Subgroups — C₅₃×#18539 #27226051

Order 13568.

1178 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#547	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#3165	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3110	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3110	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1654	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1647	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1556	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1490	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1490	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₅₃	—	2	1	no	—
11	C₅₃×#547	—	2	2	no	—
12	C₅₃×#547	—	2	2	no	—
13	C₅₃×#547	—	2	2	no	—
14	C₅₃×#547	C₂²	1	4	yes	—
15	C₅₃×#533	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#123×C₅₃	—	2	2	no	—
18	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
19	C₂×#121×C₅₃	—	2	1	no	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
23	C₅₃×#501	—	2	2	no	—
24	C₅₃×#500	C₂²	1	2	yes	complemented
25	C₅₃×#498	C₂²	1	2	yes	complemented
26	C₅₃×#484	—	2	1	no	—
27	C₅₃×#484	—	2	1	no	—
28	C₅₃×#474	C₂²	1	2	yes	complemented
29	C₅₃×#474	C₂²	1	2	yes	complemented
30	C₅₃×#461	C₂²	1	2	yes	complemented
31	C₅₃×#461	C₂²	1	2	yes	complemented
32	C₅₃×#450	—	2	1	no	—
33	C₅₃×#450	—	2	1	no	—
34	C₅₃×#440	—	2	2	no	—
35	C₅₃×#440	—	2	2	no	—
36	C₅₃×#432	C₂²	1	1	yes	characteristic
37	C₅₃×#432	C₂²	1	1	yes	characteristic
38	C₅₃×#428	—	2	2	no	—
39	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#332	C₂²	1	4	yes	complemented
41	C₅₃×D₄⋊C₈	C₂²	1	4	yes	complemented
42	C₅₃×D₄⋊C₈	C₂²	1	4	yes	complemented
43	C₂×#41×C₅₃	—	2	4	no	—
44	C₂×#41×C₅₃	—	2	4	no	—
45	C₂×#41×C₅₃	—	4	8	no	—
46	C₂²×Q₈×C₅₃	—	2	1	no	—
47	C₂²×D₄×C₅₃	—	2	2	no	—
48	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
49	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
50	C₂³×C₄×C₅₃	—	2	2	no	—
51	C₂³×C₄×C₅₃	—	2	1	no	—
52	C₂³×C₄×C₅₃	—	2	1	no	—
53	C₂×SD₁₆×C₅₃	—	2	2	no	—
54	C₂×SD₁₆×C₅₃	—	2	2	no	—
55	C₂×M₁₆×C₅₃	—	2	1	no	—
56	C₄⋊Q₈×C₅₃	—	2	1	no	—
57	C₄⋊Q₈×C₅₃	—	2	1	no	—
58	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
59	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
62	#125×C₅₃	—	2	1	no	—
63	#123×C₅₃	—	2	8	no	—
64	#123×C₅₃	—	2	8	no	—
65	C₂²⋊Q₈×C₅₃	—	2	4	no	—
66	C₂²⋊Q₈×C₅₃	—	2	4	no	—
67	C₂²⋊Q₈×C₅₃	—	2	1	no	—
68	C₂²⋊Q₈×C₅₃	—	2	1	no	—
69	#121×C₅₃	—	2	4	no	—
70	#121×C₅₃	—	2	4	no	—
71	C₄×Q₈×C₅₃	—	2	1	no	—
72	C₄×Q₈×C₅₃	—	2	1	no	—
73	C₄×D₄×C₅₃	—	2	1	no	—
74	C₄×D₄×C₅₃	—	2	1	no	—
75	C₄×D₄×C₅₃	—	2	1	no	—
76	C₄×D₄×C₅₃	—	2	1	no	—
77	C₄×D₄×C₅₃	D₄	1	2	yes	—
78	C₄×D₄×C₅₃	D₄	1	2	yes	—
79	C₄×D₄×C₅₃	C₂³	1	2	yes	—
80	C₄×D₄×C₅₃	C₂³	1	2	yes	—
81	#117×C₅₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
83	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
84	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
85	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
89	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
92	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
93	#107×C₅₃	—	2	4	no	—
94	#107×C₅₃	—	2	2	no	—
95	#107×C₅₃	—	2	2	no	—
96	#106×C₅₃	—	2	4	no	—
97	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
98	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
99	#103×C₅₃	—	2	2	no	—
100	#103×C₅₃	—	2	2	no	—
101	#102×C₅₃	—	2	2	no	—
102	#102×C₅₃	—	2	2	no	—
103	C₂²⋊C₈×C₅₃	—	2	1	no	—
104	C₂²⋊C₈×C₅₃	—	2	1	no	—
105	#97×C₅₃	—	2	1	no	—
106	#97×C₅₃	—	2	1	no	—
107	#97×C₅₃	—	2	1	no	—
108	C₄×C₈×C₅₃	C₂³	1	2	yes	—
109	C₂⁴×C₅₃	—	2	1	no	—
110	C₂⁴×C₅₃	—	2	1	no	—
111	#41×C₅₃	—	4	16	no	—
112	#41×C₅₃	—	4	8	no	—
113	#41×C₅₃	—	4	8	no	—
114	C₂×Q₈×C₅₃	—	2	2	no	—
115	C₂×Q₈×C₅₃	—	2	2	no	—
116	C₂×Q₈×C₅₃	—	2	1	no	—
117	C₂×Q₈×C₅₃	—	2	1	no	—
118	C₂×Q₈×C₅₃	—	4	4	no	—
119	C₂×D₄×C₅₃	—	2	4	no	—
120	C₂×D₄×C₅₃	—	2	8	no	—
121	C₂×D₄×C₅₃	—	2	1	no	—
122	C₂×D₄×C₅₃	—	2	1	no	—
123	C₂×D₄×C₅₃	—	4	2	no	—
124	C₂×D₄×C₅₃	—	4	8	no	—
125	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
126	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₅₃	—	2	8	no	—
128	C₂²×C₄×C₅₃	—	2	1	no	—
129	C₂²×C₄×C₅₃	—	2	1	no	—
130	C₂²×C₄×C₅₃	—	2	2	no	—
131	C₂²×C₄×C₅₃	—	2	2	no	—
132	C₂²×C₄×C₅₃	—	2	2	no	—
133	C₂²×C₄×C₅₃	—	2	2	no	—
134	C₂²×C₄×C₅₃	—	2	2	no	—
135	C₂²×C₄×C₅₃	—	4	4	no	—
136	C₂²×C₄×C₅₃	—	4	4	no	—
137	C₂²×C₄×C₅₃	—	4	8	no	—
138	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
141	SD₁₆×C₅₃	—	4	4	no	—
142	SD₁₆×C₅₃	—	4	4	no	—
143	M₁₆×C₅₃	—	2	4	no	—
144	C₂×C₈×C₅₃	—	2	2	no	—
145	C₂×C₈×C₅₃	—	2	2	no	—
146	C₂×C₈×C₅₃	—	2	2	no	—
147	C₄⋊C₄×C₅₃	—	2	8	no	—
148	C₄⋊C₄×C₅₃	—	2	1	no	—
149	C₄⋊C₄×C₅₃	—	2	2	no	—
150	C₄⋊C₄×C₅₃	—	2	1	no	—
151	C₄⋊C₄×C₅₃	—	2	2	no	—
152	C₄⋊C₄×C₅₃	—	2	2	no	—
153	C₄⋊C₄×C₅₃	—	2	1	no	—
154	C₄⋊C₄×C₅₃	—	2	1	no	—
155	C₄⋊C₄×C₅₃	—	2	1	no	—
156	C₄⋊C₄×C₅₃	—	2	1	no	—
157	C₄⋊C₄×C₅₃	—	4	1	no	—
158	C₄⋊C₄×C₅₃	—	4	1	no	—
159	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
160	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
161	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
162	C₂²⋊C₄×C₅₃	—	2	8	no	—
163	C₂²⋊C₄×C₅₃	—	2	2	no	—
164	C₂²⋊C₄×C₅₃	—	2	2	no	—
165	C₂²⋊C₄×C₅₃	—	2	2	no	—
166	C₂²⋊C₄×C₅₃	—	2	2	no	—
167	C₂²⋊C₄×C₅₃	—	4	1	no	—
168	C₄²×C₅₃	—	4	1	no	—
169	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
171	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
173	C₂³×C₅₃	—	2	4	no	—
174	C₂³×C₅₃	—	2	2	no	—
175	C₂³×C₅₃	—	2	2	no	—
176	C₂³×C₅₃	—	4	2	no	—
177	C₂³×C₅₃	—	4	2	no	—
178	C₂³×C₅₃	—	4	2	no	—
179	C₂³×C₅₃	—	4	2	no	—
180	C₂³×C₅₃	—	4	2	no	—
181	C₂³×C₅₃	#120	1	1	yes	characteristic
182	Q₈×C₅₃	—	4	4	no	—
183	Q₈×C₅₃	—	4	4	no	—
184	Q₈×C₅₃	—	4	2	no	—
185	Q₈×C₅₃	—	4	2	no	—
186	D₄×C₅₃	—	2	4	no	—
187	D₄×C₅₃	—	2	2	no	—
188	D₄×C₅₃	—	2	2	no	—
189	D₄×C₅₃	—	2	2	no	—
190	D₄×C₅₃	—	4	2	no	—
191	D₄×C₅₃	—	4	16	no	—
192	D₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
193	C₂×C₄×C₅₃	—	2	2	no	—
194	C₂×C₄×C₅₃	—	2	2	no	—
195	C₂×C₄×C₅₃	—	2	2	no	—
196	C₂×C₄×C₅₃	—	2	2	no	—
197	C₂×C₄×C₅₃	—	2	4	no	—
198	C₂×C₄×C₅₃	—	2	2	no	—
199	C₂×C₄×C₅₃	—	4	4	no	—
200	C₂×C₄×C₅₃	—	4	4	no	—
201	C₂×C₄×C₅₃	—	4	2	no	—
202	C₂×C₄×C₅₃	—	4	2	no	—
203	C₂×C₄×C₅₃	—	4	4	no	—
204	C₂×C₄×C₅₃	—	4	16	no	—
205	C₂×C₄×C₅₃	—	4	4	no	—
206	C₂×C₄×C₅₃	—	4	4	no	—
207	C₂×C₄×C₅₃	—	4	1	no	—
208	C₂×C₄×C₅₃	—	4	1	no	—
209	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
210	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
211	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
212	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
215	C₈×C₅₃	—	2	4	no	—
216	C₈×C₅₃	—	4	2	no	—
217	C₈×C₅₃	—	4	2	no	—
218	#18539	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂²×C₅₃	—	2	1	no	—
220	C₂²×C₅₃	—	2	1	no	—
221	C₂²×C₅₃	—	2	4	no	—
222	C₂²×C₅₃	—	4	8	no	—
223	C₂²×C₅₃	—	4	2	no	—
224	C₂²×C₅₃	—	4	2	no	—
225	C₂²×C₅₃	—	4	2	no	—
226	C₂²×C₅₃	—	4	2	no	—
227	C₂²×C₅₃	—	4	2	no	—
228	C₂²×C₅₃	—	8	4	no	—
229	C₂²×C₅₃	#456	1	2	yes	—
230	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
231	C₄×C₅₃	—	2	1	no	—
232	C₄×C₅₃	—	2	1	no	—
233	C₄×C₅₃	—	2	1	no	—
234	C₄×C₅₃	—	2	1	no	—
235	C₄×C₅₃	—	4	2	no	—
236	C₄×C₅₃	—	4	4	no	—
237	C₄×C₅₃	—	4	2	no	—
238	C₄×C₅₃	—	8	1	no	—
239	C₄×C₅₃	—	8	1	no	—
240	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
241	C₄×C₅₃	#575	1	2	yes	—
242	C₂×#547	C₂×C₅₃	1	1	yes	characteristic
243	#3165	C₂×C₅₃	1	1	yes	characteristic, complemented
244	#3110	C₂×C₅₃	1	1	yes	characteristic, complemented
245	#3110	C₂×C₅₃	1	1	yes	characteristic, complemented
246	#1654	C₂×C₅₃	1	4	yes	complemented
247	#1647	C₂×C₅₃	1	4	yes	complemented
248	#1556	C₂×C₅₃	1	1	yes	characteristic, complemented
249	#1490	C₂×C₅₃	1	1	yes	characteristic, complemented
250	#1490	C₂×C₅₃	1	1	yes	characteristic, complemented
251	C₂×C₅₃	—	2	2	no	—
252	C₂×C₅₃	—	4	4	no	—
253	C₂×C₅₃	—	8	2	no	—
254	C₂×C₅₃	#3003	1	1	yes	characteristic
255	C₂×C₅₃	#3005	1	1	yes	characteristic
256	C₂×C₅₃	C₂×#456	1	1	yes	characteristic
257	C₂²×#41	—	2	1	no	—
258	#547	—	2	2	no	—
259	#547	—	2	2	no	—
260	#547	—	2	2	no	—
261	#547	C₂²×C₅₃	1	4	yes	—
262	#533	C₂²×C₅₃	1	1	yes	characteristic
263	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
264	C₂×#123	—	2	2	no	—
265	C₂×C₂²⋊Q₈	—	2	1	no	—
266	C₂×#121	—	2	1	no	—
267	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
268	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
269	C₂²×C₄⋊C₄	—	2	1	no	—
270	#501	—	2	2	no	—
271	#500	C₂²×C₅₃	1	2	yes	complemented
272	#498	C₂²×C₅₃	1	2	yes	complemented
273	#484	—	2	1	no	—
274	#484	—	2	1	no	—
275	#474	C₂²×C₅₃	1	2	yes	complemented
276	#474	C₂²×C₅₃	1	2	yes	complemented
277	#461	C₂²×C₅₃	1	2	yes	complemented
278	#461	C₂²×C₅₃	1	2	yes	complemented
279	#450	—	2	1	no	—
280	#450	—	2	1	no	—
281	#440	—	2	2	no	—
282	#440	—	2	2	no	—
283	#432	C₂²×C₅₃	1	1	yes	characteristic
284	#432	C₂²×C₅₃	1	1	yes	characteristic
285	#428	—	2	2	no	—
286	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
287	#332	C₂²×C₅₃	1	4	yes	complemented
288	D₄⋊C₈	C₂²×C₅₃	1	4	yes	complemented
289	D₄⋊C₈	C₂²×C₅₃	1	4	yes	complemented
290	C₅₃	#18539	1	1	yes	characteristic, direct product factor, Sylow
291	C₂×#41	—	2	4	no	—
292	C₂×#41	—	2	4	no	—
293	C₂×#41	—	4	8	no	—
294	C₂²×Q₈	—	2	1	no	—
295	C₂²×D₄	—	2	2	no	—
296	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
297	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
298	C₂³×C₄	—	2	2	no	—
299	C₂³×C₄	—	2	1	no	—
300	C₂³×C₄	—	2	1	no	—
301	C₂×SD₁₆	—	2	2	no	—
302	C₂×SD₁₆	—	2	2	no	—
303	C₂×M₁₆	—	2	1	no	—
304	C₄⋊Q₈	—	2	1	no	—
305	C₄⋊Q₈	—	2	1	no	—
306	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
307	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
308	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
309	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
310	#125	—	2	1	no	—
311	#123	—	2	8	no	—
312	#123	—	2	8	no	—
313	C₂²⋊Q₈	—	2	4	no	—
314	C₂²⋊Q₈	—	2	4	no	—
315	C₂²⋊Q₈	—	2	1	no	—
316	C₂²⋊Q₈	—	2	1	no	—
317	#121	—	2	4	no	—
318	#121	—	2	4	no	—
319	C₄×Q₈	—	2	1	no	—
320	C₄×Q₈	—	2	1	no	—
321	C₄×D₄	—	2	1	no	—
322	C₄×D₄	—	2	1	no	—
323	C₄×D₄	—	2	1	no	—
324	C₄×D₄	—	2	1	no	—
325	C₄×D₄	D₄×C₅₃	1	2	yes	—
326	C₄×D₄	D₄×C₅₃	1	2	yes	—
327	C₄×D₄	C₂³×C₅₃	1	2	yes	—
328	C₄×D₄	C₂³×C₅₃	1	2	yes	—
329	#117	—	2	1	no	—
330	C₂×C₄⋊C₄	—	2	2	no	—
331	C₂×C₄⋊C₄	—	2	4	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	4	no	—
334	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
335	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
336	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
337	C₂×C₂²⋊C₄	—	2	2	no	—
338	C₂×C₂²⋊C₄	—	2	1	no	—
339	C₂×C₂²⋊C₄	—	2	1	no	—
340	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
341	#107	—	2	4	no	—
342	#107	—	2	2	no	—
343	#107	—	2	2	no	—
344	#106	—	2	4	no	—
345	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
346	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
347	#103	—	2	2	no	—
348	#103	—	2	2	no	—
349	#102	—	2	2	no	—
350	#102	—	2	2	no	—
351	C₂²⋊C₈	—	2	1	no	—
352	C₂²⋊C₈	—	2	1	no	—
353	#97	—	2	1	no	—
354	#97	—	2	1	no	—
355	#97	—	2	1	no	—
356	C₄×C₈	C₂³×C₅₃	1	2	yes	—
357	C₂⁴	—	2	1	no	—
358	C₂⁴	—	2	1	no	—
359	#41	—	4	16	no	—
360	#41	—	4	8	no	—
361	#41	—	4	8	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	—	2	1	no	—
365	C₂×Q₈	—	2	1	no	—
366	C₂×Q₈	—	4	4	no	—
367	C₂×D₄	—	2	4	no	—
368	C₂×D₄	—	2	8	no	—
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	—	4	2	no	—
372	C₂×D₄	—	4	8	no	—
373	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
374	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
375	C₂²×C₄	—	2	8	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	4	4	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	—	4	8	no	—
386	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
387	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
389	SD₁₆	—	4	4	no	—
390	SD₁₆	—	4	4	no	—
391	M₁₆	—	2	4	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₄⋊C₄	—	2	8	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	4	1	no	—
406	C₄⋊C₄	—	4	1	no	—
407	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
408	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
409	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
410	C₂²⋊C₄	—	2	8	no	—
411	C₂²⋊C₄	—	2	2	no	—
412	C₂²⋊C₄	—	2	2	no	—
413	C₂²⋊C₄	—	2	2	no	—
414	C₂²⋊C₄	—	2	2	no	—
415	C₂²⋊C₄	—	4	1	no	—
416	C₄²	—	4	1	no	—
417	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
418	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
419	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
420	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
421	C₂³	—	2	4	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	2	no	—
424	C₂³	—	4	2	no	—
425	C₂³	—	4	2	no	—
426	C₂³	—	4	2	no	—
427	C₂³	—	4	2	no	—
428	C₂³	—	4	2	no	—
429	C₂³	#120×C₅₃	1	1	yes	characteristic
430	Q₈	—	4	4	no	—
431	Q₈	—	4	4	no	—
432	Q₈	—	4	2	no	—
433	Q₈	—	4	2	no	—
434	D₄	—	2	4	no	—
435	D₄	—	2	2	no	—
436	D₄	—	2	2	no	—
437	D₄	—	2	2	no	—
438	D₄	—	4	2	no	—
439	D₄	—	4	16	no	—
440	D₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	4	4	no	—
448	C₂×C₄	—	4	4	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	—	4	16	no	—
453	C₂×C₄	—	4	4	no	—
454	C₂×C₄	—	4	4	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
458	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
459	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
460	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
461	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
462	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
463	C₈	—	2	4	no	—
464	C₈	—	4	2	no	—
465	C₈	—	4	2	no	—
466	C₂²	—	2	1	no	—
467	C₂²	—	2	1	no	—
468	C₂²	—	2	4	no	—
469	C₂²	—	4	8	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	4	2	no	—
474	C₂²	—	4	2	no	—
475	C₂²	—	8	4	no	—
476	C₂²	C₅₃×#456	1	2	yes	—
477	C₂²	C₂×#120×C₅₃	1	1	yes	characteristic
478	C₄	—	2	1	no	—
479	C₄	—	2	1	no	—
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	4	2	no	—
483	C₄	—	4	4	no	—
484	C₄	—	4	2	no	—
485	C₄	—	8	1	no	—
486	C₄	—	8	1	no	—
487	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
488	C₄	C₅₃×#575	1	2	yes	—
489	C₂	—	2	2	no	—
490	C₂	—	4	4	no	—
491	C₂	—	8	2	no	—
492	C₂	C₅₃×#3003	1	1	yes	characteristic
493	C₂	C₅₃×#3005	1	1	yes	characteristic
494	C₂	C₂×C₅₃×#456	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂×C₅₃×#547	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×#3165	C₂	semidirect
3. C₅₃×#3110	C₂	semidirect
4. C₅₃×#3110	C₂	semidirect
5. C₅₃×#1654	C₂	semidirect
6. C₅₃×#1647	C₂	semidirect
7. C₅₃×#1556	C₂	semidirect
8. C₅₃×#1490	C₂	semidirect
9. C₅₃×#1490	C₂	semidirect
24. C₅₃×#500	C₂²	semidirect
25. C₅₃×#498	C₂²	semidirect
28. C₅₃×#474	C₂²	semidirect
29. C₅₃×#474	C₂²	semidirect
30. C₅₃×#461	C₂²	semidirect
31. C₅₃×#461	C₂²	semidirect
40. C₅₃×#332	C₂²	semidirect
41. C₅₃×D₄⋊C₈	C₂²	semidirect
42. C₅₃×D₄⋊C₈	C₂²	semidirect
218. #18539	C₅₃	direct (N × H)
243. #3165	C₂×C₅₃	semidirect
244. #3110	C₂×C₅₃	semidirect
245. #3110	C₂×C₅₃	semidirect
246. #1654	C₂×C₅₃	semidirect
247. #1647	C₂×C₅₃	semidirect
248. #1556	C₂×C₅₃	semidirect
249. #1490	C₂×C₅₃	semidirect
250. #1490	C₂×C₅₃	semidirect
271. #500	C₂²×C₅₃	semidirect
272. #498	C₂²×C₅₃	semidirect
275. #474	C₂²×C₅₃	semidirect
276. #474	C₂²×C₅₃	semidirect
277. #461	C₂²×C₅₃	semidirect
278. #461	C₂²×C₅₃	semidirect
287. #332	C₂²×C₅₃	semidirect
288. D₄⋊C₈	C₂²×C₅₃	semidirect
289. D₄⋊C₈	C₂²×C₅₃	semidirect
290. C₅₃	#18539	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅₃×#547	2
C₅₃×#3165	2
C₅₃×#3110	2
C₅₃×#1654	2
C₅₃×#1647	2
C₅₃×#1556	2
C₅₃×#1490	2
#18539	53

Subgroups — C53×#18539 #27226051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18539 #27226051