Subgroups — C₂×#23781493 #35216316

Order 20864.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19104669	C₂	1	2	yes	maximal
2	#23781493	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₁₆×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₁₆₃	—	2	2	no	—
5	#19104669	—	2	2	no	—
6	#19104669	C₂²	1	4	yes	—
7	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×C₁₆₃	C₄	1	4	yes	complemented
9	C₂×C₁₆×C₁₆₃	C₂²	1	2	yes	—
10	C₂²×Dic₁₆₃	—	4	2	no	—
11	C₄⋊Dic₁₆₃	—	2	4	no	—
12	C₄⋊Dic₁₆₃	—	4	2	no	—
13	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
14	C₂×C₈×C₁₆₃	C₂×C₄	1	2	yes	—
15	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
16	C₂×C₈×C₁₆₃	Q₈	1	2	yes	—
17	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
18	C₁₆×C₁₆₃	C₂×C₄	1	4	yes	complemented
19	C₂×Dic₁₆₃	—	4	4	no	—
20	C₂×Dic₁₆₃	—	8	4	no	—
21	C₂³×C₁₆₃	D₈	1	1	yes	characteristic
22	C₂×C₄×C₁₆₃	C₄⋊C₄	1	2	yes	—
23	C₂×C₄×C₁₆₃	D₈	1	1	yes	characteristic
24	C₂×C₄×C₁₆₃	Q₁₆	1	2	yes	—
25	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
26	C₈×C₁₆₃	C₄⋊C₄	1	2	yes	—
27	C₈×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
28	C₈×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
29	C₂²×C₁₆₃	#107	1	2	yes	—
30	C₂²×C₁₆₃	SD₃₂	1	4	yes	—
31	C₂²×C₁₆₃	C₂×D₈	1	1	yes	characteristic
32	C₄×C₁₆₃	#107	1	2	yes	—
33	C₄×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₄×C₁₆₃	C₂×Q₁₆	1	1	yes	characteristic
35	Dic₁₆₃	—	8	4	no	—
36	C₂×C₁₆₃	#367	1	4	yes	complemented
37	C₂×C₁₆₃	C₂×#107	1	1	yes	characteristic
38	C₂×C₁₆₃	C₂×SD₃₂	1	2	yes	—
39	C₁₆₃	C₂×#367	1	1	yes	characteristic, complemented, Sylow
40	C₂×#367	—	163	1	no	maximal, Sylow
41	C₂²×C₁₆	D₁₆₃	1	1	yes	characteristic
42	C₂×#107	—	163	2	no	—
43	#367	—	163	4	no	—
44	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
45	C₂×C₄⋊C₄	—	326	2	no	—
46	C₂×C₁₆	Dic₁₆₃	1	4	yes	complemented
47	C₂×C₁₆	C₂×D₁₆₃	1	2	yes	—
48	#107	—	326	2	no	—
49	#107	—	163	4	no	—
50	C₂²×C₄	—	652	2	no	—
51	C₂²×C₄	D₆₅₂	1	1	yes	characteristic
52	C₂×C₈	C₂×Dic₁₆₃	1	2	yes	—
53	C₂×C₈	D₆₅₂	1	1	yes	characteristic
54	C₂×C₈	Dic₃₂₆	1	2	yes	—
55	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
56	C₄⋊C₄	—	326	4	no	—
57	C₄⋊C₄	—	652	2	no	—
58	C₁₆	C₂×Dic₁₆₃	1	4	yes	complemented
59	C₂³	D₁₃₀₄	1	1	yes	characteristic
60	C₂×C₄	—	1304	4	no	—
61	C₂×C₄	—	652	4	no	—
62	C₂×C₄	C₄⋊Dic₁₆₃	1	2	yes	—
63	C₂×C₄	D₁₃₀₄	1	1	yes	characteristic
64	C₂×C₄	Dic₆₅₂	1	2	yes	—
65	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
66	C₈	C₄⋊Dic₁₆₃	1	2	yes	—
67	C₈	C₂²×Dic₁₆₃	1	1	yes	characteristic
68	C₈	C₂×Dic₃₂₆	1	1	yes	characteristic
69	C₂²	#19104669	1	2	yes	—
70	C₂²	#19104680	1	4	yes	—
71	C₂²	C₂×D₁₃₀₄	1	1	yes	characteristic
72	C₄	—	1304	4	no	—
73	C₄	#19104669	1	2	yes	—
74	C₄	C₂×C₄⋊Dic₁₆₃	1	1	yes	characteristic
75	C₄	C₂×Dic₆₅₂	1	1	yes	characteristic
76	C₂	#23781493	1	4	yes	direct product factor
77	C₂	C₂×#19104669	1	1	yes	characteristic
78	C₂	C₂×#19104680	1	2	yes	—

	subgroup	G/H
center	C₂³	D₁₃₀₄
commutator	C₈×C₁₆₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₆₃
Fitting	C₂²×C₁₆×C₁₆₃	C₂
socle, prime radical	C₂³×C₁₆₃	D₈
cosocle	C₂×C₈×C₁₆₃	C₂³
hypercenter	C₂²×C₁₆	D₁₆₃
hypocenter	C₁₆₃	C₂×#367
LCS	G ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
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
2. #23781493	C₂	direct (N × H)
8. C₂×C₁₆×C₁₆₃	C₄	semidirect
18. C₁₆×C₁₆₃	C₂×C₄	semidirect
36. C₂×C₁₆₃	#367	semidirect
39. C₁₆₃	C₂×#367	semidirect
46. C₂×C₁₆	Dic₁₆₃	semidirect
58. C₁₆	C₂×Dic₁₆₃	semidirect
76. C₂	#23781493	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×#19104669	2
#23781493	2
C₂²×C₁₆×C₁₆₃	2
C₂×#367	163

Subgroups — C2×#23781493 #35216316

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23781493 #35216316