Subgroups — C₃₇×H(3,𝔽₉) #41004698

Order 26973.

1334 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×C₃²⋊H(3,𝔽₃)	C₃	1	40	yes	maximal, complemented
2	C₃⁴×C₃₇	C₃²	1	10	yes	complemented
3	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	120	yes	complemented
4	C₃³×C₃₇	—	9	40	no	—
5	C₃³×C₃₇	C₃³	1	40	yes	—
6	H(3,𝔽₃)×C₃₇	—	9	120	no	—
7	H(3,𝔽₉)	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₃²×C₃₇	—	3	160	no	—
9	C₃²×C₃₇	—	9	90	no	—
10	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
11	C₃²⋊H(3,𝔽₃)	C₃×C₃₇	1	40	yes	complemented
12	C₃×C₃₇	—	9	40	no	—
13	C₃×C₃₇	H(5,𝔽₃)	1	4	yes	—
14	C₃⁴	C₃²×C₃₇	1	10	yes	complemented
15	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	120	yes	complemented
16	C₃₇	H(3,𝔽₉)	1	1	yes	characteristic, direct product factor, Sylow
17	C₃³	—	9	40	no	—
18	C₃³	C₃³×C₃₇	1	40	yes	—
19	H(3,𝔽₃)	—	9	120	no	—
20	C₃²	—	3	160	no	—
21	C₃²	—	9	90	no	—
22	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
23	C₃	—	9	40	no	—
24	C₃	C₃₇×H(5,𝔽₃)	1	4	yes	—

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ 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
C₃₇×C₃²⋊H(3,𝔽₃)	3
H(3,𝔽₉)	37