Subgroups — #16810981

Order 3392.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133216	C₂	1	1	yes	maximal, characteristic, complemented
2	#13133204	C₂	1	1	yes	maximal, characteristic
3	#13133142	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133140	C₂	1	1	yes	maximal, characteristic, complemented
5	#13133138	C₂	1	1	yes	maximal, characteristic
6	#13133127	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×D₄×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₀₆	—	2	1	no	—
10	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₅₃	—	2	1	no	—
14	C₂²⋊Dic₅₃	—	2	1	no	—
15	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
16	#11719919	—	2	1	no	—
17	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂×#88322	C₂²	1	1	yes	characteristic
19	C₂³×C₅₃	—	2	1	no	—
20	D₄×C₅₃	—	2	1	no	—
21	D₄×C₅₃	—	2	1	no	—
22	C₂×C₄×C₅₃	—	2	1	no	—
23	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×Dic₅₃	—	2	1	no	—
25	C₂×Dic₅₃	—	2	1	no	—
26	C₂×Dic₅₃	D₄	1	1	yes	characteristic
27	C₂×Dic₅₃	D₄	1	1	yes	characteristic
28	Dic₁₀₆	—	2	1	no	—
29	Dic₁₀₆	—	2	1	no	—
30	Dic₁₀₆	—	4	2	no	—
31	C₈×C₅₃	—	2	1	no	—
32	#88322	—	2	1	no	—
33	C₂²×C₅₃	—	4	1	no	—
34	C₂²×C₅₃	—	4	1	no	—
35	C₂²×C₅₃	—	4	1	no	—
36	C₂²×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₄×C₅₃	—	4	1	no	—
38	C₄×C₅₃	#41	1	1	yes	characteristic
39	C₄×C₅₃	#41	1	1	yes	characteristic
40	Dic₅₃	—	2	1	no	—
41	Dic₅₃	—	2	1	no	—
42	Dic₅₃	—	4	1	no	—
43	Dic₅₃	—	4	1	no	—
44	C₂×C₅₃	—	8	1	no	—
45	C₂×C₅₃	#124	1	1	yes	characteristic
46	C₂×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
47	C₂×C₅₃	#137	1	1	yes	characteristic
48	#489	—	53	1	no	maximal, Sylow
49	C₅₃	#489	1	1	yes	characteristic, complemented, Sylow
50	C₄⋊Q₈	—	53	1	no	—
51	#124	—	53	1	no	—
52	#103	—	53	1	no	—
53	#103	—	53	1	no	—
54	#102	—	53	1	no	—
55	#102	D₅₃	1	1	yes	characteristic
56	#97	—	53	1	no	—
57	C₂×Q₈	—	53	1	no	—
58	C₂×Q₈	—	106	1	no	—
59	C₂×D₄	C₂×D₅₃	1	1	yes	characteristic
60	C₂×C₈	—	53	1	no	—
61	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
62	C₄⋊C₄	—	53	1	no	—
63	C₄⋊C₄	—	106	1	no	—
64	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
65	C₂²⋊C₄	—	106	1	no	—
66	C₂²⋊C₄	—	106	1	no	—
67	C₄²	—	53	1	no	—
68	C₂³	—	2	1	no	—
69	Q₈	—	106	1	no	—
70	Q₈	—	106	1	no	—
71	Q₈	—	212	2	no	—
72	D₄	—	2	1	no	—
73	D₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	53	1	no	—
76	C₂×C₄	—	53	1	no	—
77	C₂×C₄	—	106	1	no	—
78	C₂×C₄	—	106	1	no	—
79	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	106	1	no	—
82	C₂²	—	4	1	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	4	1	no	—
85	C₂²	D₄×D₅₃	1	1	yes	characteristic
86	C₄	—	4	1	no	—
87	C₄	—	106	1	no	—
88	C₄	—	106	1	no	—
89	C₄	—	212	1	no	—
90	C₄	—	212	1	no	—
91	C₄	#11719945	1	1	yes	characteristic
92	C₄	#11719947	1	1	yes	characteristic
93	C₂	—	8	1	no	—
94	C₂	#13133203	1	1	yes	characteristic
95	C₂	#13133241	1	1	yes	characteristic
96	C₂	#13133246	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	#102×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₄
prime radical	C₂×D₄×C₅₃	C₂²
hypercenter	#102	D₅₃
hypocenter	C₅₃	#489
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
1. #13133216	C₂	semidirect
3. #13133142	C₂	semidirect
4. #13133140	C₂	semidirect
6. #13133127	C₂	semidirect
49. C₅₃	#489	semidirect

Maximal subgroups

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

subgroup	index
#13133216	2
#13133204	2
#13133142	2
#13133140	2
#13133138	2
#13133127	2
#102×C₅₃	2
#489	53