Subgroups — C₇×GL(2,𝔽₃) #66647

Order 336.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×SD₁₆	—	3	1	no	maximal, Hall
3	C₂×D₃×C₇	—	4	1	no	maximal
4	C₇×Q₈	D₃	1	1	yes	characteristic, complemented
5	C₇×D₄	—	3	1	no	—
6	C₇×C₈	—	3	1	no	—
7	GL(2,𝔽₃)	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×C₇	—	4	1	no	—
9	D₃×C₇	—	4	2	no	—
10	C₂²×C₇	—	6	1	no	—
11	C₄×C₇	—	3	1	no	—
12	SL(2,𝔽₃)	C₂×C₇	1	1	yes	characteristic, complemented
13	C₃×C₇	—	4	1	no	Hall
14	SD₁₆	—	3	1	no	Sylow
15	C₂×C₇	—	12	1	no	—
16	C₂×C₇	S₄	1	1	yes	characteristic
17	C₂×D₃	—	4	1	no	—
18	Q₈	D₃×C₇	1	1	yes	characteristic, complemented
19	D₄	—	3	1	no	—
20	C₈	—	3	1	no	—
21	C₇	GL(2,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
22	C₂×C₃	—	4	1	no	—
23	D₃	—	4	2	no	—
24	C₂²	—	6	1	no	—
25	C₄	—	3	1	no	—
26	C₃	—	4	1	no	Sylow
27	C₂	—	12	1	no	—
28	C₂	C₇×S₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂×C₇	S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂×C₇
Frattini	C₂	C₇×S₄
Fitting	C₇×Q₈	D₃
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₇

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.