Subgroups — #27273447

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274460	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810345	C₂	1	2	yes	maximal
4	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#88322	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	4	yes	—
9	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
10	#16810377	C₄	1	8	yes	—
11	#16810345	C₄	1	8	yes	complemented
12	C₈×#88322	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₈²×C₅₃	C₄	1	4	yes	complemented
16	C₂²×#88322	C₂×C₄	1	4	yes	—
17	C₂²×#88322	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
20	C₄×#88322	C₂×C₄	1	4	yes	—
21	C₄×#88322	C₂×C₄	1	4	yes	—
22	C₄×#88322	C₂×C₄	1	4	yes	—
23	C₈×Dic₅₃	C₈	1	16	yes	complemented
24	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
33	C₂²×Dic₅₃	C₄²	1	4	yes	—
34	C₂×C₈×C₅₃	—	2	8	no	—
35	C₂×C₈×C₅₃	C₄²	1	8	yes	—
36	C₂×C₈×C₅₃	C₄²	1	4	yes	—
37	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
38	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
39	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₄×Dic₅₃	—	2	8	no	—
42	C₄×Dic₅₃	C₂×C₈	1	8	yes	—
43	C₂×#88322	—	2	8	no	—
44	C₂×#88322	C₄²	1	8	yes	—
45	C₂×#88322	C₄²	1	4	yes	—
46	C₂×#88322	C₂×C₈	1	8	yes	—
47	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
52	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
53	C₂×C₄×C₅₃	#131	1	8	yes	—
54	C₂×Dic₅₃	—	2	8	no	—
55	C₂×Dic₅₃	C₄×C₈	1	8	yes	—
56	C₈×C₅₃	—	2	8	no	—
57	C₈×C₅₃	C₄×C₈	1	8	yes	—
58	#88322	—	2	8	no	—
59	#88322	C₄×C₈	1	8	yes	—
60	#7841	—	53	1	no	maximal, Sylow
61	C₂²×C₅₃	C₄³	1	1	yes	characteristic
62	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
63	C₂²×C₅₃	#405	1	4	yes	—
64	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
65	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
66	C₄×C₅₃	#405	1	4	yes	—
67	C₄×C₅₃	#405	1	4	yes	—
68	Dic₅₃	—	2	8	no	—
69	#1744	—	53	2	no	—
70	C₄²×C₈	—	53	1	no	—
71	C₂×#322	—	53	2	no	—
72	C₂×C₈²	—	53	1	no	—
73	C₂×C₈²	D₅₃	1	1	yes	characteristic
74	C₂×C₅₃	#1450	1	4	yes	—
75	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
76	C₂×C₅₃	#1726	1	2	yes	—
77	C₂×C₄×C₈	—	53	4	no	—
78	C₂×C₄×C₈	—	53	2	no	—
79	C₂×C₄×C₈	—	53	1	no	—
80	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
81	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
82	C₄³	—	53	1	no	—
83	#336	—	53	8	no	—
84	#322	—	53	8	no	—
85	C₈²	—	53	4	no	—
86	C₈²	Dic₅₃	1	4	yes	complemented
87	C₅₃	#7841	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	53	2	no	—
89	C₂²×C₈	—	53	4	no	—
90	C₂²×C₈	C₄×D₅₃	1	4	yes	—
91	C₂²×C₈	C₄×D₅₃	1	2	yes	—
92	C₂×C₄²	—	53	2	no	—
93	C₂×C₄²	—	53	4	no	—
94	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
95	C₄×C₈	—	53	16	no	—
96	C₄×C₈	—	53	4	no	—
97	C₄×C₈	—	53	4	no	—
98	C₄×C₈	—	53	4	no	—
99	C₄×C₈	C₄×D₅₃	1	2	yes	—
100	C₄×C₈	C₄×D₅₃	1	4	yes	—
101	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
102	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
103	C₂²×C₄	—	53	4	no	—
104	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
106	C₂×C₈	—	2	8	no	—
107	C₂×C₈	—	53	4	no	—
108	C₂×C₈	—	53	8	no	—
109	C₂×C₈	—	53	8	no	—
110	C₂×C₈	—	106	8	no	—
111	C₂×C₈	C₈×D₅₃	1	8	yes	—
112	C₂×C₈	C₄×Dic₅₃	1	8	yes	—
113	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
114	C₄²	—	53	8	no	—
115	C₄²	—	106	8	no	—
116	C₄²	C₂×C₄×D₅₃	1	2	yes	—
117	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
118	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
119	C₂³	C₄²×D₅₃	1	1	yes	characteristic
120	C₂×C₄	—	53	8	no	—
121	C₂×C₄	—	106	8	no	—
122	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
123	C₂×C₄	C₄²×D₅₃	1	2	yes	—
124	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
126	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
127	C₂×C₄	#13133223	1	8	yes	—
128	C₈	—	2	8	no	—
129	C₈	—	53	8	no	—
130	C₈	—	106	8	no	—
131	C₈	C₈×Dic₅₃	1	8	yes	—
132	C₂²	C₄²×Dic₅₃	1	1	yes	characteristic
133	C₂²	C₄×C₈×D₅₃	1	2	yes	—
134	C₂²	#16810577	1	4	yes	—
135	C₄	—	106	8	no	—
136	C₄	C₄×C₈×D₅₃	1	2	yes	—
137	C₄	C₂×C₈×Dic₅₃	1	2	yes	—
138	C₄	#16810577	1	4	yes	—
139	C₄	#16810578	1	4	yes	—
140	C₂	#20272997	1	4	yes	—
141	C₂	C₄×C₈×Dic₅₃	1	1	yes	characteristic
142	C₂	#20274401	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₄²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	#7841
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
11. #16810345	C₄	semidirect
12. C₈×#88322	C₄	semidirect
15. C₈²×C₅₃	C₄	semidirect
23. C₈×Dic₅₃	C₈	semidirect
86. C₈²	Dic₅₃	semidirect
87. C₅₃	#7841	semidirect

Maximal subgroups

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

subgroup	index
#20274460	2
C₄×C₈×Dic₅₃	2
C₂×#16810345	2
C₂×C₈×#88322	2
C₂×C₈²×C₅₃	2
#7841	53