Subgroups — C₅₃×#21282 #27228794

Order 13568.

862 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2319	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1970	C₂	1	2	yes	maximal
4	C₅₃×#1942	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1936	C₂	1	4	yes	maximal, complemented
6	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1762	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
11	C₄×D₈×C₅₃	C₄	1	8	yes	complemented
12	C₂×#107×C₅₃	C₂²	1	2	yes	—
13	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#102×C₅₃	C₂²	1	4	yes	—
15	C₂×#102×C₅₃	C₂²	1	2	yes	—
16	C₂×#97×C₅₃	C₂²	1	2	yes	—
17	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#390	C₂²	1	4	yes	—
21	C₅₃×#389	C₂²	1	4	yes	—
22	C₅₃×#388	—	2	4	no	—
23	C₅₃×#381	—	2	2	no	—
24	C₅₃×#380	—	2	2	no	—
25	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₅₃×#376	C₂²	1	1	yes	characteristic
27	C₅₃×#340	C₂²	1	2	yes	—
28	C₅₃×#340	C₂²	1	4	yes	—
29	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
30	C₂³×C₄×C₅₃	—	2	2	no	—
31	C₂×D₈×C₅₃	—	2	4	no	—
32	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₄×D₄×C₅₃	—	2	4	no	—
37	C₄×D₄×C₅₃	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
41	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
43	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
44	C₂×C₄²×C₅₃	—	2	2	no	—
45	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	#107×C₅₃	—	2	4	no	—
48	#107×C₅₃	C₂×C₄	1	4	yes	—
49	#102×C₅₃	—	2	8	no	—
50	#102×C₅₃	C₂×C₄	1	8	yes	—
51	#97×C₅₃	—	2	4	no	—
52	#97×C₅₃	—	2	2	no	—
53	C₄×C₈×C₅₃	—	2	4	no	—
54	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
55	#95×C₅₃	—	2	2	no	—
56	#95×C₅₃	—	2	4	no	—
57	#95×C₅₃	D₄	1	1	yes	characteristic
58	#95×C₅₃	D₄	1	1	yes	characteristic
59	#95×C₅₃	D₄	1	2	yes	—
60	C₂⁴×C₅₃	—	2	2	no	—
61	C₂×D₄×C₅₃	—	2	4	no	—
62	C₂×D₄×C₅₃	—	2	4	no	—
63	C₂×D₄×C₅₃	—	2	2	no	—
64	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₅₃	—	2	2	no	—
66	C₂²×C₄×C₅₃	—	2	8	no	—
67	C₂²×C₄×C₅₃	—	2	4	no	—
68	C₂²×C₄×C₅₃	—	8	4	no	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
74	D₈×C₅₃	—	2	8	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₅₃	#41	1	4	yes	—
81	C₂×C₈×C₅₃	#41	1	4	yes	—
82	C₄⋊C₄×C₅₃	—	2	2	no	—
83	C₄⋊C₄×C₅₃	—	2	4	no	—
84	C₄⋊C₄×C₅₃	—	2	4	no	—
85	C₄⋊C₄×C₅₃	—	8	4	no	—
86	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
87	C₂²⋊C₄×C₅₃	—	2	8	no	—
88	C₂²⋊C₄×C₅₃	—	8	4	no	—
89	C₂²⋊C₄×C₅₃	—	8	4	no	—
90	C₄²×C₅₃	—	2	2	no	—
91	C₄²×C₅₃	—	2	2	no	—
92	C₄²×C₅₃	—	8	2	no	—
93	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₅₃	#41	1	2	yes	—
95	C₂³×C₅₃	—	2	4	no	—
96	C₂³×C₅₃	—	4	4	no	—
97	C₂³×C₅₃	—	4	4	no	—
98	C₂³×C₅₃	—	4	2	no	—
99	C₂³×C₅₃	—	4	2	no	—
100	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
101	D₄×C₅₃	—	2	4	no	—
102	D₄×C₅₃	—	2	4	no	—
103	D₄×C₅₃	—	4	4	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	2	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	—	4	4	no	—
115	C₂×C₄×C₅₃	—	4	4	no	—
116	C₂×C₄×C₅₃	—	4	4	no	—
117	C₂×C₄×C₅₃	—	4	4	no	—
118	C₂×C₄×C₅₃	—	8	8	no	—
119	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
124	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
125	C₈×C₅₃	—	2	2	no	—
126	C₈×C₅₃	—	2	4	no	—
127	C₈×C₅₃	#117	1	4	yes	—
128	#21282	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂²×C₅₃	—	4	4	no	—
130	C₂²×C₅₃	—	4	4	no	—
131	C₂²×C₅₃	—	4	4	no	—
132	C₂²×C₅₃	—	8	4	no	—
133	C₂²×C₅₃	—	8	4	no	—
134	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂²×C₅₃	#532	1	2	yes	—
136	C₂²×C₅₃	#576	1	2	yes	—
137	C₂²×C₅₃	#576	1	1	yes	characteristic
138	C₂²×C₅₃	#578	1	1	yes	characteristic
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	—	2	2	no	—
141	C₄×C₅₃	—	4	2	no	—
142	C₄×C₅₃	—	4	2	no	—
143	C₄×C₅₃	—	4	4	no	—
144	C₄×C₅₃	—	8	4	no	—
145	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
146	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
147	C₄×C₅₃	#529	1	2	yes	—
148	C₂×C₄×D₈	C₂×C₅₃	1	1	yes	characteristic
149	#2319	C₂×C₅₃	1	2	yes	complemented
150	#1970	C₂×C₅₃	1	2	yes	—
151	#1942	C₂×C₅₃	1	2	yes	complemented
152	#1936	C₂×C₅₃	1	4	yes	complemented
153	C₄×#107	C₂×C₅₃	1	1	yes	characteristic, complemented
154	#1762	C₂×C₅₃	1	2	yes	complemented
155	#1726	C₂×C₅₃	1	1	yes	characteristic, complemented
156	C₂×C₅₃	—	8	4	no	—
157	C₂×C₅₃	#2293	1	1	yes	characteristic
158	C₂×C₅₃	#2948	1	1	yes	characteristic
159	C₂×C₅₃	#2951	1	1	yes	characteristic
160	C₂×C₅₃	#3175	1	2	yes	—
161	C₂×C₅₃	#3176	1	2	yes	—
162	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
163	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
164	C₄×D₈	C₄×C₅₃	1	8	yes	complemented
165	C₂×#107	C₂²×C₅₃	1	2	yes	—
166	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
167	C₂×#102	C₂²×C₅₃	1	4	yes	—
168	C₂×#102	C₂²×C₅₃	1	2	yes	—
169	C₂×#97	C₂²×C₅₃	1	2	yes	—
170	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
172	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
173	#390	C₂²×C₅₃	1	4	yes	—
174	#389	C₂²×C₅₃	1	4	yes	—
175	#388	—	2	4	no	—
176	#381	—	2	2	no	—
177	#380	—	2	2	no	—
178	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
179	#376	C₂²×C₅₃	1	1	yes	characteristic
180	#340	C₂²×C₅₃	1	2	yes	—
181	#340	C₂²×C₅₃	1	4	yes	—
182	C₅₃	#21282	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
184	C₂³×C₄	—	2	2	no	—
185	C₂×D₈	—	2	4	no	—
186	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	—
187	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
189	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
190	C₄×D₄	—	2	4	no	—
191	C₄×D₄	C₂×C₄×C₅₃	1	8	yes	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
194	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	—	2	4	no	—
197	C₂×C₂²⋊C₄	—	2	4	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
200	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
201	#107	—	2	4	no	—
202	#107	C₂×C₄×C₅₃	1	4	yes	—
203	#102	—	2	8	no	—
204	#102	C₂×C₄×C₅₃	1	8	yes	—
205	#97	—	2	4	no	—
206	#97	—	2	2	no	—
207	C₄×C₈	—	2	4	no	—
208	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
209	#95	—	2	2	no	—
210	#95	—	2	4	no	—
211	#95	D₄×C₅₃	1	1	yes	characteristic
212	#95	D₄×C₅₃	1	1	yes	characteristic
213	#95	D₄×C₅₃	1	2	yes	—
214	C₂⁴	—	2	2	no	—
215	C₂×D₄	—	2	4	no	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	8	4	no	—
223	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
227	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
228	D₈	—	2	8	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
233	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
234	C₂×C₈	#41×C₅₃	1	4	yes	—
235	C₂×C₈	#41×C₅₃	1	4	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	8	4	no	—
240	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₂²⋊C₄	—	8	4	no	—
243	C₂²⋊C₄	—	8	4	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	8	2	no	—
247	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
248	C₄²	#41×C₅₃	1	2	yes	—
249	C₂³	—	2	4	no	—
250	C₂³	—	4	4	no	—
251	C₂³	—	4	2	no	—
252	C₂³	—	4	4	no	—
253	C₂³	—	4	2	no	—
254	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
255	D₄	—	2	4	no	—
256	D₄	—	2	4	no	—
257	D₄	—	4	4	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	8	8	no	—
273	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
274	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
275	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
276	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
277	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
278	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	2	4	no	—
281	C₈	#117×C₅₃	1	4	yes	—
282	C₂²	—	4	4	no	—
283	C₂²	—	4	4	no	—
284	C₂²	—	4	4	no	—
285	C₂²	—	8	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
288	C₂²	C₅₃×#532	1	2	yes	—
289	C₂²	C₅₃×#576	1	2	yes	—
290	C₂²	C₅₃×#576	1	1	yes	characteristic
291	C₂²	C₅₃×#578	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	4	no	—
295	C₄	—	4	2	no	—
296	C₄	—	4	2	no	—
297	C₄	—	8	4	no	—
298	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
299	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
300	C₄	C₅₃×#529	1	2	yes	—
301	C₂	—	8	4	no	—
302	C₂	C₅₃×#2293	1	1	yes	characteristic
303	C₂	C₅₃×#2948	1	1	yes	characteristic
304	C₂	C₅₃×#2951	1	1	yes	characteristic
305	C₂	C₅₃×#3175	1	2	yes	—
306	C₂	C₅₃×#3176	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×D₈×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₅₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2319	C₂	semidirect
4. C₅₃×#1942	C₂	semidirect
5. C₅₃×#1936	C₂	semidirect
6. C₄×#107×C₅₃	C₂	semidirect
7. C₅₃×#1762	C₂	semidirect
8. C₅₃×#1726	C₂	semidirect
11. C₄×D₈×C₅₃	C₄	semidirect
128. #21282	C₅₃	direct (N × H)
149. #2319	C₂×C₅₃	semidirect
151. #1942	C₂×C₅₃	semidirect
152. #1936	C₂×C₅₃	semidirect
153. C₄×#107	C₂×C₅₃	semidirect
154. #1762	C₂×C₅₃	semidirect
155. #1726	C₂×C₅₃	semidirect
164. C₄×D₈	C₄×C₅₃	semidirect
182. C₅₃	#21282	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×C₄×D₈×C₅₃	2
C₅₃×#2319	2
C₅₃×#1970	2
C₅₃×#1942	2
C₅₃×#1936	2
C₄×#107×C₅₃	2
C₅₃×#1762	2
C₅₃×#1726	2
#21282	53

Subgroups — C53×#21282 #27228794

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21282 #27228794