Subgroups — #46809082

Order 34432.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31770439	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₂₆₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#16864377	C₄	1	1	yes	characteristic
5	C₈×Dic₂₆₉	C₄	1	1	yes	characteristic
6	#22575359	—	2	1	no	—
7	C₂×C₁₆×C₂₆₉	C₂²	1	1	yes	characteristic
8	C₃₂×C₂₆₉	—	2	1	no	—
9	C₂×#14524130	C₈	1	1	yes	characteristic
10	C₄×Dic₂₆₉	C₈	1	1	yes	characteristic
11	#16864377	—	2	1	no	—
12	C₂×C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₂₆₉	D₄	1	1	yes	characteristic
14	C₁₆×C₂₆₉	Q₈	1	1	yes	characteristic
15	C₂×Dic₂₆₉	—	2	1	no	—
16	#14524130	C₁₆	1	2	yes	—
17	C₂×C₄×C₂₆₉	C₂×C₈	1	1	yes	characteristic
18	C₈×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
19	C₈×C₂₆₉	M₁₆	1	1	yes	characteristic
20	C₂²×C₂₆₉	M₃₂	1	1	yes	characteristic
21	C₄×C₂₆₉	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₂₆₉	C₂×C₁₆	1	1	yes	characteristic
23	Dic₂₆₉	—	2	2	no	—
24	C₂×C₂₆₉	—	2	1	no	—
25	C₂×C₂₆₉	C₄⋊C₁₆	1	1	yes	characteristic
26	C₂₆₉	#1422	1	1	yes	characteristic, complemented, Sylow
27	#1422	—	269	1	no	maximal, Sylow
28	M₆₄	—	269	1	no	—
29	M₆₄	D₂₆₉	1	1	yes	characteristic
30	C₄×C₁₆	—	269	1	no	—
31	C₂×C₁₆	—	269	1	no	—
32	C₂×C₁₆	C₂×D₂₆₉	1	1	yes	characteristic
33	C₄×C₈	—	269	1	no	—
34	C₃₂	—	2	1	no	—
35	C₃₂	—	538	1	no	—
36	C₂×C₈	—	269	1	no	—
37	C₂×C₈	C₄×D₂₆₉	1	1	yes	characteristic
38	C₄²	—	269	1	no	—
39	C₁₆	—	538	1	no	—
40	C₁₆	C₂²⋊D₂₆₉	1	1	yes	characteristic
41	C₁₆	Dic₅₃₈	1	1	yes	characteristic
42	C₂×C₄	—	538	1	no	—
43	C₂×C₄	C₈×D₂₆₉	1	1	yes	characteristic
44	C₈	—	269	2	no	—
45	C₈	#16864381	1	1	yes	characteristic
46	C₈	#16864385	1	1	yes	characteristic
47	C₂²	#22575392	1	1	yes	characteristic
48	C₄	—	538	2	no	—
49	C₄	#22575382	1	1	yes	characteristic
50	C₄	C₁₆×D₂₆₉	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	#31770423	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₂₆₉
commutator	C₄×C₂₆₉	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₂₆₉
Fitting	M₆₄×C₂₆₉	C₂
socle	C₂×C₂₆₉	C₄⋊C₁₆
prime radical	C₂²×C₂₆₉	M₃₂
cosocle	C₂×C₁₆×C₂₆₉	C₂²
hypercenter	M₆₄	D₂₆₉
hypocenter	C₂₆₉	#1422
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
26. C₂₆₉	#1422	semidirect

Maximal subgroups

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

subgroup	index
#31770439	2
C₁₆×Dic₂₆₉	2
M₆₄×C₂₆₉	2
#1422	269