Subgroups — C₅₃×#22219 #27229731

Order 13568.

990 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1957	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1926	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1864	C₂	1	4	yes	maximal, complemented
8	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
13	C₂×C₄×D₄×C₅₃	—	2	1	no	—
14	C₂×C₄×D₄×C₅₃	—	2	1	no	—
15	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#486	C₄	1	8	yes	complemented
17	C₂×#106×C₅₃	C₂²	1	4	yes	—
18	C₂×#102×C₅₃	C₂²	1	4	yes	—
19	C₂×#102×C₅₃	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#391	—	2	1	no	—
23	C₅₃×#390	—	2	1	no	—
24	C₅₃×#390	—	2	1	no	—
25	C₅₃×#390	C₂²	1	2	yes	—
26	C₅₃×#389	—	2	1	no	—
27	C₅₃×#389	C₂²	1	2	yes	—
28	C₅₃×#384	C₂²	1	1	yes	characteristic
29	C₅₃×#384	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
32	C₄³×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#340	C₂²	1	4	yes	—
34	C₅₃×#336	C₂²	1	2	yes	—
35	C₅₃×#336	C₂²	1	2	yes	—
36	C₂²×Q₈×C₅₃	—	2	1	no	—
37	C₂²×D₄×C₅₃	—	2	1	no	—
38	C₂²×D₄×C₅₃	—	2	1	no	—
39	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
40	C₂³×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
43	C₄⋊Q₈×C₅₃	—	2	2	no	—
44	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	complemented
45	#127×C₅₃	—	2	2	no	—
46	#127×C₅₃	C₂×C₄	1	4	yes	—
47	C₄×Q₈×C₅₃	—	4	4	no	—
48	C₄×D₄×C₅₃	—	4	4	no	—
49	C₄×D₄×C₅₃	—	4	4	no	—
50	C₄×D₄×C₅₃	—	4	4	no	—
51	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
60	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₄²×C₅₃	—	2	2	no	—
64	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	#106×C₅₃	—	4	4	no	—
72	#102×C₅₃	—	4	4	no	—
73	#102×C₅₃	C₂×C₄	1	16	yes	complemented
74	C₄×C₈×C₅₃	—	2	2	no	—
75	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
76	#95×C₅₃	—	2	2	no	—
77	#95×C₅₃	—	2	2	no	—
78	C₂⁴×C₅₃	—	2	2	no	—
79	C₂×Q₈×C₅₃	—	2	4	no	—
80	C₂×Q₈×C₅₃	—	4	2	no	—
81	C₂×Q₈×C₅₃	—	4	2	no	—
82	C₂×D₄×C₅₃	—	2	4	no	—
83	C₂×D₄×C₅₃	—	2	4	no	—
84	C₂×D₄×C₅₃	—	2	2	no	—
85	C₂×D₄×C₅₃	—	2	4	no	—
86	C₂×D₄×C₅₃	—	2	4	no	—
87	C₂×D₄×C₅₃	—	4	2	no	—
88	C₂×D₄×C₅₃	—	4	2	no	—
89	C₂×D₄×C₅₃	—	4	2	no	—
90	C₂×D₄×C₅₃	—	4	2	no	—
91	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
92	C₂²×C₄×C₅₃	—	2	2	no	—
93	C₂²×C₄×C₅₃	—	2	2	no	—
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂²×C₄×C₅₃	—	4	4	no	—
96	C₂²×C₄×C₅₃	—	4	4	no	—
97	C₂²×C₄×C₅₃	—	4	4	no	—
98	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
111	C₄⋊C₄×C₅₃	—	2	4	no	—
112	C₄⋊C₄×C₅₃	—	2	2	no	—
113	C₄⋊C₄×C₅₃	—	2	4	no	—
114	C₄⋊C₄×C₅₃	—	2	4	no	—
115	C₄⋊C₄×C₅₃	—	4	4	no	—
116	C₄⋊C₄×C₅₃	—	4	2	no	—
117	C₄⋊C₄×C₅₃	—	4	2	no	—
118	C₄⋊C₄×C₅₃	—	4	2	no	—
119	C₄⋊C₄×C₅₃	—	4	2	no	—
120	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
121	C₂²⋊C₄×C₅₃	—	4	4	no	—
122	C₂²⋊C₄×C₅₃	—	4	4	no	—
123	C₂²⋊C₄×C₅₃	—	4	4	no	—
124	C₄²×C₅₃	—	2	2	no	—
125	C₄²×C₅₃	—	2	2	no	—
126	C₄²×C₅₃	—	2	2	no	—
127	C₄²×C₅₃	—	2	2	no	—
128	C₄²×C₅₃	—	2	2	no	—
129	C₄²×C₅₃	—	4	4	no	—
130	C₄²×C₅₃	SD₁₆	1	4	yes	—
131	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
132	C₄²×C₅₃	#41	1	2	yes	—
133	C₂³×C₅₃	—	2	4	no	—
134	C₂³×C₅₃	—	4	2	no	—
135	C₂³×C₅₃	—	4	2	no	—
136	C₂³×C₅₃	—	4	2	no	—
137	C₂³×C₅₃	—	4	2	no	—
138	C₂³×C₅₃	—	4	2	no	—
139	C₂³×C₅₃	—	4	2	no	—
140	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
141	Q₈×C₅₃	—	4	8	no	—
142	D₄×C₅₃	—	2	8	no	—
143	D₄×C₅₃	—	4	4	no	—
144	D₄×C₅₃	—	4	8	no	—
145	D₄×C₅₃	—	4	8	no	—
146	C₂×C₄×C₅₃	—	2	1	no	—
147	C₂×C₄×C₅₃	—	2	1	no	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	—	2	1	no	—
150	C₂×C₄×C₅₃	—	2	4	no	—
151	C₂×C₄×C₅₃	—	2	1	no	—
152	C₂×C₄×C₅₃	—	2	1	no	—
153	C₂×C₄×C₅₃	—	2	1	no	—
154	C₂×C₄×C₅₃	—	2	1	no	—
155	C₂×C₄×C₅₃	—	2	1	no	—
156	C₂×C₄×C₅₃	—	2	1	no	—
157	C₂×C₄×C₅₃	—	4	2	no	—
158	C₂×C₄×C₅₃	—	4	2	no	—
159	C₂×C₄×C₅₃	—	4	2	no	—
160	C₂×C₄×C₅₃	—	4	2	no	—
161	C₂×C₄×C₅₃	—	4	2	no	—
162	C₂×C₄×C₅₃	—	4	2	no	—
163	C₂×C₄×C₅₃	—	4	2	no	—
164	C₂×C₄×C₅₃	—	4	2	no	—
165	C₂×C₄×C₅₃	—	8	8	no	—
166	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
167	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
168	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
169	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
170	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
171	C₂×C₄×C₅₃	#135	1	2	yes	—
172	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
173	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
174	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
176	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
177	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
178	C₈×C₅₃	—	2	4	no	—
179	C₈×C₅₃	—	4	2	no	—
180	#22219	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂²×C₅₃	—	4	4	no	—
182	C₂²×C₅₃	—	4	4	no	—
183	C₂²×C₅₃	—	4	4	no	—
184	C₂²×C₅₃	—	8	2	no	—
185	C₂²×C₅₃	—	8	2	no	—
186	C₂²×C₅₃	—	8	2	no	—
187	C₂²×C₅₃	—	8	2	no	—
188	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
189	C₂²×C₅₃	#535	1	1	yes	characteristic
190	C₂²×C₅₃	#536	1	1	yes	characteristic
191	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
192	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
193	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
194	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
195	C₄×C₅₃	—	2	2	no	—
196	C₄×C₅₃	—	2	2	no	—
197	C₄×C₅₃	—	4	4	no	—
198	C₄×C₅₃	—	4	1	no	—
199	C₄×C₅₃	—	4	1	no	—
200	C₄×C₅₃	—	8	2	no	—
201	C₄×C₅₃	—	8	2	no	—
202	C₄×C₅₃	C₄×SD₁₆	1	4	yes	—
203	C₄×C₅₃	#442	1	4	yes	—
204	C₄×C₅₃	#520	1	2	yes	—
205	C₄×C₅₃	#542	1	2	yes	—
206	C₂×#486	C₂×C₅₃	1	1	yes	characteristic
207	C₄×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
208	C₄×#127	C₂×C₅₃	1	1	yes	characteristic
209	#1990	C₂×C₅₃	1	1	yes	characteristic, complemented
210	#1957	C₂×C₅₃	1	1	yes	characteristic
211	#1926	C₂×C₅₃	1	4	yes	complemented
212	#1864	C₂×C₅₃	1	4	yes	complemented
213	#1844	C₂×C₅₃	1	1	yes	characteristic, complemented
214	#1744	C₂×C₅₃	1	1	yes	characteristic, complemented
215	C₂×C₅₃	—	8	4	no	—
216	C₂×C₅₃	#2326	1	1	yes	characteristic
217	C₂×C₅₃	C₂×C₄×SD₁₆	1	1	yes	characteristic
218	C₂×C₅₃	C₂×#442	1	1	yes	characteristic
219	C₂×C₅₃	#3207	1	1	yes	characteristic
220	C₂×C₅₃	#3208	1	1	yes	characteristic
221	C₂×C₅₃	#3227	1	1	yes	characteristic
222	C₂×C₅₃	#3228	1	1	yes	characteristic
223	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
224	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
225	C₂×C₄×Q₈	—	2	1	no	—
226	C₂×C₄×D₄	—	2	1	no	—
227	C₂×C₄×D₄	—	2	1	no	—
228	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
229	#486	C₄×C₅₃	1	8	yes	complemented
230	C₂×#106	C₂²×C₅₃	1	4	yes	—
231	C₂×#102	C₂²×C₅₃	1	4	yes	—
232	C₂×#102	C₂²×C₅₃	1	4	yes	—
233	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
234	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
235	#391	—	2	1	no	—
236	#390	—	2	1	no	—
237	#390	—	2	1	no	—
238	#390	C₂²×C₅₃	1	2	yes	—
239	#389	—	2	1	no	—
240	#389	C₂²×C₅₃	1	2	yes	—
241	#384	C₂²×C₅₃	1	1	yes	characteristic
242	#384	C₂²×C₅₃	1	1	yes	characteristic
243	C₄×C₄⋊C₄	—	2	1	no	—
244	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
245	C₄³	C₂²×C₅₃	1	1	yes	characteristic
246	#340	C₂²×C₅₃	1	4	yes	—
247	#336	C₂²×C₅₃	1	2	yes	—
248	#336	C₂²×C₅₃	1	2	yes	—
249	C₅₃	#22219	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²×Q₈	—	2	1	no	—
251	C₂²×D₄	—	2	1	no	—
252	C₂²×D₄	—	2	1	no	—
253	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
254	C₂³×C₄	—	2	2	no	—
255	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
256	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
257	C₄⋊Q₈	—	2	2	no	—
258	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	complemented
259	#127	—	2	2	no	—
260	#127	C₂×C₄×C₅₃	1	4	yes	—
261	C₄×Q₈	—	4	4	no	—
262	C₄×D₄	—	4	4	no	—
263	C₄×D₄	—	4	4	no	—
264	C₄×D₄	—	4	4	no	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
272	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
273	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
274	C₂×C₂²⋊C₄	—	2	2	no	—
275	C₂×C₂²⋊C₄	—	2	2	no	—
276	C₂×C₂²⋊C₄	—	2	2	no	—
277	C₂×C₄²	—	2	2	no	—
278	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
279	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
280	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
281	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
282	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
283	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
284	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
285	#106	—	4	4	no	—
286	#102	—	4	4	no	—
287	#102	C₂×C₄×C₅₃	1	16	yes	complemented
288	C₄×C₈	—	2	2	no	—
289	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
290	#95	—	2	2	no	—
291	#95	—	2	2	no	—
292	C₂⁴	—	2	2	no	—
293	C₂×Q₈	—	2	4	no	—
294	C₂×Q₈	—	4	2	no	—
295	C₂×Q₈	—	4	2	no	—
296	C₂×D₄	—	2	4	no	—
297	C₂×D₄	—	2	4	no	—
298	C₂×D₄	—	2	2	no	—
299	C₂×D₄	—	2	4	no	—
300	C₂×D₄	—	2	4	no	—
301	C₂×D₄	—	4	2	no	—
302	C₂×D₄	—	4	2	no	—
303	C₂×D₄	—	4	2	no	—
304	C₂×D₄	—	4	2	no	—
305	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	4	4	no	—
310	C₂²×C₄	—	4	4	no	—
311	C₂²×C₄	—	4	4	no	—
312	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
314	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
315	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
316	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
318	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
325	C₄⋊C₄	—	2	4	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	4	no	—
328	C₄⋊C₄	—	2	4	no	—
329	C₄⋊C₄	—	4	4	no	—
330	C₄⋊C₄	—	4	2	no	—
331	C₄⋊C₄	—	4	2	no	—
332	C₄⋊C₄	—	4	2	no	—
333	C₄⋊C₄	—	4	2	no	—
334	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
335	C₂²⋊C₄	—	4	4	no	—
336	C₂²⋊C₄	—	4	4	no	—
337	C₂²⋊C₄	—	4	4	no	—
338	C₄²	—	2	2	no	—
339	C₄²	—	2	2	no	—
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	4	4	no	—
344	C₄²	SD₁₆×C₅₃	1	4	yes	—
345	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
346	C₄²	#41×C₅₃	1	2	yes	—
347	C₂³	—	2	4	no	—
348	C₂³	—	4	2	no	—
349	C₂³	—	4	2	no	—
350	C₂³	—	4	2	no	—
351	C₂³	—	4	2	no	—
352	C₂³	—	4	2	no	—
353	C₂³	—	4	2	no	—
354	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
355	Q₈	—	4	8	no	—
356	D₄	—	2	8	no	—
357	D₄	—	4	4	no	—
358	D₄	—	4	8	no	—
359	D₄	—	4	8	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	8	8	no	—
380	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
381	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
382	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
383	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
384	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
385	C₂×C₄	#135×C₅₃	1	2	yes	—
386	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
387	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
388	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
389	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
390	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
391	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
392	C₈	—	2	4	no	—
393	C₈	—	4	2	no	—
394	C₂²	—	4	4	no	—
395	C₂²	—	4	4	no	—
396	C₂²	—	4	4	no	—
397	C₂²	—	8	2	no	—
398	C₂²	—	8	2	no	—
399	C₂²	—	8	2	no	—
400	C₂²	—	8	2	no	—
401	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
402	C₂²	C₅₃×#535	1	1	yes	characteristic
403	C₂²	C₅₃×#536	1	1	yes	characteristic
404	C₂²	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
405	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
406	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
407	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
408	C₄	—	2	2	no	—
409	C₄	—	2	2	no	—
410	C₄	—	4	4	no	—
411	C₄	—	4	1	no	—
412	C₄	—	4	1	no	—
413	C₄	—	8	2	no	—
414	C₄	—	8	2	no	—
415	C₄	C₄×SD₁₆×C₅₃	1	4	yes	—
416	C₄	C₅₃×#442	1	4	yes	—
417	C₄	C₅₃×#520	1	2	yes	—
418	C₄	C₅₃×#542	1	2	yes	—
419	C₂	—	8	4	no	—
420	C₂	C₅₃×#2326	1	1	yes	characteristic
421	C₂	C₂×C₄×SD₁₆×C₅₃	1	1	yes	characteristic
422	C₂	C₂×C₅₃×#442	1	1	yes	characteristic
423	C₂	C₅₃×#3207	1	1	yes	characteristic
424	C₂	C₅₃×#3208	1	1	yes	characteristic
425	C₂	C₅₃×#3227	1	1	yes	characteristic
426	C₂	C₅₃×#3228	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊Q₈×C₅₃	C₂	semidirect
4. C₅₃×#1990	C₂	semidirect
6. C₅₃×#1926	C₂	semidirect
7. C₅₃×#1864	C₂	semidirect
8. C₅₃×#1844	C₂	semidirect
9. C₅₃×#1744	C₂	semidirect
16. C₅₃×#486	C₄	semidirect
44. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
73. #102×C₅₃	C₂×C₄	semidirect
75. C₄×C₈×C₅₃	C₂×C₄	semidirect
180. #22219	C₅₃	direct (N × H)
207. C₄×C₄⋊Q₈	C₂×C₅₃	semidirect
209. #1990	C₂×C₅₃	semidirect
211. #1926	C₂×C₅₃	semidirect
212. #1864	C₂×C₅₃	semidirect
213. #1844	C₂×C₅₃	semidirect
214. #1744	C₂×C₅₃	semidirect
229. #486	C₄×C₅₃	semidirect
249. C₅₃	#22219	direct (N × H)
258. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
287. #102	C₂×C₄×C₅₃	semidirect
289. C₄×C₈	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×#486	2
C₄×C₄⋊Q₈×C₅₃	2
C₄×#127×C₅₃	2
C₅₃×#1990	2
C₅₃×#1957	2
C₅₃×#1926	2
C₅₃×#1864	2
C₅₃×#1844	2
C₅₃×#1744	2
#22219	53

Subgroups — C53×#22219 #27229731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22219 #27229731