Subgroups — #27300472

Order 13568.

342 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274875	C₂	1	2	yes	maximal
2	#20272807	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133151	—	2	2	no	—
5	C₂×#13133151	C₂²	1	4	yes	—
6	#16810493	—	2	2	no	—
7	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₅₃	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₅₃	—	2	4	no	—
11	C₂×C₄⋊Dic₅₃	—	4	2	no	—
12	C₂×C₄⋊Dic₅₃	—	4	2	no	—
13	#13133151	—	2	8	no	—
14	#13133151	—	4	4	no	—
15	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₃	—	2	4	no	—
19	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
20	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₅₃	Q₈	1	2	yes	—
22	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
25	C₂²×Dic₅₃	—	4	4	no	—
26	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
28	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₈×C₅₃	#41	1	4	yes	—
31	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
32	C₄²×C₅₃	Q₁₆	1	2	yes	—
33	C₄⋊Dic₅₃	—	4	4	no	—
34	C₄⋊Dic₅₃	—	4	4	no	—
35	C₄⋊Dic₅₃	—	8	4	no	—
36	C₄⋊Dic₅₃	—	8	4	no	—
37	C₁₆×C₅₃	—	2	4	no	—
38	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
39	C₂×C₄×C₅₃	#107	1	2	yes	—
40	C₂×C₄×C₅₃	#107	1	2	yes	—
41	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
42	C₂×C₄×C₅₃	SD₃₂	1	4	yes	—
43	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#124	1	2	yes	—
46	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
48	C₂×Dic₅₃	—	8	4	no	—
49	C₂×Dic₅₃	—	8	4	no	—
50	C₂×Dic₅₃	—	8	4	no	—
51	C₈×C₅₃	#106	1	4	yes	—
52	C₈×C₅₃	#117	1	2	yes	—
53	C₈×C₅₃	#125	1	2	yes	—
54	#12888	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
56	C₂²×C₅₃	#486	1	2	yes	—
57	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
58	C₂²×C₅₃	C₂×SD₃₂	1	2	yes	—
59	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
60	C₄×C₅₃	#366	1	4	yes	—
61	C₄×C₅₃	#367	1	4	yes	—
62	C₄×C₅₃	#382	1	1	yes	characteristic
63	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
64	C₄×C₅₃	#487	1	2	yes	—
65	Dic₅₃	—	16	4	no	—
66	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
67	#1845	—	53	2	no	—
68	#1380	—	53	4	no	—
69	C₂×C₅₃	#1834	1	1	yes	characteristic
70	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
71	C₂×C₅₃	C₂×#367	1	1	yes	characteristic
72	C₂×C₅₃	#2240	1	2	yes	—
73	C₂×C₅₃	#2241	1	2	yes	—
74	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
75	C₂×#107	—	53	4	no	—
76	C₂×#107	—	106	2	no	—
77	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
78	#384	—	106	2	no	—
79	C₄×C₁₆	Dic₅₃	1	4	yes	complemented
80	C₅₃	#12888	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
82	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
83	C₂×C₄⋊C₄	—	106	4	no	—
84	C₂×C₄⋊C₄	—	212	2	no	—
85	C₂×C₄⋊C₄	—	212	2	no	—
86	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
87	C₂×C₁₆	—	2	4	no	—
88	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
89	#107	—	106	8	no	—
90	#107	—	212	4	no	—
91	C₄×C₈	Dic₁₀₆	1	2	yes	—
92	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
93	C₂²×C₄	—	212	4	no	—
94	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
95	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
96	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
97	C₂×C₈	#11719913	1	4	yes	—
98	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
99	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
100	C₂×C₈	#11719945	1	4	yes	—
101	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
102	C₄⋊C₄	—	424	4	no	—
103	C₄⋊C₄	—	424	4	no	—
104	C₄⋊C₄	—	212	4	no	—
105	C₄⋊C₄	—	212	4	no	—
106	C₄²	Dic₂₁₂	1	2	yes	—
107	C₄²	C₄⋊Dic₅₃	1	2	yes	—
108	C₁₆	—	2	4	no	—
109	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
110	C₂×C₄	—	424	4	no	—
111	C₂×C₄	—	424	4	no	—
112	C₂×C₄	—	424	4	no	—
113	C₂×C₄	#13133151	1	2	yes	—
114	C₂×C₄	#13133151	1	2	yes	—
115	C₂×C₄	D₈₄₈	1	2	yes	—
116	C₂×C₄	#13133162	1	4	yes	—
117	C₂×C₄	Dic₄₂₄	1	2	yes	—
118	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
119	C₂×C₄	#13133205	1	2	yes	—
120	C₂×C₄	C₂×#11719913	1	2	yes	—
121	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
122	C₈	#13133149	1	4	yes	—
123	C₈	#13133176	1	2	yes	—
124	C₈	#13133209	1	2	yes	—
125	C₂²	C₂×#13133151	1	1	yes	characteristic
126	C₂²	#16810970	1	2	yes	—
127	C₂²	C₂×D₈₄₈	1	1	yes	characteristic
128	C₂²	C₂×#13133162	1	2	yes	—
129	C₂²	C₂×Dic₄₂₄	1	1	yes	characteristic
130	C₄	—	848	4	no	—
131	C₄	#16810449	1	4	yes	—
132	C₄	#16810451	1	4	yes	—
133	C₄	#16810489	1	1	yes	characteristic
134	C₄	C₂×#13133149	1	1	yes	characteristic
135	C₄	#16810974	1	2	yes	—
136	C₂	#20274843	1	1	yes	characteristic
137	C₂	C₂×#16810449	1	1	yes	characteristic
138	C₂	C₂×#16810451	1	1	yes	characteristic
139	C₂	#20276939	1	2	yes	—
140	C₂	#20276943	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#12888
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
9. C₄×C₁₆×C₅₃	C₄	semidirect
79. C₄×C₁₆	Dic₅₃	semidirect
80. C₅₃	#12888	semidirect

Maximal subgroups

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

subgroup	index
#20274875	2
#20272807	2
C₂×C₄×C₁₆×C₅₃	2
#12888	53