Subgroups — C₅₃×#19234 #27226746

Order 13568.

704 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3246	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3205	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3138	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2935	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1701	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1697	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1681	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1677	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1542	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#331	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#131×C₅₃	—	2	1	no	—
13	C₅₃×#533	C₂²	1	1	yes	characteristic
14	C₅₃×#532	C₂²	1	1	yes	characteristic
15	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#493	—	2	1	no	—
17	C₅₃×#490	—	2	2	no	—
18	C₅₃×#486	—	2	1	no	—
19	C₅₃×#486	C₂²	1	2	yes	—
20	C₅₃×#486	C₂²	1	2	yes	—
21	C₅₃×#476	C₂²	1	2	yes	complemented
22	C₅₃×#475	—	2	1	no	—
23	C₅₃×#465	—	2	2	no	—
24	C₅₃×#461	C₂²	1	2	yes	complemented
25	C₅₃×#460	—	2	1	no	—
26	C₅₃×#459	C₂²	1	2	yes	—
27	C₅₃×#459	C₂²	1	2	yes	—
28	C₅₃×#447	—	2	2	no	—
29	C₅₃×#436	C₂²	1	2	yes	complemented
30	C₅₃×#435	—	2	2	no	—
31	C₈×D₄×C₅₃	C₂²	1	2	yes	complemented
32	C₅₃×#431	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#418	—	2	1	no	—
36	C₅₃×#418	—	2	1	no	—
37	C₅₃×#408	—	2	1	no	—
38	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
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×#332	C₂²	1	1	yes	characteristic, complemented
44	C₅₃×#332	C₂²	1	1	yes	characteristic, complemented
45	C₅₃×#331	C₂²	1	2	yes	—
46	C₅₃×#331	C₂²	1	2	yes	—
47	C₂×#41×C₅₃	—	2	1	no	—
48	C₂²×D₄×C₅₃	—	2	1	no	—
49	C₂²×D₄×C₅₃	—	2	1	no	—
50	C₂²×D₄×C₅₃	—	2	1	no	—
51	C₂×SD₁₆×C₅₃	—	2	2	no	—
52	C₂×D₈×C₅₃	—	2	2	no	—
53	C₂×D₈×C₅₃	—	2	2	no	—
54	#131×C₅₃	—	4	4	no	—
55	C₂×M₁₆×C₅₃	—	2	2	no	—
56	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	—	2	2	no	—
59	C₂²×C₈×C₅₃	—	2	1	no	—
60	C₂²×C₈×C₅₃	—	2	1	no	—
61	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
62	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
64	#127×C₅₃	—	2	1	no	—
65	#127×C₅₃	—	2	1	no	—
66	#127×C₅₃	—	2	1	no	—
67	#127×C₅₃	D₄	1	1	yes	characteristic
68	#127×C₅₃	C₂³	1	2	yes	—
69	#125×C₅₃	—	2	1	no	—
70	#124×C₅₃	—	2	1	no	—
71	C₂²⋊Q₈×C₅₃	—	2	1	no	—
72	C₂²⋊Q₈×C₅₃	—	2	1	no	—
73	#121×C₅₃	—	2	1	no	—
74	#121×C₅₃	D₄	1	2	yes	—
75	C₄×Q₈×C₅₃	—	2	1	no	—
76	C₄×Q₈×C₅₃	—	2	1	no	—
77	C₄×D₄×C₅₃	—	2	1	no	—
78	C₄×D₄×C₅₃	C₂³	1	2	yes	—
79	#117×C₅₃	—	2	1	no	—
80	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	#106×C₅₃	—	2	2	no	—
82	C₄⋊C₈×C₅₃	—	2	1	no	—
83	C₄⋊C₈×C₅₃	—	2	1	no	—
84	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
87	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
89	#103×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	#102×C₅₃	—	2	2	no	—
95	#102×C₅₃	—	2	2	no	—
96	#102×C₅₃	—	2	2	no	—
97	C₂²⋊C₈×C₅₃	—	2	2	no	—
98	C₂²⋊C₈×C₅₃	—	2	2	no	—
99	C₂²⋊C₈×C₅₃	—	2	1	no	—
100	#97×C₅₃	D₄	1	2	yes	—
101	C₄×C₈×C₅₃	—	2	1	no	—
102	C₄×C₈×C₅₃	C₂³	1	2	yes	—
103	C₂⁴×C₅₃	—	4	1	no	—
104	#41×C₅₃	—	4	2	no	—
105	#41×C₅₃	—	4	2	no	—
106	C₂×Q₈×C₅₃	—	2	1	no	—
107	C₂×Q₈×C₅₃	—	2	1	no	—
108	C₂×Q₈×C₅₃	—	2	1	no	—
109	C₂×D₄×C₅₃	—	2	2	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	2	no	—
115	C₂×D₄×C₅₃	—	4	1	no	—
116	C₂×D₄×C₅₃	—	4	1	no	—
117	C₂×D₄×C₅₃	—	4	1	no	—
118	C₂×D₄×C₅₃	—	4	2	no	—
119	C₂×D₄×C₅₃	—	4	2	no	—
120	C₂×D₄×C₅₃	—	8	1	no	—
121	C₂×D₄×C₅₃	—	8	1	no	—
122	C₂×D₄×C₅₃	—	8	2	no	—
123	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₅₃	—	2	2	no	—
125	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
128	SD₁₆×C₅₃	—	8	2	no	—
129	D₈×C₅₃	—	8	2	no	—
130	D₈×C₅₃	—	8	2	no	—
131	M₁₆×C₅₃	—	2	2	no	—
132	M₁₆×C₅₃	—	2	2	no	—
133	M₁₆×C₅₃	—	4	4	no	—
134	C₂×C₈×C₅₃	—	2	1	no	—
135	C₂×C₈×C₅₃	—	2	1	no	—
136	C₂×C₈×C₅₃	—	2	1	no	—
137	C₂×C₈×C₅₃	—	2	1	no	—
138	C₂×C₈×C₅₃	—	2	2	no	—
139	C₂×C₈×C₅₃	—	4	4	no	—
140	C₂×C₈×C₅₃	—	4	2	no	—
141	C₂×C₈×C₅₃	—	8	1	no	—
142	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
143	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
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	1	no	—
149	C₄⋊C₄×C₅₃	—	4	1	no	—
150	C₄⋊C₄×C₅₃	—	4	1	no	—
151	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
152	C₂²⋊C₄×C₅₃	—	2	2	no	—
153	C₂²⋊C₄×C₅₃	—	2	2	no	—
154	C₂²⋊C₄×C₅₃	—	4	1	no	—
155	C₄²×C₅₃	—	4	1	no	—
156	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
158	C₄²×C₅₃	#41	1	1	yes	characteristic
159	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
160	C₂³×C₅₃	—	2	2	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	—	4	2	no	—
167	Q₈×C₅₃	—	4	1	no	—
168	Q₈×C₅₃	—	4	1	no	—
169	Q₈×C₅₃	—	8	1	no	—
170	D₄×C₅₃	—	2	2	no	—
171	D₄×C₅₃	—	2	2	no	—
172	D₄×C₅₃	—	4	4	no	—
173	D₄×C₅₃	—	4	2	no	—
174	D₄×C₅₃	—	4	2	no	—
175	D₄×C₅₃	—	4	2	no	—
176	D₄×C₅₃	—	4	2	no	—
177	D₄×C₅₃	—	4	2	no	—
178	D₄×C₅₃	—	8	1	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	1	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	4	1	no	—
187	C₂×C₄×C₅₃	—	4	2	no	—
188	C₂×C₄×C₅₃	—	4	2	no	—
189	C₂×C₄×C₅₃	—	4	1	no	—
190	C₂×C₄×C₅₃	—	4	1	no	—
191	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
192	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
193	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
194	C₂×C₄×C₅₃	#135	1	2	yes	—
195	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
196	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
199	C₈×C₅₃	—	2	2	no	—
200	C₈×C₅₃	—	2	2	no	—
201	C₈×C₅₃	—	4	1	no	—
202	C₈×C₅₃	—	4	1	no	—
203	C₈×C₅₃	—	4	2	no	—
204	C₈×C₅₃	—	4	1	no	—
205	C₈×C₅₃	—	4	1	no	—
206	#19234	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₂²×C₅₃	—	2	1	no	—
208	C₂²×C₅₃	—	2	1	no	—
209	C₂²×C₅₃	—	4	2	no	—
210	C₂²×C₅₃	—	4	2	no	—
211	C₂²×C₅₃	—	4	2	no	—
212	C₂²×C₅₃	—	8	2	no	—
213	C₂²×C₅₃	—	8	2	no	—
214	C₂²×C₅₃	—	8	2	no	—
215	C₂²×C₅₃	—	16	4	no	—
216	C₂²×C₅₃	#463	1	2	yes	—
217	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
218	C₄×C₅₃	—	2	1	no	—
219	C₄×C₅₃	—	2	1	no	—
220	C₄×C₅₃	—	2	1	no	—
221	C₄×C₅₃	—	4	2	no	—
222	C₄×C₅₃	—	8	1	no	—
223	C₄×C₅₃	—	8	1	no	—
224	C₄×C₅₃	#463	1	2	yes	—
225	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
226	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
227	C₄×C₅₃	#576	1	1	yes	characteristic
228	C₄×C₅₃	#577	1	1	yes	characteristic
229	#3246	C₂×C₅₃	1	1	yes	characteristic, complemented
230	#3205	C₂×C₅₃	1	1	yes	characteristic
231	#3138	C₂×C₅₃	1	1	yes	characteristic, complemented
232	#2935	C₂×C₅₃	1	1	yes	characteristic, complemented
233	#1701	C₂×C₅₃	1	2	yes	complemented
234	#1697	C₂×C₅₃	1	2	yes	complemented
235	#1681	C₂×C₅₃	1	2	yes	complemented
236	#1677	C₂×C₅₃	1	2	yes	complemented
237	#1565	C₂×C₅₃	1	1	yes	characteristic, complemented
238	#1542	C₂×C₅₃	1	1	yes	characteristic, complemented
239	C₂×#331	C₂×C₅₃	1	1	yes	characteristic, complemented
240	C₂×C₅₃	—	2	2	no	—
241	C₂×C₅₃	—	8	2	no	—
242	C₂×C₅₃	—	16	2	no	—
243	C₂×C₅₃	C₂×#463	1	1	yes	characteristic
244	C₂×C₅₃	#3062	1	1	yes	characteristic
245	C₂×C₅₃	#3068	1	1	yes	characteristic
246	C₂×#131	—	2	1	no	—
247	#533	C₂²×C₅₃	1	1	yes	characteristic
248	#532	C₂²×C₅₃	1	1	yes	characteristic
249	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
250	#493	—	2	1	no	—
251	#490	—	2	2	no	—
252	#486	—	2	1	no	—
253	#486	C₂²×C₅₃	1	2	yes	—
254	#486	C₂²×C₅₃	1	2	yes	—
255	#476	C₂²×C₅₃	1	2	yes	complemented
256	#475	—	2	1	no	—
257	#465	—	2	2	no	—
258	#461	C₂²×C₅₃	1	2	yes	complemented
259	#460	—	2	1	no	—
260	#459	C₂²×C₅₃	1	2	yes	—
261	#459	C₂²×C₅₃	1	2	yes	—
262	#447	—	2	2	no	—
263	#436	C₂²×C₅₃	1	2	yes	complemented
264	#435	—	2	2	no	—
265	C₈×D₄	C₂²×C₅₃	1	2	yes	complemented
266	#431	C₂²×C₅₃	1	1	yes	characteristic
267	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
268	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
269	#418	—	2	1	no	—
270	#418	—	2	1	no	—
271	#408	—	2	1	no	—
272	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
273	#335	C₂²×C₅₃	1	2	yes	complemented
274	#334	C₂²×C₅₃	1	2	yes	complemented
275	#333	C₂²×C₅₃	1	1	yes	characteristic, complemented
276	#333	C₂²×C₅₃	1	1	yes	characteristic, complemented
277	#332	C₂²×C₅₃	1	1	yes	characteristic, complemented
278	#332	C₂²×C₅₃	1	1	yes	characteristic, complemented
279	#331	C₂²×C₅₃	1	2	yes	—
280	#331	C₂²×C₅₃	1	2	yes	—
281	C₅₃	#19234	1	1	yes	characteristic, direct product factor, Sylow
282	C₂×#41	—	2	1	no	—
283	C₂²×D₄	—	2	1	no	—
284	C₂²×D₄	—	2	1	no	—
285	C₂²×D₄	—	2	1	no	—
286	C₂×SD₁₆	—	2	2	no	—
287	C₂×D₈	—	2	2	no	—
288	C₂×D₈	—	2	2	no	—
289	#131	—	4	4	no	—
290	C₂×M₁₆	—	2	2	no	—
291	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
292	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
293	C₂²×C₈	—	2	1	no	—
294	C₂²×C₈	—	2	1	no	—
295	C₂²×C₈	—	2	2	no	—
296	C₄⋊Q₈	D₄×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	D₄×C₅₃	1	1	yes	characteristic
303	#127	C₂³×C₅₃	1	2	yes	—
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	D₄×C₅₃	1	2	yes	—
310	C₄×Q₈	—	2	1	no	—
311	C₄×Q₈	—	2	1	no	—
312	C₄×D₄	—	2	1	no	—
313	C₄×D₄	C₂³×C₅₃	1	2	yes	—
314	#117	—	2	1	no	—
315	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
316	#106	—	2	2	no	—
317	C₄⋊C₈	—	2	1	no	—
318	C₄⋊C₈	—	2	1	no	—
319	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
320	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
321	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
322	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
323	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
324	#103	—	2	2	no	—
325	#102	—	2	2	no	—
326	#102	—	2	2	no	—
327	#102	—	2	2	no	—
328	#102	—	2	2	no	—
329	#102	—	2	2	no	—
330	#102	—	2	2	no	—
331	#102	—	2	2	no	—
332	C₂²⋊C₈	—	2	1	no	—
333	C₂²⋊C₈	—	2	2	no	—
334	C₂²⋊C₈	—	2	2	no	—
335	#97	D₄×C₅₃	1	2	yes	—
336	C₄×C₈	—	2	1	no	—
337	C₄×C₈	C₂³×C₅₃	1	2	yes	—
338	C₂⁴	—	4	1	no	—
339	#41	—	4	2	no	—
340	#41	—	4	2	no	—
341	C₂×Q₈	—	2	1	no	—
342	C₂×Q₈	—	2	1	no	—
343	C₂×Q₈	—	2	1	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	2	1	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	2	2	no	—
350	C₂×D₄	—	4	1	no	—
351	C₂×D₄	—	4	1	no	—
352	C₂×D₄	—	4	1	no	—
353	C₂×D₄	—	4	2	no	—
354	C₂×D₄	—	4	2	no	—
355	C₂×D₄	—	8	1	no	—
356	C₂×D₄	—	8	1	no	—
357	C₂×D₄	—	8	2	no	—
358	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
361	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
362	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
363	SD₁₆	—	8	2	no	—
364	D₈	—	8	2	no	—
365	D₈	—	8	2	no	—
366	M₁₆	—	2	2	no	—
367	M₁₆	—	2	2	no	—
368	M₁₆	—	4	4	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	4	4	no	—
375	C₂×C₈	—	4	2	no	—
376	C₂×C₈	—	8	1	no	—
377	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
378	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	4	1	no	—
385	C₄⋊C₄	—	4	1	no	—
386	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
387	C₂²⋊C₄	—	2	2	no	—
388	C₂²⋊C₄	—	2	2	no	—
389	C₂²⋊C₄	—	4	1	no	—
390	C₄²	—	4	1	no	—
391	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
392	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
393	C₄²	#41×C₅₃	1	1	yes	characteristic
394	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
395	C₂³	—	2	2	no	—
396	C₂³	—	4	2	no	—
397	C₂³	—	8	2	no	—
398	C₂³	—	8	2	no	—
399	C₂³	—	8	2	no	—
400	C₂³	#121×C₅₃	1	1	yes	characteristic
401	Q₈	—	4	2	no	—
402	Q₈	—	4	1	no	—
403	Q₈	—	4	1	no	—
404	Q₈	—	8	1	no	—
405	D₄	—	2	2	no	—
406	D₄	—	2	2	no	—
407	D₄	—	4	4	no	—
408	D₄	—	4	2	no	—
409	D₄	—	4	2	no	—
410	D₄	—	4	2	no	—
411	D₄	—	4	2	no	—
412	D₄	—	4	2	no	—
413	D₄	—	8	1	no	—
414	D₄	—	8	2	no	—
415	D₄	—	8	2	no	—
416	D₄	—	8	2	no	—
417	D₄	—	8	2	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
427	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
428	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
429	C₂×C₄	#135×C₅₃	1	2	yes	—
430	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
431	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
432	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
433	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
434	C₈	—	2	2	no	—
435	C₈	—	2	2	no	—
436	C₈	—	4	1	no	—
437	C₈	—	4	1	no	—
438	C₈	—	4	2	no	—
439	C₈	—	4	1	no	—
440	C₈	—	4	1	no	—
441	C₂²	—	2	1	no	—
442	C₂²	—	2	1	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	2	no	—
445	C₂²	—	4	2	no	—
446	C₂²	—	8	2	no	—
447	C₂²	—	8	2	no	—
448	C₂²	—	8	2	no	—
449	C₂²	—	16	4	no	—
450	C₂²	C₅₃×#463	1	2	yes	—
451	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
452	C₄	—	2	1	no	—
453	C₄	—	2	1	no	—
454	C₄	—	2	1	no	—
455	C₄	—	4	2	no	—
456	C₄	—	8	1	no	—
457	C₄	—	8	1	no	—
458	C₄	C₅₃×#463	1	2	yes	—
459	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
460	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
461	C₄	C₅₃×#576	1	1	yes	characteristic
462	C₄	C₅₃×#577	1	1	yes	characteristic
463	C₂	—	2	2	no	—
464	C₂	—	8	2	no	—
465	C₂	—	16	2	no	—
466	C₂	C₂×C₅₃×#463	1	1	yes	characteristic
467	C₂	C₅₃×#3062	1	1	yes	characteristic
468	C₂	C₅₃×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3205	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×#3246	C₂	semidirect
3. C₅₃×#3138	C₂	semidirect
4. C₅₃×#2935	C₂	semidirect
5. C₅₃×#1701	C₂	semidirect
6. C₅₃×#1697	C₂	semidirect
7. C₅₃×#1681	C₂	semidirect
8. C₅₃×#1677	C₂	semidirect
9. C₅₃×#1565	C₂	semidirect
10. C₅₃×#1542	C₂	semidirect
11. C₂×C₅₃×#331	C₂	semidirect
21. C₅₃×#476	C₂²	semidirect
24. C₅₃×#461	C₂²	semidirect
29. C₅₃×#436	C₂²	semidirect
31. 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
43. C₅₃×#332	C₂²	semidirect
44. C₅₃×#332	C₂²	semidirect
206. #19234	C₅₃	direct (N × H)
229. #3246	C₂×C₅₃	semidirect
231. #3138	C₂×C₅₃	semidirect
232. #2935	C₂×C₅₃	semidirect
233. #1701	C₂×C₅₃	semidirect
234. #1697	C₂×C₅₃	semidirect
235. #1681	C₂×C₅₃	semidirect
236. #1677	C₂×C₅₃	semidirect
237. #1565	C₂×C₅₃	semidirect
238. #1542	C₂×C₅₃	semidirect
239. C₂×#331	C₂×C₅₃	semidirect
255. #476	C₂²×C₅₃	semidirect
258. #461	C₂²×C₅₃	semidirect
263. #436	C₂²×C₅₃	semidirect
265. C₈×D₄	C₂²×C₅₃	semidirect
273. #335	C₂²×C₅₃	semidirect
274. #334	C₂²×C₅₃	semidirect
275. #333	C₂²×C₅₃	semidirect
276. #333	C₂²×C₅₃	semidirect
277. #332	C₂²×C₅₃	semidirect
278. #332	C₂²×C₅₃	semidirect
281. C₅₃	#19234	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×#3246	2
C₅₃×#3205	2
C₅₃×#3138	2
C₅₃×#2935	2
C₅₃×#1701	2
C₅₃×#1697	2
C₅₃×#1681	2
C₅₃×#1677	2
C₅₃×#1565	2
C₅₃×#1542	2
C₂×C₅₃×#331	2
#19234	53

Subgroups — C53×#19234 #27226746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19234 #27226746