Subgroups — C₅₃×#62150 #27269662

Order 13568.

1996 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3483	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×SD₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
4	C₅₃×#3358	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3354	C₂	1	6	yes	maximal, complemented
6	C₅₃×#3332	C₂	1	6	yes	maximal, complemented
7	C₅₃×#3253	C₂	1	3	yes	maximal, complemented
8	C₅₃×#3227	C₂	1	3	yes	maximal, complemented
9	C₅₃×#3106	C₂	1	3	yes	maximal, complemented
10	C₅₃×#3079	C₂	1	3	yes	maximal, complemented
11	C₅₃×#2991	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₅₃	—	2	1	no	—
13	Q₈²×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#555	—	2	3	no	—
15	C₅₃×#555	C₂²	1	3	yes	—
16	C₅₃×#550	—	2	1	no	—
17	C₅₃×#550	C₂²	1	3	yes	—
18	C₅₃×#550	C₂²	1	1	yes	characteristic
19	D₄×Q₈×C₅₃	—	2	1	no	—
20	D₄×Q₈×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#547	—	2	3	no	—
22	C₅₃×#547	C₂²	1	6	yes	—
23	C₅₃×#536	—	2	3	no	—
24	C₅₃×#532	—	2	3	no	—
25	C₅₃×#532	C₂²	1	3	yes	—
26	C₅₃×#532	C₂²	1	3	yes	—
27	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
28	C₄×#41×C₅₃	—	2	1	no	—
29	C₄×#41×C₅₃	C₂²	1	3	yes	—
30	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#498	C₂²	1	6	yes	—
32	C₅₃×#486	C₂²	1	6	yes	—
33	C₅₃×#482	C₂²	1	6	yes	—
34	C₅₃×#481	C₂²	1	12	yes	—
35	C₅₃×#477	C₂²	1	3	yes	—
36	C₅₃×#476	C₂²	1	3	yes	—
37	C₅₃×#476	C₂²	1	6	yes	—
38	C₅₃×#476	C₂²	1	6	yes	—
39	C₅₃×#475	C₂²	1	6	yes	—
40	C₅₃×#474	C₂²	1	6	yes	—
41	C₅₃×#469	C₂²	1	12	yes	—
42	C₅₃×#460	C₂²	1	1	yes	characteristic
43	C₅₃×#459	C₂²	1	3	yes	—
44	C₅₃×#459	C₂²	1	6	yes	—
45	C₅₃×#449	C₂²	1	6	yes	—
46	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
47	C₄×SD₁₆×C₅₃	C₂²	1	6	yes	—
48	C₄×D₈×C₅₃	C₂²	1	6	yes	—
49	C₅₃×#436	C₂²	1	6	yes	—
50	C₅₃×#431	C₂²	1	3	yes	—
51	C₅₃×#428	C₂²	1	6	yes	—
52	C₅₃×#423	C₂²	1	3	yes	—
53	C₅₃×#417	C₂²	1	6	yes	—
54	2₋¹⁺⁴×C₅₃	—	4	8	no	—
55	C₂×#41×C₅₃	—	2	1	no	—
56	C₂×#41×C₅₃	—	2	6	no	—
57	C₂×#41×C₅₃	—	2	3	no	—
58	C₂×#41×C₅₃	—	2	3	no	—
59	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
60	C₂×#41×C₅₃	C₂³	1	3	yes	—
61	C₂²×Q₈×C₅₃	—	2	2	no	—
62	C₂²×Q₈×C₅₃	—	2	3	no	—
63	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
64	C₂×D₈×C₅₃	C₂³	1	6	yes	—
65	C₂×M₁₆×C₅₃	C₂³	1	6	yes	—
66	C₄⋊Q₈×C₅₃	—	2	3	no	—
67	C₄⋊Q₈×C₅₃	—	2	3	no	—
68	C₄⋊Q₈×C₅₃	—	2	3	no	—
69	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
70	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
71	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
72	#127×C₅₃	—	2	3	no	—
73	#127×C₅₃	C₂³	1	3	yes	—
74	#127×C₅₃	C₂³	1	1	yes	characteristic
75	#124×C₅₃	—	2	6	no	—
76	#124×C₅₃	—	2	3	no	—
77	#124×C₅₃	—	2	3	no	—
78	#123×C₅₃	—	2	12	no	—
79	C₂²⋊Q₈×C₅₃	—	2	6	no	—
80	C₂²⋊Q₈×C₅₃	—	2	6	no	—
81	#121×C₅₃	—	2	3	no	—
82	#121×C₅₃	—	2	6	no	—
83	#121×C₅₃	—	2	3	no	—
84	#121×C₅₃	—	2	3	no	—
85	#121×C₅₃	—	2	3	no	—
86	#121×C₅₃	C₂³	1	6	yes	—
87	#121×C₅₃	C₂³	1	6	yes	—
88	C₄×Q₈×C₅₃	—	2	1	no	—
89	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
90	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₄×Q₈×C₅₃	C₂³	1	3	yes	—
93	C₄×D₄×C₅₃	—	2	3	no	—
94	C₄×D₄×C₅₃	—	2	3	no	—
95	C₄×D₄×C₅₃	—	2	3	no	—
96	C₄×D₄×C₅₃	—	2	3	no	—
97	C₄×D₄×C₅₃	—	2	3	no	—
98	C₄×D₄×C₅₃	D₄	1	3	yes	—
99	C₄×D₄×C₅₃	C₂³	1	3	yes	—
100	C₄×D₄×C₅₃	C₂³	1	3	yes	—
101	C₄×D₄×C₅₃	C₂³	1	6	yes	—
102	C₄×D₄×C₅₃	C₂³	1	3	yes	—
103	#117×C₅₃	—	2	3	no	—
104	#117×C₅₃	—	2	3	no	—
105	#117×C₅₃	D₄	1	3	yes	—
106	#117×C₅₃	C₂³	1	3	yes	—
107	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
108	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
109	C₂×C₄²×C₅₃	—	2	3	no	—
110	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
111	#107×C₅₃	C₂³	1	6	yes	—
112	#106×C₅₃	C₂³	1	6	yes	—
113	#106×C₅₃	C₂³	1	12	yes	—
114	C₄⋊C₈×C₅₃	C₂³	1	3	yes	—
115	C₄⋊C₈×C₅₃	C₂³	1	6	yes	—
116	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
117	#103×C₅₃	C₂³	1	6	yes	—
118	#102×C₅₃	C₂³	1	6	yes	—
119	#102×C₅₃	C₂³	1	12	yes	—
120	#102×C₅₃	C₂³	1	2	yes	—
121	#102×C₅₃	C₂³	1	6	yes	—
122	C₂²⋊C₈×C₅₃	C₂³	1	6	yes	—
123	C₄×C₈×C₅₃	C₂³	1	6	yes	—
124	#41×C₅₃	—	2	4	no	—
125	#41×C₅₃	—	2	6	no	—
126	#41×C₅₃	—	4	3	no	—
127	#41×C₅₃	—	4	12	no	—
128	#41×C₅₃	—	4	24	no	—
129	#41×C₅₃	—	4	4	no	—
130	#41×C₅₃	—	8	6	no	—
131	C₂×Q₈×C₅₃	—	2	6	no	—
132	C₂×Q₈×C₅₃	—	2	3	no	—
133	C₂×Q₈×C₅₃	—	2	3	no	—
134	C₂×Q₈×C₅₃	—	4	12	no	—
135	C₂×Q₈×C₅₃	—	4	8	no	—
136	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
137	C₂×Q₈×C₅₃	C₂⁴	1	1	yes	characteristic
138	C₂×D₄×C₅₃	—	2	3	no	—
139	C₂×D₄×C₅₃	—	2	6	no	—
140	C₂×D₄×C₅₃	—	2	6	no	—
141	C₂×D₄×C₅₃	—	2	1	no	—
142	C₂×D₄×C₅₃	—	2	3	no	—
143	C₂×D₄×C₅₃	—	2	1	no	—
144	C₂×D₄×C₅₃	—	2	3	no	—
145	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
146	C₂×D₄×C₅₃	C₂⁴	1	3	yes	—
147	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
148	C₂²×C₄×C₅₃	—	2	6	no	—
149	C₂²×C₄×C₅₃	—	2	3	no	—
150	C₂²×C₄×C₅₃	—	2	3	no	—
151	C₂²×C₄×C₅₃	—	2	3	no	—
152	C₂²×C₄×C₅₃	—	2	6	no	—
153	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
154	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
156	SD₁₆×C₅₃	—	2	4	no	—
157	D₈×C₅₃	—	4	6	no	—
158	M₁₆×C₅₃	—	4	6	no	—
159	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
160	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
161	C₄⋊C₄×C₅₃	—	2	3	no	—
162	C₄⋊C₄×C₅₃	—	2	6	no	—
163	C₄⋊C₄×C₅₃	—	2	6	no	—
164	C₄⋊C₄×C₅₃	—	2	1	no	—
165	C₄⋊C₄×C₅₃	—	2	3	no	—
166	C₄⋊C₄×C₅₃	—	2	3	no	—
167	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
168	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
169	C₄⋊C₄×C₅₃	C₂⁴	1	6	yes	—
170	C₄⋊C₄×C₅₃	C₂⁴	1	3	yes	—
171	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
172	C₂²⋊C₄×C₅₃	—	2	3	no	—
173	C₂²⋊C₄×C₅₃	—	2	6	no	—
174	C₂²⋊C₄×C₅₃	—	2	3	no	—
175	C₂²⋊C₄×C₅₃	—	2	3	no	—
176	C₂²⋊C₄×C₅₃	—	2	6	no	—
177	C₂²⋊C₄×C₅₃	—	2	6	no	—
178	C₂²⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
179	C₄²×C₅₃	—	2	1	no	—
180	C₄²×C₅₃	—	2	3	no	—
181	C₄²×C₅₃	—	2	3	no	—
182	C₄²×C₅₃	C₂×D₄	1	3	yes	—
183	C₄²×C₅₃	C₂×D₄	1	3	yes	—
184	C₄²×C₅₃	C₂⁴	1	3	yes	—
185	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
186	C₂³×C₅₃	—	2	3	no	—
187	C₂³×C₅₃	—	2	2	no	—
188	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
189	Q₈×C₅₃	—	2	1	no	—
190	Q₈×C₅₃	—	4	6	no	—
191	Q₈×C₅₃	—	4	12	no	—
192	Q₈×C₅₃	—	8	3	no	—
193	Q₈×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
194	Q₈×C₅₃	2₋¹⁺⁴	1	2	yes	complemented
195	D₄×C₅₃	—	2	6	no	—
196	D₄×C₅₃	—	2	3	no	—
197	D₄×C₅₃	—	2	2	no	—
198	D₄×C₅₃	—	2	3	no	—
199	D₄×C₅₃	—	2	2	no	—
200	D₄×C₅₃	—	4	6	no	—
201	D₄×C₅₃	—	4	2	no	—
202	D₄×C₅₃	—	4	12	no	—
203	D₄×C₅₃	—	4	6	no	—
204	D₄×C₅₃	—	4	2	no	—
205	D₄×C₅₃	—	8	3	no	—
206	D₄×C₅₃	—	8	6	no	—
207	C₂×C₄×C₅₃	—	2	1	no	—
208	C₂×C₄×C₅₃	—	2	6	no	—
209	C₂×C₄×C₅₃	—	2	6	no	—
210	C₂×C₄×C₅₃	—	2	3	no	—
211	C₂×C₄×C₅₃	—	2	3	no	—
212	C₂×C₄×C₅₃	—	2	6	no	—
213	C₂×C₄×C₅₃	—	2	1	no	—
214	C₂×C₄×C₅₃	—	2	3	no	—
215	C₂×C₄×C₅₃	—	4	6	no	—
216	C₂×C₄×C₅₃	—	4	12	no	—
217	C₂×C₄×C₅₃	—	4	12	no	—
218	C₂×C₄×C₅₃	—	4	6	no	—
219	C₂×C₄×C₅₃	—	4	6	no	—
220	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
221	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
223	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
224	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
226	C₈×C₅₃	—	2	2	no	—
227	C₈×C₅₃	—	2	6	no	—
228	#62150	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂²×C₅₃	—	2	3	no	—
230	C₂²×C₅₃	—	2	3	no	—
231	C₂²×C₅₃	—	2	3	no	—
232	C₂²×C₅₃	—	4	3	no	—
233	C₂²×C₅₃	—	4	2	no	—
234	C₂²×C₅₃	—	4	2	no	—
235	C₂²×C₅₃	—	4	3	no	—
236	C₂²×C₅₃	—	4	3	no	—
237	C₂²×C₅₃	—	4	2	no	—
238	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
239	C₄×C₅₃	—	2	3	no	—
240	C₄×C₅₃	—	2	2	no	—
241	C₄×C₅₃	—	2	1	no	—
242	C₄×C₅₃	—	2	3	no	—
243	C₄×C₅₃	—	4	1	no	—
244	C₄×C₅₃	—	4	6	no	—
245	C₄×C₅₃	—	4	3	no	—
246	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	6	yes	—
247	C₄×C₅₃	#577	1	2	yes	—
248	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
249	C₄×C₅₃	#585	1	1	yes	characteristic
250	#3499	C₂×C₅₃	1	1	yes	characteristic, complemented
251	#3483	C₂×C₅₃	1	1	yes	characteristic, complemented
252	Q₈×SD₁₆	C₂×C₅₃	1	2	yes	complemented
253	#3358	C₂×C₅₃	1	2	yes	complemented
254	#3354	C₂×C₅₃	1	6	yes	complemented
255	#3332	C₂×C₅₃	1	6	yes	complemented
256	#3253	C₂×C₅₃	1	3	yes	complemented
257	#3227	C₂×C₅₃	1	3	yes	complemented
258	#3106	C₂×C₅₃	1	3	yes	complemented
259	#3079	C₂×C₅₃	1	3	yes	complemented
260	#2991	C₂×C₅₃	1	1	yes	characteristic, complemented
261	C₂×C₅₃	—	4	3	no	—
262	C₂×C₅₃	—	8	3	no	—
263	C₂×C₅₃	—	8	2	no	—
264	C₂×C₅₃	#3489	1	1	yes	characteristic
265	C₂×C₅₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
266	C₂×C₅₃	C₂×#577	1	1	yes	characteristic
267	C₂×2₋¹⁺⁴	—	2	1	no	—
268	Q₈²	C₂²×C₅₃	1	1	yes	characteristic
269	#555	—	2	3	no	—
270	#555	C₂²×C₅₃	1	3	yes	—
271	#550	—	2	1	no	—
272	#550	C₂²×C₅₃	1	3	yes	—
273	#550	C₂²×C₅₃	1	1	yes	characteristic
274	D₄×Q₈	—	2	1	no	—
275	D₄×Q₈	C₂²×C₅₃	1	2	yes	—
276	#547	—	2	3	no	—
277	#547	C₂²×C₅₃	1	6	yes	—
278	#536	—	2	3	no	—
279	#532	—	2	3	no	—
280	#532	C₂²×C₅₃	1	3	yes	—
281	#532	C₂²×C₅₃	1	3	yes	—
282	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	—
283	C₄×#41	—	2	1	no	—
284	C₄×#41	C₂²×C₅₃	1	3	yes	—
285	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
286	#498	C₂²×C₅₃	1	6	yes	—
287	#486	C₂²×C₅₃	1	6	yes	—
288	#482	C₂²×C₅₃	1	6	yes	—
289	#481	C₂²×C₅₃	1	12	yes	—
290	#477	C₂²×C₅₃	1	3	yes	—
291	#476	C₂²×C₅₃	1	6	yes	—
292	#476	C₂²×C₅₃	1	6	yes	—
293	#476	C₂²×C₅₃	1	3	yes	—
294	#475	C₂²×C₅₃	1	6	yes	—
295	#474	C₂²×C₅₃	1	6	yes	—
296	#469	C₂²×C₅₃	1	12	yes	—
297	#460	C₂²×C₅₃	1	1	yes	characteristic
298	#459	C₂²×C₅₃	1	6	yes	—
299	#459	C₂²×C₅₃	1	3	yes	—
300	#449	C₂²×C₅₃	1	6	yes	—
301	C₈×Q₈	C₂²×C₅₃	1	2	yes	—
302	C₄×SD₁₆	C₂²×C₅₃	1	6	yes	—
303	C₄×D₈	C₂²×C₅₃	1	6	yes	—
304	#436	C₂²×C₅₃	1	6	yes	—
305	#431	C₂²×C₅₃	1	3	yes	—
306	#428	C₂²×C₅₃	1	6	yes	—
307	#423	C₂²×C₅₃	1	3	yes	—
308	#417	C₂²×C₅₃	1	6	yes	—
309	C₅₃	#62150	1	1	yes	characteristic, direct product factor, Sylow
310	2₋¹⁺⁴	—	4	8	no	—
311	C₂×#41	—	2	1	no	—
312	C₂×#41	—	2	6	no	—
313	C₂×#41	—	2	3	no	—
314	C₂×#41	—	2	3	no	—
315	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
316	C₂×#41	C₂³×C₅₃	1	3	yes	—
317	C₂²×Q₈	—	2	2	no	—
318	C₂²×Q₈	—	2	3	no	—
319	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
320	C₂×D₈	C₂³×C₅₃	1	6	yes	—
321	C₂×M₁₆	C₂³×C₅₃	1	6	yes	—
322	C₄⋊Q₈	—	2	3	no	—
323	C₄⋊Q₈	—	2	3	no	—
324	C₄⋊Q₈	—	2	3	no	—
325	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
326	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
327	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
328	#127	—	2	3	no	—
329	#127	C₂³×C₅₃	1	1	yes	characteristic
330	#127	C₂³×C₅₃	1	3	yes	—
331	#124	—	2	6	no	—
332	#124	—	2	3	no	—
333	#124	—	2	3	no	—
334	#123	—	2	12	no	—
335	C₂²⋊Q₈	—	2	6	no	—
336	C₂²⋊Q₈	—	2	6	no	—
337	#121	—	2	3	no	—
338	#121	—	2	3	no	—
339	#121	—	2	3	no	—
340	#121	—	2	3	no	—
341	#121	—	2	6	no	—
342	#121	C₂³×C₅₃	1	6	yes	—
343	#121	C₂³×C₅₃	1	6	yes	—
344	C₄×Q₈	—	2	1	no	—
345	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
346	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
347	C₄×Q₈	C₂³×C₅₃	1	3	yes	—
348	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
349	C₄×D₄	—	2	3	no	—
350	C₄×D₄	—	2	3	no	—
351	C₄×D₄	—	2	3	no	—
352	C₄×D₄	—	2	3	no	—
353	C₄×D₄	—	2	3	no	—
354	C₄×D₄	D₄×C₅₃	1	3	yes	—
355	C₄×D₄	C₂³×C₅₃	1	3	yes	—
356	C₄×D₄	C₂³×C₅₃	1	3	yes	—
357	C₄×D₄	C₂³×C₅₃	1	6	yes	—
358	C₄×D₄	C₂³×C₅₃	1	3	yes	—
359	#117	—	2	3	no	—
360	#117	—	2	3	no	—
361	#117	D₄×C₅₃	1	3	yes	—
362	#117	C₂³×C₅₃	1	3	yes	—
363	C₂×C₄⋊C₄	—	2	3	no	—
364	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
365	C₂×C₄²	—	2	3	no	—
366	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
367	#107	C₂³×C₅₃	1	6	yes	—
368	#106	C₂³×C₅₃	1	12	yes	—
369	#106	C₂³×C₅₃	1	6	yes	—
370	C₄⋊C₈	C₂³×C₅₃	1	3	yes	—
371	C₄⋊C₈	C₂³×C₅₃	1	6	yes	—
372	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
373	#103	C₂³×C₅₃	1	6	yes	—
374	#102	C₂³×C₅₃	1	12	yes	—
375	#102	C₂³×C₅₃	1	2	yes	—
376	#102	C₂³×C₅₃	1	6	yes	—
377	#102	C₂³×C₅₃	1	6	yes	—
378	C₂²⋊C₈	C₂³×C₅₃	1	6	yes	—
379	C₄×C₈	C₂³×C₅₃	1	6	yes	—
380	#41	—	2	4	no	—
381	#41	—	2	6	no	—
382	#41	—	4	3	no	—
383	#41	—	4	12	no	—
384	#41	—	4	24	no	—
385	#41	—	4	4	no	—
386	#41	—	8	6	no	—
387	C₂×Q₈	—	2	6	no	—
388	C₂×Q₈	—	2	3	no	—
389	C₂×Q₈	—	2	3	no	—
390	C₂×Q₈	—	4	12	no	—
391	C₂×Q₈	—	4	8	no	—
392	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
393	C₂×Q₈	C₂⁴×C₅₃	1	1	yes	characteristic
394	C₂×D₄	—	2	3	no	—
395	C₂×D₄	—	2	6	no	—
396	C₂×D₄	—	2	6	no	—
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	2	3	no	—
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	3	no	—
401	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
402	C₂×D₄	C₂⁴×C₅₃	1	3	yes	—
403	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
404	C₂²×C₄	—	2	6	no	—
405	C₂²×C₄	—	2	3	no	—
406	C₂²×C₄	—	2	3	no	—
407	C₂²×C₄	—	2	3	no	—
408	C₂²×C₄	—	2	6	no	—
409	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
410	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
411	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
412	SD₁₆	—	2	4	no	—
413	D₈	—	4	6	no	—
414	M₁₆	—	4	6	no	—
415	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
416	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
417	C₄⋊C₄	—	2	3	no	—
418	C₄⋊C₄	—	2	6	no	—
419	C₄⋊C₄	—	2	6	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	3	no	—
422	C₄⋊C₄	—	2	3	no	—
423	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
424	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
425	C₄⋊C₄	C₂⁴×C₅₃	1	6	yes	—
426	C₄⋊C₄	C₂⁴×C₅₃	1	3	yes	—
427	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
428	C₂²⋊C₄	—	2	3	no	—
429	C₂²⋊C₄	—	2	6	no	—
430	C₂²⋊C₄	—	2	3	no	—
431	C₂²⋊C₄	—	2	3	no	—
432	C₂²⋊C₄	—	2	6	no	—
433	C₂²⋊C₄	—	2	6	no	—
434	C₂²⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
435	C₄²	—	2	1	no	—
436	C₄²	—	2	3	no	—
437	C₄²	—	2	3	no	—
438	C₄²	C₂×D₄×C₅₃	1	3	yes	—
439	C₄²	C₂×D₄×C₅₃	1	3	yes	—
440	C₄²	C₂⁴×C₅₃	1	3	yes	—
441	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
442	C₂³	—	2	2	no	—
443	C₂³	—	2	3	no	—
444	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
445	Q₈	—	2	1	no	—
446	Q₈	—	4	6	no	—
447	Q₈	—	4	12	no	—
448	Q₈	—	8	3	no	—
449	Q₈	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
450	Q₈	2₋¹⁺⁴×C₅₃	1	2	yes	complemented
451	D₄	—	2	6	no	—
452	D₄	—	2	2	no	—
453	D₄	—	2	3	no	—
454	D₄	—	2	2	no	—
455	D₄	—	2	3	no	—
456	D₄	—	4	6	no	—
457	D₄	—	4	2	no	—
458	D₄	—	4	12	no	—
459	D₄	—	4	6	no	—
460	D₄	—	4	2	no	—
461	D₄	—	8	3	no	—
462	D₄	—	8	6	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	6	no	—
465	C₂×C₄	—	2	6	no	—
466	C₂×C₄	—	2	3	no	—
467	C₂×C₄	—	2	3	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	3	no	—
470	C₂×C₄	—	2	6	no	—
471	C₂×C₄	—	4	6	no	—
472	C₂×C₄	—	4	12	no	—
473	C₂×C₄	—	4	12	no	—
474	C₂×C₄	—	4	6	no	—
475	C₂×C₄	—	4	6	no	—
476	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
477	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
478	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
479	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
480	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
481	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
482	C₈	—	2	2	no	—
483	C₈	—	2	6	no	—
484	C₂²	—	2	3	no	—
485	C₂²	—	2	3	no	—
486	C₂²	—	2	3	no	—
487	C₂²	—	4	3	no	—
488	C₂²	—	4	2	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	3	no	—
491	C₂²	—	4	3	no	—
492	C₂²	—	4	2	no	—
493	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
494	C₄	—	2	3	no	—
495	C₄	—	2	2	no	—
496	C₄	—	2	1	no	—
497	C₄	—	2	3	no	—
498	C₄	—	4	6	no	—
499	C₄	—	4	1	no	—
500	C₄	—	4	3	no	—
501	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	6	yes	—
502	C₄	C₅₃×#577	1	2	yes	—
503	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
504	C₄	C₅₃×#585	1	1	yes	characteristic
505	C₂	—	4	3	no	—
506	C₂	—	8	3	no	—
507	C₂	—	8	2	no	—
508	C₂	C₅₃×#3489	1	1	yes	characteristic
509	C₂	C₂²×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
510	C₂	C₂×C₅₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3499	C₂	semidirect
2. C₅₃×#3483	C₂	semidirect
3. Q₈×SD₁₆×C₅₃	C₂	semidirect
4. C₅₃×#3358	C₂	semidirect
5. C₅₃×#3354	C₂	semidirect
6. C₅₃×#3332	C₂	semidirect
7. C₅₃×#3253	C₂	semidirect
8. C₅₃×#3227	C₂	semidirect
9. C₅₃×#3106	C₂	semidirect
10. C₅₃×#3079	C₂	semidirect
11. C₅₃×#2991	C₂	semidirect
194. Q₈×C₅₃	2₋¹⁺⁴	semidirect
228. #62150	C₅₃	direct (N × H)
250. #3499	C₂×C₅₃	semidirect
251. #3483	C₂×C₅₃	semidirect
252. Q₈×SD₁₆	C₂×C₅₃	semidirect
253. #3358	C₂×C₅₃	semidirect
254. #3354	C₂×C₅₃	semidirect
255. #3332	C₂×C₅₃	semidirect
256. #3253	C₂×C₅₃	semidirect
257. #3227	C₂×C₅₃	semidirect
258. #3106	C₂×C₅₃	semidirect
259. #3079	C₂×C₅₃	semidirect
260. #2991	C₂×C₅₃	semidirect
309. C₅₃	#62150	direct (N × H)
450. Q₈	2₋¹⁺⁴×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3499	2
C₅₃×#3483	2
Q₈×SD₁₆×C₅₃	2
C₅₃×#3358	2
C₅₃×#3354	2
C₅₃×#3332	2
C₅₃×#3253	2
C₅₃×#3227	2
C₅₃×#3106	2
C₅₃×#3079	2
C₅₃×#2991	2
#62150	53

Subgroups — C53×#62150 #27269662

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#62150 #27269662