Subgroups — #27275264

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133127	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×#13133127	C₄	1	2	yes	—
8	C₂×#13133127	C₄	1	2	yes	—
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
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	#13133127	—	2	4	no	—
26	#13133127	—	2	4	no	—
27	C₄×#88322	—	2	2	no	—
28	C₄×#88322	D₄	1	2	yes	—
29	C₄×#88322	Q₈	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₂²⋊C₈×C₅₃	—	2	2	no	—
35	C₄×C₈×C₅₃	—	2	2	no	—
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
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	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
43	C₂²×Dic₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	—	2	4	no	—
46	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
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	M₁₆	1	4	yes	—
51	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₄×Dic₅₃	—	2	2	no	—
56	C₄×Dic₅₃	—	4	2	no	—
57	C₄×Dic₅₃	—	4	2	no	—
58	C₄×Dic₅₃	M₁₆	1	4	yes	—
59	C₂×#88322	—	2	2	no	—
60	C₂×#88322	—	2	2	no	—
61	C₂×#88322	—	2	4	no	—
62	C₂×#88322	C₄⋊C₄	1	4	yes	—
63	C₂³×C₅₃	#117	1	1	yes	characteristic
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
74	C₂×Dic₅₃	—	2	4	no	—
75	C₂×Dic₅₃	—	4	2	no	—
76	C₂×Dic₅₃	—	4	2	no	—
77	C₈×C₅₃	—	2	1	no	—
78	C₈×C₅₃	—	2	1	no	—
79	C₈×C₅₃	—	4	4	no	—
80	C₈×C₅₃	#97	1	2	yes	—
81	C₈×C₅₃	#97	1	2	yes	—
82	#88322	—	2	4	no	—
83	#88322	—	4	4	no	—
84	#8269	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	1	no	—
86	C₂²×C₅₃	—	2	1	no	—
87	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₅₃	#432	1	2	yes	—
89	C₄×C₅₃	—	2	1	no	—
90	C₄×C₅₃	—	2	1	no	—
91	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
92	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
93	C₄×C₅₃	#423	1	2	yes	—
94	C₄×C₅₃	#444	1	2	yes	—
95	C₄×C₅₃	#444	1	2	yes	—
96	Dic₅₃	—	4	4	no	—
97	C₄×#97	—	53	1	no	—
98	#1564	—	53	1	no	—
99	#1564	D₅₃	1	1	yes	characteristic
100	#1452	—	53	2	no	—
101	#1276	—	53	2	no	—
102	C₂×C₅₃	—	2	2	no	—
103	C₂×C₅₃	#1767	1	1	yes	characteristic
104	C₂×C₅₃	#1773	1	1	yes	characteristic
105	C₂×C₅₃	#1773	1	1	yes	characteristic
106	#431	—	53	2	no	—
107	#431	C₂×D₅₃	1	2	yes	—
108	C₂×#97	—	53	2	no	—
109	C₂×#97	—	53	2	no	—
110	C₂×C₄×C₈	—	53	1	no	—
111	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
112	C₄³	—	53	1	no	—
113	#335	—	53	2	no	—
114	#335	Dic₅₃	1	2	yes	complemented
115	#334	—	53	2	no	—
116	#334	Dic₅₃	1	2	yes	complemented
117	#322	—	106	2	no	—
118	#322	—	106	2	no	—
119	C₅₃	#8269	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₈	—	53	2	no	—
121	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
122	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
123	C₂×C₄²	—	53	2	no	—
124	C₂×C₄²	—	106	1	no	—
125	C₂×C₄²	—	106	1	no	—
126	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
127	C₄⋊C₈	—	53	4	no	—
128	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
129	C₂²⋊C₈	—	2	2	no	—
130	C₂²⋊C₈	—	106	2	no	—
131	#97	—	106	4	no	—
132	#97	—	106	4	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	53	2	no	—
135	C₄×C₈	—	53	2	no	—
136	C₄×C₈	—	106	2	no	—
137	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
138	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
139	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
140	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
141	C₂²×C₄	—	106	2	no	—
142	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
143	C₂²×C₄	#11719947	1	1	yes	characteristic
144	C₂²×C₄	#11719949	1	1	yes	characteristic
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	53	4	no	—
148	C₂×C₈	—	106	2	no	—
149	C₂×C₈	—	106	2	no	—
150	C₂×C₈	—	106	4	no	—
151	C₂×C₈	#11719912	1	4	yes	—
152	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
153	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
154	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
155	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
156	C₄²	—	53	4	no	—
157	C₄²	—	106	2	no	—
158	C₄²	—	212	2	no	—
159	C₄²	—	212	2	no	—
160	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
161	C₄²	D₄×D₅₃	1	1	yes	characteristic
162	C₄²	Q₈×D₅₃	1	1	yes	characteristic
163	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
164	C₂³	#13133175	1	1	yes	characteristic
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	106	4	no	—
168	C₂×C₄	—	212	2	no	—
169	C₂×C₄	—	212	2	no	—
170	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
171	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
172	C₂×C₄	#13133179	1	1	yes	characteristic
173	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
174	C₂×C₄	#13133182	1	1	yes	characteristic
175	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
176	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
177	C₂×C₄	C₂×#11719912	1	2	yes	—
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	4	4	no	—
181	C₈	—	106	4	no	—
182	C₈	—	212	4	no	—
183	C₈	#13133127	1	2	yes	—
184	C₈	#13133127	1	2	yes	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
188	C₂²	#16810679	1	2	yes	—
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	212	4	no	—
192	C₄	C₂×#13133127	1	1	yes	characteristic
193	C₄	M₁₆×Dic₅₃	1	1	yes	characteristic
194	C₄	#16810645	1	2	yes	—
195	C₄	#16810744	1	2	yes	—
196	C₄	#16810745	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	#20274570	1	1	yes	characteristic
199	C₂	#20274601	1	1	yes	characteristic
200	C₂	#20274602	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	#8269
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
114. #335	Dic₅₃	semidirect
116. #334	Dic₅₃	semidirect
119. C₅₃	#8269	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133127	2
#20273463	2
#20273008	2
#20272547	2
C₅₃×#1564	2
#8269	53