Subgroups — C₅₃×#60564 #27268076

Order 13568.

2218 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3455	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3123	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3122	C₂	1	4	yes	maximal, complemented
5	C₅₃×#3118	C₂	1	8	yes	maximal, complemented
6	C₅₃×#3093	C₂	1	2	yes	maximal, complemented
7	C₅₃×#3077	C₂	1	2	yes	maximal, complemented
8	C₅₃×#3036	C₂	1	2	yes	maximal, complemented
9	C₅₃×#3003	C₂	1	2	yes	maximal, complemented
10	C₄×AGL(1,ℤ/8)×C₅₃	C₂	1	4	yes	maximal, complemented
11	C₅₃×#2921	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	4	yes	—
14	C₅₃×#542	C₂²	1	4	yes	complemented
15	C₅₃×#541	C₂²	1	2	yes	complemented
16	C₅₃×#540	C₂²	1	2	yes	complemented
17	C₅₃×#539	—	2	2	no	—
18	C₅₃×#538	—	2	2	no	—
19	C₅₃×#533	C₂²	1	1	yes	characteristic, complemented
20	C₅₃×#532	C₂²	1	1	yes	characteristic, complemented
21	C₅₃×#529	—	2	4	no	—
22	C₅₃×#525	C₂²	1	2	yes	—
23	C₅₃×#518	—	2	4	no	—
24	C₅₃×#518	C₂²	1	4	yes	—
25	C₅₃×#518	C₂²	1	4	yes	—
26	C₄×#41×C₅₃	C₂²	1	2	yes	complemented
27	C₂×C₄×D₄×C₅₃	—	2	2	no	—
28	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
30	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
32	C₂²×C₂²⋊C₄×C₅₃	—	2	2	no	—
33	C₅₃×#485	C₂²	1	8	yes	complemented
34	C₅₃×#482	C₂²	1	8	yes	complemented
35	C₅₃×#478	C₂²	1	8	yes	complemented
36	C₅₃×#476	C₂²	1	4	yes	complemented
37	C₅₃×#474	C₂²	1	4	yes	complemented
38	C₅₃×#463	C₂²	1	8	yes	complemented
39	C₅₃×#461	C₂²	1	4	yes	complemented
40	C₅₃×#459	C₂²	1	4	yes	complemented
41	C₅₃×#450	C₂²	1	8	yes	—
42	C₅₃×#447	C₂²	1	8	yes	—
43	C₅₃×#442	C₂²	1	8	yes	complemented
44	C₅₃×#440	C₂²	1	8	yes	complemented
45	C₄×SD₁₆×C₅₃	C₂²	1	8	yes	complemented
46	C₄×D₈×C₅₃	C₂²	1	8	yes	complemented
47	C₅₃×#433	C₂²	1	4	yes	complemented
48	C₅₃×#433	C₂²	1	4	yes	complemented
49	C₅₃×#428	C₂²	1	4	yes	complemented
50	C₅₃×#423	C₂²	1	2	yes	complemented
51	C₅₃×#418	C₂²	1	4	yes	—
52	C₅₃×#417	C₂²	1	4	yes	complemented
53	C₅₃×#407	C₂²	1	2	yes	—
54	C₄×M₁₆×C₅₃	C₂²	1	2	yes	complemented
55	C₂⁵×C₅₃	—	2	1	no	—
56	C₂×#41×C₅₃	C₂³	1	2	yes	—
57	C₂²×D₄×C₅₃	—	2	8	no	—
58	C₂²×D₄×C₅₃	—	2	1	no	—
59	C₂²×D₄×C₅₃	—	2	2	no	—
60	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
61	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
62	C₂³×C₄×C₅₃	—	2	2	no	—
63	C₂³×C₄×C₅₃	—	2	2	no	—
64	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
65	AGL(1,ℤ/8)×C₅₃	—	2	16	no	—
66	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	—
67	C₂×D₈×C₅₃	C₂³	1	8	yes	—
68	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
69	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	complemented
70	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic, complemented
71	#127×C₅₃	C₂³	1	1	yes	characteristic, complemented
72	#126×C₅₃	—	2	4	no	—
73	#126×C₅₃	—	2	4	no	—
74	#125×C₅₃	—	2	4	no	—
75	#125×C₅₃	C₂³	1	2	yes	complemented
76	#124×C₅₃	—	2	4	no	—
77	#124×C₅₃	C₂³	1	2	yes	complemented
78	#123×C₅₃	—	2	4	no	—
79	#123×C₅₃	—	2	4	no	—
80	#123×C₅₃	—	2	2	no	—
81	C₂²⋊Q₈×C₅₃	—	2	4	no	—
82	C₂²⋊Q₈×C₅₃	—	2	2	no	—
83	C₂²⋊Q₈×C₅₃	C₂³	1	4	yes	—
84	#121×C₅₃	—	2	4	no	—
85	#121×C₅₃	—	2	2	no	—
86	#121×C₅₃	C₂³	1	4	yes	—
87	#120×C₅₃	—	2	2	no	—
88	C₄×Q₈×C₅₃	—	2	2	no	—
89	C₄×Q₈×C₅₃	C₂³	1	4	yes	complemented
90	C₄×D₄×C₅₃	—	2	8	no	—
91	C₄×D₄×C₅₃	—	2	8	no	—
92	C₄×D₄×C₅₃	—	2	4	no	—
93	C₄×D₄×C₅₃	—	2	2	no	—
94	C₄×D₄×C₅₃	—	2	2	no	—
95	C₄×D₄×C₅₃	—	2	2	no	—
96	C₄×D₄×C₅₃	—	2	4	no	—
97	C₄×D₄×C₅₃	—	2	4	no	—
98	C₄×D₄×C₅₃	C₂³	1	8	yes	—
99	C₄×D₄×C₅₃	C₂³	1	4	yes	—
100	C₄×D₄×C₅₃	C₂³	1	4	yes	—
101	C₄×D₄×C₅₃	C₂³	1	4	yes	complemented
102	#117×C₅₃	—	2	2	no	—
103	#117×C₅₃	—	2	2	no	—
104	#117×C₅₃	D₄	1	4	yes	—
105	#117×C₅₃	C₂³	1	4	yes	—
106	#117×C₅₃	C₂³	1	2	yes	—
107	#117×C₅₃	C₂³	1	2	yes	—
108	#117×C₅₃	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
110	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
112	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
113	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
115	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
117	C₂×C₄²×C₅₃	—	2	2	no	—
118	C₂×C₄²×C₅₃	—	2	1	no	—
119	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
121	#107×C₅₃	C₂³	1	8	yes	complemented
122	#106×C₅₃	C₂³	1	8	yes	complemented
123	C₄⋊C₈×C₅₃	C₂³	1	4	yes	complemented
124	C₄⋊C₈×C₅₃	C₂³	1	4	yes	complemented
125	#103×C₅₃	C₂³	1	8	yes	complemented
126	#102×C₅₃	C₂³	1	8	yes	—
127	#102×C₅₃	C₂³	1	8	yes	—
128	#102×C₅₃	C₂³	1	8	yes	complemented
129	C₂²⋊C₈×C₅₃	C₂³	1	8	yes	—
130	#97×C₅₃	C₂³	1	4	yes	complemented
131	C₄×C₈×C₅₃	C₂³	1	4	yes	complemented
132	C₂⁴×C₅₃	—	2	8	no	—
133	C₂⁴×C₅₃	—	2	2	no	—
134	C₂⁴×C₅₃	—	2	4	no	—
135	C₂⁴×C₅₃	—	8	4	no	—
136	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
137	#41×C₅₃	—	2	8	no	—
138	C₂×Q₈×C₅₃	—	2	2	no	—
139	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
140	C₂×D₄×C₅₃	—	2	8	no	—
141	C₂×D₄×C₅₃	—	2	16	no	—
142	C₂×D₄×C₅₃	—	2	16	no	—
143	C₂×D₄×C₅₃	—	2	2	no	—
144	C₂×D₄×C₅₃	—	2	4	no	—
145	C₂×D₄×C₅₃	—	2	4	no	—
146	C₂×D₄×C₅₃	—	2	2	no	—
147	C₂×D₄×C₅₃	—	2	2	no	—
148	C₂×D₄×C₅₃	—	2	4	no	—
149	C₂×D₄×C₅₃	#41	1	8	yes	—
150	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
151	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₅₃	—	2	4	no	—
153	C₂²×C₄×C₅₃	—	2	2	no	—
154	C₂²×C₄×C₅₃	—	2	2	no	—
155	C₂²×C₄×C₅₃	—	2	4	no	—
156	C₂²×C₄×C₅₃	—	2	8	no	—
157	C₂²×C₄×C₅₃	—	2	4	no	—
158	C₂²×C₄×C₅₃	—	2	4	no	—
159	C₂²×C₄×C₅₃	—	8	4	no	—
160	C₂²×C₄×C₅₃	—	8	4	no	—
161	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
167	SD₁₆×C₅₃	—	2	16	no	—
168	D₈×C₅₃	—	2	16	no	—
169	M₁₆×C₅₃	—	2	8	no	—
170	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
171	C₄⋊C₄×C₅₃	—	2	2	no	—
172	C₄⋊C₄×C₅₃	—	2	4	no	—
173	C₄⋊C₄×C₅₃	—	2	2	no	—
174	C₄⋊C₄×C₅₃	—	2	4	no	—
175	C₄⋊C₄×C₅₃	—	2	4	no	—
176	C₄⋊C₄×C₅₃	—	2	2	no	—
177	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
178	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
179	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
180	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
181	C₂²⋊C₄×C₅₃	—	2	4	no	—
182	C₂²⋊C₄×C₅₃	—	2	2	no	—
183	C₂²⋊C₄×C₅₃	—	2	4	no	—
184	C₂²⋊C₄×C₅₃	—	2	8	no	—
185	C₂²⋊C₄×C₅₃	—	2	4	no	—
186	C₂²⋊C₄×C₅₃	—	2	4	no	—
187	C₂²⋊C₄×C₅₃	—	2	4	no	—
188	C₂²⋊C₄×C₅₃	—	2	4	no	—
189	C₂²⋊C₄×C₅₃	—	2	4	no	—
190	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
191	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
192	C₄²×C₅₃	—	2	4	no	—
193	C₄²×C₅₃	—	2	2	no	—
194	C₄²×C₅₃	—	2	4	no	—
195	C₄²×C₅₃	C₂×D₄	1	2	yes	—
196	C₄²×C₅₃	C₂×D₄	1	2	yes	—
197	C₄²×C₅₃	C₂⁴	1	2	yes	—
198	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
199	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
200	C₂³×C₅₃	—	2	16	no	—
201	C₂³×C₅₃	—	2	2	no	—
202	C₂³×C₅₃	—	2	8	no	—
203	C₂³×C₅₃	—	2	4	no	—
204	C₂³×C₅₃	—	2	2	no	—
205	C₂³×C₅₃	—	4	2	no	—
206	C₂³×C₅₃	—	4	4	no	—
207	C₂³×C₅₃	—	4	4	no	—
208	C₂³×C₅₃	—	8	8	no	—
209	C₂³×C₅₃	—	8	16	no	—
210	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
211	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
212	Q₈×C₅₃	—	2	4	no	—
213	Q₈×C₅₃	—	8	2	no	—
214	D₄×C₅₃	—	2	16	no	—
215	D₄×C₅₃	—	2	8	no	—
216	D₄×C₅₃	—	2	4	no	—
217	D₄×C₅₃	—	2	4	no	—
218	D₄×C₅₃	—	4	4	no	—
219	D₄×C₅₃	—	8	2	no	—
220	D₄×C₅₃	—	8	4	no	—
221	D₄×C₅₃	C₂×#41	1	8	yes	—
222	C₂×C₄×C₅₃	—	2	2	no	—
223	C₂×C₄×C₅₃	—	2	4	no	—
224	C₂×C₄×C₅₃	—	2	4	no	—
225	C₂×C₄×C₅₃	—	2	4	no	—
226	C₂×C₄×C₅₃	—	2	4	no	—
227	C₂×C₄×C₅₃	—	2	2	no	—
228	C₂×C₄×C₅₃	—	4	2	no	—
229	C₂×C₄×C₅₃	—	4	2	no	—
230	C₂×C₄×C₅₃	—	4	4	no	—
231	C₂×C₄×C₅₃	—	4	4	no	—
232	C₂×C₄×C₅₃	—	8	4	no	—
233	C₂×C₄×C₅₃	—	8	8	no	—
234	C₂×C₄×C₅₃	—	8	4	no	—
235	C₂×C₄×C₅₃	—	8	4	no	—
236	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
237	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
238	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
239	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
240	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
242	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
243	C₈×C₅₃	—	2	8	no	—
244	#60564	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
245	C₂²×C₅₃	—	2	2	no	—
246	C₂²×C₅₃	—	2	4	no	—
247	C₂²×C₅₃	—	2	2	no	—
248	C₂²×C₅₃	—	2	8	no	—
249	C₂²×C₅₃	—	4	16	no	—
250	C₂²×C₅₃	—	4	4	no	—
251	C₂²×C₅₃	—	4	4	no	—
252	C₂²×C₅₃	—	4	4	no	—
253	C₂²×C₅₃	—	4	4	no	—
254	C₂²×C₅₃	—	4	2	no	—
255	C₂²×C₅₃	—	8	16	no	—
256	C₂²×C₅₃	#525	1	2	yes	—
257	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
258	C₄×C₅₃	—	2	4	no	—
259	C₄×C₅₃	—	2	2	no	—
260	C₄×C₅₃	—	2	2	no	—
261	C₄×C₅₃	—	2	2	no	—
262	C₄×C₅₃	—	4	4	no	—
263	C₄×C₅₃	—	4	4	no	—
264	C₄×C₅₃	—	4	2	no	—
265	C₄×C₅₃	#525	1	2	yes	—
266	C₄×C₅₃	C₂²×#41	1	2	yes	—
267	#3455	C₂×C₅₃	1	1	yes	characteristic, complemented
268	C₂×#518	C₂×C₅₃	1	1	yes	characteristic, complemented
269	#3123	C₂×C₅₃	1	4	yes	complemented
270	#3122	C₂×C₅₃	1	4	yes	complemented
271	#3118	C₂×C₅₃	1	8	yes	complemented
272	#3093	C₂×C₅₃	1	2	yes	complemented
273	#3077	C₂×C₅₃	1	2	yes	complemented
274	#3036	C₂×C₅₃	1	2	yes	complemented
275	#3003	C₂×C₅₃	1	2	yes	complemented
276	C₄×AGL(1,ℤ/8)	C₂×C₅₃	1	4	yes	complemented
277	#2921	C₂×C₅₃	1	1	yes	characteristic, complemented
278	C₂×C₅₃	—	2	2	no	—
279	C₂×C₅₃	—	4	2	no	—
280	C₂×C₅₃	—	4	8	no	—
281	C₂×C₅₃	—	8	2	no	—
282	C₂×C₅₃	C₂×#525	1	1	yes	characteristic
283	C₂×C₅₃	#3585	1	2	yes	—
284	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
285	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	4	yes	—
286	#542	C₂²×C₅₃	1	4	yes	complemented
287	#541	C₂²×C₅₃	1	2	yes	complemented
288	#540	C₂²×C₅₃	1	2	yes	complemented
289	#539	—	2	2	no	—
290	#538	—	2	2	no	—
291	#533	C₂²×C₅₃	1	1	yes	characteristic, complemented
292	#532	C₂²×C₅₃	1	1	yes	characteristic, complemented
293	#529	—	2	4	no	—
294	#525	C₂²×C₅₃	1	2	yes	—
295	#518	—	2	4	no	—
296	#518	C₂²×C₅₃	1	4	yes	—
297	#518	C₂²×C₅₃	1	4	yes	—
298	C₄×#41	C₂²×C₅₃	1	2	yes	complemented
299	C₂×C₄×D₄	—	2	2	no	—
300	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
301	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
302	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
303	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
304	C₂²×C₂²⋊C₄	—	2	2	no	—
305	#485	C₂²×C₅₃	1	8	yes	complemented
306	#482	C₂²×C₅₃	1	8	yes	complemented
307	#478	C₂²×C₅₃	1	8	yes	complemented
308	#476	C₂²×C₅₃	1	4	yes	complemented
309	#474	C₂²×C₅₃	1	4	yes	complemented
310	#463	C₂²×C₅₃	1	8	yes	complemented
311	#461	C₂²×C₅₃	1	4	yes	complemented
312	#459	C₂²×C₅₃	1	4	yes	complemented
313	#450	C₂²×C₅₃	1	8	yes	—
314	#447	C₂²×C₅₃	1	8	yes	—
315	#442	C₂²×C₅₃	1	8	yes	complemented
316	#440	C₂²×C₅₃	1	8	yes	complemented
317	C₄×SD₁₆	C₂²×C₅₃	1	8	yes	complemented
318	C₄×D₈	C₂²×C₅₃	1	8	yes	complemented
319	#433	C₂²×C₅₃	1	4	yes	complemented
320	#433	C₂²×C₅₃	1	4	yes	complemented
321	#428	C₂²×C₅₃	1	4	yes	complemented
322	#423	C₂²×C₅₃	1	2	yes	complemented
323	#418	C₂²×C₅₃	1	4	yes	—
324	#417	C₂²×C₅₃	1	4	yes	complemented
325	#407	C₂²×C₅₃	1	2	yes	—
326	C₄×M₁₆	C₂²×C₅₃	1	2	yes	complemented
327	C₅₃	#60564	1	1	yes	characteristic, direct product factor, Sylow
328	C₂⁵	—	2	1	no	—
329	C₂×#41	C₂³×C₅₃	1	2	yes	—
330	C₂²×D₄	—	2	8	no	—
331	C₂²×D₄	—	2	1	no	—
332	C₂²×D₄	—	2	2	no	—
333	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
334	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
335	C₂³×C₄	—	2	2	no	—
336	C₂³×C₄	—	2	2	no	—
337	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
338	AGL(1,ℤ/8)	—	2	16	no	—
339	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	—
340	C₂×D₈	C₂³×C₅₃	1	8	yes	—
341	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
342	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic, complemented
343	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	complemented
344	#127	C₂³×C₅₃	1	1	yes	characteristic, complemented
345	#126	—	2	4	no	—
346	#126	—	2	4	no	—
347	#125	—	2	4	no	—
348	#125	C₂³×C₅₃	1	2	yes	complemented
349	#124	—	2	4	no	—
350	#124	C₂³×C₅₃	1	2	yes	complemented
351	#123	—	2	2	no	—
352	#123	—	2	4	no	—
353	#123	—	2	4	no	—
354	C₂²⋊Q₈	—	2	2	no	—
355	C₂²⋊Q₈	—	2	4	no	—
356	C₂²⋊Q₈	C₂³×C₅₃	1	4	yes	—
357	#121	—	2	2	no	—
358	#121	—	2	4	no	—
359	#121	C₂³×C₅₃	1	4	yes	—
360	#120	—	2	2	no	—
361	C₄×Q₈	—	2	2	no	—
362	C₄×Q₈	C₂³×C₅₃	1	4	yes	complemented
363	C₄×D₄	—	2	4	no	—
364	C₄×D₄	—	2	4	no	—
365	C₄×D₄	—	2	8	no	—
366	C₄×D₄	—	2	8	no	—
367	C₄×D₄	—	2	4	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	2	2	no	—
370	C₄×D₄	—	2	2	no	—
371	C₄×D₄	C₂³×C₅₃	1	8	yes	—
372	C₄×D₄	C₂³×C₅₃	1	4	yes	—
373	C₄×D₄	C₂³×C₅₃	1	4	yes	—
374	C₄×D₄	C₂³×C₅₃	1	4	yes	complemented
375	#117	—	2	2	no	—
376	#117	—	2	2	no	—
377	#117	D₄×C₅₃	1	4	yes	—
378	#117	C₂³×C₅₃	1	4	yes	—
379	#117	C₂³×C₅₃	1	2	yes	—
380	#117	C₂³×C₅₃	1	2	yes	—
381	#117	C₂³×C₅₃	1	2	yes	—
382	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
383	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
384	C₂×C₂²⋊C₄	—	2	1	no	—
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₂²⋊C₄	—	2	2	no	—
387	C₂×C₂²⋊C₄	—	2	8	no	—
388	C₂×C₂²⋊C₄	—	2	8	no	—
389	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
390	C₂×C₄²	—	2	1	no	—
391	C₂×C₄²	—	2	2	no	—
392	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
393	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
394	#107	C₂³×C₅₃	1	8	yes	complemented
395	#106	C₂³×C₅₃	1	8	yes	complemented
396	C₄⋊C₈	C₂³×C₅₃	1	4	yes	complemented
397	C₄⋊C₈	C₂³×C₅₃	1	4	yes	complemented
398	#103	C₂³×C₅₃	1	8	yes	complemented
399	#102	C₂³×C₅₃	1	8	yes	—
400	#102	C₂³×C₅₃	1	8	yes	—
401	#102	C₂³×C₅₃	1	8	yes	complemented
402	C₂²⋊C₈	C₂³×C₅₃	1	8	yes	—
403	#97	C₂³×C₅₃	1	4	yes	complemented
404	C₄×C₈	C₂³×C₅₃	1	4	yes	complemented
405	C₂⁴	—	2	8	no	—
406	C₂⁴	—	2	2	no	—
407	C₂⁴	—	2	4	no	—
408	C₂⁴	—	8	4	no	—
409	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
410	#41	—	2	8	no	—
411	C₂×Q₈	—	2	2	no	—
412	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
413	C₂×D₄	—	2	8	no	—
414	C₂×D₄	—	2	16	no	—
415	C₂×D₄	—	2	16	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	4	no	—
418	C₂×D₄	—	2	4	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	#41×C₅₃	1	8	yes	—
423	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
424	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	8	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	8	4	no	—
433	C₂²×C₄	—	8	4	no	—
434	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
436	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
437	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
438	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
440	SD₁₆	—	2	16	no	—
441	D₈	—	2	16	no	—
442	M₁₆	—	2	8	no	—
443	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	—	2	4	no	—
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
451	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
452	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
453	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	2	2	no	—
456	C₂²⋊C₄	—	2	4	no	—
457	C₂²⋊C₄	—	2	8	no	—
458	C₂²⋊C₄	—	2	4	no	—
459	C₂²⋊C₄	—	2	4	no	—
460	C₂²⋊C₄	—	2	4	no	—
461	C₂²⋊C₄	—	2	4	no	—
462	C₂²⋊C₄	—	2	4	no	—
463	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
464	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
465	C₄²	—	2	4	no	—
466	C₄²	—	2	2	no	—
467	C₄²	—	2	4	no	—
468	C₄²	C₂×D₄×C₅₃	1	2	yes	—
469	C₄²	C₂×D₄×C₅₃	1	2	yes	—
470	C₄²	C₂⁴×C₅₃	1	2	yes	—
471	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
472	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
473	C₂³	—	2	16	no	—
474	C₂³	—	2	2	no	—
475	C₂³	—	2	8	no	—
476	C₂³	—	2	4	no	—
477	C₂³	—	2	2	no	—
478	C₂³	—	4	2	no	—
479	C₂³	—	4	4	no	—
480	C₂³	—	4	4	no	—
481	C₂³	—	8	8	no	—
482	C₂³	—	8	16	no	—
483	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
484	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
485	Q₈	—	2	4	no	—
486	Q₈	—	8	2	no	—
487	D₄	—	2	16	no	—
488	D₄	—	2	8	no	—
489	D₄	—	2	4	no	—
490	D₄	—	2	4	no	—
491	D₄	—	4	4	no	—
492	D₄	—	8	2	no	—
493	D₄	—	8	4	no	—
494	D₄	C₂×#41×C₅₃	1	8	yes	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	4	no	—
504	C₂×C₄	—	4	4	no	—
505	C₂×C₄	—	8	4	no	—
506	C₂×C₄	—	8	8	no	—
507	C₂×C₄	—	8	4	no	—
508	C₂×C₄	—	8	4	no	—
509	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
510	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
511	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
512	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
513	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
514	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
515	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
516	C₈	—	2	8	no	—
517	C₂²	—	2	2	no	—
518	C₂²	—	2	4	no	—
519	C₂²	—	2	2	no	—
520	C₂²	—	2	8	no	—
521	C₂²	—	4	4	no	—
522	C₂²	—	4	16	no	—
523	C₂²	—	4	4	no	—
524	C₂²	—	4	4	no	—
525	C₂²	—	4	4	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	8	16	no	—
528	C₂²	C₅₃×#525	1	2	yes	—
529	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
530	C₄	—	2	4	no	—
531	C₄	—	2	2	no	—
532	C₄	—	2	2	no	—
533	C₄	—	2	2	no	—
534	C₄	—	4	4	no	—
535	C₄	—	4	4	no	—
536	C₄	—	4	2	no	—
537	C₄	C₅₃×#525	1	2	yes	—
538	C₄	C₂²×#41×C₅₃	1	2	yes	—
539	C₂	—	2	2	no	—
540	C₂	—	4	2	no	—
541	C₂	—	4	8	no	—
542	C₂	—	8	2	no	—
543	C₂	C₂×C₅₃×#525	1	1	yes	characteristic
544	C₂	C₅₃×#3585	1	2	yes	—

	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×#117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3455	C₂	semidirect
2. C₂×C₅₃×#518	C₂	semidirect
3. C₅₃×#3123	C₂	semidirect
4. C₅₃×#3122	C₂	semidirect
5. C₅₃×#3118	C₂	semidirect
6. C₅₃×#3093	C₂	semidirect
7. C₅₃×#3077	C₂	semidirect
8. C₅₃×#3036	C₂	semidirect
9. C₅₃×#3003	C₂	semidirect
10. C₄×AGL(1,ℤ/8)×C₅₃	C₂	semidirect
11. C₅₃×#2921	C₂	semidirect
14. C₅₃×#542	C₂²	semidirect
15. C₅₃×#541	C₂²	semidirect
16. C₅₃×#540	C₂²	semidirect
19. C₅₃×#533	C₂²	semidirect
20. C₅₃×#532	C₂²	semidirect
26. C₄×#41×C₅₃	C₂²	semidirect
33. C₅₃×#485	C₂²	semidirect
34. C₅₃×#482	C₂²	semidirect
35. C₅₃×#478	C₂²	semidirect
36. C₅₃×#476	C₂²	semidirect
37. C₅₃×#474	C₂²	semidirect
38. C₅₃×#463	C₂²	semidirect
39. C₅₃×#461	C₂²	semidirect
40. C₅₃×#459	C₂²	semidirect
43. C₅₃×#442	C₂²	semidirect
44. C₅₃×#440	C₂²	semidirect
45. C₄×SD₁₆×C₅₃	C₂²	semidirect
46. C₄×D₈×C₅₃	C₂²	semidirect
47. C₅₃×#433	C₂²	semidirect
48. C₅₃×#433	C₂²	semidirect
49. C₅₃×#428	C₂²	semidirect
50. C₅₃×#423	C₂²	semidirect
52. C₅₃×#417	C₂²	semidirect
54. C₄×M₁₆×C₅₃	C₂²	semidirect
69. C₄⋊Q₈×C₅₃	C₂³	semidirect
70. C₄⋊Q₈×C₅₃	C₂³	semidirect
71. #127×C₅₃	C₂³	semidirect
75. #125×C₅₃	C₂³	semidirect
77. #124×C₅₃	C₂³	semidirect
89. C₄×Q₈×C₅₃	C₂³	semidirect
101. C₄×D₄×C₅₃	C₂³	semidirect
121. #107×C₅₃	C₂³	semidirect
122. #106×C₅₃	C₂³	semidirect
123. C₄⋊C₈×C₅₃	C₂³	semidirect
124. C₄⋊C₈×C₅₃	C₂³	semidirect
125. #103×C₅₃	C₂³	semidirect
128. #102×C₅₃	C₂³	semidirect
130. #97×C₅₃	C₂³	semidirect
131. C₄×C₈×C₅₃	C₂³	semidirect
244. #60564	C₅₃	direct (N × H)
267. #3455	C₂×C₅₃	semidirect
268. C₂×#518	C₂×C₅₃	semidirect
269. #3123	C₂×C₅₃	semidirect
270. #3122	C₂×C₅₃	semidirect
271. #3118	C₂×C₅₃	semidirect
272. #3093	C₂×C₅₃	semidirect
273. #3077	C₂×C₅₃	semidirect
274. #3036	C₂×C₅₃	semidirect
275. #3003	C₂×C₅₃	semidirect
276. C₄×AGL(1,ℤ/8)	C₂×C₅₃	semidirect
277. #2921	C₂×C₅₃	semidirect
286. #542	C₂²×C₅₃	semidirect
287. #541	C₂²×C₅₃	semidirect
288. #540	C₂²×C₅₃	semidirect
291. #533	C₂²×C₅₃	semidirect
292. #532	C₂²×C₅₃	semidirect
298. C₄×#41	C₂²×C₅₃	semidirect
305. #485	C₂²×C₅₃	semidirect
306. #482	C₂²×C₅₃	semidirect
307. #478	C₂²×C₅₃	semidirect
308. #476	C₂²×C₅₃	semidirect
309. #474	C₂²×C₅₃	semidirect
310. #463	C₂²×C₅₃	semidirect
311. #461	C₂²×C₅₃	semidirect
312. #459	C₂²×C₅₃	semidirect
315. #442	C₂²×C₅₃	semidirect
316. #440	C₂²×C₅₃	semidirect
317. C₄×SD₁₆	C₂²×C₅₃	semidirect
318. C₄×D₈	C₂²×C₅₃	semidirect
319. #433	C₂²×C₅₃	semidirect
320. #433	C₂²×C₅₃	semidirect
321. #428	C₂²×C₅₃	semidirect
322. #423	C₂²×C₅₃	semidirect
324. #417	C₂²×C₅₃	semidirect
326. C₄×M₁₆	C₂²×C₅₃	semidirect
327. C₅₃	#60564	direct (N × H)
342. C₄⋊Q₈	C₂³×C₅₃	semidirect
343. C₄⋊Q₈	C₂³×C₅₃	semidirect
344. #127	C₂³×C₅₃	semidirect
348. #125	C₂³×C₅₃	semidirect
350. #124	C₂³×C₅₃	semidirect
362. C₄×Q₈	C₂³×C₅₃	semidirect
374. C₄×D₄	C₂³×C₅₃	semidirect
394. #107	C₂³×C₅₃	semidirect
395. #106	C₂³×C₅₃	semidirect
396. C₄⋊C₈	C₂³×C₅₃	semidirect
397. C₄⋊C₈	C₂³×C₅₃	semidirect
398. #103	C₂³×C₅₃	semidirect
401. #102	C₂³×C₅₃	semidirect
403. #97	C₂³×C₅₃	semidirect
404. C₄×C₈	C₂³×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3455	2
C₂×C₅₃×#518	2
C₅₃×#3123	2
C₅₃×#3122	2
C₅₃×#3118	2
C₅₃×#3093	2
C₅₃×#3077	2
C₅₃×#3036	2
C₅₃×#3003	2
C₄×AGL(1,ℤ/8)×C₅₃	2
C₅₃×#2921	2
#60564	53

Subgroups — C53×#60564 #27268076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60564 #27268076