Subgroups — C₅₃×#23048 #27230560

Order 13568.

892 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#472	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2948	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2946	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2912	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1972	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1951	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1931	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1883	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1856	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1773	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1765	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	2	yes	—
13	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#520	C₂²	1	2	yes	—
16	C₅₃×#518	C₂²	1	2	yes	—
17	C₅₃×#472	C₄	1	8	yes	complemented
18	C₅₃×#442	—	2	1	no	—
19	C₅₃×#442	—	2	1	no	—
20	C₅₃×#442	—	2	2	no	—
21	C₅₃×#440	—	2	2	no	—
22	C₄×SD₁₆×C₅₃	—	2	2	no	—
23	C₄×D₈×C₅₃	—	2	2	no	—
24	C₄×D₈×C₅₃	—	2	2	no	—
25	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#428	—	2	2	no	—
27	C₅₃×#428	C₂²	1	2	yes	—
28	C₅₃×#428	C₂²	1	2	yes	—
29	C₅₃×#427	—	2	1	no	—
30	C₅₃×#427	C₂²	1	1	yes	characteristic
31	C₂×#106×C₅₃	—	2	1	no	—
32	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#418	C₂²	1	2	yes	—
35	C₅₃×#416	C₂²	1	1	yes	characteristic
36	C₂×#102×C₅₃	C₂²	1	2	yes	—
37	C₂×#100×C₅₃	C₂²	1	2	yes	—
38	C₅₃×#405	C₂²	1	2	yes	—
39	C₅₃×#405	C₂²	1	1	yes	characteristic
40	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#390	C₂²	1	2	yes	—
42	C₅₃×#343	C₂²	1	2	yes	—
43	C₅₃×#341	C₂²	1	1	yes	characteristic
44	C₅₃×#339	C₂²	1	1	yes	characteristic
45	C₅₃×#339	C₂²	1	2	yes	—
46	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
48	AGL(1,ℤ/8)×C₅₃	—	2	4	no	—
49	AGL(1,ℤ/8)×C₅₃	C₂×C₄	1	8	yes	complemented
50	C₂×SD₁₆×C₅₃	—	2	2	no	—
51	C₂×D₈×C₅₃	—	2	1	no	—
52	C₂×D₈×C₅₃	—	2	1	no	—
53	C₂×D₈×C₅₃	—	2	2	no	—
54	C₂×D₈×C₅₃	—	2	2	no	—
55	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	complemented
56	#131×C₅₃	—	2	2	no	—
57	#131×C₅₃	C₂×C₄	1	4	yes	complemented
58	C₂×M₁₆×C₅₃	—	2	1	no	—
59	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
62	C₂²×C₈×C₅₃	—	2	1	no	—
63	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₄×Q₈×C₅₃	—	2	2	no	—
66	C₄×D₄×C₅₃	—	2	2	no	—
67	C₄×D₄×C₅₃	—	2	2	no	—
68	C₄×D₄×C₅₃	—	2	2	no	—
69	C₄×D₄×C₅₃	—	2	2	no	—
70	C₄×D₄×C₅₃	—	2	2	no	—
71	C₄×D₄×C₅₃	—	4	2	no	—
72	#117×C₅₃	—	2	2	no	—
73	#117×C₅₃	C₂³	1	1	yes	characteristic
74	#117×C₅₃	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
76	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
77	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
78	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
79	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
80	#108×C₅₃	C₂×C₄	1	4	yes	complemented
81	#107×C₅₃	—	2	2	no	—
82	#107×C₅₃	—	2	2	no	—
83	#107×C₅₃	D₄	1	2	yes	—
84	#106×C₅₃	—	2	1	no	—
85	#106×C₅₃	—	2	1	no	—
86	#106×C₅₃	—	2	2	no	—
87	#106×C₅₃	—	2	2	no	—
88	#106×C₅₃	D₄	1	1	yes	characteristic
89	#106×C₅₃	D₄	1	1	yes	characteristic
90	#104×C₅₃	—	2	2	no	—
91	#103×C₅₃	D₄	1	2	yes	—
92	#102×C₅₃	—	2	2	no	—
93	#102×C₅₃	—	2	2	no	—
94	#102×C₅₃	—	2	2	no	—
95	#102×C₅₃	—	2	2	no	—
96	#102×C₅₃	D₄	1	2	yes	—
97	#100×C₅₃	C₂×C₄	1	8	yes	complemented
98	#97×C₅₃	—	2	2	no	—
99	#97×C₅₃	—	2	1	no	—
100	#97×C₅₃	—	2	2	no	—
101	C₄×C₈×C₅₃	—	2	1	no	—
102	C₄×C₈×C₅₃	—	2	2	no	—
103	C₂⁴×C₅₃	—	2	2	no	—
104	#41×C₅₃	—	2	1	no	—
105	#41×C₅₃	—	2	1	no	—
106	#41×C₅₃	C₂²×C₄	1	2	yes	—
107	#41×C₅₃	#41	1	2	yes	—
108	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂×D₄×C₅₃	—	2	1	no	—
110	C₂×D₄×C₅₃	—	2	2	no	—
111	C₂×D₄×C₅₃	—	2	2	no	—
112	C₂×D₄×C₅₃	—	2	2	no	—
113	C₂×D₄×C₅₃	—	2	2	no	—
114	C₂×D₄×C₅₃	—	2	2	no	—
115	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
116	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	—	2	1	no	—
118	C₂²×C₄×C₅₃	—	2	1	no	—
119	C₂²×C₄×C₅₃	—	2	2	no	—
120	C₂²×C₄×C₅₃	—	4	2	no	—
121	C₂²×C₄×C₅₃	—	4	2	no	—
122	C₂²×C₄×C₅₃	—	4	2	no	—
123	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
124	SD₁₆×C₅₃	—	2	8	no	—
125	D₈×C₅₃	—	2	8	no	—
126	D₈×C₅₃	—	2	4	no	—
127	D₈×C₅₃	—	2	4	no	—
128	M₁₆×C₅₃	—	2	1	no	—
129	M₁₆×C₅₃	—	2	2	no	—
130	M₁₆×C₅₃	—	2	4	no	—
131	M₁₆×C₅₃	—	2	1	no	—
132	M₁₆×C₅₃	—	2	1	no	—
133	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
134	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
135	C₂×C₈×C₅₃	—	2	2	no	—
136	C₂×C₈×C₅₃	—	2	1	no	—
137	C₂×C₈×C₅₃	—	2	1	no	—
138	C₂×C₈×C₅₃	—	2	1	no	—
139	C₂×C₈×C₅₃	—	2	4	no	—
140	C₂×C₈×C₅₃	—	2	1	no	—
141	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
142	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
143	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
145	C₂×C₈×C₅₃	#41	1	2	yes	—
146	C₄⋊C₄×C₅₃	—	2	2	no	—
147	C₄⋊C₄×C₅₃	—	2	2	no	—
148	C₄⋊C₄×C₅₃	—	4	2	no	—
149	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
150	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
151	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
152	C₂²⋊C₄×C₅₃	—	2	1	no	—
153	C₂²⋊C₄×C₅₃	—	2	2	no	—
154	C₂²⋊C₄×C₅₃	—	4	2	no	—
155	C₂²⋊C₄×C₅₃	—	4	2	no	—
156	C₂²⋊C₄×C₅₃	—	4	2	no	—
157	C₂²⋊C₄×C₅₃	—	8	4	no	—
158	C₄²×C₅₃	—	2	1	no	—
159	C₄²×C₅₃	—	2	2	no	—
160	C₄²×C₅₃	—	2	1	no	—
161	C₄²×C₅₃	—	2	1	no	—
162	C₄²×C₅₃	—	4	2	no	—
163	C₂³×C₅₃	—	2	1	no	—
164	C₂³×C₅₃	—	2	4	no	—
165	C₂³×C₅₃	—	4	2	no	—
166	C₂³×C₅₃	—	4	2	no	—
167	C₂³×C₅₃	—	8	2	no	—
168	C₂³×C₅₃	—	8	2	no	—
169	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
170	Q₈×C₅₃	—	2	1	no	—
171	Q₈×C₅₃	C₄×D₄	1	2	yes	—
172	D₄×C₅₃	—	2	2	no	—
173	D₄×C₅₃	—	2	2	no	—
174	D₄×C₅₃	—	2	4	no	—
175	D₄×C₅₃	—	2	4	no	—
176	D₄×C₅₃	—	2	1	no	—
177	D₄×C₅₃	—	4	2	no	—
178	D₄×C₅₃	—	4	2	no	—
179	D₄×C₅₃	C₄×D₄	1	2	yes	—
180	C₂×C₄×C₅₃	—	2	1	no	—
181	C₂×C₄×C₅₃	—	2	2	no	—
182	C₂×C₄×C₅₃	—	2	2	no	—
183	C₂×C₄×C₅₃	—	2	2	no	—
184	C₂×C₄×C₅₃	—	2	2	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	4	2	no	—
187	C₂×C₄×C₅₃	—	4	2	no	—
188	C₂×C₄×C₅₃	—	4	1	no	—
189	C₂×C₄×C₅₃	—	8	2	no	—
190	C₂×C₄×C₅₃	—	16	2	no	—
191	C₂×C₄×C₅₃	—	16	2	no	—
192	C₂×C₄×C₅₃	—	16	2	no	—
193	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
199	C₈×C₅₃	—	2	2	no	—
200	C₈×C₅₃	—	2	4	no	—
201	C₈×C₅₃	—	2	2	no	—
202	C₈×C₅₃	—	4	1	no	—
203	C₈×C₅₃	C₄×D₄	1	2	yes	—
204	C₈×C₅₃	C₄×D₄	1	2	yes	—
205	#23048	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂²×C₅₃	—	2	2	no	—
207	C₂²×C₅₃	—	2	2	no	—
208	C₂²×C₅₃	—	4	2	no	—
209	C₂²×C₅₃	—	4	4	no	—
210	C₂²×C₅₃	—	8	4	no	—
211	C₂²×C₅₃	—	8	2	no	—
212	C₂²×C₅₃	—	8	2	no	—
213	C₂²×C₅₃	—	16	2	no	—
214	C₂²×C₅₃	#520	1	1	yes	characteristic
215	C₂²×C₅₃	D₄²	1	1	yes	characteristic
216	C₂²×C₅₃	#553	1	1	yes	characteristic
217	C₄×C₅₃	—	2	2	no	—
218	C₄×C₅₃	—	4	2	no	—
219	C₄×C₅₃	—	4	2	no	—
220	C₄×C₅₃	—	4	2	no	—
221	C₄×C₅₃	—	8	1	no	—
222	C₄×C₅₃	—	8	2	no	—
223	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
224	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
225	C₄×C₅₃	#547	1	1	yes	characteristic
226	C₄×C₅₃	#548	1	1	yes	characteristic
227	C₂×#472	C₂×C₅₃	1	1	yes	characteristic
228	#2948	C₂×C₅₃	1	1	yes	characteristic, complemented
229	#2946	C₂×C₅₃	1	2	yes	complemented
230	#2912	C₂×C₅₃	1	1	yes	characteristic, complemented
231	#1972	C₂×C₅₃	1	1	yes	characteristic, complemented
232	#1951	C₂×C₅₃	1	1	yes	characteristic, complemented
233	#1931	C₂×C₅₃	1	2	yes	complemented
234	#1883	C₂×C₅₃	1	2	yes	complemented
235	#1856	C₂×C₅₃	1	2	yes	complemented
236	#1773	C₂×C₅₃	1	1	yes	characteristic, complemented
237	#1765	C₂×C₅₃	1	1	yes	characteristic, complemented
238	C₂×C₅₃	—	2	1	no	—
239	C₂×C₅₃	—	2	1	no	—
240	C₂×C₅₃	—	4	2	no	—
241	C₂×C₅₃	—	8	4	no	—
242	C₂×C₅₃	#2361	1	1	yes	characteristic
243	C₂×C₅₃	#3288	1	2	yes	—
244	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	2	yes	—
245	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
246	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
247	#520	C₂²×C₅₃	1	2	yes	—
248	#518	C₂²×C₅₃	1	2	yes	—
249	#472	C₄×C₅₃	1	8	yes	complemented
250	#442	—	2	1	no	—
251	#442	—	2	1	no	—
252	#442	—	2	2	no	—
253	#440	—	2	2	no	—
254	C₄×SD₁₆	—	2	2	no	—
255	C₄×D₈	—	2	2	no	—
256	C₄×D₈	—	2	2	no	—
257	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
258	#428	—	2	2	no	—
259	#428	C₂²×C₅₃	1	2	yes	—
260	#428	C₂²×C₅₃	1	2	yes	—
261	#427	—	2	1	no	—
262	#427	C₂²×C₅₃	1	1	yes	characteristic
263	C₂×#106	—	2	1	no	—
264	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
265	C₂×#104	C₂²×C₅₃	1	1	yes	characteristic
266	#418	C₂²×C₅₃	1	2	yes	—
267	#416	C₂²×C₅₃	1	1	yes	characteristic
268	C₂×#102	C₂²×C₅₃	1	2	yes	—
269	C₂×#100	C₂²×C₅₃	1	2	yes	—
270	#405	C₂²×C₅₃	1	2	yes	—
271	#405	C₂²×C₅₃	1	1	yes	characteristic
272	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
273	#390	C₂²×C₅₃	1	2	yes	—
274	#343	C₂²×C₅₃	1	2	yes	—
275	#341	C₂²×C₅₃	1	1	yes	characteristic
276	#339	C₂²×C₅₃	1	1	yes	characteristic
277	#339	C₂²×C₅₃	1	2	yes	—
278	C₅₃	#23048	1	1	yes	characteristic, direct product factor, Sylow
279	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
280	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
281	AGL(1,ℤ/8)	—	2	4	no	—
282	AGL(1,ℤ/8)	C₂×C₄×C₅₃	1	8	yes	complemented
283	C₂×SD₁₆	—	2	2	no	—
284	C₂×D₈	—	2	2	no	—
285	C₂×D₈	—	2	1	no	—
286	C₂×D₈	—	2	1	no	—
287	C₂×D₈	—	2	2	no	—
288	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	complemented
289	#131	—	2	2	no	—
290	#131	C₂×C₄×C₅₃	1	4	yes	complemented
291	C₂×M₁₆	—	2	1	no	—
292	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
293	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
294	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
295	C₂²×C₈	—	2	1	no	—
296	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
297	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
298	C₄×Q₈	—	2	2	no	—
299	C₄×D₄	—	2	2	no	—
300	C₄×D₄	—	2	2	no	—
301	C₄×D₄	—	2	2	no	—
302	C₄×D₄	—	2	2	no	—
303	C₄×D₄	—	2	2	no	—
304	C₄×D₄	—	4	2	no	—
305	#117	—	2	2	no	—
306	#117	C₂³×C₅₃	1	1	yes	characteristic
307	#117	C₂³×C₅₃	1	2	yes	—
308	C₂×C₄⋊C₄	—	2	2	no	—
309	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
310	C₂×C₂²⋊C₄	—	4	2	no	—
311	C₂×C₂²⋊C₄	—	4	2	no	—
312	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
313	#108	C₂×C₄×C₅₃	1	4	yes	complemented
314	#107	—	2	2	no	—
315	#107	—	2	2	no	—
316	#107	D₄×C₅₃	1	2	yes	—
317	#106	—	2	1	no	—
318	#106	—	2	1	no	—
319	#106	—	2	2	no	—
320	#106	—	2	2	no	—
321	#106	D₄×C₅₃	1	1	yes	characteristic
322	#106	D₄×C₅₃	1	1	yes	characteristic
323	#104	—	2	2	no	—
324	#103	D₄×C₅₃	1	2	yes	—
325	#102	—	2	2	no	—
326	#102	—	2	2	no	—
327	#102	—	2	2	no	—
328	#102	—	2	2	no	—
329	#102	D₄×C₅₃	1	2	yes	—
330	#100	C₂×C₄×C₅₃	1	8	yes	complemented
331	#97	—	2	2	no	—
332	#97	—	2	1	no	—
333	#97	—	2	2	no	—
334	C₄×C₈	—	2	1	no	—
335	C₄×C₈	—	2	2	no	—
336	C₂⁴	—	2	2	no	—
337	#41	—	2	1	no	—
338	#41	—	2	1	no	—
339	#41	C₂²×C₄×C₅₃	1	2	yes	—
340	#41	#41×C₅₃	1	2	yes	—
341	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
349	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
350	C₂²×C₄	—	2	1	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	4	2	no	—
354	C₂²×C₄	—	4	2	no	—
355	C₂²×C₄	—	4	2	no	—
356	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
357	SD₁₆	—	2	8	no	—
358	D₈	—	2	8	no	—
359	D₈	—	2	4	no	—
360	D₈	—	2	4	no	—
361	M₁₆	—	2	1	no	—
362	M₁₆	—	2	2	no	—
363	M₁₆	—	2	4	no	—
364	M₁₆	—	2	1	no	—
365	M₁₆	—	2	1	no	—
366	M₁₆	C₂²×C₄×C₅₃	1	2	yes	—
367	M₁₆	C₂²×C₄×C₅₃	1	4	yes	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	4	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
375	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
376	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
377	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
378	C₂×C₈	#41×C₅₃	1	2	yes	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	4	2	no	—
382	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
383	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
384	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
385	C₂²⋊C₄	—	2	1	no	—
386	C₂²⋊C₄	—	2	2	no	—
387	C₂²⋊C₄	—	4	2	no	—
388	C₂²⋊C₄	—	4	2	no	—
389	C₂²⋊C₄	—	4	2	no	—
390	C₂²⋊C₄	—	8	4	no	—
391	C₄²	—	2	1	no	—
392	C₄²	—	2	2	no	—
393	C₄²	—	2	1	no	—
394	C₄²	—	2	1	no	—
395	C₄²	—	4	2	no	—
396	C₂³	—	2	1	no	—
397	C₂³	—	2	4	no	—
398	C₂³	—	4	2	no	—
399	C₂³	—	4	2	no	—
400	C₂³	—	8	2	no	—
401	C₂³	—	8	2	no	—
402	C₂³	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
403	Q₈	—	2	1	no	—
404	Q₈	C₄×D₄×C₅₃	1	2	yes	—
405	D₄	—	2	2	no	—
406	D₄	—	2	2	no	—
407	D₄	—	2	4	no	—
408	D₄	—	2	4	no	—
409	D₄	—	2	1	no	—
410	D₄	—	4	2	no	—
411	D₄	—	4	2	no	—
412	D₄	C₄×D₄×C₅₃	1	2	yes	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	8	2	no	—
423	C₂×C₄	—	16	2	no	—
424	C₂×C₄	—	16	2	no	—
425	C₂×C₄	—	16	2	no	—
426	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
427	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
428	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
429	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
430	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
431	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
432	C₈	—	2	2	no	—
433	C₈	—	2	4	no	—
434	C₈	—	2	2	no	—
435	C₈	—	4	1	no	—
436	C₈	C₄×D₄×C₅₃	1	2	yes	—
437	C₈	C₄×D₄×C₅₃	1	2	yes	—
438	C₂²	—	2	2	no	—
439	C₂²	—	2	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	4	no	—
442	C₂²	—	8	4	no	—
443	C₂²	—	8	2	no	—
444	C₂²	—	8	2	no	—
445	C₂²	—	16	2	no	—
446	C₂²	C₅₃×#520	1	1	yes	characteristic
447	C₂²	D₄²×C₅₃	1	1	yes	characteristic
448	C₂²	C₅₃×#553	1	1	yes	characteristic
449	C₄	—	2	2	no	—
450	C₄	—	4	2	no	—
451	C₄	—	4	2	no	—
452	C₄	—	4	2	no	—
453	C₄	—	8	1	no	—
454	C₄	—	8	2	no	—
455	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
456	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
457	C₄	C₅₃×#547	1	1	yes	characteristic
458	C₄	C₅₃×#548	1	1	yes	characteristic
459	C₂	—	2	1	no	—
460	C₂	—	2	1	no	—
461	C₂	—	4	2	no	—
462	C₂	—	8	4	no	—
463	C₂	C₅₃×#2361	1	1	yes	characteristic
464	C₂	C₅₃×#3288	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	D₄²
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂×C₅₃×#472	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2948	C₂	semidirect
3. C₅₃×#2946	C₂	semidirect
4. C₅₃×#2912	C₂	semidirect
5. C₅₃×#1972	C₂	semidirect
6. C₅₃×#1951	C₂	semidirect
7. C₅₃×#1931	C₂	semidirect
8. C₅₃×#1883	C₂	semidirect
9. C₅₃×#1856	C₂	semidirect
10. C₅₃×#1773	C₂	semidirect
11. C₅₃×#1765	C₂	semidirect
17. C₅₃×#472	C₄	semidirect
49. AGL(1,ℤ/8)×C₅₃	C₂×C₄	semidirect
55. C₂×D₈×C₅₃	C₂×C₄	semidirect
57. #131×C₅₃	C₂×C₄	semidirect
80. #108×C₅₃	C₂×C₄	semidirect
97. #100×C₅₃	C₂×C₄	semidirect
205. #23048	C₅₃	direct (N × H)
228. #2948	C₂×C₅₃	semidirect
229. #2946	C₂×C₅₃	semidirect
230. #2912	C₂×C₅₃	semidirect
231. #1972	C₂×C₅₃	semidirect
232. #1951	C₂×C₅₃	semidirect
233. #1931	C₂×C₅₃	semidirect
234. #1883	C₂×C₅₃	semidirect
235. #1856	C₂×C₅₃	semidirect
236. #1773	C₂×C₅₃	semidirect
237. #1765	C₂×C₅₃	semidirect
249. #472	C₄×C₅₃	semidirect
278. C₅₃	#23048	direct (N × H)
282. AGL(1,ℤ/8)	C₂×C₄×C₅₃	semidirect
288. C₂×D₈	C₂×C₄×C₅₃	semidirect
290. #131	C₂×C₄×C₅₃	semidirect
313. #108	C₂×C₄×C₅₃	semidirect
330. #100	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#472	2
C₅₃×#2948	2
C₅₃×#2946	2
C₅₃×#2912	2
C₅₃×#1972	2
C₅₃×#1951	2
C₅₃×#1931	2
C₅₃×#1883	2
C₅₃×#1856	2
C₅₃×#1773	2
C₅₃×#1765	2
#23048	53

Subgroups — C53×#23048 #27230560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23048 #27230560