Subgroups — C₄³×C₂₈₉ #72196301

Order 18496.

387 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₂₈₉	C₂	1	7	yes	maximal
2	C₂²×C₄×C₂₈₉	C₂²	1	7	yes	—
3	C₄²×C₂₈₉	C₄	1	28	yes	direct product factor
4	C₂³×C₂₈₉	C₂³	1	1	yes	characteristic
5	C₂×C₄×C₂₈₉	C₂×C₄	1	42	yes	—
6	C₂²×C₂₈₉	C₂²×C₄	1	7	yes	—
7	C₄×C₂₈₉	C₄²	1	28	yes	direct product factor
8	C₄³×C₁₇	C₁₇	1	1	yes	maximal, characteristic
9	C₂×C₂₈₉	C₂×C₄²	1	7	yes	—
10	C₂×C₄²×C₁₇	C₂×C₁₇	1	7	yes	—
11	C₂₈₉	C₄³	1	1	yes	characteristic, direct product factor, Sylow
12	C₂²×C₄×C₁₇	C₂²×C₁₇	1	7	yes	—
13	C₄²×C₁₇	C₄×C₁₇	1	28	yes	—
14	C₂³×C₁₇	C₂³×C₁₇	1	1	yes	characteristic
15	C₂×C₄×C₁₇	C₂×C₄×C₁₇	1	42	yes	—
16	C₂²×C₁₇	C₂²×C₄×C₁₇	1	7	yes	—
17	C₄×C₁₇	C₄²×C₁₇	1	28	yes	—
18	C₄³	C₂₈₉	1	1	yes	characteristic, direct product factor, Sylow
19	C₂×C₁₇	C₂×C₄²×C₁₇	1	7	yes	—
20	C₂×C₄²	C₂×C₂₈₉	1	7	yes	—
21	C₁₇	C₄³×C₁₇	1	1	yes	characteristic
22	C₂²×C₄	C₂²×C₂₈₉	1	7	yes	—
23	C₄²	C₄×C₂₈₉	1	28	yes	direct product factor
24	C₂³	C₂³×C₂₈₉	1	1	yes	characteristic
25	C₂×C₄	C₂×C₄×C₂₈₉	1	42	yes	—
26	C₂²	C₂²×C₄×C₂₈₉	1	7	yes	—
27	C₄	C₄²×C₂₈₉	1	28	yes	direct product factor
28	C₂	C₂×C₄²×C₂₈₉	1	7	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, socle, prime radical, cosocle	C₂³×C₁₇	C₂³×C₁₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

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

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

subgroup	index
C₂×C₄²×C₂₈₉	2
C₄³×C₁₇	17