Subgroups — C₅₃×#21281 #27228793

Order 13568.

766 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2320	C₂	1	2	yes	maximal
3	C₅₃×#1969	C₂	1	2	yes	maximal
4	C₅₃×#1942	C₂	1	2	yes	maximal
5	C₅₃×#1935	C₂	1	4	yes	maximal
6	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1763	C₂	1	2	yes	maximal
8	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
11	C₄×Q₁₆×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×#103×C₅₃	C₂²	1	4	yes	—
15	C₂×#103×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×#391	C₂²	1	4	yes	—
21	C₅₃×#389	C₂²	1	4	yes	—
22	C₅₃×#387	—	2	4	no	—
23	C₅₃×#383	—	2	2	no	—
24	C₅₃×#382	—	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×Q₈×C₅₃	C₂³	1	2	yes	—
30	C₂×Q₁₆×C₅₃	—	2	4	no	—
31	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₅₃	—	2	4	no	—
36	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	—
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
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×C₄²×C₅₃	—	2	2	no	—
42	C₂×C₄²×C₅₃	—	2	2	no	—
43	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	#107×C₅₃	—	2	4	no	—
46	#107×C₅₃	C₂×C₄	1	4	yes	—
47	#103×C₅₃	—	2	8	no	—
48	#103×C₅₃	C₂×C₄	1	8	yes	—
49	#97×C₅₃	—	2	4	no	—
50	#97×C₅₃	—	2	2	no	—
51	C₄×C₈×C₅₃	—	2	4	no	—
52	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
53	#95×C₅₃	—	2	2	no	—
54	#95×C₅₃	—	2	4	no	—
55	#95×C₅₃	—	2	4	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	C₂×Q₈×C₅₃	—	2	2	no	—
61	C₂×Q₈×C₅₃	—	2	4	no	—
62	C₂×Q₈×C₅₃	—	2	4	no	—
63	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₅₃	—	2	2	no	—
65	C₂²×C₄×C₅₃	—	2	2	no	—
66	C₂²×C₄×C₅₃	—	2	4	no	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
72	Q₁₆×C₅₃	—	2	8	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₅₃	#41	1	4	yes	—
79	C₂×C₈×C₅₃	#41	1	4	yes	—
80	C₄⋊C₄×C₅₃	—	2	8	no	—
81	C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₄⋊C₄×C₅₃	—	2	4	no	—
83	C₄⋊C₄×C₅₃	—	2	4	no	—
84	C₄⋊C₄×C₅₃	—	8	4	no	—
85	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
86	C₄²×C₅₃	—	2	8	no	—
87	C₄²×C₅₃	—	2	2	no	—
88	C₄²×C₅₃	—	2	2	no	—
89	C₄²×C₅₃	—	8	2	no	—
90	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
91	C₄²×C₅₃	#41	1	2	yes	—
92	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
93	Q₈×C₅₃	—	2	4	no	—
94	Q₈×C₅₃	—	2	4	no	—
95	Q₈×C₅₃	—	4	4	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	4	4	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	—	4	4	no	—
112	C₂×C₄×C₅₃	—	4	4	no	—
113	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
117	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
118	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
119	C₈×C₅₃	—	2	2	no	—
120	C₈×C₅₃	—	2	4	no	—
121	C₈×C₅₃	#117	1	4	yes	—
122	#21281	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂²×C₅₃	#532	1	2	yes	—
125	C₂²×C₅₃	#576	1	1	yes	characteristic
126	C₂²×C₅₃	#578	1	2	yes	—
127	C₂²×C₅₃	#578	1	1	yes	characteristic
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	2	2	no	—
130	C₄×C₅₃	—	4	2	no	—
131	C₄×C₅₃	—	4	2	no	—
132	C₄×C₅₃	—	4	4	no	—
133	C₄×C₅₃	—	4	4	no	—
134	C₄×C₅₃	—	8	4	no	—
135	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
136	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
137	C₄×C₅₃	#529	1	2	yes	—
138	C₂×C₄×Q₁₆	C₂×C₅₃	1	1	yes	characteristic
139	#2320	C₂×C₅₃	1	2	yes	—
140	#1969	C₂×C₅₃	1	2	yes	—
141	#1942	C₂×C₅₃	1	2	yes	—
142	#1935	C₂×C₅₃	1	4	yes	—
143	C₄×#107	C₂×C₅₃	1	1	yes	characteristic
144	#1763	C₂×C₅₃	1	2	yes	—
145	#1726	C₂×C₅₃	1	1	yes	characteristic
146	C₂×C₅₃	#2293	1	1	yes	characteristic
147	C₂×C₅₃	#2950	1	1	yes	characteristic
148	C₂×C₅₃	#2951	1	1	yes	characteristic
149	C₂×C₅₃	#3175	1	2	yes	—
150	C₂×C₅₃	#3177	1	2	yes	—
151	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
152	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
153	C₄×Q₁₆	C₄×C₅₃	1	8	yes	complemented
154	C₂×#107	C₂²×C₅₃	1	2	yes	—
155	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
156	C₂×#103	C₂²×C₅₃	1	4	yes	—
157	C₂×#103	C₂²×C₅₃	1	2	yes	—
158	C₂×#97	C₂²×C₅₃	1	2	yes	—
159	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
161	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
162	#391	C₂²×C₅₃	1	4	yes	—
163	#389	C₂²×C₅₃	1	4	yes	—
164	#387	—	2	4	no	—
165	#383	—	2	2	no	—
166	#382	—	2	2	no	—
167	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
168	#376	C₂²×C₅₃	1	1	yes	characteristic
169	#340	C₂²×C₅₃	1	2	yes	—
170	#340	C₂²×C₅₃	1	4	yes	—
171	C₅₃	#21281	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
173	C₂×Q₁₆	—	2	4	no	—
174	C₂×Q₁₆	C₂×C₄×C₅₃	1	4	yes	—
175	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
177	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
178	C₄×Q₈	—	2	4	no	—
179	C₄×Q₈	C₂×C₄×C₅₃	1	8	yes	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
183	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
187	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
188	#107	—	2	4	no	—
189	#107	C₂×C₄×C₅₃	1	4	yes	—
190	#103	—	2	8	no	—
191	#103	C₂×C₄×C₅₃	1	8	yes	—
192	#97	—	2	4	no	—
193	#97	—	2	2	no	—
194	C₄×C₈	—	2	4	no	—
195	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
196	#95	—	2	2	no	—
197	#95	—	2	4	no	—
198	#95	—	2	4	no	—
199	#95	—	2	4	no	—
200	#95	D₄×C₅₃	1	1	yes	characteristic
201	#95	D₄×C₅₃	1	1	yes	characteristic
202	#95	D₄×C₅₃	1	2	yes	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	2	4	no	—
205	C₂×Q₈	—	2	4	no	—
206	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
211	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
212	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
215	Q₁₆	—	2	8	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
220	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
221	C₂×C₈	#41×C₅₃	1	4	yes	—
222	C₂×C₈	#41×C₅₃	1	4	yes	—
223	C₄⋊C₄	—	2	8	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	8	4	no	—
228	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
229	C₄²	—	2	8	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	8	2	no	—
233	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
234	C₄²	#41×C₅₃	1	2	yes	—
235	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
236	Q₈	—	2	4	no	—
237	Q₈	—	2	4	no	—
238	Q₈	—	4	4	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
257	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
258	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
260	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
261	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
262	C₈	—	2	2	no	—
263	C₈	—	2	4	no	—
264	C₈	#117×C₅₃	1	4	yes	—
265	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
266	C₂²	C₅₃×#532	1	2	yes	—
267	C₂²	C₅₃×#576	1	1	yes	characteristic
268	C₂²	C₅₃×#578	1	2	yes	—
269	C₂²	C₅₃×#578	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	4	no	—
273	C₄	—	4	4	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	2	no	—
276	C₄	—	8	4	no	—
277	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
278	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
279	C₄	C₅₃×#529	1	2	yes	—
280	C₂	C₅₃×#2293	1	1	yes	characteristic
281	C₂	C₅₃×#2950	1	1	yes	characteristic
282	C₂	C₅₃×#2951	1	1	yes	characteristic
283	C₂	C₅₃×#3175	1	2	yes	—
284	C₂	C₅₃×#3177	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
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
11. C₄×Q₁₆×C₅₃	C₄	semidirect
122. #21281	C₅₃	direct (N × H)
153. C₄×Q₁₆	C₄×C₅₃	semidirect
171. C₅₃	#21281	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×Q₁₆×C₅₃	2
C₅₃×#2320	2
C₅₃×#1969	2
C₅₃×#1942	2
C₅₃×#1935	2
C₄×#107×C₅₃	2
C₅₃×#1763	2
C₅₃×#1726	2
#21281	53

Subgroups — C53×#21281 #27228793

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21281 #27228793