Subgroups — #16810988

Order 3392.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133214	C₂	1	1	yes	maximal, characteristic
2	#13133209	C₂	1	1	yes	maximal, characteristic, complemented
3	#13133139	C₂	1	4	yes	maximal, complemented
4	#97×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₂₁₂	—	2	2	no	—
6	C₂×D₂₁₂	C₂²	1	2	yes	—
7	C₂×C₈×C₅₃	C₂²	1	2	yes	—
8	C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₅₃	C₂²	1	2	yes	—
10	#11719919	—	2	2	no	—
11	C₂²×D₅₃	—	2	2	no	—
12	C₂×C₄×C₅₃	D₄	1	2	yes	—
13	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
14	C₂×Dic₅₃	—	2	2	no	—
15	D₂₁₂	—	2	4	no	—
16	D₂₁₂	—	4	4	no	—
17	C₈×C₅₃	—	2	2	no	—
18	C₂²×C₅₃	C₂×D₄	1	1	yes	characteristic
19	C₂×D₅₃	—	4	2	no	—
20	C₂×D₅₃	—	4	4	no	—
21	C₄×C₅₃	—	2	2	no	—
22	C₄×C₅₃	#41	1	2	yes	—
23	Dic₅₃	—	4	2	no	—
24	C₂×C₅₃	#124	1	1	yes	characteristic
25	C₂×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
26	D₅₃	—	8	2	no	—
27	#490	—	53	1	no	maximal, Sylow
28	C₅₃	#490	1	1	yes	characteristic, complemented, Sylow
29	#127	—	53	1	no	—
30	#125	—	53	1	no	—
31	#102	—	53	4	no	—
32	#97	D₅₃	1	1	yes	characteristic, complemented
33	C₂×D₄	—	53	2	no	—
34	C₂×D₄	—	106	2	no	—
35	C₂×C₈	C₂×D₅₃	1	2	yes	—
36	C₄⋊C₄	—	53	2	no	—
37	C₄⋊C₄	—	106	2	no	—
38	C₄²	C₂×D₅₃	1	1	yes	characteristic
39	C₂³	—	106	2	no	—
40	D₄	—	106	4	no	—
41	D₄	—	212	4	no	—
42	C₂×C₄	—	106	2	no	—
43	C₂×C₄	D₂₁₂	1	2	yes	—
44	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
45	C₈	—	2	2	no	—
46	C₂²	—	212	2	no	—
47	C₂²	—	212	4	no	—
48	C₂²	C₂×D₂₁₂	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	212	2	no	—
51	C₄	#11719945	1	2	yes	—
52	C₂	—	424	2	no	—
53	C₂	#13133205	1	1	yes	characteristic
54	C₂	#13133245	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₁₂
commutator, cosocle	C₂×C₄×C₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	#97×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₄
prime radical	#13133214	C₂
hypercenter	#97	D₅₃
hypocenter	C₅₃	#490
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
2. #13133209	C₂	semidirect
3. #13133139	C₂	semidirect
4. #97×C₅₃	C₂	semidirect
28. C₅₃	#490	semidirect
32. #97	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#13133214	2
#13133209	2
#13133139	2
#97×C₅₃	2
#490	53