Subgroups — #39910287

Order 26176.

108 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	#27189785	C₂	1	1	yes	maximal, characteristic
3	#27189780	C₂	1	2	yes	maximal
4	#27189771	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	2	yes	—
18	#15692550	—	2	1	no	—
19	Dic₈₁₈	—	2	2	no	—
20	Dic₈₁₈	C₂×C₄	1	2	yes	—
21	C₄×D₄₀₉	—	2	2	no	—
22	C₄×D₄₀₉	C₂³	1	1	yes	characteristic
23	#15692543	C₂×C₄	1	1	yes	characteristic
24	Q₈×C₄₀₉	C₂×C₄	1	2	yes	—
25	C₈×C₄₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄₀₉	C₂×D₄	1	1	yes	characteristic
27	#13109888	—	2	2	no	—
28	#13109888	—	4	2	no	—
29	C₄×C₄₀₉	—	2	2	no	—
30	C₄×C₄₀₉	C₂²×C₄	1	1	yes	characteristic
31	Dic₄₀₉	—	2	2	no	—
32	Dic₄₀₉	#41	1	1	yes	characteristic
33	C₂×C₄₀₉	C₄×D₄	1	1	yes	characteristic
34	D₄₀₉	#137	1	2	yes	—
35	C₄₀₉	#441	1	1	yes	characteristic, complemented, Sylow
36	#441	—	409	1	no	maximal, Sylow
37	C₂×Q₁₆	—	409	1	no	—
38	C₄×Q₈	—	409	2	no	—
39	#106	—	409	1	no	—
40	#103	—	409	2	no	—
41	#97	—	409	1	no	—
42	C₂×Q₈	—	409	2	no	—
43	Q₁₆	—	409	2	no	—
44	Q₁₆	—	409	1	no	—
45	Q₁₆	#13109888	1	1	yes	characteristic, complemented
46	C₂×C₈	—	409	1	no	—
47	C₂×C₈	—	409	1	no	—
48	C₄⋊C₄	—	818	2	no	—
49	C₄⋊C₄	—	409	2	no	—
50	C₄²	—	818	2	no	—
51	C₄²	—	409	1	no	—
52	Q₈	—	818	2	no	—
53	Q₈	—	409	2	no	—
54	Q₈	C₂×#13109888	1	2	yes	—
55	C₂×C₄	—	818	2	no	—
56	C₂×C₄	—	818	2	no	—
57	C₂×C₄	—	409	2	no	—
58	C₂×C₄	—	409	1	no	—
59	C₈	—	818	1	no	—
60	C₈	—	409	1	no	—
61	C₈	C₂×#13109888	1	1	yes	characteristic
62	C₂²	—	409	1	no	—
63	C₄	—	2	2	no	—
64	C₄	—	818	2	no	—
65	C₄	—	818	2	no	—
66	C₄	—	1636	2	no	—
67	C₄	—	409	1	no	—
68	C₄	C₂²×#13109888	1	1	yes	characteristic
69	C₂	—	409	2	no	—
70	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	#441
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
35. C₄₀₉	#441	semidirect
45. Q₁₆	#13109888	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#13109888	2
#27189785	2
#27189780	2
#27189771	2
Q₁₆×D₄₀₉	2
#441	409