Subgroups — C₂×#19128628 #24922608

Order 10668.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128628	C₂	1	2	yes	maximal, direct product factor
2	C₂×#14581888	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₇×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	#14581888	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₇×C₁₂₇	C₂×C₃	1	1	yes	characteristic, complemented
6	D₇×C₁₂₇	C₂×C₃	1	2	yes	complemented
7	C₂²×C₁₂₇⋊C₃	—	7	1	no	maximal, Hall
8	C₇×C₁₂₇	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
9	C₂×C₁₂₇⋊C₃	—	7	2	no	—
10	C₂×C₁₂₇⋊C₃	—	7	1	no	—
11	C₂²×C₁₂₇	—	7	1	no	Hall
12	C₁₂₇⋊C₃	—	7	1	no	Hall
13	C₂×C₁₂₇	—	7	2	no	—
14	C₂×C₁₂₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
15	C₁₂₇	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
16	C₂×AGL(1,𝔽₇)	—	127	1	no	maximal, Hall
17	C₂×ΓL(1,𝔽₈)	—	127	1	no	—
18	AGL(1,𝔽₇)	—	127	2	no	—
19	C₂×D₇	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented, Hall
20	ΓL(1,𝔽₈)	—	127	1	no	Hall
21	C₂×C₇	C₂×C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
22	D₇	C₂×C₁₂₇⋊C₃	1	2	yes	complemented
23	C₂²×C₃	—	889	1	no	Hall
24	C₇	C₂²×C₁₂₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₃	—	889	2	no	—
26	C₂×C₃	—	889	1	no	—
27	C₂²	—	7	1	no	Sylow
28	C₃	—	889	1	no	Sylow
29	C₂	—	7	2	no	—
30	C₂	#19128628	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	#19128628
commutator, cosocle, hypocenter	C₇×C₁₂₇	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₂₇	C₂×C₃
LCS	G ▷ C₇×C₁₂₇
UCS	C₁ ◁ C₂

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