Subgroups — #16811017

Order 3392.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₁₂	C₂	1	2	yes	maximal, complemented
2	C₂×#11719913	C₂	1	2	yes	maximal
3	#13133217	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133205	C₂	1	1	yes	maximal, characteristic
5	#97×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₁₂	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic
8	C₂×Dic₁₀₆	C₂²	1	2	yes	—
9	C₂×C₈×C₅₃	C₂²	1	2	yes	—
10	C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	#11719921	—	2	2	no	—
12	C₄⋊Dic₅₃	—	2	2	no	—
13	Dic₂₁₂	—	2	4	no	—
14	#11719913	—	2	4	no	—
15	C₂²×D₅₃	—	2	1	no	—
16	C₂×C₄×C₅₃	D₄	1	2	yes	—
17	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×Dic₅₃	—	2	2	no	—
19	C₂×Dic₅₃	—	2	1	no	—
20	D₂₁₂	—	2	2	no	—
21	Dic₁₀₆	—	2	2	no	—
22	Dic₁₀₆	—	2	2	no	—
23	Dic₁₀₆	—	2	2	no	—
24	C₈×C₅₃	D₄	1	4	yes	—
25	C₂²×C₅₃	C₂×D₄	1	1	yes	characteristic
26	C₂×D₅₃	—	4	1	no	—
27	C₂×D₅₃	—	4	2	no	—
28	C₄×C₅₃	—	2	2	no	—
29	C₄×C₅₃	C₂×D₄	1	2	yes	—
30	Dic₅₃	—	4	2	no	—
31	Dic₅₃	—	4	1	no	—
32	C₂×C₅₃	#127	1	1	yes	characteristic
33	C₂×C₅₃	#137	1	2	yes	—
34	D₅₃	—	8	1	no	—
35	#497	—	53	1	no	maximal, Sylow
36	C₅₃	#497	1	1	yes	characteristic, complemented, Sylow
37	C₂×Q₁₆	—	53	2	no	—
38	C₂×SD₁₆	—	53	2	no	—
39	C₄⋊Q₈	—	53	1	no	—
40	#124	—	53	1	no	—
41	#97	D₅₃	1	1	yes	characteristic, complemented
42	C₂×Q₈	—	53	1	no	—
43	C₂×Q₈	—	53	2	no	—
44	C₂×D₄	—	53	1	no	—
45	Q₁₆	—	106	4	no	—
46	SD₁₆	—	106	4	no	—
47	C₂×C₈	C₂×D₅₃	1	2	yes	—
48	C₄⋊C₄	—	106	2	no	—
49	C₂²⋊C₄	—	106	2	no	—
50	C₄²	C₂×D₅₃	1	1	yes	characteristic
51	C₂³	—	106	1	no	—
52	Q₈	—	106	2	no	—
53	Q₈	—	106	2	no	—
54	Q₈	—	106	2	no	—
55	D₄	—	106	2	no	—
56	C₂×C₄	—	106	1	no	—
57	C₂×C₄	—	106	2	no	—
58	C₂×C₄	D₂₁₂	1	2	yes	—
59	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
60	C₈	D₂₁₂	1	4	yes	—
61	C₂²	—	212	1	no	—
62	C₂²	—	212	2	no	—
63	C₂²	C₂×D₂₁₂	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	212	1	no	—
66	C₄	—	212	2	no	—
67	C₄	C₂×D₂₁₂	1	2	yes	—
68	C₂	—	424	1	no	—
69	C₂	#13133214	1	1	yes	characteristic
70	C₂	#13133250	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	C₂×D₂₁₂	C₂²
hypercenter	#97	D₅₃
hypocenter	C₅₃	#497
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
1. C₂×Dic₂₁₂	C₂	semidirect
3. #13133217	C₂	semidirect
5. #97×C₅₃	C₂	semidirect
36. C₅₃	#497	semidirect
41. #97	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₁₂	2
C₂×#11719913	2
#13133217	2
#13133205	2
#97×C₅₃	2
#497	53