Subgroups — #35214135

Order 20864.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782063	C₂	1	1	yes	maximal, characteristic, complemented
2	#23782017	C₂	1	2	yes	maximal, complemented
3	#23782011	C₂	1	2	yes	maximal
4	C₈×C₁₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#492×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	#19104735	C₂²	1	2	yes	—
7	#19104667	—	2	2	no	—
8	#19104658	—	2	4	no	—
9	#19104656	—	2	4	no	—
10	C₄×C₁₆₃⋊C₈	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₁₆₃	—	2	2	no	—
12	C₄⋊Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
13	#127×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₂₆	—	2	2	no	—
16	C₂×C₁₆₃⋊C₈	D₄	1	4	yes	—
17	C₄⋊Dic₁₆₃	—	2	2	no	—
18	C₄⋊Dic₁₆₃	—	2	2	no	—
19	C₂×Q₈×C₁₆₃	—	2	1	no	—
20	C₂×D₄×C₁₆₃	—	2	1	no	—
21	C₂×D₄×C₁₆₃	—	2	2	no	—
22	SD₁₆×C₁₆₃	—	4	4	no	—
23	C₂×C₈×C₁₆₃	D₄	1	2	yes	—
24	C₄⋊C₄×C₁₆₃	—	2	2	no	—
25	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
26	Dic₃₂₆	—	4	4	no	—
27	C₂×Dic₁₆₃	—	4	2	no	—
28	C₁₆₃⋊C₈	—	2	4	no	—
29	C₂³×C₁₆₃	—	4	1	no	—
30	Q₈×C₁₆₃	—	4	2	no	—
31	D₄×C₁₆₃	—	4	2	no	—
32	D₄×C₁₆₃	—	4	4	no	—
33	C₂×C₄×C₁₆₃	—	4	1	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	—	8	1	no	—
38	C₂²×C₁₆₃	—	8	2	no	—
39	C₂²×C₁₆₃	#127	1	1	yes	characteristic
40	C₄×C₁₆₃	—	8	1	no	—
41	C₄×C₁₆₃	C₂×SD₁₆	1	2	yes	—
42	C₄×C₁₆₃	#135	1	4	yes	—
43	Dic₁₆₃	—	8	2	no	—
44	C₂×C₁₆₃	—	16	1	no	—
45	C₂×C₁₆₃	#492	1	1	yes	characteristic
46	C₂×C₁₆₃	#495	1	2	yes	—
47	C₁₆₃	#1708	1	1	yes	characteristic, complemented, Sylow
48	#1708	—	163	1	no	maximal, Sylow
49	#498	—	163	1	no	—
50	#492	D₁₆₃	1	1	yes	characteristic
51	#487	—	163	2	no	—
52	#486	—	163	2	no	—
53	C₈²	—	163	1	no	—
54	C₂×SD₁₆	—	2	2	no	—
55	C₄⋊Q₈	—	163	2	no	—
56	C₄⋊Q₈	C₂×D₁₆₃	1	1	yes	characteristic
57	#127	C₂×D₁₆₃	1	1	yes	characteristic
58	#106	—	326	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	1	no	—
64	C₂×Q₈	—	326	2	no	—
65	C₂×D₄	—	2	2	no	—
66	C₂×D₄	—	2	1	no	—
67	SD₁₆	—	4	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	—	326	2	no	—
72	C₄⋊C₄	—	326	2	no	—
73	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
74	C₂³	—	4	1	no	—
75	Q₈	—	4	2	no	—
76	Q₈	—	652	4	no	—
77	D₄	—	4	2	no	—
78	D₄	—	4	4	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	—	652	2	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	#14579687	1	4	yes	—
85	C₂²	—	8	1	no	—
86	C₂²	—	8	2	no	—
87	C₂²	#19104731	1	1	yes	characteristic
88	C₄	—	8	1	no	—
89	C₄	—	1304	2	no	—
90	C₄	C₂×#14579687	1	2	yes	—
91	C₄	#19104756	1	4	yes	—
92	C₂	—	16	1	no	—
93	C₂	#23782037	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	#492×C₁₆₃	C₂
socle	C₂²×C₁₆₃	#127
prime radical	#127×C₁₆₃	C₂²
hypercenter	#492	D₁₆₃
hypocenter	C₁₆₃	#1708
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

Complements

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

N	H ≅ G/N	type
1. #23782063	C₂	semidirect
2. #23782017	C₂	semidirect
4. C₈×C₁₆₃⋊C₈	C₂	semidirect
47. C₁₆₃	#1708	semidirect

Maximal subgroups

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

subgroup	index
#23782063	2
#23782017	2
#23782011	2
C₈×C₁₆₃⋊C₈	2
#492×C₁₆₃	2
#1708	163