Subgroups — C₅₃×#17895 #27225407

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2975	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1594	C₂	1	4	yes	maximal, complemented
4	C₅₃×#1589	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1478	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
10	C₄×D₈×C₅₃	C₄	1	2	yes	—
11	C₄×D₈×C₅₃	C₄	1	2	yes	—
12	C₄×D₈×C₅₃	C₂²	1	2	yes	—
13	C₄×D₈×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#436	—	2	4	no	—
15	C₅₃×#436	C₂²	1	8	yes	complemented
16	C₅₃×#431	C₂²	1	2	yes	—
17	C₅₃×#431	C₂²	1	1	yes	characteristic
18	C₅₃×#431	C₂²	1	1	yes	characteristic
19	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
20	C₂×#102×C₅₃	C₄	1	2	yes	—
21	C₅₃×#407	—	2	4	no	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#335	C₂²	1	2	yes	complemented
24	C₅₃×#334	C₂²	1	2	yes	complemented
25	C₅₃×D₄⋊C₈	C₂²	1	8	yes	complemented
26	C₅₃×#322	C₂²	1	2	yes	complemented
27	C₈²×C₅₃	C₂²	1	2	yes	complemented
28	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₅₃	—	2	2	no	—
30	C₂×D₈×C₅₃	—	2	4	no	—
31	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×M₁₆×C₅₃	—	2	8	no	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×D₄×C₅₃	—	2	2	no	—
37	C₄×D₄×C₅₃	—	2	2	no	—
38	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
39	C₄×D₄×C₅₃	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	#107×C₅₃	C₂×C₄	1	2	yes	—
44	#107×C₅₃	C₂×C₄	1	2	yes	—
45	C₄⋊C₈×C₅₃	—	2	1	no	—
46	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
47	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
48	#102×C₅₃	C₂×C₄	1	4	yes	—
49	#102×C₅₃	C₂×C₄	1	4	yes	—
50	C₂²⋊C₈×C₅₃	—	2	8	no	—
51	C₂²⋊C₈×C₅₃	—	2	2	no	—
52	C₂²⋊C₈×C₅₃	—	2	1	no	—
53	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
54	C₄×C₈×C₅₃	—	2	2	no	—
55	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₅₃	C₂³	1	2	yes	—
58	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₅₃	C₂³	1	2	yes	—
61	C₂⁴×C₅₃	—	2	2	no	—
62	C₂×D₄×C₅₃	—	2	2	no	—
63	C₂×D₄×C₅₃	—	2	4	no	—
64	C₂×D₄×C₅₃	—	2	4	no	—
65	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	—	2	4	no	—
68	C₂²×C₄×C₅₃	—	2	4	no	—
69	C₂²×C₄×C₅₃	—	8	4	no	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
73	D₈×C₅₃	—	2	4	no	—
74	D₈×C₅₃	—	2	4	no	—
75	M₁₆×C₅₃	—	8	8	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	4	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	M₁₆	1	4	yes	complemented
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
85	C₄⋊C₄×C₅₃	—	2	2	no	—
86	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
87	C₂²⋊C₄×C₅₃	—	2	4	no	—
88	C₂²⋊C₄×C₅₃	—	2	4	no	—
89	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₃	#41	1	1	yes	characteristic
92	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₂³×C₅₃	—	2	4	no	—
94	C₂³×C₅₃	—	4	4	no	—
95	C₂³×C₅₃	—	8	2	no	—
96	C₂³×C₅₃	—	8	2	no	—
97	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
98	D₄×C₅₃	—	2	4	no	—
99	D₄×C₅₃	—	2	4	no	—
100	D₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	4	1	no	—
104	C₂×C₄×C₅₃	—	4	8	no	—
105	C₂×C₄×C₅₃	—	8	4	no	—
106	C₂×C₄×C₅₃	—	8	2	no	—
107	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₅₃	#131	1	2	yes	—
112	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
115	C₈×C₅₃	—	2	1	no	—
116	C₈×C₅₃	—	2	1	no	—
117	C₈×C₅₃	—	2	4	no	—
118	C₈×C₅₃	—	4	2	no	—
119	C₈×C₅₃	—	4	4	no	—
120	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
121	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
122	#17895	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	4	4	no	—
126	C₂²×C₅₃	—	4	4	no	—
127	C₂²×C₅₃	—	4	4	no	—
128	C₂²×C₅₃	—	8	8	no	—
129	C₂²×C₅₃	#436	1	2	yes	—
130	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	4	4	no	—
135	C₄×C₅₃	#436	1	2	yes	—
136	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
137	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
138	C₄×C₅₃	#576	1	2	yes	—
139	#2975	C₂×C₅₃	1	2	yes	complemented
140	C₂×C₄×D₈	C₂×C₅₃	1	1	yes	characteristic
141	#1594	C₂×C₅₃	1	4	yes	complemented
142	#1589	C₂×C₅₃	1	4	yes	complemented
143	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
144	#1478	C₂×C₅₃	1	2	yes	complemented
145	#1450	C₂×C₅₃	1	1	yes	characteristic, complemented
146	C₂×C₅₃	—	2	2	no	—
147	C₂×C₅₃	—	8	4	no	—
148	C₂×C₅₃	C₂×#436	1	1	yes	characteristic
149	C₂×C₅₃	#2948	1	1	yes	characteristic
150	C₂×C₅₃	#2957	1	1	yes	characteristic
151	C₂²×D₈	C₄×C₅₃	1	1	yes	characteristic
152	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
153	C₄×D₈	C₄×C₅₃	1	2	yes	—
154	C₄×D₈	C₄×C₅₃	1	2	yes	—
155	C₄×D₈	C₂²×C₅₃	1	2	yes	—
156	C₄×D₈	C₂²×C₅₃	1	2	yes	—
157	#436	—	2	4	no	—
158	#436	C₂²×C₅₃	1	8	yes	complemented
159	#431	C₂²×C₅₃	1	2	yes	—
160	#431	C₂²×C₅₃	1	1	yes	characteristic
161	#431	C₂²×C₅₃	1	1	yes	characteristic
162	C₂×#107	C₄×C₅₃	1	1	yes	characteristic
163	C₂×#102	C₄×C₅₃	1	2	yes	—
164	#407	—	2	4	no	—
165	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
166	#335	C₂²×C₅₃	1	2	yes	complemented
167	#334	C₂²×C₅₃	1	2	yes	complemented
168	D₄⋊C₈	C₂²×C₅₃	1	8	yes	complemented
169	#322	C₂²×C₅₃	1	2	yes	complemented
170	C₈²	C₂²×C₅₃	1	2	yes	complemented
171	C₅₃	#17895	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
173	C₂³×C₄	—	2	2	no	—
174	C₂×D₈	—	2	4	no	—
175	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
176	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
177	C₂×M₁₆	—	2	8	no	—
178	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
179	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
180	C₄×D₄	—	2	2	no	—
181	C₄×D₄	—	2	2	no	—
182	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
183	C₄×D₄	C₂³×C₅₃	1	4	yes	—
184	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
187	#107	C₂×C₄×C₅₃	1	2	yes	—
188	#107	C₂×C₄×C₅₃	1	2	yes	—
189	C₄⋊C₈	—	2	1	no	—
190	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
191	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
192	#102	C₂×C₄×C₅₃	1	4	yes	—
193	#102	C₂×C₄×C₅₃	1	4	yes	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	2	1	no	—
196	C₂²⋊C₈	—	2	8	no	—
197	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
200	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
201	C₄×C₈	C₂³×C₅₃	1	2	yes	—
202	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
203	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
204	C₄×C₈	C₂³×C₅₃	1	2	yes	—
205	C₂⁴	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	8	4	no	—
214	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
216	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
217	D₈	—	2	4	no	—
218	D₈	—	2	4	no	—
219	M₁₆	—	8	8	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	M₁₆×C₅₃	1	4	yes	complemented
224	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
226	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
234	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
235	C₄²	#41×C₅₃	1	1	yes	characteristic
236	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
237	C₂³	—	2	4	no	—
238	C₂³	—	4	4	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
242	D₄	—	2	4	no	—
243	D₄	—	2	4	no	—
244	D₄	—	4	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	8	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	—	8	2	no	—
251	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
252	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
253	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
255	C₂×C₄	#131×C₅₃	1	2	yes	—
256	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
257	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
258	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	4	no	—
262	C₈	—	4	2	no	—
263	C₈	—	4	4	no	—
264	C₈	C₂×M₁₆×C₅₃	1	2	yes	—
265	C₈	C₂×M₁₆×C₅₃	1	2	yes	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	8	8	no	—
272	C₂²	C₅₃×#436	1	2	yes	—
273	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	4	4	no	—
278	C₄	C₅₃×#436	1	2	yes	—
279	C₄	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
280	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
281	C₄	C₅₃×#576	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	8	4	no	—
284	C₂	C₂×C₅₃×#436	1	1	yes	characteristic
285	C₂	C₅₃×#2948	1	1	yes	characteristic
286	C₂	C₅₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₈×C₅₃	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₅₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#2975	C₂	semidirect
3. C₅₃×#1594	C₂	semidirect
4. C₅₃×#1589	C₂	semidirect
5. C₅₃×#1564	C₂	semidirect
6. C₅₃×#1478	C₂	semidirect
7. C₅₃×#1450	C₂	semidirect
15. C₅₃×#436	C₂²	semidirect
23. C₅₃×#335	C₂²	semidirect
24. C₅₃×#334	C₂²	semidirect
25. C₅₃×D₄⋊C₈	C₂²	semidirect
26. C₅₃×#322	C₂²	semidirect
27. C₈²×C₅₃	C₂²	semidirect
79. C₂×C₈×C₅₃	M₁₆	semidirect
122. #17895	C₅₃	direct (N × H)
139. #2975	C₂×C₅₃	semidirect
141. #1594	C₂×C₅₃	semidirect
142. #1589	C₂×C₅₃	semidirect
143. #1564	C₂×C₅₃	semidirect
144. #1478	C₂×C₅₃	semidirect
145. #1450	C₂×C₅₃	semidirect
158. #436	C₂²×C₅₃	semidirect
166. #335	C₂²×C₅₃	semidirect
167. #334	C₂²×C₅₃	semidirect
168. D₄⋊C₈	C₂²×C₅₃	semidirect
169. #322	C₂²×C₅₃	semidirect
170. C₈²	C₂²×C₅₃	semidirect
171. C₅₃	#17895	direct (N × H)
223. C₂×C₈	M₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2975	2
C₂×C₄×D₈×C₅₃	2
C₅₃×#1594	2
C₅₃×#1589	2
C₅₃×#1564	2
C₅₃×#1478	2
C₅₃×#1450	2
#17895	53

Subgroups — C53×#17895 #27225407

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17895 #27225407