Subgroups — C₅₃×#36635 #27244147

Order 13568.

2006 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2864	C₂	1	2	yes	maximal, complemented
2	C₅₃×#2863	C₂	1	4	yes	maximal, complemented
3	C₅₃×#2862	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2859	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2857	C₂	1	2	yes	maximal
6	C₅₃×#2855	C₂	1	4	yes	maximal
7	C₅₃×#2851	C₂	1	2	yes	maximal
8	C₄×C₄⋊Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
9	C₄×#125×C₅₃	C₂	1	2	yes	maximal, complemented
10	C₄×C₂²⋊Q₈×C₅₃	C₂	1	4	yes	maximal
11	C₅₃×#2295	C₂	1	4	yes	maximal, complemented
12	C₅₃×#2283	C₂	1	2	yes	maximal
13	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
15	C₂×#125×C₅₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	—
17	C₂×C₄×Q₈×C₅₃	C₂²	1	4	yes	—
18	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
21	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#391	C₂²	1	4	yes	—
24	C₅₃×#391	C₂²	1	4	yes	—
25	C₅₃×#391	C₂²	1	4	yes	—
26	C₅₃×#391	C₂²	1	4	yes	—
27	C₅₃×#389	C₂²	1	4	yes	—
28	C₅₃×#389	C₂²	1	4	yes	—
29	C₅₃×#389	C₂²	1	4	yes	—
30	C₅₃×#389	C₂²	1	4	yes	—
31	C₅₃×#389	C₂²	1	4	yes	—
32	C₅₃×#389	C₂²	1	4	yes	—
33	C₅₃×#387	C₂²	1	8	yes	—
34	C₅₃×#387	C₂²	1	8	yes	—
35	C₅₃×#387	C₂²	1	8	yes	—
36	C₅₃×#387	C₂²	1	8	yes	—
37	C₅₃×#385	C₂²	1	8	yes	—
38	C₅₃×#384	C₂²	1	2	yes	—
39	C₅₃×#383	C₂²	1	4	yes	—
40	C₅₃×#382	C₂²	1	2	yes	—
41	C₅₃×#382	C₂²	1	2	yes	—
42	C₅₃×#382	C₂²	1	2	yes	—
43	C₅₃×#381	C₂²	1	4	yes	—
44	C₅₃×#380	C₂²	1	4	yes	—
45	C₅₃×#380	C₂²	1	4	yes	—
46	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
47	C₄×C₄⋊C₄×C₅₃	C₂²	1	8	yes	—
48	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
49	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
50	C₅₃×#376	C₂²	1	4	yes	—
51	C₄³×C₅₃	C₂²	1	2	yes	—
52	C₄³×C₅₃	C₂²	1	1	yes	characteristic
53	C₄³×C₅₃	C₂²	1	1	yes	characteristic
54	C₄³×C₅₃	C₂²	1	1	yes	characteristic
55	C₄³×C₅₃	C₂²	1	2	yes	—
56	C₄³×C₅₃	C₂²	1	1	yes	characteristic
57	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
58	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
59	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₅₃	—	2	8	no	—
64	#125×C₅₃	—	2	8	no	—
65	C₂²⋊Q₈×C₅₃	—	2	16	no	—
66	C₄×Q₈×C₅₃	—	2	16	no	—
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
69	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
70	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
71	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
72	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
73	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
74	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
75	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
76	C₂×C₄²×C₅₃	—	2	8	no	—
77	C₂×C₄²×C₅₃	—	2	4	no	—
78	C₂×C₄²×C₅₃	—	2	4	no	—
79	C₂×C₄²×C₅₃	—	2	8	no	—
80	C₂×C₄²×C₅₃	Q₈	1	8	yes	complemented
81	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
82	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
83	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
84	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
85	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
88	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
89	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
90	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
91	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
93	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
96	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
100	#95×C₅₃	C₂³	1	8	yes	—
101	#95×C₅₃	C₂³	1	4	yes	—
102	#95×C₅₃	C₂³	1	4	yes	—
103	#95×C₅₃	C₂³	1	4	yes	—
104	#95×C₅₃	C₂³	1	4	yes	—
105	#95×C₅₃	C₂³	1	8	yes	—
106	#95×C₅₃	C₂³	1	8	yes	—
107	#95×C₅₃	C₂³	1	8	yes	—
108	#95×C₅₃	C₂³	1	8	yes	—
109	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
110	C₂×Q₈×C₅₃	—	2	8	no	—
111	C₂×Q₈×C₅₃	—	2	8	no	—
112	C₂×Q₈×C₅₃	—	2	8	no	—
113	C₂²×C₄×C₅₃	—	2	4	no	—
114	C₂²×C₄×C₅₃	—	2	4	no	—
115	C₂²×C₄×C₅₃	—	2	4	no	—
116	C₂²×C₄×C₅₃	—	2	4	no	—
117	C₂²×C₄×C₅₃	—	2	4	no	—
118	C₂²×C₄×C₅₃	—	2	4	no	—
119	C₂²×C₄×C₅₃	—	2	2	no	—
120	C₂²×C₄×C₅₃	—	2	2	no	—
121	C₂²×C₄×C₅₃	—	2	4	no	—
122	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
123	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
124	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
125	C₂²×C₄×C₅₃	#41	1	8	yes	—
126	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
127	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
128	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
129	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
132	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
133	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
134	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
135	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
136	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
137	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
138	C₄⋊C₄×C₅₃	—	2	16	no	—
139	C₄⋊C₄×C₅₃	—	2	8	no	—
140	C₄⋊C₄×C₅₃	—	2	16	no	—
141	C₄⋊C₄×C₅₃	—	2	8	no	—
142	C₄⋊C₄×C₅₃	—	2	16	no	—
143	C₄⋊C₄×C₅₃	—	2	8	no	—
144	C₄⋊C₄×C₅₃	—	2	8	no	—
145	C₄⋊C₄×C₅₃	—	2	8	no	—
146	C₂²⋊C₄×C₅₃	—	2	16	no	—
147	C₄²×C₅₃	—	2	4	no	—
148	C₄²×C₅₃	—	2	4	no	—
149	C₄²×C₅₃	—	2	4	no	—
150	C₄²×C₅₃	—	2	4	no	—
151	C₄²×C₅₃	—	2	4	no	—
152	C₄²×C₅₃	—	2	4	no	—
153	C₄²×C₅₃	—	2	4	no	—
154	C₄²×C₅₃	—	2	4	no	—
155	C₄²×C₅₃	—	2	2	no	—
156	C₄²×C₅₃	—	2	2	no	—
157	C₄²×C₅₃	—	2	4	no	—
158	C₄²×C₅₃	—	2	16	no	—
159	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
160	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
161	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
162	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
163	C₄²×C₅₃	#41	1	8	yes	—
164	C₄²×C₅₃	#41	1	4	yes	—
165	C₄²×C₅₃	#41	1	4	yes	—
166	C₂³×C₅₃	—	2	1	no	—
167	C₂³×C₅₃	—	2	2	no	—
168	C₂³×C₅₃	—	2	1	no	—
169	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
170	C₂³×C₅₃	C₂×#41	1	4	yes	—
171	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
172	Q₈×C₅₃	—	4	16	no	—
173	C₂×C₄×C₅₃	—	2	16	no	—
174	C₂×C₄×C₅₃	—	2	8	no	—
175	C₂×C₄×C₅₃	—	2	2	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	2	1	no	—
178	C₂×C₄×C₅₃	—	2	2	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	—	2	1	no	—
182	C₂×C₄×C₅₃	—	2	2	no	—
183	C₂×C₄×C₅₃	—	2	2	no	—
184	C₂×C₄×C₅₃	—	2	2	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	2	16	no	—
187	C₂×C₄×C₅₃	—	2	8	no	—
188	C₂×C₄×C₅₃	—	2	8	no	—
189	C₂×C₄×C₅₃	—	2	8	no	—
190	C₂×C₄×C₅₃	—	2	8	no	—
191	C₂×C₄×C₅₃	—	2	8	no	—
192	C₂×C₄×C₅₃	—	2	8	no	—
193	C₂×C₄×C₅₃	—	2	8	no	—
194	C₂×C₄×C₅₃	—	2	8	no	—
195	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
196	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
197	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
198	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
199	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
200	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
201	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
202	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
203	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
204	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
205	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
206	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
207	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
208	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
209	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
210	#36635	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂²×C₅₃	—	2	4	no	—
212	C₂²×C₅₃	—	2	2	no	—
213	C₂²×C₅₃	—	2	2	no	—
214	C₂²×C₅₃	—	2	4	no	—
215	C₂²×C₅₃	#533	1	4	yes	—
216	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
217	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
218	C₂²×C₅₃	C₂²×#41	1	2	yes	—
219	C₂²×C₅₃	C₂²×#41	1	2	yes	—
220	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
221	C₄×C₅₃	—	2	4	no	—
222	C₄×C₅₃	—	2	2	no	—
223	C₄×C₅₃	—	2	2	no	—
224	C₄×C₅₃	—	2	2	no	—
225	C₄×C₅₃	—	2	2	no	—
226	C₄×C₅₃	—	2	4	no	—
227	C₄×C₅₃	—	2	4	no	—
228	C₄×C₅₃	—	4	8	no	—
229	C₄×C₅₃	—	4	8	no	—
230	C₄×C₅₃	#529	1	8	yes	—
231	C₄×C₅₃	#533	1	4	yes	—
232	C₄×C₅₃	#533	1	4	yes	—
233	#2864	C₂×C₅₃	1	2	yes	complemented
234	#2863	C₂×C₅₃	1	4	yes	complemented
235	#2862	C₂×C₅₃	1	1	yes	characteristic, complemented
236	#2859	C₂×C₅₃	1	1	yes	characteristic, complemented
237	#2857	C₂×C₅₃	1	2	yes	—
238	#2855	C₂×C₅₃	1	4	yes	—
239	#2851	C₂×C₅₃	1	2	yes	—
240	C₄×C₄⋊Q₈	C₂×C₅₃	1	2	yes	complemented
241	C₄×#125	C₂×C₅₃	1	2	yes	complemented
242	C₄×C₂²⋊Q₈	C₂×C₅₃	1	4	yes	—
243	#2295	C₂×C₅₃	1	4	yes	complemented
244	#2283	C₂×C₅₃	1	2	yes	—
245	C₂×C₄³	C₂×C₅₃	1	1	yes	characteristic
246	C₂×C₅₃	—	2	4	no	—
247	C₂×C₅₃	C₂×#529	1	2	yes	—
248	C₂×C₅₃	C₂×#533	1	1	yes	characteristic
249	C₂×C₅₃	C₂×#533	1	1	yes	characteristic
250	C₂×C₅₃	C₂×#533	1	1	yes	characteristic
251	C₂×C₅₃	#3444	1	2	yes	—
252	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
253	C₂×#125	C₂²×C₅₃	1	2	yes	—
254	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	—
255	C₂×C₄×Q₈	C₂²×C₅₃	1	4	yes	—
256	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
257	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
258	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
259	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
260	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
261	#391	C₂²×C₅₃	1	4	yes	—
262	#391	C₂²×C₅₃	1	4	yes	—
263	#391	C₂²×C₅₃	1	4	yes	—
264	#391	C₂²×C₅₃	1	4	yes	—
265	#389	C₂²×C₅₃	1	4	yes	—
266	#389	C₂²×C₅₃	1	4	yes	—
267	#389	C₂²×C₅₃	1	4	yes	—
268	#389	C₂²×C₅₃	1	4	yes	—
269	#389	C₂²×C₅₃	1	4	yes	—
270	#389	C₂²×C₅₃	1	4	yes	—
271	#387	C₂²×C₅₃	1	8	yes	—
272	#387	C₂²×C₅₃	1	8	yes	—
273	#387	C₂²×C₅₃	1	8	yes	—
274	#387	C₂²×C₅₃	1	8	yes	—
275	#385	C₂²×C₅₃	1	8	yes	—
276	#384	C₂²×C₅₃	1	2	yes	—
277	#383	C₂²×C₅₃	1	4	yes	—
278	#382	C₂²×C₅₃	1	2	yes	—
279	#382	C₂²×C₅₃	1	2	yes	—
280	#382	C₂²×C₅₃	1	2	yes	—
281	#381	C₂²×C₅₃	1	4	yes	—
282	#380	C₂²×C₅₃	1	4	yes	—
283	#380	C₂²×C₅₃	1	4	yes	—
284	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
285	C₄×C₄⋊C₄	C₂²×C₅₃	1	8	yes	—
286	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
287	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
288	#376	C₂²×C₅₃	1	4	yes	—
289	C₄³	C₂²×C₅₃	1	1	yes	characteristic
290	C₄³	C₂²×C₅₃	1	1	yes	characteristic
291	C₄³	C₂²×C₅₃	1	1	yes	characteristic
292	C₄³	C₂²×C₅₃	1	2	yes	—
293	C₄³	C₂²×C₅₃	1	1	yes	characteristic
294	C₄³	C₂²×C₅₃	1	2	yes	—
295	C₅₃	#36635	1	1	yes	characteristic, direct product factor, Sylow
296	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
297	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
298	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
299	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
300	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
301	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
302	C₄⋊Q₈	—	2	8	no	—
303	#125	—	2	8	no	—
304	C₂²⋊Q₈	—	2	16	no	—
305	C₄×Q₈	—	2	16	no	—
306	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
307	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
308	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
309	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
310	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
311	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
312	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
313	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
314	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
315	C₂×C₄²	—	2	8	no	—
316	C₂×C₄²	—	2	8	no	—
317	C₂×C₄²	—	2	4	no	—
318	C₂×C₄²	—	2	4	no	—
319	C₂×C₄²	Q₈×C₅₃	1	8	yes	complemented
320	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
321	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
322	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
323	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
324	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
325	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
326	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
327	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
328	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
329	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
330	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
331	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
332	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
333	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
334	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
335	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
336	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
337	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
338	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
339	#95	C₂³×C₅₃	1	8	yes	—
340	#95	C₂³×C₅₃	1	4	yes	—
341	#95	C₂³×C₅₃	1	4	yes	—
342	#95	C₂³×C₅₃	1	4	yes	—
343	#95	C₂³×C₅₃	1	4	yes	—
344	#95	C₂³×C₅₃	1	8	yes	—
345	#95	C₂³×C₅₃	1	8	yes	—
346	#95	C₂³×C₅₃	1	8	yes	—
347	#95	C₂³×C₅₃	1	8	yes	—
348	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
349	C₂×Q₈	—	2	8	no	—
350	C₂×Q₈	—	2	8	no	—
351	C₂×Q₈	—	2	8	no	—
352	C₂²×C₄	—	2	4	no	—
353	C₂²×C₄	—	2	4	no	—
354	C₂²×C₄	—	2	4	no	—
355	C₂²×C₄	—	2	4	no	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	2	4	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
362	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
363	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
364	C₂²×C₄	#41×C₅₃	1	8	yes	—
365	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
366	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
367	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
368	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
369	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
370	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
371	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
372	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
373	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
374	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
375	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
376	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
377	C₄⋊C₄	—	2	8	no	—
378	C₄⋊C₄	—	2	16	no	—
379	C₄⋊C₄	—	2	8	no	—
380	C₄⋊C₄	—	2	16	no	—
381	C₄⋊C₄	—	2	16	no	—
382	C₄⋊C₄	—	2	8	no	—
383	C₄⋊C₄	—	2	8	no	—
384	C₄⋊C₄	—	2	8	no	—
385	C₂²⋊C₄	—	2	16	no	—
386	C₄²	—	2	4	no	—
387	C₄²	—	2	4	no	—
388	C₄²	—	2	4	no	—
389	C₄²	—	2	4	no	—
390	C₄²	—	2	4	no	—
391	C₄²	—	2	4	no	—
392	C₄²	—	2	4	no	—
393	C₄²	—	2	4	no	—
394	C₄²	—	2	4	no	—
395	C₄²	—	2	2	no	—
396	C₄²	—	2	2	no	—
397	C₄²	—	2	16	no	—
398	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
399	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
400	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
401	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
402	C₄²	#41×C₅₃	1	8	yes	—
403	C₄²	#41×C₅₃	1	4	yes	—
404	C₄²	#41×C₅₃	1	4	yes	—
405	C₂³	—	2	1	no	—
406	C₂³	—	2	2	no	—
407	C₂³	—	2	1	no	—
408	C₂³	C₂²×Q₈×C₅₃	1	2	yes	—
409	C₂³	C₂×#41×C₅₃	1	4	yes	—
410	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
411	Q₈	—	4	16	no	—
412	C₂×C₄	—	2	16	no	—
413	C₂×C₄	—	2	8	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	16	no	—
426	C₂×C₄	—	2	8	no	—
427	C₂×C₄	—	2	8	no	—
428	C₂×C₄	—	2	8	no	—
429	C₂×C₄	—	2	8	no	—
430	C₂×C₄	—	2	8	no	—
431	C₂×C₄	—	2	8	no	—
432	C₂×C₄	—	2	8	no	—
433	C₂×C₄	—	2	8	no	—
434	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
435	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
436	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
437	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
438	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
439	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
440	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
441	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
442	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
443	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
444	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
445	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
446	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
447	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
448	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
449	C₂²	—	2	4	no	—
450	C₂²	—	2	2	no	—
451	C₂²	—	2	2	no	—
452	C₂²	—	2	4	no	—
453	C₂²	C₅₃×#533	1	4	yes	—
454	C₂²	C₂³×Q₈×C₅₃	1	1	yes	characteristic
455	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
456	C₂²	C₂²×#41×C₅₃	1	2	yes	—
457	C₂²	C₂²×#41×C₅₃	1	2	yes	—
458	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
459	C₄	—	2	4	no	—
460	C₄	—	2	2	no	—
461	C₄	—	2	2	no	—
462	C₄	—	2	2	no	—
463	C₄	—	2	2	no	—
464	C₄	—	2	4	no	—
465	C₄	—	2	4	no	—
466	C₄	—	4	8	no	—
467	C₄	—	4	8	no	—
468	C₄	C₅₃×#529	1	8	yes	—
469	C₄	C₅₃×#533	1	4	yes	—
470	C₄	C₅₃×#533	1	4	yes	—
471	C₂	—	2	4	no	—
472	C₂	C₂×C₅₃×#529	1	2	yes	—
473	C₂	C₂×C₅₃×#533	1	1	yes	characteristic
474	C₂	C₂×C₅₃×#533	1	1	yes	characteristic
475	C₂	C₂×C₅₃×#533	1	1	yes	characteristic
476	C₂	C₅₃×#3444	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
socle	C₂³×C₅₃	C₂⁵
prime radical	C₂⁴×C₅₃	C₂⁴
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. C₅₃×#2864	C₂	semidirect
2. C₅₃×#2863	C₂	semidirect
3. C₅₃×#2862	C₂	semidirect
4. C₅₃×#2859	C₂	semidirect
8. C₄×C₄⋊Q₈×C₅₃	C₂	semidirect
9. C₄×#125×C₅₃	C₂	semidirect
11. C₅₃×#2295	C₂	semidirect
80. C₂×C₄²×C₅₃	Q₈	semidirect
210. #36635	C₅₃	direct (N × H)
233. #2864	C₂×C₅₃	semidirect
234. #2863	C₂×C₅₃	semidirect
235. #2862	C₂×C₅₃	semidirect
236. #2859	C₂×C₅₃	semidirect
240. C₄×C₄⋊Q₈	C₂×C₅₃	semidirect
241. C₄×#125	C₂×C₅₃	semidirect
243. #2295	C₂×C₅₃	semidirect
295. C₅₃	#36635	direct (N × H)
319. C₂×C₄²	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2864	2
C₅₃×#2863	2
C₅₃×#2862	2
C₅₃×#2859	2
C₅₃×#2857	2
C₅₃×#2855	2
C₅₃×#2851	2
C₄×C₄⋊Q₈×C₅₃	2
C₄×#125×C₅₃	2
C₄×C₂²⋊Q₈×C₅₃	2
C₅₃×#2295	2
C₅₃×#2283	2
C₂×C₄³×C₅₃	2
#36635	53

Subgroups — C53×#36635 #27244147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36635 #27244147