Subgroups — C₅₃×#23116 #27230628

Order 13568.

1076 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₅₃	C₂	1	2	yes	maximal, complemented
2	C₄×#121×C₅₃	C₂	1	1	yes	maximal, characteristic
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	C₈×C₂²⋊C₄×C₅₃	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
11	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×#121×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₈×D₄×C₅₃	—	2	8	no	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
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	1	yes	characteristic
37	C₅₃×#336	C₂²	1	1	yes	characteristic
38	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	—	2	4	no	—
45	C₂²×C₈×C₅₃	—	2	4	no	—
46	C₂²×C₈×C₅₃	—	2	4	no	—
47	C₂²×C₈×C₅₃	—	2	4	no	—
48	C₂²×C₈×C₅₃	—	4	2	no	—
49	C₂²×C₈×C₅₃	—	4	2	no	—
50	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
53	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
54	#121×C₅₃	C₈	1	8	yes	complemented
55	C₄×D₄×C₅₃	—	2	4	no	—
56	C₄×D₄×C₅₃	—	2	4	no	—
57	C₄×D₄×C₅₃	—	4	2	no	—
58	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
63	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
65	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
66	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₅₃	—	4	2	no	—
68	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
73	C₄⋊C₈×C₅₃	—	2	4	no	—
74	C₂²⋊C₈×C₅₃	—	2	4	no	—
75	C₂²⋊C₈×C₅₃	—	2	4	no	—
76	C₂²⋊C₈×C₅₃	—	2	2	no	—
77	C₂²⋊C₈×C₅₃	—	4	2	no	—
78	C₂²⋊C₈×C₅₃	D₄	1	4	yes	complemented
79	C₄×C₈×C₅₃	—	2	4	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	—	2	4	no	—
87	C₂×D₄×C₅₃	—	2	4	no	—
88	C₂×D₄×C₅₃	—	4	1	no	—
89	C₂×D₄×C₅₃	—	4	1	no	—
90	C₂×D₄×C₅₃	C₂×C₈	1	8	yes	complemented
91	C₂×D₄×C₅₃	C₂×C₈	1	4	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	2	no	—
97	C₂²×C₄×C₅₃	—	2	2	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	—	4	1	no	—
103	C₂²×C₄×C₅₃	—	4	1	no	—
104	C₂²×C₄×C₅₃	—	4	1	no	—
105	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	complemented
106	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₅₃	#41	1	2	yes	—
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	—	4	8	no	—
128	C₂×C₈×C₅₃	—	4	8	no	—
129	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
130	C₂×C₈×C₅₃	#41	1	4	yes	—
131	C₄⋊C₄×C₅₃	—	2	4	no	—
132	C₄⋊C₄×C₅₃	—	4	1	no	—
133	C₄⋊C₄×C₅₃	—	4	1	no	—
134	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	complemented
135	C₂²⋊C₄×C₅₃	—	2	4	no	—
136	C₂²⋊C₄×C₅₃	—	2	4	no	—
137	C₂²⋊C₄×C₅₃	—	2	4	no	—
138	C₂²⋊C₄×C₅₃	—	4	2	no	—
139	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	complemented
140	C₄²×C₅₃	—	2	2	no	—
141	C₄²×C₅₃	—	2	2	no	—
142	C₄²×C₅₃	—	2	2	no	—
143	C₄²×C₅₃	—	2	2	no	—
144	C₄²×C₅₃	—	4	1	no	—
145	C₄²×C₅₃	—	4	1	no	—
146	C₂³×C₅₃	—	2	1	no	—
147	C₂³×C₅₃	—	2	1	no	—
148	C₂³×C₅₃	—	2	1	no	—
149	C₂³×C₅₃	—	2	1	no	—
150	C₂³×C₅₃	—	2	2	no	—
151	C₂³×C₅₃	—	2	2	no	—
152	C₂³×C₅₃	—	2	2	no	—
153	C₂³×C₅₃	—	2	2	no	—
154	C₂³×C₅₃	—	2	2	no	—
155	C₂³×C₅₃	—	2	2	no	—
156	C₂³×C₅₃	C₄×D₄	1	2	yes	—
157	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
158	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
159	C₂³×C₅₃	#131	1	2	yes	—
160	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
161	D₄×C₅₃	—	2	16	no	—
162	D₄×C₅₃	—	4	4	no	—
163	C₂×C₄×C₅₃	—	2	2	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	2	no	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	4	no	—
174	C₂×C₄×C₅₃	—	2	4	no	—
175	C₂×C₄×C₅₃	—	2	2	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	2	2	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	2	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	C₄×D₄	1	2	yes	—
191	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
192	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
193	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
194	C₂×C₄×C₅₃	#131	1	2	yes	—
195	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
197	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
199	C₈×C₅₃	—	2	4	no	—
200	C₈×C₅₃	—	2	4	no	—
201	C₈×C₅₃	—	4	2	no	—
202	C₈×C₅₃	—	4	2	no	—
203	#23116	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂²×C₅₃	—	2	2	no	—
205	C₂²×C₅₃	—	2	2	no	—
206	C₂²×C₅₃	—	2	2	no	—
207	C₂²×C₅₃	—	2	2	no	—
208	C₂²×C₅₃	—	2	2	no	—
209	C₂²×C₅₃	—	2	2	no	—
210	C₂²×C₅₃	—	2	4	no	—
211	C₂²×C₅₃	—	2	4	no	—
212	C₂²×C₅₃	—	2	4	no	—
213	C₂²×C₅₃	—	4	2	no	—
214	C₂²×C₅₃	—	4	2	no	—
215	C₂²×C₅₃	—	4	2	no	—
216	C₂²×C₅₃	—	4	2	no	—
217	C₂²×C₅₃	C₈×D₄	1	4	yes	—
218	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
219	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
220	C₂²×C₅₃	#518	1	1	yes	characteristic
221	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
222	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
223	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
224	C₂²×C₅₃	#568	1	1	yes	characteristic
225	C₄×C₅₃	—	2	2	no	—
226	C₄×C₅₃	—	2	4	no	—
227	C₄×C₅₃	—	2	2	no	—
228	C₄×C₅₃	—	2	2	no	—
229	C₄×C₅₃	—	2	2	no	—
230	C₄×C₅₃	—	4	2	no	—
231	C₄×C₅₃	—	4	2	no	—
232	C₄×C₅₃	—	4	1	no	—
233	C₄×C₅₃	#546	1	2	yes	—
234	C₄×C₅₃	#553	1	2	yes	—
235	C₂×C₈×D₄	C₂×C₅₃	1	2	yes	complemented
236	C₄×#121	C₂×C₅₃	1	1	yes	characteristic
237	#1923	C₂×C₅₃	1	2	yes	complemented
238	#1916	C₂×C₅₃	1	2	yes	complemented
239	#1869	C₂×C₅₃	1	2	yes	complemented
240	#1869	C₂×C₅₃	1	2	yes	complemented
241	C₈×C₂²⋊C₄	C₂×C₅₃	1	2	yes	complemented
242	C₄×C₂²⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
243	C₂×#336	C₂×C₅₃	1	1	yes	characteristic, complemented
244	C₂×C₅₃	—	2	4	no	—
245	C₂×C₅₃	—	4	4	no	—
246	C₂×C₅₃	#2364	1	1	yes	characteristic
247	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
248	C₂×C₅₃	#2935	1	1	yes	characteristic
249	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
250	C₂×C₅₃	#2968	1	1	yes	characteristic
251	C₂×C₅₃	#2972	1	1	yes	characteristic
252	C₂×C₅₃	#2977	1	1	yes	characteristic
253	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
254	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
255	C₂×#121	C₄×C₅₃	1	1	yes	characteristic
256	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
257	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
258	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
259	C₈×D₄	—	2	8	no	—
260	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
261	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
262	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
263	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
264	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
265	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
266	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
267	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
268	#390	C₄×C₅₃	1	1	yes	characteristic
269	#389	C₄×C₅₃	1	1	yes	characteristic
270	#388	C₄×C₅₃	1	2	yes	—
271	#386	C₄×C₅₃	1	2	yes	—
272	#380	C₄×C₅₃	1	1	yes	characteristic
273	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
274	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
275	#336	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	1	yes	characteristic
280	#336	C₂²×C₅₃	1	1	yes	characteristic
281	C₅₃	#23116	1	1	yes	characteristic, direct product factor, Sylow
282	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
283	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
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	—	2	4	no	—
290	C₂²×C₈	—	2	4	no	—
291	C₂²×C₈	—	2	4	no	—
292	C₂²×C₈	—	4	2	no	—
293	C₂²×C₈	—	4	2	no	—
294	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
295	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
296	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
297	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
298	#121	C₈×C₅₃	1	8	yes	complemented
299	C₄×D₄	—	2	4	no	—
300	C₄×D₄	—	2	4	no	—
301	C₄×D₄	—	4	2	no	—
302	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
303	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
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×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
310	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
311	C₂×C₄²	—	4	2	no	—
312	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
313	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
314	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
315	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
316	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
317	C₄⋊C₈	—	2	4	no	—
318	C₂²⋊C₈	—	2	2	no	—
319	C₂²⋊C₈	—	2	4	no	—
320	C₂²⋊C₈	—	2	4	no	—
321	C₂²⋊C₈	—	4	2	no	—
322	C₂²⋊C₈	D₄×C₅₃	1	4	yes	complemented
323	C₄×C₈	—	2	4	no	—
324	C₄×C₈	—	4	2	no	—
325	#95	C₂×C₄×C₅₃	1	2	yes	—
326	#95	C₂×C₄×C₅₃	1	2	yes	—
327	#95	C₂×C₄×C₅₃	1	2	yes	—
328	C₂⁴	C₂²×C₄×C₅₃	1	2	yes	—
329	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
330	C₂×D₄	—	2	4	no	—
331	C₂×D₄	—	2	4	no	—
332	C₂×D₄	—	4	1	no	—
333	C₂×D₄	—	4	1	no	—
334	C₂×D₄	C₂×C₈×C₅₃	1	8	yes	complemented
335	C₂×D₄	C₂×C₈×C₅₃	1	4	yes	complemented
336	C₂²×C₄	—	2	2	no	—
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	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	4	1	no	—
347	C₂²×C₄	—	4	1	no	—
348	C₂²×C₄	—	4	1	no	—
349	C₂²×C₄	C₂×C₈×C₅₃	1	4	yes	complemented
350	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
351	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
352	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
353	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
354	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
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	C₂×D₄×C₅₃	1	2	yes	—
359	C₂²×C₄	#41×C₅₃	1	2	yes	—
360	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
361	C₂×C₈	—	2	2	no	—
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	—	4	8	no	—
372	C₂×C₈	—	4	8	no	—
373	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
374	C₂×C₈	#41×C₅₃	1	4	yes	—
375	C₄⋊C₄	—	2	4	no	—
376	C₄⋊C₄	—	4	1	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	—	2	4	no	—
381	C₂²⋊C₄	—	2	4	no	—
382	C₂²⋊C₄	—	4	2	no	—
383	C₂²⋊C₄	C₂×C₈×C₅₃	1	8	yes	complemented
384	C₄²	—	2	2	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	4	1	no	—
389	C₄²	—	4	1	no	—
390	C₂³	—	2	1	no	—
391	C₂³	—	2	1	no	—
392	C₂³	—	2	1	no	—
393	C₂³	—	2	1	no	—
394	C₂³	—	2	2	no	—
395	C₂³	—	2	2	no	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	2	no	—
398	C₂³	—	2	2	no	—
399	C₂³	—	2	2	no	—
400	C₂³	C₄×D₄×C₅₃	1	2	yes	—
401	C₂³	C₂²×C₈×C₅₃	1	4	yes	—
402	C₂³	C₂²×C₈×C₅₃	1	2	yes	—
403	C₂³	#131×C₅₃	1	2	yes	—
404	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
405	D₄	—	2	16	no	—
406	D₄	—	4	4	no	—
407	C₂×C₄	—	2	2	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	2	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	4	4	no	—
429	C₂×C₄	—	4	4	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
435	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
436	C₂×C₄	C₂²×C₈×C₅₃	1	4	yes	—
437	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
438	C₂×C₄	#131×C₅₃	1	2	yes	—
439	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
440	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
441	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
442	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
443	C₈	—	2	4	no	—
444	C₈	—	2	4	no	—
445	C₈	—	4	2	no	—
446	C₈	—	4	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	4	no	—
454	C₂²	—	2	4	no	—
455	C₂²	—	2	4	no	—
456	C₂²	—	4	2	no	—
457	C₂²	—	4	2	no	—
458	C₂²	—	4	2	no	—
459	C₂²	—	4	2	no	—
460	C₂²	C₈×D₄×C₅₃	1	4	yes	—
461	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
462	C₂²	C₄×#41×C₅₃	1	1	yes	characteristic
463	C₂²	C₅₃×#518	1	1	yes	characteristic
464	C₂²	C₂³×C₈×C₅₃	1	1	yes	characteristic
465	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
466	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
467	C₂²	C₅₃×#568	1	1	yes	characteristic
468	C₄	—	2	2	no	—
469	C₄	—	2	2	no	—
470	C₄	—	2	2	no	—
471	C₄	—	2	2	no	—
472	C₄	—	2	4	no	—
473	C₄	—	4	2	no	—
474	C₄	—	4	1	no	—
475	C₄	—	4	2	no	—
476	C₄	C₅₃×#546	1	2	yes	—
477	C₄	C₅₃×#553	1	2	yes	—
478	C₂	—	2	4	no	—
479	C₂	—	4	4	no	—
480	C₂	C₅₃×#2364	1	1	yes	characteristic
481	C₂	C₂×C₈×D₄×C₅₃	1	1	yes	characteristic
482	C₂	C₅₃×#2935	1	1	yes	characteristic
483	C₂	C₈×#41×C₅₃	1	1	yes	characteristic
484	C₂	C₅₃×#2968	1	1	yes	characteristic
485	C₂	C₅₃×#2972	1	1	yes	characteristic
486	C₂	C₅₃×#2977	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
1. C₂×C₈×D₄×C₅₃	C₂	semidirect
3. C₅₃×#1923	C₂	semidirect
4. C₅₃×#1916	C₂	semidirect
5. C₅₃×#1869	C₂	semidirect
6. C₅₃×#1869	C₂	semidirect
7. C₈×C₂²⋊C₄×C₅₃	C₂	semidirect
8. C₄×C₂²⋊C₈×C₅₃	C₂	semidirect
9. C₂×C₅₃×#336	C₂	semidirect
54. #121×C₅₃	C₈	semidirect
78. C₂²⋊C₈×C₅₃	D₄	semidirect
90. C₂×D₄×C₅₃	C₂×C₈	semidirect
91. C₂×D₄×C₅₃	C₂×C₈	semidirect
105. C₂²×C₄×C₅₃	C₂×C₈	semidirect
134. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
139. C₂²⋊C₄×C₅₃	C₂×C₈	semidirect
203. #23116	C₅₃	direct (N × H)
235. C₂×C₈×D₄	C₂×C₅₃	semidirect
237. #1923	C₂×C₅₃	semidirect
238. #1916	C₂×C₅₃	semidirect
239. #1869	C₂×C₅₃	semidirect
240. #1869	C₂×C₅₃	semidirect
241. C₈×C₂²⋊C₄	C₂×C₅₃	semidirect
242. C₄×C₂²⋊C₈	C₂×C₅₃	semidirect
243. C₂×#336	C₂×C₅₃	semidirect
281. C₅₃	#23116	direct (N × H)
298. #121	C₈×C₅₃	semidirect
322. C₂²⋊C₈	D₄×C₅₃	semidirect
334. C₂×D₄	C₂×C₈×C₅₃	semidirect
335. C₂×D₄	C₂×C₈×C₅₃	semidirect
349. 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
C₂×C₈×D₄×C₅₃	2
C₄×#121×C₅₃	2
C₅₃×#1923	2
C₅₃×#1916	2
C₅₃×#1869	2
C₈×C₂²⋊C₄×C₅₃	2
C₄×C₂²⋊C₈×C₅₃	2
C₂×C₅₃×#336	2
#23116	53

Subgroups — C53×#23116 #27230628

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23116 #27230628