Subgroups — C₅₃×#21597 #27229109

Order 13568.

934 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1984	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1939	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1916	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1850	C₂	1	2	yes	maximal
8	C₅₃×#1797	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1771	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1752	C₂	1	2	yes	maximal
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#446	—	2	4	no	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
23	C₂×#97×C₅₃	C₂²	1	2	yes	—
24	C₂×#97×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₅₃	—	2	4	no	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#391	C₄	1	1	yes	characteristic
31	C₅₃×#389	C₄	1	1	yes	characteristic
32	C₅₃×#387	C₄	1	2	yes	—
33	C₅₃×#385	C₄	1	2	yes	—
34	C₅₃×#380	C₄	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
38	C₅₃×#336	C₂²	1	2	yes	—
39	C₅₃×#336	C₂²	1	2	yes	—
40	C₅₃×#336	C₂²	1	2	yes	—
41	C₅₃×#336	C₂²	1	2	yes	—
42	C₅₃×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	—	2	2	no	—
48	C₂²×C₈×C₅₃	—	2	2	no	—
49	C₂²×C₈×C₅₃	—	2	2	no	—
50	C₂²×C₈×C₅₃	—	2	2	no	—
51	C₂²×C₈×C₅₃	—	2	2	no	—
52	C₂²×C₈×C₅₃	Q₈	1	4	yes	complemented
53	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₂²⋊Q₈×C₅₃	—	2	4	no	—
60	C₄×Q₈×C₅₃	—	2	2	no	—
61	C₄×Q₈×C₅₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
65	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
69	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
70	C₂×C₄²×C₅₃	—	2	2	no	—
71	C₂×C₄²×C₅₃	—	2	2	no	—
72	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
76	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
77	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₅₃	—	2	4	no	—
79	C₄⋊C₈×C₅₃	—	2	4	no	—
80	C₄⋊C₈×C₅₃	—	2	2	no	—
81	C₄⋊C₈×C₅₃	—	2	2	no	—
82	C₂²⋊C₈×C₅₃	—	2	4	no	—
83	#97×C₅₃	—	2	4	no	—
84	#97×C₅₃	—	2	4	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	D₄	1	4	yes	—
89	#95×C₅₃	C₂×C₄	1	2	yes	—
90	#95×C₅₃	C₂×C₄	1	2	yes	—
91	#95×C₅₃	C₂×C₄	1	2	yes	—
92	#95×C₅₃	C₂×C₄	1	2	yes	—
93	#95×C₅₃	C₂×C₄	1	2	yes	—
94	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×Q₈×C₅₃	—	2	2	no	—
96	C₂×Q₈×C₅₃	—	2	2	no	—
97	C₂×Q₈×C₅₃	—	2	2	no	—
98	C₂²×C₄×C₅₃	—	2	2	no	—
99	C₂²×C₄×C₅₃	—	2	2	no	—
100	C₂²×C₄×C₅₃	—	2	2	no	—
101	C₂²×C₄×C₅₃	—	2	2	no	—
102	C₂²×C₄×C₅₃	—	2	2	no	—
103	C₂²×C₄×C₅₃	—	2	1	no	—
104	C₂²×C₄×C₅₃	—	2	1	no	—
105	C₂²×C₄×C₅₃	—	2	1	no	—
106	C₂²×C₄×C₅₃	—	2	1	no	—
107	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
108	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
118	C₂²×C₄×C₅₃	#41	1	2	yes	—
119	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
120	C₂×C₈×C₅₃	—	2	8	no	—
121	C₂×C₈×C₅₃	—	2	1	no	—
122	C₂×C₈×C₅₃	—	2	1	no	—
123	C₂×C₈×C₅₃	—	2	1	no	—
124	C₂×C₈×C₅₃	—	2	1	no	—
125	C₂×C₈×C₅₃	—	2	1	no	—
126	C₂×C₈×C₅₃	—	2	1	no	—
127	C₂×C₈×C₅₃	—	2	8	no	—
128	C₂×C₈×C₅₃	—	2	2	no	—
129	C₂×C₈×C₅₃	—	2	2	no	—
130	C₂×C₈×C₅₃	—	2	2	no	—
131	C₂×C₈×C₅₃	—	2	2	no	—
132	C₂×C₈×C₅₃	—	2	2	no	—
133	C₂×C₈×C₅₃	—	2	2	no	—
134	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
135	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
136	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
137	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
138	C₂×C₈×C₅₃	#41	1	2	yes	—
139	C₂×C₈×C₅₃	#41	1	2	yes	—
140	C₄⋊C₄×C₅₃	—	2	4	no	—
141	C₄⋊C₄×C₅₃	—	2	2	no	—
142	C₄⋊C₄×C₅₃	—	2	4	no	—
143	C₄⋊C₄×C₅₃	—	2	2	no	—
144	C₄⋊C₄×C₅₃	—	2	4	no	—
145	C₄⋊C₄×C₅₃	—	2	2	no	—
146	C₂²⋊C₄×C₅₃	—	2	4	no	—
147	C₂²⋊C₄×C₅₃	—	2	4	no	—
148	C₄²×C₅₃	—	2	2	no	—
149	C₄²×C₅₃	—	2	2	no	—
150	C₄²×C₅₃	—	2	1	no	—
151	C₄²×C₅₃	—	2	2	no	—
152	C₄²×C₅₃	—	2	2	no	—
153	C₄²×C₅₃	—	2	2	no	—
154	C₄²×C₅₃	—	2	1	no	—
155	C₄²×C₅₃	—	2	2	no	—
156	C₄²×C₅₃	C₂×D₄	1	2	yes	—
157	C₄²×C₅₃	#41	1	2	yes	—
158	C₂³×C₅₃	—	2	1	no	—
159	C₂³×C₅₃	—	2	1	no	—
160	C₂³×C₅₃	—	2	1	no	—
161	C₂³×C₅₃	—	2	1	no	—
162	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
163	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
164	C₂³×C₅₃	#131	1	2	yes	—
165	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
166	Q₈×C₅₃	—	4	4	no	—
167	C₂×C₄×C₅₃	—	2	8	no	—
168	C₂×C₄×C₅₃	—	2	1	no	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	—	2	1	no	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	2	2	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	2	8	no	—
181	C₂×C₄×C₅₃	—	2	2	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	2	no	—
187	C₂×C₄×C₅₃	—	2	2	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	—	2	2	no	—
192	C₂×C₄×C₅₃	—	2	2	no	—
193	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
194	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
195	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
196	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
197	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
198	C₂×C₄×C₅₃	#131	1	2	yes	—
199	C₂×C₄×C₅₃	#131	1	2	yes	—
200	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
202	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
203	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
204	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
205	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
206	C₈×C₅₃	—	2	2	no	—
207	C₈×C₅₃	—	2	2	no	—
208	C₈×C₅₃	—	2	2	no	—
209	C₈×C₅₃	—	4	2	no	—
210	C₈×C₅₃	—	4	2	no	—
211	C₈×C₅₃	C₂²⋊Q₈	1	4	yes	complemented
212	#21597	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂²×C₅₃	—	2	2	no	—
214	C₂²×C₅₃	—	2	2	no	—
215	C₂²×C₅₃	—	2	2	no	—
216	C₂²×C₅₃	—	2	2	no	—
217	C₂²×C₅₃	—	2	2	no	—
218	C₂²×C₅₃	—	2	2	no	—
219	C₂²×C₅₃	#446	1	4	yes	complemented
220	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
221	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
222	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
223	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
224	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
225	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
226	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
227	C₄×C₅₃	—	2	2	no	—
228	C₄×C₅₃	—	2	2	no	—
229	C₄×C₅₃	—	2	2	no	—
230	C₄×C₅₃	—	2	2	no	—
231	C₄×C₅₃	—	4	2	no	—
232	C₄×C₅₃	—	4	2	no	—
233	C₄×C₅₃	—	4	2	no	—
234	C₄×C₅₃	—	4	2	no	—
235	C₄×C₅₃	#435	1	4	yes	—
236	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
237	C₄×C₅₃	#525	1	1	yes	characteristic
238	C₄×C₅₃	#529	1	1	yes	characteristic
239	C₄×C₅₃	#533	1	1	yes	characteristic
240	C₂×#446	C₂×C₅₃	1	1	yes	characteristic, complemented
241	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
242	C₄×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
243	#1984	C₂×C₅₃	1	1	yes	characteristic, complemented
244	#1939	C₂×C₅₃	1	1	yes	characteristic, complemented
245	#1916	C₂×C₅₃	1	2	yes	complemented
246	#1850	C₂×C₅₃	1	2	yes	—
247	#1797	C₂×C₅₃	1	1	yes	characteristic
248	#1771	C₂×C₅₃	1	2	yes	complemented
249	#1770	C₂×C₅₃	1	1	yes	characteristic, complemented
250	#1752	C₂×C₅₃	1	2	yes	—
251	C₂×C₅₃	—	2	4	no	—
252	C₂×C₅₃	C₄×C₂²⋊Q₈	1	1	yes	characteristic
253	C₂×C₅₃	C₂×#435	1	1	yes	characteristic
254	C₂×C₅₃	#2929	1	1	yes	characteristic
255	C₂×C₅₃	C₂×#446	1	1	yes	characteristic
256	C₂×C₅₃	#2960	1	1	yes	characteristic
257	C₂×C₅₃	#2965	1	1	yes	characteristic
258	C₂×C₅₃	#2966	1	1	yes	characteristic
259	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
260	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
261	C₂×C₂²⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
262	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
263	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
264	#446	—	2	4	no	—
265	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
266	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
267	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
268	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
269	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
270	C₂×#97	C₂²×C₅₃	1	2	yes	—
271	C₂×#97	C₂²×C₅₃	1	2	yes	—
272	C₂×C₄×C₈	—	2	4	no	—
273	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
274	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
275	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
276	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
277	#391	C₄×C₅₃	1	1	yes	characteristic
278	#389	C₄×C₅₃	1	1	yes	characteristic
279	#387	C₄×C₅₃	1	2	yes	—
280	#385	C₄×C₅₃	1	2	yes	—
281	#380	C₄×C₅₃	1	1	yes	characteristic
282	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
283	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
284	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
285	#336	C₂²×C₅₃	1	2	yes	—
286	#336	C₂²×C₅₃	1	2	yes	—
287	#336	C₂²×C₅₃	1	2	yes	—
288	#336	C₂²×C₅₃	1	2	yes	—
289	#336	C₂²×C₅₃	1	2	yes	—
290	C₅₃	#21597	1	1	yes	characteristic, direct product factor, Sylow
291	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
292	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
293	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
294	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
295	C₂²×C₈	—	2	2	no	—
296	C₂²×C₈	—	2	2	no	—
297	C₂²×C₈	—	2	2	no	—
298	C₂²×C₈	—	2	2	no	—
299	C₂²×C₈	—	2	2	no	—
300	C₂²×C₈	Q₈×C₅₃	1	4	yes	complemented
301	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
302	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
303	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
304	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
305	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
306	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
307	C₂²⋊Q₈	—	2	4	no	—
308	C₄×Q₈	—	2	2	no	—
309	C₄×Q₈	—	2	2	no	—
310	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
311	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
312	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
313	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
314	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
315	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
316	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
317	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
318	C₂×C₄²	—	2	2	no	—
319	C₂×C₄²	—	2	2	no	—
320	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
321	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
322	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
323	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
324	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
325	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
326	C₄⋊C₈	—	2	4	no	—
327	C₄⋊C₈	—	2	4	no	—
328	C₄⋊C₈	—	2	2	no	—
329	C₄⋊C₈	—	2	2	no	—
330	C₂²⋊C₈	—	2	4	no	—
331	#97	—	2	4	no	—
332	#97	—	2	4	no	—
333	C₄×C₈	—	2	2	no	—
334	C₄×C₈	—	2	2	no	—
335	C₄×C₈	—	2	2	no	—
336	C₄×C₈	D₄×C₅₃	1	4	yes	—
337	#95	C₂×C₄×C₅₃	1	2	yes	—
338	#95	C₂×C₄×C₅₃	1	2	yes	—
339	#95	C₂×C₄×C₅₃	1	2	yes	—
340	#95	C₂×C₄×C₅₃	1	2	yes	—
341	#95	C₂×C₄×C₅₃	1	2	yes	—
342	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
343	C₂×Q₈	—	2	2	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	M₁₆×C₅₃	1	4	yes	—
356	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
357	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
358	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
359	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
360	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
361	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
362	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
363	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
364	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
365	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
366	C₂²×C₄	#41×C₅₃	1	2	yes	—
367	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
368	C₂×C₈	—	2	8	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	8	no	—
376	C₂×C₈	—	2	2	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	2	no	—
382	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
383	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
384	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
385	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
386	C₂×C₈	#41×C₅₃	1	2	yes	—
387	C₂×C₈	#41×C₅₃	1	2	yes	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	4	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	4	no	—
392	C₄⋊C₄	—	2	4	no	—
393	C₄⋊C₄	—	2	2	no	—
394	C₂²⋊C₄	—	2	4	no	—
395	C₂²⋊C₄	—	2	4	no	—
396	C₄²	—	2	2	no	—
397	C₄²	—	2	2	no	—
398	C₄²	—	2	1	no	—
399	C₄²	—	2	2	no	—
400	C₄²	—	2	2	no	—
401	C₄²	—	2	2	no	—
402	C₄²	—	2	1	no	—
403	C₄²	—	2	2	no	—
404	C₄²	C₂×D₄×C₅₃	1	2	yes	—
405	C₄²	#41×C₅₃	1	2	yes	—
406	C₂³	—	2	1	no	—
407	C₂³	—	2	1	no	—
408	C₂³	—	2	1	no	—
409	C₂³	—	2	1	no	—
410	C₂³	C₄×Q₈×C₅₃	1	2	yes	—
411	C₂³	C₂×M₁₆×C₅₃	1	2	yes	—
412	C₂³	#131×C₅₃	1	2	yes	—
413	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
414	Q₈	—	4	4	no	—
415	C₂×C₄	—	2	8	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	8	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
442	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
443	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
444	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
445	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
446	C₂×C₄	#131×C₅₃	1	2	yes	—
447	C₂×C₄	#131×C₅₃	1	2	yes	—
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	C₂×#41×C₅₃	1	1	yes	characteristic
452	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
453	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
454	C₈	—	2	2	no	—
455	C₈	—	2	2	no	—
456	C₈	—	2	2	no	—
457	C₈	—	4	2	no	—
458	C₈	—	4	2	no	—
459	C₈	C₂²⋊Q₈×C₅₃	1	4	yes	complemented
460	C₂²	—	2	2	no	—
461	C₂²	—	2	2	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	2	2	no	—
464	C₂²	—	2	2	no	—
465	C₂²	—	2	2	no	—
466	C₂²	C₅₃×#446	1	4	yes	complemented
467	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
468	C₂²	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
469	C₂²	C₄×#41×C₅₃	1	1	yes	characteristic
470	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
471	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
472	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
473	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	2	2	no	—
477	C₄	—	2	2	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	2	no	—
480	C₄	—	4	2	no	—
481	C₄	—	4	2	no	—
482	C₄	C₅₃×#435	1	4	yes	—
483	C₄	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
484	C₄	C₅₃×#525	1	1	yes	characteristic
485	C₄	C₅₃×#529	1	1	yes	characteristic
486	C₄	C₅₃×#533	1	1	yes	characteristic
487	C₂	—	2	4	no	—
488	C₂	C₄×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
489	C₂	C₂×C₅₃×#435	1	1	yes	characteristic
490	C₂	C₅₃×#2929	1	1	yes	characteristic
491	C₂	C₂×C₅₃×#446	1	1	yes	characteristic
492	C₂	C₅₃×#2960	1	1	yes	characteristic
493	C₂	C₅₃×#2965	1	1	yes	characteristic
494	C₂	C₅₃×#2966	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#446	C₂	semidirect
4. C₅₃×#1984	C₂	semidirect
5. C₅₃×#1939	C₂	semidirect
6. C₅₃×#1916	C₂	semidirect
9. C₅₃×#1771	C₂	semidirect
10. C₅₃×#1770	C₂	semidirect
52. C₂²×C₈×C₅₃	Q₈	semidirect
211. C₈×C₅₃	C₂²⋊Q₈	semidirect
212. #21597	C₅₃	direct (N × H)
219. C₂²×C₅₃	#446	semidirect
240. C₂×#446	C₂×C₅₃	semidirect
243. #1984	C₂×C₅₃	semidirect
244. #1939	C₂×C₅₃	semidirect
245. #1916	C₂×C₅₃	semidirect
248. #1771	C₂×C₅₃	semidirect
249. #1770	C₂×C₅₃	semidirect
290. C₅₃	#21597	direct (N × H)
300. C₂²×C₈	Q₈×C₅₃	semidirect
459. C₈	C₂²⋊Q₈×C₅₃	semidirect
466. C₂²	C₅₃×#446	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#446	2
C₂²×C₄×C₈×C₅₃	2
C₄×C₂²⋊Q₈×C₅₃	2
C₅₃×#1984	2
C₅₃×#1939	2
C₅₃×#1916	2
C₅₃×#1850	2
C₅₃×#1797	2
C₅₃×#1771	2
C₅₃×#1770	2
C₅₃×#1752	2
#21597	53

Subgroups — C53×#21597 #27229109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21597 #27229109