Subgroups — #35216423

Order 20864.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782097	C₂	1	1	yes	maximal, characteristic
2	#23781783	C₂	1	2	yes	maximal, complemented
3	#23781495	C₂	1	1	yes	maximal, characteristic, complemented
4	#23781473	C₂	1	2	yes	maximal, complemented
5	#346×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19104757	C₂²	1	2	yes	—
7	#19104741	—	2	2	no	—
8	Dic₁₃₀₄	C₄	1	1	yes	characteristic, complemented
9	#19104680	C₄	1	2	yes	complemented
10	D₂₆₀₈	C₄	1	1	yes	characteristic, complemented
11	#19104671	C₂²	1	2	yes	—
12	#19104663	—	2	2	no	—
13	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
16	#14579699	—	2	2	no	—
17	Dic₆₅₂	C₂×C₄	1	2	yes	—
18	#14579688	—	2	2	no	—
19	D₁₃₀₄	C₂×C₄	1	2	yes	—
20	#14579683	—	2	2	no	—
21	#14579682	—	4	2	no	—
22	C₈×D₁₆₃	—	4	2	no	—
23	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
25	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
26	C₁₆×C₁₆₃	—	2	1	no	—
27	C₁₆×C₁₆₃	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₁₆₃	C₂×C₄	1	1	yes	characteristic
29	Dic₃₂₆	—	2	2	no	—
30	D₆₅₂	—	2	2	no	—
31	C₂²⋊D₁₆₃	—	4	2	no	—
32	C₄×D₁₆₃	—	4	2	no	—
33	C₁₆₃⋊C₈	—	4	2	no	—
34	C₂×C₄×C₁₆₃	—	2	1	no	—
35	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
36	C₈×C₁₆₃	D₈	1	2	yes	—
37	C₈×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₆₃	#41	1	1	yes	characteristic
39	C₂²×C₁₆₃	#135	1	1	yes	characteristic
40	C₂×D₁₆₃	—	4	2	no	—
41	C₄×C₁₆₃	—	4	1	no	—
42	C₄×C₁₆₃	C₄×D₄	1	1	yes	characteristic
43	C₄×C₁₆₃	C₂×D₈	1	1	yes	characteristic
44	Dic₁₆₃	—	4	2	no	—
45	C₂×C₁₆₃	—	2	1	no	—
46	C₂×C₁₆₃	C₄×D₈	1	1	yes	characteristic
47	D₁₆₃	—	8	2	no	—
48	C₁₆₃	#2179	1	1	yes	characteristic, complemented, Sylow
49	#2179	—	163	1	no	maximal, Sylow
50	#508	—	163	1	no	—
51	#443	—	163	2	no	—
52	#368	—	163	1	no	—
53	#359	—	163	2	no	—
54	#346	D₁₆₃	1	1	yes	characteristic, complemented
55	#135	—	163	2	no	—
56	#131	—	326	2	no	—
57	Q₃₂	—	163	1	no	—
58	SD₃₂	—	163	2	no	—
59	D₁₆	—	163	1	no	—
60	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
61	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
62	#108	—	163	2	no	—
63	#104	—	326	2	no	—
64	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
65	#41	—	326	2	no	—
66	Q₁₆	—	163	2	no	—
67	SD₁₆	—	326	2	no	—
68	D₈	—	163	2	no	—
69	M₁₆	—	326	2	no	—
70	M₁₆	—	652	2	no	—
71	C₂×C₈	—	652	2	no	—
72	C₂×C₈	D₆₅₂	1	1	yes	characteristic
73	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
74	C₄²	D₆₅₂	1	1	yes	characteristic
75	C₁₆	—	2	1	no	—
76	C₁₆	C₄×D₁₆₃	1	1	yes	characteristic
77	C₁₆	C₄×D₁₆₃	1	1	yes	characteristic
78	Q₈	—	326	2	no	—
79	D₄	—	326	2	no	—
80	D₄	—	652	2	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	652	2	no	—
83	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
84	C₈	—	652	2	no	—
85	C₈	D₁₃₀₄	1	2	yes	—
86	C₈	C₂×C₄×D₁₆₃	1	1	yes	characteristic
87	C₈	#14579699	1	1	yes	characteristic
88	C₂²	—	652	2	no	—
89	C₂²	#19104757	1	1	yes	characteristic
90	C₄	—	4	1	no	—
91	C₄	—	652	2	no	—
92	C₄	C₄×D₆₅₂	1	1	yes	characteristic
93	C₄	C₂×D₁₃₀₄	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	1304	2	no	—
96	C₂	C₄×D₁₃₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₁₃₀₄
commutator	C₈×C₁₆₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₆₃
Fitting	#346×C₁₆₃	C₂
socle	C₂×C₁₆₃	C₄×D₈
prime radical	#23782097	C₂
cosocle	C₂×C₈×C₁₆₃	C₂³
hypercenter	#346	D₁₆₃
hypocenter	C₁₆₃	#2179
LCS	G ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. #23781783	C₂	semidirect
3. #23781495	C₂	semidirect
4. #23781473	C₂	semidirect
5. #346×C₁₆₃	C₂	semidirect
8. Dic₁₃₀₄	C₄	semidirect
9. #19104680	C₄	semidirect
10. D₂₆₀₈	C₄	semidirect
48. C₁₆₃	#2179	semidirect
54. #346	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
#23781783	2
#23781495	2
#23781473	2
#346×C₁₆₃	2
#2179	163