Subgroups — C₂×#22553147 #30668258

Order 16768.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₀₉₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#22553147	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₁₃₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₀₄₈	—	2	1	no	—
6	D₂₀₉₆	—	2	1	no	—
7	D₂₀₉₆	C₂²	1	2	yes	complemented
8	C₁₃₁⋊C₃₂	C₂²	1	2	yes	complemented
9	C₂×D₈×C₁₃₁	—	2	1	no	—
10	D₁₆×C₁₃₁	—	2	1	no	—
11	D₁₆×C₁₃₁	C₂²	1	2	yes	complemented
12	C₂×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×D₅₂₄	—	4	1	no	—
14	D₁₀₄₈	—	2	2	no	—
15	D₁₀₄₈	—	4	1	no	—
16	C₂×D₄×C₁₃₁	—	4	1	no	—
17	D₈×C₁₃₁	—	2	2	no	—
18	D₈×C₁₃₁	—	4	1	no	—
19	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
20	C₁₆×C₁₃₁	D₄	1	1	yes	characteristic
21	C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
22	C₂²×D₁₃₁	—	8	1	no	—
23	D₅₂₄	—	4	2	no	—
24	D₅₂₄	—	8	1	no	—
25	C₂³×C₁₃₁	—	8	1	no	—
26	D₄×C₁₃₁	—	4	2	no	—
27	D₄×C₁₃₁	—	8	1	no	—
28	C₂×C₄×C₁₃₁	D₈	1	1	yes	characteristic
29	C₈×C₁₃₁	D₈	1	1	yes	characteristic
30	C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₁₃₁	—	8	2	no	—
32	C₂²×C₁₃₁	—	16	2	no	—
33	C₂²×C₁₃₁	D₁₆	1	1	yes	characteristic
34	C₂×D₁₃₁	—	8	2	no	—
35	C₂×D₁₃₁	—	16	2	no	—
36	C₄×C₁₃₁	D₁₆	1	1	yes	characteristic
37	C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
38	C₂×C₁₃₁	—	16	2	no	—
39	C₂×C₁₃₁	D₃₂	1	2	yes	complemented
40	C₂×C₁₃₁	C₂×D₁₆	1	1	yes	characteristic
41	D₁₃₁	—	16	2	no	—
42	C₁₃₁	C₂×D₃₂	1	1	yes	characteristic, complemented, Sylow
43	C₂×D₃₂	—	131	1	no	maximal, Sylow
44	C₂×D₁₆	—	131	1	no	—
45	C₂×D₁₆	D₁₃₁	1	1	yes	characteristic, complemented
46	D₃₂	—	131	4	no	—
47	C₂×C₃₂	—	131	1	no	—
48	C₂×D₈	—	2	1	no	—
49	C₂×D₈	—	262	1	no	—
50	D₁₆	—	2	1	no	—
51	D₁₆	—	262	1	no	—
52	D₁₆	—	131	2	no	—
53	D₁₆	C₂×D₁₃₁	1	2	yes	complemented
54	C₂×C₁₆	C₂×D₁₃₁	1	1	yes	characteristic
55	C₃₂	—	131	2	no	—
56	C₂×D₄	—	4	1	no	—
57	C₂×D₄	—	524	1	no	—
58	D₈	—	2	2	no	—
59	D₈	—	4	1	no	—
60	D₈	—	262	2	no	—
61	D₈	—	524	1	no	—
62	C₂×C₈	C₂²⋊D₁₃₁	1	1	yes	characteristic
63	C₁₆	C₂²⋊D₁₃₁	1	1	yes	characteristic
64	C₁₆	C₂²×D₁₃₁	1	1	yes	characteristic
65	C₂³	—	8	1	no	—
66	C₂³	—	1048	1	no	—
67	D₄	—	4	2	no	—
68	D₄	—	8	1	no	—
69	D₄	—	524	2	no	—
70	D₄	—	1048	1	no	—
71	C₂×C₄	#14523498	1	1	yes	characteristic
72	C₈	#14523498	1	1	yes	characteristic
73	C₈	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
74	C₂²	—	8	2	no	—
75	C₂²	—	16	2	no	—
76	C₂²	—	1048	2	no	—
77	C₂²	—	2096	2	no	—
78	C₂²	#16862187	1	1	yes	characteristic
79	C₄	#16862187	1	1	yes	characteristic
80	C₄	C₂×#14523498	1	1	yes	characteristic
81	C₂	—	16	2	no	—
82	C₂	—	2096	2	no	—
83	C₂	#22553147	1	2	yes	direct product factor
84	C₂	C₂×#16862187	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16862187
commutator, cosocle	C₁₆×C₁₃₁	C₂³
Frattini	C₁₆	C₂²×D₁₃₁
Fitting	C₂×D₁₆×C₁₃₁	C₂
socle	C₂²×C₁₃₁	D₁₆
hypercenter	C₂×D₁₆	D₁₃₁
hypocenter	C₁₃₁	C₂×D₃₂
LCS	G ▷ C₁₆×C₁₃₁ ▷ C₈×C₁₃₁ ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₀₉₆	C₂	semidirect
2. #22553147	C₂	direct (N × H)
3. C₂×C₁₃₁⋊C₃₂	C₂	semidirect
4. C₂×D₁₆×C₁₃₁	C₂	semidirect
7. D₂₀₉₆	C₂²	semidirect
8. C₁₃₁⋊C₃₂	C₂²	semidirect
11. D₁₆×C₁₃₁	C₂²	semidirect
39. C₂×C₁₃₁	D₃₂	semidirect
42. C₁₃₁	C₂×D₃₂	semidirect
45. C₂×D₁₆	D₁₃₁	semidirect
53. D₁₆	C₂×D₁₃₁	semidirect
83. C₂	#22553147	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₂₀₉₆	2
#22553147	2
C₂×C₁₃₁⋊C₃₂	2
C₂×D₁₆×C₁₃₁	2
C₂×D₃₂	131

Subgroups — C2×#22553147 #30668258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22553147 #30668258