Subgroups — C₅₃×#21365 #27228877

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#432	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1948	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1947	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1853	C₂	1	2	yes	maximal
7	C₅₃×#1852	C₂	1	2	yes	maximal
8	C₅₃×#1776	C₂	1	4	yes	maximal, complemented
9	C₂×#117×C₅₃	—	2	1	no	—
10	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₅₃	—	2	1	no	—
13	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#432	C₄	1	8	yes	complemented
15	C₂×#107×C₅₃	C₂²	1	4	yes	—
16	C₂×#106×C₅₃	C₂²	1	4	yes	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×#97×C₅₃	C₂²	1	2	yes	—
20	C₅₃×#384	C₂²	1	1	yes	characteristic
21	C₅₃×#384	C₂²	1	1	yes	characteristic
22	C₅₃×#382	—	2	1	no	—
23	C₅₃×#380	—	2	1	no	—
24	C₅₃×#380	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
26	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
27	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
28	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
31	C₂×#95×C₅₃	—	2	1	no	—
32	C₅₃×#340	C₂²	1	4	yes	—
33	C₅₃×#340	C₂²	1	4	yes	—
34	C₂³×C₄×C₅₃	—	2	1	no	—
35	C₂³×C₄×C₅₃	—	2	1	no	—
36	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
40	#117×C₅₃	—	4	4	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
47	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
48	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
49	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
53	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
54	C₂×C₄²×C₅₃	—	2	2	no	—
55	C₂×C₄²×C₅₃	—	2	2	no	—
56	C₂×C₄²×C₅₃	—	2	2	no	—
57	C₂×C₄²×C₅₃	—	4	4	no	—
58	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
59	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	#107×C₅₃	—	2	8	no	—
65	#106×C₅₃	—	2	8	no	—
66	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	complemented
67	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
68	#97×C₅₃	C₂×C₄	1	8	yes	complemented
69	#95×C₅₃	—	2	2	no	—
70	#95×C₅₃	—	2	2	no	—
71	#95×C₅₃	—	2	2	no	—
72	#95×C₅₃	—	2	2	no	—
73	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	—	2	2	no	—
75	C₂²×C₄×C₅₃	—	2	2	no	—
76	C₂²×C₄×C₅₃	—	2	2	no	—
77	C₂²×C₄×C₅₃	—	2	2	no	—
78	C₂²×C₄×C₅₃	—	2	2	no	—
79	C₂²×C₄×C₅₃	—	2	2	no	—
80	C₂²×C₄×C₅₃	—	2	4	no	—
81	C₂²×C₄×C₅₃	—	2	4	no	—
82	C₂²×C₄×C₅₃	—	2	2	no	—
83	C₂²×C₄×C₅₃	—	2	2	no	—
84	C₂²×C₄×C₅₃	—	2	2	no	—
85	C₂²×C₄×C₅₃	—	2	2	no	—
86	C₂²×C₄×C₅₃	—	4	2	no	—
87	C₂²×C₄×C₅₃	—	4	2	no	—
88	C₂²×C₄×C₅₃	—	4	2	no	—
89	C₂²×C₄×C₅₃	—	4	2	no	—
90	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
91	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
92	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
99	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₂×C₈×C₅₃	—	2	4	no	—
101	C₂×C₈×C₅₃	—	2	4	no	—
102	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
103	C₄⋊C₄×C₅₃	—	2	4	no	—
104	C₄⋊C₄×C₅₃	—	2	4	no	—
105	C₄⋊C₄×C₅₃	—	2	2	no	—
106	C₄⋊C₄×C₅₃	—	2	2	no	—
107	C₄⋊C₄×C₅₃	—	2	4	no	—
108	C₄⋊C₄×C₅₃	—	2	4	no	—
109	C₄⋊C₄×C₅₃	—	4	2	no	—
110	C₄⋊C₄×C₅₃	—	4	2	no	—
111	C₄⋊C₄×C₅₃	—	4	2	no	—
112	C₄⋊C₄×C₅₃	—	4	2	no	—
113	C₄⋊C₄×C₅₃	#41	1	4	yes	—
114	C₄⋊C₄×C₅₃	#41	1	4	yes	—
115	C₂²⋊C₄×C₅₃	—	4	4	no	—
116	C₄²×C₅₃	—	4	4	no	—
117	C₄²×C₅₃	—	4	4	no	—
118	C₄²×C₅₃	—	4	4	no	—
119	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
120	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
121	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
122	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
123	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
124	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
125	C₂³×C₅₃	—	2	1	no	—
126	C₂³×C₅₃	—	2	1	no	—
127	C₂³×C₅₃	—	2	1	no	—
128	C₂³×C₅₃	—	2	1	no	—
129	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂³×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
131	C₂³×C₅₃	#137	1	2	yes	—
132	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	1	no	—
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	—	2	1	no	—
139	C₂×C₄×C₅₃	—	2	1	no	—
140	C₂×C₄×C₅₃	—	2	1	no	—
141	C₂×C₄×C₅₃	—	2	8	no	—
142	C₂×C₄×C₅₃	—	2	4	no	—
143	C₂×C₄×C₅₃	—	2	4	no	—
144	C₂×C₄×C₅₃	—	2	8	no	—
145	C₂×C₄×C₅₃	—	2	4	no	—
146	C₂×C₄×C₅₃	—	2	4	no	—
147	C₂×C₄×C₅₃	—	4	8	no	—
148	C₂×C₄×C₅₃	—	4	8	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	2	no	—
156	C₂×C₄×C₅₃	—	4	2	no	—
157	C₂×C₄×C₅₃	—	4	2	no	—
158	C₂×C₄×C₅₃	—	4	2	no	—
159	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
160	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
161	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
162	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
163	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
165	C₂×C₄×C₅₃	#137	1	2	yes	—
166	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
169	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
170	C₈×C₅₃	—	2	8	no	—
171	#21365	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂²×C₅₃	—	2	2	no	—
173	C₂²×C₅₃	—	2	2	no	—
174	C₂²×C₅₃	—	2	2	no	—
175	C₂²×C₅₃	—	2	2	no	—
176	C₂²×C₅₃	—	2	2	no	—
177	C₂²×C₅₃	—	2	2	no	—
178	C₂²×C₅₃	#428	1	4	yes	—
179	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
180	C₂²×C₅₃	#525	1	1	yes	characteristic
181	C₂²×C₅₃	#525	1	1	yes	characteristic
182	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₂²×C₅₃	C₂×#137	1	1	yes	characteristic
185	C₂²×C₅₃	C₂×#137	1	1	yes	characteristic
186	C₄×C₅₃	—	2	2	no	—
187	C₄×C₅₃	—	2	2	no	—
188	C₄×C₅₃	—	2	2	no	—
189	C₄×C₅₃	—	2	2	no	—
190	C₄×C₅₃	—	4	4	no	—
191	C₄×C₅₃	—	4	4	no	—
192	C₄×C₅₃	—	8	2	no	—
193	C₄×C₅₃	—	8	2	no	—
194	C₄×C₅₃	#428	1	4	yes	—
195	C₄×C₅₃	C₄×#41	1	2	yes	—
196	C₄×C₅₃	#533	1	2	yes	—
197	C₂×#432	C₂×C₅₃	1	1	yes	characteristic
198	#2285	C₂×C₅₃	1	1	yes	characteristic
199	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
200	#1948	C₂×C₅₃	1	2	yes	complemented
201	#1947	C₂×C₅₃	1	2	yes	complemented
202	#1853	C₂×C₅₃	1	2	yes	—
203	#1852	C₂×C₅₃	1	2	yes	—
204	#1776	C₂×C₅₃	1	4	yes	complemented
205	C₂×C₅₃	—	2	4	no	—
206	C₂×C₅₃	#2296	1	1	yes	characteristic
207	C₂×C₅₃	C₂×#428	1	1	yes	characteristic
208	C₂×C₅₃	C₂×#428	1	1	yes	characteristic
209	C₂×C₅₃	#3109	1	1	yes	characteristic
210	C₂×C₅₃	#3110	1	1	yes	characteristic
211	C₂×C₅₃	#3112	1	1	yes	characteristic
212	C₂×C₅₃	#3113	1	1	yes	characteristic
213	C₂×#117	—	2	1	no	—
214	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
215	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
216	C₂²×C₄²	—	2	1	no	—
217	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
218	#432	C₄×C₅₃	1	8	yes	complemented
219	C₂×#107	C₂²×C₅₃	1	4	yes	—
220	C₂×#106	C₂²×C₅₃	1	4	yes	—
221	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
222	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
223	C₂×#97	C₂²×C₅₃	1	2	yes	—
224	#384	C₂²×C₅₃	1	1	yes	characteristic
225	#384	C₂²×C₅₃	1	1	yes	characteristic
226	#382	—	2	1	no	—
227	#380	—	2	1	no	—
228	#380	C₂²×C₅₃	1	2	yes	—
229	C₄×C₄⋊C₄	—	2	1	no	—
230	C₄×C₄⋊C₄	—	2	1	no	—
231	C₄×C₄⋊C₄	—	2	1	no	—
232	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
233	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
234	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
235	C₂×#95	—	2	1	no	—
236	#340	C₂²×C₅₃	1	4	yes	—
237	#340	C₂²×C₅₃	1	4	yes	—
238	C₅₃	#21365	1	1	yes	characteristic, direct product factor, Sylow
239	C₂³×C₄	—	2	1	no	—
240	C₂³×C₄	—	2	1	no	—
241	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
242	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
243	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
244	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
245	#117	—	4	4	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄⋊C₄	—	2	4	no	—
249	C₂×C₄⋊C₄	—	2	4	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
255	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
256	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
257	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
258	C₂×C₂²⋊C₄	—	2	2	no	—
259	C₂×C₄²	—	2	2	no	—
260	C₂×C₄²	—	2	2	no	—
261	C₂×C₄²	—	2	2	no	—
262	C₂×C₄²	—	4	4	no	—
263	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
264	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
265	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
266	C₂×C₄²	Q₈×C₅₃	1	4	yes	—
267	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
268	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
269	#107	—	2	8	no	—
270	#106	—	2	8	no	—
271	C₄⋊C₈	C₂×C₄×C₅₃	1	8	yes	complemented
272	C₂²⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
273	#97	C₂×C₄×C₅₃	1	8	yes	complemented
274	#95	—	2	2	no	—
275	#95	—	2	2	no	—
276	#95	—	2	2	no	—
277	#95	—	2	2	no	—
278	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	4	no	—
286	C₂²×C₄	—	2	4	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	4	2	no	—
292	C₂²×C₄	—	4	2	no	—
293	C₂²×C₄	—	4	2	no	—
294	C₂²×C₄	—	4	2	no	—
295	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
296	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
297	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
298	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
300	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
301	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
302	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
303	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
304	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
305	C₂×C₈	—	2	4	no	—
306	C₂×C₈	—	2	4	no	—
307	C₂×C₈	C₂²×C₄×C₅₃	1	8	yes	—
308	C₄⋊C₄	—	2	4	no	—
309	C₄⋊C₄	—	2	4	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	4	no	—
313	C₄⋊C₄	—	2	4	no	—
314	C₄⋊C₄	—	4	2	no	—
315	C₄⋊C₄	—	4	2	no	—
316	C₄⋊C₄	—	4	2	no	—
317	C₄⋊C₄	—	4	2	no	—
318	C₄⋊C₄	#41×C₅₃	1	4	yes	—
319	C₄⋊C₄	#41×C₅₃	1	4	yes	—
320	C₂²⋊C₄	—	4	4	no	—
321	C₄²	—	4	4	no	—
322	C₄²	—	4	4	no	—
323	C₄²	—	4	4	no	—
324	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
325	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
326	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
327	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
328	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
329	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
330	C₂³	—	2	1	no	—
331	C₂³	—	2	1	no	—
332	C₂³	—	2	1	no	—
333	C₂³	—	2	1	no	—
334	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
335	C₂³	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
336	C₂³	#137×C₅₃	1	2	yes	—
337	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	8	no	—
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	2	4	no	—
349	C₂×C₄	—	2	8	no	—
350	C₂×C₄	—	2	4	no	—
351	C₂×C₄	—	2	4	no	—
352	C₂×C₄	—	4	8	no	—
353	C₂×C₄	—	4	8	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	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
365	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
366	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
367	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
368	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
369	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
370	C₂×C₄	#137×C₅₃	1	2	yes	—
371	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
372	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
373	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
374	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
375	C₈	—	2	8	no	—
376	C₂²	—	2	2	no	—
377	C₂²	—	2	2	no	—
378	C₂²	—	2	2	no	—
379	C₂²	—	2	2	no	—
380	C₂²	—	2	2	no	—
381	C₂²	—	2	2	no	—
382	C₂²	C₅₃×#428	1	4	yes	—
383	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
384	C₂²	C₅₃×#525	1	1	yes	characteristic
385	C₂²	C₅₃×#525	1	1	yes	characteristic
386	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
387	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
388	C₂²	C₂×#137×C₅₃	1	1	yes	characteristic
389	C₂²	C₂×#137×C₅₃	1	1	yes	characteristic
390	C₄	—	2	2	no	—
391	C₄	—	2	2	no	—
392	C₄	—	2	2	no	—
393	C₄	—	2	2	no	—
394	C₄	—	4	4	no	—
395	C₄	—	4	4	no	—
396	C₄	—	8	2	no	—
397	C₄	—	8	2	no	—
398	C₄	C₅₃×#428	1	4	yes	—
399	C₄	C₄×#41×C₅₃	1	2	yes	—
400	C₄	C₅₃×#533	1	2	yes	—
401	C₂	—	2	4	no	—
402	C₂	C₅₃×#2296	1	1	yes	characteristic
403	C₂	C₂×C₅₃×#428	1	1	yes	characteristic
404	C₂	C₂×C₅₃×#428	1	1	yes	characteristic
405	C₂	C₅₃×#3109	1	1	yes	characteristic
406	C₂	C₅₃×#3110	1	1	yes	characteristic
407	C₂	C₅₃×#3112	1	1	yes	characteristic
408	C₂	C₅₃×#3113	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	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1948	C₂	semidirect
5. C₅₃×#1947	C₂	semidirect
8. C₅₃×#1776	C₂	semidirect
14. C₅₃×#432	C₄	semidirect
66. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
68. #97×C₅₃	C₂×C₄	semidirect
171. #21365	C₅₃	direct (N × H)
200. #1948	C₂×C₅₃	semidirect
201. #1947	C₂×C₅₃	semidirect
204. #1776	C₂×C₅₃	semidirect
218. #432	C₄×C₅₃	semidirect
238. C₅₃	#21365	direct (N × H)
271. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
273. #97	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×#432	2
C₅₃×#2285	2
C₂×C₄×C₄⋊C₄×C₅₃	2
C₅₃×#1948	2
C₅₃×#1947	2
C₅₃×#1853	2
C₅₃×#1852	2
C₅₃×#1776	2
#21365	53

Subgroups — C53×#21365 #27228877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21365 #27228877