Subgroups — C₂×C₅₃×#1581 #27225371

Order 13568.

838 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1581	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₅₃	C₄	1	1	yes	characteristic
10	C₂²×M₁₆×C₅₃	—	2	1	no	—
11	C₂³×C₈×C₅₃	—	2	1	no	—
12	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×D₈×C₅₃	C₄	1	4	yes	—
15	C₄×D₈×C₅₃	C₂²	1	4	yes	—
16	C₅₃×#436	—	2	2	no	—
17	C₅₃×#436	C₂²	1	4	yes	complemented
18	C₈×D₄×C₅₃	—	2	2	no	—
19	C₈×D₄×C₅₃	C₂²	1	4	yes	complemented
20	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#102×C₅₃	C₄	1	1	yes	characteristic
24	C₂×#102×C₅₃	C₄	1	1	yes	characteristic
25	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
26	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#335	C₂²	1	4	yes	complemented
31	C₅₃×D₄⋊C₈	C₂²	1	4	yes	complemented
32	C₅₃×D₄⋊C₈	C₂²	1	4	yes	complemented
33	C₅₃×#322	C₂²	1	4	yes	complemented
34	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	—	2	1	no	—
37	C₂³×C₄×C₅₃	—	2	1	no	—
38	C₂×D₈×C₅₃	—	2	4	no	—
39	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
40	C₂×M₁₆×C₅₃	—	2	4	no	—
41	C₂×M₁₆×C₅₃	—	4	4	no	—
42	C₂²×C₈×C₅₃	—	2	4	no	—
43	C₂²×C₈×C₅₃	—	2	1	no	—
44	C₂²×C₈×C₅₃	—	2	1	no	—
45	C₂²×C₈×C₅₃	—	2	1	no	—
46	C₂²×C₈×C₅₃	—	4	4	no	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₄×D₄×C₅₃	—	2	1	no	—
52	C₄×D₄×C₅₃	—	2	1	no	—
53	C₄×D₄×C₅₃	—	2	1	no	—
54	C₄×D₄×C₅₃	—	2	1	no	—
55	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
56	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
57	C₄×D₄×C₅₃	C₂³	1	2	yes	—
58	C₄×D₄×C₅₃	C₂³	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×C₂²⋊C₄×C₅₃	—	2	1	no	—
62	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	#107×C₅₃	C₂×C₄	1	4	yes	—
65	C₄⋊C₈×C₅₃	—	2	1	no	—
66	C₄⋊C₈×C₅₃	—	2	1	no	—
67	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
68	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
69	#102×C₅₃	C₂×C₄	1	4	yes	—
70	#102×C₅₃	C₂×C₄	1	4	yes	—
71	C₂²⋊C₈×C₅₃	—	2	4	no	—
72	C₂²⋊C₈×C₅₃	—	2	4	no	—
73	C₄×C₈×C₅₃	—	2	1	no	—
74	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₅₃	D₄	1	2	yes	—
76	C₄×C₈×C₅₃	C₂³	1	2	yes	—
77	C₄×C₈×C₅₃	C₂³	1	2	yes	—
78	C₄×C₈×C₅₃	C₂³	1	2	yes	—
79	C₂⁴×C₅₃	—	2	1	no	—
80	C₂⁴×C₅₃	—	2	1	no	—
81	C₂×D₄×C₅₃	—	2	1	no	—
82	C₂×D₄×C₅₃	—	2	1	no	—
83	C₂×D₄×C₅₃	—	2	2	no	—
84	C₂×D₄×C₅₃	—	2	2	no	—
85	C₂×D₄×C₅₃	—	2	2	no	—
86	C₂×D₄×C₅₃	M₁₆	1	4	yes	complemented
87	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅₃	—	2	1	no	—
90	C₂²×C₄×C₅₃	—	2	1	no	—
91	C₂²×C₄×C₅₃	—	2	2	no	—
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	—	4	2	no	—
96	C₂²×C₄×C₅₃	—	4	2	no	—
97	C₂²×C₄×C₅₃	—	4	2	no	—
98	C₂²×C₄×C₅₃	—	4	2	no	—
99	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
102	D₈×C₅₃	—	2	8	no	—
103	M₁₆×C₅₃	—	4	8	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	—	4	2	no	—
111	C₂×C₈×C₅₃	—	4	2	no	—
112	C₂×C₈×C₅₃	—	4	2	no	—
113	C₂×C₈×C₅₃	—	4	8	no	—
114	C₂×C₈×C₅₃	D₈	1	4	yes	complemented
115	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
116	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
119	C₄⋊C₄×C₅₃	—	2	1	no	—
120	C₄⋊C₄×C₅₃	—	2	1	no	—
121	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
122	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
123	C₂²⋊C₄×C₅₃	—	2	2	no	—
124	C₂²⋊C₄×C₅₃	—	2	2	no	—
125	C₂²⋊C₄×C₅₃	—	2	2	no	—
126	C₂²⋊C₄×C₅₃	—	2	2	no	—
127	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₅₃	#41	1	1	yes	characteristic
130	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
131	C₂³×C₅₃	—	2	2	no	—
132	C₂³×C₅₃	—	2	4	no	—
133	C₂³×C₅₃	—	2	2	no	—
134	C₂³×C₅₃	—	4	2	no	—
135	C₂³×C₅₃	—	4	2	no	—
136	C₂³×C₅₃	—	4	2	no	—
137	C₂³×C₅₃	—	4	2	no	—
138	C₂³×C₅₃	—	4	2	no	—
139	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
140	D₄×C₅₃	—	2	2	no	—
141	D₄×C₅₃	—	2	2	no	—
142	D₄×C₅₃	—	2	2	no	—
143	D₄×C₅₃	—	2	4	no	—
144	D₄×C₅₃	—	4	2	no	—
145	D₄×C₅₃	C₂×M₁₆	1	4	yes	complemented
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	2	2	no	—
149	C₂×C₄×C₅₃	—	4	2	no	—
150	C₂×C₄×C₅₃	—	4	2	no	—
151	C₂×C₄×C₅₃	—	4	2	no	—
152	C₂×C₄×C₅₃	—	4	2	no	—
153	C₂×C₄×C₅₃	—	4	2	no	—
154	C₂×C₄×C₅₃	—	4	2	no	—
155	C₂×C₄×C₅₃	—	4	4	no	—
156	C₂×C₄×C₅₃	—	4	4	no	—
157	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
161	C₂×C₄×C₅₃	#131	1	2	yes	—
162	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
163	C₂×C₄×C₅₃	#135	1	2	yes	—
164	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
167	C₈×C₅₃	—	2	2	no	—
168	C₈×C₅₃	—	2	2	no	—
169	C₈×C₅₃	—	2	2	no	—
170	C₈×C₅₃	—	4	2	no	—
171	C₈×C₅₃	—	4	2	no	—
172	C₈×C₅₃	—	4	2	no	—
173	C₈×C₅₃	C₂×D₈	1	4	yes	complemented
174	C₂×#1581	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂²×C₅₃	—	2	4	no	—
176	C₂²×C₅₃	—	4	8	no	—
177	C₂²×C₅₃	—	4	2	no	—
178	C₂²×C₅₃	—	4	2	no	—
179	C₂²×C₅₃	—	4	2	no	—
180	C₂²×C₅₃	—	4	2	no	—
181	C₂²×C₅₃	—	4	2	no	—
182	C₂²×C₅₃	—	8	4	no	—
183	C₂²×C₅₃	#435	1	2	yes	—
184	C₂²×C₅₃	C₄×D₈	1	2	yes	—
185	C₂²×C₅₃	#444	1	2	yes	—
186	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
187	C₄×C₅₃	—	2	1	no	—
188	C₄×C₅₃	—	2	1	no	—
189	C₄×C₅₃	—	4	2	no	—
190	C₄×C₅₃	—	4	2	no	—
191	C₄×C₅₃	#435	1	2	yes	—
192	C₄×C₅₃	C₄×D₈	1	2	yes	—
193	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
194	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
195	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
196	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
197	C₂×C₄×D₈	C₂×C₅₃	1	1	yes	characteristic
198	C₂×#436	C₂×C₅₃	1	1	yes	characteristic, complemented
199	C₂×C₈×D₄	C₂×C₅₃	1	1	yes	characteristic, complemented
200	#1581	C₂×C₅₃	1	8	yes	direct product factor
201	C₂×#335	C₂×C₅₃	1	1	yes	characteristic, complemented
202	C₂×D₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
203	C₂×D₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
204	C₂×#322	C₂×C₅₃	1	1	yes	characteristic, complemented
205	C₂×C₅₃	—	4	4	no	—
206	C₂×C₅₃	—	8	2	no	—
207	C₂×C₅₃	#1581	1	4	yes	direct product factor
208	C₂×C₅₃	C₂×#435	1	1	yes	characteristic
209	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
210	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
211	C₂²×D₈	C₄×C₅₃	1	1	yes	characteristic
212	C₂²×M₁₆	—	2	1	no	—
213	C₂³×C₈	—	2	1	no	—
214	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
215	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
216	C₄×D₈	C₄×C₅₃	1	4	yes	—
217	C₄×D₈	C₂²×C₅₃	1	4	yes	—
218	#436	—	2	2	no	—
219	#436	C₂²×C₅₃	1	4	yes	complemented
220	C₈×D₄	—	2	2	no	—
221	C₈×D₄	C₂²×C₅₃	1	4	yes	complemented
222	C₂×#107	C₄×C₅₃	1	1	yes	characteristic
223	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
224	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
225	C₂×#102	C₄×C₅₃	1	1	yes	characteristic
226	C₂×#102	C₄×C₅₃	1	1	yes	characteristic
227	C₂×C₂²⋊C₈	—	2	1	no	—
228	C₂×C₂²⋊C₈	—	2	1	no	—
229	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
230	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
231	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
232	#335	C₂²×C₅₃	1	4	yes	complemented
233	D₄⋊C₈	C₂²×C₅₃	1	4	yes	complemented
234	D₄⋊C₈	C₂²×C₅₃	1	4	yes	complemented
235	#322	C₂²×C₅₃	1	4	yes	complemented
236	C₅₃	C₂×#1581	1	1	yes	characteristic, direct product factor, Sylow
237	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
238	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
239	C₂³×C₄	—	2	1	no	—
240	C₂³×C₄	—	2	1	no	—
241	C₂×D₈	—	2	4	no	—
242	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	—
243	C₂×M₁₆	—	2	4	no	—
244	C₂×M₁₆	—	4	4	no	—
245	C₂²×C₈	—	2	1	no	—
246	C₂²×C₈	—	2	1	no	—
247	C₂²×C₈	—	2	1	no	—
248	C₂²×C₈	—	2	4	no	—
249	C₂²×C₈	—	4	4	no	—
250	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
251	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
252	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
253	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
254	C₄×D₄	—	2	1	no	—
255	C₄×D₄	—	2	1	no	—
256	C₄×D₄	—	2	1	no	—
257	C₄×D₄	—	2	1	no	—
258	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
259	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
260	C₄×D₄	C₂³×C₅₃	1	2	yes	—
261	C₄×D₄	C₂³×C₅₃	1	2	yes	—
262	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
263	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
264	C₂×C₂²⋊C₄	—	2	1	no	—
265	C₂×C₂²⋊C₄	—	2	1	no	—
266	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
267	#107	C₂×C₄×C₅₃	1	4	yes	—
268	C₄⋊C₈	—	2	1	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
271	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
272	#102	C₂×C₄×C₅₃	1	4	yes	—
273	#102	C₂×C₄×C₅₃	1	4	yes	—
274	C₂²⋊C₈	—	2	4	no	—
275	C₂²⋊C₈	—	2	4	no	—
276	C₄×C₈	—	2	1	no	—
277	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
278	C₄×C₈	D₄×C₅₃	1	2	yes	—
279	C₄×C₈	C₂³×C₅₃	1	2	yes	—
280	C₄×C₈	C₂³×C₅₃	1	2	yes	—
281	C₄×C₈	C₂³×C₅₃	1	2	yes	—
282	C₂⁴	—	2	1	no	—
283	C₂⁴	—	2	1	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	2	1	no	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	2	no	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	M₁₆×C₅₃	1	4	yes	complemented
290	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
291	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
292	C₂²×C₄	—	2	1	no	—
293	C₂²×C₄	—	2	1	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	4	2	no	—
299	C₂²×C₄	—	4	2	no	—
300	C₂²×C₄	—	4	2	no	—
301	C₂²×C₄	—	4	2	no	—
302	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
303	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
304	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
305	D₈	—	2	8	no	—
306	M₁₆	—	4	8	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	4	2	no	—
314	C₂×C₈	—	4	2	no	—
315	C₂×C₈	—	4	2	no	—
316	C₂×C₈	—	4	8	no	—
317	C₂×C₈	D₈×C₅₃	1	4	yes	complemented
318	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
319	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
320	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
321	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
325	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	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	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
331	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
332	C₄²	#41×C₅₃	1	1	yes	characteristic
333	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
334	C₂³	—	2	2	no	—
335	C₂³	—	2	4	no	—
336	C₂³	—	2	2	no	—
337	C₂³	—	4	2	no	—
338	C₂³	—	4	2	no	—
339	C₂³	—	4	2	no	—
340	C₂³	—	4	2	no	—
341	C₂³	—	4	2	no	—
342	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
343	D₄	—	2	2	no	—
344	D₄	—	2	2	no	—
345	D₄	—	2	2	no	—
346	D₄	—	2	4	no	—
347	D₄	—	4	2	no	—
348	D₄	C₂×M₁₆×C₅₃	1	4	yes	complemented
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	4	no	—
359	C₂×C₄	—	4	4	no	—
360	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
361	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
362	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
363	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
364	C₂×C₄	#131×C₅₃	1	2	yes	—
365	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
366	C₂×C₄	#135×C₅₃	1	2	yes	—
367	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
368	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
369	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
370	C₈	—	2	2	no	—
371	C₈	—	2	2	no	—
372	C₈	—	2	2	no	—
373	C₈	—	4	2	no	—
374	C₈	—	4	2	no	—
375	C₈	—	4	2	no	—
376	C₈	C₂×D₈×C₅₃	1	4	yes	complemented
377	C₂²	—	2	4	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	4	2	no	—
380	C₂²	—	4	8	no	—
381	C₂²	—	4	2	no	—
382	C₂²	—	4	2	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	8	4	no	—
385	C₂²	C₅₃×#435	1	2	yes	—
386	C₂²	C₄×D₈×C₅₃	1	2	yes	—
387	C₂²	C₅₃×#444	1	2	yes	—
388	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
389	C₄	—	2	1	no	—
390	C₄	—	2	1	no	—
391	C₄	—	4	2	no	—
392	C₄	—	4	2	no	—
393	C₄	C₅₃×#435	1	2	yes	—
394	C₄	C₄×D₈×C₅₃	1	2	yes	—
395	C₄	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
396	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
397	C₄	C₂²×D₈×C₅₃	1	1	yes	characteristic
398	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
399	C₂	—	4	4	no	—
400	C₂	—	8	2	no	—
401	C₂	C₅₃×#1581	1	4	yes	direct product factor
402	C₂	C₂×C₅₃×#435	1	1	yes	characteristic
403	C₂	C₂×C₄×D₈×C₅₃	1	1	yes	characteristic
404	C₂	C₂×C₅₃×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂²×D₈×C₅₃	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#436	C₂	semidirect
3. C₂×C₈×D₄×C₅₃	C₂	semidirect
4. C₅₃×#1581	C₂	direct (N × H)
5. C₂×C₅₃×#335	C₂	semidirect
6. C₂×C₅₃×D₄⋊C₈	C₂	semidirect
7. C₂×C₅₃×D₄⋊C₈	C₂	semidirect
8. C₂×C₅₃×#322	C₂	semidirect
17. C₅₃×#436	C₂²	semidirect
19. C₈×D₄×C₅₃	C₂²	semidirect
30. C₅₃×#335	C₂²	semidirect
31. C₅₃×D₄⋊C₈	C₂²	semidirect
32. C₅₃×D₄⋊C₈	C₂²	semidirect
33. C₅₃×#322	C₂²	semidirect
86. C₂×D₄×C₅₃	M₁₆	semidirect
114. C₂×C₈×C₅₃	D₈	semidirect
145. D₄×C₅₃	C₂×M₁₆	semidirect
173. C₈×C₅₃	C₂×D₈	semidirect
174. C₂×#1581	C₅₃	direct (N × H)
198. C₂×#436	C₂×C₅₃	semidirect
199. C₂×C₈×D₄	C₂×C₅₃	semidirect
200. #1581	C₂×C₅₃	direct (N × H)
201. C₂×#335	C₂×C₅₃	semidirect
202. C₂×D₄⋊C₈	C₂×C₅₃	semidirect
203. C₂×D₄⋊C₈	C₂×C₅₃	semidirect
204. C₂×#322	C₂×C₅₃	semidirect
207. C₂×C₅₃	#1581	direct (N × H)
219. #436	C₂²×C₅₃	semidirect
221. C₈×D₄	C₂²×C₅₃	semidirect
232. #335	C₂²×C₅₃	semidirect
233. D₄⋊C₈	C₂²×C₅₃	semidirect
234. D₄⋊C₈	C₂²×C₅₃	semidirect
235. #322	C₂²×C₅₃	semidirect
236. C₅₃	C₂×#1581	direct (N × H)
289. C₂×D₄	M₁₆×C₅₃	semidirect
317. C₂×C₈	D₈×C₅₃	semidirect
348. D₄	C₂×M₁₆×C₅₃	semidirect
376. C₈	C₂×D₈×C₅₃	semidirect
401. C₂	C₅₃×#1581	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₅₃	2
C₂×C₅₃×#436	2
C₂×C₈×D₄×C₅₃	2
C₅₃×#1581	2
C₂×C₅₃×#335	2
C₂×C₅₃×D₄⋊C₈	2
C₂×C₅₃×#322	2
C₂×#1581	53

Subgroups — C2×C53×#1581 #27225371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1581 #27225371