Subgroups — C₅₃×#22309 #27229821

Order 13568.

686 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1985	C₂	1	2	yes	maximal
3	C₅₃×#1984	C₂	1	2	yes	maximal
4	C₅₃×#1939	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1939	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1917	C₂	1	2	yes	maximal
7	C₅₃×#1916	C₂	1	2	yes	maximal
8	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic
10	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×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	2	yes	—
18	C₂×#97×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#391	C₄	1	2	yes	—
23	C₅₃×#389	C₄	1	1	yes	characteristic
24	C₅₃×#389	C₄	1	2	yes	—
25	C₅₃×#389	C₄	1	1	yes	characteristic
26	C₅₃×#384	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×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
30	C₄³×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#336	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
36	C₅₃×#336	C₂²	1	2	yes	—
37	C₅₃×#336	C₂²	1	2	yes	—
38	C₅₃×#336	C₂²	1	1	yes	characteristic
39	C₅₃×#336	C₂²	1	2	yes	—
40	C₅₃×#336	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
45	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₅₃	—	2	4	no	—
49	C₄×Q₈×C₅₃	—	4	2	no	—
50	C₄×Q₈×C₅₃	—	4	2	no	—
51	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
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	2	yes	—
58	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
66	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₅₃	—	2	2	no	—
70	C₄⋊C₈×C₅₃	—	2	2	no	—
71	C₄⋊C₈×C₅₃	—	4	2	no	—
72	C₄⋊C₈×C₅₃	—	4	2	no	—
73	#97×C₅₃	—	4	2	no	—
74	C₄×C₈×C₅₃	—	2	2	no	—
75	C₄×C₈×C₅₃	—	4	2	no	—
76	C₄×C₈×C₅₃	Q₈	1	4	yes	complemented
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	C₂×Q₈×C₅₃	—	2	4	no	—
82	C₂×Q₈×C₅₃	—	2	4	no	—
83	C₂×Q₈×C₅₃	—	4	2	no	—
84	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₅₃	C₂²×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	C₂⁴	1	1	yes	characteristic
95	C₂×C₈×C₅₃	—	2	1	no	—
96	C₂×C₈×C₅₃	—	2	1	no	—
97	C₂×C₈×C₅₃	—	2	1	no	—
98	C₂×C₈×C₅₃	—	2	1	no	—
99	C₂×C₈×C₅₃	—	2	1	no	—
100	C₂×C₈×C₅₃	—	2	1	no	—
101	C₂×C₈×C₅₃	—	2	2	no	—
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	1	no	—
106	C₂×C₈×C₅₃	—	2	1	no	—
107	C₂×C₈×C₅₃	—	2	1	no	—
108	C₂×C₈×C₅₃	—	2	1	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	—	2	2	no	—
111	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
112	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
113	C₄⋊C₄×C₅₃	—	2	4	no	—
114	C₄⋊C₄×C₅₃	—	2	2	no	—
115	C₄⋊C₄×C₅₃	—	2	4	no	—
116	C₄⋊C₄×C₅₃	—	2	2	no	—
117	C₄⋊C₄×C₅₃	—	2	2	no	—
118	C₄⋊C₄×C₅₃	—	4	2	no	—
119	C₄⋊C₄×C₅₃	—	4	2	no	—
120	C₄⋊C₄×C₅₃	—	4	2	no	—
121	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
122	C₄²×C₅₃	—	2	2	no	—
123	C₄²×C₅₃	—	2	2	no	—
124	C₄²×C₅₃	—	2	2	no	—
125	C₄²×C₅₃	—	2	2	no	—
126	C₄²×C₅₃	—	2	2	no	—
127	C₄²×C₅₃	—	2	2	no	—
128	C₄²×C₅₃	—	2	2	no	—
129	C₄²×C₅₃	—	2	2	no	—
130	C₄²×C₅₃	—	2	1	no	—
131	C₄²×C₅₃	—	2	1	no	—
132	C₄²×C₅₃	—	2	1	no	—
133	C₄²×C₅₃	—	2	1	no	—
134	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
135	C₄²×C₅₃	#41	1	2	yes	—
136	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
137	Q₈×C₅₃	—	4	8	no	—
138	C₂×C₄×C₅₃	—	2	1	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	1	no	—
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	1	no	—
145	C₂×C₄×C₅₃	—	2	1	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	1	no	—
150	C₂×C₄×C₅₃	—	2	1	no	—
151	C₂×C₄×C₅₃	—	2	2	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	1	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	C₄×Q₈	1	2	yes	—
162	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
163	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
164	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
165	C₂×C₄×C₅₃	#131	1	2	yes	—
166	C₂×C₄×C₅₃	#131	1	2	yes	—
167	C₂×C₄×C₅₃	#131	1	2	yes	—
168	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
170	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
171	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
172	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
173	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
174	C₈×C₅₃	—	2	2	no	—
175	C₈×C₅₃	—	2	2	no	—
176	C₈×C₅₃	—	4	1	no	—
177	C₈×C₅₃	—	4	1	no	—
178	C₈×C₅₃	—	4	2	no	—
179	C₈×C₅₃	—	4	2	no	—
180	#22309	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
182	C₂²×C₅₃	#518	1	1	yes	characteristic
183	C₂²×C₅₃	#519	1	1	yes	characteristic
184	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
185	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
186	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
187	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
188	C₄×C₅₃	—	2	2	no	—
189	C₄×C₅₃	—	2	4	no	—
190	C₄×C₅₃	—	2	2	no	—
191	C₄×C₅₃	—	2	2	no	—
192	C₄×C₅₃	—	4	2	no	—
193	C₄×C₅₃	—	4	2	no	—
194	C₄×C₅₃	—	4	2	no	—
195	C₄×C₅₃	—	4	1	no	—
196	C₄×C₅₃	—	4	1	no	—
197	C₄×C₅₃	#446	1	4	yes	—
198	C₄×C₅₃	#542	1	1	yes	characteristic
199	C₄×C₅₃	#542	1	1	yes	characteristic
200	C₄×C₅₃	#544	1	1	yes	characteristic
201	C₄×C₅₃	#544	1	1	yes	characteristic
202	C₄×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
203	#1985	C₂×C₅₃	1	2	yes	—
204	#1984	C₂×C₅₃	1	2	yes	—
205	#1939	C₂×C₅₃	1	1	yes	characteristic
206	#1939	C₂×C₅₃	1	1	yes	characteristic
207	#1917	C₂×C₅₃	1	2	yes	—
208	#1916	C₂×C₅₃	1	2	yes	—
209	#1831	C₂×C₅₃	1	1	yes	characteristic
210	#1770	C₂×C₅₃	1	1	yes	characteristic
211	C₈×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
212	#1744	C₂×C₅₃	1	1	yes	characteristic
213	C₂×C₅₃	#2331	1	1	yes	characteristic
214	C₂×C₅₃	C₂×#446	1	1	yes	characteristic
215	C₂×C₅₃	#2960	1	1	yes	characteristic
216	C₂×C₅₃	#2974	1	1	yes	characteristic
217	C₂×C₅₃	#2979	1	1	yes	characteristic
218	C₂×C₅₃	#2984	1	1	yes	characteristic
219	C₂×C₅₃	#2986	1	1	yes	characteristic
220	C₂×C₄⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
221	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
222	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
223	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
224	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
225	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
226	C₂×#97	C₂²×C₅₃	1	2	yes	—
227	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
228	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
229	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
230	#391	C₄×C₅₃	1	2	yes	—
231	#389	C₄×C₅₃	1	1	yes	characteristic
232	#389	C₄×C₅₃	1	2	yes	—
233	#389	C₄×C₅₃	1	1	yes	characteristic
234	#384	C₄×C₅₃	1	1	yes	characteristic
235	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
236	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
237	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
238	C₄³	C₂²×C₅₃	1	1	yes	characteristic
239	#336	C₂²×C₅₃	1	2	yes	—
240	#336	C₂²×C₅₃	1	2	yes	—
241	#336	C₂²×C₅₃	1	1	yes	characteristic
242	#336	C₂²×C₅₃	1	2	yes	—
243	#336	C₂²×C₅₃	1	2	yes	—
244	#336	C₂²×C₅₃	1	1	yes	characteristic
245	#336	C₂²×C₅₃	1	2	yes	—
246	#336	C₂²×C₅₃	1	2	yes	—
247	#336	C₂²×C₅₃	1	1	yes	characteristic
248	#336	C₂²×C₅₃	1	1	yes	characteristic
249	C₅₃	#22309	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
251	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
252	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
253	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
254	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
255	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
256	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
257	C₄⋊Q₈	—	2	4	no	—
258	C₄×Q₈	—	4	2	no	—
259	C₄×Q₈	—	4	2	no	—
260	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
261	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
262	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
263	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
264	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
265	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
266	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
267	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
268	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
269	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
270	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
271	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
272	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
273	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
274	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
275	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
276	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
277	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
278	C₄⋊C₈	—	2	2	no	—
279	C₄⋊C₈	—	2	2	no	—
280	C₄⋊C₈	—	4	2	no	—
281	C₄⋊C₈	—	4	2	no	—
282	#97	—	4	2	no	—
283	C₄×C₈	—	2	2	no	—
284	C₄×C₈	—	4	2	no	—
285	C₄×C₈	Q₈×C₅₃	1	4	yes	complemented
286	#95	C₂×C₄×C₅₃	1	2	yes	—
287	#95	C₂×C₄×C₅₃	1	2	yes	—
288	#95	C₂×C₄×C₅₃	1	2	yes	—
289	#95	C₂×C₄×C₅₃	1	2	yes	—
290	C₂×Q₈	—	2	4	no	—
291	C₂×Q₈	—	2	4	no	—
292	C₂×Q₈	—	4	2	no	—
293	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
294	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
295	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
296	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
297	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
298	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
299	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
300	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
301	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
302	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
303	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
321	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
322	C₄⋊C₄	—	2	4	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	4	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	4	2	no	—
328	C₄⋊C₄	—	4	2	no	—
329	C₄⋊C₄	—	4	2	no	—
330	C₄⋊C₄	M₁₆×C₅₃	1	4	yes	—
331	C₄²	—	2	2	no	—
332	C₄²	—	2	2	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	2	2	no	—
339	C₄²	—	2	1	no	—
340	C₄²	—	2	1	no	—
341	C₄²	—	2	1	no	—
342	C₄²	—	2	1	no	—
343	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
344	C₄²	#41×C₅₃	1	2	yes	—
345	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
346	Q₈	—	4	8	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
371	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
372	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
373	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
374	C₂×C₄	#131×C₅₃	1	2	yes	—
375	C₂×C₄	#131×C₅₃	1	2	yes	—
376	C₂×C₄	#131×C₅₃	1	2	yes	—
377	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
378	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
379	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
380	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
381	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
382	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
383	C₈	—	2	2	no	—
384	C₈	—	2	2	no	—
385	C₈	—	4	1	no	—
386	C₈	—	4	1	no	—
387	C₈	—	4	2	no	—
388	C₈	—	4	2	no	—
389	C₂²	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
390	C₂²	C₅₃×#518	1	1	yes	characteristic
391	C₂²	C₅₃×#519	1	1	yes	characteristic
392	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
393	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
394	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
395	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
396	C₄	—	2	2	no	—
397	C₄	—	2	4	no	—
398	C₄	—	2	2	no	—
399	C₄	—	2	2	no	—
400	C₄	—	4	2	no	—
401	C₄	—	4	1	no	—
402	C₄	—	4	2	no	—
403	C₄	—	4	2	no	—
404	C₄	—	4	1	no	—
405	C₄	C₅₃×#446	1	4	yes	—
406	C₄	C₅₃×#542	1	1	yes	characteristic
407	C₄	C₅₃×#542	1	1	yes	characteristic
408	C₄	C₅₃×#544	1	1	yes	characteristic
409	C₄	C₅₃×#544	1	1	yes	characteristic
410	C₂	C₅₃×#2331	1	1	yes	characteristic
411	C₂	C₂×C₅₃×#446	1	1	yes	characteristic
412	C₂	C₅₃×#2960	1	1	yes	characteristic
413	C₂	C₅₃×#2974	1	1	yes	characteristic
414	C₂	C₅₃×#2979	1	1	yes	characteristic
415	C₂	C₅₃×#2984	1	1	yes	characteristic
416	C₂	C₅₃×#2986	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle, prime radical	C₂³×C₅₃	C₂³×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
76. C₄×C₈×C₅₃	Q₈	semidirect
180. #22309	C₅₃	direct (N × H)
249. C₅₃	#22309	direct (N × H)
285. C₄×C₈	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₅₃	2
C₅₃×#1985	2
C₅₃×#1984	2
C₅₃×#1939	2
C₅₃×#1917	2
C₅₃×#1916	2
C₅₃×#1831	2
C₅₃×#1770	2
C₈×C₄⋊C₄×C₅₃	2
C₅₃×#1744	2
#22309	53

Subgroups — C53×#22309 #27229821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22309 #27229821