅₃	C₂⁴	1	1	yes	characteristic
190	C₂×D₄×C₅₃	—	2	1	no	—
191	C₂×D₄×C₅₃	—	2	1	no	—
192	C₂×D₄×C₅₃	—	2	2	no	—
193	C₂×D₄×C₅₃	—	2	4	no	—
194	C₂×D₄×C₅₃	—	2	4	no	—
195	C₂×D₄×C₅₃	—	2	4	no	—
196	C₂×D₄×C₅₃	—	2	4	no	—
197	C₂×D₄×C₅₃	—	2	4	no	—
198	C₂×D₄×C₅₃	—	2	4	no	—
199	C₂×D₄×C₅₃	—	8	4	no	—
200	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
201	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
202	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
203	C₂²×C₄×C₅₃	—	2	1	no	—
204	C₂²×C₄×C₅₃	—	2	2	no	—
205	C₂²×C₄×C₅₃	—	2	1	no	—
206	C₂²×C₄×C₅₃	—	2	2	no	—
207	C₂²×C₄×C₅₃	—	2	2	no	—
208	C₂²×C₄×C₅₃	—	2	4	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	C₂⁴	1	1	yes	characteristic
213	Q₁₆×C₅₃	—	2	2	no	—
214	SD₁₆×C₅₃	—	2	2	no	—
215	SD₁₆×C₅₃	—	2	8	no	—
216	SD₁₆×C₅₃	—	2	2	no	—
217	D₈×C₅₃	—	2	2	no	—
218	D₈×C₅₃	—	2	8	no	—
219	M₁₆×C₅₃	—	2	4	no	—
220	C₂×C₈×C₅₃	—	2	2	no	—
221	C₂×C₈×C₅₃	—	2	2	no	—
222	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
223	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
224	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
225	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
226	C₄⋊C₄×C₅₃	—	2	4	no	—
227	C₄⋊C₄×C₅₃	—	2	2	no	—
228	C₄⋊C₄×C₅₃	—	2	4	no	—
229	C₄⋊C₄×C₅₃	—	2	4	no	—
230	C₄⋊C₄×C₅₃	—	2	4	no	—
231	C₄⋊C₄×C₅₃	—	2	4	no	—
232	C₄⋊C₄×C₅₃	—	2	2	no	—
233	C₄⋊C₄×C₅₃	—	2	2	no	—
234	C₄⋊C₄×C₅₃	—	2	4	no	—
235	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
236	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
237	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
238	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
239	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
240	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
241	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
242	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
243	C₂²⋊C₄×C₅₃	—	2	1	no	—
244	C₂²⋊C₄×C₅₃	—	2	1	no	—
245	C₂²⋊C₄×C₅₃	—	2	4	no	—
246	C₂²⋊C₄×C₅₃	—	2	4	no	—
247	C₂²⋊C₄×C₅₃	—	2	2	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	C₂×D₄	1	4	yes	—
253	C₄²×C₅₃	—	2	1	no	—
254	C₄²×C₅₃	—	2	1	no	—
255	C₄²×C₅₃	—	2	2	no	—
256	C₄²×C₅₃	—	2	2	no	—
257	C₄²×C₅₃	—	2	4	no	—
258	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
259	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
260	C₄²×C₅₃	C₂⁴	1	4	yes	—
261	C₄²×C₅₃	C₂⁴	1	2	yes	—
262	C₂³×C₅₃	—	2	1	no	—
263	C₂³×C₅₃	—	2	1	no	—
264	C₂³×C₅₃	—	2	4	no	—
265	C₂³×C₅₃	—	4	4	no	—
266	C₂³×C₅₃	—	8	2	no	—
267	C₂³×C₅₃	—	8	2	no	—
268	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
269	Q₈×C₅₃	—	2	1	no	—
270	Q₈×C₅₃	—	2	1	no	—
271	Q₈×C₅₃	—	2	1	no	—
272	Q₈×C₅₃	—	4	4	no	—
273	Q₈×C₅₃	—	8	4	no	—
274	Q₈×C₅₃	C₂×#41	1	2	yes	—
275	D₄×C₅₃	—	2	4	no	—
276	D₄×C₅₃	—	2	4	no	—
277	D₄×C₅₃	—	2	1	no	—
278	D₄×C₅₃	—	2	1	no	—
279	D₄×C₅₃	—	2	4	no	—
280	D₄×C₅₃	—	2	1	no	—
281	D₄×C₅₃	—	4	8	no	—
282	D₄×C₅₃	—	4	2	no	—
283	D₄×C₅₃	—	4	4	no	—
284	D₄×C₅₃	—	8	4	no	—
285	D₄×C₅₃	—	8	4	no	—
286	D₄×C₅₃	—	8	4	no	—
287	D₄×C₅₃	C₂×#41	1	2	yes	—
288	C₂×C₄×C₅₃	—	2	1	no	—
289	C₂×C₄×C₅₃	—	2	4	no	—
290	C₂×C₄×C₅₃	—	2	2	no	—
291	C₂×C₄×C₅₃	—	2	4	no	—
292	C₂×C₄×C₅₃	—	2	1	no	—
293	C₂×C₄×C₅₃	—	2	2	no	—
294	C₂×C₄×C₅₃	—	4	2	no	—
295	C₂×C₄×C₅₃	—	4	4	no	—
296	C₂×C₄×C₅₃	—	4	2	no	—
297	C₂×C₄×C₅₃	—	8	2	no	—
298	C₂×C₄×C₅₃	—	8	4	no	—
299	C₂×C₄×C₅₃	—	8	4	no	—
300	C₂×C₄×C₅₃	—	8	4	no	—
301	C₂×C₄×C₅₃	—	8	2	no	—
302	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
303	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
304	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
305	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
306	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
307	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
308	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
309	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
310	C₈×C₅₃	—	2	1	no	—
311	C₈×C₅₃	—	2	1	no	—
312	C₈×C₅₃	—	2	4	no	—
313	C₈×C₅₃	—	2	2	no	—
314	#61337	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
315	C₂²×C₅₃	—	2	2	no	—
316	C₂²×C₅₃	—	2	2	no	—
317	C₂²×C₅₃	—	4	2	no	—
318	C₂²×C₅₃	—	4	2	no	—
319	C₂²×C₅₃	—	4	4	no	—
320	C₂²×C₅₃	—	4	4	no	—
321	C₂²×C₅₃	—	4	1	no	—
322	C₂²×C₅₃	—	4	4	no	—
323	C₂²×C₅₃	—	16	4	no	—
324	C₂²×C₅₃	#548	1	2	yes	—
325	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
326	C₄×C₅₃	—	2	2	no	—
327	C₄×C₅₃	—	2	2	no	—
328	C₄×C₅₃	—	2	4	no	—
329	C₄×C₅₃	—	2	4	no	—
330	C₄×C₅₃	—	2	2	no	—
331	C₄×C₅₃	—	4	2	no	—
332	C₄×C₅₃	—	4	4	no	—
333	C₄×C₅₃	—	4	2	no	—
334	C₄×C₅₃	—	4	1	no	—
335	C₄×C₅₃	#548	1	2	yes	—
336	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
337	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
338	#3508	C₂×C₅₃	1	1	yes	characteristic, complemented
339	#3504	C₂×C₅₃	1	1	yes	characteristic, complemented
340	#3371	C₂×C₅₃	1	4	yes	complemented
341	#3370	C₂×C₅₃	1	4	yes	complemented
342	#3368	C₂×C₅₃	1	4	yes	complemented
343	#3366	C₂×C₅₃	1	4	yes	complemented
344	#3136	C₂×C₅₃	1	2	yes	complemented
345	C₂×#490	C₂×C₅₃	1	1	yes	characteristic, complemented
346	#3081	C₂×C₅₃	1	2	yes	complemented
347	#3080	C₂×C₅₃	1	1	yes	characteristic, complemented
348	#3045	C₂×C₅₃	1	2	yes	complemented
349	#3038	C₂×C₅₃	1	1	yes	characteristic, complemented
350	#2961	C₂×C₅₃	1	1	yes	characteristic, complemented
351	#2946	C₂×C₅₃	1	2	yes	complemented
352	#2943	C₂×C₅₃	1	1	yes	characteristic, complemented
353	C₂×C₅₃	—	2	2	no	—
354	C₂×C₅₃	—	4	2	no	—
355	C₂×C₅₃	—	8	1	no	—
356	C₂×C₅₃	—	8	4	no	—
357	C₂×C₅₃	C₂×#548	1	1	yes	characteristic
358	C₂×C₅₃	#3585	1	2	yes	—
359	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	2	yes	—
360	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
361	#559	C₂²×C₅₃	1	4	yes	—
362	#557	C₂²×C₅₃	1	4	yes	complemented
363	#554	—	2	4	no	—
364	#554	C₂²×C₅₃	1	4	yes	complemented
365	#550	C₂²×C₅₃	1	4	yes	—
366	#546	—	2	4	no	—
367	#544	—	2	2	no	—
368	#544	C₂²×C₅₃	1	2	yes	—
369	#540	—	2	1	no	—
370	#540	—	2	1	no	—
371	#535	C₂²×C₅₃	1	2	yes	—
372	#533	—	2	1	no	—
373	#533	C₂²×C₅₃	1	1	yes	characteristic
374	#532	—	2	2	no	—
375	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
376	C₂×#125	—	2	1	no	—
377	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic
378	#520	C₂²×C₅₃	1	2	yes	—
379	#518	C₂²×C₅₃	1	2	yes	—
380	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
381	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
382	#490	C₂²×C₅₃	1	4	yes	—
383	#490	C₂²×C₅₃	1	4	yes	—
384	#490	C₂²×C₅₃	1	4	yes	complemented
385	#489	C₂²×C₅₃	1	4	yes	complemented
386	#488	C₂²×C₅₃	1	4	yes	complemented
387	#486	C₂²×C₅₃	1	4	yes	complemented
388	#479	C₂²×C₅₃	1	4	yes	complemented
389	#478	C₂²×C₅₃	1	4	yes	complemented
390	#478	C₂²×C₅₃	1	4	yes	complemented
391	#478	C₂²×C₅₃	1	4	yes	complemented
392	#476	C₂²×C₅₃	1	4	yes	complemented
393	#474	C₂²×C₅₃	1	4	yes	complemented
394	#464	C₂²×C₅₃	1	4	yes	complemented
395	#463	C₂²×C₅₃	1	4	yes	complemented
396	#460	C₂²×C₅₃	1	4	yes	—
397	#460	C₂²×C₅₃	1	4	yes	complemented
398	#459	C₂²×C₅₃	1	4	yes	—
399	#459	C₂²×C₅₃	1	4	yes	complemented
400	#446	C₂²×C₅₃	1	4	yes	complemented
401	#446	C₂²×C₅₃	1	4	yes	complemented
402	#442	C₂²×C₅₃	1	2	yes	complemented
403	#442	C₂²×C₅₃	1	4	yes	complemented
404	#441	C₂²×C₅₃	1	2	yes	complemented
405	#440	C₂²×C₅₃	1	2	yes	complemented
406	#440	C₂²×C₅₃	1	4	yes	complemented
407	#440	C₂²×C₅₃	1	2	yes	complemented
408	C₄×SD₁₆	C₂²×C₅₃	1	4	yes	complemented
409	C₄×D₈	C₂²×C₅₃	1	4	yes	complemented
410	#428	C₂²×C₅₃	1	2	yes	—
411	#427	C₂²×C₅₃	1	1	yes	characteristic
412	#424	C₂²×C₅₃	1	2	yes	—
413	#423	C₂²×C₅₃	1	1	yes	characteristic
414	#418	C₂²×C₅₃	1	2	yes	—
415	#417	C₂²×C₅₃	1	2	yes	—
416	#416	C₂²×C₅₃	1	1	yes	characteristic
417	#416	C₂²×C₅₃	1	1	yes	characteristic
418	C₂×#102	C₂²×C₅₃	1	2	yes	—
419	C₂×#102	C₂²×C₅₃	1	2	yes	—
420	#405	C₂²×C₅₃	1	2	yes	—
421	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
422	C₅₃	#61337	1	1	yes	characteristic, direct product factor, Sylow
423	C₂×#41	—	2	4	no	—
424	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
425	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
426	C₂²×D₄	—	2	2	no	—
427	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
428	AGL(1,ℤ/8)	—	2	8	no	—
429	#135	—	2	4	no	—
430	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
431	C₂×SD₁₆	C₂³×C₅₃	1	1	yes	characteristic
432	C₂×SD₁₆	C₂³×C₅₃	1	1	yes	characteristic
433	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
434	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
435	C₂×D₈	C₂³×C₅₃	1	4	yes	—
436	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
437	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
438	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
439	C₄⋊Q₈	—	2	4	no	—
440	C₄⋊Q₈	—	2	2	no	—
441	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
442	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
443	#127	—	2	1	no	—
444	#127	—	2	1	no	—
445	#127	C₂³×C₅₃	1	4	yes	—
446	#127	C₂³×C₅₃	1	4	yes	—
447	#125	—	2	2	no	—
448	#125	—	2	4	no	—
449	#125	—	2	4	no	—
450	#125	—	2	1	no	—
451	#125	—	2	1	no	—
452	#125	D₄×C₅₃	1	1	yes	characteristic
453	#125	D₄×C₅₃	1	1	yes	characteristic
454	#125	C₂³×C₅₃	1	4	yes	—
455	#125	C₂³×C₅₃	1	4	yes	—
456	#124	—	2	4	no	—
457	#124	C₂³×C₅₃	1	4	yes	—
458	#123	—	2	4	no	—
459	#123	—	2	4	no	—
460	C₂²⋊Q₈	—	2	4	no	—
461	C₂²⋊Q₈	—	2	4	no	—
462	C₂²⋊Q₈	—	2	4	no	—
463	C₂²⋊Q₈	—	2	4	no	—
464	C₂²⋊Q₈	D₄×C₅₃	1	4	yes	—
465	#121	—	2	2	no	—
466	#121	—	2	2	no	—
467	#121	—	2	4	no	—
468	#121	—	2	4	no	—
469	#121	—	2	4	no	—
470	#120	—	2	4	no	—
471	C₄×Q₈	—	2	4	no	—
472	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
473	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
474	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
475	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
476	C₄×D₄	—	2	2	no	—
477	C₄×D₄	—	2	2	no	—
478	C₄×D₄	—	2	2	no	—
479	C₄×D₄	—	2	2	no	—
480	C₄×D₄	—	2	4	no	—
481	C₄×D₄	—	2	4	no	—
482	C₄×D₄	C₂³×C₅₃	1	2	yes	—
483	C₄×D₄	C₂³×C₅₃	1	2	yes	—
484	C₄×D₄	C₂³×C₅₃	1	4	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	1	no	—
489	#117	—	2	1	no	—
490	#117	C₂³×C₅₃	1	1	yes	characteristic
491	#117	C₂³×C₅₃	1	1	yes	characteristic
492	#117	C₂³×C₅₃	1	2	yes	—
493	C₂×C₄⋊C₄	—	2	2	no	—
494	C₂×C₄⋊C₄	—	2	2	no	—
495	C₂×C₄⋊C₄	—	2	4	no	—
496	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
497	C₂×C₂²⋊C₄	—	2	4	no	—
498	C₂×C₄²	—	2	1	no	—
499	C₂×C₄²	—	2	1	no	—
500	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
501	#107	C₂³×C₅₃	1	2	yes	—
502	#107	C₂³×C₅₃	1	4	yes	—
503	#106	C₂³×C₅₃	1	2	yes	—
504	#106	C₂³×C₅₃	1	4	yes	—
505	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
506	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
507	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
508	#103	C₂³×C₅₃	1	2	yes	—
509	#103	C₂³×C₅₃	1	2	yes	—
510	#103	C₂³×C₅₃	1	4	yes	—
511	#102	C₂³×C₅₃	1	4	yes	—
512	#102	C₂³×C₅₃	1	4	yes	—
513	#102	C₂³×C₅₃	1	4	yes	—
514	#102	C₂³×C₅₃	1	4	yes	—
515	#102	C₂³×C₅₃	1	2	yes	—
516	#102	C₂³×C₅₃	1	2	yes	—
517	#102	C₂³×C₅₃	1	4	yes	—
518	#102	C₂³×C₅₃	1	4	yes	—
519	#102	C₂³×C₅₃	1	4	yes	—
520	#97	C₂³×C₅₃	1	2	yes	—
521	#97	C₂³×C₅₃	1	2	yes	—
522	#97	C₂³×C₅₃	1	4	yes	—
523	C₄×C₈	C₂³×C₅₃	1	4	yes	—
524	C₂⁴	—	2	2	no	—
525	#41	—	2	2	no	—
526	#41	—	2	2	no	—
527	#41	—	2	2	no	—
528	#41	—	8	8	no	—
529	#41	#41×C₅₃	1	4	yes	complemented
530	C₂×Q₈	—	2	4	no	—
531	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
532	C₂×Q₈	C₂⁴×C₅₃	1	1	yes	characteristic
533	C₂×Q₈	C₂⁴×C₅₃	1	1	yes	characteristic
534	C₂×D₄	—	2	1	no	—
535	C₂×D₄	—	2	1	no	—
536	C₂×D₄	—	2	2	no	—
537	C₂×D₄	—	2	4	no	—
538	C₂×D₄	—	2	4	no	—
539	C₂×D₄	—	2	4	no	—
540	C₂×D₄	—	2	4	no	—
541	C₂×D₄	—	2	4	no	—
542	C₂×D₄	—	2	4	no	—
543	C₂×D₄	—	8	4	no	—
544	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
545	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
546	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
547	C₂²×C₄	—	2	1	no	—
548	C₂²×C₄	—	2	2	no	—
549	C₂²×C₄	—	2	1	no	—
550	C₂²×C₄	—	2	2	no	—
551	C₂²×C₄	—	2	2	no	—
552	C₂²×C₄	—	2	4	no	—
553	C₂²×C₄	—	2	4	no	—
554	C₂²×C₄	—	2	4	no	—
555	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
556	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
557	Q₁₆	—	2	2	no	—
558	SD₁₆	—	2	2	no	—
559	SD₁₆	—	2	8	no	—
560	SD₁₆	—	2	2	no	—
561	D₈	—	2	2	no	—
562	D₈	—	2	8	no	—
563	M₁₆	—	2	4	no	—
564	C₂×C₈	—	2	2	no	—
565	C₂×C₈	—	2	2	no	—
566	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
567	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
568	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
569	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
570	C₄⋊C₄	—	2	4	no	—
571	C₄⋊C₄	—	2	2	no	—
572	C₄⋊C₄	—	2	4	no	—
573	C₄⋊C₄	—	2	4	no	—
574	C₄⋊C₄	—	2	4	no	—
575	C₄⋊C₄	—	2	4	no	—
576	C₄⋊C₄	—	2	2	no	—
577	C₄⋊C₄	—	2	2	no	—
578	C₄⋊C₄	—	2	4	no	—
579	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
580	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
581	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
582	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
583	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
584	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
585	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
586	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
587	C₂²⋊C₄	—	2	1	no	—
588	C₂²⋊C₄	—	2	1	no	—
589	C₂²⋊C₄	—	2	4	no	—
590	C₂²⋊C₄	—	2	4	no	—
591	C₂²⋊C₄	—	2	2	no	—
592	C₂²⋊C₄	—	2	2	no	—
593	C₂²⋊C₄	—	2	4	no	—
594	C₂²⋊C₄	—	2	4	no	—
595	C₂²⋊C₄	—	2	4	no	—
596	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
597	C₄²	—	2	1	no	—
598	C₄²	—	2	1	no	—
599	C₄²	—	2	2	no	—
600	C₄²	—	2	2	no	—
601	C₄²	—	2	4	no	—
602	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
603	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
604	C₄²	C₂⁴×C₅₃	1	4	yes	—
605	C₄²	C₂⁴×C₅₃	1	2	yes	—
606	C₂³	—	2	1	no	—
607	C₂³	—	2	4	no	—
608	C₂³	—	2	1	no	—
609	C₂³	—	4	4	no	—
610	C₂³	—	8	2	no	—
611	C₂³	—	8	2	no	—
612	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
613	Q₈	—	2	1	no	—
614	Q₈	—	2	1	no	—
615	Q₈	—	2	1	no	—
616	Q₈	—	4	4	no	—
617	Q₈	—	8	4	no	—
618	Q₈	C₂×#41×C₅₃	1	2	yes	—
619	D₄	—	2	4	no	—
620	D₄	—	2	4	no	—
621	D₄	—	2	1	no	—
622	D₄	—	2	1	no	—
623	D₄	—	2	4	no	—
624	D₄	—	2	1	no	—
625	D₄	—	4	8	no	—
626	D₄	—	4	2	no	—
627	D₄	—	4	4	no	—
628	D₄	—	8	4	no	—
629	D₄	—	8	4	no	—
630	D₄	—	8	4	no	—
631	D₄	C₂×#41×C₅₃	1	2	yes	—
632	C₂×C₄	—	2	1	no	—
633	C₂×C₄	—	2	4	no	—
634	C₂×C₄	—	2	1	no	—
635	C₂×C₄	—	2	2	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	2	4	no	—
638	C₂×C₄	—	4	2	no	—
639	C₂×C₄	—	4	4	no	—
640	C₂×C₄	—	4	2	no	—
641	C₂×C₄	—	8	2	no	—
642	C₂×C₄	—	8	4	no	—
643	C₂×C₄	—	8	4	no	—
644	C₂×C₄	—	8	4	no	—
645	C₂×C₄	—	8	2	no	—
646	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
647	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
648	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
649	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
650	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
651	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
652	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
653	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
654	C₈	—	2	1	no	—
655	C₈	—	2	1	no	—
656	C₈	—	2	4	no	—
657	C₈	—	2	2	no	—
658	C₂²	—	2	2	no	—
659	C₂²	—	2	2	no	—
660	C₂²	—	4	2	no	—
661	C₂²	—	4	1	no	—
662	C₂²	—	4	4	no	—
663	C₂²	—	4	2	no	—
664	C₂²	—	4	4	no	—
665	C₂²	—	4	4	no	—
666	C₂²	—	16	4	no	—
667	C₂²	C₅₃×#548	1	2	yes	—
668	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
669	C₄	—	2	2	no	—
670	C₄	—	2	2	no	—
671	C₄	—	2	2	no	—
672	C₄	—	2	4	no	—
673	C₄	—	2	4	no	—
674	C₄	—	4	1	no	—
675	C₄	—	4	2	no	—
676	C₄	—	4	4	no	—
677	C₄	—	4	2	no	—
678	C₄	C₅₃×#548	1	2	yes	—
679	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
680	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
681	C₂	—	2	2	no	—
682	C₂	—	4	2	no	—
683	C₂	—	8	4	no	—
684	C₂	—	8	1	no	—
685	C₂	C₂×C₅₃×#548	1	1	yes	characteristic
686	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×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3508	C₂	semidirect
2. C₅₃×#3504	C₂	semidirect
3. C₅₃×#3371	C₂	semidirect
4. C₅₃×#3370	C₂	semidirect
5. C₅₃×#3368	C₂	semidirect
6. C₅₃×#3366	C₂	semidirect
7. C₅₃×#3136	C₂	semidirect
8. C₂×C₅₃×#490	C₂	semidirect
9. C₅₃×#3081	C₂	semidirect
10. C₅₃×#3080	C₂	semidirect
11. C₅₃×#3045	C₂	semidirect
12. C₅₃×#3038	C₂	semidirect
13. C₅₃×#2961	C₂	semidirect
14. C₅₃×#2946	C₂	semidirect
15. C₅₃×#2943	C₂	semidirect
19. C₅₃×#557	C₂²	semidirect
21. C₅₃×#554	C₂²	semidirect
41. C₅₃×#490	C₂²	semidirect
42. C₅₃×#489	C₂²	semidirect
43. C₅₃×#488	C₂²	semidirect
44. C₅₃×#486	C₂²	semidirect
45. C₅₃×#479	C₂²	semidirect
46. C₅₃×#478	C₂²	semidirect
47. C₅₃×#478	C₂²	semidirect
48. C₅₃×#478	C₂²	semidirect
49. C₅₃×#476	C₂²	semidirect
50. C₅₃×#474	C₂²	semidirect
51. C₅₃×#464	C₂²	semidirect
52. C₅₃×#463	C₂²	semidirect
54. C₅₃×#460	C₂²	semidirect
56. C₅₃×#459	C₂²	semidirect
57. C₅₃×#446	C₂²	semidirect
58. C₅₃×#446	C₂²	semidirect
59. C₅₃×#442	C₂²	semidirect
60. C₅₃×#442	C₂²	semidirect
61. C₅₃×#441	C₂²	semidirect
62. C₅₃×#440	C₂²	semidirect
63. C₅₃×#440	C₂²	semidirect
64. C₅₃×#440	C₂²	semidirect
65. C₄×SD₁₆×C₅₃	C₂²	semidirect
66. C₄×D₈×C₅₃	C₂²	semidirect
185. #41×C₅₃	#41	semidirect
314. #61337	C₅₃	direct (N × H)
338. #3508	C₂×C₅₃	semidirect
339. #3504	C₂×C₅₃	semidirect
340. #3371	C₂×C₅₃	semidirect
341. #3370	C₂×C₅₃	semidirect
342. #3368	C₂×C₅₃	semidirect
343. #3366	C₂×C₅₃	semidirect
344. #3136	C₂×C₅₃	semidirect
345. C₂×#490	C₂×C₅₃	semidirect
346. #3081	C₂×C₅₃	semidirect
347. #3080	C₂×C₅₃	semidirect
348. #3045	C₂×C₅₃	semidirect
349. #3038	C₂×C₅₃	semidirect
350. #2961	C₂×C₅₃	semidirect
351. #2946	C₂×C₅₃	semidirect
352. #2943	C₂×C₅₃	semidirect
362. #557	C₂²×C₅₃	semidirect
364. #554	C₂²×C₅₃	semidirect
384. #490	C₂²×C₅₃	semidirect
385. #489	C₂²×C₅₃	semidirect
386. #488	C₂²×C₅₃	semidirect
387. #486	C₂²×C₅₃	semidirect
388. #479	C₂²×C₅₃	semidirect
389. #478	C₂²×C₅₃	semidirect
390. #478	C₂²×C₅₃	semidirect
391. #478	C₂²×C₅₃	semidirect
392. #476	C₂²×C₅₃	semidirect
393. #474	C₂²×C₅₃	semidirect
394. #464	C₂²×C₅₃	semidirect
395. #463	C₂²×C₅₃	semidirect
397. #460	C₂²×C₅₃	semidirect
399. #459	C₂²×C₅₃	semidirect
400. #446	C₂²×C₅₃	semidirect
401. #446	C₂²×C₅₃	semidirect
402. #442	C₂²×C₅₃	semidirect
403. #442	C₂²×C₅₃	semidirect
404. #441	C₂²×C₅₃	semidirect
405. #440	C₂²×C₅₃	semidirect
406. #440	C₂²×C₅₃	semidirect
407. #440	C₂²×C₅₃	semidirect
408. C₄×SD₁₆	C₂²×C₅₃	semidirect
409. C₄×D₈	C₂²×C₅₃	semidirect
422. C₅₃	#61337	direct (N × H)
529. #41	#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×#3508	2
C₅₃×#3504	2
C₅₃×#3371	2
C₅₃×#3370	2
C₅₃×#3368	2
C₅₃×#3366	2
C₅₃×#3136	2
C₂×C₅₃×#490	2
C₅₃×#3081	2
C₅₃×#3080	2
C₅₃×#3045	2
C₅₃×#3038	2
C₅₃×#2961	2
C₅₃×#2946	2
C₅₃×#2943	2
#61337	53

Subgroups — C53×#61337 #27268849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#61337 #27268849