Subgroups — C₅₃×#17793 #27225305

Order 13568.

762 conjugacy classes of subgroups.

The subgroup lattice has 374 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	2	yes	maximal
2	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1568	C₂	1	4	yes	maximal, complemented
4	C₅₃×#1567	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1566	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1564	C₂	1	4	yes	maximal, complemented
7	C₂³×C₈×C₅₃	C₄	1	2	yes	—
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#432	—	2	2	no	—
10	C₅₃×#432	C₂²	1	4	yes	—
11	C₅₃×#431	—	2	2	no	—
12	C₅₃×#431	C₂²	1	4	yes	—
13	C₅₃×#423	C₂²	1	4	yes	—
14	C₅₃×#407	—	2	2	no	—
15	C₄×M₁₆×C₅₃	—	2	2	no	—
16	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#335	C₂²	1	8	yes	complemented
26	C₅₃×#334	C₂²	1	8	yes	complemented
27	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
30	C₂×M₁₆×C₅₃	—	2	4	no	—
31	C₂²×C₈×C₅₃	—	2	8	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₅₃	—	2	4	no	—
51	C₄⋊C₈×C₅₃	C₂³	1	8	yes	—
52	C₂²⋊C₈×C₅₃	—	2	4	no	—
53	C₂²⋊C₈×C₅₃	—	2	4	no	—
54	#97×C₅₃	—	2	4	no	—
55	C₄×C₈×C₅₃	—	2	4	no	—
56	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₅₃	C₂³	1	2	yes	—
65	C₄×C₈×C₅₃	C₂³	1	2	yes	—
66	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	—	2	4	no	—
68	C₂²×C₄×C₅₃	—	2	4	no	—
69	C₂²×C₄×C₅₃	—	2	4	no	—
70	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
88	M₁₆×C₅₃	—	4	8	no	—
89	C₂×C₈×C₅₃	—	2	4	no	—
90	C₂×C₈×C₅₃	—	2	4	no	—
91	C₂×C₈×C₅₃	—	2	4	no	—
92	C₂×C₈×C₅₃	—	2	4	no	—
93	C₂×C₈×C₅₃	—	2	4	no	—
94	C₂×C₈×C₅₃	—	2	8	no	—
95	C₂×C₈×C₅₃	M₁₆	1	8	yes	complemented
96	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
113	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
114	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
115	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
116	C₂³×C₅₃	—	2	2	no	—
117	C₂³×C₅₃	—	2	2	no	—
118	C₂³×C₅₃	—	2	2	no	—
119	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	#108	1	4	yes	—
138	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
148	C₂×C₄×C₅₃	#135	1	4	yes	—
149	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
152	C₈×C₅₃	—	2	2	no	—
153	C₈×C₅₃	—	2	2	no	—
154	C₈×C₅₃	—	4	8	no	—
155	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
156	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
157	#17793	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂²×C₅₃	—	2	8	no	—
159	C₂²×C₅₃	—	2	1	no	—
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	—	2	1	no	—
164	C₂²×C₅₃	—	2	1	no	—
165	C₂²×C₅₃	#423	1	2	yes	—
166	C₂²×C₅₃	#427	1	2	yes	—
167	C₂²×C₅₃	C₂×#108	1	2	yes	—
168	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	#423	1	2	yes	—
176	C₄×C₅₃	#427	1	2	yes	—
177	C₄×C₅₃	C₂×#108	1	2	yes	—
178	C₄×C₅₃	C₂×#108	1	2	yes	—
179	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
180	C₄×C₅₃	C₂×#135	1	2	yes	—
181	#2920	C₂×C₅₃	1	2	yes	—
182	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
183	#1568	C₂×C₅₃	1	4	yes	complemented
184	#1567	C₂×C₅₃	1	2	yes	complemented
185	#1566	C₂×C₅₃	1	2	yes	complemented
186	#1564	C₂×C₅₃	1	4	yes	complemented
187	C₂×C₅₃	—	2	2	no	—
188	C₂×C₅₃	—	2	2	no	—
189	C₂×C₅₃	—	2	2	no	—
190	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
191	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
192	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
193	C₂³×C₈	C₄×C₅₃	1	2	yes	—
194	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
195	#432	—	2	2	no	—
196	#432	C₂²×C₅₃	1	4	yes	—
197	#431	—	2	2	no	—
198	#431	C₂²×C₅₃	1	4	yes	—
199	#423	C₂²×C₅₃	1	4	yes	—
200	#407	—	2	2	no	—
201	C₄×M₁₆	—	2	2	no	—
202	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
203	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
204	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
205	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
206	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
207	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
208	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
209	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
210	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
211	#335	C₂²×C₅₃	1	8	yes	complemented
212	#334	C₂²×C₅₃	1	8	yes	complemented
213	C₅₃	#17793	1	1	yes	characteristic, direct product factor, Sylow
214	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
215	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
216	C₂³×C₄	Q₈×C₅₃	1	1	yes	characteristic
217	C₂×M₁₆	—	2	4	no	—
218	C₂²×C₈	—	2	8	no	—
219	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
220	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
221	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
222	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
223	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
224	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
225	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
226	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
227	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
228	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
229	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
230	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
231	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
232	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
233	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
234	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
235	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
236	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
237	C₄⋊C₈	—	2	4	no	—
238	C₄⋊C₈	C₂³×C₅₃	1	8	yes	—
239	C₂²⋊C₈	—	2	4	no	—
240	C₂²⋊C₈	—	2	4	no	—
241	#97	—	2	4	no	—
242	C₄×C₈	—	2	4	no	—
243	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
244	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
245	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
246	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
247	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
248	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
249	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
250	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
251	C₄×C₈	C₂³×C₅₃	1	2	yes	—
252	C₄×C₈	C₂³×C₅₃	1	2	yes	—
253	C₂⁴	C₄⋊C₄×C₅₃	1	1	yes	characteristic
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
258	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
259	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
260	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
261	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
262	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
263	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
264	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
265	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
266	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
267	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
272	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
274	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
275	M₁₆	—	4	8	no	—
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	2	8	no	—
282	C₂×C₈	M₁₆×C₅₃	1	8	yes	complemented
283	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
284	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
285	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
286	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
287	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
288	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
289	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
290	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
291	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
292	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
293	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
294	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
295	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
296	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
297	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
298	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
299	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
300	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
301	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
302	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
303	C₂³	—	2	2	no	—
304	C₂³	—	2	2	no	—
305	C₂³	—	2	2	no	—
306	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
307	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
308	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	#108×C₅₃	1	4	yes	—
325	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
326	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
327	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
328	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
329	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
330	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
331	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
332	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
333	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
334	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
335	C₂×C₄	#135×C₅₃	1	4	yes	—
336	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
337	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
338	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	4	8	no	—
342	C₈	C₂×M₁₆×C₅₃	1	4	yes	—
343	C₈	C₂×M₁₆×C₅₃	1	4	yes	—
344	C₂²	—	2	8	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	C₅₃×#423	1	2	yes	—
352	C₂²	C₅₃×#427	1	2	yes	—
353	C₂²	C₂×#108×C₅₃	1	2	yes	—
354	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	C₅₃×#423	1	2	yes	—
362	C₄	C₅₃×#427	1	2	yes	—
363	C₄	C₂×#108×C₅₃	1	2	yes	—
364	C₄	C₂×#108×C₅₃	1	2	yes	—
365	C₄	C₂²×M₁₆×C₅₃	1	2	yes	—
366	C₄	C₂×#135×C₅₃	1	2	yes	—
367	C₂	—	2	2	no	—
368	C₂	—	2	2	no	—
369	C₂	—	2	2	no	—
370	C₂	C₂×C₅₃×#423	1	1	yes	characteristic
371	C₂	C₂×C₅₃×#427	1	1	yes	characteristic
372	C₂	C₂²×#108×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₅₃	C₂×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 ◁ C₄²×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1568	C₂	semidirect
4. C₅₃×#1567	C₂	semidirect
5. C₅₃×#1566	C₂	semidirect
6. C₅₃×#1564	C₂	semidirect
25. C₅₃×#335	C₂²	semidirect
26. C₅₃×#334	C₂²	semidirect
95. C₂×C₈×C₅₃	M₁₆	semidirect
157. #17793	C₅₃	direct (N × H)
183. #1568	C₂×C₅₃	semidirect
184. #1567	C₂×C₅₃	semidirect
185. #1566	C₂×C₅₃	semidirect
186. #1564	C₂×C₅₃	semidirect
211. #335	C₂²×C₅₃	semidirect
212. #334	C₂²×C₅₃	semidirect
213. C₅₃	#17793	direct (N × H)
282. C₂×C₈	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
C₂²×C₄×C₈×C₅₃	2
C₅₃×#1568	2
C₅₃×#1567	2
C₅₃×#1566	2
C₅₃×#1564	2
#17793	53

Subgroups — C53×#17793 #27225305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17793 #27225305