Subgroups — C₁₂₁⋊#5997 #37557366

Order 23716.

42 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇⋊D₇×C₁₂₁	C₂	1	1	yes	maximal, characteristic
2	C₇²×C₁₂₁	C₄	1	1	yes	characteristic, complemented, Hall
3	C₁₁⋊#5997	—	11	1	no	maximal
4	D₇×C₁₂₁	—	14	4	no	—
5	C₁₁×C₇⋊D₇	D₁₁	1	1	yes	characteristic
6	C₇×C₁₂₁	—	2	4	no	—
7	C₇²×C₁₁	Dic₁₁	1	1	yes	characteristic
8	Dic₁₂₁	—	49	1	no	maximal, Hall
9	C₂×C₁₂₁	—	49	1	no	—
10	#5997	—	121	1	no	Hall
11	C₁₁×D₇	—	14	4	no	—
12	C₁₂₁	#5997	1	1	yes	characteristic, complemented, Sylow
13	C₇⋊D₇	D₁₂₁	1	1	yes	characteristic
14	C₇×C₁₁	—	2	4	no	—
15	C₇²	Dic₁₂₁	1	1	yes	characteristic, complemented, Sylow
16	Dic₁₁	—	539	1	no	—
17	C₂×C₁₁	—	49	1	no	—
18	D₇	—	14	4	no	—
19	C₁₁	C₁₁⋊#5997	1	1	yes	characteristic
20	C₇	—	2	4	no	—
21	C₄	—	5929	1	no	Sylow
22	C₂	—	49	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, hypocenter	C₇²×C₁₂₁	C₄
Frattini	C₁₁	C₁₁⋊#5997
socle	C₇²×C₁₁	Dic₁₁
prime radical	C₁₁×C₇⋊D₇	D₁₁
cosocle	C₇⋊D₇×C₁₂₁	C₂
LCS	G ▷ C₇²×C₁₂₁
UCS	C₁

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.