Subgroups — C₅₃×#22215 #27229727

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2398	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1985	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1955	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1924	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1851	C₂	1	4	yes	maximal, complemented
8	C₅₃×#1843	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#446	—	2	4	no	—
14	C₅₃×#436	—	2	4	no	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₈×C₅₃	C₂²	1	4	yes	—
21	C₂×#97×C₅₃	C₂²	1	2	yes	—
22	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#400	C₄	1	2	yes	—
25	C₅₃×#397	C₄	1	2	yes	—
26	C₅₃×#394	C₄	1	2	yes	—
27	C₅₃×#393	C₄	1	2	yes	—
28	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
29	C₅₃×#376	C₂²	1	1	yes	characteristic
30	C₅₃×#336	C₂²	1	2	yes	—
31	C₅₃×#336	C₂²	1	4	yes	—
32	C₅₃×#336	C₂²	1	2	yes	—
33	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
36	C₂×M₁₆×C₅₃	—	2	4	no	—
37	C₂×M₁₆×C₅₃	—	4	4	no	—
38	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₅₃	—	2	2	no	—
44	C₄×Q₈×C₅₃	—	2	2	no	—
45	C₄×D₄×C₅₃	—	2	2	no	—
46	C₄×D₄×C₅₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
52	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
54	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₅₃	—	2	4	no	—
57	C₄⋊C₈×C₅₃	—	2	2	no	—
58	C₄⋊C₈×C₅₃	—	2	2	no	—
59	C₄⋊C₈×C₅₃	—	4	4	no	—
60	C₂²⋊C₈×C₅₃	—	2	4	no	—
61	C₂²⋊C₈×C₅₃	—	4	4	no	—
62	#97×C₅₃	—	2	4	no	—
63	#97×C₅₃	—	2	2	no	—
64	C₄×C₈×C₅₃	D₄	1	4	yes	—
65	#95×C₅₃	C₂×C₄	1	2	yes	—
66	#95×C₅₃	C₂×C₄	1	2	yes	—
67	#95×C₅₃	C₂×C₄	1	4	yes	—
68	#95×C₅₃	C₂×C₄	1	4	yes	—
69	C₂⁴×C₅₃	C₂²×C₄	1	2	yes	—
70	C₂×Q₈×C₅₃	—	2	2	no	—
71	C₂×Q₈×C₅₃	—	4	2	no	—
72	C₂×D₄×C₅₃	—	2	2	no	—
73	C₂×D₄×C₅₃	—	4	2	no	—
74	C₂²×C₄×C₅₃	—	2	2	no	—
75	C₂²×C₄×C₅₃	—	2	2	no	—
76	C₂²×C₄×C₅₃	—	2	4	no	—
77	C₂²×C₄×C₅₃	—	2	4	no	—
78	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
85	M₁₆×C₅₃	—	4	8	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	—	2	4	no	—
93	C₂×C₈×C₅₃	—	2	4	no	—
94	C₂×C₈×C₅₃	—	2	2	no	—
95	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
97	C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₄⋊C₄×C₅₃	—	2	4	no	—
99	C₄⋊C₄×C₅₃	—	2	2	no	—
100	C₂²⋊C₄×C₅₃	—	2	4	no	—
101	C₂²⋊C₄×C₅₃	—	4	4	no	—
102	C₂²⋊C₄×C₅₃	—	4	4	no	—
103	C₄²×C₅₃	—	2	2	no	—
104	C₄²×C₅₃	—	2	2	no	—
105	C₄²×C₅₃	—	2	2	no	—
106	C₄²×C₅₃	—	2	2	no	—
107	C₄²×C₅₃	C₂×D₄	1	2	yes	—
108	C₄²×C₅₃	#41	1	2	yes	—
109	C₂³×C₅₃	—	2	2	no	—
110	C₂³×C₅₃	—	2	4	no	—
111	C₂³×C₅₃	—	2	2	no	—
112	C₂³×C₅₃	—	2	2	no	—
113	C₂³×C₅₃	—	2	4	no	—
114	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
115	Q₈×C₅₃	—	4	4	no	—
116	D₄×C₅₃	—	4	4	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	4	no	—
123	C₂×C₄×C₅₃	—	2	4	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	4	no	—
127	C₂×C₄×C₅₃	—	2	4	no	—
128	C₂×C₄×C₅₃	—	2	4	no	—
129	C₂×C₄×C₅₃	—	4	2	no	—
130	C₂×C₄×C₅₃	—	4	2	no	—
131	C₂×C₄×C₅₃	—	4	4	no	—
132	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
137	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
138	C₈×C₅₃	—	2	2	no	—
139	C₈×C₅₃	—	2	2	no	—
140	C₈×C₅₃	—	4	2	no	—
141	C₈×C₅₃	—	4	2	no	—
142	C₈×C₅₃	—	4	2	no	—
143	#22215	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₅₃	—	4	2	no	—
145	C₂²×C₅₃	—	4	2	no	—
146	C₂²×C₅₃	—	4	2	no	—
147	C₂²×C₅₃	—	4	4	no	—
148	C₂²×C₅₃	—	4	4	no	—
149	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
150	C₂²×C₅₃	#520	1	2	yes	—
151	C₂²×C₅₃	#568	1	2	yes	—
152	C₂²×C₅₃	#568	1	2	yes	—
153	C₄×C₅₃	—	2	2	no	—
154	C₄×C₅₃	—	2	2	no	—
155	C₄×C₅₃	—	4	2	no	—
156	C₄×C₅₃	—	4	2	no	—
157	C₄×C₅₃	—	4	2	no	—
158	C₄×C₅₃	—	4	4	no	—
159	C₄×C₅₃	#535	1	1	yes	characteristic
160	C₄×C₅₃	#536	1	1	yes	characteristic
161	C₄×C₅₃	#542	1	2	yes	—
162	C₂×#446	C₂×C₅₃	1	1	yes	characteristic, complemented
163	C₂×#436	C₂×C₅₃	1	1	yes	characteristic
164	#2398	C₂×C₅₃	1	1	yes	characteristic
165	#1985	C₂×C₅₃	1	1	yes	characteristic, complemented
166	#1955	C₂×C₅₃	1	1	yes	characteristic
167	#1924	C₂×C₅₃	1	4	yes	complemented
168	#1851	C₂×C₅₃	1	4	yes	complemented
169	#1843	C₂×C₅₃	1	1	yes	characteristic, complemented
170	#1746	C₂×C₅₃	1	1	yes	characteristic, complemented
171	C₂×C₅₃	—	8	2	no	—
172	C₂×C₅₃	#2326	1	1	yes	characteristic
173	C₂×C₅₃	#2930	1	1	yes	characteristic
174	C₂×C₅₃	#2930	1	1	yes	characteristic
175	C₂×C₅₃	#2995	1	4	yes	—
176	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
177	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
178	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
179	#446	—	2	4	no	—
180	#436	—	2	4	no	—
181	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
182	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
183	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
184	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
185	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
186	C₂×C₂²⋊C₈	C₂²×C₅₃	1	4	yes	—
187	C₂×#97	C₂²×C₅₃	1	2	yes	—
188	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
190	#400	C₄×C₅₃	1	2	yes	—
191	#397	C₄×C₅₃	1	2	yes	—
192	#394	C₄×C₅₃	1	2	yes	—
193	#393	C₄×C₅₃	1	2	yes	—
194	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
195	#376	C₂²×C₅₃	1	1	yes	characteristic
196	#336	C₂²×C₅₃	1	2	yes	—
197	#336	C₂²×C₅₃	1	4	yes	—
198	#336	C₂²×C₅₃	1	2	yes	—
199	C₅₃	#22215	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
201	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
202	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
203	C₂×M₁₆	—	2	4	no	—
204	C₂×M₁₆	—	4	4	no	—
205	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
206	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
207	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
208	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	—	2	2	no	—
214	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
216	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
218	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
219	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
220	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
221	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
222	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
223	C₄⋊C₈	—	2	4	no	—
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	—	4	4	no	—
227	C₂²⋊C₈	—	2	4	no	—
228	C₂²⋊C₈	—	4	4	no	—
229	#97	—	2	4	no	—
230	#97	—	2	2	no	—
231	C₄×C₈	D₄×C₅₃	1	4	yes	—
232	#95	C₂×C₄×C₅₃	1	2	yes	—
233	#95	C₂×C₄×C₅₃	1	2	yes	—
234	#95	C₂×C₄×C₅₃	1	4	yes	—
235	#95	C₂×C₄×C₅₃	1	4	yes	—
236	C₂⁴	C₂²×C₄×C₅₃	1	2	yes	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
246	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
247	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
248	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
249	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
251	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
252	M₁₆	—	4	8	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
263	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₂²⋊C₄	—	2	4	no	—
268	C₂²⋊C₄	—	4	4	no	—
269	C₂²⋊C₄	—	4	4	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	C₂×D₄×C₅₃	1	2	yes	—
275	C₄²	#41×C₅₃	1	2	yes	—
276	C₂³	—	2	2	no	—
277	C₂³	—	2	4	no	—
278	C₂³	—	2	2	no	—
279	C₂³	—	2	4	no	—
280	C₂³	—	2	2	no	—
281	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
282	Q₈	—	4	4	no	—
283	D₄	—	4	4	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	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
300	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
301	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
302	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
303	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
304	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
305	C₈	—	2	2	no	—
306	C₈	—	2	2	no	—
307	C₈	—	4	2	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	2	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
316	C₂²	C₅₃×#520	1	2	yes	—
317	C₂²	C₅₃×#568	1	2	yes	—
318	C₂²	C₅₃×#568	1	2	yes	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	4	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	C₅₃×#535	1	1	yes	characteristic
326	C₄	C₅₃×#536	1	1	yes	characteristic
327	C₄	C₅₃×#542	1	2	yes	—
328	C₂	—	8	2	no	—
329	C₂	C₅₃×#2326	1	1	yes	characteristic
330	C₂	C₅₃×#2930	1	1	yes	characteristic
331	C₂	C₅₃×#2930	1	1	yes	characteristic
332	C₂	C₅₃×#2995	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂²×D₄×C₅₃	C₂×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
1. C₂×C₅₃×#446	C₂	semidirect
4. C₅₃×#1985	C₂	semidirect
6. C₅₃×#1924	C₂	semidirect
7. C₅₃×#1851	C₂	semidirect
8. C₅₃×#1843	C₂	semidirect
9. C₅₃×#1746	C₂	semidirect
143. #22215	C₅₃	direct (N × H)
162. C₂×#446	C₂×C₅₃	semidirect
165. #1985	C₂×C₅₃	semidirect
167. #1924	C₂×C₅₃	semidirect
168. #1851	C₂×C₅₃	semidirect
169. #1843	C₂×C₅₃	semidirect
170. #1746	C₂×C₅₃	semidirect
199. C₅₃	#22215	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×#446	2
C₂×C₅₃×#436	2
C₅₃×#2398	2
C₅₃×#1985	2
C₅₃×#1955	2
C₅₃×#1924	2
C₅₃×#1851	2
C₅₃×#1843	2
C₅₃×#1746	2
#22215	53

Subgroups — C53×#22215 #27229727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22215 #27229727