Subgroups — C₂²⋊(C₁₉₉⋊C₁₆) #27163798

Order 12736.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉₉⋊C₁₆	C₂	1	2	yes	maximal, complemented
2	C₂²×C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉₉⋊C₁₆	—	2	2	no	—
4	C₂²×C₄×C₁₉₉	C₄	1	1	yes	characteristic
5	C₂×C₈×C₁₉₉	—	2	2	no	—
6	C₂×C₈×C₁₉₉	C₄	1	1	yes	characteristic
7	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
8	C₂³×C₁₉₉	C₈	1	1	yes	characteristic
9	C₂×C₄×C₁₉₉	—	2	1	no	—
10	C₂×C₄×C₁₉₉	—	2	1	no	—
11	C₂×C₄×C₁₉₉	C₈	1	1	yes	characteristic
12	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
13	C₈×C₁₉₉	—	2	1	no	—
14	C₈×C₁₉₉	D₄	1	2	yes	—
15	C₂²×C₁₉₉	—	2	1	no	—
16	C₂²×C₁₉₉	—	2	1	no	—
17	C₂²×C₁₉₉	C₁₆	1	2	yes	complemented
18	C₂²×C₁₉₉	C₂×C₈	1	1	yes	characteristic
19	C₄×C₁₉₉	—	2	1	no	—
20	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₁₉₉	M₁₆	1	1	yes	characteristic
22	C₂×C₁₉₉	—	2	2	no	—
23	C₂×C₁₉₉	C₂²⋊C₈	1	1	yes	characteristic
24	C₂×C₁₉₉	C₂×C₁₆	1	1	yes	characteristic
25	C₂×C₁₉₉	M₃₂	1	1	yes	characteristic
26	C₁₉₉	C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
27	C₂²⋊C₁₆	—	199	1	no	maximal, Sylow
28	C₂²×C₈	D₁₉₉	1	1	yes	characteristic
29	C₂×C₁₆	—	199	2	no	—
30	C₂²×C₄	Dic₁₉₉	1	1	yes	characteristic
31	C₂×C₈	—	2	2	no	—
32	C₂×C₈	Dic₁₉₉	1	1	yes	characteristic
33	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
34	C₁₆	—	398	2	no	—
35	C₂³	C₁₉₉⋊C₈	1	1	yes	characteristic
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	C₁₉₉⋊C₈	1	1	yes	characteristic
39	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
40	C₈	—	2	1	no	—
41	C₈	C₂²⋊D₁₉₉	1	2	yes	—
42	C₂²	—	2	1	no	—
43	C₂²	—	2	1	no	—
44	C₂²	C₁₉₉⋊C₁₆	1	2	yes	complemented
45	C₂²	C₂×C₁₉₉⋊C₈	1	1	yes	characteristic
46	C₄	—	2	1	no	—
47	C₄	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
48	C₄	#15691870	1	1	yes	characteristic
49	C₂	—	2	2	no	—
50	C₂	C₂²⋊(C₁₉₉⋊C₈)	1	1	yes	characteristic
51	C₂	C₂×C₁₉₉⋊C₁₆	1	1	yes	characteristic
52	C₂	#20263895	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₁₉₉
commutator	C₂×C₁₉₉	C₂×C₁₆
Fitting	C₂²×C₈×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×C₈
prime radical	C₂³×C₁₉₉	C₈
cosocle	C₂×C₈×C₁₉₉	C₂²
hypercenter	C₂²×C₈	D₁₉₉
hypocenter	C₁₉₉	C₂²⋊C₁₆
LCS	G ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₉₉⋊C₁₆	C₂	semidirect
17. C₂²×C₁₉₉	C₁₆	semidirect
26. C₁₉₉	C₂²⋊C₁₆	semidirect
44. C₂²	C₁₉₉⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉₉⋊C₁₆	2
C₂²×C₈×C₁₉₉	2
C₂²⋊C₁₆	199

Subgroups — C22⋊(C199⋊C16) #27163798

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊(C₁₉₉⋊C₁₆) #27163798