Subgroups — C₅₃×#22983 #27230495

Order 13568.

708 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3076	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2962	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2906	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1950	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1946	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1922	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1866	C₂	1	2	yes	maximal
9	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1765	C₂	1	2	yes	maximal
11	C₄×#104×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×#131×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#533	C₂²	1	1	yes	characteristic
14	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#479	C₄	1	2	yes	complemented
16	C₅₃×#478	C₄	1	2	yes	complemented
17	C₅₃×#477	C₄	1	1	yes	characteristic, complemented
18	C₅₃×#476	C₄	1	1	yes	characteristic, complemented
19	C₅₃×#475	C₄	1	1	yes	characteristic, complemented
20	C₅₃×#474	C₄	1	1	yes	characteristic, complemented
21	C₅₃×#446	—	2	2	no	—
22	C₅₃×#446	—	2	1	no	—
23	C₈×Q₈×C₅₃	—	2	1	no	—
24	C₂×#108×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#427	C₂²	1	2	yes	—
26	C₅₃×#424	—	2	2	no	—
27	C₅₃×#424	C₂²	1	2	yes	—
28	C₅₃×#424	C₂²	1	2	yes	—
29	C₅₃×#423	—	2	1	no	—
30	C₅₃×#423	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
32	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#416	C₂²	1	2	yes	—
37	C₅₃×#405	C₂²	1	2	yes	—
38	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×#97×C₅₃	C₂²	1	2	yes	—
40	C₄³×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#341	C₂²	1	2	yes	—
42	C₅₃×#339	C₂²	1	1	yes	characteristic
43	C₅₃×#339	C₂²	1	2	yes	—
44	C₅₃×#339	C₂²	1	1	yes	characteristic
45	C₅₃×#339	C₂²	1	2	yes	—
46	C₅₃×#336	C₂²	1	2	yes	—
47	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
48	#131×C₅₃	—	2	4	no	—
49	#131×C₅₃	—	4	2	no	—
50	C₂×M₁₆×C₅₃	—	2	2	no	—
51	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
52	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	—	2	2	no	—
55	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
56	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
57	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	#125×C₅₃	C₂×C₄	1	2	yes	—
60	C₂²⋊Q₈×C₅₃	—	2	1	no	—
61	C₂²⋊Q₈×C₅₃	—	2	1	no	—
62	C₄×Q₈×C₅₃	—	2	1	no	—
63	C₄×Q₈×C₅₃	—	2	1	no	—
64	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄×D₄×C₅₃	—	2	1	no	—
67	C₄×D₄×C₅₃	—	2	1	no	—
68	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
70	#117×C₅₃	—	2	1	no	—
71	#117×C₅₃	C₂³	1	1	yes	characteristic
72	#117×C₅₃	C₂³	1	2	yes	—
73	C₂×C₄²×C₅₃	—	2	1	no	—
74	C₂×C₄²×C₅₃	—	2	1	no	—
75	C₂×C₄²×C₅₃	—	2	1	no	—
76	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
79	#108×C₅₃	—	4	2	no	—
80	#107×C₅₃	C₂×C₄	1	2	yes	—
81	#107×C₅₃	C₂×C₄	1	2	yes	—
82	#106×C₅₃	C₂×C₄	1	2	yes	—
83	#106×C₅₃	C₂×C₄	1	2	yes	—
84	C₄⋊C₈×C₅₃	—	2	2	no	—
85	C₄⋊C₈×C₅₃	—	2	2	no	—
86	C₄⋊C₈×C₅₃	—	2	2	no	—
87	C₄⋊C₈×C₅₃	—	2	2	no	—
88	C₄⋊C₈×C₅₃	—	2	1	no	—
89	C₄⋊C₈×C₅₃	—	2	1	no	—
90	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
93	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
94	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
95	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
96	#104×C₅₃	—	2	2	no	—
97	#104×C₅₃	Q₈	1	4	yes	complemented
98	#103×C₅₃	C₂×C₄	1	2	yes	—
99	#103×C₅₃	C₂×C₄	1	2	yes	—
100	#102×C₅₃	C₂×C₄	1	2	yes	—
101	#102×C₅₃	C₂×C₄	1	2	yes	—
102	#97×C₅₃	—	2	2	no	—
103	#97×C₅₃	—	2	1	no	—
104	#97×C₅₃	—	4	2	no	—
105	C₄×C₈×C₅₃	—	2	2	no	—
106	C₄×C₈×C₅₃	—	2	1	no	—
107	#41×C₅₃	—	2	1	no	—
108	#41×C₅₃	—	2	1	no	—
109	#41×C₅₃	C₂×Q₈	1	2	yes	—
110	#41×C₅₃	#41	1	2	yes	—
111	C₂×Q₈×C₅₃	—	2	1	no	—
112	C₂×Q₈×C₅₃	—	2	1	no	—
113	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×D₄×C₅₃	—	2	1	no	—
115	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	—	2	1	no	—
117	C₂²×C₄×C₅₃	—	2	1	no	—
118	C₂²×C₄×C₅₃	—	2	1	no	—
119	C₂²×C₄×C₅₃	—	2	1	no	—
120	C₂²×C₄×C₅₃	—	2	1	no	—
121	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
122	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
123	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
124	M₁₆×C₅₃	—	2	1	no	—
125	M₁₆×C₅₃	—	2	1	no	—
126	M₁₆×C₅₃	—	2	2	no	—
127	M₁₆×C₅₃	—	2	1	no	—
128	M₁₆×C₅₃	—	2	2	no	—
129	M₁₆×C₅₃	—	4	4	no	—
130	M₁₆×C₅₃	C₂×Q₈	1	2	yes	—
131	C₂×C₈×C₅₃	—	2	1	no	—
132	C₂×C₈×C₅₃	—	2	2	no	—
133	C₂×C₈×C₅₃	—	2	2	no	—
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	—	2	2	no	—
136	C₂×C₈×C₅₃	—	2	1	no	—
137	C₂×C₈×C₅₃	—	2	2	no	—
138	C₂×C₈×C₅₃	—	4	4	no	—
139	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
140	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
141	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
143	C₄⋊C₄×C₅₃	—	2	1	no	—
144	C₄⋊C₄×C₅₃	—	2	1	no	—
145	C₄⋊C₄×C₅₃	—	2	2	no	—
146	C₄⋊C₄×C₅₃	—	2	2	no	—
147	C₄⋊C₄×C₅₃	—	2	1	no	—
148	C₄⋊C₄×C₅₃	—	2	1	no	—
149	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
150	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
151	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
153	C₂²⋊C₄×C₅₃	—	2	1	no	—
154	C₂²⋊C₄×C₅₃	—	2	2	no	—
155	C₂²⋊C₄×C₅₃	—	2	1	no	—
156	C₂²⋊C₄×C₅₃	—	2	1	no	—
157	C₄²×C₅₃	—	2	1	no	—
158	C₄²×C₅₃	—	2	1	no	—
159	C₄²×C₅₃	—	2	2	no	—
160	C₄²×C₅₃	—	2	1	no	—
161	C₄²×C₅₃	—	2	1	no	—
162	C₄²×C₅₃	—	2	1	no	—
163	C₄²×C₅₃	—	2	1	no	—
164	C₄²×C₅₃	—	4	2	no	—
165	C₄²×C₅₃	—	4	2	no	—
166	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
167	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
168	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
171	C₂³×C₅₃	—	2	1	no	—
172	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
173	Q₈×C₅₃	—	2	1	no	—
174	Q₈×C₅₃	—	4	2	no	—
175	Q₈×C₅₃	—	8	1	no	—
176	Q₈×C₅₃	C₄×Q₈	1	2	yes	—
177	D₄×C₅₃	—	2	2	no	—
178	D₄×C₅₃	—	2	2	no	—
179	D₄×C₅₃	—	2	1	no	—
180	D₄×C₅₃	C₄×Q₈	1	2	yes	—
181	C₂×C₄×C₅₃	—	2	1	no	—
182	C₂×C₄×C₅₃	—	2	2	no	—
183	C₂×C₄×C₅₃	—	2	2	no	—
184	C₂×C₄×C₅₃	—	2	2	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	2	1	no	—
187	C₂×C₄×C₅₃	—	2	1	no	—
188	C₂×C₄×C₅₃	—	2	2	no	—
189	C₂×C₄×C₅₃	—	2	2	no	—
190	C₂×C₄×C₅₃	—	2	2	no	—
191	C₂×C₄×C₅₃	—	4	1	no	—
192	C₂×C₄×C₅₃	—	4	1	no	—
193	C₂×C₄×C₅₃	—	4	1	no	—
194	C₂×C₄×C₅₃	—	4	1	no	—
195	C₂×C₄×C₅₃	—	4	1	no	—
196	C₂×C₄×C₅₃	—	4	2	no	—
197	C₂×C₄×C₅₃	—	8	1	no	—
198	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
199	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
200	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
201	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
202	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
203	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
205	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
206	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
207	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
208	C₈×C₅₃	—	2	2	no	—
209	C₈×C₅₃	—	2	2	no	—
210	C₈×C₅₃	—	2	2	no	—
211	C₈×C₅₃	—	2	2	no	—
212	C₈×C₅₃	—	2	2	no	—
213	C₈×C₅₃	—	4	2	no	—
214	C₈×C₅₃	—	4	1	no	—
215	#22983	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂²×C₅₃	—	2	1	no	—
217	C₂²×C₅₃	—	2	1	no	—
218	C₂²×C₅₃	—	2	2	no	—
219	C₂²×C₅₃	—	4	1	no	—
220	C₂²×C₅₃	—	4	1	no	—
221	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂²×C₅₃	#552	1	1	yes	characteristic
223	C₂²×C₅₃	#554	1	1	yes	characteristic
224	C₄×C₅₃	—	2	2	no	—
225	C₄×C₅₃	—	2	1	no	—
226	C₄×C₅₃	—	2	1	no	—
227	C₄×C₅₃	—	2	1	no	—
228	C₄×C₅₃	—	4	2	no	—
229	C₄×C₅₃	—	4	2	no	—
230	C₄×C₅₃	—	4	1	no	—
231	C₄×C₅₃	—	4	2	no	—
232	C₄×C₅₃	—	4	1	no	—
233	C₄×C₅₃	—	4	2	no	—
234	C₄×C₅₃	#444	1	2	yes	—
235	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
236	C₄×C₅₃	#520	1	1	yes	characteristic
237	C₄×C₅₃	#546	1	1	yes	characteristic
238	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
239	#3076	C₂×C₅₃	1	1	yes	characteristic
240	#2962	C₂×C₅₃	1	1	yes	characteristic, complemented
241	#2906	C₂×C₅₃	1	1	yes	characteristic
242	#1994	C₂×C₅₃	1	1	yes	characteristic, complemented
243	#1950	C₂×C₅₃	1	1	yes	characteristic, complemented
244	#1946	C₂×C₅₃	1	2	yes	complemented
245	#1922	C₂×C₅₃	1	2	yes	complemented
246	#1866	C₂×C₅₃	1	2	yes	—
247	#1767	C₂×C₅₃	1	1	yes	characteristic, complemented
248	#1765	C₂×C₅₃	1	2	yes	—
249	C₄×#104	C₂×C₅₃	1	1	yes	characteristic
250	C₂×C₅₃	—	2	1	no	—
251	C₂×C₅₃	—	2	1	no	—
252	C₂×C₅₃	—	4	2	no	—
253	C₂×C₅₃	#2360	1	1	yes	characteristic
254	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
255	C₂×C₅₃	#2956	1	1	yes	characteristic
256	C₂×#131	C₂²×C₅₃	1	2	yes	—
257	#533	C₂²×C₅₃	1	1	yes	characteristic
258	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
259	#479	C₄×C₅₃	1	2	yes	complemented
260	#478	C₄×C₅₃	1	2	yes	complemented
261	#477	C₄×C₅₃	1	1	yes	characteristic, complemented
262	#476	C₄×C₅₃	1	1	yes	characteristic, complemented
263	#475	C₄×C₅₃	1	1	yes	characteristic, complemented
264	#474	C₄×C₅₃	1	1	yes	characteristic, complemented
265	#446	—	2	2	no	—
266	#446	—	2	1	no	—
267	C₈×Q₈	—	2	1	no	—
268	C₂×#108	C₂²×C₅₃	1	2	yes	—
269	#427	C₂²×C₅₃	1	2	yes	—
270	#424	—	2	2	no	—
271	#424	C₂²×C₅₃	1	2	yes	—
272	#424	C₂²×C₅₃	1	2	yes	—
273	#423	—	2	1	no	—
274	#423	C₂²×C₅₃	1	1	yes	characteristic
275	C₂×C₄⋊C₈	—	2	1	no	—
276	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
277	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
278	C₂×#104	C₂²×C₅₃	1	1	yes	characteristic
279	C₂×#104	C₂²×C₅₃	1	1	yes	characteristic
280	#416	C₂²×C₅₃	1	2	yes	—
281	#405	C₂²×C₅₃	1	2	yes	—
282	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
283	C₂×#97	C₂²×C₅₃	1	2	yes	—
284	C₄³	C₂²×C₅₃	1	1	yes	characteristic
285	#341	C₂²×C₅₃	1	2	yes	—
286	#339	C₂²×C₅₃	1	2	yes	—
287	#339	C₂²×C₅₃	1	1	yes	characteristic
288	#339	C₂²×C₅₃	1	1	yes	characteristic
289	#339	C₂²×C₅₃	1	2	yes	—
290	#336	C₂²×C₅₃	1	2	yes	—
291	C₅₃	#22983	1	1	yes	characteristic, direct product factor, Sylow
292	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
293	#131	—	2	4	no	—
294	#131	—	4	2	no	—
295	C₂×M₁₆	—	2	2	no	—
296	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
297	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
298	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
299	C₂²×C₈	—	2	2	no	—
300	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
301	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
302	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
303	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
304	#125	C₂×C₄×C₅₃	1	2	yes	—
305	C₂²⋊Q₈	—	2	1	no	—
306	C₂²⋊Q₈	—	2	1	no	—
307	C₄×Q₈	—	2	1	no	—
308	C₄×Q₈	—	2	1	no	—
309	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
310	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
311	C₄×D₄	—	2	1	no	—
312	C₄×D₄	—	2	1	no	—
313	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
314	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
315	#117	—	2	1	no	—
316	#117	C₂³×C₅₃	1	1	yes	characteristic
317	#117	C₂³×C₅₃	1	2	yes	—
318	C₂×C₄²	—	2	1	no	—
319	C₂×C₄²	—	2	1	no	—
320	C₂×C₄²	—	2	1	no	—
321	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
322	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
323	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
324	#108	—	4	2	no	—
325	#107	C₂×C₄×C₅₃	1	2	yes	—
326	#107	C₂×C₄×C₅₃	1	2	yes	—
327	#106	C₂×C₄×C₅₃	1	2	yes	—
328	#106	C₂×C₄×C₅₃	1	2	yes	—
329	C₄⋊C₈	—	2	2	no	—
330	C₄⋊C₈	—	2	2	no	—
331	C₄⋊C₈	—	2	2	no	—
332	C₄⋊C₈	—	2	2	no	—
333	C₄⋊C₈	—	2	1	no	—
334	C₄⋊C₈	—	2	1	no	—
335	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
336	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
337	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
338	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
339	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
340	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
341	#104	—	2	2	no	—
342	#104	Q₈×C₅₃	1	4	yes	complemented
343	#103	C₂×C₄×C₅₃	1	2	yes	—
344	#103	C₂×C₄×C₅₃	1	2	yes	—
345	#102	C₂×C₄×C₅₃	1	2	yes	—
346	#102	C₂×C₄×C₅₃	1	2	yes	—
347	#97	—	2	2	no	—
348	#97	—	2	1	no	—
349	#97	—	4	2	no	—
350	C₄×C₈	—	2	2	no	—
351	C₄×C₈	—	2	1	no	—
352	#41	—	2	1	no	—
353	#41	—	2	1	no	—
354	#41	C₂×Q₈×C₅₃	1	2	yes	—
355	#41	#41×C₅₃	1	2	yes	—
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
359	C₂×D₄	—	2	1	no	—
360	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
367	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
368	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
369	M₁₆	—	2	1	no	—
370	M₁₆	—	2	1	no	—
371	M₁₆	—	2	2	no	—
372	M₁₆	—	2	1	no	—
373	M₁₆	—	2	2	no	—
374	M₁₆	—	4	4	no	—
375	M₁₆	C₂×Q₈×C₅₃	1	2	yes	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	4	4	no	—
384	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
385	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
386	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
387	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
395	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
396	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
397	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
398	C₂²⋊C₄	—	2	1	no	—
399	C₂²⋊C₄	—	2	2	no	—
400	C₂²⋊C₄	—	2	1	no	—
401	C₂²⋊C₄	—	2	1	no	—
402	C₄²	—	2	1	no	—
403	C₄²	—	2	1	no	—
404	C₄²	—	2	2	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	2	1	no	—
407	C₄²	—	2	1	no	—
408	C₄²	—	2	1	no	—
409	C₄²	—	4	2	no	—
410	C₄²	—	4	2	no	—
411	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
412	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
413	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
414	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
415	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
416	C₂³	—	2	1	no	—
417	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
418	Q₈	—	2	1	no	—
419	Q₈	—	4	2	no	—
420	Q₈	—	8	1	no	—
421	Q₈	C₄×Q₈×C₅₃	1	2	yes	—
422	D₄	—	2	2	no	—
423	D₄	—	2	2	no	—
424	D₄	—	2	1	no	—
425	D₄	C₄×Q₈×C₅₃	1	2	yes	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	8	1	no	—
443	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
444	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
445	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
446	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
447	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
448	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
449	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
450	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
451	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
452	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
453	C₈	—	2	2	no	—
454	C₈	—	2	2	no	—
455	C₈	—	2	2	no	—
456	C₈	—	2	2	no	—
457	C₈	—	2	2	no	—
458	C₈	—	4	2	no	—
459	C₈	—	4	1	no	—
460	C₂²	—	2	1	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	4	1	no	—
464	C₂²	—	4	1	no	—
465	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
466	C₂²	C₅₃×#552	1	1	yes	characteristic
467	C₂²	C₅₃×#554	1	1	yes	characteristic
468	C₄	—	2	1	no	—
469	C₄	—	2	1	no	—
470	C₄	—	2	1	no	—
471	C₄	—	2	2	no	—
472	C₄	—	4	2	no	—
473	C₄	—	4	2	no	—
474	C₄	—	4	1	no	—
475	C₄	—	4	2	no	—
476	C₄	—	4	1	no	—
477	C₄	—	4	2	no	—
478	C₄	C₅₃×#444	1	2	yes	—
479	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
480	C₄	C₅₃×#520	1	1	yes	characteristic
481	C₄	C₅₃×#546	1	1	yes	characteristic
482	C₄	D₄×Q₈×C₅₃	1	1	yes	characteristic
483	C₂	—	2	1	no	—
484	C₂	—	2	1	no	—
485	C₂	—	4	2	no	—
486	C₂	C₅₃×#2360	1	1	yes	characteristic
487	C₂	C₂×C₅₃×#444	1	1	yes	characteristic
488	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	C₂×#41×C₅₃	C₂³
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. C₅₃×#2962	C₂	semidirect
4. C₅₃×#1994	C₂	semidirect
5. C₅₃×#1950	C₂	semidirect
6. C₅₃×#1946	C₂	semidirect
7. C₅₃×#1922	C₂	semidirect
9. C₅₃×#1767	C₂	semidirect
15. C₅₃×#479	C₄	semidirect
16. C₅₃×#478	C₄	semidirect
17. C₅₃×#477	C₄	semidirect
18. C₅₃×#476	C₄	semidirect
19. C₅₃×#475	C₄	semidirect
20. C₅₃×#474	C₄	semidirect
97. #104×C₅₃	Q₈	semidirect
215. #22983	C₅₃	direct (N × H)
240. #2962	C₂×C₅₃	semidirect
242. #1994	C₂×C₅₃	semidirect
243. #1950	C₂×C₅₃	semidirect
244. #1946	C₂×C₅₃	semidirect
245. #1922	C₂×C₅₃	semidirect
247. #1767	C₂×C₅₃	semidirect
259. #479	C₄×C₅₃	semidirect
260. #478	C₄×C₅₃	semidirect
261. #477	C₄×C₅₃	semidirect
262. #476	C₄×C₅₃	semidirect
263. #475	C₄×C₅₃	semidirect
264. #474	C₄×C₅₃	semidirect
291. C₅₃	#22983	direct (N × H)
342. #104	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3076	2
C₅₃×#2962	2
C₅₃×#2906	2
C₅₃×#1994	2
C₅₃×#1950	2
C₅₃×#1946	2
C₅₃×#1922	2
C₅₃×#1866	2
C₅₃×#1767	2
C₅₃×#1765	2
C₄×#104×C₅₃	2
#22983	53

Subgroups — C53×#22983 #27230495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22983 #27230495