Subgroups — C₅₃×#21601 #27229113

Order 13568.

862 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2396	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1985	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1940	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1917	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1851	C₂	1	2	yes	maximal
8	C₅₃×#1798	C₂	1	1	yes	maximal, characteristic
9	C₈×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal, complemented
10	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₂²⋊C₄×C₅₃	C₂	1	2	yes	maximal
12	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
16	C₈×Q₈×C₅₃	—	2	4	no	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#405	—	2	4	no	—
23	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#400	C₄	1	1	yes	characteristic
30	C₅₃×#399	C₄	1	1	yes	characteristic
31	C₅₃×#398	C₄	1	2	yes	—
32	C₅₃×#396	C₄	1	2	yes	—
33	C₅₃×#395	C₄	1	1	yes	characteristic
34	C₅₃×#393	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	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₅₃	—	2	2	no	—
46	C₂×M₁₆×C₅₃	—	2	2	no	—
47	C₂×M₁₆×C₅₃	—	2	2	no	—
48	C₂²×C₈×C₅₃	—	2	2	no	—
49	C₂²×C₈×C₅₃	—	2	2	no	—
50	C₂²×C₈×C₅₃	Q₈	1	4	yes	complemented
51	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₄×Q₈×C₅₃	—	2	2	no	—
58	C₄×Q₈×C₅₃	—	2	2	no	—
59	#117×C₅₃	—	2	2	no	—
60	#117×C₅₃	—	2	2	no	—
61	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
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	1	yes	characteristic
65	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
67	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
71	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
74	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
75	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₅₃	—	2	4	no	—
77	C₄⋊C₈×C₅₃	—	2	4	no	—
78	C₄⋊C₈×C₅₃	—	2	2	no	—
79	C₄⋊C₈×C₅₃	—	2	2	no	—
80	C₂²⋊C₈×C₅₃	—	2	4	no	—
81	#97×C₅₃	—	2	2	no	—
82	#97×C₅₃	—	2	2	no	—
83	C₄×C₈×C₅₃	—	2	2	no	—
84	C₄×C₈×C₅₃	—	2	4	no	—
85	C₄×C₈×C₅₃	—	2	4	no	—
86	C₄×C₈×C₅₃	D₄	1	4	yes	—
87	#95×C₅₃	C₂×C₄	1	2	yes	—
88	#95×C₅₃	C₂×C₄	1	2	yes	—
89	#95×C₅₃	C₂×C₄	1	2	yes	—
90	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
91	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
92	#95×C₅₃	C₂×C₄	1	2	yes	—
93	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×Q₈×C₅₃	—	2	2	no	—
95	C₂×Q₈×C₅₃	—	2	4	no	—
96	C₂²×C₄×C₅₃	—	2	1	no	—
97	C₂²×C₄×C₅₃	—	2	1	no	—
98	C₂²×C₄×C₅₃	—	2	1	no	—
99	C₂²×C₄×C₅₃	—	2	1	no	—
100	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
110	C₂²×C₄×C₅₃	#41	1	2	yes	—
111	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
112	M₁₆×C₅₃	—	2	8	no	—
113	C₂×C₈×C₅₃	—	2	1	no	—
114	C₂×C₈×C₅₃	—	2	1	no	—
115	C₂×C₈×C₅₃	—	2	1	no	—
116	C₂×C₈×C₅₃	—	2	1	no	—
117	C₂×C₈×C₅₃	—	2	1	no	—
118	C₂×C₈×C₅₃	—	2	1	no	—
119	C₂×C₈×C₅₃	—	2	8	no	—
120	C₂×C₈×C₅₃	—	2	2	no	—
121	C₂×C₈×C₅₃	—	2	2	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	—	2	2	no	—
125	C₂×C₈×C₅₃	—	2	2	no	—
126	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
127	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
129	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
130	C₂×C₈×C₅₃	#41	1	2	yes	—
131	C₂×C₈×C₅₃	#41	1	2	yes	—
132	C₄⋊C₄×C₅₃	—	2	2	no	—
133	C₄⋊C₄×C₅₃	—	2	2	no	—
134	C₄⋊C₄×C₅₃	—	2	4	no	—
135	C₄⋊C₄×C₅₃	—	2	2	no	—
136	C₄⋊C₄×C₅₃	—	2	4	no	—
137	C₄⋊C₄×C₅₃	—	2	4	no	—
138	C₄⋊C₄×C₅₃	—	2	4	no	—
139	C₂²⋊C₄×C₅₃	—	2	2	no	—
140	C₂²⋊C₄×C₅₃	—	2	2	no	—
141	C₂²⋊C₄×C₅₃	—	2	4	no	—
142	C₂²⋊C₄×C₅₃	—	2	4	no	—
143	C₄²×C₅₃	—	2	1	no	—
144	C₄²×C₅₃	—	2	2	no	—
145	C₄²×C₅₃	—	2	2	no	—
146	C₄²×C₅₃	—	2	2	no	—
147	C₄²×C₅₃	—	2	1	no	—
148	C₄²×C₅₃	—	2	2	no	—
149	C₄²×C₅₃	C₂×D₄	1	2	yes	—
150	C₄²×C₅₃	#41	1	2	yes	—
151	C₂³×C₅₃	—	2	1	no	—
152	C₂³×C₅₃	—	2	1	no	—
153	C₂³×C₅₃	—	2	2	no	—
154	C₂³×C₅₃	—	2	2	no	—
155	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
156	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
157	Q₈×C₅₃	—	2	8	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	1	no	—
161	C₂×C₄×C₅₃	—	2	2	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	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	2	1	no	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	2	2	no	—
173	C₂×C₄×C₅₃	—	2	2	no	—
174	C₂×C₄×C₅₃	—	2	2	no	—
175	C₂×C₄×C₅₃	—	2	2	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	C₄×D₄	1	2	yes	—
181	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
182	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
183	C₂×C₄×C₅₃	#131	1	4	yes	—
184	C₂×C₄×C₅₃	#131	1	4	yes	—
185	C₂×C₄×C₅₃	C₂²×D₄	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	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
191	C₈×C₅₃	—	2	2	no	—
192	C₈×C₅₃	—	2	2	no	—
193	C₈×C₅₃	—	2	2	no	—
194	C₈×C₅₃	—	4	2	no	—
195	C₈×C₅₃	—	4	2	no	—
196	C₈×C₅₃	C₂²⋊Q₈	1	4	yes	—
197	#21601	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂²×C₅₃	—	2	4	no	—
199	C₂²×C₅₃	—	2	2	no	—
200	C₂²×C₅₃	—	4	1	no	—
201	C₂²×C₅₃	—	4	1	no	—
202	C₂²×C₅₃	—	4	2	no	—
203	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
204	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
205	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
206	C₂²×C₅₃	C₂×#131	1	2	yes	—
207	C₂²×C₅₃	C₂×#131	1	2	yes	—
208	C₄×C₅₃	—	2	2	no	—
209	C₄×C₅₃	—	2	4	no	—
210	C₄×C₅₃	—	2	2	no	—
211	C₄×C₅₃	—	2	4	no	—
212	C₄×C₅₃	—	2	2	no	—
213	C₄×C₅₃	—	4	1	no	—
214	C₄×C₅₃	—	4	1	no	—
215	C₄×C₅₃	—	4	2	no	—
216	C₄×C₅₃	—	4	2	no	—
217	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
218	C₄×C₅₃	#525	1	1	yes	characteristic
219	C₄×C₅₃	#529	1	1	yes	characteristic
220	C₄×C₅₃	#533	1	1	yes	characteristic
221	C₂×C₈×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
222	C₂×#405	C₂×C₅₃	1	1	yes	characteristic
223	#2396	C₂×C₅₃	1	1	yes	characteristic
224	#1985	C₂×C₅₃	1	1	yes	characteristic, complemented
225	#1940	C₂×C₅₃	1	1	yes	characteristic, complemented
226	#1917	C₂×C₅₃	1	2	yes	complemented
227	#1851	C₂×C₅₃	1	2	yes	—
228	#1798	C₂×C₅₃	1	1	yes	characteristic
229	C₈×C₄⋊C₄	C₂×C₅₃	1	2	yes	complemented
230	C₈×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
231	C₈×C₂²⋊C₄	C₂×C₅₃	1	2	yes	—
232	C₂×C₅₃	—	4	2	no	—
233	C₂×C₅₃	C₄×C₂²⋊Q₈	1	1	yes	characteristic
234	C₂×C₅₃	#2929	1	1	yes	characteristic
235	C₂×C₅₃	#2931	1	1	yes	characteristic
236	C₂×C₅₃	#2960	1	2	yes	—
237	C₂×C₅₃	#2966	1	2	yes	—
238	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
239	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
240	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
241	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
242	C₈×Q₈	—	2	4	no	—
243	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
244	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
245	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
246	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
247	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
248	#405	—	2	4	no	—
249	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
250	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
251	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
252	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
253	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
254	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
255	#400	C₄×C₅₃	1	1	yes	characteristic
256	#399	C₄×C₅₃	1	1	yes	characteristic
257	#398	C₄×C₅₃	1	2	yes	—
258	#396	C₄×C₅₃	1	2	yes	—
259	#395	C₄×C₅₃	1	1	yes	characteristic
260	#393	C₄×C₅₃	1	1	yes	characteristic
261	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
262	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
263	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
264	#336	C₂²×C₅₃	1	2	yes	—
265	#336	C₂²×C₅₃	1	2	yes	—
266	#336	C₂²×C₅₃	1	2	yes	—
267	#336	C₂²×C₅₃	1	2	yes	—
268	#336	C₂²×C₅₃	1	2	yes	—
269	C₅₃	#21601	1	1	yes	characteristic, direct product factor, Sylow
270	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
271	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
272	C₂×M₁₆	—	2	2	no	—
273	C₂×M₁₆	—	2	2	no	—
274	C₂×M₁₆	—	2	2	no	—
275	C₂²×C₈	—	2	2	no	—
276	C₂²×C₈	—	2	2	no	—
277	C₂²×C₈	Q₈×C₅₃	1	4	yes	complemented
278	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
279	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
280	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
281	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
282	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
283	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
284	C₄×Q₈	—	2	2	no	—
285	C₄×Q₈	—	2	2	no	—
286	#117	—	2	2	no	—
287	#117	—	2	2	no	—
288	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
289	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
291	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
292	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
293	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
294	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
295	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
296	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
297	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
298	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
299	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
300	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
301	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
302	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
303	C₄⋊C₈	—	2	4	no	—
304	C₄⋊C₈	—	2	4	no	—
305	C₄⋊C₈	—	2	2	no	—
306	C₄⋊C₈	—	2	2	no	—
307	C₂²⋊C₈	—	2	4	no	—
308	#97	—	2	2	no	—
309	#97	—	2	2	no	—
310	C₄×C₈	—	2	2	no	—
311	C₄×C₈	—	2	4	no	—
312	C₄×C₈	—	2	4	no	—
313	C₄×C₈	D₄×C₅₃	1	4	yes	—
314	#95	C₂×C₄×C₅₃	1	2	yes	—
315	#95	C₂×C₄×C₅₃	1	2	yes	—
316	#95	C₂×C₄×C₅₃	1	2	yes	—
317	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
318	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
319	#95	C₂×C₄×C₅₃	1	2	yes	—
320	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
321	C₂×Q₈	—	2	2	no	—
322	C₂×Q₈	—	2	4	no	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
328	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
329	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
330	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
331	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
332	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
333	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
334	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
335	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
336	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
337	C₂²×C₄	#41×C₅₃	1	2	yes	—
338	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
339	M₁₆	—	2	8	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	1	no	—
346	C₂×C₈	—	2	8	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
354	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
355	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
356	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
357	C₂×C₈	#41×C₅₃	1	2	yes	—
358	C₂×C₈	#41×C₅₃	1	2	yes	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	4	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	4	no	—
365	C₄⋊C₄	—	2	4	no	—
366	C₂²⋊C₄	—	2	2	no	—
367	C₂²⋊C₄	—	2	2	no	—
368	C₂²⋊C₄	—	2	4	no	—
369	C₂²⋊C₄	—	2	4	no	—
370	C₄²	—	2	1	no	—
371	C₄²	—	2	2	no	—
372	C₄²	—	2	2	no	—
373	C₄²	—	2	2	no	—
374	C₄²	—	2	1	no	—
375	C₄²	—	2	2	no	—
376	C₄²	C₂×D₄×C₅₃	1	2	yes	—
377	C₄²	#41×C₅₃	1	2	yes	—
378	C₂³	—	2	1	no	—
379	C₂³	—	2	1	no	—
380	C₂³	—	2	2	no	—
381	C₂³	—	2	2	no	—
382	C₂³	C₄×Q₈×C₅₃	1	2	yes	—
383	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
384	Q₈	—	2	8	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	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	2	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
408	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
409	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
410	C₂×C₄	#131×C₅₃	1	4	yes	—
411	C₂×C₄	#131×C₅₃	1	4	yes	—
412	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
413	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
415	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
416	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
417	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
418	C₈	—	2	2	no	—
419	C₈	—	2	2	no	—
420	C₈	—	2	2	no	—
421	C₈	—	4	2	no	—
422	C₈	—	4	2	no	—
423	C₈	C₂²⋊Q₈×C₅₃	1	4	yes	—
424	C₂²	—	2	4	no	—
425	C₂²	—	2	2	no	—
426	C₂²	—	4	2	no	—
427	C₂²	—	4	1	no	—
428	C₂²	—	4	1	no	—
429	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
430	C₂²	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
431	C₂²	C₄×#41×C₅₃	1	1	yes	characteristic
432	C₂²	C₂×#131×C₅₃	1	2	yes	—
433	C₂²	C₂×#131×C₅₃	1	2	yes	—
434	C₄	—	2	4	no	—
435	C₄	—	2	2	no	—
436	C₄	—	2	2	no	—
437	C₄	—	2	4	no	—
438	C₄	—	2	2	no	—
439	C₄	—	4	2	no	—
440	C₄	—	4	1	no	—
441	C₄	—	4	1	no	—
442	C₄	—	4	2	no	—
443	C₄	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
444	C₄	C₅₃×#525	1	1	yes	characteristic
445	C₄	C₅₃×#529	1	1	yes	characteristic
446	C₄	C₅₃×#533	1	1	yes	characteristic
447	C₂	—	4	2	no	—
448	C₂	C₄×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
449	C₂	C₅₃×#2929	1	1	yes	characteristic
450	C₂	C₅₃×#2931	1	1	yes	characteristic
451	C₂	C₅₃×#2960	1	2	yes	—
452	C₂	C₅₃×#2966	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₅₃	C₂³
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×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×Q₈×C₅₃	C₂	semidirect
4. C₅₃×#1985	C₂	semidirect
5. C₅₃×#1940	C₂	semidirect
6. C₅₃×#1917	C₂	semidirect
9. C₈×C₄⋊C₄×C₅₃	C₂	semidirect
10. C₈×C₄⋊C₄×C₅₃	C₂	semidirect
50. C₂²×C₈×C₅₃	Q₈	semidirect
197. #21601	C₅₃	direct (N × H)
221. C₂×C₈×Q₈	C₂×C₅₃	semidirect
224. #1985	C₂×C₅₃	semidirect
225. #1940	C₂×C₅₃	semidirect
226. #1917	C₂×C₅₃	semidirect
229. C₈×C₄⋊C₄	C₂×C₅₃	semidirect
230. C₈×C₄⋊C₄	C₂×C₅₃	semidirect
269. C₅₃	#21601	direct (N × H)
277. C₂²×C₈	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₅₃	2
C₂×C₅₃×#405	2
C₅₃×#2396	2
C₅₃×#1985	2
C₅₃×#1940	2
C₅₃×#1917	2
C₅₃×#1851	2
C₅₃×#1798	2
C₈×C₄⋊C₄×C₅₃	2
C₈×C₂²⋊C₄×C₅₃	2
#21601	53

Subgroups — C53×#21601 #27229113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21601 #27229113