Subgroups — #29508018

Order 15424.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21440634	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₉₆₄	C₂	1	1	yes	maximal, characteristic
3	#21440576	C₂	1	1	yes	maximal, characteristic, complemented
4	#21440574	C₂	1	1	yes	maximal, characteristic, complemented
5	#21440564	C₂	1	1	yes	maximal, characteristic
6	#21440564	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₈×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₉₆₄	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₂₄₁	—	2	1	no	—
10	C₄⋊Dic₂₄₁	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₄₁	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₄₁	C₂²	1	1	yes	characteristic
13	#16838749	—	2	1	no	—
14	#16838749	—	2	1	no	—
15	#16838747	—	2	1	no	—
16	C₂×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
18	C₄²×C₂₄₁	C₂²	1	1	yes	characteristic
19	C₂²×D₂₄₁	—	2	1	no	—
20	D₉₆₄	—	2	1	no	—
21	D₉₆₄	Q₈	1	2	yes	—
22	C₂×Dic₂₄₁	—	2	1	no	—
23	C₂×Dic₂₄₁	—	2	1	no	—
24	C₂×Dic₂₄₁	—	2	1	no	—
25	C₄×D₂₄₁	—	4	2	no	—
26	C₂×C₄×C₂₄₁	D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₄₁	D₄	1	1	yes	characteristic
28	C₂×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
29	C₈×C₂₄₁	—	2	1	no	—
30	C₈×C₂₄₁	—	2	1	no	—
31	C₂²×C₂₄₁	C₂×D₄	1	1	yes	characteristic
32	C₂×D₂₄₁	—	2	2	no	—
33	C₂×D₂₄₁	—	4	1	no	—
34	C₂×D₂₄₁	—	4	1	no	—
35	C₄×C₂₄₁	—	2	1	no	—
36	C₄×C₂₄₁	—	2	1	no	—
37	C₄×C₂₄₁	C₂×Q₈	1	1	yes	characteristic
38	C₄×C₂₄₁	#41	1	1	yes	characteristic
39	Dic₂₄₁	—	4	1	no	—
40	Dic₂₄₁	—	4	1	no	—
41	Dic₂₄₁	—	4	1	no	—
42	C₂×C₂₄₁	C₂²⋊Q₈	1	1	yes	characteristic
43	C₂×C₂₄₁	#135	1	1	yes	characteristic
44	C₂×C₂₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
45	D₂₄₁	—	4	2	no	—
46	C₂₄₁	#478	1	1	yes	characteristic, complemented, Sylow
47	#478	—	241	1	no	maximal, Sylow
48	#125	—	241	1	no	—
49	C₄×D₄	—	241	1	no	—
50	#107	—	241	1	no	—
51	#106	—	241	1	no	—
52	C₄⋊C₈	D₂₄₁	1	1	yes	characteristic, complemented
53	#102	—	241	1	no	—
54	#102	—	241	1	no	—
55	C₂×D₄	—	241	1	no	—
56	C₂²×C₄	—	482	1	no	—
57	C₂×C₈	C₂×D₂₄₁	1	1	yes	characteristic
58	C₂×C₈	C₂×D₂₄₁	1	1	yes	characteristic
59	C₄⋊C₄	—	482	1	no	—
60	C₄⋊C₄	—	482	1	no	—
61	C₄⋊C₄	—	241	1	no	—
62	C₄⋊C₄	—	241	1	no	—
63	C₄⋊C₄	—	241	1	no	—
64	C₂²⋊C₄	—	482	1	no	—
65	C₄²	C₂×D₂₄₁	1	1	yes	characteristic
66	C₂³	—	482	1	no	—
67	D₄	—	482	1	no	—
68	D₄	—	241	2	no	—
69	C₂×C₄	—	964	2	no	—
70	C₂×C₄	—	482	1	no	—
71	C₂×C₄	—	482	1	no	—
72	C₂×C₄	—	482	1	no	—
73	C₂×C₄	D₉₆₄	1	1	yes	characteristic
74	C₂×C₄	D₉₆₄	1	1	yes	characteristic
75	C₂×C₄	C₂²×D₂₄₁	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	2	1	no	—
78	C₂²	—	964	1	no	—
79	C₂²	—	964	1	no	—
80	C₂²	—	482	2	no	—
81	C₂²	C₂×D₉₆₄	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	964	1	no	—
85	C₄	—	964	1	no	—
86	C₄	—	964	1	no	—
87	C₄	Q₈×D₂₄₁	1	1	yes	characteristic
88	C₄	#16838769	1	1	yes	characteristic
89	C₂	—	964	2	no	—
90	C₂	#21440622	1	1	yes	characteristic
91	C₂	#21440664	1	1	yes	characteristic
92	C₂	#21440670	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₉₆₄
commutator, cosocle	C₂×C₄×C₂₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₄₁
Fitting	C₄⋊C₈×C₂₄₁	C₂
socle	C₂²×C₂₄₁	C₂×D₄
prime radical	C₂×D₉₆₄	C₂²
hypercenter	C₄⋊C₈	D₂₄₁
hypocenter	C₂₄₁	#478
LCS	G ▷ C₂×C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. #21440634	C₂	semidirect
3. #21440576	C₂	semidirect
4. #21440574	C₂	semidirect
7. C₄⋊C₈×C₂₄₁	C₂	semidirect
46. C₂₄₁	#478	semidirect
52. C₄⋊C₈	D₂₄₁	semidirect

Maximal subgroups

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

subgroup	index
#21440634	2
C₄×D₉₆₄	2
#21440576	2
#21440574	2
#21440564	2
C₄⋊C₈×C₂₄₁	2
#478	241