Subgroups — C₅₃×#25876 #27233388

Order 13568.

1222 conjugacy classes of subgroups.

The subgroup lattice has 562 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
2	C₅₃×#3141	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3079	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2862	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2038	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1966	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1962	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1793	C₂	1	4	yes	maximal, complemented
9	C₄×#104×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	2	yes	—
11	D₄×Q₈×C₅₃	—	2	1	no	—
12	D₄×Q₈×C₅₃	—	2	1	no	—
13	C₅₃×#547	—	2	1	no	—
14	C₅₃×#547	—	2	1	no	—
15	C₅₃×#547	—	2	2	no	—
16	C₅₃×#547	—	2	2	no	—
17	C₅₃×#536	—	2	2	no	—
18	C₅₃×#536	C₂²	1	2	yes	—
19	C₅₃×#532	—	2	1	no	—
20	C₅₃×#532	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#489	—	2	2	no	—
26	C₅₃×#487	—	2	1	no	—
27	C₅₃×#486	—	2	1	no	—
28	C₅₃×#477	—	2	1	no	—
29	C₅₃×#476	—	2	1	no	—
30	C₅₃×#475	—	2	1	no	—
31	C₅₃×#474	—	2	1	no	—
32	C₅₃×#428	C₂²	1	2	yes	—
33	C₅₃×#423	C₂²	1	1	yes	characteristic
34	C₂×#104×C₅₃	C₂²	1	2	yes	—
35	C₂×#104×C₅₃	C₂²	1	4	yes	—
36	C₅₃×#417	C₂²	1	4	yes	—
37	C₅₃×#417	C₂²	1	2	yes	—
38	C₂×#101×C₅₃	C₂²	1	4	yes	—
39	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#391	—	2	2	no	—
41	C₅₃×#391	C₂²	1	4	yes	—
42	C₅₃×#382	—	2	2	no	—
43	C₄³×C₅₃	C₂²	1	1	yes	characteristic
44	C₅₃×#339	C₂²	1	2	yes	—
45	2₋¹⁺⁴×C₅₃	—	2	8	no	—
46	2₋¹⁺⁴×C₅₃	—	4	4	no	—
47	C₂×#41×C₅₃	—	2	2	no	—
48	C₂×#41×C₅₃	—	2	2	no	—
49	C₂×#41×C₅₃	—	2	4	no	—
50	C₂×#41×C₅₃	C₂³	1	2	yes	—
51	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×Q₈×C₅₃	—	2	2	no	—
53	C₂²×Q₈×C₅₃	—	2	2	no	—
54	C₂²×Q₈×C₅₃	—	2	1	no	—
55	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
56	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
57	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
58	C₄⋊Q₈×C₅₃	—	2	2	no	—
59	C₄⋊Q₈×C₅₃	—	2	2	no	—
60	C₄⋊Q₈×C₅₃	—	2	1	no	—
61	C₄⋊Q₈×C₅₃	—	2	1	no	—
62	C₄⋊Q₈×C₅₃	—	2	1	no	—
63	C₄⋊Q₈×C₅₃	—	2	1	no	—
64	C₄⋊Q₈×C₅₃	—	2	2	no	—
65	C₄⋊Q₈×C₅₃	—	2	1	no	—
66	C₄⋊Q₈×C₅₃	—	2	1	no	—
67	C₄⋊Q₈×C₅₃	—	2	1	no	—
68	C₄⋊Q₈×C₅₃	—	2	1	no	—
69	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
70	#127×C₅₃	D₄	1	1	yes	characteristic
71	#124×C₅₃	—	2	2	no	—
72	#124×C₅₃	D₄	1	2	yes	—
73	#123×C₅₃	—	2	4	no	—
74	#123×C₅₃	—	2	4	no	—
75	#123×C₅₃	—	2	4	no	—
76	C₂²⋊Q₈×C₅₃	—	2	4	no	—
77	C₂²⋊Q₈×C₅₃	—	2	4	no	—
78	C₂²⋊Q₈×C₅₃	—	2	2	no	—
79	C₂²⋊Q₈×C₅₃	—	2	2	no	—
80	#121×C₅₃	—	2	2	no	—
81	#121×C₅₃	—	2	2	no	—
82	#121×C₅₃	—	2	1	no	—
83	#121×C₅₃	—	2	1	no	—
84	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
85	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
86	C₄×D₄×C₅₃	—	2	2	no	—
87	C₄×D₄×C₅₃	—	2	2	no	—
88	C₄×D₄×C₅₃	—	2	1	no	—
89	C₄×D₄×C₅₃	—	2	1	no	—
90	C₄×D₄×C₅₃	—	2	1	no	—
91	C₄×D₄×C₅₃	—	2	1	no	—
92	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
93	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
94	#117×C₅₃	—	2	1	no	—
95	#117×C₅₃	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
100	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
101	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
102	C₂×C₄²×C₅₃	—	2	1	no	—
103	C₂×C₄²×C₅₃	—	2	1	no	—
104	C₂×C₄²×C₅₃	—	2	1	no	—
105	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
106	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
107	#107×C₅₃	—	2	2	no	—
108	#106×C₅₃	—	2	2	no	—
109	C₄⋊C₈×C₅₃	—	2	1	no	—
110	C₄⋊C₈×C₅₃	—	2	1	no	—
111	#104×C₅₃	—	2	4	no	—
112	#104×C₅₃	—	4	4	no	—
113	#103×C₅₃	—	2	4	no	—
114	#103×C₅₃	—	2	2	no	—
115	#102×C₅₃	—	2	4	no	—
116	#102×C₅₃	—	2	2	no	—
117	#101×C₅₃	—	4	4	no	—
118	#97×C₅₃	—	2	1	no	—
119	C₄×C₈×C₅₃	—	2	1	no	—
120	#95×C₅₃	—	2	8	no	—
121	#41×C₅₃	—	2	2	no	—
122	#41×C₅₃	—	2	4	no	—
123	#41×C₅₃	—	2	4	no	—
124	#41×C₅₃	—	4	8	no	—
125	#41×C₅₃	—	4	16	no	—
126	#41×C₅₃	—	4	4	no	—
127	#41×C₅₃	—	4	4	no	—
128	#41×C₅₃	#41	1	4	yes	—
129	C₂×Q₈×C₅₃	—	2	4	no	—
130	C₂×Q₈×C₅₃	—	2	2	no	—
131	C₂×Q₈×C₅₃	—	2	2	no	—
132	C₂×Q₈×C₅₃	—	2	4	no	—
133	C₂×Q₈×C₅₃	—	2	4	no	—
134	C₂×Q₈×C₅₃	—	4	8	no	—
135	C₂×Q₈×C₅₃	—	4	4	no	—
136	C₂×Q₈×C₅₃	—	4	4	no	—
137	C₂×Q₈×C₅₃	—	4	4	no	—
138	C₂×Q₈×C₅₃	—	4	1	no	—
139	C₂×Q₈×C₅₃	—	4	1	no	—
140	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
141	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
142	C₂×D₄×C₅₃	—	2	1	no	—
143	C₂×D₄×C₅₃	—	2	4	no	—
144	C₂×D₄×C₅₃	—	2	2	no	—
145	C₂×D₄×C₅₃	—	2	1	no	—
146	C₂×D₄×C₅₃	—	2	1	no	—
147	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
148	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₅₃	—	2	1	no	—
150	C₂²×C₄×C₅₃	—	2	4	no	—
151	C₂²×C₄×C₅₃	—	2	4	no	—
152	C₂²×C₄×C₅₃	—	2	2	no	—
153	C₂²×C₄×C₅₃	—	2	1	no	—
154	C₂²×C₄×C₅₃	—	2	1	no	—
155	C₂²×C₄×C₅₃	—	2	2	no	—
156	C₂²×C₄×C₅₃	—	2	2	no	—
157	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
158	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
161	M₁₆×C₅₃	—	2	2	no	—
162	M₁₆×C₅₃	—	2	2	no	—
163	M₁₆×C₅₃	—	2	4	no	—
164	C₂×C₈×C₅₃	—	2	2	no	—
165	C₂×C₈×C₅₃	—	2	2	no	—
166	C₄⋊C₄×C₅₃	—	2	4	no	—
167	C₄⋊C₄×C₅₃	—	2	4	no	—
168	C₄⋊C₄×C₅₃	—	2	1	no	—
169	C₄⋊C₄×C₅₃	—	2	1	no	—
170	C₄⋊C₄×C₅₃	—	2	2	no	—
171	C₄⋊C₄×C₅₃	—	2	2	no	—
172	C₄⋊C₄×C₅₃	—	2	2	no	—
173	C₄⋊C₄×C₅₃	—	2	2	no	—
174	C₄⋊C₄×C₅₃	—	2	1	no	—
175	C₄⋊C₄×C₅₃	—	2	1	no	—
176	C₄⋊C₄×C₅₃	—	2	2	no	—
177	C₄⋊C₄×C₅₃	—	2	2	no	—
178	C₄⋊C₄×C₅₃	—	4	1	no	—
179	C₄⋊C₄×C₅₃	—	4	1	no	—
180	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
181	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
182	C₂²⋊C₄×C₅₃	—	2	4	no	—
183	C₂²⋊C₄×C₅₃	—	2	4	no	—
184	C₂²⋊C₄×C₅₃	—	2	1	no	—
185	C₂²⋊C₄×C₅₃	—	2	1	no	—
186	C₂²⋊C₄×C₅₃	—	2	1	no	—
187	C₂²⋊C₄×C₅₃	—	2	2	no	—
188	C₂²⋊C₄×C₅₃	—	2	2	no	—
189	C₄²×C₅₃	—	2	4	no	—
190	C₄²×C₅₃	—	2	1	no	—
191	C₄²×C₅₃	—	2	1	no	—
192	C₄²×C₅₃	—	2	1	no	—
193	C₄²×C₅₃	—	2	2	no	—
194	C₄²×C₅₃	—	2	2	no	—
195	C₄²×C₅₃	—	4	2	no	—
196	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
197	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
199	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
200	C₂³×C₅₃	—	2	1	no	—
201	C₂³×C₅₃	—	2	2	no	—
202	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
203	Q₈×C₅₃	—	2	1	no	—
204	Q₈×C₅₃	—	2	4	no	—
205	Q₈×C₅₃	—	4	8	no	—
206	Q₈×C₅₃	—	4	2	no	—
207	Q₈×C₅₃	—	4	2	no	—
208	Q₈×C₅₃	—	4	4	no	—
209	Q₈×C₅₃	—	8	2	no	—
210	Q₈×C₅₃	—	8	2	no	—
211	Q₈×C₅₃	#124	1	2	yes	—
212	D₄×C₅₃	—	2	4	no	—
213	D₄×C₅₃	—	2	1	no	—
214	D₄×C₅₃	—	2	4	no	—
215	D₄×C₅₃	—	4	8	no	—
216	D₄×C₅₃	—	4	4	no	—
217	D₄×C₅₃	—	4	2	no	—
218	D₄×C₅₃	—	4	2	no	—
219	D₄×C₅₃	#124	1	2	yes	—
220	C₂×C₄×C₅₃	—	2	1	no	—
221	C₂×C₄×C₅₃	—	2	4	no	—
222	C₂×C₄×C₅₃	—	2	2	no	—
223	C₂×C₄×C₅₃	—	2	2	no	—
224	C₂×C₄×C₅₃	—	2	4	no	—
225	C₂×C₄×C₅₃	—	2	2	no	—
226	C₂×C₄×C₅₃	—	2	4	no	—
227	C₂×C₄×C₅₃	—	2	1	no	—
228	C₂×C₄×C₅₃	—	2	1	no	—
229	C₂×C₄×C₅₃	—	2	8	no	—
230	C₂×C₄×C₅₃	—	4	8	no	—
231	C₂×C₄×C₅₃	—	4	8	no	—
232	C₂×C₄×C₅₃	—	4	2	no	—
233	C₂×C₄×C₅₃	—	4	2	no	—
234	C₂×C₄×C₅₃	—	4	2	no	—
235	C₂×C₄×C₅₃	—	4	2	no	—
236	C₂×C₄×C₅₃	—	4	2	no	—
237	C₂×C₄×C₅₃	—	4	2	no	—
238	C₂×C₄×C₅₃	—	4	4	no	—
239	C₂×C₄×C₅₃	—	4	2	no	—
240	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
241	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
242	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
243	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
244	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
247	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
248	C₈×C₅₃	—	4	2	no	—
249	C₈×C₅₃	—	4	2	no	—
250	#25876	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
251	C₂²×C₅₃	—	2	1	no	—
252	C₂²×C₅₃	—	2	1	no	—
253	C₂²×C₅₃	—	2	2	no	—
254	C₂²×C₅₃	—	4	1	no	—
255	C₂²×C₅₃	—	4	1	no	—
256	C₂²×C₅₃	—	4	2	no	—
257	C₂²×C₅₃	—	4	2	no	—
258	C₂²×C₅₃	—	4	2	no	—
259	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
260	C₂²×C₅₃	#546	1	2	yes	—
261	C₄×C₅₃	—	2	2	no	—
262	C₄×C₅₃	—	2	1	no	—
263	C₄×C₅₃	—	2	1	no	—
264	C₄×C₅₃	—	4	4	no	—
265	C₄×C₅₃	—	4	4	no	—
266	C₄×C₅₃	—	4	2	no	—
267	C₄×C₅₃	—	4	1	no	—
268	C₄×C₅₃	—	4	1	no	—
269	C₄×C₅₃	—	8	2	no	—
270	C₄×C₅₃	#454	1	2	yes	—
271	C₄×C₅₃	#456	1	2	yes	—
272	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
273	C₄×C₅₃	#535	1	1	yes	characteristic
274	C₄×C₅₃	#546	1	2	yes	—
275	#3499	C₂×C₅₃	1	1	yes	characteristic
276	#3141	C₂×C₅₃	1	1	yes	characteristic, complemented
277	#3079	C₂×C₅₃	1	1	yes	characteristic, complemented
278	#2862	C₂×C₅₃	1	1	yes	characteristic, complemented
279	#2038	C₂×C₅₃	1	4	yes	complemented
280	#1966	C₂×C₅₃	1	1	yes	characteristic, complemented
281	#1962	C₂×C₅₃	1	1	yes	characteristic, complemented
282	#1793	C₂×C₅₃	1	4	yes	complemented
283	C₄×#104	C₂×C₅₃	1	1	yes	characteristic, complemented
284	C₂×C₅₃	—	2	2	no	—
285	C₂×C₅₃	—	4	2	no	—
286	C₂×C₅₃	—	8	2	no	—
287	C₂×C₅₃	#2435	1	1	yes	characteristic
288	C₂×C₅₃	C₂×#454	1	1	yes	characteristic
289	C₂×C₅₃	C₂×#456	1	1	yes	characteristic
290	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	2	yes	—
291	D₄×Q₈	—	2	1	no	—
292	D₄×Q₈	—	2	1	no	—
293	#547	—	2	1	no	—
294	#547	—	2	1	no	—
295	#547	—	2	2	no	—
296	#547	—	2	2	no	—
297	#536	—	2	2	no	—
298	#536	C₂²×C₅₃	1	2	yes	—
299	#532	—	2	1	no	—
300	#532	C₂²×C₅₃	1	1	yes	characteristic
301	C₂×C₄⋊Q₈	—	2	1	no	—
302	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
303	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
304	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
305	#489	—	2	2	no	—
306	#487	—	2	1	no	—
307	#486	—	2	1	no	—
308	#477	—	2	1	no	—
309	#476	—	2	1	no	—
310	#475	—	2	1	no	—
311	#474	—	2	1	no	—
312	#428	C₂²×C₅₃	1	2	yes	—
313	#423	C₂²×C₅₃	1	1	yes	characteristic
314	C₂×#104	C₂²×C₅₃	1	2	yes	—
315	C₂×#104	C₂²×C₅₃	1	4	yes	—
316	#417	C₂²×C₅₃	1	2	yes	—
317	#417	C₂²×C₅₃	1	4	yes	—
318	C₂×#101	C₂²×C₅₃	1	4	yes	—
319	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
320	#391	—	2	2	no	—
321	#391	C₂²×C₅₃	1	4	yes	—
322	#382	—	2	2	no	—
323	C₄³	C₂²×C₅₃	1	1	yes	characteristic
324	#339	C₂²×C₅₃	1	2	yes	—
325	C₅₃	#25876	1	1	yes	characteristic, direct product factor, Sylow
326	2₋¹⁺⁴	—	2	8	no	—
327	2₋¹⁺⁴	—	4	4	no	—
328	C₂×#41	—	2	2	no	—
329	C₂×#41	—	2	2	no	—
330	C₂×#41	—	2	4	no	—
331	C₂×#41	C₂³×C₅₃	1	2	yes	—
332	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
333	C₂²×Q₈	—	2	2	no	—
334	C₂²×Q₈	—	2	2	no	—
335	C₂²×Q₈	—	2	1	no	—
336	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
337	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
338	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
339	C₄⋊Q₈	—	2	2	no	—
340	C₄⋊Q₈	—	2	2	no	—
341	C₄⋊Q₈	—	2	2	no	—
342	C₄⋊Q₈	—	2	1	no	—
343	C₄⋊Q₈	—	2	1	no	—
344	C₄⋊Q₈	—	2	1	no	—
345	C₄⋊Q₈	—	2	1	no	—
346	C₄⋊Q₈	—	2	1	no	—
347	C₄⋊Q₈	—	2	1	no	—
348	C₄⋊Q₈	—	2	1	no	—
349	C₄⋊Q₈	—	2	1	no	—
350	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
351	#127	D₄×C₅₃	1	1	yes	characteristic
352	#124	—	2	2	no	—
353	#124	D₄×C₅₃	1	2	yes	—
354	#123	—	2	4	no	—
355	#123	—	2	4	no	—
356	#123	—	2	4	no	—
357	C₂²⋊Q₈	—	2	4	no	—
358	C₂²⋊Q₈	—	2	4	no	—
359	C₂²⋊Q₈	—	2	2	no	—
360	C₂²⋊Q₈	—	2	2	no	—
361	#121	—	2	2	no	—
362	#121	—	2	2	no	—
363	#121	—	2	1	no	—
364	#121	—	2	1	no	—
365	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
366	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
367	C₄×D₄	—	2	2	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	2	1	no	—
370	C₄×D₄	—	2	1	no	—
371	C₄×D₄	—	2	1	no	—
372	C₄×D₄	—	2	1	no	—
373	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
374	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
375	#117	—	2	1	no	—
376	#117	C₂³×C₅₃	1	1	yes	characteristic
377	C₂×C₄⋊C₄	—	2	2	no	—
378	C₂×C₄⋊C₄	—	2	2	no	—
379	C₂×C₄⋊C₄	—	2	1	no	—
380	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
381	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
382	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
383	C₂×C₄²	—	2	1	no	—
384	C₂×C₄²	—	2	1	no	—
385	C₂×C₄²	—	2	1	no	—
386	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
387	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
388	#107	—	2	2	no	—
389	#106	—	2	2	no	—
390	C₄⋊C₈	—	2	1	no	—
391	C₄⋊C₈	—	2	1	no	—
392	#104	—	2	4	no	—
393	#104	—	4	4	no	—
394	#103	—	2	2	no	—
395	#103	—	2	4	no	—
396	#102	—	2	2	no	—
397	#102	—	2	4	no	—
398	#101	—	4	4	no	—
399	#97	—	2	1	no	—
400	C₄×C₈	—	2	1	no	—
401	#95	—	2	8	no	—
402	#41	—	2	2	no	—
403	#41	—	2	4	no	—
404	#41	—	2	4	no	—
405	#41	—	4	8	no	—
406	#41	—	4	16	no	—
407	#41	—	4	4	no	—
408	#41	—	4	4	no	—
409	#41	#41×C₅₃	1	4	yes	—
410	C₂×Q₈	—	2	4	no	—
411	C₂×Q₈	—	2	2	no	—
412	C₂×Q₈	—	2	2	no	—
413	C₂×Q₈	—	2	4	no	—
414	C₂×Q₈	—	2	4	no	—
415	C₂×Q₈	—	4	8	no	—
416	C₂×Q₈	—	4	4	no	—
417	C₂×Q₈	—	4	4	no	—
418	C₂×Q₈	—	4	4	no	—
419	C₂×Q₈	—	4	1	no	—
420	C₂×Q₈	—	4	1	no	—
421	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
422	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
423	C₂×D₄	—	2	1	no	—
424	C₂×D₄	—	2	4	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	2	1	no	—
427	C₂×D₄	—	2	1	no	—
428	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
429	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
439	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
441	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
442	M₁₆	—	2	2	no	—
443	M₁₆	—	2	2	no	—
444	M₁₆	—	2	4	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	—	2	2	no	—
447	C₄⋊C₄	—	2	4	no	—
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	2	no	—
452	C₄⋊C₄	—	2	2	no	—
453	C₄⋊C₄	—	2	2	no	—
454	C₄⋊C₄	—	2	2	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	2	2	no	—
459	C₄⋊C₄	—	4	1	no	—
460	C₄⋊C₄	—	4	1	no	—
461	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
462	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
463	C₂²⋊C₄	—	2	4	no	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	1	no	—
466	C₂²⋊C₄	—	2	1	no	—
467	C₂²⋊C₄	—	2	1	no	—
468	C₂²⋊C₄	—	2	2	no	—
469	C₂²⋊C₄	—	2	2	no	—
470	C₄²	—	2	4	no	—
471	C₄²	—	2	1	no	—
472	C₄²	—	2	1	no	—
473	C₄²	—	2	1	no	—
474	C₄²	—	2	2	no	—
475	C₄²	—	2	2	no	—
476	C₄²	—	4	2	no	—
477	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
478	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
479	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
480	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
481	C₂³	—	2	1	no	—
482	C₂³	—	2	2	no	—
483	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
484	Q₈	—	2	1	no	—
485	Q₈	—	2	4	no	—
486	Q₈	—	4	8	no	—
487	Q₈	—	4	2	no	—
488	Q₈	—	4	2	no	—
489	Q₈	—	4	4	no	—
490	Q₈	—	8	2	no	—
491	Q₈	—	8	2	no	—
492	Q₈	#124×C₅₃	1	2	yes	—
493	D₄	—	2	4	no	—
494	D₄	—	2	1	no	—
495	D₄	—	2	4	no	—
496	D₄	—	4	8	no	—
497	D₄	—	4	4	no	—
498	D₄	—	4	2	no	—
499	D₄	—	4	2	no	—
500	D₄	#124×C₅₃	1	2	yes	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	8	no	—
511	C₂×C₄	—	4	8	no	—
512	C₂×C₄	—	4	2	no	—
513	C₂×C₄	—	4	8	no	—
514	C₂×C₄	—	4	2	no	—
515	C₂×C₄	—	4	2	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	—	4	2	no	—
518	C₂×C₄	—	4	2	no	—
519	C₂×C₄	—	4	4	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
522	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
523	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
524	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
525	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
526	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
527	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
528	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
529	C₈	—	4	2	no	—
530	C₈	—	4	2	no	—
531	C₂²	—	2	1	no	—
532	C₂²	—	2	1	no	—
533	C₂²	—	2	2	no	—
534	C₂²	—	4	1	no	—
535	C₂²	—	4	1	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	2	no	—
538	C₂²	—	4	2	no	—
539	C₂²	C₂×#120×C₅₃	1	1	yes	characteristic
540	C₂²	C₅₃×#546	1	2	yes	—
541	C₄	—	2	2	no	—
542	C₄	—	2	1	no	—
543	C₄	—	2	1	no	—
544	C₄	—	4	4	no	—
545	C₄	—	4	4	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	1	no	—
548	C₄	—	4	1	no	—
549	C₄	—	8	2	no	—
550	C₄	C₅₃×#454	1	2	yes	—
551	C₄	C₅₃×#456	1	2	yes	—
552	C₄	C₂×#124×C₅₃	1	1	yes	characteristic
553	C₄	C₅₃×#535	1	1	yes	characteristic
554	C₄	C₅₃×#546	1	2	yes	—
555	C₂	—	2	2	no	—
556	C₂	—	4	2	no	—
557	C₂	—	8	2	no	—
558	C₂	C₅₃×#2435	1	1	yes	characteristic
559	C₂	C₂×C₅₃×#454	1	1	yes	characteristic
560	C₂	C₂×C₅₃×#456	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₅₃×#3499	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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
2. C₅₃×#3141	C₂	semidirect
3. C₅₃×#3079	C₂	semidirect
4. C₅₃×#2862	C₂	semidirect
5. C₅₃×#2038	C₂	semidirect
6. C₅₃×#1966	C₂	semidirect
7. C₅₃×#1962	C₂	semidirect
8. C₅₃×#1793	C₂	semidirect
9. C₄×#104×C₅₃	C₂	semidirect
250. #25876	C₅₃	direct (N × H)
276. #3141	C₂×C₅₃	semidirect
277. #3079	C₂×C₅₃	semidirect
278. #2862	C₂×C₅₃	semidirect
279. #2038	C₂×C₅₃	semidirect
280. #1966	C₂×C₅₃	semidirect
281. #1962	C₂×C₅₃	semidirect
282. #1793	C₂×C₅₃	semidirect
283. C₄×#104	C₂×C₅₃	semidirect
325. C₅₃	#25876	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×#3499	2
C₅₃×#3141	2
C₅₃×#3079	2
C₅₃×#2862	2
C₅₃×#2038	2
C₅₃×#1966	2
C₅₃×#1962	2
C₅₃×#1793	2
C₄×#104×C₅₃	2
#25876	53

Subgroups — C53×#25876 #27233388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#25876 #27233388