Subgroups — C₂²×C₂²⋊Dic₁₉₉ #27164466

Order 12736.

674 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₁₉₉	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊Dic₁₉₉	C₂	1	12	yes	maximal, direct product factor
3	C₂⁵×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₉₉	—	2	8	no	—
5	C₂²×Dic₁₉₉	C₂²	1	12	yes	complemented
6	C₂²⋊Dic₁₉₉	C₂²	1	16	yes	direct product factor
7	C₂⁴×C₁₉₉	—	2	8	no	—
8	C₂⁴×C₁₉₉	C₄	1	8	yes	complemented
9	C₂⁴×C₁₉₉	C₂²	1	6	yes	—
10	C₂⁴×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₉₉	—	2	24	no	—
12	C₂×Dic₁₉₉	C₂³	1	8	yes	complemented
13	C₂³×C₁₉₉	—	2	32	no	—
14	C₂³×C₁₉₉	—	2	24	no	—
15	C₂³×C₁₉₉	C₂×C₄	1	4	yes	—
16	C₂³×C₁₉₉	C₂×C₄	1	24	yes	complemented
17	C₂³×C₁₉₉	D₄	1	8	yes	—
18	C₂³×C₁₉₉	C₂³	1	4	yes	—
19	C₂³×C₁₉₉	C₂³	1	3	yes	—
20	C₂²×C₁₉₉	—	2	48	no	—
21	C₂²×C₁₉₉	—	2	8	no	—
22	C₂²×C₁₉₉	C₂²⋊C₄	1	16	yes	complemented
23	C₂²×C₁₉₉	C₂²×C₄	1	6	yes	—
24	C₂²×C₁₉₉	C₂²×C₄	1	8	yes	complemented
25	C₂²×C₁₉₉	C₂×D₄	1	12	yes	—
26	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
27	Dic₁₉₉	—	2	8	no	—
28	C₂×C₁₉₉	—	2	8	no	—
29	C₂×C₁₉₉	C₂×C₂²⋊C₄	1	12	yes	complemented
30	C₂×C₁₉₉	C₂³×C₄	1	1	yes	characteristic
31	C₂×C₁₉₉	C₂²×D₄	1	2	yes	—
32	C₁₉₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
33	C₂²×C₂²⋊C₄	—	199	1	no	maximal, Sylow
34	C₂⁵	D₁₉₉	1	1	yes	characteristic
35	C₂³×C₄	—	199	2	no	—
36	C₂×C₂²⋊C₄	—	199	12	no	—
37	C₂⁴	—	2	8	no	—
38	C₂⁴	Dic₁₉₉	1	8	yes	complemented
39	C₂⁴	C₂×D₁₉₉	1	6	yes	—
40	C₂⁴	C₂×D₁₉₉	1	1	yes	characteristic
41	C₂²×C₄	—	398	8	no	—
42	C₂²×C₄	—	199	12	no	—
43	C₂²⋊C₄	—	199	16	no	—
44	C₂³	—	2	32	no	—
45	C₂³	—	2	24	no	—
46	C₂³	C₂×Dic₁₉₉	1	4	yes	—
47	C₂³	C₂×Dic₁₉₉	1	24	yes	complemented
48	C₂³	C₂²⋊D₁₉₉	1	8	yes	—
49	C₂³	C₂²×D₁₉₉	1	4	yes	—
50	C₂³	C₂²×D₁₉₉	1	3	yes	—
51	C₂×C₄	—	398	24	no	—
52	C₂×C₄	—	199	8	no	—
53	C₂²	—	2	8	no	—
54	C₂²	—	2	48	no	—
55	C₂²	C₂²⋊Dic₁₉₉	1	16	yes	direct product factor
56	C₂²	C₂²×Dic₁₉₉	1	6	yes	—
57	C₂²	C₂²×Dic₁₉₉	1	8	yes	complemented
58	C₂²	C₂×C₂²⋊D₁₉₉	1	12	yes	—
59	C₂²	C₂³×D₁₉₉	1	1	yes	characteristic
60	C₄	—	398	8	no	—
61	C₂	—	2	8	no	—
62	C₂	C₂×C₂²⋊Dic₁₉₉	1	12	yes	direct product factor
63	C₂	C₂³×Dic₁₉₉	1	1	yes	characteristic
64	C₂	C₂²×C₂²⋊D₁₉₉	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₉₉
commutator	C₂×C₁₉₉	C₂³×C₄
Frattini	C₂²	C₂³×D₁₉₉
Fitting, prime radical	C₂⁵×C₁₉₉	C₂
socle	C₂⁴×C₁₉₉	C₂²
cosocle	C₂²×C₁₉₉	C₂⁴
hypercenter	C₂⁵	D₁₉₉
hypocenter	C₁₉₉	C₂²×C₂²⋊C₄
LCS	G ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₁₉₉	C₂	semidirect
2. C₂×C₂²⋊Dic₁₉₉	C₂	direct (N × H)
5. C₂²×Dic₁₉₉	C₂²	semidirect
6. C₂²⋊Dic₁₉₉	C₂²	direct (N × H)
8. C₂⁴×C₁₉₉	C₄	semidirect
12. C₂×Dic₁₉₉	C₂³	semidirect
16. C₂³×C₁₉₉	C₂×C₄	semidirect
22. C₂²×C₁₉₉	C₂²⋊C₄	semidirect
24. C₂²×C₁₉₉	C₂²×C₄	semidirect
29. C₂×C₁₉₉	C₂×C₂²⋊C₄	semidirect
32. C₁₉₉	C₂²×C₂²⋊C₄	semidirect
38. C₂⁴	Dic₁₉₉	semidirect
47. C₂³	C₂×Dic₁₉₉	semidirect
55. C₂²	C₂²⋊Dic₁₉₉	direct (N × H)
57. C₂²	C₂²×Dic₁₉₉	semidirect
62. C₂	C₂×C₂²⋊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×C₂²⋊Dic₁₉₉	2
C₂⁵×C₁₉₉	2
C₂²×C₂²⋊C₄	199

Subgroups — C22×C22⋊Dic199 #27164466

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊Dic₁₉₉ #27164466