Subgroups — C₂²×D₆₈₈ #19147701

Order 5504.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₃₄₄	C₂	1	2	yes	maximal, complemented
2	C₂×D₆₈₈	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₁₇₂	—	2	2	no	—
5	C₂×D₃₄₄	—	2	6	no	—
6	C₂×D₃₄₄	C₂²	1	12	yes	complemented
7	D₆₈₈	C₂²	1	16	yes	direct product factor
8	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₄₃	C₂²	1	6	yes	complemented
10	C₂³×D₄₃	—	4	2	no	—
11	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
12	C₂×D₁₇₂	—	2	12	no	—
13	C₂×D₁₇₂	—	4	6	no	—
14	C₂×C₈×C₄₃	D₄	1	3	yes	—
15	C₂×C₈×C₄₃	C₂³	1	3	yes	—
16	D₃₄₄	—	2	12	no	—
17	D₃₄₄	C₂³	1	8	yes	complemented
18	C₁₆×C₄₃	C₂³	1	4	yes	complemented
19	C₂³×C₄₃	D₈	1	1	yes	characteristic
20	C₂²×D₄₃	—	4	12	no	—
21	C₂²×D₄₃	—	8	8	no	—
22	C₂×C₄×C₄₃	D₈	1	3	yes	—
23	C₂×C₄×C₄₃	C₂×D₄	1	3	yes	—
24	D₁₇₂	—	2	8	no	—
25	D₁₇₂	—	4	12	no	—
26	C₈×C₄₃	C₂×D₄	1	3	yes	—
27	C₈×C₄₃	C₂⁴	1	1	yes	characteristic
28	C₂²×C₄₃	D₁₆	1	4	yes	complemented
29	C₂²×C₄₃	C₂×D₈	1	3	yes	—
30	C₂×D₄₃	—	4	8	no	—
31	C₂×D₄₃	—	8	24	no	—
32	C₄×C₄₃	C₂×D₈	1	3	yes	—
33	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
34	C₂²×D₁₆	—	43	1	no	maximal, Sylow
35	C₂×C₄₃	C₂×D₁₆	1	6	yes	complemented
36	C₂×C₄₃	C₂²×D₈	1	1	yes	characteristic
37	D₄₃	—	8	8	no	—
38	C₂²×D₈	—	43	2	no	—
39	C₂×D₁₆	—	43	12	no	—
40	C₂²×C₁₆	D₄₃	1	1	yes	characteristic, complemented
41	C₄₃	C₂²×D₁₆	1	1	yes	characteristic, complemented, Sylow
42	C₂²×D₄	—	86	2	no	—
43	C₂×D₈	—	43	12	no	—
44	C₂×D₈	—	86	6	no	—
45	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
46	D₁₆	—	43	16	no	—
47	C₂×C₁₆	C₂×D₄₃	1	6	yes	complemented
48	C₂⁴	—	172	2	no	—
49	C₂×D₄	—	86	12	no	—
50	C₂×D₄	—	172	6	no	—
51	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
52	D₈	—	43	8	no	—
53	D₈	—	86	12	no	—
54	C₂×C₈	D₁₇₂	1	3	yes	—
55	C₂×C₈	C₂²×D₄₃	1	3	yes	—
56	C₁₆	C₂²×D₄₃	1	4	yes	complemented
57	C₂³	—	344	8	no	—
58	C₂³	—	172	12	no	—
59	C₂³	D₃₄₄	1	1	yes	characteristic
60	D₄	—	86	8	no	—
61	D₄	—	172	12	no	—
62	C₂×C₄	D₃₄₄	1	3	yes	—
63	C₂×C₄	C₂×D₁₇₂	1	3	yes	—
64	C₈	C₂×D₁₇₂	1	3	yes	—
65	C₈	C₂³×D₄₃	1	1	yes	characteristic
66	C₂²	—	344	24	no	—
67	C₂²	—	172	8	no	—
68	C₂²	D₆₈₈	1	4	yes	direct product factor
69	C₂²	C₂×D₃₄₄	1	3	yes	—
70	C₄	C₂×D₃₄₄	1	3	yes	—
71	C₄	C₂²×D₁₇₂	1	1	yes	characteristic
72	C₂	—	344	8	no	—
73	C₂	C₂×D₆₈₈	1	6	yes	direct product factor
74	C₂	C₂²×D₃₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₃₄₄
commutator	C₈×C₄₃	C₂⁴
Frattini	C₈	C₂³×D₄₃
Fitting	C₂²×C₁₆×C₄₃	C₂
socle	C₂³×C₄₃	D₈
hypercenter	C₂²×C₁₆	D₄₃
hypocenter	C₄₃	C₂²×D₁₆
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
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×D₃₄₄	C₂	semidirect
2. C₂×D₆₈₈	C₂	direct (N × H)
3. C₂²×C₁₆×C₄₃	C₂	semidirect
6. C₂×D₃₄₄	C₂²	semidirect
7. D₆₈₈	C₂²	direct (N × H)
9. C₂×C₁₆×C₄₃	C₂²	semidirect
17. D₃₄₄	C₂³	semidirect
18. C₁₆×C₄₃	C₂³	semidirect
28. C₂²×C₄₃	D₁₆	semidirect
35. C₂×C₄₃	C₂×D₁₆	semidirect
40. C₂²×C₁₆	D₄₃	semidirect
41. C₄₃	C₂²×D₁₆	semidirect
47. C₂×C₁₆	C₂×D₄₃	semidirect
56. C₁₆	C₂²×D₄₃	semidirect
68. C₂²	D₆₈₈	direct (N × H)
73. C₂	C₂×D₆₈₈	direct (N × H)

Subgroups — C22×D688 #19147701

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×D₆₈₈ #19147701