Subgroups — #27289005

Order 13568.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275474	C₂	1	2	yes	maximal
2	#20275433	C₂	1	1	yes	maximal, characteristic
3	#20274875	C₂	1	1	yes	maximal, characteristic
4	#20272599	C₂	1	2	yes	maximal
5	C₅₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133151	—	2	2	no	—
7	C₂×#13133151	—	2	1	no	—
8	C₂×#13133151	—	2	1	no	—
9	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
10	C₂×#13133146	—	2	1	no	—
11	#16810493	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
13	#16810386	—	2	2	no	—
14	#16810377	—	2	1	no	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×#88322	—	2	1	no	—
18	C₂×C₄⋊Dic₅₃	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅₃	—	2	1	no	—
24	#13133151	—	4	4	no	—
25	#13133151	—	4	2	no	—
26	#13133151	—	4	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	#13133146	—	4	2	no	—
29	#13133124	—	2	1	no	—
30	C₂²×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄⋊C₈×C₅₃	Q₈	1	4	yes	—
36	C₄×C₈×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
40	C₂²×Dic₅₃	—	2	1	no	—
41	C₂²×Dic₅₃	—	2	1	no	—
42	C₂²×Dic₅₃	—	4	2	no	—
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	—	2	4	no	—
48	C₂×C₈×C₅₃	—	4	2	no	—
49	C₂×C₈×C₅₃	—	4	2	no	—
50	C₂×C₈×C₅₃	D₈	1	2	yes	—
51	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
52	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
53	C₄²×C₅₃	#41	1	2	yes	—
54	C₄⋊Dic₅₃	—	2	1	no	—
55	C₄⋊Dic₅₃	—	2	2	no	—
56	C₄⋊Dic₅₃	—	4	2	no	—
57	C₄⋊Dic₅₃	—	4	2	no	—
58	C₄⋊Dic₅₃	—	4	2	no	—
59	C₄⋊Dic₅₃	—	4	2	no	—
60	C₄⋊Dic₅₃	—	8	2	no	—
61	C₄⋊Dic₅₃	Q₁₆	1	2	yes	—
62	C₄×Dic₅₃	—	4	1	no	—
63	C₄×Dic₅₃	—	4	1	no	—
64	C₂×#88322	—	4	1	no	—
65	C₂×#88322	—	4	1	no	—
66	C₂×#88322	—	4	1	no	—
67	C₂³×C₅₃	#121	1	1	yes	characteristic
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#125	1	2	yes	—
73	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
78	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
81	C₂×Dic₅₃	—	2	2	no	—
82	C₂×Dic₅₃	—	4	1	no	—
83	C₂×Dic₅₃	—	4	1	no	—
84	C₂×Dic₅₃	—	4	1	no	—
85	C₂×Dic₅₃	—	4	1	no	—
86	C₂×Dic₅₃	—	4	1	no	—
87	C₂×Dic₅₃	—	8	2	no	—
88	C₂×Dic₅₃	—	8	2	no	—
89	C₂×Dic₅₃	—	8	2	no	—
90	C₈×C₅₃	—	2	2	no	—
91	C₈×C₅₃	—	2	2	no	—
92	C₈×C₅₃	—	4	4	no	—
93	#88322	—	8	1	no	—
94	#10650	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	#398	1	1	yes	characteristic
96	C₂²×C₅₃	#459	1	1	yes	characteristic
97	C₂²×C₅₃	#462	1	1	yes	characteristic
98	C₂²×C₅₃	#467	1	1	yes	characteristic
99	C₂²×C₅₃	#469	1	1	yes	characteristic
100	C₂²×C₅₃	#472	1	1	yes	characteristic
101	C₂²×C₅₃	#473	1	1	yes	characteristic
102	C₄×C₅₃	—	2	2	no	—
103	C₄×C₅₃	#477	1	2	yes	—
104	C₄×C₅₃	#478	1	2	yes	—
105	C₄×C₅₃	#480	1	1	yes	characteristic
106	C₄×C₅₃	#484	1	1	yes	characteristic
107	C₄×C₅₃	#500	1	1	yes	characteristic
108	C₄×C₅₃	#501	1	1	yes	characteristic
109	Dic₅₃	—	4	2	no	—
110	Dic₅₃	—	8	1	no	—
111	Dic₅₃	—	16	2	no	—
112	#1947	—	53	2	no	—
113	#1939	—	53	1	no	—
114	#1845	—	53	1	no	—
115	#1296	—	53	2	no	—
116	#1277	D₅₃	1	1	yes	characteristic
117	C₂×C₅₃	#1669	1	1	yes	characteristic
118	C₂×C₅₃	#1676	1	1	yes	characteristic
119	C₂×C₅₃	#1687	1	1	yes	characteristic
120	C₂×C₅₃	#1694	1	1	yes	characteristic
121	C₂×C₅₃	#2083	1	1	yes	characteristic
122	C₂×C₅₃	#2087	1	1	yes	characteristic
123	C₂×C₅₃	#2089	1	1	yes	characteristic
124	C₂×#107	—	106	2	no	—
125	C₂×#107	—	106	1	no	—
126	C₂×#107	—	106	1	no	—
127	C₂×C₄⋊C₈	—	53	1	no	—
128	C₂×C₄⋊C₈	—	106	1	no	—
129	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
130	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
131	#384	—	53	2	no	—
132	C₄×C₄⋊C₄	—	53	1	no	—
133	#340	—	106	2	no	—
134	#336	—	106	1	no	—
135	C₅₃	#10650	1	1	yes	characteristic, complemented, Sylow
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	106	1	no	—
138	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
139	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	53	1	no	—
141	C₂×C₄⋊C₄	—	53	1	no	—
142	C₂×C₄⋊C₄	—	106	2	no	—
143	C₂×C₄⋊C₄	—	106	2	no	—
144	C₂×C₄⋊C₄	—	106	2	no	—
145	C₂×C₄²	—	106	1	no	—
146	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
147	#107	—	212	4	no	—
148	#107	—	212	2	no	—
149	#107	—	212	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	106	2	no	—
152	C₄⋊C₈	—	212	2	no	—
153	C₄⋊C₈	Dic₁₀₆	1	4	yes	—
154	C₄×C₈	—	2	2	no	—
155	#95	—	106	1	no	—
156	C₂²×C₄	—	106	1	no	—
157	C₂²×C₄	—	106	1	no	—
158	C₂²×C₄	—	212	2	no	—
159	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
160	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
161	C₂²×C₄	#11719949	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	212	1	no	—
170	C₂×C₈	—	212	1	no	—
171	C₂×C₈	—	212	1	no	—
172	C₂×C₈	D₄₂₄	1	2	yes	—
173	C₂×C₈	Dic₂₁₂	1	2	yes	—
174	C₄⋊C₄	—	53	2	no	—
175	C₄⋊C₄	—	106	1	no	—
176	C₄⋊C₄	—	106	2	no	—
177	C₄⋊C₄	—	424	2	no	—
178	C₄⋊C₄	—	212	2	no	—
179	C₄⋊C₄	—	212	2	no	—
180	C₄⋊C₄	—	212	2	no	—
181	C₄⋊C₄	—	212	2	no	—
182	C₄²	—	212	1	no	—
183	C₄²	—	212	1	no	—
184	C₄²	C₂×Dic₁₀₆	1	2	yes	—
185	C₄²	#11719947	1	2	yes	—
186	C₂³	#13133192	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	106	2	no	—
190	C₂×C₄	—	424	2	no	—
191	C₂×C₄	—	424	2	no	—
192	C₂×C₄	—	424	2	no	—
193	C₂×C₄	—	212	1	no	—
194	C₂×C₄	—	212	1	no	—
195	C₂×C₄	—	212	1	no	—
196	C₂×C₄	—	212	1	no	—
197	C₂×C₄	—	212	1	no	—
198	C₂×C₄	#13133194	1	2	yes	—
199	C₂×C₄	#13133201	1	1	yes	characteristic
200	C₂×C₄	#13133208	1	2	yes	—
201	C₂×C₄	#13133217	1	1	yes	characteristic
202	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
203	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
204	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
205	C₂×C₄	#13133236	1	1	yes	characteristic
206	C₂×C₄	#13133244	1	2	yes	—
207	C₂×C₄	#13133245	1	1	yes	characteristic
208	C₂×C₄	#13133250	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	4	4	no	—
212	C₈	—	424	1	no	—
213	C₂²	#16810555	1	1	yes	characteristic
214	C₂²	#16810819	1	1	yes	characteristic
215	C₂²	#16810834	1	1	yes	characteristic
216	C₂²	#16810865	1	1	yes	characteristic
217	C₂²	#16810877	1	1	yes	characteristic
218	C₂²	#16810894	1	1	yes	characteristic
219	C₂²	#16810899	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	848	2	no	—
222	C₄	—	424	1	no	—
223	C₄	—	212	2	no	—
224	C₄	#16810916	1	2	yes	—
225	C₄	#16810921	1	2	yes	—
226	C₄	#16810937	1	1	yes	characteristic
227	C₄	#16810959	1	1	yes	characteristic
228	C₄	#16811028	1	1	yes	characteristic
229	C₄	#16811032	1	1	yes	characteristic
230	C₂	#20274081	1	1	yes	characteristic
231	C₂	#20274120	1	1	yes	characteristic
232	C₂	#20274195	1	1	yes	characteristic
233	C₂	#20274234	1	1	yes	characteristic
234	C₂	#20276188	1	1	yes	characteristic
235	C₂	#20276210	1	1	yes	characteristic
236	C₂	#20276222	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133192
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1277	C₂
socle, prime radical	C₂³×C₅₃	#121
hypercenter	#1277	D₅₃
hypocenter	C₅₃	#10650
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
135. C₅₃	#10650	semidirect

Maximal subgroups

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

subgroup	index
#20275474	2
#20275433	2
#20274875	2
#20272599	2
C₅₃×#1277	2
#10650	53