Subgroups — C₅₃×#23015 #27230527

Order 13568.

892 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3035	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#443	C₂	1	2	yes	maximal, complemented
3	C₅₃×#2935	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2906	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1962	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1950	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1937	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1876	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1866	C₂	1	2	yes	maximal, complemented
10	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#104×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₅₃	C₂²	1	2	yes	—
13	C₂×#131×C₅₃	C₂²	1	2	yes	—
14	C₂×#131×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#532	C₂²	1	1	yes	characteristic
16	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#464	C₄	1	2	yes	complemented
18	C₅₃×#463	C₄	1	2	yes	complemented
19	C₅₃×#462	C₄	1	1	yes	characteristic, complemented
20	C₅₃×#461	C₄	1	1	yes	characteristic, complemented
21	C₅₃×#460	C₄	1	1	yes	characteristic, complemented
22	C₅₃×#459	C₄	1	1	yes	characteristic, complemented
23	C₅₃×#443	—	2	8	no	—
24	C₅₃×#436	—	2	1	no	—
25	C₅₃×#435	—	2	2	no	—
26	C₈×D₄×C₅₃	—	2	1	no	—
27	C₂×#108×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#424	—	2	2	no	—
29	C₅₃×#424	C₂²	1	2	yes	—
30	C₅₃×#423	—	2	1	no	—
31	C₅₃×#423	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
33	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#104×C₅₃	C₂²	1	2	yes	—
36	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×#104×C₅₃	C₂²	1	2	yes	—
38	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#416	C₂²	1	2	yes	—
40	C₅₃×#408	C₂²	1	2	yes	—
41	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
43	C₄³×C₅₃	C₂²	1	1	yes	characteristic
44	C₅₃×#341	C₂²	1	2	yes	—
45	C₅₃×#339	C₂²	1	1	yes	characteristic
46	C₅₃×#339	C₂²	1	1	yes	characteristic
47	C₅₃×#336	C₂²	1	2	yes	—
48	C₂×#41×C₅₃	C₂³	1	2	yes	—
49	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
50	#135×C₅₃	—	2	4	no	—
51	#135×C₅₃	—	2	4	no	—
52	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
55	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
56	#131×C₅₃	—	2	4	no	—
57	#131×C₅₃	—	2	4	no	—
58	#131×C₅₃	—	4	2	no	—
59	#131×C₅₃	—	4	2	no	—
60	C₂×M₁₆×C₅₃	—	2	2	no	—
61	C₂×M₁₆×C₅₃	—	2	2	no	—
62	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
63	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₅₃	—	2	2	no	—
66	C₂²×C₈×C₅₃	—	2	2	no	—
67	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
68	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
70	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
71	#124×C₅₃	C₂×C₄	1	2	yes	—
72	#121×C₅₃	—	2	1	no	—
73	#121×C₅₃	—	2	1	no	—
74	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₄×D₄×C₅₃	—	2	1	no	—
77	C₄×D₄×C₅₃	—	2	1	no	—
78	C₄×D₄×C₅₃	—	2	1	no	—
79	C₄×D₄×C₅₃	—	2	1	no	—
80	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
82	#117×C₅₃	—	2	1	no	—
83	#117×C₅₃	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₅₃	—	2	1	no	—
85	C₂×C₄²×C₅₃	—	2	1	no	—
86	C₂×C₄²×C₅₃	—	2	1	no	—
87	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
90	#108×C₅₃	—	2	4	no	—
91	C₄⋊C₈×C₅₃	—	2	2	no	—
92	C₄⋊C₈×C₅₃	—	2	2	no	—
93	C₄⋊C₈×C₅₃	—	2	1	no	—
94	C₄⋊C₈×C₅₃	—	2	1	no	—
95	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
98	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
99	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
100	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
101	#104×C₅₃	—	2	4	no	—
102	#104×C₅₃	—	2	2	no	—
103	#104×C₅₃	—	4	2	no	—
104	#104×C₅₃	D₄	1	4	yes	complemented
105	#103×C₅₃	C₂×C₄	1	2	yes	—
106	#103×C₅₃	C₂×C₄	1	2	yes	—
107	#102×C₅₃	C₂×C₄	1	2	yes	—
108	#102×C₅₃	C₂×C₄	1	2	yes	—
109	C₂²⋊C₈×C₅₃	—	2	2	no	—
110	C₂²⋊C₈×C₅₃	—	2	2	no	—
111	#97×C₅₃	—	2	1	no	—
112	C₄×C₈×C₅₃	—	2	4	no	—
113	C₄×C₈×C₅₃	—	2	1	no	—
114	#41×C₅₃	—	2	1	no	—
115	#41×C₅₃	—	2	1	no	—
116	#41×C₅₃	—	2	2	no	—
117	#41×C₅₃	—	2	2	no	—
118	#41×C₅₃	—	2	2	no	—
119	#41×C₅₃	—	2	2	no	—
120	#41×C₅₃	C₂×D₄	1	2	yes	—
121	#41×C₅₃	#41	1	2	yes	—
122	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
123	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×D₄×C₅₃	—	2	1	no	—
125	C₂×D₄×C₅₃	—	2	2	no	—
126	C₂×D₄×C₅₃	—	2	1	no	—
127	C₂×D₄×C₅₃	—	2	1	no	—
128	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
129	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	—	2	1	no	—
131	C₂²×C₄×C₅₃	—	2	1	no	—
132	C₂²×C₄×C₅₃	—	2	1	no	—
133	C₂²×C₄×C₅₃	—	2	1	no	—
134	C₂²×C₄×C₅₃	—	2	2	no	—
135	C₂²×C₄×C₅₃	—	2	1	no	—
136	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
137	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
138	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
139	Q₁₆×C₅₃	—	2	4	no	—
140	SD₁₆×C₅₃	—	2	4	no	—
141	SD₁₆×C₅₃	—	2	4	no	—
142	D₈×C₅₃	—	2	4	no	—
143	M₁₆×C₅₃	—	2	1	no	—
144	M₁₆×C₅₃	—	2	2	no	—
145	M₁₆×C₅₃	—	2	1	no	—
146	M₁₆×C₅₃	—	2	2	no	—
147	M₁₆×C₅₃	—	2	1	no	—
148	M₁₆×C₅₃	—	4	4	no	—
149	M₁₆×C₅₃	—	4	4	no	—
150	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
151	C₂×C₈×C₅₃	—	2	2	no	—
152	C₂×C₈×C₅₃	—	2	2	no	—
153	C₂×C₈×C₅₃	—	2	1	no	—
154	C₂×C₈×C₅₃	—	2	2	no	—
155	C₂×C₈×C₅₃	—	2	2	no	—
156	C₂×C₈×C₅₃	—	2	1	no	—
157	C₂×C₈×C₅₃	—	2	2	no	—
158	C₂×C₈×C₅₃	—	4	4	no	—
159	C₂×C₈×C₅₃	—	4	4	no	—
160	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
161	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
162	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
163	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
164	C₄⋊C₄×C₅₃	—	2	1	no	—
165	C₄⋊C₄×C₅₃	—	2	1	no	—
166	C₄⋊C₄×C₅₃	—	2	1	no	—
167	C₄⋊C₄×C₅₃	—	2	1	no	—
168	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
169	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
170	C₂²⋊C₄×C₅₃	—	2	1	no	—
171	C₂²⋊C₄×C₅₃	—	2	1	no	—
172	C₂²⋊C₄×C₅₃	—	2	1	no	—
173	C₂²⋊C₄×C₅₃	—	2	2	no	—
174	C₂²⋊C₄×C₅₃	—	2	2	no	—
175	C₄²×C₅₃	—	2	4	no	—
176	C₄²×C₅₃	—	2	1	no	—
177	C₄²×C₅₃	—	2	1	no	—
178	C₄²×C₅₃	—	2	1	no	—
179	C₄²×C₅₃	—	2	1	no	—
180	C₄²×C₅₃	—	2	1	no	—
181	C₄²×C₅₃	—	2	1	no	—
182	C₄²×C₅₃	—	4	2	no	—
183	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
184	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
185	C₄²×C₅₃	C₂×D₄	1	2	yes	—
186	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
187	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
188	C₂³×C₅₃	—	2	1	no	—
189	C₂³×C₅₃	—	2	2	no	—
190	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
191	Q₈×C₅₃	—	2	1	no	—
192	Q₈×C₅₃	—	2	2	no	—
193	Q₈×C₅₃	—	2	2	no	—
194	Q₈×C₅₃	C₄×D₄	1	2	yes	—
195	D₄×C₅₃	—	2	2	no	—
196	D₄×C₅₃	—	2	2	no	—
197	D₄×C₅₃	—	2	2	no	—
198	D₄×C₅₃	—	2	1	no	—
199	D₄×C₅₃	—	2	2	no	—
200	D₄×C₅₃	—	4	2	no	—
201	D₄×C₅₃	—	4	2	no	—
202	D₄×C₅₃	—	4	2	no	—
203	D₄×C₅₃	—	8	1	no	—
204	D₄×C₅₃	C₄×D₄	1	2	yes	—
205	C₂×C₄×C₅₃	—	2	2	no	—
206	C₂×C₄×C₅₃	—	2	2	no	—
207	C₂×C₄×C₅₃	—	2	1	no	—
208	C₂×C₄×C₅₃	—	2	2	no	—
209	C₂×C₄×C₅₃	—	2	2	no	—
210	C₂×C₄×C₅₃	—	2	2	no	—
211	C₂×C₄×C₅₃	—	2	2	no	—
212	C₂×C₄×C₅₃	—	2	2	no	—
213	C₂×C₄×C₅₃	—	2	2	no	—
214	C₂×C₄×C₅₃	—	2	1	no	—
215	C₂×C₄×C₅₃	—	2	1	no	—
216	C₂×C₄×C₅₃	—	4	1	no	—
217	C₂×C₄×C₅₃	—	4	1	no	—
218	C₂×C₄×C₅₃	—	4	1	no	—
219	C₂×C₄×C₅₃	—	4	2	no	—
220	C₂×C₄×C₅₃	—	4	2	no	—
221	C₂×C₄×C₅₃	—	4	2	no	—
222	C₂×C₄×C₅₃	—	8	1	no	—
223	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
224	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
225	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
226	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
227	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
228	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
231	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
232	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
233	C₈×C₅₃	—	2	2	no	—
234	C₈×C₅₃	—	2	2	no	—
235	C₈×C₅₃	—	2	2	no	—
236	C₈×C₅₃	—	2	2	no	—
237	C₈×C₅₃	—	2	2	no	—
238	C₈×C₅₃	—	4	2	no	—
239	C₈×C₅₃	—	4	1	no	—
240	#23015	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
241	C₂²×C₅₃	—	2	1	no	—
242	C₂²×C₅₃	—	2	1	no	—
243	C₂²×C₅₃	—	2	2	no	—
244	C₂²×C₅₃	—	4	1	no	—
245	C₂²×C₅₃	—	4	1	no	—
246	C₂²×C₅₃	—	4	2	no	—
247	C₂²×C₅₃	—	4	2	no	—
248	C₂²×C₅₃	—	4	2	no	—
249	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
250	C₂²×C₅₃	#547	1	1	yes	characteristic
251	C₂²×C₅₃	#553	1	1	yes	characteristic
252	C₄×C₅₃	—	2	4	no	—
253	C₄×C₅₃	—	2	2	no	—
254	C₄×C₅₃	—	2	1	no	—
255	C₄×C₅₃	—	2	1	no	—
256	C₄×C₅₃	—	2	1	no	—
257	C₄×C₅₃	—	4	2	no	—
258	C₄×C₅₃	—	4	2	no	—
259	C₄×C₅₃	—	4	1	no	—
260	C₄×C₅₃	—	4	1	no	—
261	C₄×C₅₃	#443	1	2	yes	complemented
262	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
263	C₄×C₅₃	#520	1	1	yes	characteristic
264	C₄×C₅₃	D₄²	1	1	yes	characteristic
265	C₄×C₅₃	#548	1	1	yes	characteristic
266	#3035	C₂×C₅₃	1	1	yes	characteristic
267	C₂×#443	C₂×C₅₃	1	2	yes	complemented
268	#2935	C₂×C₅₃	1	1	yes	characteristic, complemented
269	#2906	C₂×C₅₃	1	1	yes	characteristic, complemented
270	#1962	C₂×C₅₃	1	1	yes	characteristic, complemented
271	#1950	C₂×C₅₃	1	1	yes	characteristic, complemented
272	#1937	C₂×C₅₃	1	2	yes	complemented
273	#1876	C₂×C₅₃	1	2	yes	complemented
274	#1866	C₂×C₅₃	1	2	yes	complemented
275	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
276	C₄×#104	C₂×C₅₃	1	1	yes	characteristic, complemented
277	C₂×C₅₃	—	2	1	no	—
278	C₂×C₅₃	—	2	1	no	—
279	C₂×C₅₃	—	4	2	no	—
280	C₂×C₅₃	—	8	2	no	—
281	C₂×C₅₃	#2361	1	1	yes	characteristic
282	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
283	C₂×C₅₃	#2956	1	1	yes	characteristic
284	C₂×#135	C₂²×C₅₃	1	2	yes	—
285	C₂×#131	C₂²×C₅₃	1	2	yes	—
286	C₂×#131	C₂²×C₅₃	1	2	yes	—
287	#532	C₂²×C₅₃	1	1	yes	characteristic
288	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
289	#464	C₄×C₅₃	1	2	yes	complemented
290	#463	C₄×C₅₃	1	2	yes	complemented
291	#462	C₄×C₅₃	1	1	yes	characteristic, complemented
292	#461	C₄×C₅₃	1	1	yes	characteristic, complemented
293	#460	C₄×C₅₃	1	1	yes	characteristic, complemented
294	#459	C₄×C₅₃	1	1	yes	characteristic, complemented
295	#443	—	2	8	no	—
296	#436	—	2	1	no	—
297	#435	—	2	2	no	—
298	C₈×D₄	—	2	1	no	—
299	C₂×#108	C₂²×C₅₃	1	2	yes	—
300	#424	—	2	2	no	—
301	#424	C₂²×C₅₃	1	2	yes	—
302	#423	—	2	1	no	—
303	#423	C₂²×C₅₃	1	1	yes	characteristic
304	C₂×C₄⋊C₈	—	2	1	no	—
305	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
306	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
307	C₂×#104	C₂²×C₅₃	1	1	yes	characteristic
308	C₂×#104	C₂²×C₅₃	1	2	yes	—
309	C₂×#104	C₂²×C₅₃	1	2	yes	—
310	C₂×#104	C₂²×C₅₃	1	1	yes	characteristic
311	#416	C₂²×C₅₃	1	2	yes	—
312	#408	C₂²×C₅₃	1	2	yes	—
313	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
314	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
315	C₄³	C₂²×C₅₃	1	1	yes	characteristic
316	#341	C₂²×C₅₃	1	2	yes	—
317	#339	C₂²×C₅₃	1	1	yes	characteristic
318	#339	C₂²×C₅₃	1	1	yes	characteristic
319	#336	C₂²×C₅₃	1	2	yes	—
320	C₅₃	#23015	1	1	yes	characteristic, direct product factor, Sylow
321	C₂×#41	C₂³×C₅₃	1	2	yes	—
322	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
323	#135	—	2	4	no	—
324	#135	—	2	4	no	—
325	C₂×Q₁₆	C₂×C₄×C₅₃	1	2	yes	—
326	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
327	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
328	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
329	#131	—	2	4	no	—
330	#131	—	2	4	no	—
331	#131	—	4	2	no	—
332	#131	—	4	2	no	—
333	C₂×M₁₆	—	2	2	no	—
334	C₂×M₁₆	—	2	2	no	—
335	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
336	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
337	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
338	C₂²×C₈	—	2	2	no	—
339	C₂²×C₈	—	2	2	no	—
340	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
341	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
342	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
343	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
344	#124	C₂×C₄×C₅₃	1	2	yes	—
345	#121	—	2	1	no	—
346	#121	—	2	1	no	—
347	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
348	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
349	C₄×D₄	—	2	1	no	—
350	C₄×D₄	—	2	1	no	—
351	C₄×D₄	—	2	1	no	—
352	C₄×D₄	—	2	1	no	—
353	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
354	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
355	#117	—	2	1	no	—
356	#117	C₂³×C₅₃	1	1	yes	characteristic
357	C₂×C₄²	—	2	1	no	—
358	C₂×C₄²	—	2	1	no	—
359	C₂×C₄²	—	2	1	no	—
360	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
361	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
362	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
363	#108	—	2	4	no	—
364	C₄⋊C₈	—	2	2	no	—
365	C₄⋊C₈	—	2	2	no	—
366	C₄⋊C₈	—	2	1	no	—
367	C₄⋊C₈	—	2	1	no	—
368	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
369	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
370	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
371	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
372	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
373	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
374	#104	—	2	2	no	—
375	#104	—	2	4	no	—
376	#104	—	4	2	no	—
377	#104	D₄×C₅₃	1	4	yes	complemented
378	#103	C₂×C₄×C₅₃	1	2	yes	—
379	#103	C₂×C₄×C₅₃	1	2	yes	—
380	#102	C₂×C₄×C₅₃	1	2	yes	—
381	#102	C₂×C₄×C₅₃	1	2	yes	—
382	C₂²⋊C₈	—	2	2	no	—
383	C₂²⋊C₈	—	2	2	no	—
384	#97	—	2	1	no	—
385	C₄×C₈	—	2	4	no	—
386	C₄×C₈	—	2	1	no	—
387	#41	—	2	1	no	—
388	#41	—	2	1	no	—
389	#41	—	2	2	no	—
390	#41	—	2	2	no	—
391	#41	—	2	2	no	—
392	#41	—	2	2	no	—
393	#41	C₂×D₄×C₅₃	1	2	yes	—
394	#41	#41×C₅₃	1	2	yes	—
395	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
396	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	1	no	—
401	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
402	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	1	no	—
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	1	no	—
409	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
410	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
411	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
412	Q₁₆	—	2	4	no	—
413	SD₁₆	—	2	4	no	—
414	SD₁₆	—	2	4	no	—
415	D₈	—	2	4	no	—
416	M₁₆	—	2	1	no	—
417	M₁₆	—	2	2	no	—
418	M₁₆	—	2	1	no	—
419	M₁₆	—	2	2	no	—
420	M₁₆	—	2	1	no	—
421	M₁₆	—	4	4	no	—
422	M₁₆	—	4	4	no	—
423	M₁₆	C₂×D₄×C₅₃	1	2	yes	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	2	1	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	2	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	2	no	—
431	C₂×C₈	—	4	4	no	—
432	C₂×C₈	—	4	4	no	—
433	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
434	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
435	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
436	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	2	1	no	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
442	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
443	C₂²⋊C₄	—	2	1	no	—
444	C₂²⋊C₄	—	2	1	no	—
445	C₂²⋊C₄	—	2	1	no	—
446	C₂²⋊C₄	—	2	2	no	—
447	C₂²⋊C₄	—	2	2	no	—
448	C₄²	—	2	4	no	—
449	C₄²	—	2	1	no	—
450	C₄²	—	2	1	no	—
451	C₄²	—	2	1	no	—
452	C₄²	—	2	1	no	—
453	C₄²	—	2	1	no	—
454	C₄²	—	2	1	no	—
455	C₄²	—	4	2	no	—
456	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
457	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
458	C₄²	C₂×D₄×C₅₃	1	2	yes	—
459	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
460	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
461	C₂³	—	2	1	no	—
462	C₂³	—	2	2	no	—
463	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
464	Q₈	—	2	1	no	—
465	Q₈	—	2	2	no	—
466	Q₈	—	2	2	no	—
467	Q₈	C₄×D₄×C₅₃	1	2	yes	—
468	D₄	—	2	2	no	—
469	D₄	—	2	2	no	—
470	D₄	—	2	2	no	—
471	D₄	—	2	1	no	—
472	D₄	—	2	2	no	—
473	D₄	—	4	2	no	—
474	D₄	—	4	2	no	—
475	D₄	—	4	2	no	—
476	D₄	—	8	1	no	—
477	D₄	C₄×D₄×C₅₃	1	2	yes	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	8	1	no	—
496	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
497	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
498	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
499	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
500	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
501	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
502	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
503	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
504	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
505	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
506	C₈	—	2	2	no	—
507	C₈	—	2	2	no	—
508	C₈	—	2	2	no	—
509	C₈	—	2	2	no	—
510	C₈	—	2	2	no	—
511	C₈	—	4	2	no	—
512	C₈	—	4	1	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	2	1	no	—
515	C₂²	—	2	2	no	—
516	C₂²	—	4	1	no	—
517	C₂²	—	4	1	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	2	no	—
521	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
522	C₂²	C₅₃×#547	1	1	yes	characteristic
523	C₂²	C₅₃×#553	1	1	yes	characteristic
524	C₄	—	2	4	no	—
525	C₄	—	2	1	no	—
526	C₄	—	2	1	no	—
527	C₄	—	2	1	no	—
528	C₄	—	2	2	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	1	no	—
531	C₄	—	4	1	no	—
532	C₄	—	4	2	no	—
533	C₄	C₅₃×#443	1	2	yes	complemented
534	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
535	C₄	C₅₃×#520	1	1	yes	characteristic
536	C₄	D₄²×C₅₃	1	1	yes	characteristic
537	C₄	C₅₃×#548	1	1	yes	characteristic
538	C₂	—	2	1	no	—
539	C₂	—	2	1	no	—
540	C₂	—	4	2	no	—
541	C₂	—	8	2	no	—
542	C₂	C₅₃×#2361	1	1	yes	characteristic
543	C₂	C₂×C₅₃×#443	1	1	yes	characteristic
544	C₂	C₅₃×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#3035	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄⋊C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#443	C₂	semidirect
3. C₅₃×#2935	C₂	semidirect
4. C₅₃×#2906	C₂	semidirect
5. C₅₃×#1962	C₂	semidirect
6. C₅₃×#1950	C₂	semidirect
7. C₅₃×#1937	C₂	semidirect
8. C₅₃×#1876	C₂	semidirect
9. C₅₃×#1866	C₂	semidirect
10. C₄×C₄⋊C₈×C₅₃	C₂	semidirect
11. C₄×#104×C₅₃	C₂	semidirect
17. C₅₃×#464	C₄	semidirect
18. C₅₃×#463	C₄	semidirect
19. C₅₃×#462	C₄	semidirect
20. C₅₃×#461	C₄	semidirect
21. C₅₃×#460	C₄	semidirect
22. C₅₃×#459	C₄	semidirect
104. #104×C₅₃	D₄	semidirect
240. #23015	C₅₃	direct (N × H)
261. C₄×C₅₃	#443	semidirect
267. C₂×#443	C₂×C₅₃	semidirect
268. #2935	C₂×C₅₃	semidirect
269. #2906	C₂×C₅₃	semidirect
270. #1962	C₂×C₅₃	semidirect
271. #1950	C₂×C₅₃	semidirect
272. #1937	C₂×C₅₃	semidirect
273. #1876	C₂×C₅₃	semidirect
274. #1866	C₂×C₅₃	semidirect
275. C₄×C₄⋊C₈	C₂×C₅₃	semidirect
276. C₄×#104	C₂×C₅₃	semidirect
289. #464	C₄×C₅₃	semidirect
290. #463	C₄×C₅₃	semidirect
291. #462	C₄×C₅₃	semidirect
292. #461	C₄×C₅₃	semidirect
293. #460	C₄×C₅₃	semidirect
294. #459	C₄×C₅₃	semidirect
320. C₅₃	#23015	direct (N × H)
377. #104	D₄×C₅₃	semidirect
533. C₄	C₅₃×#443	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3035	2
C₂×C₅₃×#443	2
C₅₃×#2935	2
C₅₃×#2906	2
C₅₃×#1962	2
C₅₃×#1950	2
C₅₃×#1937	2
C₅₃×#1876	2
C₅₃×#1866	2
C₄×C₄⋊C₈×C₅₃	2
C₄×#104×C₅₃	2
#23015	53

Subgroups — C53×#23015 #27230527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23015 #27230527