Subgroups — C₅₃×#61317 #27268829

Order 13568.

2064 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3480	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3373	C₂	1	6	yes	maximal, complemented
4	C₅₃×#3367	C₂	1	6	yes	maximal, complemented
5	C₅₃×#3359	C₂	1	2	yes	maximal, complemented
6	C₅₃×#3358	C₂	1	2	yes	maximal, complemented
7	C₅₃×#3141	C₂	1	3	yes	maximal, complemented
8	C₅₃×#3079	C₂	1	3	yes	maximal, complemented
9	C₅₃×#3036	C₂	1	3	yes	maximal, complemented
10	Q₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×AGL(1,ℤ/8)×C₅₃	C₂	1	3	yes	maximal, complemented
12	C₂³×Q₈×C₅₃	—	2	1	no	—
13	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	3	yes	—
14	C₅₃×#559	C₂²	1	3	yes	complemented
15	C₅₃×#557	C₂²	1	1	yes	characteristic, complemented
16	C₅₃×#556	C₂²	1	3	yes	complemented
17	C₅₃×#553	—	2	3	no	—
18	C₅₃×#550	C₂²	1	1	yes	characteristic, complemented
19	D₄×Q₈×C₅₃	—	2	1	no	—
20	D₄×Q₈×C₅₃	—	2	3	no	—
21	D₄×Q₈×C₅₃	C₂²	1	2	yes	—
22	D₄×Q₈×C₅₃	C₂²	1	6	yes	—
23	C₅₃×#548	—	2	1	no	—
24	C₅₃×#532	C₂²	1	3	yes	—
25	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
26	C₅₃×#529	—	2	3	no	—
27	C₂×C₂²⋊Q₈×C₅₃	—	2	3	no	—
28	C₄×#41×C₅₃	—	2	1	no	—
29	C₄×#41×C₅₃	C₂²	1	3	yes	—
30	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₅₃	C₂²	1	3	yes	—
32	C₅₃×#489	C₂²	1	12	yes	complemented
33	C₅₃×#487	C₂²	1	6	yes	complemented
34	C₅₃×#486	C₂²	1	6	yes	complemented
35	C₅₃×#477	C₂²	1	6	yes	complemented
36	C₅₃×#476	C₂²	1	6	yes	complemented
37	C₅₃×#475	C₂²	1	6	yes	complemented
38	C₅₃×#474	C₂²	1	6	yes	complemented
39	C₅₃×#463	C₂²	1	12	yes	—
40	C₅₃×#461	C₂²	1	6	yes	—
41	C₅₃×#459	C₂²	1	6	yes	—
42	C₅₃×#446	C₂²	1	6	yes	complemented
43	C₈×Q₈×C₅₃	C₂²	1	2	yes	complemented
44	C₅₃×#442	C₂²	1	6	yes	—
45	C₅₃×#440	C₂²	1	6	yes	—
46	C₄×SD₁₆×C₅₃	C₂²	1	6	yes	—
47	C₄×D₈×C₅₃	C₂²	1	6	yes	—
48	C₅₃×#428	C₂²	1	3	yes	—
49	C₅₃×#423	C₂²	1	3	yes	—
50	C₅₃×#418	C₂²	1	1	yes	characteristic
51	C₅₃×#417	C₂²	1	3	yes	—
52	C₅₃×#417	C₂²	1	6	yes	—
53	C₄×M₁₆×C₅₃	C₂²	1	3	yes	—
54	C₂×#41×C₅₃	—	2	1	no	—
55	C₂×#41×C₅₃	C₂³	1	3	yes	—
56	C₂²×Q₈×C₅₃	—	2	6	no	—
57	C₂²×Q₈×C₅₃	—	2	2	no	—
58	C₂²×Q₈×C₅₃	—	2	3	no	—
59	C₂²×Q₈×C₅₃	—	4	8	no	—
60	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
61	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₅₃	—	2	3	no	—
63	AGL(1,ℤ/8)×C₅₃	—	2	12	no	—
64	C₂×SD₁₆×C₅₃	C₂³	1	6	yes	—
65	C₂×D₈×C₅₃	C₂³	1	6	yes	—
66	C₂×M₁₆×C₅₃	C₂³	1	3	yes	—
67	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₅₃	—	2	3	no	—
69	C₄⋊Q₈×C₅₃	—	2	3	no	—
70	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
71	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
72	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
73	#127×C₅₃	C₂³	1	3	yes	—
74	#126×C₅₃	—	2	6	no	—
75	#125×C₅₃	—	2	3	no	—
76	#124×C₅₃	—	2	3	no	—
77	#124×C₅₃	C₂³	1	6	yes	—
78	#123×C₅₃	—	2	6	no	—
79	C₂²⋊Q₈×C₅₃	—	2	6	no	—
80	C₂²⋊Q₈×C₅₃	—	2	6	no	—
81	C₂²⋊Q₈×C₅₃	—	2	12	no	—
82	C₂²⋊Q₈×C₅₃	—	2	3	no	—
83	#121×C₅₃	—	2	3	no	—
84	#121×C₅₃	—	2	3	no	—
85	#121×C₅₃	—	2	3	no	—
86	C₄×Q₈×C₅₃	—	2	1	no	—
87	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
88	C₄×Q₈×C₅₃	D₄	1	3	yes	—
89	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₄×Q₈×C₅₃	C₂³	1	3	yes	—
91	C₄×Q₈×C₅₃	C₂³	1	3	yes	—
92	C₄×Q₈×C₅₃	C₂³	1	3	yes	—
93	C₄×D₄×C₅₃	—	2	3	no	—
94	C₄×D₄×C₅₃	—	2	3	no	—
95	C₄×D₄×C₅₃	—	2	3	no	—
96	C₄×D₄×C₅₃	—	2	3	no	—
97	C₄×D₄×C₅₃	—	2	3	no	—
98	C₄×D₄×C₅₃	—	2	3	no	—
99	C₄×D₄×C₅₃	—	2	3	no	—
100	C₄×D₄×C₅₃	C₂³	1	6	yes	—
101	C₄×D₄×C₅₃	C₂³	1	6	yes	—
102	C₄×D₄×C₅₃	C₂³	1	3	yes	—
103	C₄×D₄×C₅₃	C₂³	1	3	yes	—
104	#117×C₅₃	—	2	3	no	—
105	#117×C₅₃	—	2	3	no	—
106	#117×C₅₃	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
108	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
109	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
110	C₂×C₂²⋊C₄×C₅₃	—	2	3	no	—
111	C₂×C₄²×C₅₃	—	2	3	no	—
112	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
113	#107×C₅₃	C₂³	1	6	yes	—
114	#106×C₅₃	C₂³	1	6	yes	—
115	C₄⋊C₈×C₅₃	C₂³	1	6	yes	—
116	C₄⋊C₈×C₅₃	C₂³	1	6	yes	—
117	#103×C₅₃	C₂³	1	6	yes	—
118	#103×C₅₃	C₂³	1	12	yes	—
119	#102×C₅₃	C₂³	1	6	yes	—
120	#102×C₅₃	C₂³	1	12	yes	—
121	#102×C₅₃	C₂³	1	2	yes	—
122	#102×C₅₃	C₂³	1	2	yes	—
123	#97×C₅₃	C₂³	1	6	yes	—
124	C₄×C₈×C₅₃	C₂³	1	6	yes	—
125	C₂⁴×C₅₃	—	2	1	no	—
126	#41×C₅₃	—	2	6	no	—
127	#41×C₅₃	—	2	6	no	—
128	#41×C₅₃	—	4	4	no	—
129	C₂×Q₈×C₅₃	—	2	4	no	—
130	C₂×Q₈×C₅₃	—	2	6	no	—
131	C₂×Q₈×C₅₃	—	2	6	no	—
132	C₂×Q₈×C₅₃	—	4	12	no	—
133	C₂×Q₈×C₅₃	—	4	24	no	—
134	C₂×Q₈×C₅₃	—	4	8	no	—
135	C₂×Q₈×C₅₃	C₂×D₄	1	3	yes	—
136	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
137	C₂×Q₈×C₅₃	C₂⁴	1	3	yes	—
138	C₂×D₄×C₅₃	—	2	3	no	—
139	C₂×D₄×C₅₃	—	2	1	no	—
140	C₂×D₄×C₅₃	—	2	3	no	—
141	C₂×D₄×C₅₃	—	2	3	no	—
142	C₂×D₄×C₅₃	—	2	2	no	—
143	C₂×D₄×C₅₃	#41	1	4	yes	complemented
144	C₂×D₄×C₅₃	C₂⁴	1	3	yes	—
145	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
146	C₂²×C₄×C₅₃	—	2	3	no	—
147	C₂²×C₄×C₅₃	—	2	3	no	—
148	C₂²×C₄×C₅₃	—	2	3	no	—
149	C₂²×C₄×C₅₃	—	2	6	no	—
150	C₂²×C₄×C₅₃	—	2	6	no	—
151	C₂²×C₄×C₅₃	—	2	3	no	—
152	C₂²×C₄×C₅₃	—	4	12	no	—
153	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
154	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
156	SD₁₆×C₅₃	—	2	12	no	—
157	D₈×C₅₃	—	2	12	no	—
158	M₁₆×C₅₃	—	2	6	no	—
159	M₁₆×C₅₃	—	2	2	no	—
160	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
161	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
162	C₄⋊C₄×C₅₃	—	2	6	no	—
163	C₄⋊C₄×C₅₃	—	2	6	no	—
164	C₄⋊C₄×C₅₃	—	2	3	no	—
165	C₄⋊C₄×C₅₃	—	2	3	no	—
166	C₄⋊C₄×C₅₃	—	2	3	no	—
167	C₄⋊C₄×C₅₃	—	2	3	no	—
168	C₄⋊C₄×C₅₃	—	2	3	no	—
169	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
170	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
171	C₄⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
172	C₄⋊C₄×C₅₃	C₂⁴	1	6	yes	—
173	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
174	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
175	C₂²⋊C₄×C₅₃	—	2	1	no	—
176	C₂²⋊C₄×C₅₃	—	2	6	no	—
177	C₂²⋊C₄×C₅₃	—	2	6	no	—
178	C₂²⋊C₄×C₅₃	—	2	6	no	—
179	C₂²⋊C₄×C₅₃	—	2	3	no	—
180	C₂²⋊C₄×C₅₃	—	2	3	no	—
181	C₂²⋊C₄×C₅₃	—	2	6	no	—
182	C₄²×C₅₃	—	2	1	no	—
183	C₄²×C₅₃	—	2	3	no	—
184	C₄²×C₅₃	—	2	3	no	—
185	C₄²×C₅₃	C₂×D₄	1	3	yes	—
186	C₄²×C₅₃	C₂×D₄	1	3	yes	—
187	C₄²×C₅₃	C₂⁴	1	3	yes	—
188	C₄²×C₅₃	C₂⁴	1	3	yes	—
189	C₂³×C₅₃	—	2	4	no	—
190	C₂³×C₅₃	—	2	2	no	—
191	C₂³×C₅₃	—	2	3	no	—
192	C₂³×C₅₃	—	4	2	no	—
193	C₂³×C₅₃	—	4	2	no	—
194	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
195	Q₈×C₅₃	—	2	6	no	—
196	Q₈×C₅₃	—	2	3	no	—
197	Q₈×C₅₃	—	2	3	no	—
198	Q₈×C₅₃	—	4	12	no	—
199	Q₈×C₅₃	—	4	12	no	—
200	Q₈×C₅₃	AGL(1,ℤ/8)	1	4	yes	complemented
201	D₄×C₅₃	—	2	4	no	—
202	D₄×C₅₃	—	2	3	no	—
203	D₄×C₅₃	—	2	2	no	—
204	D₄×C₅₃	—	2	3	no	—
205	D₄×C₅₃	—	4	6	no	—
206	D₄×C₅₃	—	4	6	no	—
207	D₄×C₅₃	—	8	3	no	—
208	D₄×C₅₃	C₂×#41	1	4	yes	—
209	C₂×C₄×C₅₃	—	2	6	no	—
210	C₂×C₄×C₅₃	—	2	6	no	—
211	C₂×C₄×C₅₃	—	2	1	no	—
212	C₂×C₄×C₅₃	—	2	1	no	—
213	C₂×C₄×C₅₃	—	2	3	no	—
214	C₂×C₄×C₅₃	—	4	3	no	—
215	C₂×C₄×C₅₃	—	4	12	no	—
216	C₂×C₄×C₅₃	—	4	6	no	—
217	C₂×C₄×C₅₃	—	4	6	no	—
218	C₂×C₄×C₅₃	—	4	12	no	—
219	C₂×C₄×C₅₃	—	4	6	no	—
220	C₂×C₄×C₅₃	—	8	3	no	—
221	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
222	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
223	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
224	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
226	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
227	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
228	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
229	C₈×C₅₃	—	2	6	no	—
230	C₈×C₅₃	—	2	2	no	—
231	#61317	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂²×C₅₃	—	2	1	no	—
233	C₂²×C₅₃	—	2	1	no	—
234	C₂²×C₅₃	—	2	4	no	—
235	C₂²×C₅₃	—	4	8	no	—
236	C₂²×C₅₃	—	4	2	no	—
237	C₂²×C₅₃	—	4	2	no	—
238	C₂²×C₅₃	—	4	3	no	—
239	C₂²×C₅₃	—	4	3	no	—
240	C₂²×C₅₃	—	4	3	no	—
241	C₂²×C₅₃	#548	1	2	yes	—
242	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
243	C₄×C₅₃	—	2	3	no	—
244	C₄×C₅₃	—	2	3	no	—
245	C₄×C₅₃	—	2	3	no	—
246	C₄×C₅₃	—	4	1	no	—
247	C₄×C₅₃	—	4	3	no	—
248	C₄×C₅₃	—	4	1	no	—
249	C₄×C₅₃	—	4	6	no	—
250	C₄×C₅₃	#548	1	2	yes	—
251	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	6	yes	—
252	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
253	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
254	#3480	C₂×C₅₃	1	1	yes	characteristic, complemented
255	C₂×D₄×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
256	#3373	C₂×C₅₃	1	6	yes	complemented
257	#3367	C₂×C₅₃	1	6	yes	complemented
258	#3359	C₂×C₅₃	1	2	yes	complemented
259	#3358	C₂×C₅₃	1	2	yes	complemented
260	#3141	C₂×C₅₃	1	3	yes	complemented
261	#3079	C₂×C₅₃	1	3	yes	complemented
262	#3036	C₂×C₅₃	1	3	yes	complemented
263	Q₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
264	C₄×AGL(1,ℤ/8)	C₂×C₅₃	1	3	yes	complemented
265	C₂×C₅₃	—	2	2	no	—
266	C₂×C₅₃	—	4	4	no	—
267	C₂×C₅₃	—	8	3	no	—
268	C₂×C₅₃	C₂×#548	1	1	yes	characteristic
269	C₂×C₅₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
270	C₂×C₅₃	#3586	1	1	yes	characteristic
271	C₂³×Q₈	—	2	1	no	—
272	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	3	yes	—
273	#559	C₂²×C₅₃	1	3	yes	complemented
274	#557	C₂²×C₅₃	1	1	yes	characteristic, complemented
275	#556	C₂²×C₅₃	1	3	yes	complemented
276	#553	—	2	3	no	—
277	#550	C₂²×C₅₃	1	1	yes	characteristic, complemented
278	D₄×Q₈	—	2	1	no	—
279	D₄×Q₈	—	2	3	no	—
280	D₄×Q₈	C₂²×C₅₃	1	2	yes	—
281	D₄×Q₈	C₂²×C₅₃	1	6	yes	—
282	#548	—	2	1	no	—
283	#532	C₂²×C₅₃	1	3	yes	—
284	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	—
285	#529	—	2	3	no	—
286	C₂×C₂²⋊Q₈	—	2	3	no	—
287	C₄×#41	—	2	1	no	—
288	C₄×#41	C₂²×C₅₃	1	3	yes	—
289	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
290	C₂×C₄×D₄	C₂²×C₅₃	1	3	yes	—
291	#489	C₂²×C₅₃	1	12	yes	complemented
292	#487	C₂²×C₅₃	1	6	yes	complemented
293	#486	C₂²×C₅₃	1	6	yes	complemented
294	#477	C₂²×C₅₃	1	6	yes	complemented
295	#476	C₂²×C₅₃	1	6	yes	complemented
296	#475	C₂²×C₅₃	1	6	yes	complemented
297	#474	C₂²×C₅₃	1	6	yes	complemented
298	#463	C₂²×C₅₃	1	12	yes	—
299	#461	C₂²×C₅₃	1	6	yes	—
300	#459	C₂²×C₅₃	1	6	yes	—
301	#446	C₂²×C₅₃	1	6	yes	complemented
302	C₈×Q₈	C₂²×C₅₃	1	2	yes	complemented
303	#442	C₂²×C₅₃	1	6	yes	—
304	#440	C₂²×C₅₃	1	6	yes	—
305	C₄×SD₁₆	C₂²×C₅₃	1	6	yes	—
306	C₄×D₈	C₂²×C₅₃	1	6	yes	—
307	#428	C₂²×C₅₃	1	3	yes	—
308	#423	C₂²×C₅₃	1	3	yes	—
309	#418	C₂²×C₅₃	1	1	yes	characteristic
310	#417	C₂²×C₅₃	1	6	yes	—
311	#417	C₂²×C₅₃	1	3	yes	—
312	C₄×M₁₆	C₂²×C₅₃	1	3	yes	—
313	C₅₃	#61317	1	1	yes	characteristic, direct product factor, Sylow
314	C₂×#41	—	2	1	no	—
315	C₂×#41	C₂³×C₅₃	1	3	yes	—
316	C₂²×Q₈	—	2	6	no	—
317	C₂²×Q₈	—	2	2	no	—
318	C₂²×Q₈	—	2	3	no	—
319	C₂²×Q₈	—	4	8	no	—
320	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
321	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
322	C₂³×C₄	—	2	3	no	—
323	AGL(1,ℤ/8)	—	2	12	no	—
324	C₂×SD₁₆	C₂³×C₅₃	1	6	yes	—
325	C₂×D₈	C₂³×C₅₃	1	6	yes	—
326	C₂×M₁₆	C₂³×C₅₃	1	3	yes	—
327	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
328	C₄⋊Q₈	—	2	3	no	—
329	C₄⋊Q₈	—	2	3	no	—
330	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
331	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
332	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
333	#127	C₂³×C₅₃	1	3	yes	—
334	#126	—	2	6	no	—
335	#125	—	2	3	no	—
336	#124	—	2	3	no	—
337	#124	C₂³×C₅₃	1	6	yes	—
338	#123	—	2	6	no	—
339	C₂²⋊Q₈	—	2	3	no	—
340	C₂²⋊Q₈	—	2	6	no	—
341	C₂²⋊Q₈	—	2	6	no	—
342	C₂²⋊Q₈	—	2	12	no	—
343	#121	—	2	3	no	—
344	#121	—	2	3	no	—
345	#121	—	2	3	no	—
346	C₄×Q₈	—	2	1	no	—
347	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
348	C₄×Q₈	D₄×C₅₃	1	3	yes	—
349	C₄×Q₈	C₂³×C₅₃	1	3	yes	—
350	C₄×Q₈	C₂³×C₅₃	1	3	yes	—
351	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
352	C₄×Q₈	C₂³×C₅₃	1	3	yes	—
353	C₄×D₄	—	2	3	no	—
354	C₄×D₄	—	2	3	no	—
355	C₄×D₄	—	2	3	no	—
356	C₄×D₄	—	2	3	no	—
357	C₄×D₄	—	2	3	no	—
358	C₄×D₄	—	2	3	no	—
359	C₄×D₄	—	2	3	no	—
360	C₄×D₄	C₂³×C₅₃	1	6	yes	—
361	C₄×D₄	C₂³×C₅₃	1	6	yes	—
362	C₄×D₄	C₂³×C₅₃	1	3	yes	—
363	C₄×D₄	C₂³×C₅₃	1	3	yes	—
364	#117	—	2	3	no	—
365	#117	—	2	3	no	—
366	#117	C₂³×C₅₃	1	1	yes	characteristic
367	C₂×C₄⋊C₄	—	2	3	no	—
368	C₂×C₄⋊C₄	D₄×C₅₃	1	3	yes	—
369	C₂×C₄⋊C₄	C₂³×C₅₃	1	3	yes	—
370	C₂×C₂²⋊C₄	—	2	3	no	—
371	C₂×C₄²	—	2	3	no	—
372	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
373	#107	C₂³×C₅₃	1	6	yes	—
374	#106	C₂³×C₅₃	1	6	yes	—
375	C₄⋊C₈	C₂³×C₅₃	1	6	yes	—
376	C₄⋊C₈	C₂³×C₅₃	1	6	yes	—
377	#103	C₂³×C₅₃	1	6	yes	—
378	#103	C₂³×C₅₃	1	12	yes	—
379	#102	C₂³×C₅₃	1	6	yes	—
380	#102	C₂³×C₅₃	1	12	yes	—
381	#102	C₂³×C₅₃	1	2	yes	—
382	#102	C₂³×C₅₃	1	2	yes	—
383	#97	C₂³×C₅₃	1	6	yes	—
384	C₄×C₈	C₂³×C₅₃	1	6	yes	—
385	C₂⁴	—	2	1	no	—
386	#41	—	2	6	no	—
387	#41	—	2	6	no	—
388	#41	—	4	4	no	—
389	C₂×Q₈	—	2	4	no	—
390	C₂×Q₈	—	2	6	no	—
391	C₂×Q₈	—	2	6	no	—
392	C₂×Q₈	—	4	12	no	—
393	C₂×Q₈	—	4	24	no	—
394	C₂×Q₈	—	4	8	no	—
395	C₂×Q₈	C₂×D₄×C₅₃	1	3	yes	—
396	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
397	C₂×Q₈	C₂⁴×C₅₃	1	3	yes	—
398	C₂×D₄	—	2	3	no	—
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	3	no	—
401	C₂×D₄	—	2	3	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	#41×C₅₃	1	4	yes	complemented
404	C₂×D₄	C₂⁴×C₅₃	1	3	yes	—
405	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
406	C₂²×C₄	—	2	3	no	—
407	C₂²×C₄	—	2	3	no	—
408	C₂²×C₄	—	2	3	no	—
409	C₂²×C₄	—	2	6	no	—
410	C₂²×C₄	—	2	6	no	—
411	C₂²×C₄	—	2	3	no	—
412	C₂²×C₄	—	4	12	no	—
413	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
414	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
415	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
416	SD₁₆	—	2	12	no	—
417	D₈	—	2	12	no	—
418	M₁₆	—	2	6	no	—
419	M₁₆	—	2	2	no	—
420	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
421	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
422	C₄⋊C₄	—	2	6	no	—
423	C₄⋊C₄	—	2	6	no	—
424	C₄⋊C₄	—	2	3	no	—
425	C₄⋊C₄	—	2	3	no	—
426	C₄⋊C₄	—	2	3	no	—
427	C₄⋊C₄	—	2	3	no	—
428	C₄⋊C₄	—	2	3	no	—
429	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
430	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
431	C₄⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
432	C₄⋊C₄	C₂⁴×C₅₃	1	6	yes	—
433	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
434	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
435	C₂²⋊C₄	—	2	1	no	—
436	C₂²⋊C₄	—	2	6	no	—
437	C₂²⋊C₄	—	2	6	no	—
438	C₂²⋊C₄	—	2	6	no	—
439	C₂²⋊C₄	—	2	3	no	—
440	C₂²⋊C₄	—	2	3	no	—
441	C₂²⋊C₄	—	2	6	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	3	no	—
444	C₄²	—	2	3	no	—
445	C₄²	C₂×D₄×C₅₃	1	3	yes	—
446	C₄²	C₂×D₄×C₅₃	1	3	yes	—
447	C₄²	C₂⁴×C₅₃	1	3	yes	—
448	C₄²	C₂⁴×C₅₃	1	3	yes	—
449	C₂³	—	2	4	no	—
450	C₂³	—	2	2	no	—
451	C₂³	—	2	3	no	—
452	C₂³	—	4	2	no	—
453	C₂³	—	4	2	no	—
454	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
455	Q₈	—	2	6	no	—
456	Q₈	—	2	3	no	—
457	Q₈	—	2	3	no	—
458	Q₈	—	4	12	no	—
459	Q₈	—	4	12	no	—
460	Q₈	AGL(1,ℤ/8)×C₅₃	1	4	yes	complemented
461	D₄	—	2	4	no	—
462	D₄	—	2	3	no	—
463	D₄	—	2	2	no	—
464	D₄	—	2	3	no	—
465	D₄	—	4	6	no	—
466	D₄	—	4	6	no	—
467	D₄	—	8	3	no	—
468	D₄	C₂×#41×C₅₃	1	4	yes	—
469	C₂×C₄	—	2	6	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	3	no	—
473	C₂×C₄	—	2	6	no	—
474	C₂×C₄	—	4	3	no	—
475	C₂×C₄	—	4	12	no	—
476	C₂×C₄	—	4	6	no	—
477	C₂×C₄	—	4	6	no	—
478	C₂×C₄	—	4	12	no	—
479	C₂×C₄	—	4	6	no	—
480	C₂×C₄	—	8	3	no	—
481	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
482	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
483	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
484	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
486	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
487	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
488	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
489	C₈	—	2	6	no	—
490	C₈	—	2	2	no	—
491	C₂²	—	2	1	no	—
492	C₂²	—	2	1	no	—
493	C₂²	—	2	4	no	—
494	C₂²	—	4	8	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	4	3	no	—
498	C₂²	—	4	3	no	—
499	C₂²	—	4	3	no	—
500	C₂²	C₅₃×#548	1	2	yes	—
501	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
502	C₄	—	2	3	no	—
503	C₄	—	2	3	no	—
504	C₄	—	2	3	no	—
505	C₄	—	4	3	no	—
506	C₄	—	4	1	no	—
507	C₄	—	4	6	no	—
508	C₄	—	4	1	no	—
509	C₄	C₅₃×#548	1	2	yes	—
510	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	6	yes	—
511	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
512	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
513	C₂	—	2	2	no	—
514	C₂	—	4	4	no	—
515	C₂	—	8	3	no	—
516	C₂	C₂×C₅₃×#548	1	1	yes	characteristic
517	C₂	C₂²×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
518	C₂	C₅₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3480	C₂	semidirect
2. C₂×D₄×Q₈×C₅₃	C₂	semidirect
3. C₅₃×#3373	C₂	semidirect
4. C₅₃×#3367	C₂	semidirect
5. C₅₃×#3359	C₂	semidirect
6. C₅₃×#3358	C₂	semidirect
7. C₅₃×#3141	C₂	semidirect
8. C₅₃×#3079	C₂	semidirect
9. C₅₃×#3036	C₂	semidirect
10. Q₈×M₁₆×C₅₃	C₂	semidirect
11. C₄×AGL(1,ℤ/8)×C₅₃	C₂	semidirect
14. C₅₃×#559	C₂²	semidirect
15. C₅₃×#557	C₂²	semidirect
16. C₅₃×#556	C₂²	semidirect
18. C₅₃×#550	C₂²	semidirect
32. C₅₃×#489	C₂²	semidirect
33. C₅₃×#487	C₂²	semidirect
34. C₅₃×#486	C₂²	semidirect
35. C₅₃×#477	C₂²	semidirect
36. C₅₃×#476	C₂²	semidirect
37. C₅₃×#475	C₂²	semidirect
38. C₅₃×#474	C₂²	semidirect
42. C₅₃×#446	C₂²	semidirect
43. C₈×Q₈×C₅₃	C₂²	semidirect
143. C₂×D₄×C₅₃	#41	semidirect
200. Q₈×C₅₃	AGL(1,ℤ/8)	semidirect
231. #61317	C₅₃	direct (N × H)
254. #3480	C₂×C₅₃	semidirect
255. C₂×D₄×Q₈	C₂×C₅₃	semidirect
256. #3373	C₂×C₅₃	semidirect
257. #3367	C₂×C₅₃	semidirect
258. #3359	C₂×C₅₃	semidirect
259. #3358	C₂×C₅₃	semidirect
260. #3141	C₂×C₅₃	semidirect
261. #3079	C₂×C₅₃	semidirect
262. #3036	C₂×C₅₃	semidirect
263. Q₈×M₁₆	C₂×C₅₃	semidirect
264. C₄×AGL(1,ℤ/8)	C₂×C₅₃	semidirect
273. #559	C₂²×C₅₃	semidirect
274. #557	C₂²×C₅₃	semidirect
275. #556	C₂²×C₅₃	semidirect
277. #550	C₂²×C₅₃	semidirect
291. #489	C₂²×C₅₃	semidirect
292. #487	C₂²×C₅₃	semidirect
293. #486	C₂²×C₅₃	semidirect
294. #477	C₂²×C₅₃	semidirect
295. #476	C₂²×C₅₃	semidirect
296. #475	C₂²×C₅₃	semidirect
297. #474	C₂²×C₅₃	semidirect
301. #446	C₂²×C₅₃	semidirect
302. C₈×Q₈	C₂²×C₅₃	semidirect
313. C₅₃	#61317	direct (N × H)
403. C₂×D₄	#41×C₅₃	semidirect
460. Q₈	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×#3480	2
C₂×D₄×Q₈×C₅₃	2
C₅₃×#3373	2
C₅₃×#3367	2
C₅₃×#3359	2
C₅₃×#3358	2
C₅₃×#3141	2
C₅₃×#3079	2
C₅₃×#3036	2
Q₈×M₁₆×C₅₃	2
C₄×AGL(1,ℤ/8)×C₅₃	2
#61317	53

Subgroups — C53×#61317 #27268829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#61317 #27268829