Subgroups — #16810893

Order 3392.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133242	C₂	1	2	yes	maximal, complemented
2	C₂×D₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#13133153	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133133	C₂	1	2	yes	maximal, complemented
5	#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×D₂₁₂	C₂²	1	2	yes	—
8	M₁₆×C₅₃	—	2	1	no	—
9	M₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₅₃	—	2	1	no	—
11	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	#11719924	—	2	2	no	—
13	#11719922	—	2	2	no	—
14	#11719917	C₂²	1	2	yes	—
15	D₄₂₄	—	2	1	no	—
16	D₄₂₄	—	2	1	no	—
17	C₂²×D₅₃	—	2	2	no	—
18	Q₈×C₅₃	D₄	1	1	yes	characteristic
19	D₄×C₅₃	—	2	1	no	—
20	D₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅₃	—	2	1	no	—
22	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	D₂₁₂	—	2	2	no	—
24	D₂₁₂	—	2	2	no	—
25	C₈×C₅₃	—	2	1	no	—
26	C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₈×C₅₃	D₄	1	1	yes	characteristic
28	#88322	—	2	2	no	—
29	C₂²×C₅₃	—	2	1	no	—
30	C₂²×C₅₃	#41	1	1	yes	characteristic
31	C₂×D₅₃	—	4	2	no	—
32	C₂×D₅₃	—	8	2	no	—
33	C₄×C₅₃	—	2	1	no	—
34	C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₅₃	—	2	1	no	—
37	C₂×C₅₃	—	4	1	no	—
38	C₂×C₅₃	#121	1	1	yes	characteristic
39	D₅₃	—	8	2	no	—
40	#472	—	53	1	no	maximal, Sylow
41	C₅₃	#472	1	1	yes	characteristic, complemented, Sylow
42	AGL(1,ℤ/8)	—	53	2	no	—
43	C₂×D₈	—	53	1	no	—
44	#131	D₅₃	1	1	yes	characteristic, complemented
45	#108	—	53	1	no	—
46	#100	—	53	2	no	—
47	#41	C₂×D₅₃	1	1	yes	characteristic
48	C₂×D₄	—	53	2	no	—
49	SD₁₆	—	106	2	no	—
50	D₈	—	106	2	no	—
51	D₈	—	106	1	no	—
52	D₈	—	106	1	no	—
53	M₁₆	—	2	1	no	—
54	M₁₆	—	53	2	no	—
55	M₁₆	C₂×D₅₃	1	1	yes	characteristic
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
58	C₂³	—	106	2	no	—
59	Q₈	D₂₁₂	1	1	yes	characteristic
60	D₄	—	2	1	no	—
61	D₄	—	106	2	no	—
62	D₄	—	106	2	no	—
63	D₄	D₂₁₂	1	1	yes	characteristic
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
66	C₈	—	2	1	no	—
67	C₈	—	106	2	no	—
68	C₈	C₂²⋊D₅₃	1	1	yes	characteristic
69	C₈	C₂²⋊D₅₃	1	1	yes	characteristic
70	C₂²	—	2	1	no	—
71	C₂²	—	424	2	no	—
72	C₂²	—	212	2	no	—
73	C₂²	#11719945	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	C₂×D₂₁₂	1	1	yes	characteristic
76	C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	4	1	no	—
79	C₂	—	424	2	no	—
80	C₂	#13133191	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13133191
commutator, cosocle	C₂×C₄×C₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	#131×C₅₃	C₂
socle	C₂×C₅₃	#121
hypercenter	#131	D₅₃
hypocenter	C₅₃	#472
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131

Complements

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

N	H ≅ G/N	type
1. #13133242	C₂	semidirect
2. C₂×D₄₂₄	C₂	semidirect
3. #13133153	C₂	semidirect
4. #13133133	C₂	semidirect
5. #131×C₅₃	C₂	semidirect
41. C₅₃	#472	semidirect
44. #131	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#13133242	2
C₂×D₄₂₄	2
#13133153	2
#13133133	2
#131×C₅₃	2
#472	53