Subgroups — C₅₃×#17541 #27225053

Order 13568.

1150 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3444	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1506	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1505	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1504	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1500	C₂	1	4	yes	maximal, complemented
8	C₅₃×#533	C₄	1	1	yes	characteristic, complemented
9	C₅₃×#533	C₄	1	1	yes	characteristic, complemented
10	C₅₃×#533	C₂²	1	1	yes	characteristic
11	C₅₃×#532	C₄	1	1	yes	characteristic, complemented
12	C₅₃×#532	C₄	1	1	yes	characteristic, complemented
13	C₅₃×#532	C₂²	1	1	yes	characteristic
14	C₅₃×#529	—	2	1	no	—
15	C₅₃×#529	—	2	1	no	—
16	C₅₃×#529	—	2	1	no	—
17	C₅₃×#529	—	2	1	no	—
18	C₅₃×#529	C₄	1	2	yes	complemented
19	C₅₃×#529	C₂²	1	2	yes	—
20	C₅₃×#525	—	2	1	no	—
21	C₅₃×#525	—	2	1	no	—
22	C₅₃×#525	C₂²	1	2	yes	—
23	C₄×#41×C₅₃	—	2	1	no	—
24	C₄×#41×C₅₃	—	2	1	no	—
25	C₄×#41×C₅₃	C₄	1	2	yes	complemented
26	C₂²×C₄²×C₅₃	—	2	1	no	—
27	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#432	—	2	1	no	—
29	C₅₃×#432	C₂²	1	2	yes	—
30	C₅₃×#431	—	2	1	no	—
31	C₅₃×#431	C₂²	1	2	yes	—
32	C₅₃×#423	—	2	1	no	—
33	C₅₃×#407	C₂²	1	2	yes	—
34	C₄×M₁₆×C₅₃	—	2	1	no	—
35	C₅₃×#376	—	2	1	no	—
36	C₅₃×#376	—	2	1	no	—
37	C₅₃×#376	C₂²	1	2	yes	—
38	C₅₃×#376	C₂²	1	2	yes	—
39	C₂×#95×C₅₃	—	2	1	no	—
40	C₂×#95×C₅₃	C₂²	1	2	yes	—
41	C₅₃×#328	C₂²	1	8	yes	—
42	C₅₃×#327	C₂²	1	8	yes	—
43	C₂×#41×C₅₃	—	2	1	no	—
44	C₂×#41×C₅₃	—	2	1	no	—
45	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
46	C₂³×C₄×C₅₃	—	2	2	no	—
47	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₅₃	—	2	2	no	—
51	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
52	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
53	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
54	#127×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
55	#126×C₅₃	—	2	2	no	—
56	#126×C₅₃	—	2	2	no	—
57	#125×C₅₃	—	2	1	no	—
58	#125×C₅₃	—	2	1	no	—
59	#125×C₅₃	C₂×C₄	1	2	yes	complemented
60	#124×C₅₃	—	2	1	no	—
61	#124×C₅₃	—	2	1	no	—
62	#124×C₅₃	C₂×C₄	1	2	yes	complemented
63	#123×C₅₃	—	2	2	no	—
64	#123×C₅₃	—	2	2	no	—
65	#123×C₅₃	—	2	2	no	—
66	C₂²⋊Q₈×C₅₃	—	2	1	no	—
67	C₂²⋊Q₈×C₅₃	—	2	1	no	—
68	C₂²⋊Q₈×C₅₃	—	2	1	no	—
69	C₂²⋊Q₈×C₅₃	—	2	1	no	—
70	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
71	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
72	#121×C₅₃	—	2	1	no	—
73	#121×C₅₃	—	2	1	no	—
74	#121×C₅₃	—	2	1	no	—
75	#121×C₅₃	—	2	1	no	—
76	#121×C₅₃	C₂×C₄	1	2	yes	—
77	#121×C₅₃	C₂³	1	2	yes	—
78	#120×C₅₃	—	2	2	no	—
79	C₄×Q₈×C₅₃	—	2	1	no	—
80	C₄×Q₈×C₅₃	—	2	1	no	—
81	C₄×Q₈×C₅₃	—	2	1	no	—
82	C₄×Q₈×C₅₃	—	2	1	no	—
83	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	complemented
84	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	complemented
85	C₄×D₄×C₅₃	—	2	1	no	—
86	C₄×D₄×C₅₃	—	2	1	no	—
87	C₄×D₄×C₅₃	—	2	1	no	—
88	C₄×D₄×C₅₃	—	2	1	no	—
89	C₄×D₄×C₅₃	—	2	2	no	—
90	C₄×D₄×C₅₃	—	2	2	no	—
91	C₄×D₄×C₅₃	—	2	2	no	—
92	C₄×D₄×C₅₃	—	2	2	no	—
93	C₄×D₄×C₅₃	—	2	2	no	—
94	C₄×D₄×C₅₃	—	2	2	no	—
95	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	complemented
96	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	complemented
97	#117×C₅₃	—	2	2	no	—
98	#117×C₅₃	—	2	2	no	—
99	#117×C₅₃	—	2	2	no	—
100	#117×C₅₃	—	2	1	no	—
101	#117×C₅₃	—	2	1	no	—
102	#117×C₅₃	C₂×C₄	1	2	yes	—
103	C₂×C₄²×C₅₃	—	2	8	no	—
104	C₂×C₄²×C₅₃	—	2	2	no	—
105	C₂×C₄²×C₅₃	—	2	2	no	—
106	C₂×C₄²×C₅₃	—	2	2	no	—
107	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
108	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
109	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
110	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
111	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
112	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
113	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
114	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
117	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
119	C₄⋊C₈×C₅₃	—	2	2	no	—
120	C₄⋊C₈×C₅₃	—	2	2	no	—
121	C₂²⋊C₈×C₅₃	—	2	2	no	—
122	C₂²⋊C₈×C₅₃	C₂³	1	4	yes	—
123	#97×C₅₃	—	2	2	no	—
124	C₄×C₈×C₅₃	—	2	2	no	—
125	#95×C₅₃	—	2	2	no	—
126	#95×C₅₃	—	2	2	no	—
127	#95×C₅₃	—	2	2	no	—
128	#95×C₅₃	C₂³	1	4	yes	—
129	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
130	#41×C₅₃	—	4	4	no	—
131	#41×C₅₃	—	4	4	no	—
132	#41×C₅₃	—	4	4	no	—
133	C₂×Q₈×C₅₃	—	2	1	no	—
134	C₂×Q₈×C₅₃	—	2	1	no	—
135	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
136	C₂×D₄×C₅₃	—	2	2	no	—
137	C₂×D₄×C₅₃	—	2	2	no	—
138	C₂×D₄×C₅₃	—	2	2	no	—
139	C₂×D₄×C₅₃	—	2	1	no	—
140	C₂×D₄×C₅₃	—	2	1	no	—
141	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
142	C₂²×C₄×C₅₃	—	2	4	no	—
143	C₂²×C₄×C₅₃	—	2	8	no	—
144	C₂²×C₄×C₅₃	—	2	4	no	—
145	C₂²×C₄×C₅₃	—	2	2	no	—
146	C₂²×C₄×C₅₃	—	2	2	no	—
147	C₂²×C₄×C₅₃	—	2	2	no	—
148	C₂²×C₄×C₅₃	—	4	2	no	—
149	C₂²×C₄×C₅₃	—	4	2	no	—
150	C₂²×C₄×C₅₃	—	4	4	no	—
151	C₂²×C₄×C₅₃	—	4	4	no	—
152	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
170	M₁₆×C₅₃	—	4	4	no	—
171	C₂×C₈×C₅₃	—	2	4	no	—
172	C₄⋊C₄×C₅₃	—	2	1	no	—
173	C₄⋊C₄×C₅₃	—	2	1	no	—
174	C₄⋊C₄×C₅₃	—	2	1	no	—
175	C₄⋊C₄×C₅₃	—	2	1	no	—
176	C₄⋊C₄×C₅₃	—	2	2	no	—
177	C₄⋊C₄×C₅₃	—	2	2	no	—
178	C₄⋊C₄×C₅₃	—	2	2	no	—
179	C₄⋊C₄×C₅₃	—	2	2	no	—
180	C₄⋊C₄×C₅₃	—	2	2	no	—
181	C₄⋊C₄×C₅₃	—	2	2	no	—
182	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
183	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
184	C₂²⋊C₄×C₅₃	—	2	2	no	—
185	C₂²⋊C₄×C₅₃	—	2	2	no	—
186	C₂²⋊C₄×C₅₃	—	2	2	no	—
187	C₂²⋊C₄×C₅₃	—	2	2	no	—
188	C₂²⋊C₄×C₅₃	—	2	2	no	—
189	C₂²⋊C₄×C₅₃	—	2	2	no	—
190	C₂²⋊C₄×C₅₃	—	2	2	no	—
191	C₂²⋊C₄×C₅₃	—	2	2	no	—
192	C₂²⋊C₄×C₅₃	—	2	2	no	—
193	C₄²×C₅₃	—	2	8	no	—
194	C₄²×C₅₃	—	2	8	no	—
195	C₄²×C₅₃	—	2	2	no	—
196	C₄²×C₅₃	—	2	2	no	—
197	C₄²×C₅₃	—	2	2	no	—
198	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
199	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
200	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
201	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
202	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
203	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
204	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
205	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
206	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
207	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
208	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
209	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
210	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
211	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
212	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
213	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
214	C₂³×C₅₃	—	2	2	no	—
215	C₂³×C₅₃	—	2	2	no	—
216	C₂³×C₅₃	—	2	2	no	—
217	C₂³×C₅₃	—	2	2	no	—
218	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
219	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
220	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
221	Q₈×C₅₃	—	4	2	no	—
222	Q₈×C₅₃	—	4	2	no	—
223	Q₈×C₅₃	—	4	2	no	—
224	D₄×C₅₃	—	4	2	no	—
225	D₄×C₅₃	—	4	2	no	—
226	D₄×C₅₃	—	4	4	no	—
227	D₄×C₅₃	—	4	4	no	—
228	D₄×C₅₃	—	4	4	no	—
229	D₄×C₅₃	—	4	2	no	—
230	C₂×C₄×C₅₃	—	2	2	no	—
231	C₂×C₄×C₅₃	—	2	2	no	—
232	C₂×C₄×C₅₃	—	2	2	no	—
233	C₂×C₄×C₅₃	—	2	2	no	—
234	C₂×C₄×C₅₃	—	2	2	no	—
235	C₂×C₄×C₅₃	—	2	2	no	—
236	C₂×C₄×C₅₃	—	2	2	no	—
237	C₂×C₄×C₅₃	—	2	2	no	—
238	C₂×C₄×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	—	2	8	no	—
240	C₂×C₄×C₅₃	—	2	2	no	—
241	C₂×C₄×C₅₃	—	2	2	no	—
242	C₂×C₄×C₅₃	—	2	2	no	—
243	C₂×C₄×C₅₃	—	4	1	no	—
244	C₂×C₄×C₅₃	—	4	1	no	—
245	C₂×C₄×C₅₃	—	4	1	no	—
246	C₂×C₄×C₅₃	—	4	1	no	—
247	C₂×C₄×C₅₃	—	4	1	no	—
248	C₂×C₄×C₅₃	—	4	1	no	—
249	C₂×C₄×C₅₃	—	4	1	no	—
250	C₂×C₄×C₅₃	—	4	1	no	—
251	C₂×C₄×C₅₃	—	4	4	no	—
252	C₂×C₄×C₅₃	—	4	4	no	—
253	C₂×C₄×C₅₃	—	4	16	no	—
254	C₂×C₄×C₅₃	—	4	4	no	—
255	C₂×C₄×C₅₃	—	4	4	no	—
256	C₂×C₄×C₅₃	—	4	4	no	—
257	C₂×C₄×C₅₃	#104	1	4	yes	—
258	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
269	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
270	C₈×C₅₃	—	4	4	no	—
271	#17541	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
272	C₂²×C₅₃	—	2	8	no	—
273	C₂²×C₅₃	—	2	1	no	—
274	C₂²×C₅₃	—	2	1	no	—
275	C₂²×C₅₃	—	2	1	no	—
276	C₂²×C₅₃	—	2	1	no	—
277	C₂²×C₅₃	—	2	1	no	—
278	C₂²×C₅₃	—	2	1	no	—
279	C₂²×C₅₃	—	4	2	no	—
280	C₂²×C₅₃	—	4	2	no	—
281	C₂²×C₅₃	—	4	2	no	—
282	C₂²×C₅₃	#410	1	2	yes	—
283	C₂²×C₅₃	C₂×#104	1	2	yes	—
284	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
285	C₂²×C₅₃	#575	1	2	yes	—
286	C₄×C₅₃	—	2	1	no	—
287	C₄×C₅₃	—	2	1	no	—
288	C₄×C₅₃	—	2	1	no	—
289	C₄×C₅₃	—	2	1	no	—
290	C₄×C₅₃	—	2	1	no	—
291	C₄×C₅₃	—	2	1	no	—
292	C₄×C₅₃	—	2	1	no	—
293	C₄×C₅₃	—	2	1	no	—
294	C₄×C₅₃	—	2	1	no	—
295	C₄×C₅₃	—	2	1	no	—
296	C₄×C₅₃	—	4	8	no	—
297	C₄×C₅₃	—	4	2	no	—
298	C₄×C₅₃	—	4	2	no	—
299	C₄×C₅₃	—	4	2	no	—
300	C₄×C₅₃	C₂×#104	1	2	yes	—
301	C₄×C₅₃	C₂×#104	1	2	yes	—
302	#3444	C₂×C₅₃	1	1	yes	characteristic
303	#2920	C₂×C₅₃	1	1	yes	characteristic, complemented
304	C₂×#376	C₂×C₅₃	1	1	yes	characteristic, complemented
305	#1506	C₂×C₅₃	1	2	yes	complemented
306	#1505	C₂×C₅₃	1	2	yes	complemented
307	#1504	C₂×C₅₃	1	4	yes	complemented
308	#1500	C₂×C₅₃	1	4	yes	complemented
309	C₂×C₅₃	—	2	2	no	—
310	C₂×C₅₃	—	2	2	no	—
311	C₂×C₅₃	—	2	2	no	—
312	C₂×C₅₃	—	8	2	no	—
313	C₂×C₅₃	C₂×#410	1	1	yes	characteristic
314	C₂×C₅₃	#2896	1	1	yes	characteristic
315	C₂×C₅₃	C₂²×#104	1	1	yes	characteristic
316	#533	C₄×C₅₃	1	1	yes	characteristic, complemented
317	#533	C₄×C₅₃	1	1	yes	characteristic, complemented
318	#533	C₂²×C₅₃	1	1	yes	characteristic
319	#532	C₄×C₅₃	1	1	yes	characteristic, complemented
320	#532	C₄×C₅₃	1	1	yes	characteristic, complemented
321	#532	C₂²×C₅₃	1	1	yes	characteristic
322	#529	—	2	1	no	—
323	#529	—	2	1	no	—
324	#529	—	2	1	no	—
325	#529	—	2	1	no	—
326	#529	C₄×C₅₃	1	2	yes	complemented
327	#529	C₂²×C₅₃	1	2	yes	—
328	#525	—	2	1	no	—
329	#525	—	2	1	no	—
330	#525	C₂²×C₅₃	1	2	yes	—
331	C₄×#41	—	2	1	no	—
332	C₄×#41	—	2	1	no	—
333	C₄×#41	C₄×C₅₃	1	2	yes	complemented
334	C₂²×C₄²	—	2	1	no	—
335	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
336	#432	—	2	1	no	—
337	#432	C₂²×C₅₃	1	2	yes	—
338	#431	—	2	1	no	—
339	#431	C₂²×C₅₃	1	2	yes	—
340	#423	—	2	1	no	—
341	#407	C₂²×C₅₃	1	2	yes	—
342	C₄×M₁₆	—	2	1	no	—
343	#376	—	2	1	no	—
344	#376	—	2	1	no	—
345	#376	C₂²×C₅₃	1	2	yes	—
346	#376	C₂²×C₅₃	1	2	yes	—
347	C₂×#95	—	2	1	no	—
348	C₂×#95	C₂²×C₅₃	1	2	yes	—
349	#328	C₂²×C₅₃	1	8	yes	—
350	#327	C₂²×C₅₃	1	8	yes	—
351	C₅₃	#17541	1	1	yes	characteristic, direct product factor, Sylow
352	C₂×#41	—	2	1	no	—
353	C₂×#41	—	2	1	no	—
354	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
355	C₂³×C₄	—	2	2	no	—
356	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
357	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
358	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
359	C₂×M₁₆	—	2	2	no	—
360	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
361	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
362	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
363	#127	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
364	#126	—	2	2	no	—
365	#126	—	2	2	no	—
366	#125	—	2	1	no	—
367	#125	—	2	1	no	—
368	#125	C₂×C₄×C₅₃	1	2	yes	complemented
369	#124	—	2	1	no	—
370	#124	—	2	1	no	—
371	#124	C₂×C₄×C₅₃	1	2	yes	complemented
372	#123	—	2	2	no	—
373	#123	—	2	2	no	—
374	#123	—	2	2	no	—
375	C₂²⋊Q₈	—	2	1	no	—
376	C₂²⋊Q₈	—	2	1	no	—
377	C₂²⋊Q₈	—	2	1	no	—
378	C₂²⋊Q₈	—	2	1	no	—
379	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
380	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
381	#121	—	2	1	no	—
382	#121	—	2	1	no	—
383	#121	—	2	1	no	—
384	#121	—	2	1	no	—
385	#121	C₂×C₄×C₅₃	1	2	yes	—
386	#121	C₂³×C₅₃	1	2	yes	—
387	#120	—	2	2	no	—
388	C₄×Q₈	—	2	1	no	—
389	C₄×Q₈	—	2	1	no	—
390	C₄×Q₈	—	2	1	no	—
391	C₄×Q₈	—	2	1	no	—
392	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	complemented
393	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	complemented
394	C₄×D₄	—	2	2	no	—
395	C₄×D₄	—	2	2	no	—
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	—	2	1	no	—
401	C₄×D₄	—	2	1	no	—
402	C₄×D₄	—	2	1	no	—
403	C₄×D₄	—	2	1	no	—
404	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	complemented
405	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	complemented
406	#117	—	2	1	no	—
407	#117	—	2	1	no	—
408	#117	—	2	2	no	—
409	#117	—	2	2	no	—
410	#117	—	2	2	no	—
411	#117	C₂×C₄×C₅₃	1	2	yes	—
412	C₂×C₄²	—	2	8	no	—
413	C₂×C₄²	—	2	2	no	—
414	C₂×C₄²	—	2	2	no	—
415	C₂×C₄²	—	2	2	no	—
416	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
417	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
418	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
419	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
420	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
421	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
422	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
423	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
424	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
425	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
426	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
427	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
428	C₄⋊C₈	—	2	2	no	—
429	C₄⋊C₈	—	2	2	no	—
430	C₂²⋊C₈	—	2	2	no	—
431	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
432	#97	—	2	2	no	—
433	C₄×C₈	—	2	2	no	—
434	#95	—	2	2	no	—
435	#95	—	2	2	no	—
436	#95	—	2	2	no	—
437	#95	C₂³×C₅₃	1	4	yes	—
438	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
439	#41	—	4	4	no	—
440	#41	—	4	4	no	—
441	#41	—	4	4	no	—
442	C₂×Q₈	—	2	1	no	—
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	8	no	—
453	C₂²×C₄	—	2	4	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	4	2	no	—
458	C₂²×C₄	—	4	2	no	—
459	C₂²×C₄	—	4	4	no	—
460	C₂²×C₄	—	4	4	no	—
461	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
462	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
463	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
464	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
465	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
466	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
467	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
468	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
469	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
470	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
474	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
478	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
479	M₁₆	—	4	4	no	—
480	C₂×C₈	—	2	4	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
492	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	2	2	no	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	2	no	—
498	C₂²⋊C₄	—	2	2	no	—
499	C₂²⋊C₄	—	2	2	no	—
500	C₂²⋊C₄	—	2	2	no	—
501	C₂²⋊C₄	—	2	2	no	—
502	C₄²	—	2	8	no	—
503	C₄²	—	2	8	no	—
504	C₄²	—	2	2	no	—
505	C₄²	—	2	2	no	—
506	C₄²	—	2	2	no	—
507	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
508	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
509	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
510	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
511	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
512	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
513	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
514	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
515	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
516	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
517	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
518	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
519	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
520	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
521	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
522	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
523	C₂³	—	2	2	no	—
524	C₂³	—	2	2	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	2	no	—
527	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
528	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
529	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
530	Q₈	—	4	2	no	—
531	Q₈	—	4	2	no	—
532	Q₈	—	4	2	no	—
533	D₄	—	4	2	no	—
534	D₄	—	4	2	no	—
535	D₄	—	4	4	no	—
536	D₄	—	4	4	no	—
537	D₄	—	4	4	no	—
538	D₄	—	4	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	8	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	4	no	—
561	C₂×C₄	—	4	4	no	—
562	C₂×C₄	—	4	16	no	—
563	C₂×C₄	—	4	4	no	—
564	C₂×C₄	—	4	4	no	—
565	C₂×C₄	—	4	4	no	—
566	C₂×C₄	#104×C₅₃	1	4	yes	—
567	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
568	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
569	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
570	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
571	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
572	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
573	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
574	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
575	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
576	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
577	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
578	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
579	C₈	—	4	4	no	—
580	C₂²	—	2	8	no	—
581	C₂²	—	2	1	no	—
582	C₂²	—	2	1	no	—
583	C₂²	—	2	1	no	—
584	C₂²	—	2	1	no	—
585	C₂²	—	2	1	no	—
586	C₂²	—	2	1	no	—
587	C₂²	—	4	2	no	—
588	C₂²	—	4	2	no	—
589	C₂²	—	4	2	no	—
590	C₂²	C₅₃×#410	1	2	yes	—
591	C₂²	C₂×#104×C₅₃	1	2	yes	—
592	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
593	C₂²	C₅₃×#575	1	2	yes	—
594	C₄	—	2	1	no	—
595	C₄	—	2	1	no	—
596	C₄	—	2	1	no	—
597	C₄	—	2	1	no	—
598	C₄	—	2	1	no	—
599	C₄	—	2	1	no	—
600	C₄	—	2	1	no	—
601	C₄	—	2	1	no	—
602	C₄	—	2	1	no	—
603	C₄	—	2	1	no	—
604	C₄	—	4	8	no	—
605	C₄	—	4	2	no	—
606	C₄	—	4	2	no	—
607	C₄	—	4	2	no	—
608	C₄	C₂×#104×C₅₃	1	2	yes	—
609	C₄	C₂×#104×C₅₃	1	2	yes	—
610	C₂	—	2	2	no	—
611	C₂	—	2	2	no	—
612	C₂	—	2	2	no	—
613	C₂	—	8	2	no	—
614	C₂	C₂×C₅₃×#410	1	1	yes	characteristic
615	C₂	C₅₃×#2896	1	1	yes	characteristic
616	C₂	C₂²×#104×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₅₃×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2920	C₂	semidirect
3. C₂×C₅₃×#376	C₂	semidirect
4. C₅₃×#1506	C₂	semidirect
5. C₅₃×#1505	C₂	semidirect
6. C₅₃×#1504	C₂	semidirect
7. C₅₃×#1500	C₂	semidirect
8. C₅₃×#533	C₄	semidirect
9. C₅₃×#533	C₄	semidirect
11. C₅₃×#532	C₄	semidirect
12. C₅₃×#532	C₄	semidirect
18. C₅₃×#529	C₄	semidirect
25. C₄×#41×C₅₃	C₄	semidirect
51. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
52. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
53. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
54. #127×C₅₃	C₂×C₄	semidirect
59. #125×C₅₃	C₂×C₄	semidirect
62. #124×C₅₃	C₂×C₄	semidirect
83. C₄×Q₈×C₅₃	C₂×C₄	semidirect
84. C₄×Q₈×C₅₃	C₂×C₄	semidirect
95. C₄×D₄×C₅₃	C₂×C₄	semidirect
96. C₄×D₄×C₅₃	C₂×C₄	semidirect
271. #17541	C₅₃	direct (N × H)
303. #2920	C₂×C₅₃	semidirect
304. C₂×#376	C₂×C₅₃	semidirect
305. #1506	C₂×C₅₃	semidirect
306. #1505	C₂×C₅₃	semidirect
307. #1504	C₂×C₅₃	semidirect
308. #1500	C₂×C₅₃	semidirect
316. #533	C₄×C₅₃	semidirect
317. #533	C₄×C₅₃	semidirect
319. #532	C₄×C₅₃	semidirect
320. #532	C₄×C₅₃	semidirect
326. #529	C₄×C₅₃	semidirect
333. C₄×#41	C₄×C₅₃	semidirect
351. C₅₃	#17541	direct (N × H)
360. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
361. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
362. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
363. #127	C₂×C₄×C₅₃	semidirect
368. #125	C₂×C₄×C₅₃	semidirect
371. #124	C₂×C₄×C₅₃	semidirect
392. C₄×Q₈	C₂×C₄×C₅₃	semidirect
393. C₄×Q₈	C₂×C₄×C₅₃	semidirect
404. C₄×D₄	C₂×C₄×C₅₃	semidirect
405. C₄×D₄	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3444	2
C₅₃×#2920	2
C₂×C₅₃×#376	2
C₅₃×#1506	2
C₅₃×#1505	2
C₅₃×#1504	2
C₅₃×#1500	2
#17541	53

Subgroups — C53×#17541 #27225053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17541 #27225053