Subgroups — #19136845

Order 5504.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₆₈₈	C₂	1	4	yes	maximal, complemented
2	#14583276	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₃₄₄	—	2	2	no	—
5	C₂×D₃₄₄	C₂²	1	4	yes	—
6	#13063322	—	2	2	no	—
7	D₆₈₈	—	2	8	no	—
8	C₂×C₁₆×C₄₃	C₂²	1	2	yes	—
9	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
10	C₂×D₁₇₂	—	2	4	no	—
11	C₂×D₁₇₂	—	4	4	no	—
12	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
13	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
14	C₄²×C₄₃	D₄	1	1	yes	characteristic
15	D₃₄₄	—	2	8	no	—
16	D₃₄₄	—	4	4	no	—
17	C₁₆×C₄₃	D₄	1	4	yes	complemented
18	C₂²×D₄₃	—	4	4	no	—
19	C₂×C₄×C₄₃	D₈	1	2	yes	—
20	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
21	D₁₇₂	—	4	4	no	—
22	D₁₇₂	—	4	4	no	—
23	D₁₇₂	—	8	8	no	—
24	C₈×C₄₃	D₈	1	2	yes	—
25	C₈×C₄₃	C₂×D₄	1	2	yes	—
26	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
27	C₂×D₄₃	—	8	4	no	—
28	C₂×D₄₃	—	8	8	no	—
29	C₄×C₄₃	D₁₆	1	4	yes	complemented
30	C₄×C₄₃	#127	1	1	yes	characteristic
31	C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
32	#2246	—	43	1	no	maximal, Sylow
33	C₂×C₄₃	#493	1	1	yes	characteristic
34	C₂×C₄₃	C₂×D₁₆	1	2	yes	—
35	D₄₃	—	16	4	no	—
36	C₂×D₁₆	—	43	4	no	—
37	#493	—	43	2	no	—
38	C₄×C₁₆	D₄₃	1	1	yes	characteristic, complemented
39	C₄₃	#2246	1	1	yes	characteristic, complemented, Sylow
40	C₂×D₈	—	43	4	no	—
41	C₂×D₈	—	86	2	no	—
42	#127	—	86	2	no	—
43	D₁₆	—	86	8	no	—
44	C₂×C₁₆	C₂×D₄₃	1	2	yes	—
45	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
46	C₂×D₄	—	86	4	no	—
47	C₂×D₄	—	172	4	no	—
48	D₈	—	86	8	no	—
49	D₈	—	172	4	no	—
50	C₂×C₈	D₁₇₂	1	1	yes	characteristic
51	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
52	C₄²	D₁₇₂	1	1	yes	characteristic
53	C₁₆	D₁₇₂	1	4	yes	complemented
54	C₂³	—	172	4	no	—
55	D₄	—	344	8	no	—
56	D₄	—	172	4	no	—
57	D₄	—	172	4	no	—
58	C₂×C₄	D₃₄₄	1	2	yes	—
59	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
60	C₈	D₃₄₄	1	2	yes	—
61	C₈	C₂×D₁₇₂	1	2	yes	—
62	C₂²	—	344	8	no	—
63	C₂²	—	344	4	no	—
64	C₂²	C₂×D₃₄₄	1	1	yes	characteristic
65	C₄	D₆₈₈	1	4	yes	complemented
66	C₄	#13063322	1	1	yes	characteristic
67	C₄	C₂×D₃₄₄	1	1	yes	characteristic
68	C₂	—	688	4	no	—
69	C₂	#14583276	1	1	yes	characteristic
70	C₂	C₂×D₆₈₈	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₃₄₄
commutator	C₂×C₈×C₄₃	C₂³
Frattini	C₂×C₈	C₂²×D₄₃
Fitting	C₄×C₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂×D₈
hypercenter	C₄×C₁₆	D₄₃
hypocenter	C₄₃	#2246
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₆₈₈	C₂	semidirect
2. #14583276	C₂	semidirect
3. C₄×C₁₆×C₄₃	C₂	semidirect
17. C₁₆×C₄₃	D₄	semidirect
29. C₄×C₄₃	D₁₆	semidirect
38. C₄×C₁₆	D₄₃	semidirect
39. C₄₃	#2246	semidirect
53. C₁₆	D₁₇₂	semidirect
65. C₄	D₆₈₈	semidirect