Subgroups — C₅₃×#18641 #27226153

Order 13568.

1114 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3499	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#3227	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3079	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1637	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1635	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1492	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1476	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×2₋¹⁺⁴×C₅₃	—	2	1	no	—
11	D₄×Q₈×C₅₃	—	2	1	no	—
12	D₄×Q₈×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#547	—	2	2	no	—
14	C₅₃×#547	—	2	1	no	—
15	C₅₃×#547	—	2	2	no	—
16	C₅₃×#547	C₂²	1	2	yes	—
17	C₅₃×#536	—	2	2	no	—
18	C₅₃×#536	—	2	1	no	—
19	C₅₃×#532	—	2	1	no	—
20	C₅₃×#532	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#498	—	2	1	no	—
26	C₅₃×#487	—	2	2	no	—
27	C₅₃×#487	C₂²	1	4	yes	—
28	C₅₃×#486	—	2	1	no	—
29	C₅₃×#481	—	2	2	no	—
30	C₅₃×#477	C₂²	1	2	yes	—
31	C₅₃×#476	C₂²	1	4	yes	—
32	C₅₃×#476	C₂²	1	2	yes	—
33	C₅₃×#475	—	2	1	no	—
34	C₅₃×#474	—	2	1	no	—
35	C₅₃×#431	C₂²	1	1	yes	characteristic
36	C₅₃×#428	—	2	1	no	—
37	C₅₃×#423	C₂²	1	1	yes	characteristic
38	C₅₃×#417	—	2	1	no	—
39	C₂×#103×C₅₃	—	2	2	no	—
40	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#332	C₂²	1	4	yes	—
42	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
43	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
44	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
45	2₋¹⁺⁴×C₅₃	—	4	4	no	—
46	2₋¹⁺⁴×C₅₃	—	4	4	no	—
47	C₂×#41×C₅₃	—	2	2	no	—
48	C₂×#41×C₅₃	—	2	4	no	—
49	C₂×#41×C₅₃	—	2	2	no	—
50	C₂×#41×C₅₃	—	2	1	no	—
51	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
52	C₂²×Q₈×C₅₃	—	2	2	no	—
53	C₂²×Q₈×C₅₃	—	2	2	no	—
54	C₂²×Q₈×C₅₃	—	2	1	no	—
55	C₂²×Q₈×C₅₃	—	2	1	no	—
56	C₂×M₁₆×C₅₃	—	2	1	no	—
57	C₂²×C₈×C₅₃	—	2	1	no	—
58	C₄⋊Q₈×C₅₃	—	2	2	no	—
59	C₄⋊Q₈×C₅₃	—	2	2	no	—
60	C₄⋊Q₈×C₅₃	—	2	2	no	—
61	C₄⋊Q₈×C₅₃	—	2	1	no	—
62	C₄⋊Q₈×C₅₃	—	2	1	no	—
63	C₄⋊Q₈×C₅₃	—	2	1	no	—
64	C₄⋊Q₈×C₅₃	—	2	1	no	—
65	C₄⋊Q₈×C₅₃	—	2	1	no	—
66	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
67	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
68	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
70	#127×C₅₃	D₄	1	1	yes	characteristic
71	#124×C₅₃	—	2	2	no	—
72	#124×C₅₃	—	2	1	no	—
73	#123×C₅₃	—	2	4	no	—
74	#123×C₅₃	—	2	4	no	—
75	#123×C₅₃	—	2	4	no	—
76	C₂²⋊Q₈×C₅₃	—	2	4	no	—
77	C₂²⋊Q₈×C₅₃	—	2	4	no	—
78	C₂²⋊Q₈×C₅₃	—	2	2	no	—
79	C₂²⋊Q₈×C₅₃	—	2	2	no	—
80	#121×C₅₃	—	2	2	no	—
81	#121×C₅₃	—	2	2	no	—
82	#121×C₅₃	—	2	1	no	—
83	#121×C₅₃	D₄	1	2	yes	—
84	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
85	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄×D₄×C₅₃	—	2	2	no	—
87	C₄×D₄×C₅₃	—	2	1	no	—
88	C₄×D₄×C₅₃	—	2	2	no	—
89	C₄×D₄×C₅₃	—	2	1	no	—
90	C₄×D₄×C₅₃	—	2	1	no	—
91	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
92	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×D₄×C₅₃	C₂³	1	2	yes	—
94	#117×C₅₃	—	2	1	no	—
95	#117×C₅₃	D₄	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
100	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
101	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
102	C₂×C₄²×C₅₃	—	2	1	no	—
103	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
104	#107×C₅₃	—	2	2	no	—
105	#106×C₅₃	—	2	4	no	—
106	#106×C₅₃	—	2	2	no	—
107	C₄⋊C₈×C₅₃	—	2	1	no	—
108	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
109	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
110	#103×C₅₃	—	2	8	no	—
111	#103×C₅₃	—	2	2	no	—
112	#102×C₅₃	—	2	4	no	—
113	#102×C₅₃	—	2	2	no	—
114	C₂²⋊C₈×C₅₃	—	2	1	no	—
115	C₄×C₈×C₅₃	—	2	1	no	—
116	C₄×C₈×C₅₃	—	2	1	no	—
117	C₄×C₈×C₅₃	C₂³	1	2	yes	—
118	#41×C₅₃	—	2	2	no	—
119	#41×C₅₃	—	4	8	no	—
120	#41×C₅₃	—	4	16	no	—
121	#41×C₅₃	—	4	4	no	—
122	#41×C₅₃	—	4	4	no	—
123	#41×C₅₃	—	4	2	no	—
124	#41×C₅₃	—	4	2	no	—
125	#41×C₅₃	SD₁₆	1	4	yes	—
126	C₂×Q₈×C₅₃	—	2	4	no	—
127	C₂×Q₈×C₅₃	—	2	2	no	—
128	C₂×Q₈×C₅₃	—	2	1	no	—
129	C₂×Q₈×C₅₃	—	2	2	no	—
130	C₂×Q₈×C₅₃	—	2	2	no	—
131	C₂×Q₈×C₅₃	—	4	8	no	—
132	C₂×Q₈×C₅₃	—	4	4	no	—
133	C₂×Q₈×C₅₃	—	4	4	no	—
134	C₂×Q₈×C₅₃	—	4	4	no	—
135	C₂×Q₈×C₅₃	—	4	1	no	—
136	C₂×Q₈×C₅₃	—	4	2	no	—
137	C₂×Q₈×C₅₃	—	4	2	no	—
138	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
139	C₂×D₄×C₅₃	—	2	1	no	—
140	C₂×D₄×C₅₃	—	2	4	no	—
141	C₂×D₄×C₅₃	—	2	2	no	—
142	C₂×D₄×C₅₃	—	2	1	no	—
143	C₂×D₄×C₅₃	—	2	1	no	—
144	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
145	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₅₃	—	2	1	no	—
147	C₂²×C₄×C₅₃	—	2	4	no	—
148	C₂²×C₄×C₅₃	—	2	4	no	—
149	C₂²×C₄×C₅₃	—	2	2	no	—
150	C₂²×C₄×C₅₃	—	2	1	no	—
151	C₂²×C₄×C₅₃	—	2	1	no	—
152	C₂²×C₄×C₅₃	—	2	2	no	—
153	C₂²×C₄×C₅₃	—	4	1	no	—
154	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
157	M₁₆×C₅₃	—	4	2	no	—
158	C₂×C₈×C₅₃	—	2	2	no	—
159	C₂×C₈×C₅₃	—	2	2	no	—
160	C₂×C₈×C₅₃	—	2	2	no	—
161	C₂×C₈×C₅₃	—	4	2	no	—
162	C₄⋊C₄×C₅₃	—	2	4	no	—
163	C₄⋊C₄×C₅₃	—	2	4	no	—
164	C₄⋊C₄×C₅₃	—	2	1	no	—
165	C₄⋊C₄×C₅₃	—	2	2	no	—
166	C₄⋊C₄×C₅₃	—	2	2	no	—
167	C₄⋊C₄×C₅₃	—	2	1	no	—
168	C₄⋊C₄×C₅₃	—	2	2	no	—
169	C₄⋊C₄×C₅₃	—	2	1	no	—
170	C₄⋊C₄×C₅₃	—	2	2	no	—
171	C₄⋊C₄×C₅₃	—	2	2	no	—
172	C₄⋊C₄×C₅₃	—	4	1	no	—
173	C₄⋊C₄×C₅₃	—	4	1	no	—
174	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
175	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
176	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
177	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
178	C₂²⋊C₄×C₅₃	—	2	4	no	—
179	C₂²⋊C₄×C₅₃	—	2	4	no	—
180	C₂²⋊C₄×C₅₃	—	2	1	no	—
181	C₂²⋊C₄×C₅₃	—	2	1	no	—
182	C₂²⋊C₄×C₅₃	—	2	2	no	—
183	C₂²⋊C₄×C₅₃	—	2	1	no	—
184	C₂²⋊C₄×C₅₃	—	2	2	no	—
185	C₄²×C₅₃	—	2	1	no	—
186	C₄²×C₅₃	—	2	1	no	—
187	C₄²×C₅₃	—	2	1	no	—
188	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
189	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
191	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
192	C₂³×C₅₃	—	2	1	no	—
193	C₂³×C₅₃	—	2	2	no	—
194	C₂³×C₅₃	#120	1	1	yes	characteristic
195	Q₈×C₅₃	—	2	1	no	—
196	Q₈×C₅₃	—	4	8	no	—
197	Q₈×C₅₃	—	4	4	no	—
198	Q₈×C₅₃	—	4	2	no	—
199	Q₈×C₅₃	—	4	2	no	—
200	Q₈×C₅₃	—	4	2	no	—
201	Q₈×C₅₃	—	4	4	no	—
202	Q₈×C₅₃	—	8	2	no	—
203	Q₈×C₅₃	C₂×SD₁₆	1	2	yes	—
204	D₄×C₅₃	—	2	4	no	—
205	D₄×C₅₃	—	2	1	no	—
206	D₄×C₅₃	—	2	4	no	—
207	D₄×C₅₃	—	4	8	no	—
208	D₄×C₅₃	—	4	4	no	—
209	D₄×C₅₃	—	4	2	no	—
210	D₄×C₅₃	—	4	2	no	—
211	D₄×C₅₃	C₂×SD₁₆	1	2	yes	—
212	C₂×C₄×C₅₃	—	2	1	no	—
213	C₂×C₄×C₅₃	—	2	2	no	—
214	C₂×C₄×C₅₃	—	2	4	no	—
215	C₂×C₄×C₅₃	—	2	2	no	—
216	C₂×C₄×C₅₃	—	2	2	no	—
217	C₂×C₄×C₅₃	—	2	4	no	—
218	C₂×C₄×C₅₃	—	2	1	no	—
219	C₂×C₄×C₅₃	—	2	1	no	—
220	C₂×C₄×C₅₃	—	4	8	no	—
221	C₂×C₄×C₅₃	—	4	8	no	—
222	C₂×C₄×C₅₃	—	4	4	no	—
223	C₂×C₄×C₅₃	—	4	2	no	—
224	C₂×C₄×C₅₃	—	4	2	no	—
225	C₂×C₄×C₅₃	—	4	2	no	—
226	C₂×C₄×C₅₃	—	4	2	no	—
227	C₂×C₄×C₅₃	—	4	2	no	—
228	C₂×C₄×C₅₃	—	8	2	no	—
229	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
230	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
231	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
232	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
233	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
234	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
237	C₈×C₅₃	—	4	2	no	—
238	C₈×C₅₃	—	4	2	no	—
239	C₈×C₅₃	—	4	2	no	—
240	#18641	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
241	C₂²×C₅₃	—	2	1	no	—
242	C₂²×C₅₃	—	2	1	no	—
243	C₂²×C₅₃	—	2	2	no	—
244	C₂²×C₅₃	—	4	1	no	—
245	C₂²×C₅₃	—	4	1	no	—
246	C₂²×C₅₃	—	4	2	no	—
247	C₂²×C₅₃	—	4	2	no	—
248	C₂²×C₅₃	—	4	2	no	—
249	C₂²×C₅₃	#450	1	2	yes	—
250	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
251	C₄×C₅₃	—	2	2	no	—
252	C₄×C₅₃	—	2	1	no	—
253	C₄×C₅₃	—	2	1	no	—
254	C₄×C₅₃	—	4	4	no	—
255	C₄×C₅₃	—	4	2	no	—
256	C₄×C₅₃	—	4	1	no	—
257	C₄×C₅₃	—	4	1	no	—
258	C₄×C₅₃	—	8	2	no	—
259	C₄×C₅₃	#450	1	2	yes	—
260	C₄×C₅₃	#454	1	2	yes	—
261	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
262	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
263	C₄×C₅₃	#578	1	2	yes	—
264	#3499	C₂×C₅₃	1	1	yes	characteristic
265	#3227	C₂×C₅₃	1	1	yes	characteristic, complemented
266	C₂×#487	C₂×C₅₃	1	1	yes	characteristic, complemented
267	#3079	C₂×C₅₃	1	1	yes	characteristic, complemented
268	#1637	C₂×C₅₃	1	4	yes	complemented
269	#1635	C₂×C₅₃	1	4	yes	complemented
270	#1546	C₂×C₅₃	1	1	yes	characteristic, complemented
271	#1492	C₂×C₅₃	1	1	yes	characteristic, complemented
272	#1476	C₂×C₅₃	1	1	yes	characteristic, complemented
273	C₂×C₅₃	—	2	2	no	—
274	C₂×C₅₃	—	4	2	no	—
275	C₂×C₅₃	—	8	2	no	—
276	C₂×C₅₃	C₂×#450	1	1	yes	characteristic
277	C₂×C₅₃	#3011	1	1	yes	characteristic
278	C₂×C₅₃	C₂×#454	1	1	yes	characteristic
279	C₂×2₋¹⁺⁴	—	2	1	no	—
280	D₄×Q₈	—	2	1	no	—
281	D₄×Q₈	C₂²×C₅₃	1	2	yes	—
282	#547	—	2	2	no	—
283	#547	—	2	1	no	—
284	#547	—	2	2	no	—
285	#547	C₂²×C₅₃	1	2	yes	—
286	#536	—	2	2	no	—
287	#536	—	2	1	no	—
288	#532	—	2	1	no	—
289	#532	C₂²×C₅₃	1	1	yes	characteristic
290	C₂×C₄⋊Q₈	—	2	1	no	—
291	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
292	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
293	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
294	#498	—	2	1	no	—
295	#487	—	2	2	no	—
296	#487	C₂²×C₅₃	1	4	yes	—
297	#486	—	2	1	no	—
298	#481	—	2	2	no	—
299	#477	C₂²×C₅₃	1	2	yes	—
300	#476	C₂²×C₅₃	1	2	yes	—
301	#476	C₂²×C₅₃	1	4	yes	—
302	#475	—	2	1	no	—
303	#474	—	2	1	no	—
304	#431	C₂²×C₅₃	1	1	yes	characteristic
305	#428	—	2	1	no	—
306	#423	C₂²×C₅₃	1	1	yes	characteristic
307	#417	—	2	1	no	—
308	C₂×#103	—	2	2	no	—
309	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
310	#332	C₂²×C₅₃	1	4	yes	—
311	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
312	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
313	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
314	C₅₃	#18641	1	1	yes	characteristic, direct product factor, Sylow
315	2₋¹⁺⁴	—	4	4	no	—
316	2₋¹⁺⁴	—	4	4	no	—
317	C₂×#41	—	2	2	no	—
318	C₂×#41	—	2	4	no	—
319	C₂×#41	—	2	2	no	—
320	C₂×#41	—	2	1	no	—
321	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
322	C₂²×Q₈	—	2	2	no	—
323	C₂²×Q₈	—	2	2	no	—
324	C₂²×Q₈	—	2	1	no	—
325	C₂²×Q₈	—	2	1	no	—
326	C₂×M₁₆	—	2	1	no	—
327	C₂²×C₈	—	2	1	no	—
328	C₄⋊Q₈	—	2	2	no	—
329	C₄⋊Q₈	—	2	2	no	—
330	C₄⋊Q₈	—	2	2	no	—
331	C₄⋊Q₈	—	2	1	no	—
332	C₄⋊Q₈	—	2	1	no	—
333	C₄⋊Q₈	—	2	1	no	—
334	C₄⋊Q₈	—	2	1	no	—
335	C₄⋊Q₈	—	2	1	no	—
336	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
337	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
338	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
339	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
340	#127	D₄×C₅₃	1	1	yes	characteristic
341	#124	—	2	2	no	—
342	#124	—	2	1	no	—
343	#123	—	2	4	no	—
344	#123	—	2	4	no	—
345	#123	—	2	4	no	—
346	C₂²⋊Q₈	—	2	4	no	—
347	C₂²⋊Q₈	—	2	2	no	—
348	C₂²⋊Q₈	—	2	4	no	—
349	C₂²⋊Q₈	—	2	2	no	—
350	#121	—	2	2	no	—
351	#121	—	2	2	no	—
352	#121	—	2	1	no	—
353	#121	D₄×C₅₃	1	2	yes	—
354	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
355	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
356	C₄×D₄	—	2	2	no	—
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	2	1	no	—
359	C₄×D₄	—	2	1	no	—
360	C₄×D₄	—	2	1	no	—
361	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
362	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
363	C₄×D₄	C₂³×C₅₃	1	2	yes	—
364	#117	—	2	1	no	—
365	#117	D₄×C₅₃	1	1	yes	characteristic
366	C₂×C₄⋊C₄	—	2	2	no	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
372	C₂×C₄²	—	2	1	no	—
373	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
374	#107	—	2	2	no	—
375	#106	—	2	2	no	—
376	#106	—	2	4	no	—
377	C₄⋊C₈	—	2	1	no	—
378	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
379	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
380	#103	—	2	2	no	—
381	#103	—	2	8	no	—
382	#102	—	2	2	no	—
383	#102	—	2	4	no	—
384	C₂²⋊C₈	—	2	1	no	—
385	C₄×C₈	—	2	1	no	—
386	C₄×C₈	—	2	1	no	—
387	C₄×C₈	C₂³×C₅₃	1	2	yes	—
388	#41	—	2	2	no	—
389	#41	—	4	8	no	—
390	#41	—	4	16	no	—
391	#41	—	4	4	no	—
392	#41	—	4	4	no	—
393	#41	—	4	2	no	—
394	#41	—	4	2	no	—
395	#41	SD₁₆×C₅₃	1	4	yes	—
396	C₂×Q₈	—	2	4	no	—
397	C₂×Q₈	—	2	2	no	—
398	C₂×Q₈	—	2	1	no	—
399	C₂×Q₈	—	2	2	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	4	8	no	—
402	C₂×Q₈	—	4	4	no	—
403	C₂×Q₈	—	4	4	no	—
404	C₂×Q₈	—	4	4	no	—
405	C₂×Q₈	—	4	1	no	—
406	C₂×Q₈	—	4	2	no	—
407	C₂×Q₈	—	4	2	no	—
408	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
409	C₂×D₄	—	2	1	no	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	2	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
415	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	4	1	no	—
424	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
425	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
426	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
427	M₁₆	—	4	2	no	—
428	C₂×C₈	—	2	2	no	—
429	C₂×C₈	—	2	2	no	—
430	C₂×C₈	—	2	2	no	—
431	C₂×C₈	—	4	2	no	—
432	C₄⋊C₄	—	2	4	no	—
433	C₄⋊C₄	—	2	4	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	2	no	—
436	C₄⋊C₄	—	2	2	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	—	2	1	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	2	no	—
442	C₄⋊C₄	—	4	1	no	—
443	C₄⋊C₄	—	4	1	no	—
444	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
445	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
446	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
447	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
448	C₂²⋊C₄	—	2	4	no	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	—	2	1	no	—
451	C₂²⋊C₄	—	2	1	no	—
452	C₂²⋊C₄	—	2	2	no	—
453	C₂²⋊C₄	—	2	1	no	—
454	C₂²⋊C₄	—	2	2	no	—
455	C₄²	—	2	1	no	—
456	C₄²	—	2	1	no	—
457	C₄²	—	2	1	no	—
458	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
459	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
460	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
461	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
462	C₂³	—	2	1	no	—
463	C₂³	—	2	2	no	—
464	C₂³	#120×C₅₃	1	1	yes	characteristic
465	Q₈	—	2	1	no	—
466	Q₈	—	4	8	no	—
467	Q₈	—	4	4	no	—
468	Q₈	—	4	2	no	—
469	Q₈	—	4	2	no	—
470	Q₈	—	4	2	no	—
471	Q₈	—	4	4	no	—
472	Q₈	—	8	2	no	—
473	Q₈	C₂×SD₁₆×C₅₃	1	2	yes	—
474	D₄	—	2	4	no	—
475	D₄	—	2	1	no	—
476	D₄	—	2	4	no	—
477	D₄	—	4	8	no	—
478	D₄	—	4	4	no	—
479	D₄	—	4	2	no	—
480	D₄	—	4	2	no	—
481	D₄	C₂×SD₁₆×C₅₃	1	2	yes	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	4	8	no	—
491	C₂×C₄	—	4	8	no	—
492	C₂×C₄	—	4	4	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	4	2	no	—
497	C₂×C₄	—	4	2	no	—
498	C₂×C₄	—	8	2	no	—
499	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
500	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
501	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
502	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
503	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
504	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
505	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
506	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
507	C₈	—	4	2	no	—
508	C₈	—	4	2	no	—
509	C₈	—	4	2	no	—
510	C₂²	—	2	1	no	—
511	C₂²	—	2	1	no	—
512	C₂²	—	2	2	no	—
513	C₂²	—	4	1	no	—
514	C₂²	—	4	1	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	4	2	no	—
517	C₂²	—	4	2	no	—
518	C₂²	C₅₃×#450	1	2	yes	—
519	C₂²	C₂×#120×C₅₃	1	1	yes	characteristic
520	C₄	—	2	2	no	—
521	C₄	—	2	1	no	—
522	C₄	—	2	1	no	—
523	C₄	—	4	4	no	—
524	C₄	—	4	2	no	—
525	C₄	—	4	1	no	—
526	C₄	—	4	1	no	—
527	C₄	—	8	2	no	—
528	C₄	C₅₃×#450	1	2	yes	—
529	C₄	C₅₃×#454	1	2	yes	—
530	C₄	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
531	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
532	C₄	C₅₃×#578	1	2	yes	—
533	C₂	—	2	2	no	—
534	C₂	—	4	2	no	—
535	C₂	—	8	2	no	—
536	C₂	C₂×C₅₃×#450	1	1	yes	characteristic
537	C₂	C₅₃×#3011	1	1	yes	characteristic
538	C₂	C₂×C₅₃×#454	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3499	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#3227	C₂	semidirect
3. C₂×C₅₃×#487	C₂	semidirect
4. C₅₃×#3079	C₂	semidirect
5. C₅₃×#1637	C₂	semidirect
6. C₅₃×#1635	C₂	semidirect
7. C₅₃×#1546	C₂	semidirect
8. C₅₃×#1492	C₂	semidirect
9. C₅₃×#1476	C₂	semidirect
240. #18641	C₅₃	direct (N × H)
265. #3227	C₂×C₅₃	semidirect
266. C₂×#487	C₂×C₅₃	semidirect
267. #3079	C₂×C₅₃	semidirect
268. #1637	C₂×C₅₃	semidirect
269. #1635	C₂×C₅₃	semidirect
270. #1546	C₂×C₅₃	semidirect
271. #1492	C₂×C₅₃	semidirect
272. #1476	C₂×C₅₃	semidirect
314. C₅₃	#18641	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×#3499	2
C₅₃×#3227	2
C₂×C₅₃×#487	2
C₅₃×#3079	2
C₅₃×#1637	2
C₅₃×#1635	2
C₅₃×#1546	2
C₅₃×#1492	2
C₅₃×#1476	2
#18641	53

Subgroups — C53×#18641 #27226153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18641 #27226153