Subgroups — C₅₃×#22286 #27229798

Order 13568.

820 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3155	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2948	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1978	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1976	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1911	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1910	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1904	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1903	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1833	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1747	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	2	yes	—
13	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#568	C₂²	1	2	yes	—
16	C₅₃×#535	C₂²	1	2	yes	—
17	C₅₃×#518	C₂²	1	2	yes	—
18	C₅₃×#496	C₄	1	4	yes	complemented
19	C₅₃×#495	C₄	1	2	yes	complemented
20	C₅₃×#493	C₄	1	2	yes	complemented
21	C₅₃×#444	—	2	2	no	—
22	C₅₃×#443	—	2	2	no	—
23	C₅₃×#442	—	2	2	no	—
24	C₄×D₈×C₅₃	—	2	2	no	—
25	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#427	C₂²	1	1	yes	characteristic
27	C₅₃×#421	C₂²	1	2	yes	—
28	C₅₃×#421	C₂²	1	2	yes	—
29	C₅₃×#418	C₂²	1	2	yes	—
30	C₅₃×#418	C₂²	1	2	yes	—
31	C₂×#100×C₅₃	C₂²	1	2	yes	—
32	C₅₃×#410	C₂²	1	2	yes	—
33	C₅₃×#405	C₂²	1	1	yes	characteristic
34	C₅₃×#405	C₂²	1	1	yes	characteristic
35	C₅₃×#405	C₂²	1	1	yes	characteristic
36	C₅₃×#344	C₂²	1	2	yes	—
37	C₅₃×#338	C₂²	1	1	yes	characteristic
38	C₅₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
39	C₅₃×H(3,ℤ/4)	C₂²	1	2	yes	—
40	C₅₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
41	C₅₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
42	C₂×#41×C₅₃	C₂³	1	2	yes	—
43	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
44	AGL(1,ℤ/8)×C₅₃	—	4	2	no	—
45	AGL(1,ℤ/8)×C₅₃	—	4	2	no	—
46	#135×C₅₃	—	2	1	no	—
47	#135×C₅₃	—	2	1	no	—
48	#135×C₅₃	—	2	1	no	—
49	#135×C₅₃	—	2	1	no	—
50	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
51	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
52	C₂×SD₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
53	C₂×D₈×C₅₃	—	2	1	no	—
54	C₂×D₈×C₅₃	—	2	1	no	—
55	C₂×D₈×C₅₃	—	2	1	no	—
56	C₂×D₈×C₅₃	—	2	1	no	—
57	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
58	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
60	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	complemented
61	#131×C₅₃	—	2	4	no	—
62	#131×C₅₃	—	4	2	no	—
63	C₂×M₁₆×C₅₃	—	4	2	no	—
64	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
65	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	#127×C₅₃	C₂×C₄	1	4	yes	complemented
70	#124×C₅₃	C₂×C₄	1	4	yes	complemented
71	#121×C₅₃	—	4	2	no	—
72	#120×C₅₃	—	4	2	no	—
73	C₄×D₄×C₅₃	—	2	4	no	—
74	C₄×D₄×C₅₃	—	4	2	no	—
75	#117×C₅₃	C₂³	1	2	yes	—
76	#117×C₅₃	C₂³	1	1	yes	characteristic
77	#117×C₅₃	C₂³	1	1	yes	characteristic
78	#117×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
80	#108×C₅₃	—	2	1	no	—
81	#108×C₅₃	—	2	1	no	—
82	#107×C₅₃	—	2	1	no	—
83	#106×C₅₃	—	2	1	no	—
84	#104×C₅₃	—	2	4	no	—
85	#104×C₅₃	—	4	2	no	—
86	#102×C₅₃	—	2	4	no	—
87	#102×C₅₃	—	4	2	no	—
88	#100×C₅₃	—	4	2	no	—
89	#99×C₅₃	—	4	2	no	—
90	#97×C₅₃	—	2	1	no	—
91	#97×C₅₃	C₂×C₄	1	2	yes	complemented
92	#97×C₅₃	D₄	1	2	yes	complemented
93	C₄×C₈×C₅₃	—	2	1	no	—
94	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	complemented
95	C₄×C₈×C₅₃	D₄	1	2	yes	complemented
96	C₂⁴×C₅₃	—	2	2	no	—
97	#41×C₅₃	—	2	2	no	—
98	#41×C₅₃	—	2	2	no	—
99	#41×C₅₃	—	2	2	no	—
100	#41×C₅₃	—	2	2	no	—
101	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
102	C₂×D₄×C₅₃	—	2	2	no	—
103	C₂×D₄×C₅₃	—	2	2	no	—
104	C₂×D₄×C₅₃	—	2	2	no	—
105	C₂×D₄×C₅₃	—	2	2	no	—
106	C₂×D₄×C₅₃	—	2	2	no	—
107	C₂×D₄×C₅₃	—	2	2	no	—
108	C₂×D₄×C₅₃	—	4	4	no	—
109	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
110	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₅₃	—	2	2	no	—
112	C₂²×C₄×C₅₃	—	4	4	no	—
113	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
114	Q₁₆×C₅₃	—	2	1	no	—
115	Q₁₆×C₅₃	—	2	1	no	—
116	SD₁₆×C₅₃	—	2	2	no	—
117	SD₁₆×C₅₃	—	2	2	no	—
118	SD₁₆×C₅₃	—	4	4	no	—
119	D₈×C₅₃	—	2	2	no	—
120	D₈×C₅₃	—	2	2	no	—
121	D₈×C₅₃	—	2	2	no	—
122	D₈×C₅₃	—	2	2	no	—
123	D₈×C₅₃	—	2	1	no	—
124	D₈×C₅₃	—	2	1	no	—
125	D₈×C₅₃	—	4	4	no	—
126	M₁₆×C₅₃	—	2	1	no	—
127	M₁₆×C₅₃	—	2	1	no	—
128	M₁₆×C₅₃	—	2	1	no	—
129	M₁₆×C₅₃	—	2	1	no	—
130	M₁₆×C₅₃	—	2	2	no	—
131	M₁₆×C₅₃	—	2	1	no	—
132	M₁₆×C₅₃	—	2	2	no	—
133	M₁₆×C₅₃	—	2	1	no	—
134	M₁₆×C₅₃	—	4	4	no	—
135	C₂×C₈×C₅₃	—	2	1	no	—
136	C₂×C₈×C₅₃	—	2	2	no	—
137	C₂×C₈×C₅₃	—	4	4	no	—
138	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
139	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
145	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
146	C₄⋊C₄×C₅₃	—	2	1	no	—
147	C₄⋊C₄×C₅₃	—	2	1	no	—
148	C₄⋊C₄×C₅₃	—	2	2	no	—
149	C₄⋊C₄×C₅₃	—	2	1	no	—
150	C₂²⋊C₄×C₅₃	—	2	1	no	—
151	C₂²⋊C₄×C₅₃	—	2	1	no	—
152	C₂²⋊C₄×C₅₃	—	2	1	no	—
153	C₂²⋊C₄×C₅₃	—	2	2	no	—
154	C₂²⋊C₄×C₅₃	—	4	4	no	—
155	C₂²⋊C₄×C₅₃	—	4	4	no	—
156	C₄²×C₅₃	—	2	1	no	—
157	C₄²×C₅₃	—	2	2	no	—
158	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
159	C₄²×C₅₃	C₂×D₄	1	2	yes	—
160	C₂³×C₅₃	—	2	2	no	—
161	C₂³×C₅₃	—	2	4	no	—
162	C₂³×C₅₃	—	4	2	no	—
163	C₂³×C₅₃	—	4	2	no	—
164	C₂³×C₅₃	—	16	2	no	—
165	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
166	Q₈×C₅₃	—	2	2	no	—
167	Q₈×C₅₃	—	2	2	no	—
168	D₄×C₅₃	—	2	4	no	—
169	D₄×C₅₃	—	2	2	no	—
170	D₄×C₅₃	—	2	4	no	—
171	D₄×C₅₃	—	2	2	no	—
172	D₄×C₅₃	—	4	2	no	—
173	D₄×C₅₃	—	4	2	no	—
174	D₄×C₅₃	—	4	2	no	—
175	D₄×C₅₃	—	4	2	no	—
176	D₄×C₅₃	—	16	4	no	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	2	2	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	4	2	no	—
181	C₂×C₄×C₅₃	—	4	2	no	—
182	C₂×C₄×C₅₃	—	4	2	no	—
183	C₂×C₄×C₅₃	—	4	1	no	—
184	C₂×C₄×C₅₃	—	16	4	no	—
185	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
186	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
189	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
190	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
191	C₈×C₅₃	—	2	2	no	—
192	C₈×C₅₃	—	2	2	no	—
193	C₈×C₅₃	—	4	2	no	—
194	C₈×C₅₃	—	4	1	no	—
195	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
196	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
197	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
198	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
199	#22286	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂²×C₅₃	—	4	1	no	—
201	C₂²×C₅₃	—	4	2	no	—
202	C₂²×C₅₃	—	4	4	no	—
203	C₂²×C₅₃	—	8	2	no	—
204	C₂²×C₅₃	—	8	4	no	—
205	C₂²×C₅₃	—	8	4	no	—
206	C₂²×C₅₃	—	16	2	no	—
207	C₂²×C₅₃	#518	1	1	yes	characteristic
208	C₂²×C₅₃	#535	1	1	yes	characteristic
209	C₂²×C₅₃	#540	1	1	yes	characteristic
210	C₄×C₅₃	—	4	2	no	—
211	C₄×C₅₃	—	4	2	no	—
212	C₄×C₅₃	—	4	2	no	—
213	C₄×C₅₃	—	8	1	no	—
214	C₄×C₅₃	—	8	2	no	—
215	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
216	C₄×C₅₃	#518	1	1	yes	characteristic
217	C₄×C₅₃	#535	1	1	yes	characteristic
218	C₄×C₅₃	#540	1	1	yes	characteristic
219	#3155	C₂×C₅₃	1	1	yes	characteristic
220	#2957	C₂×C₅₃	1	1	yes	characteristic, complemented
221	#2948	C₂×C₅₃	1	1	yes	characteristic, complemented
222	#1978	C₂×C₅₃	1	1	yes	characteristic, complemented
223	#1976	C₂×C₅₃	1	1	yes	characteristic, complemented
224	#1911	C₂×C₅₃	1	2	yes	complemented
225	#1910	C₂×C₅₃	1	2	yes	complemented
226	#1904	C₂×C₅₃	1	2	yes	complemented
227	#1903	C₂×C₅₃	1	2	yes	complemented
228	#1833	C₂×C₅₃	1	1	yes	characteristic, complemented
229	#1747	C₂×C₅₃	1	1	yes	characteristic, complemented
230	C₂×C₅₃	—	2	1	no	—
231	C₂×C₅₃	—	2	1	no	—
232	C₂×C₅₃	—	2	1	no	—
233	C₂×C₅₃	—	8	2	no	—
234	C₂×C₅₃	—	8	4	no	—
235	C₂×C₅₃	#2328	1	1	yes	characteristic
236	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	2	yes	—
237	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
238	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
239	#568	C₂²×C₅₃	1	2	yes	—
240	#535	C₂²×C₅₃	1	2	yes	—
241	#518	C₂²×C₅₃	1	2	yes	—
242	#496	C₄×C₅₃	1	4	yes	complemented
243	#495	C₄×C₅₃	1	2	yes	complemented
244	#493	C₄×C₅₃	1	2	yes	complemented
245	#444	—	2	2	no	—
246	#443	—	2	2	no	—
247	#442	—	2	2	no	—
248	C₄×D₈	—	2	2	no	—
249	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
250	#427	C₂²×C₅₃	1	1	yes	characteristic
251	#421	C₂²×C₅₃	1	2	yes	—
252	#421	C₂²×C₅₃	1	2	yes	—
253	#418	C₂²×C₅₃	1	2	yes	—
254	#418	C₂²×C₅₃	1	2	yes	—
255	C₂×#100	C₂²×C₅₃	1	2	yes	—
256	#410	C₂²×C₅₃	1	2	yes	—
257	#405	C₂²×C₅₃	1	1	yes	characteristic
258	#405	C₂²×C₅₃	1	1	yes	characteristic
259	#405	C₂²×C₅₃	1	1	yes	characteristic
260	#344	C₂²×C₅₃	1	2	yes	—
261	#338	C₂²×C₅₃	1	1	yes	characteristic
262	H(3,ℤ/4)	C₂²×C₅₃	1	2	yes	—
263	H(3,ℤ/4)	C₂²×C₅₃	1	1	yes	characteristic
264	H(3,ℤ/4)	C₂²×C₅₃	1	1	yes	characteristic
265	H(3,ℤ/4)	C₂²×C₅₃	1	1	yes	characteristic
266	C₅₃	#22286	1	1	yes	characteristic, direct product factor, Sylow
267	C₂×#41	C₂³×C₅₃	1	2	yes	—
268	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
269	AGL(1,ℤ/8)	—	4	2	no	—
270	AGL(1,ℤ/8)	—	4	2	no	—
271	#135	—	2	1	no	—
272	#135	—	2	1	no	—
273	#135	—	2	1	no	—
274	#135	—	2	1	no	—
275	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
276	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	complemented
277	C₂×SD₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
278	C₂×D₈	—	2	1	no	—
279	C₂×D₈	—	2	1	no	—
280	C₂×D₈	—	2	1	no	—
281	C₂×D₈	—	2	1	no	—
282	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
283	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
284	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
285	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	complemented
286	#131	—	2	4	no	—
287	#131	—	4	2	no	—
288	C₂×M₁₆	—	4	2	no	—
289	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
290	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
291	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
292	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
293	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
294	#127	C₂×C₄×C₅₃	1	4	yes	complemented
295	#124	C₂×C₄×C₅₃	1	4	yes	complemented
296	#121	—	4	2	no	—
297	#120	—	4	2	no	—
298	C₄×D₄	—	2	4	no	—
299	C₄×D₄	—	4	2	no	—
300	#117	C₂³×C₅₃	1	2	yes	—
301	#117	C₂³×C₅₃	1	1	yes	characteristic
302	#117	C₂³×C₅₃	1	1	yes	characteristic
303	#117	C₂³×C₅₃	1	1	yes	characteristic
304	C₂×C₂²⋊C₄	—	4	2	no	—
305	#108	—	2	1	no	—
306	#108	—	2	1	no	—
307	#107	—	2	1	no	—
308	#106	—	2	1	no	—
309	#104	—	2	4	no	—
310	#104	—	4	2	no	—
311	#102	—	2	4	no	—
312	#102	—	4	2	no	—
313	#100	—	4	2	no	—
314	#99	—	4	2	no	—
315	#97	—	2	1	no	—
316	#97	C₂×C₄×C₅₃	1	2	yes	complemented
317	#97	D₄×C₅₃	1	2	yes	complemented
318	C₄×C₈	—	2	1	no	—
319	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	complemented
320	C₄×C₈	D₄×C₅₃	1	2	yes	complemented
321	C₂⁴	—	2	2	no	—
322	#41	—	2	2	no	—
323	#41	—	2	2	no	—
324	#41	—	2	2	no	—
325	#41	—	2	2	no	—
326	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	4	4	no	—
334	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
335	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	4	4	no	—
338	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
339	Q₁₆	—	2	1	no	—
340	Q₁₆	—	2	1	no	—
341	SD₁₆	—	2	2	no	—
342	SD₁₆	—	2	2	no	—
343	SD₁₆	—	4	4	no	—
344	D₈	—	2	2	no	—
345	D₈	—	2	2	no	—
346	D₈	—	2	2	no	—
347	D₈	—	2	2	no	—
348	D₈	—	2	1	no	—
349	D₈	—	2	1	no	—
350	D₈	—	4	4	no	—
351	M₁₆	—	2	1	no	—
352	M₁₆	—	2	1	no	—
353	M₁₆	—	2	1	no	—
354	M₁₆	—	2	1	no	—
355	M₁₆	—	2	2	no	—
356	M₁₆	—	2	1	no	—
357	M₁₆	—	2	2	no	—
358	M₁₆	—	2	1	no	—
359	M₁₆	—	4	4	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	4	4	no	—
363	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
364	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
365	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
366	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
367	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
368	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
369	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
370	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	1	no	—
375	C₂²⋊C₄	—	2	1	no	—
376	C₂²⋊C₄	—	2	1	no	—
377	C₂²⋊C₄	—	2	1	no	—
378	C₂²⋊C₄	—	2	2	no	—
379	C₂²⋊C₄	—	4	4	no	—
380	C₂²⋊C₄	—	4	4	no	—
381	C₄²	—	2	1	no	—
382	C₄²	—	2	2	no	—
383	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
384	C₄²	C₂×D₄×C₅₃	1	2	yes	—
385	C₂³	—	2	4	no	—
386	C₂³	—	2	2	no	—
387	C₂³	—	4	2	no	—
388	C₂³	—	4	2	no	—
389	C₂³	—	16	2	no	—
390	C₂³	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
391	Q₈	—	2	2	no	—
392	Q₈	—	2	2	no	—
393	D₄	—	2	4	no	—
394	D₄	—	2	2	no	—
395	D₄	—	2	2	no	—
396	D₄	—	2	4	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	2	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	2	no	—
401	D₄	—	16	4	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	16	4	no	—
410	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
411	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
412	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
413	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
414	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
415	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
416	C₈	—	2	2	no	—
417	C₈	—	2	2	no	—
418	C₈	—	4	2	no	—
419	C₈	—	4	1	no	—
420	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
421	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
422	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
423	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
424	C₂²	—	4	1	no	—
425	C₂²	—	4	2	no	—
426	C₂²	—	4	4	no	—
427	C₂²	—	8	2	no	—
428	C₂²	—	8	4	no	—
429	C₂²	—	8	4	no	—
430	C₂²	—	16	2	no	—
431	C₂²	C₅₃×#518	1	1	yes	characteristic
432	C₂²	C₅₃×#535	1	1	yes	characteristic
433	C₂²	C₅₃×#540	1	1	yes	characteristic
434	C₄	—	4	2	no	—
435	C₄	—	4	2	no	—
436	C₄	—	4	2	no	—
437	C₄	—	8	2	no	—
438	C₄	—	8	1	no	—
439	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
440	C₄	C₅₃×#518	1	1	yes	characteristic
441	C₄	C₅₃×#535	1	1	yes	characteristic
442	C₄	C₅₃×#540	1	1	yes	characteristic
443	C₂	—	2	1	no	—
444	C₂	—	2	1	no	—
445	C₂	—	2	1	no	—
446	C₂	—	8	4	no	—
447	C₂	—	8	2	no	—
448	C₂	C₅₃×#2328	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#2328
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₅₃×#3155	C₂
LCS	G ▷ C₂×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
2. C₅₃×#2957	C₂	semidirect
3. C₅₃×#2948	C₂	semidirect
4. C₅₃×#1978	C₂	semidirect
5. C₅₃×#1976	C₂	semidirect
6. C₅₃×#1911	C₂	semidirect
7. C₅₃×#1910	C₂	semidirect
8. C₅₃×#1904	C₂	semidirect
9. C₅₃×#1903	C₂	semidirect
10. C₅₃×#1833	C₂	semidirect
11. C₅₃×#1747	C₂	semidirect
18. C₅₃×#496	C₄	semidirect
19. C₅₃×#495	C₄	semidirect
20. C₅₃×#493	C₄	semidirect
50. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
51. C₂×SD₁₆×C₅₃	C₂×C₄	semidirect
52. C₂×SD₁₆×C₅₃	C₂×C₄	semidirect
59. C₂×D₈×C₅₃	C₂×C₄	semidirect
60. C₂×D₈×C₅₃	C₂×C₄	semidirect
69. #127×C₅₃	C₂×C₄	semidirect
70. #124×C₅₃	C₂×C₄	semidirect
91. #97×C₅₃	C₂×C₄	semidirect
92. #97×C₅₃	D₄	semidirect
94. C₄×C₈×C₅₃	C₂×C₄	semidirect
95. C₄×C₈×C₅₃	D₄	semidirect
199. #22286	C₅₃	direct (N × H)
220. #2957	C₂×C₅₃	semidirect
221. #2948	C₂×C₅₃	semidirect
222. #1978	C₂×C₅₃	semidirect
223. #1976	C₂×C₅₃	semidirect
224. #1911	C₂×C₅₃	semidirect
225. #1910	C₂×C₅₃	semidirect
226. #1904	C₂×C₅₃	semidirect
227. #1903	C₂×C₅₃	semidirect
228. #1833	C₂×C₅₃	semidirect
229. #1747	C₂×C₅₃	semidirect
242. #496	C₄×C₅₃	semidirect
243. #495	C₄×C₅₃	semidirect
244. #493	C₄×C₅₃	semidirect
266. C₅₃	#22286	direct (N × H)
275. C₂×Q₁₆	C₂×C₄×C₅₃	semidirect
276. C₂×SD₁₆	C₂×C₄×C₅₃	semidirect
277. C₂×SD₁₆	C₂×C₄×C₅₃	semidirect
284. C₂×D₈	C₂×C₄×C₅₃	semidirect
285. C₂×D₈	C₂×C₄×C₅₃	semidirect
294. #127	C₂×C₄×C₅₃	semidirect
295. #124	C₂×C₄×C₅₃	semidirect
316. #97	C₂×C₄×C₅₃	semidirect
317. #97	D₄×C₅₃	semidirect
319. C₄×C₈	C₂×C₄×C₅₃	semidirect
320. C₄×C₈	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×#3155	2
C₅₃×#2957	2
C₅₃×#2948	2
C₅₃×#1978	2
C₅₃×#1976	2
C₅₃×#1911	2
C₅₃×#1910	2
C₅₃×#1904	2
C₅₃×#1903	2
C₅₃×#1833	2
C₅₃×#1747	2
#22286	53

Subgroups — C53×#22286 #27229798

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22286 #27229798