Subgroups — #35212508

Order 20864.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781783	C₂	1	1	yes	maximal, characteristic
2	C₈×C₁₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#19104757	C₄	1	1	yes	characteristic
5	#19104741	—	2	1	no	—
6	#19104671	C₄	1	1	yes	characteristic
7	#19104663	—	2	1	no	—
8	C₄×C₁₆₃⋊C₈	—	2	1	no	—
9	M₃₂×C₁₆₃	—	2	1	no	—
10	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
11	#14579699	—	2	1	no	—
12	Dic₆₅₂	C₈	1	1	yes	characteristic, complemented
13	#14579688	—	2	1	no	—
14	D₁₃₀₄	C₈	1	1	yes	characteristic, complemented
15	#14579683	—	2	1	no	—
16	#14579682	—	4	1	no	—
17	C₂×C₁₆₃⋊C₈	—	2	2	no	—
18	C₈×D₁₆₃	—	4	1	no	—
19	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
21	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
22	C₁₆×C₁₆₃	—	4	1	no	—
23	Dic₃₂₆	—	2	1	no	—
24	D₆₅₂	—	2	1	no	—
25	C₂²⋊D₁₆₃	—	4	1	no	—
26	C₄×D₁₆₃	—	4	1	no	—
27	C₁₆₃⋊C₈	—	2	4	no	—
28	C₁₆₃⋊C₈	—	4	1	no	—
29	C₂×C₄×C₁₆₃	—	2	1	no	—
30	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₈×C₁₆₃	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₆₃	M₁₆	1	1	yes	characteristic
33	C₈×C₁₆₃	D₈	1	1	yes	characteristic
34	C₈×C₁₆₃	SD₁₆	1	1	yes	characteristic
35	C₂²×C₁₆₃	#104	1	1	yes	characteristic
36	C₂×D₁₆₃	—	4	1	no	—
37	C₄×C₁₆₃	—	2	2	no	—
38	C₄×C₁₆₃	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₆₃	#102	1	1	yes	characteristic
40	Dic₁₆₃	—	4	1	no	—
41	C₂×C₁₆₃	—	2	1	no	—
42	C₂×C₁₆₃	D₄⋊C₈	1	1	yes	characteristic
43	D₁₆₃	—	8	1	no	—
44	C₁₆₃	#1335	1	1	yes	characteristic, complemented, Sylow
45	#1335	—	163	1	no	maximal, Sylow
46	#443	—	163	1	no	—
47	#364	D₁₆₃	1	1	yes	characteristic, complemented
48	C₈²	—	163	1	no	—
49	#135	—	163	1	no	—
50	#131	—	326	1	no	—
51	M₃₂	—	2	1	no	—
52	#108	—	163	1	no	—
53	#104	—	326	1	no	—
54	C₄×C₈	—	326	1	no	—
55	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
56	#41	—	326	1	no	—
57	Q₁₆	—	163	1	no	—
58	SD₁₆	—	326	1	no	—
59	D₈	—	163	1	no	—
60	M₁₆	—	326	1	no	—
61	M₁₆	—	652	1	no	—
62	C₂×C₈	—	326	2	no	—
63	C₂×C₈	—	652	1	no	—
64	C₂×C₈	C₄×D₁₆₃	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
66	C₄²	D₆₅₂	1	1	yes	characteristic
67	C₁₆	—	4	1	no	—
68	Q₈	—	326	1	no	—
69	D₄	—	326	1	no	—
70	D₄	—	652	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	652	1	no	—
73	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
74	C₈	—	326	4	no	—
75	C₈	—	652	1	no	—
76	C₈	C₈×D₁₆₃	1	1	yes	characteristic
77	C₈	#14579682	1	1	yes	characteristic
78	C₈	#14579684	1	1	yes	characteristic
79	C₈	#14579686	1	1	yes	characteristic
80	C₂²	—	652	1	no	—
81	C₂²	#19104661	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	652	1	no	—
84	C₄	(C₂×D₁₆₃)⋊C₈	1	1	yes	characteristic
85	C₄	#19104655	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	1304	1	no	—
88	C₂	#23781392	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14579684
commutator	C₈×C₁₆₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₆₃
Fitting	#364×C₁₆₃	C₂
socle	C₂×C₁₆₃	D₄⋊C₈
prime radical	#19104757	C₄
cosocle	C₄×C₈×C₁₆₃	C₂²
hypercenter	#364	D₁₆₃
hypocenter	C₁₆₃	#1335
LCS	G ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. C₈×C₁₆₃⋊C₈	C₂	semidirect
3. #364×C₁₆₃	C₂	semidirect
12. Dic₆₅₂	C₈	semidirect
14. D₁₃₀₄	C₈	semidirect
44. C₁₆₃	#1335	semidirect
47. #364	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23781783	2
C₈×C₁₆₃⋊C₈	2
#364×C₁₆₃	2
#1335	163