Subgroups — C₅₃×#17517 #27225029

Order 13568.

1096 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3481	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1494	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1490	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1489	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1484	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1483	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1476	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
12	C₂×#131×C₅₃	—	2	1	no	—
13	C₅₃×#559	C₄	1	1	yes	characteristic
14	C₅₃×#554	C₄	1	1	yes	characteristic
15	C₅₃×#554	C₄	1	1	yes	characteristic
16	C₅₃×#553	—	2	1	no	—
17	C₅₃×#553	C₄	1	2	yes	—
18	C₅₃×#551	—	2	2	no	—
19	C₅₃×#546	—	2	1	no	—
20	C₅₃×#546	C₄	1	2	yes	—
21	D₄²×C₅₃	C₄	1	1	yes	characteristic
22	C₅₃×#532	C₂²	1	1	yes	characteristic
23	C₅₃×#529	—	2	1	no	—
24	C₅₃×#529	C₂²	1	2	yes	—
25	C₅₃×#525	—	2	2	no	—
26	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₅₃	—	2	1	no	—
29	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
30	C₅₃×#446	—	2	1	no	—
31	C₅₃×#435	—	2	2	no	—
32	C₈×D₄×C₅₃	—	2	1	no	—
33	C₅₃×#432	C₂²	1	2	yes	complemented
34	C₅₃×#431	C₂²	1	1	yes	characteristic, complemented
35	C₅₃×#423	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	complemented
37	C₅₃×#405	—	2	2	no	—
38	C₅₃×#405	—	2	1	no	—
39	C₄×M₁₆×C₅₃	C₂²	1	2	yes	complemented
40	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
41	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	complemented
42	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
44	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
45	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
46	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
47	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
48	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
49	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
50	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
51	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
52	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
53	C₂×#41×C₅₃	—	2	1	no	—
54	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
56	C₂²×D₄×C₅₃	—	2	1	no	—
57	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
58	C₂³×C₄×C₅₃	—	2	2	no	—
59	#131×C₅₃	—	2	8	no	—
60	C₂×M₁₆×C₅₃	—	2	2	no	—
61	C₂×M₁₆×C₅₃	—	2	1	no	—
62	C₂×M₁₆×C₅₃	—	2	1	no	—
63	C₂²×C₈×C₅₃	—	2	2	no	—
64	C₂²×C₈×C₅₃	—	2	1	no	—
65	C₂²×C₈×C₅₃	—	2	2	no	—
66	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
68	#126×C₅₃	—	2	2	no	—
69	#126×C₅₃	—	2	2	no	—
70	#125×C₅₃	—	2	1	no	—
71	#125×C₅₃	C₂×C₄	1	2	yes	—
72	#124×C₅₃	—	2	1	no	—
73	#124×C₅₃	—	2	1	no	—
74	#124×C₅₃	C₂×C₄	1	2	yes	—
75	#123×C₅₃	—	2	2	no	—
76	#123×C₅₃	—	2	2	no	—
77	#123×C₅₃	—	2	2	no	—
78	#123×C₅₃	—	2	2	no	—
79	C₂²⋊Q₈×C₅₃	—	2	2	no	—
80	C₂²⋊Q₈×C₅₃	—	2	2	no	—
81	C₂²⋊Q₈×C₅₃	—	2	1	no	—
82	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
83	#121×C₅₃	—	2	1	no	—
84	#121×C₅₃	—	2	2	no	—
85	#121×C₅₃	—	2	2	no	—
86	#121×C₅₃	—	2	1	no	—
87	#121×C₅₃	C₂×C₄	1	2	yes	—
88	#121×C₅₃	C₂×C₄	1	2	yes	—
89	#120×C₅₃	—	2	2	no	—
90	#120×C₅₃	—	2	2	no	—
91	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
94	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₄×D₄×C₅₃	—	2	1	no	—
96	C₄×D₄×C₅₃	—	2	2	no	—
97	C₄×D₄×C₅₃	—	2	1	no	—
98	C₄×D₄×C₅₃	—	2	2	no	—
99	C₄×D₄×C₅₃	—	2	2	no	—
100	C₄×D₄×C₅₃	—	2	1	no	—
101	C₄×D₄×C₅₃	—	2	2	no	—
102	C₄×D₄×C₅₃	—	2	1	no	—
103	C₄×D₄×C₅₃	—	2	1	no	—
104	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
105	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
106	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
107	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
108	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
109	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
110	C₄×D₄×C₅₃	D₄	1	2	yes	—
111	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
112	C₄×D₄×C₅₃	C₂³	1	2	yes	—
113	#117×C₅₃	—	2	1	no	—
114	#117×C₅₃	—	2	2	no	—
115	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
116	#117×C₅₃	D₄	1	2	yes	—
117	#117×C₅₃	D₄	1	1	yes	characteristic
118	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
119	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
120	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
121	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
122	C₂×C₄²×C₅₃	—	2	1	no	—
123	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
124	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
125	C₄⋊C₈×C₅₃	—	2	2	no	—
126	C₄⋊C₈×C₅₃	—	2	1	no	—
127	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
128	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
129	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
130	C₂²⋊C₈×C₅₃	—	2	1	no	—
131	C₂²⋊C₈×C₅₃	—	2	2	no	—
132	#97×C₅₃	—	2	1	no	—
133	#97×C₅₃	—	2	2	no	—
134	#97×C₅₃	—	2	2	no	—
135	C₄×C₈×C₅₃	—	2	1	no	—
136	C₄×C₈×C₅₃	—	2	2	no	—
137	C₄×C₈×C₅₃	—	2	1	no	—
138	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₅₃	C₂³	1	2	yes	—
140	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
141	C₂⁴×C₅₃	—	2	2	no	—
142	#41×C₅₃	—	2	4	no	—
143	#41×C₅₃	—	2	2	no	—
144	#41×C₅₃	—	4	1	no	—
145	#41×C₅₃	—	8	1	no	—
146	#41×C₅₃	—	8	1	no	—
147	C₂×Q₈×C₅₃	—	2	1	no	—
148	C₂×Q₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
149	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₂×D₄×C₅₃	—	2	4	no	—
151	C₂×D₄×C₅₃	—	2	1	no	—
152	C₂×D₄×C₅₃	—	2	2	no	—
153	C₂×D₄×C₅₃	—	2	2	no	—
154	C₂×D₄×C₅₃	—	2	2	no	—
155	C₂×D₄×C₅₃	—	2	1	no	—
156	C₂×D₄×C₅₃	—	2	1	no	—
157	C₂×D₄×C₅₃	—	4	2	no	—
158	C₂×D₄×C₅₃	—	4	4	no	—
159	C₂×D₄×C₅₃	C₂²⋊C₄	1	2	yes	—
160	C₂×D₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
161	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
162	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₅₃	—	2	2	no	—
164	C₂²×C₄×C₅₃	—	2	1	no	—
165	C₂²×C₄×C₅₃	—	2	1	no	—
166	C₂²×C₄×C₅₃	—	2	1	no	—
167	C₂²×C₄×C₅₃	—	2	2	no	—
168	C₂²×C₄×C₅₃	—	2	2	no	—
169	C₂²×C₄×C₅₃	—	2	2	no	—
170	C₂²×C₄×C₅₃	—	8	2	no	—
171	C₂²×C₄×C₅₃	—	8	2	no	—
172	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
173	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
174	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
178	M₁₆×C₅₃	—	2	8	no	—
179	M₁₆×C₅₃	—	2	4	no	—
180	M₁₆×C₅₃	—	8	1	no	—
181	C₂×C₈×C₅₃	—	2	8	no	—
182	C₂×C₈×C₅₃	—	2	4	no	—
183	C₂×C₈×C₅₃	—	2	1	no	—
184	C₂×C₈×C₅₃	—	2	1	no	—
185	C₂×C₈×C₅₃	—	2	2	no	—
186	C₂×C₈×C₅₃	—	2	1	no	—
187	C₂×C₈×C₅₃	—	2	1	no	—
188	C₂×C₈×C₅₃	—	2	2	no	—
189	C₂×C₈×C₅₃	—	8	2	no	—
190	C₄⋊C₄×C₅₃	—	2	1	no	—
191	C₄⋊C₄×C₅₃	—	2	2	no	—
192	C₄⋊C₄×C₅₃	—	2	2	no	—
193	C₄⋊C₄×C₅₃	—	2	1	no	—
194	C₄⋊C₄×C₅₃	—	2	2	no	—
195	C₄⋊C₄×C₅₃	—	2	2	no	—
196	C₄⋊C₄×C₅₃	—	2	1	no	—
197	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
198	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
199	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
200	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
201	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
202	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
203	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
204	C₂²⋊C₄×C₅₃	—	2	2	no	—
205	C₂²⋊C₄×C₅₃	—	2	1	no	—
206	C₂²⋊C₄×C₅₃	—	2	2	no	—
207	C₂²⋊C₄×C₅₃	—	2	2	no	—
208	C₂²⋊C₄×C₅₃	—	2	2	no	—
209	C₂²⋊C₄×C₅₃	—	2	2	no	—
210	C₂²⋊C₄×C₅₃	—	2	1	no	—
211	C₂²⋊C₄×C₅₃	—	2	2	no	—
212	C₂²⋊C₄×C₅₃	—	2	2	no	—
213	C₂²⋊C₄×C₅₃	—	2	1	no	—
214	C₂²⋊C₄×C₅₃	—	2	2	no	—
215	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
216	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
217	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
218	C₄²×C₅₃	—	2	1	no	—
219	C₄²×C₅₃	—	2	1	no	—
220	C₄²×C₅₃	—	2	1	no	—
221	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
222	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
223	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
224	C₄²×C₅₃	C₂×D₄	1	2	yes	—
225	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
226	C₄²×C₅₃	C₂×D₄	1	2	yes	—
227	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
228	C₂³×C₅₃	—	2	1	no	—
229	C₂³×C₅₃	—	2	2	no	—
230	C₂³×C₅₃	—	4	4	no	—
231	C₂³×C₅₃	—	4	4	no	—
232	C₂³×C₅₃	—	8	2	no	—
233	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
234	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
235	Q₈×C₅₃	—	2	2	no	—
236	Q₈×C₅₃	—	2	1	no	—
237	Q₈×C₅₃	—	2	1	no	—
238	Q₈×C₅₃	—	8	1	no	—
239	D₄×C₅₃	—	2	4	no	—
240	D₄×C₅₃	—	2	2	no	—
241	D₄×C₅₃	—	2	1	no	—
242	D₄×C₅₃	—	2	2	no	—
243	D₄×C₅₃	—	4	4	no	—
244	D₄×C₅₃	—	4	2	no	—
245	D₄×C₅₃	—	4	4	no	—
246	D₄×C₅₃	—	4	2	no	—
247	D₄×C₅₃	—	8	1	no	—
248	D₄×C₅₃	—	8	2	no	—
249	D₄×C₅₃	#104	1	4	yes	—
250	D₄×C₅₃	#131	1	2	yes	complemented
251	C₂×C₄×C₅₃	—	2	4	no	—
252	C₂×C₄×C₅₃	—	2	2	no	—
253	C₂×C₄×C₅₃	—	2	1	no	—
254	C₂×C₄×C₅₃	—	2	2	no	—
255	C₂×C₄×C₅₃	—	2	1	no	—
256	C₂×C₄×C₅₃	—	2	2	no	—
257	C₂×C₄×C₅₃	—	2	1	no	—
258	C₂×C₄×C₅₃	—	2	2	no	—
259	C₂×C₄×C₅₃	—	2	1	no	—
260	C₂×C₄×C₅₃	—	2	2	no	—
261	C₂×C₄×C₅₃	—	2	1	no	—
262	C₂×C₄×C₅₃	—	2	2	no	—
263	C₂×C₄×C₅₃	—	2	1	no	—
264	C₂×C₄×C₅₃	—	2	1	no	—
265	C₂×C₄×C₅₃	—	4	2	no	—
266	C₂×C₄×C₅₃	—	4	1	no	—
267	C₂×C₄×C₅₃	—	4	2	no	—
268	C₂×C₄×C₅₃	—	4	1	no	—
269	C₂×C₄×C₅₃	—	4	2	no	—
270	C₂×C₄×C₅₃	—	4	2	no	—
271	C₂×C₄×C₅₃	—	8	2	no	—
272	C₂×C₄×C₅₃	—	8	2	no	—
273	C₂×C₄×C₅₃	—	8	2	no	—
274	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
275	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
277	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
278	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
279	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
280	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
281	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
283	C₈×C₅₃	—	2	2	no	—
284	C₈×C₅₃	—	2	4	no	—
285	C₈×C₅₃	—	2	2	no	—
286	C₈×C₅₃	—	4	1	no	—
287	C₈×C₅₃	—	4	2	no	—
288	C₈×C₅₃	—	4	1	no	—
289	#17517	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
290	C₂²×C₅₃	—	2	2	no	—
291	C₂²×C₅₃	—	2	2	no	—
292	C₂²×C₅₃	—	2	1	no	—
293	C₂²×C₅₃	—	2	2	no	—
294	C₂²×C₅₃	—	2	1	no	—
295	C₂²×C₅₃	—	2	1	no	—
296	C₂²×C₅₃	—	4	1	no	—
297	C₂²×C₅₃	—	4	1	no	—
298	C₂²×C₅₃	—	4	2	no	—
299	C₂²×C₅₃	—	4	2	no	—
300	C₂²×C₅₃	—	4	2	no	—
301	C₂²×C₅₃	—	8	8	no	—
302	C₂²×C₅₃	C₂×#104	1	4	yes	—
303	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
304	C₄×C₅₃	—	2	1	no	—
305	C₄×C₅₃	—	2	2	no	—
306	C₄×C₅₃	—	2	2	no	—
307	C₄×C₅₃	—	2	1	no	—
308	C₄×C₅₃	—	2	1	no	—
309	C₄×C₅₃	—	2	2	no	—
310	C₄×C₅₃	—	4	1	no	—
311	C₄×C₅₃	—	4	1	no	—
312	C₄×C₅₃	—	4	2	no	—
313	C₄×C₅₃	—	4	1	no	—
314	C₄×C₅₃	C₂×#104	1	2	yes	—
315	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
316	C₄×C₅₃	#568	1	1	yes	characteristic
317	C₄×C₅₃	#576	1	1	yes	characteristic
318	C₄×C₅₃	#577	1	1	yes	characteristic
319	#3481	C₂×C₅₃	1	1	yes	characteristic
320	#2990	C₂×C₅₃	1	1	yes	characteristic, complemented
321	C₄×#131	C₂×C₅₃	1	1	yes	characteristic, complemented
322	#1494	C₂×C₅₃	1	2	yes	complemented
323	#1492	C₂×C₅₃	1	1	yes	characteristic, complemented
324	#1490	C₂×C₅₃	1	2	yes	complemented
325	#1489	C₂×C₅₃	1	1	yes	characteristic, complemented
326	#1484	C₂×C₅₃	1	1	yes	characteristic, complemented
327	#1483	C₂×C₅₃	1	2	yes	complemented
328	#1476	C₂×C₅₃	1	1	yes	characteristic, complemented
329	C₂×D₄⋊C₈	C₂×C₅₃	1	2	yes	complemented
330	C₂×C₅₃	—	2	4	no	—
331	C₂×C₅₃	—	4	1	no	—
332	C₂×C₅₃	—	4	2	no	—
333	C₂×C₅₃	—	8	2	no	—
334	C₂×C₅₃	#2880	1	1	yes	characteristic
335	C₂×C₅₃	#2897	1	1	yes	characteristic
336	C₂×C₅₃	C₂²×#104	1	1	yes	characteristic
337	C₂×#131	—	2	1	no	—
338	#559	C₄×C₅₃	1	1	yes	characteristic
339	#554	C₄×C₅₃	1	1	yes	characteristic
340	#554	C₄×C₅₃	1	1	yes	characteristic
341	#553	—	2	1	no	—
342	#553	C₄×C₅₃	1	2	yes	—
343	#551	—	2	2	no	—
344	#546	—	2	1	no	—
345	#546	C₄×C₅₃	1	2	yes	—
346	D₄²	C₄×C₅₃	1	1	yes	characteristic
347	#532	C₂²×C₅₃	1	1	yes	characteristic
348	#529	—	2	1	no	—
349	#529	C₂²×C₅₃	1	2	yes	—
350	#525	—	2	2	no	—
351	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
352	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
353	C₂×C₄×D₄	—	2	1	no	—
354	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
355	#446	—	2	1	no	—
356	#435	—	2	2	no	—
357	C₈×D₄	—	2	1	no	—
358	#432	C₂²×C₅₃	1	2	yes	complemented
359	#431	C₂²×C₅₃	1	1	yes	characteristic, complemented
360	#423	C₂²×C₅₃	1	1	yes	characteristic, complemented
361	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
362	#405	—	2	2	no	—
363	#405	—	2	1	no	—
364	C₄×M₁₆	C₂²×C₅₃	1	2	yes	complemented
365	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic, complemented
366	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	complemented
367	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
368	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
369	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
370	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	complemented
371	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
372	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
373	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
374	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
375	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
376	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
377	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
378	C₅₃	#17517	1	1	yes	characteristic, direct product factor, Sylow
379	C₂×#41	—	2	1	no	—
380	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
381	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
382	C₂²×D₄	—	2	1	no	—
383	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
384	C₂³×C₄	—	2	2	no	—
385	#131	—	2	8	no	—
386	C₂×M₁₆	—	2	2	no	—
387	C₂×M₁₆	—	2	1	no	—
388	C₂×M₁₆	—	2	1	no	—
389	C₂²×C₈	—	2	2	no	—
390	C₂²×C₈	—	2	1	no	—
391	C₂²×C₈	—	2	2	no	—
392	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
393	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
394	#126	—	2	2	no	—
395	#126	—	2	2	no	—
396	#125	—	2	1	no	—
397	#125	C₂×C₄×C₅₃	1	2	yes	—
398	#124	—	2	1	no	—
399	#124	—	2	1	no	—
400	#124	C₂×C₄×C₅₃	1	2	yes	—
401	#123	—	2	2	no	—
402	#123	—	2	2	no	—
403	#123	—	2	2	no	—
404	#123	—	2	2	no	—
405	C₂²⋊Q₈	—	2	1	no	—
406	C₂²⋊Q₈	—	2	2	no	—
407	C₂²⋊Q₈	—	2	2	no	—
408	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
409	#121	—	2	2	no	—
410	#121	—	2	1	no	—
411	#121	—	2	1	no	—
412	#121	—	2	2	no	—
413	#121	C₂×C₄×C₅₃	1	2	yes	—
414	#121	C₂×C₄×C₅₃	1	2	yes	—
415	#120	—	2	2	no	—
416	#120	—	2	2	no	—
417	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
418	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
419	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
420	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
421	C₄×D₄	—	2	1	no	—
422	C₄×D₄	—	2	2	no	—
423	C₄×D₄	—	2	1	no	—
424	C₄×D₄	—	2	1	no	—
425	C₄×D₄	—	2	1	no	—
426	C₄×D₄	—	2	2	no	—
427	C₄×D₄	—	2	1	no	—
428	C₄×D₄	—	2	2	no	—
429	C₄×D₄	—	2	2	no	—
430	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
431	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
432	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
433	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
434	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
435	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
436	C₄×D₄	D₄×C₅₃	1	2	yes	—
437	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
438	C₄×D₄	C₂³×C₅₃	1	2	yes	—
439	#117	—	2	1	no	—
440	#117	—	2	2	no	—
441	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
442	#117	D₄×C₅₃	1	2	yes	—
443	#117	D₄×C₅₃	1	1	yes	characteristic
444	C₂×C₄⋊C₄	—	2	1	no	—
445	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
446	C₂×C₂²⋊C₄	—	2	2	no	—
447	C₂×C₂²⋊C₄	—	2	2	no	—
448	C₂×C₄²	—	2	1	no	—
449	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
450	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
451	C₄⋊C₈	—	2	2	no	—
452	C₄⋊C₈	—	2	1	no	—
453	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
454	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
455	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
456	C₂²⋊C₈	—	2	1	no	—
457	C₂²⋊C₈	—	2	2	no	—
458	#97	—	2	1	no	—
459	#97	—	2	2	no	—
460	#97	—	2	2	no	—
461	C₄×C₈	—	2	1	no	—
462	C₄×C₈	—	2	2	no	—
463	C₄×C₈	—	2	1	no	—
464	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
465	C₄×C₈	C₂³×C₅₃	1	2	yes	—
466	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
467	C₂⁴	—	2	2	no	—
468	#41	—	2	4	no	—
469	#41	—	2	2	no	—
470	#41	—	4	1	no	—
471	#41	—	8	1	no	—
472	#41	—	8	1	no	—
473	C₂×Q₈	—	2	1	no	—
474	C₂×Q₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
475	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
476	C₂×D₄	—	2	4	no	—
477	C₂×D₄	—	2	1	no	—
478	C₂×D₄	—	2	2	no	—
479	C₂×D₄	—	2	2	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	2	1	no	—
482	C₂×D₄	—	2	1	no	—
483	C₂×D₄	—	4	2	no	—
484	C₂×D₄	—	4	4	no	—
485	C₂×D₄	C₂²⋊C₄×C₅₃	1	2	yes	—
486	C₂×D₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
487	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
488	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	2	no	—
494	C₂²×C₄	—	2	2	no	—
495	C₂²×C₄	—	2	2	no	—
496	C₂²×C₄	—	8	2	no	—
497	C₂²×C₄	—	8	2	no	—
498	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
499	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
500	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
501	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
502	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
503	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
504	M₁₆	—	2	8	no	—
505	M₁₆	—	2	4	no	—
506	M₁₆	—	8	1	no	—
507	C₂×C₈	—	2	8	no	—
508	C₂×C₈	—	2	4	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	—	2	2	no	—
512	C₂×C₈	—	2	1	no	—
513	C₂×C₈	—	2	1	no	—
514	C₂×C₈	—	2	2	no	—
515	C₂×C₈	—	8	2	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	2	no	—
518	C₄⋊C₄	—	2	2	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	2	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
524	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
525	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
526	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
527	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
528	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
529	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
530	C₂²⋊C₄	—	2	2	no	—
531	C₂²⋊C₄	—	2	1	no	—
532	C₂²⋊C₄	—	2	2	no	—
533	C₂²⋊C₄	—	2	2	no	—
534	C₂²⋊C₄	—	2	2	no	—
535	C₂²⋊C₄	—	2	2	no	—
536	C₂²⋊C₄	—	2	1	no	—
537	C₂²⋊C₄	—	2	2	no	—
538	C₂²⋊C₄	—	2	2	no	—
539	C₂²⋊C₄	—	2	1	no	—
540	C₂²⋊C₄	—	2	2	no	—
541	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
542	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
543	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
544	C₄²	—	2	1	no	—
545	C₄²	—	2	1	no	—
546	C₄²	—	2	1	no	—
547	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
548	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
549	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
550	C₄²	C₂×D₄×C₅₃	1	2	yes	—
551	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
552	C₄²	C₂×D₄×C₅₃	1	2	yes	—
553	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
554	C₂³	—	2	1	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	4	4	no	—
557	C₂³	—	4	4	no	—
558	C₂³	—	8	2	no	—
559	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
560	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
561	Q₈	—	2	2	no	—
562	Q₈	—	2	1	no	—
563	Q₈	—	2	1	no	—
564	Q₈	—	8	1	no	—
565	D₄	—	2	4	no	—
566	D₄	—	2	2	no	—
567	D₄	—	2	1	no	—
568	D₄	—	2	2	no	—
569	D₄	—	4	4	no	—
570	D₄	—	4	2	no	—
571	D₄	—	4	4	no	—
572	D₄	—	4	2	no	—
573	D₄	—	8	1	no	—
574	D₄	—	8	2	no	—
575	D₄	#104×C₅₃	1	4	yes	—
576	D₄	#131×C₅₃	1	2	yes	complemented
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	4	2	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	2	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	2	no	—
596	C₂×C₄	—	4	2	no	—
597	C₂×C₄	—	8	2	no	—
598	C₂×C₄	—	8	2	no	—
599	C₂×C₄	—	8	2	no	—
600	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
601	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
602	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
603	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
604	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
605	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
606	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
607	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
608	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
609	C₈	—	2	2	no	—
610	C₈	—	2	4	no	—
611	C₈	—	2	2	no	—
612	C₈	—	4	1	no	—
613	C₈	—	4	2	no	—
614	C₈	—	4	1	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	2	2	no	—
617	C₂²	—	2	1	no	—
618	C₂²	—	2	2	no	—
619	C₂²	—	2	1	no	—
620	C₂²	—	2	1	no	—
621	C₂²	—	4	1	no	—
622	C₂²	—	4	1	no	—
623	C₂²	—	4	2	no	—
624	C₂²	—	4	2	no	—
625	C₂²	—	4	2	no	—
626	C₂²	—	8	8	no	—
627	C₂²	C₂×#104×C₅₃	1	4	yes	—
628	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
629	C₄	—	2	1	no	—
630	C₄	—	2	2	no	—
631	C₄	—	2	2	no	—
632	C₄	—	2	1	no	—
633	C₄	—	2	1	no	—
634	C₄	—	2	2	no	—
635	C₄	—	4	1	no	—
636	C₄	—	4	1	no	—
637	C₄	—	4	2	no	—
638	C₄	—	4	1	no	—
639	C₄	C₂×#104×C₅₃	1	2	yes	—
640	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
641	C₄	C₅₃×#568	1	1	yes	characteristic
642	C₄	C₅₃×#576	1	1	yes	characteristic
643	C₄	C₅₃×#577	1	1	yes	characteristic
644	C₂	—	2	4	no	—
645	C₂	—	4	1	no	—
646	C₂	—	4	2	no	—
647	C₂	—	8	2	no	—
648	C₂	C₅₃×#2880	1	1	yes	characteristic
649	C₂	C₅₃×#2897	1	1	yes	characteristic
650	C₂	C₂²×#104×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₅₃×#3481	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2990	C₂	semidirect
3. C₄×#131×C₅₃	C₂	semidirect
4. C₅₃×#1494	C₂	semidirect
5. C₅₃×#1492	C₂	semidirect
6. C₅₃×#1490	C₂	semidirect
7. C₅₃×#1489	C₂	semidirect
8. C₅₃×#1484	C₂	semidirect
9. C₅₃×#1483	C₂	semidirect
10. C₅₃×#1476	C₂	semidirect
11. C₂×C₅₃×D₄⋊C₈	C₂	semidirect
33. C₅₃×#432	C₂²	semidirect
34. C₅₃×#431	C₂²	semidirect
35. C₅₃×#423	C₂²	semidirect
36. C₂×C₄⋊C₈×C₅₃	C₂²	semidirect
39. C₄×M₁₆×C₅₃	C₂²	semidirect
40. C₄×M₁₆×C₅₃	C₂²	semidirect
41. C₂×C₄×C₈×C₅₃	C₂²	semidirect
42. C₂×C₄×C₈×C₅₃	C₂²	semidirect
43. C₅₃×Q₈⋊C₈	C₂²	semidirect
44. C₅₃×Q₈⋊C₈	C₂²	semidirect
45. C₅₃×Q₈⋊C₈	C₂²	semidirect
49. C₅₃×D₄⋊C₈	C₂²	semidirect
50. C₅₃×D₄⋊C₈	C₂²	semidirect
51. C₅₃×D₄⋊C₈	C₂²	semidirect
52. C₅₃×D₄⋊C₈	C₂²	semidirect
250. D₄×C₅₃	#131	semidirect
289. #17517	C₅₃	direct (N × H)
320. #2990	C₂×C₅₃	semidirect
321. C₄×#131	C₂×C₅₃	semidirect
322. #1494	C₂×C₅₃	semidirect
323. #1492	C₂×C₅₃	semidirect
324. #1490	C₂×C₅₃	semidirect
325. #1489	C₂×C₅₃	semidirect
326. #1484	C₂×C₅₃	semidirect
327. #1483	C₂×C₅₃	semidirect
328. #1476	C₂×C₅₃	semidirect
329. C₂×D₄⋊C₈	C₂×C₅₃	semidirect
358. #432	C₂²×C₅₃	semidirect
359. #431	C₂²×C₅₃	semidirect
360. #423	C₂²×C₅₃	semidirect
361. C₂×C₄⋊C₈	C₂²×C₅₃	semidirect
364. C₄×M₁₆	C₂²×C₅₃	semidirect
365. C₄×M₁₆	C₂²×C₅₃	semidirect
366. C₂×C₄×C₈	C₂²×C₅₃	semidirect
367. C₂×C₄×C₈	C₂²×C₅₃	semidirect
368. Q₈⋊C₈	C₂²×C₅₃	semidirect
369. Q₈⋊C₈	C₂²×C₅₃	semidirect
370. Q₈⋊C₈	C₂²×C₅₃	semidirect
374. D₄⋊C₈	C₂²×C₅₃	semidirect
375. D₄⋊C₈	C₂²×C₅₃	semidirect
376. D₄⋊C₈	C₂²×C₅₃	semidirect
377. D₄⋊C₈	C₂²×C₅₃	semidirect
378. C₅₃	#17517	direct (N × H)
576. D₄	#131×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3481	2
C₅₃×#2990	2
C₄×#131×C₅₃	2
C₅₃×#1494	2
C₅₃×#1492	2
C₅₃×#1490	2
C₅₃×#1489	2
C₅₃×#1484	2
C₅₃×#1483	2
C₅₃×#1476	2
C₂×C₅₃×D₄⋊C₈	2
#17517	53

Subgroups — C53×#17517 #27225029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17517 #27225029