Subgroups — #27163775

Order 12736.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263914	C₂	1	2	yes	maximal
2	C₂×M₁₆×C₁₉₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₁₉₉	—	2	2	no	—
4	C₂²⋊Dic₁₉₉	—	2	2	no	—
5	C₄×Dic₁₉₉	C₄	1	4	yes	complemented
6	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
7	M₁₆×C₁₉₉	—	2	2	no	—
8	C₂×C₈×C₁₉₉	C₄	1	2	yes	complemented
9	C₂×Dic₁₉₉	—	2	4	no	—
10	C₂³×C₁₉₉	D₄	1	1	yes	characteristic
11	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
14	C₂×C₄×C₁₉₉	Q₈	1	1	yes	characteristic
15	C₈×C₁₉₉	—	2	2	no	—
16	C₂²×C₁₉₉	—	4	1	no	—
17	C₂²×C₁₉₉	C₂²⋊C₄	1	2	yes	—
18	C₂²×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
19	C₄×C₁₉₉	C₄²	1	1	yes	characteristic
20	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₁₉₉	C₄⋊C₄	1	2	yes	—
22	Dic₁₉₉	—	4	4	no	—
23	C₂×C₁₉₉	—	2	2	no	—
24	C₂×C₁₉₉	—	2	1	no	—
25	C₂×C₁₉₉	#95	1	1	yes	characteristic
26	C₁₉₉	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
27	H(3,ℤ/4)	—	199	1	no	maximal, Sylow
28	C₂×M₁₆	D₁₉₉	1	1	yes	characteristic
29	#117	—	199	2	no	—
30	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
31	M₁₆	—	2	2	no	—
32	C₂×C₈	Dic₁₉₉	1	2	yes	complemented
33	C₄⋊C₄	—	398	2	no	—
34	C₂²⋊C₄	—	398	2	no	—
35	C₄²	—	199	4	no	—
36	C₂³	D₇₉₆	1	1	yes	characteristic
37	C₂×C₄	—	398	4	no	—
38	C₂×C₄	C₄×D₁₉₉	1	2	yes	—
39	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
40	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
41	C₂×C₄	Dic₃₉₈	1	1	yes	characteristic
42	C₈	—	2	2	no	—
43	C₂²	—	4	1	no	—
44	C₂²	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
45	C₂²	C₄⋊Dic₁₉₉	1	1	yes	characteristic
46	C₄	—	796	4	no	—
47	C₄	C₄×Dic₁₉₉	1	1	yes	characteristic
48	C₄	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
49	C₄	Dic₁₉₉⋊C₄	1	2	yes	—
50	C₂	—	2	2	no	—
51	C₂	—	2	1	no	—
52	C₂	C₁₉₉⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×Dic₁₉₉
commutator	C₄×C₁₉₉	C₄²
Frattini	C₂²×C₄	C₂×D₁₉₉
Fitting	C₂×M₁₆×C₁₉₉	C₂
socle	C₂×C₁₉₉	#95
prime radical	C₂³×C₁₉₉	D₄
cosocle	C₂²×C₄×C₁₉₉	C₂²
hypercenter	C₂×M₁₆	D₁₉₉
hypocenter	C₁₉₉	H(3,ℤ/4)
LCS	G ▷ C₄×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
5. C₄×Dic₁₉₉	C₄	semidirect
8. C₂×C₈×C₁₉₉	C₄	semidirect
26. C₁₉₉	H(3,ℤ/4)	semidirect
32. C₂×C₈	Dic₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263914	2
C₂×M₁₆×C₁₉₉	2
H(3,ℤ/4)	199