Subgroups — C₅₃×#23093 #27230605

Order 13568.

1042 conjugacy classes of subgroups.

The subgroup lattice has 528 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	C₅₃×#2366	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2354	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1970	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1957	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1925	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1879	C₂	1	2	yes	maximal, complemented
9	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#102×C₅₃	C₂	1	2	yes	maximal, complemented
11	C₄×#102×C₅₃	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	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#117×C₅₃	—	2	1	no	—
18	C₅₃×#459	C₄	1	8	yes	complemented
19	C₂×#107×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#102×C₅₃	C₂²	1	2	yes	—
22	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#102×C₅₃	C₂²	1	2	yes	—
25	C₂×#102×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#390	—	2	1	no	—
29	C₅₃×#390	C₂²	1	2	yes	—
30	C₅₃×#390	C₂²	1	2	yes	—
31	C₅₃×#388	—	2	2	no	—
32	C₅₃×#386	—	2	2	no	—
33	C₅₃×#384	C₂²	1	1	yes	characteristic
34	C₅₃×#380	—	2	1	no	—
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#340	C₂²	1	1	yes	characteristic
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₅₃×#340	C₂²	1	1	yes	characteristic
41	C₅₃×#336	C₂²	1	2	yes	—
42	C₂²×D₄×C₅₃	—	2	1	no	—
43	C₂²×D₄×C₅₃	—	2	1	no	—
44	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
45	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	—	2	1	no	—
47	C₂³×C₄×C₅₃	—	2	2	no	—
48	C₂×D₈×C₅₃	—	4	2	no	—
49	C₂×D₈×C₅₃	—	4	2	no	—
50	C₂×D₈×C₅₃	C₂×C₄	1	8	yes	complemented
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	#127×C₅₃	—	2	2	no	—
55	#127×C₅₃	C₂×C₄	1	4	yes	—
56	C₄×D₄×C₅₃	—	2	2	no	—
57	C₄×D₄×C₅₃	—	2	4	no	—
58	C₄×D₄×C₅₃	—	4	2	no	—
59	C₄×D₄×C₅₃	—	4	4	no	—
60	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	complemented
61	#117×C₅₃	—	4	4	no	—
62	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
63	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
64	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
75	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
76	C₂×C₄²×C₅₃	—	2	2	no	—
77	C₂×C₄²×C₅₃	—	2	1	no	—
78	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
79	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
80	#107×C₅₃	—	4	2	no	—
81	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
82	#102×C₅₃	—	2	4	no	—
83	#102×C₅₃	—	2	2	no	—
84	#102×C₅₃	—	2	2	no	—
85	#102×C₅₃	—	4	2	no	—
86	#102×C₅₃	C₂×C₄	1	8	yes	complemented
87	C₄×C₈×C₅₃	—	2	2	no	—
88	C₄×C₈×C₅₃	—	4	2	no	—
89	#95×C₅₃	—	2	2	no	—
90	#95×C₅₃	—	2	1	no	—
91	#95×C₅₃	D₄	1	2	yes	—
92	C₂⁴×C₅₃	—	2	1	no	—
93	C₂⁴×C₅₃	—	2	2	no	—
94	C₂×D₄×C₅₃	—	2	4	no	—
95	C₂×D₄×C₅₃	—	2	4	no	—
96	C₂×D₄×C₅₃	—	2	2	no	—
97	C₂×D₄×C₅₃	—	2	1	no	—
98	C₂×D₄×C₅₃	—	2	1	no	—
99	C₂×D₄×C₅₃	—	2	2	no	—
100	C₂×D₄×C₅₃	—	2	1	no	—
101	C₂×D₄×C₅₃	—	2	2	no	—
102	C₂×D₄×C₅₃	—	2	2	no	—
103	C₂×D₄×C₅₃	—	2	2	no	—
104	C₂×D₄×C₅₃	—	2	1	no	—
105	C₂×D₄×C₅₃	—	2	1	no	—
106	C₂×D₄×C₅₃	—	4	2	no	—
107	C₂×D₄×C₅₃	—	4	2	no	—
108	C₂×D₄×C₅₃	—	8	1	no	—
109	C₂×D₄×C₅₃	—	8	1	no	—
110	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
111	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₅₃	—	2	1	no	—
113	C₂²×C₄×C₅₃	—	2	1	no	—
114	C₂²×C₄×C₅₃	—	2	4	no	—
115	C₂²×C₄×C₅₃	—	2	1	no	—
116	C₂²×C₄×C₅₃	—	2	2	no	—
117	C₂²×C₄×C₅₃	—	4	4	no	—
118	C₂²×C₄×C₅₃	—	4	2	no	—
119	C₂²×C₄×C₅₃	—	4	2	no	—
120	C₂²×C₄×C₅₃	—	4	4	no	—
121	C₂²×C₄×C₅₃	—	8	2	no	—
122	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
127	D₈×C₅₃	—	4	8	no	—
128	C₂×C₈×C₅₃	—	2	1	no	—
129	C₂×C₈×C₅₃	—	2	1	no	—
130	C₂×C₈×C₅₃	—	2	2	no	—
131	C₂×C₈×C₅₃	—	2	2	no	—
132	C₂×C₈×C₅₃	—	2	1	no	—
133	C₂×C₈×C₅₃	—	2	1	no	—
134	C₂×C₈×C₅₃	—	2	1	no	—
135	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
136	C₂×C₈×C₅₃	#41	1	2	yes	—
137	C₄⋊C₄×C₅₃	—	2	2	no	—
138	C₄⋊C₄×C₅₃	—	2	1	no	—
139	C₄⋊C₄×C₅₃	—	2	1	no	—
140	C₄⋊C₄×C₅₃	—	2	2	no	—
141	C₄⋊C₄×C₅₃	—	2	1	no	—
142	C₄⋊C₄×C₅₃	—	2	2	no	—
143	C₄⋊C₄×C₅₃	—	2	1	no	—
144	C₄⋊C₄×C₅₃	—	4	2	no	—
145	C₄⋊C₄×C₅₃	—	8	1	no	—
146	C₄⋊C₄×C₅₃	D₈	1	4	yes	complemented
147	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
148	C₄⋊C₄×C₅₃	#41	1	2	yes	—
149	C₂²⋊C₄×C₅₃	—	2	4	no	—
150	C₂²⋊C₄×C₅₃	—	4	4	no	—
151	C₂²⋊C₄×C₅₃	—	4	4	no	—
152	C₂²⋊C₄×C₅₃	—	4	4	no	—
153	C₂²⋊C₄×C₅₃	—	8	2	no	—
154	C₂²⋊C₄×C₅₃	—	8	2	no	—
155	C₄²×C₅₃	—	2	2	no	—
156	C₄²×C₅₃	—	2	2	no	—
157	C₄²×C₅₃	—	4	4	no	—
158	C₄²×C₅₃	—	4	2	no	—
159	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
160	C₄²×C₅₃	#41	1	2	yes	—
161	C₂³×C₅₃	—	2	2	no	—
162	C₂³×C₅₃	—	2	4	no	—
163	C₂³×C₅₃	—	4	2	no	—
164	C₂³×C₅₃	—	4	1	no	—
165	C₂³×C₅₃	—	4	2	no	—
166	C₂³×C₅₃	—	4	1	no	—
167	C₂³×C₅₃	—	4	2	no	—
168	C₂³×C₅₃	—	4	1	no	—
169	C₂³×C₅₃	—	4	1	no	—
170	C₂³×C₅₃	—	4	2	no	—
171	C₂³×C₅₃	—	4	2	no	—
172	C₂³×C₅₃	—	4	1	no	—
173	C₂³×C₅₃	—	4	2	no	—
174	C₂³×C₅₃	—	4	1	no	—
175	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
176	D₄×C₅₃	—	2	2	no	—
177	D₄×C₅₃	—	2	4	no	—
178	D₄×C₅₃	—	2	4	no	—
179	D₄×C₅₃	—	2	2	no	—
180	D₄×C₅₃	—	4	1	no	—
181	D₄×C₅₃	—	4	2	no	—
182	D₄×C₅₃	—	4	1	no	—
183	D₄×C₅₃	—	4	2	no	—
184	D₄×C₅₃	—	4	8	no	—
185	D₄×C₅₃	—	8	4	no	—
186	C₂×C₄×C₅₃	—	2	2	no	—
187	C₂×C₄×C₅₃	—	2	1	no	—
188	C₂×C₄×C₅₃	—	2	2	no	—
189	C₂×C₄×C₅₃	—	2	1	no	—
190	C₂×C₄×C₅₃	—	2	2	no	—
191	C₂×C₄×C₅₃	—	2	2	no	—
192	C₂×C₄×C₅₃	—	2	2	no	—
193	C₂×C₄×C₅₃	—	2	2	no	—
194	C₂×C₄×C₅₃	—	2	2	no	—
195	C₂×C₄×C₅₃	—	4	1	no	—
196	C₂×C₄×C₅₃	—	4	1	no	—
197	C₂×C₄×C₅₃	—	4	1	no	—
198	C₂×C₄×C₅₃	—	4	1	no	—
199	C₂×C₄×C₅₃	—	4	8	no	—
200	C₂×C₄×C₅₃	—	4	1	no	—
201	C₂×C₄×C₅₃	—	4	2	no	—
202	C₂×C₄×C₅₃	—	4	1	no	—
203	C₂×C₄×C₅₃	—	4	2	no	—
204	C₂×C₄×C₅₃	—	4	1	no	—
205	C₂×C₄×C₅₃	—	4	2	no	—
206	C₂×C₄×C₅₃	—	8	8	no	—
207	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
208	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
209	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
210	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
211	C₂×C₄×C₅₃	#135	1	2	yes	—
212	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
214	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
217	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
218	C₈×C₅₃	—	2	4	no	—
219	C₈×C₅₃	—	2	2	no	—
220	C₈×C₅₃	—	4	1	no	—
221	#23093	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂²×C₅₃	—	4	4	no	—
223	C₂²×C₅₃	—	4	2	no	—
224	C₂²×C₅₃	—	4	2	no	—
225	C₂²×C₅₃	—	4	4	no	—
226	C₂²×C₅₃	—	4	4	no	—
227	C₂²×C₅₃	—	4	2	no	—
228	C₂²×C₅₃	—	8	2	no	—
229	C₂²×C₅₃	—	8	1	no	—
230	C₂²×C₅₃	—	8	1	no	—
231	C₂²×C₅₃	—	8	2	no	—
232	C₂²×C₅₃	—	8	2	no	—
233	C₂²×C₅₃	—	8	1	no	—
234	C₂²×C₅₃	—	8	2	no	—
235	C₂²×C₅₃	—	8	1	no	—
236	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
237	C₂²×C₅₃	#546	1	1	yes	characteristic
238	C₂²×C₅₃	#550	1	1	yes	characteristic
239	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
240	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
241	C₂²×C₅₃	#576	1	1	yes	characteristic
242	C₂²×C₅₃	#577	1	1	yes	characteristic
243	C₄×C₅₃	—	2	2	no	—
244	C₄×C₅₃	—	2	4	no	—
245	C₄×C₅₃	—	4	2	no	—
246	C₄×C₅₃	—	4	2	no	—
247	C₄×C₅₃	—	4	2	no	—
248	C₄×C₅₃	—	8	1	no	—
249	C₄×C₅₃	—	8	2	no	—
250	C₄×C₅₃	C₄×D₈	1	4	yes	—
251	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
252	C₄×C₅₃	#518	1	1	yes	characteristic
253	C₄×C₅₃	#553	1	1	yes	characteristic
254	C₄×C₅₃	#555	1	1	yes	characteristic
255	C₂×#459	C₂×C₅₃	1	1	yes	characteristic
256	#2366	C₂×C₅₃	1	1	yes	characteristic, complemented
257	#2354	C₂×C₅₃	1	1	yes	characteristic, complemented
258	#1970	C₂×C₅₃	1	2	yes	complemented
259	#1957	C₂×C₅₃	1	1	yes	characteristic, complemented
260	#1947	C₂×C₅₃	1	1	yes	characteristic, complemented
261	#1925	C₂×C₅₃	1	2	yes	complemented
262	#1879	C₂×C₅₃	1	2	yes	complemented
263	C₈×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
264	C₄×#102	C₂×C₅₃	1	2	yes	complemented
265	C₄×#102	C₂×C₅₃	1	1	yes	characteristic, complemented
266	C₂×C₅₃	—	8	2	no	—
267	C₂×C₅₃	—	8	4	no	—
268	C₂×C₅₃	#2363	1	1	yes	characteristic
269	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
270	C₂×C₅₃	#2946	1	1	yes	characteristic
271	C₂×C₅₃	#3294	1	1	yes	characteristic
272	C₂×C₅₃	#3305	1	1	yes	characteristic
273	C₂×C₅₃	#3391	1	1	yes	characteristic
274	C₂×C₅₃	#3394	1	1	yes	characteristic
275	C₂²×D₈	C₂²×C₅₃	1	2	yes	—
276	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
277	C₂×C₄×D₄	—	2	1	no	—
278	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
279	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
280	C₂×#117	—	2	1	no	—
281	#459	C₄×C₅₃	1	8	yes	complemented
282	C₂×#107	C₂²×C₅₃	1	2	yes	—
283	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
284	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
285	C₂×#102	C₂²×C₅₃	1	2	yes	—
286	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
287	C₂×#102	C₂²×C₅₃	1	2	yes	—
288	C₂×#102	C₂²×C₅₃	1	2	yes	—
289	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
290	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
291	#390	—	2	1	no	—
292	#390	C₂²×C₅₃	1	2	yes	—
293	#390	C₂²×C₅₃	1	2	yes	—
294	#388	—	2	2	no	—
295	#386	—	2	2	no	—
296	#384	C₂²×C₅₃	1	1	yes	characteristic
297	#380	—	2	1	no	—
298	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
299	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
300	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
301	#340	C₂²×C₅₃	1	2	yes	—
302	#340	C₂²×C₅₃	1	1	yes	characteristic
303	#340	C₂²×C₅₃	1	1	yes	characteristic
304	#336	C₂²×C₅₃	1	2	yes	—
305	C₅₃	#23093	1	1	yes	characteristic, direct product factor, Sylow
306	C₂²×D₄	—	2	1	no	—
307	C₂²×D₄	—	2	1	no	—
308	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
309	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
310	C₂³×C₄	—	2	1	no	—
311	C₂³×C₄	—	2	2	no	—
312	C₂×D₈	—	4	2	no	—
313	C₂×D₈	—	4	2	no	—
314	C₂×D₈	C₂×C₄×C₅₃	1	8	yes	complemented
315	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
316	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
317	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
318	#127	—	2	2	no	—
319	#127	C₂×C₄×C₅₃	1	4	yes	—
320	C₄×D₄	—	2	4	no	—
321	C₄×D₄	—	2	2	no	—
322	C₄×D₄	—	4	2	no	—
323	C₄×D₄	—	4	4	no	—
324	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	complemented
325	#117	—	4	4	no	—
326	C₂×C₄⋊C₄	—	2	1	no	—
327	C₂×C₄⋊C₄	—	2	1	no	—
328	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
329	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
330	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
331	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
332	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
333	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
334	C₂×C₂²⋊C₄	—	2	1	no	—
335	C₂×C₂²⋊C₄	—	2	2	no	—
336	C₂×C₂²⋊C₄	—	2	2	no	—
337	C₂×C₂²⋊C₄	—	2	2	no	—
338	C₂×C₂²⋊C₄	—	2	2	no	—
339	C₂×C₂²⋊C₄	—	2	2	no	—
340	C₂×C₄²	—	2	2	no	—
341	C₂×C₄²	—	2	1	no	—
342	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
343	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
344	#107	—	4	2	no	—
345	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
346	#102	—	2	4	no	—
347	#102	—	2	2	no	—
348	#102	—	2	2	no	—
349	#102	—	4	2	no	—
350	#102	C₂×C₄×C₅₃	1	8	yes	complemented
351	C₄×C₈	—	2	2	no	—
352	C₄×C₈	—	4	2	no	—
353	#95	—	2	2	no	—
354	#95	—	2	1	no	—
355	#95	D₄×C₅₃	1	2	yes	—
356	C₂⁴	—	2	1	no	—
357	C₂⁴	—	2	2	no	—
358	C₂×D₄	—	2	4	no	—
359	C₂×D₄	—	2	4	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	1	no	—
362	C₂×D₄	—	2	1	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	1	no	—
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	4	2	no	—
371	C₂×D₄	—	4	2	no	—
372	C₂×D₄	—	8	1	no	—
373	C₂×D₄	—	8	1	no	—
374	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
375	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	4	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	4	4	no	—
382	C₂²×C₄	—	4	2	no	—
383	C₂²×C₄	—	4	2	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	—	8	2	no	—
386	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
387	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
390	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
391	D₈	—	4	8	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
400	C₂×C₈	#41×C₅₃	1	2	yes	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	4	2	no	—
409	C₄⋊C₄	—	8	1	no	—
410	C₄⋊C₄	D₈×C₅₃	1	4	yes	complemented
411	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
412	C₄⋊C₄	#41×C₅₃	1	2	yes	—
413	C₂²⋊C₄	—	2	4	no	—
414	C₂²⋊C₄	—	4	4	no	—
415	C₂²⋊C₄	—	4	4	no	—
416	C₂²⋊C₄	—	4	4	no	—
417	C₂²⋊C₄	—	8	2	no	—
418	C₂²⋊C₄	—	8	2	no	—
419	C₄²	—	2	2	no	—
420	C₄²	—	2	2	no	—
421	C₄²	—	4	4	no	—
422	C₄²	—	4	2	no	—
423	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
424	C₄²	#41×C₅₃	1	2	yes	—
425	C₂³	—	2	2	no	—
426	C₂³	—	2	4	no	—
427	C₂³	—	4	2	no	—
428	C₂³	—	4	1	no	—
429	C₂³	—	4	2	no	—
430	C₂³	—	4	1	no	—
431	C₂³	—	4	1	no	—
432	C₂³	—	4	2	no	—
433	C₂³	—	4	1	no	—
434	C₂³	—	4	2	no	—
435	C₂³	—	4	1	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	4	1	no	—
439	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
440	D₄	—	2	2	no	—
441	D₄	—	2	4	no	—
442	D₄	—	2	2	no	—
443	D₄	—	2	4	no	—
444	D₄	—	4	1	no	—
445	D₄	—	4	2	no	—
446	D₄	—	4	1	no	—
447	D₄	—	4	2	no	—
448	D₄	—	4	8	no	—
449	D₄	—	8	4	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	8	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	8	8	no	—
471	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
472	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
473	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
474	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
475	C₂×C₄	#135×C₅₃	1	2	yes	—
476	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
477	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
478	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
479	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
480	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
481	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
482	C₈	—	2	4	no	—
483	C₈	—	2	2	no	—
484	C₈	—	4	1	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	4	no	—
487	C₂²	—	4	4	no	—
488	C₂²	—	4	2	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	4	no	—
491	C₂²	—	8	2	no	—
492	C₂²	—	8	1	no	—
493	C₂²	—	8	2	no	—
494	C₂²	—	8	1	no	—
495	C₂²	—	8	1	no	—
496	C₂²	—	8	2	no	—
497	C₂²	—	8	1	no	—
498	C₂²	—	8	2	no	—
499	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
500	C₂²	C₅₃×#546	1	1	yes	characteristic
501	C₂²	C₅₃×#550	1	1	yes	characteristic
502	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
503	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
504	C₂²	C₅₃×#576	1	1	yes	characteristic
505	C₂²	C₅₃×#577	1	1	yes	characteristic
506	C₄	—	2	2	no	—
507	C₄	—	2	4	no	—
508	C₄	—	4	2	no	—
509	C₄	—	4	2	no	—
510	C₄	—	4	2	no	—
511	C₄	—	8	1	no	—
512	C₄	—	8	2	no	—
513	C₄	C₄×D₈×C₅₃	1	4	yes	—
514	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
515	C₄	C₅₃×#518	1	1	yes	characteristic
516	C₄	C₅₃×#553	1	1	yes	characteristic
517	C₄	C₅₃×#555	1	1	yes	characteristic
518	C₂	—	8	2	no	—
519	C₂	—	8	4	no	—
520	C₂	C₅₃×#2363	1	1	yes	characteristic
521	C₂	C₂×C₄×D₈×C₅₃	1	1	yes	characteristic
522	C₂	C₅₃×#2946	1	1	yes	characteristic
523	C₂	C₅₃×#3294	1	1	yes	characteristic
524	C₂	C₅₃×#3305	1	1	yes	characteristic
525	C₂	C₅₃×#3391	1	1	yes	characteristic
526	C₂	C₅₃×#3394	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 ◁ C₄×C₄⋊C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2366	C₂	semidirect
3. C₅₃×#2354	C₂	semidirect
4. C₅₃×#1970	C₂	semidirect
5. C₅₃×#1957	C₂	semidirect
6. C₅₃×#1947	C₂	semidirect
7. C₅₃×#1925	C₂	semidirect
8. C₅₃×#1879	C₂	semidirect
9. C₈×C₄⋊C₄×C₅₃	C₂	semidirect
10. C₄×#102×C₅₃	C₂	semidirect
11. C₄×#102×C₅₃	C₂	semidirect
18. C₅₃×#459	C₄	semidirect
50. C₂×D₈×C₅₃	C₂×C₄	semidirect
60. C₄×D₄×C₅₃	C₂×C₄	semidirect
81. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
86. #102×C₅₃	C₂×C₄	semidirect
146. C₄⋊C₄×C₅₃	D₈	semidirect
221. #23093	C₅₃	direct (N × H)
256. #2366	C₂×C₅₃	semidirect
257. #2354	C₂×C₅₃	semidirect
258. #1970	C₂×C₅₃	semidirect
259. #1957	C₂×C₅₃	semidirect
260. #1947	C₂×C₅₃	semidirect
261. #1925	C₂×C₅₃	semidirect
262. #1879	C₂×C₅₃	semidirect
263. C₈×C₄⋊C₄	C₂×C₅₃	semidirect
264. C₄×#102	C₂×C₅₃	semidirect
265. C₄×#102	C₂×C₅₃	semidirect
281. #459	C₄×C₅₃	semidirect
305. C₅₃	#23093	direct (N × H)
314. C₂×D₈	C₂×C₄×C₅₃	semidirect
324. C₄×D₄	C₂×C₄×C₅₃	semidirect
345. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
350. #102	C₂×C₄×C₅₃	semidirect
410. C₄⋊C₄	D₈×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
C₅₃×#2366	2
C₅₃×#2354	2
C₅₃×#1970	2
C₅₃×#1957	2
C₅₃×#1947	2
C₅₃×#1925	2
C₅₃×#1879	2
C₈×C₄⋊C₄×C₅₃	2
C₄×#102×C₅₃	2
#23093	53

Subgroups — C53×#23093 #27230605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23093 #27230605