Subgroups — #27270625

Order 13568.

294 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274832	C₂	1	1	yes	maximal, characteristic
2	#20274461	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88322	—	2	1	no	—
5	C₂×C₄×#88322	C₄	1	2	yes	—
6	#16810378	—	2	1	no	—
7	#16810378	—	2	1	no	—
8	#16810378	—	2	1	no	—
9	#16810378	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
11	C₄³×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#336	—	2	1	no	—
13	C₅₃×#336	C₄	1	2	yes	—
14	C₂²×#88322	—	2	2	no	—
15	C₂²×#88322	—	2	2	no	—
16	C₄⋊#88322	—	4	1	no	—
17	C₄⋊#88322	—	4	1	no	—
18	C₄×#88322	—	2	2	no	—
19	C₄×#88322	C₈	1	4	yes	complemented
20	C₂²×C₈×C₅₃	—	2	2	no	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	—	4	1	no	—
29	C₄⋊C₈×C₅₃	—	4	1	no	—
30	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	—	4	2	no	—
40	C₄²×C₅₃	—	2	2	no	—
41	C₄²×C₅₃	—	4	1	no	—
42	C₄²×C₅₃	—	4	1	no	—
43	C₄²×C₅₃	—	4	1	no	—
44	C₄²×C₅₃	—	4	1	no	—
45	C₄²×C₅₃	C₂×C₈	1	2	yes	—
46	C₄²×C₅₃	M₁₆	1	2	yes	—
47	C₄²×C₅₃	D₈	1	1	yes	characteristic
48	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
49	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
50	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
51	C₂×#88322	—	2	4	no	—
52	C₂×#88322	—	4	2	no	—
53	C₂×#88322	—	4	2	no	—
54	C₂×#88322	—	4	2	no	—
55	C₂×#88322	—	4	2	no	—
56	C₂×#88322	—	4	2	no	—
57	C₂³×C₅₃	#95	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
75	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
80	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#108	1	2	yes	—
83	C₈×C₅₃	—	8	2	no	—
84	#88322	—	4	4	no	—
85	#88322	—	8	2	no	—
86	#7018	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	#336	1	1	yes	characteristic
88	C₂²×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
89	C₂²×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
90	C₂²×C₅₃	#340	1	1	yes	characteristic
91	C₂²×C₅₃	#341	1	1	yes	characteristic
92	C₂²×C₅₃	#344	1	1	yes	characteristic
93	C₂²×C₅₃	#344	1	1	yes	characteristic
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	2	1	no	—
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	2	no	—
101	C₄×C₅₃	—	4	1	no	—
102	C₄×C₅₃	—	4	1	no	—
103	C₄×C₅₃	#334	1	2	yes	—
104	C₄×C₅₃	#335	1	2	yes	—
105	#1831	—	53	1	no	—
106	#1831	D₅₃	1	1	yes	characteristic
107	#1744	—	53	1	no	—
108	C₂×C₅₃	#1274	1	1	yes	characteristic
109	C₂×C₅₃	#1277	1	1	yes	characteristic
110	C₂×C₅₃	#1281	1	1	yes	characteristic
111	C₂×C₅₃	#1286	1	1	yes	characteristic
112	C₂×C₅₃	#1286	1	1	yes	characteristic
113	C₂×C₅₃	#1287	1	1	yes	characteristic
114	C₂×C₅₃	#1287	1	1	yes	characteristic
115	C₂×C₄⋊C₈	—	2	1	no	—
116	C₂×C₄⋊C₈	—	106	1	no	—
117	C₂×C₄×C₈	—	53	2	no	—
118	C₄³	C₂×D₅₃	1	1	yes	characteristic
119	#336	—	2	1	no	—
120	#336	—	53	2	no	—
121	#336	—	106	1	no	—
122	#336	—	106	1	no	—
123	#336	—	106	1	no	—
124	#336	Dic₅₃	1	2	yes	—
125	C₅₃	#7018	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	106	2	no	—
128	C₂²×C₈	—	106	2	no	—
129	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
130	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
131	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
132	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
133	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
136	C₄⋊C₈	—	4	1	no	—
137	C₄⋊C₈	—	4	1	no	—
138	C₄⋊C₈	—	212	1	no	—
139	C₄⋊C₈	—	212	1	no	—
140	C₄×C₈	—	53	4	no	—
141	C₄×C₈	—	106	2	no	—
142	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
143	C₂²×C₄	#11719919	1	1	yes	characteristic
144	C₂²×C₄	#11719919	1	1	yes	characteristic
145	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
146	C₂²×C₄	#11719921	1	1	yes	characteristic
147	C₂²×C₄	#11719921	1	1	yes	characteristic
148	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	106	4	no	—
153	C₂×C₈	—	212	2	no	—
154	C₂×C₈	—	212	2	no	—
155	C₂×C₈	—	212	2	no	—
156	C₂×C₈	—	212	2	no	—
157	C₂×C₈	—	212	2	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	4	1	no	—
160	C₄²	—	4	1	no	—
161	C₄²	—	4	1	no	—
162	C₄²	—	4	1	no	—
163	C₄²	C₈×D₅₃	1	2	yes	—
164	C₄²	#11719912	1	2	yes	—
165	C₄²	#11719922	1	1	yes	characteristic
166	C₄²	#11719923	1	1	yes	characteristic
167	C₄²	#11719924	1	1	yes	characteristic
168	C₄²	#11719925	1	1	yes	characteristic
169	C₂³	#13133124	1	1	yes	characteristic
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
185	C₂×C₄	#13133127	1	1	yes	characteristic
186	C₂×C₄	#13133129	1	2	yes	—
187	C₂×C₄	#13133137	1	1	yes	characteristic
188	C₂×C₄	#13133138	1	1	yes	characteristic
189	C₂×C₄	#13133140	1	1	yes	characteristic
190	C₂×C₄	#13133141	1	1	yes	characteristic
191	C₂×C₄	#13133146	1	2	yes	—
192	C₂×C₄	#13133148	1	1	yes	characteristic
193	C₂×C₄	#13133150	1	1	yes	characteristic
194	C₂×C₄	#13133152	1	2	yes	—
195	C₈	—	8	2	no	—
196	C₈	—	424	2	no	—
197	C₈	—	212	4	no	—
198	C₂²	#16810377	1	1	yes	characteristic
199	C₂²	#16810379	1	1	yes	characteristic
200	C₂²	#16810380	1	1	yes	characteristic
201	C₂²	#16810385	1	1	yes	characteristic
202	C₂²	#16810387	1	1	yes	characteristic
203	C₂²	#16810393	1	1	yes	characteristic
204	C₂²	#16810395	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	4	1	no	—
213	C₄	—	4	1	no	—
214	C₄	#16810373	1	2	yes	—
215	C₄	#16810375	1	2	yes	—
216	C₂	#20272543	1	1	yes	characteristic
217	C₂	#20272549	1	1	yes	characteristic
218	C₂	#20272559	1	1	yes	characteristic
219	C₂	#20272573	1	1	yes	characteristic
220	C₂	#20272575	1	1	yes	characteristic
221	C₂	#20272576	1	1	yes	characteristic
222	C₂	#20272578	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133124
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₄³	C₂×D₅₃
Fitting	C₅₃×#1831	C₂
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄³×C₅₃	C₂²
hypercenter	#1831	D₅₃
hypocenter	C₅₃	#7018
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
19. C₄×#88322	C₈	semidirect
125. C₅₃	#7018	semidirect

Maximal subgroups

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

subgroup	index
#20274832	2
#20274461	2
C₅₃×#1831	2
#7018	53