Subgroups — C₅₃×#19554 #27227066

Order 13568.

688 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3253	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3205	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3205	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2979	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1697	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1687	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1677	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1667	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1574	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1545	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1543	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#532	C₂²	1	1	yes	characteristic
13	C₅₃×#532	C₂²	1	1	yes	characteristic
14	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#493	—	2	1	no	—
16	C₅₃×#493	C₂²	1	2	yes	—
17	C₅₃×#486	—	2	1	no	—
18	C₅₃×#486	C₂²	1	2	yes	—
19	C₅₃×#476	—	2	1	no	—
20	C₅₃×#476	C₂²	1	2	yes	complemented
21	C₅₃×#469	—	2	2	no	—
22	C₅₃×#459	—	2	1	no	—
23	C₅₃×#459	—	2	1	no	—
24	C₅₃×#459	—	2	1	no	—
25	C₅₃×#459	C₂²	1	2	yes	—
26	C₅₃×#459	C₂²	1	2	yes	—
27	C₅₃×#459	C₂²	1	2	yes	complemented
28	C₅₃×#447	—	2	2	no	—
29	C₅₃×#447	—	2	2	no	—
30	C₅₃×#436	—	2	1	no	—
31	C₅₃×#436	C₂²	1	2	yes	complemented
32	C₈×D₄×C₅₃	—	2	1	no	—
33	C₈×D₄×C₅₃	C₂²	1	2	yes	complemented
34	C₅₃×#431	C₂²	1	1	yes	characteristic
35	C₅₃×#431	C₂²	1	1	yes	characteristic
36	C₅₃×#431	C₂²	1	1	yes	characteristic
37	C₅₃×#423	C₂²	1	1	yes	characteristic
38	C₅₃×#408	—	2	2	no	—
39	C₅₃×#335	C₂²	1	2	yes	complemented
40	C₅₃×#334	C₂²	1	2	yes	complemented
41	C₅₃×#333	C₂²	1	1	yes	characteristic, complemented
42	C₅₃×#333	C₂²	1	1	yes	characteristic, complemented
43	C₅₃×#331	C₂²	1	2	yes	—
44	C₅₃×#331	C₂²	1	2	yes	—
45	C₅₃×#331	C₂²	1	1	yes	characteristic, complemented
46	C₅₃×#331	C₂²	1	1	yes	characteristic, complemented
47	C₂×#41×C₅₃	—	2	1	no	—
48	C₂×#41×C₅₃	—	2	1	no	—
49	C₂²×D₄×C₅₃	—	2	1	no	—
50	C₂²×D₄×C₅₃	—	2	1	no	—
51	C₂²×D₄×C₅₃	—	2	1	no	—
52	C₂×D₈×C₅₃	—	2	2	no	—
53	C₂×D₈×C₅₃	—	2	2	no	—
54	C₂×D₈×C₅₃	—	2	2	no	—
55	C₂×D₈×C₅₃	—	2	2	no	—
56	C₂×D₈×C₅₃	—	2	2	no	—
57	C₂×M₁₆×C₅₃	—	2	2	no	—
58	C₂×M₁₆×C₅₃	—	2	1	no	—
59	C₂²×C₈×C₅₃	—	2	2	no	—
60	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
62	#127×C₅₃	—	2	1	no	—
63	#127×C₅₃	—	2	1	no	—
64	#127×C₅₃	—	2	1	no	—
65	#127×C₅₃	D₄	1	1	yes	characteristic
66	#127×C₅₃	C₂³	1	2	yes	—
67	#127×C₅₃	C₂³	1	1	yes	characteristic
68	#124×C₅₃	—	2	1	no	—
69	#124×C₅₃	—	2	1	no	—
70	#121×C₅₃	—	2	1	no	—
71	#121×C₅₃	—	2	1	no	—
72	#121×C₅₃	—	2	1	no	—
73	#121×C₅₃	D₄	1	2	yes	—
74	C₄×D₄×C₅₃	—	2	1	no	—
75	C₄×D₄×C₅₃	—	2	1	no	—
76	C₄×D₄×C₅₃	—	2	1	no	—
77	C₄×D₄×C₅₃	C₂³	1	2	yes	—
78	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
79	#106×C₅₃	—	2	2	no	—
80	C₄⋊C₈×C₅₃	—	2	1	no	—
81	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
82	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
83	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
86	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
88	#102×C₅₃	—	2	2	no	—
89	#102×C₅₃	—	2	2	no	—
90	#102×C₅₃	—	2	2	no	—
91	#102×C₅₃	—	2	2	no	—
92	#102×C₅₃	—	2	2	no	—
93	#102×C₅₃	—	2	2	no	—
94	C₂²⋊C₈×C₅₃	—	2	2	no	—
95	C₂²⋊C₈×C₅₃	—	2	2	no	—
96	C₂²⋊C₈×C₅₃	—	2	1	no	—
97	C₂²⋊C₈×C₅₃	—	2	1	no	—
98	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
99	C₄×C₈×C₅₃	—	2	1	no	—
100	C₄×C₈×C₅₃	—	2	1	no	—
101	C₄×C₈×C₅₃	C₂³	1	2	yes	—
102	C₂⁴×C₅₃	—	4	1	no	—
103	#41×C₅₃	—	8	1	no	—
104	#41×C₅₃	—	8	1	no	—
105	#41×C₅₃	—	8	2	no	—
106	C₂×Q₈×C₅₃	—	2	1	no	—
107	C₂×Q₈×C₅₃	—	2	1	no	—
108	C₂×D₄×C₅₃	—	2	2	no	—
109	C₂×D₄×C₅₃	—	2	1	no	—
110	C₂×D₄×C₅₃	—	2	1	no	—
111	C₂×D₄×C₅₃	—	2	1	no	—
112	C₂×D₄×C₅₃	—	2	2	no	—
113	C₂×D₄×C₅₃	—	2	2	no	—
114	C₂×D₄×C₅₃	—	2	1	no	—
115	C₂×D₄×C₅₃	—	2	1	no	—
116	C₂×D₄×C₅₃	—	2	2	no	—
117	C₂×D₄×C₅₃	—	4	1	no	—
118	C₂×D₄×C₅₃	—	4	1	no	—
119	C₂×D₄×C₅₃	—	4	1	no	—
120	C₂×D₄×C₅₃	—	4	1	no	—
121	C₂×D₄×C₅₃	—	8	1	no	—
122	C₂×D₄×C₅₃	—	8	1	no	—
123	C₂×D₄×C₅₃	—	8	1	no	—
124	C₂×D₄×C₅₃	—	8	1	no	—
125	C₂×D₄×C₅₃	—	8	2	no	—
126	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₅₃	—	2	2	no	—
128	C₂²×C₄×C₅₃	—	4	1	no	—
129	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
132	D₈×C₅₃	—	8	2	no	—
133	D₈×C₅₃	—	8	2	no	—
134	D₈×C₅₃	—	8	2	no	—
135	D₈×C₅₃	—	8	2	no	—
136	D₈×C₅₃	—	8	2	no	—
137	M₁₆×C₅₃	—	8	2	no	—
138	M₁₆×C₅₃	—	8	1	no	—
139	C₂×C₈×C₅₃	—	2	1	no	—
140	C₂×C₈×C₅₃	—	2	1	no	—
141	C₂×C₈×C₅₃	—	2	1	no	—
142	C₂×C₈×C₅₃	—	2	1	no	—
143	C₂×C₈×C₅₃	—	2	2	no	—
144	C₂×C₈×C₅₃	—	8	2	no	—
145	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
146	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
147	C₄⋊C₄×C₅₃	—	2	1	no	—
148	C₄⋊C₄×C₅₃	—	2	1	no	—
149	C₄⋊C₄×C₅₃	—	2	1	no	—
150	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
151	C₂²⋊C₄×C₅₃	—	2	2	no	—
152	C₂²⋊C₄×C₅₃	—	2	2	no	—
153	C₂²⋊C₄×C₅₃	—	4	1	no	—
154	C₂²⋊C₄×C₅₃	—	4	1	no	—
155	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
156	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₅₃	#41	1	1	yes	characteristic
158	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
159	C₂³×C₅₃	—	2	2	no	—
160	C₂³×C₅₃	—	4	1	no	—
161	C₂³×C₅₃	—	4	2	no	—
162	C₂³×C₅₃	—	8	2	no	—
163	C₂³×C₅₃	—	8	2	no	—
164	C₂³×C₅₃	—	8	2	no	—
165	C₂³×C₅₃	#121	1	1	yes	characteristic
166	Q₈×C₅₃	—	8	1	no	—
167	Q₈×C₅₃	—	8	1	no	—
168	D₄×C₅₃	—	2	2	no	—
169	D₄×C₅₃	—	2	2	no	—
170	D₄×C₅₃	—	4	2	no	—
171	D₄×C₅₃	—	4	2	no	—
172	D₄×C₅₃	—	4	2	no	—
173	D₄×C₅₃	—	4	1	no	—
174	D₄×C₅₃	—	4	1	no	—
175	D₄×C₅₃	—	4	1	no	—
176	D₄×C₅₃	—	4	1	no	—
177	D₄×C₅₃	—	4	2	no	—
178	D₄×C₅₃	—	4	2	no	—
179	D₄×C₅₃	—	8	1	no	—
180	D₄×C₅₃	—	8	2	no	—
181	D₄×C₅₃	—	8	1	no	—
182	D₄×C₅₃	—	8	2	no	—
183	D₄×C₅₃	—	8	2	no	—
184	D₄×C₅₃	—	8	2	no	—
185	D₄×C₅₃	—	8	2	no	—
186	D₄×C₅₃	—	8	2	no	—
187	C₂×C₄×C₅₃	—	2	1	no	—
188	C₂×C₄×C₅₃	—	2	1	no	—
189	C₂×C₄×C₅₃	—	2	1	no	—
190	C₂×C₄×C₅₃	—	2	1	no	—
191	C₂×C₄×C₅₃	—	2	2	no	—
192	C₂×C₄×C₅₃	—	2	2	no	—
193	C₂×C₄×C₅₃	—	4	2	no	—
194	C₂×C₄×C₅₃	—	4	2	no	—
195	C₂×C₄×C₅₃	—	4	1	no	—
196	C₂×C₄×C₅₃	—	8	2	no	—
197	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
198	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
199	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
200	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
203	C₈×C₅₃	—	2	2	no	—
204	C₈×C₅₃	—	2	2	no	—
205	C₈×C₅₃	—	4	1	no	—
206	C₈×C₅₃	—	4	1	no	—
207	C₈×C₅₃	—	4	2	no	—
208	C₈×C₅₃	—	4	1	no	—
209	C₈×C₅₃	—	4	1	no	—
210	#19554	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂²×C₅₃	—	2	1	no	—
212	C₂²×C₅₃	—	2	1	no	—
213	C₂²×C₅₃	—	2	1	no	—
214	C₂²×C₅₃	—	4	2	no	—
215	C₂²×C₅₃	—	4	2	no	—
216	C₂²×C₅₃	—	4	2	no	—
217	C₂²×C₅₃	—	8	2	no	—
218	C₂²×C₅₃	—	8	1	no	—
219	C₂²×C₅₃	—	8	2	no	—
220	C₂²×C₅₃	—	8	1	no	—
221	C₂²×C₅₃	—	8	1	no	—
222	C₂²×C₅₃	—	8	2	no	—
223	C₂²×C₅₃	—	16	4	no	—
224	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
225	C₄×C₅₃	—	2	1	no	—
226	C₄×C₅₃	—	2	1	no	—
227	C₄×C₅₃	—	2	1	no	—
228	C₄×C₅₃	—	4	2	no	—
229	C₄×C₅₃	—	8	1	no	—
230	C₄×C₅₃	#472	1	2	yes	—
231	C₄×C₅₃	#576	1	1	yes	characteristic
232	C₄×C₅₃	#576	1	1	yes	characteristic
233	C₄×C₅₃	#576	1	1	yes	characteristic
234	C₄×C₅₃	#577	1	1	yes	characteristic
235	#3253	C₂×C₅₃	1	1	yes	characteristic, complemented
236	#3205	C₂×C₅₃	1	1	yes	characteristic, complemented
237	#3205	C₂×C₅₃	1	1	yes	characteristic, complemented
238	#2979	C₂×C₅₃	1	1	yes	characteristic, complemented
239	#1697	C₂×C₅₃	1	2	yes	complemented
240	#1687	C₂×C₅₃	1	2	yes	complemented
241	#1677	C₂×C₅₃	1	2	yes	complemented
242	#1667	C₂×C₅₃	1	2	yes	complemented
243	#1574	C₂×C₅₃	1	1	yes	characteristic, complemented
244	#1545	C₂×C₅₃	1	1	yes	characteristic, complemented
245	#1543	C₂×C₅₃	1	1	yes	characteristic, complemented
246	C₂×C₅₃	—	4	1	no	—
247	C₂×C₅₃	—	8	2	no	—
248	C₂×C₅₃	—	16	1	no	—
249	C₂×C₅₃	—	16	2	no	—
250	C₂×C₅₃	#3041	1	1	yes	characteristic
251	C₂×C₅₃	#3062	1	1	yes	characteristic
252	C₂×C₅₃	C₂×#472	1	1	yes	characteristic
253	#532	C₂²×C₅₃	1	1	yes	characteristic
254	#532	C₂²×C₅₃	1	1	yes	characteristic
255	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
256	#493	—	2	1	no	—
257	#493	C₂²×C₅₃	1	2	yes	—
258	#486	—	2	1	no	—
259	#486	C₂²×C₅₃	1	2	yes	—
260	#476	—	2	1	no	—
261	#476	C₂²×C₅₃	1	2	yes	complemented
262	#469	—	2	2	no	—
263	#459	—	2	1	no	—
264	#459	—	2	1	no	—
265	#459	—	2	1	no	—
266	#459	C₂²×C₅₃	1	2	yes	—
267	#459	C₂²×C₅₃	1	2	yes	—
268	#459	C₂²×C₅₃	1	2	yes	complemented
269	#447	—	2	2	no	—
270	#447	—	2	2	no	—
271	#436	—	2	1	no	—
272	#436	C₂²×C₅₃	1	2	yes	complemented
273	C₈×D₄	—	2	1	no	—
274	C₈×D₄	C₂²×C₅₃	1	2	yes	complemented
275	#431	C₂²×C₅₃	1	1	yes	characteristic
276	#431	C₂²×C₅₃	1	1	yes	characteristic
277	#431	C₂²×C₅₃	1	1	yes	characteristic
278	#423	C₂²×C₅₃	1	1	yes	characteristic
279	#408	—	2	2	no	—
280	#335	C₂²×C₅₃	1	2	yes	complemented
281	#334	C₂²×C₅₃	1	2	yes	complemented
282	#333	C₂²×C₅₃	1	1	yes	characteristic, complemented
283	#333	C₂²×C₅₃	1	1	yes	characteristic, complemented
284	#331	C₂²×C₅₃	1	2	yes	—
285	#331	C₂²×C₅₃	1	2	yes	—
286	#331	C₂²×C₅₃	1	1	yes	characteristic, complemented
287	#331	C₂²×C₅₃	1	1	yes	characteristic, complemented
288	C₅₃	#19554	1	1	yes	characteristic, direct product factor, Sylow
289	C₂×#41	—	2	1	no	—
290	C₂×#41	—	2	1	no	—
291	C₂²×D₄	—	2	1	no	—
292	C₂²×D₄	—	2	1	no	—
293	C₂²×D₄	—	2	1	no	—
294	C₂×D₈	—	2	2	no	—
295	C₂×D₈	—	2	2	no	—
296	C₂×D₈	—	2	2	no	—
297	C₂×D₈	—	2	2	no	—
298	C₂×D₈	—	2	2	no	—
299	C₂×M₁₆	—	2	1	no	—
300	C₂×M₁₆	—	2	2	no	—
301	C₂²×C₈	—	2	2	no	—
302	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
303	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
304	#127	—	2	1	no	—
305	#127	—	2	1	no	—
306	#127	—	2	1	no	—
307	#127	D₄×C₅₃	1	1	yes	characteristic
308	#127	C₂³×C₅₃	1	2	yes	—
309	#127	C₂³×C₅₃	1	1	yes	characteristic
310	#124	—	2	1	no	—
311	#124	—	2	1	no	—
312	#121	—	2	1	no	—
313	#121	—	2	1	no	—
314	#121	—	2	1	no	—
315	#121	D₄×C₅₃	1	2	yes	—
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	—	2	1	no	—
319	C₄×D₄	C₂³×C₅₃	1	2	yes	—
320	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
321	#106	—	2	2	no	—
322	C₄⋊C₈	—	2	1	no	—
323	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
324	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
325	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
326	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
327	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
328	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
329	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
330	#102	—	2	2	no	—
331	#102	—	2	2	no	—
332	#102	—	2	2	no	—
333	#102	—	2	2	no	—
334	#102	—	2	2	no	—
335	#102	—	2	2	no	—
336	C₂²⋊C₈	—	2	1	no	—
337	C₂²⋊C₈	—	2	1	no	—
338	C₂²⋊C₈	—	2	2	no	—
339	C₂²⋊C₈	—	2	2	no	—
340	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
341	C₄×C₈	—	2	1	no	—
342	C₄×C₈	—	2	1	no	—
343	C₄×C₈	C₂³×C₅₃	1	2	yes	—
344	C₂⁴	—	4	1	no	—
345	#41	—	8	1	no	—
346	#41	—	8	1	no	—
347	#41	—	8	2	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	1	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	1	no	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	2	1	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	4	1	no	—
360	C₂×D₄	—	4	1	no	—
361	C₂×D₄	—	4	1	no	—
362	C₂×D₄	—	4	1	no	—
363	C₂×D₄	—	8	1	no	—
364	C₂×D₄	—	8	1	no	—
365	C₂×D₄	—	8	1	no	—
366	C₂×D₄	—	8	1	no	—
367	C₂×D₄	—	8	2	no	—
368	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	4	1	no	—
371	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
373	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
374	D₈	—	8	2	no	—
375	D₈	—	8	2	no	—
376	D₈	—	8	2	no	—
377	D₈	—	8	2	no	—
378	D₈	—	8	2	no	—
379	M₁₆	—	8	2	no	—
380	M₁₆	—	8	1	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	8	2	no	—
387	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
388	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
393	C₂²⋊C₄	—	2	2	no	—
394	C₂²⋊C₄	—	2	2	no	—
395	C₂²⋊C₄	—	4	1	no	—
396	C₂²⋊C₄	—	4	1	no	—
397	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
398	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
399	C₄²	#41×C₅₃	1	1	yes	characteristic
400	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
401	C₂³	—	2	2	no	—
402	C₂³	—	4	1	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	8	2	no	—
405	C₂³	—	8	2	no	—
406	C₂³	—	8	2	no	—
407	C₂³	#121×C₅₃	1	1	yes	characteristic
408	Q₈	—	8	1	no	—
409	Q₈	—	8	1	no	—
410	D₄	—	2	2	no	—
411	D₄	—	2	2	no	—
412	D₄	—	4	1	no	—
413	D₄	—	4	1	no	—
414	D₄	—	4	1	no	—
415	D₄	—	4	2	no	—
416	D₄	—	4	2	no	—
417	D₄	—	4	2	no	—
418	D₄	—	4	2	no	—
419	D₄	—	4	1	no	—
420	D₄	—	4	2	no	—
421	D₄	—	8	1	no	—
422	D₄	—	8	2	no	—
423	D₄	—	8	1	no	—
424	D₄	—	8	2	no	—
425	D₄	—	8	2	no	—
426	D₄	—	8	2	no	—
427	D₄	—	8	2	no	—
428	D₄	—	8	2	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	8	2	no	—
439	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
440	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
441	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
442	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
443	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
444	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
445	C₈	—	2	2	no	—
446	C₈	—	2	2	no	—
447	C₈	—	4	1	no	—
448	C₈	—	4	1	no	—
449	C₈	—	4	2	no	—
450	C₈	—	4	1	no	—
451	C₈	—	4	1	no	—
452	C₂²	—	2	1	no	—
453	C₂²	—	2	1	no	—
454	C₂²	—	2	1	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	4	2	no	—
457	C₂²	—	4	2	no	—
458	C₂²	—	8	2	no	—
459	C₂²	—	8	1	no	—
460	C₂²	—	8	2	no	—
461	C₂²	—	8	1	no	—
462	C₂²	—	8	1	no	—
463	C₂²	—	8	2	no	—
464	C₂²	—	16	4	no	—
465	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
466	C₄	—	2	1	no	—
467	C₄	—	2	1	no	—
468	C₄	—	2	1	no	—
469	C₄	—	4	2	no	—
470	C₄	—	8	1	no	—
471	C₄	C₅₃×#472	1	2	yes	—
472	C₄	C₅₃×#576	1	1	yes	characteristic
473	C₄	C₅₃×#576	1	1	yes	characteristic
474	C₄	C₅₃×#576	1	1	yes	characteristic
475	C₄	C₅₃×#577	1	1	yes	characteristic
476	C₂	—	4	1	no	—
477	C₂	—	8	2	no	—
478	C₂	—	16	2	no	—
479	C₂	—	16	1	no	—
480	C₂	C₅₃×#3041	1	1	yes	characteristic
481	C₂	C₅₃×#3062	1	1	yes	characteristic
482	C₂	C₂×C₅₃×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3253	C₂	semidirect
2. C₅₃×#3205	C₂	semidirect
3. C₅₃×#3205	C₂	semidirect
4. C₅₃×#2979	C₂	semidirect
5. C₅₃×#1697	C₂	semidirect
6. C₅₃×#1687	C₂	semidirect
7. C₅₃×#1677	C₂	semidirect
8. C₅₃×#1667	C₂	semidirect
9. C₅₃×#1574	C₂	semidirect
10. C₅₃×#1545	C₂	semidirect
11. C₅₃×#1543	C₂	semidirect
20. C₅₃×#476	C₂²	semidirect
27. C₅₃×#459	C₂²	semidirect
31. C₅₃×#436	C₂²	semidirect
33. C₈×D₄×C₅₃	C₂²	semidirect
39. C₅₃×#335	C₂²	semidirect
40. C₅₃×#334	C₂²	semidirect
41. C₅₃×#333	C₂²	semidirect
42. C₅₃×#333	C₂²	semidirect
45. C₅₃×#331	C₂²	semidirect
46. C₅₃×#331	C₂²	semidirect
210. #19554	C₅₃	direct (N × H)
235. #3253	C₂×C₅₃	semidirect
236. #3205	C₂×C₅₃	semidirect
237. #3205	C₂×C₅₃	semidirect
238. #2979	C₂×C₅₃	semidirect
239. #1697	C₂×C₅₃	semidirect
240. #1687	C₂×C₅₃	semidirect
241. #1677	C₂×C₅₃	semidirect
242. #1667	C₂×C₅₃	semidirect
243. #1574	C₂×C₅₃	semidirect
244. #1545	C₂×C₅₃	semidirect
245. #1543	C₂×C₅₃	semidirect
261. #476	C₂²×C₅₃	semidirect
268. #459	C₂²×C₅₃	semidirect
272. #436	C₂²×C₅₃	semidirect
274. C₈×D₄	C₂²×C₅₃	semidirect
280. #335	C₂²×C₅₃	semidirect
281. #334	C₂²×C₅₃	semidirect
282. #333	C₂²×C₅₃	semidirect
283. #333	C₂²×C₅₃	semidirect
286. #331	C₂²×C₅₃	semidirect
287. #331	C₂²×C₅₃	semidirect
288. C₅₃	#19554	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×#3253	2
C₅₃×#3205	2
C₅₃×#2979	2
C₅₃×#1697	2
C₅₃×#1687	2
C₅₃×#1677	2
C₅₃×#1667	2
C₅₃×#1574	2
C₅₃×#1545	2
C₅₃×#1543	2
#19554	53

Subgroups — C53×#19554 #27227066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19554 #27227066