Subgroups — C₅₃×#26091 #27233603

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2571	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#2354	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2092	C₂	1	2	yes	maximal
4	C₅₃×#2065	C₂	1	2	yes	maximal
5	C₅₃×#1949	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1939	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1919	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1802	C₂	1	1	yes	maximal, characteristic
11	C₄×#102×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#117×C₅₃	—	2	1	no	—
14	C₂×#107×C₅₃	C₂²	1	2	yes	—
15	C₂×#106×C₅₃	C₂²	1	2	yes	—
16	C₂×#106×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#102×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#400	—	2	2	no	—
24	C₅₃×#390	C₂²	1	2	yes	—
25	C₅₃×#389	—	2	1	no	—
26	C₅₃×#389	C₂²	1	2	yes	—
27	C₅₃×#388	—	2	2	no	—
28	C₅₃×#384	C₂²	1	1	yes	characteristic
29	C₅₃×#384	C₂²	1	1	yes	characteristic
30	C₅₃×#382	—	2	1	no	—
31	C₅₃×#382	C₂²	1	2	yes	—
32	C₅₃×#380	—	2	1	no	—
33	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#340	C₂²	1	1	yes	characteristic
37	C₅₃×#340	C₂²	1	2	yes	—
38	C₅₃×#340	C₂²	1	2	yes	—
39	C₅₃×#340	C₂²	1	1	yes	characteristic
40	C₅₃×#336	C₂²	1	2	yes	—
41	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₄×D₄×C₅₃	—	2	2	no	—
47	C₄×D₄×C₅₃	Q₈	1	4	yes	—
48	#117×C₅₃	—	4	4	no	—
49	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
50	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
62	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
64	C₂×C₄²×C₅₃	—	2	2	no	—
65	C₂×C₄²×C₅₃	—	2	1	no	—
66	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	#107×C₅₃	—	4	2	no	—
69	#106×C₅₃	—	2	4	no	—
70	#106×C₅₃	—	4	2	no	—
71	C₄⋊C₈×C₅₃	—	2	4	no	—
72	C₄⋊C₈×C₅₃	—	2	2	no	—
73	#102×C₅₃	—	2	2	no	—
74	#102×C₅₃	—	2	2	no	—
75	#102×C₅₃	—	4	2	no	—
76	C₄×C₈×C₅₃	—	2	2	no	—
77	#95×C₅₃	—	2	2	no	—
78	#95×C₅₃	—	2	1	no	—
79	#95×C₅₃	—	2	2	no	—
80	#95×C₅₃	—	2	2	no	—
81	#95×C₅₃	—	2	2	no	—
82	#95×C₅₃	D₄	1	2	yes	—
83	C₂⁴×C₅₃	—	2	1	no	—
84	C₂×D₄×C₅₃	—	2	1	no	—
85	C₂×D₄×C₅₃	—	2	1	no	—
86	C₂×D₄×C₅₃	—	2	1	no	—
87	C₂×D₄×C₅₃	—	2	1	no	—
88	C₂×D₄×C₅₃	—	2	1	no	—
89	C₂×D₄×C₅₃	C₂×Q₈	1	2	yes	—
90	C₂²×C₄×C₅₃	—	2	1	no	—
91	C₂²×C₄×C₅₃	—	2	4	no	—
92	C₂²×C₄×C₅₃	—	2	1	no	—
93	C₂²×C₄×C₅₃	—	2	1	no	—
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	—	4	2	no	—
97	C₂²×C₄×C₅₃	—	4	2	no	—
98	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
103	C₂×C₈×C₅₃	—	2	1	no	—
104	C₂×C₈×C₅₃	—	2	1	no	—
105	C₂×C₈×C₅₃	—	2	1	no	—
106	C₂×C₈×C₅₃	—	2	1	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	#41	1	2	yes	—
111	C₂×C₈×C₅₃	#41	1	2	yes	—
112	C₄⋊C₄×C₅₃	—	2	2	no	—
113	C₄⋊C₄×C₅₃	—	2	2	no	—
114	C₄⋊C₄×C₅₃	—	2	1	no	—
115	C₄⋊C₄×C₅₃	—	2	1	no	—
116	C₄⋊C₄×C₅₃	—	2	1	no	—
117	C₄⋊C₄×C₅₃	—	2	2	no	—
118	C₄⋊C₄×C₅₃	—	2	2	no	—
119	C₄⋊C₄×C₅₃	—	2	2	no	—
120	C₄⋊C₄×C₅₃	—	2	2	no	—
121	C₄⋊C₄×C₅₃	—	2	1	no	—
122	C₄⋊C₄×C₅₃	—	2	1	no	—
123	C₄⋊C₄×C₅₃	—	4	2	no	—
124	C₄⋊C₄×C₅₃	—	4	2	no	—
125	C₄⋊C₄×C₅₃	—	8	1	no	—
126	C₄⋊C₄×C₅₃	—	8	1	no	—
127	C₄⋊C₄×C₅₃	—	8	2	no	—
128	C₄⋊C₄×C₅₃	C₂×Q₈	1	2	yes	—
129	C₂²⋊C₄×C₅₃	—	2	4	no	—
130	C₂²⋊C₄×C₅₃	—	4	4	no	—
131	C₂²⋊C₄×C₅₃	—	8	2	no	—
132	C₄²×C₅₃	—	2	2	no	—
133	C₄²×C₅₃	—	2	2	no	—
134	C₄²×C₅₃	—	4	4	no	—
135	C₄²×C₅₃	—	4	2	no	—
136	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
137	C₄²×C₅₃	#41	1	2	yes	—
138	C₂³×C₅₃	—	2	2	no	—
139	C₂³×C₅₃	—	4	1	no	—
140	C₂³×C₅₃	—	4	1	no	—
141	C₂³×C₅₃	—	4	1	no	—
142	C₂³×C₅₃	—	4	1	no	—
143	C₂³×C₅₃	—	4	1	no	—
144	C₂³×C₅₃	—	4	1	no	—
145	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
146	D₄×C₅₃	—	2	2	no	—
147	D₄×C₅₃	—	2	2	no	—
148	D₄×C₅₃	—	4	1	no	—
149	D₄×C₅₃	—	4	1	no	—
150	C₂×C₄×C₅₃	—	2	1	no	—
151	C₂×C₄×C₅₃	—	2	1	no	—
152	C₂×C₄×C₅₃	—	2	2	no	—
153	C₂×C₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	2	2	no	—
158	C₂×C₄×C₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	—	4	1	no	—
160	C₂×C₄×C₅₃	—	4	1	no	—
161	C₂×C₄×C₅₃	—	4	8	no	—
162	C₂×C₄×C₅₃	—	4	1	no	—
163	C₂×C₄×C₅₃	—	4	2	no	—
164	C₂×C₄×C₅₃	—	4	1	no	—
165	C₂×C₄×C₅₃	—	4	2	no	—
166	C₂×C₄×C₅₃	—	4	2	no	—
167	C₂×C₄×C₅₃	—	4	1	no	—
168	C₂×C₄×C₅₃	—	4	1	no	—
169	C₂×C₄×C₅₃	—	4	2	no	—
170	C₂×C₄×C₅₃	—	4	1	no	—
171	C₂×C₄×C₅₃	—	4	2	no	—
172	C₂×C₄×C₅₃	—	4	1	no	—
173	C₂×C₄×C₅₃	—	4	2	no	—
174	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
175	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
176	C₂×C₄×C₅₃	#135	1	2	yes	—
177	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
178	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
182	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
184	C₈×C₅₃	—	2	2	no	—
185	C₈×C₅₃	—	2	2	no	—
186	C₈×C₅₃	—	4	2	no	—
187	#26091	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂²×C₅₃	—	4	2	no	—
189	C₂²×C₅₃	—	4	2	no	—
190	C₂²×C₅₃	—	4	2	no	—
191	C₂²×C₅₃	—	8	1	no	—
192	C₂²×C₅₃	—	8	1	no	—
193	C₂²×C₅₃	—	8	1	no	—
194	C₂²×C₅₃	—	8	1	no	—
195	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
196	C₂²×C₅₃	#547	1	1	yes	characteristic
197	C₂²×C₅₃	#550	1	1	yes	characteristic
198	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
199	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₂²×C₅₃	#577	1	1	yes	characteristic
201	C₂²×C₅₃	#577	1	1	yes	characteristic
202	C₄×C₅₃	—	2	2	no	—
203	C₄×C₅₃	—	4	2	no	—
204	C₄×C₅₃	—	4	2	no	—
205	C₄×C₅₃	—	4	2	no	—
206	C₄×C₅₃	—	8	1	no	—
207	C₄×C₅₃	—	8	2	no	—
208	C₄×C₅₃	—	8	1	no	—
209	C₄×C₅₃	—	8	2	no	—
210	C₄×C₅₃	#478	1	4	yes	—
211	C₄×C₅₃	#533	1	1	yes	characteristic
212	C₄×C₅₃	#544	1	1	yes	characteristic
213	C₄×C₅₃	#552	1	1	yes	characteristic
214	C₄×C₅₃	#553	1	1	yes	characteristic
215	#2571	C₂×C₅₃	1	1	yes	characteristic, complemented
216	#2354	C₂×C₅₃	1	1	yes	characteristic
217	#2092	C₂×C₅₃	1	2	yes	—
218	#2065	C₂×C₅₃	1	2	yes	—
219	#1949	C₂×C₅₃	1	2	yes	complemented
220	#1947	C₂×C₅₃	1	1	yes	characteristic, complemented
221	#1939	C₂×C₅₃	1	1	yes	characteristic, complemented
222	#1919	C₂×C₅₃	1	2	yes	complemented
223	#1844	C₂×C₅₃	1	1	yes	characteristic, complemented
224	#1802	C₂×C₅₃	1	1	yes	characteristic
225	C₄×#102	C₂×C₅₃	1	1	yes	characteristic
226	C₂×C₅₃	—	8	2	no	—
227	C₂×C₅₃	#2441	1	1	yes	characteristic
228	C₂×C₅₃	C₂×#478	1	1	yes	characteristic
229	C₂×C₅₃	#3084	1	1	yes	characteristic
230	C₂×C₅₃	#3340	1	1	yes	characteristic
231	C₂×C₅₃	#3354	1	1	yes	characteristic
232	C₂×C₅₃	#3394	1	1	yes	characteristic
233	C₂×C₅₃	#3397	1	1	yes	characteristic
234	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
235	C₂×#117	—	2	1	no	—
236	C₂×#107	C₂²×C₅₃	1	2	yes	—
237	C₂×#106	C₂²×C₅₃	1	2	yes	—
238	C₂×#106	C₂²×C₅₃	1	2	yes	—
239	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
240	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
241	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
242	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
243	C₂×#102	C₂²×C₅₃	1	2	yes	—
244	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
245	#400	—	2	2	no	—
246	#390	C₂²×C₅₃	1	2	yes	—
247	#389	—	2	1	no	—
248	#389	C₂²×C₅₃	1	2	yes	—
249	#388	—	2	2	no	—
250	#384	C₂²×C₅₃	1	1	yes	characteristic
251	#384	C₂²×C₅₃	1	1	yes	characteristic
252	#382	—	2	1	no	—
253	#382	C₂²×C₅₃	1	2	yes	—
254	#380	—	2	1	no	—
255	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
256	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
257	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
258	#340	C₂²×C₅₃	1	2	yes	—
259	#340	C₂²×C₅₃	1	2	yes	—
260	#340	C₂²×C₅₃	1	1	yes	characteristic
261	#340	C₂²×C₅₃	1	1	yes	characteristic
262	#336	C₂²×C₅₃	1	2	yes	—
263	C₅₃	#26091	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
265	C₂³×C₄	—	2	1	no	—
266	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
267	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
269	C₄×D₄	—	2	2	no	—
270	C₄×D₄	Q₈×C₅₃	1	4	yes	—
271	#117	—	4	4	no	—
272	C₂×C₄⋊C₄	—	2	1	no	—
273	C₂×C₄⋊C₄	—	2	1	no	—
274	C₂×C₄⋊C₄	—	2	1	no	—
275	C₂×C₄⋊C₄	—	2	1	no	—
276	C₂×C₄⋊C₄	—	2	2	no	—
277	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
278	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
279	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
280	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
281	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
282	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
283	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
284	C₂×C₂²⋊C₄	—	2	1	no	—
285	C₂×C₂²⋊C₄	—	2	2	no	—
286	C₂×C₂²⋊C₄	—	2	2	no	—
287	C₂×C₄²	—	2	2	no	—
288	C₂×C₄²	—	2	1	no	—
289	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
290	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
291	#107	—	4	2	no	—
292	#106	—	2	4	no	—
293	#106	—	4	2	no	—
294	C₄⋊C₈	—	2	4	no	—
295	C₄⋊C₈	—	2	2	no	—
296	#102	—	2	2	no	—
297	#102	—	2	2	no	—
298	#102	—	4	2	no	—
299	C₄×C₈	—	2	2	no	—
300	#95	—	2	2	no	—
301	#95	—	2	1	no	—
302	#95	—	2	2	no	—
303	#95	—	2	2	no	—
304	#95	—	2	2	no	—
305	#95	D₄×C₅₃	1	2	yes	—
306	C₂⁴	—	2	1	no	—
307	C₂×D₄	—	2	1	no	—
308	C₂×D₄	—	2	1	no	—
309	C₂×D₄	—	2	1	no	—
310	C₂×D₄	—	2	1	no	—
311	C₂×D₄	—	2	1	no	—
312	C₂×D₄	C₂×Q₈×C₅₃	1	2	yes	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	4	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	4	2	no	—
320	C₂²×C₄	—	4	2	no	—
321	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
322	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
323	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
324	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
325	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	#41×C₅₃	1	2	yes	—
334	C₂×C₈	#41×C₅₃	1	2	yes	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	4	2	no	—
347	C₄⋊C₄	—	4	2	no	—
348	C₄⋊C₄	—	8	1	no	—
349	C₄⋊C₄	—	8	1	no	—
350	C₄⋊C₄	—	8	2	no	—
351	C₄⋊C₄	C₂×Q₈×C₅₃	1	2	yes	—
352	C₂²⋊C₄	—	2	4	no	—
353	C₂²⋊C₄	—	4	4	no	—
354	C₂²⋊C₄	—	8	2	no	—
355	C₄²	—	2	2	no	—
356	C₄²	—	2	2	no	—
357	C₄²	—	4	4	no	—
358	C₄²	—	4	2	no	—
359	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
360	C₄²	#41×C₅₃	1	2	yes	—
361	C₂³	—	2	2	no	—
362	C₂³	—	4	1	no	—
363	C₂³	—	4	1	no	—
364	C₂³	—	4	1	no	—
365	C₂³	—	4	1	no	—
366	C₂³	—	4	1	no	—
367	C₂³	—	4	1	no	—
368	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
369	D₄	—	2	2	no	—
370	D₄	—	2	2	no	—
371	D₄	—	4	1	no	—
372	D₄	—	4	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	8	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
398	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
399	C₂×C₄	#135×C₅₃	1	2	yes	—
400	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
401	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
402	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
403	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
404	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
405	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
406	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
407	C₈	—	2	2	no	—
408	C₈	—	2	2	no	—
409	C₈	—	4	2	no	—
410	C₂²	—	4	2	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	4	2	no	—
413	C₂²	—	8	1	no	—
414	C₂²	—	8	1	no	—
415	C₂²	—	8	1	no	—
416	C₂²	—	8	1	no	—
417	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
418	C₂²	C₅₃×#547	1	1	yes	characteristic
419	C₂²	C₅₃×#550	1	1	yes	characteristic
420	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
421	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
422	C₂²	C₅₃×#577	1	1	yes	characteristic
423	C₂²	C₅₃×#577	1	1	yes	characteristic
424	C₄	—	2	2	no	—
425	C₄	—	4	2	no	—
426	C₄	—	4	2	no	—
427	C₄	—	4	2	no	—
428	C₄	—	8	1	no	—
429	C₄	—	8	2	no	—
430	C₄	—	8	2	no	—
431	C₄	—	8	1	no	—
432	C₄	C₅₃×#478	1	4	yes	—
433	C₄	C₅₃×#533	1	1	yes	characteristic
434	C₄	C₅₃×#544	1	1	yes	characteristic
435	C₄	C₅₃×#552	1	1	yes	characteristic
436	C₄	C₅₃×#553	1	1	yes	characteristic
437	C₂	—	8	2	no	—
438	C₂	C₅₃×#2441	1	1	yes	characteristic
439	C₂	C₂×C₅₃×#478	1	1	yes	characteristic
440	C₂	C₅₃×#3084	1	1	yes	characteristic
441	C₂	C₅₃×#3340	1	1	yes	characteristic
442	C₂	C₅₃×#3354	1	1	yes	characteristic
443	C₂	C₅₃×#3394	1	1	yes	characteristic
444	C₂	C₅₃×#3397	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂³
LCS	G ▷ C₂²×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
1. C₅₃×#2571	C₂	semidirect
5. C₅₃×#1949	C₂	semidirect
6. C₅₃×#1947	C₂	semidirect
7. C₅₃×#1939	C₂	semidirect
8. C₅₃×#1919	C₂	semidirect
9. C₅₃×#1844	C₂	semidirect
187. #26091	C₅₃	direct (N × H)
215. #2571	C₂×C₅₃	semidirect
219. #1949	C₂×C₅₃	semidirect
220. #1947	C₂×C₅₃	semidirect
221. #1939	C₂×C₅₃	semidirect
222. #1919	C₂×C₅₃	semidirect
223. #1844	C₂×C₅₃	semidirect
263. C₅₃	#26091	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2571	2
C₅₃×#2354	2
C₅₃×#2092	2
C₅₃×#2065	2
C₅₃×#1949	2
C₅₃×#1947	2
C₅₃×#1939	2
C₅₃×#1919	2
C₅₃×#1844	2
C₅₃×#1802	2
C₄×#102×C₅₃	2
#26091	53

Subgroups — C53×#26091 #27233603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#26091 #27233603