Subgroups — #27289911

Order 13568.

1126 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810493	C₂	1	2	yes	maximal
2	C₂×#16810386	C₂	1	4	yes	maximal
3	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₅₃	—	2	4	no	—
5	C₂²×C₄⋊Dic₅₃	C₂²	1	4	yes	—
6	#16810493	—	2	4	no	—
7	#16810493	C₄	1	8	yes	complemented
8	#16810386	—	2	8	no	—
9	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	—	2	4	no	—
12	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	complemented
13	C₂³×Dic₅₃	—	2	4	no	—
14	C₂×C₄⋊Dic₅₃	—	2	16	no	—
15	C₂×C₄⋊Dic₅₃	—	2	4	no	—
16	C₂×C₄⋊Dic₅₃	—	2	8	no	—
17	C₂×C₄⋊Dic₅₃	—	2	8	no	—
18	C₂×C₄⋊Dic₅₃	—	4	8	no	—
19	C₂×C₄⋊Dic₅₃	—	4	8	no	—
20	C₂×C₄⋊Dic₅₃	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₅₃	D₄	1	2	yes	—
22	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	—	2	4	no	—
24	C₂²×C₈×C₅₃	—	2	4	no	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₅₃	D₄	1	4	yes	—
27	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₃	D₄	1	4	yes	—
31	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
32	C₄×C₈×C₅₃	—	2	8	no	—
33	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
36	C₂²×C₄×C₅₃	D₈	1	4	yes	—
37	C₂²×C₄×C₅₃	SD₁₆	1	8	yes	—
38	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₅₃	#41	1	4	yes	—
46	C₂²×Dic₅₃	—	2	8	no	—
47	C₂²×Dic₅₃	—	4	4	no	—
48	C₂²×Dic₅₃	—	4	4	no	—
49	C₂²×Dic₅₃	—	4	8	no	—
50	C₂²×Dic₅₃	—	4	8	no	—
51	C₂×C₈×C₅₃	—	2	4	no	—
52	C₂×C₈×C₅₃	—	2	4	no	—
53	C₂×C₈×C₅₃	—	2	8	no	—
54	C₂×C₈×C₅₃	—	2	8	no	—
55	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	complemented
56	C₄²×C₅₃	C₄²	1	4	yes	—
57	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
58	C₄²×C₅₃	C₄⋊C₄	1	8	yes	—
59	C₄⋊Dic₅₃	—	2	16	no	—
60	C₄⋊Dic₅₃	—	4	32	no	—
61	C₄⋊Dic₅₃	—	4	8	no	—
62	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
63	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂³×C₅₃	#124	1	2	yes	—
65	C₂³×C₅₃	#127	1	1	yes	characteristic
66	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂³×C₅₃	C₂×D₈	1	2	yes	—
68	C₂³×C₅₃	C₂×SD₁₆	1	4	yes	—
69	C₂³×C₅₃	C₂×Q₁₆	1	2	yes	—
70	C₂×C₄×C₅₃	#95	1	8	yes	—
71	C₂×C₄×C₅₃	#102	1	16	yes	—
72	C₂×C₄×C₅₃	#103	1	16	yes	—
73	C₂×C₄×C₅₃	#106	1	8	yes	—
74	C₂×C₄×C₅₃	#107	1	8	yes	—
75	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₅₃	#117	1	2	yes	—
80	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
81	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
82	C₂×C₄×C₅₃	#121	1	4	yes	—
83	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
84	C₂×C₄×C₅₃	#125	1	2	yes	—
85	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
86	C₂×Dic₅₃	—	4	8	no	—
87	C₂×Dic₅₃	—	4	16	no	—
88	C₂×Dic₅₃	—	8	8	no	—
89	C₂×Dic₅₃	—	8	4	no	—
90	C₂×Dic₅₃	—	8	4	no	—
91	C₈×C₅₃	—	2	8	no	—
92	#10841	—	53	1	no	maximal, Sylow
93	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
94	C₂²×C₅₃	#382	1	1	yes	characteristic
95	C₂²×C₅₃	#384	1	1	yes	characteristic
96	C₂²×C₅₃	#390	1	2	yes	—
97	C₂²×C₅₃	#391	1	2	yes	—
98	C₂²×C₅₃	C₂×#102	1	4	yes	—
99	C₂²×C₅₃	C₂×#103	1	4	yes	—
100	C₂²×C₅₃	C₂×#106	1	2	yes	—
101	C₂²×C₅₃	C₂×#107	1	2	yes	—
102	C₂²×C₅₃	#486	1	4	yes	—
103	C₂²×C₅₃	#487	1	4	yes	—
104	C₂²×C₅₃	#492	1	2	yes	—
105	C₂²×C₅₃	#493	1	1	yes	characteristic
106	C₂²×C₅₃	#494	1	1	yes	characteristic
107	C₂²×C₅₃	#498	1	2	yes	—
108	C₂²×C₅₃	#500	1	2	yes	—
109	C₄×C₅₃	#340	1	16	yes	complemented
110	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
111	C₄×C₅₃	#380	1	2	yes	—
112	C₄×C₅₃	#389	1	4	yes	—
113	Dic₅₃	—	8	8	no	—
114	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
115	C₂×#384	—	53	2	no	—
116	C₂×#340	—	53	4	no	—
117	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
118	C₂×C₅₃	C₂×#340	1	2	yes	—
119	C₂×C₅₃	#1834	1	2	yes	—
120	C₂×C₅₃	#1844	1	1	yes	characteristic
121	C₂×C₅₃	#1845	1	1	yes	characteristic
122	C₂×C₅₃	#1956	1	2	yes	—
123	C₂×C₅₃	#1957	1	2	yes	—
124	C₂×C₅₃	#1990	1	4	yes	—
125	C₂³×C₈	C₂×D₅₃	1	2	yes	—
126	C₂²×C₄⋊C₄	—	53	4	no	—
127	C₂²×C₄⋊C₄	—	106	4	no	—
128	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
129	C₂×C₄×C₈	—	2	4	no	—
130	C₂×C₄×C₈	Dic₅₃	1	4	yes	complemented
131	#384	—	53	8	no	—
132	#384	—	106	4	no	—
133	#340	—	106	8	no	—
134	C₅₃	#10841	1	1	yes	characteristic, complemented, Sylow
135	C₂³×C₄	—	106	4	no	—
136	C₂³×C₄	D₂₁₂	1	2	yes	—
137	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
138	C₂²×C₈	—	2	4	no	—
139	C₂²×C₈	—	2	4	no	—
140	C₂²×C₈	Dic₁₀₆	1	4	yes	—
141	C₂²×C₈	D₂₁₂	1	4	yes	—
142	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
143	C₂×C₄⋊C₄	—	53	8	no	—
144	C₂×C₄⋊C₄	—	106	8	no	—
145	C₂×C₄⋊C₄	—	106	8	no	—
146	C₂×C₄⋊C₄	—	106	4	no	—
147	C₂×C₄⋊C₄	—	106	16	no	—
148	C₂×C₄⋊C₄	—	212	8	no	—
149	C₂×C₄⋊C₄	—	212	8	no	—
150	C₂×C₄²	Dic₁₀₆	1	2	yes	—
151	C₂×C₄²	C₄×D₅₃	1	4	yes	—
152	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
153	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
154	C₄×C₈	—	2	8	no	—
155	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
156	C₂²×C₄	—	106	8	no	—
157	C₂²×C₄	—	212	4	no	—
158	C₂²×C₄	—	212	4	no	—
159	C₂²×C₄	—	212	8	no	—
160	C₂²×C₄	—	212	8	no	—
161	C₂²×C₄	#11719913	1	8	yes	—
162	C₂²×C₄	D₄₂₄	1	4	yes	—
163	C₂²×C₄	Dic₂₁₂	1	4	yes	—
164	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
165	C₂²×C₄	#11719921	1	8	yes	—
166	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
167	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
168	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
169	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
170	C₂²×C₄	#11719945	1	4	yes	—
171	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	—	2	8	no	—
177	C₂×C₈	C₄⋊Dic₅₃	1	8	yes	complemented
178	C₄⋊C₄	—	106	16	no	—
179	C₄⋊C₄	—	212	32	no	—
180	C₄⋊C₄	—	212	8	no	—
181	C₄²	C₄×Dic₅₃	1	4	yes	—
182	C₄²	#11719919	1	8	yes	—
183	C₄²	C₂²⋊Dic₅₃	1	4	yes	—
184	C₂³	C₂×#11719921	1	2	yes	—
185	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
186	C₂³	#13133205	1	2	yes	—
187	C₂³	#13133214	1	1	yes	characteristic
188	C₂³	#13133217	1	1	yes	characteristic
189	C₂³	C₂×D₄₂₄	1	2	yes	—
190	C₂³	C₂×#11719913	1	4	yes	—
191	C₂³	C₂×Dic₂₁₂	1	2	yes	—
192	C₂×C₄	—	424	8	no	—
193	C₂×C₄	—	424	4	no	—
194	C₂×C₄	—	424	4	no	—
195	C₂×C₄	—	212	8	no	—
196	C₂×C₄	—	212	16	no	—
197	C₂×C₄	#13133124	1	8	yes	—
198	C₂×C₄	#13133139	1	16	yes	—
199	C₂×C₄	#13133142	1	16	yes	—
200	C₂×C₄	#13133149	1	8	yes	—
201	C₂×C₄	#13133151	1	8	yes	—
202	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
203	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
204	C₂×C₄	C₂×#11719919	1	2	yes	—
205	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
206	C₂×C₄	#13133176	1	2	yes	—
207	C₂×C₄	C₄×D₂₁₂	1	4	yes	—
208	C₂×C₄	C₄×Dic₁₀₆	1	4	yes	—
209	C₂×C₄	#13133191	1	4	yes	—
210	C₂×C₄	#13133196	1	4	yes	—
211	C₂×C₄	#13133209	1	2	yes	—
212	C₂×C₄	#13133217	1	2	yes	—
213	C₈	—	2	8	no	—
214	C₂²	C₂×#13133124	1	1	yes	characteristic
215	C₂²	#16810489	1	1	yes	characteristic
216	C₂²	#16810493	1	1	yes	characteristic
217	C₂²	#16810525	1	2	yes	—
218	C₂²	#16810529	1	2	yes	—
219	C₂²	C₂×#13133139	1	4	yes	—
220	C₂²	C₂×#13133142	1	4	yes	—
221	C₂²	C₂×#13133149	1	2	yes	—
222	C₂²	C₂×#13133151	1	2	yes	—
223	C₂²	#16810970	1	4	yes	—
224	C₂²	#16810974	1	4	yes	—
225	C₂²	#16810996	1	2	yes	—
226	C₂²	#16811000	1	1	yes	characteristic
227	C₂²	#16811003	1	1	yes	characteristic
228	C₂²	#16811021	1	2	yes	—
229	C₂²	#16811028	1	2	yes	—
230	C₄	—	424	8	no	—
231	C₄	#16810386	1	16	yes	complemented
232	C₄	C₄×C₄⋊Dic₅₃	1	2	yes	—
233	C₄	#16810483	1	2	yes	—
234	C₄	#16810521	1	4	yes	—
235	C₂	#20272945	1	1	yes	characteristic
236	C₂	C₂×#16810386	1	2	yes	—
237	C₂	#20274843	1	2	yes	—
238	C₂	#20274872	1	1	yes	characteristic
239	C₂	#20274875	1	1	yes	characteristic
240	C₂	#20275517	1	2	yes	—
241	C₂	#20275521	1	2	yes	—
242	C₂	#20275676	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	#10841
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. #16810493	C₄	semidirect
12. C₂×C₄×C₈×C₅₃	C₄	semidirect
55. C₂×C₈×C₅₃	C₄⋊C₄	semidirect
109. C₄×C₅₃	#340	semidirect
130. C₂×C₄×C₈	Dic₅₃	semidirect
134. C₅₃	#10841	semidirect
177. C₂×C₈	C₄⋊Dic₅₃	semidirect
231. C₄	#16810386	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810493	2
C₂×#16810386	2
C₂²×C₄×C₈×C₅₃	2
#10841	53