Subgroups — #29508636

Order 15424.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21440717	C₂	1	1	yes	maximal, characteristic
2	#21440712	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×D₂₄₁	C₂	1	1	yes	maximal, characteristic
4	#16838780	—	2	1	no	—
5	#16838773	—	2	1	no	—
6	Q₈×D₂₄₁	—	2	1	no	—
7	Dic₉₆₄	C₄	1	1	yes	characteristic
8	#16838760	—	2	1	no	—
9	C₈×D₂₄₁	C₂²	1	1	yes	characteristic
10	Q₁₆×C₂₄₁	C₄	1	1	yes	characteristic
11	#14515356	—	4	1	no	—
12	Dic₄₈₂	—	2	1	no	—
13	Dic₄₈₂	—	4	1	no	—
14	C₄×D₂₄₁	—	4	1	no	—
15	C₄×D₂₄₁	D₄	1	1	yes	characteristic
16	#14515349	D₄	1	1	yes	characteristic
17	Q₈×C₂₄₁	—	2	1	no	—
18	C₈×C₂₄₁	C₂×C₄	1	1	yes	characteristic
19	C₂×D₂₄₁	SD₁₆	1	1	yes	characteristic
20	C₄×C₂₄₁	—	4	1	no	—
21	C₄×C₂₄₁	C₂²⋊C₄	1	1	yes	characteristic
22	Dic₂₄₁	—	4	1	no	—
23	Dic₂₄₁	D₈	1	1	yes	characteristic
24	C₂×C₂₄₁	#102	1	1	yes	characteristic
25	D₂₄₁	—	2	1	no	—
26	C₂₄₁	#362	1	1	yes	characteristic, complemented, Sylow
27	#362	—	241	1	no	maximal, Sylow
28	C₂×Q₁₆	—	241	1	no	—
29	M₃₂	—	241	1	no	—
30	#108	—	241	1	no	—
31	C₂×Q₈	—	482	1	no	—
32	Q₁₆	—	482	1	no	—
33	Q₁₆	—	241	1	no	—
34	Q₁₆	#11757210	1	1	yes	characteristic
35	M₁₆	—	482	1	no	—
36	C₂×C₈	—	241	1	no	—
37	C₁₆	—	482	1	no	—
38	Q₈	—	2	1	no	—
39	Q₈	—	964	1	no	—
40	Q₈	—	482	1	no	—
41	C₂×C₄	—	964	1	no	—
42	C₂×C₄	—	241	1	no	—
43	C₈	—	964	1	no	—
44	C₈	—	241	1	no	—
45	C₈	C₂×#11757210	1	1	yes	characteristic
46	C₂²	—	241	1	no	—
47	C₄	—	4	1	no	—
48	C₄	—	964	1	no	—
49	C₄	—	241	1	no	—
50	C₄	C₂²⋊#11757210	1	1	yes	characteristic
51	C₂	—	482	1	no	—
52	C₂	#21440704	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21440704
commutator	C₈×C₂₄₁	C₂×C₄
Frattini	C₈	C₂×#11757210
Fitting	Q₁₆×C₂₄₁	C₄
socle	C₂×C₂₄₁	#102
prime radical	C₂×D₂₄₁	SD₁₆
cosocle	C₈×D₂₄₁	C₂²
hypercenter	Q₁₆	#11757210
hypocenter	C₂₄₁	#362
LCS	G ▷ C₈×C₂₄₁ ▷ C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
26. C₂₄₁	#362	semidirect

Maximal subgroups

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

subgroup	index
#21440717	2
#21440712	2
Q₁₆×D₂₄₁	2
#362	241