Subgroups — #29508576

Order 15424.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21440709	C₂	1	1	yes	maximal, characteristic
2	#21440709	C₂	1	1	yes	maximal, characteristic
3	#21440641	C₂	1	1	yes	maximal, characteristic
4	C₂×#14515356	—	2	1	no	—
5	C₂×#14515356	—	2	1	no	—
6	C₂×Dic₄₈₂	—	2	1	no	—
7	C₄⋊Dic₂₄₁	C₄	1	1	yes	characteristic
8	#16838749	—	2	1	no	—
9	C₄×Dic₂₄₁	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₂₄₁	C₄	1	1	yes	characteristic
11	#14515356	—	4	1	no	—
12	#14515356	—	4	1	no	—
13	Dic₄₈₂	—	4	1	no	—
14	Dic₄₈₂	—	4	1	no	—
15	C₂×Dic₂₄₁	—	2	1	no	—
16	C₂×Dic₂₄₁	D₄	1	1	yes	characteristic
17	C₂×Dic₂₄₁	D₄	1	1	yes	characteristic
18	C₂×C₄×C₂₄₁	—	2	1	no	—
19	C₂×C₄×C₂₄₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₂₄₁	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₂₄₁	—	2	1	no	—
22	C₄×C₂₄₁	—	4	1	no	—
23	Dic₂₄₁	—	4	1	no	—
24	Dic₂₄₁	D₈	1	1	yes	characteristic
25	Dic₂₄₁	SD₁₆	1	1	yes	characteristic
26	Dic₂₄₁	SD₁₆	1	1	yes	characteristic
27	Dic₂₄₁	Q₁₆	1	1	yes	characteristic
28	C₂×C₂₄₁	#101	1	1	yes	characteristic
29	C₂×C₂₄₁	#102	1	1	yes	characteristic
30	C₂×C₂₄₁	#103	1	1	yes	characteristic
31	C₂₄₁	#332	1	1	yes	characteristic, complemented, Sylow
32	#332	—	241	1	no	maximal, Sylow
33	C₄⋊Q₈	—	241	1	no	—
34	C₄⋊C₈	—	241	1	no	—
35	C₄⋊C₈	—	241	1	no	—
36	C₂×Q₈	—	482	1	no	—
37	C₂×C₈	—	482	1	no	—
38	C₂×C₈	—	482	1	no	—
39	C₄⋊C₄	—	482	1	no	—
40	C₄⋊C₄	—	241	1	no	—
41	C₄⋊C₄	#11757210	1	1	yes	characteristic
42	C₄²	—	241	1	no	—
43	Q₈	—	964	1	no	—
44	Q₈	—	964	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	482	1	no	—
47	C₂×C₄	—	241	1	no	—
48	C₂×C₄	—	241	1	no	—
49	C₂×C₄	C₂×#11757210	1	1	yes	characteristic
50	C₈	—	964	1	no	—
51	C₈	—	964	1	no	—
52	C₂²	C₂²⋊#11757210	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	4	1	no	—
55	C₄	—	964	1	no	—
56	C₄	—	241	1	no	—
57	C₄	—	241	1	no	—
58	C₄	—	241	1	no	—
59	C₄	—	241	1	no	—
60	C₂	#21440703	1	1	yes	characteristic
61	C₂	#21440704	1	1	yes	characteristic
62	C₂	#21440705	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11757210
commutator	C₂×C₄×C₂₄₁	C₂×C₄
Frattini	C₂×C₄	C₂×#11757210
Fitting	C₄⋊C₄×C₂₄₁	C₄
socle, prime radical	C₂²×C₂₄₁	C₂²⋊C₄
cosocle	C₄×Dic₂₄₁	C₂²
hypercenter	C₄⋊C₄	#11757210
hypocenter	C₂₄₁	#332
LCS	G ▷ C₂×C₄×C₂₄₁ ▷ C₂²×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
31. C₂₄₁	#332	semidirect

Maximal subgroups

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

subgroup	index
#21440709	2
#21440641	2
#332	241