Subgroups — #46834556

Order 34688.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₁₆₈	C₂	1	1	yes	maximal, characteristic
2	#31773731	C₂	1	1	yes	maximal, characteristic
3	#368×C₂₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₀₈₄	—	2	1	no	—
5	Dic₂₁₆₈	—	2	1	no	—
6	Dic₂₁₆₈	C₄	1	2	yes	—
7	C₂₇₁⋊C₃₂	—	2	1	no	—
8	C₂×C₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
9	#108×C₂₇₁	—	2	1	no	—
10	C₂×Dic₅₄₂	—	4	1	no	—
11	Dic₁₀₈₄	—	2	2	no	—
12	Dic₁₀₈₄	—	4	1	no	—
13	M₁₆×C₂₇₁	—	4	1	no	—
14	C₂×C₈×C₂₇₁	D₄	1	1	yes	characteristic
15	C₁₆×C₂₇₁	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₇₁	D₄	1	1	yes	characteristic
17	Dic₅₄₂	—	4	2	no	—
18	Dic₅₄₂	—	8	1	no	—
19	C₂×Dic₂₇₁	—	8	1	no	—
20	C₂×C₄×C₂₇₁	D₈	1	1	yes	characteristic
21	C₈×C₂₇₁	—	8	1	no	—
22	C₈×C₂₇₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₂₇₁	SD₁₆	1	1	yes	characteristic
24	C₂²×C₂₇₁	SD₃₂	1	1	yes	characteristic
25	C₄×C₂₇₁	#102	1	1	yes	characteristic
26	C₄×C₂₇₁	D₁₆	1	1	yes	characteristic
27	Dic₂₇₁	—	8	2	no	—
28	C₂×C₂₇₁	—	2	1	no	—
29	C₂×C₂₇₁	#357	1	1	yes	characteristic
30	C₂₇₁	#1420	1	1	yes	characteristic, complemented, Sylow
31	#1420	—	271	1	no	maximal, Sylow
32	C₂×Q₃₂	—	271	1	no	—
33	M₆₄	—	271	1	no	—
34	#368	D₂₇₁	1	1	yes	characteristic
35	C₂×Q₁₆	—	542	1	no	—
36	Q₃₂	—	271	2	no	—
37	Q₃₂	—	542	1	no	—
38	C₂×C₁₆	C₂×D₂₇₁	1	1	yes	characteristic
39	#108	—	2	1	no	—
40	C₃₂	—	542	1	no	—
41	C₂×Q₈	—	1084	1	no	—
42	Q₁₆	—	542	2	no	—
43	Q₁₆	—	1084	1	no	—
44	M₁₆	—	4	1	no	—
45	C₂×C₈	C₂²⋊D₂₇₁	1	1	yes	characteristic
46	C₁₆	C₄×D₂₇₁	1	1	yes	characteristic
47	C₁₆	D₁₀₈₄	1	1	yes	characteristic
48	Q₈	—	1084	2	no	—
49	Q₈	—	2168	1	no	—
50	C₂×C₄	—	2168	1	no	—
51	C₂×C₄	#16864639	1	1	yes	characteristic
52	C₈	—	8	1	no	—
53	C₈	(C₂×D₂₇₁)⋊C₄	1	1	yes	characteristic
54	C₈	#16864641	1	1	yes	characteristic
55	C₂²	#22576103	1	1	yes	characteristic
56	C₄	—	2168	2	no	—
57	C₄	#22576080	1	1	yes	characteristic
58	C₄	#22576101	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	#31773697	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#31773697
commutator	C₁₆×C₂₇₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₇₁
Fitting	#368×C₂₇₁	C₂
socle	C₂×C₂₇₁	#357
prime radical	C₂²×C₂₇₁	SD₃₂
cosocle	C₂×C₁₆×C₂₇₁	C₂²
hypercenter	#368	D₂₇₁
hypocenter	C₂₇₁	#1420
LCS	G ▷ C₁₆×C₂₇₁ ▷ C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
30. C₂₇₁	#1420	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₁₆₈	2
#31773731	2
#368×C₂₇₁	2
#1420	271