Subgroups — C₅₃×#12614 #27220126

Order 13568.

426 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#504	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#350	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1764	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1379	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#504	—	2	2	no	—
10	C₅₃×#504	C₄	1	4	yes	complemented
11	C₂×M₃₂×C₅₃	—	2	1	no	—
12	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₅₃	—	2	1	no	—
15	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#104×C₅₃	C₄	1	1	yes	characteristic
18	C₅₃×#416	C₄	1	1	yes	characteristic
19	C₅₃×#405	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#350	C₄	1	4	yes	complemented
22	C₅₃×#346	—	2	1	no	—
23	C₄×C₁₆×C₅₃	—	2	2	no	—
24	C₄×C₁₆×C₅₃	—	2	1	no	—
25	C₅₃×#341	C₄	1	1	yes	characteristic
26	C₅₃×#339	C₄	1	1	yes	characteristic
27	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
28	#131×C₅₃	—	2	1	no	—
29	#131×C₅₃	—	2	1	no	—
30	#131×C₅₃	C₂×C₄	1	2	yes	—
31	#131×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₅₃	—	2	1	no	—
33	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	—	2	1	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₅₃	—	2	1	no	—
41	M₃₂×C₅₃	—	2	1	no	—
42	M₃₂×C₅₃	—	2	4	no	—
43	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
44	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₁₆×C₅₃	—	2	1	no	—
46	C₂×C₁₆×C₅₃	—	2	1	no	—
47	C₂×C₁₆×C₅₃	—	2	1	no	—
48	C₂×C₁₆×C₅₃	—	2	1	no	—
49	C₂×C₁₆×C₅₃	—	2	4	no	—
50	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
54	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
55	#104×C₅₃	C₈	1	4	yes	—
56	#103×C₅₃	C₈	1	2	yes	—
57	#102×C₅₃	C₈	1	2	yes	—
58	#97×C₅₃	—	2	1	no	—
59	C₄×C₈×C₅₃	—	2	1	no	—
60	C₄×C₈×C₅₃	—	2	1	no	—
61	C₄×C₈×C₅₃	—	2	1	no	—
62	#41×C₅₃	—	2	1	no	—
63	#41×C₅₃	—	2	1	no	—
64	#41×C₅₃	C₄²	1	2	yes	—
65	#41×C₅₃	C₂×C₈	1	2	yes	—
66	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
67	C₂×D₄×C₅₃	—	2	1	no	—
68	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	—	2	1	no	—
70	C₂²×C₄×C₅₃	—	2	1	no	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	M₁₆×C₅₃	—	2	1	no	—
73	M₁₆×C₅₃	—	2	1	no	—
74	M₁₆×C₅₃	—	2	2	no	—
75	M₁₆×C₅₃	—	2	2	no	—
76	M₁₆×C₅₃	C₄²	1	2	yes	—
77	M₁₆×C₅₃	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₅₃	—	2	1	no	—
79	C₂×C₈×C₅₃	—	2	1	no	—
80	C₂×C₈×C₅₃	—	2	1	no	—
81	C₂×C₈×C₅₃	—	2	1	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
90	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
91	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
92	C₂²⋊C₄×C₅₃	—	2	1	no	—
93	C₄²×C₅₃	—	2	1	no	—
94	C₄²×C₅₃	—	2	1	no	—
95	C₄²×C₅₃	C₂×C₈	1	2	yes	—
96	C₁₆×C₅₃	—	2	2	no	—
97	C₁₆×C₅₃	—	2	2	no	—
98	C₁₆×C₅₃	—	2	2	no	—
99	C₁₆×C₅₃	C₂²⋊C₄	1	2	yes	—
100	C₁₆×C₅₃	C₂²⋊C₄	1	2	yes	—
101	C₂³×C₅₃	—	2	1	no	—
102	C₂³×C₅₃	#131	1	1	yes	characteristic
103	Q₈×C₅₃	—	2	1	no	—
104	Q₈×C₅₃	C₄×C₈	1	2	yes	—
105	D₄×C₅₃	—	2	2	no	—
106	D₄×C₅₃	—	2	2	no	—
107	D₄×C₅₃	—	2	1	no	—
108	D₄×C₅₃	C₄×C₈	1	2	yes	—
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	—	2	1	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	—	2	2	no	—
115	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
121	C₈×C₅₃	—	2	2	no	—
122	C₈×C₅₃	—	2	2	no	—
123	C₈×C₅₃	—	2	2	no	—
124	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₈×C₅₃	#117	1	1	yes	characteristic
126	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
127	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
128	#12614	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	—	4	2	no	—
132	C₂²×C₅₃	#405	1	1	yes	characteristic
133	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
134	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
135	C₄×C₅₃	—	2	4	no	—
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	4	2	no	—
138	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
140	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
141	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
142	C₂×#504	C₂×C₅₃	1	1	yes	characteristic
143	C₂×#350	C₂×C₅₃	1	1	yes	characteristic
144	#2108	C₂×C₅₃	1	1	yes	characteristic, complemented
145	C₂×C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
146	#1764	C₂×C₅₃	1	1	yes	characteristic
147	#1379	C₂×C₅₃	1	1	yes	characteristic, complemented
148	#1376	C₂×C₅₃	1	1	yes	characteristic, complemented
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	—	2	1	no	—
151	C₂×C₅₃	—	4	2	no	—
152	C₂×C₅₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₅₃	#2170	1	2	yes	—
154	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
155	#504	—	2	2	no	—
156	#504	C₄×C₅₃	1	4	yes	complemented
157	C₂×M₃₂	—	2	1	no	—
158	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
159	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
160	C₂²×C₁₆	—	2	1	no	—
161	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
162	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
163	C₂×#104	C₄×C₅₃	1	1	yes	characteristic
164	#416	C₄×C₅₃	1	1	yes	characteristic
165	#405	C₂²×C₅₃	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
167	#350	C₄×C₅₃	1	4	yes	complemented
168	#346	—	2	1	no	—
169	C₄×C₁₆	—	2	2	no	—
170	C₄×C₁₆	—	2	1	no	—
171	#341	C₄×C₅₃	1	1	yes	characteristic
172	#339	C₄×C₅₃	1	1	yes	characteristic
173	C₅₃	#12614	1	1	yes	characteristic, direct product factor, Sylow
174	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
175	#131	—	2	1	no	—
176	#131	—	2	1	no	—
177	#131	C₂×C₄×C₅₃	1	2	yes	—
178	#131	C₂×C₄×C₅₃	1	2	yes	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
181	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
185	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
186	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	M₃₂	—	2	4	no	—
190	M₃₂	C₂×C₄×C₅₃	1	2	yes	—
191	M₃₂	C₂×C₄×C₅₃	1	2	yes	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	—	2	1	no	—
196	C₂×C₁₆	—	2	4	no	—
197	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
198	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
199	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
200	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
201	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
202	#104	C₈×C₅₃	1	4	yes	—
203	#103	C₈×C₅₃	1	2	yes	—
204	#102	C₈×C₅₃	1	2	yes	—
205	#97	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	2	1	no	—
209	#41	—	2	1	no	—
210	#41	—	2	1	no	—
211	#41	C₄²×C₅₃	1	2	yes	—
212	#41	C₂×C₈×C₅₃	1	2	yes	—
213	C₂×Q₈	C₂×C₈×C₅₃	1	1	yes	characteristic
214	C₂×D₄	—	2	1	no	—
215	C₂×D₄	C₂×C₈×C₅₃	1	1	yes	characteristic
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
219	M₁₆	—	2	1	no	—
220	M₁₆	—	2	1	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	C₄²×C₅₃	1	2	yes	—
224	M₁₆	C₂×C₈×C₅₃	1	2	yes	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
232	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
233	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
234	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
235	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
236	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
237	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
238	C₄⋊C₄	C₂×C₈×C₅₃	1	2	yes	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	1	no	—
242	C₄²	C₂×C₈×C₅₃	1	2	yes	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	—	2	2	no	—
246	C₁₆	C₂²⋊C₄×C₅₃	1	2	yes	—
247	C₁₆	C₂²⋊C₄×C₅₃	1	2	yes	—
248	C₂³	—	2	1	no	—
249	C₂³	#131×C₅₃	1	1	yes	characteristic
250	Q₈	—	2	1	no	—
251	Q₈	C₄×C₈×C₅₃	1	2	yes	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	1	no	—
255	D₄	C₄×C₈×C₅₃	1	2	yes	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
263	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
264	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
267	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	2	2	no	—
270	C₈	—	2	2	no	—
271	C₈	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
272	C₈	#117×C₅₃	1	1	yes	characteristic
273	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
274	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	C₅₃×#405	1	1	yes	characteristic
279	C₂²	C₈×D₄×C₅₃	1	1	yes	characteristic
280	C₂²	C₈×D₄×C₅₃	1	1	yes	characteristic
281	C₄	—	2	4	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	2	no	—
284	C₄	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
285	C₄	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
286	C₄	C₈×D₄×C₅₃	1	1	yes	characteristic
287	C₄	C₈×D₄×C₅₃	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	4	2	no	—
291	C₂	C₈×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
292	C₂	C₅₃×#2170	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆×C₅₃	D₄
commutator	C₄	C₂×C₄×C₈×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×#41×C₅₃	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₅₃ ◁ C₂²×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2108	C₂	semidirect
4. C₂×C₄×C₁₆×C₅₃	C₂	semidirect
6. C₅₃×#1379	C₂	semidirect
7. C₅₃×#1376	C₂	semidirect
10. C₅₃×#504	C₄	semidirect
21. C₅₃×#350	C₄	semidirect
128. #12614	C₅₃	direct (N × H)
144. #2108	C₂×C₅₃	semidirect
145. C₂×C₄×C₁₆	C₂×C₅₃	semidirect
147. #1379	C₂×C₅₃	semidirect
148. #1376	C₂×C₅₃	semidirect
156. #504	C₄×C₅₃	semidirect
167. #350	C₄×C₅₃	semidirect
173. C₅₃	#12614	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×#504	2
C₂×C₅₃×#350	2
C₅₃×#2108	2
C₂×C₄×C₁₆×C₅₃	2
C₅₃×#1764	2
C₅₃×#1379	2
C₅₃×#1376	2
#12614	53

Subgroups — C53×#12614 #27220126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12614 #27220126