Subgroups — C₅₃×#21478 #27228990

Order 13568.

1282 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3573	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3018	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2955	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2932	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1884	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1881	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1861	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1858	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1794	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1765	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1757	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#585	C₂²	1	1	yes	characteristic
13	C₅₃×#575	C₂²	1	2	yes	—
14	C₅₃×#568	—	2	4	no	—
15	C₅₃×#568	—	2	2	no	—
16	C₅₃×#568	C₂²	1	1	yes	characteristic
17	C₅₃×#568	C₂²	1	2	yes	—
18	C₂×#131×C₅₃	—	2	4	no	—
19	C₂×#131×C₅₃	—	2	2	no	—
20	C₂×#131×C₅₃	C₂²	1	2	yes	—
21	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic, complemented
23	C₅₃×#532	C₂²	1	1	yes	characteristic
24	C₅₃×#456	C₄	1	2	yes	complemented
25	C₅₃×#455	C₄	1	2	yes	complemented
26	C₅₃×#454	C₄	1	2	yes	complemented
27	C₅₃×#453	C₄	1	2	yes	complemented
28	C₅₃×#444	—	2	4	no	—
29	C₅₃×#443	—	2	4	no	—
30	C₅₃×#435	—	2	2	no	—
31	C₅₃×#435	—	2	2	no	—
32	C₅₃×#430	C₂²	1	2	yes	complemented
33	C₅₃×#423	C₂²	1	1	yes	characteristic, complemented
34	C₅₃×#421	C₂²	1	2	yes	—
35	C₅₃×#421	C₂²	1	2	yes	complemented
36	C₂×#104×C₅₃	C₂²	1	2	yes	—
37	C₅₃×#416	C₂²	1	2	yes	—
38	C₅₃×#413	C₂²	1	1	yes	characteristic
39	C₅₃×#410	C₂²	1	1	yes	characteristic
40	C₅₃×#408	C₂²	1	1	yes	characteristic
41	C₅₃×#408	C₂²	1	1	yes	characteristic, complemented
42	C₅₃×#405	C₂²	1	2	yes	complemented
43	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic, complemented
44	C₅₃×#344	C₂²	1	2	yes	complemented
45	C₅₃×#341	C₂²	1	2	yes	complemented
46	C₅₃×#339	C₂²	1	2	yes	complemented
47	2₋¹⁺⁴×C₅₃	—	2	2	no	—
48	2₋¹⁺⁴×C₅₃	C₂×C₄	1	2	yes	—
49	H(5,𝔽₂)×C₅₃	—	2	2	no	—
50	H(5,𝔽₂)×C₅₃	—	2	2	no	—
51	H(5,𝔽₂)×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×#41×C₅₃	—	2	4	no	—
53	C₂×#41×C₅₃	—	2	2	no	—
54	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×#41×C₅₃	C₂³	1	2	yes	—
56	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
57	#137×C₅₃	—	2	2	no	—
58	#137×C₅₃	C₂×C₄	1	4	yes	—
59	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
60	AGL(1,ℤ/8)×C₅₃	C₂×C₄	1	4	yes	—
61	#135×C₅₃	—	2	4	no	—
62	#131×C₅₃	—	2	4	no	—
63	#131×C₅₃	—	2	8	no	—
64	#131×C₅₃	—	2	4	no	—
65	#131×C₅₃	—	2	4	no	—
66	#131×C₅₃	—	2	2	no	—
67	#131×C₅₃	—	2	2	no	—
68	#131×C₅₃	—	2	4	no	—
69	#131×C₅₃	—	2	2	no	—
70	#131×C₅₃	—	2	2	no	—
71	#131×C₅₃	—	2	2	no	—
72	#131×C₅₃	—	2	2	no	—
73	#131×C₅₃	—	2	4	no	—
74	#131×C₅₃	D₄	1	4	yes	complemented
75	#131×C₅₃	D₄	1	4	yes	complemented
76	C₂×M₁₆×C₅₃	—	2	8	no	—
77	C₂×M₁₆×C₅₃	—	2	4	no	—
78	C₂×M₁₆×C₅₃	—	2	4	no	—
79	C₂×M₁₆×C₅₃	—	2	2	no	—
80	C₂×M₁₆×C₅₃	—	2	1	no	—
81	C₂×M₁₆×C₅₃	—	2	2	no	—
82	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
83	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
84	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
85	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
86	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
87	C₂²×C₈×C₅₃	—	2	4	no	—
88	C₂²×C₈×C₅₃	—	2	2	no	—
89	C₂²×C₈×C₅₃	—	2	1	no	—
90	C₂²×C₈×C₅₃	D₄	1	2	yes	—
91	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
94	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
95	#124×C₅₃	C₂×C₄	1	1	yes	characteristic
96	#124×C₅₃	C₂×C₄	1	1	yes	characteristic
97	#121×C₅₃	—	2	2	no	—
98	C₄×D₄×C₅₃	—	2	2	no	—
99	#117×C₅₃	C₂³	1	2	yes	—
100	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
101	#108×C₅₃	—	2	4	no	—
102	C₄⋊C₈×C₅₃	—	2	2	no	—
103	#104×C₅₃	—	2	4	no	—
104	#104×C₅₃	—	2	4	no	—
105	#104×C₅₃	C₂×C₄	1	4	yes	—
106	#104×C₅₃	C₂×C₄	1	4	yes	—
107	#103×C₅₃	—	2	2	no	—
108	#102×C₅₃	—	2	2	no	—
109	#101×C₅₃	C₂×C₄	1	2	yes	—
110	#100×C₅₃	C₂×C₄	1	2	yes	—
111	#99×C₅₃	—	2	2	no	—
112	C₂²⋊C₈×C₅₃	—	2	2	no	—
113	C₂²⋊C₈×C₅₃	—	2	2	no	—
114	#97×C₅₃	—	2	1	no	—
115	#97×C₅₃	—	2	1	no	—
116	#97×C₅₃	—	2	2	no	—
117	C₄×C₈×C₅₃	—	2	2	no	—
118	#41×C₅₃	—	2	4	no	—
119	#41×C₅₃	—	2	8	no	—
120	#41×C₅₃	—	2	4	no	—
121	#41×C₅₃	—	2	4	no	—
122	#41×C₅₃	—	2	4	no	—
123	#41×C₅₃	—	2	4	no	—
124	#41×C₅₃	—	2	2	no	—
125	#41×C₅₃	—	2	2	no	—
126	#41×C₅₃	—	2	2	no	—
127	#41×C₅₃	—	2	2	no	—
128	#41×C₅₃	—	2	2	no	—
129	#41×C₅₃	—	2	2	no	—
130	#41×C₅₃	C₂²×C₄	1	4	yes	—
131	#41×C₅₃	C₂×D₄	1	4	yes	—
132	C₂×Q₈×C₅₃	—	2	2	no	—
133	C₂×Q₈×C₅₃	—	2	4	no	—
134	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₂×Q₈×C₅₃	#41	1	2	yes	—
137	C₂×D₄×C₅₃	—	2	8	no	—
138	C₂×D₄×C₅₃	—	2	2	no	—
139	C₂×D₄×C₅₃	—	2	4	no	—
140	C₂×D₄×C₅₃	—	2	2	no	—
141	C₂×D₄×C₅₃	—	2	4	no	—
142	C₂×D₄×C₅₃	—	2	1	no	—
143	C₂×D₄×C₅₃	—	2	2	no	—
144	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂×D₄×C₅₃	#41	1	2	yes	—
147	C₂×D₄×C₅₃	#41	1	2	yes	—
148	C₂²×C₄×C₅₃	—	2	4	no	—
149	C₂²×C₄×C₅₃	—	2	2	no	—
150	C₂²×C₄×C₅₃	—	2	1	no	—
151	C₂²×C₄×C₅₃	—	2	1	no	—
152	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
154	Q₁₆×C₅₃	—	2	4	no	—
155	SD₁₆×C₅₃	—	2	4	no	—
156	SD₁₆×C₅₃	—	2	4	no	—
157	D₈×C₅₃	—	2	4	no	—
158	M₁₆×C₅₃	—	2	4	no	—
159	M₁₆×C₅₃	—	2	8	no	—
160	M₁₆×C₅₃	—	2	2	no	—
161	M₁₆×C₅₃	—	2	2	no	—
162	M₁₆×C₅₃	—	2	2	no	—
163	M₁₆×C₅₃	—	2	4	no	—
164	M₁₆×C₅₃	—	2	4	no	—
165	M₁₆×C₅₃	—	2	4	no	—
166	M₁₆×C₅₃	—	2	2	no	—
167	M₁₆×C₅₃	—	4	1	no	—
168	M₁₆×C₅₃	—	4	1	no	—
169	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
170	M₁₆×C₅₃	C₂×D₄	1	4	yes	—
171	C₂×C₈×C₅₃	—	2	4	no	—
172	C₂×C₈×C₅₃	—	2	8	no	—
173	C₂×C₈×C₅₃	—	2	2	no	—
174	C₂×C₈×C₅₃	—	2	2	no	—
175	C₂×C₈×C₅₃	—	2	4	no	—
176	C₂×C₈×C₅₃	—	2	2	no	—
177	C₂×C₈×C₅₃	—	2	4	no	—
178	C₂×C₈×C₅₃	—	2	2	no	—
179	C₂×C₈×C₅₃	—	2	1	no	—
180	C₂×C₈×C₅₃	—	2	1	no	—
181	C₂×C₈×C₅₃	—	2	2	no	—
182	C₂×C₈×C₅₃	—	4	2	no	—
183	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
184	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
185	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
186	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
187	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
188	C₄⋊C₄×C₅₃	—	2	2	no	—
189	C₄⋊C₄×C₅₃	—	2	2	no	—
190	C₂²⋊C₄×C₅₃	—	2	2	no	—
191	C₂²⋊C₄×C₅₃	—	2	2	no	—
192	C₂²⋊C₄×C₅₃	—	2	2	no	—
193	C₄²×C₅₃	—	2	1	no	—
194	C₄²×C₅₃	—	2	2	no	—
195	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
196	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
197	C₂³×C₅₃	—	2	4	no	—
198	C₂³×C₅₃	—	2	2	no	—
199	C₂³×C₅₃	—	2	1	no	—
200	C₂³×C₅₃	C₄×D₄	1	2	yes	—
201	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
202	Q₈×C₅₃	—	2	4	no	—
203	Q₈×C₅₃	—	2	2	no	—
204	Q₈×C₅₃	—	2	2	no	—
205	Q₈×C₅₃	—	2	2	no	—
206	Q₈×C₅₃	C₄×D₄	1	4	yes	—
207	D₄×C₅₃	—	2	8	no	—
208	D₄×C₅₃	—	2	4	no	—
209	D₄×C₅₃	—	2	4	no	—
210	D₄×C₅₃	—	2	4	no	—
211	D₄×C₅₃	—	2	4	no	—
212	D₄×C₅₃	—	2	2	no	—
213	D₄×C₅₃	—	2	2	no	—
214	D₄×C₅₃	—	2	2	no	—
215	D₄×C₅₃	—	4	2	no	—
216	D₄×C₅₃	—	8	2	no	—
217	D₄×C₅₃	C₄×D₄	1	4	yes	—
218	C₂×C₄×C₅₃	—	2	4	no	—
219	C₂×C₄×C₅₃	—	2	8	no	—
220	C₂×C₄×C₅₃	—	2	2	no	—
221	C₂×C₄×C₅₃	—	2	4	no	—
222	C₂×C₄×C₅₃	—	2	4	no	—
223	C₂×C₄×C₅₃	—	2	2	no	—
224	C₂×C₄×C₅₃	—	2	2	no	—
225	C₂×C₄×C₅₃	—	2	1	no	—
226	C₂×C₄×C₅₃	—	2	1	no	—
227	C₂×C₄×C₅₃	—	2	1	no	—
228	C₂×C₄×C₅₃	—	4	1	no	—
229	C₂×C₄×C₅₃	—	4	1	no	—
230	C₂×C₄×C₅₃	—	4	2	no	—
231	C₂×C₄×C₅₃	—	8	1	no	—
232	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
233	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
234	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
235	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
237	C₈×C₅₃	—	2	4	no	—
238	C₈×C₅₃	—	2	4	no	—
239	C₈×C₅₃	—	2	2	no	—
240	C₈×C₅₃	—	4	1	no	—
241	C₈×C₅₃	—	4	1	no	—
242	C₈×C₅₃	#120	1	1	yes	characteristic
243	C₈×C₅₃	#120	1	2	yes	—
244	C₈×C₅₃	#120	1	1	yes	characteristic
245	#21478	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
246	C₂²×C₅₃	—	2	2	no	—
247	C₂²×C₅₃	—	2	4	no	—
248	C₂²×C₅₃	—	4	1	no	—
249	C₂²×C₅₃	—	4	2	no	—
250	C₂²×C₅₃	—	4	2	no	—
251	C₂²×C₅₃	—	4	1	no	—
252	C₂²×C₅₃	—	8	2	no	—
253	C₂²×C₅₃	—	8	4	no	—
254	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
255	C₂²×C₅₃	#525	1	2	yes	—
256	C₄×C₅₃	—	2	4	no	—
257	C₄×C₅₃	—	2	2	no	—
258	C₄×C₅₃	—	4	1	no	—
259	C₄×C₅₃	—	4	1	no	—
260	C₄×C₅₃	—	4	1	no	—
261	C₄×C₅₃	—	8	2	no	—
262	C₄×C₅₃	C₂×C₄×D₄	1	2	yes	—
263	C₄×C₅₃	C₂×#120	1	1	yes	characteristic
264	C₄×C₅₃	#525	1	1	yes	characteristic
265	#3573	C₂×C₅₃	1	1	yes	characteristic, complemented
266	#3018	C₂×C₅₃	1	1	yes	characteristic
267	#2955	C₂×C₅₃	1	2	yes	complemented
268	#2932	C₂×C₅₃	1	1	yes	characteristic, complemented
269	#1884	C₂×C₅₃	1	1	yes	characteristic, complemented
270	#1881	C₂×C₅₃	1	2	yes	complemented
271	#1861	C₂×C₅₃	1	2	yes	complemented
272	#1858	C₂×C₅₃	1	1	yes	characteristic, complemented
273	#1794	C₂×C₅₃	1	1	yes	characteristic, complemented
274	#1765	C₂×C₅₃	1	2	yes	complemented
275	#1757	C₂×C₅₃	1	1	yes	characteristic, complemented
276	C₂×C₅₃	—	2	1	no	—
277	C₂×C₅₃	—	2	2	no	—
278	C₂×C₅₃	—	4	2	no	—
279	C₂×C₅₃	—	4	4	no	—
280	C₂×C₅₃	—	8	1	no	—
281	C₂×C₅₃	C₄×#120	1	1	yes	characteristic
282	#585	C₂²×C₅₃	1	1	yes	characteristic
283	#575	C₂²×C₅₃	1	2	yes	—
284	#568	—	2	4	no	—
285	#568	—	2	2	no	—
286	#568	C₂²×C₅₃	1	1	yes	characteristic
287	#568	C₂²×C₅₃	1	2	yes	—
288	C₂×#131	—	2	4	no	—
289	C₂×#131	—	2	2	no	—
290	C₂×#131	C₂²×C₅₃	1	2	yes	—
291	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
292	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic, complemented
293	#532	C₂²×C₅₃	1	1	yes	characteristic
294	#456	C₄×C₅₃	1	2	yes	complemented
295	#455	C₄×C₅₃	1	2	yes	complemented
296	#454	C₄×C₅₃	1	2	yes	complemented
297	#453	C₄×C₅₃	1	2	yes	complemented
298	#444	—	2	4	no	—
299	#443	—	2	4	no	—
300	#435	—	2	2	no	—
301	#435	—	2	2	no	—
302	#430	C₂²×C₅₃	1	2	yes	complemented
303	#423	C₂²×C₅₃	1	1	yes	characteristic, complemented
304	#421	C₂²×C₅₃	1	2	yes	—
305	#421	C₂²×C₅₃	1	2	yes	complemented
306	C₂×#104	C₂²×C₅₃	1	2	yes	—
307	#416	C₂²×C₅₃	1	2	yes	—
308	#413	C₂²×C₅₃	1	1	yes	characteristic
309	#410	C₂²×C₅₃	1	1	yes	characteristic
310	#408	C₂²×C₅₃	1	1	yes	characteristic
311	#408	C₂²×C₅₃	1	1	yes	characteristic, complemented
312	#405	C₂²×C₅₃	1	2	yes	complemented
313	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic, complemented
314	#344	C₂²×C₅₃	1	2	yes	complemented
315	#341	C₂²×C₅₃	1	2	yes	complemented
316	#339	C₂²×C₅₃	1	2	yes	complemented
317	C₅₃	#21478	1	1	yes	characteristic, direct product factor, Sylow
318	2₋¹⁺⁴	—	2	2	no	—
319	2₋¹⁺⁴	C₂×C₄×C₅₃	1	2	yes	—
320	H(5,𝔽₂)	—	2	2	no	—
321	H(5,𝔽₂)	—	2	2	no	—
322	H(5,𝔽₂)	C₂×C₄×C₅₃	1	2	yes	—
323	C₂×#41	—	2	4	no	—
324	C₂×#41	—	2	2	no	—
325	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
326	C₂×#41	C₂³×C₅₃	1	2	yes	—
327	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
328	#137	—	2	2	no	—
329	#137	C₂×C₄×C₅₃	1	4	yes	—
330	AGL(1,ℤ/8)	—	2	2	no	—
331	AGL(1,ℤ/8)	C₂×C₄×C₅₃	1	4	yes	—
332	#135	—	2	4	no	—
333	#131	—	2	4	no	—
334	#131	—	2	8	no	—
335	#131	—	2	4	no	—
336	#131	—	2	4	no	—
337	#131	—	2	4	no	—
338	#131	—	2	2	no	—
339	#131	—	2	2	no	—
340	#131	—	2	4	no	—
341	#131	—	2	2	no	—
342	#131	—	2	2	no	—
343	#131	—	2	2	no	—
344	#131	—	2	2	no	—
345	#131	D₄×C₅₃	1	4	yes	complemented
346	#131	D₄×C₅₃	1	4	yes	complemented
347	C₂×M₁₆	—	2	8	no	—
348	C₂×M₁₆	—	2	4	no	—
349	C₂×M₁₆	—	2	2	no	—
350	C₂×M₁₆	—	2	4	no	—
351	C₂×M₁₆	—	2	2	no	—
352	C₂×M₁₆	—	2	1	no	—
353	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
354	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
355	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
356	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
357	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
358	C₂²×C₈	—	2	4	no	—
359	C₂²×C₈	—	2	2	no	—
360	C₂²×C₈	—	2	1	no	—
361	C₂²×C₈	D₄×C₅₃	1	2	yes	—
362	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
363	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
364	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
365	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
366	#124	C₂×C₄×C₅₃	1	1	yes	characteristic
367	#124	C₂×C₄×C₅₃	1	1	yes	characteristic
368	#121	—	2	2	no	—
369	C₄×D₄	—	2	2	no	—
370	#117	C₂³×C₅₃	1	2	yes	—
371	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
372	#108	—	2	4	no	—
373	C₄⋊C₈	—	2	2	no	—
374	#104	—	2	4	no	—
375	#104	—	2	4	no	—
376	#104	C₂×C₄×C₅₃	1	4	yes	—
377	#104	C₂×C₄×C₅₃	1	4	yes	—
378	#103	—	2	2	no	—
379	#102	—	2	2	no	—
380	#101	C₂×C₄×C₅₃	1	2	yes	—
381	#100	C₂×C₄×C₅₃	1	2	yes	—
382	#99	—	2	2	no	—
383	C₂²⋊C₈	—	2	2	no	—
384	C₂²⋊C₈	—	2	2	no	—
385	#97	—	2	1	no	—
386	#97	—	2	1	no	—
387	#97	—	2	2	no	—
388	C₄×C₈	—	2	2	no	—
389	#41	—	2	4	no	—
390	#41	—	2	8	no	—
391	#41	—	2	4	no	—
392	#41	—	2	4	no	—
393	#41	—	2	4	no	—
394	#41	—	2	4	no	—
395	#41	—	2	2	no	—
396	#41	—	2	2	no	—
397	#41	—	2	2	no	—
398	#41	—	2	2	no	—
399	#41	—	2	2	no	—
400	#41	—	2	2	no	—
401	#41	C₂²×C₄×C₅₃	1	4	yes	—
402	#41	C₂×D₄×C₅₃	1	4	yes	—
403	C₂×Q₈	—	2	2	no	—
404	C₂×Q₈	—	2	4	no	—
405	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
406	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
407	C₂×Q₈	#41×C₅₃	1	2	yes	—
408	C₂×D₄	—	2	8	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	2	no	—
412	C₂×D₄	—	2	4	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	2	2	no	—
415	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
416	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
417	C₂×D₄	#41×C₅₃	1	2	yes	—
418	C₂×D₄	#41×C₅₃	1	2	yes	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
424	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
425	Q₁₆	—	2	4	no	—
426	SD₁₆	—	2	4	no	—
427	SD₁₆	—	2	4	no	—
428	D₈	—	2	4	no	—
429	M₁₆	—	2	4	no	—
430	M₁₆	—	2	8	no	—
431	M₁₆	—	2	2	no	—
432	M₁₆	—	2	2	no	—
433	M₁₆	—	2	2	no	—
434	M₁₆	—	2	4	no	—
435	M₁₆	—	2	4	no	—
436	M₁₆	—	2	4	no	—
437	M₁₆	—	2	2	no	—
438	M₁₆	—	4	1	no	—
439	M₁₆	—	4	1	no	—
440	M₁₆	C₂²×C₄×C₅₃	1	4	yes	—
441	M₁₆	C₂×D₄×C₅₃	1	4	yes	—
442	C₂×C₈	—	2	4	no	—
443	C₂×C₈	—	2	8	no	—
444	C₂×C₈	—	2	2	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	—	2	4	no	—
447	C₂×C₈	—	2	2	no	—
448	C₂×C₈	—	2	4	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	2	no	—
453	C₂×C₈	—	4	2	no	—
454	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
455	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
456	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
457	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
458	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₂²⋊C₄	—	2	2	no	—
462	C₂²⋊C₄	—	2	2	no	—
463	C₂²⋊C₄	—	2	2	no	—
464	C₄²	—	2	1	no	—
465	C₄²	—	2	2	no	—
466	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
467	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
468	C₂³	—	2	4	no	—
469	C₂³	—	2	2	no	—
470	C₂³	—	2	1	no	—
471	C₂³	C₄×D₄×C₅₃	1	2	yes	—
472	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
473	Q₈	—	2	4	no	—
474	Q₈	—	2	2	no	—
475	Q₈	—	2	2	no	—
476	Q₈	—	2	2	no	—
477	Q₈	C₄×D₄×C₅₃	1	4	yes	—
478	D₄	—	2	8	no	—
479	D₄	—	2	4	no	—
480	D₄	—	2	4	no	—
481	D₄	—	2	4	no	—
482	D₄	—	2	4	no	—
483	D₄	—	2	2	no	—
484	D₄	—	2	2	no	—
485	D₄	—	2	2	no	—
486	D₄	—	4	2	no	—
487	D₄	—	8	2	no	—
488	D₄	C₄×D₄×C₅₃	1	4	yes	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	8	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	8	1	no	—
503	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
504	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
505	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
506	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
507	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
508	C₈	—	2	4	no	—
509	C₈	—	2	4	no	—
510	C₈	—	2	2	no	—
511	C₈	—	4	1	no	—
512	C₈	—	4	1	no	—
513	C₈	#120×C₅₃	1	1	yes	characteristic
514	C₈	#120×C₅₃	1	2	yes	—
515	C₈	#120×C₅₃	1	1	yes	characteristic
516	C₂²	—	2	2	no	—
517	C₂²	—	2	4	no	—
518	C₂²	—	4	1	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	2	no	—
521	C₂²	—	4	1	no	—
522	C₂²	—	8	2	no	—
523	C₂²	—	8	4	no	—
524	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
525	C₂²	C₅₃×#525	1	2	yes	—
526	C₄	—	2	4	no	—
527	C₄	—	2	2	no	—
528	C₄	—	4	1	no	—
529	C₄	—	4	1	no	—
530	C₄	—	4	1	no	—
531	C₄	—	8	2	no	—
532	C₄	C₂×C₄×D₄×C₅₃	1	2	yes	—
533	C₄	C₂×#120×C₅₃	1	1	yes	characteristic
534	C₄	C₅₃×#525	1	1	yes	characteristic
535	C₂	—	2	1	no	—
536	C₂	—	2	2	no	—
537	C₂	—	4	2	no	—
538	C₂	—	4	4	no	—
539	C₂	—	8	1	no	—
540	C₂	C₄×#120×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	C₂×#120
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂×C₅₃	C₄×#120
prime radical	C₅₃×#3018	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3573	C₂	semidirect
3. C₅₃×#2955	C₂	semidirect
4. C₅₃×#2932	C₂	semidirect
5. C₅₃×#1884	C₂	semidirect
6. C₅₃×#1881	C₂	semidirect
7. C₅₃×#1861	C₂	semidirect
8. C₅₃×#1858	C₂	semidirect
9. C₅₃×#1794	C₂	semidirect
10. C₅₃×#1765	C₂	semidirect
11. C₅₃×#1757	C₂	semidirect
22. C₂×#131×C₅₃	C₂²	semidirect
24. C₅₃×#456	C₄	semidirect
25. C₅₃×#455	C₄	semidirect
26. C₅₃×#454	C₄	semidirect
27. C₅₃×#453	C₄	semidirect
32. C₅₃×#430	C₂²	semidirect
33. C₅₃×#423	C₂²	semidirect
35. C₅₃×#421	C₂²	semidirect
41. C₅₃×#408	C₂²	semidirect
42. C₅₃×#405	C₂²	semidirect
43. C₂×#97×C₅₃	C₂²	semidirect
44. C₅₃×#344	C₂²	semidirect
45. C₅₃×#341	C₂²	semidirect
46. C₅₃×#339	C₂²	semidirect
74. #131×C₅₃	D₄	semidirect
75. #131×C₅₃	D₄	semidirect
245. #21478	C₅₃	direct (N × H)
265. #3573	C₂×C₅₃	semidirect
267. #2955	C₂×C₅₃	semidirect
268. #2932	C₂×C₅₃	semidirect
269. #1884	C₂×C₅₃	semidirect
270. #1881	C₂×C₅₃	semidirect
271. #1861	C₂×C₅₃	semidirect
272. #1858	C₂×C₅₃	semidirect
273. #1794	C₂×C₅₃	semidirect
274. #1765	C₂×C₅₃	semidirect
275. #1757	C₂×C₅₃	semidirect
292. C₂×#131	C₂²×C₅₃	semidirect
294. #456	C₄×C₅₃	semidirect
295. #455	C₄×C₅₃	semidirect
296. #454	C₄×C₅₃	semidirect
297. #453	C₄×C₅₃	semidirect
302. #430	C₂²×C₅₃	semidirect
303. #423	C₂²×C₅₃	semidirect
305. #421	C₂²×C₅₃	semidirect
311. #408	C₂²×C₅₃	semidirect
312. #405	C₂²×C₅₃	semidirect
313. C₂×#97	C₂²×C₅₃	semidirect
314. #344	C₂²×C₅₃	semidirect
315. #341	C₂²×C₅₃	semidirect
316. #339	C₂²×C₅₃	semidirect
317. C₅₃	#21478	direct (N × H)
345. #131	D₄×C₅₃	semidirect
346. #131	D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3573	2
C₅₃×#3018	2
C₅₃×#2955	2
C₅₃×#2932	2
C₅₃×#1884	2
C₅₃×#1881	2
C₅₃×#1861	2
C₅₃×#1858	2
C₅₃×#1794	2
C₅₃×#1765	2
C₅₃×#1757	2
#21478	53

Subgroups — C53×#21478 #27228990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21478 #27228990