Subgroups — #27275256

Order 13568.

390 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
2	#20273463	C₂	1	1	yes	maximal, characteristic
3	#20273008	C₂	1	2	yes	maximal
4	#20272547	C₂	1	2	yes	maximal
5	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
9	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
10	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
11	#16810376	C₄	1	2	yes	complemented
12	#16810374	C₄	1	2	yes	complemented
13	#16810345	—	2	2	no	—
14	#16810344	—	2	2	no	—
15	C₅₃×#431	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#335	C₄	1	2	yes	complemented
18	C₅₃×#334	C₄	1	2	yes	complemented
19	C₂²×#88322	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₅₃	—	2	1	no	—
21	C₂×C₄×Dic₅₃	—	2	1	no	—
22	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
23	C₄⋊#88322	C₂×C₄	1	4	yes	—
24	C₂²⋊#88322	—	2	2	no	—
25	C₄×#88322	—	2	2	no	—
26	C₄×#88322	D₄	1	1	yes	characteristic
27	C₄×#88322	D₄	1	1	yes	characteristic
28	C₄×#88322	Q₈	1	2	yes	—
29	C₈×Dic₅₃	—	2	4	no	—
30	C₈×Dic₅₃	—	2	4	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
35	C₂²⋊C₈×C₅₃	—	2	2	no	—
36	C₄×C₈×C₅₃	—	2	2	no	—
37	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
44	C₂²×Dic₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	4	no	—
47	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
52	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₄×Dic₅₃	—	2	4	no	—
57	C₄×Dic₅₃	—	2	4	no	—
58	C₄×Dic₅₃	—	2	2	no	—
59	C₄×Dic₅₃	M₁₆	1	4	yes	—
60	C₂×#88322	—	2	2	no	—
61	C₂×#88322	—	2	2	no	—
62	C₂×#88322	—	2	4	no	—
63	C₂×#88322	C₄⋊C₄	1	2	yes	—
64	C₂×#88322	C₄⋊C₄	1	2	yes	—
65	C₂³×C₅₃	#117	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
76	C₂×Dic₅₃	—	2	4	no	—
77	C₂×Dic₅₃	—	2	4	no	—
78	C₂×Dic₅₃	—	2	4	no	—
79	C₈×C₅₃	—	2	1	no	—
80	C₈×C₅₃	—	2	1	no	—
81	C₈×C₅₃	—	4	4	no	—
82	C₈×C₅₃	#97	1	2	yes	—
83	C₈×C₅₃	#97	1	2	yes	—
84	#88322	—	2	2	no	—
85	#88322	—	2	2	no	—
86	#88322	—	4	4	no	—
87	#8267	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	—	2	1	no	—
89	C₂²×C₅₃	—	2	1	no	—
90	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₅₃	#432	1	2	yes	—
92	C₄×C₅₃	—	2	1	no	—
93	C₄×C₅₃	—	2	1	no	—
94	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
95	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
96	C₄×C₅₃	#423	1	2	yes	—
97	C₄×C₅₃	#443	1	2	yes	—
98	C₄×C₅₃	#443	1	2	yes	—
99	Dic₅₃	—	2	8	no	—
100	C₄²×C₈	—	53	1	no	—
101	#1564	—	53	1	no	—
102	#1564	D₅₃	1	1	yes	characteristic
103	#1452	—	53	2	no	—
104	#1276	—	53	2	no	—
105	C₂×C₅₃	—	2	2	no	—
106	C₂×C₅₃	#1767	1	1	yes	characteristic
107	C₂×C₅₃	#1772	1	1	yes	characteristic
108	C₂×C₅₃	#1772	1	1	yes	characteristic
109	#431	—	53	2	no	—
110	#431	C₂×D₅₃	1	2	yes	—
111	C₂×C₄×C₈	—	53	1	no	—
112	C₂×C₄×C₈	—	53	2	no	—
113	C₂×C₄×C₈	—	53	2	no	—
114	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
115	C₄³	—	53	1	no	—
116	#335	—	53	2	no	—
117	#335	Dic₅₃	1	2	yes	complemented
118	#334	—	53	2	no	—
119	#334	Dic₅₃	1	2	yes	complemented
120	#322	—	106	2	no	—
121	#322	—	106	2	no	—
122	C₅₃	#8267	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₈	—	53	2	no	—
124	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
125	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
126	C₂×C₄²	—	53	2	no	—
127	C₂×C₄²	—	106	1	no	—
128	C₂×C₄²	—	106	1	no	—
129	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
130	C₄⋊C₈	—	53	4	no	—
131	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
132	C₂²⋊C₈	—	2	2	no	—
133	C₂²⋊C₈	—	106	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	—	53	1	no	—
136	C₄×C₈	—	53	1	no	—
137	C₄×C₈	—	53	2	no	—
138	C₄×C₈	—	106	4	no	—
139	C₄×C₈	—	106	4	no	—
140	C₄×C₈	—	106	2	no	—
141	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
142	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
143	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
144	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
145	C₂²×C₄	—	106	2	no	—
146	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
147	C₂²×C₄	#11719947	1	1	yes	characteristic
148	C₂²×C₄	#11719949	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	53	2	no	—
152	C₂×C₈	—	53	2	no	—
153	C₂×C₈	—	106	2	no	—
154	C₂×C₈	—	106	2	no	—
155	C₂×C₈	—	106	4	no	—
156	C₂×C₈	#11719912	1	4	yes	—
157	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
158	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
159	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
160	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
161	C₄²	—	53	4	no	—
162	C₄²	—	106	4	no	—
163	C₄²	—	106	4	no	—
164	C₄²	—	106	2	no	—
165	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
166	C₄²	D₄×D₅₃	1	1	yes	characteristic
167	C₄²	Q₈×D₅₃	1	1	yes	characteristic
168	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
169	C₂³	#13133175	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	106	4	no	—
173	C₂×C₄	—	106	4	no	—
174	C₂×C₄	—	106	4	no	—
175	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
176	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
177	C₂×C₄	#13133179	1	1	yes	characteristic
178	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
179	C₂×C₄	#13133182	1	1	yes	characteristic
180	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
181	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
182	C₂×C₄	C₂×#11719912	1	2	yes	—
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	4	4	no	—
186	C₈	—	106	2	no	—
187	C₈	—	106	2	no	—
188	C₈	—	212	4	no	—
189	C₈	#13133127	1	2	yes	—
190	C₈	#13133127	1	2	yes	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
194	C₂²	#16810679	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	106	8	no	—
198	C₄	C₂×#13133127	1	1	yes	characteristic
199	C₄	M₁₆×Dic₅₃	1	1	yes	characteristic
200	C₄	#16810645	1	2	yes	—
201	C₄	#16810739	1	2	yes	—
202	C₄	#16810740	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	#20274570	1	1	yes	characteristic
205	C₂	#20274596	1	1	yes	characteristic
206	C₂	#20274597	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#8267
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
11. #16810376	C₄	semidirect
12. #16810374	C₄	semidirect
17. C₅₃×#335	C₄	semidirect
18. C₅₃×#334	C₄	semidirect
117. #335	Dic₅₃	semidirect
119. #334	Dic₅₃	semidirect
122. C₅₃	#8267	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₈×Dic₅₃	2
#20273463	2
#20273008	2
#20272547	2
C₅₃×#1564	2
#8267	53