Subgroups — #39909613

Order 26176.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27189714	C₂	1	1	yes	maximal, characteristic
2	#27189637	C₂	1	4	yes	maximal, complemented
3	#27189628	C₂	1	1	yes	maximal, characteristic, complemented
4	#125×C₄₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₆₃₆	—	2	2	no	—
6	C₂×D₁₆₃₆	C₂²	1	2	yes	—
7	C₂×#15692543	C₂²	1	2	yes	—
8	C₄⋊C₄×C₄₀₉	—	2	2	no	—
9	C₄⋊C₄×C₄₀₉	C₂²	1	2	yes	—
10	C₄²×C₄₀₉	C₂²	1	1	yes	characteristic
11	C₂²×D₄₀₉	—	2	2	no	—
12	D₁₆₃₆	—	2	4	no	—
13	D₁₆₃₆	—	4	4	no	—
14	#15692543	—	2	2	no	—
15	C₂×C₄×C₄₀₉	—	2	2	no	—
16	C₂×C₄×C₄₀₉	D₄	1	2	yes	—
17	C₂×C₄×C₄₀₉	C₂³	1	1	yes	characteristic
18	C₂²×C₄₀₉	C₂×D₄	1	1	yes	characteristic
19	C₂×D₄₀₉	—	4	2	no	—
20	C₂×D₄₀₉	—	4	4	no	—
21	C₄×C₄₀₉	—	2	2	no	—
22	C₄×C₄₀₉	—	4	2	no	—
23	C₄×C₄₀₉	#41	1	2	yes	—
24	C₂×C₄₀₉	#124	1	1	yes	characteristic
25	C₂×C₄₀₉	AGL(1,ℤ/8)	1	2	yes	—
26	D₄₀₉	—	8	2	no	—
27	C₄₀₉	#490	1	1	yes	characteristic, complemented, Sylow
28	#490	—	409	1	no	maximal, Sylow
29	#127	—	409	1	no	—
30	#125	D₄₀₉	1	1	yes	characteristic, complemented
31	#102	—	409	4	no	—
32	#97	—	409	1	no	—
33	C₂×D₄	—	818	2	no	—
34	C₂×D₄	—	409	2	no	—
35	C₂×C₈	—	409	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	C₂×D₄₀₉	1	2	yes	—
38	C₄²	C₂×D₄₀₉	1	1	yes	characteristic
39	C₂³	—	818	2	no	—
40	D₄	—	818	4	no	—
41	D₄	—	1636	4	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	C₂²⋊D₄₀₉	1	2	yes	—
44	C₂×C₄	C₂²×D₄₀₉	1	1	yes	characteristic
45	C₈	—	818	2	no	—
46	C₂²	—	1636	2	no	—
47	C₂²	—	1636	4	no	—
48	C₂²	C₂×C₂²⋊D₄₀₉	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	4	2	no	—
51	C₄	#20266989	1	2	yes	—
52	C₂	—	3272	2	no	—
53	C₂	#27189706	1	1	yes	characteristic
54	C₂	#27189742	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₀₉
commutator, cosocle	C₂×C₄×C₄₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₄₀₉
Fitting	#125×C₄₀₉	C₂
socle	C₂²×C₄₀₉	C₂×D₄
prime radical	#27189714	C₂
hypercenter	#125	D₄₀₉
hypocenter	C₄₀₉	#490
LCS	G ▷ C₂×C₄×C₄₀₉ ▷ C₂×C₄₀₉ ▷ C₄₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #27189637	C₂	semidirect
3. #27189628	C₂	semidirect
4. #125×C₄₀₉	C₂	semidirect
27. C₄₀₉	#490	semidirect
30. #125	D₄₀₉	semidirect

Maximal subgroups

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

subgroup	index
#27189714	2
#27189637	2
#27189628	2
#125×C₄₀₉	2
#490	409