Subgroups — C₅₃×#20748 #27228260

Order 13568.

1336 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1826	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1806	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1789	C₂	1	2	yes	maximal
7	C₅₃×#1759	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
9	C₄×#41×C₅₃	—	2	1	no	—
10	C₄×#41×C₅₃	—	2	1	no	—
11	C₄×#41×C₅₃	—	2	1	no	—
12	C₄×#41×C₅₃	—	2	1	no	—
13	C₄×#41×C₅₃	C₄	1	2	yes	complemented
14	C₄×#41×C₅₃	C₄	1	2	yes	complemented
15	C₄×#41×C₅₃	C₂²	1	2	yes	—
16	C₄×#41×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
18	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
19	C₂×C₄×D₄×C₅₃	—	2	1	no	—
20	C₂×C₄×D₄×C₅₃	—	2	1	no	—
21	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic, complemented
22	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic, complemented
23	C₂×#117×C₅₃	—	2	1	no	—
24	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄²×C₅₃	—	2	1	no	—
26	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#432	—	2	1	no	—
28	C₅₃×#432	—	2	1	no	—
29	C₅₃×#431	—	2	1	no	—
30	C₅₃×#431	—	2	1	no	—
31	C₅₃×#423	C₂²	1	2	yes	complemented
32	C₂×#104×C₅₃	C₂²	1	8	yes	—
33	C₅₃×#407	C₂²	1	2	yes	—
34	C₄×M₁₆×C₅₃	C₂²	1	2	yes	complemented
35	C₅₃×#383	—	2	1	no	—
36	C₅₃×#382	—	2	1	no	—
37	C₅₃×#381	—	2	1	no	—
38	C₅₃×#380	—	2	1	no	—
39	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
40	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
41	C₅₃×#376	C₂²	1	2	yes	complemented
42	C₅₃×#339	C₂²	1	8	yes	complemented
43	C₂×#41×C₅₃	—	2	8	no	—
44	C₂×#41×C₅₃	—	2	1	no	—
45	C₂×#41×C₅₃	—	2	1	no	—
46	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×#41×C₅₃	C₂³	1	2	yes	—
48	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×D₄×C₅₃	—	2	1	no	—
50	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₅₃	—	2	1	no	—
52	C₂³×C₄×C₅₃	—	2	1	no	—
53	C₂³×C₄×C₅₃	—	2	1	no	—
54	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
56	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
58	C₄×Q₈×C₅₃	—	2	1	no	—
59	C₄×Q₈×C₅₃	—	2	1	no	—
60	C₄×Q₈×C₅₃	—	2	1	no	—
61	C₄×Q₈×C₅₃	—	2	1	no	—
62	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
63	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
64	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
65	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
66	C₄×D₄×C₅₃	—	2	1	no	—
67	C₄×D₄×C₅₃	—	2	1	no	—
68	C₄×D₄×C₅₃	—	2	1	no	—
69	C₄×D₄×C₅₃	—	2	1	no	—
70	C₄×D₄×C₅₃	—	2	2	no	—
71	C₄×D₄×C₅₃	—	2	2	no	—
72	C₄×D₄×C₅₃	—	2	2	no	—
73	C₄×D₄×C₅₃	—	2	2	no	—
74	C₄×D₄×C₅₃	—	2	2	no	—
75	C₄×D₄×C₅₃	—	2	2	no	—
76	C₄×D₄×C₅₃	—	2	2	no	—
77	C₄×D₄×C₅₃	—	2	2	no	—
78	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
79	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
80	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
81	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
82	#117×C₅₃	—	2	2	no	—
83	#117×C₅₃	—	2	2	no	—
84	#117×C₅₃	—	2	2	no	—
85	#117×C₅₃	—	2	2	no	—
86	#117×C₅₃	—	2	1	no	—
87	#117×C₅₃	—	2	1	no	—
88	#117×C₅₃	C₂×C₄	1	2	yes	—
89	#117×C₅₃	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
91	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
92	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
95	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
97	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
99	C₂×C₄²×C₅₃	—	2	2	no	—
100	C₂×C₄²×C₅₃	—	2	2	no	—
101	C₂×C₄²×C₅₃	—	2	2	no	—
102	C₂×C₄²×C₅₃	—	2	2	no	—
103	C₂×C₄²×C₅₃	—	2	1	no	—
104	C₂×C₄²×C₅₃	—	2	1	no	—
105	C₂×C₄²×C₅₃	—	2	1	no	—
106	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
107	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
108	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
109	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
110	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
111	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
112	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
113	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
114	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
117	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
119	C₄⋊C₈×C₅₃	—	2	2	no	—
120	C₄⋊C₈×C₅₃	—	2	2	no	—
121	#104×C₅₃	—	2	16	no	—
122	C₂²⋊C₈×C₅₃	—	2	2	no	—
123	C₂²⋊C₈×C₅₃	—	2	2	no	—
124	#97×C₅₃	—	2	2	no	—
125	C₄×C₈×C₅₃	—	2	2	no	—
126	#95×C₅₃	—	2	2	no	—
127	#95×C₅₃	—	2	2	no	—
128	#95×C₅₃	—	2	2	no	—
129	#95×C₅₃	—	2	2	no	—
130	C₂⁴×C₅₃	—	2	1	no	—
131	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
132	#41×C₅₃	—	2	8	no	—
133	#41×C₅₃	—	2	8	no	—
134	#41×C₅₃	—	2	4	no	—
135	#41×C₅₃	—	2	4	no	—
136	#41×C₅₃	—	2	4	no	—
137	#41×C₅₃	#41	1	8	yes	—
138	C₂×Q₈×C₅₃	—	2	4	no	—
139	C₂×Q₈×C₅₃	—	2	1	no	—
140	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
141	C₂×D₄×C₅₃	—	2	8	no	—
142	C₂×D₄×C₅₃	—	2	2	no	—
143	C₂×D₄×C₅₃	—	2	2	no	—
144	C₂×D₄×C₅₃	—	2	4	no	—
145	C₂×D₄×C₅₃	—	2	1	no	—
146	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
147	C₂²×C₄×C₅₃	—	2	8	no	—
148	C₂²×C₄×C₅₃	—	2	1	no	—
149	C₂²×C₄×C₅₃	—	2	2	no	—
150	C₂²×C₄×C₅₃	—	2	2	no	—
151	C₂²×C₄×C₅₃	—	2	4	no	—
152	C₂²×C₄×C₅₃	—	2	1	no	—
153	C₂²×C₄×C₅₃	—	2	1	no	—
154	C₂²×C₄×C₅₃	—	2	2	no	—
155	C₂²×C₄×C₅₃	—	2	2	no	—
156	C₂²×C₄×C₅₃	—	2	2	no	—
157	C₂²×C₄×C₅₃	—	2	2	no	—
158	C₂²×C₄×C₅₃	—	2	4	no	—
159	C₂²×C₄×C₅₃	—	4	2	no	—
160	C₂²×C₄×C₅₃	—	4	2	no	—
161	C₂²×C₄×C₅₃	—	4	4	no	—
162	C₂²×C₄×C₅₃	—	4	4	no	—
163	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
164	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
165	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
170	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
171	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
172	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
173	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
181	M₁₆×C₅₃	—	2	8	no	—
182	C₂×C₈×C₅₃	—	2	4	no	—
183	C₄⋊C₄×C₅₃	—	2	2	no	—
184	C₄⋊C₄×C₅₃	—	2	2	no	—
185	C₄⋊C₄×C₅₃	—	2	2	no	—
186	C₄⋊C₄×C₅₃	—	2	2	no	—
187	C₄⋊C₄×C₅₃	—	2	1	no	—
188	C₄⋊C₄×C₅₃	—	2	1	no	—
189	C₄⋊C₄×C₅₃	—	4	4	no	—
190	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
191	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
192	C₂²⋊C₄×C₅₃	—	2	2	no	—
193	C₂²⋊C₄×C₅₃	—	2	2	no	—
194	C₂²⋊C₄×C₅₃	—	2	2	no	—
195	C₂²⋊C₄×C₅₃	—	2	2	no	—
196	C₂²⋊C₄×C₅₃	—	2	2	no	—
197	C₂²⋊C₄×C₅₃	—	2	2	no	—
198	C₂²⋊C₄×C₅₃	—	4	4	no	—
199	C₄²×C₅₃	—	2	2	no	—
200	C₄²×C₅₃	—	2	2	no	—
201	C₄²×C₅₃	—	2	2	no	—
202	C₄²×C₅₃	—	2	2	no	—
203	C₄²×C₅₃	—	2	2	no	—
204	C₄²×C₅₃	—	2	2	no	—
205	C₄²×C₅₃	—	2	8	no	—
206	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
207	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
208	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
209	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
210	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
211	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
212	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
213	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
214	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
215	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
216	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
217	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
218	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
219	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
221	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
222	C₂³×C₅₃	—	2	4	no	—
223	C₂³×C₅₃	—	2	2	no	—
224	C₂³×C₅₃	—	2	2	no	—
225	C₂³×C₅₃	—	2	2	no	—
226	C₂³×C₅₃	—	4	1	no	—
227	C₂³×C₅₃	—	4	2	no	—
228	C₂³×C₅₃	—	4	2	no	—
229	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
230	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
231	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
232	Q₈×C₅₃	—	2	4	no	—
233	Q₈×C₅₃	—	2	2	no	—
234	Q₈×C₅₃	#117	1	4	yes	—
235	D₄×C₅₃	—	2	8	no	—
236	D₄×C₅₃	—	2	4	no	—
237	D₄×C₅₃	—	2	8	no	—
238	D₄×C₅₃	—	2	2	no	—
239	D₄×C₅₃	#117	1	4	yes	—
240	C₂×C₄×C₅₃	—	2	4	no	—
241	C₂×C₄×C₅₃	—	2	8	no	—
242	C₂×C₄×C₅₃	—	2	2	no	—
243	C₂×C₄×C₅₃	—	2	2	no	—
244	C₂×C₄×C₅₃	—	2	2	no	—
245	C₂×C₄×C₅₃	—	2	2	no	—
246	C₂×C₄×C₅₃	—	2	2	no	—
247	C₂×C₄×C₅₃	—	2	2	no	—
248	C₂×C₄×C₅₃	—	2	2	no	—
249	C₂×C₄×C₅₃	—	2	8	no	—
250	C₂×C₄×C₅₃	—	2	2	no	—
251	C₂×C₄×C₅₃	—	2	2	no	—
252	C₂×C₄×C₅₃	—	2	2	no	—
253	C₂×C₄×C₅₃	—	2	2	no	—
254	C₂×C₄×C₅₃	—	2	2	no	—
255	C₂×C₄×C₅₃	—	2	2	no	—
256	C₂×C₄×C₅₃	—	4	1	no	—
257	C₂×C₄×C₅₃	—	4	1	no	—
258	C₂×C₄×C₅₃	—	4	1	no	—
259	C₂×C₄×C₅₃	—	4	1	no	—
260	C₂×C₄×C₅₃	—	4	1	no	—
261	C₂×C₄×C₅₃	—	4	1	no	—
262	C₂×C₄×C₅₃	—	4	1	no	—
263	C₂×C₄×C₅₃	—	4	4	no	—
264	C₂×C₄×C₅₃	—	4	4	no	—
265	C₂×C₄×C₅₃	—	4	4	no	—
266	C₂×C₄×C₅₃	—	4	2	no	—
267	C₂×C₄×C₅₃	—	4	2	no	—
268	C₂×C₄×C₅₃	—	4	4	no	—
269	C₂×C₄×C₅₃	—	4	4	no	—
270	C₂×C₄×C₅₃	—	4	4	no	—
271	C₂×C₄×C₅₃	—	4	4	no	—
272	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
274	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
277	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
278	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
280	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
281	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
282	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
283	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
285	C₈×C₅₃	—	4	4	no	—
286	#20748	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
287	C₂²×C₅₃	—	2	1	no	—
288	C₂²×C₅₃	—	2	1	no	—
289	C₂²×C₅₃	—	2	1	no	—
290	C₂²×C₅₃	—	2	1	no	—
291	C₂²×C₅₃	—	2	1	no	—
292	C₂²×C₅₃	—	2	1	no	—
293	C₂²×C₅₃	—	2	1	no	—
294	C₂²×C₅₃	—	2	4	no	—
295	C₂²×C₅₃	—	4	4	no	—
296	C₂²×C₅₃	—	4	8	no	—
297	C₂²×C₅₃	—	4	2	no	—
298	C₂²×C₅₃	—	4	2	no	—
299	C₂²×C₅₃	#421	1	2	yes	—
300	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
301	C₂²×C₅₃	#525	1	2	yes	—
302	C₄×C₅₃	—	2	1	no	—
303	C₄×C₅₃	—	2	1	no	—
304	C₄×C₅₃	—	2	1	no	—
305	C₄×C₅₃	—	2	1	no	—
306	C₄×C₅₃	—	2	1	no	—
307	C₄×C₅₃	—	2	1	no	—
308	C₄×C₅₃	—	2	1	no	—
309	C₄×C₅₃	—	2	1	no	—
310	C₄×C₅₃	—	2	4	no	—
311	C₄×C₅₃	—	2	1	no	—
312	C₄×C₅₃	—	4	2	no	—
313	C₄×C₅₃	—	4	2	no	—
314	C₄×C₅₃	—	8	4	no	—
315	C₄×C₅₃	#421	1	2	yes	—
316	C₄×C₅₃	C₂×#117	1	2	yes	—
317	C₄×C₅₃	#525	1	2	yes	—
318	C₂×C₄×#41	C₂×C₅₃	1	1	yes	characteristic
319	#2920	C₂×C₅₃	1	1	yes	characteristic, complemented
320	#2283	C₂×C₅₃	1	1	yes	characteristic, complemented
321	#1826	C₂×C₅₃	1	2	yes	complemented
322	#1806	C₂×C₅₃	1	4	yes	complemented
323	#1789	C₂×C₅₃	1	2	yes	—
324	#1759	C₂×C₅₃	1	4	yes	complemented
325	C₂×C₅₃	—	2	2	no	—
326	C₂×C₅₃	—	2	2	no	—
327	C₂×C₅₃	—	4	1	no	—
328	C₂×C₅₃	—	4	4	no	—
329	C₂×C₅₃	#2280	1	1	yes	characteristic
330	C₂×C₅₃	C₂×#421	1	1	yes	characteristic
331	C₂×C₅₃	C₂×#421	1	1	yes	characteristic
332	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
333	C₄×#41	—	2	1	no	—
334	C₄×#41	—	2	1	no	—
335	C₄×#41	—	2	1	no	—
336	C₄×#41	—	2	1	no	—
337	C₄×#41	C₄×C₅₃	1	2	yes	complemented
338	C₄×#41	C₄×C₅₃	1	2	yes	complemented
339	C₄×#41	C₂²×C₅₃	1	2	yes	—
340	C₄×#41	C₂²×C₅₃	1	2	yes	—
341	C₂×C₄×Q₈	C₄×C₅₃	1	1	yes	characteristic, complemented
342	C₂×C₄×Q₈	C₄×C₅₃	1	1	yes	characteristic, complemented
343	C₂×C₄×D₄	—	2	1	no	—
344	C₂×C₄×D₄	—	2	1	no	—
345	C₂×C₄×D₄	C₄×C₅₃	1	1	yes	characteristic, complemented
346	C₂×C₄×D₄	C₄×C₅₃	1	1	yes	characteristic, complemented
347	C₂×#117	—	2	1	no	—
348	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
349	C₂²×C₄²	—	2	1	no	—
350	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
351	#432	—	2	1	no	—
352	#432	—	2	1	no	—
353	#431	—	2	1	no	—
354	#431	—	2	1	no	—
355	#423	C₂²×C₅₃	1	2	yes	complemented
356	C₂×#104	C₂²×C₅₃	1	8	yes	—
357	#407	C₂²×C₅₃	1	2	yes	—
358	C₄×M₁₆	C₂²×C₅₃	1	2	yes	complemented
359	#383	—	2	1	no	—
360	#382	—	2	1	no	—
361	#381	—	2	1	no	—
362	#380	—	2	1	no	—
363	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
364	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
365	#376	C₂²×C₅₃	1	2	yes	complemented
366	#339	C₂²×C₅₃	1	8	yes	complemented
367	C₅₃	#20748	1	1	yes	characteristic, direct product factor, Sylow
368	C₂×#41	—	2	8	no	—
369	C₂×#41	—	2	1	no	—
370	C₂×#41	—	2	1	no	—
371	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
372	C₂×#41	C₂³×C₅₃	1	2	yes	—
373	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
374	C₂²×D₄	—	2	1	no	—
375	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
376	C₂³×C₄	—	2	1	no	—
377	C₂³×C₄	—	2	1	no	—
378	C₂³×C₄	—	2	1	no	—
379	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
380	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
381	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
382	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
383	C₄×Q₈	—	2	1	no	—
384	C₄×Q₈	—	2	1	no	—
385	C₄×Q₈	—	2	1	no	—
386	C₄×Q₈	—	2	1	no	—
387	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
388	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
389	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
390	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
391	C₄×D₄	—	2	2	no	—
392	C₄×D₄	—	2	2	no	—
393	C₄×D₄	—	2	2	no	—
394	C₄×D₄	—	2	2	no	—
395	C₄×D₄	—	2	2	no	—
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	—	2	1	no	—
400	C₄×D₄	—	2	1	no	—
401	C₄×D₄	—	2	1	no	—
402	C₄×D₄	—	2	1	no	—
403	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
404	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
405	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
406	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
407	#117	—	2	1	no	—
408	#117	—	2	1	no	—
409	#117	—	2	2	no	—
410	#117	—	2	2	no	—
411	#117	—	2	2	no	—
412	#117	—	2	2	no	—
413	#117	C₂×C₄×C₅₃	1	2	yes	—
414	#117	C₂³×C₅₃	1	2	yes	—
415	C₂×C₄⋊C₄	—	2	1	no	—
416	C₂×C₄⋊C₄	—	2	1	no	—
417	C₂×C₄⋊C₄	—	2	2	no	—
418	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
419	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
420	C₂×C₂²⋊C₄	—	2	1	no	—
421	C₂×C₂²⋊C₄	—	2	1	no	—
422	C₂×C₂²⋊C₄	—	2	1	no	—
423	C₂×C₂²⋊C₄	—	2	2	no	—
424	C₂×C₄²	—	2	1	no	—
425	C₂×C₄²	—	2	1	no	—
426	C₂×C₄²	—	2	1	no	—
427	C₂×C₄²	—	2	2	no	—
428	C₂×C₄²	—	2	2	no	—
429	C₂×C₄²	—	2	2	no	—
430	C₂×C₄²	—	2	2	no	—
431	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
432	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
433	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
434	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
435	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
436	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
437	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
438	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
439	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
440	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
441	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
442	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
443	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
444	C₄⋊C₈	—	2	2	no	—
445	C₄⋊C₈	—	2	2	no	—
446	#104	—	2	16	no	—
447	C₂²⋊C₈	—	2	2	no	—
448	C₂²⋊C₈	—	2	2	no	—
449	#97	—	2	2	no	—
450	C₄×C₈	—	2	2	no	—
451	#95	—	2	2	no	—
452	#95	—	2	2	no	—
453	#95	—	2	2	no	—
454	#95	—	2	2	no	—
455	C₂⁴	—	2	1	no	—
456	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
457	#41	—	2	8	no	—
458	#41	—	2	8	no	—
459	#41	—	2	4	no	—
460	#41	—	2	4	no	—
461	#41	—	2	4	no	—
462	#41	#41×C₅₃	1	8	yes	—
463	C₂×Q₈	—	2	4	no	—
464	C₂×Q₈	—	2	1	no	—
465	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
466	C₂×D₄	—	2	8	no	—
467	C₂×D₄	—	2	2	no	—
468	C₂×D₄	—	2	2	no	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
472	C₂²×C₄	—	2	8	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	2	no	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	2	4	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	4	2	no	—
485	C₂²×C₄	—	4	2	no	—
486	C₂²×C₄	—	4	4	no	—
487	C₂²×C₄	—	4	4	no	—
488	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
489	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
490	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
491	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
492	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
493	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
494	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
495	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
496	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
497	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
498	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
499	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
500	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
501	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
502	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
503	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
504	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
505	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
506	M₁₆	—	2	8	no	—
507	C₂×C₈	—	2	4	no	—
508	C₄⋊C₄	—	2	2	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	2	1	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	4	4	no	—
515	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
516	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	2	2	no	—
519	C₂²⋊C₄	—	2	2	no	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	2	2	no	—
523	C₂²⋊C₄	—	4	4	no	—
524	C₄²	—	2	2	no	—
525	C₄²	—	2	2	no	—
526	C₄²	—	2	2	no	—
527	C₄²	—	2	2	no	—
528	C₄²	—	2	2	no	—
529	C₄²	—	2	2	no	—
530	C₄²	—	2	8	no	—
531	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
532	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
533	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
534	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
535	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
536	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
537	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
538	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
539	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
540	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
541	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
542	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
543	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
544	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
545	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
546	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
547	C₂³	—	2	4	no	—
548	C₂³	—	2	2	no	—
549	C₂³	—	2	2	no	—
550	C₂³	—	2	2	no	—
551	C₂³	—	4	1	no	—
552	C₂³	—	4	2	no	—
553	C₂³	—	4	2	no	—
554	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
555	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
556	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
557	Q₈	—	2	4	no	—
558	Q₈	—	2	2	no	—
559	Q₈	#117×C₅₃	1	4	yes	—
560	D₄	—	2	8	no	—
561	D₄	—	2	4	no	—
562	D₄	—	2	8	no	—
563	D₄	—	2	2	no	—
564	D₄	#117×C₅₃	1	4	yes	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	8	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	8	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	4	no	—
589	C₂×C₄	—	4	4	no	—
590	C₂×C₄	—	4	4	no	—
591	C₂×C₄	—	4	2	no	—
592	C₂×C₄	—	4	2	no	—
593	C₂×C₄	—	4	4	no	—
594	C₂×C₄	—	4	4	no	—
595	C₂×C₄	—	4	4	no	—
596	C₂×C₄	—	4	4	no	—
597	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
598	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
599	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
600	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
601	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
602	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
603	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
604	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
605	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
606	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
607	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
608	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
609	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
610	C₈	—	4	4	no	—
611	C₂²	—	2	1	no	—
612	C₂²	—	2	1	no	—
613	C₂²	—	2	1	no	—
614	C₂²	—	2	1	no	—
615	C₂²	—	2	1	no	—
616	C₂²	—	2	4	no	—
617	C₂²	—	2	1	no	—
618	C₂²	—	2	1	no	—
619	C₂²	—	4	4	no	—
620	C₂²	—	4	8	no	—
621	C₂²	—	4	2	no	—
622	C₂²	—	4	2	no	—
623	C₂²	C₅₃×#421	1	2	yes	—
624	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
625	C₂²	C₅₃×#525	1	2	yes	—
626	C₄	—	2	1	no	—
627	C₄	—	2	4	no	—
628	C₄	—	2	1	no	—
629	C₄	—	2	1	no	—
630	C₄	—	2	1	no	—
631	C₄	—	2	1	no	—
632	C₄	—	2	1	no	—
633	C₄	—	2	1	no	—
634	C₄	—	2	1	no	—
635	C₄	—	2	1	no	—
636	C₄	—	4	2	no	—
637	C₄	—	4	2	no	—
638	C₄	—	8	4	no	—
639	C₄	C₅₃×#421	1	2	yes	—
640	C₄	C₂×#117×C₅₃	1	2	yes	—
641	C₄	C₅₃×#525	1	2	yes	—
642	C₂	—	2	2	no	—
643	C₂	—	2	2	no	—
644	C₂	—	4	1	no	—
645	C₂	—	4	4	no	—
646	C₂	C₅₃×#2280	1	1	yes	characteristic
647	C₂	C₂×C₅₃×#421	1	1	yes	characteristic
648	C₂	C₂×C₅₃×#421	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	C₂²×C₂²⋊C₄
prime radical	C₂²×#41×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2920	C₂	semidirect
3. C₅₃×#2283	C₂	semidirect
4. C₅₃×#1826	C₂	semidirect
5. C₅₃×#1806	C₂	semidirect
7. C₅₃×#1759	C₂	semidirect
13. C₄×#41×C₅₃	C₄	semidirect
14. C₄×#41×C₅₃	C₄	semidirect
17. C₂×C₄×Q₈×C₅₃	C₄	semidirect
18. C₂×C₄×Q₈×C₅₃	C₄	semidirect
21. C₂×C₄×D₄×C₅₃	C₄	semidirect
22. C₂×C₄×D₄×C₅₃	C₄	semidirect
31. C₅₃×#423	C₂²	semidirect
34. C₄×M₁₆×C₅₃	C₂²	semidirect
39. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
41. C₅₃×#376	C₂²	semidirect
42. C₅₃×#339	C₂²	semidirect
286. #20748	C₅₃	direct (N × H)
319. #2920	C₂×C₅₃	semidirect
320. #2283	C₂×C₅₃	semidirect
321. #1826	C₂×C₅₃	semidirect
322. #1806	C₂×C₅₃	semidirect
324. #1759	C₂×C₅₃	semidirect
337. C₄×#41	C₄×C₅₃	semidirect
338. C₄×#41	C₄×C₅₃	semidirect
341. C₂×C₄×Q₈	C₄×C₅₃	semidirect
342. C₂×C₄×Q₈	C₄×C₅₃	semidirect
345. C₂×C₄×D₄	C₄×C₅₃	semidirect
346. C₂×C₄×D₄	C₄×C₅₃	semidirect
355. #423	C₂²×C₅₃	semidirect
358. C₄×M₁₆	C₂²×C₅₃	semidirect
363. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
365. #376	C₂²×C₅₃	semidirect
366. #339	C₂²×C₅₃	semidirect
367. C₅₃	#20748	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×C₄×#41×C₅₃	2
C₅₃×#2920	2
C₅₃×#2283	2
C₅₃×#1826	2
C₅₃×#1806	2
C₅₃×#1789	2
C₅₃×#1759	2
#20748	53

Subgroups — C53×#20748 #27228260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#20748 #27228260