Subgroups — M₁₆×Dic₁₉₉ #27163969

Order 12736.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691870	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
3	#20263865	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₁₉₉	C₂	1	2	yes	maximal, complemented
5	C₂×M₁₆×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₉₉	C₄	1	2	yes	—
7	#15691870	C₄	1	4	yes	complemented
8	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
9	C₄×Dic₁₉₉	C₄	1	2	yes	—
10	C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
13	M₁₆×C₁₉₉	C₄	1	4	yes	complemented
14	C₂×C₈×C₁₉₉	C₂²	1	2	yes	—
15	C₂×Dic₁₉₉	—	2	4	no	—
16	C₂×Dic₁₉₉	C₂×C₄	1	2	yes	—
17	C₂×Dic₁₉₉	C₂×C₄	1	2	yes	—
18	C₁₉₉⋊C₈	C₂×C₄	1	4	yes	complemented
19	C₂³×C₁₉₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
24	C₈×C₁₉₉	C₂×C₄	1	4	yes	complemented
25	C₂²×C₁₉₉	—	2	2	no	—
26	C₂²×C₁₉₉	C₄²	1	2	yes	—
27	C₂²×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₁₉₉	C₄²	1	2	yes	—
29	C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
31	Dic₁₉₉	—	2	2	no	—
32	Dic₁₉₉	M₁₆	1	4	yes	direct product factor
33	C₂×C₁₉₉	—	2	2	no	—
34	C₂×C₁₉₉	C₂×C₄²	1	1	yes	characteristic
35	C₂×C₁₉₉	C₂×M₁₆	1	2	yes	—
36	C₁₉₉	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
37	C₄×M₁₆	—	199	1	no	maximal, Sylow
38	C₂×M₁₆	—	199	1	no	—
39	C₂×M₁₆	D₁₉₉	1	1	yes	characteristic
40	C₂×C₄²	—	199	1	no	—
41	#97	—	199	2	no	—
42	C₄×C₈	—	199	2	no	—
43	C₂²×C₄	—	199	2	no	—
44	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
45	M₁₆	—	199	4	no	—
46	M₁₆	Dic₁₉₉	1	4	yes	direct product factor
47	C₂×C₈	—	199	2	no	—
48	C₂×C₈	C₂×D₁₉₉	1	2	yes	—
49	C₄²	—	199	2	no	—
50	C₄²	—	199	1	no	—
51	C₄²	—	199	1	no	—
52	C₂³	C₄×D₁₉₉	1	1	yes	characteristic
53	C₂×C₄	—	398	4	no	—
54	C₂×C₄	—	199	2	no	—
55	C₂×C₄	—	199	2	no	—
56	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
57	C₂×C₄	C₄×D₁₉₉	1	2	yes	—
58	C₂×C₄	C₂×Dic₁₉₉	1	2	yes	—
59	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
60	C₈	—	199	4	no	—
61	C₈	C₂×Dic₁₉₉	1	4	yes	complemented
62	C₂²	—	2	2	no	—
63	C₂²	C₄×Dic₁₉₉	1	2	yes	—
64	C₂²	C₂×C₄×D₁₉₉	1	1	yes	characteristic
65	C₄	—	398	2	no	—
66	C₄	—	199	4	no	—
67	C₄	C₄×Dic₁₉₉	1	2	yes	—
68	C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
69	C₄	C₂²×Dic₁₉₉	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	C₂×C₄×Dic₁₉₉	1	1	yes	characteristic
72	C₂	M₁₆×D₁₉₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₉₉
commutator	C₂×C₁₉₉	C₂×C₄²
Fitting	C₂×M₁₆×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂²×C₄
prime radical	C₂³×C₁₉₉	C₂×C₄
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	C₂×M₁₆	D₁₉₉
hypocenter	C₁₉₉	C₄×M₁₆
LCS	G ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20263865	C₂	semidirect
4. C₈×Dic₁₉₉	C₂	semidirect
7. #15691870	C₄	semidirect
13. M₁₆×C₁₉₉	C₄	semidirect
18. C₁₉₉⋊C₈	C₂×C₄	semidirect
24. C₈×C₁₉₉	C₂×C₄	semidirect
32. Dic₁₉₉	M₁₆	direct (N × H)
36. C₁₉₉	C₄×M₁₆	semidirect
46. M₁₆	Dic₁₉₉	direct (N × H)
61. C₈	C₂×Dic₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691870	2
C₂×C₄×Dic₁₉₉	2
#20263865	2
C₈×Dic₁₉₉	2
C₂×M₁₆×C₁₉₉	2
C₄×M₁₆	199

Subgroups — M16×Dic199 #27163969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×Dic₁₉₉ #27163969