Subgroups — #16811002

Order 3392.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719925	C₂	1	4	yes	maximal
2	#13133217	C₂	1	1	yes	maximal, characteristic
3	C₄×#88322	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×C₅₃	C₂²	1	2	yes	—
6	C₂×Dic₁₀₆	C₂²	1	2	yes	—
7	C₄⋊C₄×C₅₃	—	2	2	no	—
8	C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	#11719925	—	2	8	no	—
10	C₄⋊Dic₅₃	—	2	2	no	—
11	C₂×#88322	C₂²	1	2	yes	—
12	Q₈×C₅₃	—	2	4	no	—
13	C₂×C₄×C₅₃	—	2	2	no	—
14	C₂×C₄×C₅₃	D₄	1	2	yes	—
15	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	C₂×Dic₅₃	—	2	2	no	—
17	Dic₁₀₆	—	2	4	no	—
18	#88322	D₄	1	4	yes	—
19	C₂²×C₅₃	C₂×D₄	1	1	yes	characteristic
20	C₄×C₅₃	—	4	2	no	—
21	C₄×C₅₃	Q₁₆	1	4	yes	complemented
22	C₄×C₅₃	C₂×D₄	1	2	yes	—
23	Dic₅₃	—	4	2	no	—
24	C₂×C₅₃	#127	1	1	yes	characteristic
25	C₂×C₅₃	C₂×Q₁₆	1	2	yes	—
26	#494	—	53	1	no	maximal, Sylow
27	C₅₃	#494	1	1	yes	characteristic, complemented, Sylow
28	C₂×Q₁₆	—	53	4	no	—
29	C₄⋊Q₈	—	53	1	no	—
30	C₄⋊Q₈	D₅₃	1	1	yes	characteristic
31	C₄×C₈	—	53	1	no	—
32	C₂×Q₈	—	53	2	no	—
33	C₂×Q₈	C₂×D₅₃	1	2	yes	—
34	Q₁₆	—	106	8	no	—
35	C₂×C₈	—	53	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	106	2	no	—
38	C₄²	C₂×D₅₃	1	1	yes	characteristic
39	Q₈	—	2	4	no	—
40	Q₈	—	106	4	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	106	2	no	—
43	C₂×C₄	C₂²⋊D₅₃	1	2	yes	—
44	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
45	C₈	—	53	4	no	—
46	C₂²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
47	C₄	—	4	2	no	—
48	C₄	—	212	2	no	—
49	C₄	#11719925	1	4	yes	complemented
50	C₄	D₄×D₅₃	1	2	yes	—
51	C₂	#13133213	1	1	yes	characteristic
52	C₂	C₂×#11719925	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	C₄⋊Q₈×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#494
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
21. C₄×C₅₃	Q₁₆	semidirect
27. C₅₃	#494	semidirect
49. C₄	#11719925	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719925	2
#13133217	2
C₄×#88322	2
C₄⋊Q₈×C₅₃	2
#494	53