Subgroups — C₂×#340 #1734

Order 128.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄	C₂	1	2	yes	maximal
3	#340	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₄	—	2	2	no	—
5	C₂³×C₄	C₂²	1	1	yes	characteristic
6	C₂²×C₈	—	2	4	no	—
7	C₂²×C₈	C₄	1	4	yes	complemented
8	C₂²×C₈	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄	—	2	2	no	—
10	C₂×C₄⋊C₄	—	2	4	no	—
11	C₂×C₄⋊C₄	C₄	1	8	yes	complemented
12	C₂×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂⁴	D₄	1	1	yes	characteristic
14	C₂²×C₄	—	2	8	no	—
15	C₂²×C₄	—	2	4	no	—
16	C₂²×C₄	—	4	4	no	—
17	C₂²×C₄	—	4	4	no	—
18	C₂²×C₄	C₂×C₄	1	2	yes	—
19	C₂²×C₄	C₂×C₄	1	4	yes	—
20	C₂²×C₄	D₄	1	1	yes	characteristic
21	C₂²×C₄	D₄	1	4	yes	—
22	C₂²×C₄	Q₈	1	2	yes	—
23	C₂²×C₄	C₂³	1	1	yes	characteristic
24	C₂×C₈	—	2	8	no	—
25	C₂×C₈	—	2	4	no	—
26	C₂×C₈	C₂×C₄	1	4	yes	complemented
27	C₄⋊C₄	—	2	8	no	—
28	C₄⋊C₄	—	2	4	no	—
29	C₄⋊C₄	C₂×C₄	1	8	yes	complemented
30	C₂³	C₂²⋊C₄	1	4	yes	—
31	C₂³	C₄⋊C₄	1	2	yes	—
32	C₂³	D₈	1	2	yes	—
33	C₂³	SD₁₆	1	4	yes	—
34	C₂³	Q₁₆	1	2	yes	—
35	C₂³	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄	—	2	8	no	—
37	C₂×C₄	—	4	16	no	—
38	C₂×C₄	—	4	4	no	—
39	C₂×C₄	—	4	4	no	—
40	C₂×C₄	C₄²	1	4	yes	—
41	C₂×C₄	C₂²⋊C₄	1	4	yes	—
42	C₂×C₄	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄	C₄⋊C₄	1	2	yes	—
44	C₂×C₄	C₄⋊C₄	1	8	yes	—
45	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄	C₂²×C₄	1	2	yes	—
47	C₂×C₄	C₂×D₄	1	2	yes	—
48	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
49	C₈	—	2	4	no	—
50	C₂²	#95	1	4	yes	—
51	C₂²	#102	1	8	yes	—
52	C₂²	#103	1	8	yes	—
53	C₂²	#106	1	4	yes	—
54	C₂²	#107	1	4	yes	—
55	C₂²	C₂×C₂²⋊C₄	1	2	yes	—
56	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²	C₂×D₈	1	1	yes	characteristic
58	C₂²	C₂×SD₁₆	1	2	yes	—
59	C₂²	C₂×Q₁₆	1	1	yes	characteristic
60	C₄	—	4	8	no	—
61	C₄	#95	1	4	yes	—
62	C₄	C₂×C₄²	1	1	yes	characteristic
63	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₄	C₂×C₄⋊C₄	1	2	yes	—
65	C₂	#340	1	8	yes	direct product factor
66	C₂	C₂×#95	1	1	yes	characteristic
67	C₂	C₂×#102	1	2	yes	—
68	C₂	C₂×#103	1	2	yes	—
69	C₂	C₂×#106	1	1	yes	characteristic
70	C₂	C₂×#107	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴	D₄
commutator	C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄	C₂³
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #340	C₂	direct (N × H)
7. C₂²×C₈	C₄	semidirect
11. C₂×C₄⋊C₄	C₄	semidirect
26. C₂×C₈	C₂×C₄	semidirect
29. C₄⋊C₄	C₂×C₄	semidirect
65. C₂	#340	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈	2
C₂²×C₄⋊C₄	2
#340	2

Subgroups — C2×#340 #1734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#340 #1734