Subgroups — C₅₃×#21205 #27228717

Order 13568.

1526 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#536	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#417	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#419	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#416	C₂	1	1	yes	maximal, characteristic
5	C₂²×#103×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2317	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1960	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1959	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1958	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1956	C₂	1	2	yes	maximal, complemented
12	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#536	—	2	4	no	—
17	C₅₃×#536	C₄	1	8	yes	complemented
18	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
19	C₂×#124×C₅₃	C₂²	1	2	yes	—
20	C₂×#123×C₅₃	—	2	2	no	—
21	C₂×C₂²⋊Q₈×C₅₃	—	2	2	no	—
22	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#419	—	2	4	no	—
26	C₅₃×#417	—	2	4	no	—
27	C₅₃×#416	—	2	4	no	—
28	C₂×#103×C₅₃	—	2	4	no	—
29	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#103×C₅₃	C₂²	1	2	yes	—
31	C₂×#103×C₅₃	C₂²	1	2	yes	—
32	C₂×#103×C₅₃	C₂²	1	2	yes	—
33	C₂×#103×C₅₃	C₂²	1	2	yes	—
34	C₂×#103×C₅₃	C₂²	1	2	yes	—
35	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×#102×C₅₃	C₂²	1	2	yes	—
38	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#405	—	2	4	no	—
40	C₂×#97×C₅₃	C₂²	1	2	yes	—
41	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
42	C₅₃×#389	—	2	2	no	—
43	C₅₃×#385	—	2	2	no	—
44	C₅₃×#384	C₂²	1	2	yes	—
45	C₅₃×#382	C₂²	1	2	yes	—
46	C₂×#41×C₅₃	—	2	2	no	—
47	C₂×#41×C₅₃	—	2	2	no	—
48	C₂×#41×C₅₃	—	2	2	no	—
49	C₂×#41×C₅₃	—	2	2	no	—
50	C₂×#41×C₅₃	—	2	2	no	—
51	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
52	C₂²×Q₈×C₅₃	—	2	1	no	—
53	C₂²×Q₈×C₅₃	—	2	2	no	—
54	C₂²×Q₈×C₅₃	—	2	2	no	—
55	C₂²×Q₈×C₅₃	—	2	2	no	—
56	C₂²×Q₈×C₅₃	—	2	2	no	—
57	C₂²×Q₈×C₅₃	—	2	2	no	—
58	C₂²×Q₈×C₅₃	C₂×C₄	1	4	yes	—
59	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
60	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₂²×D₄×C₅₃	—	2	1	no	—
62	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₅₃	—	2	1	no	—
64	C₂³×C₄×C₅₃	—	2	1	no	—
65	C₂³×C₄×C₅₃	—	2	1	no	—
66	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₅₃	—	2	2	no	—
68	C₂×M₁₆×C₅₃	—	2	2	no	—
69	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
70	C₂²×C₈×C₅₃	—	2	2	no	—
71	C₂²×C₈×C₅₃	—	2	2	no	—
72	C₂²×C₈×C₅₃	D₄	1	4	yes	—
73	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
76	C₄⋊Q₈×C₅₃	—	2	4	no	—
77	C₄⋊Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
78	#124×C₅₃	—	2	4	no	—
79	#124×C₅₃	C₂×C₄	1	8	yes	complemented
80	#123×C₅₃	—	2	16	no	—
81	C₂²⋊Q₈×C₅₃	—	2	16	no	—
82	#117×C₅₃	—	2	2	no	—
83	#117×C₅₃	—	2	2	no	—
84	#117×C₅₃	C₂×C₄	1	4	yes	—
85	#117×C₅₃	D₄	1	4	yes	—
86	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
87	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
89	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
90	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
91	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
92	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
95	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
97	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
98	C₂×C₄²×C₅₃	—	2	1	no	—
99	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
100	#103×C₅₃	—	2	4	no	—
101	#103×C₅₃	—	2	4	no	—
102	#103×C₅₃	—	2	2	no	—
103	#103×C₅₃	—	2	2	no	—
104	#103×C₅₃	—	2	4	no	—
105	#103×C₅₃	—	2	4	no	—
106	#103×C₅₃	—	2	4	no	—
107	#102×C₅₃	—	2	2	no	—
108	#102×C₅₃	—	2	2	no	—
109	#102×C₅₃	—	2	4	no	—
110	#97×C₅₃	—	2	4	no	—
111	C₄×C₈×C₅₃	—	2	4	no	—
112	#95×C₅₃	—	2	4	no	—
113	C₂⁴×C₅₃	—	2	1	no	—
114	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
115	#41×C₅₃	—	2	8	no	—
116	#41×C₅₃	—	2	8	no	—
117	#41×C₅₃	—	4	4	no	—
118	#41×C₅₃	—	4	4	no	—
119	C₂×Q₈×C₅₃	—	2	4	no	—
120	C₂×Q₈×C₅₃	—	2	2	no	—
121	C₂×Q₈×C₅₃	—	2	1	no	—
122	C₂×Q₈×C₅₃	—	2	8	no	—
123	C₂×Q₈×C₅₃	—	2	8	no	—
124	C₂×Q₈×C₅₃	—	2	2	no	—
125	C₂×Q₈×C₅₃	—	2	2	no	—
126	C₂×Q₈×C₅₃	—	2	1	no	—
127	C₂×Q₈×C₅₃	—	2	1	no	—
128	C₂×Q₈×C₅₃	—	2	2	no	—
129	C₂×Q₈×C₅₃	—	2	2	no	—
130	C₂×Q₈×C₅₃	—	2	1	no	—
131	C₂×Q₈×C₅₃	—	2	1	no	—
132	C₂×Q₈×C₅₃	—	4	4	no	—
133	C₂×Q₈×C₅₃	—	4	4	no	—
134	C₂×Q₈×C₅₃	—	4	2	no	—
135	C₂×Q₈×C₅₃	—	4	2	no	—
136	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
137	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
138	C₂×D₄×C₅₃	—	2	4	no	—
139	C₂×D₄×C₅₃	—	2	1	no	—
140	C₂×D₄×C₅₃	—	2	1	no	—
141	C₂×D₄×C₅₃	—	2	1	no	—
142	C₂×D₄×C₅₃	—	2	1	no	—
143	C₂×D₄×C₅₃	—	2	1	no	—
144	C₂×D₄×C₅₃	—	4	2	no	—
145	C₂×D₄×C₅₃	—	4	2	no	—
146	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
147	C₂²×C₄×C₅₃	—	2	4	no	—
148	C₂²×C₄×C₅₃	—	2	2	no	—
149	C₂²×C₄×C₅₃	—	2	1	no	—
150	C₂²×C₄×C₅₃	—	2	4	no	—
151	C₂²×C₄×C₅₃	—	2	1	no	—
152	C₂²×C₄×C₅₃	—	2	2	no	—
153	C₂²×C₄×C₅₃	—	2	1	no	—
154	C₂²×C₄×C₅₃	—	4	2	no	—
155	C₂²×C₄×C₅₃	—	4	2	no	—
156	C₂²×C₄×C₅₃	—	4	2	no	—
157	C₂²×C₄×C₅₃	—	4	2	no	—
158	C₂²×C₄×C₅₃	—	4	2	no	—
159	C₂²×C₄×C₅₃	—	8	2	no	—
160	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
161	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
162	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₅₃	#41	1	2	yes	—
167	C₂²×C₄×C₅₃	#41	1	2	yes	—
168	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
169	M₁₆×C₅₃	—	2	8	no	—
170	C₂×C₈×C₅₃	—	2	8	no	—
171	C₂×C₈×C₅₃	—	2	1	no	—
172	C₂×C₈×C₅₃	—	2	2	no	—
173	C₂×C₈×C₅₃	—	2	1	no	—
174	C₂×C₈×C₅₃	—	2	1	no	—
175	C₂×C₈×C₅₃	—	2	1	no	—
176	C₂×C₈×C₅₃	—	2	2	no	—
177	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
178	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
179	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
180	C₄⋊C₄×C₅₃	—	2	16	no	—
181	C₄⋊C₄×C₅₃	—	2	2	no	—
182	C₄⋊C₄×C₅₃	—	2	1	no	—
183	C₄⋊C₄×C₅₃	—	2	1	no	—
184	C₄⋊C₄×C₅₃	—	2	2	no	—
185	C₄⋊C₄×C₅₃	—	2	1	no	—
186	C₄⋊C₄×C₅₃	—	2	1	no	—
187	C₄⋊C₄×C₅₃	—	4	2	no	—
188	C₄⋊C₄×C₅₃	—	8	4	no	—
189	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
190	C₂²⋊C₄×C₅₃	—	2	16	no	—
191	C₂²⋊C₄×C₅₃	—	4	2	no	—
192	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
193	C₄²×C₅₃	—	4	2	no	—
194	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
195	C₄²×C₅₃	C₂×D₄	1	4	yes	—
196	C₂³×C₅₃	—	2	2	no	—
197	C₂³×C₅₃	—	2	2	no	—
198	C₂³×C₅₃	—	2	2	no	—
199	C₂³×C₅₃	—	4	1	no	—
200	C₂³×C₅₃	—	4	1	no	—
201	C₂³×C₅₃	—	4	2	no	—
202	C₂³×C₅₃	—	4	2	no	—
203	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
204	C₂³×C₅₃	#124	1	2	yes	—
205	C₂³×C₅₃	#127	1	2	yes	—
206	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
207	Q₈×C₅₃	—	2	4	no	—
208	Q₈×C₅₃	—	2	4	no	—
209	Q₈×C₅₃	—	2	2	no	—
210	Q₈×C₅₃	—	2	2	no	—
211	Q₈×C₅₃	—	4	2	no	—
212	Q₈×C₅₃	—	4	2	no	—
213	Q₈×C₅₃	—	4	1	no	—
214	Q₈×C₅₃	—	4	1	no	—
215	Q₈×C₅₃	—	4	8	no	—
216	D₄×C₅₃	—	2	2	no	—
217	D₄×C₅₃	—	2	2	no	—
218	D₄×C₅₃	—	4	8	no	—
219	D₄×C₅₃	—	4	1	no	—
220	D₄×C₅₃	—	4	1	no	—
221	C₂×C₄×C₅₃	—	2	4	no	—
222	C₂×C₄×C₅₃	—	2	4	no	—
223	C₂×C₄×C₅₃	—	2	2	no	—
224	C₂×C₄×C₅₃	—	2	4	no	—
225	C₂×C₄×C₅₃	—	4	1	no	—
226	C₂×C₄×C₅₃	—	4	2	no	—
227	C₂×C₄×C₅₃	—	4	1	no	—
228	C₂×C₄×C₅₃	—	4	2	no	—
229	C₂×C₄×C₅₃	—	4	8	no	—
230	C₂×C₄×C₅₃	—	4	8	no	—
231	C₂×C₄×C₅₃	—	4	2	no	—
232	C₂×C₄×C₅₃	—	4	1	no	—
233	C₂×C₄×C₅₃	—	4	1	no	—
234	C₂×C₄×C₅₃	—	4	4	no	—
235	C₂×C₄×C₅₃	—	4	2	no	—
236	C₂×C₄×C₅₃	—	4	2	no	—
237	C₂×C₄×C₅₃	—	8	4	no	—
238	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	8	yes	—
239	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
240	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
241	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
242	C₂×C₄×C₅₃	#121	1	4	yes	—
243	C₂×C₄×C₅₃	#121	1	4	yes	—
244	C₂×C₄×C₅₃	#124	1	2	yes	—
245	C₂×C₄×C₅₃	#127	1	2	yes	—
246	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
247	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
251	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
252	C₈×C₅₃	—	2	2	no	—
253	C₈×C₅₃	—	2	2	no	—
254	C₈×C₅₃	—	2	4	no	—
255	#21205	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
256	C₂²×C₅₃	—	2	4	no	—
257	C₂²×C₅₃	—	2	4	no	—
258	C₂²×C₅₃	—	2	2	no	—
259	C₂²×C₅₃	—	2	2	no	—
260	C₂²×C₅₃	—	4	4	no	—
261	C₂²×C₅₃	—	4	2	no	—
262	C₂²×C₅₃	—	8	2	no	—
263	C₂²×C₅₃	—	8	1	no	—
264	C₂²×C₅₃	—	8	1	no	—
265	C₂²×C₅₃	#390	1	4	yes	—
266	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
267	C₂²×C₅₃	C₂×#124	1	1	yes	characteristic
268	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
269	C₂²×C₅₃	#577	1	2	yes	—
270	C₂²×C₅₃	#578	1	2	yes	—
271	C₄×C₅₃	—	2	8	no	—
272	C₄×C₅₃	—	4	2	no	—
273	C₄×C₅₃	—	4	4	no	—
274	C₄×C₅₃	—	8	1	no	—
275	C₄×C₅₃	—	8	2	no	—
276	C₄×C₅₃	—	8	1	no	—
277	C₄×C₅₃	#390	1	4	yes	—
278	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
279	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
280	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
281	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
282	C₂×#536	C₂×C₅₃	1	1	yes	characteristic
283	C₂×#417	C₂×C₅₃	1	1	yes	characteristic
284	C₂×#419	C₂×C₅₃	1	1	yes	characteristic, complemented
285	C₂×#416	C₂×C₅₃	1	1	yes	characteristic
286	C₂²×#103	C₂×C₅₃	1	1	yes	characteristic, complemented
287	C₂×#405	C₂×C₅₃	1	1	yes	characteristic, complemented
288	#2317	C₂×C₅₃	1	1	yes	characteristic, complemented
289	#1960	C₂×C₅₃	1	2	yes	complemented
290	#1959	C₂×C₅₃	1	2	yes	complemented
291	#1958	C₂×C₅₃	1	2	yes	complemented
292	#1956	C₂×C₅₃	1	2	yes	complemented
293	C₂×C₅₃	—	2	4	no	—
294	C₂×C₅₃	—	8	2	no	—
295	C₂×C₅₃	C₂×#390	1	1	yes	characteristic
296	C₂×C₅₃	#2898	1	2	yes	—
297	C₂×C₅₃	#3137	1	2	yes	—
298	C₂×C₅₃	#3154	1	1	yes	characteristic
299	C₂×C₅₃	#3156	1	1	yes	characteristic
300	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
301	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
302	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
303	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
304	#536	—	2	4	no	—
305	#536	C₄×C₅₃	1	8	yes	complemented
306	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
307	C₂×#124	C₂²×C₅₃	1	2	yes	—
308	C₂×#123	—	2	2	no	—
309	C₂×C₂²⋊Q₈	—	2	2	no	—
310	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
311	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
312	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
313	#419	—	2	4	no	—
314	#417	—	2	4	no	—
315	#416	—	2	4	no	—
316	C₂×#103	—	2	4	no	—
317	C₂×#103	C₂²×C₅₃	1	2	yes	—
318	C₂×#103	C₂²×C₅₃	1	2	yes	—
319	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
320	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
321	C₂×#103	C₂²×C₅₃	1	2	yes	—
322	C₂×#103	C₂²×C₅₃	1	2	yes	—
323	C₂×#103	C₂²×C₅₃	1	2	yes	—
324	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
325	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
326	C₂×#102	C₂²×C₅₃	1	2	yes	—
327	#405	—	2	4	no	—
328	C₂×#97	C₂²×C₅₃	1	2	yes	—
329	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
330	#389	—	2	2	no	—
331	#385	—	2	2	no	—
332	#384	C₂²×C₅₃	1	2	yes	—
333	#382	C₂²×C₅₃	1	2	yes	—
334	C₅₃	#21205	1	1	yes	characteristic, direct product factor, Sylow
335	C₂×#41	—	2	2	no	—
336	C₂×#41	—	2	2	no	—
337	C₂×#41	—	2	2	no	—
338	C₂×#41	—	2	2	no	—
339	C₂×#41	—	2	2	no	—
340	C₂×#41	C₂×C₄×C₅₃	1	4	yes	—
341	C₂²×Q₈	—	2	1	no	—
342	C₂²×Q₈	—	2	2	no	—
343	C₂²×Q₈	—	2	2	no	—
344	C₂²×Q₈	—	2	2	no	—
345	C₂²×Q₈	—	2	2	no	—
346	C₂²×Q₈	—	2	2	no	—
347	C₂²×Q₈	C₂×C₄×C₅₃	1	4	yes	—
348	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
349	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
350	C₂²×D₄	—	2	1	no	—
351	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
352	C₂³×C₄	—	2	1	no	—
353	C₂³×C₄	—	2	1	no	—
354	C₂³×C₄	—	2	1	no	—
355	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
356	C₂×M₁₆	—	2	2	no	—
357	C₂×M₁₆	—	2	2	no	—
358	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
359	C₂²×C₈	—	2	2	no	—
360	C₂²×C₈	—	2	2	no	—
361	C₂²×C₈	D₄×C₅₃	1	4	yes	—
362	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
363	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
364	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
365	C₄⋊Q₈	—	2	4	no	—
366	C₄⋊Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
367	#124	—	2	4	no	—
368	#124	C₂×C₄×C₅₃	1	8	yes	complemented
369	#123	—	2	16	no	—
370	C₂²⋊Q₈	—	2	16	no	—
371	#117	—	2	2	no	—
372	#117	—	2	2	no	—
373	#117	C₂×C₄×C₅₃	1	4	yes	—
374	#117	D₄×C₅₃	1	4	yes	—
375	C₂×C₄⋊C₄	—	2	4	no	—
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	—	2	2	no	—
378	C₂×C₄⋊C₄	—	2	1	no	—
379	C₂×C₄⋊C₄	—	2	4	no	—
380	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
381	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
382	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
383	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
384	C₂×C₂²⋊C₄	—	2	4	no	—
385	C₂×C₂²⋊C₄	—	2	1	no	—
386	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
387	C₂×C₄²	—	2	1	no	—
388	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
389	#103	—	2	4	no	—
390	#103	—	2	4	no	—
391	#103	—	2	2	no	—
392	#103	—	2	2	no	—
393	#103	—	2	4	no	—
394	#103	—	2	4	no	—
395	#103	—	2	4	no	—
396	#102	—	2	2	no	—
397	#102	—	2	2	no	—
398	#102	—	2	4	no	—
399	#97	—	2	4	no	—
400	C₄×C₈	—	2	4	no	—
401	#95	—	2	4	no	—
402	C₂⁴	—	2	1	no	—
403	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
404	#41	—	2	8	no	—
405	#41	—	2	8	no	—
406	#41	—	4	4	no	—
407	#41	—	4	4	no	—
408	C₂×Q₈	—	2	4	no	—
409	C₂×Q₈	—	2	2	no	—
410	C₂×Q₈	—	2	1	no	—
411	C₂×Q₈	—	2	8	no	—
412	C₂×Q₈	—	2	8	no	—
413	C₂×Q₈	—	2	2	no	—
414	C₂×Q₈	—	2	2	no	—
415	C₂×Q₈	—	2	1	no	—
416	C₂×Q₈	—	2	1	no	—
417	C₂×Q₈	—	2	2	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	—	2	1	no	—
420	C₂×Q₈	—	2	1	no	—
421	C₂×Q₈	—	4	4	no	—
422	C₂×Q₈	—	4	4	no	—
423	C₂×Q₈	—	4	2	no	—
424	C₂×Q₈	—	4	2	no	—
425	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
426	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
427	C₂×D₄	—	2	4	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	4	2	no	—
434	C₂×D₄	—	4	2	no	—
435	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	4	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	4	2	no	—
444	C₂²×C₄	—	4	2	no	—
445	C₂²×C₄	—	4	2	no	—
446	C₂²×C₄	—	4	2	no	—
447	C₂²×C₄	—	4	2	no	—
448	C₂²×C₄	—	8	2	no	—
449	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
450	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
451	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
453	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
454	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
455	C₂²×C₄	#41×C₅₃	1	2	yes	—
456	C₂²×C₄	#41×C₅₃	1	2	yes	—
457	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
458	M₁₆	—	2	8	no	—
459	C₂×C₈	—	2	8	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	2	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	2	no	—
466	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
467	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
468	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
469	C₄⋊C₄	—	2	16	no	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	4	2	no	—
477	C₄⋊C₄	—	8	4	no	—
478	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	8	yes	—
479	C₂²⋊C₄	—	2	16	no	—
480	C₂²⋊C₄	—	4	2	no	—
481	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	8	yes	—
482	C₄²	—	4	2	no	—
483	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
484	C₄²	C₂×D₄×C₅₃	1	4	yes	—
485	C₂³	—	2	2	no	—
486	C₂³	—	2	2	no	—
487	C₂³	—	2	2	no	—
488	C₂³	—	4	1	no	—
489	C₂³	—	4	1	no	—
490	C₂³	—	4	2	no	—
491	C₂³	—	4	2	no	—
492	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
493	C₂³	#124×C₅₃	1	2	yes	—
494	C₂³	#127×C₅₃	1	2	yes	—
495	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
496	Q₈	—	2	4	no	—
497	Q₈	—	2	4	no	—
498	Q₈	—	2	2	no	—
499	Q₈	—	2	2	no	—
500	Q₈	—	4	2	no	—
501	Q₈	—	4	2	no	—
502	Q₈	—	4	1	no	—
503	Q₈	—	4	1	no	—
504	Q₈	—	4	8	no	—
505	D₄	—	2	2	no	—
506	D₄	—	2	2	no	—
507	D₄	—	4	8	no	—
508	D₄	—	4	1	no	—
509	D₄	—	4	1	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	4	1	no	—
515	C₂×C₄	—	4	2	no	—
516	C₂×C₄	—	4	1	no	—
517	C₂×C₄	—	4	2	no	—
518	C₂×C₄	—	4	8	no	—
519	C₂×C₄	—	4	8	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	4	2	no	—
524	C₂×C₄	—	4	4	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	—	8	4	no	—
527	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	8	yes	—
528	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
529	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
530	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
531	C₂×C₄	#121×C₅₃	1	4	yes	—
532	C₂×C₄	#121×C₅₃	1	4	yes	—
533	C₂×C₄	#124×C₅₃	1	2	yes	—
534	C₂×C₄	#127×C₅₃	1	2	yes	—
535	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
536	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
537	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
538	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
539	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
540	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
541	C₈	—	2	2	no	—
542	C₈	—	2	2	no	—
543	C₈	—	2	4	no	—
544	C₂²	—	2	4	no	—
545	C₂²	—	2	4	no	—
546	C₂²	—	2	2	no	—
547	C₂²	—	2	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	4	4	no	—
550	C₂²	—	8	2	no	—
551	C₂²	—	8	1	no	—
552	C₂²	—	8	1	no	—
553	C₂²	C₅₃×#390	1	4	yes	—
554	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
555	C₂²	C₂×#124×C₅₃	1	1	yes	characteristic
556	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
557	C₂²	C₅₃×#577	1	2	yes	—
558	C₂²	C₅₃×#578	1	2	yes	—
559	C₄	—	2	8	no	—
560	C₄	—	4	2	no	—
561	C₄	—	4	4	no	—
562	C₄	—	8	2	no	—
563	C₄	—	8	1	no	—
564	C₄	—	8	1	no	—
565	C₄	C₅₃×#390	1	4	yes	—
566	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
567	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
568	C₄	C₂×#121×C₅₃	1	1	yes	characteristic
569	C₄	C₂×#121×C₅₃	1	1	yes	characteristic
570	C₂	—	2	4	no	—
571	C₂	—	8	2	no	—
572	C₂	C₂×C₅₃×#390	1	1	yes	characteristic
573	C₂	C₅₃×#2898	1	2	yes	—
574	C₂	C₅₃×#3137	1	2	yes	—
575	C₂	C₅₃×#3154	1	1	yes	characteristic
576	C₂	C₅₃×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×#41×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×#117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅₃×#419	C₂	semidirect
5. C₂²×#103×C₅₃	C₂	semidirect
6. C₂×C₅₃×#405	C₂	semidirect
7. C₅₃×#2317	C₂	semidirect
8. C₅₃×#1960	C₂	semidirect
9. C₅₃×#1959	C₂	semidirect
10. C₅₃×#1958	C₂	semidirect
11. C₅₃×#1956	C₂	semidirect
17. C₅₃×#536	C₄	semidirect
77. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
79. #124×C₅₃	C₂×C₄	semidirect
255. #21205	C₅₃	direct (N × H)
284. C₂×#419	C₂×C₅₃	semidirect
286. C₂²×#103	C₂×C₅₃	semidirect
287. C₂×#405	C₂×C₅₃	semidirect
288. #2317	C₂×C₅₃	semidirect
289. #1960	C₂×C₅₃	semidirect
290. #1959	C₂×C₅₃	semidirect
291. #1958	C₂×C₅₃	semidirect
292. #1956	C₂×C₅₃	semidirect
305. #536	C₄×C₅₃	semidirect
334. C₅₃	#21205	direct (N × H)
366. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
368. #124	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
C₂×C₅₃×#536	2
C₂×C₅₃×#417	2
C₂×C₅₃×#419	2
C₂×C₅₃×#416	2
C₂²×#103×C₅₃	2
C₂×C₅₃×#405	2
C₅₃×#2317	2
C₅₃×#1960	2
C₅₃×#1959	2
C₅₃×#1958	2
C₅₃×#1956	2
#21205	53

Subgroups — C53×#21205 #27228717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21205 #27228717