Subgroups — C₅₃×#34631 #27242143

Order 13568.

2138 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×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#529	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2664	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2663	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2660	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#2658	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2642	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2640	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2621	C₂	1	2	yes	maximal, complemented
10	C₅₃×#2620	C₂	1	2	yes	maximal
11	C₅₃×#2396	C₂	1	4	yes	maximal, complemented
12	C₅₃×#2395	C₂	1	4	yes	maximal, complemented
13	C₄×C₄⋊Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
14	C₄×#124×C₅₃	C₂	1	2	yes	maximal, complemented
15	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅₃×#533	—	2	8	no	—
17	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#529	—	2	8	no	—
21	C₂×#126×C₅₃	C₂²	1	2	yes	—
22	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#125×C₅₃	C₂²	1	2	yes	—
24	C₂×#124×C₅₃	C₂²	1	2	yes	—
25	C₂×#124×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#123×C₅₃	C₂²	1	2	yes	—
27	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
28	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
30	C₂×#121×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
32	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
34	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
36	C₂×#117×C₅₃	C₂²	1	2	yes	—
37	C₂×#117×C₅₃	C₂²	1	2	yes	—
38	C₂×#117×C₅₃	C₂²	1	2	yes	—
39	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#400	C₂²	1	4	yes	—
41	C₅₃×#400	C₂²	1	4	yes	—
42	C₅₃×#399	C₂²	1	4	yes	—
43	C₅₃×#399	C₂²	1	4	yes	—
44	C₅₃×#398	C₂²	1	8	yes	—
45	C₅₃×#398	C₂²	1	4	yes	—
46	C₅₃×#397	C₂²	1	8	yes	—
47	C₅₃×#397	C₂²	1	4	yes	—
48	C₅₃×#396	C₂²	1	8	yes	—
49	C₅₃×#395	C₂²	1	4	yes	—
50	C₅₃×#394	C₂²	1	8	yes	—
51	C₅₃×#393	C₂²	1	4	yes	—
52	C₅₃×#391	C₂²	1	2	yes	—
53	C₅₃×#391	C₂²	1	2	yes	—
54	C₅₃×#391	C₂²	1	2	yes	—
55	C₅₃×#390	C₂²	1	2	yes	—
56	C₅₃×#389	C₂²	1	8	yes	—
57	C₅₃×#388	C₂²	1	8	yes	—
58	C₅₃×#384	C₂²	1	2	yes	—
59	C₅₃×#382	C₂²	1	2	yes	—
60	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
62	C₄×C₄⋊C₄×C₅₃	C₂²	1	8	yes	—
63	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
64	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
65	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
66	C₅₃×#376	C₂²	1	1	yes	characteristic
67	C₅₃×#376	C₂²	1	1	yes	characteristic
68	C₄³×C₅₃	C₂²	1	2	yes	—
69	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
71	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
72	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
73	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
74	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
75	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
78	C₄⋊Q₈×C₅₃	—	2	4	no	—
79	C₄⋊Q₈×C₅₃	—	2	4	no	—
80	C₄⋊Q₈×C₅₃	—	2	8	no	—
81	#126×C₅₃	—	2	8	no	—
82	#125×C₅₃	—	2	4	no	—
83	#125×C₅₃	—	2	8	no	—
84	#124×C₅₃	—	2	4	no	—
85	#124×C₅₃	—	2	8	no	—
86	#123×C₅₃	—	2	8	no	—
87	C₂²⋊Q₈×C₅₃	—	2	4	no	—
88	C₂²⋊Q₈×C₅₃	—	2	8	no	—
89	C₂²⋊Q₈×C₅₃	—	2	8	no	—
90	#121×C₅₃	—	2	4	no	—
91	C₄×Q₈×C₅₃	—	2	4	no	—
92	C₄×Q₈×C₅₃	—	2	8	no	—
93	C₄×Q₈×C₅₃	—	2	8	no	—
94	C₄×D₄×C₅₃	—	2	4	no	—
95	C₄×D₄×C₅₃	—	2	8	no	—
96	#117×C₅₃	—	2	8	no	—
97	#117×C₅₃	—	2	8	no	—
98	#117×C₅₃	—	2	8	no	—
99	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
108	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
109	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
113	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
114	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
118	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
120	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
121	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
122	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
123	C₂×C₄²×C₅₃	D₄	1	8	yes	—
124	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
125	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
126	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
129	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
130	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
131	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
132	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
134	#95×C₅₃	C₂³	1	2	yes	—
135	#95×C₅₃	C₂³	1	2	yes	—
136	#95×C₅₃	C₂³	1	2	yes	—
137	#95×C₅₃	C₂³	1	2	yes	—
138	#95×C₅₃	C₂³	1	8	yes	—
139	#95×C₅₃	C₂³	1	8	yes	—
140	#95×C₅₃	C₂³	1	8	yes	—
141	#95×C₅₃	C₂³	1	8	yes	—
142	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
143	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
144	C₂×Q₈×C₅₃	—	2	2	no	—
145	C₂×Q₈×C₅₃	—	2	4	no	—
146	C₂×Q₈×C₅₃	—	2	4	no	—
147	C₂×Q₈×C₅₃	—	2	4	no	—
148	C₂×Q₈×C₅₃	—	2	4	no	—
149	C₂×Q₈×C₅₃	—	2	4	no	—
150	C₂×Q₈×C₅₃	—	4	4	no	—
151	C₂×D₄×C₅₃	—	2	4	no	—
152	C₂×D₄×C₅₃	—	2	2	no	—
153	C₂×D₄×C₅₃	—	2	4	no	—
154	C₂×D₄×C₅₃	—	4	4	no	—
155	C₂²×C₄×C₅₃	—	2	4	no	—
156	C₂²×C₄×C₅₃	—	2	4	no	—
157	C₂²×C₄×C₅₃	—	2	4	no	—
158	C₂²×C₄×C₅₃	—	2	8	no	—
159	C₂²×C₄×C₅₃	—	4	2	no	—
160	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
162	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
163	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
170	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
174	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
175	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
176	C₄⋊C₄×C₅₃	—	2	4	no	—
177	C₄⋊C₄×C₅₃	—	2	4	no	—
178	C₄⋊C₄×C₅₃	—	2	4	no	—
179	C₄⋊C₄×C₅₃	—	2	4	no	—
180	C₄⋊C₄×C₅₃	—	2	2	no	—
181	C₄⋊C₄×C₅₃	—	2	4	no	—
182	C₄⋊C₄×C₅₃	—	2	2	no	—
183	C₄⋊C₄×C₅₃	—	2	8	no	—
184	C₄⋊C₄×C₅₃	—	2	8	no	—
185	C₄⋊C₄×C₅₃	—	2	4	no	—
186	C₄⋊C₄×C₅₃	—	2	4	no	—
187	C₄⋊C₄×C₅₃	—	2	4	no	—
188	C₄⋊C₄×C₅₃	—	2	8	no	—
189	C₄⋊C₄×C₅₃	—	2	16	no	—
190	C₄⋊C₄×C₅₃	—	4	8	no	—
191	C₂²⋊C₄×C₅₃	—	2	8	no	—
192	C₂²⋊C₄×C₅₃	—	2	4	no	—
193	C₂²⋊C₄×C₅₃	—	2	4	no	—
194	C₂²⋊C₄×C₅₃	—	2	4	no	—
195	C₂²⋊C₄×C₅₃	—	2	4	no	—
196	C₂²⋊C₄×C₅₃	—	2	4	no	—
197	C₂²⋊C₄×C₅₃	—	2	8	no	—
198	C₂²⋊C₄×C₅₃	—	2	16	no	—
199	C₂²⋊C₄×C₅₃	—	4	8	no	—
200	C₄²×C₅₃	—	2	4	no	—
201	C₄²×C₅₃	—	2	8	no	—
202	C₄²×C₅₃	—	2	4	no	—
203	C₄²×C₅₃	—	2	8	no	—
204	C₄²×C₅₃	—	4	4	no	—
205	C₄²×C₅₃	C₂×D₄	1	4	yes	—
206	C₄²×C₅₃	C₂×D₄	1	4	yes	—
207	C₄²×C₅₃	C₂×D₄	1	4	yes	—
208	C₄²×C₅₃	C₂×D₄	1	4	yes	—
209	C₄²×C₅₃	#41	1	8	yes	—
210	C₂³×C₅₃	—	2	2	no	—
211	C₂³×C₅₃	—	2	2	no	—
212	C₂³×C₅₃	—	2	2	no	—
213	C₂³×C₅₃	—	2	2	no	—
214	C₂³×C₅₃	—	2	4	no	—
215	C₂³×C₅₃	—	2	4	no	—
216	C₂³×C₅₃	—	2	4	no	—
217	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
218	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
219	Q₈×C₅₃	—	4	8	no	—
220	Q₈×C₅₃	—	4	4	no	—
221	Q₈×C₅₃	—	4	8	no	—
222	D₄×C₅₃	—	4	8	no	—
223	D₄×C₅₃	—	4	4	no	—
224	C₂×C₄×C₅₃	—	2	2	no	—
225	C₂×C₄×C₅₃	—	2	2	no	—
226	C₂×C₄×C₅₃	—	2	4	no	—
227	C₂×C₄×C₅₃	—	2	2	no	—
228	C₂×C₄×C₅₃	—	2	2	no	—
229	C₂×C₄×C₅₃	—	2	4	no	—
230	C₂×C₄×C₅₃	—	2	8	no	—
231	C₂×C₄×C₅₃	—	2	8	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	4	no	—
236	C₂×C₄×C₅₃	—	2	4	no	—
237	C₂×C₄×C₅₃	—	2	4	no	—
238	C₂×C₄×C₅₃	—	2	4	no	—
239	C₂×C₄×C₅₃	—	2	4	no	—
240	C₂×C₄×C₅₃	—	2	4	no	—
241	C₂×C₄×C₅₃	—	2	4	no	—
242	C₂×C₄×C₅₃	—	2	4	no	—
243	C₂×C₄×C₅₃	—	2	4	no	—
244	C₂×C₄×C₅₃	—	2	8	no	—
245	C₂×C₄×C₅₃	—	2	4	no	—
246	C₂×C₄×C₅₃	—	4	4	no	—
247	C₂×C₄×C₅₃	—	4	8	no	—
248	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
249	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
250	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
251	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
253	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
254	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
255	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
256	C₂×C₄×C₅₃	2₋¹⁺⁴	1	4	yes	—
257	#34631	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
258	C₂²×C₅₃	—	2	2	no	—
259	C₂²×C₅₃	—	2	2	no	—
260	C₂²×C₅₃	—	2	2	no	—
261	C₂²×C₅₃	—	4	4	no	—
262	C₂²×C₅₃	—	4	2	no	—
263	C₂²×C₅₃	—	4	2	no	—
264	C₂²×C₅₃	—	4	2	no	—
265	C₂²×C₅₃	—	4	8	no	—
266	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
267	C₂²×C₅₃	C₂²×#41	1	2	yes	—
268	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
269	C₂²×C₅₃	#585	1	2	yes	—
270	C₄×C₅₃	—	2	2	no	—
271	C₄×C₅₃	—	2	2	no	—
272	C₄×C₅₃	—	2	2	no	—
273	C₄×C₅₃	—	2	2	no	—
274	C₄×C₅₃	—	2	8	no	—
275	C₄×C₅₃	—	4	2	no	—
276	C₄×C₅₃	—	4	4	no	—
277	C₄×C₅₃	—	4	2	no	—
278	C₄×C₅₃	—	4	2	no	—
279	C₄×C₅₃	—	4	4	no	—
280	C₄×C₅₃	—	4	4	no	—
281	C₄×C₅₃	#532	1	4	yes	—
282	C₄×C₅₃	#536	1	4	yes	—
283	C₂×#533	C₂×C₅₃	1	1	yes	characteristic, complemented
284	C₂×#529	C₂×C₅₃	1	1	yes	characteristic
285	#2664	C₂×C₅₃	1	4	yes	complemented
286	#2663	C₂×C₅₃	1	4	yes	complemented
287	#2660	C₂×C₅₃	1	1	yes	characteristic, complemented
288	#2658	C₂×C₅₃	1	1	yes	characteristic, complemented
289	#2642	C₂×C₅₃	1	1	yes	characteristic, complemented
290	#2640	C₂×C₅₃	1	1	yes	characteristic, complemented
291	#2621	C₂×C₅₃	1	2	yes	complemented
292	#2620	C₂×C₅₃	1	2	yes	—
293	#2396	C₂×C₅₃	1	4	yes	complemented
294	#2395	C₂×C₅₃	1	4	yes	complemented
295	C₄×C₄⋊Q₈	C₂×C₅₃	1	2	yes	complemented
296	C₄×#124	C₂×C₅₃	1	2	yes	complemented
297	C₄×#117	C₂×C₅₃	1	1	yes	characteristic, complemented
298	C₂×C₅₃	—	4	2	no	—
299	C₂×C₅₃	—	8	2	no	—
300	C₂×C₅₃	C₂×#532	1	1	yes	characteristic
301	C₂×C₅₃	C₂×#536	1	1	yes	characteristic
302	C₂×C₅₃	#3449	1	1	yes	characteristic
303	C₂×C₅₃	#3461	1	4	yes	—
304	#533	—	2	8	no	—
305	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
306	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
307	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
308	#529	—	2	8	no	—
309	C₂×#126	C₂²×C₅₃	1	2	yes	—
310	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic
311	C₂×#125	C₂²×C₅₃	1	2	yes	—
312	C₂×#124	C₂²×C₅₃	1	2	yes	—
313	C₂×#124	C₂²×C₅₃	1	1	yes	characteristic
314	C₂×#123	C₂²×C₅₃	1	2	yes	—
315	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
316	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
317	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
318	C₂×#121	C₂²×C₅₃	1	1	yes	characteristic
319	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
320	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
321	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
322	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
323	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
324	C₂×#117	C₂²×C₅₃	1	2	yes	—
325	C₂×#117	C₂²×C₅₃	1	2	yes	—
326	C₂×#117	C₂²×C₅₃	1	2	yes	—
327	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
328	#400	C₂²×C₅₃	1	4	yes	—
329	#400	C₂²×C₅₃	1	4	yes	—
330	#399	C₂²×C₅₃	1	4	yes	—
331	#399	C₂²×C₅₃	1	4	yes	—
332	#398	C₂²×C₅₃	1	8	yes	—
333	#398	C₂²×C₅₃	1	4	yes	—
334	#397	C₂²×C₅₃	1	8	yes	—
335	#397	C₂²×C₅₃	1	4	yes	—
336	#396	C₂²×C₅₃	1	8	yes	—
337	#395	C₂²×C₅₃	1	4	yes	—
338	#394	C₂²×C₅₃	1	8	yes	—
339	#393	C₂²×C₅₃	1	4	yes	—
340	#391	C₂²×C₅₃	1	2	yes	—
341	#391	C₂²×C₅₃	1	2	yes	—
342	#391	C₂²×C₅₃	1	2	yes	—
343	#390	C₂²×C₅₃	1	2	yes	—
344	#389	C₂²×C₅₃	1	8	yes	—
345	#388	C₂²×C₅₃	1	8	yes	—
346	#384	C₂²×C₅₃	1	2	yes	—
347	#382	C₂²×C₅₃	1	2	yes	—
348	C₄×C₄⋊C₄	C₂²×C₅₃	1	8	yes	—
349	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
350	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
351	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
352	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
353	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
354	#376	C₂²×C₅₃	1	1	yes	characteristic
355	#376	C₂²×C₅₃	1	1	yes	characteristic
356	C₄³	C₂²×C₅₃	1	2	yes	—
357	C₅₃	#34631	1	1	yes	characteristic, direct product factor, Sylow
358	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
359	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
360	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
361	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
362	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
363	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
364	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
365	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
366	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
367	C₄⋊Q₈	—	2	4	no	—
368	C₄⋊Q₈	—	2	4	no	—
369	C₄⋊Q₈	—	2	8	no	—
370	#126	—	2	8	no	—
371	#125	—	2	4	no	—
372	#125	—	2	8	no	—
373	#124	—	2	4	no	—
374	#124	—	2	8	no	—
375	#123	—	2	8	no	—
376	C₂²⋊Q₈	—	2	4	no	—
377	C₂²⋊Q₈	—	2	8	no	—
378	C₂²⋊Q₈	—	2	8	no	—
379	#121	—	2	4	no	—
380	C₄×Q₈	—	2	4	no	—
381	C₄×Q₈	—	2	8	no	—
382	C₄×Q₈	—	2	8	no	—
383	C₄×D₄	—	2	8	no	—
384	C₄×D₄	—	2	4	no	—
385	#117	—	2	8	no	—
386	#117	—	2	8	no	—
387	#117	—	2	8	no	—
388	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
389	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
390	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
391	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
392	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
393	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
394	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
395	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
396	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
397	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
398	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
399	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
400	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
401	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
402	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
403	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
404	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
406	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
407	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
408	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
409	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
410	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
411	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
412	C₂×C₄²	D₄×C₅₃	1	8	yes	—
413	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
414	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
415	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
416	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
417	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
418	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
419	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
420	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
421	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
422	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
423	#95	C₂³×C₅₃	1	2	yes	—
424	#95	C₂³×C₅₃	1	2	yes	—
425	#95	C₂³×C₅₃	1	2	yes	—
426	#95	C₂³×C₅₃	1	2	yes	—
427	#95	C₂³×C₅₃	1	8	yes	—
428	#95	C₂³×C₅₃	1	8	yes	—
429	#95	C₂³×C₅₃	1	8	yes	—
430	#95	C₂³×C₅₃	1	8	yes	—
431	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
432	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
433	C₂×Q₈	—	2	2	no	—
434	C₂×Q₈	—	2	4	no	—
435	C₂×Q₈	—	2	4	no	—
436	C₂×Q₈	—	2	4	no	—
437	C₂×Q₈	—	2	4	no	—
438	C₂×Q₈	—	2	4	no	—
439	C₂×Q₈	—	4	4	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	4	4	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	4	no	—
447	C₂²×C₄	—	2	8	no	—
448	C₂²×C₄	—	4	2	no	—
449	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
450	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
451	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
452	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
453	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
454	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
455	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
456	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
457	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
458	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
459	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
460	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
461	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
462	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
463	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
464	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	4	no	—
468	C₄⋊C₄	—	2	4	no	—
469	C₄⋊C₄	—	2	2	no	—
470	C₄⋊C₄	—	2	4	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	2	8	no	—
473	C₄⋊C₄	—	2	8	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	—	2	8	no	—
478	C₄⋊C₄	—	2	16	no	—
479	C₄⋊C₄	—	4	8	no	—
480	C₂²⋊C₄	—	2	8	no	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	2	4	no	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	2	8	no	—
487	C₂²⋊C₄	—	2	16	no	—
488	C₂²⋊C₄	—	4	8	no	—
489	C₄²	—	2	4	no	—
490	C₄²	—	2	8	no	—
491	C₄²	—	2	4	no	—
492	C₄²	—	2	8	no	—
493	C₄²	—	4	4	no	—
494	C₄²	C₂×D₄×C₅₃	1	4	yes	—
495	C₄²	C₂×D₄×C₅₃	1	4	yes	—
496	C₄²	C₂×D₄×C₅₃	1	4	yes	—
497	C₄²	C₂×D₄×C₅₃	1	4	yes	—
498	C₄²	#41×C₅₃	1	8	yes	—
499	C₂³	—	2	2	no	—
500	C₂³	—	2	2	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	2	2	no	—
503	C₂³	—	2	4	no	—
504	C₂³	—	2	4	no	—
505	C₂³	—	2	4	no	—
506	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
507	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
508	Q₈	—	4	8	no	—
509	Q₈	—	4	4	no	—
510	Q₈	—	4	8	no	—
511	D₄	—	4	8	no	—
512	D₄	—	4	4	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	8	no	—
520	C₂×C₄	—	2	8	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	8	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	4	4	no	—
536	C₂×C₄	—	4	8	no	—
537	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
538	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
539	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
540	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
541	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
542	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
543	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
544	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
545	C₂×C₄	2₋¹⁺⁴×C₅₃	1	4	yes	—
546	C₂²	—	2	2	no	—
547	C₂²	—	2	2	no	—
548	C₂²	—	2	2	no	—
549	C₂²	—	4	4	no	—
550	C₂²	—	4	2	no	—
551	C₂²	—	4	2	no	—
552	C₂²	—	4	8	no	—
553	C₂²	—	4	2	no	—
554	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
555	C₂²	C₂²×#41×C₅₃	1	2	yes	—
556	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	2	yes	—
557	C₂²	C₅₃×#585	1	2	yes	—
558	C₄	—	2	2	no	—
559	C₄	—	2	2	no	—
560	C₄	—	2	2	no	—
561	C₄	—	2	8	no	—
562	C₄	—	2	2	no	—
563	C₄	—	4	2	no	—
564	C₄	—	4	2	no	—
565	C₄	—	4	4	no	—
566	C₄	—	4	4	no	—
567	C₄	—	4	2	no	—
568	C₄	—	4	4	no	—
569	C₄	C₅₃×#532	1	4	yes	—
570	C₄	C₅₃×#536	1	4	yes	—
571	C₂	—	4	2	no	—
572	C₂	—	8	2	no	—
573	C₂	C₂×C₅₃×#532	1	1	yes	characteristic
574	C₂	C₂×C₅₃×#536	1	1	yes	characteristic
575	C₂	C₅₃×#3449	1	1	yes	characteristic
576	C₂	C₅₃×#3461	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
socle	C₂³×C₅₃	C₂⁵
prime radical	C₂×#121×C₅₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#533	C₂	semidirect
3. C₅₃×#2664	C₂	semidirect
4. C₅₃×#2663	C₂	semidirect
5. C₅₃×#2660	C₂	semidirect
6. C₅₃×#2658	C₂	semidirect
7. C₅₃×#2642	C₂	semidirect
8. C₅₃×#2640	C₂	semidirect
9. C₅₃×#2621	C₂	semidirect
11. C₅₃×#2396	C₂	semidirect
12. C₅₃×#2395	C₂	semidirect
13. C₄×C₄⋊Q₈×C₅₃	C₂	semidirect
14. C₄×#124×C₅₃	C₂	semidirect
15. C₄×#117×C₅₃	C₂	semidirect
257. #34631	C₅₃	direct (N × H)
283. C₂×#533	C₂×C₅₃	semidirect
285. #2664	C₂×C₅₃	semidirect
286. #2663	C₂×C₅₃	semidirect
287. #2660	C₂×C₅₃	semidirect
288. #2658	C₂×C₅₃	semidirect
289. #2642	C₂×C₅₃	semidirect
290. #2640	C₂×C₅₃	semidirect
291. #2621	C₂×C₅₃	semidirect
293. #2396	C₂×C₅₃	semidirect
294. #2395	C₂×C₅₃	semidirect
295. C₄×C₄⋊Q₈	C₂×C₅₃	semidirect
296. C₄×#124	C₂×C₅₃	semidirect
297. C₄×#117	C₂×C₅₃	semidirect
357. C₅₃	#34631	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#533	2
C₂×C₅₃×#529	2
C₅₃×#2664	2
C₅₃×#2663	2
C₅₃×#2660	2
C₅₃×#2658	2
C₅₃×#2642	2
C₅₃×#2640	2
C₅₃×#2621	2
C₅₃×#2620	2
C₅₃×#2396	2
C₅₃×#2395	2
C₄×C₄⋊Q₈×C₅₃	2
C₄×#124×C₅₃	2
C₄×#117×C₅₃	2
#34631	53

Subgroups — C53×#34631 #27242143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34631 #27242143