Subgroups — #35212568

Order 20864.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782030	C₂	1	1	yes	maximal, characteristic
2	#23781443	C₂	1	1	yes	maximal, characteristic, complemented
3	#347×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#19104732	C₄	1	1	yes	characteristic
5	#19104727	C₄	1	1	yes	characteristic
6	#19104675	—	2	1	no	—
7	#19104657	—	2	1	no	—
8	#19104657	—	2	1	no	—
9	M₃₂×C₁₆₃	—	2	1	no	—
10	#97×C₁₆₃	C₂²	1	1	yes	characteristic
11	C₂×D₆₅₂	—	2	1	no	—
12	C₂×D₆₅₂	—	4	1	no	—
13	C₄⋊Dic₁₆₃	—	2	1	no	—
14	Dic₁₆₃⋊C₄	—	4	1	no	—
15	C₁₆₃⋊C₁₆	—	2	2	no	—
16	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
17	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
18	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₁₆₃	—	2	2	no	—
20	C₂²×D₁₆₃	—	4	1	no	—
21	D₆₅₂	—	4	1	no	—
22	D₆₅₂	—	4	1	no	—
23	D₆₅₂	—	8	2	no	—
24	C₂×Dic₁₆₃	—	4	1	no	—
25	C₂×C₄×C₁₆₃	—	2	1	no	—
26	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₈×C₁₆₃	—	2	1	no	—
28	C₈×C₁₆₃	—	2	1	no	—
29	C₂²×C₁₆₃	#100	1	1	yes	characteristic
30	C₂×D₁₆₃	—	8	2	no	—
31	C₂×D₁₆₃	—	8	1	no	—
32	C₄×C₁₆₃	—	4	1	no	—
33	C₄×C₁₆₃	#104	1	1	yes	characteristic
34	C₄×C₁₆₃	#104	1	1	yes	characteristic
35	Dic₁₆₃	—	8	1	no	—
36	C₂×C₁₆₃	—	2	1	no	—
37	C₂×C₁₆₃	#329	1	1	yes	characteristic
38	D₁₆₃	—	16	1	no	—
39	C₁₆₃	#1355	1	1	yes	characteristic, complemented, Sylow
40	#1355	—	163	1	no	maximal, Sylow
41	#490	—	163	1	no	—
42	#347	—	163	1	no	—
43	#347	D₁₆₃	1	1	yes	characteristic, complemented
44	#127	—	163	1	no	—
45	#125	—	163	1	no	—
46	M₃₂	—	2	1	no	—
47	M₃₂	—	326	1	no	—
48	#102	—	326	1	no	—
49	#102	—	326	1	no	—
50	#97	C₂×D₁₆₃	1	1	yes	characteristic
51	C₂×D₄	—	326	1	no	—
52	C₂×D₄	—	652	1	no	—
53	C₂×C₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
54	C₂×C₈	D₆₅₂	1	1	yes	characteristic
55	C₄⋊C₄	—	326	1	no	—
56	C₄⋊C₄	—	652	1	no	—
57	C₄²	C₄×D₁₆₃	1	1	yes	characteristic
58	C₁₆	—	2	2	no	—
59	C₁₆	—	326	2	no	—
60	C₂³	—	652	1	no	—
61	D₄	—	1304	2	no	—
62	D₄	—	652	1	no	—
63	D₄	—	652	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	652	1	no	—
66	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	2	1	no	—
69	C₂²	—	1304	2	no	—
70	C₂²	—	1304	1	no	—
71	C₂²	#19104651	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	1304	1	no	—
74	C₄	#19104661	1	1	yes	characteristic
75	C₄	#19104663	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	2608	1	no	—
78	C₂	#23781404	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23781404
commutator	C₄²×C₁₆₃	C₂×C₄
Frattini	#97	C₂×D₁₆₃
Fitting	#347×C₁₆₃	C₂
socle	C₂×C₁₆₃	#329
prime radical	#19104732	C₄
cosocle	#97×C₁₆₃	C₂²
hypercenter	#347	D₁₆₃
hypocenter	C₁₆₃	#1355
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
2. #23781443	C₂	semidirect
3. #347×C₁₆₃	C₂	semidirect
39. C₁₆₃	#1355	semidirect
43. #347	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782030	2
#23781443	2
#347×C₁₆₃	2
#1355	163