Subgroups — C₂²×C₄⋊Dic₁₉₉ #27164469

Order 12736.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₁₉₉	C₂	1	2	yes	maximal
2	C₂×C₄⋊Dic₁₉₉	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₄×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₉₉	—	2	8	no	—
5	C₂²×Dic₁₉₉	C₂²	1	12	yes	—
6	C₄⋊Dic₁₉₉	C₂²	1	16	yes	direct product factor
7	C₂⁴×C₁₉₉	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₁₉₉	C₄	1	8	yes	complemented
9	C₂²×C₄×C₁₉₉	C₂²	1	6	yes	—
10	C₂×Dic₁₉₉	—	2	24	no	—
11	C₂×Dic₁₉₉	C₂³	1	8	yes	—
12	C₂³×C₁₉₉	C₂×C₄	1	4	yes	—
13	C₂³×C₁₉₉	D₄	1	4	yes	—
14	C₂³×C₁₉₉	Q₈	1	4	yes	—
15	C₂³×C₁₉₉	C₂³	1	3	yes	—
16	C₂×C₄×C₁₉₉	C₂×C₄	1	24	yes	complemented
17	C₂×C₄×C₁₉₉	C₂³	1	4	yes	—
18	C₂²×C₁₉₉	C₄⋊C₄	1	16	yes	complemented
19	C₂²×C₁₉₉	C₂²×C₄	1	6	yes	—
20	C₂²×C₁₉₉	C₂×D₄	1	6	yes	—
21	C₂²×C₁₉₉	C₂×Q₈	1	6	yes	—
22	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
23	C₄×C₁₉₉	C₂²×C₄	1	8	yes	complemented
24	Dic₁₉₉	—	2	8	no	—
25	C₂×C₁₉₉	C₂×C₄⋊C₄	1	12	yes	complemented
26	C₂×C₁₉₉	C₂³×C₄	1	1	yes	characteristic
27	C₂×C₁₉₉	C₂²×D₄	1	1	yes	characteristic
28	C₂×C₁₉₉	C₂²×Q₈	1	1	yes	characteristic
29	C₁₉₉	C₂²×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
30	C₂²×C₄⋊C₄	—	199	1	no	maximal, Sylow
31	C₂³×C₄	—	199	2	no	—
32	C₂³×C₄	D₁₉₉	1	1	yes	characteristic
33	C₂×C₄⋊C₄	—	199	12	no	—
34	C₂⁴	C₂×D₁₉₉	1	1	yes	characteristic
35	C₂²×C₄	—	398	8	no	—
36	C₂²×C₄	—	199	12	no	—
37	C₂²×C₄	Dic₁₉₉	1	8	yes	complemented
38	C₂²×C₄	C₂×D₁₉₉	1	6	yes	—
39	C₄⋊C₄	—	199	16	no	—
40	C₂³	C₂×Dic₁₉₉	1	4	yes	—
41	C₂³	D₇₉₆	1	4	yes	—
42	C₂³	Dic₃₉₈	1	4	yes	—
43	C₂³	C₂²×D₁₉₉	1	3	yes	—
44	C₂×C₄	—	398	24	no	—
45	C₂×C₄	—	199	8	no	—
46	C₂×C₄	C₂×Dic₁₉₉	1	24	yes	complemented
47	C₂×C₄	C₂²×D₁₉₉	1	4	yes	—
48	C₂²	C₄⋊Dic₁₉₉	1	16	yes	direct product factor
49	C₂²	C₂²×Dic₁₉₉	1	6	yes	—
50	C₂²	C₂×D₇₉₆	1	6	yes	—
51	C₂²	C₂×Dic₃₉₈	1	6	yes	—
52	C₂²	C₂³×D₁₉₉	1	1	yes	characteristic
53	C₄	—	398	8	no	—
54	C₄	C₂²×Dic₁₉₉	1	8	yes	complemented
55	C₂	C₂×C₄⋊Dic₁₉₉	1	12	yes	direct product factor
56	C₂	C₂³×Dic₁₉₉	1	1	yes	characteristic
57	C₂	C₂²×D₇₉₆	1	1	yes	characteristic
58	C₂	C₂²×Dic₃₉₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₁₉₉
commutator	C₂×C₁₉₉	C₂³×C₄
Frattini	C₂²	C₂³×D₁₉₉
Fitting	C₂³×C₄×C₁₉₉	C₂
socle, prime radical	C₂⁴×C₁₉₉	C₂²
cosocle	C₂²×C₁₉₉	C₂⁴
hypercenter	C₂³×C₄	D₁₉₉
hypocenter	C₁₉₉	C₂²×C₄⋊C₄
LCS	G ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊Dic₁₉₉	C₂	direct (N × H)
6. C₄⋊Dic₁₉₉	C₂²	direct (N × H)
8. C₂²×C₄×C₁₉₉	C₄	semidirect
16. C₂×C₄×C₁₉₉	C₂×C₄	semidirect
18. C₂²×C₁₉₉	C₄⋊C₄	semidirect
23. C₄×C₁₉₉	C₂²×C₄	semidirect
25. C₂×C₁₉₉	C₂×C₄⋊C₄	semidirect
29. C₁₉₉	C₂²×C₄⋊C₄	semidirect
37. C₂²×C₄	Dic₁₉₉	semidirect
46. C₂×C₄	C₂×Dic₁₉₉	semidirect
48. C₂²	C₄⋊Dic₁₉₉	direct (N × H)
54. C₄	C₂²×Dic₁₉₉	semidirect
55. C₂	C₂×C₄⋊Dic₁₉₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₁₉₉	2
C₂×C₄⋊Dic₁₉₉	2
C₂³×C₄×C₁₉₉	2
C₂²×C₄⋊C₄	199

Subgroups — C22×C4⋊Dic199 #27164469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄⋊Dic₁₉₉ #27164469