Subgroups — Q₁₆×#13109888 #39910285

Order 26176.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#13109888	C₂	1	2	yes	maximal
2	#27189786	C₂	1	1	yes	maximal, characteristic
3	#27189780	C₂	1	2	yes	maximal
4	C₈×#13109888	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×D₄₀₉	C₂	1	1	yes	maximal, characteristic
6	#20266996	C₂²	1	1	yes	characteristic
7	C₄⋊#13109888	—	2	2	no	—
8	C₄⋊#13109888	C₂²	1	2	yes	—
9	C₄×#13109888	—	2	2	no	—
10	C₄×#13109888	C₂²	1	1	yes	characteristic
11	Q₈×D₄₀₉	C₂²	1	2	yes	—
12	Dic₁₆₃₆	C₄	1	1	yes	characteristic, complemented
13	#20266978	C₄	1	2	yes	complemented
14	C₈×D₄₀₉	C₂²	1	1	yes	characteristic
15	Q₁₆×C₄₀₉	C₄	1	1	yes	characteristic, complemented
16	C₂×#13109888	—	2	2	no	—
17	C₂×#13109888	D₄	1	1	yes	characteristic
18	C₂×#13109888	D₄	1	1	yes	characteristic
19	#15692550	—	2	1	no	—
20	Dic₈₁₈	—	2	2	no	—
21	Dic₈₁₈	C₂×C₄	1	2	yes	—
22	C₄×D₄₀₉	—	2	2	no	—
23	C₄×D₄₀₉	C₂³	1	1	yes	characteristic
24	#15692543	C₂×C₄	1	1	yes	characteristic
25	Q₈×C₄₀₉	C₂×C₄	1	2	yes	—
26	C₈×C₄₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂×D₄₀₉	C₂×D₄	1	1	yes	characteristic
28	#13109888	—	2	1	no	—
29	#13109888	—	4	2	no	—
30	#13109888	Q₁₆	1	2	yes	direct product factor
31	C₄×C₄₀₉	—	2	2	no	—
32	C₄×C₄₀₉	C₂²×C₄	1	1	yes	characteristic
33	Dic₄₀₉	—	2	2	no	—
34	Dic₄₀₉	#41	1	1	yes	characteristic
35	C₂×C₄₀₉	C₄×D₄	1	1	yes	characteristic
36	D₄₀₉	C₂×Q₁₆	1	1	yes	characteristic
37	D₄₀₉	#135	1	1	yes	characteristic
38	C₄₀₉	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
39	C₄×Q₁₆	—	409	1	no	maximal, Sylow
40	C₂×Q₁₆	—	409	1	no	—
41	C₄×Q₈	—	409	2	no	—
42	#107	—	409	1	no	—
43	#103	—	409	2	no	—
44	C₄×C₈	—	409	1	no	—
45	C₂×Q₈	—	409	2	no	—
46	Q₁₆	—	409	2	no	—
47	Q₁₆	—	409	1	no	—
48	Q₁₆	#13109888	1	1	yes	characteristic, direct product factor
49	C₂×C₈	—	409	1	no	—
50	C₂×C₈	—	409	1	no	—
51	C₄⋊C₄	—	818	2	no	—
52	C₄⋊C₄	—	409	2	no	—
53	C₄²	—	818	2	no	—
54	C₄²	—	409	1	no	—
55	Q₈	—	818	2	no	—
56	Q₈	—	409	2	no	—
57	Q₈	C₂×#13109888	1	2	yes	—
58	C₂×C₄	—	818	2	no	—
59	C₂×C₄	—	818	2	no	—
60	C₂×C₄	—	409	1	no	—
61	C₂×C₄	—	409	1	no	—
62	C₂×C₄	—	409	1	no	—
63	C₈	—	818	1	no	—
64	C₈	—	409	1	no	—
65	C₈	C₂×#13109888	1	1	yes	characteristic
66	C₂²	—	409	1	no	—
67	C₄	—	2	2	no	—
68	C₄	—	818	1	no	—
69	C₄	—	818	2	no	—
70	C₄	—	1636	2	no	—
71	C₄	—	409	2	no	—
72	C₄	—	409	1	no	—
73	C₄	C₂²×#13109888	1	1	yes	characteristic
74	C₂	—	409	1	no	—
75	C₂	—	409	1	no	—
76	C₂	D₄×#13109888	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#13109888
commutator	C₄×C₄₀₉	C₂²×C₄
Frattini	C₄	C₂²×#13109888
Fitting	Q₁₆×C₄₀₉	C₄
socle	C₂×C₄₀₉	C₄×D₄
prime radical	C₂×D₄₀₉	C₂×D₄
cosocle	C₄×D₄₀₉	C₂³
hypercenter	Q₁₆	#13109888
hypocenter	C₄₀₉	C₄×Q₁₆
LCS	G ▷ C₄×C₄₀₉ ▷ C₂×C₄₀₉ ▷ C₄₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
12. Dic₁₆₃₆	C₄	semidirect
13. #20266978	C₄	semidirect
15. Q₁₆×C₄₀₉	C₄	semidirect
30. #13109888	Q₁₆	direct (N × H)
38. C₄₀₉	C₄×Q₁₆	semidirect
48. Q₁₆	#13109888	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#13109888	2
#27189786	2
#27189780	2
C₈×#13109888	2
Q₁₆×D₄₀₉	2
C₄×Q₁₆	409

Subgroups — Q16×#13109888 #39910285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×#13109888 #39910285