Subgroups — #29507828

Order 15424.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₉₂₈	C₂	1	1	yes	maximal, characteristic
2	#21440604	C₂	1	2	yes	maximal, complemented
3	#21440562	C₂	1	2	yes	maximal, complemented
4	#21440552	C₂	1	1	yes	maximal, characteristic, complemented
5	#106×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₉₆₄	C₂²	1	2	yes	—
7	C₂×C₄×D₂₄₁	—	2	2	no	—
8	D₁₉₂₈	C₄	1	4	yes	complemented
9	C₂×#14515349	C₂²	1	1	yes	characteristic
10	#16838747	—	2	2	no	—
11	C₄×Dic₂₄₁	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₄₁	C₂²	1	2	yes	—
14	C₂²×D₂₄₁	—	2	2	no	—
15	D₉₆₄	—	2	2	no	—
16	D₉₆₄	C₂×C₄	1	4	yes	complemented
17	C₂×Dic₂₄₁	D₄	1	2	yes	—
18	C₄×D₂₄₁	—	4	4	no	—
19	#14515349	—	2	1	no	—
20	C₂×C₄×C₂₄₁	—	2	2	no	—
21	C₂×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
22	C₈×C₂₄₁	C₂×C₄	1	2	yes	complemented
23	C₂²×C₂₄₁	C₂×D₄	1	1	yes	characteristic
24	C₂×D₂₄₁	—	2	4	no	—
25	C₂×D₂₄₁	—	4	2	no	—
26	C₂×D₂₄₁	—	4	2	no	—
27	C₄×C₂₄₁	—	4	2	no	—
28	C₄×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₂₄₁	#41	1	1	yes	characteristic
30	Dic₂₄₁	—	2	2	no	—
31	C₂×C₂₄₁	C₄×D₄	1	1	yes	characteristic
32	C₂×C₂₄₁	AGL(1,ℤ/8)	1	2	yes	—
33	D₂₄₁	—	4	4	no	—
34	C₂₄₁	#442	1	1	yes	characteristic, complemented, Sylow
35	#442	—	241	1	no	maximal, Sylow
36	C₂×D₈	—	241	1	no	—
37	C₄×D₄	—	241	2	no	—
38	#106	D₂₄₁	1	1	yes	characteristic, complemented
39	#102	—	241	2	no	—
40	#97	—	241	1	no	—
41	C₂×D₄	—	241	2	no	—
42	C₂²×C₄	—	482	2	no	—
43	D₈	—	241	4	no	—
44	C₂×C₈	—	241	1	no	—
45	C₂×C₈	C₂×D₂₄₁	1	1	yes	characteristic
46	C₄⋊C₄	C₂×D₂₄₁	1	2	yes	—
47	C₂²⋊C₄	—	482	2	no	—
48	C₄²	—	241	1	no	—
49	C₂³	—	482	2	no	—
50	D₄	—	482	2	no	—
51	D₄	—	241	4	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	964	4	no	—
54	C₂×C₄	—	241	2	no	—
55	C₂×C₄	C₂²×D₂₄₁	1	1	yes	characteristic
56	C₈	—	482	1	no	—
57	C₈	C₄×D₂₄₁	1	2	yes	complemented
58	C₂²	—	964	2	no	—
59	C₂²	—	964	2	no	—
60	C₂²	—	482	4	no	—
61	C₂²	D₄×D₂₄₁	1	1	yes	characteristic
62	C₄	—	4	2	no	—
63	C₄	—	482	2	no	—
64	C₄	C₂×C₄×D₂₄₁	1	1	yes	characteristic
65	C₄	#16838771	1	1	yes	characteristic
66	C₂	—	964	4	no	—
67	C₂	#21440604	1	1	yes	characteristic
68	C₂	#21440669	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₄₁
commutator	C₄×C₂₄₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₄₁
Fitting	#106×C₂₄₁	C₂
socle	C₂²×C₂₄₁	C₂×D₄
prime radical	C₂×D₁₉₂₈	C₂
cosocle	C₂×C₄×C₂₄₁	C₂³
hypercenter	#106	D₂₄₁
hypocenter	C₂₄₁	#442
LCS	G ▷ C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
2. #21440604	C₂	semidirect
3. #21440562	C₂	semidirect
4. #21440552	C₂	semidirect
5. #106×C₂₄₁	C₂	semidirect
8. D₁₉₂₈	C₄	semidirect
16. D₉₆₄	C₂×C₄	semidirect
22. C₈×C₂₄₁	C₂×C₄	semidirect
34. C₂₄₁	#442	semidirect
38. #106	D₂₄₁	semidirect
57. C₈	C₄×D₂₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₉₂₈	2
#21440604	2
#21440562	2
#21440552	2
#106×C₂₄₁	2
#442	241