Subgroups — C₅₃×#22329 #27229841

Order 13568.

918 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#121×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1954	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1923	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1916	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1869	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1869	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1817	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×C₂²⋊C₄×C₅₃	C₂	1	2	yes	maximal, complemented
9	C₄×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×#121×C₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#390	C₄	1	1	yes	characteristic
26	C₅₃×#389	C₄	1	1	yes	characteristic
27	C₅₃×#388	C₄	1	2	yes	—
28	C₅₃×#386	C₄	1	2	yes	—
29	C₅₃×#380	C₄	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊C₄×C₅₃	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×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
39	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₅₃	—	2	4	no	—
44	C₂²×C₈×C₅₃	—	4	2	no	—
45	C₂²×C₈×C₅₃	—	4	2	no	—
46	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
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	#121×C₅₃	C₈	1	8	yes	complemented
51	C₄×D₄×C₅₃	—	4	1	no	—
52	C₄×D₄×C₅₃	—	4	2	no	—
53	C₄×D₄×C₅₃	—	4	2	no	—
54	C₄×D₄×C₅₃	—	4	1	no	—
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	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
63	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
64	C₂×C₄²×C₅₃	—	4	1	no	—
65	C₂×C₄²×C₅₃	—	4	1	no	—
66	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
71	C₄⋊C₈×C₅₃	—	4	2	no	—
72	C₂²⋊C₈×C₅₃	—	2	4	no	—
73	C₂²⋊C₈×C₅₃	—	2	4	no	—
74	C₂²⋊C₈×C₅₃	—	4	1	no	—
75	C₂²⋊C₈×C₅₃	—	4	1	no	—
76	C₂²⋊C₈×C₅₃	—	4	2	no	—
77	C₂²⋊C₈×C₅₃	—	4	2	no	—
78	C₄×C₈×C₅₃	—	4	1	no	—
79	C₄×C₈×C₅₃	—	4	1	no	—
80	C₄×C₈×C₅₃	—	4	2	no	—
81	#95×C₅₃	C₂×C₄	1	2	yes	—
82	#95×C₅₃	C₂×C₄	1	2	yes	—
83	#95×C₅₃	C₂×C₄	1	2	yes	—
84	C₂⁴×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×D₄×C₅₃	—	4	1	no	—
87	C₂×D₄×C₅₃	—	4	2	no	—
88	C₂×D₄×C₅₃	—	4	1	no	—
89	C₂×D₄×C₅₃	—	4	2	no	—
90	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
91	C₂×D₄×C₅₃	C₂×C₈	1	8	yes	complemented
92	C₂²×C₄×C₅₃	—	2	2	no	—
93	C₂²×C₄×C₅₃	—	2	2	no	—
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	—	2	1	no	—
97	C₂²×C₄×C₅₃	—	2	1	no	—
98	C₂²×C₄×C₅₃	—	2	2	no	—
99	C₂²×C₄×C₅₃	—	2	2	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	—	4	1	no	—
104	C₂²×C₄×C₅₃	—	4	1	no	—
105	C₂²×C₄×C₅₃	—	4	1	no	—
106	C₂²×C₄×C₅₃	—	4	1	no	—
107	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	complemented
108	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
114	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
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	—	2	2	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	—	2	2	no	—
125	C₂×C₈×C₅₃	—	2	2	no	—
126	C₂×C₈×C₅₃	—	2	2	no	—
127	C₂×C₈×C₅₃	—	2	2	no	—
128	C₂×C₈×C₅₃	—	4	8	no	—
129	C₂×C₈×C₅₃	#41	1	4	yes	—
130	C₄⋊C₄×C₅₃	—	4	1	no	—
131	C₄⋊C₄×C₅₃	—	4	1	no	—
132	C₄⋊C₄×C₅₃	—	4	2	no	—
133	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	complemented
134	C₂²⋊C₄×C₅₃	—	2	4	no	—
135	C₂²⋊C₄×C₅₃	—	4	2	no	—
136	C₂²⋊C₄×C₅₃	—	4	2	no	—
137	C₂²⋊C₄×C₅₃	—	4	2	no	—
138	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	complemented
139	C₄²×C₅₃	—	2	1	no	—
140	C₄²×C₅₃	—	2	1	no	—
141	C₄²×C₅₃	—	2	2	no	—
142	C₄²×C₅₃	—	2	2	no	—
143	C₄²×C₅₃	—	2	1	no	—
144	C₄²×C₅₃	—	2	1	no	—
145	C₄²×C₅₃	—	4	1	no	—
146	C₄²×C₅₃	—	4	1	no	—
147	C₂³×C₅₃	—	2	1	no	—
148	C₂³×C₅₃	—	2	2	no	—
149	C₂³×C₅₃	—	2	1	no	—
150	C₂³×C₅₃	—	2	1	no	—
151	C₂³×C₅₃	—	2	2	no	—
152	C₂³×C₅₃	—	2	2	no	—
153	C₂³×C₅₃	—	2	1	no	—
154	C₂³×C₅₃	—	2	2	no	—
155	C₂³×C₅₃	—	2	1	no	—
156	C₂³×C₅₃	—	2	1	no	—
157	C₂³×C₅₃	—	2	2	no	—
158	C₂³×C₅₃	—	2	2	no	—
159	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
160	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
161	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
162	D₄×C₅₃	—	4	4	no	—
163	D₄×C₅₃	—	4	8	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	2	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	2	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	2	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	2	1	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	2	1	no	—
181	C₂×C₄×C₅₃	—	2	2	no	—
182	C₂×C₄×C₅₃	—	2	1	no	—
183	C₂×C₄×C₅₃	—	2	2	no	—
184	C₂×C₄×C₅₃	—	4	4	no	—
185	C₂×C₄×C₅₃	—	4	4	no	—
186	C₂×C₄×C₅₃	—	4	2	no	—
187	C₂×C₄×C₅₃	—	4	2	no	—
188	C₂×C₄×C₅₃	—	4	2	no	—
189	C₂×C₄×C₅₃	—	4	2	no	—
190	C₂×C₄×C₅₃	—	4	2	no	—
191	C₂×C₄×C₅₃	—	4	2	no	—
192	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
193	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
194	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
195	C₂×C₄×C₅₃	#131	1	2	yes	—
196	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
197	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
198	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
199	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
200	C₈×C₅₃	—	2	4	no	—
201	C₈×C₅₃	—	4	2	no	—
202	C₈×C₅₃	—	4	2	no	—
203	C₈×C₅₃	—	4	2	no	—
204	#22329	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂²×C₅₃	—	2	2	no	—
206	C₂²×C₅₃	—	2	4	no	—
207	C₂²×C₅₃	—	2	2	no	—
208	C₂²×C₅₃	—	2	2	no	—
209	C₂²×C₅₃	—	2	4	no	—
210	C₂²×C₅₃	—	2	4	no	—
211	C₂²×C₅₃	—	4	1	no	—
212	C₂²×C₅₃	—	4	2	no	—
213	C₂²×C₅₃	—	4	1	no	—
214	C₂²×C₅₃	—	4	2	no	—
215	C₂²×C₅₃	—	4	1	no	—
216	C₂²×C₅₃	—	4	2	no	—
217	C₂²×C₅₃	—	4	1	no	—
218	C₂²×C₅₃	—	4	2	no	—
219	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
220	C₂²×C₅₃	#518	1	1	yes	characteristic
221	C₂²×C₅₃	#518	1	1	yes	characteristic
222	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
223	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
224	C₂²×C₅₃	#568	1	1	yes	characteristic
225	C₂²×C₅₃	#568	1	1	yes	characteristic
226	C₄×C₅₃	—	2	2	no	—
227	C₄×C₅₃	—	2	2	no	—
228	C₄×C₅₃	—	2	4	no	—
229	C₄×C₅₃	—	2	2	no	—
230	C₄×C₅₃	—	4	1	no	—
231	C₄×C₅₃	—	4	2	no	—
232	C₄×C₅₃	—	4	1	no	—
233	C₄×C₅₃	—	4	2	no	—
234	C₄×C₅₃	#538	1	2	yes	—
235	C₄×C₅₃	#540	1	2	yes	—
236	C₄×#121	C₂×C₅₃	1	1	yes	characteristic
237	#1954	C₂×C₅₃	1	2	yes	complemented
238	#1923	C₂×C₅₃	1	2	yes	complemented
239	#1916	C₂×C₅₃	1	2	yes	complemented
240	#1869	C₂×C₅₃	1	2	yes	complemented
241	#1869	C₂×C₅₃	1	2	yes	complemented
242	#1817	C₂×C₅₃	1	1	yes	characteristic, complemented
243	C₈×C₂²⋊C₄	C₂×C₅₃	1	2	yes	complemented
244	C₄×C₂²⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
245	C₂×C₅₃	—	4	2	no	—
246	C₂×C₅₃	—	4	4	no	—
247	C₂×C₅₃	#2333	1	1	yes	characteristic
248	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
249	C₂×C₅₃	#2967	1	1	yes	characteristic
250	C₂×C₅₃	#2972	1	1	yes	characteristic
251	C₂×C₅₃	#2972	1	1	yes	characteristic
252	C₂×C₅₃	#2976	1	1	yes	characteristic
253	C₂×C₅₃	#2981	1	1	yes	characteristic
254	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
255	C₂×#121	C₄×C₅₃	1	1	yes	characteristic
256	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
257	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
258	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
259	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
260	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
261	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
262	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
263	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
264	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
265	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
266	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
267	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
268	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
269	#390	C₄×C₅₃	1	1	yes	characteristic
270	#389	C₄×C₅₃	1	1	yes	characteristic
271	#388	C₄×C₅₃	1	2	yes	—
272	#386	C₄×C₅₃	1	2	yes	—
273	#380	C₄×C₅₃	1	1	yes	characteristic
274	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
275	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
276	#336	C₂²×C₅₃	1	2	yes	—
277	#336	C₂²×C₅₃	1	2	yes	—
278	#336	C₂²×C₅₃	1	2	yes	—
279	#336	C₂²×C₅₃	1	2	yes	—
280	#336	C₂²×C₅₃	1	2	yes	—
281	C₅₃	#22329	1	1	yes	characteristic, direct product factor, Sylow
282	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
283	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
284	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
285	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
286	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
287	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
288	C₂²×C₈	—	2	4	no	—
289	C₂²×C₈	—	4	2	no	—
290	C₂²×C₈	—	4	2	no	—
291	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
292	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
293	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
294	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
295	#121	C₈×C₅₃	1	8	yes	complemented
296	C₄×D₄	—	4	2	no	—
297	C₄×D₄	—	4	2	no	—
298	C₄×D₄	—	4	1	no	—
299	C₄×D₄	—	4	1	no	—
300	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
301	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
302	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
304	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
305	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
306	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
307	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
308	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
309	C₂×C₄²	—	4	1	no	—
310	C₂×C₄²	—	4	1	no	—
311	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
312	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
314	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
315	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
316	C₄⋊C₈	—	4	2	no	—
317	C₂²⋊C₈	—	2	4	no	—
318	C₂²⋊C₈	—	2	4	no	—
319	C₂²⋊C₈	—	4	2	no	—
320	C₂²⋊C₈	—	4	1	no	—
321	C₂²⋊C₈	—	4	2	no	—
322	C₂²⋊C₈	—	4	1	no	—
323	C₄×C₈	—	4	1	no	—
324	C₄×C₈	—	4	1	no	—
325	C₄×C₈	—	4	2	no	—
326	#95	C₂×C₄×C₅₃	1	2	yes	—
327	#95	C₂×C₄×C₅₃	1	2	yes	—
328	#95	C₂×C₄×C₅₃	1	2	yes	—
329	C₂⁴	C₂²×C₄×C₅₃	1	2	yes	—
330	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
331	C₂×D₄	—	4	1	no	—
332	C₂×D₄	—	4	2	no	—
333	C₂×D₄	—	4	1	no	—
334	C₂×D₄	—	4	2	no	—
335	C₂×D₄	C₂×C₈×C₅₃	1	4	yes	—
336	C₂×D₄	C₂×C₈×C₅₃	1	8	yes	complemented
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	4	1	no	—
349	C₂²×C₄	—	4	1	no	—
350	C₂²×C₄	—	4	1	no	—
351	C₂²×C₄	—	4	1	no	—
352	C₂²×C₄	C₂×C₈×C₅₃	1	4	yes	complemented
353	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
354	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
355	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
356	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
357	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
358	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
359	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
360	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
361	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	4	8	no	—
374	C₂×C₈	#41×C₅₃	1	4	yes	—
375	C₄⋊C₄	—	4	1	no	—
376	C₄⋊C₄	—	4	2	no	—
377	C₄⋊C₄	—	4	1	no	—
378	C₄⋊C₄	C₂×C₈×C₅₃	1	4	yes	complemented
379	C₂²⋊C₄	—	2	4	no	—
380	C₂²⋊C₄	—	4	2	no	—
381	C₂²⋊C₄	—	4	2	no	—
382	C₂²⋊C₄	—	4	2	no	—
383	C₂²⋊C₄	C₂×C₈×C₅₃	1	8	yes	complemented
384	C₄²	—	2	1	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	1	no	—
390	C₄²	—	4	1	no	—
391	C₄²	—	4	1	no	—
392	C₂³	—	2	1	no	—
393	C₂³	—	2	2	no	—
394	C₂³	—	2	1	no	—
395	C₂³	—	2	2	no	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	1	no	—
398	C₂³	—	2	2	no	—
399	C₂³	—	2	1	no	—
400	C₂³	—	2	1	no	—
401	C₂³	—	2	1	no	—
402	C₂³	—	2	2	no	—
403	C₂³	—	2	2	no	—
404	C₂³	C₂²×C₈×C₅₃	1	4	yes	—
405	C₂³	C₂²×C₈×C₅₃	1	2	yes	—
406	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
407	D₄	—	4	4	no	—
408	D₄	—	4	8	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	4	4	no	—
430	C₂×C₄	—	4	4	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
438	C₂×C₄	C₂²×C₈×C₅₃	1	4	yes	—
439	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
440	C₂×C₄	#131×C₅₃	1	2	yes	—
441	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
442	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
443	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
444	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
445	C₈	—	2	4	no	—
446	C₈	—	4	2	no	—
447	C₈	—	4	2	no	—
448	C₈	—	4	2	no	—
449	C₂²	—	2	4	no	—
450	C₂²	—	2	2	no	—
451	C₂²	—	2	2	no	—
452	C₂²	—	2	2	no	—
453	C₂²	—	2	4	no	—
454	C₂²	—	2	4	no	—
455	C₂²	—	4	1	no	—
456	C₂²	—	4	2	no	—
457	C₂²	—	4	1	no	—
458	C₂²	—	4	2	no	—
459	C₂²	—	4	1	no	—
460	C₂²	—	4	2	no	—
461	C₂²	—	4	2	no	—
462	C₂²	—	4	1	no	—
463	C₂²	C₄×#41×C₅₃	1	1	yes	characteristic
464	C₂²	C₅₃×#518	1	1	yes	characteristic
465	C₂²	C₅₃×#518	1	1	yes	characteristic
466	C₂²	C₂³×C₈×C₅₃	1	1	yes	characteristic
467	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
468	C₂²	C₅₃×#568	1	1	yes	characteristic
469	C₂²	C₅₃×#568	1	1	yes	characteristic
470	C₄	—	2	2	no	—
471	C₄	—	2	2	no	—
472	C₄	—	2	4	no	—
473	C₄	—	2	2	no	—
474	C₄	—	4	1	no	—
475	C₄	—	4	2	no	—
476	C₄	—	4	1	no	—
477	C₄	—	4	2	no	—
478	C₄	C₅₃×#538	1	2	yes	—
479	C₄	C₅₃×#540	1	2	yes	—
480	C₂	—	4	2	no	—
481	C₂	—	4	4	no	—
482	C₂	C₅₃×#2333	1	1	yes	characteristic
483	C₂	C₈×#41×C₅₃	1	1	yes	characteristic
484	C₂	C₅₃×#2967	1	1	yes	characteristic
485	C₂	C₅₃×#2972	1	1	yes	characteristic
486	C₂	C₅₃×#2972	1	1	yes	characteristic
487	C₂	C₅₃×#2976	1	1	yes	characteristic
488	C₂	C₅₃×#2981	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂³×C₈×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂×#121×C₅₃	C₄
LCS	G ▷ C₂² ▷ 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×#1954	C₂	semidirect
3. C₅₃×#1923	C₂	semidirect
4. C₅₃×#1916	C₂	semidirect
5. C₅₃×#1869	C₂	semidirect
6. C₅₃×#1869	C₂	semidirect
7. C₅₃×#1817	C₂	semidirect
8. C₈×C₂²⋊C₄×C₅₃	C₂	semidirect
9. C₄×C₂²⋊C₈×C₅₃	C₂	semidirect
50. #121×C₅₃	C₈	semidirect
91. C₂×D₄×C₅₃	C₂×C₈	semidirect
107. C₂²×C₄×C₅₃	C₂×C₈	semidirect
133. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
138. C₂²⋊C₄×C₅₃	C₂×C₈	semidirect
204. #22329	C₅₃	direct (N × H)
237. #1954	C₂×C₅₃	semidirect
238. #1923	C₂×C₅₃	semidirect
239. #1916	C₂×C₅₃	semidirect
240. #1869	C₂×C₅₃	semidirect
241. #1869	C₂×C₅₃	semidirect
242. #1817	C₂×C₅₃	semidirect
243. C₈×C₂²⋊C₄	C₂×C₅₃	semidirect
244. C₄×C₂²⋊C₈	C₂×C₅₃	semidirect
281. C₅₃	#22329	direct (N × H)
295. #121	C₈×C₅₃	semidirect
336. C₂×D₄	C₂×C₈×C₅₃	semidirect
352. C₂²×C₄	C₂×C₈×C₅₃	semidirect
378. C₄⋊C₄	C₂×C₈×C₅₃	semidirect
383. C₂²⋊C₄	C₂×C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#121×C₅₃	2
C₅₃×#1954	2
C₅₃×#1923	2
C₅₃×#1916	2
C₅₃×#1869	2
C₅₃×#1817	2
C₈×C₂²⋊C₄×C₅₃	2
C₄×C₂²⋊C₈×C₅₃	2
#22329	53

Subgroups — C53×#22329 #27229841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22329 #27229841