Subgroups — #30668211

Order 16768.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₀₉₆	C₂	1	4	yes	maximal, complemented
2	#22553689	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₀₄₈	—	2	2	no	—
5	C₂×D₁₀₄₈	C₂²	1	4	yes	—
6	#16862243	—	2	2	no	—
7	D₂₀₉₆	—	2	8	no	—
8	C₂×C₁₆×C₁₃₁	C₂²	1	2	yes	—
9	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×D₅₂₄	—	2	4	no	—
11	C₂×D₅₂₄	—	4	4	no	—
12	D₁₀₄₈	—	2	8	no	—
13	D₁₀₄₈	—	4	4	no	—
14	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
15	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
16	C₄²×C₁₃₁	D₄	1	1	yes	characteristic
17	C₁₆×C₁₃₁	D₄	1	4	yes	complemented
18	C₂²×D₁₃₁	—	4	4	no	—
19	D₅₂₄	—	4	4	no	—
20	D₅₂₄	—	4	4	no	—
21	D₅₂₄	—	8	8	no	—
22	C₂×C₄×C₁₃₁	D₈	1	2	yes	—
23	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
24	C₈×C₁₃₁	D₈	1	2	yes	—
25	C₈×C₁₃₁	C₂×D₄	1	2	yes	—
26	C₂²×C₁₃₁	C₂×D₈	1	1	yes	characteristic
27	C₂×D₁₃₁	—	8	4	no	—
28	C₂×D₁₃₁	—	8	8	no	—
29	C₄×C₁₃₁	D₁₆	1	4	yes	complemented
30	C₄×C₁₃₁	#127	1	1	yes	characteristic
31	C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
32	C₂×C₁₃₁	#493	1	1	yes	characteristic
33	C₂×C₁₃₁	C₂×D₁₆	1	2	yes	—
34	D₁₃₁	—	16	4	no	—
35	C₁₃₁	#2246	1	1	yes	characteristic, complemented, Sylow
36	#2246	—	131	1	no	maximal, Sylow
37	C₂×D₁₆	—	131	4	no	—
38	#493	—	131	2	no	—
39	C₄×C₁₆	D₁₃₁	1	1	yes	characteristic, complemented
40	C₂×D₈	—	262	2	no	—
41	C₂×D₈	—	131	4	no	—
42	#127	—	262	2	no	—
43	D₁₆	—	262	8	no	—
44	C₂×C₁₆	C₂×D₁₃₁	1	2	yes	—
45	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
46	C₂×D₄	—	262	4	no	—
47	C₂×D₄	—	524	4	no	—
48	D₈	—	262	8	no	—
49	D₈	—	524	4	no	—
50	C₂×C₈	D₅₂₄	1	1	yes	characteristic
51	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
52	C₄²	D₅₂₄	1	1	yes	characteristic
53	C₁₆	D₅₂₄	1	4	yes	complemented
54	C₂³	—	524	4	no	—
55	D₄	—	524	4	no	—
56	D₄	—	524	4	no	—
57	D₄	—	1048	8	no	—
58	C₂×C₄	D₁₀₄₈	1	2	yes	—
59	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
60	C₈	D₁₀₄₈	1	2	yes	—
61	C₈	C₂×D₅₂₄	1	2	yes	—
62	C₂²	—	1048	4	no	—
63	C₂²	—	1048	8	no	—
64	C₂²	C₂×D₁₀₄₈	1	1	yes	characteristic
65	C₄	D₂₀₉₆	1	4	yes	complemented
66	C₄	#16862243	1	1	yes	characteristic
67	C₄	C₂×D₁₀₄₈	1	1	yes	characteristic
68	C₂	—	2096	4	no	—
69	C₂	#22553689	1	1	yes	characteristic
70	C₂	C₂×D₂₀₉₆	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₀₄₈
commutator, cosocle	C₂×C₈×C₁₃₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₁
Fitting	C₄×C₁₆×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×D₈
hypercenter	C₄×C₁₆	D₁₃₁
hypocenter	C₁₃₁	#2246
LCS	G ▷ C₂×C₈×C₁₃₁ ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
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×D₂₀₉₆	C₂	semidirect
2. #22553689	C₂	semidirect
3. C₄×C₁₆×C₁₃₁	C₂	semidirect
17. C₁₆×C₁₃₁	D₄	semidirect
29. C₄×C₁₃₁	D₁₆	semidirect
35. C₁₃₁	#2246	semidirect
39. C₄×C₁₆	D₁₃₁	semidirect
53. C₁₆	D₅₂₄	semidirect
65. C₄	D₂₀₉₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₀₉₆	2
#22553689	2
C₄×C₁₆×C₁₃₁	2
#2246	131