Subgroups — C₅₃×#60625 #27268137

Order 13568.

2066 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×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3500	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3141	C₂	1	3	yes	maximal, complemented
4	C₅₃×#3140	C₂	1	3	yes	maximal, complemented
5	C₅₃×#3137	C₂	1	6	yes	maximal, complemented
6	C₅₃×#3136	C₂	1	6	yes	maximal, complemented
7	C₅₃×#3135	C₂	1	3	yes	maximal, complemented
8	C₂×C₅₃×#489	C₂	1	3	yes	maximal, complemented
9	C₅₃×#2898	C₂	1	2	yes	maximal, complemented
10	C₅₃×#2897	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2875	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#131×C₅₃	C₂²	1	2	yes	—
15	Q₈²×C₅₃	—	2	1	no	—
16	C₅₃×#557	—	2	1	no	—
17	C₅₃×#554	—	2	3	no	—
18	D₄×Q₈×C₅₃	—	2	3	no	—
19	C₅₃×#548	—	2	1	no	—
20	C₅₃×#548	—	2	1	no	—
21	C₅₃×#546	—	2	3	no	—
22	C₅₃×#546	—	2	3	no	—
23	C₅₃×#544	C₂²	1	6	yes	complemented
24	C₅₃×#536	C₂²	1	3	yes	—
25	C₅₃×#535	C₂²	1	3	yes	—
26	C₅₃×#533	C₂²	1	3	yes	complemented
27	C₅₃×#532	C₂²	1	3	yes	—
28	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	complemented
29	C₂×#124×C₅₃	C₂²	1	3	yes	—
30	C₅₃×#520	C₂²	1	2	yes	—
31	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#491	C₂²	1	6	yes	complemented
33	C₅₃×#490	C₂²	1	6	yes	complemented
34	C₅₃×#489	C₂²	1	3	yes	complemented
35	C₅₃×#489	C₂²	1	12	yes	complemented
36	C₅₃×#489	C₂²	1	6	yes	complemented
37	C₅₃×#489	C₂²	1	1	yes	characteristic, complemented
38	C₅₃×#489	C₂²	1	3	yes	complemented
39	C₅₃×#489	C₂²	1	3	yes	complemented
40	C₅₃×#488	C₂²	1	12	yes	complemented
41	C₅₃×#487	C₂²	1	6	yes	complemented
42	C₅₃×#486	C₂²	1	6	yes	complemented
43	C₅₃×#419	C₂²	1	6	yes	complemented
44	C₅₃×#418	C₂²	1	6	yes	—
45	C₅₃×#417	C₂²	1	6	yes	—
46	C₅₃×#417	C₂²	1	6	yes	complemented
47	C₅₃×#416	C₂²	1	6	yes	—
48	C₅₃×#416	C₂²	1	6	yes	complemented
49	C₂×#103×C₅₃	C₂²	1	6	yes	complemented
50	C₂×#102×C₅₃	C₂²	1	6	yes	—
51	C₅₃×#405	C₂²	1	6	yes	complemented
52	C₄×M₁₆×C₅₃	C₂²	1	3	yes	complemented
53	C₂×#97×C₅₃	C₂²	1	3	yes	complemented
54	2₋¹⁺⁴×C₅₃	—	2	8	no	—
55	H(5,𝔽₂)×C₅₃	—	2	8	no	—
56	C₂×#41×C₅₃	—	2	1	no	—
57	C₂×#41×C₅₃	—	2	3	no	—
58	C₂×#41×C₅₃	C₂³	1	3	yes	—
59	C₂×#41×C₅₃	C₂³	1	3	yes	—
60	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
61	C₂²×Q₈×C₅₃	—	2	1	no	—
62	C₂²×Q₈×C₅₃	—	2	3	no	—
63	C₂²×Q₈×C₅₃	C₂³	1	3	yes	—
64	C₂²×D₄×C₅₃	—	2	3	no	—
65	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
66	#131×C₅₃	—	2	8	no	—
67	C₂×M₁₆×C₅₃	C₂³	1	6	yes	—
68	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
69	C₄⋊Q₈×C₅₃	—	2	6	no	—
70	C₄⋊Q₈×C₅₃	—	2	3	no	—
71	C₄⋊Q₈×C₅₃	—	2	3	no	—
72	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
73	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
74	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
76	#127×C₅₃	C₂³	1	3	yes	—
77	#125×C₅₃	—	2	6	no	—
78	#125×C₅₃	C₂³	1	6	yes	—
79	#124×C₅₃	—	2	3	no	—
80	#124×C₅₃	—	2	3	no	—
81	#124×C₅₃	C₂³	1	6	yes	—
82	#124×C₅₃	C₂³	1	3	yes	—
83	#124×C₅₃	C₂³	1	1	yes	characteristic
84	#123×C₅₃	—	2	6	no	—
85	C₂²⋊Q₈×C₅₃	—	2	3	no	—
86	C₂²⋊Q₈×C₅₃	—	2	3	no	—
87	C₂²⋊Q₈×C₅₃	—	2	12	no	—
88	C₂²⋊Q₈×C₅₃	—	2	3	no	—
89	C₂²⋊Q₈×C₅₃	—	2	3	no	—
90	#121×C₅₃	—	2	3	no	—
91	#121×C₅₃	—	2	3	no	—
92	#121×C₅₃	—	2	3	no	—
93	#121×C₅₃	—	2	3	no	—
94	#120×C₅₃	—	2	6	no	—
95	C₄×Q₈×C₅₃	—	2	3	no	—
96	C₄×Q₈×C₅₃	—	2	3	no	—
97	C₄×Q₈×C₅₃	—	2	2	no	—
98	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
99	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
100	C₄×D₄×C₅₃	—	2	3	no	—
101	C₄×D₄×C₅₃	—	2	3	no	—
102	C₄×D₄×C₅₃	—	2	6	no	—
103	C₄×D₄×C₅₃	D₄	1	3	yes	—
104	C₄×D₄×C₅₃	D₄	1	3	yes	—
105	#117×C₅₃	C₂³	1	6	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×C₂²⋊C₄×C₅₃	—	2	3	no	—
110	C₂×C₂²⋊C₄×C₅₃	—	2	3	no	—
111	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
112	#103×C₅₃	C₂³	1	6	yes	—
113	#103×C₅₃	C₂³	1	12	yes	—
114	#103×C₅₃	C₂³	1	6	yes	—
115	#103×C₅₃	C₂³	1	6	yes	—
116	#103×C₅₃	C₂³	1	2	yes	—
117	#102×C₅₃	C₂³	1	6	yes	—
118	#102×C₅₃	C₂³	1	12	yes	—
119	#102×C₅₃	C₂³	1	6	yes	—
120	#102×C₅₃	C₂³	1	6	yes	—
121	#102×C₅₃	C₂³	1	2	yes	—
122	#97×C₅₃	C₂³	1	1	yes	characteristic
123	#97×C₅₃	C₂³	1	3	yes	—
124	#97×C₅₃	C₂³	1	6	yes	—
125	C₄×C₈×C₅₃	C₂³	1	6	yes	—
126	C₂⁴×C₅₃	—	2	3	no	—
127	#41×C₅₃	—	2	12	no	—
128	#41×C₅₃	—	2	12	no	—
129	#41×C₅₃	—	4	4	no	—
130	#41×C₅₃	—	4	12	no	—
131	#41×C₅₃	#41	1	8	yes	—
132	C₂×Q₈×C₅₃	—	2	1	no	—
133	C₂×Q₈×C₅₃	—	2	3	no	—
134	C₂×Q₈×C₅₃	—	2	3	no	—
135	C₂×Q₈×C₅₃	—	2	6	no	—
136	C₂×Q₈×C₅₃	—	2	12	no	—
137	C₂×Q₈×C₅₃	—	4	4	no	—
138	C₂×Q₈×C₅₃	—	8	6	no	—
139	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₂×Q₈×C₅₃	C₂⁴	1	3	yes	—
141	C₂×Q₈×C₅₃	C₂⁴	1	1	yes	characteristic
142	C₂×D₄×C₅₃	—	2	3	no	—
143	C₂×D₄×C₅₃	—	2	3	no	—
144	C₂×D₄×C₅₃	—	2	3	no	—
145	C₂×D₄×C₅₃	—	2	3	no	—
146	C₂×D₄×C₅₃	—	2	3	no	—
147	C₂×D₄×C₅₃	—	2	12	no	—
148	C₂×D₄×C₅₃	—	4	12	no	—
149	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
150	C₂×D₄×C₅₃	C₂⁴	1	3	yes	—
151	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
152	C₂²×C₄×C₅₃	—	2	3	no	—
153	C₂²×C₄×C₅₃	—	2	6	no	—
154	C₂²×C₄×C₅₃	—	2	3	no	—
155	C₂²×C₄×C₅₃	—	2	3	no	—
156	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
157	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
158	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
159	M₁₆×C₅₃	—	2	12	no	—
160	C₂×C₈×C₅₃	—	2	12	no	—
161	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
162	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
163	C₄⋊C₄×C₅₃	—	2	3	no	—
164	C₄⋊C₄×C₅₃	—	2	6	no	—
165	C₄⋊C₄×C₅₃	—	2	6	no	—
166	C₄⋊C₄×C₅₃	—	2	3	no	—
167	C₄⋊C₄×C₅₃	—	2	3	no	—
168	C₄⋊C₄×C₅₃	—	2	1	no	—
169	C₄⋊C₄×C₅₃	—	2	6	no	—
170	C₄⋊C₄×C₅₃	—	2	6	no	—
171	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
172	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
173	C₄⋊C₄×C₅₃	C₂⁴	1	6	yes	—
174	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
175	C₂²⋊C₄×C₅₃	—	2	6	no	—
176	C₂²⋊C₄×C₅₃	—	2	3	no	—
177	C₂²⋊C₄×C₅₃	—	2	3	no	—
178	C₂²⋊C₄×C₅₃	—	2	6	no	—
179	C₂²⋊C₄×C₅₃	—	2	6	no	—
180	C₂²⋊C₄×C₅₃	—	2	1	no	—
181	C₂²⋊C₄×C₅₃	—	2	1	no	—
182	C₂²⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
183	C₄²×C₅₃	—	2	6	no	—
184	C₄²×C₅₃	C₂×D₄	1	3	yes	—
185	C₄²×C₅₃	C₂×D₄	1	3	yes	—
186	C₄²×C₅₃	C₂⁴	1	3	yes	—
187	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
188	C₂³×C₅₃	—	2	1	no	—
189	C₂³×C₅₃	—	2	3	no	—
190	C₂³×C₅₃	—	4	6	no	—
191	C₂³×C₅₃	—	8	3	no	—
192	C₂³×C₅₃	—	8	3	no	—
193	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
194	Q₈×C₅₃	—	2	6	no	—
195	Q₈×C₅₃	—	2	2	no	—
196	Q₈×C₅₃	—	4	2	no	—
197	Q₈×C₅₃	—	4	6	no	—
198	Q₈×C₅₃	—	4	6	no	—
199	Q₈×C₅₃	—	8	6	no	—
200	Q₈×C₅₃	#124	1	4	yes	—
201	D₄×C₅₃	—	2	6	no	—
202	D₄×C₅₃	—	2	2	no	—
203	D₄×C₅₃	—	4	6	no	—
204	D₄×C₅₃	—	4	6	no	—
205	D₄×C₅₃	—	4	6	no	—
206	D₄×C₅₃	—	8	3	no	—
207	D₄×C₅₃	—	8	3	no	—
208	D₄×C₅₃	#124	1	12	yes	—
209	C₂×C₄×C₅₃	—	2	1	no	—
210	C₂×C₄×C₅₃	—	2	3	no	—
211	C₂×C₄×C₅₃	—	2	6	no	—
212	C₂×C₄×C₅₃	—	2	2	no	—
213	C₂×C₄×C₅₃	—	4	3	no	—
214	C₂×C₄×C₅₃	—	4	3	no	—
215	C₂×C₄×C₅₃	—	4	6	no	—
216	C₂×C₄×C₅₃	—	4	6	no	—
217	C₂×C₄×C₅₃	—	4	6	no	—
218	C₂×C₄×C₅₃	—	8	6	no	—
219	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
222	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
223	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
224	C₂×C₄×C₅₃	C₂×#41	1	12	yes	—
225	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
226	C₈×C₅₃	—	2	2	no	—
227	C₈×C₅₃	—	2	6	no	—
228	#60625	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	—	4	1	no	—
232	C₂²×C₅₃	—	4	3	no	—
233	C₂²×C₅₃	—	4	1	no	—
234	C₂²×C₅₃	—	4	3	no	—
235	C₂²×C₅₃	—	4	1	no	—
236	C₂²×C₅₃	—	4	3	no	—
237	C₂²×C₅₃	—	16	6	no	—
238	C₂²×C₅₃	C₂×#124	1	6	yes	—
239	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
240	C₄×C₅₃	—	2	1	no	—
241	C₄×C₅₃	—	2	3	no	—
242	C₄×C₅₃	—	2	1	no	—
243	C₄×C₅₃	—	2	3	no	—
244	C₄×C₅₃	—	4	6	no	—
245	C₄×C₅₃	—	4	2	no	—
246	C₄×C₅₃	—	4	1	no	—
247	C₄×C₅₃	—	4	3	no	—
248	C₄×C₅₃	C₂×#124	1	6	yes	—
249	C₄×C₅₃	C₂²×#41	1	2	yes	—
250	#3503	C₂×C₅₃	1	1	yes	characteristic, complemented
251	#3500	C₂×C₅₃	1	1	yes	characteristic
252	#3141	C₂×C₅₃	1	3	yes	complemented
253	#3140	C₂×C₅₃	1	3	yes	complemented
254	#3137	C₂×C₅₃	1	6	yes	complemented
255	#3136	C₂×C₅₃	1	6	yes	complemented
256	#3135	C₂×C₅₃	1	3	yes	complemented
257	C₂×#489	C₂×C₅₃	1	3	yes	complemented
258	#2898	C₂×C₅₃	1	2	yes	complemented
259	#2897	C₂×C₅₃	1	2	yes	complemented
260	#2875	C₂×C₅₃	1	1	yes	characteristic, complemented
261	C₂×C₅₃	—	2	6	no	—
262	C₂×C₅₃	—	8	1	no	—
263	C₂×C₅₃	—	8	3	no	—
264	C₂×C₅₃	C₂²×#124	1	1	yes	characteristic
265	C₂×C₅₃	#3585	1	1	yes	characteristic
266	C₂×C₅₃	#3586	1	1	yes	characteristic
267	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	1	yes	characteristic
268	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	1	yes	characteristic
269	C₂×#131	C₂²×C₅₃	1	2	yes	—
270	Q₈²	—	2	1	no	—
271	#557	—	2	1	no	—
272	#554	—	2	3	no	—
273	D₄×Q₈	—	2	3	no	—
274	#548	—	2	1	no	—
275	#548	—	2	1	no	—
276	#546	—	2	3	no	—
277	#546	—	2	3	no	—
278	#544	C₂²×C₅₃	1	6	yes	complemented
279	#536	C₂²×C₅₃	1	3	yes	—
280	#535	C₂²×C₅₃	1	3	yes	—
281	#533	C₂²×C₅₃	1	3	yes	complemented
282	#532	C₂²×C₅₃	1	3	yes	—
283	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	complemented
284	C₂×#124	C₂²×C₅₃	1	3	yes	—
285	#520	C₂²×C₅₃	1	2	yes	—
286	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
287	#491	C₂²×C₅₃	1	6	yes	complemented
288	#490	C₂²×C₅₃	1	6	yes	complemented
289	#489	C₂²×C₅₃	1	3	yes	complemented
290	#489	C₂²×C₅₃	1	12	yes	complemented
291	#489	C₂²×C₅₃	1	6	yes	complemented
292	#489	C₂²×C₅₃	1	1	yes	characteristic, complemented
293	#489	C₂²×C₅₃	1	3	yes	complemented
294	#489	C₂²×C₅₃	1	3	yes	complemented
295	#488	C₂²×C₅₃	1	12	yes	complemented
296	#487	C₂²×C₅₃	1	6	yes	complemented
297	#486	C₂²×C₅₃	1	6	yes	complemented
298	#419	C₂²×C₅₃	1	6	yes	complemented
299	#418	C₂²×C₅₃	1	6	yes	—
300	#417	C₂²×C₅₃	1	6	yes	—
301	#417	C₂²×C₅₃	1	6	yes	complemented
302	#416	C₂²×C₅₃	1	6	yes	—
303	#416	C₂²×C₅₃	1	6	yes	complemented
304	C₂×#103	C₂²×C₅₃	1	6	yes	complemented
305	C₂×#102	C₂²×C₅₃	1	6	yes	—
306	#405	C₂²×C₅₃	1	6	yes	complemented
307	C₄×M₁₆	C₂²×C₅₃	1	3	yes	complemented
308	C₂×#97	C₂²×C₅₃	1	3	yes	complemented
309	C₅₃	#60625	1	1	yes	characteristic, direct product factor, Sylow
310	2₋¹⁺⁴	—	2	8	no	—
311	H(5,𝔽₂)	—	2	8	no	—
312	C₂×#41	—	2	1	no	—
313	C₂×#41	—	2	3	no	—
314	C₂×#41	C₂³×C₅₃	1	3	yes	—
315	C₂×#41	C₂³×C₅₃	1	3	yes	—
316	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
317	C₂²×Q₈	—	2	1	no	—
318	C₂²×Q₈	—	2	3	no	—
319	C₂²×Q₈	C₂³×C₅₃	1	3	yes	—
320	C₂²×D₄	—	2	3	no	—
321	C₂²×D₄	C₂³×C₅₃	1	3	yes	—
322	#131	—	2	8	no	—
323	C₂×M₁₆	C₂³×C₅₃	1	6	yes	—
324	C₂²×C₈	C₂³×C₅₃	1	6	yes	—
325	C₄⋊Q₈	—	2	6	no	—
326	C₄⋊Q₈	—	2	3	no	—
327	C₄⋊Q₈	—	2	3	no	—
328	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
329	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
330	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
331	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
332	#127	C₂³×C₅₃	1	3	yes	—
333	#125	—	2	6	no	—
334	#125	C₂³×C₅₃	1	6	yes	—
335	#124	—	2	3	no	—
336	#124	—	2	3	no	—
337	#124	C₂³×C₅₃	1	3	yes	—
338	#124	C₂³×C₅₃	1	6	yes	—
339	#124	C₂³×C₅₃	1	1	yes	characteristic
340	#123	—	2	6	no	—
341	C₂²⋊Q₈	—	2	3	no	—
342	C₂²⋊Q₈	—	2	3	no	—
343	C₂²⋊Q₈	—	2	3	no	—
344	C₂²⋊Q₈	—	2	3	no	—
345	C₂²⋊Q₈	—	2	12	no	—
346	#121	—	2	3	no	—
347	#121	—	2	3	no	—
348	#121	—	2	3	no	—
349	#121	—	2	3	no	—
350	#120	—	2	6	no	—
351	C₄×Q₈	—	2	2	no	—
352	C₄×Q₈	—	2	3	no	—
353	C₄×Q₈	—	2	3	no	—
354	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
355	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
356	C₄×D₄	—	2	6	no	—
357	C₄×D₄	—	2	3	no	—
358	C₄×D₄	—	2	3	no	—
359	C₄×D₄	D₄×C₅₃	1	3	yes	—
360	C₄×D₄	D₄×C₅₃	1	3	yes	—
361	#117	C₂³×C₅₃	1	6	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×C₂²⋊C₄	—	2	3	no	—
366	C₂×C₂²⋊C₄	—	2	3	no	—
367	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
368	#103	C₂³×C₅₃	1	6	yes	—
369	#103	C₂³×C₅₃	1	12	yes	—
370	#103	C₂³×C₅₃	1	2	yes	—
371	#103	C₂³×C₅₃	1	6	yes	—
372	#103	C₂³×C₅₃	1	6	yes	—
373	#102	C₂³×C₅₃	1	6	yes	—
374	#102	C₂³×C₅₃	1	12	yes	—
375	#102	C₂³×C₅₃	1	6	yes	—
376	#102	C₂³×C₅₃	1	2	yes	—
377	#102	C₂³×C₅₃	1	6	yes	—
378	#97	C₂³×C₅₃	1	1	yes	characteristic
379	#97	C₂³×C₅₃	1	3	yes	—
380	#97	C₂³×C₅₃	1	6	yes	—
381	C₄×C₈	C₂³×C₅₃	1	6	yes	—
382	C₂⁴	—	2	3	no	—
383	#41	—	2	12	no	—
384	#41	—	2	12	no	—
385	#41	—	4	4	no	—
386	#41	—	4	12	no	—
387	#41	#41×C₅₃	1	8	yes	—
388	C₂×Q₈	—	2	1	no	—
389	C₂×Q₈	—	2	3	no	—
390	C₂×Q₈	—	2	3	no	—
391	C₂×Q₈	—	2	6	no	—
392	C₂×Q₈	—	2	12	no	—
393	C₂×Q₈	—	4	4	no	—
394	C₂×Q₈	—	8	6	no	—
395	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
396	C₂×Q₈	C₂⁴×C₅₃	1	3	yes	—
397	C₂×Q₈	C₂⁴×C₅₃	1	1	yes	characteristic
398	C₂×D₄	—	2	3	no	—
399	C₂×D₄	—	2	3	no	—
400	C₂×D₄	—	2	3	no	—
401	C₂×D₄	—	2	3	no	—
402	C₂×D₄	—	2	3	no	—
403	C₂×D₄	—	2	12	no	—
404	C₂×D₄	—	4	12	no	—
405	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
406	C₂×D₄	C₂⁴×C₅₃	1	3	yes	—
407	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
408	C₂²×C₄	—	2	3	no	—
409	C₂²×C₄	—	2	6	no	—
410	C₂²×C₄	—	2	3	no	—
411	C₂²×C₄	—	2	3	no	—
412	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
413	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
414	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
415	M₁₆	—	2	12	no	—
416	C₂×C₈	—	2	12	no	—
417	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
418	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
419	C₄⋊C₄	—	2	3	no	—
420	C₄⋊C₄	—	2	6	no	—
421	C₄⋊C₄	—	2	6	no	—
422	C₄⋊C₄	—	2	3	no	—
423	C₄⋊C₄	—	2	3	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	6	no	—
426	C₄⋊C₄	—	2	6	no	—
427	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
428	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
429	C₄⋊C₄	C₂⁴×C₅₃	1	6	yes	—
430	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
431	C₂²⋊C₄	—	2	6	no	—
432	C₂²⋊C₄	—	2	3	no	—
433	C₂²⋊C₄	—	2	3	no	—
434	C₂²⋊C₄	—	2	6	no	—
435	C₂²⋊C₄	—	2	6	no	—
436	C₂²⋊C₄	—	2	1	no	—
437	C₂²⋊C₄	—	2	1	no	—
438	C₂²⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
439	C₄²	—	2	6	no	—
440	C₄²	C₂×D₄×C₅₃	1	3	yes	—
441	C₄²	C₂×D₄×C₅₃	1	3	yes	—
442	C₄²	C₂⁴×C₅₃	1	3	yes	—
443	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
444	C₂³	—	2	3	no	—
445	C₂³	—	2	1	no	—
446	C₂³	—	4	6	no	—
447	C₂³	—	8	3	no	—
448	C₂³	—	8	3	no	—
449	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
450	Q₈	—	2	6	no	—
451	Q₈	—	2	2	no	—
452	Q₈	—	4	2	no	—
453	Q₈	—	4	6	no	—
454	Q₈	—	4	6	no	—
455	Q₈	—	8	6	no	—
456	Q₈	#124×C₅₃	1	4	yes	—
457	D₄	—	2	6	no	—
458	D₄	—	2	2	no	—
459	D₄	—	4	6	no	—
460	D₄	—	4	6	no	—
461	D₄	—	4	6	no	—
462	D₄	—	8	3	no	—
463	D₄	—	8	3	no	—
464	D₄	#124×C₅₃	1	12	yes	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	6	no	—
467	C₂×C₄	—	2	3	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	4	3	no	—
470	C₂×C₄	—	4	3	no	—
471	C₂×C₄	—	4	6	no	—
472	C₂×C₄	—	4	6	no	—
473	C₂×C₄	—	4	6	no	—
474	C₂×C₄	—	8	6	no	—
475	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
476	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
477	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
478	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
479	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
480	C₂×C₄	C₂×#41×C₅₃	1	12	yes	—
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	—	4	1	no	—
487	C₂²	—	4	3	no	—
488	C₂²	—	4	1	no	—
489	C₂²	—	4	3	no	—
490	C₂²	—	4	1	no	—
491	C₂²	—	4	3	no	—
492	C₂²	—	16	6	no	—
493	C₂²	C₂×#124×C₅₃	1	6	yes	—
494	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
495	C₄	—	2	1	no	—
496	C₄	—	2	3	no	—
497	C₄	—	2	1	no	—
498	C₄	—	2	3	no	—
499	C₄	—	4	6	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	1	no	—
502	C₄	—	4	3	no	—
503	C₄	C₂×#124×C₅₃	1	6	yes	—
504	C₄	C₂²×#41×C₅₃	1	2	yes	—
505	C₂	—	2	6	no	—
506	C₂	—	8	3	no	—
507	C₂	—	8	1	no	—
508	C₂	C₂²×#124×C₅₃	1	1	yes	characteristic
509	C₂	C₅₃×#3585	1	1	yes	characteristic
510	C₂	C₅₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₅₃×#3500	C₂
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×#3503	C₂	semidirect
3. C₅₃×#3141	C₂	semidirect
4. C₅₃×#3140	C₂	semidirect
5. C₅₃×#3137	C₂	semidirect
6. C₅₃×#3136	C₂	semidirect
7. C₅₃×#3135	C₂	semidirect
8. C₂×C₅₃×#489	C₂	semidirect
9. C₅₃×#2898	C₂	semidirect
10. C₅₃×#2897	C₂	semidirect
11. C₅₃×#2875	C₂	semidirect
23. C₅₃×#544	C₂²	semidirect
26. C₅₃×#533	C₂²	semidirect
28. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
32. C₅₃×#491	C₂²	semidirect
33. C₅₃×#490	C₂²	semidirect
34. C₅₃×#489	C₂²	semidirect
35. C₅₃×#489	C₂²	semidirect
36. C₅₃×#489	C₂²	semidirect
37. C₅₃×#489	C₂²	semidirect
38. C₅₃×#489	C₂²	semidirect
39. C₅₃×#489	C₂²	semidirect
40. C₅₃×#488	C₂²	semidirect
41. C₅₃×#487	C₂²	semidirect
42. C₅₃×#486	C₂²	semidirect
43. C₅₃×#419	C₂²	semidirect
46. C₅₃×#417	C₂²	semidirect
48. C₅₃×#416	C₂²	semidirect
49. C₂×#103×C₅₃	C₂²	semidirect
51. C₅₃×#405	C₂²	semidirect
52. C₄×M₁₆×C₅₃	C₂²	semidirect
53. C₂×#97×C₅₃	C₂²	semidirect
228. #60625	C₅₃	direct (N × H)
250. #3503	C₂×C₅₃	semidirect
252. #3141	C₂×C₅₃	semidirect
253. #3140	C₂×C₅₃	semidirect
254. #3137	C₂×C₅₃	semidirect
255. #3136	C₂×C₅₃	semidirect
256. #3135	C₂×C₅₃	semidirect
257. C₂×#489	C₂×C₅₃	semidirect
258. #2898	C₂×C₅₃	semidirect
259. #2897	C₂×C₅₃	semidirect
260. #2875	C₂×C₅₃	semidirect
278. #544	C₂²×C₅₃	semidirect
281. #533	C₂²×C₅₃	semidirect
283. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
287. #491	C₂²×C₅₃	semidirect
288. #490	C₂²×C₅₃	semidirect
289. #489	C₂²×C₅₃	semidirect
290. #489	C₂²×C₅₃	semidirect
291. #489	C₂²×C₅₃	semidirect
292. #489	C₂²×C₅₃	semidirect
293. #489	C₂²×C₅₃	semidirect
294. #489	C₂²×C₅₃	semidirect
295. #488	C₂²×C₅₃	semidirect
296. #487	C₂²×C₅₃	semidirect
297. #486	C₂²×C₅₃	semidirect
298. #419	C₂²×C₅₃	semidirect
301. #417	C₂²×C₅₃	semidirect
303. #416	C₂²×C₅₃	semidirect
304. C₂×#103	C₂²×C₅₃	semidirect
306. #405	C₂²×C₅₃	semidirect
307. C₄×M₁₆	C₂²×C₅₃	semidirect
308. C₂×#97	C₂²×C₅₃	semidirect
309. C₅₃	#60625	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×#3503	2
C₅₃×#3500	2
C₅₃×#3141	2
C₅₃×#3140	2
C₅₃×#3137	2
C₅₃×#3136	2
C₅₃×#3135	2
C₂×C₅₃×#489	2
C₅₃×#2898	2
C₅₃×#2897	2
C₅₃×#2875	2
#60625	53

Subgroups — C53×#60625 #27268137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60625 #27268137