Subgroups — #27484881

Order 13568.

408 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276293	C₂	1	4	yes	maximal
2	#20275505	C₂	1	2	yes	maximal
3	#20275503	C₂	1	2	yes	maximal
4	#20274887	C₂	1	1	yes	maximal, characteristic
5	#20274839	C₂	1	2	yes	maximal
6	#20274838	C₂	1	2	yes	maximal
7	#20274594	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1772	C₂	1	1	yes	maximal, characteristic
9	#16810671	C₂²	1	4	yes	—
10	C₂×#13133153	C₂²	1	2	yes	—
11	C₂×#13133153	C₂²	1	1	yes	characteristic
12	C₂×#13133153	C₂²	1	2	yes	—
13	#16810661	C₂²	1	1	yes	characteristic
14	#16810647	C₂²	1	2	yes	—
15	#16810578	C₂²	1	2	yes	—
16	#16810394	C₂²	1	4	yes	—
17	#16810393	C₂²	1	4	yes	—
18	#16810383	C₂²	1	2	yes	—
19	#16810379	C₂²	1	4	yes	—
20	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#427	C₂²	1	1	yes	characteristic
22	C₅₃×#405	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#339	C₂²	1	2	yes	—
25	C₂×#11719917	C₂³	1	2	yes	—
26	C₂×#11719917	C₂³	1	4	yes	—
27	#13133176	C₂³	1	2	yes	—
28	#13133153	—	2	2	no	—
29	#13133153	—	2	2	no	—
30	#13133153	—	2	1	no	—
31	#13133153	—	2	1	no	—
32	#13133153	—	2	2	no	—
33	#13133153	—	2	2	no	—
34	#13133152	—	4	4	no	—
35	#13133151	—	2	1	no	—
36	#13133149	—	2	1	no	—
37	C₄⋊#88322	—	4	2	no	—
38	#13133127	—	2	2	no	—
39	C₈×Dic₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	#117×C₅₃	C₂³	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	#108×C₅₃	—	2	1	no	—
46	#108×C₅₃	—	2	1	no	—
47	#107×C₅₃	—	2	1	no	—
48	#106×C₅₃	—	2	1	no	—
49	#97×C₅₃	—	2	2	no	—
50	C₄×C₈×C₅₃	—	2	2	no	—
51	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
52	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
53	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
54	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	—	2	1	no	—
56	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
57	M₁₆×C₅₃	—	2	2	no	—
58	M₁₆×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
65	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
66	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
67	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
68	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
69	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
70	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
71	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
72	C₄⋊C₄×C₅₃	—	2	2	no	—
73	C₂²⋊C₄×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	—	2	1	no	—
75	C₄²×C₅₃	—	2	2	no	—
76	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
78	C₂²⋊Dic₅₃	—	2	2	no	—
79	C₄⋊Dic₅₃	—	2	2	no	—
80	C₄×Dic₅₃	—	2	2	no	—
81	#11719917	—	2	2	no	—
82	#11719917	—	2	4	no	—
83	#11719917	—	2	4	no	—
84	#11719917	—	2	2	no	—
85	C₂×#88322	—	2	2	no	—
86	C₂×#88322	—	2	4	no	—
87	C₂³×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
97	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
98	C₂×Dic₅₃	—	4	2	no	—
99	C₈×C₅₃	—	4	2	no	—
100	C₈×C₅₃	#125	1	1	yes	characteristic
101	C₈×C₅₃	#125	1	1	yes	characteristic
102	C₈×C₅₃	#125	1	1	yes	characteristic
103	C₈×C₅₃	#125	1	1	yes	characteristic
104	C₈×C₅₃	#125	1	1	yes	characteristic
105	C₈×C₅₃	#125	1	1	yes	characteristic
106	C₈×C₅₃	#125	1	1	yes	characteristic
107	C₈×C₅₃	#125	1	1	yes	characteristic
108	#88322	—	4	2	no	—
109	#88322	—	4	4	no	—
110	#29230	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	—	4	1	no	—
112	C₂²×C₅₃	#544	1	1	yes	characteristic
113	C₂²×C₅₃	#556	1	1	yes	characteristic
114	C₂²×C₅₃	#556	1	1	yes	characteristic
115	C₄×C₅₃	—	2	2	no	—
116	C₄×C₅₃	—	4	1	no	—
117	C₄×C₅₃	—	8	2	no	—
118	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
119	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
120	C₄×C₅₃	#555	1	1	yes	characteristic
121	C₄×C₅₃	#559	1	1	yes	characteristic
122	Dic₅₃	—	8	2	no	—
123	#2102	—	53	4	no	—
124	#1953	—	53	2	no	—
125	#1953	—	53	2	no	—
126	#1848	—	53	1	no	—
127	#1833	—	53	2	no	—
128	#1833	—	53	2	no	—
129	#1772	—	53	1	no	—
130	#1772	D₅₃	1	1	yes	characteristic
131	C₂×C₅₃	—	2	1	no	—
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	#2571	1	1	yes	characteristic
135	#430	—	53	4	no	—
136	C₂×#108	—	53	2	no	—
137	C₂×#108	—	53	2	no	—
138	C₂×#108	—	53	1	no	—
139	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
140	#427	—	53	1	no	—
141	#427	C₂×D₅₃	1	1	yes	characteristic
142	#423	—	53	2	no	—
143	#405	—	53	2	no	—
144	#405	C₂×D₅₃	1	2	yes	—
145	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
146	#344	—	53	4	no	—
147	#344	—	53	4	no	—
148	#339	—	53	2	no	—
149	#339	C₂×D₅₃	1	2	yes	—
150	H(3,ℤ/4)	—	53	4	no	—
151	C₅₃	#29230	1	1	yes	characteristic, complemented, Sylow
152	C₂×M₁₆	—	53	4	no	—
153	C₂×M₁₆	—	53	2	no	—
154	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
155	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
156	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
157	#117	—	53	2	no	—
158	#117	C₂²×D₅₃	1	2	yes	—
159	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
160	#108	—	2	1	no	—
161	#108	—	2	1	no	—
162	#108	—	106	2	no	—
163	#108	—	106	2	no	—
164	#108	—	106	1	no	—
165	#108	—	106	1	no	—
166	#108	—	106	2	no	—
167	#108	—	106	2	no	—
168	#108	—	212	4	no	—
169	#107	—	2	1	no	—
170	#107	—	106	1	no	—
171	#106	—	2	1	no	—
172	#106	—	106	1	no	—
173	C₄⋊C₈	—	212	2	no	—
174	#97	—	2	2	no	—
175	#97	—	106	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	106	2	no	—
178	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
179	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
180	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
181	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
184	M₁₆	—	2	2	no	—
185	M₁₆	—	2	2	no	—
186	M₁₆	—	106	2	no	—
187	M₁₆	—	106	4	no	—
188	M₁₆	—	106	4	no	—
189	M₁₆	—	106	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	106	2	no	—
192	C₂×C₈	—	106	4	no	—
193	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
194	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
195	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
196	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
197	C₂×C₈	#11719945	1	1	yes	characteristic
198	C₂×C₈	#11719945	1	1	yes	characteristic
199	C₂×C₈	#11719945	1	1	yes	characteristic
200	C₂×C₈	#11719945	1	1	yes	characteristic
201	C₂×C₈	#11719947	1	1	yes	characteristic
202	C₂×C₈	#11719947	1	1	yes	characteristic
203	C₂×C₈	#11719949	1	1	yes	characteristic
204	C₂×C₈	#11719949	1	1	yes	characteristic
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	106	2	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	106	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	106	2	no	—
212	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
213	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
214	C₂³	#13133264	1	1	yes	characteristic
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	212	2	no	—
220	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
221	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
222	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
223	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
224	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
225	C₂×C₄	#13133263	1	1	yes	characteristic
226	C₈	—	4	2	no	—
227	C₈	—	212	2	no	—
228	C₈	—	212	4	no	—
229	C₈	#13133208	1	1	yes	characteristic
230	C₈	#13133208	1	1	yes	characteristic
231	C₈	#13133208	1	1	yes	characteristic
232	C₈	#13133208	1	1	yes	characteristic
233	C₈	#13133209	1	1	yes	characteristic
234	C₈	#13133209	1	1	yes	characteristic
235	C₈	#13133209	1	1	yes	characteristic
236	C₈	#13133209	1	1	yes	characteristic
237	C₂²	—	4	1	no	—
238	C₂²	#16811251	1	1	yes	characteristic
239	C₂²	#16811343	1	1	yes	characteristic
240	C₂²	#16811346	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	8	2	no	—
244	C₄	—	424	2	no	—
245	C₄	C₂×#13133208	1	1	yes	characteristic
246	C₄	C₂×#13133209	1	1	yes	characteristic
247	C₄	#16811338	1	1	yes	characteristic
248	C₄	#16811358	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	#20279954	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16811338
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1772	C₂
socle	C₂×C₅₃	#2571
prime radical	C₂³×C₅₃	2₋¹⁺⁴
hypercenter	#1772	D₅₃
hypocenter	C₅₃	#29230
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772

Complements

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

N	H ≅ G/N	type
151. C₅₃	#29230	semidirect

Maximal subgroups

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

subgroup	index
#20276293	2
#20275505	2
#20275503	2
#20274887	2
#20274839	2
#20274838	2
#20274594	2
C₅₃×#1772	2
#29230	53