Subgroups — C₅₃×#61466 #27268978

Order 13568.

2102 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3482	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3392	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3359	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#3358	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#3353	C₂	1	3	yes	maximal, complemented
8	C₅₃×#3330	C₂	1	3	yes	maximal, complemented
9	C₅₃×#3324	C₂	1	3	yes	maximal, complemented
10	C₅₃×#3272	C₂	1	3	yes	maximal, complemented
11	C₅₃×#3182	C₂	1	3	yes	maximal, complemented
12	C₅₃×#3115	C₂	1	3	yes	maximal, complemented
13	C₅₃×#3110	C₂	1	3	yes	maximal, complemented
14	C₅₃×#2971	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×AGL(1,ℤ/8)×C₅₃	C₂	1	3	yes	maximal, complemented
16	C₂²×#41×C₅₃	—	2	1	no	—
17	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	3	yes	—
18	C₅₃×#557	C₂²	1	1	yes	characteristic, complemented
19	C₅₃×#556	C₂²	1	3	yes	complemented
20	C₅₃×#552	—	2	3	no	—
21	C₅₃×#550	C₂²	1	1	yes	characteristic
22	D₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	D₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
24	C₅₃×#548	—	2	1	no	—
25	C₅₃×#548	—	2	1	no	—
26	C₅₃×#547	C₂²	1	3	yes	—
27	C₅₃×#547	C₂²	1	3	yes	complemented
28	C₅₃×#546	—	2	3	no	—
29	D₄²×C₅₃	C₂²	1	3	yes	—
30	C₅₃×#533	C₂²	1	3	yes	complemented
31	C₅₃×#532	C₂²	1	3	yes	—
32	C₅₃×#529	—	2	3	no	—
33	C₅₃×#525	—	2	3	no	—
34	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×#41×C₅₃	C₂²	1	3	yes	complemented
36	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
37	C₂×C₄×D₄×C₅₃	C₂²	1	3	yes	—
38	C₅₃×#500	C₂²	1	3	yes	complemented
39	C₅₃×#492	C₂²	1	3	yes	complemented
40	C₅₃×#487	C₂²	1	3	yes	complemented
41	C₅₃×#486	C₂²	1	3	yes	complemented
42	C₅₃×#484	C₂²	1	6	yes	complemented
43	C₅₃×#482	C₂²	1	6	yes	complemented
44	C₅₃×#475	C₂²	1	3	yes	complemented
45	C₅₃×#474	C₂²	1	3	yes	—
46	C₅₃×#474	C₂²	1	3	yes	complemented
47	C₅₃×#474	C₂²	1	3	yes	complemented
48	C₅₃×#468	C₂²	1	6	yes	complemented
49	C₅₃×#468	C₂²	1	6	yes	complemented
50	C₅₃×#461	C₂²	1	3	yes	—
51	C₅₃×#461	C₂²	1	3	yes	complemented
52	C₅₃×#460	C₂²	1	3	yes	complemented
53	C₅₃×#459	C₂²	1	3	yes	complemented
54	C₅₃×#450	C₂²	1	6	yes	—
55	C₅₃×#449	C₂²	1	6	yes	complemented
56	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
57	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
58	C₅₃×#442	C₂²	1	1	yes	characteristic
59	C₅₃×#442	C₂²	1	3	yes	complemented
60	C₅₃×#440	C₂²	1	3	yes	—
61	C₅₃×#440	C₂²	1	3	yes	complemented
62	C₅₃×#440	C₂²	1	6	yes	complemented
63	C₄×SD₁₆×C₅₃	C₂²	1	3	yes	—
64	C₄×SD₁₆×C₅₃	C₂²	1	3	yes	complemented
65	C₄×D₈×C₅₃	C₂²	1	3	yes	—
66	C₄×D₈×C₅₃	C₂²	1	3	yes	complemented
67	C₅₃×#436	C₂²	1	3	yes	complemented
68	C₅₃×#436	C₂²	1	3	yes	complemented
69	C₅₃×#432	C₂²	1	3	yes	complemented
70	C₅₃×#428	C₂²	1	3	yes	complemented
71	C₅₃×#418	C₂²	1	3	yes	—
72	C₅₃×#417	C₂²	1	3	yes	complemented
73	C₄×M₁₆×C₅₃	C₂²	1	3	yes	complemented
74	C₂×#41×C₅₃	—	2	1	no	—
75	C₂×#41×C₅₃	—	2	3	no	—
76	C₂×#41×C₅₃	—	2	3	no	—
77	C₂×#41×C₅₃	—	4	8	no	—
78	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
79	C₂×#41×C₅₃	C₂³	1	3	yes	—
80	C₂²×Q₈×C₅₃	—	2	1	no	—
81	C₂²×Q₈×C₅₃	—	2	1	no	—
82	C₂²×D₄×C₅₃	—	2	3	no	—
83	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
84	C₂³×C₄×C₅₃	—	2	3	no	—
85	AGL(1,ℤ/8)×C₅₃	—	2	12	no	—
86	C₂×SD₁₆×C₅₃	C₂³	1	3	yes	—
87	C₂×SD₁₆×C₅₃	C₂³	1	3	yes	—
88	C₂×SD₁₆×C₅₃	C₂³	1	6	yes	—
89	C₂×D₈×C₅₃	C₂³	1	3	yes	—
90	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₂×M₁₆×C₅₃	C₂³	1	3	yes	—
92	C₂×M₁₆×C₅₃	C₂³	1	3	yes	—
93	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
94	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
95	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
96	#127×C₅₃	C₂³	1	3	yes	—
97	#126×C₅₃	—	2	6	no	—
98	#125×C₅₃	—	2	3	no	—
99	#125×C₅₃	—	2	3	no	—
100	#124×C₅₃	—	2	3	no	—
101	#124×C₅₃	—	2	3	no	—
102	#123×C₅₃	—	2	6	no	—
103	#123×C₅₃	—	2	6	no	—
104	C₂²⋊Q₈×C₅₃	—	2	3	no	—
105	C₂²⋊Q₈×C₅₃	—	2	3	no	—
106	C₂²⋊Q₈×C₅₃	—	2	3	no	—
107	C₂²⋊Q₈×C₅₃	—	2	3	no	—
108	C₂²⋊Q₈×C₅₃	C₂³	1	6	yes	—
109	#121×C₅₃	—	2	3	no	—
110	#121×C₅₃	—	2	3	no	—
111	#121×C₅₃	—	2	3	no	—
112	#121×C₅₃	—	2	3	no	—
113	#121×C₅₃	C₂³	1	6	yes	—
114	#120×C₅₃	—	2	6	no	—
115	C₄×Q₈×C₅₃	—	2	3	no	—
116	C₄×Q₈×C₅₃	—	2	3	no	—
117	C₄×Q₈×C₅₃	—	2	1	no	—
118	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
119	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
120	C₄×Q₈×C₅₃	C₂³	1	3	yes	—
121	C₄×Q₈×C₅₃	C₂³	1	3	yes	—
122	C₄×D₄×C₅₃	—	2	3	no	—
123	C₄×D₄×C₅₃	—	2	3	no	—
124	C₄×D₄×C₅₃	—	2	3	no	—
125	C₄×D₄×C₅₃	—	2	3	no	—
126	C₄×D₄×C₅₃	—	2	3	no	—
127	C₄×D₄×C₅₃	D₄	1	3	yes	—
128	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
129	C₄×D₄×C₅₃	C₂³	1	3	yes	—
130	C₄×D₄×C₅₃	C₂³	1	3	yes	—
131	C₄×D₄×C₅₃	C₂³	1	3	yes	—
132	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
133	C₄×D₄×C₅₃	C₂³	1	3	yes	—
134	C₄×D₄×C₅₃	C₂³	1	3	yes	—
135	#117×C₅₃	—	2	3	no	—
136	#117×C₅₃	—	2	3	no	—
137	#117×C₅₃	D₄	1	3	yes	—
138	#117×C₅₃	C₂³	1	3	yes	—
139	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
140	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
141	C₂×C₂²⋊C₄×C₅₃	—	2	3	no	—
142	C₂×C₄²×C₅₃	—	2	3	no	—
143	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
144	#107×C₅₃	C₂³	1	3	yes	—
145	#107×C₅₃	C₂³	1	6	yes	—
146	#107×C₅₃	C₂³	1	3	yes	—
147	#106×C₅₃	C₂³	1	1	yes	characteristic
148	#106×C₅₃	C₂³	1	3	yes	—
149	C₄⋊C₈×C₅₃	C₂³	1	3	yes	—
150	C₄⋊C₈×C₅₃	C₂³	1	3	yes	—
151	#103×C₅₃	C₂³	1	3	yes	—
152	#103×C₅₃	C₂³	1	6	yes	—
153	#103×C₅₃	C₂³	1	3	yes	—
154	#102×C₅₃	C₂³	1	3	yes	—
155	#102×C₅₃	C₂³	1	6	yes	—
156	#102×C₅₃	C₂³	1	1	yes	characteristic
157	#102×C₅₃	C₂³	1	3	yes	—
158	#102×C₅₃	C₂³	1	1	yes	characteristic
159	#102×C₅₃	C₂³	1	3	yes	—
160	#102×C₅₃	C₂³	1	3	yes	—
161	C₂²⋊C₈×C₅₃	C₂³	1	6	yes	—
162	#97×C₅₃	C₂³	1	1	yes	characteristic
163	#97×C₅₃	C₂³	1	3	yes	—
164	C₄×C₈×C₅₃	C₂³	1	3	yes	—
165	C₄×C₈×C₅₃	C₂³	1	3	yes	—
166	C₂⁴×C₅₃	—	2	3	no	—
167	#41×C₅₃	—	2	4	no	—
168	#41×C₅₃	—	2	6	no	—
169	#41×C₅₃	—	4	3	no	—
170	#41×C₅₃	—	4	12	no	—
171	#41×C₅₃	—	4	12	no	—
172	#41×C₅₃	—	4	4	no	—
173	C₂×Q₈×C₅₃	—	2	3	no	—
174	C₂×Q₈×C₅₃	—	2	3	no	—
175	C₂×Q₈×C₅₃	—	4	4	no	—
176	C₂×Q₈×C₅₃	—	4	4	no	—
177	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
178	C₂×Q₈×C₅₃	C₂⁴	1	3	yes	—
179	C₂×D₄×C₅₃	—	2	6	no	—
180	C₂×D₄×C₅₃	—	2	3	no	—
181	C₂×D₄×C₅₃	—	2	6	no	—
182	C₂×D₄×C₅₃	—	2	3	no	—
183	C₂×D₄×C₅₃	—	2	3	no	—
184	C₂×D₄×C₅₃	—	2	3	no	—
185	C₂×D₄×C₅₃	—	4	3	no	—
186	C₂×D₄×C₅₃	—	4	12	no	—
187	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
188	C₂×D₄×C₅₃	C₂⁴	1	3	yes	—
189	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
190	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
191	C₂²×C₄×C₅₃	—	2	3	no	—
192	C₂²×C₄×C₅₃	—	2	1	no	—
193	C₂²×C₄×C₅₃	—	2	3	no	—
194	C₂²×C₄×C₅₃	—	2	3	no	—
195	C₂²×C₄×C₅₃	—	2	1	no	—
196	C₂²×C₄×C₅₃	—	2	3	no	—
197	C₂²×C₄×C₅₃	—	2	3	no	—
198	C₂²×C₄×C₅₃	—	2	3	no	—
199	C₂²×C₄×C₅₃	—	4	12	no	—
200	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
201	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
202	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
203	SD₁₆×C₅₃	—	2	6	no	—
204	SD₁₆×C₅₃	—	2	6	no	—
205	SD₁₆×C₅₃	—	4	6	no	—
206	D₈×C₅₃	—	2	6	no	—
207	D₈×C₅₃	#41	1	4	yes	complemented
208	M₁₆×C₅₃	—	2	6	no	—
209	M₁₆×C₅₃	—	4	3	no	—
210	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
211	C₂×C₈×C₅₃	C₂⁴	1	3	yes	—
212	C₂×C₈×C₅₃	C₂⁴	1	3	yes	—
213	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
214	C₄⋊C₄×C₅₃	—	2	3	no	—
215	C₄⋊C₄×C₅₃	—	2	6	no	—
216	C₄⋊C₄×C₅₃	—	2	3	no	—
217	C₄⋊C₄×C₅₃	—	2	3	no	—
218	C₄⋊C₄×C₅₃	—	2	3	no	—
219	C₄⋊C₄×C₅₃	—	2	3	no	—
220	C₄⋊C₄×C₅₃	—	2	3	no	—
221	C₄⋊C₄×C₅₃	—	2	3	no	—
222	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
223	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
224	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
225	C₄⋊C₄×C₅₃	C₂⁴	1	3	yes	—
226	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
227	C₄⋊C₄×C₅₃	C₂⁴	1	3	yes	—
228	C₂²⋊C₄×C₅₃	—	2	6	no	—
229	C₂²⋊C₄×C₅₃	—	2	3	no	—
230	C₂²⋊C₄×C₅₃	—	2	1	no	—
231	C₂²⋊C₄×C₅₃	—	2	3	no	—
232	C₂²⋊C₄×C₅₃	—	2	3	no	—
233	C₂²⋊C₄×C₅₃	—	2	3	no	—
234	C₂²⋊C₄×C₅₃	—	2	1	no	—
235	C₂²⋊C₄×C₅₃	—	2	3	no	—
236	C₂²⋊C₄×C₅₃	—	2	6	no	—
237	C₂²⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
238	C₄²×C₅₃	—	2	3	no	—
239	C₄²×C₅₃	—	2	3	no	—
240	C₄²×C₅₃	—	2	3	no	—
241	C₄²×C₅₃	C₂×D₄	1	3	yes	—
242	C₄²×C₅₃	C₂×D₄	1	3	yes	—
243	C₄²×C₅₃	C₂⁴	1	3	yes	—
244	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
245	C₂³×C₅₃	—	2	1	no	—
246	C₂³×C₅₃	—	2	1	no	—
247	C₂³×C₅₃	—	2	3	no	—
248	C₂³×C₅₃	—	4	6	no	—
249	C₂³×C₅₃	—	4	6	no	—
250	C₂³×C₅₃	—	8	3	no	—
251	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
252	Q₈×C₅₃	—	2	3	no	—
253	Q₈×C₅₃	—	2	3	no	—
254	Q₈×C₅₃	—	4	6	no	—
255	Q₈×C₅₃	—	4	6	no	—
256	Q₈×C₅₃	AGL(1,ℤ/8)	1	4	yes	complemented
257	D₄×C₅₃	—	2	6	no	—
258	D₄×C₅₃	—	2	3	no	—
259	D₄×C₅₃	—	2	1	no	—
260	D₄×C₅₃	—	2	1	no	—
261	D₄×C₅₃	—	2	3	no	—
262	D₄×C₅₃	—	4	6	no	—
263	D₄×C₅₃	—	4	6	no	—
264	D₄×C₅₃	—	4	12	no	—
265	D₄×C₅₃	—	4	6	no	—
266	D₄×C₅₃	—	8	3	no	—
267	D₄×C₅₃	C₂×#41	1	2	yes	—
268	D₄×C₅₃	C₂×#41	1	2	yes	complemented
269	C₂×C₄×C₅₃	—	2	6	no	—
270	C₂×C₄×C₅₃	—	2	3	no	—
271	C₂×C₄×C₅₃	—	2	3	no	—
272	C₂×C₄×C₅₃	—	2	1	no	—
273	C₂×C₄×C₅₃	—	2	3	no	—
274	C₂×C₄×C₅₃	—	2	3	no	—
275	C₂×C₄×C₅₃	—	2	3	no	—
276	C₂×C₄×C₅₃	—	2	1	no	—
277	C₂×C₄×C₅₃	—	4	3	no	—
278	C₂×C₄×C₅₃	—	4	6	no	—
279	C₂×C₄×C₅₃	—	4	2	no	—
280	C₂×C₄×C₅₃	—	4	2	no	—
281	C₂×C₄×C₅₃	—	4	6	no	—
282	C₂×C₄×C₅₃	—	4	6	no	—
283	C₂×C₄×C₅₃	—	4	6	no	—
284	C₂×C₄×C₅₃	—	4	6	no	—
285	C₂×C₄×C₅₃	—	4	6	no	—
286	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
288	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
289	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
290	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
291	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
292	C₈×C₅₃	—	2	3	no	—
293	C₈×C₅₃	—	2	1	no	—
294	C₈×C₅₃	—	2	3	no	—
295	C₈×C₅₃	C₂×#41	1	2	yes	complemented
296	#61466	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
297	C₂²×C₅₃	—	2	3	no	—
298	C₂²×C₅₃	—	2	3	no	—
299	C₂²×C₅₃	—	2	2	no	—
300	C₂²×C₅₃	—	2	2	no	—
301	C₂²×C₅₃	—	2	3	no	—
302	C₂²×C₅₃	—	4	1	no	—
303	C₂²×C₅₃	—	4	1	no	—
304	C₂²×C₅₃	—	4	1	no	—
305	C₂²×C₅₃	—	4	1	no	—
306	C₂²×C₅₃	—	4	3	no	—
307	C₂²×C₅₃	—	4	3	no	—
308	C₂²×C₅₃	—	4	3	no	—
309	C₂²×C₅₃	—	8	12	no	—
310	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
311	C₄×C₅₃	—	2	1	no	—
312	C₄×C₅₃	—	2	3	no	—
313	C₄×C₅₃	—	2	3	no	—
314	C₄×C₅₃	—	2	1	no	—
315	C₄×C₅₃	—	4	1	no	—
316	C₄×C₅₃	—	4	3	no	—
317	C₄×C₅₃	—	4	1	no	—
318	C₄×C₅₃	—	4	3	no	—
319	C₄×C₅₃	—	4	3	no	—
320	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	6	yes	—
321	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
322	C₄×C₅₃	#585	1	1	yes	characteristic
323	#3482	C₂×C₅₃	1	1	yes	characteristic, complemented
324	D₄×#41	C₂×C₅₃	1	1	yes	characteristic, complemented
325	#3392	C₂×C₅₃	1	1	yes	characteristic, complemented
326	Q₈×D₈	C₂×C₅₃	1	1	yes	characteristic, complemented
327	#3359	C₂×C₅₃	1	1	yes	characteristic, complemented
328	#3358	C₂×C₅₃	1	1	yes	characteristic, complemented
329	#3353	C₂×C₅₃	1	3	yes	complemented
330	#3330	C₂×C₅₃	1	3	yes	complemented
331	#3324	C₂×C₅₃	1	3	yes	complemented
332	#3272	C₂×C₅₃	1	3	yes	complemented
333	#3182	C₂×C₅₃	1	3	yes	complemented
334	#3115	C₂×C₅₃	1	3	yes	complemented
335	#3110	C₂×C₅₃	1	3	yes	complemented
336	#2971	C₂×C₅₃	1	1	yes	characteristic, complemented
337	C₄×AGL(1,ℤ/8)	C₂×C₅₃	1	3	yes	complemented
338	C₂×C₅₃	—	4	3	no	—
339	C₂×C₅₃	—	4	2	no	—
340	C₂×C₅₃	—	4	2	no	—
341	C₂×C₅₃	—	8	3	no	—
342	C₂×C₅₃	D₄×#41	1	1	yes	characteristic
343	C₂×C₅₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
344	C₂×C₅₃	#3586	1	1	yes	characteristic
345	C₂²×#41	—	2	1	no	—
346	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	3	yes	—
347	#557	C₂²×C₅₃	1	1	yes	characteristic, complemented
348	#556	C₂²×C₅₃	1	3	yes	complemented
349	#552	—	2	3	no	—
350	#550	C₂²×C₅₃	1	1	yes	characteristic
351	D₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
352	D₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
353	#548	—	2	1	no	—
354	#548	—	2	1	no	—
355	#547	C₂²×C₅₃	1	3	yes	—
356	#547	C₂²×C₅₃	1	3	yes	complemented
357	#546	—	2	3	no	—
358	D₄²	C₂²×C₅₃	1	3	yes	—
359	#533	C₂²×C₅₃	1	3	yes	complemented
360	#532	C₂²×C₅₃	1	3	yes	—
361	#529	—	2	3	no	—
362	#525	—	2	3	no	—
363	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
364	C₄×#41	C₂²×C₅₃	1	3	yes	complemented
365	C₂×C₄×Q₈	—	2	1	no	—
366	C₂×C₄×D₄	C₂²×C₅₃	1	3	yes	—
367	#500	C₂²×C₅₃	1	3	yes	complemented
368	#492	C₂²×C₅₃	1	3	yes	complemented
369	#487	C₂²×C₅₃	1	3	yes	complemented
370	#486	C₂²×C₅₃	1	3	yes	complemented
371	#484	C₂²×C₅₃	1	6	yes	complemented
372	#482	C₂²×C₅₃	1	6	yes	complemented
373	#475	C₂²×C₅₃	1	3	yes	complemented
374	#474	C₂²×C₅₃	1	3	yes	—
375	#474	C₂²×C₅₃	1	3	yes	complemented
376	#474	C₂²×C₅₃	1	3	yes	complemented
377	#468	C₂²×C₅₃	1	6	yes	complemented
378	#468	C₂²×C₅₃	1	6	yes	complemented
379	#461	C₂²×C₅₃	1	3	yes	—
380	#461	C₂²×C₅₃	1	3	yes	complemented
381	#460	C₂²×C₅₃	1	3	yes	complemented
382	#459	C₂²×C₅₃	1	3	yes	complemented
383	#450	C₂²×C₅₃	1	6	yes	—
384	#449	C₂²×C₅₃	1	6	yes	complemented
385	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
386	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
387	#442	C₂²×C₅₃	1	1	yes	characteristic
388	#442	C₂²×C₅₃	1	3	yes	complemented
389	#440	C₂²×C₅₃	1	3	yes	—
390	#440	C₂²×C₅₃	1	6	yes	complemented
391	#440	C₂²×C₅₃	1	3	yes	complemented
392	C₄×SD₁₆	C₂²×C₅₃	1	3	yes	—
393	C₄×SD₁₆	C₂²×C₅₃	1	3	yes	complemented
394	C₄×D₈	C₂²×C₅₃	1	3	yes	—
395	C₄×D₈	C₂²×C₅₃	1	3	yes	complemented
396	#436	C₂²×C₅₃	1	3	yes	complemented
397	#436	C₂²×C₅₃	1	3	yes	complemented
398	#432	C₂²×C₅₃	1	3	yes	complemented
399	#428	C₂²×C₅₃	1	3	yes	complemented
400	#418	C₂²×C₅₃	1	3	yes	—
401	#417	C₂²×C₅₃	1	3	yes	complemented
402	C₄×M₁₆	C₂²×C₅₃	1	3	yes	complemented
403	C₅₃	#61466	1	1	yes	characteristic, direct product factor, Sylow
404	C₂×#41	—	2	1	no	—
405	C₂×#41	—	2	3	no	—
406	C₂×#41	—	2	3	no	—
407	C₂×#41	—	4	8	no	—
408	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
409	C₂×#41	C₂³×C₅₃	1	3	yes	—
410	C₂²×Q₈	—	2	1	no	—
411	C₂²×Q₈	—	2	1	no	—
412	C₂²×D₄	—	2	3	no	—
413	C₂²×D₄	C₂³×C₅₃	1	3	yes	—
414	C₂³×C₄	—	2	3	no	—
415	AGL(1,ℤ/8)	—	2	12	no	—
416	C₂×SD₁₆	C₂³×C₅₃	1	3	yes	—
417	C₂×SD₁₆	C₂³×C₅₃	1	3	yes	—
418	C₂×SD₁₆	C₂³×C₅₃	1	6	yes	—
419	C₂×D₈	C₂³×C₅₃	1	3	yes	—
420	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
421	C₂×M₁₆	C₂³×C₅₃	1	3	yes	—
422	C₂×M₁₆	C₂³×C₅₃	1	3	yes	—
423	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
424	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
425	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
426	#127	C₂³×C₅₃	1	3	yes	—
427	#126	—	2	6	no	—
428	#125	—	2	3	no	—
429	#125	—	2	3	no	—
430	#124	—	2	3	no	—
431	#124	—	2	3	no	—
432	#123	—	2	6	no	—
433	#123	—	2	6	no	—
434	C₂²⋊Q₈	—	2	3	no	—
435	C₂²⋊Q₈	—	2	3	no	—
436	C₂²⋊Q₈	—	2	3	no	—
437	C₂²⋊Q₈	—	2	3	no	—
438	C₂²⋊Q₈	C₂³×C₅₃	1	6	yes	—
439	#121	—	2	3	no	—
440	#121	—	2	3	no	—
441	#121	—	2	3	no	—
442	#121	—	2	3	no	—
443	#121	C₂³×C₅₃	1	6	yes	—
444	#120	—	2	6	no	—
445	C₄×Q₈	—	2	1	no	—
446	C₄×Q₈	—	2	3	no	—
447	C₄×Q₈	—	2	3	no	—
448	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
449	C₄×Q₈	C₂³×C₅₃	1	3	yes	—
450	C₄×Q₈	C₂³×C₅₃	1	3	yes	—
451	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
452	C₄×D₄	—	2	3	no	—
453	C₄×D₄	—	2	3	no	—
454	C₄×D₄	—	2	3	no	—
455	C₄×D₄	—	2	3	no	—
456	C₄×D₄	—	2	3	no	—
457	C₄×D₄	D₄×C₅₃	1	3	yes	—
458	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
459	C₄×D₄	C₂³×C₅₃	1	3	yes	—
460	C₄×D₄	C₂³×C₅₃	1	3	yes	—
461	C₄×D₄	C₂³×C₅₃	1	3	yes	—
462	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
463	C₄×D₄	C₂³×C₅₃	1	3	yes	—
464	C₄×D₄	C₂³×C₅₃	1	3	yes	—
465	#117	—	2	3	no	—
466	#117	—	2	3	no	—
467	#117	D₄×C₅₃	1	3	yes	—
468	#117	C₂³×C₅₃	1	3	yes	—
469	C₂×C₄⋊C₄	—	2	3	no	—
470	C₂×C₄⋊C₄	C₂³×C₅₃	1	3	yes	—
471	C₂×C₂²⋊C₄	—	2	3	no	—
472	C₂×C₄²	—	2	3	no	—
473	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
474	#107	C₂³×C₅₃	1	3	yes	—
475	#107	C₂³×C₅₃	1	3	yes	—
476	#107	C₂³×C₅₃	1	6	yes	—
477	#106	C₂³×C₅₃	1	1	yes	characteristic
478	#106	C₂³×C₅₃	1	3	yes	—
479	C₄⋊C₈	C₂³×C₅₃	1	3	yes	—
480	C₄⋊C₈	C₂³×C₅₃	1	3	yes	—
481	#103	C₂³×C₅₃	1	3	yes	—
482	#103	C₂³×C₅₃	1	3	yes	—
483	#103	C₂³×C₅₃	1	6	yes	—
484	#102	C₂³×C₅₃	1	1	yes	characteristic
485	#102	C₂³×C₅₃	1	3	yes	—
486	#102	C₂³×C₅₃	1	1	yes	characteristic
487	#102	C₂³×C₅₃	1	3	yes	—
488	#102	C₂³×C₅₃	1	3	yes	—
489	#102	C₂³×C₅₃	1	3	yes	—
490	#102	C₂³×C₅₃	1	6	yes	—
491	C₂²⋊C₈	C₂³×C₅₃	1	6	yes	—
492	#97	C₂³×C₅₃	1	1	yes	characteristic
493	#97	C₂³×C₅₃	1	3	yes	—
494	C₄×C₈	C₂³×C₅₃	1	3	yes	—
495	C₄×C₈	C₂³×C₅₃	1	3	yes	—
496	C₂⁴	—	2	3	no	—
497	#41	—	2	4	no	—
498	#41	—	2	6	no	—
499	#41	—	4	3	no	—
500	#41	—	4	12	no	—
501	#41	—	4	12	no	—
502	#41	—	4	4	no	—
503	C₂×Q₈	—	2	3	no	—
504	C₂×Q₈	—	2	3	no	—
505	C₂×Q₈	—	4	4	no	—
506	C₂×Q₈	—	4	4	no	—
507	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
508	C₂×Q₈	C₂⁴×C₅₃	1	3	yes	—
509	C₂×D₄	—	2	6	no	—
510	C₂×D₄	—	2	3	no	—
511	C₂×D₄	—	2	6	no	—
512	C₂×D₄	—	2	3	no	—
513	C₂×D₄	—	2	3	no	—
514	C₂×D₄	—	2	3	no	—
515	C₂×D₄	—	4	3	no	—
516	C₂×D₄	—	4	12	no	—
517	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
518	C₂×D₄	C₂⁴×C₅₃	1	3	yes	—
519	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
520	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
521	C₂²×C₄	—	2	3	no	—
522	C₂²×C₄	—	2	1	no	—
523	C₂²×C₄	—	2	3	no	—
524	C₂²×C₄	—	2	3	no	—
525	C₂²×C₄	—	2	1	no	—
526	C₂²×C₄	—	2	3	no	—
527	C₂²×C₄	—	2	3	no	—
528	C₂²×C₄	—	2	3	no	—
529	C₂²×C₄	—	4	12	no	—
530	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
531	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
532	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
533	SD₁₆	—	2	6	no	—
534	SD₁₆	—	2	6	no	—
535	SD₁₆	—	4	6	no	—
536	D₈	—	2	6	no	—
537	D₈	#41×C₅₃	1	4	yes	complemented
538	M₁₆	—	2	6	no	—
539	M₁₆	—	4	3	no	—
540	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
541	C₂×C₈	C₂⁴×C₅₃	1	3	yes	—
542	C₂×C₈	C₂⁴×C₅₃	1	3	yes	—
543	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
544	C₄⋊C₄	—	2	3	no	—
545	C₄⋊C₄	—	2	6	no	—
546	C₄⋊C₄	—	2	3	no	—
547	C₄⋊C₄	—	2	3	no	—
548	C₄⋊C₄	—	2	3	no	—
549	C₄⋊C₄	—	2	3	no	—
550	C₄⋊C₄	—	2	3	no	—
551	C₄⋊C₄	—	2	3	no	—
552	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
553	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
554	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
555	C₄⋊C₄	C₂⁴×C₅₃	1	3	yes	—
556	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
557	C₄⋊C₄	C₂⁴×C₅₃	1	3	yes	—
558	C₂²⋊C₄	—	2	6	no	—
559	C₂²⋊C₄	—	2	3	no	—
560	C₂²⋊C₄	—	2	1	no	—
561	C₂²⋊C₄	—	2	3	no	—
562	C₂²⋊C₄	—	2	3	no	—
563	C₂²⋊C₄	—	2	3	no	—
564	C₂²⋊C₄	—	2	1	no	—
565	C₂²⋊C₄	—	2	3	no	—
566	C₂²⋊C₄	—	2	6	no	—
567	C₂²⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
568	C₄²	—	2	3	no	—
569	C₄²	—	2	3	no	—
570	C₄²	—	2	3	no	—
571	C₄²	C₂×D₄×C₅₃	1	3	yes	—
572	C₄²	C₂×D₄×C₅₃	1	3	yes	—
573	C₄²	C₂⁴×C₅₃	1	3	yes	—
574	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
575	C₂³	—	2	1	no	—
576	C₂³	—	2	1	no	—
577	C₂³	—	2	3	no	—
578	C₂³	—	4	6	no	—
579	C₂³	—	4	6	no	—
580	C₂³	—	8	3	no	—
581	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
582	Q₈	—	2	3	no	—
583	Q₈	—	2	3	no	—
584	Q₈	—	4	6	no	—
585	Q₈	—	4	6	no	—
586	Q₈	AGL(1,ℤ/8)×C₅₃	1	4	yes	complemented
587	D₄	—	2	6	no	—
588	D₄	—	2	3	no	—
589	D₄	—	2	1	no	—
590	D₄	—	2	1	no	—
591	D₄	—	2	3	no	—
592	D₄	—	4	6	no	—
593	D₄	—	4	6	no	—
594	D₄	—	4	12	no	—
595	D₄	—	4	6	no	—
596	D₄	—	8	3	no	—
597	D₄	C₂×#41×C₅₃	1	2	yes	—
598	D₄	C₂×#41×C₅₃	1	2	yes	complemented
599	C₂×C₄	—	2	6	no	—
600	C₂×C₄	—	2	3	no	—
601	C₂×C₄	—	2	3	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	2	3	no	—
604	C₂×C₄	—	2	3	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	3	no	—
607	C₂×C₄	—	4	3	no	—
608	C₂×C₄	—	4	6	no	—
609	C₂×C₄	—	4	2	no	—
610	C₂×C₄	—	4	2	no	—
611	C₂×C₄	—	4	6	no	—
612	C₂×C₄	—	4	6	no	—
613	C₂×C₄	—	4	6	no	—
614	C₂×C₄	—	4	6	no	—
615	C₂×C₄	—	4	6	no	—
616	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
617	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
618	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
619	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
620	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
621	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
622	C₈	—	2	3	no	—
623	C₈	—	2	1	no	—
624	C₈	—	2	3	no	—
625	C₈	C₂×#41×C₅₃	1	2	yes	complemented
626	C₂²	—	2	3	no	—
627	C₂²	—	2	2	no	—
628	C₂²	—	2	2	no	—
629	C₂²	—	2	3	no	—
630	C₂²	—	2	3	no	—
631	C₂²	—	4	1	no	—
632	C₂²	—	4	1	no	—
633	C₂²	—	4	1	no	—
634	C₂²	—	4	1	no	—
635	C₂²	—	4	3	no	—
636	C₂²	—	4	3	no	—
637	C₂²	—	4	3	no	—
638	C₂²	—	8	12	no	—
639	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
640	C₄	—	2	3	no	—
641	C₄	—	2	1	no	—
642	C₄	—	2	1	no	—
643	C₄	—	2	3	no	—
644	C₄	—	4	3	no	—
645	C₄	—	4	1	no	—
646	C₄	—	4	3	no	—
647	C₄	—	4	3	no	—
648	C₄	—	4	1	no	—
649	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	6	yes	—
650	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
651	C₄	C₅₃×#585	1	1	yes	characteristic
652	C₂	—	4	3	no	—
653	C₂	—	4	2	no	—
654	C₂	—	4	2	no	—
655	C₂	—	8	3	no	—
656	C₂	D₄×#41×C₅₃	1	1	yes	characteristic
657	C₂	C₂²×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
658	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 ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3482	C₂	semidirect
2. D₄×#41×C₅₃	C₂	semidirect
3. C₅₃×#3392	C₂	semidirect
4. Q₈×D₈×C₅₃	C₂	semidirect
5. C₅₃×#3359	C₂	semidirect
6. C₅₃×#3358	C₂	semidirect
7. C₅₃×#3353	C₂	semidirect
8. C₅₃×#3330	C₂	semidirect
9. C₅₃×#3324	C₂	semidirect
10. C₅₃×#3272	C₂	semidirect
11. C₅₃×#3182	C₂	semidirect
12. C₅₃×#3115	C₂	semidirect
13. C₅₃×#3110	C₂	semidirect
14. C₅₃×#2971	C₂	semidirect
15. C₄×AGL(1,ℤ/8)×C₅₃	C₂	semidirect
18. C₅₃×#557	C₂²	semidirect
19. C₅₃×#556	C₂²	semidirect
23. D₄×Q₈×C₅₃	C₂²	semidirect
27. C₅₃×#547	C₂²	semidirect
30. C₅₃×#533	C₂²	semidirect
35. C₄×#41×C₅₃	C₂²	semidirect
38. C₅₃×#500	C₂²	semidirect
39. C₅₃×#492	C₂²	semidirect
40. C₅₃×#487	C₂²	semidirect
41. C₅₃×#486	C₂²	semidirect
42. C₅₃×#484	C₂²	semidirect
43. C₅₃×#482	C₂²	semidirect
44. C₅₃×#475	C₂²	semidirect
46. C₅₃×#474	C₂²	semidirect
47. C₅₃×#474	C₂²	semidirect
48. C₅₃×#468	C₂²	semidirect
49. C₅₃×#468	C₂²	semidirect
51. C₅₃×#461	C₂²	semidirect
52. C₅₃×#460	C₂²	semidirect
53. C₅₃×#459	C₂²	semidirect
55. C₅₃×#449	C₂²	semidirect
56. C₈×Q₈×C₅₃	C₂²	semidirect
57. C₈×Q₈×C₅₃	C₂²	semidirect
59. C₅₃×#442	C₂²	semidirect
61. C₅₃×#440	C₂²	semidirect
62. C₅₃×#440	C₂²	semidirect
64. C₄×SD₁₆×C₅₃	C₂²	semidirect
66. C₄×D₈×C₅₃	C₂²	semidirect
67. C₅₃×#436	C₂²	semidirect
68. C₅₃×#436	C₂²	semidirect
69. C₅₃×#432	C₂²	semidirect
70. C₅₃×#428	C₂²	semidirect
72. C₅₃×#417	C₂²	semidirect
73. C₄×M₁₆×C₅₃	C₂²	semidirect
207. D₈×C₅₃	#41	semidirect
256. Q₈×C₅₃	AGL(1,ℤ/8)	semidirect
268. D₄×C₅₃	C₂×#41	semidirect
295. C₈×C₅₃	C₂×#41	semidirect
296. #61466	C₅₃	direct (N × H)
323. #3482	C₂×C₅₃	semidirect
324. D₄×#41	C₂×C₅₃	semidirect
325. #3392	C₂×C₅₃	semidirect
326. Q₈×D₈	C₂×C₅₃	semidirect
327. #3359	C₂×C₅₃	semidirect
328. #3358	C₂×C₅₃	semidirect
329. #3353	C₂×C₅₃	semidirect
330. #3330	C₂×C₅₃	semidirect
331. #3324	C₂×C₅₃	semidirect
332. #3272	C₂×C₅₃	semidirect
333. #3182	C₂×C₅₃	semidirect
334. #3115	C₂×C₅₃	semidirect
335. #3110	C₂×C₅₃	semidirect
336. #2971	C₂×C₅₃	semidirect
337. C₄×AGL(1,ℤ/8)	C₂×C₅₃	semidirect
347. #557	C₂²×C₅₃	semidirect
348. #556	C₂²×C₅₃	semidirect
352. D₄×Q₈	C₂²×C₅₃	semidirect
356. #547	C₂²×C₅₃	semidirect
359. #533	C₂²×C₅₃	semidirect
364. C₄×#41	C₂²×C₅₃	semidirect
367. #500	C₂²×C₅₃	semidirect
368. #492	C₂²×C₅₃	semidirect
369. #487	C₂²×C₅₃	semidirect
370. #486	C₂²×C₅₃	semidirect
371. #484	C₂²×C₅₃	semidirect
372. #482	C₂²×C₅₃	semidirect
373. #475	C₂²×C₅₃	semidirect
375. #474	C₂²×C₅₃	semidirect
376. #474	C₂²×C₅₃	semidirect
377. #468	C₂²×C₅₃	semidirect
378. #468	C₂²×C₅₃	semidirect
380. #461	C₂²×C₅₃	semidirect
381. #460	C₂²×C₅₃	semidirect
382. #459	C₂²×C₅₃	semidirect
384. #449	C₂²×C₅₃	semidirect
385. C₈×Q₈	C₂²×C₅₃	semidirect
386. C₈×Q₈	C₂²×C₅₃	semidirect
388. #442	C₂²×C₅₃	semidirect
390. #440	C₂²×C₅₃	semidirect
391. #440	C₂²×C₅₃	semidirect
393. C₄×SD₁₆	C₂²×C₅₃	semidirect
395. C₄×D₈	C₂²×C₅₃	semidirect
396. #436	C₂²×C₅₃	semidirect
397. #436	C₂²×C₅₃	semidirect
398. #432	C₂²×C₅₃	semidirect
399. #428	C₂²×C₅₃	semidirect
401. #417	C₂²×C₅₃	semidirect
402. C₄×M₁₆	C₂²×C₅₃	semidirect
403. C₅₃	#61466	direct (N × H)
537. D₈	#41×C₅₃	semidirect
586. Q₈	AGL(1,ℤ/8)×C₅₃	semidirect
598. D₄	C₂×#41×C₅₃	semidirect
625. C₈	C₂×#41×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3482	2
D₄×#41×C₅₃	2
C₅₃×#3392	2
Q₈×D₈×C₅₃	2
C₅₃×#3359	2
C₅₃×#3358	2
C₅₃×#3353	2
C₅₃×#3330	2
C₅₃×#3324	2
C₅₃×#3272	2
C₅₃×#3182	2
C₅₃×#3115	2
C₅₃×#3110	2
C₅₃×#2971	2
C₄×AGL(1,ℤ/8)×C₅₃	2
#61466	53

Subgroups — C53×#61466 #27268978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#61466 #27268978