Subgroups — C₅₃×#34982 #27242494

Order 13568.

4046 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#543	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2848	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2626	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2437	C₂	1	16	yes	maximal, complemented
6	C₅₃×#2315	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2315	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#379	C₂	1	2	yes	maximal
9	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#543	—	2	8	no	—
12	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	8	yes	complemented
13	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
14	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	8	yes	complemented
15	C₂²×C₂²⋊C₄×C₅₃	—	2	8	no	—
16	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
18	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
22	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#399	C₂²	1	8	yes	complemented
24	C₅₃×#396	C₂²	1	16	yes	complemented
25	C₅₃×#393	C₂²	1	8	yes	complemented
26	C₅₃×#385	C₂²	1	8	yes	complemented
27	C₅₃×#385	C₂²	1	8	yes	complemented
28	C₅₃×#385	C₂²	1	16	yes	complemented
29	C₅₃×#385	C₂²	1	8	yes	complemented
30	C₅₃×#380	C₂²	1	8	yes	complemented
31	C₅₃×#380	C₂²	1	8	yes	complemented
32	C₅₃×#380	C₂²	1	4	yes	complemented
33	C₅₃×#379	C₂²	1	8	yes	—
34	C₅₃×#379	C₂²	1	4	yes	—
35	C₅₃×#379	C₂²	1	4	yes	—
36	C₂⁵×C₅₃	—	2	4	no	—
37	C₂⁵×C₅₃	—	2	4	no	—
38	C₂⁵×C₅₃	—	4	8	no	—
39	C₂⁵×C₅₃	Q₈	1	8	yes	complemented
40	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
41	C₂⁵×C₅₃	C₂³	1	4	yes	—
42	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
43	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×Q₈×C₅₃	C₂³	1	4	yes	complemented
45	C₂³×C₄×C₅₃	—	2	8	no	—
46	C₂³×C₄×C₅₃	—	4	8	no	—
47	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
48	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
49	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
50	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
51	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
52	C₂²⋊Q₈×C₅₃	—	2	32	no	—
53	C₂²⋊Q₈×C₅₃	—	2	16	no	—
54	C₂²⋊Q₈×C₅₃	—	4	16	no	—
55	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	complemented
56	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	complemented
57	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	complemented
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	complemented
59	C₂×C₂²⋊C₄×C₅₃	—	2	32	no	—
60	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
61	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
62	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
63	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
64	C₂×C₂²⋊C₄×C₅₃	—	4	16	no	—
65	C₂×C₂²⋊C₄×C₅₃	D₄	1	8	yes	—
66	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
67	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
68	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
69	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
70	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
71	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
72	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
73	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
74	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
75	#95×C₅₃	C₂³	1	16	yes	complemented
76	#95×C₅₃	C₂³	1	8	yes	complemented
77	C₂⁴×C₅₃	—	2	2	no	—
78	C₂⁴×C₅₃	—	2	2	no	—
79	C₂⁴×C₅₃	—	2	8	no	—
80	C₂⁴×C₅₃	—	2	32	no	—
81	C₂⁴×C₅₃	—	2	8	no	—
82	C₂⁴×C₅₃	—	2	2	no	—
83	C₂⁴×C₅₃	—	2	8	no	—
84	C₂⁴×C₅₃	—	2	2	no	—
85	C₂⁴×C₅₃	—	2	16	no	—
86	C₂⁴×C₅₃	—	4	32	no	—
87	C₂⁴×C₅₃	—	4	4	no	—
88	C₂⁴×C₅₃	—	4	8	no	—
89	C₂⁴×C₅₃	—	4	4	no	—
90	C₂⁴×C₅₃	—	4	32	no	—
91	C₂⁴×C₅₃	—	4	32	no	—
92	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
93	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
94	C₂⁴×C₅₃	C₂×Q₈	1	16	yes	—
95	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
96	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
97	C₂⁴×C₅₃	#41	1	4	yes	—
98	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
99	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
101	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
102	C₂×Q₈×C₅₃	—	2	8	no	—
103	C₂×Q₈×C₅₃	—	4	8	no	—
104	C₂²×C₄×C₅₃	—	2	4	no	—
105	C₂²×C₄×C₅₃	—	2	16	no	—
106	C₂²×C₄×C₅₃	—	2	8	no	—
107	C₂²×C₄×C₅₃	—	2	8	no	—
108	C₂²×C₄×C₅₃	—	2	16	no	—
109	C₂²×C₄×C₅₃	—	4	4	no	—
110	C₂²×C₄×C₅₃	—	4	8	no	—
111	C₂²×C₄×C₅₃	—	4	4	no	—
112	C₂²×C₄×C₅₃	—	4	8	no	—
113	C₂²×C₄×C₅₃	—	4	8	no	—
114	C₂²×C₄×C₅₃	—	4	32	no	—
115	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
116	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
117	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
118	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
119	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
120	C₄⋊C₄×C₅₃	—	2	8	no	—
121	C₄⋊C₄×C₅₃	—	2	16	no	—
122	C₄⋊C₄×C₅₃	—	4	8	no	—
123	C₄⋊C₄×C₅₃	—	4	16	no	—
124	C₂²⋊C₄×C₅₃	—	2	8	no	—
125	C₂²⋊C₄×C₅₃	—	2	8	no	—
126	C₂²⋊C₄×C₅₃	—	2	8	no	—
127	C₂²⋊C₄×C₅₃	—	2	8	no	—
128	C₂²⋊C₄×C₅₃	—	2	8	no	—
129	C₂²⋊C₄×C₅₃	—	2	32	no	—
130	C₂²⋊C₄×C₅₃	—	2	16	no	—
131	C₂²⋊C₄×C₅₃	—	4	16	no	—
132	C₂²⋊C₄×C₅₃	C₂×D₄	1	16	yes	—
133	C₂³×C₅₃	—	2	16	no	—
134	C₂³×C₅₃	—	2	16	no	—
135	C₂³×C₅₃	—	2	8	no	—
136	C₂³×C₅₃	—	2	16	no	—
137	C₂³×C₅₃	—	2	2	no	—
138	C₂³×C₅₃	—	2	2	no	—
139	C₂³×C₅₃	—	2	4	no	—
140	C₂³×C₅₃	—	2	4	no	—
141	C₂³×C₅₃	—	2	8	no	—
142	C₂³×C₅₃	—	2	32	no	—
143	C₂³×C₅₃	—	2	8	no	—
144	C₂³×C₅₃	—	2	32	no	—
145	C₂³×C₅₃	—	2	2	no	—
146	C₂³×C₅₃	—	2	2	no	—
147	C₂³×C₅₃	—	2	8	no	—
148	C₂³×C₅₃	—	4	8	no	—
149	C₂³×C₅₃	—	4	8	no	—
150	C₂³×C₅₃	—	4	16	no	—
151	C₂³×C₅₃	—	4	8	no	—
152	C₂³×C₅₃	—	4	8	no	—
153	C₂³×C₅₃	—	4	32	no	—
154	C₂³×C₅₃	—	4	32	no	—
155	C₂³×C₅₃	—	4	128	no	—
156	C₂³×C₅₃	—	4	8	no	—
157	C₂³×C₅₃	—	4	8	no	—
158	C₂³×C₅₃	C₂²⋊Q₈	1	16	yes	complemented
159	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
160	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
161	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
162	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
163	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
164	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
165	C₂³×C₅₃	C₂×#41	1	4	yes	—
166	C₂³×C₅₃	C₂×#41	1	2	yes	—
167	C₂³×C₅₃	H(5,𝔽₂)	1	8	yes	—
168	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
169	Q₈×C₅₃	—	8	8	no	—
170	C₂×C₄×C₅₃	—	2	8	no	—
171	C₂×C₄×C₅₃	—	2	4	no	—
172	C₂×C₄×C₅₃	—	2	8	no	—
173	C₂×C₄×C₅₃	—	2	8	no	—
174	C₂×C₄×C₅₃	—	2	8	no	—
175	C₂×C₄×C₅₃	—	2	16	no	—
176	C₂×C₄×C₅₃	—	2	16	no	—
177	C₂×C₄×C₅₃	—	4	8	no	—
178	C₂×C₄×C₅₃	—	4	32	no	—
179	C₂×C₄×C₅₃	—	4	8	no	—
180	C₂×C₄×C₅₃	—	8	8	no	—
181	C₂×C₄×C₅₃	—	8	8	no	—
182	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
183	#34982	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂²×C₅₃	—	2	32	no	—
185	C₂²×C₅₃	—	2	4	no	—
186	C₂²×C₅₃	—	2	4	no	—
187	C₂²×C₅₃	—	2	4	no	—
188	C₂²×C₅₃	—	2	8	no	—
189	C₂²×C₅₃	—	2	4	no	—
190	C₂²×C₅₃	—	2	8	no	—
191	C₂²×C₅₃	—	2	8	no	—
192	C₂²×C₅₃	—	2	8	no	—
193	C₂²×C₅₃	—	2	4	no	—
194	C₂²×C₅₃	—	2	8	no	—
195	C₂²×C₅₃	—	2	4	no	—
196	C₂²×C₅₃	—	2	8	no	—
197	C₂²×C₅₃	—	4	16	no	—
198	C₂²×C₅₃	—	4	64	no	—
199	C₂²×C₅₃	—	4	16	no	—
200	C₂²×C₅₃	—	4	4	no	—
201	C₂²×C₅₃	—	4	4	no	—
202	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	4	yes	—
203	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	8	yes	—
204	C₂²×C₅₃	#543	1	8	yes	complemented
205	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
206	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
207	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
208	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
209	C₄×C₅₃	—	2	8	no	—
210	C₄×C₅₃	—	4	4	no	—
211	C₄×C₅₃	—	4	8	no	—
212	C₄×C₅₃	—	4	8	no	—
213	C₂×#543	C₂×C₅₃	1	1	yes	characteristic, complemented
214	C₂³×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
215	#2848	C₂×C₅₃	1	4	yes	complemented
216	#2626	C₂×C₅₃	1	4	yes	complemented
217	#2437	C₂×C₅₃	1	16	yes	complemented
218	#2315	C₂×C₅₃	1	2	yes	complemented
219	#2315	C₂×C₅₃	1	1	yes	characteristic, complemented
220	C₂×#379	C₂×C₅₃	1	2	yes	—
221	C₂×C₅₃	—	2	4	no	—
222	C₂×C₅₃	—	2	8	no	—
223	C₂×C₅₃	—	2	8	no	—
224	C₂×C₅₃	—	4	4	no	—
225	C₂×C₅₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
226	C₂×C₅₃	C₂×#543	1	2	yes	—
227	C₂×C₅₃	#3484	1	2	yes	—
228	C₂×C₅₃	#3490	1	2	yes	—
229	C₂⁶	C₂²×C₅₃	1	1	yes	characteristic
230	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
231	#543	—	2	8	no	—
232	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
233	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	8	yes	complemented
234	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	8	yes	complemented
235	C₂²×C₂²⋊C₄	—	2	8	no	—
236	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
237	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
238	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
239	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
240	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
241	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
242	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
243	#399	C₂²×C₅₃	1	8	yes	complemented
244	#396	C₂²×C₅₃	1	16	yes	complemented
245	#393	C₂²×C₅₃	1	8	yes	complemented
246	#385	C₂²×C₅₃	1	8	yes	complemented
247	#385	C₂²×C₅₃	1	8	yes	complemented
248	#385	C₂²×C₅₃	1	8	yes	complemented
249	#385	C₂²×C₅₃	1	16	yes	complemented
250	#380	C₂²×C₅₃	1	4	yes	complemented
251	#380	C₂²×C₅₃	1	8	yes	complemented
252	#380	C₂²×C₅₃	1	8	yes	complemented
253	#379	C₂²×C₅₃	1	4	yes	—
254	#379	C₂²×C₅₃	1	4	yes	—
255	#379	C₂²×C₅₃	1	8	yes	—
256	C₅₃	#34982	1	1	yes	characteristic, direct product factor, Sylow
257	C₂⁵	—	2	4	no	—
258	C₂⁵	—	2	4	no	—
259	C₂⁵	—	4	8	no	—
260	C₂⁵	Q₈×C₅₃	1	8	yes	complemented
261	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
262	C₂⁵	C₂³×C₅₃	1	4	yes	—
263	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
264	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
265	C₂²×Q₈	C₂³×C₅₃	1	4	yes	complemented
266	C₂³×C₄	—	2	8	no	—
267	C₂³×C₄	—	4	8	no	—
268	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
269	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
270	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
271	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
272	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
273	C₂²⋊Q₈	—	2	16	no	—
274	C₂²⋊Q₈	—	2	32	no	—
275	C₂²⋊Q₈	—	4	16	no	—
276	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	complemented
277	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	complemented
278	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	complemented
279	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	complemented
280	C₂×C₂²⋊C₄	—	2	4	no	—
281	C₂×C₂²⋊C₄	—	2	8	no	—
282	C₂×C₂²⋊C₄	—	2	8	no	—
283	C₂×C₂²⋊C₄	—	2	8	no	—
284	C₂×C₂²⋊C₄	—	2	32	no	—
285	C₂×C₂²⋊C₄	—	4	16	no	—
286	C₂×C₂²⋊C₄	D₄×C₅₃	1	8	yes	—
287	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
288	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
289	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
290	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
291	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
292	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
293	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
294	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
295	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
296	#95	C₂³×C₅₃	1	16	yes	complemented
297	#95	C₂³×C₅₃	1	8	yes	complemented
298	C₂⁴	—	2	2	no	—
299	C₂⁴	—	2	2	no	—
300	C₂⁴	—	2	8	no	—
301	C₂⁴	—	2	32	no	—
302	C₂⁴	—	2	8	no	—
303	C₂⁴	—	2	2	no	—
304	C₂⁴	—	2	8	no	—
305	C₂⁴	—	2	2	no	—
306	C₂⁴	—	2	16	no	—
307	C₂⁴	—	4	32	no	—
308	C₂⁴	—	4	4	no	—
309	C₂⁴	—	4	8	no	—
310	C₂⁴	—	4	4	no	—
311	C₂⁴	—	4	32	no	—
312	C₂⁴	—	4	32	no	—
313	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
314	C₂⁴	C₂×Q₈×C₅₃	1	4	yes	—
315	C₂⁴	C₂×Q₈×C₅₃	1	16	yes	—
316	C₂⁴	C₂×Q₈×C₅₃	1	4	yes	—
317	C₂⁴	C₂×Q₈×C₅₃	1	4	yes	—
318	C₂⁴	#41×C₅₃	1	4	yes	—
319	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
320	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
321	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
322	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
323	C₂×Q₈	—	2	8	no	—
324	C₂×Q₈	—	4	8	no	—
325	C₂²×C₄	—	2	4	no	—
326	C₂²×C₄	—	2	16	no	—
327	C₂²×C₄	—	2	8	no	—
328	C₂²×C₄	—	2	8	no	—
329	C₂²×C₄	—	2	16	no	—
330	C₂²×C₄	—	4	4	no	—
331	C₂²×C₄	—	4	8	no	—
332	C₂²×C₄	—	4	4	no	—
333	C₂²×C₄	—	4	8	no	—
334	C₂²×C₄	—	4	8	no	—
335	C₂²×C₄	—	4	32	no	—
336	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
337	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
338	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
339	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
340	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
341	C₄⋊C₄	—	2	8	no	—
342	C₄⋊C₄	—	2	16	no	—
343	C₄⋊C₄	—	4	8	no	—
344	C₄⋊C₄	—	4	16	no	—
345	C₂²⋊C₄	—	2	8	no	—
346	C₂²⋊C₄	—	2	8	no	—
347	C₂²⋊C₄	—	2	8	no	—
348	C₂²⋊C₄	—	2	8	no	—
349	C₂²⋊C₄	—	2	8	no	—
350	C₂²⋊C₄	—	2	32	no	—
351	C₂²⋊C₄	—	2	16	no	—
352	C₂²⋊C₄	—	4	16	no	—
353	C₂²⋊C₄	C₂×D₄×C₅₃	1	16	yes	—
354	C₂³	—	2	16	no	—
355	C₂³	—	2	16	no	—
356	C₂³	—	2	8	no	—
357	C₂³	—	2	16	no	—
358	C₂³	—	2	2	no	—
359	C₂³	—	2	2	no	—
360	C₂³	—	2	4	no	—
361	C₂³	—	2	4	no	—
362	C₂³	—	2	8	no	—
363	C₂³	—	2	32	no	—
364	C₂³	—	2	8	no	—
365	C₂³	—	2	32	no	—
366	C₂³	—	2	2	no	—
367	C₂³	—	2	2	no	—
368	C₂³	—	2	8	no	—
369	C₂³	—	4	8	no	—
370	C₂³	—	4	8	no	—
371	C₂³	—	4	16	no	—
372	C₂³	—	4	8	no	—
373	C₂³	—	4	8	no	—
374	C₂³	—	4	32	no	—
375	C₂³	—	4	32	no	—
376	C₂³	—	4	128	no	—
377	C₂³	—	4	8	no	—
378	C₂³	—	4	8	no	—
379	C₂³	C₂²⋊Q₈×C₅₃	1	16	yes	complemented
380	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
381	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
382	C₂³	C₂²×Q₈×C₅₃	1	4	yes	—
383	C₂³	C₂²×Q₈×C₅₃	1	2	yes	—
384	C₂³	C₂²×Q₈×C₅₃	1	4	yes	—
385	C₂³	C₂²×Q₈×C₅₃	1	4	yes	—
386	C₂³	C₂×#41×C₅₃	1	2	yes	—
387	C₂³	C₂×#41×C₅₃	1	4	yes	—
388	C₂³	H(5,𝔽₂)×C₅₃	1	8	yes	—
389	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
390	Q₈	—	8	8	no	—
391	C₂×C₄	—	2	8	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	2	8	no	—
394	C₂×C₄	—	2	8	no	—
395	C₂×C₄	—	2	16	no	—
396	C₂×C₄	—	2	16	no	—
397	C₂×C₄	—	2	8	no	—
398	C₂×C₄	—	4	8	no	—
399	C₂×C₄	—	4	32	no	—
400	C₂×C₄	—	4	8	no	—
401	C₂×C₄	—	8	8	no	—
402	C₂×C₄	—	8	8	no	—
403	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
404	C₂²	—	2	32	no	—
405	C₂²	—	2	4	no	—
406	C₂²	—	2	4	no	—
407	C₂²	—	2	8	no	—
408	C₂²	—	2	4	no	—
409	C₂²	—	2	8	no	—
410	C₂²	—	2	8	no	—
411	C₂²	—	2	8	no	—
412	C₂²	—	2	4	no	—
413	C₂²	—	2	8	no	—
414	C₂²	—	2	4	no	—
415	C₂²	—	2	8	no	—
416	C₂²	—	2	4	no	—
417	C₂²	—	4	16	no	—
418	C₂²	—	4	64	no	—
419	C₂²	—	4	16	no	—
420	C₂²	—	4	4	no	—
421	C₂²	—	4	4	no	—
422	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	4	yes	—
423	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	8	yes	—
424	C₂²	C₅₃×#543	1	8	yes	complemented
425	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
426	C₂²	C₂³×Q₈×C₅₃	1	1	yes	characteristic
427	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
428	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	4	yes	—
429	C₄	—	2	8	no	—
430	C₄	—	4	4	no	—
431	C₄	—	4	8	no	—
432	C₄	—	4	8	no	—
433	C₂	—	2	4	no	—
434	C₂	—	2	8	no	—
435	C₂	—	2	8	no	—
436	C₂	—	4	4	no	—
437	C₂	C₂²×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
438	C₂	C₂×C₅₃×#543	1	2	yes	—
439	C₂	C₅₃×#3484	1	2	yes	—
440	C₂	C₅₃×#3490	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂⁶×C₅₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#543	C₂	semidirect
3. C₅₃×#2848	C₂	semidirect
4. C₅₃×#2626	C₂	semidirect
5. C₅₃×#2437	C₂	semidirect
6. C₅₃×#2315	C₂	semidirect
7. C₅₃×#2315	C₂	semidirect
12. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
13. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
14. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
23. C₅₃×#399	C₂²	semidirect
24. C₅₃×#396	C₂²	semidirect
25. C₅₃×#393	C₂²	semidirect
26. C₅₃×#385	C₂²	semidirect
27. C₅₃×#385	C₂²	semidirect
28. C₅₃×#385	C₂²	semidirect
29. C₅₃×#385	C₂²	semidirect
30. C₅₃×#380	C₂²	semidirect
31. C₅₃×#380	C₂²	semidirect
32. C₅₃×#380	C₂²	semidirect
39. C₂⁵×C₅₃	Q₈	semidirect
44. C₂²×Q₈×C₅₃	C₂³	semidirect
55. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
56. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
57. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
58. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
75. #95×C₅₃	C₂³	semidirect
76. #95×C₅₃	C₂³	semidirect
158. C₂³×C₅₃	C₂²⋊Q₈	semidirect
183. #34982	C₅₃	direct (N × H)
204. C₂²×C₅₃	#543	semidirect
213. C₂×#543	C₂×C₅₃	semidirect
215. #2848	C₂×C₅₃	semidirect
216. #2626	C₂×C₅₃	semidirect
217. #2437	C₂×C₅₃	semidirect
218. #2315	C₂×C₅₃	semidirect
219. #2315	C₂×C₅₃	semidirect
232. C₂×C₂²⋊Q₈	C₂²×C₅₃	semidirect
233. C₂×C₂²⋊Q₈	C₂²×C₅₃	semidirect
234. C₂×C₂²⋊Q₈	C₂²×C₅₃	semidirect
243. #399	C₂²×C₅₃	semidirect
244. #396	C₂²×C₅₃	semidirect
245. #393	C₂²×C₅₃	semidirect
246. #385	C₂²×C₅₃	semidirect
247. #385	C₂²×C₅₃	semidirect
248. #385	C₂²×C₅₃	semidirect
249. #385	C₂²×C₅₃	semidirect
250. #380	C₂²×C₅₃	semidirect
251. #380	C₂²×C₅₃	semidirect
252. #380	C₂²×C₅₃	semidirect
256. C₅₃	#34982	direct (N × H)
260. C₂⁵	Q₈×C₅₃	semidirect
265. C₂²×Q₈	C₂³×C₅₃	semidirect
276. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
277. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
278. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
279. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
296. #95	C₂³×C₅₃	semidirect
297. #95	C₂³×C₅₃	semidirect
379. C₂³	C₂²⋊Q₈×C₅₃	semidirect
424. C₂²	C₅₃×#543	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#543	2
C₂³×C₂²⋊C₄×C₅₃	2
C₅₃×#2848	2
C₅₃×#2626	2
C₅₃×#2437	2
C₅₃×#2315	2
C₂×C₅₃×#379	2
#34982	53

Subgroups — C53×#34982 #27242494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34982 #27242494