Subgroups — #39910327

Order 26176.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27189805	C₂	1	8	yes	maximal, complemented
2	D₄₀₉⋊M₁₆	C₂	1	6	yes	maximal, complemented
3	C₂×#20266989	C₂	1	1	yes	maximal, characteristic
4	#20266998	C₂²	1	12	yes	complemented
5	#20266997	C₂²	1	4	yes	—
6	#20266996	C₂²	1	12	yes	complemented
7	#20266989	C₄	1	4	yes	—
8	#20266989	C₂²	1	4	yes	—
9	C₂×D₁₆₃₆	C₄	1	3	yes	—
10	C₂×C₄×D₄₀₉	C₂²	1	3	yes	—
11	C₂×Q₈×C₄₀₉	C₄	1	1	yes	characteristic
12	#15692550	C₂³	1	8	yes	—
13	C₂²×D₄₀₉	C₂×C₄	1	3	yes	—
14	D₁₆₃₆	C₂×C₄	1	12	yes	—
15	C₂×Dic₄₀₉	C₂³	1	1	yes	characteristic
16	C₄×D₄₀₉	C₂×C₄	1	6	yes	—
17	C₄×D₄₀₉	C₂³	1	6	yes	—
18	Q₈×C₄₀₉	C₂×C₄	1	4	yes	—
19	C₂×C₄×C₄₀₉	C₂×C₄	1	3	yes	—
20	C₂²×C₄₀₉	C₂²×C₄	1	1	yes	characteristic
21	C₂×D₄₀₉	—	4	3	no	—
22	C₂×D₄₀₉	C₂²×C₄	1	6	yes	—
23	C₄×C₄₀₉	C₂²×C₄	1	6	yes	—
24	Dic₄₀₉	C₂²×C₄	1	1	yes	characteristic
25	Dic₄₀₉	C₂⁴	1	1	yes	characteristic
26	C₂×C₄₀₉	—	2	1	no	—
27	C₂×C₄₀₉	C₂³×C₄	1	1	yes	characteristic
28	D₄₀₉	—	2	6	no	—
29	C₄₀₉	#568	1	1	yes	characteristic, complemented, Sylow
30	#568	—	409	1	no	maximal, Sylow
31	C₂×#41	—	409	1	no	—
32	#131	—	409	8	no	—
33	C₂×M₁₆	—	409	6	no	—
34	#41	—	409	4	no	—
35	#41	—	409	4	no	—
36	C₂×Q₈	#13109888	1	1	yes	characteristic
37	C₂×D₄	—	409	3	no	—
38	C₂²×C₄	—	409	3	no	—
39	M₁₆	—	409	4	no	—
40	M₁₆	—	409	12	no	—
41	C₂×C₈	—	409	12	no	—
42	C₂³	—	409	3	no	—
43	Q₈	C₂×#13109888	1	4	yes	—
44	D₄	—	409	12	no	—
45	C₂×C₄	—	409	1	no	—
46	C₂×C₄	—	409	6	no	—
47	C₂×C₄	—	409	6	no	—
48	C₂×C₄	C₂×#13109888	1	3	yes	—
49	C₈	—	409	8	no	—
50	C₂²	—	1636	3	no	—
51	C₂²	—	409	6	no	—
52	C₂²	C₂²×#13109888	1	1	yes	characteristic
53	C₄	—	409	1	no	—
54	C₄	—	409	1	no	—
55	C₄	C₂²×#13109888	1	6	yes	—
56	C₂	—	2	1	no	—
57	C₂	—	818	6	no	—
58	C₂	C₂³×#13109888	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#13109888
commutator, socle	C₂×C₄₀₉	C₂³×C₄
Fitting	C₂×Q₈×C₄₀₉	C₄
prime radical	C₂×#20266989	C₂
cosocle	Dic₄₀₉	C₂⁴
hypercenter	C₂×Q₈	#13109888
hypocenter	C₄₀₉	#568
LCS	G ▷ C₂×C₄₀₉ ▷ C₄₀₉
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #27189805	C₂	semidirect
2. D₄₀₉⋊M₁₆	C₂	semidirect
4. #20266998	C₂²	semidirect
6. #20266996	C₂²	semidirect
29. C₄₀₉	#568	semidirect

Maximal subgroups

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

subgroup	index
#27189805	2
D₄₀₉⋊M₁₆	2
C₂×#20266989	2
#568	409