Subgroups — C₅₃×#2418 #20271355

Order 6784.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#123×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#120×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#399	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#397	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#394	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#394	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#388	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#388	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#386	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#386	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#385	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#379	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂⁵×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
23	#123×C₅₃	—	2	4	no	—
24	#120×C₅₃	—	2	4	no	—
25	C₄×D₄×C₅₃	—	2	4	no	—
26	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
32	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
46	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
47	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
48	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
49	#95×C₅₃	C₂²	1	1	yes	characteristic, complemented
50	#95×C₅₃	C₂²	1	1	yes	characteristic, complemented
51	#95×C₅₃	C₂²	1	1	yes	characteristic, complemented
52	#95×C₅₃	C₂²	1	1	yes	characteristic, complemented
53	#95×C₅₃	C₂²	1	1	yes	characteristic, complemented
54	#95×C₅₃	C₂²	1	1	yes	characteristic, complemented
55	C₂⁴×C₅₃	—	2	2	no	—
56	C₂⁴×C₅₃	—	2	2	no	—
57	C₂⁴×C₅₃	—	2	2	no	—
58	C₂⁴×C₅₃	—	2	2	no	—
59	C₂⁴×C₅₃	—	2	2	no	—
60	C₂⁴×C₅₃	—	4	1	no	—
61	C₂⁴×C₅₃	—	4	1	no	—
62	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
63	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
64	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
65	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₅₃	—	2	2	no	—
67	C₂×D₄×C₅₃	—	2	2	no	—
68	C₂×D₄×C₅₃	—	2	2	no	—
69	C₂×D₄×C₅₃	—	2	2	no	—
70	C₂×D₄×C₅₃	—	2	2	no	—
71	C₂×D₄×C₅₃	—	2	2	no	—
72	C₂×D₄×C₅₃	—	4	1	no	—
73	C₂×D₄×C₅₃	—	4	1	no	—
74	C₂×D₄×C₅₃	D₄	1	4	yes	—
75	C₂²×C₄×C₅₃	—	2	2	no	—
76	C₂²×C₄×C₅₃	—	2	2	no	—
77	C₂²×C₄×C₅₃	—	2	2	no	—
78	C₂²×C₄×C₅₃	—	2	2	no	—
79	C₂²×C₄×C₅₃	—	2	2	no	—
80	C₂²×C₄×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	—	4	1	no	—
82	C₂²×C₄×C₅₃	—	4	1	no	—
83	C₂²×C₄×C₅₃	—	4	1	no	—
84	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₄⋊C₄×C₅₃	—	4	1	no	—
99	C₄⋊C₄×C₅₃	—	4	1	no	—
100	C₂²⋊C₄×C₅₃	—	2	2	no	—
101	C₂²⋊C₄×C₅₃	—	2	2	no	—
102	C₂²⋊C₄×C₅₃	—	2	2	no	—
103	C₂²⋊C₄×C₅₃	—	2	2	no	—
104	C₂²⋊C₄×C₅₃	—	2	2	no	—
105	C₂²⋊C₄×C₅₃	—	2	2	no	—
106	C₂²⋊C₄×C₅₃	—	2	2	no	—
107	C₂²⋊C₄×C₅₃	—	2	2	no	—
108	C₂²⋊C₄×C₅₃	—	2	2	no	—
109	C₂²⋊C₄×C₅₃	—	2	2	no	—
110	C₂²⋊C₄×C₅₃	—	2	2	no	—
111	C₂²⋊C₄×C₅₃	—	2	2	no	—
112	C₂²⋊C₄×C₅₃	—	4	1	no	—
113	C₂²⋊C₄×C₅₃	—	4	1	no	—
114	C₂²⋊C₄×C₅₃	—	4	1	no	—
115	C₂²⋊C₄×C₅₃	D₄	1	4	yes	complemented
116	C₄²×C₅₃	—	2	2	no	—
117	C₂³×C₅₃	—	2	1	no	—
118	C₂³×C₅₃	—	2	1	no	—
119	C₂³×C₅₃	—	2	1	no	—
120	C₂³×C₅₃	—	2	1	no	—
121	C₂³×C₅₃	—	2	8	no	—
122	C₂³×C₅₃	—	2	1	no	—
123	C₂³×C₅₃	—	2	1	no	—
124	C₂³×C₅₃	—	2	1	no	—
125	C₂³×C₅₃	—	2	1	no	—
126	C₂³×C₅₃	—	2	1	no	—
127	C₂³×C₅₃	—	2	1	no	—
128	C₂³×C₅₃	—	2	1	no	—
129	C₂³×C₅₃	—	2	1	no	—
130	C₂³×C₅₃	—	2	1	no	—
131	C₂³×C₅₃	—	2	1	no	—
132	C₂³×C₅₃	—	2	1	no	—
133	C₂³×C₅₃	—	2	1	no	—
134	C₂³×C₅₃	—	2	1	no	—
135	C₂³×C₅₃	—	2	1	no	—
136	C₂³×C₅₃	—	2	1	no	—
137	C₂³×C₅₃	—	2	1	no	—
138	C₂³×C₅₃	—	2	1	no	—
139	C₂³×C₅₃	—	2	1	no	—
140	C₂³×C₅₃	—	4	4	no	—
141	C₂³×C₅₃	—	4	4	no	—
142	C₂³×C₅₃	—	4	4	no	—
143	C₂³×C₅₃	—	4	4	no	—
144	C₂³×C₅₃	—	4	4	no	—
145	C₂³×C₅₃	—	4	4	no	—
146	C₂³×C₅₃	C₂×D₄	1	2	yes	—
147	C₂³×C₅₃	C₂×D₄	1	2	yes	—
148	C₂³×C₅₃	C₂×D₄	1	2	yes	—
149	C₂³×C₅₃	#41	1	2	yes	—
150	C₂³×C₅₃	#41	1	2	yes	—
151	C₂³×C₅₃	#41	1	2	yes	—
152	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
153	D₄×C₅₃	—	2	8	no	—
154	D₄×C₅₃	—	4	4	no	—
155	D₄×C₅₃	—	4	4	no	—
156	C₂×C₄×C₅₃	—	2	1	no	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	2	1	no	—
160	C₂×C₄×C₅₃	—	2	1	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	1	no	—
163	C₂×C₄×C₅₃	—	2	1	no	—
164	C₂×C₄×C₅₃	—	2	1	no	—
165	C₂×C₄×C₅₃	—	2	1	no	—
166	C₂×C₄×C₅₃	—	2	1	no	—
167	C₂×C₄×C₅₃	—	2	1	no	—
168	C₂×C₄×C₅₃	—	2	1	no	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	—	2	1	no	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	2	1	no	—
177	C₂×C₄×C₅₃	—	2	1	no	—
178	C₂×C₄×C₅₃	—	2	1	no	—
179	C₂×C₄×C₅₃	—	2	1	no	—
180	C₂×C₄×C₅₃	—	2	1	no	—
181	C₂×C₄×C₅₃	—	2	1	no	—
182	C₂×C₄×C₅₃	—	2	1	no	—
183	C₂×C₄×C₅₃	—	2	1	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	1	no	—
186	C₂×C₄×C₅₃	—	4	4	no	—
187	C₂×C₄×C₅₃	—	4	4	no	—
188	C₂×C₄×C₅₃	—	4	4	no	—
189	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
190	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
191	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
192	C₂²×C₅₃	—	2	2	no	—
193	C₂²×C₅₃	—	2	2	no	—
194	C₂²×C₅₃	—	2	2	no	—
195	C₂²×C₅₃	—	2	2	no	—
196	C₂²×C₅₃	—	2	2	no	—
197	C₂²×C₅₃	—	2	2	no	—
198	C₂²×C₅₃	—	2	2	no	—
199	C₂²×C₅₃	—	2	2	no	—
200	C₂²×C₅₃	—	2	2	no	—
201	C₂²×C₅₃	—	2	2	no	—
202	C₂²×C₅₃	—	2	2	no	—
203	C₂²×C₅₃	—	2	2	no	—
204	C₂²×C₅₃	—	2	2	no	—
205	C₂²×C₅₃	—	2	2	no	—
206	C₂²×C₅₃	—	2	2	no	—
207	C₂²×C₅₃	—	4	16	no	—
208	C₂²×C₅₃	—	4	1	no	—
209	C₂²×C₅₃	—	4	1	no	—
210	C₂²×C₅₃	—	4	1	no	—
211	C₂²×C₅₃	—	4	1	no	—
212	C₂²×C₅₃	—	4	1	no	—
213	C₂²×C₅₃	—	4	1	no	—
214	C₂²×C₅₃	—	4	1	no	—
215	C₂²×C₅₃	—	4	1	no	—
216	C₂²×C₅₃	—	4	1	no	—
217	C₂²×C₅₃	—	4	1	no	—
218	C₂²×C₅₃	—	4	1	no	—
219	C₂²×C₅₃	—	4	1	no	—
220	C₂²×C₅₃	#123	1	4	yes	complemented
221	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
222	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
223	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
224	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
225	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
226	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
227	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
228	C₄×C₅₃	—	2	2	no	—
229	C₄×C₅₃	—	2	2	no	—
230	C₄×C₅₃	—	2	2	no	—
231	C₄×C₅₃	—	4	1	no	—
232	C₄×C₅₃	—	4	1	no	—
233	C₄×C₅₃	—	4	1	no	—
234	C₄×C₅₃	—	4	1	no	—
235	C₄×C₅₃	—	4	1	no	—
236	C₄×C₅₃	—	4	1	no	—
237	C₄×C₅₃	—	4	1	no	—
238	C₄×C₅₃	—	4	1	no	—
239	#2418	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₂×C₅₃	—	2	4	no	—
241	C₂×C₅₃	—	4	2	no	—
242	C₂×C₅₃	—	4	2	no	—
243	C₂×C₅₃	—	4	2	no	—
244	C₂×C₅₃	C₂×#123	1	1	yes	characteristic
245	C₂×C₅₃	#525	1	1	yes	characteristic
246	C₂×C₅₃	#538	1	1	yes	characteristic
247	C₂×C₅₃	D₄²	1	1	yes	characteristic
248	C₂×C₅₃	#546	1	1	yes	characteristic
249	C₂×C₅₃	#546	1	1	yes	characteristic
250	C₂×C₅₃	#551	1	1	yes	characteristic
251	C₂×#123	C₂×C₅₃	1	1	yes	characteristic, complemented
252	C₂×#120	C₂×C₅₃	1	1	yes	characteristic
253	C₂×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic, complemented
254	C₂²×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
255	#399	C₂×C₅₃	1	1	yes	characteristic, complemented
256	#397	C₂×C₅₃	1	1	yes	characteristic, complemented
257	#394	C₂×C₅₃	1	1	yes	characteristic, complemented
258	#394	C₂×C₅₃	1	1	yes	characteristic, complemented
259	#388	C₂×C₅₃	1	1	yes	characteristic, complemented
260	#388	C₂×C₅₃	1	1	yes	characteristic, complemented
261	#386	C₂×C₅₃	1	1	yes	characteristic, complemented
262	#386	C₂×C₅₃	1	1	yes	characteristic, complemented
263	#385	C₂×C₅₃	1	1	yes	characteristic, complemented
264	#380	C₂×C₅₃	1	1	yes	characteristic, complemented
265	#379	C₂×C₅₃	1	1	yes	characteristic, complemented
266	C₅₃	#2418	1	1	yes	characteristic, direct product factor, Sylow
267	C₂⁵	C₂²×C₅₃	1	1	yes	characteristic
268	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
269	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
270	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
271	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
272	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
273	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
274	#123	—	2	4	no	—
275	#120	—	2	4	no	—
276	C₄×D₄	—	2	4	no	—
277	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
278	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
279	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
280	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
281	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
282	C₂×C₂²⋊C₄	—	2	4	no	—
283	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
284	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
285	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
286	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
287	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
288	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
289	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
290	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
291	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
292	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
293	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
294	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
295	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
296	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
297	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
298	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
299	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic, complemented
300	#95	C₂²×C₅₃	1	1	yes	characteristic, complemented
301	#95	C₂²×C₅₃	1	1	yes	characteristic, complemented
302	#95	C₂²×C₅₃	1	1	yes	characteristic, complemented
303	#95	C₂²×C₅₃	1	1	yes	characteristic, complemented
304	#95	C₂²×C₅₃	1	1	yes	characteristic, complemented
305	#95	C₂²×C₅₃	1	1	yes	characteristic, complemented
306	C₂⁴	—	2	2	no	—
307	C₂⁴	—	2	2	no	—
308	C₂⁴	—	2	2	no	—
309	C₂⁴	—	2	2	no	—
310	C₂⁴	—	2	2	no	—
311	C₂⁴	—	4	1	no	—
312	C₂⁴	—	4	1	no	—
313	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
314	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
315	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
316	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	—	2	2	no	—
320	C₂×D₄	—	2	2	no	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	2	2	no	—
323	C₂×D₄	—	4	1	no	—
324	C₂×D₄	—	4	1	no	—
325	C₂×D₄	D₄×C₅₃	1	4	yes	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	4	1	no	—
333	C₂²×C₄	—	4	1	no	—
334	C₂²×C₄	—	4	1	no	—
335	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
336	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
337	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
338	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
339	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
340	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
341	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
342	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
343	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
344	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
345	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	4	1	no	—
350	C₄⋊C₄	—	4	1	no	—
351	C₂²⋊C₄	—	2	2	no	—
352	C₂²⋊C₄	—	2	2	no	—
353	C₂²⋊C₄	—	2	2	no	—
354	C₂²⋊C₄	—	2	2	no	—
355	C₂²⋊C₄	—	2	2	no	—
356	C₂²⋊C₄	—	2	2	no	—
357	C₂²⋊C₄	—	2	2	no	—
358	C₂²⋊C₄	—	2	2	no	—
359	C₂²⋊C₄	—	2	2	no	—
360	C₂²⋊C₄	—	2	2	no	—
361	C₂²⋊C₄	—	2	2	no	—
362	C₂²⋊C₄	—	2	2	no	—
363	C₂²⋊C₄	—	4	1	no	—
364	C₂²⋊C₄	—	4	1	no	—
365	C₂²⋊C₄	—	4	1	no	—
366	C₂²⋊C₄	D₄×C₅₃	1	4	yes	complemented
367	C₄²	—	2	2	no	—
368	C₂³	—	2	1	no	—
369	C₂³	—	2	1	no	—
370	C₂³	—	2	1	no	—
371	C₂³	—	2	1	no	—
372	C₂³	—	2	8	no	—
373	C₂³	—	2	1	no	—
374	C₂³	—	2	1	no	—
375	C₂³	—	2	1	no	—
376	C₂³	—	2	1	no	—
377	C₂³	—	2	1	no	—
378	C₂³	—	2	1	no	—
379	C₂³	—	2	1	no	—
380	C₂³	—	2	1	no	—
381	C₂³	—	2	1	no	—
382	C₂³	—	2	1	no	—
383	C₂³	—	2	1	no	—
384	C₂³	—	2	1	no	—
385	C₂³	—	2	1	no	—
386	C₂³	—	2	1	no	—
387	C₂³	—	2	1	no	—
388	C₂³	—	2	1	no	—
389	C₂³	—	2	1	no	—
390	C₂³	—	2	1	no	—
391	C₂³	—	4	4	no	—
392	C₂³	—	4	4	no	—
393	C₂³	—	4	4	no	—
394	C₂³	—	4	4	no	—
395	C₂³	—	4	4	no	—
396	C₂³	—	4	4	no	—
397	C₂³	C₂×D₄×C₅₃	1	2	yes	—
398	C₂³	C₂×D₄×C₅₃	1	2	yes	—
399	C₂³	C₂×D₄×C₅₃	1	2	yes	—
400	C₂³	#41×C₅₃	1	2	yes	—
401	C₂³	#41×C₅₃	1	2	yes	—
402	C₂³	#41×C₅₃	1	2	yes	—
403	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
404	D₄	—	2	8	no	—
405	D₄	—	4	4	no	—
406	D₄	—	4	4	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	4	4	no	—
438	C₂×C₄	—	4	4	no	—
439	C₂×C₄	—	4	4	no	—
440	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
441	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
442	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
443	C₂²	—	2	2	no	—
444	C₂²	—	2	2	no	—
445	C₂²	—	2	2	no	—
446	C₂²	—	2	2	no	—
447	C₂²	—	2	2	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	2	2	no	—
450	C₂²	—	2	2	no	—
451	C₂²	—	2	2	no	—
452	C₂²	—	2	2	no	—
453	C₂²	—	2	2	no	—
454	C₂²	—	2	2	no	—
455	C₂²	—	2	2	no	—
456	C₂²	—	2	2	no	—
457	C₂²	—	2	2	no	—
458	C₂²	—	4	16	no	—
459	C₂²	—	4	1	no	—
460	C₂²	—	4	1	no	—
461	C₂²	—	4	1	no	—
462	C₂²	—	4	1	no	—
463	C₂²	—	4	1	no	—
464	C₂²	—	4	1	no	—
465	C₂²	—	4	1	no	—
466	C₂²	—	4	1	no	—
467	C₂²	—	4	1	no	—
468	C₂²	—	4	1	no	—
469	C₂²	—	4	1	no	—
470	C₂²	—	4	1	no	—
471	C₂²	#123×C₅₃	1	4	yes	complemented
472	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
473	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
474	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
475	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
476	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
477	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
478	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
479	C₄	—	2	2	no	—
480	C₄	—	2	2	no	—
481	C₄	—	2	2	no	—
482	C₄	—	4	1	no	—
483	C₄	—	4	1	no	—
484	C₄	—	4	1	no	—
485	C₄	—	4	1	no	—
486	C₄	—	4	1	no	—
487	C₄	—	4	1	no	—
488	C₄	—	4	1	no	—
489	C₄	—	4	1	no	—
490	C₂	—	2	4	no	—
491	C₂	—	4	2	no	—
492	C₂	—	4	2	no	—
493	C₂	—	4	2	no	—
494	C₂	C₂×#123×C₅₃	1	1	yes	characteristic
495	C₂	C₅₃×#525	1	1	yes	characteristic
496	C₂	C₅₃×#538	1	1	yes	characteristic
497	C₂	D₄²×C₅₃	1	1	yes	characteristic
498	C₂	C₅₃×#546	1	1	yes	characteristic
499	C₂	C₅₃×#546	1	1	yes	characteristic
500	C₂	C₅₃×#551	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂×#120×C₅₃	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#123×C₅₃	C₂	semidirect
3. C₂×C₄×D₄×C₅₃	C₂	semidirect
4. C₂²×C₂²⋊C₄×C₅₃	C₂	semidirect
5. C₅₃×#399	C₂	semidirect
6. C₅₃×#397	C₂	semidirect
7. C₅₃×#394	C₂	semidirect
8. C₅₃×#394	C₂	semidirect
9. C₅₃×#388	C₂	semidirect
10. C₅₃×#388	C₂	semidirect
11. C₅₃×#386	C₂	semidirect
12. C₅₃×#386	C₂	semidirect
13. C₅₃×#385	C₂	semidirect
14. C₅₃×#380	C₂	semidirect
15. C₅₃×#379	C₂	semidirect
22. C₂³×C₄×C₅₃	C₂²	semidirect
26. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
27. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
28. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
29. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
30. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
45. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
46. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
47. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
48. C₂×C₄²×C₅₃	C₂²	semidirect
49. #95×C₅₃	C₂²	semidirect
50. #95×C₅₃	C₂²	semidirect
51. #95×C₅₃	C₂²	semidirect
52. #95×C₅₃	C₂²	semidirect
53. #95×C₅₃	C₂²	semidirect
54. #95×C₅₃	C₂²	semidirect
115. C₂²⋊C₄×C₅₃	D₄	semidirect
220. C₂²×C₅₃	#123	semidirect
239. #2418	C₅₃	direct (N × H)
251. C₂×#123	C₂×C₅₃	semidirect
253. C₂×C₄×D₄	C₂×C₅₃	semidirect
254. C₂²×C₂²⋊C₄	C₂×C₅₃	semidirect
255. #399	C₂×C₅₃	semidirect
256. #397	C₂×C₅₃	semidirect
257. #394	C₂×C₅₃	semidirect
258. #394	C₂×C₅₃	semidirect
259. #388	C₂×C₅₃	semidirect
260. #388	C₂×C₅₃	semidirect
261. #386	C₂×C₅₃	semidirect
262. #386	C₂×C₅₃	semidirect
263. #385	C₂×C₅₃	semidirect
264. #380	C₂×C₅₃	semidirect
265. #379	C₂×C₅₃	semidirect
266. C₅₃	#2418	direct (N × H)
273. C₂³×C₄	C₂²×C₅₃	semidirect
277. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
278. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
279. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
280. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
281. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
296. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
297. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
298. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
299. C₂×C₄²	C₂²×C₅₃	semidirect
300. #95	C₂²×C₅₃	semidirect
301. #95	C₂²×C₅₃	semidirect
302. #95	C₂²×C₅₃	semidirect
303. #95	C₂²×C₅₃	semidirect
304. #95	C₂²×C₅₃	semidirect
305. #95	C₂²×C₅₃	semidirect
366. C₂²⋊C₄	D₄×C₅₃	semidirect
471. C₂²	#123×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#123×C₅₃	2
C₂×#120×C₅₃	2
C₂×C₄×D₄×C₅₃	2
C₂²×C₂²⋊C₄×C₅₃	2
C₅₃×#399	2
C₅₃×#397	2
C₅₃×#394	2
C₅₃×#388	2
C₅₃×#386	2
C₅₃×#385	2
C₅₃×#380	2
C₅₃×#379	2
#2418	53

Subgroups — C53×#2418 #20271355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2418 #20271355