Subgroups — #27300529

Order 13568.

374 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810451	C₂	1	4	yes	maximal
2	#20274875	C₂	1	2	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	#16810451	—	2	8	no	—
8	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₅₃	C₄	1	4	yes	complemented
11	C₂×C₄⋊Dic₅₃	—	2	4	no	—
12	C₂×C₄⋊Dic₅₃	—	4	4	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	C₂×C₄	1	4	yes	—
19	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
20	C₂×C₁₆×C₅₃	Q₈	1	4	yes	—
21	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₃	Q₈	1	2	yes	—
23	C₂²×C₄×C₅₃	D₈	1	2	yes	—
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	D₈	1	2	yes	—
28	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
29	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
31	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
32	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
34	C₄²×C₅₃	Q₁₆	1	2	yes	—
35	C₄⋊Dic₅₃	—	4	4	no	—
36	C₄⋊Dic₅₃	—	4	4	no	—
37	C₄⋊Dic₅₃	—	8	8	no	—
38	C₁₆×C₅₃	C₄⋊C₄	1	8	yes	complemented
39	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#107	1	4	yes	—
41	C₂×C₄×C₅₃	SD₃₂	1	8	yes	—
42	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
48	C₂×Dic₅₃	—	8	4	no	—
49	C₂×Dic₅₃	—	8	8	no	—
50	C₈×C₅₃	#107	1	4	yes	—
51	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
52	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
53	#12906	—	53	1	no	maximal, Sylow
54	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
55	C₂²×C₅₃	#493	1	1	yes	characteristic
56	C₂²×C₅₃	#500	1	1	yes	characteristic
57	C₂²×C₅₃	C₂×SD₃₂	1	4	yes	—
58	C₄×C₅₃	#367	1	8	yes	complemented
59	C₄×C₅₃	#384	1	1	yes	characteristic
60	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
61	C₄×C₅₃	#494	1	1	yes	characteristic
62	C₄×C₅₃	#500	1	1	yes	characteristic
63	Dic₅₃	—	16	4	no	—
64	C₂×#367	—	53	4	no	—
65	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
66	#1845	—	53	2	no	—
67	C₂×C₅₃	#1845	1	1	yes	characteristic
68	C₂×C₅₃	C₂×#367	1	2	yes	—
69	C₂×C₅₃	#2248	1	2	yes	—
70	C₂×C₅₃	#2254	1	2	yes	—
71	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
72	C₂×#107	—	53	4	no	—
73	C₂×#107	—	106	2	no	—
74	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
75	#384	—	106	2	no	—
76	#367	—	106	8	no	—
77	C₄×C₁₆	Dic₅₃	1	4	yes	complemented
78	C₅₃	#12906	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
80	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
81	C₂×C₄⋊C₄	—	106	4	no	—
82	C₂×C₄⋊C₄	—	212	4	no	—
83	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
84	C₂×C₁₆	Dic₁₀₆	1	4	yes	—
85	C₂×C₁₆	D₂₁₂	1	4	yes	—
86	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
87	#107	—	106	8	no	—
88	#107	—	212	4	no	—
89	C₄×C₈	Dic₁₀₆	1	2	yes	—
90	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
91	C₂²×C₄	—	212	4	no	—
92	C₂²×C₄	D₄₂₄	1	2	yes	—
93	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
94	C₂×C₈	D₄₂₄	1	2	yes	—
95	C₂×C₈	Dic₂₁₂	1	2	yes	—
96	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
97	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
98	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
99	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
100	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
101	C₄⋊C₄	—	424	8	no	—
102	C₄⋊C₄	—	212	4	no	—
103	C₄⋊C₄	—	212	4	no	—
104	C₄²	Dic₂₁₂	1	2	yes	—
105	C₄²	C₄⋊Dic₅₃	1	2	yes	—
106	C₁₆	C₄⋊Dic₅₃	1	8	yes	complemented
107	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
108	C₂×C₄	—	424	8	no	—
109	C₂×C₄	—	424	4	no	—
110	C₂×C₄	#13133151	1	4	yes	—
111	C₂×C₄	#13133162	1	8	yes	—
112	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
113	C₂×C₄	#13133214	1	1	yes	characteristic
114	C₂×C₄	#13133217	1	1	yes	characteristic
115	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
116	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
117	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
118	C₈	#13133151	1	4	yes	—
119	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
120	C₈	#13133217	1	2	yes	—
121	C₂²	C₂×#13133151	1	1	yes	characteristic
122	C₂²	#16811000	1	1	yes	characteristic
123	C₂²	#16811028	1	1	yes	characteristic
124	C₂²	C₂×#13133162	1	4	yes	—
125	C₄	—	848	4	no	—
126	C₄	#16810451	1	8	yes	complemented
127	C₄	#16810493	1	1	yes	characteristic
128	C₄	C₂×#13133151	1	1	yes	characteristic
129	C₄	#16811003	1	1	yes	characteristic
130	C₄	#16811028	1	1	yes	characteristic
131	C₂	#20274875	1	1	yes	characteristic
132	C₂	C₂×#16810451	1	2	yes	—
133	C₂	#20276971	1	2	yes	—
134	C₂	#20276997	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	#12906
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
10. C₄×C₁₆×C₅₃	C₄	semidirect
38. C₁₆×C₅₃	C₄⋊C₄	semidirect
58. C₄×C₅₃	#367	semidirect
77. C₄×C₁₆	Dic₅₃	semidirect
78. C₅₃	#12906	semidirect
106. C₁₆	C₄⋊Dic₅₃	semidirect
126. C₄	#16810451	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810451	2
#20274875	2
C₂×C₄×C₁₆×C₅₃	2
#12906	53