Subgroups — C₅₃×#17796 #27225308

Order 13568.

756 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1569	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1567	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1566	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1565	C₂	1	4	yes	maximal, complemented
8	C₂²×M₁₆×C₅₃	—	2	1	no	—
9	C₂²×M₁₆×C₅₃	C₄	1	2	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#432	—	2	1	no	—
12	C₅₃×#432	C₂²	1	2	yes	—
13	C₅₃×#431	—	2	1	no	—
14	C₅₃×#431	C₂²	1	2	yes	—
15	C₅₃×#423	—	2	1	no	—
16	C₅₃×#423	—	2	1	no	—
17	C₅₃×#423	C₂²	1	1	yes	characteristic
18	C₅₃×#423	C₂²	1	2	yes	—
19	C₅₃×#423	C₂²	1	1	yes	characteristic
20	C₅₃×#423	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#407	—	2	1	no	—
24	C₄×M₁₆×C₅₃	—	2	1	no	—
25	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
26	C₄×M₁₆×C₅₃	C₄	1	2	yes	—
27	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
28	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
29	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
30	C₂×#97×C₅₃	C₄	1	2	yes	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
32	C₅₃×#335	C₂²	1	8	yes	complemented
33	C₅₃×#334	C₂²	1	8	yes	complemented
34	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
37	C₂×M₁₆×C₅₃	—	2	8	no	—
38	C₂×M₁₆×C₅₃	—	2	2	no	—
39	C₂×M₁₆×C₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	—	2	1	no	—
41	C₂×M₁₆×C₅₃	—	2	1	no	—
42	C₂×M₁₆×C₅₃	—	4	4	no	—
43	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
44	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₅₃	—	2	2	no	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₅₃	—	2	2	no	—
61	C₄⋊C₈×C₅₃	—	2	2	no	—
62	C₄⋊C₈×C₅₃	—	2	2	no	—
63	C₄⋊C₈×C₅₃	—	2	1	no	—
64	C₄⋊C₈×C₅₃	—	2	1	no	—
65	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
66	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
67	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
68	C₂²⋊C₈×C₅₃	—	2	2	no	—
69	C₂²⋊C₈×C₅₃	—	2	2	no	—
70	#97×C₅₃	—	2	2	no	—
71	#97×C₅₃	C₂×C₄	1	2	yes	—
72	#97×C₅₃	C₂×C₄	1	4	yes	—
73	#97×C₅₃	C₂×C₄	1	2	yes	—
74	C₄×C₈×C₅₃	—	2	2	no	—
75	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
76	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
77	C₄×C₈×C₅₃	C₂³	1	4	yes	—
78	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	—	2	4	no	—
80	C₂²×C₄×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	—	2	2	no	—
82	C₂²×C₄×C₅₃	—	4	2	no	—
83	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
101	M₁₆×C₅₃	—	2	4	no	—
102	M₁₆×C₅₃	—	2	8	no	—
103	M₁₆×C₅₃	—	4	4	no	—
104	M₁₆×C₅₃	—	4	2	no	—
105	M₁₆×C₅₃	—	4	4	no	—
106	M₁₆×C₅₃	—	4	2	no	—
107	M₁₆×C₅₃	M₁₆	1	8	yes	complemented
108	C₂×C₈×C₅₃	—	2	8	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	—	2	2	no	—
111	C₂×C₈×C₅₃	—	2	4	no	—
112	C₂×C₈×C₅₃	—	4	4	no	—
113	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
114	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
115	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
117	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
118	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
119	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
120	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
121	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
128	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
129	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
130	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
131	C₂³×C₅₃	—	2	2	no	—
132	C₂³×C₅₃	—	2	2	no	—
133	C₂³×C₅₃	—	4	1	no	—
134	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	2	1	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	2	2	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	2	2	no	—
152	C₂×C₄×C₅₃	—	4	1	no	—
153	C₂×C₄×C₅₃	—	4	1	no	—
154	C₂×C₄×C₅₃	—	4	1	no	—
155	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
165	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
166	C₂×C₄×C₅₃	#137	1	2	yes	—
167	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
170	C₈×C₅₃	—	2	4	no	—
171	C₈×C₅₃	—	4	4	no	—
172	C₈×C₅₃	—	4	2	no	—
173	C₈×C₅₃	—	4	2	no	—
174	C₈×C₅₃	C₂×M₁₆	1	8	yes	complemented
175	#17796	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂²×C₅₃	—	2	1	no	—
177	C₂²×C₅₃	—	2	1	no	—
178	C₂²×C₅₃	—	2	1	no	—
179	C₂²×C₅₃	—	2	1	no	—
180	C₂²×C₅₃	—	2	1	no	—
181	C₂²×C₅₃	—	2	1	no	—
182	C₂²×C₅₃	—	2	1	no	—
183	C₂²×C₅₃	—	4	4	no	—
184	C₂²×C₅₃	#423	1	2	yes	—
185	C₂²×C₅₃	#428	1	2	yes	—
186	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
187	C₄×C₅₃	—	2	1	no	—
188	C₄×C₅₃	—	2	1	no	—
189	C₄×C₅₃	—	2	1	no	—
190	C₄×C₅₃	—	2	1	no	—
191	C₄×C₅₃	—	2	1	no	—
192	C₄×C₅₃	—	2	1	no	—
193	C₄×C₅₃	—	2	1	no	—
194	C₄×C₅₃	#423	1	2	yes	—
195	C₄×C₅₃	#428	1	2	yes	—
196	C₄×C₅₃	#430	1	2	yes	—
197	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
198	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
199	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
200	#2920	C₂×C₅₃	1	1	yes	characteristic
201	C₂×#423	C₂×C₅₃	1	1	yes	characteristic
202	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic
203	#1569	C₂×C₅₃	1	4	yes	complemented
204	#1567	C₂×C₅₃	1	2	yes	complemented
205	#1566	C₂×C₅₃	1	2	yes	complemented
206	#1565	C₂×C₅₃	1	4	yes	complemented
207	C₂×C₅₃	—	2	2	no	—
208	C₂×C₅₃	—	2	2	no	—
209	C₂×C₅₃	—	4	1	no	—
210	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
211	C₂×C₅₃	C₂×#428	1	1	yes	characteristic
212	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
213	C₂²×M₁₆	—	2	1	no	—
214	C₂²×M₁₆	C₄×C₅₃	1	2	yes	—
215	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
216	#432	—	2	1	no	—
217	#432	C₂²×C₅₃	1	2	yes	—
218	#431	—	2	1	no	—
219	#431	C₂²×C₅₃	1	2	yes	—
220	#423	—	2	1	no	—
221	#423	—	2	1	no	—
222	#423	C₂²×C₅₃	1	1	yes	characteristic
223	#423	C₂²×C₅₃	1	2	yes	—
224	#423	C₂²×C₅₃	1	1	yes	characteristic
225	#423	C₂²×C₅₃	1	2	yes	—
226	C₂×C₄⋊C₈	—	2	1	no	—
227	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
228	#407	—	2	1	no	—
229	C₄×M₁₆	—	2	1	no	—
230	C₄×M₁₆	C₄×C₅₃	1	1	yes	characteristic
231	C₄×M₁₆	C₄×C₅₃	1	2	yes	—
232	C₄×M₁₆	C₄×C₅₃	1	1	yes	characteristic
233	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
234	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
235	C₂×#97	C₄×C₅₃	1	2	yes	—
236	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
237	#335	C₂²×C₅₃	1	8	yes	complemented
238	#334	C₂²×C₅₃	1	8	yes	complemented
239	C₅₃	#17796	1	1	yes	characteristic, direct product factor, Sylow
240	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
241	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
242	C₂³×C₄	Q₈×C₅₃	1	1	yes	characteristic
243	C₂×M₁₆	—	2	8	no	—
244	C₂×M₁₆	—	2	2	no	—
245	C₂×M₁₆	—	2	2	no	—
246	C₂×M₁₆	—	2	1	no	—
247	C₂×M₁₆	—	2	1	no	—
248	C₂×M₁₆	—	4	4	no	—
249	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
250	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
251	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
252	C₂²×C₈	—	2	2	no	—
253	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
254	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
255	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
256	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
257	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
258	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
259	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
260	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
261	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
262	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
263	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
264	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
265	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
266	C₄⋊C₈	—	2	2	no	—
267	C₄⋊C₈	—	2	2	no	—
268	C₄⋊C₈	—	2	2	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	2	1	no	—
271	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
272	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
273	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
274	C₂²⋊C₈	—	2	2	no	—
275	C₂²⋊C₈	—	2	2	no	—
276	#97	—	2	2	no	—
277	#97	C₂×C₄×C₅₃	1	2	yes	—
278	#97	C₂×C₄×C₅₃	1	4	yes	—
279	#97	C₂×C₄×C₅₃	1	2	yes	—
280	C₄×C₈	—	2	2	no	—
281	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
282	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
283	C₄×C₈	C₂³×C₅₃	1	4	yes	—
284	C₂⁴	C₄⋊C₄×C₅₃	1	1	yes	characteristic
285	C₂²×C₄	—	2	4	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	4	2	no	—
289	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
290	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
291	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
292	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
293	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
294	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
295	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
296	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
297	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
298	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
299	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
300	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
301	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
302	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
303	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
304	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
305	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
306	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
307	M₁₆	—	2	4	no	—
308	M₁₆	—	2	8	no	—
309	M₁₆	—	4	4	no	—
310	M₁₆	—	4	4	no	—
311	M₁₆	—	4	2	no	—
312	M₁₆	—	4	2	no	—
313	M₁₆	M₁₆×C₅₃	1	8	yes	complemented
314	C₂×C₈	—	2	8	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	4	no	—
318	C₂×C₈	—	4	4	no	—
319	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
320	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
321	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
322	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
323	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
324	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
325	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
326	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
327	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
328	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
329	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
330	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
331	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
332	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
333	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
334	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
335	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
336	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
337	C₂³	—	2	2	no	—
338	C₂³	—	2	2	no	—
339	C₂³	—	4	1	no	—
340	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
341	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
342	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
362	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
363	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
364	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
365	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
366	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
367	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
368	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
369	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
370	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
371	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
372	C₂×C₄	#137×C₅₃	1	2	yes	—
373	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
375	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
376	C₈	—	2	4	no	—
377	C₈	—	4	4	no	—
378	C₈	—	4	2	no	—
379	C₈	—	4	2	no	—
380	C₈	C₂×M₁₆×C₅₃	1	8	yes	complemented
381	C₂²	—	2	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	1	no	—
386	C₂²	—	2	1	no	—
387	C₂²	—	2	1	no	—
388	C₂²	—	4	4	no	—
389	C₂²	C₅₃×#423	1	2	yes	—
390	C₂²	C₅₃×#428	1	2	yes	—
391	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
392	C₄	—	2	1	no	—
393	C₄	—	2	1	no	—
394	C₄	—	2	1	no	—
395	C₄	—	2	1	no	—
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	C₅₃×#423	1	2	yes	—
400	C₄	C₅₃×#428	1	2	yes	—
401	C₄	C₅₃×#430	1	2	yes	—
402	C₄	C₂²×M₁₆×C₅₃	1	2	yes	—
403	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
404	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
405	C₂	—	2	2	no	—
406	C₂	—	2	2	no	—
407	C₂	—	4	1	no	—
408	C₂	C₂×C₅₃×#423	1	1	yes	characteristic
409	C₂	C₂×C₅₃×#428	1	1	yes	characteristic
410	C₂	C₂×C₅₃×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1569	C₂	semidirect
5. C₅₃×#1567	C₂	semidirect
6. C₅₃×#1566	C₂	semidirect
7. C₅₃×#1565	C₂	semidirect
32. C₅₃×#335	C₂²	semidirect
33. C₅₃×#334	C₂²	semidirect
107. M₁₆×C₅₃	M₁₆	semidirect
174. C₈×C₅₃	C₂×M₁₆	semidirect
175. #17796	C₅₃	direct (N × H)
203. #1569	C₂×C₅₃	semidirect
204. #1567	C₂×C₅₃	semidirect
205. #1566	C₂×C₅₃	semidirect
206. #1565	C₂×C₅₃	semidirect
237. #335	C₂²×C₅₃	semidirect
238. #334	C₂²×C₅₃	semidirect
239. C₅₃	#17796	direct (N × H)
313. M₁₆	M₁₆×C₅₃	semidirect
380. C₈	C₂×M₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2920	2
C₂×C₅₃×#423	2
C₂×C₄×M₁₆×C₅₃	2
C₅₃×#1569	2
C₅₃×#1567	2
C₅₃×#1566	2
C₅₃×#1565	2
#17796	53

Subgroups — C53×#17796 #27225308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17796 #27225308