Subgroups — C₅₃×#19621 #27227133

Order 13568.

900 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3227	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3167	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3165	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3151	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1715	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1714	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1706	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1705	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
13	C₂²×SD₁₆×C₅₃	—	2	2	no	—
14	C₅₃×#533	C₂²	1	1	yes	characteristic
15	C₅₃×#532	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#501	—	2	2	no	—
19	C₅₃×#500	C₂²	1	2	yes	complemented
20	C₅₃×#498	C₂²	1	2	yes	complemented
21	C₅₃×#498	C₂²	1	1	yes	characteristic, complemented
22	C₅₃×#498	C₂²	1	2	yes	complemented
23	C₅₃×#496	—	2	2	no	—
24	C₅₃×#493	C₂²	1	2	yes	—
25	C₅₃×#492	—	2	2	no	—
26	C₅₃×#492	C₂²	1	2	yes	—
27	C₅₃×#492	C₂²	1	2	yes	—
28	C₅₃×#492	C₂²	1	2	yes	—
29	C₅₃×#492	C₂²	1	2	yes	—
30	C₅₃×#492	C₂²	1	1	yes	characteristic, complemented
31	C₅₃×#487	C₂²	1	1	yes	characteristic, complemented
32	C₅₃×#486	C₂²	1	2	yes	—
33	C₅₃×#486	C₂²	1	2	yes	complemented
34	C₅₃×#486	C₂²	1	1	yes	characteristic, complemented
35	C₅₃×#481	—	2	2	no	—
36	C₅₃×#476	—	2	2	no	—
37	C₅₃×#465	—	2	2	no	—
38	C₅₃×#460	—	2	2	no	—
39	C₅₃×#450	—	2	2	no	—
40	C₄×SD₁₆×C₅₃	—	2	2	no	—
41	C₅₃×#431	C₂²	1	1	yes	characteristic
42	C₅₃×#428	—	2	2	no	—
43	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#322	C₂²	1	2	yes	complemented
46	C₈²×C₅₃	C₂²	1	2	yes	complemented
47	C₂×#41×C₅₃	—	2	1	no	—
48	C₂²×Q₈×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	AGL(1,ℤ/8)×C₅₃	—	4	8	no	—
53	C₂×SD₁₆×C₅₃	—	2	4	no	—
54	C₂×SD₁₆×C₅₃	—	2	4	no	—
55	C₂×SD₁₆×C₅₃	—	2	4	no	—
56	C₂×SD₁₆×C₅₃	—	2	2	no	—
57	C₂×SD₁₆×C₅₃	—	2	4	no	—
58	C₂×SD₁₆×C₅₃	—	4	8	no	—
59	C₂×D₈×C₅₃	—	2	4	no	—
60	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
61	C₂²×C₈×C₅₃	D₄	1	2	yes	—
62	C₄⋊Q₈×C₅₃	—	2	1	no	—
63	C₄⋊Q₈×C₅₃	—	2	1	no	—
64	C₄⋊Q₈×C₅₃	—	2	1	no	—
65	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
68	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
69	#127×C₅₃	—	2	1	no	—
70	#127×C₅₃	—	2	1	no	—
71	#127×C₅₃	—	2	1	no	—
72	#127×C₅₃	C₂³	1	2	yes	—
73	#127×C₅₃	C₂³	1	1	yes	characteristic
74	#125×C₅₃	—	2	1	no	—
75	#124×C₅₃	—	2	1	no	—
76	C₂²⋊Q₈×C₅₃	—	2	1	no	—
77	C₂²⋊Q₈×C₅₃	—	2	1	no	—
78	#121×C₅₃	—	2	1	no	—
79	#121×C₅₃	—	2	1	no	—
80	C₄×Q₈×C₅₃	—	2	1	no	—
81	C₄×Q₈×C₅₃	—	2	1	no	—
82	C₄×D₄×C₅₃	—	2	1	no	—
83	C₄×D₄×C₅₃	—	2	1	no	—
84	#117×C₅₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
87	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
88	#107×C₅₃	—	2	4	no	—
89	#106×C₅₃	—	2	4	no	—
90	#106×C₅₃	—	2	2	no	—
91	#106×C₅₃	—	2	4	no	—
92	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
93	#103×C₅₃	—	2	2	no	—
94	#102×C₅₃	—	2	4	no	—
95	#102×C₅₃	—	2	2	no	—
96	#102×C₅₃	—	2	4	no	—
97	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
98	#97×C₅₃	D₄	1	2	yes	—
99	C₄×C₈×C₅₃	D₄	1	2	yes	—
100	C₄×C₈×C₅₃	C₂³	1	2	yes	—
101	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
102	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
103	C₄×C₈×C₅₃	C₂³	1	2	yes	—
104	C₂⁴×C₅₃	—	4	1	no	—
105	#41×C₅₃	—	4	4	no	—
106	C₂×Q₈×C₅₃	—	2	1	no	—
107	C₂×Q₈×C₅₃	—	2	1	no	—
108	C₂×Q₈×C₅₃	—	2	1	no	—
109	C₂×Q₈×C₅₃	—	2	2	no	—
110	C₂×Q₈×C₅₃	—	4	4	no	—
111	C₂×Q₈×C₅₃	—	4	1	no	—
112	C₂×D₄×C₅₃	—	2	1	no	—
113	C₂×D₄×C₅₃	—	2	1	no	—
114	C₂×D₄×C₅₃	—	2	2	no	—
115	C₂×D₄×C₅₃	—	2	1	no	—
116	C₂×D₄×C₅₃	—	2	2	no	—
117	C₂×D₄×C₅₃	—	2	2	no	—
118	C₂×D₄×C₅₃	—	4	4	no	—
119	C₂×D₄×C₅₃	—	4	1	no	—
120	C₂×D₄×C₅₃	—	4	1	no	—
121	C₂×D₄×C₅₃	—	4	1	no	—
122	C₂×D₄×C₅₃	—	4	1	no	—
123	C₂×D₄×C₅₃	—	4	4	no	—
124	C₂×D₄×C₅₃	—	8	2	no	—
125	C₂²×C₄×C₅₃	—	4	1	no	—
126	C₂²×C₄×C₅₃	—	4	1	no	—
127	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
130	SD₁₆×C₅₃	—	4	8	no	—
131	SD₁₆×C₅₃	—	4	8	no	—
132	SD₁₆×C₅₃	—	4	4	no	—
133	SD₁₆×C₅₃	—	4	8	no	—
134	SD₁₆×C₅₃	—	8	4	no	—
135	D₈×C₅₃	—	4	8	no	—
136	M₁₆×C₅₃	SD₁₆	1	8	yes	complemented
137	C₂×C₈×C₅₃	—	2	4	no	—
138	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
140	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
141	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
142	C₄⋊C₄×C₅₃	—	2	1	no	—
143	C₄⋊C₄×C₅₃	—	2	2	no	—
144	C₄⋊C₄×C₅₃	—	2	1	no	—
145	C₄⋊C₄×C₅₃	—	2	1	no	—
146	C₄⋊C₄×C₅₃	—	2	1	no	—
147	C₄⋊C₄×C₅₃	—	2	1	no	—
148	C₄⋊C₄×C₅₃	—	2	2	no	—
149	C₄⋊C₄×C₅₃	—	2	1	no	—
150	C₄⋊C₄×C₅₃	—	4	1	no	—
151	C₄⋊C₄×C₅₃	—	4	1	no	—
152	C₄⋊C₄×C₅₃	—	4	1	no	—
153	C₄⋊C₄×C₅₃	—	4	1	no	—
154	C₂²⋊C₄×C₅₃	—	4	1	no	—
155	C₂²⋊C₄×C₅₃	—	4	1	no	—
156	C₂²⋊C₄×C₅₃	—	4	1	no	—
157	C₄²×C₅₃	—	4	1	no	—
158	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
159	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
160	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
161	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
162	C₂³×C₅₃	—	4	1	no	—
163	C₂³×C₅₃	—	4	2	no	—
164	C₂³×C₅₃	—	8	2	no	—
165	C₂³×C₅₃	—	8	2	no	—
166	C₂³×C₅₃	—	8	2	no	—
167	C₂³×C₅₃	#127	1	1	yes	characteristic
168	Q₈×C₅₃	—	4	2	no	—
169	Q₈×C₅₃	—	4	4	no	—
170	Q₈×C₅₃	—	4	2	no	—
171	Q₈×C₅₃	—	4	2	no	—
172	Q₈×C₅₃	—	8	2	no	—
173	D₄×C₅₃	—	4	4	no	—
174	D₄×C₅₃	—	4	2	no	—
175	D₄×C₅₃	—	4	4	no	—
176	D₄×C₅₃	—	4	2	no	—
177	D₄×C₅₃	—	4	2	no	—
178	D₄×C₅₃	—	4	4	no	—
179	D₄×C₅₃	—	8	2	no	—
180	D₄×C₅₃	—	8	2	no	—
181	D₄×C₅₃	—	8	2	no	—
182	D₄×C₅₃	—	8	2	no	—
183	C₂×C₄×C₅₃	—	2	2	no	—
184	C₂×C₄×C₅₃	—	2	2	no	—
185	C₂×C₄×C₅₃	—	4	1	no	—
186	C₂×C₄×C₅₃	—	4	2	no	—
187	C₂×C₄×C₅₃	—	4	1	no	—
188	C₂×C₄×C₅₃	—	4	1	no	—
189	C₂×C₄×C₅₃	—	8	2	no	—
190	C₂×C₄×C₅₃	—	8	2	no	—
191	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
192	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
193	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
195	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
198	C₈×C₅₃	—	2	2	no	—
199	C₈×C₅₃	—	2	4	no	—
200	C₈×C₅₃	C₂×SD₁₆	1	8	yes	complemented
201	C₈×C₅₃	AGL(1,ℤ/8)	1	4	yes	complemented
202	#19621	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂²×C₅₃	—	2	1	no	—
204	C₂²×C₅₃	—	2	1	no	—
205	C₂²×C₅₃	—	8	1	no	—
206	C₂²×C₅₃	—	8	2	no	—
207	C₂²×C₅₃	—	8	2	no	—
208	C₂²×C₅₃	—	8	1	no	—
209	C₂²×C₅₃	—	8	2	no	—
210	C₂²×C₅₃	—	8	1	no	—
211	C₂²×C₅₃	—	16	4	no	—
212	C₂²×C₅₃	#492	1	2	yes	—
213	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
214	C₄×C₅₃	—	2	1	no	—
215	C₄×C₅₃	—	2	1	no	—
216	C₄×C₅₃	—	8	1	no	—
217	C₄×C₅₃	—	8	2	no	—
218	C₄×C₅₃	—	8	1	no	—
219	C₄×C₅₃	—	8	1	no	—
220	C₄×C₅₃	#492	1	2	yes	—
221	C₄×C₅₃	C₂²×SD₁₆	1	2	yes	—
222	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
223	C₄×C₅₃	#577	1	2	yes	—
224	#3227	C₂×C₅₃	1	1	yes	characteristic, complemented
225	#3208	C₂×C₅₃	1	1	yes	characteristic, complemented
226	#3167	C₂×C₅₃	1	1	yes	characteristic, complemented
227	#3165	C₂×C₅₃	1	1	yes	characteristic, complemented
228	#3151	C₂×C₅₃	1	1	yes	characteristic
229	C₂×#492	C₂×C₅₃	1	1	yes	characteristic, complemented
230	#1715	C₂×C₅₃	1	2	yes	complemented
231	#1714	C₂×C₅₃	1	2	yes	complemented
232	#1706	C₂×C₅₃	1	2	yes	complemented
233	#1705	C₂×C₅₃	1	2	yes	complemented
234	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
235	C₂×C₅₃	—	2	2	no	—
236	C₂×C₅₃	—	16	1	no	—
237	C₂×C₅₃	—	16	2	no	—
238	C₂×C₅₃	C₂×#492	1	1	yes	characteristic
239	C₂×C₅₃	#3151	1	1	yes	characteristic
240	C₂×C₅₃	#3154	1	1	yes	characteristic
241	C₂×AGL(1,ℤ/8)	—	2	2	no	—
242	C₂²×SD₁₆	—	2	2	no	—
243	#533	C₂²×C₅₃	1	1	yes	characteristic
244	#532	C₂²×C₅₃	1	1	yes	characteristic
245	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
246	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
247	#501	—	2	2	no	—
248	#500	C₂²×C₅₃	1	2	yes	complemented
249	#498	C₂²×C₅₃	1	2	yes	complemented
250	#498	C₂²×C₅₃	1	1	yes	characteristic, complemented
251	#498	C₂²×C₅₃	1	2	yes	complemented
252	#496	—	2	2	no	—
253	#493	C₂²×C₅₃	1	2	yes	—
254	#492	—	2	2	no	—
255	#492	C₂²×C₅₃	1	2	yes	—
256	#492	C₂²×C₅₃	1	2	yes	—
257	#492	C₂²×C₅₃	1	2	yes	—
258	#492	C₂²×C₅₃	1	2	yes	—
259	#492	C₂²×C₅₃	1	1	yes	characteristic, complemented
260	#487	C₂²×C₅₃	1	1	yes	characteristic, complemented
261	#486	C₂²×C₅₃	1	2	yes	—
262	#486	C₂²×C₅₃	1	1	yes	characteristic, complemented
263	#486	C₂²×C₅₃	1	2	yes	complemented
264	#481	—	2	2	no	—
265	#476	—	2	2	no	—
266	#465	—	2	2	no	—
267	#460	—	2	2	no	—
268	#450	—	2	2	no	—
269	C₄×SD₁₆	—	2	2	no	—
270	#431	C₂²×C₅₃	1	1	yes	characteristic
271	#428	—	2	2	no	—
272	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
273	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
274	#322	C₂²×C₅₃	1	2	yes	complemented
275	C₈²	C₂²×C₅₃	1	2	yes	complemented
276	C₅₃	#19621	1	1	yes	characteristic, direct product factor, Sylow
277	C₂×#41	—	2	1	no	—
278	C₂²×Q₈	—	2	1	no	—
279	C₂²×D₄	—	2	1	no	—
280	C₂²×D₄	—	2	1	no	—
281	C₂²×D₄	—	2	1	no	—
282	AGL(1,ℤ/8)	—	4	8	no	—
283	C₂×SD₁₆	—	2	4	no	—
284	C₂×SD₁₆	—	2	4	no	—
285	C₂×SD₁₆	—	2	4	no	—
286	C₂×SD₁₆	—	2	2	no	—
287	C₂×SD₁₆	—	2	4	no	—
288	C₂×SD₁₆	—	4	8	no	—
289	C₂×D₈	—	2	4	no	—
290	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
291	C₂²×C₈	D₄×C₅₃	1	2	yes	—
292	C₄⋊Q₈	—	2	1	no	—
293	C₄⋊Q₈	—	2	1	no	—
294	C₄⋊Q₈	—	2	1	no	—
295	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
296	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
297	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
298	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
299	#127	—	2	1	no	—
300	#127	—	2	1	no	—
301	#127	—	2	1	no	—
302	#127	C₂³×C₅₃	1	2	yes	—
303	#127	C₂³×C₅₃	1	1	yes	characteristic
304	#125	—	2	1	no	—
305	#124	—	2	1	no	—
306	C₂²⋊Q₈	—	2	1	no	—
307	C₂²⋊Q₈	—	2	1	no	—
308	#121	—	2	1	no	—
309	#121	—	2	1	no	—
310	C₄×Q₈	—	2	1	no	—
311	C₄×Q₈	—	2	1	no	—
312	C₄×D₄	—	2	1	no	—
313	C₄×D₄	—	2	1	no	—
314	#117	—	2	1	no	—
315	C₂×C₄⋊C₄	—	2	1	no	—
316	C₂×C₄⋊C₄	—	2	1	no	—
317	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
318	#107	—	2	4	no	—
319	#106	—	2	4	no	—
320	#106	—	2	2	no	—
321	#106	—	2	4	no	—
322	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
323	#103	—	2	2	no	—
324	#102	—	2	2	no	—
325	#102	—	2	4	no	—
326	#102	—	2	4	no	—
327	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
328	#97	D₄×C₅₃	1	2	yes	—
329	C₄×C₈	D₄×C₅₃	1	2	yes	—
330	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
331	C₄×C₈	C₂³×C₅₃	1	2	yes	—
332	C₄×C₈	C₂³×C₅₃	1	2	yes	—
333	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
334	C₂⁴	—	4	1	no	—
335	#41	—	4	4	no	—
336	C₂×Q₈	—	2	1	no	—
337	C₂×Q₈	—	2	1	no	—
338	C₂×Q₈	—	2	1	no	—
339	C₂×Q₈	—	2	2	no	—
340	C₂×Q₈	—	4	4	no	—
341	C₂×Q₈	—	4	1	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	1	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	4	4	no	—
349	C₂×D₄	—	4	1	no	—
350	C₂×D₄	—	4	1	no	—
351	C₂×D₄	—	4	1	no	—
352	C₂×D₄	—	4	1	no	—
353	C₂×D₄	—	4	4	no	—
354	C₂×D₄	—	8	2	no	—
355	C₂²×C₄	—	4	1	no	—
356	C₂²×C₄	—	4	1	no	—
357	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
358	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
360	SD₁₆	—	4	8	no	—
361	SD₁₆	—	4	8	no	—
362	SD₁₆	—	4	8	no	—
363	SD₁₆	—	4	4	no	—
364	SD₁₆	—	8	4	no	—
365	D₈	—	4	8	no	—
366	M₁₆	SD₁₆×C₅₃	1	8	yes	complemented
367	C₂×C₈	—	2	4	no	—
368	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
369	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
370	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
371	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	1	no	—
375	C₄⋊C₄	—	2	1	no	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	4	1	no	—
381	C₄⋊C₄	—	4	1	no	—
382	C₄⋊C₄	—	4	1	no	—
383	C₄⋊C₄	—	4	1	no	—
384	C₂²⋊C₄	—	4	1	no	—
385	C₂²⋊C₄	—	4	1	no	—
386	C₂²⋊C₄	—	4	1	no	—
387	C₄²	—	4	1	no	—
388	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
389	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
390	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
391	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
392	C₂³	—	4	1	no	—
393	C₂³	—	4	2	no	—
394	C₂³	—	8	2	no	—
395	C₂³	—	8	2	no	—
396	C₂³	—	8	2	no	—
397	C₂³	#127×C₅₃	1	1	yes	characteristic
398	Q₈	—	4	2	no	—
399	Q₈	—	4	4	no	—
400	Q₈	—	4	2	no	—
401	Q₈	—	4	2	no	—
402	Q₈	—	8	2	no	—
403	D₄	—	4	2	no	—
404	D₄	—	4	4	no	—
405	D₄	—	4	4	no	—
406	D₄	—	4	2	no	—
407	D₄	—	4	4	no	—
408	D₄	—	4	2	no	—
409	D₄	—	8	2	no	—
410	D₄	—	8	2	no	—
411	D₄	—	8	2	no	—
412	D₄	—	8	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	8	2	no	—
420	C₂×C₄	—	8	2	no	—
421	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
422	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
423	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
424	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
425	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
426	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
427	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
428	C₈	—	2	2	no	—
429	C₈	—	2	4	no	—
430	C₈	C₂×SD₁₆×C₅₃	1	8	yes	complemented
431	C₈	AGL(1,ℤ/8)×C₅₃	1	4	yes	complemented
432	C₂²	—	2	1	no	—
433	C₂²	—	2	1	no	—
434	C₂²	—	8	1	no	—
435	C₂²	—	8	2	no	—
436	C₂²	—	8	2	no	—
437	C₂²	—	8	1	no	—
438	C₂²	—	8	2	no	—
439	C₂²	—	8	1	no	—
440	C₂²	—	16	4	no	—
441	C₂²	C₅₃×#492	1	2	yes	—
442	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
443	C₄	—	2	1	no	—
444	C₄	—	2	1	no	—
445	C₄	—	8	1	no	—
446	C₄	—	8	2	no	—
447	C₄	—	8	1	no	—
448	C₄	—	8	1	no	—
449	C₄	C₅₃×#492	1	2	yes	—
450	C₄	C₂²×SD₁₆×C₅₃	1	2	yes	—
451	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
452	C₄	C₅₃×#577	1	2	yes	—
453	C₂	—	2	2	no	—
454	C₂	—	16	2	no	—
455	C₂	—	16	1	no	—
456	C₂	C₂×C₅₃×#492	1	1	yes	characteristic
457	C₂	C₅₃×#3151	1	1	yes	characteristic
458	C₂	C₅₃×#3154	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3151	C₂
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×#3227	C₂	semidirect
2. C₅₃×#3208	C₂	semidirect
3. C₅₃×#3167	C₂	semidirect
4. C₅₃×#3165	C₂	semidirect
6. C₂×C₅₃×#492	C₂	semidirect
7. C₅₃×#1715	C₂	semidirect
8. C₅₃×#1714	C₂	semidirect
9. C₅₃×#1706	C₂	semidirect
10. C₅₃×#1705	C₂	semidirect
11. C₈×M₁₆×C₅₃	C₂	semidirect
19. C₅₃×#500	C₂²	semidirect
20. C₅₃×#498	C₂²	semidirect
21. C₅₃×#498	C₂²	semidirect
22. C₅₃×#498	C₂²	semidirect
30. C₅₃×#492	C₂²	semidirect
31. C₅₃×#487	C₂²	semidirect
33. C₅₃×#486	C₂²	semidirect
34. C₅₃×#486	C₂²	semidirect
45. C₅₃×#322	C₂²	semidirect
46. C₈²×C₅₃	C₂²	semidirect
136. M₁₆×C₅₃	SD₁₆	semidirect
200. C₈×C₅₃	C₂×SD₁₆	semidirect
201. C₈×C₅₃	AGL(1,ℤ/8)	semidirect
202. #19621	C₅₃	direct (N × H)
224. #3227	C₂×C₅₃	semidirect
225. #3208	C₂×C₅₃	semidirect
226. #3167	C₂×C₅₃	semidirect
227. #3165	C₂×C₅₃	semidirect
229. C₂×#492	C₂×C₅₃	semidirect
230. #1715	C₂×C₅₃	semidirect
231. #1714	C₂×C₅₃	semidirect
232. #1706	C₂×C₅₃	semidirect
233. #1705	C₂×C₅₃	semidirect
234. C₈×M₁₆	C₂×C₅₃	semidirect
248. #500	C₂²×C₅₃	semidirect
249. #498	C₂²×C₅₃	semidirect
250. #498	C₂²×C₅₃	semidirect
251. #498	C₂²×C₅₃	semidirect
259. #492	C₂²×C₅₃	semidirect
260. #487	C₂²×C₅₃	semidirect
262. #486	C₂²×C₅₃	semidirect
263. #486	C₂²×C₅₃	semidirect
274. #322	C₂²×C₅₃	semidirect
275. C₈²	C₂²×C₅₃	semidirect
276. C₅₃	#19621	direct (N × H)
366. M₁₆	SD₁₆×C₅₃	semidirect
430. C₈	C₂×SD₁₆×C₅₃	semidirect
431. C₈	AGL(1,ℤ/8)×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3227	2
C₅₃×#3208	2
C₅₃×#3167	2
C₅₃×#3165	2
C₅₃×#3151	2
C₂×C₅₃×#492	2
C₅₃×#1715	2
C₅₃×#1714	2
C₅₃×#1706	2
C₅₃×#1705	2
C₈×M₁₆×C₅₃	2
#19621	53

Subgroups — C53×#19621 #27227133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19621 #27227133