Subgroups — #66558

Order 336.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₃×C₇	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#1136	—	3	1	no	maximal, Hall
5	C₂×D₃×C₇	C₄	1	1	yes	characteristic
6	C₃×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₇×Dic₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₇	—	3	1	no	—
9	C₇⋊C₈	—	3	1	no	—
10	C₇⋊C₈	D₃	1	1	yes	characteristic, complemented
11	ΓL(1,𝔽₂₅)	—	7	1	no	maximal, Hall
12	C₂×C₃×C₇	C₂×C₄	1	1	yes	characteristic
13	D₃×C₇	—	2	1	no	—
14	C₂²×C₇	—	3	1	no	—
15	C₄×C₇	—	3	1	no	—
16	C₄×C₇	C₂×D₃	1	1	yes	characteristic
17	C₄×D₃	D₇	1	1	yes	characteristic
18	C₃×C₈	—	7	1	no	—
19	C₃⋊C₈	—	7	1	no	—
20	C₃×C₇	M₁₆	1	1	yes	characteristic, complemented, Hall
21	M₁₆	—	21	1	no	Sylow
22	C₂×C₇	—	6	1	no	—
23	C₂×C₇	C₄×D₃	1	1	yes	characteristic
24	C₂×D₃	Dic₇	1	1	yes	characteristic
25	C₃×C₄	C₂×D₇	1	1	yes	characteristic
26	Dic₃	Dic₇	1	1	yes	characteristic
27	C₂×C₄	—	3	1	no	—
28	C₈	—	7	1	no	—
29	C₈	—	21	1	no	—
30	C₇	ΓL(1,𝔽₂₅)	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₃	C₂×Dic₇	1	1	yes	characteristic
32	D₃	—	2	1	no	—
33	C₂²	—	3	1	no	—
34	C₄	—	3	1	no	—
35	C₄	D₃×D₇	1	1	yes	characteristic
36	C₃	#1136	1	1	yes	characteristic, complemented, Sylow
37	C₂	—	6	1	no	—
38	C₂	D₃×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₃×D₇
commutator, socle	C₂×C₃×C₇	C₂×C₄
Fitting, cosocle	C₃×C₄×C₇	C₂²
prime radical	C₂×D₃×C₇	C₄
hypocenter	C₃×C₇	M₁₆
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₄

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.