Subgroups — #46838617

Order 34688.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₁₆₈	C₂	1	4	yes	maximal
2	#31774277	C₂	1	2	yes	maximal
3	C₄×C₁₆×C₂₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₀₈₄	—	2	2	no	—
5	C₂×Dic₁₀₈₄	C₂²	1	4	yes	—
6	#22576160	—	2	2	no	—
7	Dic₂₁₆₈	—	2	8	no	—
8	C₂×C₁₆×C₂₇₁	C₂²	1	2	yes	—
9	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
10	C₂×Dic₅₄₂	—	2	4	no	—
11	Dic₁₀₈₄	—	2	8	no	—
12	Dic₁₀₈₄	—	4	4	no	—
13	C₄⋊Dic₂₇₁	—	4	4	no	—
14	C₂×C₈×C₂₇₁	D₄	1	1	yes	characteristic
15	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
16	C₄²×C₂₇₁	D₄	1	1	yes	characteristic
17	C₁₆×C₂₇₁	D₄	1	4	yes	—
18	Dic₅₄₂	—	4	4	no	—
19	Dic₅₄₂	—	4	4	no	—
20	C₂×Dic₂₇₁	—	4	4	no	—
21	C₂×C₄×C₂₇₁	D₈	1	2	yes	—
22	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
23	C₈×C₂₇₁	D₈	1	2	yes	—
24	C₈×C₂₇₁	C₂×D₄	1	2	yes	—
25	C₂²×C₂₇₁	C₂×D₈	1	1	yes	characteristic
26	C₄×C₂₇₁	Q₃₂	1	4	yes	complemented
27	C₄×C₂₇₁	#127	1	1	yes	characteristic
28	C₄×C₂₇₁	C₂×D₈	1	1	yes	characteristic
29	Dic₂₇₁	—	8	4	no	—
30	C₂×C₂₇₁	#493	1	1	yes	characteristic
31	C₂×C₂₇₁	C₂×Q₃₂	1	2	yes	—
32	C₂₇₁	#2247	1	1	yes	characteristic, complemented, Sylow
33	#2247	—	271	1	no	maximal, Sylow
34	C₂×Q₃₂	—	271	4	no	—
35	#494	—	271	2	no	—
36	C₄×C₁₆	D₂₇₁	1	1	yes	characteristic
37	C₂×Q₁₆	—	271	4	no	—
38	C₂×Q₁₆	—	542	2	no	—
39	C₄⋊Q₈	—	542	2	no	—
40	Q₃₂	—	542	8	no	—
41	C₂×C₁₆	C₂×D₂₇₁	1	2	yes	—
42	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic
43	C₂×Q₈	—	542	4	no	—
44	Q₁₆	—	542	8	no	—
45	Q₁₆	—	1084	4	no	—
46	C₂×C₈	D₁₀₈₄	1	1	yes	characteristic
47	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
48	C₄⋊C₄	—	1084	4	no	—
49	C₄²	D₁₀₈₄	1	1	yes	characteristic
50	C₁₆	D₁₀₈₄	1	4	yes	—
51	Q₈	—	1084	4	no	—
52	Q₈	—	1084	4	no	—
53	C₂×C₄	—	1084	4	no	—
54	C₂×C₄	D₂₁₆₈	1	2	yes	—
55	C₂×C₄	C₂×D₁₀₈₄	1	1	yes	characteristic
56	C₈	D₂₁₆₈	1	2	yes	—
57	C₈	C₂×D₁₀₈₄	1	2	yes	—
58	C₂²	C₂×D₂₁₆₈	1	1	yes	characteristic
59	C₄	—	2168	4	no	—
60	C₄	Dic₂₁₆₈	1	4	yes	complemented
61	C₄	#22576157	1	1	yes	characteristic
62	C₄	C₂×D₂₁₆₈	1	1	yes	characteristic
63	C₂	#31774274	1	1	yes	characteristic
64	C₂	C₂×Dic₂₁₆₈	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₁₆₈
commutator, cosocle	C₂×C₈×C₂₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₇₁
Fitting	C₄×C₁₆×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	C₂×D₈
hypercenter	C₄×C₁₆	D₂₇₁
hypocenter	C₂₇₁	#2247
LCS	G ▷ C₂×C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
26. C₄×C₂₇₁	Q₃₂	semidirect
32. C₂₇₁	#2247	semidirect
60. C₄	Dic₂₁₆₈	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
#31774277	2
C₄×C₁₆×C₂₇₁	2
#2247	271