Subgroups — C₄×#15691870 #27163971

Order 12736.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691870	C₂	1	2	yes	maximal
2	#20263866	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₉₉⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂×C₄²×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	#15691870	C₄	1	8	yes	direct product factor
6	C₂×C₁₉₉⋊C₈	C₂²	1	4	yes	—
7	C₂²×C₄×C₁₉₉	C₄	1	2	yes	—
8	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
9	C₄²×C₁₉₉	C₄	1	2	yes	—
10	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₁₉₉⋊C₈	C₂×C₄	1	8	yes	complemented
13	C₂³×C₁₉₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₉₉	—	2	4	no	—
15	C₂×C₄×C₁₉₉	C₂×C₄	1	4	yes	—
16	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
20	C₂²×C₁₉₉	—	2	2	no	—
21	C₂²×C₁₉₉	C₄²	1	2	yes	—
22	C₂²×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
23	C₄×C₁₉₉	—	2	2	no	—
24	C₄×C₁₉₉	C₄²	1	2	yes	—
25	C₄×C₁₉₉	M₁₆	1	4	yes	complemented
26	C₄×C₁₉₉	C₂²×C₄	1	2	yes	—
27	C₂×C₁₉₉	—	2	2	no	—
28	C₂×C₁₉₉	C₂×C₄²	1	1	yes	characteristic
29	C₂×C₁₉₉	C₂×M₁₆	1	2	yes	—
30	C₁₉₉	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
31	C₄×M₁₆	—	199	1	no	maximal, Sylow
32	C₂×M₁₆	—	199	2	no	—
33	C₂×C₄²	D₁₉₉	1	1	yes	characteristic
34	#97	—	199	2	no	—
35	C₄×C₈	—	199	2	no	—
36	C₂²×C₄	Dic₁₉₉	1	2	yes	—
37	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
38	M₁₆	—	199	8	no	—
39	C₂×C₈	—	199	4	no	—
40	C₄²	Dic₁₉₉	1	2	yes	—
41	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
42	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
43	C₂³	C₂×Dic₁₉₉	1	1	yes	characteristic
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	C₄×D₁₉₉	1	4	yes	—
46	C₂×C₄	C₂×Dic₁₉₉	1	2	yes	—
47	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
48	C₂×C₄	C₂×Dic₁₉₉	1	2	yes	—
49	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
50	C₈	—	199	8	no	—
51	C₂²	—	2	2	no	—
52	C₂²	C₄×Dic₁₉₉	1	2	yes	—
53	C₂²	C₂²×Dic₁₉₉	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	C₄×Dic₁₉₉	1	2	yes	—
56	C₄	#15691870	1	4	yes	direct product factor
57	C₄	C₂×C₄×D₁₉₉	1	2	yes	—
58	C₂	—	2	2	no	—
59	C₂	C₂×C₄×Dic₁₉₉	1	1	yes	characteristic
60	C₂	C₂×#15691870	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. #20263866	C₂	semidirect
3. C₄×C₁₉₉⋊C₈	C₂	semidirect
5. #15691870	C₄	direct (N × H)
12. C₁₉₉⋊C₈	C₂×C₄	semidirect
25. C₄×C₁₉₉	M₁₆	semidirect
30. C₁₉₉	C₄×M₁₆	semidirect
56. C₄	#15691870	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#15691870	2
#20263866	2
C₄×C₁₉₉⋊C₈	2
C₂×C₄²×C₁₉₉	2
C₄×M₁₆	199

Subgroups — C4×#15691870 #27163971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15691870 #27163971