Subgroups — #27164423

Order 12736.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263955	C₂	1	2	yes	maximal
2	#20263889	C₂	1	4	yes	maximal
3	C₄×C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₉₈	—	2	2	no	—
5	C₄⋊Dic₁₉₉	—	2	2	no	—
6	C₄⋊Dic₁₉₉	C₂²	1	4	yes	—
7	C₂×C₈×C₁₉₉	C₂²	1	2	yes	—
8	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
9	Dic₃₉₈	—	4	4	no	—
10	C₂×Dic₁₉₉	—	2	4	no	—
11	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
12	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
13	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
14	C₈×C₁₉₉	Q₈	1	4	yes	complemented
15	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
16	C₄×C₁₉₉	D₈	1	2	yes	—
17	C₄×C₁₉₉	Q₁₆	1	2	yes	—
18	C₄×C₁₉₉	C₂×Q₈	1	2	yes	—
19	Dic₁₉₉	—	4	4	no	—
20	C₂×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
21	C₂×C₁₉₉	C₂×D₈	1	1	yes	characteristic
22	C₂×C₁₉₉	C₂×Q₁₆	1	1	yes	characteristic
23	C₁₉₉	#500	1	1	yes	characteristic, complemented, Sylow
24	#500	—	199	1	no	maximal, Sylow
25	C₄⋊Q₈	—	199	2	no	—
26	#107	—	199	4	no	—
27	C₄×C₈	D₁₉₉	1	1	yes	characteristic
28	C₂×Q₈	—	398	2	no	—
29	C₂×C₈	C₂×D₁₉₉	1	2	yes	—
30	C₄⋊C₄	—	398	2	no	—
31	C₄⋊C₄	—	199	4	no	—
32	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
33	Q₈	—	796	4	no	—
34	C₂×C₄	—	398	4	no	—
35	C₂×C₄	D₇₉₆	1	1	yes	characteristic
36	C₂×C₄	D₇₉₆	1	1	yes	characteristic
37	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
38	C₈	Dic₃₉₈	1	4	yes	complemented
39	C₂²	C₂×D₇₉₆	1	1	yes	characteristic
40	C₄	—	796	4	no	—
41	C₄	D₁₅₉₂	1	2	yes	—
42	C₄	Dic₇₉₆	1	2	yes	—
43	C₄	C₂×Dic₃₉₈	1	2	yes	—
44	C₂	#20263955	1	1	yes	characteristic
45	C₂	C₂×D₁₅₉₂	1	1	yes	characteristic
46	C₂	C₂×Dic₇₉₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₇₉₆
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	C₄×C₈×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂×D₄
hypercenter	C₄×C₈	D₁₉₉
hypocenter	C₁₉₉	#500
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
14. C₈×C₁₉₉	Q₈	semidirect
23. C₁₉₉	#500	semidirect
38. C₈	Dic₃₉₈	semidirect

Maximal subgroups

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

subgroup	index
#20263955	2
#20263889	2
C₄×C₈×C₁₉₉	2
#500	199