Subgroups — C₅₃×#17595 #27225107

Order 13568.

1274 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1519	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1514	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1505	C₂	1	4	yes	maximal, complemented
7	C₂²×#41×C₅₃	—	2	1	no	—
8	C₅₃×#532	—	2	1	no	—
9	C₅₃×#532	—	2	1	no	—
10	C₅₃×#532	—	2	1	no	—
11	C₅₃×#532	—	2	1	no	—
12	C₅₃×#532	C₄	1	2	yes	complemented
13	C₅₃×#532	C₄	1	2	yes	complemented
14	C₅₃×#532	C₂²	1	2	yes	—
15	C₅₃×#532	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
17	C₂×#127×C₅₃	C₄	1	1	yes	characteristic, complemented
18	C₂×#124×C₅₃	—	2	1	no	—
19	C₂×#121×C₅₃	—	2	1	no	—
20	C₂×#121×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₅₃	—	2	1	no	—
22	C₂×C₄×D₄×C₅₃	C₄	1	2	yes	complemented
23	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#432	—	2	1	no	—
25	C₅₃×#432	—	2	1	no	—
26	C₅₃×#432	C₂²	1	2	yes	complemented
27	C₅₃×#432	C₂²	1	2	yes	complemented
28	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
29	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
30	C₂×#97×C₅₃	—	2	1	no	—
31	C₅₃×#383	C₂²	1	2	yes	complemented
32	C₅₃×#382	—	2	1	no	—
33	C₅₃×#381	—	2	1	no	—
34	C₅₃×#380	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
37	C₅₃×#376	C₂²	1	2	yes	complemented
38	C₅₃×#327	C₂²	1	16	yes	complemented
39	C₂×#41×C₅₃	—	2	2	no	—
40	C₂×#41×C₅₃	—	2	2	no	—
41	C₂×#41×C₅₃	—	2	2	no	—
42	C₂×#41×C₅₃	—	2	2	no	—
43	C₂×#41×C₅₃	—	4	8	no	—
44	C₂²×Q₈×C₅₃	—	2	1	no	—
45	C₂²×D₄×C₅₃	—	2	2	no	—
46	C₂²×D₄×C₅₃	—	2	1	no	—
47	C₂²×D₄×C₅₃	—	2	1	no	—
48	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₂³×C₄×C₅₃	—	2	2	no	—
50	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	—	2	2	no	—
54	C₄⋊Q₈×C₅₃	—	2	1	no	—
55	C₄⋊Q₈×C₅₃	—	2	1	no	—
56	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
57	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
58	#127×C₅₃	—	2	1	no	—
59	#127×C₅₃	—	2	1	no	—
60	#127×C₅₃	C₂×C₄	1	2	yes	—
61	#127×C₅₃	C₂×C₄	1	2	yes	—
62	#124×C₅₃	—	2	2	no	—
63	#124×C₅₃	—	2	2	no	—
64	#124×C₅₃	—	2	2	no	—
65	#124×C₅₃	—	2	2	no	—
66	#121×C₅₃	—	2	2	no	—
67	#121×C₅₃	—	2	2	no	—
68	#121×C₅₃	—	2	2	no	—
69	#121×C₅₃	—	2	2	no	—
70	#121×C₅₃	—	2	2	no	—
71	#121×C₅₃	—	2	2	no	—
72	#121×C₅₃	C₂×C₄	1	4	yes	—
73	#121×C₅₃	C₂³	1	4	yes	—
74	C₄×D₄×C₅₃	—	2	2	no	—
75	C₄×D₄×C₅₃	—	2	2	no	—
76	C₄×D₄×C₅₃	—	2	2	no	—
77	C₄×D₄×C₅₃	—	2	2	no	—
78	C₄×D₄×C₅₃	—	2	2	no	—
79	C₄×D₄×C₅₃	—	2	2	no	—
80	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
81	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
82	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
84	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
85	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
88	C₂×C₄²×C₅₃	—	2	2	no	—
89	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
96	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₈×C₅₃	—	2	2	no	—
102	C₄⋊C₈×C₅₃	—	2	2	no	—
103	C₂²⋊C₈×C₅₃	—	2	2	no	—
104	C₂²⋊C₈×C₅₃	C₂³	1	4	yes	—
105	#97×C₅₃	—	2	2	no	—
106	#97×C₅₃	—	2	2	no	—
107	#95×C₅₃	—	2	2	no	—
108	#95×C₅₃	—	2	2	no	—
109	#95×C₅₃	—	2	2	no	—
110	#95×C₅₃	C₂³	1	4	yes	—
111	C₂⁴×C₅₃	—	2	2	no	—
112	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
113	#41×C₅₃	—	4	8	no	—
114	#41×C₅₃	—	4	8	no	—
115	#41×C₅₃	—	4	8	no	—
116	#41×C₅₃	—	4	8	no	—
117	C₂×Q₈×C₅₃	—	2	2	no	—
118	C₂×Q₈×C₅₃	—	2	2	no	—
119	C₂×Q₈×C₅₃	—	4	4	no	—
120	C₂×D₄×C₅₃	—	2	4	no	—
121	C₂×D₄×C₅₃	—	2	4	no	—
122	C₂×D₄×C₅₃	—	2	2	no	—
123	C₂×D₄×C₅₃	—	2	2	no	—
124	C₂×D₄×C₅₃	—	2	2	no	—
125	C₂×D₄×C₅₃	—	2	2	no	—
126	C₂×D₄×C₅₃	—	2	2	no	—
127	C₂×D₄×C₅₃	—	4	4	no	—
128	C₂×D₄×C₅₃	—	4	4	no	—
129	C₂×D₄×C₅₃	—	4	4	no	—
130	C₂×D₄×C₅₃	—	4	8	no	—
131	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
132	C₂²×C₄×C₅₃	—	2	4	no	—
133	C₂²×C₄×C₅₃	—	2	2	no	—
134	C₂²×C₄×C₅₃	—	2	4	no	—
135	C₂²×C₄×C₅₃	—	2	4	no	—
136	C₂²×C₄×C₅₃	—	2	4	no	—
137	C₂²×C₄×C₅₃	—	4	2	no	—
138	C₂²×C₄×C₅₃	—	4	2	no	—
139	C₂²×C₄×C₅₃	—	4	8	no	—
140	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
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	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
158	C₂×C₈×C₅₃	—	2	4	no	—
159	C₂×C₈×C₅₃	—	4	4	no	—
160	C₄⋊C₄×C₅₃	—	2	2	no	—
161	C₄⋊C₄×C₅₃	—	2	2	no	—
162	C₄⋊C₄×C₅₃	—	2	2	no	—
163	C₄⋊C₄×C₅₃	—	4	4	no	—
164	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
165	C₂²⋊C₄×C₅₃	—	2	4	no	—
166	C₂²⋊C₄×C₅₃	—	2	4	no	—
167	C₂²⋊C₄×C₅₃	—	2	4	no	—
168	C₂²⋊C₄×C₅₃	—	2	4	no	—
169	C₂²⋊C₄×C₅₃	—	4	4	no	—
170	C₄²×C₅₃	—	4	4	no	—
171	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
172	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
173	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
174	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
175	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
176	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
177	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
178	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
179	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
180	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
181	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
182	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
183	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
184	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
185	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
186	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
187	C₂³×C₅₃	—	2	4	no	—
188	C₂³×C₅₃	—	4	1	no	—
189	C₂³×C₅₃	—	4	1	no	—
190	C₂³×C₅₃	—	4	4	no	—
191	C₂³×C₅₃	—	4	4	no	—
192	C₂³×C₅₃	—	4	4	no	—
193	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
194	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
195	C₂³×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
196	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
197	Q₈×C₅₃	—	4	4	no	—
198	Q₈×C₅₃	—	4	4	no	—
199	D₄×C₅₃	—	4	4	no	—
200	D₄×C₅₃	—	4	4	no	—
201	D₄×C₅₃	—	4	4	no	—
202	D₄×C₅₃	—	4	4	no	—
203	D₄×C₅₃	—	4	8	no	—
204	D₄×C₅₃	—	4	8	no	—
205	D₄×C₅₃	—	4	4	no	—
206	D₄×C₅₃	—	4	4	no	—
207	C₂×C₄×C₅₃	—	2	2	no	—
208	C₂×C₄×C₅₃	—	2	2	no	—
209	C₂×C₄×C₅₃	—	2	2	no	—
210	C₂×C₄×C₅₃	—	2	2	no	—
211	C₂×C₄×C₅₃	—	2	2	no	—
212	C₂×C₄×C₅₃	—	2	2	no	—
213	C₂×C₄×C₅₃	—	2	2	no	—
214	C₂×C₄×C₅₃	—	2	2	no	—
215	C₂×C₄×C₅₃	—	2	2	no	—
216	C₂×C₄×C₅₃	—	2	2	no	—
217	C₂×C₄×C₅₃	—	2	2	no	—
218	C₂×C₄×C₅₃	—	2	2	no	—
219	C₂×C₄×C₅₃	—	2	4	no	—
220	C₂×C₄×C₅₃	—	4	1	no	—
221	C₂×C₄×C₅₃	—	4	1	no	—
222	C₂×C₄×C₅₃	—	4	1	no	—
223	C₂×C₄×C₅₃	—	4	1	no	—
224	C₂×C₄×C₅₃	—	4	1	no	—
225	C₂×C₄×C₅₃	—	4	1	no	—
226	C₂×C₄×C₅₃	—	4	8	no	—
227	C₂×C₄×C₅₃	—	4	8	no	—
228	C₂×C₄×C₅₃	—	4	4	no	—
229	C₂×C₄×C₅₃	—	4	4	no	—
230	C₂×C₄×C₅₃	—	4	4	no	—
231	C₂×C₄×C₅₃	—	4	4	no	—
232	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
233	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
235	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
238	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
239	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
240	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
241	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
242	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
243	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
244	C₈×C₅₃	—	4	4	no	—
245	#17595	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
246	C₂²×C₅₃	—	2	1	no	—
247	C₂²×C₅₃	—	2	1	no	—
248	C₂²×C₅₃	—	2	1	no	—
249	C₂²×C₅₃	—	2	1	no	—
250	C₂²×C₅₃	—	2	1	no	—
251	C₂²×C₅₃	—	2	1	no	—
252	C₂²×C₅₃	—	4	4	no	—
253	C₂²×C₅₃	—	4	4	no	—
254	C₂²×C₅₃	—	4	4	no	—
255	C₂²×C₅₃	—	4	4	no	—
256	C₂²×C₅₃	—	8	8	no	—
257	C₂²×C₅₃	#418	1	2	yes	—
258	C₂²×C₅₃	#418	1	2	yes	—
259	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
260	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
261	C₄×C₅₃	—	2	1	no	—
262	C₄×C₅₃	—	2	1	no	—
263	C₄×C₅₃	—	2	1	no	—
264	C₄×C₅₃	—	2	1	no	—
265	C₄×C₅₃	—	2	1	no	—
266	C₄×C₅₃	—	2	1	no	—
267	C₄×C₅₃	—	2	1	no	—
268	C₄×C₅₃	—	2	1	no	—
269	C₄×C₅₃	—	2	1	no	—
270	C₄×C₅₃	—	2	1	no	—
271	C₄×C₅₃	—	4	4	no	—
272	C₄×C₅₃	—	8	4	no	—
273	C₄×C₅₃	#410	1	2	yes	—
274	C₄×C₅₃	#421	1	2	yes	—
275	C₂×#532	C₂×C₅₃	1	1	yes	characteristic
276	C₂×#432	C₂×C₅₃	1	1	yes	characteristic, complemented
277	#2283	C₂×C₅₃	1	1	yes	characteristic, complemented
278	#1519	C₂×C₅₃	1	4	yes	complemented
279	#1514	C₂×C₅₃	1	4	yes	complemented
280	#1505	C₂×C₅₃	1	4	yes	complemented
281	C₂×C₅₃	—	2	2	no	—
282	C₂×C₅₃	—	2	2	no	—
283	C₂×C₅₃	—	2	2	no	—
284	C₂×C₅₃	—	8	4	no	—
285	C₂×C₅₃	C₂×#410	1	1	yes	characteristic
286	C₂×C₅₃	C₂×#418	1	1	yes	characteristic
287	C₂×C₅₃	C₂×#421	1	1	yes	characteristic
288	C₂²×#41	—	2	1	no	—
289	#532	—	2	1	no	—
290	#532	—	2	1	no	—
291	#532	—	2	1	no	—
292	#532	—	2	1	no	—
293	#532	C₄×C₅₃	1	2	yes	complemented
294	#532	C₄×C₅₃	1	2	yes	complemented
295	#532	C₂²×C₅₃	1	2	yes	—
296	#532	C₂²×C₅₃	1	2	yes	—
297	C₂×C₄⋊Q₈	C₄×C₅₃	1	1	yes	characteristic, complemented
298	C₂×#127	C₄×C₅₃	1	1	yes	characteristic, complemented
299	C₂×#124	—	2	1	no	—
300	C₂×#121	—	2	1	no	—
301	C₂×#121	C₂²×C₅₃	1	2	yes	—
302	C₂×C₄×D₄	—	2	1	no	—
303	C₂×C₄×D₄	C₄×C₅₃	1	2	yes	complemented
304	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
305	#432	—	2	1	no	—
306	#432	—	2	1	no	—
307	#432	C₂²×C₅₃	1	2	yes	complemented
308	#432	C₂²×C₅₃	1	2	yes	complemented
309	C₂×C₄⋊C₈	—	2	1	no	—
310	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
311	C₂×#97	—	2	1	no	—
312	#383	C₂²×C₅₃	1	2	yes	complemented
313	#382	—	2	1	no	—
314	#381	—	2	1	no	—
315	#380	C₂²×C₅₃	1	2	yes	—
316	C₄×C₄⋊C₄	—	2	1	no	—
317	C₄×C₂²⋊C₄	—	2	1	no	—
318	#376	C₂²×C₅₃	1	2	yes	complemented
319	#327	C₂²×C₅₃	1	16	yes	complemented
320	C₅₃	#17595	1	1	yes	characteristic, direct product factor, Sylow
321	C₂×#41	—	2	2	no	—
322	C₂×#41	—	2	2	no	—
323	C₂×#41	—	2	2	no	—
324	C₂×#41	—	2	2	no	—
325	C₂×#41	—	4	8	no	—
326	C₂²×Q₈	—	2	1	no	—
327	C₂²×D₄	—	2	2	no	—
328	C₂²×D₄	—	2	1	no	—
329	C₂²×D₄	—	2	1	no	—
330	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
331	C₂³×C₄	—	2	2	no	—
332	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
333	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
334	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
335	C₂²×C₈	—	2	2	no	—
336	C₄⋊Q₈	—	2	1	no	—
337	C₄⋊Q₈	—	2	1	no	—
338	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
339	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
340	#127	—	2	1	no	—
341	#127	—	2	1	no	—
342	#127	C₂×C₄×C₅₃	1	2	yes	—
343	#127	C₂×C₄×C₅₃	1	2	yes	—
344	#124	—	2	2	no	—
345	#124	—	2	2	no	—
346	#124	—	2	2	no	—
347	#124	—	2	2	no	—
348	#121	—	2	2	no	—
349	#121	—	2	2	no	—
350	#121	—	2	2	no	—
351	#121	—	2	2	no	—
352	#121	—	2	2	no	—
353	#121	—	2	2	no	—
354	#121	C₂×C₄×C₅₃	1	4	yes	—
355	#121	C₂³×C₅₃	1	4	yes	—
356	C₄×D₄	—	2	2	no	—
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	—	2	2	no	—
361	C₄×D₄	—	2	2	no	—
362	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
363	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	C₂×C₄⋊C₄	—	2	2	no	—
366	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
367	C₂×C₂²⋊C₄	—	2	2	no	—
368	C₂×C₂²⋊C₄	—	2	2	no	—
369	C₂×C₂²⋊C₄	—	2	2	no	—
370	C₂×C₄²	—	2	2	no	—
371	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
372	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
373	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
374	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
375	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
376	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
377	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
378	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
379	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
380	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
381	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
382	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
383	C₄⋊C₈	—	2	2	no	—
384	C₄⋊C₈	—	2	2	no	—
385	C₂²⋊C₈	—	2	2	no	—
386	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
387	#97	—	2	2	no	—
388	#97	—	2	2	no	—
389	#95	—	2	2	no	—
390	#95	—	2	2	no	—
391	#95	—	2	2	no	—
392	#95	C₂³×C₅₃	1	4	yes	—
393	C₂⁴	—	2	2	no	—
394	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
395	#41	—	4	8	no	—
396	#41	—	4	8	no	—
397	#41	—	4	8	no	—
398	#41	—	4	8	no	—
399	C₂×Q₈	—	2	2	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	4	4	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	2	4	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	4	4	no	—
410	C₂×D₄	—	4	4	no	—
411	C₂×D₄	—	4	4	no	—
412	C₂×D₄	—	4	8	no	—
413	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	4	2	no	—
420	C₂²×C₄	—	4	2	no	—
421	C₂²×C₄	—	4	8	no	—
422	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
423	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
424	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
425	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
426	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
427	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
428	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
429	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
430	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
431	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
433	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
434	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
436	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
437	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
438	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
439	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
440	C₂×C₈	—	2	4	no	—
441	C₂×C₈	—	4	4	no	—
442	C₄⋊C₄	—	2	2	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	4	4	no	—
446	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
447	C₂²⋊C₄	—	2	4	no	—
448	C₂²⋊C₄	—	2	4	no	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	4	4	no	—
452	C₄²	—	4	4	no	—
453	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
454	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
455	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
456	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
457	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
458	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
459	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
460	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
461	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
462	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
463	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
464	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
465	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
466	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
467	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
468	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
469	C₂³	—	2	4	no	—
470	C₂³	—	4	1	no	—
471	C₂³	—	4	1	no	—
472	C₂³	—	4	4	no	—
473	C₂³	—	4	4	no	—
474	C₂³	—	4	4	no	—
475	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
476	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
477	C₂³	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
478	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
479	Q₈	—	4	4	no	—
480	Q₈	—	4	4	no	—
481	D₄	—	4	4	no	—
482	D₄	—	4	4	no	—
483	D₄	—	4	4	no	—
484	D₄	—	4	4	no	—
485	D₄	—	4	8	no	—
486	D₄	—	4	8	no	—
487	D₄	—	4	4	no	—
488	D₄	—	4	4	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	4	1	no	—
505	C₂×C₄	—	4	1	no	—
506	C₂×C₄	—	4	1	no	—
507	C₂×C₄	—	4	1	no	—
508	C₂×C₄	—	4	8	no	—
509	C₂×C₄	—	4	8	no	—
510	C₂×C₄	—	4	4	no	—
511	C₂×C₄	—	4	4	no	—
512	C₂×C₄	—	4	4	no	—
513	C₂×C₄	—	4	4	no	—
514	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
515	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
516	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
517	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
518	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
519	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
520	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
521	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
522	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
523	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
524	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
525	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
526	C₈	—	4	4	no	—
527	C₂²	—	2	1	no	—
528	C₂²	—	2	1	no	—
529	C₂²	—	2	1	no	—
530	C₂²	—	2	1	no	—
531	C₂²	—	2	1	no	—
532	C₂²	—	2	1	no	—
533	C₂²	—	4	4	no	—
534	C₂²	—	4	4	no	—
535	C₂²	—	4	4	no	—
536	C₂²	—	4	4	no	—
537	C₂²	—	8	8	no	—
538	C₂²	C₅₃×#418	1	2	yes	—
539	C₂²	C₅₃×#418	1	2	yes	—
540	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
541	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
542	C₄	—	2	1	no	—
543	C₄	—	2	1	no	—
544	C₄	—	2	1	no	—
545	C₄	—	2	1	no	—
546	C₄	—	2	1	no	—
547	C₄	—	2	1	no	—
548	C₄	—	2	1	no	—
549	C₄	—	2	1	no	—
550	C₄	—	2	1	no	—
551	C₄	—	2	1	no	—
552	C₄	—	4	4	no	—
553	C₄	—	8	4	no	—
554	C₄	C₅₃×#410	1	2	yes	—
555	C₄	C₅₃×#421	1	2	yes	—
556	C₂	—	2	2	no	—
557	C₂	—	2	2	no	—
558	C₂	—	2	2	no	—
559	C₂	—	8	4	no	—
560	C₂	C₂×C₅₃×#410	1	1	yes	characteristic
561	C₂	C₂×C₅₃×#418	1	1	yes	characteristic
562	C₂	C₂×C₅₃×#421	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂×C₅₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#432	C₂	semidirect
3. C₅₃×#2283	C₂	semidirect
4. C₅₃×#1519	C₂	semidirect
5. C₅₃×#1514	C₂	semidirect
6. C₅₃×#1505	C₂	semidirect
12. C₅₃×#532	C₄	semidirect
13. C₅₃×#532	C₄	semidirect
16. C₂×C₄⋊Q₈×C₅₃	C₄	semidirect
17. C₂×#127×C₅₃	C₄	semidirect
22. C₂×C₄×D₄×C₅₃	C₄	semidirect
26. C₅₃×#432	C₂²	semidirect
27. C₅₃×#432	C₂²	semidirect
31. C₅₃×#383	C₂²	semidirect
37. C₅₃×#376	C₂²	semidirect
38. C₅₃×#327	C₂²	semidirect
245. #17595	C₅₃	direct (N × H)
276. C₂×#432	C₂×C₅₃	semidirect
277. #2283	C₂×C₅₃	semidirect
278. #1519	C₂×C₅₃	semidirect
279. #1514	C₂×C₅₃	semidirect
280. #1505	C₂×C₅₃	semidirect
293. #532	C₄×C₅₃	semidirect
294. #532	C₄×C₅₃	semidirect
297. C₂×C₄⋊Q₈	C₄×C₅₃	semidirect
298. C₂×#127	C₄×C₅₃	semidirect
303. C₂×C₄×D₄	C₄×C₅₃	semidirect
307. #432	C₂²×C₅₃	semidirect
308. #432	C₂²×C₅₃	semidirect
312. #383	C₂²×C₅₃	semidirect
318. #376	C₂²×C₅₃	semidirect
319. #327	C₂²×C₅₃	semidirect
320. C₅₃	#17595	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×#532	2
C₂×C₅₃×#432	2
C₅₃×#2283	2
C₅₃×#1519	2
C₅₃×#1514	2
C₅₃×#1505	2
#17595	53

Subgroups — C53×#17595 #27225107

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17595 #27225107