Subgroups — #27410681

Order 13568.

664 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133217	C₂	1	1	yes	maximal, characteristic
2	#20275649	C₂	1	4	yes	maximal
3	#20275273	C₂	1	4	yes	maximal
4	#20275271	C₂	1	4	yes	maximal
5	#20274858	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1839	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133217	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
9	C₂×#13133146	C₂²	1	4	yes	—
10	C₂×C₈×Dic₅₃	C₂²	1	4	yes	—
11	#16810529	C₄	1	2	yes	—
12	#16810521	C₄	1	2	yes	—
13	#16810521	C₄	1	2	yes	—
14	#16810493	C₄	1	1	yes	characteristic
15	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
16	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
17	#16810378	C₂²	1	2	yes	—
18	#16810378	C₂²	1	2	yes	—
19	#16810378	C₂²	1	2	yes	—
20	#16810377	C₂²	1	8	yes	—
21	C₄³×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#336	C₂²	1	2	yes	—
23	C₅₃×#336	C₂²	1	2	yes	—
24	C₅₃×#336	C₂²	1	2	yes	—
25	C₂²×Dic₁₀₆	C₂×C₄	1	2	yes	—
26	C₂²×#88322	C₂³	1	4	yes	—
27	#13133217	C₈	1	8	yes	complemented
28	C₄×Dic₁₀₆	—	4	2	no	—
29	C₄×Dic₁₀₆	—	4	2	no	—
30	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
35	C₂×#11719919	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₅₃	C₂³	1	4	yes	—
37	#13133146	—	4	4	no	—
38	C₈×Dic₅₃	—	4	4	no	—
39	#13133124	C₂×C₄	1	4	yes	—
40	#13133124	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₂²×Dic₅₃	C₂²×C₄	1	4	yes	—
57	C₂²×Dic₅₃	C₂²×C₄	1	4	yes	—
58	C₂×Dic₁₀₆	—	4	2	no	—
59	C₂×Dic₁₀₆	—	4	2	no	—
60	C₂×Dic₁₀₆	C₂×C₈	1	8	yes	—
61	C₂×C₈×C₅₃	—	2	4	no	—
62	C₂×C₈×C₅₃	—	2	4	no	—
63	C₂×C₈×C₅₃	—	2	4	no	—
64	C₄²×C₅₃	—	2	2	no	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	—	4	1	no	—
68	C₄²×C₅₃	—	4	1	no	—
69	C₄²×C₅₃	—	4	1	no	—
70	C₄²×C₅₃	C₂×C₈	1	4	yes	—
71	C₄⋊Dic₅₃	—	4	2	no	—
72	C₄⋊Dic₅₃	—	4	2	no	—
73	C₄⋊Dic₅₃	—	4	2	no	—
74	C₄⋊Dic₅₃	C₂×C₈	1	8	yes	—
75	C₄⋊Dic₅₃	C₂×C₈	1	8	yes	—
76	#11719919	—	4	4	no	—
77	C₄×Dic₅₃	—	2	4	no	—
78	C₄×Dic₅₃	—	2	4	no	—
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	—	2	4	no	—
81	C₂×#88322	—	2	4	no	—
82	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
99	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
101	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
102	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
103	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
104	C₂×Dic₅₃	—	2	4	no	—
105	C₂×Dic₅₃	—	2	4	no	—
106	C₂×Dic₅₃	—	2	4	no	—
107	C₂×Dic₅₃	—	2	4	no	—
108	C₂×Dic₅₃	—	2	4	no	—
109	C₂×Dic₅₃	C₂²×C₈	1	8	yes	—
110	Dic₁₀₆	—	4	8	no	—
111	C₈×C₅₃	—	4	4	no	—
112	#88322	—	4	4	no	—
113	#23363	—	53	1	no	maximal, Sylow
114	C₂²×C₅₃	#518	1	1	yes	characteristic
115	C₂²×C₅₃	#519	1	1	yes	characteristic
116	C₂²×C₅₃	#519	1	1	yes	characteristic
117	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
118	C₂²×C₅₃	#568	1	1	yes	characteristic
119	C₂²×C₅₃	#568	1	1	yes	characteristic
120	C₂²×C₅₃	#568	1	1	yes	characteristic
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	—	2	2	no	—
124	C₄×C₅₃	—	4	1	no	—
125	C₄×C₅₃	—	4	1	no	—
126	C₄×C₅₃	—	4	1	no	—
127	C₄×C₅₃	#563	1	4	yes	—
128	Dic₅₃	—	2	8	no	—
129	Dic₅₃	—	4	4	no	—
130	C₄×C₄⋊Q₈	—	53	1	no	—
131	#1984	—	53	4	no	—
132	#1916	—	53	4	no	—
133	#1916	—	53	4	no	—
134	#1839	—	53	1	no	—
135	#1839	D₅₃	1	1	yes	characteristic
136	C₂×C₅₃	#2381	1	1	yes	characteristic
137	C₂×C₅₃	#2972	1	1	yes	characteristic
138	C₂×C₅₃	#2980	1	1	yes	characteristic
139	C₂×C₅₃	#2982	1	1	yes	characteristic
140	C₂×C₅₃	#2982	1	1	yes	characteristic
141	C₂×C₅₃	#2987	1	1	yes	characteristic
142	C₂×C₅₃	#2987	1	1	yes	characteristic
143	C₂×C₄⋊Q₈	—	53	1	no	—
144	C₂×C₄×Q₈	—	53	2	no	—
145	C₂×C₄⋊C₈	—	53	4	no	—
146	C₂×C₄×C₈	—	53	4	no	—
147	#391	—	53	2	no	—
148	#389	—	53	2	no	—
149	#389	—	53	2	no	—
150	#384	—	53	1	no	—
151	C₄×C₄⋊C₄	—	53	2	no	—
152	C₄×C₄⋊C₄	—	53	2	no	—
153	C₄³	C₂×D₅₃	1	1	yes	characteristic
154	#336	—	53	8	no	—
155	#336	—	53	2	no	—
156	#336	—	53	2	no	—
157	#336	—	53	2	no	—
158	#336	C₂×D₅₃	1	2	yes	—
159	#336	C₂×D₅₃	1	2	yes	—
160	#336	C₂×D₅₃	1	2	yes	—
161	C₅₃	#23363	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₈	—	53	2	no	—
163	C₂²×C₈	—	53	4	no	—
164	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
165	C₄⋊Q₈	—	53	8	no	—
166	C₄×Q₈	—	212	2	no	—
167	C₄×Q₈	—	212	2	no	—
168	C₂×C₄⋊C₄	—	53	2	no	—
169	C₂×C₄⋊C₄	—	53	2	no	—
170	C₂×C₄⋊C₄	—	53	4	no	—
171	C₂×C₄⋊C₄	—	53	2	no	—
172	C₂×C₄⋊C₄	—	53	2	no	—
173	C₂×C₄⋊C₄	—	53	2	no	—
174	C₂×C₄²	—	53	4	no	—
175	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
176	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
177	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
179	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
180	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
181	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
182	C₄⋊C₈	—	212	4	no	—
183	C₄×C₈	—	212	4	no	—
184	#95	—	53	4	no	—
185	#95	—	53	4	no	—
186	C₂×Q₈	—	53	8	no	—
187	C₂×Q₈	—	212	2	no	—
188	C₂×Q₈	—	212	2	no	—
189	C₂²×C₄	—	53	4	no	—
190	C₂²×C₄	—	53	4	no	—
191	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
197	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	106	4	no	—
202	C₂×C₈	—	106	4	no	—
203	C₂×C₈	—	106	4	no	—
204	C₄⋊C₄	—	53	8	no	—
205	C₄⋊C₄	—	53	8	no	—
206	C₄⋊C₄	—	212	4	no	—
207	C₄⋊C₄	—	212	2	no	—
208	C₄⋊C₄	—	212	2	no	—
209	C₄⋊C₄	—	212	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	4	1	no	—
214	C₄²	—	4	1	no	—
215	C₄²	—	4	1	no	—
216	C₄²	—	106	4	no	—
217	C₄²	—	106	4	no	—
218	C₄²	C₈×D₅₃	1	4	yes	—
219	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
220	Q₈	—	212	8	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	53	8	no	—
237	C₂×C₄	—	106	4	no	—
238	C₂×C₄	—	106	4	no	—
239	C₂×C₄	—	106	4	no	—
240	C₂×C₄	—	106	4	no	—
241	C₂×C₄	—	106	4	no	—
242	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
243	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
244	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
245	C₂×C₄	#13133263	1	2	yes	—
246	C₂×C₄	#13133264	1	2	yes	—
247	C₂×C₄	#13133264	1	2	yes	—
248	C₈	—	4	4	no	—
249	C₈	—	212	4	no	—
250	C₂²	#16811097	1	1	yes	characteristic
251	C₂²	#16811100	1	1	yes	characteristic
252	C₂²	#16811100	1	1	yes	characteristic
253	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
254	C₂²	#16811389	1	1	yes	characteristic
255	C₂²	#16811389	1	1	yes	characteristic
256	C₂²	#16811389	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	1	no	—
263	C₄	—	106	8	no	—
264	C₄	—	212	4	no	—
265	C₄	#16811375	1	4	yes	—
266	C₂	#20277895	1	1	yes	characteristic
267	C₂	#20283738	1	1	yes	characteristic
268	C₂	#20283785	1	1	yes	characteristic
269	C₂	#20283795	1	1	yes	characteristic
270	C₂	#20283795	1	1	yes	characteristic
271	C₂	#20283820	1	1	yes	characteristic
272	C₂	#20283820	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#1839	C₂
socle, prime radical	C₂³×C₅₃	C₂³×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1839	D₅₃
hypocenter	C₅₃	#23363
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1839

Complements

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

N	H ≅ G/N	type
27. #13133217	C₈	semidirect
161. C₅₃	#23363	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133217	2
#20275649	2
#20275273	2
#20275271	2
#20274858	2
C₅₃×#1839	2
#23363	53