Subgroups — C₂²×D₂₂₂₄ #31819510

Order 17792.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₁₁₂	C₂	1	2	yes	maximal, complemented
2	C₂×D₂₂₂₄	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₆×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₅₅₆	—	2	2	no	—
5	C₂×D₁₁₁₂	—	2	6	no	—
6	C₂×D₁₁₁₂	C₂²	1	12	yes	complemented
7	D₂₂₂₄	C₂²	1	16	yes	direct product factor
8	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₁₃₉	C₂²	1	6	yes	complemented
10	C₂³×D₁₃₉	—	4	2	no	—
11	C₂×D₅₅₆	—	2	12	no	—
12	C₂×D₅₅₆	—	4	6	no	—
13	D₁₁₁₂	—	2	12	no	—
14	D₁₁₁₂	C₂³	1	8	yes	complemented
15	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
16	C₂×C₈×C₁₃₉	D₄	1	3	yes	—
17	C₂×C₈×C₁₃₉	C₂³	1	3	yes	—
18	C₁₆×C₁₃₉	C₂³	1	4	yes	complemented
19	C₂²×D₁₃₉	—	4	12	no	—
20	C₂²×D₁₃₉	—	8	8	no	—
21	D₅₅₆	—	2	8	no	—
22	D₅₅₆	—	4	12	no	—
23	C₂³×C₁₃₉	D₈	1	1	yes	characteristic
24	C₂×C₄×C₁₃₉	D₈	1	3	yes	—
25	C₂×C₄×C₁₃₉	C₂×D₄	1	3	yes	—
26	C₈×C₁₃₉	C₂×D₄	1	3	yes	—
27	C₈×C₁₃₉	C₂⁴	1	1	yes	characteristic
28	C₂²×C₁₃₉	D₁₆	1	4	yes	complemented
29	C₂²×C₁₃₉	C₂×D₈	1	3	yes	—
30	C₂×D₁₃₉	—	4	8	no	—
31	C₂×D₁₃₉	—	8	24	no	—
32	C₄×C₁₃₉	C₂×D₈	1	3	yes	—
33	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
34	C₂×C₁₃₉	C₂×D₁₆	1	6	yes	complemented
35	C₂×C₁₃₉	C₂²×D₈	1	1	yes	characteristic
36	D₁₃₉	—	8	8	no	—
37	C₁₃₉	C₂²×D₁₆	1	1	yes	characteristic, complemented, Sylow
38	C₂²×D₁₆	—	139	1	no	maximal, Sylow
39	C₂²×D₈	—	139	2	no	—
40	C₂×D₁₆	—	139	12	no	—
41	C₂²×C₁₆	D₁₃₉	1	1	yes	characteristic, complemented
42	C₂²×D₄	—	278	2	no	—
43	C₂×D₈	—	278	6	no	—
44	C₂×D₈	—	139	12	no	—
45	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
46	D₁₆	—	139	16	no	—
47	C₂×C₁₆	C₂×D₁₃₉	1	6	yes	complemented
48	C₂⁴	—	556	2	no	—
49	C₂×D₄	—	278	12	no	—
50	C₂×D₄	—	556	6	no	—
51	C₂²×C₄	D₅₅₆	1	1	yes	characteristic
52	D₈	—	278	12	no	—
53	D₈	—	139	8	no	—
54	C₂×C₈	D₅₅₆	1	3	yes	—
55	C₂×C₈	C₂²×D₁₃₉	1	3	yes	—
56	C₁₆	C₂²×D₁₃₉	1	4	yes	complemented
57	C₂³	—	556	12	no	—
58	C₂³	—	1112	8	no	—
59	C₂³	D₁₁₁₂	1	1	yes	characteristic
60	D₄	—	278	8	no	—
61	D₄	—	556	12	no	—
62	C₂×C₄	D₁₁₁₂	1	3	yes	—
63	C₂×C₄	C₂×D₅₅₆	1	3	yes	—
64	C₈	C₂×D₅₅₆	1	3	yes	—
65	C₈	C₂³×D₁₃₉	1	1	yes	characteristic
66	C₂²	—	556	8	no	—
67	C₂²	—	1112	24	no	—
68	C₂²	D₂₂₂₄	1	4	yes	direct product factor
69	C₂²	C₂×D₁₁₁₂	1	3	yes	—
70	C₄	C₂×D₁₁₁₂	1	3	yes	—
71	C₄	C₂²×D₅₅₆	1	1	yes	characteristic
72	C₂	—	1112	8	no	—
73	C₂	C₂×D₂₂₂₄	1	6	yes	direct product factor
74	C₂	C₂²×D₁₁₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₁₁₁₂
commutator, cosocle	C₈×C₁₃₉	C₂⁴
Frattini	C₈	C₂³×D₁₃₉
Fitting	C₂²×C₁₆×C₁₃₉	C₂
socle	C₂³×C₁₃₉	D₈
hypercenter	C₂²×C₁₆	D₁₃₉
hypocenter	C₁₃₉	C₂²×D₁₆
LCS	G ▷ C₈×C₁₃₉ ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
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×D₁₁₁₂	C₂	semidirect
2. C₂×D₂₂₂₄	C₂	direct (N × H)
3. C₂²×C₁₆×C₁₃₉	C₂	semidirect
6. C₂×D₁₁₁₂	C₂²	semidirect
7. D₂₂₂₄	C₂²	direct (N × H)
9. C₂×C₁₆×C₁₃₉	C₂²	semidirect
14. D₁₁₁₂	C₂³	semidirect
18. C₁₆×C₁₃₉	C₂³	semidirect
28. C₂²×C₁₃₉	D₁₆	semidirect
34. C₂×C₁₃₉	C₂×D₁₆	semidirect
37. C₁₃₉	C₂²×D₁₆	semidirect
41. C₂²×C₁₆	D₁₃₉	semidirect
47. C₂×C₁₆	C₂×D₁₃₉	semidirect
56. C₁₆	C₂²×D₁₃₉	semidirect
68. C₂²	D₂₂₂₄	direct (N × H)
73. C₂	C₂×D₂₂₂₄	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×D₁₁₁₂	2
C₂×D₂₂₂₄	2
C₂²×C₁₆×C₁₃₉	2
C₂²×D₁₆	139

Subgroups — C22×D2224 #31819510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₂₂₂₄ #31819510