Subgroups — C₂²×D₂₅₁₂ #34113242

Order 20096.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₂₅₆	C₂	1	2	yes	maximal, complemented
2	C₂×D₂₅₁₂	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₆×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₆₂₈	—	2	2	no	—
5	C₂×D₁₂₅₆	—	2	6	no	—
6	C₂×D₁₂₅₆	C₂²	1	12	yes	complemented
7	D₂₅₁₂	C₂²	1	16	yes	direct product factor
8	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₁₅₇	C₂²	1	6	yes	complemented
10	C₂³×D₁₅₇	—	4	2	no	—
11	C₂×D₆₂₈	—	2	12	no	—
12	C₂×D₆₂₈	—	4	6	no	—
13	D₁₂₅₆	—	2	12	no	—
14	D₁₂₅₆	C₂³	1	8	yes	complemented
15	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
16	C₂×C₈×C₁₅₇	D₄	1	3	yes	—
17	C₂×C₈×C₁₅₇	C₂³	1	3	yes	—
18	C₁₆×C₁₅₇	C₂³	1	4	yes	complemented
19	C₂²×D₁₅₇	—	4	12	no	—
20	C₂²×D₁₅₇	—	8	8	no	—
21	D₆₂₈	—	2	8	no	—
22	D₆₂₈	—	4	12	no	—
23	C₂³×C₁₅₇	D₈	1	1	yes	characteristic
24	C₂×C₄×C₁₅₇	D₈	1	3	yes	—
25	C₂×C₄×C₁₅₇	C₂×D₄	1	3	yes	—
26	C₈×C₁₅₇	C₂×D₄	1	3	yes	—
27	C₈×C₁₅₇	C₂⁴	1	1	yes	characteristic
28	C₂²×C₁₅₇	D₁₆	1	4	yes	complemented
29	C₂²×C₁₅₇	C₂×D₈	1	3	yes	—
30	C₂×D₁₅₇	—	4	8	no	—
31	C₂×D₁₅₇	—	8	24	no	—
32	C₄×C₁₅₇	C₂×D₈	1	3	yes	—
33	C₄×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
34	C₂×C₁₅₇	C₂×D₁₆	1	6	yes	complemented
35	C₂×C₁₅₇	C₂²×D₈	1	1	yes	characteristic
36	D₁₅₇	—	8	8	no	—
37	C₁₅₇	C₂²×D₁₆	1	1	yes	characteristic, complemented, Sylow
38	C₂²×D₁₆	—	157	1	no	maximal, Sylow
39	C₂²×D₈	—	157	2	no	—
40	C₂×D₁₆	—	157	12	no	—
41	C₂²×C₁₆	D₁₅₇	1	1	yes	characteristic, complemented
42	C₂²×D₄	—	314	2	no	—
43	C₂×D₈	—	314	6	no	—
44	C₂×D₈	—	157	12	no	—
45	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
46	D₁₆	—	157	16	no	—
47	C₂×C₁₆	C₂×D₁₅₇	1	6	yes	complemented
48	C₂⁴	—	628	2	no	—
49	C₂×D₄	—	314	12	no	—
50	C₂×D₄	—	628	6	no	—
51	C₂²×C₄	D₆₂₈	1	1	yes	characteristic
52	D₈	—	314	12	no	—
53	D₈	—	157	8	no	—
54	C₂×C₈	D₆₂₈	1	3	yes	—
55	C₂×C₈	C₂²×D₁₅₇	1	3	yes	—
56	C₁₆	C₂²×D₁₅₇	1	4	yes	complemented
57	C₂³	—	628	12	no	—
58	C₂³	—	1256	8	no	—
59	C₂³	D₁₂₅₆	1	1	yes	characteristic
60	D₄	—	314	8	no	—
61	D₄	—	628	12	no	—
62	C₂×C₄	D₁₂₅₆	1	3	yes	—
63	C₂×C₄	C₂×D₆₂₈	1	3	yes	—
64	C₈	C₂×D₆₂₈	1	3	yes	—
65	C₈	C₂³×D₁₅₇	1	1	yes	characteristic
66	C₂²	—	628	8	no	—
67	C₂²	—	1256	24	no	—
68	C₂²	D₂₅₁₂	1	4	yes	direct product factor
69	C₂²	C₂×D₁₂₅₆	1	3	yes	—
70	C₄	C₂×D₁₂₅₆	1	3	yes	—
71	C₄	C₂²×D₆₂₈	1	1	yes	characteristic
72	C₂	—	1256	8	no	—
73	C₂	C₂×D₂₅₁₂	1	6	yes	direct product factor
74	C₂	C₂²×D₁₂₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₁₂₅₆
commutator, cosocle	C₈×C₁₅₇	C₂⁴
Frattini	C₈	C₂³×D₁₅₇
Fitting	C₂²×C₁₆×C₁₅₇	C₂
socle	C₂³×C₁₅₇	D₈
hypercenter	C₂²×C₁₆	D₁₅₇
hypocenter	C₁₅₇	C₂²×D₁₆
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₂₅₆	C₂	semidirect
2. C₂×D₂₅₁₂	C₂	direct (N × H)
3. C₂²×C₁₆×C₁₅₇	C₂	semidirect
6. C₂×D₁₂₅₆	C₂²	semidirect
7. D₂₅₁₂	C₂²	direct (N × H)
9. C₂×C₁₆×C₁₅₇	C₂²	semidirect
14. D₁₂₅₆	C₂³	semidirect
18. C₁₆×C₁₅₇	C₂³	semidirect
28. C₂²×C₁₅₇	D₁₆	semidirect
34. C₂×C₁₅₇	C₂×D₁₆	semidirect
37. C₁₅₇	C₂²×D₁₆	semidirect
41. C₂²×C₁₆	D₁₅₇	semidirect
47. C₂×C₁₆	C₂×D₁₅₇	semidirect
56. C₁₆	C₂²×D₁₅₇	semidirect
68. C₂²	D₂₅₁₂	direct (N × H)
73. C₂	C₂×D₂₅₁₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₂₅₆	2
C₂×D₂₅₁₂	2
C₂²×C₁₆×C₁₅₇	2
C₂²×D₁₆	157

Subgroups — C22×D2512 #34113242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₂₅₁₂ #34113242