Subgroups — C₅₃×#17468 #27224980

Order 13568.

984 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3460	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2988	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2965	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1484	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1483	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1482	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1480	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1478	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1477	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1476	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#544	C₄	1	1	yes	characteristic
13	C₅₃×#542	—	2	1	no	—
14	C₅₃×#542	C₄	1	2	yes	—
15	C₅₃×#540	—	2	1	no	—
16	C₅₃×#540	C₄	1	2	yes	—
17	C₅₃×#538	—	2	2	no	—
18	C₅₃×#537	C₄	1	1	yes	characteristic
19	C₅₃×#535	C₄	1	1	yes	characteristic
20	C₅₃×#535	C₄	1	1	yes	characteristic
21	C₅₃×#532	—	2	1	no	—
22	C₅₃×#532	C₂²	1	2	yes	—
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	C₂²	1	1	yes	characteristic
29	C₅₃×#446	—	2	1	no	—
30	C₈×Q₈×C₅₃	—	2	1	no	—
31	C₅₃×#436	—	2	1	no	—
32	C₅₃×#435	—	2	2	no	—
33	C₈×D₄×C₅₃	—	2	1	no	—
34	C₈×D₄×C₅₃	—	2	2	no	—
35	C₅₃×#431	C₂²	1	1	yes	characteristic
36	C₅₃×#431	C₂²	1	1	yes	characteristic
37	C₅₃×#431	C₂²	1	2	yes	complemented
38	C₅₃×#431	C₂²	1	2	yes	complemented
39	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	complemented
41	C₄×M₁₆×C₅₃	C₂²	1	2	yes	complemented
42	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
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	2	yes	—
47	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
48	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
49	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
50	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
51	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
52	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
53	C₂×#41×C₅₃	—	2	2	no	—
54	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂³×C₄×C₅₃	—	2	1	no	—
58	C₂×M₁₆×C₅₃	—	2	2	no	—
59	C₂²×C₈×C₅₃	—	2	1	no	—
60	C₂²×C₈×C₅₃	—	2	1	no	—
61	C₂²×C₈×C₅₃	—	2	2	no	—
62	C₄⋊Q₈×C₅₃	—	2	1	no	—
63	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
64	#127×C₅₃	—	2	1	no	—
65	#127×C₅₃	C₂×C₄	1	2	yes	—
66	#126×C₅₃	—	2	2	no	—
67	#126×C₅₃	—	2	2	no	—
68	#125×C₅₃	—	2	1	no	—
69	#125×C₅₃	C₂×C₄	1	2	yes	—
70	#124×C₅₃	—	2	2	no	—
71	#124×C₅₃	—	2	2	no	—
72	#124×C₅₃	—	2	1	no	—
73	#124×C₅₃	C₂×C₄	1	2	yes	—
74	#123×C₅₃	—	2	2	no	—
75	#123×C₅₃	—	2	2	no	—
76	#123×C₅₃	—	2	2	no	—
77	#123×C₅₃	—	2	2	no	—
78	C₂²⋊Q₈×C₅₃	—	2	2	no	—
79	C₂²⋊Q₈×C₅₃	—	2	2	no	—
80	C₂²⋊Q₈×C₅₃	—	2	1	no	—
81	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
82	#121×C₅₃	—	2	1	no	—
83	#121×C₅₃	—	2	1	no	—
84	#121×C₅₃	—	2	2	no	—
85	#121×C₅₃	—	2	2	no	—
86	#121×C₅₃	—	2	2	no	—
87	#121×C₅₃	—	2	1	no	—
88	#121×C₅₃	—	2	2	no	—
89	#121×C₅₃	C₂×C₄	1	2	yes	—
90	#121×C₅₃	C₂×C₄	1	2	yes	—
91	#121×C₅₃	C₂×C₄	1	2	yes	—
92	#120×C₅₃	—	2	2	no	—
93	#120×C₅₃	—	2	2	no	—
94	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
95	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
96	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
97	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
98	C₄×D₄×C₅₃	—	2	1	no	—
99	C₄×D₄×C₅₃	—	2	1	no	—
100	C₄×D₄×C₅₃	—	2	2	no	—
101	C₄×D₄×C₅₃	—	2	1	no	—
102	C₄×D₄×C₅₃	—	2	1	no	—
103	C₄×D₄×C₅₃	—	2	2	no	—
104	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
105	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
106	C₄×D₄×C₅₃	D₄	1	2	yes	—
107	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
108	C₄×D₄×C₅₃	D₄	1	2	yes	—
109	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
110	C₄×D₄×C₅₃	C₂³	1	2	yes	—
111	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
112	#117×C₅₃	—	2	1	no	—
113	#117×C₅₃	—	2	2	no	—
114	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
115	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
116	#117×C₅₃	C₂×C₄	1	2	yes	—
117	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
118	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
119	C₂×C₄²×C₅₃	—	2	1	no	—
120	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
122	C₄⋊C₈×C₅₃	—	2	1	no	—
123	C₄⋊C₈×C₅₃	—	2	2	no	—
124	C₄⋊C₈×C₅₃	—	2	1	no	—
125	C₄⋊C₈×C₅₃	—	2	2	no	—
126	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
127	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
128	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
129	C₂²⋊C₈×C₅₃	—	2	1	no	—
130	C₂²⋊C₈×C₅₃	—	2	1	no	—
131	C₂²⋊C₈×C₅₃	—	2	2	no	—
132	C₂²⋊C₈×C₅₃	—	2	2	no	—
133	#97×C₅₃	—	2	2	no	—
134	C₄×C₈×C₅₃	—	2	1	no	—
135	C₄×C₈×C₅₃	—	2	1	no	—
136	C₄×C₈×C₅₃	—	2	2	no	—
137	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₅₃	C₂³	1	2	yes	—
140	C₂⁴×C₅₃	—	2	1	no	—
141	#41×C₅₃	—	2	4	no	—
142	#41×C₅₃	—	2	4	no	—
143	#41×C₅₃	—	8	2	no	—
144	#41×C₅₃	—	8	2	no	—
145	C₂×Q₈×C₅₃	—	2	2	no	—
146	C₂×Q₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
147	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₂×D₄×C₅₃	—	2	1	no	—
149	C₂×D₄×C₅₃	—	2	2	no	—
150	C₂×D₄×C₅₃	—	2	2	no	—
151	C₂×D₄×C₅₃	—	2	2	no	—
152	C₂×D₄×C₅₃	—	2	1	no	—
153	C₂×D₄×C₅₃	—	2	2	no	—
154	C₂×D₄×C₅₃	—	2	2	no	—
155	C₂×D₄×C₅₃	—	2	4	no	—
156	C₂×D₄×C₅₃	C₂²⋊C₄	1	2	yes	—
157	C₂×D₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
158	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
159	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₅₃	—	2	1	no	—
161	C₂²×C₄×C₅₃	—	2	1	no	—
162	C₂²×C₄×C₅₃	—	2	1	no	—
163	C₂²×C₄×C₅₃	—	2	2	no	—
164	C₂²×C₄×C₅₃	—	2	1	no	—
165	C₂²×C₄×C₅₃	—	2	2	no	—
166	C₂²×C₄×C₅₃	—	8	1	no	—
167	C₂²×C₄×C₅₃	—	8	1	no	—
168	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
169	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
170	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
174	M₁₆×C₅₃	—	4	4	no	—
175	C₂×C₈×C₅₃	—	2	1	no	—
176	C₂×C₈×C₅₃	—	2	1	no	—
177	C₂×C₈×C₅₃	—	2	2	no	—
178	C₂×C₈×C₅₃	—	2	1	no	—
179	C₂×C₈×C₅₃	—	2	1	no	—
180	C₂×C₈×C₅₃	—	2	2	no	—
181	C₂×C₈×C₅₃	—	4	2	no	—
182	C₂×C₈×C₅₃	—	8	1	no	—
183	C₂×C₈×C₅₃	—	8	2	no	—
184	C₄⋊C₄×C₅₃	—	2	1	no	—
185	C₄⋊C₄×C₅₃	—	2	1	no	—
186	C₄⋊C₄×C₅₃	—	2	2	no	—
187	C₄⋊C₄×C₅₃	—	2	2	no	—
188	C₄⋊C₄×C₅₃	—	2	2	no	—
189	C₄⋊C₄×C₅₃	—	2	2	no	—
190	C₄⋊C₄×C₅₃	—	2	1	no	—
191	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
192	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
193	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
194	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
195	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
196	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
197	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
198	C₂²⋊C₄×C₅₃	—	2	1	no	—
199	C₂²⋊C₄×C₅₃	—	2	2	no	—
200	C₂²⋊C₄×C₅₃	—	2	2	no	—
201	C₂²⋊C₄×C₅₃	—	2	1	no	—
202	C₂²⋊C₄×C₅₃	—	2	2	no	—
203	C₂²⋊C₄×C₅₃	—	2	2	no	—
204	C₂²⋊C₄×C₅₃	—	2	2	no	—
205	C₂²⋊C₄×C₅₃	—	2	2	no	—
206	C₂²⋊C₄×C₅₃	—	2	2	no	—
207	C₂²⋊C₄×C₅₃	—	2	2	no	—
208	C₂²⋊C₄×C₅₃	—	2	1	no	—
209	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
210	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
211	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
212	C₄²×C₅₃	—	2	1	no	—
213	C₄²×C₅₃	—	2	1	no	—
214	C₄²×C₅₃	—	2	1	no	—
215	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
216	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
217	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
218	C₄²×C₅₃	C₂×D₄	1	2	yes	—
219	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₅₃	C₂×D₄	1	2	yes	—
221	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
222	C₂³×C₅₃	—	2	1	no	—
223	C₂³×C₅₃	—	2	2	no	—
224	C₂³×C₅₃	—	4	2	no	—
225	C₂³×C₅₃	—	8	1	no	—
226	C₂³×C₅₃	—	8	1	no	—
227	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
228	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
229	Q₈×C₅₃	—	2	2	no	—
230	Q₈×C₅₃	—	2	2	no	—
231	Q₈×C₅₃	—	8	2	no	—
232	D₄×C₅₃	—	2	4	no	—
233	D₄×C₅₃	—	2	4	no	—
234	D₄×C₅₃	—	2	2	no	—
235	D₄×C₅₃	—	2	2	no	—
236	D₄×C₅₃	—	4	2	no	—
237	D₄×C₅₃	—	4	2	no	—
238	D₄×C₅₃	—	8	2	no	—
239	D₄×C₅₃	—	8	2	no	—
240	D₄×C₅₃	—	8	2	no	—
241	D₄×C₅₃	—	8	2	no	—
242	D₄×C₅₃	—	8	2	no	—
243	C₂×C₄×C₅₃	—	2	4	no	—
244	C₂×C₄×C₅₃	—	2	1	no	—
245	C₂×C₄×C₅₃	—	2	2	no	—
246	C₂×C₄×C₅₃	—	2	1	no	—
247	C₂×C₄×C₅₃	—	2	2	no	—
248	C₂×C₄×C₅₃	—	2	2	no	—
249	C₂×C₄×C₅₃	—	2	1	no	—
250	C₂×C₄×C₅₃	—	2	1	no	—
251	C₂×C₄×C₅₃	—	2	1	no	—
252	C₂×C₄×C₅₃	—	4	1	no	—
253	C₂×C₄×C₅₃	—	4	2	no	—
254	C₂×C₄×C₅₃	—	4	2	no	—
255	C₂×C₄×C₅₃	—	4	1	no	—
256	C₂×C₄×C₅₃	—	4	2	no	—
257	C₂×C₄×C₅₃	—	4	1	no	—
258	C₂×C₄×C₅₃	—	4	2	no	—
259	C₂×C₄×C₅₃	—	8	1	no	—
260	C₂×C₄×C₅₃	—	8	2	no	—
261	C₂×C₄×C₅₃	—	8	1	no	—
262	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
265	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
266	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
267	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄×C₅₃	#131	1	4	yes	—
269	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
270	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
272	C₈×C₅₃	—	2	2	no	—
273	C₈×C₅₃	—	4	1	no	—
274	C₈×C₅₃	—	4	2	no	—
275	C₈×C₅₃	—	4	1	no	—
276	C₈×C₅₃	—	4	2	no	—
277	C₈×C₅₃	—	4	1	no	—
278	#17468	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
279	C₂²×C₅₃	—	2	1	no	—
280	C₂²×C₅₃	—	2	1	no	—
281	C₂²×C₅₃	—	2	2	no	—
282	C₂²×C₅₃	—	2	2	no	—
283	C₂²×C₅₃	—	2	2	no	—
284	C₂²×C₅₃	—	4	1	no	—
285	C₂²×C₅₃	—	4	2	no	—
286	C₂²×C₅₃	—	4	2	no	—
287	C₂²×C₅₃	—	4	1	no	—
288	C₂²×C₅₃	—	4	1	no	—
289	C₂²×C₅₃	—	4	2	no	—
290	C₂²×C₅₃	—	8	4	no	—
291	C₂²×C₅₃	#408	1	2	yes	—
292	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
293	C₄×C₅₃	—	2	2	no	—
294	C₄×C₅₃	—	2	1	no	—
295	C₄×C₅₃	—	2	2	no	—
296	C₄×C₅₃	—	2	1	no	—
297	C₄×C₅₃	—	2	1	no	—
298	C₄×C₅₃	—	2	2	no	—
299	C₄×C₅₃	—	4	1	no	—
300	C₄×C₅₃	—	4	2	no	—
301	C₄×C₅₃	—	4	1	no	—
302	C₄×C₅₃	—	4	1	no	—
303	C₄×C₅₃	#408	1	2	yes	—
304	C₄×C₅₃	C₂×#131	1	2	yes	—
305	C₄×C₅₃	#576	1	1	yes	characteristic
306	C₄×C₅₃	#577	1	1	yes	characteristic
307	#3460	C₂×C₅₃	1	1	yes	characteristic
308	#2988	C₂×C₅₃	1	1	yes	characteristic, complemented
309	#2965	C₂×C₅₃	1	1	yes	characteristic, complemented
310	#1484	C₂×C₅₃	1	2	yes	complemented
311	#1483	C₂×C₅₃	1	2	yes	complemented
312	#1482	C₂×C₅₃	1	2	yes	complemented
313	#1480	C₂×C₅₃	1	2	yes	complemented
314	#1478	C₂×C₅₃	1	1	yes	characteristic, complemented
315	#1477	C₂×C₅₃	1	1	yes	characteristic, complemented
316	#1476	C₂×C₅₃	1	1	yes	characteristic, complemented
317	C₂×D₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
318	C₂×C₅₃	—	2	2	no	—
319	C₂×C₅₃	—	4	2	no	—
320	C₂×C₅₃	—	8	1	no	—
321	C₂×C₅₃	—	8	2	no	—
322	C₂×C₅₃	C₂×#408	1	1	yes	characteristic
323	C₂×C₅₃	#2897	1	1	yes	characteristic
324	C₂×C₅₃	#2901	1	1	yes	characteristic
325	#544	C₄×C₅₃	1	1	yes	characteristic
326	#542	—	2	1	no	—
327	#542	C₄×C₅₃	1	2	yes	—
328	#540	—	2	1	no	—
329	#540	C₄×C₅₃	1	2	yes	—
330	#538	—	2	2	no	—
331	#537	C₄×C₅₃	1	1	yes	characteristic
332	#535	C₄×C₅₃	1	1	yes	characteristic
333	#535	C₄×C₅₃	1	1	yes	characteristic
334	#532	—	2	1	no	—
335	#532	C₂²×C₅₃	1	2	yes	—
336	#529	—	2	1	no	—
337	#529	C₂²×C₅₃	1	2	yes	—
338	#525	—	2	2	no	—
339	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
340	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
341	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
342	#446	—	2	1	no	—
343	C₈×Q₈	—	2	1	no	—
344	#436	—	2	1	no	—
345	#435	—	2	2	no	—
346	C₈×D₄	—	2	1	no	—
347	C₈×D₄	—	2	2	no	—
348	#431	C₂²×C₅₃	1	1	yes	characteristic
349	#431	C₂²×C₅₃	1	1	yes	characteristic
350	#431	C₂²×C₅₃	1	2	yes	complemented
351	#431	C₂²×C₅₃	1	2	yes	complemented
352	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
353	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
354	C₄×M₁₆	C₂²×C₅₃	1	2	yes	complemented
355	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
356	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
357	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
358	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	complemented
359	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
360	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
361	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
362	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
363	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
364	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
365	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
366	C₅₃	#17468	1	1	yes	characteristic, direct product factor, Sylow
367	C₂×#41	—	2	2	no	—
368	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
369	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
370	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
371	C₂³×C₄	—	2	1	no	—
372	C₂×M₁₆	—	2	2	no	—
373	C₂²×C₈	—	2	1	no	—
374	C₂²×C₈	—	2	1	no	—
375	C₂²×C₈	—	2	2	no	—
376	C₄⋊Q₈	—	2	1	no	—
377	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
378	#127	—	2	1	no	—
379	#127	C₂×C₄×C₅₃	1	2	yes	—
380	#126	—	2	2	no	—
381	#126	—	2	2	no	—
382	#125	—	2	1	no	—
383	#125	C₂×C₄×C₅₃	1	2	yes	—
384	#124	—	2	1	no	—
385	#124	—	2	2	no	—
386	#124	—	2	2	no	—
387	#124	C₂×C₄×C₅₃	1	2	yes	—
388	#123	—	2	2	no	—
389	#123	—	2	2	no	—
390	#123	—	2	2	no	—
391	#123	—	2	2	no	—
392	C₂²⋊Q₈	—	2	1	no	—
393	C₂²⋊Q₈	—	2	2	no	—
394	C₂²⋊Q₈	—	2	2	no	—
395	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
396	#121	—	2	2	no	—
397	#121	—	2	1	no	—
398	#121	—	2	2	no	—
399	#121	—	2	1	no	—
400	#121	—	2	1	no	—
401	#121	—	2	2	no	—
402	#121	—	2	2	no	—
403	#121	C₂×C₄×C₅₃	1	2	yes	—
404	#121	C₂×C₄×C₅₃	1	2	yes	—
405	#121	C₂×C₄×C₅₃	1	2	yes	—
406	#120	—	2	2	no	—
407	#120	—	2	2	no	—
408	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
409	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
410	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
411	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
412	C₄×D₄	—	2	1	no	—
413	C₄×D₄	—	2	1	no	—
414	C₄×D₄	—	2	2	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	—	2	1	no	—
417	C₄×D₄	—	2	2	no	—
418	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
419	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
420	C₄×D₄	D₄×C₅₃	1	2	yes	—
421	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
422	C₄×D₄	D₄×C₅₃	1	2	yes	—
423	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
424	C₄×D₄	C₂³×C₅₃	1	2	yes	—
425	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
426	#117	—	2	1	no	—
427	#117	—	2	2	no	—
428	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
429	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
430	#117	C₂×C₄×C₅₃	1	2	yes	—
431	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
432	C₂×C₂²⋊C₄	—	2	1	no	—
433	C₂×C₄²	—	2	1	no	—
434	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
435	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
436	C₄⋊C₈	—	2	1	no	—
437	C₄⋊C₈	—	2	2	no	—
438	C₄⋊C₈	—	2	1	no	—
439	C₄⋊C₈	—	2	2	no	—
440	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
441	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
442	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
443	C₂²⋊C₈	—	2	1	no	—
444	C₂²⋊C₈	—	2	1	no	—
445	C₂²⋊C₈	—	2	2	no	—
446	C₂²⋊C₈	—	2	2	no	—
447	#97	—	2	2	no	—
448	C₄×C₈	—	2	1	no	—
449	C₄×C₈	—	2	1	no	—
450	C₄×C₈	—	2	2	no	—
451	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
452	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
453	C₄×C₈	C₂³×C₅₃	1	2	yes	—
454	C₂⁴	—	2	1	no	—
455	#41	—	2	4	no	—
456	#41	—	2	4	no	—
457	#41	—	8	2	no	—
458	#41	—	8	2	no	—
459	C₂×Q₈	—	2	2	no	—
460	C₂×Q₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
461	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
462	C₂×D₄	—	2	1	no	—
463	C₂×D₄	—	2	2	no	—
464	C₂×D₄	—	2	2	no	—
465	C₂×D₄	—	2	2	no	—
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	—	2	2	no	—
468	C₂×D₄	—	2	2	no	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	C₂²⋊C₄×C₅₃	1	2	yes	—
471	C₂×D₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
472	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
473	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	2	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	8	1	no	—
481	C₂²×C₄	—	8	1	no	—
482	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
483	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
484	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
485	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
486	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
487	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
488	M₁₆	—	4	4	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	2	2	no	—
494	C₂×C₈	—	2	2	no	—
495	C₂×C₈	—	4	2	no	—
496	C₂×C₈	—	8	1	no	—
497	C₂×C₈	—	8	2	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
506	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
507	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
508	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
509	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
510	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
511	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
512	C₂²⋊C₄	—	2	1	no	—
513	C₂²⋊C₄	—	2	2	no	—
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	—	2	1	no	—
516	C₂²⋊C₄	—	2	2	no	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	2	2	no	—
519	C₂²⋊C₄	—	2	2	no	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	2	1	no	—
523	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
524	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
525	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
526	C₄²	—	2	1	no	—
527	C₄²	—	2	1	no	—
528	C₄²	—	2	1	no	—
529	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
530	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
531	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
532	C₄²	C₂×D₄×C₅₃	1	2	yes	—
533	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
534	C₄²	C₂×D₄×C₅₃	1	2	yes	—
535	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
536	C₂³	—	2	2	no	—
537	C₂³	—	2	1	no	—
538	C₂³	—	4	2	no	—
539	C₂³	—	8	1	no	—
540	C₂³	—	8	1	no	—
541	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
542	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
543	Q₈	—	2	2	no	—
544	Q₈	—	2	2	no	—
545	Q₈	—	8	2	no	—
546	D₄	—	2	4	no	—
547	D₄	—	2	4	no	—
548	D₄	—	2	2	no	—
549	D₄	—	2	2	no	—
550	D₄	—	4	2	no	—
551	D₄	—	4	2	no	—
552	D₄	—	8	2	no	—
553	D₄	—	8	2	no	—
554	D₄	—	8	2	no	—
555	D₄	—	8	2	no	—
556	D₄	—	8	2	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	2	no	—
568	C₂×C₄	—	4	2	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	2	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	2	no	—
573	C₂×C₄	—	8	1	no	—
574	C₂×C₄	—	8	2	no	—
575	C₂×C₄	—	8	1	no	—
576	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
577	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
578	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
579	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
580	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
581	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
582	C₂×C₄	#131×C₅₃	1	4	yes	—
583	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
584	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
585	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
586	C₈	—	2	2	no	—
587	C₈	—	4	1	no	—
588	C₈	—	4	2	no	—
589	C₈	—	4	1	no	—
590	C₈	—	4	2	no	—
591	C₈	—	4	1	no	—
592	C₂²	—	2	1	no	—
593	C₂²	—	2	1	no	—
594	C₂²	—	2	2	no	—
595	C₂²	—	2	2	no	—
596	C₂²	—	2	2	no	—
597	C₂²	—	4	1	no	—
598	C₂²	—	4	2	no	—
599	C₂²	—	4	2	no	—
600	C₂²	—	4	1	no	—
601	C₂²	—	4	1	no	—
602	C₂²	—	4	2	no	—
603	C₂²	—	8	4	no	—
604	C₂²	C₅₃×#408	1	2	yes	—
605	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
606	C₄	—	2	2	no	—
607	C₄	—	2	1	no	—
608	C₄	—	2	2	no	—
609	C₄	—	2	1	no	—
610	C₄	—	2	1	no	—
611	C₄	—	2	2	no	—
612	C₄	—	4	1	no	—
613	C₄	—	4	2	no	—
614	C₄	—	4	1	no	—
615	C₄	—	4	1	no	—
616	C₄	C₅₃×#408	1	2	yes	—
617	C₄	C₂×#131×C₅₃	1	2	yes	—
618	C₄	C₅₃×#576	1	1	yes	characteristic
619	C₄	C₅₃×#577	1	1	yes	characteristic
620	C₂	—	2	2	no	—
621	C₂	—	4	2	no	—
622	C₂	—	8	2	no	—
623	C₂	—	8	1	no	—
624	C₂	C₂×C₅₃×#408	1	1	yes	characteristic
625	C₂	C₅₃×#2897	1	1	yes	characteristic
626	C₂	C₅₃×#2901	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×#3460	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×#2988	C₂	semidirect
3. C₅₃×#2965	C₂	semidirect
4. C₅₃×#1484	C₂	semidirect
5. C₅₃×#1483	C₂	semidirect
6. C₅₃×#1482	C₂	semidirect
7. C₅₃×#1480	C₂	semidirect
8. C₅₃×#1478	C₂	semidirect
9. C₅₃×#1477	C₂	semidirect
10. C₅₃×#1476	C₂	semidirect
11. C₂×C₅₃×D₄⋊C₈	C₂	semidirect
37. C₅₃×#431	C₂²	semidirect
38. C₅₃×#431	C₂²	semidirect
40. C₂×C₄⋊C₈×C₅₃	C₂²	semidirect
41. C₄×M₁₆×C₅₃	C₂²	semidirect
43. C₅₃×Q₈⋊C₈	C₂²	semidirect
44. C₅₃×Q₈⋊C₈	C₂²	semidirect
45. C₅₃×Q₈⋊C₈	C₂²	semidirect
50. C₅₃×D₄⋊C₈	C₂²	semidirect
51. C₅₃×D₄⋊C₈	C₂²	semidirect
52. C₅₃×D₄⋊C₈	C₂²	semidirect
278. #17468	C₅₃	direct (N × H)
308. #2988	C₂×C₅₃	semidirect
309. #2965	C₂×C₅₃	semidirect
310. #1484	C₂×C₅₃	semidirect
311. #1483	C₂×C₅₃	semidirect
312. #1482	C₂×C₅₃	semidirect
313. #1480	C₂×C₅₃	semidirect
314. #1478	C₂×C₅₃	semidirect
315. #1477	C₂×C₅₃	semidirect
316. #1476	C₂×C₅₃	semidirect
317. C₂×D₄⋊C₈	C₂×C₅₃	semidirect
350. #431	C₂²×C₅₃	semidirect
351. #431	C₂²×C₅₃	semidirect
353. C₂×C₄⋊C₈	C₂²×C₅₃	semidirect
354. C₄×M₁₆	C₂²×C₅₃	semidirect
356. Q₈⋊C₈	C₂²×C₅₃	semidirect
357. Q₈⋊C₈	C₂²×C₅₃	semidirect
358. Q₈⋊C₈	C₂²×C₅₃	semidirect
363. D₄⋊C₈	C₂²×C₅₃	semidirect
364. D₄⋊C₈	C₂²×C₅₃	semidirect
365. D₄⋊C₈	C₂²×C₅₃	semidirect
366. C₅₃	#17468	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×#3460	2
C₅₃×#2988	2
C₅₃×#2965	2
C₅₃×#1484	2
C₅₃×#1483	2
C₅₃×#1482	2
C₅₃×#1480	2
C₅₃×#1478	2
C₅₃×#1477	2
C₅₃×#1476	2
C₂×C₅₃×D₄⋊C₈	2
#17468	53

Subgroups — C53×#17468 #27224980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17468 #27224980