Subgroups — #27299175

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810402	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810399	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16810398	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×#11719921	C₂	1	1	yes	maximal, characteristic
6	#20272792	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	C₂²×#11719908	C₂²	1	1	yes	characteristic
10	C₂²×#11719908	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×D₅₃	—	2	2	no	—
12	C₂×C₁₆×D₅₃	—	2	2	no	—
13	C₂×C₁₆×D₅₃	C₄	1	4	yes	complemented
14	C₂×#13133129	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₄×#11719921	C₄	1	1	yes	characteristic
17	#16810402	C₄	1	4	yes	complemented
18	#16810399	—	2	2	no	—
19	#16810398	—	2	2	no	—
20	#16810378	C₄	1	1	yes	characteristic
21	#16810377	C₄	1	1	yes	characteristic
22	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₅₃	—	2	1	no	—
29	C₂×C₈×D₅₃	—	2	1	no	—
30	C₂×C₈×D₅₃	—	2	1	no	—
31	C₂×C₈×D₅₃	—	2	1	no	—
32	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
34	C₂×#11719921	C₈	1	2	yes	—
35	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×#11719908	—	2	1	no	—
37	C₂×#11719908	—	2	1	no	—
38	C₂×#11719908	C₂×C₄	1	2	yes	—
39	C₂×#11719908	C₂×C₄	1	2	yes	—
40	C₂×#11719908	D₄	1	2	yes	—
41	C₂×#11719908	D₄	1	2	yes	—
42	C₁₆×D₅₃	—	2	8	no	—
43	#13133129	C₈	1	4	yes	—
44	C₈×Dic₅₃	—	2	1	no	—
45	C₈×Dic₅₃	—	2	1	no	—
46	#13133124	C₈	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₁₆×C₅₃	—	2	1	no	—
51	C₂×C₁₆×C₅₃	—	2	1	no	—
52	C₂×C₁₆×C₅₃	—	2	1	no	—
53	C₂×C₁₆×C₅₃	—	2	1	no	—
54	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₅₃	—	2	1	no	—
57	C₄×C₈×C₅₃	—	2	1	no	—
58	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
62	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
63	C₂×C₄×D₅₃	—	2	1	no	—
64	C₂×C₄×D₅₃	—	2	1	no	—
65	C₂×C₄×D₅₃	—	2	1	no	—
66	C₂×C₄×D₅₃	—	2	1	no	—
67	C₂×C₄×D₅₃	C₄²	1	2	yes	—
68	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	—	2	1	no	—
71	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
77	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
78	C₄²×C₅₃	—	2	1	no	—
79	C₄²×C₅₃	—	2	1	no	—
80	#11719921	—	2	4	no	—
81	C₄×Dic₅₃	—	2	1	no	—
82	C₄×Dic₅₃	—	2	1	no	—
83	C₂×#88322	—	2	1	no	—
84	C₂×#88322	—	2	1	no	—
85	C₂×#88322	—	2	1	no	—
86	C₂×#88322	—	2	1	no	—
87	C₂×#88322	C₄²	1	2	yes	—
88	C₂×#88322	C₂×C₈	1	2	yes	—
89	C₈×D₅₃	—	2	2	no	—
90	C₈×D₅₃	—	2	2	no	—
91	C₈×D₅₃	—	2	2	no	—
92	C₈×D₅₃	—	2	2	no	—
93	C₁₆×C₅₃	—	2	2	no	—
94	C₁₆×C₅₃	—	2	2	no	—
95	#11719908	—	2	2	no	—
96	#11719908	C₂²⋊C₄	1	4	yes	complemented
97	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
98	C₂²×D₅₃	—	2	1	no	—
99	C₂²×D₅₃	—	2	1	no	—
100	C₂²×D₅₃	—	2	2	no	—
101	C₂²×D₅₃	C₄×C₈	1	2	yes	—
102	C₂²×D₅₃	M₃₂	1	4	yes	—
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	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
112	C₂×Dic₅₃	—	2	1	no	—
113	C₂×Dic₅₃	—	2	1	no	—
114	C₂×Dic₅₃	—	2	2	no	—
115	C₂×Dic₅₃	—	2	2	no	—
116	C₂×Dic₅₃	—	2	2	no	—
117	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
118	C₄×D₅₃	—	2	2	no	—
119	C₄×D₅₃	—	2	2	no	—
120	C₄×D₅₃	—	2	2	no	—
121	C₄×D₅₃	—	2	2	no	—
122	C₈×C₅₃	—	2	2	no	—
123	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₈×C₅₃	#117	1	1	yes	characteristic
125	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
126	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
127	#88322	—	2	2	no	—
128	#88322	—	2	2	no	—
129	#12611	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
131	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
132	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
134	C₂²×C₅₃	#504	1	2	yes	—
135	C₂×D₅₃	—	2	2	no	—
136	C₂×D₅₃	—	2	4	no	—
137	C₂×D₅₃	—	2	4	no	—
138	C₂×D₅₃	—	2	2	no	—
139	C₂×D₅₃	#346	1	4	yes	complemented
140	C₄×C₅₃	—	4	2	no	—
141	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₅₃	#405	1	1	yes	characteristic
143	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
144	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
145	Dic₅₃	—	2	2	no	—
146	Dic₅₃	—	4	2	no	—
147	C₂³×C₁₆	—	53	1	no	—
148	C₂×C₂²⋊C₁₆	—	53	1	no	—
149	C₂×#346	—	53	1	no	—
150	C₂×#346	—	53	1	no	—
151	C₈×C₂²⋊C₄	—	53	1	no	—
152	#1374	—	53	1	no	—
153	#1374	D₅₃	1	1	yes	characteristic, complemented
154	C₂×C₅₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
156	C₂×C₅₃	#2108	1	1	yes	characteristic
157	C₂×C₅₃	#2168	1	2	yes	—
158	C₂×C₅₃	#2168	1	2	yes	—
159	D₅₃	—	2	4	no	—
160	C₂³×C₈	—	53	1	no	—
161	C₂²×C₁₆	—	53	1	no	—
162	C₂²×C₁₆	—	53	1	no	—
163	C₂²×C₁₆	—	53	4	no	—
164	C₂²×C₁₆	—	106	2	no	—
165	C₂²×C₁₆	—	106	2	no	—
166	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
167	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
168	C₂×C₂²⋊C₈	—	53	1	no	—
169	C₂×C₄×C₈	—	53	1	no	—
170	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
171	C₄×C₂²⋊C₄	—	53	1	no	—
172	C₂²⋊C₁₆	—	53	4	no	—
173	#346	—	106	2	no	—
174	#346	—	106	2	no	—
175	#336	—	53	1	no	—
176	#336	—	53	1	no	—
177	C₅₃	#12611	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₄	—	53	1	no	—
179	C₂²×C₈	—	53	1	no	—
180	C₂²×C₈	—	53	1	no	—
181	C₂²×C₈	—	53	2	no	—
182	C₂²×C₈	—	53	2	no	—
183	C₂²×C₈	—	106	1	no	—
184	C₂²×C₈	—	106	1	no	—
185	C₂²×C₈	—	106	1	no	—
186	C₂²×C₈	—	106	1	no	—
187	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
189	C₂×C₂²⋊C₄	—	53	2	no	—
190	C₂×C₄²	—	53	1	no	—
191	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
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	—	53	2	no	—
197	C₂×C₁₆	—	53	2	no	—
198	C₂×C₁₆	—	53	2	no	—
199	C₂×C₁₆	—	53	2	no	—
200	C₂×C₁₆	—	106	8	no	—
201	C₂×C₁₆	—	106	1	no	—
202	C₂×C₁₆	—	106	1	no	—
203	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
204	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
205	C₂²⋊C₈	—	53	4	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	106	1	no	—
209	C₄×C₈	—	106	1	no	—
210	#95	—	53	2	no	—
211	C₂⁴	—	53	1	no	—
212	C₂²×C₄	—	53	1	no	—
213	C₂²×C₄	—	53	2	no	—
214	C₂²×C₄	—	53	2	no	—
215	C₂²×C₄	—	53	2	no	—
216	C₂²×C₄	—	106	1	no	—
217	C₂²×C₄	—	106	1	no	—
218	C₂²×C₄	—	106	1	no	—
219	C₂²×C₄	—	106	1	no	—
220	C₂²×C₄	C₈×D₅₃	1	2	yes	—
221	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	53	2	no	—
225	C₂×C₈	—	53	2	no	—
226	C₂×C₈	—	106	1	no	—
227	C₂×C₈	—	106	1	no	—
228	C₂×C₈	—	106	1	no	—
229	C₂×C₈	—	106	1	no	—
230	C₂×C₈	—	106	2	no	—
231	C₂×C₈	—	106	2	no	—
232	C₂×C₈	—	106	2	no	—
233	C₂×C₈	—	106	2	no	—
234	C₂×C₈	C₈×D₅₃	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
237	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
238	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
239	C₂×C₈	#11719947	1	1	yes	characteristic
240	C₂×C₈	#11719949	1	1	yes	characteristic
241	C₂²⋊C₄	—	106	4	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	106	1	no	—
245	C₄²	—	106	1	no	—
246	C₁₆	—	2	2	no	—
247	C₁₆	—	2	2	no	—
248	C₁₆	—	53	4	no	—
249	C₁₆	—	106	2	no	—
250	C₂³	—	53	4	no	—
251	C₂³	—	53	2	no	—
252	C₂³	—	106	2	no	—
253	C₂³	—	106	1	no	—
254	C₂³	—	106	1	no	—
255	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	53	2	no	—
260	C₂×C₄	—	106	1	no	—
261	C₂×C₄	—	106	1	no	—
262	C₂×C₄	—	106	2	no	—
263	C₂×C₄	—	106	2	no	—
264	C₂×C₄	—	106	2	no	—
265	C₂×C₄	—	106	2	no	—
266	C₂×C₄	—	106	2	no	—
267	C₂×C₄	—	106	2	no	—
268	C₂×C₄	—	106	2	no	—
269	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
270	C₂×C₄	#13133177	1	1	yes	characteristic
271	C₂×C₄	#13133179	1	1	yes	characteristic
272	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
273	C₂×C₄	#13133225	1	1	yes	characteristic
274	C₂×C₄	#13133225	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	106	2	no	—
277	C₈	—	106	2	no	—
278	C₈	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
279	C₈	#13133175	1	1	yes	characteristic
280	C₈	#13133177	1	1	yes	characteristic
281	C₈	#13133179	1	1	yes	characteristic
282	C₂²	—	53	4	no	—
283	C₂²	—	106	2	no	—
284	C₂²	—	106	4	no	—
285	C₂²	—	106	4	no	—
286	C₂²	—	106	2	no	—
287	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
288	C₂²	#16810686	1	1	yes	characteristic
289	C₂²	#16810688	1	1	yes	characteristic
290	C₂²	M₃₂×D₅₃	1	2	yes	—
291	C₂²	#16811039	1	2	yes	—
292	C₄	—	4	2	no	—
293	C₄	—	106	2	no	—
294	C₄	—	212	2	no	—
295	C₄	#16810470	1	1	yes	characteristic
296	C₄	#16810580	1	1	yes	characteristic
297	C₄	#16810686	1	1	yes	characteristic
298	C₄	#16810688	1	1	yes	characteristic
299	C₂	—	106	4	no	—
300	C₂	#20274485	1	1	yes	characteristic
301	C₂	#346×D₅₃	1	1	yes	characteristic
302	C₂	#20276310	1	1	yes	characteristic
303	C₂	#20276539	1	2	yes	—
304	C₂	#20276543	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×#1374	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1374	D₅₃
hypocenter	C₅₃	#12611
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
3. C₂×#16810399	C₂	semidirect
4. C₂×#16810398	C₂	semidirect
6. #20272792	C₂	semidirect
7. C₅₃×#1374	C₂	semidirect
13. C₂×C₁₆×D₅₃	C₄	semidirect
17. #16810402	C₄	semidirect
96. #11719908	C₂²⋊C₄	semidirect
139. C₂×D₅₃	#346	semidirect
153. #1374	D₅₃	semidirect
177. C₅₃	#12611	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₅₃	2
C₂×#16810402	2
C₂×#16810399	2
C₂×#16810398	2
C₈×#11719921	2
#20272792	2
C₅₃×#1374	2
#12611	53