Subgroups — C₂×#13737 #513307

Order 512.

286 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2258	C₂	1	1	yes	maximal, characteristic
2	#13737	C₂	1	8	yes	maximal, direct product factor
3	C₂×M₁₂₈	C₂	1	3	yes	maximal, complemented
4	C₂²×C₆₄	C₂	1	3	yes	maximal, complemented
5	C₂×#504	C₄	1	1	yes	characteristic
6	#2258	C₄	1	4	yes	—
7	#2258	C₂²	1	4	yes	—
8	C₂×M₆₄	C₄	1	3	yes	—
9	C₂²×C₃₂	C₂²	1	3	yes	—
10	M₁₂₈	C₂²	1	12	yes	complemented
11	C₂×C₆₄	C₂²	1	3	yes	—
12	C₂×C₆₄	C₂²	1	1	yes	characteristic
13	C₂×C₆₄	C₂²	1	12	yes	complemented
14	C₂×#131	C₈	1	1	yes	characteristic
15	#504	C₈	1	4	yes	—
16	#504	C₂×C₄	1	4	yes	—
17	C₂×M₃₂	C₈	1	3	yes	—
18	C₂²×C₁₆	C₂×C₄	1	3	yes	—
19	M₆₄	C₂×C₄	1	12	yes	—
20	C₂×C₃₂	C₂×C₄	1	3	yes	—
21	C₂×C₃₂	C₂×C₄	1	6	yes	—
22	C₂×C₃₂	C₂³	1	1	yes	characteristic
23	C₂×C₃₂	C₂³	1	6	yes	—
24	C₆₄	C₂³	1	6	yes	—
25	C₆₄	C₂³	1	2	yes	—
26	C₂×#41	C₁₆	1	1	yes	characteristic
27	#131	C₁₆	1	4	yes	—
28	#131	C₂×C₈	1	4	yes	—
29	C₂×M₁₆	C₁₆	1	3	yes	—
30	C₂²×C₈	C₂×C₈	1	3	yes	—
31	M₃₂	C₂×C₈	1	12	yes	—
32	C₂×C₁₆	C₂×C₈	1	3	yes	—
33	C₂×C₁₆	C₂×C₈	1	6	yes	—
34	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₁₆	C₂²×C₄	1	6	yes	—
36	C₃₂	C₂²×C₄	1	1	yes	characteristic
37	C₃₂	C₂²×C₄	1	6	yes	—
38	C₃₂	C₂⁴	1	1	yes	characteristic
39	#41	C₃₂	1	4	yes	—
40	#41	C₂×C₁₆	1	4	yes	—
41	C₂×Q₈	C₃₂	1	1	yes	characteristic
42	C₂×D₄	C₃₂	1	3	yes	—
43	C₂²×C₄	C₂×C₁₆	1	3	yes	—
44	M₁₆	C₂×C₁₆	1	12	yes	—
45	C₂×C₈	C₂×C₁₆	1	3	yes	—
46	C₂×C₈	C₂×C₁₆	1	6	yes	—
47	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₈	C₂²×C₈	1	6	yes	—
49	C₁₆	C₂²×C₈	1	1	yes	characteristic
50	C₁₆	C₂²×C₈	1	6	yes	—
51	C₁₆	C₂³×C₄	1	1	yes	characteristic
52	C₂³	C₂×C₃₂	1	3	yes	—
53	Q₈	C₂×C₃₂	1	4	yes	—
54	D₄	C₂×C₃₂	1	12	yes	—
55	C₂×C₄	C₂×C₃₂	1	3	yes	—
56	C₂×C₄	C₂×C₃₂	1	6	yes	—
57	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
58	C₂×C₄	C₂²×C₁₆	1	6	yes	—
59	C₈	C₂²×C₁₆	1	1	yes	characteristic
60	C₈	C₂²×C₁₆	1	6	yes	—
61	C₈	C₂³×C₈	1	1	yes	characteristic
62	C₂²	—	2	6	no	—
63	C₂²	C₂²×C₃₂	1	6	yes	—
64	C₂²	C₂²×C₃₂	1	1	yes	characteristic
65	C₄	C₂²×C₃₂	1	1	yes	characteristic
66	C₄	C₂²×C₃₂	1	6	yes	—
67	C₄	C₂³×C₁₆	1	1	yes	characteristic
68	C₂	—	2	6	no	—
69	C₂	#13737	1	2	yes	direct product factor
70	C₂	C₂³×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₆₄	C₂²
commutator	C₂	C₂³×C₃₂
Frattini, cosocle	C₃₂	C₂⁴
socle	C₂²	C₂²×C₃₂
prime radical	C₂×#41	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₄ ◁ G
LECS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #13737	C₂	direct (N × H)
3. C₂×M₁₂₈	C₂	semidirect
4. C₂²×C₆₄	C₂	semidirect
10. M₁₂₈	C₂²	semidirect
13. C₂×C₆₄	C₂²	semidirect
69. C₂	#13737	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#2258	2
#13737	2
C₂×M₁₂₈	2
C₂²×C₆₄	2

Subgroups — C2×#13737 #513307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13737 #513307