Subgroups — #44544643

Order 32004.

36 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22547877	C₂	1	1	yes	maximal, characteristic
2	C₃×Dic₇×C₁₂₇	C₃	1	1	yes	maximal, characteristic
3	#22547877	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₃×C₇×C₁₂₇	C₂×C₃	1	1	yes	characteristic
5	C₄×#11765917	—	7	1	no	maximal, Hall
6	Dic₇×C₁₂₇	C₉	1	1	yes	characteristic, complemented, Hall
7	C₃×C₇×C₁₂₇	C₃×C₄	1	1	yes	characteristic
8	C₂×#11765917	—	7	1	no	—
9	C₂×C₇×C₁₂₇	C₂×C₉	1	1	yes	characteristic
10	C₃×C₄×C₁₂₇	—	7	1	no	—
11	#11765917	—	7	1	no	Hall
12	C₇×C₁₂₇	C₄×C₉	1	1	yes	characteristic, complemented, Hall
13	C₂×C₃×C₁₂₇	AGL(1,𝔽₇)	1	1	yes	characteristic
14	C₄×C₁₂₇	—	7	1	no	Hall
15	C₃×C₁₂₇	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
16	C₂×C₁₂₇	D₇⋊C₉	1	1	yes	characteristic
17	Dic₇⋊C₉	—	127	1	no	maximal, Hall
18	C₁₂₇	Dic₇⋊C₉	1	1	yes	characteristic, complemented, Sylow
19	C₂×C₇⋊C₉	—	127	1	no	—
20	C₃×Dic₇	C₁₂₇⋊C₃	1	1	yes	characteristic
21	C₇⋊C₉	—	127	1	no	Hall
22	C₂×C₃×C₇	C₂×C₁₂₇⋊C₃	1	1	yes	characteristic
23	C₄×C₉	—	889	1	no	Hall
24	Dic₇	#11765917	1	1	yes	characteristic, complemented, Hall
25	C₃×C₇	C₄×C₁₂₇⋊C₃	1	1	yes	characteristic
26	C₂×C₉	—	889	1	no	—
27	C₂×C₇	C₂×#11765917	1	1	yes	characteristic
28	C₃×C₄	—	7	1	no	—
29	C₉	—	889	1	no	Sylow
30	C₇	C₄×#11765917	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₃	#19128629	1	1	yes	characteristic
32	C₄	—	7	1	no	Sylow
33	C₃	#24922584	1	1	yes	characteristic
34	C₂	#30630668	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₃	#19128629
commutator, hypocenter	C₇×C₁₂₇	C₄×C₉
Fitting, socle, prime radical, cosocle	C₂×C₃×C₇×C₁₂₇	C₂×C₃
LCS	G ▷ C₇×C₁₂₇
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
3. #22547877	C₄	semidirect
6. Dic₇×C₁₂₇	C₉	semidirect
12. C₇×C₁₂₇	C₄×C₉	semidirect
18. C₁₂₇	Dic₇⋊C₉	semidirect
24. Dic₇	#11765917	semidirect
30. C₇	C₄×#11765917	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22547877	2
C₃×Dic₇×C₁₂₇	3
C₄×#11765917	7
Dic₇⋊C₉	127