Subgroups — #27292535

Order 13568.

578 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283279	C₂	1	1	yes	maximal, characteristic
2	#20274442	C₂	1	2	yes	maximal
3	C₂×#16810387	C₂	1	2	yes	maximal
4	#20274412	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1742	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂²×#11719917	C₂²	1	1	yes	characteristic
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	#16810650	—	2	4	no	—
10	#16810647	—	2	2	no	—
11	#16810388	—	2	4	no	—
12	#16810387	—	2	4	no	—
13	#16810378	—	2	2	no	—
14	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#343	C₄	1	4	yes	—
17	C₂×#11719917	—	2	2	no	—
18	C₂×#11719917	—	2	2	no	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	—	2	1	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	D₄	1	4	yes	—
26	C₂×#11719917	Q₈	1	2	yes	—
27	C₂²×#88322	—	2	1	no	—
28	C₂²×#88322	—	2	1	no	—
29	C₂²×#88322	—	2	2	no	—
30	C₂²×#88322	—	2	2	no	—
31	C₂²×#88322	—	2	2	no	—
32	C₂²×#88322	—	2	1	no	—
33	C₂²×#88322	D₄	1	2	yes	—
34	C₄⋊#88322	—	2	8	no	—
35	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₅₃	—	2	2	no	—
37	C₂×M₁₆×C₅₃	—	2	2	no	—
38	C₂×M₁₆×C₅₃	—	2	2	no	—
39	C₂×M₁₆×C₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	—	2	2	no	—
41	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₅₃	—	2	2	no	—
43	#117×C₅₃	—	2	2	no	—
44	#117×C₅₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₅₃	Q₈	1	2	yes	—
47	C₂×C₄²×C₅₃	D₄	1	2	yes	—
48	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	#41	1	2	yes	—
56	C₂²×C₄×C₅₃	#41	1	2	yes	—
57	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	#41	1	2	yes	—
59	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
60	M₁₆×C₅₃	—	2	4	no	—
61	M₁₆×C₅₃	—	2	4	no	—
62	M₁₆×C₅₃	—	4	2	no	—
63	M₁₆×C₅₃	—	4	2	no	—
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	4	2	no	—
66	C₂×C₈×C₅₃	—	4	2	no	—
67	C₄⋊C₄×C₅₃	—	2	4	no	—
68	C₂²⋊C₄×C₅₃	—	2	4	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	#11719917	—	2	4	no	—
72	#11719917	—	2	4	no	—
73	#11719917	—	2	4	no	—
74	#11719917	—	4	2	no	—
75	#11719917	—	4	2	no	—
76	#11719917	—	4	2	no	—
77	C₂×#88322	—	2	4	no	—
78	C₂×#88322	—	2	2	no	—
79	C₂×#88322	—	2	2	no	—
80	C₂×#88322	—	2	4	no	—
81	C₂×#88322	—	4	1	no	—
82	C₂×#88322	—	4	2	no	—
83	C₂×#88322	—	4	1	no	—
84	C₂×#88322	—	4	2	no	—
85	C₂×#88322	—	4	1	no	—
86	C₂×#88322	—	4	1	no	—
87	C₂×#88322	—	4	2	no	—
88	C₂³×C₅₃	—	4	2	no	—
89	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
91	C₂³×C₅₃	#125	1	1	yes	characteristic
92	C₂³×C₅₃	#125	1	2	yes	—
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	#117	1	2	yes	—
97	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#121	1	2	yes	—
102	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#123	1	2	yes	—
106	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#123	1	2	yes	—
108	C₂×C₄×C₅₃	#124	1	2	yes	—
109	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#125	1	2	yes	—
112	C₂×C₄×C₅₃	#126	1	2	yes	—
113	C₂×C₄×C₅₃	#126	1	2	yes	—
114	C₈×C₅₃	—	4	4	no	—
115	#88322	—	4	2	no	—
116	#88322	—	4	4	no	—
117	#88322	—	4	2	no	—
118	#11297	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	2	4	no	—
121	C₂²×C₅₃	—	4	1	no	—
122	C₂²×C₅₃	—	4	2	no	—
123	C₂²×C₅₃	—	4	1	no	—
124	C₂²×C₅₃	#382	1	1	yes	characteristic
125	C₂²×C₅₃	#385	1	1	yes	characteristic
126	C₂²×C₅₃	#389	1	1	yes	characteristic
127	C₂²×C₅₃	#398	1	2	yes	—
128	C₂²×C₅₃	#400	1	2	yes	—
129	C₄×C₅₃	—	4	4	no	—
130	C₄×C₅₃	#387	1	2	yes	—
131	C₄×C₅₃	#388	1	2	yes	—
132	C₄×C₅₃	#393	1	1	yes	characteristic
133	C₄×C₅₃	#394	1	1	yes	characteristic
134	C₄×C₅₃	#397	1	1	yes	characteristic
135	C₄×C₅₃	#400	1	1	yes	characteristic
136	#2905	—	53	1	no	—
137	#1742	D₅₃	1	1	yes	characteristic
138	#1738	—	53	2	no	—
139	C₂×#341	—	53	2	no	—
140	#1729	—	53	1	no	—
141	C₂×C₅₃	—	2	1	no	—
142	C₂×C₅₃	—	2	1	no	—
143	C₂×C₅₃	—	2	2	no	—
144	C₂×C₅₃	—	2	2	no	—
145	C₂×C₅₃	#1442	1	1	yes	characteristic
146	C₂×C₅₃	#1858	1	2	yes	—
147	C₂²×M₁₆	—	53	2	no	—
148	C₂²×M₁₆	—	53	1	no	—
149	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
150	C₂³×C₈	—	53	1	no	—
151	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
152	#424	—	106	4	no	—
153	#423	—	106	2	no	—
154	#343	Dic₅₃	1	4	yes	—
155	#341	—	106	4	no	—
156	#341	—	106	4	no	—
157	#336	—	106	2	no	—
158	C₅₃	#11297	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	53	4	no	—
166	C₂×M₁₆	—	53	2	no	—
167	C₂×M₁₆	—	106	2	no	—
168	C₂×M₁₆	—	106	1	no	—
169	C₂×M₁₆	—	106	2	no	—
170	C₂×M₁₆	—	106	2	no	—
171	C₂×M₁₆	—	106	2	no	—
172	C₂×M₁₆	—	106	2	no	—
173	C₂×M₁₆	—	106	2	no	—
174	C₂×M₁₆	—	106	2	no	—
175	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	53	2	no	—
178	C₂²×C₈	—	106	1	no	—
179	C₂²×C₈	—	106	1	no	—
180	C₂²×C₈	—	106	2	no	—
181	C₂²×C₈	—	106	2	no	—
182	C₂²×C₈	—	106	2	no	—
183	C₂²×C₈	—	106	1	no	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
187	C₂×C₂²⋊C₄	Dic₁₀₆	1	2	yes	—
188	C₂×C₄²	D₂₁₂	1	2	yes	—
189	C₄⋊C₈	—	106	8	no	—
190	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
191	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
192	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
193	C₂²×C₄	#11719945	1	2	yes	—
194	C₂²×C₄	#11719945	1	2	yes	—
195	C₂²×C₄	D₄×D₅₃	1	2	yes	—
196	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
197	C₂²×C₄	#11719947	1	1	yes	characteristic
198	C₂²×C₄	#11719947	1	2	yes	—
199	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
200	C₂²×C₄	#11719949	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	4	2	no	—
205	M₁₆	—	4	2	no	—
206	M₁₆	—	106	4	no	—
207	M₁₆	—	106	4	no	—
208	M₁₆	—	106	4	no	—
209	M₁₆	—	212	2	no	—
210	M₁₆	—	212	2	no	—
211	M₁₆	—	212	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	106	4	no	—
216	C₂×C₈	—	106	2	no	—
217	C₂×C₈	—	106	2	no	—
218	C₂×C₈	—	106	4	no	—
219	C₂×C₈	—	212	1	no	—
220	C₂×C₈	—	212	2	no	—
221	C₂×C₈	—	212	1	no	—
222	C₂×C₈	—	212	2	no	—
223	C₂×C₈	—	212	1	no	—
224	C₂×C₈	—	212	1	no	—
225	C₂×C₈	—	212	2	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₂³	—	4	2	no	—
231	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
232	C₂³	#13133194	1	2	yes	—
233	C₂³	#13133208	1	1	yes	characteristic
234	C₂³	#13133209	1	2	yes	—
235	C₂³	#13133216	1	1	yes	characteristic
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	#13133176	1	2	yes	—
239	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
240	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
241	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
242	C₂×C₄	#13133187	1	1	yes	characteristic
243	C₂×C₄	#13133190	1	2	yes	—
244	C₂×C₄	#13133192	1	1	yes	characteristic
245	C₂×C₄	#13133195	1	2	yes	—
246	C₂×C₄	#13133197	1	1	yes	characteristic
247	C₂×C₄	#13133199	1	2	yes	—
248	C₂×C₄	#13133201	1	1	yes	characteristic
249	C₂×C₄	#13133202	1	2	yes	—
250	C₂×C₄	#13133203	1	2	yes	—
251	C₂×C₄	#13133205	1	1	yes	characteristic
252	C₂×C₄	#13133205	1	1	yes	characteristic
253	C₂×C₄	#13133207	1	2	yes	—
254	C₂×C₄	#13133210	1	2	yes	—
255	C₂×C₄	#13133212	1	2	yes	—
256	C₈	—	4	4	no	—
257	C₈	—	212	2	no	—
258	C₈	—	212	4	no	—
259	C₈	—	212	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	#16810489	1	1	yes	characteristic
266	C₂²	#16810498	1	1	yes	characteristic
267	C₂²	#16810522	1	1	yes	characteristic
268	C₂²	#16810557	1	2	yes	—
269	C₂²	#16810565	1	2	yes	—
270	C₄	—	4	4	no	—
271	C₄	#16810509	1	2	yes	—
272	C₄	#16810516	1	2	yes	—
273	C₄	#16810536	1	1	yes	characteristic
274	C₄	#16810541	1	1	yes	characteristic
275	C₄	#16810550	1	1	yes	characteristic
276	C₄	#16810563	1	1	yes	characteristic
277	C₂	—	2	1	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	2	no	—
280	C₂	—	2	1	no	—
281	C₂	#20272972	1	1	yes	characteristic
282	C₂	#20274937	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133187
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1742	C₂
socle	C₂²×C₅₃	#385
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1742	D₅₃
hypocenter	C₅₃	#11297
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1742

Complements

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

N	H ≅ G/N	type
158. C₅₃	#11297	semidirect

Maximal subgroups

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

subgroup	index
#20283279	2
#20274442	2
C₂×#16810387	2
#20274412	2
C₅₃×#1742	2
#11297	53