Subgroups — #31774005

Order 17344.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₀₈₄	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₅₄₂	C₂	1	2	yes	maximal
3	#22576092	C₂	1	1	yes	maximal, characteristic
4	#22576085	C₂	1	2	yes	maximal
5	#97×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₄₂	C₂²	1	2	yes	—
7	Dic₁₀₈₄	C₄	1	4	yes	complemented
8	C₄⋊Dic₂₇₁	C₂²	1	2	yes	—
9	Dic₂₇₁⋊C₄	—	2	2	no	—
10	C₄×Dic₂₇₁	—	2	2	no	—
11	C₂×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
13	C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
14	Dic₅₄₂	—	2	2	no	—
15	Dic₅₄₂	C₂×C₄	1	4	yes	—
16	C₂×Dic₂₇₁	—	2	2	no	—
17	C₂×Dic₂₇₁	—	2	2	no	—
18	C₂×C₄×C₂₇₁	D₄	1	2	yes	—
19	C₂×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
20	C₈×C₂₇₁	—	2	1	no	—
21	C₈×C₂₇₁	C₂×C₄	1	2	yes	—
22	C₂²×C₂₇₁	C₂×D₄	1	1	yes	characteristic
23	C₄×C₂₇₁	—	2	2	no	—
24	C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₂₇₁	#41	1	1	yes	characteristic
26	Dic₂₇₁	—	2	4	no	—
27	Dic₂₇₁	—	4	2	no	—
28	C₂×C₂₇₁	C₄×D₄	1	1	yes	characteristic
29	C₂×C₂₇₁	#137	1	2	yes	—
30	C₂₇₁	#441	1	1	yes	characteristic, complemented, Sylow
31	#441	—	271	1	no	maximal, Sylow
32	C₂×Q₁₆	—	271	1	no	—
33	C₄×Q₈	—	271	2	no	—
34	#106	—	271	1	no	—
35	#103	—	271	2	no	—
36	#97	D₂₇₁	1	1	yes	characteristic
37	C₂×Q₈	—	271	2	no	—
38	Q₁₆	—	271	4	no	—
39	C₂×C₈	C₂×D₂₇₁	1	1	yes	characteristic
40	C₂×C₈	C₂×D₂₇₁	1	1	yes	characteristic
41	C₄⋊C₄	—	271	2	no	—
42	C₄⋊C₄	—	542	2	no	—
43	C₄²	—	542	2	no	—
44	C₄²	C₂×D₂₇₁	1	1	yes	characteristic
45	Q₈	—	271	4	no	—
46	Q₈	—	542	2	no	—
47	C₂×C₄	—	542	2	no	—
48	C₂×C₄	—	542	2	no	—
49	C₂×C₄	D₁₀₈₄	1	2	yes	—
50	C₂×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
51	C₈	—	2	1	no	—
52	C₈	C₄×D₂₇₁	1	2	yes	—
53	C₂²	C₂×D₁₀₈₄	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	—	542	4	no	—
56	C₄	—	1084	2	no	—
57	C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
58	C₄	#16864654	1	1	yes	characteristic
59	C₂	C₄×D₁₀₈₄	1	1	yes	characteristic
60	C₂	#22576193	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₀₈₄
commutator	C₄×C₂₇₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₇₁
Fitting	#97×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	C₂×D₄
cosocle	C₂×C₄×C₂₇₁	C₂³
hypercenter	#97	D₂₇₁
hypocenter	C₂₇₁	#441
LCS	G ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
7. Dic₁₀₈₄	C₄	semidirect
30. C₂₇₁	#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₀₈₄	2
C₄×Dic₅₄₂	2
#22576092	2
#22576085	2
#97×C₂₇₁	2
#441	271