Subgroups — C₅₃×#34493 #27242005

Order 13568.

3076 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×D₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2375	C₂	1	4	yes	maximal, complemented
3	C₅₃×#2366	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2363	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2354	C₂	1	2	yes	maximal, complemented
6	D₄×C₂²⋊C₄×C₅₃	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2324	C₂	1	4	yes	maximal, complemented
8	C₅₃×#2321	C₂	1	4	yes	maximal, complemented
9	C₅₃×#2309	C₂	1	4	yes	maximal, complemented
10	C₄×#127×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2293	C₂	1	2	yes	maximal, complemented
12	C₂³×D₄×C₅₃	C₂²	1	4	yes	—
13	D₄²×C₅₃	C₄	1	16	yes	complemented
14	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#121×C₅₃	C₂²	1	4	yes	—
16	C₂×#120×C₅₃	C₂²	1	4	yes	—
17	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
19	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
20	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
21	C₂×#117×C₅₃	C₂²	1	4	yes	complemented
22	C₂×#117×C₅₃	C₂²	1	4	yes	complemented
23	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
24	C₅₃×#390	C₂²	1	2	yes	—
25	C₅₃×#390	C₂²	1	2	yes	—
26	C₅₃×#390	C₂²	1	4	yes	—
27	C₅₃×#390	C₂²	1	4	yes	complemented
28	C₅₃×#390	C₂²	1	2	yes	complemented
29	C₅₃×#390	C₂²	1	8	yes	complemented
30	C₅₃×#390	C₂²	1	4	yes	complemented
31	C₅₃×#390	C₂²	1	8	yes	complemented
32	C₅₃×#389	C₂²	1	2	yes	complemented
33	C₅₃×#389	C₂²	1	2	yes	complemented
34	C₅₃×#388	C₂²	1	4	yes	complemented
35	C₅₃×#388	C₂²	1	4	yes	complemented
36	C₅₃×#388	C₂²	1	8	yes	complemented
37	C₅₃×#388	C₂²	1	8	yes	complemented
38	C₅₃×#386	C₂²	1	8	yes	complemented
39	C₅₃×#386	C₂²	1	4	yes	complemented
40	C₅₃×#386	C₂²	1	8	yes	complemented
41	C₅₃×#385	C₂²	1	4	yes	complemented
42	C₅₃×#384	C₂²	1	1	yes	characteristic, complemented
43	C₅₃×#380	C₂²	1	4	yes	complemented
44	C₅₃×#379	C₂²	1	4	yes	complemented
45	C₅₃×#379	C₂²	1	8	yes	complemented
46	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
47	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	complemented
48	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
49	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
50	C₄³×C₅₃	C₂²	1	1	yes	characteristic, complemented
51	C₂⁵×C₅₃	C₂³	1	4	yes	—
52	C₂²×D₄×C₅₃	—	4	8	no	—
53	C₂²×D₄×C₅₃	—	4	8	no	—
54	C₂²×D₄×C₅₃	C₂×C₄	1	32	yes	complemented
55	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
56	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
57	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
58	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
59	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
60	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
61	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
62	#127×C₅₃	C₂×C₄	1	8	yes	—
63	#121×C₅₃	C₂×C₄	1	32	yes	complemented
64	#120×C₅₃	C₂×C₄	1	32	yes	complemented
65	C₄×D₄×C₅₃	—	4	4	no	—
66	C₄×D₄×C₅₃	—	4	8	no	—
67	C₄×D₄×C₅₃	—	4	8	no	—
68	C₄×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
69	#117×C₅₃	—	4	8	no	—
70	#117×C₅₃	—	4	8	no	—
71	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
73	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
75	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
76	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
78	C₂×C₂²⋊C₄×C₅₃	—	4	8	no	—
79	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
80	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
82	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
84	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
85	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
87	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
91	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
92	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
93	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
94	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
96	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	#95×C₅₃	C₂³	1	4	yes	—
99	#95×C₅₃	C₂³	1	4	yes	—
100	#95×C₅₃	C₂³	1	8	yes	—
101	#95×C₅₃	C₂³	1	4	yes	—
102	C₂⁴×C₅₃	—	4	4	no	—
103	C₂⁴×C₅₃	—	4	8	no	—
104	C₂⁴×C₅₃	—	4	8	no	—
105	C₂⁴×C₅₃	—	4	4	no	—
106	C₂⁴×C₅₃	C₂²×C₄	1	16	yes	—
107	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
108	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
109	C₂×D₄×C₅₃	—	2	8	no	—
110	C₂×D₄×C₅₃	—	2	8	no	—
111	C₂×D₄×C₅₃	—	2	4	no	—
112	C₂×D₄×C₅₃	—	2	4	no	—
113	C₂×D₄×C₅₃	—	4	64	no	—
114	C₂×D₄×C₅₃	—	4	2	no	—
115	C₂×D₄×C₅₃	—	4	8	no	—
116	C₂×D₄×C₅₃	—	4	8	no	—
117	C₂×D₄×C₅₃	—	4	2	no	—
118	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
119	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	—
120	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
121	C₂×D₄×C₅₃	C₂²×C₄	1	32	yes	—
122	C₂²×C₄×C₅₃	—	2	8	no	—
123	C₂²×C₄×C₅₃	—	2	8	no	—
124	C₂²×C₄×C₅₃	—	2	8	no	—
125	C₂²×C₄×C₅₃	—	4	4	no	—
126	C₂²×C₄×C₅₃	—	4	8	no	—
127	C₂²×C₄×C₅₃	—	4	8	no	—
128	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	—
129	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
131	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
132	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
133	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
134	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
135	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
136	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
137	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
138	C₄⋊C₄×C₅₃	—	2	4	no	—
139	C₄⋊C₄×C₅₃	—	2	8	no	—
140	C₄⋊C₄×C₅₃	—	2	4	no	—
141	C₄⋊C₄×C₅₃	—	4	4	no	—
142	C₄⋊C₄×C₅₃	—	4	4	no	—
143	C₄⋊C₄×C₅₃	—	4	2	no	—
144	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
145	C₂²⋊C₄×C₅₃	—	2	8	no	—
146	C₂²⋊C₄×C₅₃	—	4	8	no	—
147	C₂²⋊C₄×C₅₃	—	4	4	no	—
148	C₂²⋊C₄×C₅₃	—	4	4	no	—
149	C₂²⋊C₄×C₅₃	—	4	4	no	—
150	C₂²⋊C₄×C₅₃	—	4	8	no	—
151	C₂²⋊C₄×C₅₃	—	4	4	no	—
152	C₂²⋊C₄×C₅₃	—	4	4	no	—
153	C₂²⋊C₄×C₅₃	—	4	8	no	—
154	C₂²⋊C₄×C₅₃	—	4	8	no	—
155	C₂²⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
156	C₂²⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
157	C₄²×C₅₃	—	2	4	no	—
158	C₄²×C₅₃	—	2	8	no	—
159	C₄²×C₅₃	—	2	4	no	—
160	C₄²×C₅₃	—	2	2	no	—
161	C₄²×C₅₃	—	4	1	no	—
162	C₄²×C₅₃	—	4	8	no	—
163	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
164	C₂³×C₅₃	—	2	4	no	—
165	C₂³×C₅₃	—	2	4	no	—
166	C₂³×C₅₃	—	2	4	no	—
167	C₂³×C₅₃	—	2	8	no	—
168	C₂³×C₅₃	—	2	8	no	—
169	C₂³×C₅₃	—	2	4	no	—
170	C₂³×C₅₃	—	2	4	no	—
171	C₂³×C₅₃	—	2	4	no	—
172	C₂³×C₅₃	—	2	4	no	—
173	C₂³×C₅₃	—	2	4	no	—
174	C₂³×C₅₃	—	4	32	no	—
175	C₂³×C₅₃	—	4	16	no	—
176	C₂³×C₅₃	—	8	8	no	—
177	C₂³×C₅₃	—	8	16	no	—
178	C₂³×C₅₃	—	8	8	no	—
179	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
180	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
181	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
182	D₄×C₅₃	—	2	16	no	—
183	D₄×C₅₃	—	2	32	no	—
184	D₄×C₅₃	—	4	32	no	—
185	D₄×C₅₃	—	4	8	no	—
186	C₂×C₄×C₅₃	—	2	4	no	—
187	C₂×C₄×C₅₃	—	2	4	no	—
188	C₂×C₄×C₅₃	—	2	2	no	—
189	C₂×C₄×C₅₃	—	2	1	no	—
190	C₂×C₄×C₅₃	—	2	1	no	—
191	C₂×C₄×C₅₃	—	2	32	no	—
192	C₂×C₄×C₅₃	—	2	4	no	—
193	C₂×C₄×C₅₃	—	2	4	no	—
194	C₂×C₄×C₅₃	—	2	4	no	—
195	C₂×C₄×C₅₃	—	2	4	no	—
196	C₂×C₄×C₅₃	—	2	8	no	—
197	C₂×C₄×C₅₃	—	2	2	no	—
198	C₂×C₄×C₅₃	—	2	1	no	—
199	C₂×C₄×C₅₃	—	2	4	no	—
200	C₂×C₄×C₅₃	—	2	2	no	—
201	C₂×C₄×C₅₃	—	2	2	no	—
202	C₂×C₄×C₅₃	—	2	4	no	—
203	C₂×C₄×C₅₃	—	2	2	no	—
204	C₂×C₄×C₅₃	—	2	4	no	—
205	C₂×C₄×C₅₃	—	8	8	no	—
206	C₂×C₄×C₅₃	—	8	8	no	—
207	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
208	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
209	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
210	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
211	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
212	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
213	#34493	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
214	C₂²×C₅₃	—	2	8	no	—
215	C₂²×C₅₃	—	2	8	no	—
216	C₂²×C₅₃	—	2	16	no	—
217	C₂²×C₅₃	—	2	8	no	—
218	C₂²×C₅₃	—	2	8	no	—
219	C₂²×C₅₃	—	4	4	no	—
220	C₂²×C₅₃	—	4	4	no	—
221	C₂²×C₅₃	—	4	8	no	—
222	C₂²×C₅₃	—	4	4	no	—
223	C₂²×C₅₃	—	4	4	no	—
224	C₂²×C₅₃	—	4	4	no	—
225	C₂²×C₅₃	—	4	4	no	—
226	C₂²×C₅₃	—	8	32	no	—
227	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
228	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
229	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
230	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
231	C₂²×C₅₃	#585	1	1	yes	characteristic
232	C₄×C₅₃	—	2	8	no	—
233	C₄×C₅₃	—	2	2	no	—
234	C₄×C₅₃	—	2	2	no	—
235	C₄×C₅₃	—	4	4	no	—
236	C₄×C₅₃	—	4	2	no	—
237	C₄×C₅₃	—	4	4	no	—
238	C₄×C₅₃	—	4	1	no	—
239	C₄×C₅₃	—	4	4	no	—
240	C₄×C₅₃	#518	1	8	yes	—
241	C₂×D₄²	C₂×C₅₃	1	1	yes	characteristic
242	#2375	C₂×C₅₃	1	4	yes	complemented
243	#2366	C₂×C₅₃	1	1	yes	characteristic, complemented
244	#2363	C₂×C₅₃	1	4	yes	complemented
245	#2354	C₂×C₅₃	1	2	yes	complemented
246	D₄×C₂²⋊C₄	C₂×C₅₃	1	4	yes	complemented
247	#2324	C₂×C₅₃	1	4	yes	complemented
248	#2321	C₂×C₅₃	1	4	yes	complemented
249	#2309	C₂×C₅₃	1	4	yes	complemented
250	C₄×#127	C₂×C₅₃	1	1	yes	characteristic, complemented
251	#2293	C₂×C₅₃	1	2	yes	complemented
252	C₂×C₅₃	—	4	8	no	—
253	C₂×C₅₃	—	4	8	no	—
254	C₂×C₅₃	C₂×#518	1	2	yes	—
255	C₂×C₅₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic
256	C₂×C₅₃	#3543	1	1	yes	characteristic
257	C₂×C₅₃	#3546	1	2	yes	—
258	C₂×C₅₃	#3549	1	1	yes	characteristic
259	C₂³×D₄	C₂²×C₅₃	1	4	yes	—
260	D₄²	C₄×C₅₃	1	16	yes	complemented
261	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
262	C₂×#121	C₂²×C₅₃	1	4	yes	—
263	C₂×#120	C₂²×C₅₃	1	4	yes	—
264	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
265	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
266	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	complemented
267	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	complemented
268	C₂×#117	C₂²×C₅₃	1	4	yes	complemented
269	C₂×#117	C₂²×C₅₃	1	4	yes	complemented
270	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
271	#390	C₂²×C₅₃	1	2	yes	—
272	#390	C₂²×C₅₃	1	4	yes	—
273	#390	C₂²×C₅₃	1	2	yes	—
274	#390	C₂²×C₅₃	1	4	yes	complemented
275	#390	C₂²×C₅₃	1	2	yes	complemented
276	#390	C₂²×C₅₃	1	8	yes	complemented
277	#390	C₂²×C₅₃	1	4	yes	complemented
278	#390	C₂²×C₅₃	1	8	yes	complemented
279	#389	C₂²×C₅₃	1	2	yes	complemented
280	#389	C₂²×C₅₃	1	2	yes	complemented
281	#388	C₂²×C₅₃	1	4	yes	complemented
282	#388	C₂²×C₅₃	1	8	yes	complemented
283	#388	C₂²×C₅₃	1	8	yes	complemented
284	#388	C₂²×C₅₃	1	4	yes	complemented
285	#386	C₂²×C₅₃	1	8	yes	complemented
286	#386	C₂²×C₅₃	1	4	yes	complemented
287	#386	C₂²×C₅₃	1	8	yes	complemented
288	#385	C₂²×C₅₃	1	4	yes	complemented
289	#384	C₂²×C₅₃	1	1	yes	characteristic, complemented
290	#380	C₂²×C₅₃	1	4	yes	complemented
291	#379	C₂²×C₅₃	1	4	yes	complemented
292	#379	C₂²×C₅₃	1	8	yes	complemented
293	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	complemented
294	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
295	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
296	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
297	C₄³	C₂²×C₅₃	1	1	yes	characteristic, complemented
298	C₅₃	#34493	1	1	yes	characteristic, direct product factor, Sylow
299	C₂⁵	C₂³×C₅₃	1	4	yes	—
300	C₂²×D₄	—	4	8	no	—
301	C₂²×D₄	—	4	8	no	—
302	C₂²×D₄	C₂×C₄×C₅₃	1	32	yes	complemented
303	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
304	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
305	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
306	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
307	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
308	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
309	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
310	#127	C₂×C₄×C₅₃	1	8	yes	—
311	#121	C₂×C₄×C₅₃	1	32	yes	complemented
312	#120	C₂×C₄×C₅₃	1	32	yes	complemented
313	C₄×D₄	—	4	8	no	—
314	C₄×D₄	—	4	8	no	—
315	C₄×D₄	—	4	4	no	—
316	C₄×D₄	C₂×C₄×C₅₃	1	16	yes	complemented
317	#117	—	4	8	no	—
318	#117	—	4	8	no	—
319	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
320	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
321	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
322	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
323	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
324	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
325	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
326	C₂×C₂²⋊C₄	—	4	8	no	—
327	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
328	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
329	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
330	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
331	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
332	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
333	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
334	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
335	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
336	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
337	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
338	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
339	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
340	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
341	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
342	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
343	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
344	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
345	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
346	#95	C₂³×C₅₃	1	4	yes	—
347	#95	C₂³×C₅₃	1	4	yes	—
348	#95	C₂³×C₅₃	1	8	yes	—
349	#95	C₂³×C₅₃	1	4	yes	—
350	C₂⁴	—	4	4	no	—
351	C₂⁴	—	4	8	no	—
352	C₂⁴	—	4	8	no	—
353	C₂⁴	—	4	4	no	—
354	C₂⁴	C₂²×C₄×C₅₃	1	16	yes	—
355	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
356	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
357	C₂×D₄	—	2	8	no	—
358	C₂×D₄	—	2	8	no	—
359	C₂×D₄	—	2	4	no	—
360	C₂×D₄	—	2	4	no	—
361	C₂×D₄	—	4	64	no	—
362	C₂×D₄	—	4	2	no	—
363	C₂×D₄	—	4	8	no	—
364	C₂×D₄	—	4	8	no	—
365	C₂×D₄	—	4	2	no	—
366	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
367	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	—
368	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
369	C₂×D₄	C₂²×C₄×C₅₃	1	32	yes	—
370	C₂²×C₄	—	2	8	no	—
371	C₂²×C₄	—	2	8	no	—
372	C₂²×C₄	—	2	8	no	—
373	C₂²×C₄	—	4	4	no	—
374	C₂²×C₄	—	4	8	no	—
375	C₂²×C₄	—	4	8	no	—
376	C₂²×C₄	C₂²×C₄×C₅₃	1	16	yes	—
377	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
378	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
379	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
380	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
381	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
382	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
383	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
384	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
385	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
386	C₄⋊C₄	—	2	4	no	—
387	C₄⋊C₄	—	2	8	no	—
388	C₄⋊C₄	—	2	4	no	—
389	C₄⋊C₄	—	4	4	no	—
390	C₄⋊C₄	—	4	4	no	—
391	C₄⋊C₄	—	4	2	no	—
392	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
393	C₂²⋊C₄	—	2	8	no	—
394	C₂²⋊C₄	—	4	8	no	—
395	C₂²⋊C₄	—	4	4	no	—
396	C₂²⋊C₄	—	4	4	no	—
397	C₂²⋊C₄	—	4	8	no	—
398	C₂²⋊C₄	—	4	8	no	—
399	C₂²⋊C₄	—	4	4	no	—
400	C₂²⋊C₄	—	4	8	no	—
401	C₂²⋊C₄	—	4	4	no	—
402	C₂²⋊C₄	—	4	4	no	—
403	C₂²⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
404	C₂²⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
405	C₄²	—	2	4	no	—
406	C₄²	—	2	8	no	—
407	C₄²	—	2	2	no	—
408	C₄²	—	2	4	no	—
409	C₄²	—	4	8	no	—
410	C₄²	—	4	1	no	—
411	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
412	C₂³	—	2	4	no	—
413	C₂³	—	2	4	no	—
414	C₂³	—	2	4	no	—
415	C₂³	—	2	4	no	—
416	C₂³	—	2	8	no	—
417	C₂³	—	2	8	no	—
418	C₂³	—	2	4	no	—
419	C₂³	—	2	4	no	—
420	C₂³	—	2	4	no	—
421	C₂³	—	2	4	no	—
422	C₂³	—	4	32	no	—
423	C₂³	—	4	16	no	—
424	C₂³	—	8	8	no	—
425	C₂³	—	8	16	no	—
426	C₂³	—	8	8	no	—
427	C₂³	C₂³×C₄×C₅₃	1	8	yes	—
428	C₂³	C₂³×C₄×C₅₃	1	8	yes	—
429	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
430	D₄	—	2	16	no	—
431	D₄	—	2	32	no	—
432	D₄	—	4	32	no	—
433	D₄	—	4	8	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	32	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	8	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	8	8	no	—
454	C₂×C₄	—	8	8	no	—
455	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
456	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
457	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
458	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
459	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
460	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
461	C₂²	—	2	8	no	—
462	C₂²	—	2	8	no	—
463	C₂²	—	2	16	no	—
464	C₂²	—	2	8	no	—
465	C₂²	—	2	8	no	—
466	C₂²	—	4	4	no	—
467	C₂²	—	4	4	no	—
468	C₂²	—	4	4	no	—
469	C₂²	—	4	4	no	—
470	C₂²	—	4	8	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	4	4	no	—
473	C₂²	—	8	32	no	—
474	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
475	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
476	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
477	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
478	C₂²	C₅₃×#585	1	1	yes	characteristic
479	C₄	—	2	2	no	—
480	C₄	—	2	8	no	—
481	C₄	—	2	2	no	—
482	C₄	—	4	4	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	4	no	—
486	C₄	—	4	1	no	—
487	C₄	C₅₃×#518	1	8	yes	—
488	C₂	—	4	8	no	—
489	C₂	—	4	8	no	—
490	C₂	C₂×C₅₃×#518	1	2	yes	—
491	C₂	C₄×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
492	C₂	C₅₃×#3543	1	1	yes	characteristic
493	C₂	C₅₃×#3546	1	2	yes	—
494	C₂	C₅₃×#3549	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂×D₄²×C₅₃	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2375	C₂	semidirect
3. C₅₃×#2366	C₂	semidirect
4. C₅₃×#2363	C₂	semidirect
5. C₅₃×#2354	C₂	semidirect
6. D₄×C₂²⋊C₄×C₅₃	C₂	semidirect
7. C₅₃×#2324	C₂	semidirect
8. C₅₃×#2321	C₂	semidirect
9. C₅₃×#2309	C₂	semidirect
10. C₄×#127×C₅₃	C₂	semidirect
11. C₅₃×#2293	C₂	semidirect
13. D₄²×C₅₃	C₄	semidirect
18. C₂×C₄×D₄×C₅₃	C₂²	semidirect
19. C₂×C₄×D₄×C₅₃	C₂²	semidirect
20. C₂×C₄×D₄×C₅₃	C₂²	semidirect
21. C₂×#117×C₅₃	C₂²	semidirect
22. C₂×#117×C₅₃	C₂²	semidirect
23. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
27. C₅₃×#390	C₂²	semidirect
28. C₅₃×#390	C₂²	semidirect
29. C₅₃×#390	C₂²	semidirect
30. C₅₃×#390	C₂²	semidirect
31. C₅₃×#390	C₂²	semidirect
32. C₅₃×#389	C₂²	semidirect
33. C₅₃×#389	C₂²	semidirect
34. C₅₃×#388	C₂²	semidirect
35. C₅₃×#388	C₂²	semidirect
36. C₅₃×#388	C₂²	semidirect
37. C₅₃×#388	C₂²	semidirect
38. C₅₃×#386	C₂²	semidirect
39. C₅₃×#386	C₂²	semidirect
40. C₅₃×#386	C₂²	semidirect
41. C₅₃×#385	C₂²	semidirect
42. C₅₃×#384	C₂²	semidirect
43. C₅₃×#380	C₂²	semidirect
44. C₅₃×#379	C₂²	semidirect
45. C₅₃×#379	C₂²	semidirect
46. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
47. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
48. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
49. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
50. C₄³×C₅₃	C₂²	semidirect
54. C₂²×D₄×C₅₃	C₂×C₄	semidirect
63. #121×C₅₃	C₂×C₄	semidirect
64. #120×C₅₃	C₂×C₄	semidirect
68. C₄×D₄×C₅₃	C₂×C₄	semidirect
213. #34493	C₅₃	direct (N × H)
242. #2375	C₂×C₅₃	semidirect
243. #2366	C₂×C₅₃	semidirect
244. #2363	C₂×C₅₃	semidirect
245. #2354	C₂×C₅₃	semidirect
246. D₄×C₂²⋊C₄	C₂×C₅₃	semidirect
247. #2324	C₂×C₅₃	semidirect
248. #2321	C₂×C₅₃	semidirect
249. #2309	C₂×C₅₃	semidirect
250. C₄×#127	C₂×C₅₃	semidirect
251. #2293	C₂×C₅₃	semidirect
260. D₄²	C₄×C₅₃	semidirect
265. C₂×C₄×D₄	C₂²×C₅₃	semidirect
266. C₂×C₄×D₄	C₂²×C₅₃	semidirect
267. C₂×C₄×D₄	C₂²×C₅₃	semidirect
268. C₂×#117	C₂²×C₅₃	semidirect
269. C₂×#117	C₂²×C₅₃	semidirect
270. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
274. #390	C₂²×C₅₃	semidirect
275. #390	C₂²×C₅₃	semidirect
276. #390	C₂²×C₅₃	semidirect
277. #390	C₂²×C₅₃	semidirect
278. #390	C₂²×C₅₃	semidirect
279. #389	C₂²×C₅₃	semidirect
280. #389	C₂²×C₅₃	semidirect
281. #388	C₂²×C₅₃	semidirect
282. #388	C₂²×C₅₃	semidirect
283. #388	C₂²×C₅₃	semidirect
284. #388	C₂²×C₅₃	semidirect
285. #386	C₂²×C₅₃	semidirect
286. #386	C₂²×C₅₃	semidirect
287. #386	C₂²×C₅₃	semidirect
288. #385	C₂²×C₅₃	semidirect
289. #384	C₂²×C₅₃	semidirect
290. #380	C₂²×C₅₃	semidirect
291. #379	C₂²×C₅₃	semidirect
292. #379	C₂²×C₅₃	semidirect
293. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
294. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
295. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
296. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
297. C₄³	C₂²×C₅₃	semidirect
298. C₅₃	#34493	direct (N × H)
302. C₂²×D₄	C₂×C₄×C₅₃	semidirect
311. #121	C₂×C₄×C₅₃	semidirect
312. #120	C₂×C₄×C₅₃	semidirect
316. C₄×D₄	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄²×C₅₃	2
C₅₃×#2375	2
C₅₃×#2366	2
C₅₃×#2363	2
C₅₃×#2354	2
D₄×C₂²⋊C₄×C₅₃	2
C₅₃×#2324	2
C₅₃×#2321	2
C₅₃×#2309	2
C₄×#127×C₅₃	2
C₅₃×#2293	2
#34493	53

Subgroups — C53×#34493 #27242005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34493 #27242005