Subgroups — #31770900

Order 17216.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575453	C₂	1	1	yes	maximal, characteristic
2	#22575386	C₂	1	2	yes	maximal
3	C₄⋊#14524130	C₂	1	1	yes	maximal, characteristic
4	#22575377	C₂	1	2	yes	maximal
5	C₄×Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₃₈	—	2	1	no	—
7	C₂×#14524130	C₂²	1	2	yes	—
8	C₄⋊Dic₂₆₉	—	2	1	no	—
9	C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
10	C₂×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₂₆₉	—	2	1	no	—
12	C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
13	C₄²×C₂₆₉	—	2	1	no	—
14	C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
15	Dic₅₃₈	—	4	2	no	—
16	C₂×Dic₂₆₉	—	2	2	no	—
17	#14524130	—	2	2	no	—
18	Q₈×C₂₆₉	—	2	1	no	—
19	Q₈×C₂₆₉	Q₈	1	2	yes	complemented
20	C₂×C₄×C₂₆₉	—	2	1	no	—
21	C₂×C₄×C₂₆₉	—	2	1	no	—
22	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
25	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
26	C₄×C₂₆₉	—	2	2	no	—
27	C₄×C₂₆₉	—	2	1	no	—
28	C₄×C₂₆₉	—	4	1	no	—
29	C₄×C₂₆₉	Q₁₆	1	2	yes	—
30	C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
31	C₄×C₂₆₉	#41	1	1	yes	characteristic
32	Dic₂₆₉	—	4	2	no	—
33	C₂×C₂₆₉	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂×C₂₆₉	C₂×Q₁₆	1	1	yes	characteristic
35	C₂×C₂₆₉	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₂₆₉	#477	1	1	yes	characteristic, complemented, Sylow
37	#477	—	269	1	no	maximal, Sylow
38	C₄⋊Q₈	—	269	1	no	—
39	C₄×Q₈	D₂₆₉	1	1	yes	characteristic
40	#107	—	269	2	no	—
41	C₄⋊C₈	—	269	1	no	—
42	#103	—	269	2	no	—
43	C₂×Q₈	—	538	1	no	—
44	C₂×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
45	C₂×C₈	—	269	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	269	2	no	—
48	C₄⋊C₄	—	538	1	no	—
49	C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
50	C₄²	—	2	1	no	—
51	C₄²	C₂×D₂₆₉	1	1	yes	characteristic
52	Q₈	—	2	1	no	—
53	Q₈	—	1076	2	no	—
54	Q₈	Dic₅₃₈	1	2	yes	complemented
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	538	2	no	—
58	C₂×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
61	C₈	—	538	2	no	—
62	C₂²	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	—	2	1	no	—
65	C₄	—	4	1	no	—
66	C₄	—	1076	2	no	—
67	C₄	#16864392	1	2	yes	—
68	C₄	C₂×Dic₅₃₈	1	1	yes	characteristic
69	C₄	#16864402	1	1	yes	characteristic
70	C₂	#22575432	1	1	yes	characteristic
71	C₂	C₂×#16864392	1	1	yes	characteristic
72	C₂	#22575478	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	C₄×Q₈×C₂₆₉	C₂
socle, prime radical	C₂²×C₂₆₉	C₂×D₄
hypercenter	C₄×Q₈	D₂₆₉
hypocenter	C₂₆₉	#477
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
19. Q₈×C₂₆₉	Q₈	semidirect
36. C₂₆₉	#477	semidirect
54. Q₈	Dic₅₃₈	semidirect

Maximal subgroups

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

subgroup	index
#22575453	2
#22575386	2
C₄⋊#14524130	2
#22575377	2
C₄×Q₈×C₂₆₉	2
#477	269