Subgroups — C₅₃×#23019 #27230531

Order 13568.

880 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3050	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2933	C₂	1	2	yes	maximal, complemented
4	C₂²×#108×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1975	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1946	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1929	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1871	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1866	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1773	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1752	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#131×C₅₃	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#525	C₂²	1	2	yes	—
17	C₅₃×#467	C₄	1	1	yes	characteristic, complemented
18	C₅₃×#467	C₄	1	1	yes	characteristic, complemented
19	C₅₃×#466	C₄	1	1	yes	characteristic, complemented
20	C₅₃×#466	C₄	1	1	yes	characteristic, complemented
21	C₅₃×#465	C₄	1	2	yes	complemented
22	C₅₃×#465	C₄	1	2	yes	complemented
23	C₅₃×#444	—	2	4	no	—
24	C₅₃×#436	—	2	2	no	—
25	C₅₃×#435	—	2	2	no	—
26	C₅₃×#435	—	2	2	no	—
27	C₈×D₄×C₅₃	—	2	2	no	—
28	C₂×#108×C₅₃	—	2	4	no	—
29	C₂×#108×C₅₃	C₂²	1	2	yes	—
30	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#427	C₂²	1	1	yes	characteristic
33	C₅₃×#424	C₂²	1	2	yes	—
34	C₂×#104×C₅₃	C₂²	1	2	yes	—
35	C₂×#104×C₅₃	C₂²	1	2	yes	—
36	C₅₃×#416	C₂²	1	2	yes	—
37	C₅₃×#408	C₂²	1	2	yes	—
38	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#405	C₂²	1	2	yes	—
40	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
42	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
43	C₅₃×#339	C₂²	1	2	yes	—
44	C₅₃×#339	C₂²	1	2	yes	—
45	C₅₃×#336	C₂²	1	1	yes	characteristic
46	C₅₃×#336	C₂²	1	1	yes	characteristic
47	C₂×#41×C₅₃	C₂³	1	2	yes	—
48	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
49	#135×C₅₃	—	2	2	no	—
50	#135×C₅₃	—	2	2	no	—
51	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	#131×C₅₃	—	2	4	no	—
55	#131×C₅₃	—	4	2	no	—
56	C₂×M₁₆×C₅₃	—	2	2	no	—
57	C₂×M₁₆×C₅₃	—	2	4	no	—
58	C₂×M₁₆×C₅₃	—	2	2	no	—
59	C₂×M₁₆×C₅₃	—	4	2	no	—
60	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
61	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
62	C₂²×C₈×C₅₃	—	2	2	no	—
63	C₂²×C₈×C₅₃	—	2	2	no	—
64	C₂²×C₈×C₅₃	—	2	2	no	—
65	C₂²×C₈×C₅₃	—	2	2	no	—
66	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	#123×C₅₃	—	2	2	no	—
75	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
76	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
77	#121×C₅₃	C₂×C₄	1	2	yes	—
78	#121×C₅₃	C₂×C₄	1	2	yes	—
79	#120×C₅₃	—	2	2	no	—
80	C₄×D₄×C₅₃	—	2	2	no	—
81	C₄×D₄×C₅₃	—	2	2	no	—
82	#117×C₅₃	C₂³	1	2	yes	—
83	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
84	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
86	#108×C₅₃	—	2	4	no	—
87	#108×C₅₃	—	2	2	no	—
88	#108×C₅₃	D₄	1	4	yes	complemented
89	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
90	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
91	#106×C₅₃	C₂×C₄	1	2	yes	—
92	C₄⋊C₈×C₅₃	—	2	2	no	—
93	C₄⋊C₈×C₅₃	—	2	2	no	—
94	#104×C₅₃	—	2	4	no	—
95	#104×C₅₃	—	4	2	no	—
96	#103×C₅₃	C₂×C₄	1	2	yes	—
97	#103×C₅₃	C₂×C₄	1	2	yes	—
98	#102×C₅₃	C₂×C₄	1	2	yes	—
99	#102×C₅₃	C₂×C₄	1	2	yes	—
100	C₂²⋊C₈×C₅₃	—	2	2	no	—
101	C₂²⋊C₈×C₅₃	—	2	2	no	—
102	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
103	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
104	#97×C₅₃	—	2	2	no	—
105	#97×C₅₃	—	2	2	no	—
106	#97×C₅₃	—	2	2	no	—
107	C₄×C₈×C₅₃	—	2	2	no	—
108	#95×C₅₃	—	2	1	no	—
109	C₂⁴×C₅₃	#41	1	1	yes	characteristic
110	#41×C₅₃	—	2	2	no	—
111	#41×C₅₃	—	2	2	no	—
112	#41×C₅₃	—	2	2	no	—
113	#41×C₅₃	—	2	2	no	—
114	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
115	C₂×D₄×C₅₃	—	2	2	no	—
116	C₂×D₄×C₅₃	—	2	2	no	—
117	C₂×D₄×C₅₃	—	2	2	no	—
118	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₅₃	—	2	1	no	—
120	C₂²×C₄×C₅₃	—	2	1	no	—
121	C₂²×C₄×C₅₃	—	2	1	no	—
122	C₂²×C₄×C₅₃	—	2	1	no	—
123	C₂²×C₄×C₅₃	—	2	2	no	—
124	C₂²×C₄×C₅₃	—	2	1	no	—
125	C₂²×C₄×C₅₃	—	2	1	no	—
126	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
132	Q₁₆×C₅₃	—	2	2	no	—
133	SD₁₆×C₅₃	—	2	4	no	—
134	D₈×C₅₃	—	2	2	no	—
135	M₁₆×C₅₃	—	2	2	no	—
136	M₁₆×C₅₃	—	2	2	no	—
137	M₁₆×C₅₃	—	2	2	no	—
138	M₁₆×C₅₃	—	4	4	no	—
139	M₁₆×C₅₃	—	4	4	no	—
140	M₁₆×C₅₃	C₂×D₄	1	4	yes	—
141	C₂×C₈×C₅₃	—	2	1	no	—
142	C₂×C₈×C₅₃	—	2	1	no	—
143	C₂×C₈×C₅₃	—	2	2	no	—
144	C₂×C₈×C₅₃	—	2	1	no	—
145	C₂×C₈×C₅₃	—	2	1	no	—
146	C₂×C₈×C₅₃	—	2	1	no	—
147	C₂×C₈×C₅₃	—	2	1	no	—
148	C₂×C₈×C₅₃	—	2	2	no	—
149	C₂×C₈×C₅₃	—	2	2	no	—
150	C₂×C₈×C₅₃	—	2	2	no	—
151	C₂×C₈×C₅₃	—	2	2	no	—
152	C₂×C₈×C₅₃	—	2	2	no	—
153	C₂×C₈×C₅₃	—	4	4	no	—
154	C₂×C₈×C₅₃	—	4	4	no	—
155	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
156	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
157	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
159	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
160	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
161	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
162	C₂×C₈×C₅₃	#41	1	2	yes	—
163	C₄⋊C₄×C₅₃	—	2	2	no	—
164	C₄⋊C₄×C₅₃	—	2	2	no	—
165	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
166	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
167	C₂²⋊C₄×C₅₃	—	2	2	no	—
168	C₂²⋊C₄×C₅₃	—	2	2	no	—
169	C₂²⋊C₄×C₅₃	—	2	2	no	—
170	C₂²⋊C₄×C₅₃	—	2	2	no	—
171	C₂²⋊C₄×C₅₃	—	2	2	no	—
172	C₂²⋊C₄×C₅₃	—	4	2	no	—
173	C₄²×C₅₃	—	2	2	no	—
174	C₄²×C₅₃	—	2	1	no	—
175	C₄²×C₅₃	—	2	1	no	—
176	C₄²×C₅₃	—	2	1	no	—
177	C₄²×C₅₃	—	2	1	no	—
178	C₂³×C₅₃	—	2	1	no	—
179	C₂³×C₅₃	—	2	1	no	—
180	C₂³×C₅₃	—	2	2	no	—
181	C₂³×C₅₃	—	2	2	no	—
182	C₂³×C₅₃	—	4	1	no	—
183	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
184	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
185	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
186	Q₈×C₅₃	—	2	2	no	—
187	Q₈×C₅₃	—	2	2	no	—
188	D₄×C₅₃	—	2	2	no	—
189	D₄×C₅₃	—	2	2	no	—
190	D₄×C₅₃	—	4	2	no	—
191	D₄×C₅₃	—	4	2	no	—
192	D₄×C₅₃	—	4	4	no	—
193	D₄×C₅₃	—	8	2	no	—
194	C₂×C₄×C₅₃	—	2	1	no	—
195	C₂×C₄×C₅₃	—	2	1	no	—
196	C₂×C₄×C₅₃	—	2	1	no	—
197	C₂×C₄×C₅₃	—	2	1	no	—
198	C₂×C₄×C₅₃	—	2	1	no	—
199	C₂×C₄×C₅₃	—	2	1	no	—
200	C₂×C₄×C₅₃	—	2	1	no	—
201	C₂×C₄×C₅₃	—	2	1	no	—
202	C₂×C₄×C₅₃	—	2	1	no	—
203	C₂×C₄×C₅₃	—	2	1	no	—
204	C₂×C₄×C₅₃	—	2	2	no	—
205	C₂×C₄×C₅₃	—	2	2	no	—
206	C₂×C₄×C₅₃	—	2	2	no	—
207	C₂×C₄×C₅₃	—	4	2	no	—
208	C₂×C₄×C₅₃	—	4	2	no	—
209	C₂×C₄×C₅₃	—	4	1	no	—
210	C₂×C₄×C₅₃	—	4	1	no	—
211	C₂×C₄×C₅₃	—	4	1	no	—
212	C₂×C₄×C₅₃	—	4	1	no	—
213	C₂×C₄×C₅₃	—	4	1	no	—
214	C₂×C₄×C₅₃	—	4	1	no	—
215	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
216	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
217	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
218	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
222	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
223	C₈×C₅₃	—	2	1	no	—
224	C₈×C₅₃	—	2	1	no	—
225	C₈×C₅₃	—	2	2	no	—
226	C₈×C₅₃	—	2	4	no	—
227	C₈×C₅₃	—	2	2	no	—
228	C₈×C₅₃	—	4	2	no	—
229	C₈×C₅₃	C₄×D₄	1	2	yes	—
230	C₈×C₅₃	C₄×D₄	1	2	yes	—
231	#23019	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂²×C₅₃	—	2	1	no	—
233	C₂²×C₅₃	—	2	1	no	—
234	C₂²×C₅₃	—	2	1	no	—
235	C₂²×C₅₃	—	2	1	no	—
236	C₂²×C₅₃	—	2	1	no	—
237	C₂²×C₅₃	—	2	1	no	—
238	C₂²×C₅₃	—	2	1	no	—
239	C₂²×C₅₃	—	4	2	no	—
240	C₂²×C₅₃	—	4	2	no	—
241	C₂²×C₅₃	—	4	2	no	—
242	C₂²×C₅₃	—	4	2	no	—
243	C₂²×C₅₃	—	4	2	no	—
244	C₂²×C₅₃	#444	1	2	yes	complemented
245	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
246	C₂²×C₅₃	#548	1	1	yes	characteristic
247	C₂²×C₅₃	#553	1	1	yes	characteristic
248	C₄×C₅₃	—	2	1	no	—
249	C₄×C₅₃	—	2	1	no	—
250	C₄×C₅₃	—	4	2	no	—
251	C₄×C₅₃	—	4	2	no	—
252	C₄×C₅₃	—	4	2	no	—
253	C₄×C₅₃	—	8	1	no	—
254	C₄×C₅₃	—	8	1	no	—
255	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
256	C₄×C₅₃	#520	1	1	yes	characteristic
257	C₄×C₅₃	D₄²	1	1	yes	characteristic
258	C₄×C₅₃	#547	1	1	yes	characteristic
259	#3050	C₂×C₅₃	1	1	yes	characteristic
260	C₂×#444	C₂×C₅₃	1	1	yes	characteristic, complemented
261	#2933	C₂×C₅₃	1	2	yes	complemented
262	C₂²×#108	C₂×C₅₃	1	1	yes	characteristic, complemented
263	#1975	C₂×C₅₃	1	1	yes	characteristic, complemented
264	#1946	C₂×C₅₃	1	1	yes	characteristic, complemented
265	#1929	C₂×C₅₃	1	2	yes	complemented
266	#1871	C₂×C₅₃	1	2	yes	complemented
267	#1866	C₂×C₅₃	1	2	yes	complemented
268	#1773	C₂×C₅₃	1	1	yes	characteristic, complemented
269	#1752	C₂×C₅₃	1	1	yes	characteristic, complemented
270	C₂×C₅₃	—	2	1	no	—
271	C₂×C₅₃	—	2	1	no	—
272	C₂×C₅₃	—	2	2	no	—
273	C₂×C₅₃	—	4	1	no	—
274	C₂×C₅₃	—	8	2	no	—
275	C₂×C₅₃	#2361	1	1	yes	characteristic
276	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
277	C₂×C₅₃	#2957	1	1	yes	characteristic
278	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
279	C₂×#131	C₂²×C₅₃	1	2	yes	—
280	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
281	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
282	#525	C₂²×C₅₃	1	2	yes	—
283	#467	C₄×C₅₃	1	1	yes	characteristic, complemented
284	#467	C₄×C₅₃	1	1	yes	characteristic, complemented
285	#466	C₄×C₅₃	1	1	yes	characteristic, complemented
286	#466	C₄×C₅₃	1	1	yes	characteristic, complemented
287	#465	C₄×C₅₃	1	2	yes	complemented
288	#465	C₄×C₅₃	1	2	yes	complemented
289	#444	—	2	4	no	—
290	#436	—	2	2	no	—
291	#435	—	2	2	no	—
292	#435	—	2	2	no	—
293	C₈×D₄	—	2	2	no	—
294	C₂×#108	—	2	4	no	—
295	C₂×#108	C₂²×C₅₃	1	2	yes	—
296	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
297	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
298	#427	C₂²×C₅₃	1	1	yes	characteristic
299	#424	C₂²×C₅₃	1	2	yes	—
300	C₂×#104	C₂²×C₅₃	1	2	yes	—
301	C₂×#104	C₂²×C₅₃	1	2	yes	—
302	#416	C₂²×C₅₃	1	2	yes	—
303	#408	C₂²×C₅₃	1	2	yes	—
304	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
305	#405	C₂²×C₅₃	1	2	yes	—
306	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
307	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
308	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
309	#339	C₂²×C₅₃	1	2	yes	—
310	#339	C₂²×C₅₃	1	2	yes	—
311	#336	C₂²×C₅₃	1	1	yes	characteristic
312	#336	C₂²×C₅₃	1	1	yes	characteristic
313	C₅₃	#23019	1	1	yes	characteristic, direct product factor, Sylow
314	C₂×#41	C₂³×C₅₃	1	2	yes	—
315	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
316	#135	—	2	2	no	—
317	#135	—	2	2	no	—
318	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
319	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
320	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
321	#131	—	2	4	no	—
322	#131	—	4	2	no	—
323	C₂×M₁₆	—	2	2	no	—
324	C₂×M₁₆	—	2	2	no	—
325	C₂×M₁₆	—	2	4	no	—
326	C₂×M₁₆	—	4	2	no	—
327	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
328	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
329	C₂²×C₈	—	2	2	no	—
330	C₂²×C₈	—	2	2	no	—
331	C₂²×C₈	—	2	2	no	—
332	C₂²×C₈	—	2	2	no	—
333	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
334	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
335	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
336	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
337	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
338	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
339	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
340	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
341	#123	—	2	2	no	—
342	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
343	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
344	#121	C₂×C₄×C₅₃	1	2	yes	—
345	#121	C₂×C₄×C₅₃	1	2	yes	—
346	#120	—	2	2	no	—
347	C₄×D₄	—	2	2	no	—
348	C₄×D₄	—	2	2	no	—
349	#117	C₂³×C₅₃	1	2	yes	—
350	C₂×C₂²⋊C₄	—	2	1	no	—
351	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
352	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
353	#108	—	2	4	no	—
354	#108	—	2	2	no	—
355	#108	D₄×C₅₃	1	4	yes	complemented
356	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
357	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
358	#106	C₂×C₄×C₅₃	1	2	yes	—
359	C₄⋊C₈	—	2	2	no	—
360	C₄⋊C₈	—	2	2	no	—
361	#104	—	2	4	no	—
362	#104	—	4	2	no	—
363	#103	C₂×C₄×C₅₃	1	2	yes	—
364	#103	C₂×C₄×C₅₃	1	2	yes	—
365	#102	C₂×C₄×C₅₃	1	2	yes	—
366	#102	C₂×C₄×C₅₃	1	2	yes	—
367	C₂²⋊C₈	—	2	2	no	—
368	C₂²⋊C₈	—	2	2	no	—
369	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
370	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
371	#97	—	2	2	no	—
372	#97	—	2	2	no	—
373	#97	—	2	2	no	—
374	C₄×C₈	—	2	2	no	—
375	#95	—	2	1	no	—
376	C₂⁴	#41×C₅₃	1	1	yes	characteristic
377	#41	—	2	2	no	—
378	#41	—	2	2	no	—
379	#41	—	2	2	no	—
380	#41	—	2	2	no	—
381	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	—	2	1	no	—
388	C₂²×C₄	—	2	1	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
394	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
395	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
397	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
398	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
399	Q₁₆	—	2	2	no	—
400	SD₁₆	—	2	4	no	—
401	D₈	—	2	2	no	—
402	M₁₆	—	2	2	no	—
403	M₁₆	—	2	2	no	—
404	M₁₆	—	2	2	no	—
405	M₁₆	—	4	4	no	—
406	M₁₆	—	4	4	no	—
407	M₁₆	C₂×D₄×C₅₃	1	4	yes	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	1	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	4	4	no	—
421	C₂×C₈	—	4	4	no	—
422	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
423	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
424	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
425	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
426	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
427	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
428	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
429	C₂×C₈	#41×C₅₃	1	2	yes	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
433	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
434	C₂²⋊C₄	—	2	2	no	—
435	C₂²⋊C₄	—	2	2	no	—
436	C₂²⋊C₄	—	2	2	no	—
437	C₂²⋊C₄	—	2	2	no	—
438	C₂²⋊C₄	—	2	2	no	—
439	C₂²⋊C₄	—	4	2	no	—
440	C₄²	—	2	2	no	—
441	C₄²	—	2	1	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	1	no	—
444	C₄²	—	2	1	no	—
445	C₂³	—	2	1	no	—
446	C₂³	—	2	1	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	4	1	no	—
450	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
451	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
452	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
453	Q₈	—	2	2	no	—
454	Q₈	—	2	2	no	—
455	D₄	—	2	2	no	—
456	D₄	—	2	2	no	—
457	D₄	—	4	2	no	—
458	D₄	—	4	2	no	—
459	D₄	—	4	4	no	—
460	D₄	—	8	2	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
483	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
484	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
486	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
487	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
488	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
489	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
490	C₈	—	2	1	no	—
491	C₈	—	2	1	no	—
492	C₈	—	2	2	no	—
493	C₈	—	2	4	no	—
494	C₈	—	2	2	no	—
495	C₈	—	4	2	no	—
496	C₈	C₄×D₄×C₅₃	1	2	yes	—
497	C₈	C₄×D₄×C₅₃	1	2	yes	—
498	C₂²	—	2	1	no	—
499	C₂²	—	2	1	no	—
500	C₂²	—	2	1	no	—
501	C₂²	—	2	1	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	2	1	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	4	2	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	C₅₃×#444	1	2	yes	complemented
511	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
512	C₂²	C₅₃×#548	1	1	yes	characteristic
513	C₂²	C₅₃×#553	1	1	yes	characteristic
514	C₄	—	2	1	no	—
515	C₄	—	2	1	no	—
516	C₄	—	4	2	no	—
517	C₄	—	4	2	no	—
518	C₄	—	4	2	no	—
519	C₄	—	8	1	no	—
520	C₄	—	8	1	no	—
521	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
522	C₄	C₅₃×#520	1	1	yes	characteristic
523	C₄	D₄²×C₅₃	1	1	yes	characteristic
524	C₄	C₅₃×#547	1	1	yes	characteristic
525	C₂	—	2	1	no	—
526	C₂	—	2	1	no	—
527	C₂	—	2	2	no	—
528	C₂	—	4	1	no	—
529	C₂	—	8	2	no	—
530	C₂	C₅₃×#2361	1	1	yes	characteristic
531	C₂	C₂×C₅₃×#444	1	1	yes	characteristic
532	C₂	C₅₃×#2957	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×#3050	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₂²⋊C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#444	C₂	semidirect
3. C₅₃×#2933	C₂	semidirect
4. C₂²×#108×C₅₃	C₂	semidirect
5. C₅₃×#1975	C₂	semidirect
6. C₅₃×#1946	C₂	semidirect
7. C₅₃×#1929	C₂	semidirect
8. C₅₃×#1871	C₂	semidirect
9. C₅₃×#1866	C₂	semidirect
10. C₅₃×#1773	C₂	semidirect
11. C₅₃×#1752	C₂	semidirect
17. C₅₃×#467	C₄	semidirect
18. C₅₃×#467	C₄	semidirect
19. C₅₃×#466	C₄	semidirect
20. C₅₃×#466	C₄	semidirect
21. C₅₃×#465	C₄	semidirect
22. C₅₃×#465	C₄	semidirect
88. #108×C₅₃	D₄	semidirect
231. #23019	C₅₃	direct (N × H)
244. C₂²×C₅₃	#444	semidirect
260. C₂×#444	C₂×C₅₃	semidirect
261. #2933	C₂×C₅₃	semidirect
262. C₂²×#108	C₂×C₅₃	semidirect
263. #1975	C₂×C₅₃	semidirect
264. #1946	C₂×C₅₃	semidirect
265. #1929	C₂×C₅₃	semidirect
266. #1871	C₂×C₅₃	semidirect
267. #1866	C₂×C₅₃	semidirect
268. #1773	C₂×C₅₃	semidirect
269. #1752	C₂×C₅₃	semidirect
283. #467	C₄×C₅₃	semidirect
284. #467	C₄×C₅₃	semidirect
285. #466	C₄×C₅₃	semidirect
286. #466	C₄×C₅₃	semidirect
287. #465	C₄×C₅₃	semidirect
288. #465	C₄×C₅₃	semidirect
313. C₅₃	#23019	direct (N × H)
355. #108	D₄×C₅₃	semidirect
510. C₂²	C₅₃×#444	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3050	2
C₂×C₅₃×#444	2
C₅₃×#2933	2
C₂²×#108×C₅₃	2
C₅₃×#1975	2
C₅₃×#1946	2
C₅₃×#1929	2
C₅₃×#1871	2
C₅₃×#1866	2
C₅₃×#1773	2
C₅₃×#1752	2
#23019	53

Subgroups — C53×#23019 #27230531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23019 #27230531