Subgroups — #39909626

Order 26176.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20266978	C₂	1	4	yes	maximal
2	#27189717	C₂	1	1	yes	maximal, characteristic
3	C₄×#15692543	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₄₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₈₁₈	C₂²	1	2	yes	—
6	#20266978	—	2	8	no	—
7	C₂×#15692543	C₂²	1	2	yes	—
8	C₄⋊Dic₄₀₉	—	2	2	no	—
9	C₂×Q₈×C₄₀₉	C₂²	1	2	yes	—
10	C₄⋊C₄×C₄₀₉	—	2	2	no	—
11	C₄²×C₄₀₉	C₂²	1	1	yes	characteristic
12	Dic₈₁₈	—	2	4	no	—
13	C₂×Dic₄₀₉	—	2	2	no	—
14	#15692543	D₄	1	4	yes	—
15	Q₈×C₄₀₉	—	2	4	no	—
16	C₂×C₄×C₄₀₉	—	2	2	no	—
17	C₂×C₄×C₄₀₉	D₄	1	2	yes	—
18	C₂×C₄×C₄₀₉	C₂³	1	1	yes	characteristic
19	C₂²×C₄₀₉	C₂×D₄	1	1	yes	characteristic
20	C₄×C₄₀₉	—	4	2	no	—
21	C₄×C₄₀₉	Q₁₆	1	4	yes	complemented
22	C₄×C₄₀₉	C₂×D₄	1	2	yes	—
23	Dic₄₀₉	—	4	2	no	—
24	C₂×C₄₀₉	#127	1	1	yes	characteristic
25	C₂×C₄₀₉	C₂×Q₁₆	1	2	yes	—
26	C₄₀₉	#494	1	1	yes	characteristic, complemented, Sylow
27	#494	—	409	1	no	maximal, Sylow
28	C₂×Q₁₆	—	409	4	no	—
29	C₄⋊Q₈	—	409	1	no	—
30	C₄⋊Q₈	D₄₀₉	1	1	yes	characteristic
31	C₄×C₈	—	409	1	no	—
32	C₂×Q₈	—	409	2	no	—
33	C₂×Q₈	C₂×D₄₀₉	1	2	yes	—
34	Q₁₆	—	818	8	no	—
35	C₂×C₈	—	409	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	818	2	no	—
38	C₄²	C₂×D₄₀₉	1	1	yes	characteristic
39	Q₈	—	2	4	no	—
40	Q₈	—	818	4	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	818	2	no	—
43	C₂×C₄	C₂²⋊D₄₀₉	1	2	yes	—
44	C₂×C₄	C₂²×D₄₀₉	1	1	yes	characteristic
45	C₈	—	409	4	no	—
46	C₂²	C₂×C₂²⋊D₄₀₉	1	1	yes	characteristic
47	C₄	—	4	2	no	—
48	C₄	—	1636	2	no	—
49	C₄	#20266978	1	4	yes	complemented
50	C₄	D₄×D₄₀₉	1	2	yes	—
51	C₂	#27189713	1	1	yes	characteristic
52	C₂	C₂×#20266978	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₀₉
commutator, cosocle	C₂×C₄×C₄₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₄₀₉
Fitting	C₄⋊Q₈×C₄₀₉	C₂
socle, prime radical	C₂²×C₄₀₉	C₂×D₄
hypercenter	C₄⋊Q₈	D₄₀₉
hypocenter	C₄₀₉	#494
LCS	G ▷ C₂×C₄×C₄₀₉ ▷ C₂×C₄₀₉ ▷ C₄₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
21. C₄×C₄₀₉	Q₁₆	semidirect
26. C₄₀₉	#494	semidirect
49. C₄	#20266978	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20266978	2
#27189717	2
C₄×#15692543	2
C₄⋊Q₈×C₄₀₉	2
#494	409