Subgroups — C₅₃×#17630 #27225142

Order 13568.

1066 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3506	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2975	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2349	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1518	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1517	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1513	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1504	C₂	1	4	yes	maximal, complemented
8	C₅₃×#1503	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1502	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#559	—	2	2	no	—
11	C₅₃×#556	C₄	1	2	yes	complemented
12	C₅₃×#555	C₄	1	2	yes	complemented
13	C₅₃×#553	—	2	2	no	—
14	C₅₃×#550	C₄	1	2	yes	complemented
15	C₅₃×#547	C₄	1	2	yes	complemented
16	C₅₃×#538	—	2	2	no	—
17	C₅₃×#535	—	2	1	no	—
18	C₅₃×#533	C₂²	1	1	yes	characteristic
19	C₅₃×#532	C₂²	1	1	yes	characteristic
20	C₅₃×#525	C₂²	1	4	yes	—
21	C₅₃×#520	—	2	1	no	—
22	C₂×C₄×D₄×C₅₃	—	2	1	no	—
23	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#436	—	2	2	no	—
25	C₅₃×#436	—	2	2	no	—
26	C₅₃×#431	C₂²	1	1	yes	characteristic
27	C₅₃×#431	C₂²	1	1	yes	characteristic
28	C₅₃×#407	C₂²	1	4	yes	—
29	C₅₃×#389	—	2	1	no	—
30	C₅₃×#385	—	2	2	no	—
31	C₅₃×#382	C₂²	1	1	yes	characteristic
32	C₅₃×#380	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#95×C₅₃	C₂²	1	2	yes	—
35	C₅₃×#328	C₂²	1	2	yes	—
36	C₅₃×#328	C₂²	1	2	yes	—
37	C₅₃×#328	C₂²	1	4	yes	—
38	C₅₃×#327	C₂²	1	2	yes	—
39	C₅₃×#327	C₂²	1	2	yes	—
40	C₅₃×#327	C₂²	1	4	yes	—
41	C₂×#41×C₅₃	—	2	1	no	—
42	C₂×#41×C₅₃	C₂×C₄	1	4	yes	complemented
43	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	—	2	2	no	—
45	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
46	C₂×M₁₆×C₅₃	—	2	4	no	—
47	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
48	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
49	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
50	#127×C₅₃	—	2	2	no	—
51	#127×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
52	#126×C₅₃	—	2	4	no	—
53	#125×C₅₃	—	2	1	no	—
54	#124×C₅₃	—	2	4	no	—
55	#124×C₅₃	—	2	1	no	—
56	#124×C₅₃	—	2	2	no	—
57	#123×C₅₃	—	2	4	no	—
58	#123×C₅₃	—	2	4	no	—
59	C₂²⋊Q₈×C₅₃	—	2	1	no	—
60	C₂²⋊Q₈×C₅₃	C₂×C₄	1	4	yes	complemented
61	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
62	#121×C₅₃	—	2	4	no	—
63	#121×C₅₃	—	2	2	no	—
64	#121×C₅₃	—	2	2	no	—
65	#121×C₅₃	—	2	1	no	—
66	#121×C₅₃	C₂×C₄	1	4	yes	complemented
67	#121×C₅₃	C₂³	1	2	yes	—
68	#120×C₅₃	—	2	4	no	—
69	C₄×Q₈×C₅₃	—	2	2	no	—
70	C₄×Q₈×C₅₃	—	2	1	no	—
71	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	complemented
72	C₄×D₄×C₅₃	—	2	2	no	—
73	C₄×D₄×C₅₃	—	2	1	no	—
74	C₄×D₄×C₅₃	—	2	4	no	—
75	C₄×D₄×C₅₃	—	2	4	no	—
76	C₄×D₄×C₅₃	—	4	4	no	—
77	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
78	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
79	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	complemented
80	C₄×D₄×C₅₃	D₄	1	2	yes	—
81	C₄×D₄×C₅₃	D₄	1	2	yes	—
82	#117×C₅₃	—	2	2	no	—
83	#117×C₅₃	—	2	1	no	—
84	#117×C₅₃	C₂×C₄	1	4	yes	complemented
85	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
86	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
92	C₂×C₄²×C₅₃	—	2	1	no	—
93	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₅₃	—	2	1	no	—
95	C₄⋊C₈×C₅₃	—	2	1	no	—
96	C₂²⋊C₈×C₅₃	—	2	4	no	—
97	C₂²⋊C₈×C₅₃	C₂³	1	2	yes	—
98	C₂²⋊C₈×C₅₃	C₂³	1	2	yes	—
99	C₄×C₈×C₅₃	—	2	1	no	—
100	C₄×C₈×C₅₃	—	2	1	no	—
101	#95×C₅₃	—	2	2	no	—
102	#95×C₅₃	—	2	1	no	—
103	#95×C₅₃	C₂³	1	1	yes	characteristic
104	#95×C₅₃	C₂³	1	2	yes	—
105	#95×C₅₃	C₂³	1	1	yes	characteristic
106	C₂⁴×C₅₃	C₂×D₄	1	2	yes	—
107	#41×C₅₃	—	4	2	no	—
108	#41×C₅₃	—	4	2	no	—
109	#41×C₅₃	—	8	4	no	—
110	C₂×Q₈×C₅₃	—	2	1	no	—
111	C₂×Q₈×C₅₃	—	2	1	no	—
112	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
113	C₂×D₄×C₅₃	—	2	2	no	—
114	C₂×D₄×C₅₃	—	2	4	no	—
115	C₂×D₄×C₅₃	—	2	4	no	—
116	C₂×D₄×C₅₃	—	2	1	no	—
117	C₂×D₄×C₅₃	—	2	1	no	—
118	C₂×D₄×C₅₃	—	2	2	no	—
119	C₂×D₄×C₅₃	—	2	2	no	—
120	C₂×D₄×C₅₃	C₂²⋊C₄	1	2	yes	—
121	C₂×D₄×C₅₃	C₂²⋊C₄	1	2	yes	—
122	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₅₃	—	2	1	no	—
124	C₂²×C₄×C₅₃	—	2	2	no	—
125	C₂²×C₄×C₅₃	—	2	4	no	—
126	C₂²×C₄×C₅₃	—	2	2	no	—
127	C₂²×C₄×C₅₃	—	2	1	no	—
128	C₂²×C₄×C₅₃	—	4	4	no	—
129	C₂²×C₄×C₅₃	—	4	4	no	—
130	C₂²×C₄×C₅₃	—	8	2	no	—
131	C₂²×C₄×C₅₃	—	8	2	no	—
132	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
133	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
136	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
138	M₁₆×C₅₃	—	8	4	no	—
139	C₂×C₈×C₅₃	—	2	2	no	—
140	C₂×C₈×C₅₃	—	2	2	no	—
141	C₄⋊C₄×C₅₃	—	2	1	no	—
142	C₄⋊C₄×C₅₃	—	2	4	no	—
143	C₄⋊C₄×C₅₃	—	2	1	no	—
144	C₄⋊C₄×C₅₃	—	2	1	no	—
145	C₄⋊C₄×C₅₃	—	2	1	no	—
146	C₄⋊C₄×C₅₃	—	2	2	no	—
147	C₄⋊C₄×C₅₃	—	2	2	no	—
148	C₄⋊C₄×C₅₃	—	4	2	no	—
149	C₄⋊C₄×C₅₃	—	4	2	no	—
150	C₄⋊C₄×C₅₃	—	8	2	no	—
151	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
152	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
153	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
154	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
155	C₂²⋊C₄×C₅₃	—	2	4	no	—
156	C₂²⋊C₄×C₅₃	—	2	4	no	—
157	C₂²⋊C₄×C₅₃	—	2	2	no	—
158	C₂²⋊C₄×C₅₃	—	2	4	no	—
159	C₂²⋊C₄×C₅₃	—	2	2	no	—
160	C₂²⋊C₄×C₅₃	—	2	2	no	—
161	C₂²⋊C₄×C₅₃	—	4	4	no	—
162	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
163	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
164	C₄²×C₅₃	—	2	4	no	—
165	C₄²×C₅₃	—	2	2	no	—
166	C₄²×C₅₃	—	4	2	no	—
167	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
168	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
169	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
171	C₂³×C₅₃	—	2	2	no	—
172	C₂³×C₅₃	—	2	2	no	—
173	C₂³×C₅₃	—	2	2	no	—
174	C₂³×C₅₃	—	4	2	no	—
175	C₂³×C₅₃	—	4	2	no	—
176	C₂³×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
177	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
178	Q₈×C₅₃	—	4	2	no	—
179	Q₈×C₅₃	—	4	2	no	—
180	Q₈×C₅₃	—	8	1	no	—
181	D₄×C₅₃	—	2	4	no	—
182	D₄×C₅₃	—	2	4	no	—
183	D₄×C₅₃	—	4	2	no	—
184	D₄×C₅₃	—	4	2	no	—
185	D₄×C₅₃	—	4	2	no	—
186	D₄×C₅₃	—	4	2	no	—
187	D₄×C₅₃	—	8	1	no	—
188	D₄×C₅₃	—	8	4	no	—
189	D₄×C₅₃	—	8	4	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	—	4	1	no	—
195	C₂×C₄×C₅₃	—	4	4	no	—
196	C₂×C₄×C₅₃	—	4	2	no	—
197	C₂×C₄×C₅₃	—	4	4	no	—
198	C₂×C₄×C₅₃	—	4	1	no	—
199	C₂×C₄×C₅₃	—	4	1	no	—
200	C₂×C₄×C₅₃	—	4	1	no	—
201	C₂×C₄×C₅₃	—	4	4	no	—
202	C₂×C₄×C₅₃	—	4	2	no	—
203	C₂×C₄×C₅₃	—	4	2	no	—
204	C₂×C₄×C₅₃	—	4	1	no	—
205	C₂×C₄×C₅₃	—	4	1	no	—
206	C₂×C₄×C₅₃	—	4	2	no	—
207	C₂×C₄×C₅₃	—	4	1	no	—
208	C₂×C₄×C₅₃	—	8	4	no	—
209	C₂×C₄×C₅₃	—	8	8	no	—
210	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
211	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
216	C₂×C₄×C₅₃	#137	1	2	yes	—
217	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
218	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
219	C₈×C₅₃	—	4	2	no	—
220	C₈×C₅₃	—	4	2	no	—
221	#17630	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂²×C₅₃	—	2	1	no	—
223	C₂²×C₅₃	—	2	1	no	—
224	C₂²×C₅₃	—	2	4	no	—
225	C₂²×C₅₃	—	2	4	no	—
226	C₂²×C₅₃	—	2	4	no	—
227	C₂²×C₅₃	—	4	4	no	—
228	C₂²×C₅₃	—	4	4	no	—
229	C₂²×C₅₃	—	4	2	no	—
230	C₂²×C₅₃	—	4	2	no	—
231	C₂²×C₅₃	—	4	2	no	—
232	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
233	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
234	C₂²×C₅₃	C₂×#137	1	1	yes	characteristic
235	C₄×C₅₃	—	2	1	no	—
236	C₄×C₅₃	—	2	1	no	—
237	C₄×C₅₃	—	2	4	no	—
238	C₄×C₅₃	—	2	1	no	—
239	C₄×C₅₃	—	2	1	no	—
240	C₄×C₅₃	—	4	2	no	—
241	C₄×C₅₃	—	4	2	no	—
242	C₄×C₅₃	—	4	2	no	—
243	C₄×C₅₃	—	4	2	no	—
244	C₄×C₅₃	—	8	1	no	—
245	C₄×C₅₃	—	8	2	no	—
246	C₄×C₅₃	#417	1	2	yes	—
247	C₄×C₅₃	#417	1	2	yes	—
248	#3506	C₂×C₅₃	1	1	yes	characteristic
249	#2975	C₂×C₅₃	1	1	yes	characteristic, complemented
250	#2349	C₂×C₅₃	1	1	yes	characteristic, complemented
251	#1518	C₂×C₅₃	1	1	yes	characteristic, complemented
252	#1517	C₂×C₅₃	1	1	yes	characteristic, complemented
253	#1513	C₂×C₅₃	1	4	yes	complemented
254	#1504	C₂×C₅₃	1	4	yes	complemented
255	#1503	C₂×C₅₃	1	1	yes	characteristic, complemented
256	#1502	C₂×C₅₃	1	1	yes	characteristic, complemented
257	C₂×C₅₃	—	2	1	no	—
258	C₂×C₅₃	—	2	1	no	—
259	C₂×C₅₃	—	4	4	no	—
260	C₂×C₅₃	—	8	2	no	—
261	C₂×C₅₃	#2883	1	1	yes	characteristic
262	C₂×C₅₃	C₂×#417	1	1	yes	characteristic
263	C₂×C₅₃	#2901	1	1	yes	characteristic
264	#559	—	2	2	no	—
265	#556	C₄×C₅₃	1	2	yes	complemented
266	#555	C₄×C₅₃	1	2	yes	complemented
267	#553	—	2	2	no	—
268	#550	C₄×C₅₃	1	2	yes	complemented
269	#547	C₄×C₅₃	1	2	yes	complemented
270	#538	—	2	2	no	—
271	#535	—	2	1	no	—
272	#533	C₂²×C₅₃	1	1	yes	characteristic
273	#532	C₂²×C₅₃	1	1	yes	characteristic
274	#525	C₂²×C₅₃	1	4	yes	—
275	#520	—	2	1	no	—
276	C₂×C₄×D₄	—	2	1	no	—
277	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
278	#436	—	2	2	no	—
279	#436	—	2	2	no	—
280	#431	C₂²×C₅₃	1	1	yes	characteristic
281	#431	C₂²×C₅₃	1	1	yes	characteristic
282	#407	C₂²×C₅₃	1	4	yes	—
283	#389	—	2	1	no	—
284	#385	—	2	2	no	—
285	#382	C₂²×C₅₃	1	1	yes	characteristic
286	#380	C₂²×C₅₃	1	2	yes	—
287	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
288	C₂×#95	C₂²×C₅₃	1	2	yes	—
289	#328	C₂²×C₅₃	1	2	yes	—
290	#328	C₂²×C₅₃	1	2	yes	—
291	#328	C₂²×C₅₃	1	4	yes	—
292	#327	C₂²×C₅₃	1	2	yes	—
293	#327	C₂²×C₅₃	1	2	yes	—
294	#327	C₂²×C₅₃	1	4	yes	—
295	C₅₃	#17630	1	1	yes	characteristic, direct product factor, Sylow
296	C₂×#41	—	2	1	no	—
297	C₂×#41	C₂×C₄×C₅₃	1	4	yes	complemented
298	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
299	C₂³×C₄	—	2	2	no	—
300	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
301	C₂×M₁₆	—	2	4	no	—
302	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
303	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
304	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
305	#127	—	2	2	no	—
306	#127	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
307	#126	—	2	4	no	—
308	#125	—	2	1	no	—
309	#124	—	2	2	no	—
310	#124	—	2	4	no	—
311	#124	—	2	1	no	—
312	#123	—	2	4	no	—
313	#123	—	2	4	no	—
314	C₂²⋊Q₈	—	2	1	no	—
315	C₂²⋊Q₈	C₂×C₄×C₅₃	1	4	yes	complemented
316	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
317	#121	—	2	2	no	—
318	#121	—	2	2	no	—
319	#121	—	2	1	no	—
320	#121	—	2	4	no	—
321	#121	C₂×C₄×C₅₃	1	4	yes	complemented
322	#121	C₂³×C₅₃	1	2	yes	—
323	#120	—	2	4	no	—
324	C₄×Q₈	—	2	2	no	—
325	C₄×Q₈	—	2	1	no	—
326	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	complemented
327	C₄×D₄	—	2	4	no	—
328	C₄×D₄	—	2	4	no	—
329	C₄×D₄	—	2	2	no	—
330	C₄×D₄	—	2	1	no	—
331	C₄×D₄	—	4	4	no	—
332	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
333	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
334	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	complemented
335	C₄×D₄	D₄×C₅₃	1	2	yes	—
336	C₄×D₄	D₄×C₅₃	1	2	yes	—
337	#117	—	2	1	no	—
338	#117	—	2	2	no	—
339	#117	C₂×C₄×C₅₃	1	4	yes	complemented
340	C₂×C₄⋊C₄	—	2	2	no	—
341	C₂×C₄⋊C₄	—	2	1	no	—
342	C₂×C₄⋊C₄	—	2	1	no	—
343	C₂×C₄⋊C₄	—	2	2	no	—
344	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
345	C₂×C₂²⋊C₄	—	2	2	no	—
346	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
347	C₂×C₄²	—	2	1	no	—
348	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
349	C₄⋊C₈	—	2	1	no	—
350	C₄⋊C₈	—	2	1	no	—
351	C₂²⋊C₈	—	2	4	no	—
352	C₂²⋊C₈	C₂³×C₅₃	1	2	yes	—
353	C₂²⋊C₈	C₂³×C₅₃	1	2	yes	—
354	C₄×C₈	—	2	1	no	—
355	C₄×C₈	—	2	1	no	—
356	#95	—	2	2	no	—
357	#95	—	2	1	no	—
358	#95	C₂³×C₅₃	1	1	yes	characteristic
359	#95	C₂³×C₅₃	1	2	yes	—
360	#95	C₂³×C₅₃	1	1	yes	characteristic
361	C₂⁴	C₂×D₄×C₅₃	1	2	yes	—
362	#41	—	4	2	no	—
363	#41	—	4	2	no	—
364	#41	—	8	4	no	—
365	C₂×Q₈	—	2	1	no	—
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	4	no	—
370	C₂×D₄	—	2	4	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	C₂²⋊C₄×C₅₃	1	2	yes	—
376	C₂×D₄	C₂²⋊C₄×C₅₃	1	2	yes	—
377	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	4	4	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	—	8	2	no	—
386	C₂²×C₄	—	8	2	no	—
387	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
388	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
389	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
390	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
391	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
392	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
393	M₁₆	—	8	4	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	4	2	no	—
404	C₄⋊C₄	—	4	2	no	—
405	C₄⋊C₄	—	8	2	no	—
406	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
407	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
408	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
409	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
410	C₂²⋊C₄	—	2	4	no	—
411	C₂²⋊C₄	—	2	4	no	—
412	C₂²⋊C₄	—	2	2	no	—
413	C₂²⋊C₄	—	2	4	no	—
414	C₂²⋊C₄	—	2	2	no	—
415	C₂²⋊C₄	—	2	2	no	—
416	C₂²⋊C₄	—	4	4	no	—
417	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
418	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
419	C₄²	—	2	4	no	—
420	C₄²	—	2	2	no	—
421	C₄²	—	4	2	no	—
422	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
423	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
424	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
425	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
426	C₂³	—	2	2	no	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	2	no	—
429	C₂³	—	4	2	no	—
430	C₂³	—	4	2	no	—
431	C₂³	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
432	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
433	Q₈	—	4	2	no	—
434	Q₈	—	4	2	no	—
435	Q₈	—	8	1	no	—
436	D₄	—	2	4	no	—
437	D₄	—	2	4	no	—
438	D₄	—	4	2	no	—
439	D₄	—	4	2	no	—
440	D₄	—	4	2	no	—
441	D₄	—	4	2	no	—
442	D₄	—	8	1	no	—
443	D₄	—	8	4	no	—
444	D₄	—	8	4	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	4	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	4	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	8	4	no	—
464	C₂×C₄	—	8	8	no	—
465	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
466	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
467	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
468	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
469	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
470	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
471	C₂×C₄	#137×C₅₃	1	2	yes	—
472	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
473	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
474	C₈	—	4	2	no	—
475	C₈	—	4	2	no	—
476	C₂²	—	2	1	no	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	4	no	—
479	C₂²	—	2	4	no	—
480	C₂²	—	2	4	no	—
481	C₂²	—	4	4	no	—
482	C₂²	—	4	4	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
487	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
488	C₂²	C₂×#137×C₅₃	1	1	yes	characteristic
489	C₄	—	2	1	no	—
490	C₄	—	2	1	no	—
491	C₄	—	2	4	no	—
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	4	2	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	8	2	no	—
499	C₄	—	8	1	no	—
500	C₄	C₅₃×#417	1	2	yes	—
501	C₄	C₅₃×#417	1	2	yes	—
502	C₂	—	2	1	no	—
503	C₂	—	2	1	no	—
504	C₂	—	4	4	no	—
505	C₂	—	8	2	no	—
506	C₂	C₅₃×#2883	1	1	yes	characteristic
507	C₂	C₂×C₅₃×#417	1	1	yes	characteristic
508	C₂	C₅₃×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₅₃×#3506	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄×D₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2975	C₂	semidirect
3. C₅₃×#2349	C₂	semidirect
4. C₅₃×#1518	C₂	semidirect
5. C₅₃×#1517	C₂	semidirect
6. C₅₃×#1513	C₂	semidirect
7. C₅₃×#1504	C₂	semidirect
8. C₅₃×#1503	C₂	semidirect
9. C₅₃×#1502	C₂	semidirect
11. C₅₃×#556	C₄	semidirect
12. C₅₃×#555	C₄	semidirect
14. C₅₃×#550	C₄	semidirect
15. C₅₃×#547	C₄	semidirect
42. C₂×#41×C₅₃	C₂×C₄	semidirect
47. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
48. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
49. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
51. #127×C₅₃	C₂×C₄	semidirect
60. C₂²⋊Q₈×C₅₃	C₂×C₄	semidirect
66. #121×C₅₃	C₂×C₄	semidirect
71. C₄×Q₈×C₅₃	C₂×C₄	semidirect
79. C₄×D₄×C₅₃	C₂×C₄	semidirect
84. #117×C₅₃	C₂×C₄	semidirect
221. #17630	C₅₃	direct (N × H)
249. #2975	C₂×C₅₃	semidirect
250. #2349	C₂×C₅₃	semidirect
251. #1518	C₂×C₅₃	semidirect
252. #1517	C₂×C₅₃	semidirect
253. #1513	C₂×C₅₃	semidirect
254. #1504	C₂×C₅₃	semidirect
255. #1503	C₂×C₅₃	semidirect
256. #1502	C₂×C₅₃	semidirect
265. #556	C₄×C₅₃	semidirect
266. #555	C₄×C₅₃	semidirect
268. #550	C₄×C₅₃	semidirect
269. #547	C₄×C₅₃	semidirect
295. C₅₃	#17630	direct (N × H)
297. C₂×#41	C₂×C₄×C₅₃	semidirect
302. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
303. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
304. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
306. #127	C₂×C₄×C₅₃	semidirect
315. C₂²⋊Q₈	C₂×C₄×C₅₃	semidirect
321. #121	C₂×C₄×C₅₃	semidirect
326. C₄×Q₈	C₂×C₄×C₅₃	semidirect
334. C₄×D₄	C₂×C₄×C₅₃	semidirect
339. #117	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
C₅₃×#3506	2
C₅₃×#2975	2
C₅₃×#2349	2
C₅₃×#1518	2
C₅₃×#1517	2
C₅₃×#1513	2
C₅₃×#1504	2
C₅₃×#1503	2
C₅₃×#1502	2
#17630	53

Subgroups — C53×#17630 #27225142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17630 #27225142