Subgroups — #46834544

Order 34688.

74 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	C₂×C₂₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	#366×C₂₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₀₈₄	—	2	1	no	—
5	Dic₂₁₆₈	—	2	1	no	—
6	Dic₂₁₆₈	C₄	1	2	yes	complemented
7	C₂₇₁⋊C₃₂	—	2	1	no	—
8	C₂×C₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
9	#107×C₂₇₁	—	2	1	no	—
10	C₂×Dic₅₄₂	—	4	1	no	—
11	Dic₁₀₈₄	—	2	2	no	—
12	Dic₁₀₈₄	—	4	1	no	—
13	C₂×C₈×C₂₇₁	D₄	1	1	yes	characteristic
14	C₄⋊C₄×C₂₇₁	—	4	1	no	—
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	—	8	1	no	—
21	C₂×C₄×C₂₇₁	D₈	1	1	yes	characteristic
22	C₈×C₂₇₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₂₇₁	SD₁₆	1	1	yes	characteristic
24	C₂²×C₂₇₁	D₁₆	1	1	yes	characteristic
25	C₄×C₂₇₁	—	16	1	no	—
26	C₄×C₂₇₁	#102	1	1	yes	characteristic
27	C₄×C₂₇₁	SD₃₂	1	1	yes	characteristic
28	Dic₂₇₁	—	8	2	no	—
29	C₂×C₂₇₁	#357	1	1	yes	characteristic
30	C₂×C₂₇₁	SD₆₄	1	1	yes	characteristic
31	C₂×C₂₇₁	Q₆₄	1	1	yes	characteristic
32	C₂₇₁	#1416	1	1	yes	characteristic, complemented, Sylow
33	#1416	—	271	1	no	maximal, Sylow
34	C₂×Q₃₂	—	271	1	no	—
35	C₂×C₃₂	—	271	1	no	—
36	#366	D₂₇₁	1	1	yes	characteristic
37	C₂×Q₁₆	—	542	1	no	—
38	Q₃₂	—	271	2	no	—
39	Q₃₂	—	542	1	no	—
40	C₂×C₁₆	C₂×D₂₇₁	1	1	yes	characteristic
41	#107	—	2	1	no	—
42	C₃₂	—	542	1	no	—
43	C₂×Q₈	—	1084	1	no	—
44	Q₁₆	—	542	2	no	—
45	Q₁₆	—	1084	1	no	—
46	C₂×C₈	C₂²⋊D₂₇₁	1	1	yes	characteristic
47	C₄⋊C₄	—	4	1	no	—
48	C₁₆	C₄×D₂₇₁	1	1	yes	characteristic
49	C₁₆	D₁₀₈₄	1	1	yes	characteristic
50	Q₈	—	1084	2	no	—
51	Q₈	—	2168	1	no	—
52	C₂×C₄	—	8	1	no	—
53	C₂×C₄	—	2168	1	no	—
54	C₂×C₄	#16864639	1	1	yes	characteristic
55	C₈	(C₂×D₂₇₁)⋊C₄	1	1	yes	characteristic
56	C₈	#16864641	1	1	yes	characteristic
57	C₂²	#22576101	1	1	yes	characteristic
58	C₄	—	16	1	no	—
59	C₄	—	2168	2	no	—
60	C₄	#22576080	1	1	yes	characteristic
61	C₄	#22576103	1	1	yes	characteristic
62	C₂	#31773697	1	1	yes	characteristic
63	C₂	#31773735	1	1	yes	characteristic
64	C₂	#31773737	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22576101
commutator	C₁₆×C₂₇₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₇₁
Fitting	#366×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	D₁₆
cosocle	C₂×C₁₆×C₂₇₁	C₂²
hypercenter	#366	D₂₇₁
hypocenter	C₂₇₁	#1416
LCS	G ▷ C₁₆×C₂₇₁ ▷ C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
6. Dic₂₁₆₈	C₄	semidirect
32. C₂₇₁	#1416	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
C₂×C₂₇₁⋊C₃₂	2
#366×C₂₇₁	2
#1416	271