Subgroups — #29508153

Order 15424.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21440664	C₂	1	1	yes	maximal, characteristic, complemented
2	#21440586	C₂	1	2	yes	maximal, complemented
3	#21440583	C₂	1	2	yes	maximal, complemented
4	#21440582	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#16838770	—	2	1	no	—
7	Dic₉₆₄	C₂²	1	1	yes	characteristic, complemented
8	#16838759	—	2	1	no	—
9	D₁₉₂₈	C₂²	1	1	yes	characteristic, complemented
10	#16838745	C₂²	1	2	yes	complemented
11	C₂×D₄×C₂₄₁	—	2	1	no	—
12	D₈×C₂₄₁	—	2	1	no	—
13	D₈×C₂₄₁	C₂²	1	2	yes	—
14	C₂×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
15	Dic₄₈₂	—	2	1	no	—
16	D₉₆₄	—	2	1	no	—
17	C₂²⋊D₂₄₁	—	4	1	no	—
18	C₄×D₂₄₁	—	4	1	no	—
19	C₂³×C₂₄₁	—	4	1	no	—
20	D₄×C₂₄₁	—	2	2	no	—
21	D₄×C₂₄₁	—	4	1	no	—
22	C₂×C₄×C₂₄₁	D₄	1	1	yes	characteristic
23	C₈×C₂₄₁	D₄	1	1	yes	characteristic
24	C₈×C₂₄₁	C₂³	1	1	yes	characteristic
25	C₂²×C₂₄₁	—	4	2	no	—
26	C₂²×C₂₄₁	—	8	2	no	—
27	C₂²×C₂₄₁	D₈	1	1	yes	characteristic
28	C₂×D₂₄₁	—	4	1	no	—
29	C₄×C₂₄₁	D₈	1	1	yes	characteristic
30	C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
31	Dic₂₄₁	—	4	1	no	—
32	C₂×C₂₄₁	—	2	1	no	—
33	C₂×C₂₄₁	—	8	2	no	—
34	C₂×C₂₄₁	C₂×D₈	1	1	yes	characteristic
35	D₂₄₁	—	8	1	no	—
36	C₂₄₁	#509	1	1	yes	characteristic, complemented, Sylow
37	#509	—	241	1	no	maximal, Sylow
38	#135	—	241	1	no	—
39	C₂×D₈	D₂₄₁	1	1	yes	characteristic, complemented
40	SD₃₂	—	241	2	no	—
41	D₁₆	—	241	2	no	—
42	M₃₂	—	241	1	no	—
43	#41	—	482	1	no	—
44	C₂×D₄	—	2	1	no	—
45	Q₁₆	—	241	1	no	—
46	SD₁₆	—	482	1	no	—
47	D₈	—	2	1	no	—
48	D₈	—	241	1	no	—
49	D₈	C₂×D₂₄₁	1	2	yes	—
50	C₂×C₈	C₂×D₂₄₁	1	1	yes	characteristic
51	C₁₆	—	241	2	no	—
52	C₂³	—	4	1	no	—
53	Q₈	—	482	1	no	—
54	D₄	—	2	2	no	—
55	D₄	—	4	1	no	—
56	D₄	—	964	1	no	—
57	D₄	—	482	1	no	—
58	C₂×C₄	—	964	1	no	—
59	C₂×C₄	C₂²⋊D₂₄₁	1	1	yes	characteristic
60	C₈	C₂²⋊D₂₄₁	1	1	yes	characteristic
61	C₈	C₂²×D₂₄₁	1	1	yes	characteristic
62	C₂²	—	4	2	no	—
63	C₂²	—	8	2	no	—
64	C₂²	—	964	1	no	—
65	C₂²	#16838755	1	1	yes	characteristic
66	C₄	—	964	1	no	—
67	C₄	#16838755	1	1	yes	characteristic
68	C₄	C₂×C₂²⋊D₂₄₁	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	8	2	no	—
71	C₂	—	1928	1	no	—
72	C₂	C₂×#16838755	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16838755
commutator, cosocle	C₈×C₂₄₁	C₂³
Frattini	C₈	C₂²×D₂₄₁
Fitting	C₂×D₈×C₂₄₁	C₂
socle	C₂×C₂₄₁	C₂×D₈
hypercenter	C₂×D₈	D₂₄₁
hypocenter	C₂₄₁	#509
LCS	G ▷ C₈×C₂₄₁ ▷ C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #21440664	C₂	semidirect
2. #21440586	C₂	semidirect
3. #21440583	C₂	semidirect
4. #21440582	C₂	semidirect
5. C₂×D₈×C₂₄₁	C₂	semidirect
7. Dic₉₆₄	C₂²	semidirect
9. D₁₉₂₈	C₂²	semidirect
10. #16838745	C₂²	semidirect
36. C₂₄₁	#509	semidirect
39. C₂×D₈	D₂₄₁	semidirect

Maximal subgroups

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

subgroup	index
#21440664	2
#21440586	2
#21440583	2
#21440582	2
C₂×D₈×C₂₄₁	2
#509	241