Subgroups — #35212709

Order 20864.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781464	C₂	1	2	yes	maximal
2	#491×C₁₆₃	C₂	1	1	yes	maximal, characteristic
3	#19104654	—	2	2	no	—
4	C₄⋊Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
5	#125×C₁₆₃	C₄	1	1	yes	characteristic
6	#103×C₁₆₃	—	2	2	no	—
7	#97×C₁₆₃	C₄	1	1	yes	characteristic
8	#14579683	—	4	2	no	—
9	C₂×Q₈×C₁₆₃	D₄	1	2	yes	—
10	C₂×C₈×C₁₆₃	—	2	1	no	—
11	C₄⋊C₄×C₁₆₃	—	2	1	no	—
12	C₄⋊C₄×C₁₆₃	—	4	1	no	—
13	C₄⋊C₄×C₁₆₃	—	4	1	no	—
14	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
15	C₁₆₃⋊C₈	—	8	2	no	—
16	Q₈×C₁₆₃	—	4	2	no	—
17	C₂×C₄×C₁₆₃	—	2	1	no	—
18	C₂×C₄×C₁₆₃	—	2	2	no	—
19	C₂×C₄×C₁₆₃	—	4	1	no	—
20	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₈×C₁₆₃	—	4	1	no	—
22	C₂²×C₁₆₃	#99	1	1	yes	characteristic
23	C₄×C₁₆₃	—	2	1	no	—
24	C₄×C₁₆₃	—	4	1	no	—
25	C₄×C₁₆₃	—	4	2	no	—
26	C₄×C₁₆₃	—	8	1	no	—
27	C₂×C₁₆₃	—	2	1	no	—
28	C₂×C₁₆₃	#354	1	1	yes	characteristic
29	C₁₆₃	#1413	1	1	yes	characteristic, complemented, Sylow
30	#1413	—	163	1	no	maximal, Sylow
31	#491	D₁₆₃	1	1	yes	characteristic
32	#356	—	163	2	no	—
33	C₄⋊Q₈	C₂×D₁₆₃	1	1	yes	characteristic
34	#125	Dic₁₆₃	1	1	yes	characteristic
35	#103	—	2	2	no	—
36	#101	—	326	2	no	—
37	#97	Dic₁₆₃	1	1	yes	characteristic
38	C₂×Q₈	C₂²⋊D₁₆₃	1	2	yes	—
39	M₁₆	—	652	2	no	—
40	C₂×C₈	—	2	1	no	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	—	4	1	no	—
43	C₄⋊C₄	—	4	1	no	—
44	C₄²	C₂×Dic₁₆₃	1	1	yes	characteristic
45	Q₈	—	4	2	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	4	1	no	—
49	C₂×C₄	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
50	C₈	—	4	1	no	—
51	C₈	—	1304	2	no	—
52	C₂²	#19104650	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	4	1	no	—
55	C₄	—	4	2	no	—
56	C₄	—	8	1	no	—
57	C₂	—	2	1	no	—
58	C₂	#23781460	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23781460
commutator	C₄²×C₁₆₃	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₁₆₃
Fitting	#491×C₁₆₃	C₂
socle	C₂×C₁₆₃	#354
prime radical	C₂²×C₁₆₃	#99
cosocle	C₄⋊Q₈×C₁₆₃	C₂²
hypercenter	#491	D₁₆₃
hypocenter	C₁₆₃	#1413
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂×C₄×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #491

Complements

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

N	H ≅ G/N	type
29. C₁₆₃	#1413	semidirect

Maximal subgroups

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

subgroup	index
#23781464	2
#491×C₁₆₃	2
#1413	163