Subgroups — C₅₃×#25452 #27232964

Order 13568.

806 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#474	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2563	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2365	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2085	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2065	C₂	1	2	yes	maximal
7	C₅₃×#1987	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1949	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1939	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1926	C₂	1	2	yes	maximal
11	C₄×#102×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
15	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#498	—	2	4	no	—
17	C₅₃×#474	—	2	4	no	—
18	C₂×#107×C₅₃	C₂²	1	2	yes	—
19	C₂×#106×C₅₃	C₂²	1	2	yes	—
20	C₂×#106×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#102×C₅₃	C₂²	1	2	yes	—
24	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#393	—	2	2	no	—
28	C₅₃×#391	—	2	1	no	—
29	C₅₃×#390	C₂²	1	2	yes	—
30	C₅₃×#389	C₂²	1	2	yes	—
31	C₅₃×#387	—	2	2	no	—
32	C₅₃×#382	C₂²	1	2	yes	—
33	C₅₃×#380	—	2	1	no	—
34	C₅₃×#380	—	2	1	no	—
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#340	C₂²	1	2	yes	—
39	C₅₃×#340	C₂²	1	1	yes	characteristic
40	C₅₃×#340	C₂²	1	2	yes	—
41	C₅₃×#340	C₂²	1	1	yes	characteristic
42	C₅₃×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₅₃	—	2	1	no	—
44	C₂²×Q₈×C₅₃	—	2	1	no	—
45	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₄⋊Q₈×C₅₃	—	2	2	no	—
51	C₄⋊Q₈×C₅₃	—	2	2	no	—
52	C₄⋊Q₈×C₅₃	—	2	2	no	—
53	C₄⋊Q₈×C₅₃	D₄	1	4	yes	—
54	C₄×Q₈×C₅₃	—	4	4	no	—
55	C₄×D₄×C₅₃	—	2	2	no	—
56	C₄×D₄×C₅₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
70	C₂×C₄²×C₅₃	—	2	2	no	—
71	C₂×C₄²×C₅₃	—	2	1	no	—
72	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
73	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	#107×C₅₃	—	2	4	no	—
75	#106×C₅₃	—	2	4	no	—
76	#106×C₅₃	—	2	4	no	—
77	C₄⋊C₈×C₅₃	—	2	4	no	—
78	C₄⋊C₈×C₅₃	—	2	2	no	—
79	#102×C₅₃	—	2	2	no	—
80	#102×C₅₃	—	2	4	no	—
81	#102×C₅₃	Q₈	1	4	yes	complemented
82	C₄×C₈×C₅₃	—	2	2	no	—
83	#95×C₅₃	—	2	1	no	—
84	#95×C₅₃	—	2	2	no	—
85	#95×C₅₃	—	2	2	no	—
86	#95×C₅₃	—	2	2	no	—
87	#95×C₅₃	—	2	2	no	—
88	#95×C₅₃	D₄	1	2	yes	—
89	C₂⁴×C₅₃	—	2	1	no	—
90	C₂×Q₈×C₅₃	—	2	4	no	—
91	C₂×Q₈×C₅₃	—	4	2	no	—
92	C₂×Q₈×C₅₃	—	4	2	no	—
93	C₂×Q₈×C₅₃	—	4	2	no	—
94	C₂×Q₈×C₅₃	—	8	1	no	—
95	C₂×Q₈×C₅₃	—	8	1	no	—
96	C₂×D₄×C₅₃	—	2	1	no	—
97	C₂×D₄×C₅₃	—	2	1	no	—
98	C₂×D₄×C₅₃	—	2	1	no	—
99	C₂×D₄×C₅₃	—	2	1	no	—
100	C₂×D₄×C₅₃	—	2	1	no	—
101	C₂×D₄×C₅₃	C₂×Q₈	1	2	yes	—
102	C₂²×C₄×C₅₃	—	2	1	no	—
103	C₂²×C₄×C₅₃	—	2	2	no	—
104	C₂²×C₄×C₅₃	—	2	1	no	—
105	C₂²×C₄×C₅₃	—	2	1	no	—
106	C₂²×C₄×C₅₃	—	2	2	no	—
107	C₂²×C₄×C₅₃	—	4	2	no	—
108	C₂²×C₄×C₅₃	—	4	2	no	—
109	C₂²×C₄×C₅₃	—	4	2	no	—
110	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
115	C₂×C₈×C₅₃	—	2	1	no	—
116	C₂×C₈×C₅₃	—	2	1	no	—
117	C₂×C₈×C₅₃	—	2	1	no	—
118	C₂×C₈×C₅₃	—	2	1	no	—
119	C₂×C₈×C₅₃	—	2	2	no	—
120	C₂×C₈×C₅₃	—	2	2	no	—
121	C₂×C₈×C₅₃	—	2	2	no	—
122	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
123	C₂×C₈×C₅₃	#41	1	2	yes	—
124	C₄⋊C₄×C₅₃	—	2	4	no	—
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	—	2	2	no	—
127	C₄⋊C₄×C₅₃	—	2	2	no	—
128	C₄⋊C₄×C₅₃	—	2	1	no	—
129	C₄⋊C₄×C₅₃	—	2	2	no	—
130	C₄⋊C₄×C₅₃	—	2	1	no	—
131	C₄⋊C₄×C₅₃	—	2	1	no	—
132	C₄⋊C₄×C₅₃	—	2	1	no	—
133	C₄⋊C₄×C₅₃	—	2	2	no	—
134	C₄⋊C₄×C₅₃	—	2	1	no	—
135	C₄⋊C₄×C₅₃	—	4	4	no	—
136	C₄⋊C₄×C₅₃	—	4	2	no	—
137	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
138	C₄⋊C₄×C₅₃	C₂×Q₈	1	2	yes	—
139	C₂²⋊C₄×C₅₃	—	4	2	no	—
140	C₂²⋊C₄×C₅₃	—	8	2	no	—
141	C₂²⋊C₄×C₅₃	—	8	2	no	—
142	C₄²×C₅₃	—	2	2	no	—
143	C₄²×C₅₃	—	2	2	no	—
144	C₄²×C₅₃	—	4	4	no	—
145	C₄²×C₅₃	—	4	2	no	—
146	C₄²×C₅₃	C₂×D₄	1	2	yes	—
147	C₄²×C₅₃	#41	1	2	yes	—
148	C₂³×C₅₃	—	2	2	no	—
149	C₂³×C₅₃	—	4	1	no	—
150	C₂³×C₅₃	—	4	1	no	—
151	C₂³×C₅₃	—	4	1	no	—
152	C₂³×C₅₃	—	4	1	no	—
153	C₂³×C₅₃	—	4	1	no	—
154	C₂³×C₅₃	—	4	1	no	—
155	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
156	Q₈×C₅₃	—	4	8	no	—
157	Q₈×C₅₃	—	8	4	no	—
158	D₄×C₅₃	—	2	2	no	—
159	D₄×C₅₃	—	2	2	no	—
160	D₄×C₅₃	—	4	1	no	—
161	D₄×C₅₃	—	4	1	no	—
162	C₂×C₄×C₅₃	—	2	1	no	—
163	C₂×C₄×C₅₃	—	2	1	no	—
164	C₂×C₄×C₅₃	—	2	4	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	2	2	no	—
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	4	2	no	—
173	C₂×C₄×C₅₃	—	4	2	no	—
174	C₂×C₄×C₅₃	—	4	1	no	—
175	C₂×C₄×C₅₃	—	4	1	no	—
176	C₂×C₄×C₅₃	—	4	1	no	—
177	C₂×C₄×C₅₃	—	4	2	no	—
178	C₂×C₄×C₅₃	—	4	1	no	—
179	C₂×C₄×C₅₃	—	4	1	no	—
180	C₂×C₄×C₅₃	—	4	1	no	—
181	C₂×C₄×C₅₃	—	4	1	no	—
182	C₂×C₄×C₅₃	—	4	1	no	—
183	C₂×C₄×C₅₃	—	4	2	no	—
184	C₂×C₄×C₅₃	—	4	2	no	—
185	C₂×C₄×C₅₃	—	8	4	no	—
186	C₂×C₄×C₅₃	#121	1	2	yes	—
187	C₂×C₄×C₅₃	#121	1	2	yes	—
188	C₂×C₄×C₅₃	#135	1	2	yes	—
189	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
190	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
195	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
196	C₈×C₅₃	—	2	2	no	—
197	C₈×C₅₃	—	2	2	no	—
198	C₈×C₅₃	—	4	2	no	—
199	#25452	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂²×C₅₃	—	4	2	no	—
201	C₂²×C₅₃	—	4	2	no	—
202	C₂²×C₅₃	—	4	2	no	—
203	C₂²×C₅₃	—	8	1	no	—
204	C₂²×C₅₃	—	8	1	no	—
205	C₂²×C₅₃	—	8	1	no	—
206	C₂²×C₅₃	—	8	1	no	—
207	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
208	C₂²×C₅₃	#547	1	1	yes	characteristic
209	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
210	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
211	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
212	C₂²×C₅₃	#577	1	1	yes	characteristic
213	C₂²×C₅₃	#577	1	1	yes	characteristic
214	C₄×C₅₃	—	2	2	no	—
215	C₄×C₅₃	—	4	2	no	—
216	C₄×C₅₃	—	4	2	no	—
217	C₄×C₅₃	—	4	2	no	—
218	C₄×C₅₃	—	4	4	no	—
219	C₄×C₅₃	—	8	1	no	—
220	C₄×C₅₃	—	8	1	no	—
221	C₄×C₅₃	—	8	2	no	—
222	C₄×C₅₃	#463	1	4	yes	—
223	C₄×C₅₃	#533	1	1	yes	characteristic
224	C₄×C₅₃	#541	1	1	yes	characteristic
225	C₄×C₅₃	#548	1	1	yes	characteristic
226	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
227	C₂×#498	C₂×C₅₃	1	1	yes	characteristic, complemented
228	C₂×#474	C₂×C₅₃	1	1	yes	characteristic
229	#2563	C₂×C₅₃	1	1	yes	characteristic
230	#2365	C₂×C₅₃	1	1	yes	characteristic, complemented
231	#2085	C₂×C₅₃	1	2	yes	complemented
232	#2065	C₂×C₅₃	1	2	yes	—
233	#1987	C₂×C₅₃	1	1	yes	characteristic, complemented
234	#1949	C₂×C₅₃	1	2	yes	complemented
235	#1939	C₂×C₅₃	1	1	yes	characteristic, complemented
236	#1926	C₂×C₅₃	1	2	yes	—
237	C₄×#102	C₂×C₅₃	1	1	yes	characteristic
238	C₂×C₅₃	—	8	2	no	—
239	C₂×C₅₃	#2429	1	1	yes	characteristic
240	C₂×C₅₃	C₂×#463	1	1	yes	characteristic
241	C₂×C₅₃	#3043	1	1	yes	characteristic
242	C₂×C₅₃	#3340	1	1	yes	characteristic
243	C₂×C₅₃	#3354	1	1	yes	characteristic
244	C₂×C₅₃	#3381	1	1	yes	characteristic
245	C₂×C₅₃	#3384	1	1	yes	characteristic
246	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
247	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
248	C₂×C₄×Q₈	—	2	1	no	—
249	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
250	#498	—	2	4	no	—
251	#474	—	2	4	no	—
252	C₂×#107	C₂²×C₅₃	1	2	yes	—
253	C₂×#106	C₂²×C₅₃	1	2	yes	—
254	C₂×#106	C₂²×C₅₃	1	2	yes	—
255	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
256	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
257	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
258	C₂×#102	C₂²×C₅₃	1	2	yes	—
259	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
260	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
261	#393	—	2	2	no	—
262	#391	—	2	1	no	—
263	#390	C₂²×C₅₃	1	2	yes	—
264	#389	C₂²×C₅₃	1	2	yes	—
265	#387	—	2	2	no	—
266	#382	C₂²×C₅₃	1	2	yes	—
267	#380	—	2	1	no	—
268	#380	—	2	1	no	—
269	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
270	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
271	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
272	#340	C₂²×C₅₃	1	2	yes	—
273	#340	C₂²×C₅₃	1	1	yes	characteristic
274	#340	C₂²×C₅₃	1	2	yes	—
275	#340	C₂²×C₅₃	1	1	yes	characteristic
276	#336	C₂²×C₅₃	1	2	yes	—
277	C₅₃	#25452	1	1	yes	characteristic, direct product factor, Sylow
278	C₂²×Q₈	—	2	1	no	—
279	C₂²×Q₈	—	2	1	no	—
280	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
281	C₂³×C₄	—	2	1	no	—
282	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
283	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
284	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
285	C₄⋊Q₈	—	2	2	no	—
286	C₄⋊Q₈	—	2	2	no	—
287	C₄⋊Q₈	—	2	2	no	—
288	C₄⋊Q₈	D₄×C₅₃	1	4	yes	—
289	C₄×Q₈	—	4	4	no	—
290	C₄×D₄	—	2	2	no	—
291	C₄×D₄	—	2	2	no	—
292	C₂×C₄⋊C₄	—	2	1	no	—
293	C₂×C₄⋊C₄	—	2	2	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
296	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
297	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
298	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
299	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
300	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
301	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
302	C₂×C₂²⋊C₄	—	2	1	no	—
303	C₂×C₂²⋊C₄	—	2	2	no	—
304	C₂×C₂²⋊C₄	—	2	2	no	—
305	C₂×C₄²	—	2	2	no	—
306	C₂×C₄²	—	2	1	no	—
307	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
308	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
309	#107	—	2	4	no	—
310	#106	—	2	4	no	—
311	#106	—	2	4	no	—
312	C₄⋊C₈	—	2	4	no	—
313	C₄⋊C₈	—	2	2	no	—
314	#102	—	2	2	no	—
315	#102	—	2	4	no	—
316	#102	Q₈×C₅₃	1	4	yes	complemented
317	C₄×C₈	—	2	2	no	—
318	#95	—	2	1	no	—
319	#95	—	2	2	no	—
320	#95	—	2	2	no	—
321	#95	—	2	2	no	—
322	#95	—	2	2	no	—
323	#95	D₄×C₅₃	1	2	yes	—
324	C₂⁴	—	2	1	no	—
325	C₂×Q₈	—	2	4	no	—
326	C₂×Q₈	—	4	2	no	—
327	C₂×Q₈	—	4	2	no	—
328	C₂×Q₈	—	4	2	no	—
329	C₂×Q₈	—	8	1	no	—
330	C₂×Q₈	—	8	1	no	—
331	C₂×D₄	—	2	1	no	—
332	C₂×D₄	—	2	1	no	—
333	C₂×D₄	—	2	1	no	—
334	C₂×D₄	—	2	1	no	—
335	C₂×D₄	—	2	1	no	—
336	C₂×D₄	C₂×Q₈×C₅₃	1	2	yes	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	4	2	no	—
343	C₂²×C₄	—	4	2	no	—
344	C₂²×C₄	—	4	2	no	—
345	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
346	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
347	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
348	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
349	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
358	C₂×C₈	#41×C₅₃	1	2	yes	—
359	C₄⋊C₄	—	2	4	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	4	4	no	—
371	C₄⋊C₄	—	4	2	no	—
372	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
373	C₄⋊C₄	C₂×Q₈×C₅₃	1	2	yes	—
374	C₂²⋊C₄	—	4	2	no	—
375	C₂²⋊C₄	—	8	2	no	—
376	C₂²⋊C₄	—	8	2	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	2	2	no	—
379	C₄²	—	4	4	no	—
380	C₄²	—	4	2	no	—
381	C₄²	C₂×D₄×C₅₃	1	2	yes	—
382	C₄²	#41×C₅₃	1	2	yes	—
383	C₂³	—	2	2	no	—
384	C₂³	—	4	1	no	—
385	C₂³	—	4	1	no	—
386	C₂³	—	4	1	no	—
387	C₂³	—	4	1	no	—
388	C₂³	—	4	1	no	—
389	C₂³	—	4	1	no	—
390	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
391	Q₈	—	4	8	no	—
392	Q₈	—	8	4	no	—
393	D₄	—	2	2	no	—
394	D₄	—	2	2	no	—
395	D₄	—	4	1	no	—
396	D₄	—	4	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	8	4	no	—
421	C₂×C₄	#121×C₅₃	1	2	yes	—
422	C₂×C₄	#121×C₅₃	1	2	yes	—
423	C₂×C₄	#135×C₅₃	1	2	yes	—
424	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
425	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
426	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
428	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
429	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
430	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
431	C₈	—	2	2	no	—
432	C₈	—	2	2	no	—
433	C₈	—	4	2	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	4	2	no	—
436	C₂²	—	4	2	no	—
437	C₂²	—	8	1	no	—
438	C₂²	—	8	1	no	—
439	C₂²	—	8	1	no	—
440	C₂²	—	8	1	no	—
441	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
442	C₂²	C₅₃×#547	1	1	yes	characteristic
443	C₂²	D₄×Q₈×C₅₃	1	1	yes	characteristic
444	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
445	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
446	C₂²	C₅₃×#577	1	1	yes	characteristic
447	C₂²	C₅₃×#577	1	1	yes	characteristic
448	C₄	—	2	2	no	—
449	C₄	—	4	4	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	—	8	1	no	—
456	C₄	C₅₃×#463	1	4	yes	—
457	C₄	C₅₃×#533	1	1	yes	characteristic
458	C₄	C₅₃×#541	1	1	yes	characteristic
459	C₄	C₅₃×#548	1	1	yes	characteristic
460	C₄	D₄×Q₈×C₅₃	1	1	yes	characteristic
461	C₂	—	8	2	no	—
462	C₂	C₅₃×#2429	1	1	yes	characteristic
463	C₂	C₂×C₅₃×#463	1	1	yes	characteristic
464	C₂	C₅₃×#3043	1	1	yes	characteristic
465	C₂	C₅₃×#3340	1	1	yes	characteristic
466	C₂	C₅₃×#3354	1	1	yes	characteristic
467	C₂	C₅₃×#3381	1	1	yes	characteristic
468	C₂	C₅₃×#3384	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#498	C₂	semidirect
4. C₅₃×#2365	C₂	semidirect
5. C₅₃×#2085	C₂	semidirect
7. C₅₃×#1987	C₂	semidirect
8. C₅₃×#1949	C₂	semidirect
9. C₅₃×#1939	C₂	semidirect
81. #102×C₅₃	Q₈	semidirect
199. #25452	C₅₃	direct (N × H)
227. C₂×#498	C₂×C₅₃	semidirect
230. #2365	C₂×C₅₃	semidirect
231. #2085	C₂×C₅₃	semidirect
233. #1987	C₂×C₅₃	semidirect
234. #1949	C₂×C₅₃	semidirect
235. #1939	C₂×C₅₃	semidirect
277. C₅₃	#25452	direct (N × H)
316. #102	Q₈×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×#498	2
C₂×C₅₃×#474	2
C₅₃×#2563	2
C₅₃×#2365	2
C₅₃×#2085	2
C₅₃×#2065	2
C₅₃×#1987	2
C₅₃×#1949	2
C₅₃×#1939	2
C₅₃×#1926	2
C₄×#102×C₅₃	2
#25452	53

Subgroups — C53×#25452 #27232964

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#25452 #27232964