Subgroups — C₂×#1726 #20327

Order 256.

807 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#97	C₂	1	3	yes	maximal
2	C₂²×C₄×C₈	C₂	1	3	yes	maximal
3	C₂×#376	C₂	1	1	yes	maximal, characteristic
4	#1726	C₂	1	8	yes	maximal, direct product factor
5	C₂³×C₈	C₂²	1	3	yes	—
6	C₂³×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄²	C₂²	1	3	yes	—
8	C₂×#97	C₄	1	24	yes	complemented
9	C₂×#97	C₂²	1	12	yes	—
10	C₂×C₄×C₈	C₄	1	24	yes	complemented
11	C₂×C₄×C₈	C₂²	1	12	yes	—
12	#376	C₄	1	4	yes	—
13	#376	C₂²	1	4	yes	—
14	C₂×#95	C₄	1	4	yes	—
15	C₂³×C₄	C₂×C₄	1	6	yes	—
16	C₂³×C₄	C₂³	1	1	yes	characteristic
17	C₂²×C₈	C₂×C₄	1	12	yes	—
18	C₂²×C₈	C₂×C₄	1	24	yes	—
19	C₂²×C₈	C₂×C₄	1	12	yes	—
20	C₂²×C₈	C₂³	1	6	yes	—
21	C₂²×C₈	C₂³	1	2	yes	—
22	C₂×C₄²	C₂×C₄	1	6	yes	—
23	C₂×C₄²	C₂×C₄	1	12	yes	—
24	C₂×C₄²	C₂×C₄	1	12	yes	—
25	C₂×C₄²	C₂³	1	6	yes	—
26	#97	C₂×C₄	1	48	yes	complemented
27	C₄×C₈	C₂×C₄	1	48	yes	complemented
28	#95	C₂×C₄	1	16	yes	—
29	C₂⁴	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄	—	2	24	no	—
31	C₂²×C₄	C₄²	1	24	yes	—
32	C₂²×C₄	C₂²×C₄	1	12	yes	—
33	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄	C₂²×C₄	1	6	yes	—
35	C₂²×C₄	C₂²×C₄	1	6	yes	—
36	C₂²×C₄	C₂⁴	1	1	yes	characteristic
37	C₂×C₈	C₄²	1	48	yes	—
38	C₂×C₈	C₄²	1	16	yes	—
39	C₂×C₈	C₂²×C₄	1	12	yes	—
40	C₂×C₈	C₂²×C₄	1	12	yes	—
41	C₂×C₈	C₂²×C₄	1	24	yes	—
42	C₄²	C₄²	1	24	yes	—
43	C₄²	C₂²×C₄	1	12	yes	—
44	C₄²	C₂²×C₄	1	12	yes	—
45	C₂³	C₂×C₄²	1	6	yes	—
46	C₂³	#131	1	8	yes	—
47	C₂³	C₂³×C₄	1	1	yes	characteristic
48	C₂×C₄	—	2	24	no	—
49	C₂×C₄	—	2	48	no	—
50	C₂×C₄	C₂×C₄²	1	24	yes	—
51	C₂×C₄	C₂×C₄²	1	6	yes	—
52	C₂×C₄	C₂×C₄²	1	12	yes	—
53	C₂×C₄	C₂×C₄²	1	4	yes	—
54	C₂×C₄	C₂³×C₄	1	3	yes	—
55	C₂×C₄	C₂³×C₄	1	3	yes	—
56	C₈	C₂×C₄²	1	8	yes	—
57	C₈	C₂×C₄²	1	24	yes	—
58	C₂²	C₄³	1	4	yes	—
59	C₂²	#405	1	24	yes	—
60	C₂²	C₂²×C₄²	1	3	yes	—
61	C₂²	C₂×#131	1	4	yes	—
62	C₄	—	2	24	no	—
63	C₄	C₄³	1	4	yes	—
64	C₄	C₂²×C₄²	1	1	yes	characteristic
65	C₄	C₂²×C₄²	1	3	yes	—
66	C₂	#1726	1	8	yes	direct product factor
67	C₂	C₂×C₄³	1	1	yes	characteristic
68	C₂	C₂×#405	1	6	yes	—

	subgroup	G/H
center	C₂³×C₈	C₂²
commutator	C₂	C₂×C₄³
Frattini, cosocle	C₂²×C₄	C₂⁴
socle, prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1726	C₂	direct (N × H)
8. C₂×#97	C₄	semidirect
10. C₂×C₄×C₈	C₄	semidirect
26. #97	C₂×C₄	semidirect
27. C₄×C₈	C₂×C₄	semidirect
66. C₂	#1726	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×#97	2
C₂²×C₄×C₈	2
C₂×#376	2
#1726	2

Subgroups — C2×#1726 #20327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1726 #20327