Subgroups — C₅₃×#36720 #27244232

Order 13568.

2450 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#543	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#123×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2848	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2687	C₂	1	8	yes	maximal, complemented
5	C₅₃×#2680	C₂	1	8	yes	maximal, complemented
6	C₅₃×#2627	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2561	C₂	1	2	yes	maximal, complemented
8	C₅₃×#2502	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2483	C₂	1	2	yes	maximal, complemented
10	C₅₃×#2405	C₂	1	2	yes	maximal
11	C₅₃×#2312	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#543	—	2	8	no	—
15	C₂×#123×C₅₃	—	2	8	no	—
16	C₂×#123×C₅₃	C₂²	1	8	yes	—
17	C₂×#123×C₅₃	C₂²	1	4	yes	—
18	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
21	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
22	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
23	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#400	C₂²	1	4	yes	complemented
30	C₅₃×#398	C₂²	1	8	yes	complemented
31	C₅₃×#398	C₂²	1	8	yes	complemented
32	C₅₃×#397	C₂²	1	4	yes	complemented
33	C₅₃×#397	C₂²	1	8	yes	complemented
34	C₅₃×#395	C₂²	1	4	yes	complemented
35	C₅₃×#394	C₂²	1	8	yes	—
36	C₅₃×#393	C₂²	1	4	yes	complemented
37	C₅₃×#391	C₂²	1	4	yes	complemented
38	C₅₃×#388	C₂²	1	8	yes	complemented
39	C₅₃×#388	C₂²	1	8	yes	complemented
40	C₅₃×#386	C₂²	1	4	yes	—
41	C₅₃×#385	C₂²	1	4	yes	—
42	C₅₃×#385	C₂²	1	4	yes	—
43	C₅₃×#385	C₂²	1	4	yes	complemented
44	C₅₃×#385	C₂²	1	8	yes	complemented
45	C₅₃×#384	C₂²	1	4	yes	complemented
46	C₅₃×#381	C₂²	1	4	yes	—
47	C₅₃×#381	C₂²	1	4	yes	complemented
48	C₅₃×#380	C₂²	1	2	yes	—
49	C₅₃×#380	C₂²	1	4	yes	complemented
50	C₅₃×#379	C₂²	1	2	yes	—
51	C₅₃×#379	C₂²	1	4	yes	—
52	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
53	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
54	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
55	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
56	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
57	C₂²×D₄×C₅₃	—	2	4	no	—
58	C₂²×D₄×C₅₃	—	4	4	no	—
59	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
60	C₂³×C₄×C₅₃	—	2	4	no	—
61	C₂³×C₄×C₅₃	—	4	4	no	—
62	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
64	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
66	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
67	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
70	#123×C₅₃	—	2	32	no	—
71	#123×C₅₃	—	4	8	no	—
72	C₂²⋊Q₈×C₅₃	—	2	16	no	—
73	C₂²⋊Q₈×C₅₃	—	2	16	no	—
74	C₂²⋊Q₈×C₅₃	—	2	8	no	—
75	C₂²⋊Q₈×C₅₃	—	2	8	no	—
76	C₂²⋊Q₈×C₅₃	—	4	8	no	—
77	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
78	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
79	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
87	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
88	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
89	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
90	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
91	C₂×C₂²⋊C₄×C₅₃	D₄	1	8	yes	—
92	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
93	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
97	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
100	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
102	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
103	#95×C₅₃	C₂³	1	4	yes	—
104	#95×C₅₃	C₂³	1	4	yes	—
105	#95×C₅₃	C₂³	1	8	yes	—
106	#95×C₅₃	C₂³	1	8	yes	—
107	#95×C₅₃	C₂³	1	8	yes	—
108	#95×C₅₃	C₂³	1	8	yes	—
109	C₂⁴×C₅₃	—	2	2	no	—
110	C₂⁴×C₅₃	—	2	4	no	—
111	C₂⁴×C₅₃	—	2	4	no	—
112	C₂⁴×C₅₃	—	2	4	no	—
113	C₂⁴×C₅₃	—	4	2	no	—
114	C₂⁴×C₅₃	—	4	2	no	—
115	C₂⁴×C₅₃	—	4	2	no	—
116	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
117	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
118	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
119	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
120	C₂×Q₈×C₅₃	—	2	4	no	—
121	C₂×Q₈×C₅₃	—	2	4	no	—
122	C₂×Q₈×C₅₃	—	4	4	no	—
123	C₂×Q₈×C₅₃	—	4	4	no	—
124	C₂×D₄×C₅₃	—	2	8	no	—
125	C₂×D₄×C₅₃	—	2	8	no	—
126	C₂×D₄×C₅₃	—	4	4	no	—
127	C₂×D₄×C₅₃	—	4	16	no	—
128	C₂²×C₄×C₅₃	—	2	2	no	—
129	C₂²×C₄×C₅₃	—	2	2	no	—
130	C₂²×C₄×C₅₃	—	2	2	no	—
131	C₂²×C₄×C₅₃	—	2	8	no	—
132	C₂²×C₄×C₅₃	—	2	4	no	—
133	C₂²×C₄×C₅₃	—	2	4	no	—
134	C₂²×C₄×C₅₃	—	2	4	no	—
135	C₂²×C₄×C₅₃	—	2	8	no	—
136	C₂²×C₄×C₅₃	—	4	2	no	—
137	C₂²×C₄×C₅₃	—	4	4	no	—
138	C₂²×C₄×C₅₃	—	4	2	no	—
139	C₂²×C₄×C₅₃	—	4	2	no	—
140	C₂²×C₄×C₅₃	—	4	2	no	—
141	C₂²×C₄×C₅₃	—	4	4	no	—
142	C₂²×C₄×C₅₃	—	4	4	no	—
143	C₂²×C₄×C₅₃	—	4	16	no	—
144	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
145	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
146	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
148	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
154	C₄⋊C₄×C₅₃	—	2	8	no	—
155	C₄⋊C₄×C₅₃	—	2	8	no	—
156	C₄⋊C₄×C₅₃	—	2	4	no	—
157	C₄⋊C₄×C₅₃	—	2	8	no	—
158	C₄⋊C₄×C₅₃	—	2	4	no	—
159	C₄⋊C₄×C₅₃	—	2	8	no	—
160	C₄⋊C₄×C₅₃	—	4	4	no	—
161	C₄⋊C₄×C₅₃	—	4	8	no	—
162	C₄⋊C₄×C₅₃	—	4	8	no	—
163	C₂²⋊C₄×C₅₃	—	2	8	no	—
164	C₂²⋊C₄×C₅₃	—	2	8	no	—
165	C₂²⋊C₄×C₅₃	—	2	8	no	—
166	C₂²⋊C₄×C₅₃	—	2	16	no	—
167	C₂²⋊C₄×C₅₃	—	2	8	no	—
168	C₂²⋊C₄×C₅₃	—	2	8	no	—
169	C₂²⋊C₄×C₅₃	—	4	8	no	—
170	C₂²⋊C₄×C₅₃	—	4	4	no	—
171	C₂²⋊C₄×C₅₃	—	4	4	no	—
172	C₂²⋊C₄×C₅₃	C₂×D₄	1	16	yes	—
173	C₄²×C₅₃	—	4	4	no	—
174	C₂³×C₅₃	—	2	2	no	—
175	C₂³×C₅₃	—	2	2	no	—
176	C₂³×C₅₃	—	2	8	no	—
177	C₂³×C₅₃	—	2	2	no	—
178	C₂³×C₅₃	—	2	2	no	—
179	C₂³×C₅₃	—	2	4	no	—
180	C₂³×C₅₃	—	2	4	no	—
181	C₂³×C₅₃	—	2	4	no	—
182	C₂³×C₅₃	—	2	4	no	—
183	C₂³×C₅₃	—	2	4	no	—
184	C₂³×C₅₃	—	2	8	no	—
185	C₂³×C₅₃	—	4	4	no	—
186	C₂³×C₅₃	—	4	4	no	—
187	C₂³×C₅₃	—	4	16	no	—
188	C₂³×C₅₃	—	4	4	no	—
189	C₂³×C₅₃	—	4	4	no	—
190	C₂³×C₅₃	—	8	8	no	—
191	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
192	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
193	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
194	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
195	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
196	Q₈×C₅₃	—	8	4	no	—
197	Q₈×C₅₃	—	8	4	no	—
198	D₄×C₅₃	—	4	16	no	—
199	C₂×C₄×C₅₃	—	2	4	no	—
200	C₂×C₄×C₅₃	—	2	4	no	—
201	C₂×C₄×C₅₃	—	2	2	no	—
202	C₂×C₄×C₅₃	—	2	2	no	—
203	C₂×C₄×C₅₃	—	2	4	no	—
204	C₂×C₄×C₅₃	—	2	4	no	—
205	C₂×C₄×C₅₃	—	2	2	no	—
206	C₂×C₄×C₅₃	—	2	4	no	—
207	C₂×C₄×C₅₃	—	2	4	no	—
208	C₂×C₄×C₅₃	—	2	8	no	—
209	C₂×C₄×C₅₃	—	2	8	no	—
210	C₂×C₄×C₅₃	—	2	4	no	—
211	C₂×C₄×C₅₃	—	2	8	no	—
212	C₂×C₄×C₅₃	—	2	8	no	—
213	C₂×C₄×C₅₃	—	2	8	no	—
214	C₂×C₄×C₅₃	—	4	4	no	—
215	C₂×C₄×C₅₃	—	4	4	no	—
216	C₂×C₄×C₅₃	—	4	16	no	—
217	C₂×C₄×C₅₃	—	4	4	no	—
218	C₂×C₄×C₅₃	—	4	4	no	—
219	C₂×C₄×C₅₃	—	4	4	no	—
220	C₂×C₄×C₅₃	—	8	4	no	—
221	C₂×C₄×C₅₃	—	8	4	no	—
222	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
223	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
224	#36720	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂²×C₅₃	—	2	2	no	—
226	C₂²×C₅₃	—	2	2	no	—
227	C₂²×C₅₃	—	2	8	no	—
228	C₂²×C₅₃	—	2	4	no	—
229	C₂²×C₅₃	—	2	4	no	—
230	C₂²×C₅₃	—	2	4	no	—
231	C₂²×C₅₃	—	2	4	no	—
232	C₂²×C₅₃	—	2	8	no	—
233	C₂²×C₅₃	—	4	16	no	—
234	C₂²×C₅₃	—	4	2	no	—
235	C₂²×C₅₃	—	4	2	no	—
236	C₂²×C₅₃	—	4	4	no	—
237	C₂²×C₅₃	—	4	8	no	—
238	C₂²×C₅₃	—	8	8	no	—
239	C₂²×C₅₃	#536	1	4	yes	—
240	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
241	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
242	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
243	C₂²×C₅₃	#585	1	2	yes	—
244	C₄×C₅₃	—	2	4	no	—
245	C₄×C₅₃	—	2	4	no	—
246	C₄×C₅₃	—	4	2	no	—
247	C₄×C₅₃	—	4	2	no	—
248	C₄×C₅₃	—	4	2	no	—
249	C₄×C₅₃	—	4	4	no	—
250	C₄×C₅₃	—	4	4	no	—
251	C₄×C₅₃	—	4	8	no	—
252	C₂×#543	C₂×C₅₃	1	1	yes	characteristic, complemented
253	C₂²×#123	C₂×C₅₃	1	1	yes	characteristic
254	#2848	C₂×C₅₃	1	2	yes	complemented
255	#2687	C₂×C₅₃	1	8	yes	complemented
256	#2680	C₂×C₅₃	1	8	yes	complemented
257	#2627	C₂×C₅₃	1	2	yes	complemented
258	#2561	C₂×C₅₃	1	2	yes	complemented
259	#2502	C₂×C₅₃	1	2	yes	complemented
260	#2483	C₂×C₅₃	1	2	yes	complemented
261	#2405	C₂×C₅₃	1	2	yes	—
262	#2312	C₂×C₅₃	1	1	yes	characteristic, complemented
263	C₂×C₅₃	—	2	4	no	—
264	C₂×C₅₃	—	4	4	no	—
265	C₂×C₅₃	—	4	4	no	—
266	C₂×C₅₃	C₂×#536	1	1	yes	characteristic
267	C₂×C₅₃	#3484	1	1	yes	characteristic
268	C₂×C₅₃	#3485	1	1	yes	characteristic
269	C₂×C₅₃	#3535	1	4	yes	—
270	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
271	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
272	#543	—	2	8	no	—
273	C₂×#123	—	2	8	no	—
274	C₂×#123	C₂²×C₅₃	1	8	yes	—
275	C₂×#123	C₂²×C₅₃	1	4	yes	—
276	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
277	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
278	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
279	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
280	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
281	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
282	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
283	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
284	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
285	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
286	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
287	#400	C₂²×C₅₃	1	4	yes	complemented
288	#398	C₂²×C₅₃	1	8	yes	complemented
289	#398	C₂²×C₅₃	1	8	yes	complemented
290	#397	C₂²×C₅₃	1	4	yes	complemented
291	#397	C₂²×C₅₃	1	8	yes	complemented
292	#395	C₂²×C₅₃	1	4	yes	complemented
293	#394	C₂²×C₅₃	1	8	yes	—
294	#393	C₂²×C₅₃	1	4	yes	complemented
295	#391	C₂²×C₅₃	1	4	yes	complemented
296	#388	C₂²×C₅₃	1	8	yes	complemented
297	#388	C₂²×C₅₃	1	8	yes	complemented
298	#386	C₂²×C₅₃	1	4	yes	—
299	#385	C₂²×C₅₃	1	4	yes	—
300	#385	C₂²×C₅₃	1	4	yes	—
301	#385	C₂²×C₅₃	1	8	yes	complemented
302	#385	C₂²×C₅₃	1	4	yes	complemented
303	#384	C₂²×C₅₃	1	4	yes	complemented
304	#381	C₂²×C₅₃	1	4	yes	—
305	#381	C₂²×C₅₃	1	4	yes	complemented
306	#380	C₂²×C₅₃	1	2	yes	—
307	#380	C₂²×C₅₃	1	4	yes	complemented
308	#379	C₂²×C₅₃	1	4	yes	—
309	#379	C₂²×C₅₃	1	2	yes	—
310	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
311	C₅₃	#36720	1	1	yes	characteristic, direct product factor, Sylow
312	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
313	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
314	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
315	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
316	C₂²×D₄	—	2	4	no	—
317	C₂²×D₄	—	4	4	no	—
318	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
319	C₂³×C₄	—	2	4	no	—
320	C₂³×C₄	—	4	4	no	—
321	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
322	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
323	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
324	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
325	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
326	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
327	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
328	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
329	#123	—	2	32	no	—
330	#123	—	4	8	no	—
331	C₂²⋊Q₈	—	2	8	no	—
332	C₂²⋊Q₈	—	2	8	no	—
333	C₂²⋊Q₈	—	2	16	no	—
334	C₂²⋊Q₈	—	2	16	no	—
335	C₂²⋊Q₈	—	4	8	no	—
336	C₂×C₄⋊C₄	—	2	4	no	—
337	C₂×C₄⋊C₄	—	2	4	no	—
338	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
339	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
340	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
341	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
342	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
343	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
344	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
345	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
346	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
347	C₂×C₂²⋊C₄	—	2	4	no	—
348	C₂×C₂²⋊C₄	—	2	8	no	—
349	C₂×C₂²⋊C₄	—	2	4	no	—
350	C₂×C₂²⋊C₄	D₄×C₅₃	1	8	yes	—
351	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
352	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
353	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
359	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
361	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
362	#95	C₂³×C₅₃	1	4	yes	—
363	#95	C₂³×C₅₃	1	4	yes	—
364	#95	C₂³×C₅₃	1	8	yes	—
365	#95	C₂³×C₅₃	1	8	yes	—
366	#95	C₂³×C₅₃	1	8	yes	—
367	#95	C₂³×C₅₃	1	8	yes	—
368	C₂⁴	—	2	2	no	—
369	C₂⁴	—	2	4	no	—
370	C₂⁴	—	2	4	no	—
371	C₂⁴	—	2	4	no	—
372	C₂⁴	—	4	2	no	—
373	C₂⁴	—	4	2	no	—
374	C₂⁴	—	4	2	no	—
375	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
376	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
377	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
378	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
379	C₂×Q₈	—	2	4	no	—
380	C₂×Q₈	—	2	4	no	—
381	C₂×Q₈	—	4	4	no	—
382	C₂×Q₈	—	4	4	no	—
383	C₂×D₄	—	2	8	no	—
384	C₂×D₄	—	2	8	no	—
385	C₂×D₄	—	4	4	no	—
386	C₂×D₄	—	4	16	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	8	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	—	2	4	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	8	no	—
395	C₂²×C₄	—	4	2	no	—
396	C₂²×C₄	—	4	4	no	—
397	C₂²×C₄	—	4	2	no	—
398	C₂²×C₄	—	4	2	no	—
399	C₂²×C₄	—	4	2	no	—
400	C₂²×C₄	—	4	4	no	—
401	C₂²×C₄	—	4	4	no	—
402	C₂²×C₄	—	4	16	no	—
403	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
404	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
405	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
406	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
407	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
408	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
409	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
410	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
411	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
412	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
413	C₄⋊C₄	—	2	8	no	—
414	C₄⋊C₄	—	2	8	no	—
415	C₄⋊C₄	—	2	4	no	—
416	C₄⋊C₄	—	2	8	no	—
417	C₄⋊C₄	—	2	4	no	—
418	C₄⋊C₄	—	2	8	no	—
419	C₄⋊C₄	—	4	4	no	—
420	C₄⋊C₄	—	4	8	no	—
421	C₄⋊C₄	—	4	8	no	—
422	C₂²⋊C₄	—	2	8	no	—
423	C₂²⋊C₄	—	2	8	no	—
424	C₂²⋊C₄	—	2	8	no	—
425	C₂²⋊C₄	—	2	16	no	—
426	C₂²⋊C₄	—	2	8	no	—
427	C₂²⋊C₄	—	2	8	no	—
428	C₂²⋊C₄	—	4	4	no	—
429	C₂²⋊C₄	—	4	4	no	—
430	C₂²⋊C₄	—	4	8	no	—
431	C₂²⋊C₄	C₂×D₄×C₅₃	1	16	yes	—
432	C₄²	—	4	4	no	—
433	C₂³	—	2	2	no	—
434	C₂³	—	2	2	no	—
435	C₂³	—	2	8	no	—
436	C₂³	—	2	2	no	—
437	C₂³	—	2	4	no	—
438	C₂³	—	2	2	no	—
439	C₂³	—	2	4	no	—
440	C₂³	—	2	4	no	—
441	C₂³	—	2	4	no	—
442	C₂³	—	2	4	no	—
443	C₂³	—	2	8	no	—
444	C₂³	—	4	4	no	—
445	C₂³	—	4	4	no	—
446	C₂³	—	4	16	no	—
447	C₂³	—	4	4	no	—
448	C₂³	—	4	4	no	—
449	C₂³	—	8	8	no	—
450	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
451	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
452	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
453	C₂³	2₋¹⁺⁴×C₅₃	1	4	yes	—
454	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
455	Q₈	—	8	4	no	—
456	Q₈	—	8	4	no	—
457	D₄	—	4	16	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	8	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	8	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	8	no	—
472	C₂×C₄	—	2	8	no	—
473	C₂×C₄	—	4	4	no	—
474	C₂×C₄	—	4	4	no	—
475	C₂×C₄	—	4	16	no	—
476	C₂×C₄	—	4	4	no	—
477	C₂×C₄	—	4	4	no	—
478	C₂×C₄	—	4	4	no	—
479	C₂×C₄	—	8	4	no	—
480	C₂×C₄	—	8	4	no	—
481	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
482	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
483	C₂²	—	2	2	no	—
484	C₂²	—	2	2	no	—
485	C₂²	—	2	8	no	—
486	C₂²	—	2	4	no	—
487	C₂²	—	2	4	no	—
488	C₂²	—	2	4	no	—
489	C₂²	—	2	4	no	—
490	C₂²	—	2	8	no	—
491	C₂²	—	4	16	no	—
492	C₂²	—	4	2	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	4	8	no	—
496	C₂²	—	8	8	no	—
497	C₂²	C₅₃×#536	1	4	yes	—
498	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
499	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
500	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	2	yes	—
501	C₂²	C₅₃×#585	1	2	yes	—
502	C₄	—	2	4	no	—
503	C₄	—	2	4	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	2	no	—
506	C₄	—	4	2	no	—
507	C₄	—	4	4	no	—
508	C₄	—	4	4	no	—
509	C₄	—	4	8	no	—
510	C₂	—	2	4	no	—
511	C₂	—	4	4	no	—
512	C₂	—	4	4	no	—
513	C₂	C₂×C₅₃×#536	1	1	yes	characteristic
514	C₂	C₅₃×#3484	1	1	yes	characteristic
515	C₂	C₅₃×#3485	1	1	yes	characteristic
516	C₂	C₅₃×#3535	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂³×D₄×C₅₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#543	C₂	semidirect
3. C₅₃×#2848	C₂	semidirect
4. C₅₃×#2687	C₂	semidirect
5. C₅₃×#2680	C₂	semidirect
6. C₅₃×#2627	C₂	semidirect
7. C₅₃×#2561	C₂	semidirect
8. C₅₃×#2502	C₂	semidirect
9. C₅₃×#2483	C₂	semidirect
11. C₅₃×#2312	C₂	semidirect
20. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
21. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
22. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
29. C₅₃×#400	C₂²	semidirect
30. C₅₃×#398	C₂²	semidirect
31. C₅₃×#398	C₂²	semidirect
32. C₅₃×#397	C₂²	semidirect
33. C₅₃×#397	C₂²	semidirect
34. C₅₃×#395	C₂²	semidirect
36. C₅₃×#393	C₂²	semidirect
37. C₅₃×#391	C₂²	semidirect
38. C₅₃×#388	C₂²	semidirect
39. C₅₃×#388	C₂²	semidirect
43. C₅₃×#385	C₂²	semidirect
44. C₅₃×#385	C₂²	semidirect
45. C₅₃×#384	C₂²	semidirect
47. C₅₃×#381	C₂²	semidirect
49. C₅₃×#380	C₂²	semidirect
224. #36720	C₅₃	direct (N × H)
252. C₂×#543	C₂×C₅₃	semidirect
254. #2848	C₂×C₅₃	semidirect
255. #2687	C₂×C₅₃	semidirect
256. #2680	C₂×C₅₃	semidirect
257. #2627	C₂×C₅₃	semidirect
258. #2561	C₂×C₅₃	semidirect
259. #2502	C₂×C₅₃	semidirect
260. #2483	C₂×C₅₃	semidirect
262. #2312	C₂×C₅₃	semidirect
278. C₂×C₂²⋊Q₈	C₂²×C₅₃	semidirect
279. C₂×C₂²⋊Q₈	C₂²×C₅₃	semidirect
280. C₂×C₂²⋊Q₈	C₂²×C₅₃	semidirect
287. #400	C₂²×C₅₃	semidirect
288. #398	C₂²×C₅₃	semidirect
289. #398	C₂²×C₅₃	semidirect
290. #397	C₂²×C₅₃	semidirect
291. #397	C₂²×C₅₃	semidirect
292. #395	C₂²×C₅₃	semidirect
294. #393	C₂²×C₅₃	semidirect
295. #391	C₂²×C₅₃	semidirect
296. #388	C₂²×C₅₃	semidirect
297. #388	C₂²×C₅₃	semidirect
301. #385	C₂²×C₅₃	semidirect
302. #385	C₂²×C₅₃	semidirect
303. #384	C₂²×C₅₃	semidirect
305. #381	C₂²×C₅₃	semidirect
307. #380	C₂²×C₅₃	semidirect
311. C₅₃	#36720	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#543	2
C₂²×#123×C₅₃	2
C₅₃×#2848	2
C₅₃×#2687	2
C₅₃×#2680	2
C₅₃×#2627	2
C₅₃×#2561	2
C₅₃×#2502	2
C₅₃×#2483	2
C₅₃×#2405	2
C₅₃×#2312	2
#36720	53

Subgroups — C53×#36720 #27244232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36720 #27244232