Subgroups — C₅₃×#19883 #27227395

Order 13568.

798 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×#3334	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3272	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1685	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1673	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1639	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1625	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
13	C₂²×D₈×C₅₃	—	2	1	no	—
14	C₅₃×#547	C₂²	1	2	yes	complemented
15	D₄²×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#500	C₂²	1	1	yes	characteristic, complemented
17	C₅₃×#493	C₂²	1	1	yes	characteristic
18	C₅₃×#492	C₂²	1	2	yes	complemented
19	C₅₃×#486	C₂²	1	2	yes	complemented
20	C₅₃×#480	—	2	2	no	—
21	C₅₃×#474	C₂²	1	2	yes	complemented
22	C₅₃×#468	—	2	2	no	—
23	C₅₃×#466	—	2	1	no	—
24	C₅₃×#466	—	2	1	no	—
25	C₅₃×#461	C₂²	1	2	yes	—
26	C₅₃×#459	C₂²	1	2	yes	—
27	C₅₃×#459	C₂²	1	2	yes	complemented
28	C₅₃×#450	—	2	2	no	—
29	C₅₃×#449	—	2	2	no	—
30	C₅₃×#447	—	2	2	no	—
31	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#436	C₂²	1	2	yes	complemented
34	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
36	C₂×#107×C₅₃	—	2	1	no	—
37	C₅₃×#335	C₂²	1	1	yes	characteristic, complemented
38	C₅₃×#335	C₂²	1	1	yes	characteristic, complemented
39	C₅₃×#332	C₂²	1	2	yes	complemented
40	C₅₃×#331	C₂²	1	2	yes	complemented
41	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
44	C₈²×C₅₃	C₂²	1	1	yes	characteristic, complemented
45	C₂×#41×C₅₃	—	2	2	no	—
46	C₂²×D₄×C₅₃	—	2	2	no	—
47	C₂²×D₄×C₅₃	—	2	2	no	—
48	AGL(1,ℤ/8)×C₅₃	—	4	8	no	—
49	C₂×SD₁₆×C₅₃	—	2	2	no	—
50	C₂×SD₁₆×C₅₃	—	2	2	no	—
51	C₂×SD₁₆×C₅₃	—	2	2	no	—
52	C₂×D₈×C₅₃	—	2	1	no	—
53	C₂×D₈×C₅₃	—	2	1	no	—
54	C₂×D₈×C₅₃	—	2	2	no	—
55	C₂×D₈×C₅₃	—	2	2	no	—
56	C₂×D₈×C₅₃	—	4	4	no	—
57	C₂×D₈×C₅₃	D₄	1	1	yes	characteristic
58	C₂×D₈×C₅₃	D₄	1	1	yes	characteristic
59	C₂×M₁₆×C₅₃	—	2	2	no	—
60	C₂²×C₈×C₅₃	—	2	1	no	—
61	C₂²×C₈×C₅₃	—	2	1	no	—
62	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
63	#127×C₅₃	C₂³	1	2	yes	—
64	#123×C₅₃	—	4	2	no	—
65	C₂²⋊Q₈×C₅₃	—	2	2	no	—
66	#121×C₅₃	—	2	2	no	—
67	#121×C₅₃	—	2	2	no	—
68	#121×C₅₃	—	2	2	no	—
69	#120×C₅₃	—	4	2	no	—
70	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
71	C₄×D₄×C₅₃	C₂³	1	2	yes	—
72	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
74	#107×C₅₃	—	2	1	no	—
75	#107×C₅₃	—	2	1	no	—
76	#107×C₅₃	—	2	2	no	—
77	#107×C₅₃	D₄	1	1	yes	characteristic
78	#107×C₅₃	D₄	1	1	yes	characteristic
79	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
81	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
82	#103×C₅₃	—	2	2	no	—
83	#102×C₅₃	—	2	2	no	—
84	#102×C₅₃	—	2	2	no	—
85	#102×C₅₃	—	2	2	no	—
86	#102×C₅₃	—	4	2	no	—
87	#102×C₅₃	D₄	1	2	yes	—
88	#102×C₅₃	D₄	1	1	yes	characteristic
89	#102×C₅₃	D₄	1	1	yes	characteristic
90	C₂²⋊C₈×C₅₃	—	2	1	no	—
91	C₂²⋊C₈×C₅₃	—	2	1	no	—
92	C₂²⋊C₈×C₅₃	—	2	2	no	—
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
96	C₂⁴×C₅₃	—	4	2	no	—
97	#41×C₅₃	—	4	4	no	—
98	#41×C₅₃	—	8	2	no	—
99	C₂×Q₈×C₅₃	—	2	2	no	—
100	C₂×D₄×C₅₃	—	2	2	no	—
101	C₂×D₄×C₅₃	—	2	2	no	—
102	C₂×D₄×C₅₃	—	2	2	no	—
103	C₂×D₄×C₅₃	—	4	2	no	—
104	C₂×D₄×C₅₃	—	4	2	no	—
105	C₂×D₄×C₅₃	—	4	4	no	—
106	C₂×D₄×C₅₃	—	4	4	no	—
107	C₂×D₄×C₅₃	—	4	2	no	—
108	C₂×D₄×C₅₃	—	8	2	no	—
109	C₂×D₄×C₅₃	—	8	2	no	—
110	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
111	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	—	2	1	no	—
114	C₂²×C₄×C₅₃	—	2	1	no	—
115	C₂²×C₄×C₅₃	—	2	2	no	—
116	C₂²×C₄×C₅₃	—	4	2	no	—
117	SD₁₆×C₅₃	—	4	4	no	—
118	SD₁₆×C₅₃	—	4	4	no	—
119	SD₁₆×C₅₃	—	8	2	no	—
120	D₈×C₅₃	—	2	2	no	—
121	D₈×C₅₃	—	4	4	no	—
122	D₈×C₅₃	—	4	4	no	—
123	D₈×C₅₃	—	4	2	no	—
124	D₈×C₅₃	—	8	1	no	—
125	D₈×C₅₃	D₈	1	4	yes	complemented
126	M₁₆×C₅₃	—	4	4	no	—
127	C₂×C₈×C₅₃	—	2	1	no	—
128	C₂×C₈×C₅₃	—	2	1	no	—
129	C₂×C₈×C₅₃	—	2	2	no	—
130	C₂×C₈×C₅₃	—	2	1	no	—
131	C₂×C₈×C₅₃	—	4	2	no	—
132	C₂×C₈×C₅₃	—	4	2	no	—
133	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄×C₅₃	—	2	2	no	—
138	C₄⋊C₄×C₅₃	—	2	2	no	—
139	C₄⋊C₄×C₅₃	—	4	2	no	—
140	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
141	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
142	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
143	C₂²⋊C₄×C₅₃	—	2	1	no	—
144	C₂²⋊C₄×C₅₃	—	2	2	no	—
145	C₂²⋊C₄×C₅₃	—	2	1	no	—
146	C₂²⋊C₄×C₅₃	—	4	2	no	—
147	C₂²⋊C₄×C₅₃	—	4	2	no	—
148	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
149	C₂³×C₅₃	—	2	1	no	—
150	C₂³×C₅₃	—	2	1	no	—
151	C₂³×C₅₃	—	2	2	no	—
152	C₂³×C₅₃	—	4	2	no	—
153	C₂³×C₅₃	—	8	4	no	—
154	C₂³×C₅₃	—	8	4	no	—
155	C₂³×C₅₃	—	16	2	no	—
156	Q₈×C₅₃	—	4	2	no	—
157	Q₈×C₅₃	—	4	2	no	—
158	D₄×C₅₃	—	2	2	no	—
159	D₄×C₅₃	—	2	1	no	—
160	D₄×C₅₃	—	2	1	no	—
161	D₄×C₅₃	—	2	2	no	—
162	D₄×C₅₃	—	4	2	no	—
163	D₄×C₅₃	—	4	2	no	—
164	D₄×C₅₃	—	4	2	no	—
165	D₄×C₅₃	—	4	2	no	—
166	D₄×C₅₃	—	8	2	no	—
167	D₄×C₅₃	—	8	4	no	—
168	D₄×C₅₃	—	8	4	no	—
169	D₄×C₅₃	—	8	4	no	—
170	D₄×C₅₃	C₂×D₈	1	2	yes	—
171	D₄×C₅₃	C₂×D₈	1	2	yes	complemented
172	C₂×C₄×C₅₃	—	2	2	no	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	4	2	no	—
176	C₂×C₄×C₅₃	—	4	1	no	—
177	C₂×C₄×C₅₃	—	4	1	no	—
178	C₂×C₄×C₅₃	—	4	2	no	—
179	C₂×C₄×C₅₃	—	4	1	no	—
180	C₂×C₄×C₅₃	—	4	1	no	—
181	C₂×C₄×C₅₃	—	4	2	no	—
182	C₂×C₄×C₅₃	—	8	4	no	—
183	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
186	C₈×C₅₃	—	2	1	no	—
187	C₈×C₅₃	—	2	1	no	—
188	C₈×C₅₃	—	4	1	no	—
189	C₈×C₅₃	—	4	2	no	—
190	C₈×C₅₃	—	4	1	no	—
191	C₈×C₅₃	—	4	1	no	—
192	C₈×C₅₃	C₂×D₈	1	2	yes	complemented
193	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	complemented
194	#19883	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂²×C₅₃	—	2	2	no	—
196	C₂²×C₅₃	—	2	2	no	—
197	C₂²×C₅₃	—	4	1	no	—
198	C₂²×C₅₃	—	4	1	no	—
199	C₂²×C₅₃	—	4	1	no	—
200	C₂²×C₅₃	—	4	1	no	—
201	C₂²×C₅₃	—	4	2	no	—
202	C₂²×C₅₃	—	4	2	no	—
203	C₂²×C₅₃	—	4	2	no	—
204	C₂²×C₅₃	—	8	2	no	—
205	C₂²×C₅₃	—	8	2	no	—
206	C₂²×C₅₃	—	8	2	no	—
207	C₂²×C₅₃	—	16	8	no	—
208	C₂²×C₅₃	D₄²	1	1	yes	characteristic
209	C₄×C₅₃	—	2	1	no	—
210	C₄×C₅₃	—	4	1	no	—
211	C₄×C₅₃	—	4	1	no	—
212	C₄×C₅₃	—	4	2	no	—
213	C₄×C₅₃	—	8	2	no	—
214	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
215	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
216	C₄×C₅₃	#576	1	1	yes	characteristic
217	C₄×C₅₃	#577	1	1	yes	characteristic
218	#3334	C₂×C₅₃	1	1	yes	characteristic, complemented
219	D₄×D₈	C₂×C₅₃	1	1	yes	characteristic, complemented
220	#3272	C₂×C₅₃	1	2	yes	complemented
221	#1706	C₂×C₅₃	1	1	yes	characteristic, complemented
222	#1685	C₂×C₅₃	1	2	yes	complemented
223	#1673	C₂×C₅₃	1	1	yes	characteristic, complemented
224	#1667	C₂×C₅₃	1	1	yes	characteristic, complemented
225	#1639	C₂×C₅₃	1	2	yes	complemented
226	#1625	C₂×C₅₃	1	2	yes	complemented
227	#1589	C₂×C₅₃	1	1	yes	characteristic, complemented
228	C₈×D₈	C₂×C₅₃	1	1	yes	characteristic, complemented
229	C₂×C₅₃	—	4	2	no	—
230	C₂×C₅₃	—	4	2	no	—
231	C₂×C₅₃	—	8	2	no	—
232	C₂×C₅₃	—	16	2	no	—
233	C₂×C₅₃	#3272	1	1	yes	characteristic
234	C₂×C₅₃	D₄×D₈	1	1	yes	characteristic
235	C₂×C₅₃	#3290	1	1	yes	characteristic
236	C₂×AGL(1,ℤ/8)	—	2	2	no	—
237	C₂²×D₈	—	2	1	no	—
238	#547	C₂²×C₅₃	1	2	yes	complemented
239	D₄²	C₂²×C₅₃	1	2	yes	—
240	#500	C₂²×C₅₃	1	1	yes	characteristic, complemented
241	#493	C₂²×C₅₃	1	1	yes	characteristic
242	#492	C₂²×C₅₃	1	2	yes	complemented
243	#486	C₂²×C₅₃	1	2	yes	complemented
244	#480	—	2	2	no	—
245	#474	C₂²×C₅₃	1	2	yes	complemented
246	#468	—	2	2	no	—
247	#466	—	2	1	no	—
248	#466	—	2	1	no	—
249	#461	C₂²×C₅₃	1	2	yes	—
250	#459	C₂²×C₅₃	1	2	yes	—
251	#459	C₂²×C₅₃	1	2	yes	complemented
252	#450	—	2	2	no	—
253	#449	—	2	2	no	—
254	#447	—	2	2	no	—
255	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
256	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
257	#436	C₂²×C₅₃	1	2	yes	complemented
258	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic
259	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
260	C₂×#107	—	2	1	no	—
261	#335	C₂²×C₅₃	1	1	yes	characteristic, complemented
262	#335	C₂²×C₅₃	1	1	yes	characteristic, complemented
263	#332	C₂²×C₅₃	1	2	yes	complemented
264	#331	C₂²×C₅₃	1	2	yes	complemented
265	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
266	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
267	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
268	C₈²	C₂²×C₅₃	1	1	yes	characteristic, complemented
269	C₅₃	#19883	1	1	yes	characteristic, direct product factor, Sylow
270	C₂×#41	—	2	2	no	—
271	C₂²×D₄	—	2	2	no	—
272	C₂²×D₄	—	2	2	no	—
273	AGL(1,ℤ/8)	—	4	8	no	—
274	C₂×SD₁₆	—	2	2	no	—
275	C₂×SD₁₆	—	2	2	no	—
276	C₂×SD₁₆	—	2	2	no	—
277	C₂×D₈	—	2	2	no	—
278	C₂×D₈	—	2	2	no	—
279	C₂×D₈	—	2	1	no	—
280	C₂×D₈	—	2	1	no	—
281	C₂×D₈	—	4	4	no	—
282	C₂×D₈	D₄×C₅₃	1	1	yes	characteristic
283	C₂×D₈	D₄×C₅₃	1	1	yes	characteristic
284	C₂×M₁₆	—	2	2	no	—
285	C₂²×C₈	—	2	1	no	—
286	C₂²×C₈	—	2	1	no	—
287	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
288	#127	C₂³×C₅₃	1	2	yes	—
289	#123	—	4	2	no	—
290	C₂²⋊Q₈	—	2	2	no	—
291	#121	—	2	2	no	—
292	#121	—	2	2	no	—
293	#121	—	2	2	no	—
294	#120	—	4	2	no	—
295	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
296	C₄×D₄	C₂³×C₅₃	1	2	yes	—
297	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
298	C₂×C₄⋊C₄	—	2	2	no	—
299	#107	—	2	1	no	—
300	#107	—	2	1	no	—
301	#107	—	2	2	no	—
302	#107	D₄×C₅₃	1	1	yes	characteristic
303	#107	D₄×C₅₃	1	1	yes	characteristic
304	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
305	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
306	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
307	#103	—	2	2	no	—
308	#102	—	2	2	no	—
309	#102	—	2	2	no	—
310	#102	—	2	2	no	—
311	#102	—	4	2	no	—
312	#102	D₄×C₅₃	1	2	yes	—
313	#102	D₄×C₅₃	1	1	yes	characteristic
314	#102	D₄×C₅₃	1	1	yes	characteristic
315	C₂²⋊C₈	—	2	1	no	—
316	C₂²⋊C₈	—	2	1	no	—
317	C₂²⋊C₈	—	2	2	no	—
318	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
319	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
320	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
321	C₂⁴	—	4	2	no	—
322	#41	—	4	4	no	—
323	#41	—	8	2	no	—
324	C₂×Q₈	—	2	2	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	4	2	no	—
329	C₂×D₄	—	4	2	no	—
330	C₂×D₄	—	4	4	no	—
331	C₂×D₄	—	4	4	no	—
332	C₂×D₄	—	4	2	no	—
333	C₂×D₄	—	8	2	no	—
334	C₂×D₄	—	8	2	no	—
335	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
336	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
337	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	4	2	no	—
342	SD₁₆	—	4	4	no	—
343	SD₁₆	—	4	4	no	—
344	SD₁₆	—	8	2	no	—
345	D₈	—	2	2	no	—
346	D₈	—	4	4	no	—
347	D₈	—	4	4	no	—
348	D₈	—	4	2	no	—
349	D₈	—	8	1	no	—
350	D₈	D₈×C₅₃	1	4	yes	complemented
351	M₁₆	—	4	4	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	4	2	no	—
357	C₂×C₈	—	4	2	no	—
358	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
359	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
360	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
361	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	4	2	no	—
365	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
366	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
367	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
368	C₂²⋊C₄	—	2	1	no	—
369	C₂²⋊C₄	—	2	2	no	—
370	C₂²⋊C₄	—	2	1	no	—
371	C₂²⋊C₄	—	4	2	no	—
372	C₂²⋊C₄	—	4	2	no	—
373	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
374	C₂³	—	2	1	no	—
375	C₂³	—	2	1	no	—
376	C₂³	—	2	2	no	—
377	C₂³	—	4	2	no	—
378	C₂³	—	8	4	no	—
379	C₂³	—	8	4	no	—
380	C₂³	—	16	2	no	—
381	Q₈	—	4	2	no	—
382	Q₈	—	4	2	no	—
383	D₄	—	2	2	no	—
384	D₄	—	2	1	no	—
385	D₄	—	2	1	no	—
386	D₄	—	2	2	no	—
387	D₄	—	4	2	no	—
388	D₄	—	4	2	no	—
389	D₄	—	4	2	no	—
390	D₄	—	4	2	no	—
391	D₄	—	8	2	no	—
392	D₄	—	8	4	no	—
393	D₄	—	8	4	no	—
394	D₄	—	8	4	no	—
395	D₄	C₂×D₈×C₅₃	1	2	yes	—
396	D₄	C₂×D₈×C₅₃	1	2	yes	complemented
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	8	4	no	—
408	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
409	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
410	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
411	C₈	—	2	1	no	—
412	C₈	—	2	1	no	—
413	C₈	—	4	1	no	—
414	C₈	—	4	2	no	—
415	C₈	—	4	1	no	—
416	C₈	—	4	1	no	—
417	C₈	C₂×D₈×C₅₃	1	2	yes	complemented
418	C₈	AGL(1,ℤ/8)×C₅₃	1	2	yes	complemented
419	C₂²	—	2	2	no	—
420	C₂²	—	2	2	no	—
421	C₂²	—	4	1	no	—
422	C₂²	—	4	1	no	—
423	C₂²	—	4	1	no	—
424	C₂²	—	4	1	no	—
425	C₂²	—	4	2	no	—
426	C₂²	—	4	2	no	—
427	C₂²	—	4	2	no	—
428	C₂²	—	8	2	no	—
429	C₂²	—	8	2	no	—
430	C₂²	—	8	2	no	—
431	C₂²	—	16	8	no	—
432	C₂²	D₄²×C₅₃	1	1	yes	characteristic
433	C₄	—	2	1	no	—
434	C₄	—	4	1	no	—
435	C₄	—	4	1	no	—
436	C₄	—	4	2	no	—
437	C₄	—	8	2	no	—
438	C₄	C₂²×D₈×C₅₃	1	1	yes	characteristic
439	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
440	C₄	C₅₃×#576	1	1	yes	characteristic
441	C₄	C₅₃×#577	1	1	yes	characteristic
442	C₂	—	4	2	no	—
443	C₂	—	4	2	no	—
444	C₂	—	8	2	no	—
445	C₂	—	16	2	no	—
446	C₂	C₅₃×#3272	1	1	yes	characteristic
447	C₂	D₄×D₈×C₅₃	1	1	yes	characteristic
448	C₂	C₅₃×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3334	C₂	semidirect
2. D₄×D₈×C₅₃	C₂	semidirect
3. C₅₃×#3272	C₂	semidirect
4. C₅₃×#1706	C₂	semidirect
5. C₅₃×#1685	C₂	semidirect
6. C₅₃×#1673	C₂	semidirect
7. C₅₃×#1667	C₂	semidirect
8. C₅₃×#1639	C₂	semidirect
9. C₅₃×#1625	C₂	semidirect
10. C₅₃×#1589	C₂	semidirect
11. C₈×D₈×C₅₃	C₂	semidirect
14. C₅₃×#547	C₂²	semidirect
16. C₅₃×#500	C₂²	semidirect
18. C₅₃×#492	C₂²	semidirect
19. C₅₃×#486	C₂²	semidirect
21. C₅₃×#474	C₂²	semidirect
27. C₅₃×#459	C₂²	semidirect
33. C₅₃×#436	C₂²	semidirect
35. C₈×D₄×C₅₃	C₂²	semidirect
37. C₅₃×#335	C₂²	semidirect
38. C₅₃×#335	C₂²	semidirect
39. C₅₃×#332	C₂²	semidirect
40. C₅₃×#331	C₂²	semidirect
42. C₅₃×D₄⋊C₈	C₂²	semidirect
43. C₅₃×D₄⋊C₈	C₂²	semidirect
44. C₈²×C₅₃	C₂²	semidirect
125. D₈×C₅₃	D₈	semidirect
171. D₄×C₅₃	C₂×D₈	semidirect
192. C₈×C₅₃	C₂×D₈	semidirect
193. C₈×C₅₃	AGL(1,ℤ/8)	semidirect
194. #19883	C₅₃	direct (N × H)
218. #3334	C₂×C₅₃	semidirect
219. D₄×D₈	C₂×C₅₃	semidirect
220. #3272	C₂×C₅₃	semidirect
221. #1706	C₂×C₅₃	semidirect
222. #1685	C₂×C₅₃	semidirect
223. #1673	C₂×C₅₃	semidirect
224. #1667	C₂×C₅₃	semidirect
225. #1639	C₂×C₅₃	semidirect
226. #1625	C₂×C₅₃	semidirect
227. #1589	C₂×C₅₃	semidirect
228. C₈×D₈	C₂×C₅₃	semidirect
238. #547	C₂²×C₅₃	semidirect
240. #500	C₂²×C₅₃	semidirect
242. #492	C₂²×C₅₃	semidirect
243. #486	C₂²×C₅₃	semidirect
245. #474	C₂²×C₅₃	semidirect
251. #459	C₂²×C₅₃	semidirect
257. #436	C₂²×C₅₃	semidirect
259. C₈×D₄	C₂²×C₅₃	semidirect
261. #335	C₂²×C₅₃	semidirect
262. #335	C₂²×C₅₃	semidirect
263. #332	C₂²×C₅₃	semidirect
264. #331	C₂²×C₅₃	semidirect
266. D₄⋊C₈	C₂²×C₅₃	semidirect
267. D₄⋊C₈	C₂²×C₅₃	semidirect
268. C₈²	C₂²×C₅₃	semidirect
269. C₅₃	#19883	direct (N × H)
350. D₈	D₈×C₅₃	semidirect
396. D₄	C₂×D₈×C₅₃	semidirect
417. C₈	C₂×D₈×C₅₃	semidirect
418. C₈	AGL(1,ℤ/8)×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3334	2
D₄×D₈×C₅₃	2
C₅₃×#3272	2
C₅₃×#1706	2
C₅₃×#1685	2
C₅₃×#1673	2
C₅₃×#1667	2
C₅₃×#1639	2
C₅₃×#1625	2
C₅₃×#1589	2
C₈×D₈×C₅₃	2
#19883	53

Subgroups — C53×#19883 #27227395

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19883 #27227395