Subgroups — #31774304

Order 17344.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22576159	C₂	1	1	yes	maximal, characteristic
2	#22576151	C₂	1	1	yes	maximal, characteristic
3	#22576094	C₂	1	1	yes	maximal, characteristic
4	#22576093	C₂	1	1	yes	maximal, characteristic
5	#22576091	C₂	1	1	yes	maximal, characteristic
6	#22576070	C₂	1	1	yes	maximal, characteristic
7	#106×C₂₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₄₂	—	2	1	no	—
9	C₂×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₇₁	—	2	1	no	—
11	C₄⋊Dic₂₇₁	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₇₁	C₂²	1	1	yes	characteristic
13	Dic₂₇₁⋊C₄	—	2	1	no	—
14	Dic₂₇₁⋊C₄	—	2	1	no	—
15	C₄×Dic₂₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₇₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₇₁	C₂²	1	1	yes	characteristic
19	Dic₅₄₂	—	4	2	no	—
20	C₂×Dic₂₇₁	—	2	1	no	—
21	C₂×Dic₂₇₁	—	2	1	no	—
22	C₂×Dic₂₇₁	D₄	1	1	yes	characteristic
23	C₂×Dic₂₇₁	D₄	1	1	yes	characteristic
24	C₂₇₁⋊C₈	Q₈	1	2	yes	complemented
25	C₂×C₄×C₂₇₁	—	2	1	no	—
26	C₂×C₄×C₂₇₁	—	2	1	no	—
27	C₂×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
28	C₈×C₂₇₁	Q₈	1	2	yes	complemented
29	C₂²×C₂₇₁	C₂×D₄	1	1	yes	characteristic
30	C₄×C₂₇₁	—	4	1	no	—
31	C₄×C₂₇₁	—	4	1	no	—
32	C₄×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
34	Dic₂₇₁	—	2	1	no	—
35	Dic₂₇₁	—	2	1	no	—
36	Dic₂₇₁	—	4	1	no	—
37	Dic₂₇₁	—	4	1	no	—
38	C₂×C₂₇₁	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₂₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
40	C₂×C₂₇₁	#137	1	1	yes	characteristic
41	C₂₇₁	#501	1	1	yes	characteristic, complemented, Sylow
42	#501	—	271	1	no	maximal, Sylow
43	C₄⋊Q₈	—	271	1	no	—
44	#125	—	271	1	no	—
45	#107	—	271	1	no	—
46	#107	—	271	1	no	—
47	#106	—	271	1	no	—
48	#106	D₂₇₁	1	1	yes	characteristic
49	#97	—	271	1	no	—
50	C₂×Q₈	—	542	1	no	—
51	C₂×C₈	—	271	1	no	—
52	C₂×C₈	C₂×D₂₇₁	1	1	yes	characteristic
53	C₄⋊C₄	—	271	1	no	—
54	C₄⋊C₄	—	271	1	no	—
55	C₄⋊C₄	—	542	1	no	—
56	C₄⋊C₄	—	542	1	no	—
57	C₄⋊C₄	—	542	1	no	—
58	C₄⋊C₄	C₂×D₂₇₁	1	1	yes	characteristic
59	C₄⋊C₄	C₂×D₂₇₁	1	1	yes	characteristic
60	C₄²	—	271	1	no	—
61	Q₈	—	1084	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	271	1	no	—
65	C₂×C₄	—	271	1	no	—
66	C₂×C₄	—	542	1	no	—
67	C₂×C₄	—	542	1	no	—
68	C₂×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
69	C₈	—	271	2	no	—
70	C₈	Dic₅₄₂	1	2	yes	complemented
71	C₂²	D₄×D₂₇₁	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	542	1	no	—
75	C₄	—	542	1	no	—
76	C₄	—	1084	1	no	—
77	C₄	—	1084	1	no	—
78	C₄	Q₈×D₂₇₁	1	1	yes	characteristic
79	C₄	C₂×Dic₅₄₂	1	1	yes	characteristic
80	C₂	#22576159	1	1	yes	characteristic
81	C₂	#22576187	1	1	yes	characteristic
82	C₂	#22576189	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₇₁
commutator, cosocle	C₂×C₄×C₂₇₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₇₁
Fitting	#106×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	C₂×D₄
hypercenter	#106	D₂₇₁
hypocenter	C₂₇₁	#501
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
24. C₂₇₁⋊C₈	Q₈	semidirect
28. C₈×C₂₇₁	Q₈	semidirect
41. C₂₇₁	#501	semidirect
70. C₈	Dic₅₄₂	semidirect

Maximal subgroups

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

subgroup	index
#22576159	2
#22576151	2
#22576094	2
#22576093	2
#22576091	2
#22576070	2
#106×C₂₇₁	2
#501	271