Subgroups — #35212704

Order 20864.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782044	C₂	1	1	yes	maximal, characteristic
2	#23781464	C₂	1	1	yes	maximal, characteristic
3	#356×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579689	—	2	1	no	—
5	C₂×#14579689	—	2	1	no	—
6	#19104735	C₄	1	1	yes	characteristic
7	#19104654	—	2	1	no	—
8	C₄×C₁₆₃⋊C₈	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
10	#101×C₁₆₃	—	2	1	no	—
11	C₂×Dic₃₂₆	—	2	1	no	—
12	#14579689	—	4	2	no	—
13	#14579689	—	4	2	no	—
14	#14579683	—	4	1	no	—
15	C₂×C₁₆₃⋊C₈	—	2	1	no	—
16	C₄⋊Dic₁₆₃	—	4	1	no	—
17	C₂×Q₈×C₁₆₃	D₄	1	1	yes	characteristic
18	C₂×Q₈×C₁₆₃	D₄	1	1	yes	characteristic
19	M₁₆×C₁₆₃	—	4	1	no	—
20	C₄⋊C₄×C₁₆₃	—	4	1	no	—
21	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
22	Dic₃₂₆	—	4	1	no	—
23	Dic₃₂₆	—	4	1	no	—
24	C₂×Dic₁₆₃	—	4	1	no	—
25	C₁₆₃⋊C₈	—	2	2	no	—
26	C₁₆₃⋊C₈	—	8	1	no	—
27	Q₈×C₁₆₃	—	4	1	no	—
28	Q₈×C₁₆₃	—	4	1	no	—
29	C₂×C₄×C₁₆₃	—	2	1	no	—
30	C₂×C₄×C₁₆₃	—	2	1	no	—
31	C₂×C₄×C₁₆₃	—	2	1	no	—
32	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₈×C₁₆₃	—	8	1	no	—
34	C₂²×C₁₆₃	#99	1	1	yes	characteristic
35	C₄×C₁₆₃	—	2	2	no	—
36	C₄×C₁₆₃	—	2	1	no	—
37	C₄×C₁₆₃	—	4	1	no	—
38	C₄×C₁₆₃	—	4	1	no	—
39	Dic₁₆₃	—	8	1	no	—
40	C₂×C₁₆₃	—	2	1	no	—
41	C₂×C₁₆₃	#353	1	1	yes	characteristic
42	C₁₆₃	#1411	1	1	yes	characteristic, complemented, Sylow
43	#1411	—	163	1	no	maximal, Sylow
44	#494	—	163	1	no	—
45	#356	—	163	1	no	—
46	#356	D₁₆₃	1	1	yes	characteristic
47	C₂×Q₁₆	—	326	1	no	—
48	C₂×Q₁₆	—	326	1	no	—
49	C₄⋊Q₈	—	163	1	no	—
50	C₄⋊Q₈	C₂×D₁₆₃	1	1	yes	characteristic
51	#101	—	2	1	no	—
52	#101	—	326	1	no	—
53	C₄×C₈	—	163	1	no	—
54	C₂×Q₈	—	326	1	no	—
55	C₂×Q₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
56	C₂×Q₈	D₆₅₂	1	1	yes	characteristic
57	Q₁₆	—	652	2	no	—
58	Q₁₆	—	652	2	no	—
59	M₁₆	—	4	1	no	—
60	M₁₆	—	652	1	no	—
61	C₂×C₈	—	326	1	no	—
62	C₄⋊C₄	—	4	1	no	—
63	C₄⋊C₄	—	652	1	no	—
64	C₄²	C₄×D₁₆₃	1	1	yes	characteristic
65	Q₈	—	4	1	no	—
66	Q₈	—	4	1	no	—
67	Q₈	—	652	1	no	—
68	Q₈	—	652	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	652	1	no	—
73	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
74	C₈	—	8	1	no	—
75	C₈	—	1304	1	no	—
76	C₈	—	326	2	no	—
77	C₂²	#19104649	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	2	1	no	—
80	C₄	—	4	1	no	—
81	C₄	—	4	1	no	—
82	C₄	—	1304	1	no	—
83	C₂	—	2	1	no	—
84	C₂	#23781457	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23781457
commutator	C₄²×C₁₆₃	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₁₆₃
Fitting	#356×C₁₆₃	C₂
socle	C₂×C₁₆₃	#353
prime radical	C₂²×C₁₆₃	#99
cosocle	C₄⋊Q₈×C₁₆₃	C₂²
hypercenter	#356	D₁₆₃
hypocenter	C₁₆₃	#1411
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂×C₄×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
42. C₁₆₃	#1411	semidirect

Maximal subgroups

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

subgroup	index
#23782044	2
#23781464	2
#356×C₁₆₃	2
#1411	163