Subgroups — C₅₃×#22314 #27229826

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2399	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1985	C₂	1	2	yes	maximal
3	C₅₃×#1984	C₂	1	2	yes	maximal
4	C₅₃×#1939	C₂	1	2	yes	maximal
5	C₅₃×#1917	C₂	1	2	yes	maximal
6	C₅₃×#1916	C₂	1	2	yes	maximal
7	C₅₃×#1832	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1771	C₂	1	2	yes	maximal
9	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×#97×C₅₃	C₂²	1	2	yes	—
15	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#398	C₄	1	2	yes	—
19	C₅₃×#398	C₄	1	2	yes	—
20	C₅₃×#395	C₄	1	2	yes	—
21	C₅₃×#395	C₄	1	2	yes	—
22	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#376	C₂²	1	1	yes	characteristic
26	C₅₃×#336	C₂²	1	2	yes	—
27	C₅₃×#336	C₂²	1	2	yes	—
28	C₅₃×#336	C₂²	1	2	yes	—
29	C₅₃×#336	C₂²	1	2	yes	—
30	C₅₃×#336	C₂²	1	2	yes	—
31	C₅₃×#336	C₂²	1	2	yes	—
32	C₅₃×#336	C₂²	1	1	yes	characteristic
33	C₅₃×#336	C₂²	1	2	yes	—
34	C₅₃×#336	C₂²	1	1	yes	characteristic
35	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
40	C₄×Q₈×C₅₃	—	4	2	no	—
41	C₄×Q₈×C₅₃	—	4	2	no	—
42	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₅₃	—	2	4	no	—
55	C₄⋊C₈×C₅₃	—	4	2	no	—
56	C₄⋊C₈×C₅₃	—	4	2	no	—
57	#97×C₅₃	—	4	2	no	—
58	#97×C₅₃	Q₈	1	4	yes	complemented
59	C₄×C₈×C₅₃	—	2	2	no	—
60	C₄×C₈×C₅₃	—	4	2	no	—
61	#95×C₅₃	C₂×C₄	1	2	yes	—
62	#95×C₅₃	C₂×C₄	1	2	yes	—
63	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
64	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
65	#95×C₅₃	C₂×C₄	1	2	yes	—
66	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
67	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
68	#95×C₅₃	C₂×C₄	1	2	yes	—
69	C₂×Q₈×C₅₃	—	2	4	no	—
70	C₂×Q₈×C₅₃	—	4	2	no	—
71	C₂×Q₈×C₅₃	—	4	2	no	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
93	C₄⋊C₄×C₅₃	—	2	4	no	—
94	C₄⋊C₄×C₅₃	—	2	4	no	—
95	C₄⋊C₄×C₅₃	—	4	2	no	—
96	C₄⋊C₄×C₅₃	—	4	2	no	—
97	C₄⋊C₄×C₅₃	—	4	2	no	—
98	C₄⋊C₄×C₅₃	—	4	2	no	—
99	C₄⋊C₄×C₅₃	—	4	2	no	—
100	C₄²×C₅₃	—	2	2	no	—
101	C₄²×C₅₃	—	2	2	no	—
102	C₄²×C₅₃	—	2	2	no	—
103	C₄²×C₅₃	—	2	2	no	—
104	C₄²×C₅₃	—	2	2	no	—
105	C₄²×C₅₃	—	2	2	no	—
106	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
107	C₄²×C₅₃	#41	1	2	yes	—
108	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
109	Q₈×C₅₃	—	4	8	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
131	C₂×C₄×C₅₃	#131	1	4	yes	—
132	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
135	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
136	C₈×C₅₃	—	2	4	no	—
137	C₈×C₅₃	—	4	2	no	—
138	C₈×C₅₃	—	4	2	no	—
139	C₈×C₅₃	—	4	2	no	—
140	#22314	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
142	C₂²×C₅₃	#518	1	1	yes	characteristic
143	C₂²×C₅₃	#519	1	1	yes	characteristic
144	C₂²×C₅₃	C₂×#131	1	2	yes	—
145	C₂²×C₅₃	#568	1	2	yes	—
146	C₄×C₅₃	—	2	4	no	—
147	C₄×C₅₃	—	2	4	no	—
148	C₄×C₅₃	—	4	2	no	—
149	C₄×C₅₃	—	4	2	no	—
150	C₄×C₅₃	—	4	2	no	—
151	C₄×C₅₃	—	4	2	no	—
152	C₄×C₅₃	—	4	1	no	—
153	C₄×C₅₃	—	4	1	no	—
154	C₄×C₅₃	#542	1	2	yes	—
155	C₄×C₅₃	#544	1	2	yes	—
156	#2399	C₂×C₅₃	1	1	yes	characteristic
157	#1985	C₂×C₅₃	1	2	yes	—
158	#1984	C₂×C₅₃	1	2	yes	—
159	#1939	C₂×C₅₃	1	2	yes	—
160	#1917	C₂×C₅₃	1	2	yes	—
161	#1916	C₂×C₅₃	1	2	yes	—
162	#1832	C₂×C₅₃	1	1	yes	characteristic
163	#1771	C₂×C₅₃	1	2	yes	—
164	#1746	C₂×C₅₃	1	1	yes	characteristic
165	C₂×C₅₃	#2331	1	1	yes	characteristic
166	C₂×C₅₃	#2962	1	2	yes	—
167	C₂×C₅₃	#2976	1	2	yes	—
168	C₂×C₅₃	#2986	1	1	yes	characteristic
169	C₂×C₅₃	#2987	1	1	yes	characteristic
170	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
171	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
172	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
173	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
174	C₂×#97	C₂²×C₅₃	1	2	yes	—
175	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
176	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
177	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
178	#398	C₄×C₅₃	1	2	yes	—
179	#398	C₄×C₅₃	1	2	yes	—
180	#395	C₄×C₅₃	1	2	yes	—
181	#395	C₄×C₅₃	1	2	yes	—
182	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
183	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
184	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
185	#376	C₂²×C₅₃	1	1	yes	characteristic
186	#336	C₂²×C₅₃	1	2	yes	—
187	#336	C₂²×C₅₃	1	1	yes	characteristic
188	#336	C₂²×C₅₃	1	1	yes	characteristic
189	#336	C₂²×C₅₃	1	2	yes	—
190	#336	C₂²×C₅₃	1	2	yes	—
191	#336	C₂²×C₅₃	1	2	yes	—
192	#336	C₂²×C₅₃	1	2	yes	—
193	#336	C₂²×C₅₃	1	2	yes	—
194	#336	C₂²×C₅₃	1	2	yes	—
195	C₅₃	#22314	1	1	yes	characteristic, direct product factor, Sylow
196	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
197	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
198	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
199	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
200	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
201	C₄×Q₈	—	4	2	no	—
202	C₄×Q₈	—	4	2	no	—
203	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
204	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
205	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
206	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
207	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
208	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
209	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
210	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
211	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
212	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
213	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
215	C₄⋊C₈	—	2	4	no	—
216	C₄⋊C₈	—	4	2	no	—
217	C₄⋊C₈	—	4	2	no	—
218	#97	—	4	2	no	—
219	#97	Q₈×C₅₃	1	4	yes	complemented
220	C₄×C₈	—	2	2	no	—
221	C₄×C₈	—	4	2	no	—
222	#95	C₂×C₄×C₅₃	1	2	yes	—
223	#95	C₂×C₄×C₅₃	1	2	yes	—
224	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
225	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
226	#95	C₂×C₄×C₅₃	1	2	yes	—
227	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
228	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
229	#95	C₂×C₄×C₅₃	1	2	yes	—
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	4	2	no	—
232	C₂×Q₈	—	4	2	no	—
233	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
234	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
235	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
236	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
237	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
238	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
239	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
240	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
268	C₄²	#41×C₅₃	1	2	yes	—
269	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
270	Q₈	—	4	8	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
292	C₂×C₄	#131×C₅₃	1	4	yes	—
293	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
294	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
295	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
296	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
297	C₈	—	2	4	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	4	2	no	—
301	C₂²	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
302	C₂²	C₅₃×#518	1	1	yes	characteristic
303	C₂²	C₅₃×#519	1	1	yes	characteristic
304	C₂²	C₂×#131×C₅₃	1	2	yes	—
305	C₂²	C₅₃×#568	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	2	4	no	—
308	C₄	—	4	2	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	C₅₃×#542	1	2	yes	—
315	C₄	C₅₃×#544	1	2	yes	—
316	C₂	C₅₃×#2331	1	1	yes	characteristic
317	C₂	C₅₃×#2962	1	2	yes	—
318	C₂	C₅₃×#2976	1	2	yes	—
319	C₂	C₅₃×#2986	1	1	yes	characteristic
320	C₂	C₅₃×#2987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle, prime radical	C₂³×C₅₃	C₂³×C₄
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
58. #97×C₅₃	Q₈	semidirect
140. #22314	C₅₃	direct (N × H)
195. C₅₃	#22314	direct (N × H)
219. #97	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×#2399	2
C₅₃×#1985	2
C₅₃×#1984	2
C₅₃×#1939	2
C₅₃×#1917	2
C₅₃×#1916	2
C₅₃×#1832	2
C₅₃×#1771	2
C₅₃×#1746	2
#22314	53

Subgroups — C53×#22314 #27229826

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22314 #27229826