Subgroups — #35212828

Order 20864.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781718	C₂	1	2	yes	maximal
2	#23781387	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₈×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₁₆₃⋊C₈)	C₄	1	4	yes	—
5	C₂²⋊(C₁₆₃⋊C₈)	C₄	1	4	yes	—
6	C₄×C₁₆₃⋊C₈	C₂²	1	4	yes	—
7	C₂²×C₈×C₁₆₃	C₄	1	2	yes	—
8	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₁₆₃	C₄	1	2	yes	—
10	C₄×C₈×C₁₆₃	C₂²	1	2	yes	—
11	C₂×C₁₆₃⋊C₈	C₂×C₄	1	8	yes	—
12	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₆₃	—	2	4	no	—
15	C₂×C₈×C₁₆₃	C₂×C₄	1	4	yes	—
16	C₄²×C₁₆₃	C₂×C₄	1	2	yes	—
17	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
18	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
19	C₁₆₃⋊C₈	—	2	8	no	—
20	C₂³×C₁₆₃	C₄²	1	1	yes	characteristic
21	C₂×C₄×C₁₆₃	—	2	4	no	—
22	C₂×C₄×C₁₆₃	C₄²	1	2	yes	—
23	C₂×C₄×C₁₆₃	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₁₆₃	M₁₆	1	4	yes	—
25	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
26	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
27	C₈×C₁₆₃	—	2	4	no	—
28	C₂²×C₁₆₃	—	2	2	no	—
29	C₂²×C₁₆₃	#97	1	2	yes	—
30	C₂²×C₁₆₃	C₂×C₄²	1	1	yes	characteristic
31	C₄×C₁₆₃	—	2	2	no	—
32	C₄×C₁₆₃	#97	1	2	yes	—
33	C₄×C₁₆₃	C₂×M₁₆	1	2	yes	—
34	C₄×C₁₆₃	#131	1	4	yes	—
35	C₂×C₁₆₃	—	2	2	no	—
36	C₂×C₁₆₃	C₂×#97	1	1	yes	characteristic
37	C₂×C₁₆₃	#405	1	2	yes	—
38	C₁₆₃	#1452	1	1	yes	characteristic, complemented, Sylow
39	#1452	—	163	1	no	maximal, Sylow
40	#431	—	163	2	no	—
41	C₂×C₄×C₈	D₁₆₃	1	1	yes	characteristic
42	#322	—	163	4	no	—
43	C₂²×C₈	Dic₁₆₃	1	2	yes	—
44	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
45	C₄⋊C₈	—	163	4	no	—
46	C₂²⋊C₈	—	163	4	no	—
47	C₄×C₈	—	163	4	no	—
48	C₄×C₈	Dic₁₆₃	1	2	yes	—
49	C₄×C₈	C₂×D₁₆₃	1	2	yes	—
50	C₂²×C₄	C₄×D₁₆₃	1	2	yes	—
51	C₂²×C₄	C₂×Dic₁₆₃	1	1	yes	characteristic
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	163	8	no	—
54	C₂×C₈	C₂×Dic₁₆₃	1	4	yes	—
55	C₄²	C₄×D₁₆₃	1	2	yes	—
56	C₄²	C₂×Dic₁₆₃	1	1	yes	characteristic
57	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
58	C₂³	C₄×Dic₁₆₃	1	1	yes	characteristic
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	C₄×Dic₁₆₃	1	2	yes	—
61	C₂×C₄	C₄×Dic₁₆₃	1	1	yes	characteristic
62	C₂×C₄	#14579683	1	4	yes	—
63	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
64	C₂×C₄	C₂²×Dic₁₆₃	1	1	yes	characteristic
65	C₈	—	2	4	no	—
66	C₈	—	326	8	no	—
67	C₂²	—	2	2	no	—
68	C₂²	#19104646	1	2	yes	—
69	C₂²	C₂×C₄×Dic₁₆₃	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	#19104646	1	2	yes	—
72	C₄	C₂×#14579683	1	2	yes	—
73	C₄	#19104741	1	4	yes	—
74	C₂	—	2	2	no	—
75	C₂	C₂×#19104646	1	1	yes	characteristic
76	C₂	#23781620	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₆₃
commutator, socle	C₂²×C₁₆₃	C₂×C₄²
Fitting	C₂×C₄×C₈×C₁₆₃	C₂
prime radical	C₂³×C₁₆₃	C₄²
cosocle	C₄²×C₁₆₃	C₂³
hypercenter	C₂×C₄×C₈	D₁₆₃
hypocenter	C₁₆₃	#1452
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #23781387	C₂	semidirect
38. C₁₆₃	#1452	semidirect

Maximal subgroups

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

subgroup	index
#23781718	2
#23781387	2
C₂×C₄×C₈×C₁₆₃	2
#1452	163