Subgroups — C₅₃×#21816 #27229328

Order 13568.

1352 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#532	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×#1980	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1963	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1789	C₂	1	4	yes	maximal, complemented
7	C₂²×#41×C₅₃	C₂²	1	2	yes	—
8	C₅₃×#532	—	2	1	no	—
9	C₅₃×#532	—	2	1	no	—
10	C₅₃×#532	—	2	1	no	—
11	C₅₃×#532	—	2	1	no	—
12	C₅₃×#532	C₄	1	2	yes	complemented
13	C₅₃×#532	C₄	1	2	yes	complemented
14	C₅₃×#532	C₂²	1	2	yes	—
15	C₅₃×#532	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
17	C₂×#127×C₅₃	C₄	1	1	yes	characteristic, complemented
18	C₂×#124×C₅₃	C₄	1	2	yes	complemented
19	C₂×#121×C₅₃	—	2	1	no	—
20	C₂×#121×C₅₃	—	2	1	no	—
21	C₂×C₄×D₄×C₅₃	—	2	1	no	—
22	C₂×C₄×D₄×C₅₃	—	2	1	no	—
23	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#432	—	2	1	no	—
25	C₅₃×#432	—	2	1	no	—
26	C₅₃×#431	—	2	1	no	—
27	C₅₃×#431	—	2	1	no	—
28	C₅₃×#423	C₂²	1	2	yes	complemented
29	C₂×#104×C₅₃	C₂²	1	16	yes	complemented
30	C₅₃×#407	C₂²	1	2	yes	—
31	C₄×M₁₆×C₅₃	C₂²	1	2	yes	complemented
32	C₅₃×#383	—	2	1	no	—
33	C₅₃×#382	—	2	1	no	—
34	C₅₃×#381	—	2	1	no	—
35	C₅₃×#380	—	2	1	no	—
36	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
37	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
38	C₅₃×#376	C₂²	1	2	yes	complemented
39	C₂×#41×C₅₃	—	2	2	no	—
40	C₂×#41×C₅₃	—	2	2	no	—
41	C₂×#41×C₅₃	—	4	8	no	—
42	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
43	C₂×#41×C₅₃	C₂³	1	4	yes	—
44	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₂²×D₄×C₅₃	—	2	2	no	—
46	C₂²×D₄×C₅₃	—	2	1	no	—
47	C₂²×D₄×C₅₃	—	2	1	no	—
48	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₂³×C₄×C₅₃	—	2	2	no	—
50	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
54	C₄⋊Q₈×C₅₃	—	2	1	no	—
55	C₄⋊Q₈×C₅₃	—	2	1	no	—
56	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
57	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
58	#127×C₅₃	—	2	1	no	—
59	#127×C₅₃	—	2	1	no	—
60	#127×C₅₃	C₂×C₄	1	2	yes	—
61	#127×C₅₃	C₂×C₄	1	2	yes	—
62	#124×C₅₃	—	2	2	no	—
63	#124×C₅₃	—	2	2	no	—
64	#124×C₅₃	C₂×C₄	1	4	yes	—
65	#124×C₅₃	C₂×C₄	1	4	yes	—
66	#121×C₅₃	—	2	2	no	—
67	#121×C₅₃	—	2	2	no	—
68	#121×C₅₃	—	2	2	no	—
69	#121×C₅₃	—	2	2	no	—
70	#121×C₅₃	—	2	2	no	—
71	#121×C₅₃	—	2	2	no	—
72	#121×C₅₃	—	2	2	no	—
73	#121×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	—	2	2	no	—
79	C₄×D₄×C₅₃	—	2	2	no	—
80	C₄×D₄×C₅₃	—	2	2	no	—
81	C₄×D₄×C₅₃	—	2	2	no	—
82	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
88	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
89	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
93	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
96	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
99	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₈×C₅₃	—	2	2	no	—
102	C₄⋊C₈×C₅₃	—	2	2	no	—
103	#104×C₅₃	—	4	16	no	—
104	C₂²⋊C₈×C₅₃	—	2	2	no	—
105	C₂²⋊C₈×C₅₃	—	2	2	no	—
106	#97×C₅₃	—	2	2	no	—
107	C₄×C₈×C₅₃	—	2	2	no	—
108	#95×C₅₃	—	2	2	no	—
109	#95×C₅₃	—	2	2	no	—
110	#95×C₅₃	—	2	2	no	—
111	#95×C₅₃	—	2	2	no	—
112	C₂⁴×C₅₃	—	2	2	no	—
113	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
114	#41×C₅₃	—	2	16	no	—
115	#41×C₅₃	—	4	8	no	—
116	#41×C₅₃	—	4	8	no	—
117	#41×C₅₃	—	4	8	no	—
118	C₂×Q₈×C₅₃	—	2	2	no	—
119	C₂×Q₈×C₅₃	—	4	4	no	—
120	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
121	C₂×D₄×C₅₃	—	2	4	no	—
122	C₂×D₄×C₅₃	—	2	4	no	—
123	C₂×D₄×C₅₃	—	2	2	no	—
124	C₂×D₄×C₅₃	—	2	2	no	—
125	C₂×D₄×C₅₃	—	2	2	no	—
126	C₂×D₄×C₅₃	—	2	2	no	—
127	C₂×D₄×C₅₃	—	2	2	no	—
128	C₂×D₄×C₅₃	—	4	4	no	—
129	C₂×D₄×C₅₃	—	4	4	no	—
130	C₂×D₄×C₅₃	—	4	4	no	—
131	C₂×D₄×C₅₃	—	4	8	no	—
132	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
133	C₂²×C₄×C₅₃	—	2	4	no	—
134	C₂²×C₄×C₅₃	—	2	2	no	—
135	C₂²×C₄×C₅₃	—	2	4	no	—
136	C₂²×C₄×C₅₃	—	2	4	no	—
137	C₂²×C₄×C₅₃	—	2	4	no	—
138	C₂²×C₄×C₅₃	—	4	2	no	—
139	C₂²×C₄×C₅₃	—	4	2	no	—
140	C₂²×C₄×C₅₃	—	4	8	no	—
141	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
159	M₁₆×C₅₃	—	2	8	no	—
160	C₂×C₈×C₅₃	—	2	4	no	—
161	C₄⋊C₄×C₅₃	—	2	2	no	—
162	C₄⋊C₄×C₅₃	—	2	2	no	—
163	C₄⋊C₄×C₅₃	—	2	2	no	—
164	C₄⋊C₄×C₅₃	—	2	2	no	—
165	C₄⋊C₄×C₅₃	—	4	4	no	—
166	C₂²⋊C₄×C₅₃	—	2	4	no	—
167	C₂²⋊C₄×C₅₃	—	2	4	no	—
168	C₂²⋊C₄×C₅₃	—	2	4	no	—
169	C₂²⋊C₄×C₅₃	—	2	4	no	—
170	C₂²⋊C₄×C₅₃	—	4	4	no	—
171	C₄²×C₅₃	—	2	8	no	—
172	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
173	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
174	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
175	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
176	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
177	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
178	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
179	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
180	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
181	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
182	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
183	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
184	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
185	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
186	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
187	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
188	C₂³×C₅₃	—	2	2	no	—
189	C₂³×C₅₃	—	2	2	no	—
190	C₂³×C₅₃	—	2	4	no	—
191	C₂³×C₅₃	—	4	1	no	—
192	C₂³×C₅₃	—	4	4	no	—
193	C₂³×C₅₃	—	4	4	no	—
194	C₂³×C₅₃	—	4	4	no	—
195	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
196	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
198	Q₈×C₅₃	—	2	8	no	—
199	Q₈×C₅₃	—	4	4	no	—
200	D₄×C₅₃	—	2	8	no	—
201	D₄×C₅₃	—	4	4	no	—
202	D₄×C₅₃	—	4	4	no	—
203	D₄×C₅₃	—	4	4	no	—
204	D₄×C₅₃	—	4	4	no	—
205	D₄×C₅₃	—	4	8	no	—
206	D₄×C₅₃	—	4	8	no	—
207	D₄×C₅₃	—	4	4	no	—
208	C₂×C₄×C₅₃	—	2	2	no	—
209	C₂×C₄×C₅₃	—	2	2	no	—
210	C₂×C₄×C₅₃	—	2	2	no	—
211	C₂×C₄×C₅₃	—	2	2	no	—
212	C₂×C₄×C₅₃	—	2	2	no	—
213	C₂×C₄×C₅₃	—	2	2	no	—
214	C₂×C₄×C₅₃	—	2	2	no	—
215	C₂×C₄×C₅₃	—	2	2	no	—
216	C₂×C₄×C₅₃	—	2	2	no	—
217	C₂×C₄×C₅₃	—	2	2	no	—
218	C₂×C₄×C₅₃	—	2	4	no	—
219	C₂×C₄×C₅₃	—	4	1	no	—
220	C₂×C₄×C₅₃	—	4	1	no	—
221	C₂×C₄×C₅₃	—	4	1	no	—
222	C₂×C₄×C₅₃	—	4	1	no	—
223	C₂×C₄×C₅₃	—	4	1	no	—
224	C₂×C₄×C₅₃	—	4	1	no	—
225	C₂×C₄×C₅₃	—	4	1	no	—
226	C₂×C₄×C₅₃	—	4	8	no	—
227	C₂×C₄×C₅₃	—	4	4	no	—
228	C₂×C₄×C₅₃	—	4	8	no	—
229	C₂×C₄×C₅₃	—	4	4	no	—
230	C₂×C₄×C₅₃	—	4	4	no	—
231	C₂×C₄×C₅₃	—	4	4	no	—
232	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
233	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
235	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
238	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
239	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
240	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
241	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
242	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
243	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
245	C₈×C₅₃	—	4	4	no	—
246	#21816	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
247	C₂²×C₅₃	—	2	1	no	—
248	C₂²×C₅₃	—	2	1	no	—
249	C₂²×C₅₃	—	2	1	no	—
250	C₂²×C₅₃	—	2	1	no	—
251	C₂²×C₅₃	—	2	1	no	—
252	C₂²×C₅₃	—	2	1	no	—
253	C₂²×C₅₃	—	2	1	no	—
254	C₂²×C₅₃	—	4	4	no	—
255	C₂²×C₅₃	—	4	4	no	—
256	C₂²×C₅₃	—	4	4	no	—
257	C₂²×C₅₃	—	4	4	no	—
258	C₂²×C₅₃	—	8	8	no	—
259	C₂²×C₅₃	#421	1	2	yes	—
260	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
261	C₂²×C₅₃	#535	1	2	yes	—
262	C₄×C₅₃	—	2	1	no	—
263	C₄×C₅₃	—	2	1	no	—
264	C₄×C₅₃	—	2	1	no	—
265	C₄×C₅₃	—	2	1	no	—
266	C₄×C₅₃	—	2	1	no	—
267	C₄×C₅₃	—	2	1	no	—
268	C₄×C₅₃	—	2	1	no	—
269	C₄×C₅₃	—	2	1	no	—
270	C₄×C₅₃	—	2	1	no	—
271	C₄×C₅₃	—	4	4	no	—
272	C₄×C₅₃	—	8	4	no	—
273	C₄×C₅₃	#421	1	2	yes	—
274	C₄×C₅₃	#518	1	2	yes	—
275	C₄×C₅₃	#535	1	2	yes	—
276	C₂×#532	C₂×C₅₃	1	1	yes	characteristic
277	#2920	C₂×C₅₃	1	1	yes	characteristic, complemented
278	#2283	C₂×C₅₃	1	1	yes	characteristic, complemented
279	#1980	C₂×C₅₃	1	4	yes	complemented
280	#1963	C₂×C₅₃	1	4	yes	complemented
281	#1789	C₂×C₅₃	1	4	yes	complemented
282	C₂×C₅₃	—	2	2	no	—
283	C₂×C₅₃	—	2	2	no	—
284	C₂×C₅₃	—	4	1	no	—
285	C₂×C₅₃	—	8	4	no	—
286	C₂×C₅₃	#2309	1	1	yes	characteristic
287	C₂×C₅₃	C₂×#421	1	1	yes	characteristic
288	C₂×C₅₃	C₂×#421	1	1	yes	characteristic
289	C₂²×#41	C₂²×C₅₃	1	2	yes	—
290	#532	—	2	1	no	—
291	#532	—	2	1	no	—
292	#532	—	2	1	no	—
293	#532	—	2	1	no	—
294	#532	C₄×C₅₃	1	2	yes	complemented
295	#532	C₄×C₅₃	1	2	yes	complemented
296	#532	C₂²×C₅₃	1	2	yes	—
297	#532	C₂²×C₅₃	1	2	yes	—
298	C₂×C₄⋊Q₈	C₄×C₅₃	1	1	yes	characteristic, complemented
299	C₂×#127	C₄×C₅₃	1	1	yes	characteristic, complemented
300	C₂×#124	C₄×C₅₃	1	2	yes	complemented
301	C₂×#121	—	2	1	no	—
302	C₂×#121	—	2	1	no	—
303	C₂×C₄×D₄	—	2	1	no	—
304	C₂×C₄×D₄	—	2	1	no	—
305	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
306	#432	—	2	1	no	—
307	#432	—	2	1	no	—
308	#431	—	2	1	no	—
309	#431	—	2	1	no	—
310	#423	C₂²×C₅₃	1	2	yes	complemented
311	C₂×#104	C₂²×C₅₃	1	16	yes	complemented
312	#407	C₂²×C₅₃	1	2	yes	—
313	C₄×M₁₆	C₂²×C₅₃	1	2	yes	complemented
314	#383	—	2	1	no	—
315	#382	—	2	1	no	—
316	#381	—	2	1	no	—
317	#380	—	2	1	no	—
318	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
319	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
320	#376	C₂²×C₅₃	1	2	yes	complemented
321	C₅₃	#21816	1	1	yes	characteristic, direct product factor, Sylow
322	C₂×#41	—	2	2	no	—
323	C₂×#41	—	2	2	no	—
324	C₂×#41	—	4	8	no	—
325	C₂×#41	C₂×C₄×C₅₃	1	4	yes	—
326	C₂×#41	C₂³×C₅₃	1	4	yes	—
327	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
328	C₂²×D₄	—	2	2	no	—
329	C₂²×D₄	—	2	1	no	—
330	C₂²×D₄	—	2	1	no	—
331	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
332	C₂³×C₄	—	2	2	no	—
333	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
334	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
335	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
336	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
337	C₄⋊Q₈	—	2	1	no	—
338	C₄⋊Q₈	—	2	1	no	—
339	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
340	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
341	#127	—	2	1	no	—
342	#127	—	2	1	no	—
343	#127	C₂×C₄×C₅₃	1	2	yes	—
344	#127	C₂×C₄×C₅₃	1	2	yes	—
345	#124	—	2	2	no	—
346	#124	—	2	2	no	—
347	#124	C₂×C₄×C₅₃	1	4	yes	—
348	#124	C₂×C₄×C₅₃	1	4	yes	—
349	#121	—	2	2	no	—
350	#121	—	2	2	no	—
351	#121	—	2	2	no	—
352	#121	—	2	2	no	—
353	#121	—	2	2	no	—
354	#121	—	2	2	no	—
355	#121	—	2	2	no	—
356	#121	—	2	2	no	—
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	—	2	2	no	—
361	C₄×D₄	—	2	2	no	—
362	C₄×D₄	—	2	2	no	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	—	2	2	no	—
365	C₂×C₄⋊C₄	—	2	1	no	—
366	C₂×C₄⋊C₄	—	2	1	no	—
367	C₂×C₄⋊C₄	—	2	2	no	—
368	C₂×C₂²⋊C₄	—	2	2	no	—
369	C₂×C₂²⋊C₄	—	2	2	no	—
370	C₂×C₂²⋊C₄	—	2	2	no	—
371	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
372	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
373	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
374	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
375	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
376	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
377	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
378	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
379	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
380	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
381	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
382	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
383	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
384	C₄⋊C₈	—	2	2	no	—
385	C₄⋊C₈	—	2	2	no	—
386	#104	—	4	16	no	—
387	C₂²⋊C₈	—	2	2	no	—
388	C₂²⋊C₈	—	2	2	no	—
389	#97	—	2	2	no	—
390	C₄×C₈	—	2	2	no	—
391	#95	—	2	2	no	—
392	#95	—	2	2	no	—
393	#95	—	2	2	no	—
394	#95	—	2	2	no	—
395	C₂⁴	—	2	2	no	—
396	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
397	#41	—	2	16	no	—
398	#41	—	4	8	no	—
399	#41	—	4	8	no	—
400	#41	—	4	8	no	—
401	C₂×Q₈	—	2	2	no	—
402	C₂×Q₈	—	4	4	no	—
403	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	4	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	2	no	—
411	C₂×D₄	—	4	4	no	—
412	C₂×D₄	—	4	4	no	—
413	C₂×D₄	—	4	4	no	—
414	C₂×D₄	—	4	8	no	—
415	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	4	2	no	—
422	C₂²×C₄	—	4	2	no	—
423	C₂²×C₄	—	4	8	no	—
424	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
425	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
426	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
427	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
428	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
429	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
430	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
431	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
432	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
433	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
434	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
436	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
437	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
438	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
439	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
441	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
442	M₁₆	—	2	8	no	—
443	C₂×C₈	—	2	4	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	4	4	no	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	—	4	4	no	—
454	C₄²	—	2	8	no	—
455	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
456	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
457	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
458	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
459	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
460	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
461	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
462	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
463	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
464	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
465	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
466	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
467	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
468	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
469	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
470	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
471	C₂³	—	2	2	no	—
472	C₂³	—	2	2	no	—
473	C₂³	—	2	4	no	—
474	C₂³	—	4	1	no	—
475	C₂³	—	4	4	no	—
476	C₂³	—	4	4	no	—
477	C₂³	—	4	4	no	—
478	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
479	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
480	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
481	Q₈	—	2	8	no	—
482	Q₈	—	4	4	no	—
483	D₄	—	2	8	no	—
484	D₄	—	4	4	no	—
485	D₄	—	4	4	no	—
486	D₄	—	4	4	no	—
487	D₄	—	4	4	no	—
488	D₄	—	4	8	no	—
489	D₄	—	4	8	no	—
490	D₄	—	4	4	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	4	1	no	—
505	C₂×C₄	—	4	1	no	—
506	C₂×C₄	—	4	1	no	—
507	C₂×C₄	—	4	1	no	—
508	C₂×C₄	—	4	1	no	—
509	C₂×C₄	—	4	8	no	—
510	C₂×C₄	—	4	4	no	—
511	C₂×C₄	—	4	8	no	—
512	C₂×C₄	—	4	4	no	—
513	C₂×C₄	—	4	4	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
516	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
517	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
518	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
519	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
520	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
521	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
522	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
523	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
524	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
525	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
526	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
527	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
528	C₈	—	4	4	no	—
529	C₂²	—	2	1	no	—
530	C₂²	—	2	1	no	—
531	C₂²	—	2	1	no	—
532	C₂²	—	2	1	no	—
533	C₂²	—	2	1	no	—
534	C₂²	—	2	1	no	—
535	C₂²	—	2	1	no	—
536	C₂²	—	4	4	no	—
537	C₂²	—	4	4	no	—
538	C₂²	—	4	4	no	—
539	C₂²	—	4	4	no	—
540	C₂²	—	8	8	no	—
541	C₂²	C₅₃×#421	1	2	yes	—
542	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
543	C₂²	C₅₃×#535	1	2	yes	—
544	C₄	—	2	1	no	—
545	C₄	—	2	1	no	—
546	C₄	—	2	1	no	—
547	C₄	—	2	1	no	—
548	C₄	—	2	1	no	—
549	C₄	—	2	1	no	—
550	C₄	—	2	1	no	—
551	C₄	—	2	1	no	—
552	C₄	—	2	1	no	—
553	C₄	—	4	4	no	—
554	C₄	—	8	4	no	—
555	C₄	C₅₃×#421	1	2	yes	—
556	C₄	C₅₃×#518	1	2	yes	—
557	C₄	C₅₃×#535	1	2	yes	—
558	C₂	—	2	2	no	—
559	C₂	—	2	2	no	—
560	C₂	—	4	1	no	—
561	C₂	—	8	4	no	—
562	C₂	C₅₃×#2309	1	1	yes	characteristic
563	C₂	C₂×C₅₃×#421	1	1	yes	characteristic
564	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	C₂×C₅₃×#532	C₂
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×#1980	C₂	semidirect
5. C₅₃×#1963	C₂	semidirect
6. C₅₃×#1789	C₂	semidirect
12. C₅₃×#532	C₄	semidirect
13. C₅₃×#532	C₄	semidirect
16. C₂×C₄⋊Q₈×C₅₃	C₄	semidirect
17. C₂×#127×C₅₃	C₄	semidirect
18. C₂×#124×C₅₃	C₄	semidirect
28. C₅₃×#423	C₂²	semidirect
29. C₂×#104×C₅₃	C₂²	semidirect
31. C₄×M₁₆×C₅₃	C₂²	semidirect
36. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
38. C₅₃×#376	C₂²	semidirect
246. #21816	C₅₃	direct (N × H)
277. #2920	C₂×C₅₃	semidirect
278. #2283	C₂×C₅₃	semidirect
279. #1980	C₂×C₅₃	semidirect
280. #1963	C₂×C₅₃	semidirect
281. #1789	C₂×C₅₃	semidirect
294. #532	C₄×C₅₃	semidirect
295. #532	C₄×C₅₃	semidirect
298. C₂×C₄⋊Q₈	C₄×C₅₃	semidirect
299. C₂×#127	C₄×C₅₃	semidirect
300. C₂×#124	C₄×C₅₃	semidirect
310. #423	C₂²×C₅₃	semidirect
311. C₂×#104	C₂²×C₅₃	semidirect
313. C₄×M₁₆	C₂²×C₅₃	semidirect
318. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
320. #376	C₂²×C₅₃	semidirect
321. C₅₃	#21816	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×#532	2
C₅₃×#2920	2
C₅₃×#2283	2
C₅₃×#1980	2
C₅₃×#1963	2
C₅₃×#1789	2
#21816	53

Subgroups — C53×#21816 #27229328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21816 #27229328