Subgroups — #35216851

Order 20864.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782097	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₂₆₀₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#23781504	C₂	1	2	yes	maximal, complemented
4	D₅₂₁₆	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19104757	—	2	1	no	—
7	C₂×D₁₃₀₄	—	2	1	no	—
8	Dic₁₃₀₄	C₂²	1	1	yes	characteristic, complemented
9	#19104680	—	2	1	no	—
10	D₂₆₀₈	—	2	1	no	—
11	D₂₆₀₈	C₂²	1	2	yes	—
12	D₂₆₀₈	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₃₂×C₁₆₃	C₂²	1	2	yes	complemented
15	#14579699	—	4	1	no	—
16	C₂×D₆₅₂	—	4	1	no	—
17	Dic₆₅₂	—	2	1	no	—
18	#14579688	—	4	1	no	—
19	D₁₃₀₄	—	2	2	no	—
20	D₁₃₀₄	—	2	1	no	—
21	D₁₃₀₄	—	4	1	no	—
22	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
23	C₁₆×C₁₆₃	D₄	1	1	yes	characteristic
24	C₁₆×C₁₆₃	C₂³	1	1	yes	characteristic
25	C₂²×D₁₆₃	—	8	1	no	—
26	Dic₃₂₆	—	4	1	no	—
27	D₆₅₂	—	4	2	no	—
28	D₆₅₂	—	4	1	no	—
29	D₆₅₂	—	8	1	no	—
30	C₂²⋊D₁₆₃	—	8	1	no	—
31	C₄×D₁₆₃	—	8	1	no	—
32	C₂×C₄×C₁₆₃	D₈	1	1	yes	characteristic
33	C₈×C₁₆₃	D₈	1	1	yes	characteristic
34	C₈×C₁₆₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₁₆₃	D₁₆	1	1	yes	characteristic
36	C₂×D₁₆₃	—	8	1	no	—
37	C₂×D₁₆₃	—	8	2	no	—
38	C₂×D₁₆₃	—	16	2	no	—
39	C₄×C₁₆₃	D₁₆	1	1	yes	characteristic
40	C₄×C₁₆₃	C₂×D₈	1	1	yes	characteristic
41	Dic₁₆₃	—	8	1	no	—
42	C₂×C₁₆₃	—	2	1	no	—
43	C₂×C₁₆₃	C₂×D₁₆	1	1	yes	characteristic
44	D₁₆₃	—	16	1	no	—
45	D₁₆₃	—	16	2	no	—
46	C₁₆₃	#2263	1	1	yes	characteristic, complemented, Sylow
47	#2263	—	163	1	no	maximal, Sylow
48	#508	—	163	1	no	—
49	C₂×D₁₆	—	163	1	no	—
50	SD₆₄	—	163	2	no	—
51	D₃₂	—	163	2	no	—
52	M₆₄	D₁₆₃	1	1	yes	characteristic, complemented
53	#135	—	326	1	no	—
54	C₂×D₈	—	326	1	no	—
55	Q₃₂	—	163	1	no	—
56	SD₃₂	—	326	1	no	—
57	D₁₆	—	326	1	no	—
58	D₁₆	—	163	1	no	—
59	D₁₆	—	163	2	no	—
60	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
61	C₃₂	C₂×D₁₆₃	1	2	yes	complemented
62	#41	—	652	1	no	—
63	C₂×D₄	—	652	1	no	—
64	Q₁₆	—	326	1	no	—
65	SD₁₆	—	652	1	no	—
66	D₈	—	326	2	no	—
67	D₈	—	326	1	no	—
68	D₈	—	652	1	no	—
69	C₂×C₈	D₆₅₂	1	1	yes	characteristic
70	C₁₆	D₆₅₂	1	1	yes	characteristic
71	C₁₆	C₂²×D₁₆₃	1	1	yes	characteristic
72	C₂³	—	1304	1	no	—
73	Q₈	—	652	1	no	—
74	D₄	—	1304	1	no	—
75	D₄	—	1304	1	no	—
76	D₄	—	652	2	no	—
77	D₄	—	652	1	no	—
78	C₂×C₄	—	1304	1	no	—
79	C₂×C₄	D₁₃₀₄	1	1	yes	characteristic
80	C₈	D₁₃₀₄	1	1	yes	characteristic
81	C₈	C₂×D₆₅₂	1	1	yes	characteristic
82	C₂²	—	1304	1	no	—
83	C₂²	—	1304	2	no	—
84	C₂²	—	2608	2	no	—
85	C₂²	D₂₆₀₈	1	1	yes	characteristic
86	C₄	—	1304	1	no	—
87	C₄	D₂₆₀₈	1	1	yes	characteristic
88	C₄	C₂×D₁₃₀₄	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	2608	2	no	—
91	C₂	—	2608	1	no	—
92	C₂	C₂×D₂₆₀₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₆₀₈
commutator, cosocle	C₁₆×C₁₆₃	C₂³
Frattini	C₁₆	C₂²×D₁₆₃
Fitting	M₆₄×C₁₆₃	C₂
socle	C₂×C₁₆₃	C₂×D₁₆
hypercenter	M₆₄	D₁₆₃
hypocenter	C₁₆₃	#2263
LCS	G ▷ C₁₆×C₁₆₃ ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
1. #23782097	C₂	semidirect
2. C₂×D₂₆₀₈	C₂	semidirect
3. #23781504	C₂	semidirect
4. D₅₂₁₆	C₂	semidirect
5. M₆₄×C₁₆₃	C₂	semidirect
8. Dic₁₃₀₄	C₂²	semidirect
12. D₂₆₀₈	C₂²	semidirect
14. C₃₂×C₁₆₃	C₂²	semidirect
46. C₁₆₃	#2263	semidirect
52. M₆₄	D₁₆₃	semidirect
61. C₃₂	C₂×D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782097	2
C₂×D₂₆₀₈	2
#23781504	2
D₅₂₁₆	2
M₆₄×C₁₆₃	2
#2263	163