Subgroups — #31774257

Order 17344.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22576160	C₂	1	1	yes	maximal, characteristic, complemented
2	#22576148	C₂	1	1	yes	maximal, characteristic
3	#22576085	C₂	1	2	yes	maximal, complemented
4	#22576082	C₂	1	2	yes	maximal
5	#97×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₅₄₂	—	2	1	no	—
7	C₂×Dic₅₄₂	C₂²	1	1	yes	characteristic
8	C₂×D₁₀₈₄	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₂₇₁	—	2	1	no	—
10	C₄⋊Dic₂₇₁	C₂²	1	2	yes	—
11	(C₂×D₂₇₁)⋊C₄	—	2	1	no	—
12	(C₂×D₂₇₁)⋊C₄	—	2	1	no	—
13	C₂×C₈×C₂₇₁	C₂²	1	2	yes	—
14	C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
15	C₂²×D₂₇₁	—	2	1	no	—
16	Dic₅₄₂	—	2	2	no	—
17	Dic₅₄₂	—	4	2	no	—
18	D₁₀₈₄	—	2	2	no	—
19	C₂×Dic₂₇₁	—	2	1	no	—
20	C₂×Dic₂₇₁	—	2	2	no	—
21	C₂×C₄×C₂₇₁	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₇₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
24	C₈×C₂₇₁	—	2	2	no	—
25	C₂²×C₂₇₁	C₂×D₄	1	1	yes	characteristic
26	C₂×D₂₇₁	—	4	1	no	—
27	C₂×D₂₇₁	—	4	1	no	—
28	C₂×D₂₇₁	—	4	1	no	—
29	C₄×C₂₇₁	—	2	1	no	—
30	C₄×C₂₇₁	—	2	1	no	—
31	C₄×C₂₇₁	#41	1	2	yes	—
32	Dic₂₇₁	—	4	1	no	—
33	Dic₂₇₁	—	4	2	no	—
34	C₂×C₂₇₁	#124	1	1	yes	characteristic
35	C₂×C₂₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₂×C₂₇₁	#137	1	1	yes	characteristic
37	D₂₇₁	—	8	1	no	—
38	C₂₇₁	#489	1	1	yes	characteristic, complemented, Sylow
39	#489	—	271	1	no	maximal, Sylow
40	C₄⋊Q₈	—	271	1	no	—
41	#124	—	271	1	no	—
42	#103	—	271	2	no	—
43	#102	—	271	2	no	—
44	#97	D₂₇₁	1	1	yes	characteristic, complemented
45	C₂×Q₈	—	271	1	no	—
46	C₂×Q₈	—	542	1	no	—
47	C₂×D₄	—	271	1	no	—
48	C₂×C₈	C₂×D₂₇₁	1	2	yes	—
49	C₄⋊C₄	—	271	2	no	—
50	C₄⋊C₄	—	542	1	no	—
51	C₂²⋊C₄	—	542	1	no	—
52	C₂²⋊C₄	—	542	1	no	—
53	C₄²	C₂×D₂₇₁	1	1	yes	characteristic
54	C₂³	—	542	1	no	—
55	Q₈	—	542	2	no	—
56	Q₈	—	1084	2	no	—
57	D₄	—	542	2	no	—
58	C₂×C₄	—	542	1	no	—
59	C₂×C₄	—	542	2	no	—
60	C₂×C₄	D₁₀₈₄	1	1	yes	characteristic
61	C₂×C₄	D₁₀₈₄	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₂²	—	1084	1	no	—
65	C₂²	—	1084	1	no	—
66	C₂²	—	1084	1	no	—
67	C₂²	C₂×D₁₀₈₄	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	1084	1	no	—
71	C₄	—	1084	2	no	—
72	C₄	#16864654	1	2	yes	—
73	C₂	—	2168	1	no	—
74	C₂	#22576148	1	1	yes	characteristic
75	C₂	#22576188	1	1	yes	characteristic
76	C₂	#22576193	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₀₈₄
commutator, cosocle	C₂×C₄×C₂₇₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₇₁
Fitting	#97×C₂₇₁	C₂
socle	C₂²×C₂₇₁	C₂×D₄
prime radical	C₂×D₁₀₈₄	C₂²
hypercenter	#97	D₂₇₁
hypocenter	C₂₇₁	#489
LCS	G ▷ C₂×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
1. #22576160	C₂	semidirect
3. #22576085	C₂	semidirect
5. #97×C₂₇₁	C₂	semidirect
38. C₂₇₁	#489	semidirect
44. #97	D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22576160	2
#22576148	2
#22576085	2
#22576082	2
#97×C₂₇₁	2
#489	271