Subgroups — #38767056

Order 25472.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27163848	C₂	1	1	yes	maximal, characteristic
2	#27163846	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	#20263891	—	2	1	no	—
5	#20263889	—	2	1	no	—
6	C₂×C₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
7	C₃₂×C₁₉₉	C₄	1	2	yes	complemented
8	#15691870	—	4	1	no	—
9	C₄⋊Dic₁₉₉	—	4	1	no	—
10	C₂×C₈×C₁₉₉	D₄	1	1	yes	characteristic
11	C₁₆×C₁₉₉	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₉₉	Q₈	1	1	yes	characteristic
13	C₂×Dic₁₉₉	—	8	1	no	—
14	C₁₉₉⋊C₈	—	8	1	no	—
15	C₂×C₄×C₁₉₉	D₈	1	1	yes	characteristic
16	C₈×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₉₉	Q₁₆	1	1	yes	characteristic
18	C₂²×C₁₉₉	SD₃₂	1	1	yes	characteristic
19	C₄×C₁₉₉	#107	1	1	yes	characteristic
20	C₄×C₁₉₉	SD₃₂	1	1	yes	characteristic
21	Dic₁₉₉	—	16	1	no	—
22	C₂×C₁₉₉	—	2	1	no	—
23	C₂×C₁₉₉	#367	1	1	yes	characteristic
24	C₁₉₉	#1426	1	1	yes	characteristic, complemented, Sylow
25	#1426	—	199	1	no	maximal, Sylow
26	M₆₄	D₁₉₉	1	1	yes	characteristic
27	#368	—	199	1	no	—
28	#367	—	199	1	no	—
29	C₂×C₁₆	C₂×D₁₉₉	1	1	yes	characteristic
30	#108	—	398	1	no	—
31	#107	—	398	1	no	—
32	C₃₂	Dic₁₉₉	1	2	yes	complemented
33	M₁₆	—	796	1	no	—
34	C₂×C₈	D₇₉₆	1	1	yes	characteristic
35	C₄⋊C₄	—	796	1	no	—
36	C₁₆	C₂×Dic₁₉₉	1	1	yes	characteristic
37	C₁₆	Dic₃₉₈	1	1	yes	characteristic
38	C₂×C₄	—	1592	1	no	—
39	C₂×C₄	D₁₅₉₂	1	1	yes	characteristic
40	C₈	—	1592	1	no	—
41	C₈	C₄⋊Dic₁₉₉	1	1	yes	characteristic
42	C₈	Dic₇₉₆	1	1	yes	characteristic
43	C₂²	#20263900	1	1	yes	characteristic
44	C₄	—	3184	1	no	—
45	C₄	#20263889	1	1	yes	characteristic
46	C₄	#20263900	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#27163846	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27163846
commutator	C₁₆×C₁₉₉	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₉₉
Fitting	M₆₄×C₁₉₉	C₂
socle	C₂×C₁₉₉	#367
prime radical	C₂²×C₁₉₉	SD₃₂
cosocle	C₂×C₁₆×C₁₉₉	C₂²
hypercenter	M₆₄	D₁₉₉
hypocenter	C₁₉₉	#1426
LCS	G ▷ C₁₆×C₁₉₉ ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
7. C₃₂×C₁₉₉	C₄	semidirect
24. C₁₉₉	#1426	semidirect
32. C₃₂	Dic₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#27163848	2
#27163846	2
M₆₄×C₁₉₉	2
#1426	199