Subgroups — C₅₃×#24408 #27231920

Order 13568.

1364 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3585	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3041	C₂	1	2	yes	maximal, complemented
3	C₅₃×#3027	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3008	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2883	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#2053	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2008	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1910	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1904	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1868	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1794	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#585	C₂²	1	1	yes	characteristic
13	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#577	—	2	4	no	—
15	C₅₃×#576	—	2	4	no	—
16	C₅₃×#575	—	2	2	no	—
17	C₅₃×#575	C₂²	1	2	yes	—
18	C₂×AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
19	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	2	yes	—
20	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	1	yes	characteristic, complemented
21	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
22	C₅₃×#568	C₂²	1	1	yes	characteristic
23	C₅₃×#535	C₂²	1	1	yes	characteristic
24	C₅₃×#535	C₂²	1	2	yes	complemented
25	C₅₃×#518	C₂²	1	2	yes	complemented
26	C₅₃×#463	—	2	4	no	—
27	C₅₃×#459	—	2	4	no	—
28	C₅₃×B(4,𝔽₂)	—	2	4	no	—
29	C₅₃×#449	—	2	2	no	—
30	C₅₃×#447	—	2	2	no	—
31	C₅₃×#424	C₂²	1	2	yes	complemented
32	C₅₃×#421	C₂²	1	2	yes	complemented
33	C₅₃×#418	C₂²	1	1	yes	characteristic, complemented
34	C₅₃×#418	C₂²	1	2	yes	complemented
35	C₅₃×#416	C₂²	1	1	yes	characteristic, complemented
36	C₅₃×#416	C₂²	1	2	yes	complemented
37	C₅₃×#410	C₂²	1	1	yes	characteristic
38	C₅₃×#410	C₂²	1	1	yes	characteristic
39	C₅₃×#410	C₂²	1	2	yes	complemented
40	C₅₃×#410	C₂²	1	2	yes	complemented
41	C₂×#99×C₅₃	—	2	4	no	—
42	C₅₃×#408	C₂²	1	1	yes	characteristic
43	C₅₃×#344	C₂²	1	2	yes	complemented
44	C₅₃×H(3,ℤ/4)	C₂²	1	2	yes	complemented
45	2₋¹⁺⁴×C₅₃	—	2	2	no	—
46	2₋¹⁺⁴×C₅₃	—	2	1	no	—
47	H(5,𝔽₂)×C₅₃	—	2	2	no	—
48	H(5,𝔽₂)×C₅₃	—	2	4	no	—
49	H(5,𝔽₂)×C₅₃	—	2	2	no	—
50	H(5,𝔽₂)×C₅₃	—	2	1	no	—
51	H(5,𝔽₂)×C₅₃	—	4	2	no	—
52	C₂×#41×C₅₃	—	2	1	no	—
53	C₂×#41×C₅₃	—	2	2	no	—
54	C₂×#41×C₅₃	—	4	2	no	—
55	C₂×#41×C₅₃	D₄	1	2	yes	complemented
56	C₂×#41×C₅₃	C₂³	1	2	yes	—
57	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
59	C₂²×D₄×C₅₃	—	2	2	no	—
60	C₂²×D₄×C₅₃	—	2	1	no	—
61	C₂²×D₄×C₅₃	D₄	1	2	yes	complemented
62	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
64	#137×C₅₃	—	2	4	no	—
65	#137×C₅₃	—	2	2	no	—
66	#137×C₅₃	—	2	2	no	—
67	AGL(1,ℤ/8)×C₅₃	—	2	8	no	—
68	AGL(1,ℤ/8)×C₅₃	—	2	4	no	—
69	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
70	AGL(1,ℤ/8)×C₅₃	—	2	4	no	—
71	AGL(1,ℤ/8)×C₅₃	—	2	4	no	—
72	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
73	AGL(1,ℤ/8)×C₅₃	—	4	2	no	—
74	#135×C₅₃	—	2	8	no	—
75	#135×C₅₃	—	2	4	no	—
76	C₂×SD₁₆×C₅₃	—	2	4	no	—
77	C₂×SD₁₆×C₅₃	—	2	1	no	—
78	C₂×SD₁₆×C₅₃	D₄	1	4	yes	complemented
79	C₂×D₈×C₅₃	—	2	4	no	—
80	C₂×D₈×C₅₃	—	2	2	no	—
81	C₂×D₈×C₅₃	—	2	4	no	—
82	C₂×D₈×C₅₃	—	2	1	no	—
83	C₂×D₈×C₅₃	D₄	1	4	yes	complemented
84	#131×C₅₃	—	2	4	no	—
85	C₂×M₁₆×C₅₃	—	2	2	no	—
86	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
87	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
88	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	#127×C₅₃	—	2	2	no	—
90	#127×C₅₃	—	2	1	no	—
91	#124×C₅₃	—	2	2	no	—
92	#124×C₅₃	—	2	1	no	—
93	#121×C₅₃	—	2	2	no	—
94	#121×C₅₃	—	4	2	no	—
95	#120×C₅₃	—	2	4	no	—
96	#120×C₅₃	—	2	2	no	—
97	C₄×D₄×C₅₃	—	2	4	no	—
98	C₄×D₄×C₅₃	—	4	2	no	—
99	#117×C₅₃	C₂³	1	2	yes	—
100	#117×C₅₃	C₂³	1	1	yes	characteristic
101	#117×C₅₃	C₂³	1	2	yes	—
102	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
103	C₄⋊C₈×C₅₃	—	2	4	no	—
104	#104×C₅₃	—	4	2	no	—
105	#103×C₅₃	—	2	1	no	—
106	#103×C₅₃	—	2	2	no	—
107	#102×C₅₃	—	2	4	no	—
108	#102×C₅₃	—	2	1	no	—
109	#102×C₅₃	—	2	2	no	—
110	#102×C₅₃	—	2	2	no	—
111	#99×C₅₃	—	2	4	no	—
112	#99×C₅₃	—	2	4	no	—
113	#99×C₅₃	—	2	2	no	—
114	#99×C₅₃	D₄	1	4	yes	—
115	C₂²⋊C₈×C₅₃	—	2	2	no	—
116	C₂⁴×C₅₃	—	2	1	no	—
117	C₂⁴×C₅₃	—	2	2	no	—
118	C₂⁴×C₅₃	—	2	1	no	—
119	#41×C₅₃	—	2	2	no	—
120	#41×C₅₃	—	2	2	no	—
121	#41×C₅₃	—	2	4	no	—
122	#41×C₅₃	—	2	4	no	—
123	#41×C₅₃	—	2	2	no	—
124	#41×C₅₃	—	2	2	no	—
125	#41×C₅₃	—	2	2	no	—
126	#41×C₅₃	—	2	2	no	—
127	#41×C₅₃	—	2	2	no	—
128	#41×C₅₃	—	2	2	no	—
129	#41×C₅₃	—	4	2	no	—
130	#41×C₅₃	—	4	4	no	—
131	#41×C₅₃	—	4	2	no	—
132	#41×C₅₃	—	4	4	no	—
133	#41×C₅₃	—	4	2	no	—
134	#41×C₅₃	—	4	2	no	—
135	#41×C₅₃	—	4	2	no	—
136	C₂×Q₈×C₅₃	—	2	2	no	—
137	C₂×Q₈×C₅₃	—	4	2	no	—
138	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
139	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₂×Q₈×C₅₃	#41	1	1	yes	characteristic
141	C₂×D₄×C₅₃	—	2	2	no	—
142	C₂×D₄×C₅₃	—	2	4	no	—
143	C₂×D₄×C₅₃	—	2	2	no	—
144	C₂×D₄×C₅₃	—	2	1	no	—
145	C₂×D₄×C₅₃	—	2	2	no	—
146	C₂×D₄×C₅₃	—	2	2	no	—
147	C₂×D₄×C₅₃	—	2	2	no	—
148	C₂×D₄×C₅₃	—	2	1	no	—
149	C₂×D₄×C₅₃	—	2	1	no	—
150	C₂×D₄×C₅₃	—	2	1	no	—
151	C₂×D₄×C₅₃	—	2	2	no	—
152	C₂×D₄×C₅₃	—	2	2	no	—
153	C₂×D₄×C₅₃	—	4	2	no	—
154	C₂×D₄×C₅₃	—	4	4	no	—
155	C₂×D₄×C₅₃	—	4	4	no	—
156	C₂×D₄×C₅₃	—	4	2	no	—
157	C₂×D₄×C₅₃	—	4	8	no	—
158	C₂×D₄×C₅₃	—	4	4	no	—
159	C₂×D₄×C₅₃	—	4	2	no	—
160	C₂×D₄×C₅₃	—	4	2	no	—
161	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
162	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
163	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
164	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
165	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
166	C₂×D₄×C₅₃	#41	1	1	yes	characteristic
167	C₂²×C₄×C₅₃	—	2	2	no	—
168	C₂²×C₄×C₅₃	—	2	1	no	—
169	C₂²×C₄×C₅₃	—	2	2	no	—
170	C₂²×C₄×C₅₃	—	4	2	no	—
171	C₂²×C₄×C₅₃	—	4	4	no	—
172	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
174	Q₁₆×C₅₃	—	2	4	no	—
175	SD₁₆×C₅₃	—	2	4	no	—
176	SD₁₆×C₅₃	—	2	4	no	—
177	SD₁₆×C₅₃	—	2	8	no	—
178	SD₁₆×C₅₃	—	4	2	no	—
179	D₈×C₅₃	—	2	4	no	—
180	D₈×C₅₃	—	2	8	no	—
181	D₈×C₅₃	—	4	2	no	—
182	D₈×C₅₃	—	4	4	no	—
183	M₁₆×C₅₃	—	2	4	no	—
184	M₁₆×C₅₃	—	2	2	no	—
185	M₁₆×C₅₃	—	2	2	no	—
186	M₁₆×C₅₃	—	2	2	no	—
187	C₂×C₈×C₅₃	—	2	4	no	—
188	C₂×C₈×C₅₃	—	2	1	no	—
189	C₂×C₈×C₅₃	—	2	1	no	—
190	C₂×C₈×C₅₃	—	2	2	no	—
191	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
192	C₄⋊C₄×C₅₃	—	2	2	no	—
193	C₄⋊C₄×C₅₃	—	2	2	no	—
194	C₄⋊C₄×C₅₃	—	2	1	no	—
195	C₂²⋊C₄×C₅₃	—	2	2	no	—
196	C₂²⋊C₄×C₅₃	—	2	1	no	—
197	C₂²⋊C₄×C₅₃	—	4	4	no	—
198	C₂²⋊C₄×C₅₃	—	4	4	no	—
199	C₂²⋊C₄×C₅₃	—	4	2	no	—
200	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
201	C₄²×C₅₃	—	2	2	no	—
202	C₄²×C₅₃	—	2	1	no	—
203	C₄²×C₅₃	—	2	2	no	—
204	C₂³×C₅₃	—	2	2	no	—
205	C₂³×C₅₃	—	2	4	no	—
206	C₂³×C₅₃	—	2	1	no	—
207	C₂³×C₅₃	—	2	2	no	—
208	C₂³×C₅₃	—	4	4	no	—
209	C₂³×C₅₃	—	4	2	no	—
210	C₂³×C₅₃	—	4	1	no	—
211	C₂³×C₅₃	—	4	2	no	—
212	C₂³×C₅₃	—	16	1	no	—
213	C₂³×C₅₃	—	16	1	no	—
214	C₂³×C₅₃	—	16	2	no	—
215	C₂³×C₅₃	#121	1	2	yes	—
216	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
217	Q₈×C₅₃	—	2	2	no	—
218	Q₈×C₅₃	—	2	2	no	—
219	Q₈×C₅₃	—	2	1	no	—
220	Q₈×C₅₃	—	2	1	no	—
221	Q₈×C₅₃	—	2	2	no	—
222	Q₈×C₅₃	—	4	2	no	—
223	D₄×C₅₃	—	2	4	no	—
224	D₄×C₅₃	—	2	1	no	—
225	D₄×C₅₃	—	2	2	no	—
226	D₄×C₅₃	—	2	2	no	—
227	D₄×C₅₃	—	2	1	no	—
228	D₄×C₅₃	—	2	4	no	—
229	D₄×C₅₃	—	2	2	no	—
230	D₄×C₅₃	—	2	2	no	—
231	D₄×C₅₃	—	4	4	no	—
232	D₄×C₅₃	—	4	2	no	—
233	D₄×C₅₃	—	4	4	no	—
234	D₄×C₅₃	—	4	4	no	—
235	D₄×C₅₃	—	4	2	no	—
236	D₄×C₅₃	—	4	2	no	—
237	D₄×C₅₃	—	4	2	no	—
238	D₄×C₅₃	—	4	1	no	—
239	D₄×C₅₃	—	4	1	no	—
240	D₄×C₅₃	—	4	2	no	—
241	D₄×C₅₃	—	16	4	no	—
242	D₄×C₅₃	—	16	2	no	—
243	D₄×C₅₃	#120	1	4	yes	—
244	C₂×C₄×C₅₃	—	2	2	no	—
245	C₂×C₄×C₅₃	—	2	2	no	—
246	C₂×C₄×C₅₃	—	2	1	no	—
247	C₂×C₄×C₅₃	—	2	2	no	—
248	C₂×C₄×C₅₃	—	2	4	no	—
249	C₂×C₄×C₅₃	—	4	4	no	—
250	C₂×C₄×C₅₃	—	4	4	no	—
251	C₂×C₄×C₅₃	—	4	2	no	—
252	C₂×C₄×C₅₃	—	4	2	no	—
253	C₂×C₄×C₅₃	—	4	2	no	—
254	C₂×C₄×C₅₃	—	4	2	no	—
255	C₂×C₄×C₅₃	—	4	1	no	—
256	C₂×C₄×C₅₃	—	4	2	no	—
257	C₂×C₄×C₅₃	—	16	4	no	—
258	C₂×C₄×C₅₃	#121	1	2	yes	—
259	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
261	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
262	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
263	C₈×C₅₃	—	2	4	no	—
264	C₈×C₅₃	—	4	1	no	—
265	C₈×C₅₃	—	4	1	no	—
266	#24408	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
267	C₂²×C₅₃	—	2	2	no	—
268	C₂²×C₅₃	—	2	4	no	—
269	C₂²×C₅₃	—	4	1	no	—
270	C₂²×C₅₃	—	4	2	no	—
271	C₂²×C₅₃	—	4	2	no	—
272	C₂²×C₅₃	—	4	1	no	—
273	C₂²×C₅₃	—	8	4	no	—
274	C₂²×C₅₃	—	8	2	no	—
275	C₂²×C₅₃	—	8	2	no	—
276	C₂²×C₅₃	—	8	1	no	—
277	C₂²×C₅₃	—	8	4	no	—
278	C₂²×C₅₃	—	8	2	no	—
279	C₂²×C₅₃	—	8	1	no	—
280	C₂²×C₅₃	—	16	4	no	—
281	C₂²×C₅₃	—	16	2	no	—
282	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
283	C₂²×C₅₃	D₄²	1	2	yes	—
284	C₄×C₅₃	—	2	2	no	—
285	C₄×C₅₃	—	4	4	no	—
286	C₄×C₅₃	—	4	1	no	—
287	C₄×C₅₃	—	4	2	no	—
288	C₄×C₅₃	—	8	2	no	—
289	C₄×C₅₃	—	8	1	no	—
290	C₄×C₅₃	C₂×#120	1	1	yes	characteristic
291	C₄×C₅₃	#534	1	1	yes	characteristic
292	C₄×C₅₃	#546	1	2	yes	—
293	#3585	C₂×C₅₃	1	1	yes	characteristic, complemented
294	#3041	C₂×C₅₃	1	2	yes	complemented
295	#3027	C₂×C₅₃	1	1	yes	characteristic, complemented
296	#3008	C₂×C₅₃	1	1	yes	characteristic, complemented
297	#2883	C₂×C₅₃	1	1	yes	characteristic, complemented
298	#2053	C₂×C₅₃	1	2	yes	complemented
299	#2008	C₂×C₅₃	1	2	yes	complemented
300	#1910	C₂×C₅₃	1	1	yes	characteristic, complemented
301	#1904	C₂×C₅₃	1	2	yes	complemented
302	#1868	C₂×C₅₃	1	1	yes	characteristic, complemented
303	#1794	C₂×C₅₃	1	1	yes	characteristic, complemented
304	C₂×C₅₃	—	2	2	no	—
305	C₂×C₅₃	—	2	1	no	—
306	C₂×C₅₃	—	4	2	no	—
307	C₂×C₅₃	—	4	4	no	—
308	C₂×C₅₃	—	8	2	no	—
309	C₂×C₅₃	—	8	1	no	—
310	C₂×C₅₃	—	16	1	no	—
311	C₂×C₅₃	#2410	1	1	yes	characteristic
312	#585	C₂²×C₅₃	1	1	yes	characteristic
313	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	1	yes	characteristic
314	#577	—	2	4	no	—
315	#576	—	2	4	no	—
316	#575	—	2	2	no	—
317	#575	C₂²×C₅₃	1	2	yes	—
318	C₂×AGL(1,ℤ/8)	—	2	2	no	—
319	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	2	yes	—
320	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	1	yes	characteristic, complemented
321	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
322	#568	C₂²×C₅₃	1	1	yes	characteristic
323	#535	C₂²×C₅₃	1	1	yes	characteristic
324	#535	C₂²×C₅₃	1	2	yes	complemented
325	#518	C₂²×C₅₃	1	2	yes	complemented
326	#463	—	2	4	no	—
327	#459	—	2	4	no	—
328	B(4,𝔽₂)	—	2	4	no	—
329	#449	—	2	2	no	—
330	#447	—	2	2	no	—
331	#424	C₂²×C₅₃	1	2	yes	complemented
332	#421	C₂²×C₅₃	1	2	yes	complemented
333	#418	C₂²×C₅₃	1	1	yes	characteristic, complemented
334	#418	C₂²×C₅₃	1	2	yes	complemented
335	#416	C₂²×C₅₃	1	2	yes	complemented
336	#416	C₂²×C₅₃	1	1	yes	characteristic, complemented
337	#410	C₂²×C₅₃	1	1	yes	characteristic
338	#410	C₂²×C₅₃	1	1	yes	characteristic
339	#410	C₂²×C₅₃	1	2	yes	complemented
340	#410	C₂²×C₅₃	1	2	yes	complemented
341	C₂×#99	—	2	4	no	—
342	#408	C₂²×C₅₃	1	1	yes	characteristic
343	#344	C₂²×C₅₃	1	2	yes	complemented
344	H(3,ℤ/4)	C₂²×C₅₃	1	2	yes	complemented
345	C₅₃	#24408	1	1	yes	characteristic, direct product factor, Sylow
346	2₋¹⁺⁴	—	2	2	no	—
347	2₋¹⁺⁴	—	2	1	no	—
348	H(5,𝔽₂)	—	2	2	no	—
349	H(5,𝔽₂)	—	2	4	no	—
350	H(5,𝔽₂)	—	2	2	no	—
351	H(5,𝔽₂)	—	2	1	no	—
352	H(5,𝔽₂)	—	4	2	no	—
353	C₂×#41	—	2	1	no	—
354	C₂×#41	—	2	2	no	—
355	C₂×#41	—	4	2	no	—
356	C₂×#41	D₄×C₅₃	1	2	yes	complemented
357	C₂×#41	C₂³×C₅₃	1	2	yes	—
358	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
359	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
360	C₂²×D₄	—	2	2	no	—
361	C₂²×D₄	—	2	1	no	—
362	C₂²×D₄	D₄×C₅₃	1	2	yes	complemented
363	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
364	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
365	#137	—	2	4	no	—
366	#137	—	2	2	no	—
367	#137	—	2	2	no	—
368	AGL(1,ℤ/8)	—	2	8	no	—
369	AGL(1,ℤ/8)	—	2	4	no	—
370	AGL(1,ℤ/8)	—	2	2	no	—
371	AGL(1,ℤ/8)	—	2	4	no	—
372	AGL(1,ℤ/8)	—	2	4	no	—
373	AGL(1,ℤ/8)	—	2	2	no	—
374	AGL(1,ℤ/8)	—	4	2	no	—
375	#135	—	2	8	no	—
376	#135	—	2	4	no	—
377	C₂×SD₁₆	—	2	4	no	—
378	C₂×SD₁₆	—	2	1	no	—
379	C₂×SD₁₆	D₄×C₅₃	1	4	yes	complemented
380	C₂×D₈	—	2	4	no	—
381	C₂×D₈	—	2	2	no	—
382	C₂×D₈	—	2	4	no	—
383	C₂×D₈	—	2	1	no	—
384	C₂×D₈	D₄×C₅₃	1	4	yes	complemented
385	#131	—	2	4	no	—
386	C₂×M₁₆	—	2	2	no	—
387	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
388	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
389	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
390	#127	—	2	2	no	—
391	#127	—	2	1	no	—
392	#124	—	2	2	no	—
393	#124	—	2	1	no	—
394	#121	—	2	2	no	—
395	#121	—	4	2	no	—
396	#120	—	2	2	no	—
397	#120	—	2	4	no	—
398	C₄×D₄	—	2	4	no	—
399	C₄×D₄	—	4	2	no	—
400	#117	C₂³×C₅₃	1	2	yes	—
401	#117	C₂³×C₅₃	1	1	yes	characteristic
402	#117	C₂³×C₅₃	1	2	yes	—
403	C₂×C₂²⋊C₄	—	2	4	no	—
404	C₄⋊C₈	—	2	4	no	—
405	#104	—	4	2	no	—
406	#103	—	2	1	no	—
407	#103	—	2	2	no	—
408	#102	—	2	1	no	—
409	#102	—	2	2	no	—
410	#102	—	2	2	no	—
411	#102	—	2	4	no	—
412	#99	—	2	2	no	—
413	#99	—	2	4	no	—
414	#99	—	2	4	no	—
415	#99	D₄×C₅₃	1	4	yes	—
416	C₂²⋊C₈	—	2	2	no	—
417	C₂⁴	—	2	1	no	—
418	C₂⁴	—	2	2	no	—
419	C₂⁴	—	2	1	no	—
420	#41	—	2	2	no	—
421	#41	—	2	2	no	—
422	#41	—	2	4	no	—
423	#41	—	2	4	no	—
424	#41	—	2	2	no	—
425	#41	—	2	2	no	—
426	#41	—	2	2	no	—
427	#41	—	2	2	no	—
428	#41	—	2	2	no	—
429	#41	—	2	2	no	—
430	#41	—	4	2	no	—
431	#41	—	4	4	no	—
432	#41	—	4	2	no	—
433	#41	—	4	4	no	—
434	#41	—	4	2	no	—
435	#41	—	4	2	no	—
436	#41	—	4	2	no	—
437	C₂×Q₈	—	2	2	no	—
438	C₂×Q₈	—	4	2	no	—
439	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
440	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
441	C₂×Q₈	#41×C₅₃	1	1	yes	characteristic
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	1	no	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	2	1	no	—
452	C₂×D₄	—	2	2	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	4	2	no	—
455	C₂×D₄	—	4	4	no	—
456	C₂×D₄	—	4	4	no	—
457	C₂×D₄	—	4	2	no	—
458	C₂×D₄	—	4	8	no	—
459	C₂×D₄	—	4	4	no	—
460	C₂×D₄	—	4	2	no	—
461	C₂×D₄	—	4	2	no	—
462	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
463	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
464	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
465	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
466	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
467	C₂×D₄	#41×C₅₃	1	1	yes	characteristic
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	4	2	no	—
472	C₂²×C₄	—	4	4	no	—
473	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
474	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
475	Q₁₆	—	2	4	no	—
476	SD₁₆	—	2	4	no	—
477	SD₁₆	—	2	4	no	—
478	SD₁₆	—	2	8	no	—
479	SD₁₆	—	4	2	no	—
480	D₈	—	2	4	no	—
481	D₈	—	2	8	no	—
482	D₈	—	4	2	no	—
483	D₈	—	4	4	no	—
484	M₁₆	—	2	4	no	—
485	M₁₆	—	2	2	no	—
486	M₁₆	—	2	2	no	—
487	M₁₆	—	2	2	no	—
488	C₂×C₈	—	2	4	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	2	no	—
492	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	1	no	—
498	C₂²⋊C₄	—	4	4	no	—
499	C₂²⋊C₄	—	4	4	no	—
500	C₂²⋊C₄	—	4	2	no	—
501	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
502	C₄²	—	2	2	no	—
503	C₄²	—	2	1	no	—
504	C₄²	—	2	2	no	—
505	C₂³	—	2	2	no	—
506	C₂³	—	2	4	no	—
507	C₂³	—	2	1	no	—
508	C₂³	—	2	2	no	—
509	C₂³	—	4	4	no	—
510	C₂³	—	4	2	no	—
511	C₂³	—	4	1	no	—
512	C₂³	—	4	2	no	—
513	C₂³	—	16	1	no	—
514	C₂³	—	16	1	no	—
515	C₂³	—	16	2	no	—
516	C₂³	#121×C₅₃	1	2	yes	—
517	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
518	Q₈	—	2	2	no	—
519	Q₈	—	2	2	no	—
520	Q₈	—	2	1	no	—
521	Q₈	—	2	1	no	—
522	Q₈	—	2	2	no	—
523	Q₈	—	4	2	no	—
524	D₄	—	2	4	no	—
525	D₄	—	2	2	no	—
526	D₄	—	2	1	no	—
527	D₄	—	2	2	no	—
528	D₄	—	2	1	no	—
529	D₄	—	2	4	no	—
530	D₄	—	2	2	no	—
531	D₄	—	2	2	no	—
532	D₄	—	4	4	no	—
533	D₄	—	4	2	no	—
534	D₄	—	4	4	no	—
535	D₄	—	4	4	no	—
536	D₄	—	4	2	no	—
537	D₄	—	4	2	no	—
538	D₄	—	4	2	no	—
539	D₄	—	4	1	no	—
540	D₄	—	4	1	no	—
541	D₄	—	4	2	no	—
542	D₄	—	16	4	no	—
543	D₄	—	16	2	no	—
544	D₄	#120×C₅₃	1	4	yes	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	4	4	no	—
551	C₂×C₄	—	4	4	no	—
552	C₂×C₄	—	4	2	no	—
553	C₂×C₄	—	4	2	no	—
554	C₂×C₄	—	4	2	no	—
555	C₂×C₄	—	4	2	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	2	no	—
558	C₂×C₄	—	16	4	no	—
559	C₂×C₄	#121×C₅₃	1	2	yes	—
560	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
561	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
562	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
563	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
564	C₈	—	2	4	no	—
565	C₈	—	4	1	no	—
566	C₈	—	4	1	no	—
567	C₂²	—	2	2	no	—
568	C₂²	—	2	4	no	—
569	C₂²	—	4	1	no	—
570	C₂²	—	4	2	no	—
571	C₂²	—	4	2	no	—
572	C₂²	—	4	1	no	—
573	C₂²	—	8	4	no	—
574	C₂²	—	8	2	no	—
575	C₂²	—	8	2	no	—
576	C₂²	—	8	1	no	—
577	C₂²	—	8	4	no	—
578	C₂²	—	8	2	no	—
579	C₂²	—	8	1	no	—
580	C₂²	—	16	4	no	—
581	C₂²	—	16	2	no	—
582	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
583	C₂²	D₄²×C₅₃	1	2	yes	—
584	C₄	—	2	2	no	—
585	C₄	—	4	4	no	—
586	C₄	—	4	1	no	—
587	C₄	—	4	2	no	—
588	C₄	—	8	2	no	—
589	C₄	—	8	1	no	—
590	C₄	C₂×#120×C₅₃	1	1	yes	characteristic
591	C₄	C₅₃×#534	1	1	yes	characteristic
592	C₄	C₅₃×#546	1	2	yes	—
593	C₂	—	2	2	no	—
594	C₂	—	2	1	no	—
595	C₂	—	4	2	no	—
596	C₂	—	4	4	no	—
597	C₂	—	8	1	no	—
598	C₂	—	8	2	no	—
599	C₂	—	16	1	no	—
600	C₂	C₅₃×#2410	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#2410
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3585	C₂	semidirect
2. C₅₃×#3041	C₂	semidirect
3. C₅₃×#3027	C₂	semidirect
4. C₅₃×#3008	C₂	semidirect
5. C₅₃×#2883	C₂	semidirect
6. C₅₃×#2053	C₂	semidirect
7. C₅₃×#2008	C₂	semidirect
8. C₅₃×#1910	C₂	semidirect
9. C₅₃×#1904	C₂	semidirect
10. C₅₃×#1868	C₂	semidirect
11. C₅₃×#1794	C₂	semidirect
20. C₂×AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
21. C₂²×D₈×C₅₃	C₂²	semidirect
24. C₅₃×#535	C₂²	semidirect
25. C₅₃×#518	C₂²	semidirect
31. C₅₃×#424	C₂²	semidirect
32. C₅₃×#421	C₂²	semidirect
33. C₅₃×#418	C₂²	semidirect
34. C₅₃×#418	C₂²	semidirect
35. C₅₃×#416	C₂²	semidirect
36. C₅₃×#416	C₂²	semidirect
39. C₅₃×#410	C₂²	semidirect
40. C₅₃×#410	C₂²	semidirect
43. C₅₃×#344	C₂²	semidirect
44. C₅₃×H(3,ℤ/4)	C₂²	semidirect
55. C₂×#41×C₅₃	D₄	semidirect
61. C₂²×D₄×C₅₃	D₄	semidirect
78. C₂×SD₁₆×C₅₃	D₄	semidirect
83. C₂×D₈×C₅₃	D₄	semidirect
266. #24408	C₅₃	direct (N × H)
293. #3585	C₂×C₅₃	semidirect
294. #3041	C₂×C₅₃	semidirect
295. #3027	C₂×C₅₃	semidirect
296. #3008	C₂×C₅₃	semidirect
297. #2883	C₂×C₅₃	semidirect
298. #2053	C₂×C₅₃	semidirect
299. #2008	C₂×C₅₃	semidirect
300. #1910	C₂×C₅₃	semidirect
301. #1904	C₂×C₅₃	semidirect
302. #1868	C₂×C₅₃	semidirect
303. #1794	C₂×C₅₃	semidirect
320. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
321. C₂²×D₈	C₂²×C₅₃	semidirect
324. #535	C₂²×C₅₃	semidirect
325. #518	C₂²×C₅₃	semidirect
331. #424	C₂²×C₅₃	semidirect
332. #421	C₂²×C₅₃	semidirect
333. #418	C₂²×C₅₃	semidirect
334. #418	C₂²×C₅₃	semidirect
335. #416	C₂²×C₅₃	semidirect
336. #416	C₂²×C₅₃	semidirect
339. #410	C₂²×C₅₃	semidirect
340. #410	C₂²×C₅₃	semidirect
343. #344	C₂²×C₅₃	semidirect
344. H(3,ℤ/4)	C₂²×C₅₃	semidirect
345. C₅₃	#24408	direct (N × H)
356. C₂×#41	D₄×C₅₃	semidirect
362. C₂²×D₄	D₄×C₅₃	semidirect
379. C₂×SD₁₆	D₄×C₅₃	semidirect
384. C₂×D₈	D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3585	2
C₅₃×#3041	2
C₅₃×#3027	2
C₅₃×#3008	2
C₅₃×#2883	2
C₅₃×#2053	2
C₅₃×#2008	2
C₅₃×#1910	2
C₅₃×#1904	2
C₅₃×#1868	2
C₅₃×#1794	2
#24408	53

Subgroups — C53×#24408 #27231920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#24408 #27231920