Subgroups — #27536614

Order 13568.

1422 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811003	C₂	1	12	yes	maximal
2	#20283113	C₂	1	2	yes	maximal
3	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₁₂	C₂²	1	16	yes	—
5	C₂×#13133217	C₂²	1	12	yes	—
6	#16811003	—	2	48	no	—
7	#16810493	—	2	8	no	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	6	yes	—
9	C₄³×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₀₆	C₂³	1	8	yes	—
11	C₂×Dic₂₁₂	—	2	96	no	—
12	#13133217	—	2	48	no	—
13	C₂×C₄⋊Dic₅₃	—	2	24	no	—
14	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
15	C₂×C₄²×C₅₃	D₄	1	4	yes	—
16	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
17	C₄×C₈×C₅₃	D₄	1	24	yes	—
18	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
19	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
20	C₂²×Dic₅₃	—	2	8	no	—
21	C₂×Dic₁₀₆	—	2	48	no	—
22	C₂×C₈×C₅₃	C₂×D₄	1	24	yes	—
23	C₄²×C₅₃	Q₁₆	1	16	yes	complemented
24	C₄²×C₅₃	C₂×D₄	1	12	yes	—
25	C₄⋊Dic₅₃	—	4	48	no	—
26	Dic₂₁₂	—	4	64	no	—
27	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	#127	1	12	yes	—
29	C₂×C₄×C₅₃	C₂×Q₁₆	1	24	yes	—
30	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
31	C₂×Dic₅₃	—	4	24	no	—
32	Dic₁₀₆	—	4	32	no	—
33	C₈×C₅₃	#127	1	16	yes	—
34	#33302	—	53	1	no	maximal, Sylow
35	C₂²×C₅₃	C₂×#127	1	3	yes	—
36	C₂²×C₅₃	C₂²×Q₁₆	1	4	yes	—
37	C₄×C₅₃	#494	1	24	yes	complemented
38	C₄×C₅₃	C₂×#127	1	4	yes	—
39	Dic₅₃	—	8	8	no	—
40	C₂×#494	—	53	12	no	—
41	#2868	—	53	2	no	—
42	C₄²×C₈	D₅₃	1	1	yes	characteristic
43	C₂×C₅₃	#2867	1	1	yes	characteristic
44	C₂×C₅₃	C₂×#494	1	6	yes	—
45	C₂²×Q₁₆	—	53	16	no	—
46	C₂×C₄⋊Q₈	—	53	12	no	—
47	#494	—	106	48	no	—
48	C₂×C₄×C₈	C₂×D₅₃	1	6	yes	—
49	#384	—	106	8	no	—
50	C₄³	C₂×D₅₃	1	1	yes	characteristic
51	C₅₃	#33302	1	1	yes	characteristic, complemented, Sylow
52	C₂²×Q₈	—	53	8	no	—
53	C₂×Q₁₆	—	106	96	no	—
54	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
55	C₄⋊Q₈	—	106	48	no	—
56	C₂×C₄⋊C₄	—	106	24	no	—
57	C₂×C₄²	D₂₁₂	1	4	yes	—
58	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
59	C₄×C₈	D₂₁₂	1	24	yes	—
60	C₂×Q₈	—	106	48	no	—
61	C₂²×C₄	—	106	8	no	—
62	C₂²×C₄	C₂×D₂₁₂	1	6	yes	—
63	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
64	Q₁₆	—	212	64	no	—
65	C₂×C₈	C₂×D₂₁₂	1	24	yes	—
66	C₄⋊C₄	—	212	48	no	—
67	C₄²	Dic₂₁₂	1	16	yes	complemented
68	C₄²	C₂×D₂₁₂	1	12	yes	—
69	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
70	Q₈	—	212	32	no	—
71	C₂×C₄	—	212	24	no	—
72	C₂×C₄	#13133214	1	12	yes	—
73	C₂×C₄	C₂×Dic₂₁₂	1	24	yes	—
74	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
75	C₈	#13133214	1	16	yes	—
76	C₂²	C₂×#13133214	1	3	yes	—
77	C₂²	C₂²×Dic₂₁₂	1	4	yes	—
78	C₄	—	424	8	no	—
79	C₄	#16811003	1	24	yes	complemented
80	C₄	C₂×#13133214	1	4	yes	—
81	C₂	#20283110	1	1	yes	characteristic
82	C₂	C₂×#16811003	1	6	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₁₂
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#33302
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
23. C₄²×C₅₃	Q₁₆	semidirect
37. C₄×C₅₃	#494	semidirect
51. C₅₃	#33302	semidirect
67. C₄²	Dic₂₁₂	semidirect
79. C₄	#16811003	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811003	2
#20283113	2
C₄²×C₈×C₅₃	2
#33302	53