Subgroups — C₂×#137 #574

Order 64.

129 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41	C₂	1	1	yes	maximal, characteristic
2	C₂²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#137	C₂	1	8	yes	maximal, direct product factor
4	C₂×Q₁₆	C₂	1	2	yes	maximal, complemented
5	C₂×SD₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#41	—	2	2	no	—
8	#41	C₂²	1	4	yes	—
9	C₂×Q₈	—	2	2	no	—
10	C₂×Q₈	—	2	1	no	—
11	C₂×Q₈	C₂²	1	2	yes	—
12	C₂×Q₈	C₂²	1	1	yes	characteristic
13	C₂×Q₈	C₂²	1	4	yes	complemented
14	C₂×D₄	—	2	1	no	—
15	C₂×D₄	C₂²	1	1	yes	characteristic
16	C₂²×C₄	—	2	1	no	—
17	C₂²×C₄	—	2	1	no	—
18	C₂²×C₄	C₂²	1	1	yes	characteristic
19	Q₁₆	C₂²	1	8	yes	complemented
20	SD₁₆	C₂²	1	8	yes	complemented
21	M₁₆	C₂²	1	4	yes	complemented
22	C₂×C₈	C₂²	1	2	yes	—
23	C₂³	—	2	1	no	—
24	C₂³	D₄	1	1	yes	characteristic
25	Q₈	—	2	2	no	—
26	Q₈	—	2	4	no	—
27	Q₈	—	2	1	no	—
28	Q₈	C₂³	1	4	yes	—
29	Q₈	C₂³	1	2	yes	—
30	D₄	—	2	4	no	—
31	D₄	—	2	1	no	—
32	D₄	C₂³	1	2	yes	—
33	C₂×C₄	—	2	4	no	—
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	—	2	2	no	—
36	C₂×C₄	—	2	4	no	—
37	C₂×C₄	D₄	1	2	yes	—
38	C₂×C₄	D₄	1	1	yes	characteristic
39	C₂×C₄	C₂³	1	2	yes	—
40	C₂×C₄	C₂³	1	1	yes	characteristic
41	C₈	C₂³	1	4	yes	—
42	C₂²	—	2	2	no	—
43	C₂²	—	2	2	no	—
44	C₂²	—	4	2	no	—
45	C₂²	C₂×D₄	1	2	yes	—
46	C₂²	C₂×D₄	1	1	yes	characteristic
47	C₄	—	2	2	no	—
48	C₄	—	2	4	no	—
49	C₄	C₂×D₄	1	2	yes	—
50	C₄	C₂×D₄	1	1	yes	characteristic
51	C₄	C₂⁴	1	1	yes	characteristic
52	C₂	—	2	2	no	—
53	C₂	—	4	2	no	—
54	C₂	#137	1	2	yes	direct product factor
55	C₂	C₂²×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁴
prime radical	C₂×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²×Q₈	C₂	semidirect
3. #137	C₂	direct (N × H)
4. C₂×Q₁₆	C₂	semidirect
5. C₂×SD₁₆	C₂	semidirect
6. C₂×M₁₆	C₂	semidirect
13. C₂×Q₈	C₂²	semidirect
19. Q₁₆	C₂²	semidirect
20. SD₁₆	C₂²	semidirect
21. M₁₆	C₂²	semidirect
54. C₂	#137	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×#41	2
C₂²×Q₈	2
#137	2
C₂×Q₁₆	2
C₂×SD₁₆	2
C₂×M₁₆	2

Subgroups — C2×#137 #574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#137 #574