Subgroups — C₅₃×#60566 #27268078

Order 13568.

2002 conjugacy classes of subgroups.

The subgroup lattice has 516 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
2	C₂×C₅₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3127	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3126	C₂	1	4	yes	maximal, complemented
5	C₅₃×#3121	C₂	1	8	yes	maximal, complemented
6	C₅₃×#3093	C₂	1	2	yes	maximal, complemented
7	C₅₃×#3078	C₂	1	2	yes	maximal, complemented
8	C₅₃×#3037	C₂	1	2	yes	maximal, complemented
9	C₅₃×#3004	C₂	1	2	yes	maximal, complemented
10	C₄×#137×C₅₃	C₂	1	4	yes	maximal, complemented
11	C₅₃×#2921	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#137×C₅₃	C₂²	1	4	yes	—
14	C₅₃×#542	C₂²	1	4	yes	—
15	C₅₃×#541	C₂²	1	2	yes	—
16	C₅₃×#540	C₂²	1	2	yes	—
17	C₅₃×#539	—	2	2	no	—
18	C₅₃×#538	—	2	2	no	—
19	C₅₃×#533	C₂²	1	1	yes	characteristic
20	C₅₃×#532	C₂²	1	1	yes	characteristic
21	C₅₃×#529	—	2	4	no	—
22	C₅₃×#525	C₂²	1	2	yes	—
23	C₅₃×#519	—	2	4	no	—
24	C₅₃×#519	C₂²	1	4	yes	—
25	C₅₃×#519	C₂²	1	4	yes	—
26	C₄×#41×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₅₃	—	2	2	no	—
28	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
30	C₂×#117×C₅₃	—	2	2	no	—
31	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#485	C₂²	1	8	yes	—
34	C₅₃×#482	C₂²	1	8	yes	—
35	C₅₃×#479	C₂²	1	8	yes	complemented
36	C₅₃×#477	C₂²	1	4	yes	complemented
37	C₅₃×#475	C₂²	1	4	yes	complemented
38	C₅₃×#464	C₂²	1	8	yes	complemented
39	C₅₃×#462	C₂²	1	4	yes	complemented
40	C₅₃×#460	C₂²	1	4	yes	complemented
41	C₅₃×#451	C₂²	1	8	yes	—
42	C₅₃×#448	C₂²	1	8	yes	—
43	C₅₃×#441	C₂²	1	8	yes	complemented
44	C₅₃×#440	C₂²	1	8	yes	complemented
45	C₄×Q₁₆×C₅₃	C₂²	1	8	yes	complemented
46	C₄×SD₁₆×C₅₃	C₂²	1	8	yes	complemented
47	C₅₃×#433	C₂²	1	4	yes	—
48	C₅₃×#433	C₂²	1	4	yes	—
49	C₅₃×#428	C₂²	1	4	yes	—
50	C₅₃×#423	C₂²	1	2	yes	—
51	C₅₃×#419	C₂²	1	4	yes	—
52	C₅₃×#417	C₂²	1	4	yes	—
53	C₅₃×#407	C₂²	1	2	yes	—
54	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
55	C₂×#41×C₅₃	C₂³	1	2	yes	—
56	C₂²×Q₈×C₅₃	—	2	8	no	—
57	C₂²×Q₈×C₅₃	—	2	1	no	—
58	C₂²×Q₈×C₅₃	—	2	2	no	—
59	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
60	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
61	C₂³×C₄×C₅₃	—	2	1	no	—
62	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
63	#137×C₅₃	—	2	16	no	—
64	C₂×Q₁₆×C₅₃	C₂³	1	8	yes	—
65	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	—
66	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
67	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
68	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
69	#127×C₅₃	C₂³	1	1	yes	characteristic
70	#126×C₅₃	—	2	4	no	—
71	#126×C₅₃	—	2	4	no	—
72	#125×C₅₃	—	2	4	no	—
73	#125×C₅₃	C₂³	1	2	yes	—
74	#124×C₅₃	—	2	4	no	—
75	#124×C₅₃	C₂³	1	2	yes	—
76	#123×C₅₃	—	2	4	no	—
77	#123×C₅₃	—	2	4	no	—
78	#123×C₅₃	—	2	2	no	—
79	C₂²⋊Q₈×C₅₃	—	2	4	no	—
80	C₂²⋊Q₈×C₅₃	—	2	2	no	—
81	C₂²⋊Q₈×C₅₃	C₂³	1	4	yes	—
82	#121×C₅₃	—	2	4	no	—
83	#121×C₅₃	—	2	2	no	—
84	#121×C₅₃	C₂³	1	4	yes	—
85	#120×C₅₃	—	2	2	no	—
86	C₄×Q₈×C₅₃	—	2	8	no	—
87	C₄×Q₈×C₅₃	—	2	8	no	—
88	C₄×Q₈×C₅₃	—	2	4	no	—
89	C₄×Q₈×C₅₃	—	2	2	no	—
90	C₄×Q₈×C₅₃	—	2	2	no	—
91	C₄×Q₈×C₅₃	—	2	2	no	—
92	C₄×Q₈×C₅₃	C₂³	1	8	yes	—
93	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
94	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
95	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
96	C₄×D₄×C₅₃	—	2	2	no	—
97	C₄×D₄×C₅₃	—	2	4	no	—
98	C₄×D₄×C₅₃	—	2	4	no	—
99	C₄×D₄×C₅₃	C₂³	1	4	yes	—
100	#117×C₅₃	—	2	8	no	—
101	#117×C₅₃	—	2	8	no	—
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	—	2	2	no	—
110	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
111	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
112	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
115	C₂×C₄²×C₅₃	—	2	2	no	—
116	C₂×C₄²×C₅₃	—	2	2	no	—
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	—
122	#106×C₅₃	C₂³	1	8	yes	—
123	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
124	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
125	#103×C₅₃	C₂³	1	8	yes	—
126	#103×C₅₃	C₂³	1	8	yes	—
127	#103×C₅₃	C₂³	1	8	yes	—
128	#102×C₅₃	C₂³	1	8	yes	—
129	C₂²⋊C₈×C₅₃	C₂³	1	8	yes	—
130	#97×C₅₃	C₂³	1	4	yes	—
131	C₄×C₈×C₅₃	C₂³	1	4	yes	—
132	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
133	#41×C₅₃	—	2	8	no	—
134	C₂×Q₈×C₅₃	—	2	16	no	—
135	C₂×Q₈×C₅₃	—	2	8	no	—
136	C₂×Q₈×C₅₃	—	2	16	no	—
137	C₂×Q₈×C₅₃	—	2	4	no	—
138	C₂×Q₈×C₅₃	—	2	2	no	—
139	C₂×Q₈×C₅₃	—	2	2	no	—
140	C₂×Q₈×C₅₃	—	2	4	no	—
141	C₂×Q₈×C₅₃	#41	1	8	yes	—
142	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
143	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
144	C₂×D₄×C₅₃	—	2	2	no	—
145	C₂×D₄×C₅₃	—	2	4	no	—
146	C₂×D₄×C₅₃	—	2	2	no	—
147	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
148	C₂²×C₄×C₅₃	—	2	8	no	—
149	C₂²×C₄×C₅₃	—	2	2	no	—
150	C₂²×C₄×C₅₃	—	2	4	no	—
151	C₂²×C₄×C₅₃	—	2	4	no	—
152	C₂²×C₄×C₅₃	—	2	2	no	—
153	C₂²×C₄×C₅₃	—	2	2	no	—
154	C₂²×C₄×C₅₃	—	2	4	no	—
155	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
157	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
158	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
161	Q₁₆×C₅₃	—	2	16	no	—
162	SD₁₆×C₅₃	—	2	16	no	—
163	M₁₆×C₅₃	—	2	8	no	—
164	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
165	C₄⋊C₄×C₅₃	—	2	8	no	—
166	C₄⋊C₄×C₅₃	—	2	4	no	—
167	C₄⋊C₄×C₅₃	—	2	4	no	—
168	C₄⋊C₄×C₅₃	—	2	2	no	—
169	C₄⋊C₄×C₅₃	—	2	4	no	—
170	C₄⋊C₄×C₅₃	—	2	2	no	—
171	C₄⋊C₄×C₅₃	—	2	4	no	—
172	C₄⋊C₄×C₅₃	—	2	4	no	—
173	C₄⋊C₄×C₅₃	—	2	2	no	—
174	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
175	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
176	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
177	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
178	C₂²⋊C₄×C₅₃	—	2	4	no	—
179	C₂²⋊C₄×C₅₃	—	2	2	no	—
180	C₂²⋊C₄×C₅₃	—	2	4	no	—
181	C₂²⋊C₄×C₅₃	—	2	4	no	—
182	C₂²⋊C₄×C₅₃	—	2	4	no	—
183	C₂²⋊C₄×C₅₃	—	2	4	no	—
184	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
185	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
186	C₄²×C₅₃	—	2	8	no	—
187	C₄²×C₅₃	—	2	4	no	—
188	C₄²×C₅₃	—	2	4	no	—
189	C₄²×C₅₃	—	2	2	no	—
190	C₄²×C₅₃	—	2	4	no	—
191	C₄²×C₅₃	—	2	4	no	—
192	C₄²×C₅₃	C₂×D₄	1	2	yes	—
193	C₄²×C₅₃	C₂×D₄	1	2	yes	—
194	C₄²×C₅₃	C₂⁴	1	2	yes	—
195	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
196	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
197	C₂³×C₅₃	—	2	2	no	—
198	C₂³×C₅₃	—	2	2	no	—
199	C₂³×C₅₃	—	4	2	no	—
200	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
201	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
202	Q₈×C₅₃	—	2	16	no	—
203	Q₈×C₅₃	—	2	8	no	—
204	Q₈×C₅₃	—	2	4	no	—
205	Q₈×C₅₃	—	2	4	no	—
206	Q₈×C₅₃	—	8	2	no	—
207	Q₈×C₅₃	C₂×#41	1	8	yes	—
208	D₄×C₅₃	—	2	4	no	—
209	D₄×C₅₃	—	4	4	no	—
210	D₄×C₅₃	—	8	2	no	—
211	D₄×C₅₃	—	8	4	no	—
212	C₂×C₄×C₅₃	—	2	16	no	—
213	C₂×C₄×C₅₃	—	2	8	no	—
214	C₂×C₄×C₅₃	—	2	2	no	—
215	C₂×C₄×C₅₃	—	2	4	no	—
216	C₂×C₄×C₅₃	—	2	4	no	—
217	C₂×C₄×C₅₃	—	2	4	no	—
218	C₂×C₄×C₅₃	—	2	4	no	—
219	C₂×C₄×C₅₃	—	2	2	no	—
220	C₂×C₄×C₅₃	—	2	4	no	—
221	C₂×C₄×C₅₃	—	4	2	no	—
222	C₂×C₄×C₅₃	—	4	2	no	—
223	C₂×C₄×C₅₃	—	4	4	no	—
224	C₂×C₄×C₅₃	—	4	4	no	—
225	C₂×C₄×C₅₃	—	8	4	no	—
226	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
227	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
228	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
229	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
230	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
232	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
233	C₈×C₅₃	—	2	8	no	—
234	#60566	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
235	C₂²×C₅₃	—	2	2	no	—
236	C₂²×C₅₃	—	2	4	no	—
237	C₂²×C₅₃	—	2	2	no	—
238	C₂²×C₅₃	—	4	4	no	—
239	C₂²×C₅₃	—	4	4	no	—
240	C₂²×C₅₃	—	4	2	no	—
241	C₂²×C₅₃	#525	1	2	yes	—
242	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
243	C₄×C₅₃	—	2	4	no	—
244	C₄×C₅₃	—	2	2	no	—
245	C₄×C₅₃	—	2	2	no	—
246	C₄×C₅₃	—	2	8	no	—
247	C₄×C₅₃	—	2	2	no	—
248	C₄×C₅₃	—	4	4	no	—
249	C₄×C₅₃	—	4	4	no	—
250	C₄×C₅₃	—	4	2	no	—
251	C₄×C₅₃	#525	1	2	yes	—
252	C₄×C₅₃	C₂²×#41	1	2	yes	—
253	#3455	C₂×C₅₃	1	1	yes	characteristic
254	C₂×#519	C₂×C₅₃	1	1	yes	characteristic, complemented
255	#3127	C₂×C₅₃	1	4	yes	complemented
256	#3126	C₂×C₅₃	1	4	yes	complemented
257	#3121	C₂×C₅₃	1	8	yes	complemented
258	#3093	C₂×C₅₃	1	2	yes	complemented
259	#3078	C₂×C₅₃	1	2	yes	complemented
260	#3037	C₂×C₅₃	1	2	yes	complemented
261	#3004	C₂×C₅₃	1	2	yes	complemented
262	C₄×#137	C₂×C₅₃	1	4	yes	complemented
263	#2921	C₂×C₅₃	1	1	yes	characteristic, complemented
264	C₂×C₅₃	—	2	2	no	—
265	C₂×C₅₃	—	4	2	no	—
266	C₂×C₅₃	—	8	2	no	—
267	C₂×C₅₃	C₂×#525	1	1	yes	characteristic
268	C₂×C₅₃	#3586	1	2	yes	—
269	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
270	C₂×#137	C₂²×C₅₃	1	4	yes	—
271	#542	C₂²×C₅₃	1	4	yes	—
272	#541	C₂²×C₅₃	1	2	yes	—
273	#540	C₂²×C₅₃	1	2	yes	—
274	#539	—	2	2	no	—
275	#538	—	2	2	no	—
276	#533	C₂²×C₅₃	1	1	yes	characteristic
277	#532	C₂²×C₅₃	1	1	yes	characteristic
278	#529	—	2	4	no	—
279	#525	C₂²×C₅₃	1	2	yes	—
280	#519	—	2	4	no	—
281	#519	C₂²×C₅₃	1	4	yes	—
282	#519	C₂²×C₅₃	1	4	yes	—
283	C₄×#41	C₂²×C₅₃	1	2	yes	—
284	C₂×C₄×Q₈	—	2	2	no	—
285	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
286	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
287	C₂×#117	—	2	2	no	—
288	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
289	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
290	#485	C₂²×C₅₃	1	8	yes	—
291	#482	C₂²×C₅₃	1	8	yes	—
292	#479	C₂²×C₅₃	1	8	yes	complemented
293	#477	C₂²×C₅₃	1	4	yes	complemented
294	#475	C₂²×C₅₃	1	4	yes	complemented
295	#464	C₂²×C₅₃	1	8	yes	complemented
296	#462	C₂²×C₅₃	1	4	yes	complemented
297	#460	C₂²×C₅₃	1	4	yes	complemented
298	#451	C₂²×C₅₃	1	8	yes	—
299	#448	C₂²×C₅₃	1	8	yes	—
300	#441	C₂²×C₅₃	1	8	yes	complemented
301	#440	C₂²×C₅₃	1	8	yes	complemented
302	C₄×Q₁₆	C₂²×C₅₃	1	8	yes	complemented
303	C₄×SD₁₆	C₂²×C₅₃	1	8	yes	complemented
304	#433	C₂²×C₅₃	1	4	yes	—
305	#433	C₂²×C₅₃	1	4	yes	—
306	#428	C₂²×C₅₃	1	4	yes	—
307	#423	C₂²×C₅₃	1	2	yes	—
308	#419	C₂²×C₅₃	1	4	yes	—
309	#417	C₂²×C₅₃	1	4	yes	—
310	#407	C₂²×C₅₃	1	2	yes	—
311	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
312	C₅₃	#60566	1	1	yes	characteristic, direct product factor, Sylow
313	C₂×#41	C₂³×C₅₃	1	2	yes	—
314	C₂²×Q₈	—	2	8	no	—
315	C₂²×Q₈	—	2	1	no	—
316	C₂²×Q₈	—	2	2	no	—
317	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
318	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
319	C₂³×C₄	—	2	1	no	—
320	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
321	#137	—	2	16	no	—
322	C₂×Q₁₆	C₂³×C₅₃	1	8	yes	—
323	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	—
324	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
325	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
326	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
327	#127	C₂³×C₅₃	1	1	yes	characteristic
328	#126	—	2	4	no	—
329	#126	—	2	4	no	—
330	#125	—	2	4	no	—
331	#125	C₂³×C₅₃	1	2	yes	—
332	#124	—	2	4	no	—
333	#124	C₂³×C₅₃	1	2	yes	—
334	#123	—	2	2	no	—
335	#123	—	2	4	no	—
336	#123	—	2	4	no	—
337	C₂²⋊Q₈	—	2	2	no	—
338	C₂²⋊Q₈	—	2	4	no	—
339	C₂²⋊Q₈	C₂³×C₅₃	1	4	yes	—
340	#121	—	2	2	no	—
341	#121	—	2	4	no	—
342	#121	C₂³×C₅₃	1	4	yes	—
343	#120	—	2	2	no	—
344	C₄×Q₈	—	2	8	no	—
345	C₄×Q₈	—	2	8	no	—
346	C₄×Q₈	—	2	4	no	—
347	C₄×Q₈	—	2	2	no	—
348	C₄×Q₈	—	2	2	no	—
349	C₄×Q₈	—	2	2	no	—
350	C₄×Q₈	C₂³×C₅₃	1	8	yes	—
351	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
352	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
353	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
354	C₄×D₄	—	2	4	no	—
355	C₄×D₄	—	2	4	no	—
356	C₄×D₄	—	2	2	no	—
357	C₄×D₄	C₂³×C₅₃	1	4	yes	—
358	#117	—	2	2	no	—
359	#117	—	2	8	no	—
360	#117	—	2	8	no	—
361	#117	—	2	2	no	—
362	#117	D₄×C₅₃	1	4	yes	—
363	#117	C₂³×C₅₃	1	4	yes	—
364	#117	C₂³×C₅₃	1	2	yes	—
365	#117	C₂³×C₅₃	1	2	yes	—
366	#117	C₂³×C₅₃	1	2	yes	—
367	C₂×C₄⋊C₄	—	2	2	no	—
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
370	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
371	C₂×C₂²⋊C₄	—	2	1	no	—
372	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
373	C₂×C₄²	—	2	2	no	—
374	C₂×C₄²	—	2	2	no	—
375	C₂×C₄²	—	2	1	no	—
376	C₂×C₄²	—	2	2	no	—
377	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
378	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
379	#107	C₂³×C₅₃	1	8	yes	—
380	#106	C₂³×C₅₃	1	8	yes	—
381	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
382	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
383	#103	C₂³×C₅₃	1	8	yes	—
384	#103	C₂³×C₅₃	1	8	yes	—
385	#103	C₂³×C₅₃	1	8	yes	—
386	#102	C₂³×C₅₃	1	8	yes	—
387	C₂²⋊C₈	C₂³×C₅₃	1	8	yes	—
388	#97	C₂³×C₅₃	1	4	yes	—
389	C₄×C₈	C₂³×C₅₃	1	4	yes	—
390	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
391	#41	—	2	8	no	—
392	C₂×Q₈	—	2	16	no	—
393	C₂×Q₈	—	2	8	no	—
394	C₂×Q₈	—	2	16	no	—
395	C₂×Q₈	—	2	4	no	—
396	C₂×Q₈	—	2	2	no	—
397	C₂×Q₈	—	2	2	no	—
398	C₂×Q₈	—	2	4	no	—
399	C₂×Q₈	#41×C₅₃	1	8	yes	—
400	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
401	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	4	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
406	C₂²×C₄	—	2	8	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	4	no	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
414	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
415	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
416	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
417	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
418	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
419	Q₁₆	—	2	16	no	—
420	SD₁₆	—	2	16	no	—
421	M₁₆	—	2	8	no	—
422	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
423	C₄⋊C₄	—	2	8	no	—
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	—	2	4	no	—
426	C₄⋊C₄	—	2	2	no	—
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	4	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
433	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
434	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
435	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
436	C₂²⋊C₄	—	2	4	no	—
437	C₂²⋊C₄	—	2	2	no	—
438	C₂²⋊C₄	—	2	4	no	—
439	C₂²⋊C₄	—	2	4	no	—
440	C₂²⋊C₄	—	2	4	no	—
441	C₂²⋊C₄	—	2	4	no	—
442	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
443	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
444	C₄²	—	2	8	no	—
445	C₄²	—	2	4	no	—
446	C₄²	—	2	4	no	—
447	C₄²	—	2	2	no	—
448	C₄²	—	2	4	no	—
449	C₄²	—	2	4	no	—
450	C₄²	C₂×D₄×C₅₃	1	2	yes	—
451	C₄²	C₂×D₄×C₅₃	1	2	yes	—
452	C₄²	C₂⁴×C₅₃	1	2	yes	—
453	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
454	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
455	C₂³	—	2	2	no	—
456	C₂³	—	2	2	no	—
457	C₂³	—	4	2	no	—
458	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
459	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
460	Q₈	—	2	16	no	—
461	Q₈	—	2	8	no	—
462	Q₈	—	2	4	no	—
463	Q₈	—	2	4	no	—
464	Q₈	—	8	2	no	—
465	Q₈	C₂×#41×C₅₃	1	8	yes	—
466	D₄	—	2	4	no	—
467	D₄	—	4	4	no	—
468	D₄	—	8	2	no	—
469	D₄	—	8	4	no	—
470	C₂×C₄	—	2	16	no	—
471	C₂×C₄	—	2	8	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	4	no	—
482	C₂×C₄	—	4	4	no	—
483	C₂×C₄	—	8	4	no	—
484	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
485	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
486	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
487	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
488	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
489	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
490	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
491	C₈	—	2	8	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	2	4	no	—
494	C₂²	—	2	2	no	—
495	C₂²	—	4	4	no	—
496	C₂²	—	4	4	no	—
497	C₂²	—	4	2	no	—
498	C₂²	C₅₃×#525	1	2	yes	—
499	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
500	C₄	—	2	4	no	—
501	C₄	—	2	2	no	—
502	C₄	—	2	2	no	—
503	C₄	—	2	8	no	—
504	C₄	—	2	2	no	—
505	C₄	—	4	4	no	—
506	C₄	—	4	4	no	—
507	C₄	—	4	2	no	—
508	C₄	C₅₃×#525	1	2	yes	—
509	C₄	C₂²×#41×C₅₃	1	2	yes	—
510	C₂	—	2	2	no	—
511	C₂	—	4	2	no	—
512	C₂	—	8	2	no	—
513	C₂	C₂×C₅₃×#525	1	1	yes	characteristic
514	C₂	C₅₃×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₅₃×#3455	C₂
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
2. C₂×C₅₃×#519	C₂	semidirect
3. C₅₃×#3127	C₂	semidirect
4. C₅₃×#3126	C₂	semidirect
5. C₅₃×#3121	C₂	semidirect
6. C₅₃×#3093	C₂	semidirect
7. C₅₃×#3078	C₂	semidirect
8. C₅₃×#3037	C₂	semidirect
9. C₅₃×#3004	C₂	semidirect
10. C₄×#137×C₅₃	C₂	semidirect
11. C₅₃×#2921	C₂	semidirect
35. C₅₃×#479	C₂²	semidirect
36. C₅₃×#477	C₂²	semidirect
37. C₅₃×#475	C₂²	semidirect
38. C₅₃×#464	C₂²	semidirect
39. C₅₃×#462	C₂²	semidirect
40. C₅₃×#460	C₂²	semidirect
43. C₅₃×#441	C₂²	semidirect
44. C₅₃×#440	C₂²	semidirect
45. C₄×Q₁₆×C₅₃	C₂²	semidirect
46. C₄×SD₁₆×C₅₃	C₂²	semidirect
234. #60566	C₅₃	direct (N × H)
254. C₂×#519	C₂×C₅₃	semidirect
255. #3127	C₂×C₅₃	semidirect
256. #3126	C₂×C₅₃	semidirect
257. #3121	C₂×C₅₃	semidirect
258. #3093	C₂×C₅₃	semidirect
259. #3078	C₂×C₅₃	semidirect
260. #3037	C₂×C₅₃	semidirect
261. #3004	C₂×C₅₃	semidirect
262. C₄×#137	C₂×C₅₃	semidirect
263. #2921	C₂×C₅₃	semidirect
292. #479	C₂²×C₅₃	semidirect
293. #477	C₂²×C₅₃	semidirect
294. #475	C₂²×C₅₃	semidirect
295. #464	C₂²×C₅₃	semidirect
296. #462	C₂²×C₅₃	semidirect
297. #460	C₂²×C₅₃	semidirect
300. #441	C₂²×C₅₃	semidirect
301. #440	C₂²×C₅₃	semidirect
302. C₄×Q₁₆	C₂²×C₅₃	semidirect
303. C₄×SD₁₆	C₂²×C₅₃	semidirect
312. C₅₃	#60566	direct (N × H)

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×#519	2
C₅₃×#3127	2
C₅₃×#3126	2
C₅₃×#3121	2
C₅₃×#3093	2
C₅₃×#3078	2
C₅₃×#3037	2
C₅₃×#3004	2
C₄×#137×C₅₃	2
C₅₃×#2921	2
#60566	53

Subgroups — C53×#60566 #27268078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60566 #27268078