Subgroups — #519×C₁₉₉ #27163669

Order 12736.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×C₁₉₉	C₂	1	8	yes	maximal, complemented
3	#117×C₁₉₉	C₂	1	6	yes	maximal
4	C₂×Q₈×C₁₉₉	C₄	1	8	yes	complemented
5	C₂×Q₈×C₁₉₉	C₂²	1	4	yes	—
6	C₂²×C₄×C₁₉₉	C₂²	1	3	yes	—
7	C₄⋊C₄×C₁₉₉	C₂²	1	12	yes	—
8	C₂²⋊C₄×C₁₉₉	C₂²	1	4	yes	—
9	C₄²×C₁₉₉	C₂²	1	12	yes	—
10	C₂³×C₁₉₉	C₂³	1	1	yes	characteristic
11	Q₈×C₁₉₉	C₂×C₄	1	16	yes	—
12	C₂×C₄×C₁₉₉	C₂×C₄	1	12	yes	—
13	C₂×C₄×C₁₉₉	C₂³	1	8	yes	—
14	C₂×C₄×C₁₉₉	C₂³	1	6	yes	—
15	C₂²×C₁₉₉	—	2	2	no	—
16	C₂²×C₁₉₉	C₂²×C₄	1	2	yes	—
17	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
18	C₄×C₁₉₉	—	2	8	no	—
19	C₄×C₁₉₉	C₂²×C₄	1	12	yes	—
20	C₂×C₁₉₉	—	2	2	no	—
21	C₂×C₁₉₉	C₂³×C₄	1	1	yes	characteristic
22	C₂×C₁₉₉	2₋¹⁺⁴	1	2	yes	—
23	C₁₉₉	#519	1	1	yes	characteristic, direct product factor, Sylow
24	#519	C₁₉₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×Q₈	C₂×C₁₉₉	1	1	yes	characteristic
26	C₄×Q₈	C₂×C₁₉₉	1	8	yes	complemented
27	#117	C₂×C₁₉₉	1	6	yes	—
28	C₂×Q₈	C₄×C₁₉₉	1	8	yes	complemented
29	C₂×Q₈	C₂²×C₁₉₉	1	4	yes	—
30	C₂²×C₄	C₂²×C₁₉₉	1	3	yes	—
31	C₄⋊C₄	C₂²×C₁₉₉	1	12	yes	—
32	C₂²⋊C₄	C₂²×C₁₉₉	1	4	yes	—
33	C₄²	C₂²×C₁₉₉	1	12	yes	—
34	C₂³	C₂³×C₁₉₉	1	1	yes	characteristic
35	Q₈	C₂×C₄×C₁₉₉	1	16	yes	—
36	C₂×C₄	C₂×C₄×C₁₉₉	1	12	yes	—
37	C₂×C₄	C₂³×C₁₉₉	1	8	yes	—
38	C₂×C₄	C₂³×C₁₉₉	1	6	yes	—
39	C₂²	—	2	2	no	—
40	C₂²	C₂²×C₄×C₁₉₉	1	2	yes	—
41	C₂²	C₂⁴×C₁₉₉	1	1	yes	characteristic
42	C₄	—	2	8	no	—
43	C₄	C₂²×C₄×C₁₉₉	1	12	yes	—
44	C₂	—	2	2	no	—
45	C₂	C₂³×C₄×C₁₉₉	1	1	yes	characteristic
46	C₂	2₋¹⁺⁴×C₁₉₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉₉	C₂⁴
commutator	C₂	C₂³×C₄×C₁₉₉
Frattini, cosocle	C₂²	C₂⁴×C₁₉₉
prime radical	C₂³×C₁₉₉	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×C₁₉₉	C₂	semidirect
4. C₂×Q₈×C₁₉₉	C₄	semidirect
23. C₁₉₉	#519	direct (N × H)
24. #519	C₁₉₉	direct (N × H)
26. C₄×Q₈	C₂×C₁₉₉	semidirect
28. C₂×Q₈	C₄×C₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₈×C₁₉₉	2
C₄×Q₈×C₁₉₉	2
#117×C₁₉₉	2
#519	199

Subgroups — #519×C199 #27163669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #519×C₁₉₉ #27163669