Subgroups — #35216116

Order 20864.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781714	C₂	1	1	yes	maximal, characteristic
2	#23781435	C₂	1	2	yes	maximal
3	#23781430	C₂	1	2	yes	maximal
4	C₁₆₃⋊#338	C₂	1	1	yes	maximal, characteristic
5	#424×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579683	C₂²	1	2	yes	—
7	C₂×#14579683	C₂²	1	2	yes	—
8	#19104671	—	2	2	no	—
9	C₂×M₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
11	#117×C₁₆₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₆₃	—	2	2	no	—
13	#14579683	—	2	2	no	—
14	#14579683	—	2	2	no	—
15	#14579683	—	2	2	no	—
16	#14579683	—	2	2	no	—
17	C₂×C₁₆₃⋊C₈	—	2	2	no	—
18	C₂×C₁₆₃⋊C₈	—	2	2	no	—
19	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
20	M₁₆×C₁₆₃	—	2	2	no	—
21	C₂×C₈×C₁₆₃	—	2	1	no	—
22	C₂×C₈×C₁₆₃	—	2	2	no	—
23	C₂×C₈×C₁₆₃	D₄	1	2	yes	—
24	C₄⋊C₄×C₁₆₃	—	2	1	no	—
25	C₂²⋊C₄×C₁₆₃	Q₈	1	2	yes	—
26	C₄²×C₁₆₃	—	2	1	no	—
27	C₁₆₃⋊C₈	—	4	2	no	—
28	C₁₆₃⋊C₈	—	4	2	no	—
29	C₂³×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₁₆₃	—	2	2	no	—
31	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₆₃	#41	1	2	yes	—
33	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₆₃	#41	1	2	yes	—
35	C₈×C₁₆₃	—	2	2	no	—
36	C₈×C₁₆₃	—	4	1	no	—
37	C₂²×C₁₆₃	—	4	1	no	—
38	C₂²×C₁₆₃	C₂²⋊Q₈	1	1	yes	characteristic
39	C₂²×C₁₆₃	#125	1	2	yes	—
40	C₄×C₁₆₃	—	4	2	no	—
41	C₄×C₁₆₃	#123	1	1	yes	characteristic
42	C₄×C₁₆₃	#124	1	1	yes	characteristic
43	C₄×C₁₆₃	#126	1	2	yes	—
44	C₂×C₁₆₃	—	2	1	no	—
45	C₂×C₁₆₃	—	2	2	no	—
46	C₂×C₁₆₃	#400	1	1	yes	characteristic
47	C₁₆₃	#2103	1	1	yes	characteristic, complemented, Sylow
48	#2103	—	163	1	no	maximal, Sylow
49	#430	—	163	1	no	—
50	#424	D₁₆₃	1	1	yes	characteristic
51	#344	—	163	2	no	—
52	#341	—	163	2	no	—
53	#338	—	163	1	no	—
54	C₂×M₁₆	—	163	2	no	—
55	C₂×M₁₆	—	163	2	no	—
56	C₂×M₁₆	C₂×D₁₆₃	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
58	#117	C₂×D₁₆₃	1	1	yes	characteristic
59	#108	—	326	2	no	—
60	C₄⋊C₈	—	2	2	no	—
61	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
62	M₁₆	—	2	2	no	—
63	M₁₆	—	326	2	no	—
64	M₁₆	—	326	2	no	—
65	M₁₆	—	326	2	no	—
66	M₁₆	—	326	2	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	326	2	no	—
70	C₂×C₈	—	326	2	no	—
71	C₂×C₈	C₂²⋊D₁₆₃	1	2	yes	—
72	C₄⋊C₄	—	2	1	no	—
73	C₂²⋊C₄	Dic₃₂₆	1	2	yes	—
74	C₄²	—	2	1	no	—
75	C₂³	C₂×Dic₃₂₆	1	1	yes	characteristic
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
78	C₂×C₄	#14579698	1	2	yes	—
79	C₂×C₄	#14579699	1	1	yes	characteristic
80	C₂×C₄	#14579700	1	2	yes	—
81	C₈	—	2	2	no	—
82	C₈	—	4	1	no	—
83	C₈	—	652	2	no	—
84	C₈	—	652	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	#19104714	1	1	yes	characteristic
87	C₂²	#19104726	1	2	yes	—
88	C₄	—	4	2	no	—
89	C₄	#19104716	1	1	yes	characteristic
90	C₄	#19104724	1	1	yes	characteristic
91	C₄	#19104730	1	2	yes	—
92	C₂	—	2	1	no	—
93	C₂	—	2	2	no	—
94	C₂	#23781604	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19104724
commutator, cosocle	C₂²×C₄×C₁₆₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#424×C₁₆₃	C₂
socle	C₂×C₁₆₃	#400
prime radical	C₂³×C₁₆₃	C₂×Q₈
hypercenter	#424	D₁₆₃
hypocenter	C₁₆₃	#2103
LCS	G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
47. C₁₆₃	#2103	semidirect

Maximal subgroups

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

subgroup	index
#23781714	2
#23781435	2
#23781430	2
C₁₆₃⋊#338	2
#424×C₁₆₃	2
#2103	163