Subgroups — C₅₃×#22151 #27229663

Order 13568.

1052 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#459	C₂	1	1	yes	maximal, characteristic
2	C₄×#127×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2293	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1982	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1972	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1936	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1879	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1864	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1802	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
13	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₃	—	2	1	no	—
15	C₂×C₄×D₄×C₅₃	—	2	1	no	—
16	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
18	C₂×#117×C₅₃	—	2	1	no	—
19	C₅₃×#459	C₄	1	8	yes	complemented
20	C₂×#106×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#102×C₅₃	C₂²	1	2	yes	—
24	C₂×#102×C₅₃	C₂²	1	2	yes	—
25	C₂×#102×C₅₃	C₂²	1	2	yes	—
26	C₂×#102×C₅₃	C₂²	1	2	yes	—
27	C₂×#97×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#390	—	2	1	no	—
29	C₅₃×#390	—	2	1	no	—
30	C₅₃×#390	C₂²	1	2	yes	—
31	C₅₃×#390	C₂²	1	1	yes	characteristic
32	C₅₃×#390	C₂²	1	1	yes	characteristic
33	C₅₃×#389	C₂²	1	1	yes	characteristic
34	C₅₃×#389	C₂²	1	1	yes	characteristic
35	C₅₃×#388	—	2	1	no	—
36	C₅₃×#388	—	2	1	no	—
37	C₅₃×#384	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
40	C₄³×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#340	C₂²	1	2	yes	—
42	C₅₃×#340	C₂²	1	2	yes	—
43	C₅₃×#336	C₂²	1	2	yes	—
44	C₂²×D₄×C₅₃	—	2	1	no	—
45	C₂²×D₄×C₅₃	—	2	1	no	—
46	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
47	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	—	2	1	no	—
49	C₂³×C₄×C₅₃	—	2	2	no	—
50	C₂×D₈×C₅₃	—	4	2	no	—
51	C₂×D₈×C₅₃	—	4	2	no	—
52	C₂×D₈×C₅₃	C₂×C₄	1	8	yes	complemented
53	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
55	#127×C₅₃	—	2	2	no	—
56	#127×C₅₃	C₂×C₄	1	4	yes	—
57	C₄×D₄×C₅₃	—	2	2	no	—
58	C₄×D₄×C₅₃	—	4	2	no	—
59	C₄×D₄×C₅₃	—	4	2	no	—
60	C₄×D₄×C₅₃	—	4	4	no	—
61	C₄×D₄×C₅₃	—	4	4	no	—
62	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	complemented
63	#117×C₅₃	—	4	4	no	—
64	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
65	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
67	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
79	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
80	C₂×C₄²×C₅₃	—	2	1	no	—
81	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
82	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
83	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
84	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
85	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
88	#106×C₅₃	—	4	2	no	—
89	C₄⋊C₈×C₅₃	—	2	2	no	—
90	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
91	#102×C₅₃	—	4	2	no	—
92	#102×C₅₃	—	4	2	no	—
93	#102×C₅₃	—	4	2	no	—
94	#102×C₅₃	C₂×C₄	1	8	yes	complemented
95	#97×C₅₃	—	4	2	no	—
96	#95×C₅₃	—	2	1	no	—
97	#95×C₅₃	—	2	1	no	—
98	#95×C₅₃	—	2	1	no	—
99	C₂⁴×C₅₃	—	2	1	no	—
100	C₂⁴×C₅₃	—	2	2	no	—
101	C₂×D₄×C₅₃	—	2	4	no	—
102	C₂×D₄×C₅₃	—	2	4	no	—
103	C₂×D₄×C₅₃	—	2	1	no	—
104	C₂×D₄×C₅₃	—	2	2	no	—
105	C₂×D₄×C₅₃	—	2	1	no	—
106	C₂×D₄×C₅₃	—	2	2	no	—
107	C₂×D₄×C₅₃	—	2	2	no	—
108	C₂×D₄×C₅₃	—	2	2	no	—
109	C₂×D₄×C₅₃	—	2	1	no	—
110	C₂×D₄×C₅₃	—	2	1	no	—
111	C₂×D₄×C₅₃	—	2	2	no	—
112	C₂×D₄×C₅₃	—	2	1	no	—
113	C₂×D₄×C₅₃	—	4	2	no	—
114	C₂×D₄×C₅₃	—	4	2	no	—
115	C₂×D₄×C₅₃	—	4	2	no	—
116	C₂×D₄×C₅₃	—	4	2	no	—
117	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
118	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₅₃	—	2	1	no	—
120	C₂²×C₄×C₅₃	—	2	4	no	—
121	C₂²×C₄×C₅₃	—	2	1	no	—
122	C₂²×C₄×C₅₃	—	2	1	no	—
123	C₂²×C₄×C₅₃	—	2	2	no	—
124	C₂²×C₄×C₅₃	—	4	4	no	—
125	C₂²×C₄×C₅₃	—	4	4	no	—
126	C₂²×C₄×C₅₃	—	4	2	no	—
127	C₂²×C₄×C₅₃	—	4	2	no	—
128	C₂²×C₄×C₅₃	—	4	4	no	—
129	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
136	D₈×C₅₃	—	4	8	no	—
137	C₂×C₈×C₅₃	—	2	2	no	—
138	C₂×C₈×C₅₃	—	2	2	no	—
139	C₂×C₈×C₅₃	—	2	2	no	—
140	C₂×C₈×C₅₃	—	2	2	no	—
141	C₂×C₈×C₅₃	—	2	2	no	—
142	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
143	C₄⋊C₄×C₅₃	—	2	1	no	—
144	C₄⋊C₄×C₅₃	—	2	1	no	—
145	C₄⋊C₄×C₅₃	—	2	1	no	—
146	C₄⋊C₄×C₅₃	—	2	2	no	—
147	C₄⋊C₄×C₅₃	—	2	1	no	—
148	C₄⋊C₄×C₅₃	—	2	2	no	—
149	C₄⋊C₄×C₅₃	—	2	1	no	—
150	C₄⋊C₄×C₅₃	—	4	2	no	—
151	C₄⋊C₄×C₅₃	—	4	2	no	—
152	C₄⋊C₄×C₅₃	—	4	2	no	—
153	C₄⋊C₄×C₅₃	—	4	2	no	—
154	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
155	C₂²⋊C₄×C₅₃	—	2	4	no	—
156	C₂²⋊C₄×C₅₃	—	4	2	no	—
157	C₂²⋊C₄×C₅₃	—	4	2	no	—
158	C₂²⋊C₄×C₅₃	—	4	2	no	—
159	C₂²⋊C₄×C₅₃	—	4	4	no	—
160	C₂²⋊C₄×C₅₃	—	4	4	no	—
161	C₂²⋊C₄×C₅₃	—	4	4	no	—
162	C₂²⋊C₄×C₅₃	—	4	2	no	—
163	C₄²×C₅₃	—	2	2	no	—
164	C₄²×C₅₃	—	2	2	no	—
165	C₄²×C₅₃	—	2	2	no	—
166	C₄²×C₅₃	—	2	1	no	—
167	C₄²×C₅₃	—	2	1	no	—
168	C₄²×C₅₃	—	2	1	no	—
169	C₄²×C₅₃	—	2	1	no	—
170	C₄²×C₅₃	—	4	1	no	—
171	C₄²×C₅₃	—	4	2	no	—
172	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
173	C₄²×C₅₃	#41	1	2	yes	—
174	C₂³×C₅₃	—	2	2	no	—
175	C₂³×C₅₃	—	2	4	no	—
176	C₂³×C₅₃	—	4	2	no	—
177	C₂³×C₅₃	—	4	1	no	—
178	C₂³×C₅₃	—	4	2	no	—
179	C₂³×C₅₃	—	4	1	no	—
180	C₂³×C₅₃	—	4	2	no	—
181	C₂³×C₅₃	—	4	2	no	—
182	C₂³×C₅₃	—	4	1	no	—
183	C₂³×C₅₃	—	4	1	no	—
184	C₂³×C₅₃	—	4	2	no	—
185	C₂³×C₅₃	—	4	1	no	—
186	C₂³×C₅₃	—	4	2	no	—
187	C₂³×C₅₃	—	4	1	no	—
188	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
189	D₄×C₅₃	—	2	4	no	—
190	D₄×C₅₃	—	2	2	no	—
191	D₄×C₅₃	—	2	2	no	—
192	D₄×C₅₃	—	2	4	no	—
193	D₄×C₅₃	—	4	8	no	—
194	D₄×C₅₃	—	4	1	no	—
195	D₄×C₅₃	—	4	2	no	—
196	D₄×C₅₃	—	4	1	no	—
197	D₄×C₅₃	—	4	2	no	—
198	D₄×C₅₃	—	4	8	no	—
199	C₂×C₄×C₅₃	—	2	1	no	—
200	C₂×C₄×C₅₃	—	2	1	no	—
201	C₂×C₄×C₅₃	—	2	1	no	—
202	C₂×C₄×C₅₃	—	2	1	no	—
203	C₂×C₄×C₅₃	—	2	2	no	—
204	C₂×C₄×C₅₃	—	2	1	no	—
205	C₂×C₄×C₅₃	—	2	1	no	—
206	C₂×C₄×C₅₃	—	2	1	no	—
207	C₂×C₄×C₅₃	—	2	1	no	—
208	C₂×C₄×C₅₃	—	2	1	no	—
209	C₂×C₄×C₅₃	—	2	1	no	—
210	C₂×C₄×C₅₃	—	2	1	no	—
211	C₂×C₄×C₅₃	—	2	1	no	—
212	C₂×C₄×C₅₃	—	2	1	no	—
213	C₂×C₄×C₅₃	—	4	1	no	—
214	C₂×C₄×C₅₃	—	4	1	no	—
215	C₂×C₄×C₅₃	—	4	8	no	—
216	C₂×C₄×C₅₃	—	4	2	no	—
217	C₂×C₄×C₅₃	—	4	1	no	—
218	C₂×C₄×C₅₃	—	4	1	no	—
219	C₂×C₄×C₅₃	—	4	2	no	—
220	C₂×C₄×C₅₃	—	4	2	no	—
221	C₂×C₄×C₅₃	—	4	1	no	—
222	C₂×C₄×C₅₃	—	4	1	no	—
223	C₂×C₄×C₅₃	—	4	1	no	—
224	C₂×C₄×C₅₃	—	4	1	no	—
225	C₂×C₄×C₅₃	—	8	8	no	—
226	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
227	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
228	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
229	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
230	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
231	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
232	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
233	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
234	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
235	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
236	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
237	C₈×C₅₃	—	2	4	no	—
238	C₈×C₅₃	—	4	2	no	—
239	#22151	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₂²×C₅₃	—	4	4	no	—
241	C₂²×C₅₃	—	4	2	no	—
242	C₂²×C₅₃	—	4	2	no	—
243	C₂²×C₅₃	—	4	4	no	—
244	C₂²×C₅₃	—	4	4	no	—
245	C₂²×C₅₃	—	4	2	no	—
246	C₂²×C₅₃	—	8	2	no	—
247	C₂²×C₅₃	—	8	1	no	—
248	C₂²×C₅₃	—	8	2	no	—
249	C₂²×C₅₃	—	8	1	no	—
250	C₂²×C₅₃	—	8	2	no	—
251	C₂²×C₅₃	—	8	1	no	—
252	C₂²×C₅₃	—	8	2	no	—
253	C₂²×C₅₃	—	8	1	no	—
254	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
255	C₂²×C₅₃	#535	1	1	yes	characteristic
256	C₂²×C₅₃	#537	1	1	yes	characteristic
257	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
258	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
259	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
260	C₂²×C₅₃	#575	1	1	yes	characteristic
261	C₄×C₅₃	—	2	2	no	—
262	C₄×C₅₃	—	2	2	no	—
263	C₄×C₅₃	—	4	2	no	—
264	C₄×C₅₃	—	4	1	no	—
265	C₄×C₅₃	—	4	1	no	—
266	C₄×C₅₃	—	4	1	no	—
267	C₄×C₅₃	—	8	1	no	—
268	C₄×C₅₃	—	8	2	no	—
269	C₄×C₅₃	—	8	1	no	—
270	C₄×C₅₃	#442	1	4	yes	—
271	C₄×C₅₃	#518	1	1	yes	characteristic
272	C₄×C₅₃	#520	1	1	yes	characteristic
273	C₄×C₅₃	#540	1	1	yes	characteristic
274	C₄×C₅₃	#542	1	1	yes	characteristic
275	C₂×#459	C₂×C₅₃	1	1	yes	characteristic
276	C₄×#127	C₂×C₅₃	1	1	yes	characteristic, complemented
277	#2293	C₂×C₅₃	1	1	yes	characteristic, complemented
278	#1982	C₂×C₅₃	1	1	yes	characteristic, complemented
279	#1972	C₂×C₅₃	1	2	yes	complemented
280	#1948	C₂×C₅₃	1	1	yes	characteristic, complemented
281	#1936	C₂×C₅₃	1	2	yes	complemented
282	#1879	C₂×C₅₃	1	2	yes	complemented
283	#1864	C₂×C₅₃	1	2	yes	complemented
284	#1802	C₂×C₅₃	1	1	yes	characteristic, complemented
285	#1767	C₂×C₅₃	1	1	yes	characteristic, complemented
286	C₂×C₅₃	—	8	2	no	—
287	C₂×C₅₃	—	8	4	no	—
288	C₂×C₅₃	#2324	1	1	yes	characteristic
289	C₂×C₅₃	C₂×#442	1	1	yes	characteristic
290	C₂×C₅₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₂×C₅₃	#3214	1	1	yes	characteristic
292	C₂×C₅₃	#3217	1	1	yes	characteristic
293	C₂×C₅₃	#3251	1	1	yes	characteristic
294	C₂×C₅₃	#3253	1	1	yes	characteristic
295	C₂²×D₈	C₂²×C₅₃	1	2	yes	—
296	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
297	C₂×C₄×D₄	—	2	1	no	—
298	C₂×C₄×D₄	—	2	1	no	—
299	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
300	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
301	C₂×#117	—	2	1	no	—
302	#459	C₄×C₅₃	1	8	yes	complemented
303	C₂×#106	C₂²×C₅₃	1	2	yes	—
304	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
305	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
306	C₂×#102	C₂²×C₅₃	1	2	yes	—
307	C₂×#102	C₂²×C₅₃	1	2	yes	—
308	C₂×#102	C₂²×C₅₃	1	2	yes	—
309	C₂×#102	C₂²×C₅₃	1	2	yes	—
310	C₂×#97	C₂²×C₅₃	1	2	yes	—
311	#390	—	2	1	no	—
312	#390	—	2	1	no	—
313	#390	C₂²×C₅₃	1	2	yes	—
314	#390	C₂²×C₅₃	1	1	yes	characteristic
315	#390	C₂²×C₅₃	1	1	yes	characteristic
316	#389	C₂²×C₅₃	1	1	yes	characteristic
317	#389	C₂²×C₅₃	1	1	yes	characteristic
318	#388	—	2	1	no	—
319	#388	—	2	1	no	—
320	#384	C₂²×C₅₃	1	1	yes	characteristic
321	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
322	C₄×C₂²⋊C₄	—	2	1	no	—
323	C₄³	C₂²×C₅₃	1	1	yes	characteristic
324	#340	C₂²×C₅₃	1	2	yes	—
325	#340	C₂²×C₅₃	1	2	yes	—
326	#336	C₂²×C₅₃	1	2	yes	—
327	C₅₃	#22151	1	1	yes	characteristic, direct product factor, Sylow
328	C₂²×D₄	—	2	1	no	—
329	C₂²×D₄	—	2	1	no	—
330	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
331	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
332	C₂³×C₄	—	2	1	no	—
333	C₂³×C₄	—	2	2	no	—
334	C₂×D₈	—	4	2	no	—
335	C₂×D₈	—	4	2	no	—
336	C₂×D₈	C₂×C₄×C₅₃	1	8	yes	complemented
337	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
338	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
339	#127	—	2	2	no	—
340	#127	C₂×C₄×C₅₃	1	4	yes	—
341	C₄×D₄	—	2	2	no	—
342	C₄×D₄	—	4	2	no	—
343	C₄×D₄	—	4	2	no	—
344	C₄×D₄	—	4	4	no	—
345	C₄×D₄	—	4	4	no	—
346	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	complemented
347	#117	—	4	4	no	—
348	C₂×C₄⋊C₄	—	2	1	no	—
349	C₂×C₄⋊C₄	—	2	1	no	—
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄⋊C₄	—	2	1	no	—
352	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
353	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
354	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
355	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
356	C₂×C₂²⋊C₄	—	2	1	no	—
357	C₂×C₂²⋊C₄	—	2	1	no	—
358	C₂×C₂²⋊C₄	—	2	1	no	—
359	C₂×C₂²⋊C₄	—	2	2	no	—
360	C₂×C₂²⋊C₄	—	2	2	no	—
361	C₂×C₂²⋊C₄	—	2	2	no	—
362	C₂×C₂²⋊C₄	—	2	1	no	—
363	C₂×C₂²⋊C₄	—	2	1	no	—
364	C₂×C₄²	—	2	1	no	—
365	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
366	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
367	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
368	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
369	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
370	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
371	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
372	#106	—	4	2	no	—
373	C₄⋊C₈	—	2	2	no	—
374	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
375	#102	—	4	2	no	—
376	#102	—	4	2	no	—
377	#102	—	4	2	no	—
378	#102	C₂×C₄×C₅₃	1	8	yes	complemented
379	#97	—	4	2	no	—
380	#95	—	2	1	no	—
381	#95	—	2	1	no	—
382	#95	—	2	1	no	—
383	C₂⁴	—	2	1	no	—
384	C₂⁴	—	2	2	no	—
385	C₂×D₄	—	2	4	no	—
386	C₂×D₄	—	2	4	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	2	no	—
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	—	2	1	no	—
397	C₂×D₄	—	4	2	no	—
398	C₂×D₄	—	4	2	no	—
399	C₂×D₄	—	4	2	no	—
400	C₂×D₄	—	4	2	no	—
401	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
402	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	1	no	—
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	4	4	no	—
409	C₂²×C₄	—	4	4	no	—
410	C₂²×C₄	—	4	2	no	—
411	C₂²×C₄	—	4	2	no	—
412	C₂²×C₄	—	4	4	no	—
413	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
414	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
415	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
416	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
417	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
418	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
419	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
420	D₈	—	4	8	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	4	2	no	—
435	C₄⋊C₄	—	4	2	no	—
436	C₄⋊C₄	—	4	2	no	—
437	C₄⋊C₄	—	4	2	no	—
438	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
439	C₂²⋊C₄	—	2	4	no	—
440	C₂²⋊C₄	—	4	2	no	—
441	C₂²⋊C₄	—	4	2	no	—
442	C₂²⋊C₄	—	4	4	no	—
443	C₂²⋊C₄	—	4	4	no	—
444	C₂²⋊C₄	—	4	4	no	—
445	C₂²⋊C₄	—	4	2	no	—
446	C₂²⋊C₄	—	4	2	no	—
447	C₄²	—	2	2	no	—
448	C₄²	—	2	2	no	—
449	C₄²	—	2	2	no	—
450	C₄²	—	2	1	no	—
451	C₄²	—	2	1	no	—
452	C₄²	—	2	1	no	—
453	C₄²	—	2	1	no	—
454	C₄²	—	4	1	no	—
455	C₄²	—	4	2	no	—
456	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
457	C₄²	#41×C₅₃	1	2	yes	—
458	C₂³	—	2	2	no	—
459	C₂³	—	2	4	no	—
460	C₂³	—	4	2	no	—
461	C₂³	—	4	1	no	—
462	C₂³	—	4	2	no	—
463	C₂³	—	4	1	no	—
464	C₂³	—	4	1	no	—
465	C₂³	—	4	2	no	—
466	C₂³	—	4	2	no	—
467	C₂³	—	4	1	no	—
468	C₂³	—	4	1	no	—
469	C₂³	—	4	2	no	—
470	C₂³	—	4	1	no	—
471	C₂³	—	4	2	no	—
472	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
473	D₄	—	2	2	no	—
474	D₄	—	2	4	no	—
475	D₄	—	2	2	no	—
476	D₄	—	2	4	no	—
477	D₄	—	4	8	no	—
478	D₄	—	4	1	no	—
479	D₄	—	4	2	no	—
480	D₄	—	4	1	no	—
481	D₄	—	4	2	no	—
482	D₄	—	4	8	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	4	1	no	—
498	C₂×C₄	—	4	1	no	—
499	C₂×C₄	—	4	8	no	—
500	C₂×C₄	—	4	2	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	4	1	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	1	no	—
508	C₂×C₄	—	4	1	no	—
509	C₂×C₄	—	8	8	no	—
510	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
511	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
512	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
513	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
514	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
515	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
516	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
517	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
518	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
519	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
520	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
521	C₈	—	2	4	no	—
522	C₈	—	4	2	no	—
523	C₂²	—	4	2	no	—
524	C₂²	—	4	4	no	—
525	C₂²	—	4	4	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	4	2	no	—
528	C₂²	—	4	4	no	—
529	C₂²	—	8	2	no	—
530	C₂²	—	8	1	no	—
531	C₂²	—	8	2	no	—
532	C₂²	—	8	1	no	—
533	C₂²	—	8	2	no	—
534	C₂²	—	8	1	no	—
535	C₂²	—	8	1	no	—
536	C₂²	—	8	2	no	—
537	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
538	C₂²	C₅₃×#535	1	1	yes	characteristic
539	C₂²	C₅₃×#537	1	1	yes	characteristic
540	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
541	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
542	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
543	C₂²	C₅₃×#575	1	1	yes	characteristic
544	C₄	—	2	2	no	—
545	C₄	—	2	2	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	1	no	—
548	C₄	—	4	1	no	—
549	C₄	—	4	1	no	—
550	C₄	—	8	1	no	—
551	C₄	—	8	2	no	—
552	C₄	—	8	1	no	—
553	C₄	C₅₃×#442	1	4	yes	—
554	C₄	C₅₃×#518	1	1	yes	characteristic
555	C₄	C₅₃×#520	1	1	yes	characteristic
556	C₄	C₅₃×#540	1	1	yes	characteristic
557	C₄	C₅₃×#542	1	1	yes	characteristic
558	C₂	—	8	2	no	—
559	C₂	—	8	4	no	—
560	C₂	C₅₃×#2324	1	1	yes	characteristic
561	C₂	C₂×C₅₃×#442	1	1	yes	characteristic
562	C₂	C₄×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
563	C₂	C₅₃×#3214	1	1	yes	characteristic
564	C₂	C₅₃×#3217	1	1	yes	characteristic
565	C₂	C₅₃×#3251	1	1	yes	characteristic
566	C₂	C₅₃×#3253	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂×C₅₃×#459	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×#127×C₅₃	C₂	semidirect
3. C₅₃×#2293	C₂	semidirect
4. C₅₃×#1982	C₂	semidirect
5. C₅₃×#1972	C₂	semidirect
6. C₅₃×#1948	C₂	semidirect
7. C₅₃×#1936	C₂	semidirect
8. C₅₃×#1879	C₂	semidirect
9. C₅₃×#1864	C₂	semidirect
10. C₅₃×#1802	C₂	semidirect
11. C₅₃×#1767	C₂	semidirect
19. C₅₃×#459	C₄	semidirect
52. C₂×D₈×C₅₃	C₂×C₄	semidirect
62. C₄×D₄×C₅₃	C₂×C₄	semidirect
90. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
94. #102×C₅₃	C₂×C₄	semidirect
239. #22151	C₅₃	direct (N × H)
276. C₄×#127	C₂×C₅₃	semidirect
277. #2293	C₂×C₅₃	semidirect
278. #1982	C₂×C₅₃	semidirect
279. #1972	C₂×C₅₃	semidirect
280. #1948	C₂×C₅₃	semidirect
281. #1936	C₂×C₅₃	semidirect
282. #1879	C₂×C₅₃	semidirect
283. #1864	C₂×C₅₃	semidirect
284. #1802	C₂×C₅₃	semidirect
285. #1767	C₂×C₅₃	semidirect
302. #459	C₄×C₅₃	semidirect
327. C₅₃	#22151	direct (N × H)
336. C₂×D₈	C₂×C₄×C₅₃	semidirect
346. C₄×D₄	C₂×C₄×C₅₃	semidirect
374. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
378. #102	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#459	2
C₄×#127×C₅₃	2
C₅₃×#2293	2
C₅₃×#1982	2
C₅₃×#1972	2
C₅₃×#1948	2
C₅₃×#1936	2
C₅₃×#1879	2
C₅₃×#1864	2
C₅₃×#1802	2
C₅₃×#1767	2
#22151	53

Subgroups — C53×#22151 #27229663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22151 #27229663