Subgroups — #35216422

Order 20864.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781782	C₂	1	2	yes	maximal, complemented
2	#23781495	C₂	1	1	yes	maximal, characteristic, complemented
3	#23781472	C₂	1	2	yes	maximal, complemented
4	#23781440	C₂	1	1	yes	maximal, characteristic, complemented
5	#508×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	#19104742	—	2	2	no	—
7	C₂×C₁₆₃⋊C₁₆	C₂²	1	1	yes	characteristic
8	#19104671	C₂²	1	2	yes	—
9	#19104662	—	2	2	no	—
10	C₈×Dic₁₆₃	C₂²	1	1	yes	characteristic
11	#135×C₁₆₃	C₂²	1	2	yes	—
12	Q₃₂×C₁₆₃	C₄	1	1	yes	characteristic, complemented
13	SD₃₂×C₁₆₃	C₄	1	2	yes	complemented
14	D₁₆×C₁₆₃	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
16	#14579683	—	2	2	no	—
17	#14579683	—	4	2	no	—
18	C₂×C₁₆₃⋊C₈	—	4	2	no	—
19	C₂×C₁₆₃⋊C₈	D₄	1	1	yes	characteristic
20	C₄×Dic₁₆₃	D₄	1	1	yes	characteristic
21	C₁₆₃⋊C₁₆	—	2	1	no	—
22	#41×C₁₆₃	—	2	2	no	—
23	Q₁₆×C₁₆₃	C₂×C₄	1	2	yes	—
24	SD₁₆×C₁₆₃	—	2	2	no	—
25	D₈×C₁₆₃	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₁₆₃	C₂³	1	1	yes	characteristic
27	C₁₆×C₁₆₃	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₁₆₃	C₂×C₄	1	1	yes	characteristic
29	C₂×Dic₁₆₃	—	2	1	no	—
30	C₁₆₃⋊C₈	—	4	2	no	—
31	C₁₆₃⋊C₈	D₈	1	2	yes	—
32	Q₈×C₁₆₃	—	2	2	no	—
33	D₄×C₁₆₃	—	2	2	no	—
34	D₄×C₁₆₃	—	4	2	no	—
35	C₂×C₄×C₁₆₃	—	4	2	no	—
36	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
37	C₈×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₆₃	#41	1	1	yes	characteristic
39	C₂²×C₁₆₃	—	4	2	no	—
40	C₂²×C₁₆₃	#135	1	1	yes	characteristic
41	C₄×C₁₆₃	—	4	2	no	—
42	C₄×C₁₆₃	C₄×D₄	1	1	yes	characteristic
43	C₄×C₁₆₃	C₂×D₈	1	1	yes	characteristic
44	Dic₁₆₃	—	4	1	no	—
45	C₂×C₁₆₃	—	2	1	no	—
46	C₂×C₁₆₃	—	8	2	no	—
47	C₂×C₁₆₃	C₄×D₈	1	1	yes	characteristic
48	C₁₆₃	#2179	1	1	yes	characteristic, complemented, Sylow
49	#2179	—	163	1	no	maximal, Sylow
50	#508	D₁₆₃	1	1	yes	characteristic
51	#443	—	163	2	no	—
52	#368	—	163	1	no	—
53	#359	—	163	2	no	—
54	#346	—	163	1	no	—
55	#135	C₂×D₁₆₃	1	2	yes	—
56	#131	—	326	2	no	—
57	Q₃₂	Dic₁₆₃	1	1	yes	characteristic, complemented
58	SD₃₂	Dic₁₆₃	1	2	yes	complemented
59	D₁₆	Dic₁₆₃	1	1	yes	characteristic, complemented
60	C₂×C₁₆	—	163	1	no	—
61	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
62	#108	—	163	2	no	—
63	#104	—	326	2	no	—
64	C₄×C₈	—	163	1	no	—
65	#41	—	2	2	no	—
66	Q₁₆	C₂×Dic₁₆₃	1	2	yes	—
67	SD₁₆	—	2	2	no	—
68	D₈	C₂×Dic₁₆₃	1	2	yes	—
69	M₁₆	—	326	2	no	—
70	M₁₆	—	652	2	no	—
71	C₂×C₈	—	652	2	no	—
72	C₂×C₈	—	163	1	no	—
73	C₂×C₈	C₂²×D₁₆₃	1	1	yes	characteristic
74	C₄²	—	163	1	no	—
75	C₁₆	—	326	1	no	—
76	C₁₆	C₂×Dic₁₆₃	1	1	yes	characteristic
77	C₁₆	C₂×Dic₁₆₃	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	D₄	—	2	2	no	—
80	D₄	—	4	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	326	1	no	—
83	C₂×C₄	D₄×D₁₆₃	1	1	yes	characteristic
84	C₈	—	652	2	no	—
85	C₈	—	163	2	no	—
86	C₈	C₂²×Dic₁₆₃	1	1	yes	characteristic
87	C₈	#14579698	1	1	yes	characteristic
88	C₂²	—	4	2	no	—
89	C₂²	#19104754	1	1	yes	characteristic
90	C₄	—	4	2	no	—
91	C₄	—	652	1	no	—
92	C₄	D₄×Dic₁₆₃	1	1	yes	characteristic
93	C₄	D₈×D₁₆₃	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	8	2	no	—
96	C₂	D₈×Dic₁₆₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₁₆₃
commutator	C₈×C₁₆₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₆₃
Fitting, prime radical	#508×C₁₆₃	C₂
socle	C₂×C₁₆₃	C₄×D₈
cosocle	C₂×C₈×C₁₆₃	C₂³
hypercenter	#508	D₁₆₃
hypocenter	C₁₆₃	#2179
LCS	G ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #23781782	C₂	semidirect
2. #23781495	C₂	semidirect
3. #23781472	C₂	semidirect
4. #23781440	C₂	semidirect
12. Q₃₂×C₁₆₃	C₄	semidirect
13. SD₃₂×C₁₆₃	C₄	semidirect
14. D₁₆×C₁₆₃	C₄	semidirect
48. C₁₆₃	#2179	semidirect
57. Q₃₂	Dic₁₆₃	semidirect
58. SD₃₂	Dic₁₆₃	semidirect
59. D₁₆	Dic₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23781782	2
#23781495	2
#23781472	2
#23781440	2
#508×C₁₆₃	2
#2179	163