Subgroups — #35214138

Order 20864.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782038	C₂	1	1	yes	maximal, characteristic
2	#23782015	C₂	1	2	yes	maximal, complemented
3	#23782013	C₂	1	2	yes	maximal
4	C₈×C₁₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#498×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14579688	—	2	2	no	—
7	#19104735	C₂²	1	1	yes	characteristic
8	#19104732	C₂²	1	1	yes	characteristic
9	#19104659	—	2	4	no	—
10	#19104655	—	2	4	no	—
11	C₄×C₁₆₃⋊C₈	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₆₃	C₂²	1	2	yes	—
13	#106×C₁₆₃	—	2	2	no	—
14	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₂₆	—	2	1	no	—
16	C₂×D₆₅₂	—	2	2	no	—
17	C₂×D₆₅₂	—	2	1	no	—
18	#14579688	—	4	4	no	—
19	C₂×C₁₆₃⋊C₈	D₄	1	4	yes	—
20	C₄⋊Dic₁₆₃	—	2	2	no	—
21	C₂×Q₈×C₁₆₃	—	2	2	no	—
22	C₂×C₈×C₁₆₃	D₄	1	2	yes	—
23	C₄⋊C₄×C₁₆₃	—	2	2	no	—
24	C₄⋊C₄×C₁₆₃	—	2	2	no	—
25	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
26	C₂²×D₁₆₃	—	4	1	no	—
27	Dic₃₂₆	—	4	2	no	—
28	D₆₅₂	—	4	2	no	—
29	D₆₅₂	—	4	4	no	—
30	C₂×Dic₁₆₃	—	4	1	no	—
31	C₁₆₃⋊C₈	—	2	4	no	—
32	Q₈×C₁₆₃	—	4	4	no	—
33	C₂×C₄×C₁₆₃	—	4	2	no	—
34	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
36	C₈×C₁₆₃	SD₁₆	1	4	yes	—
37	C₂²×C₁₆₃	#127	1	1	yes	characteristic
38	C₂×D₁₆₃	—	8	1	no	—
39	C₂×D₁₆₃	—	8	2	no	—
40	C₄×C₁₆₃	—	8	2	no	—
41	C₄×C₁₆₃	C₂×SD₁₆	1	2	yes	—
42	C₄×C₁₆₃	#135	1	4	yes	—
43	Dic₁₆₃	—	8	1	no	—
44	C₂×C₁₆₃	#492	1	1	yes	characteristic
45	C₂×C₁₆₃	#495	1	2	yes	—
46	D₁₆₃	—	16	1	no	—
47	C₁₆₃	#1708	1	1	yes	characteristic, complemented, Sylow
48	#1708	—	163	1	no	maximal, Sylow
49	#498	D₁₆₃	1	1	yes	characteristic, complemented
50	#492	—	163	1	no	—
51	#487	—	163	2	no	—
52	#486	—	163	2	no	—
53	C₈²	—	163	1	no	—
54	C₂×SD₁₆	—	326	2	no	—
55	C₄⋊Q₈	—	163	1	no	—
56	C₄⋊Q₈	C₂×D₁₆₃	1	2	yes	—
57	#127	—	163	1	no	—
58	#106	—	2	2	no	—
59	#103	—	326	4	no	—
60	#102	—	326	4	no	—
61	C₄×C₈	—	163	2	no	—
62	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
63	C₂×Q₈	—	2	2	no	—
64	C₂×Q₈	—	326	1	no	—
65	C₂×D₄	—	326	2	no	—
66	C₂×D₄	—	326	1	no	—
67	SD₁₆	—	652	4	no	—
68	C₂×C₈	—	163	4	no	—
69	C₂×C₈	C₂²⋊D₁₆₃	1	2	yes	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	326	2	no	—
73	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
74	C₂³	—	652	1	no	—
75	Q₈	—	4	4	no	—
76	Q₈	—	652	2	no	—
77	D₄	—	652	2	no	—
78	D₄	—	652	4	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	—	652	1	no	—
81	C₂×C₄	D₄×D₁₆₃	1	2	yes	—
82	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
83	C₈	—	326	4	no	—
84	C₈	#14579686	1	4	yes	—
85	C₂²	—	1304	1	no	—
86	C₂²	—	1304	2	no	—
87	C₂²	#19104731	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	—	1304	1	no	—
90	C₄	C₂×#14579686	1	2	yes	—
91	C₄	#19104756	1	4	yes	—
92	C₂	—	2608	1	no	—
93	C₂	#23782039	1	1	yes	characteristic
94	C₂	#23782046	1	2	yes	—

	subgroup	G/H
center	C₂²	#19104731
commutator, cosocle	C₄²×C₁₆₃	C₂³
Frattini	C₄²	C₂²×D₁₆₃
Fitting	#498×C₁₆₃	C₂
socle	C₂²×C₁₆₃	#127
prime radical	#19104732	C₂²
hypercenter	#498	D₁₆₃
hypocenter	C₁₆₃	#1708
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

Complements

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

N	H ≅ G/N	type
2. #23782015	C₂	semidirect
4. C₈×C₁₆₃⋊C₈	C₂	semidirect
5. #498×C₁₆₃	C₂	semidirect
47. C₁₆₃	#1708	semidirect
49. #498	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782038	2
#23782015	2
#23782013	2
C₈×C₁₆₃⋊C₈	2
#498×C₁₆₃	2
#1708	163