Subgroups — C₅₃×#22289 #27229801

Order 13568.

754 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1984	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1940	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1917	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1916	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1851	C₂	1	2	yes	maximal
7	C₅₃×#1797	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1753	C₂	1	2	yes	maximal
9	C₅₃×#1728	C₂	1	1	yes	maximal, characteristic
10	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
20	C₂×#97×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#391	C₄	1	1	yes	characteristic
23	C₅₃×#389	C₄	1	1	yes	characteristic
24	C₅₃×#387	C₄	1	2	yes	—
25	C₅₃×#385	C₄	1	2	yes	—
26	C₅₃×#380	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
30	C₅₃×#336	C₂²	1	2	yes	—
31	C₅₃×#336	C₂²	1	2	yes	—
32	C₅₃×#336	C₂²	1	2	yes	—
33	C₅₃×#336	C₂²	1	2	yes	—
34	C₅₃×#336	C₂²	1	2	yes	—
35	C₅₃×#336	C₂²	1	2	yes	—
36	C₅₃×#336	C₂²	1	2	yes	—
37	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₅₃	—	2	2	no	—
42	C₂×M₁₆×C₅₃	—	4	1	no	—
43	C₂×M₁₆×C₅₃	—	4	1	no	—
44	C₂×M₁₆×C₅₃	—	4	1	no	—
45	C₂×M₁₆×C₅₃	—	4	1	no	—
46	C₂×M₁₆×C₅₃	Q₈	1	4	yes	complemented
47	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
51	C₂²⋊Q₈×C₅₃	—	2	4	no	—
52	C₄×Q₈×C₅₃	—	4	1	no	—
53	C₄×Q₈×C₅₃	—	4	1	no	—
54	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₅₃	—	4	1	no	—
63	C₂×C₄²×C₅₃	—	4	1	no	—
64	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
69	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
70	C₄⋊C₈×C₅₃	—	4	2	no	—
71	C₄⋊C₈×C₅₃	—	4	1	no	—
72	C₄⋊C₈×C₅₃	—	4	1	no	—
73	C₄⋊C₈×C₅₃	—	4	2	no	—
74	C₂²⋊C₈×C₅₃	—	2	4	no	—
75	#97×C₅₃	—	4	2	no	—
76	C₄×C₈×C₅₃	—	4	2	no	—
77	#95×C₅₃	C₂×C₄	1	2	yes	—
78	#95×C₅₃	C₂×C₄	1	2	yes	—
79	#95×C₅₃	C₂×C₄	1	2	yes	—
80	#95×C₅₃	C₂×C₄	1	2	yes	—
81	#95×C₅₃	C₂×C₄	1	2	yes	—
82	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×Q₈×C₅₃	—	4	1	no	—
84	C₂×Q₈×C₅₃	—	4	1	no	—
85	C₂×Q₈×C₅₃	M₁₆	1	4	yes	complemented
86	C₂²×C₄×C₅₃	—	2	1	no	—
87	C₂²×C₄×C₅₃	—	2	1	no	—
88	C₂²×C₄×C₅₃	—	2	1	no	—
89	C₂²×C₄×C₅₃	—	2	1	no	—
90	C₂²×C₄×C₅₃	—	2	2	no	—
91	C₂²×C₄×C₅₃	—	2	2	no	—
92	C₂²×C₄×C₅₃	—	4	1	no	—
93	C₂²×C₄×C₅₃	—	4	1	no	—
94	C₂²×C₄×C₅₃	—	4	1	no	—
95	C₂²×C₄×C₅₃	—	4	1	no	—
96	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
106	C₂²×C₄×C₅₃	#41	1	2	yes	—
107	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
108	M₁₆×C₅₃	—	4	4	no	—
109	M₁₆×C₅₃	—	4	4	no	—
110	C₂×C₈×C₅₃	—	2	2	no	—
111	C₂×C₈×C₅₃	—	2	2	no	—
112	C₂×C₈×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	2	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	—	2	2	no	—
120	C₂×C₈×C₅₃	—	2	2	no	—
121	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
122	C₄⋊C₄×C₅₃	—	2	4	no	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	4	no	—
125	C₄⋊C₄×C₅₃	—	4	1	no	—
126	C₄⋊C₄×C₅₃	—	4	1	no	—
127	C₄⋊C₄×C₅₃	—	4	2	no	—
128	C₂²⋊C₄×C₅₃	—	2	4	no	—
129	C₂²⋊C₄×C₅₃	—	4	2	no	—
130	C₄²×C₅₃	—	2	2	no	—
131	C₄²×C₅₃	—	2	2	no	—
132	C₄²×C₅₃	—	2	1	no	—
133	C₄²×C₅₃	—	2	1	no	—
134	C₄²×C₅₃	—	2	2	no	—
135	C₄²×C₅₃	—	2	1	no	—
136	C₄²×C₅₃	—	2	1	no	—
137	C₄²×C₅₃	—	2	2	no	—
138	C₄²×C₅₃	—	4	1	no	—
139	C₄²×C₅₃	—	4	1	no	—
140	C₂³×C₅₃	—	2	1	no	—
141	C₂³×C₅₃	—	2	1	no	—
142	C₂³×C₅₃	—	2	1	no	—
143	C₂³×C₅₃	—	2	1	no	—
144	C₂³×C₅₃	—	2	1	no	—
145	C₂³×C₅₃	—	2	1	no	—
146	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
147	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
148	Q₈×C₅₃	—	4	4	no	—
149	C₂×C₄×C₅₃	—	2	1	no	—
150	C₂×C₄×C₅₃	—	2	1	no	—
151	C₂×C₄×C₅₃	—	2	1	no	—
152	C₂×C₄×C₅₃	—	2	2	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	2	no	—
157	C₂×C₄×C₅₃	—	2	2	no	—
158	C₂×C₄×C₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	1	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	1	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	1	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	2	2	no	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	2	no	—
174	C₂×C₄×C₅₃	—	2	2	no	—
175	C₂×C₄×C₅₃	—	2	2	no	—
176	C₂×C₄×C₅₃	—	4	4	no	—
177	C₂×C₄×C₅₃	—	4	4	no	—
178	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
179	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
180	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
181	C₂×C₄×C₅₃	#131	1	2	yes	—
182	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
184	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
185	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
186	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
188	C₈×C₅₃	—	2	4	no	—
189	C₈×C₅₃	—	4	2	no	—
190	C₈×C₅₃	—	4	2	no	—
191	C₈×C₅₃	—	4	2	no	—
192	#22289	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂²×C₅₃	—	2	2	no	—
194	C₂²×C₅₃	—	2	2	no	—
195	C₂²×C₅₃	—	2	2	no	—
196	C₂²×C₅₃	—	4	1	no	—
197	C₂²×C₅₃	—	4	1	no	—
198	C₂²×C₅₃	—	4	1	no	—
199	C₂²×C₅₃	—	4	1	no	—
200	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
201	C₂²×C₅₃	#518	1	1	yes	characteristic
202	C₂²×C₅₃	#520	1	1	yes	characteristic
203	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
204	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
205	C₂²×C₅₃	#568	1	1	yes	characteristic
206	C₂²×C₅₃	#568	1	1	yes	characteristic
207	C₄×C₅₃	—	2	2	no	—
208	C₄×C₅₃	—	2	2	no	—
209	C₄×C₅₃	—	2	4	no	—
210	C₄×C₅₃	—	2	2	no	—
211	C₄×C₅₃	—	4	1	no	—
212	C₄×C₅₃	—	4	2	no	—
213	C₄×C₅₃	—	4	1	no	—
214	C₄×C₅₃	—	4	2	no	—
215	C₄×C₅₃	—	4	2	no	—
216	C₄×C₅₃	#538	1	1	yes	characteristic
217	C₄×C₅₃	#539	1	1	yes	characteristic
218	C₄×C₅₃	#543	1	1	yes	characteristic
219	C₄×C₅₃	#544	1	1	yes	characteristic
220	C₄×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
221	#1984	C₂×C₅₃	1	2	yes	complemented
222	#1940	C₂×C₅₃	1	2	yes	complemented
223	#1917	C₂×C₅₃	1	2	yes	complemented
224	#1916	C₂×C₅₃	1	2	yes	complemented
225	#1851	C₂×C₅₃	1	2	yes	—
226	#1797	C₂×C₅₃	1	1	yes	characteristic
227	#1753	C₂×C₅₃	1	2	yes	—
228	#1728	C₂×C₅₃	1	1	yes	characteristic
229	C₂×C₅₃	—	4	2	no	—
230	C₂×C₅₃	#2329	1	1	yes	characteristic
231	C₂×C₅₃	#2962	1	1	yes	characteristic
232	C₂×C₅₃	Q₈×M₁₆	1	1	yes	characteristic
233	C₂×C₅₃	#2973	1	1	yes	characteristic
234	C₂×C₅₃	#2976	1	1	yes	characteristic
235	C₂×C₅₃	#2991	1	1	yes	characteristic
236	C₂×C₅₃	#2995	1	1	yes	characteristic
237	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
238	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
239	C₂×C₂²⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
240	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
241	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
242	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
243	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
244	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
245	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
246	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
247	C₂×#97	C₂²×C₅₃	1	2	yes	—
248	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
249	#391	C₄×C₅₃	1	1	yes	characteristic
250	#389	C₄×C₅₃	1	1	yes	characteristic
251	#387	C₄×C₅₃	1	2	yes	—
252	#385	C₄×C₅₃	1	2	yes	—
253	#380	C₄×C₅₃	1	1	yes	characteristic
254	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
255	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
256	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
257	#336	C₂²×C₅₃	1	2	yes	—
258	#336	C₂²×C₅₃	1	2	yes	—
259	#336	C₂²×C₅₃	1	2	yes	—
260	#336	C₂²×C₅₃	1	2	yes	—
261	#336	C₂²×C₅₃	1	2	yes	—
262	#336	C₂²×C₅₃	1	2	yes	—
263	#336	C₂²×C₅₃	1	2	yes	—
264	C₅₃	#22289	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
266	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
267	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
268	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
269	C₂×M₁₆	—	2	2	no	—
270	C₂×M₁₆	—	4	1	no	—
271	C₂×M₁₆	—	4	1	no	—
272	C₂×M₁₆	—	4	1	no	—
273	C₂×M₁₆	—	4	1	no	—
274	C₂×M₁₆	Q₈×C₅₃	1	4	yes	complemented
275	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
276	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
277	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
278	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
279	C₂²⋊Q₈	—	2	4	no	—
280	C₄×Q₈	—	4	1	no	—
281	C₄×Q₈	—	4	1	no	—
282	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
283	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
284	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
285	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
286	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
287	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
288	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
289	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
290	C₂×C₄²	—	4	1	no	—
291	C₂×C₄²	—	4	1	no	—
292	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
293	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
294	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
295	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
296	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
297	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
298	C₄⋊C₈	—	4	2	no	—
299	C₄⋊C₈	—	4	1	no	—
300	C₄⋊C₈	—	4	1	no	—
301	C₄⋊C₈	—	4	2	no	—
302	C₂²⋊C₈	—	2	4	no	—
303	#97	—	4	2	no	—
304	C₄×C₈	—	4	2	no	—
305	#95	C₂×C₄×C₅₃	1	2	yes	—
306	#95	C₂×C₄×C₅₃	1	2	yes	—
307	#95	C₂×C₄×C₅₃	1	2	yes	—
308	#95	C₂×C₄×C₅₃	1	2	yes	—
309	#95	C₂×C₄×C₅₃	1	2	yes	—
310	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
311	C₂×Q₈	—	4	1	no	—
312	C₂×Q₈	—	4	1	no	—
313	C₂×Q₈	M₁₆×C₅₃	1	4	yes	complemented
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	4	1	no	—
321	C₂²×C₄	—	4	1	no	—
322	C₂²×C₄	—	4	1	no	—
323	C₂²×C₄	—	4	1	no	—
324	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
325	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
326	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
327	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
328	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
329	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
330	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
331	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
332	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
333	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
334	C₂²×C₄	#41×C₅₃	1	2	yes	—
335	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
336	M₁₆	—	4	4	no	—
337	M₁₆	—	4	4	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	4	no	—
352	C₄⋊C₄	—	2	4	no	—
353	C₄⋊C₄	—	4	1	no	—
354	C₄⋊C₄	—	4	2	no	—
355	C₄⋊C₄	—	4	1	no	—
356	C₂²⋊C₄	—	2	4	no	—
357	C₂²⋊C₄	—	4	2	no	—
358	C₄²	—	2	2	no	—
359	C₄²	—	2	2	no	—
360	C₄²	—	2	1	no	—
361	C₄²	—	2	1	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	2	1	no	—
364	C₄²	—	2	1	no	—
365	C₄²	—	2	2	no	—
366	C₄²	—	4	1	no	—
367	C₄²	—	4	1	no	—
368	C₂³	—	2	1	no	—
369	C₂³	—	2	1	no	—
370	C₂³	—	2	1	no	—
371	C₂³	—	2	1	no	—
372	C₂³	—	2	1	no	—
373	C₂³	—	2	1	no	—
374	C₂³	C₄×Q₈×C₅₃	1	2	yes	—
375	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
376	Q₈	—	4	4	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	4	4	no	—
405	C₂×C₄	—	4	4	no	—
406	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
407	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
408	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
409	C₂×C₄	#131×C₅₃	1	2	yes	—
410	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
411	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
412	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
413	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
414	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
415	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
416	C₈	—	2	4	no	—
417	C₈	—	4	2	no	—
418	C₈	—	4	2	no	—
419	C₈	—	4	2	no	—
420	C₂²	—	2	2	no	—
421	C₂²	—	2	2	no	—
422	C₂²	—	2	2	no	—
423	C₂²	—	4	1	no	—
424	C₂²	—	4	1	no	—
425	C₂²	—	4	1	no	—
426	C₂²	—	4	1	no	—
427	C₂²	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
428	C₂²	C₅₃×#518	1	1	yes	characteristic
429	C₂²	C₅₃×#520	1	1	yes	characteristic
430	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
431	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
432	C₂²	C₅₃×#568	1	1	yes	characteristic
433	C₂²	C₅₃×#568	1	1	yes	characteristic
434	C₄	—	2	2	no	—
435	C₄	—	2	4	no	—
436	C₄	—	2	2	no	—
437	C₄	—	2	2	no	—
438	C₄	—	4	1	no	—
439	C₄	—	4	1	no	—
440	C₄	—	4	2	no	—
441	C₄	—	4	2	no	—
442	C₄	—	4	2	no	—
443	C₄	C₅₃×#538	1	1	yes	characteristic
444	C₄	C₅₃×#539	1	1	yes	characteristic
445	C₄	C₅₃×#543	1	1	yes	characteristic
446	C₄	C₅₃×#544	1	1	yes	characteristic
447	C₂	—	4	2	no	—
448	C₂	C₅₃×#2329	1	1	yes	characteristic
449	C₂	C₅₃×#2962	1	1	yes	characteristic
450	C₂	Q₈×M₁₆×C₅₃	1	1	yes	characteristic
451	C₂	C₅₃×#2973	1	1	yes	characteristic
452	C₂	C₅₃×#2976	1	1	yes	characteristic
453	C₂	C₅₃×#2991	1	1	yes	characteristic
454	C₂	C₅₃×#2995	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1984	C₂	semidirect
3. C₅₃×#1940	C₂	semidirect
4. C₅₃×#1917	C₂	semidirect
5. C₅₃×#1916	C₂	semidirect
46. C₂×M₁₆×C₅₃	Q₈	semidirect
85. C₂×Q₈×C₅₃	M₁₆	semidirect
192. #22289	C₅₃	direct (N × H)
221. #1984	C₂×C₅₃	semidirect
222. #1940	C₂×C₅₃	semidirect
223. #1917	C₂×C₅₃	semidirect
224. #1916	C₂×C₅₃	semidirect
264. C₅₃	#22289	direct (N × H)
274. C₂×M₁₆	Q₈×C₅₃	semidirect
313. C₂×Q₈	M₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊Q₈×C₅₃	2
C₅₃×#1984	2
C₅₃×#1940	2
C₅₃×#1917	2
C₅₃×#1916	2
C₅₃×#1851	2
C₅₃×#1797	2
C₅₃×#1753	2
C₅₃×#1728	2
#22289	53

Subgroups — C53×#22289 #27229801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22289 #27229801