Subgroups — C₅₃×#26065 #27233577

Order 13568.

894 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2440	C₂	1	2	yes	maximal
4	C₅₃×#2083	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2057	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1942	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1942	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1852	C₂	1	2	yes	maximal
9	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1808	C₂	1	1	yes	maximal, characteristic
11	C₈×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₂²⋊Q₈×C₅₃	—	2	2	no	—
15	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#500	—	2	4	no	—
17	C₂×#107×C₅₃	—	2	4	no	—
18	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#107×C₅₃	C₂²	1	2	yes	—
21	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#107×C₅₃	C₂²	1	2	yes	—
23	C₂×#107×C₅₃	C₂²	1	2	yes	—
24	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#398	—	2	2	no	—
29	C₅₃×#395	—	2	2	no	—
30	C₅₃×#389	C₂²	1	2	yes	—
31	C₅₃×#389	C₂²	1	2	yes	—
32	C₅₃×#385	—	2	2	no	—
33	C₅₃×#384	C₂²	1	2	yes	—
34	C₅₃×#380	C₂²	1	2	yes	—
35	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#340	C₂²	1	2	yes	—
37	C₅₃×#340	C₂²	1	2	yes	—
38	C₅₃×#340	C₂²	1	2	yes	—
39	C₅₃×#336	C₂²	1	1	yes	characteristic
40	C₅₃×#336	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₅₃	—	2	2	no	—
42	C₂³×C₄×C₅₃	—	2	2	no	—
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	—	2	2	no	—
45	C₂²×C₈×C₅₃	—	2	2	no	—
46	C₂²×C₈×C₅₃	Q₈	1	4	yes	complemented
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₅₃	—	2	4	no	—
52	C₄⋊Q₈×C₅₃	—	4	2	no	—
53	C₂²⋊Q₈×C₅₃	—	4	8	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
61	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
62	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	#107×C₅₃	—	2	2	no	—
72	#107×C₅₃	—	2	2	no	—
73	#107×C₅₃	—	2	2	no	—
74	#107×C₅₃	—	2	4	no	—
75	#107×C₅₃	—	2	4	no	—
76	#107×C₅₃	—	4	2	no	—
77	#107×C₅₃	D₄	1	4	yes	—
78	C₂²⋊C₈×C₅₃	Q₈	1	4	yes	complemented
79	C₄×C₈×C₅₃	—	2	2	no	—
80	C₄×C₈×C₅₃	—	2	2	no	—
81	#95×C₅₃	—	2	2	no	—
82	#95×C₅₃	—	2	2	no	—
83	#95×C₅₃	—	2	2	no	—
84	#95×C₅₃	D₄	1	1	yes	characteristic
85	#95×C₅₃	D₄	1	1	yes	characteristic
86	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂×Q₈×C₅₃	—	4	4	no	—
88	C₂×Q₈×C₅₃	—	4	4	no	—
89	C₂×Q₈×C₅₃	—	8	2	no	—
90	C₂²×C₄×C₅₃	—	2	1	no	—
91	C₂²×C₄×C₅₃	—	2	1	no	—
92	C₂²×C₄×C₅₃	—	2	1	no	—
93	C₂²×C₄×C₅₃	—	2	1	no	—
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	—	2	2	no	—
97	C₂²×C₄×C₅₃	—	2	2	no	—
98	C₂²×C₄×C₅₃	—	4	4	no	—
99	C₂²×C₄×C₅₃	—	4	4	no	—
100	C₂²×C₄×C₅₃	—	8	2	no	—
101	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
107	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
108	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
109	C₂×C₈×C₅₃	—	2	1	no	—
110	C₂×C₈×C₅₃	—	2	1	no	—
111	C₂×C₈×C₅₃	—	2	1	no	—
112	C₂×C₈×C₅₃	—	2	1	no	—
113	C₂×C₈×C₅₃	—	2	1	no	—
114	C₂×C₈×C₅₃	—	2	1	no	—
115	C₂×C₈×C₅₃	—	2	8	no	—
116	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
118	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
119	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
120	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
121	C₂×C₈×C₅₃	#41	1	2	yes	—
122	C₄⋊C₄×C₅₃	—	2	2	no	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	2	no	—
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	—	2	2	no	—
127	C₄⋊C₄×C₅₃	—	2	2	no	—
128	C₄⋊C₄×C₅₃	—	2	2	no	—
129	C₄⋊C₄×C₅₃	—	4	4	no	—
130	C₄⋊C₄×C₅₃	—	4	4	no	—
131	C₄⋊C₄×C₅₃	—	4	4	no	—
132	C₄⋊C₄×C₅₃	—	4	4	no	—
133	C₄⋊C₄×C₅₃	—	4	4	no	—
134	C₄⋊C₄×C₅₃	—	8	2	no	—
135	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
136	C₂²⋊C₄×C₅₃	—	4	4	no	—
137	C₂²⋊C₄×C₅₃	D₈	1	4	yes	—
138	C₂²⋊C₄×C₅₃	Q₁₆	1	4	yes	—
139	C₄²×C₅₃	—	2	1	no	—
140	C₄²×C₅₃	—	2	1	no	—
141	C₄²×C₅₃	—	2	1	no	—
142	C₄²×C₅₃	—	2	1	no	—
143	C₂³×C₅₃	—	2	1	no	—
144	C₂³×C₅₃	—	2	1	no	—
145	C₂³×C₅₃	—	2	1	no	—
146	C₂³×C₅₃	—	2	1	no	—
147	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
148	C₂³×C₅₃	C₂×D₈	1	2	yes	—
149	C₂³×C₅₃	C₂×Q₁₆	1	2	yes	—
150	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
151	Q₈×C₅₃	—	8	8	no	—
152	C₂×C₄×C₅₃	—	2	1	no	—
153	C₂×C₄×C₅₃	—	2	1	no	—
154	C₂×C₄×C₅₃	—	2	1	no	—
155	C₂×C₄×C₅₃	—	2	1	no	—
156	C₂×C₄×C₅₃	—	2	1	no	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	2	1	no	—
160	C₂×C₄×C₅₃	—	2	1	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	4	no	—
167	C₂×C₄×C₅₃	—	4	2	no	—
168	C₂×C₄×C₅₃	—	4	2	no	—
169	C₂×C₄×C₅₃	—	4	2	no	—
170	C₂×C₄×C₅₃	—	4	2	no	—
171	C₂×C₄×C₅₃	—	4	2	no	—
172	C₂×C₄×C₅₃	—	4	2	no	—
173	C₂×C₄×C₅₃	—	4	2	no	—
174	C₂×C₄×C₅₃	—	4	2	no	—
175	C₂×C₄×C₅₃	—	4	2	no	—
176	C₂×C₄×C₅₃	—	4	2	no	—
177	C₂×C₄×C₅₃	—	4	2	no	—
178	C₂×C₄×C₅₃	—	8	8	no	—
179	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
180	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
181	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
182	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
183	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
184	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
186	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
189	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
190	C₈×C₅₃	—	2	2	no	—
191	C₈×C₅₃	—	2	2	no	—
192	C₈×C₅₃	—	2	2	no	—
193	C₈×C₅₃	C₂²⋊Q₈	1	4	yes	complemented
194	#26065	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂²×C₅₃	—	2	2	no	—
196	C₂²×C₅₃	—	2	2	no	—
197	C₂²×C₅₃	—	2	2	no	—
198	C₂²×C₅₃	—	2	2	no	—
199	C₂²×C₅₃	—	2	2	no	—
200	C₂²×C₅₃	—	2	2	no	—
201	C₂²×C₅₃	#500	1	4	yes	complemented
202	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
203	C₂²×C₅₃	D₄²	1	1	yes	characteristic
204	C₂²×C₅₃	#547	1	1	yes	characteristic
205	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
206	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
207	C₂²×C₅₃	#576	1	1	yes	characteristic
208	C₂²×C₅₃	#578	1	1	yes	characteristic
209	C₄×C₅₃	—	2	2	no	—
210	C₄×C₅₃	—	2	2	no	—
211	C₄×C₅₃	—	2	2	no	—
212	C₄×C₅₃	—	2	2	no	—
213	C₄×C₅₃	—	2	2	no	—
214	C₄×C₅₃	—	4	4	no	—
215	C₄×C₅₃	—	8	2	no	—
216	C₄×C₅₃	—	8	2	no	—
217	C₄×C₅₃	—	8	2	no	—
218	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
219	C₄×C₅₃	#544	1	1	yes	characteristic
220	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
221	C₄×C₅₃	#554	1	1	yes	characteristic
222	C₂×#500	C₂×C₅₃	1	1	yes	characteristic, complemented
223	C₂²×#107	C₂×C₅₃	1	1	yes	characteristic
224	#2440	C₂×C₅₃	1	2	yes	—
225	#2083	C₂×C₅₃	1	2	yes	complemented
226	#2057	C₂×C₅₃	1	2	yes	complemented
227	#1942	C₂×C₅₃	1	1	yes	characteristic, complemented
228	#1942	C₂×C₅₃	1	1	yes	characteristic, complemented
229	#1852	C₂×C₅₃	1	2	yes	—
230	#1845	C₂×C₅₃	1	1	yes	characteristic, complemented
231	#1808	C₂×C₅₃	1	1	yes	characteristic
232	C₈×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
233	C₂×C₅₃	—	2	4	no	—
234	C₂×C₅₃	#2440	1	1	yes	characteristic
235	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
236	C₂×C₅₃	#3164	1	1	yes	characteristic
237	C₂×C₅₃	D₄×D₈	1	1	yes	characteristic
238	C₂×C₅₃	D₄×Q₁₆	1	1	yes	characteristic
239	C₂×C₅₃	#3334	1	1	yes	characteristic
240	C₂×C₅₃	#3338	1	1	yes	characteristic
241	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
242	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
243	C₂×C₂²⋊Q₈	—	2	2	no	—
244	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
245	#500	—	2	4	no	—
246	C₂×#107	—	2	4	no	—
247	C₂×#107	C₂²×C₅₃	1	2	yes	—
248	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
249	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
250	C₂×#107	C₂²×C₅₃	1	2	yes	—
251	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
252	C₂×#107	C₂²×C₅₃	1	2	yes	—
253	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
254	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
255	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
256	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
257	#398	—	2	2	no	—
258	#395	—	2	2	no	—
259	#389	C₂²×C₅₃	1	2	yes	—
260	#389	C₂²×C₅₃	1	2	yes	—
261	#385	—	2	2	no	—
262	#384	C₂²×C₅₃	1	2	yes	—
263	#380	C₂²×C₅₃	1	2	yes	—
264	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
265	#340	C₂²×C₅₃	1	2	yes	—
266	#340	C₂²×C₅₃	1	2	yes	—
267	#340	C₂²×C₅₃	1	2	yes	—
268	#336	C₂²×C₅₃	1	1	yes	characteristic
269	#336	C₂²×C₅₃	1	1	yes	characteristic
270	C₅₃	#26065	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²×Q₈	—	2	2	no	—
272	C₂³×C₄	—	2	2	no	—
273	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
274	C₂²×C₈	—	2	2	no	—
275	C₂²×C₈	—	2	2	no	—
276	C₂²×C₈	Q₈×C₅₃	1	4	yes	complemented
277	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
278	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
279	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
280	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
281	C₄⋊Q₈	—	2	4	no	—
282	C₄⋊Q₈	—	4	2	no	—
283	C₂²⋊Q₈	—	4	8	no	—
284	C₂×C₄⋊C₄	—	2	4	no	—
285	C₂×C₄⋊C₄	—	2	2	no	—
286	C₂×C₄⋊C₄	—	2	2	no	—
287	C₂×C₄⋊C₄	—	2	2	no	—
288	C₂×C₄⋊C₄	—	2	2	no	—
289	C₂×C₄⋊C₄	—	2	2	no	—
290	C₂×C₄⋊C₄	—	2	2	no	—
291	C₂×C₄⋊C₄	—	4	2	no	—
292	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
293	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
294	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
295	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
296	C₂×C₂²⋊C₄	—	2	2	no	—
297	C₂×C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
298	C₂×C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
299	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
300	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
301	#107	—	2	2	no	—
302	#107	—	2	2	no	—
303	#107	—	2	2	no	—
304	#107	—	2	4	no	—
305	#107	—	2	4	no	—
306	#107	—	4	2	no	—
307	#107	D₄×C₅₃	1	4	yes	—
308	C₂²⋊C₈	Q₈×C₅₃	1	4	yes	complemented
309	C₄×C₈	—	2	2	no	—
310	C₄×C₈	—	2	2	no	—
311	#95	—	2	2	no	—
312	#95	—	2	2	no	—
313	#95	—	2	2	no	—
314	#95	D₄×C₅₃	1	1	yes	characteristic
315	#95	D₄×C₅₃	1	1	yes	characteristic
316	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
317	C₂×Q₈	—	4	4	no	—
318	C₂×Q₈	—	4	4	no	—
319	C₂×Q₈	—	8	2	no	—
320	C₂²×C₄	—	2	1	no	—
321	C₂²×C₄	—	2	1	no	—
322	C₂²×C₄	—	2	1	no	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	4	4	no	—
329	C₂²×C₄	—	4	4	no	—
330	C₂²×C₄	—	8	2	no	—
331	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
332	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
335	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
336	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
337	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
338	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	2	1	no	—
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	2	8	no	—
346	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
347	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
348	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
349	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
350	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
351	C₂×C₈	#41×C₅₃	1	2	yes	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	4	4	no	—
360	C₄⋊C₄	—	4	4	no	—
361	C₄⋊C₄	—	4	4	no	—
362	C₄⋊C₄	—	4	4	no	—
363	C₄⋊C₄	—	4	4	no	—
364	C₄⋊C₄	—	8	2	no	—
365	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
366	C₂²⋊C₄	—	4	4	no	—
367	C₂²⋊C₄	D₈×C₅₃	1	4	yes	—
368	C₂²⋊C₄	Q₁₆×C₅₃	1	4	yes	—
369	C₄²	—	2	1	no	—
370	C₄²	—	2	1	no	—
371	C₄²	—	2	1	no	—
372	C₄²	—	2	1	no	—
373	C₂³	—	2	1	no	—
374	C₂³	—	2	1	no	—
375	C₂³	—	2	1	no	—
376	C₂³	—	2	1	no	—
377	C₂³	C₄⋊Q₈×C₅₃	1	2	yes	—
378	C₂³	C₂×D₈×C₅₃	1	2	yes	—
379	C₂³	C₂×Q₁₆×C₅₃	1	2	yes	—
380	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
381	Q₈	—	8	8	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	8	8	no	—
409	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
410	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
411	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
412	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
413	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
414	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
415	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
416	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
417	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
418	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
419	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
420	C₈	—	2	2	no	—
421	C₈	—	2	2	no	—
422	C₈	—	2	2	no	—
423	C₈	C₂²⋊Q₈×C₅₃	1	4	yes	complemented
424	C₂²	—	2	2	no	—
425	C₂²	—	2	2	no	—
426	C₂²	—	2	2	no	—
427	C₂²	—	2	2	no	—
428	C₂²	—	2	2	no	—
429	C₂²	—	2	2	no	—
430	C₂²	C₅₃×#500	1	4	yes	complemented
431	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
432	C₂²	D₄²×C₅₃	1	1	yes	characteristic
433	C₂²	C₅₃×#547	1	1	yes	characteristic
434	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
435	C₂²	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
436	C₂²	C₅₃×#576	1	1	yes	characteristic
437	C₂²	C₅₃×#578	1	1	yes	characteristic
438	C₄	—	2	2	no	—
439	C₄	—	2	2	no	—
440	C₄	—	2	2	no	—
441	C₄	—	2	2	no	—
442	C₄	—	2	2	no	—
443	C₄	—	4	4	no	—
444	C₄	—	8	2	no	—
445	C₄	—	8	2	no	—
446	C₄	—	8	2	no	—
447	C₄	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
448	C₄	C₅₃×#544	1	1	yes	characteristic
449	C₄	D₄×Q₈×C₅₃	1	1	yes	characteristic
450	C₄	C₅₃×#554	1	1	yes	characteristic
451	C₂	—	2	4	no	—
452	C₂	C₅₃×#2440	1	1	yes	characteristic
453	C₂	C₂×C₅₃×#500	1	1	yes	characteristic
454	C₂	C₅₃×#3164	1	1	yes	characteristic
455	C₂	D₄×D₈×C₅₃	1	1	yes	characteristic
456	C₂	D₄×Q₁₆×C₅₃	1	1	yes	characteristic
457	C₂	C₅₃×#3334	1	1	yes	characteristic
458	C₂	C₅₃×#3338	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#500	C₂	semidirect
4. C₅₃×#2083	C₂	semidirect
5. C₅₃×#2057	C₂	semidirect
6. C₅₃×#1942	C₂	semidirect
7. C₅₃×#1942	C₂	semidirect
9. C₅₃×#1845	C₂	semidirect
46. C₂²×C₈×C₅₃	Q₈	semidirect
78. C₂²⋊C₈×C₅₃	Q₈	semidirect
193. C₈×C₅₃	C₂²⋊Q₈	semidirect
194. #26065	C₅₃	direct (N × H)
201. C₂²×C₅₃	#500	semidirect
222. C₂×#500	C₂×C₅₃	semidirect
225. #2083	C₂×C₅₃	semidirect
226. #2057	C₂×C₅₃	semidirect
227. #1942	C₂×C₅₃	semidirect
228. #1942	C₂×C₅₃	semidirect
230. #1845	C₂×C₅₃	semidirect
270. C₅₃	#26065	direct (N × H)
276. C₂²×C₈	Q₈×C₅₃	semidirect
308. C₂²⋊C₈	Q₈×C₅₃	semidirect
423. C₈	C₂²⋊Q₈×C₅₃	semidirect
430. C₂²	C₅₃×#500	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#500	2
C₂²×#107×C₅₃	2
C₅₃×#2440	2
C₅₃×#2083	2
C₅₃×#2057	2
C₅₃×#1942	2
C₅₃×#1852	2
C₅₃×#1845	2
C₅₃×#1808	2
C₈×C₂²⋊C₄×C₅₃	2
#26065	53

Subgroups — C53×#26065 #27233577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#26065 #27233577