Subgroups — C₅₃×#10866 #27218378

Order 13568.

746 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#427	C₂	1	2	yes	maximal
2	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1740	C₂	1	2	yes	maximal
4	C₂×C₅₃×H(3,ℤ/4)	C₂	1	2	yes	maximal
5	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#117×C₅₃	C₂²	1	4	yes	—
8	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#427	—	2	4	no	—
10	C₅₃×#427	C₄	1	8	yes	complemented
11	C₂×#107×C₅₃	—	2	2	no	—
12	C₂×#106×C₅₃	—	2	2	no	—
13	C₅₃×#405	—	2	2	no	—
14	C₅₃×#405	—	2	2	no	—
15	C₂×#97×C₅₃	C₄	1	2	yes	complemented
16	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	complemented
17	C₅₃×#342	—	2	4	no	—
18	C₅₃×H(3,ℤ/4)	—	2	4	no	—
19	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₅₃	—	2	2	no	—
21	C₂×M₁₆×C₅₃	—	2	2	no	—
22	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
23	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₅₃	D₄	1	2	yes	—
29	C₂²×C₈×C₅₃	D₄	1	2	yes	—
30	C₂²×C₈×C₅₃	D₄	1	2	yes	—
31	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
32	#117×C₅₃	—	2	2	no	—
33	#117×C₅₃	—	2	4	no	—
34	#117×C₅₃	—	2	4	no	—
35	#117×C₅₃	—	2	2	no	—
36	#117×C₅₃	—	2	4	no	—
37	#117×C₅₃	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
40	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
41	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
42	C₂×C₄²×C₅₃	—	2	4	no	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
44	#107×C₅₃	—	2	8	no	—
45	#106×C₅₃	—	2	8	no	—
46	#97×C₅₃	—	2	4	no	—
47	C₄×C₈×C₅₃	—	2	2	no	—
48	C₄×C₈×C₅₃	—	2	2	no	—
49	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	—	4	4	no	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	#41	1	2	yes	—
61	C₂²×C₄×C₅₃	#41	1	2	yes	—
62	M₁₆×C₅₃	—	2	4	no	—
63	M₁₆×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	—	2	2	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	C₄²	1	2	yes	—
67	C₂×C₈×C₅₃	C₄²	1	2	yes	—
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
73	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
74	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
77	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
78	C₄⋊C₄×C₅₃	—	2	4	no	—
79	C₄⋊C₄×C₅₃	—	2	8	no	—
80	C₄⋊C₄×C₅₃	—	4	4	no	—
81	C₂²⋊C₄×C₅₃	—	2	4	no	—
82	C₂²⋊C₄×C₅₃	—	2	8	no	—
83	C₂²⋊C₄×C₅₃	—	4	4	no	—
84	C₄²×C₅₃	—	2	2	no	—
85	C₄²×C₅₃	—	2	2	no	—
86	C₄²×C₅₃	—	2	8	no	—
87	C₄²×C₅₃	—	4	4	no	—
88	C₂³×C₅₃	—	4	2	no	—
89	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂³×C₅₃	C₄×D₄	1	2	yes	—
91	C₂³×C₅₃	#121	1	2	yes	—
92	C₂³×C₅₃	#127	1	1	yes	characteristic
93	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	—	2	4	no	—
95	C₂×C₄×C₅₃	—	4	4	no	—
96	C₂×C₄×C₅₃	—	4	8	no	—
97	C₂×C₄×C₅₃	—	8	8	no	—
98	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
109	C₂×C₄×C₅₃	#121	1	2	yes	—
110	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₅₃	#124	1	2	yes	—
113	C₂×C₄×C₅₃	#125	1	2	yes	—
114	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
116	C₈×C₅₃	—	2	4	no	—
117	C₈×C₅₃	#95	1	2	yes	—
118	C₈×C₅₃	#95	1	2	yes	—
119	C₈×C₅₃	#95	1	4	yes	—
120	#10866	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	—	2	4	no	—
123	C₂²×C₅₃	—	4	2	no	—
124	C₂²×C₅₃	—	4	1	no	—
125	C₂²×C₅₃	—	4	1	no	—
126	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₅₃	#380	1	1	yes	characteristic
128	C₂²×C₅₃	#384	1	1	yes	characteristic
129	C₂²×C₅₃	#389	1	2	yes	—
130	C₂²×C₅₃	#390	1	2	yes	—
131	C₄×C₅₃	—	4	4	no	—
132	C₄×C₅₃	—	8	8	no	—
133	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
134	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
135	C₄×C₅₃	#380	1	1	yes	characteristic
136	C₄×C₅₃	#382	1	1	yes	characteristic
137	C₄×C₅₃	#389	1	2	yes	—
138	C₄×C₅₃	#391	1	2	yes	—
139	C₂×#427	C₂×C₅₃	1	2	yes	—
140	C₂×#405	C₂×C₅₃	1	1	yes	characteristic
141	#1740	C₂×C₅₃	1	2	yes	—
142	C₂×H(3,ℤ/4)	C₂×C₅₃	1	2	yes	—
143	C₂×C₅₃	—	2	2	no	—
144	C₂×C₅₃	—	2	1	no	—
145	C₂×C₅₃	—	2	1	no	—
146	C₂×C₅₃	—	2	2	no	—
147	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
148	C₂×C₅₃	#1812	1	2	yes	—
149	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
150	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
151	C₂×#117	C₂²×C₅₃	1	4	yes	—
152	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
153	#427	—	2	4	no	—
154	#427	C₄×C₅₃	1	8	yes	complemented
155	C₂×#107	—	2	2	no	—
156	C₂×#106	—	2	2	no	—
157	#405	—	2	2	no	—
158	#405	—	2	2	no	—
159	C₂×#97	C₄×C₅₃	1	2	yes	complemented
160	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	complemented
161	#342	—	2	4	no	—
162	H(3,ℤ/4)	—	2	4	no	—
163	C₅₃	#10866	1	1	yes	characteristic, direct product factor, Sylow
164	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
168	C₂×M₁₆	Q₈×C₅₃	1	2	yes	—
169	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
170	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
171	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
172	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
173	C₂²×C₈	D₄×C₅₃	1	2	yes	—
174	C₂²×C₈	D₄×C₅₃	1	2	yes	—
175	C₂²×C₈	D₄×C₅₃	1	2	yes	—
176	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
177	#117	—	2	2	no	—
178	#117	—	2	2	no	—
179	#117	—	2	4	no	—
180	#117	—	2	4	no	—
181	#117	—	2	4	no	—
182	#117	C₂×C₄×C₅₃	1	8	yes	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	Q₈×C₅₃	1	2	yes	—
185	C₂×C₂²⋊C₄	—	2	4	no	—
186	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
187	C₂×C₄²	—	2	4	no	—
188	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
189	#107	—	2	8	no	—
190	#106	—	2	8	no	—
191	#97	—	2	4	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	—	2	2	no	—
194	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
195	C₂²×C₄	—	4	4	no	—
196	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
197	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
198	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
199	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
200	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
205	C₂²×C₄	#41×C₅₃	1	2	yes	—
206	C₂²×C₄	#41×C₅₃	1	2	yes	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₄²×C₅₃	1	2	yes	—
212	C₂×C₈	C₄²×C₅₃	1	2	yes	—
213	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
214	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
215	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
216	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
217	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
218	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
219	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
220	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
221	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
222	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	8	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	2	8	no	—
228	C₂²⋊C₄	—	4	4	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	8	no	—
232	C₄²	—	4	4	no	—
233	C₂³	—	4	2	no	—
234	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
235	C₂³	C₄×D₄×C₅₃	1	2	yes	—
236	C₂³	#121×C₅₃	1	2	yes	—
237	C₂³	#127×C₅₃	1	1	yes	characteristic
238	C₂³	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	—	8	8	no	—
243	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
244	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
245	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
246	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
247	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
248	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
249	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
250	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
251	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
252	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
253	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
254	C₂×C₄	#121×C₅₃	1	2	yes	—
255	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
256	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
257	C₂×C₄	#124×C₅₃	1	2	yes	—
258	C₂×C₄	#125×C₅₃	1	2	yes	—
259	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
260	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
261	C₈	—	2	4	no	—
262	C₈	#95×C₅₃	1	2	yes	—
263	C₈	#95×C₅₃	1	2	yes	—
264	C₈	#95×C₅₃	1	4	yes	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
271	C₂²	C₅₃×#380	1	1	yes	characteristic
272	C₂²	C₅₃×#384	1	1	yes	characteristic
273	C₂²	C₅₃×#389	1	2	yes	—
274	C₂²	C₅₃×#390	1	2	yes	—
275	C₄	—	4	4	no	—
276	C₄	—	8	8	no	—
277	C₄	C₂×#95×C₅₃	1	1	yes	characteristic
278	C₄	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
279	C₄	C₅₃×#380	1	1	yes	characteristic
280	C₄	C₅₃×#382	1	1	yes	characteristic
281	C₄	C₅₃×#389	1	2	yes	—
282	C₄	C₅₃×#391	1	2	yes	—
283	C₂	—	2	2	no	—
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	2	no	—
287	C₂	C₅₃×C₄⋊#95	1	1	yes	characteristic
288	C₂	C₅₃×#1812	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	#117
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂²×C₅₃	#380
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×M₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₅₃×#427	C₄	semidirect
15. C₂×#97×C₅₃	C₄	semidirect
16. C₂×C₄×C₈×C₅₃	C₄	semidirect
120. #10866	C₅₃	direct (N × H)
154. #427	C₄×C₅₃	semidirect
159. C₂×#97	C₄×C₅₃	semidirect
160. C₂×C₄×C₈	C₄×C₅₃	semidirect
163. C₅₃	#10866	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×#427	2
C₂×C₅₃×#405	2
C₅₃×#1740	2
C₂×C₅₃×H(3,ℤ/4)	2
#10866	53

Subgroups — C53×#10866 #27218378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10866 #27218378