Subgroups — #29507515

Order 15424.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21440640	C₂	1	1	yes	maximal, characteristic
2	#21440561	C₂	1	1	yes	maximal, characteristic
3	#101×C₂₄₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₈₂	C₄	1	1	yes	characteristic
5	#16838754	—	2	1	no	—
6	#16838749	—	2	1	no	—
7	#16838749	—	2	1	no	—
8	C₄×Dic₂₄₁	C₄	1	1	yes	characteristic
9	C₂×Q₈×C₂₄₁	C₂²	1	1	yes	characteristic
10	M₁₆×C₂₄₁	—	2	1	no	—
11	Dic₄₈₂	—	4	1	no	—
12	C₂×Dic₂₄₁	—	2	1	no	—
13	C₂×Dic₂₄₁	—	2	1	no	—
14	#14515349	—	4	1	no	—
15	Q₈×C₂₄₁	—	4	1	no	—
16	C₂×C₄×C₂₄₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₄₁	D₄	1	1	yes	characteristic
18	C₂×C₄×C₂₄₁	D₄	1	1	yes	characteristic
19	C₈×C₂₄₁	—	4	1	no	—
20	C₂²×C₂₄₁	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₂₄₁	—	2	1	no	—
22	C₄×C₂₄₁	—	2	1	no	—
23	C₄×C₂₄₁	—	2	1	no	—
24	Dic₂₄₁	—	2	2	no	—
25	Dic₂₄₁	—	4	1	no	—
26	C₂×C₂₄₁	—	2	1	no	—
27	C₂×C₂₄₁	#99	1	1	yes	characteristic
28	C₂₄₁	#356	1	1	yes	characteristic, complemented, Sylow
29	#356	—	241	1	no	maximal, Sylow
30	C₄⋊Q₈	—	241	1	no	—
31	#101	—	241	1	no	—
32	#101	D₂₄₁	1	1	yes	characteristic
33	C₂×Q₈	—	241	1	no	—
34	C₂×Q₈	C₂×D₂₄₁	1	1	yes	characteristic
35	M₁₆	—	2	1	no	—
36	M₁₆	—	482	1	no	—
37	C₄⋊C₄	—	482	1	no	—
38	C₄⋊C₄	—	482	1	no	—
39	C₄²	—	241	1	no	—
40	Q₈	—	4	1	no	—
41	Q₈	—	964	1	no	—
42	C₂×C₄	—	482	1	no	—
43	C₂×C₄	—	482	1	no	—
44	C₂×C₄	C₄×D₂₄₁	1	1	yes	characteristic
45	C₂×C₄	C₂²⋊D₂₄₁	1	1	yes	characteristic
46	C₂×C₄	D₉₆₄	1	1	yes	characteristic
47	C₈	—	4	1	no	—
48	C₈	—	964	1	no	—
49	C₂²	#16838747	1	1	yes	characteristic
50	C₄	—	2	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	964	1	no	—
54	C₄	—	482	2	no	—
55	C₂	—	2	1	no	—
56	C₂	#21440556	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21440556
commutator	C₂×C₄×C₂₄₁	C₂×C₄
Frattini	C₂×Q₈	C₂×D₂₄₁
Fitting	#101×C₂₄₁	C₂
socle	C₂×C₂₄₁	#99
prime radical	C₂²×C₂₄₁	C₂²⋊C₄
cosocle	C₂×Q₈×C₂₄₁	C₂²
hypercenter	#101	D₂₄₁
hypocenter	C₂₄₁	#356
LCS	G ▷ C₂×C₄×C₂₄₁ ▷ C₂²×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ #101

Complements

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

N	H ≅ G/N	type
28. C₂₄₁	#356	semidirect

Maximal subgroups

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

subgroup	index
#21440640	2
#21440561	2
#101×C₂₄₁	2
#356	241