Subgroups — C₂×#7545 #153571

Order 512.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#1423	C₂	1	2	yes	maximal
3	#7545	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₃₂	C₂²	1	1	yes	characteristic
5	C₂×#366	—	2	2	no	—
6	C₂×C₆₄	C₄	1	4	yes	complemented
7	C₂×C₆₄	C₂²	1	2	yes	—
8	#1423	—	2	2	no	—
9	#1423	C₂²	1	4	yes	—
10	C₂²×C₁₆	D₄	1	1	yes	characteristic
11	C₂×#107	—	4	2	no	—
12	C₂×C₃₂	C₂×C₄	1	2	yes	—
13	C₂×C₃₂	D₄	1	1	yes	characteristic
14	C₂×C₃₂	Q₈	1	2	yes	—
15	C₂×C₃₂	C₂³	1	1	yes	characteristic
16	#366	—	2	4	no	—
17	#366	—	4	2	no	—
18	C₆₄	C₂×C₄	1	4	yes	complemented
19	C₂²×C₈	D₈	1	1	yes	characteristic
20	C₂×C₄⋊C₄	—	8	2	no	—
21	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
22	C₂×C₁₆	D₈	1	1	yes	characteristic
23	C₂×C₁₆	Q₁₆	1	2	yes	—
24	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
25	#107	—	4	4	no	—
26	#107	—	8	2	no	—
27	C₃₂	C₄⋊C₄	1	2	yes	—
28	C₃₂	C₂²×C₄	1	1	yes	characteristic
29	C₃₂	C₂×Q₈	1	1	yes	characteristic
30	C₂²×C₄	—	16	2	no	—
31	C₂²×C₄	D₁₆	1	1	yes	characteristic
32	C₂×C₈	#107	1	2	yes	—
33	C₂×C₈	D₁₆	1	1	yes	characteristic
34	C₂×C₈	Q₃₂	1	2	yes	—
35	C₂×C₈	C₂×D₈	1	1	yes	characteristic
36	C₄⋊C₄	—	8	4	no	—
37	C₄⋊C₄	—	16	2	no	—
38	C₁₆	#107	1	2	yes	—
39	C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₁₆	C₂×Q₁₆	1	1	yes	characteristic
41	C₂³	D₃₂	1	1	yes	characteristic
42	C₂×C₄	—	16	4	no	—
43	C₂×C₄	—	32	4	no	—
44	C₂×C₄	#366	1	2	yes	—
45	C₂×C₄	D₃₂	1	1	yes	characteristic
46	C₂×C₄	Q₆₄	1	2	yes	—
47	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
48	C₈	#366	1	2	yes	—
49	C₈	C₂×#107	1	1	yes	characteristic
50	C₈	C₂×Q₃₂	1	1	yes	characteristic
51	C₂²	#1423	1	2	yes	—
52	C₂²	D₆₄	1	2	yes	—
53	C₂²	Q₁₂₈	1	2	yes	—
54	C₂²	C₂×D₃₂	1	1	yes	characteristic
55	C₄	—	32	4	no	—
56	C₄	#1423	1	2	yes	—
57	C₄	C₂×#366	1	1	yes	characteristic
58	C₄	C₂×Q₆₄	1	1	yes	characteristic
59	C₂	#7545	1	4	yes	direct product factor
60	C₂	C₂×#1423	1	1	yes	characteristic
61	C₂	C₂×D₆₄	1	1	yes	characteristic
62	C₂	C₂×Q₁₂₈	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	D₃₂
commutator	C₃₂	C₂²×C₄
Frattini, cosocle	C₂×C₃₂	C₂³
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #7545	C₂	direct (N × H)
6. C₂×C₆₄	C₄	semidirect
18. C₆₄	C₂×C₄	semidirect
59. C₂	#7545	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₆₄	2
C₂×#1423	2
#7545	2

Subgroups — C2×#7545 #153571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#7545 #153571