Subgroups — C₂²⋊(C₇⋊(C₁₁²⋊C₉)) #44478977

Order 30492.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×C₁₁²	C₃	1	1	yes	maximal, characteristic
2	C₇⋊(C₁₁²⋊C₉)	—	4	1	no	maximal, Hall
3	C₂×C₃×C₇×C₁₁²	—	3	1	no	—
4	C₂²⋊(C₁₁²⋊C₉)	—	7	1	no	maximal, Hall
5	C₂²×C₇×C₁₁²	C₉	1	1	yes	characteristic, complemented, Hall
6	C₃×C₇×C₁₁²	A₄	1	1	yes	characteristic
7	C₂×C₇×C₁₁²	—	3	1	no	—
8	C₂²×C₃×C₁₁²	ΓL(1,𝔽₈)	1	1	yes	characteristic
9	C₁₁²⋊C₉	—	28	1	no	Hall
10	C₂²×C₃×C₇×C₁₁	—	3	4	no	—
11	C₇×C₁₁²	C₂²⋊C₉	1	1	yes	characteristic, complemented, Hall
12	C₂×C₃×C₁₁²	—	3	1	no	—
13	C₂²×C₁₁²	C₇⋊C₉	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₇×C₁₁	—	3	12	no	—
15	C₃×C₁₁²	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
16	C₂²×C₇×C₁₁	—	3	4	no	—
17	C₂²⋊(C₇⋊C₉)	—	121	1	no	maximal, Hall
18	C₂×C₁₁²	—	3	1	no	—
19	C₃×C₇×C₁₁	—	3	4	no	—
20	C₂×C₇×C₁₁	—	3	12	no	—
21	C₂²×C₃×C₁₁	—	3	4	no	—
22	C₁₁²	C₂²⋊(C₇⋊C₉)	1	1	yes	characteristic, complemented, Sylow
23	C₂²×C₃×C₇	C₁₁²⋊C₃	1	1	yes	characteristic
24	C₇×C₁₁	—	3	4	no	—
25	C₂×C₃×C₁₁	—	3	12	no	—
26	C₇⋊C₉	—	484	1	no	Hall
27	C₂²×C₁₁	—	3	4	no	—
28	C₂×C₃×C₇	—	3	1	no	—
29	C₂²⋊C₉	—	847	1	no	Hall
30	C₃×C₁₁	—	3	4	no	—
31	C₂²×C₇	C₁₁²⋊C₉	1	1	yes	characteristic, complemented, Hall
32	C₂×C₁₁	—	3	12	no	—
33	C₃×C₇	C₂²⋊(C₁₁²⋊C₃)	1	1	yes	characteristic
34	C₂×C₇	—	3	1	no	—
35	C₂²×C₃	C₇⋊(C₁₁²⋊C₃)	1	1	yes	characteristic
36	C₁₁	—	3	4	no	—
37	C₉	—	3388	1	no	Sylow
38	C₇	C₂²⋊(C₁₁²⋊C₉)	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₃	—	3	1	no	—
40	C₂²	C₇⋊(C₁₁²⋊C₉)	1	1	yes	characteristic, complemented, Sylow
41	C₃	C₂²⋊(C₇⋊(C₁₁²⋊C₃))	1	1	yes	characteristic
42	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₂²⋊(C₇⋊(C₁₁²⋊C₃))
commutator, hypocenter	C₂²×C₇×C₁₁²	C₉
Fitting, socle, prime radical, cosocle	C₂²×C₃×C₇×C₁₁²	C₃
LCS	G ▷ C₂²×C₇×C₁₁²
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
5. C₂²×C₇×C₁₁²	C₉	semidirect
11. C₇×C₁₁²	C₂²⋊C₉	semidirect
13. C₂²×C₁₁²	C₇⋊C₉	semidirect
22. C₁₁²	C₂²⋊(C₇⋊C₉)	semidirect
31. C₂²×C₇	C₁₁²⋊C₉	semidirect
38. C₇	C₂²⋊(C₁₁²⋊C₉)	semidirect
40. C₂²	C₇⋊(C₁₁²⋊C₉)	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₇×C₁₁²	3
C₇⋊(C₁₁²⋊C₉)	4
C₂²⋊(C₁₁²⋊C₉)	7
C₂²⋊(C₇⋊C₉)	121

Subgroups — C22⋊(C7⋊(C112⋊C9)) #44478977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊(C₇⋊(C₁₁²⋊C₉)) #44478977