Subgroups — C₂×#544 #3457

Order 128.

540 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#544	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₄⋊Q₈	C₂	1	4	yes	maximal, complemented
3	C₂×#125	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊Q₈	C₂	1	4	yes	maximal
5	C₂×#117	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈	C₂²	1	4	yes	—
8	C₂³×C₄	C₂²	1	2	yes	—
9	C₂³×C₄	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈	C₂²	1	32	yes	complemented
11	#125	C₂²	1	32	yes	complemented
12	C₂²⋊Q₈	C₂²	1	32	yes	—
13	#117	C₂²	1	16	yes	—
14	C₂×C₄⋊C₄	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄	C₂²	1	8	yes	—
16	C₂×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄	C₂²	1	4	yes	—
19	C₂×C₄⋊C₄	C₂²	1	8	yes	—
20	C₂×C₂²⋊C₄	C₂²	1	4	yes	—
21	C₂×C₄²	C₂²	1	4	yes	—
22	C₂⁴	C₂³	1	1	yes	characteristic
23	C₂×Q₈	—	4	8	no	—
24	C₂×Q₈	C₂³	1	16	yes	—
25	C₂²×C₄	—	4	4	no	—
26	C₂²×C₄	Q₈	1	8	yes	complemented
27	C₂²×C₄	C₂³	1	4	yes	—
28	C₂²×C₄	C₂³	1	4	yes	—
29	C₂²×C₄	C₂³	1	4	yes	—
30	C₂²×C₄	C₂³	1	8	yes	—
31	C₂²×C₄	C₂³	1	2	yes	—
32	C₂²×C₄	C₂³	1	4	yes	—
33	C₄⋊C₄	C₂³	1	16	yes	—
34	C₄⋊C₄	C₂³	1	8	yes	—
35	C₄⋊C₄	C₂³	1	8	yes	—
36	C₄⋊C₄	C₂³	1	16	yes	—
37	C₄⋊C₄	C₂³	1	32	yes	—
38	C₂²⋊C₄	C₂³	1	16	yes	—
39	C₄²	C₂³	1	16	yes	—
40	C₂³	—	2	2	no	—
41	C₂³	—	2	2	no	—
42	C₂³	C₂×Q₈	1	4	yes	—
43	C₂³	C₂⁴	1	2	yes	—
44	C₂³	C₂⁴	1	1	yes	characteristic
45	Q₈	—	4	16	no	—
46	C₂×C₄	—	2	8	no	—
47	C₂×C₄	—	2	8	no	—
48	C₂×C₄	—	2	8	no	—
49	C₂×C₄	—	4	8	no	—
50	C₂×C₄	C₂×Q₈	1	8	yes	—
51	C₂×C₄	C₂×Q₈	1	16	yes	complemented
52	C₂×C₄	C₂⁴	1	8	yes	—
53	C₂×C₄	C₂⁴	1	8	yes	—
54	C₂×C₄	C₂⁴	1	8	yes	—
55	C₂×C₄	C₂⁴	1	4	yes	—
56	C₂²	—	2	8	no	—
57	C₂²	—	2	2	no	—
58	C₂²	—	2	2	no	—
59	C₂²	C₂²×Q₈	1	4	yes	—
60	C₂²	C₂²×Q₈	1	2	yes	—
61	C₂²	H(5,𝔽₂)	1	2	yes	—
62	C₂²	2₋¹⁺⁴	1	2	yes	—
63	C₂²	C₂⁵	1	1	yes	characteristic
64	C₄	—	2	8	no	—
65	C₄	—	2	8	no	—
66	C₄	—	2	8	no	—
67	C₄	C₂²×Q₈	1	8	yes	—
68	C₂	—	2	4	no	—
69	C₂	#544	1	4	yes	direct product factor
70	C₂	C₂³×Q₈	1	1	yes	characteristic
71	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
72	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵
prime radical	C₂⁴	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #544	C₂	direct (N × H)
2. C₂×C₄⋊Q₈	C₂	semidirect
3. C₂×#125	C₂	semidirect
10. C₄⋊Q₈	C₂²	semidirect
11. #125	C₂²	semidirect
26. C₂²×C₄	Q₈	semidirect
51. C₂×C₄	C₂×Q₈	semidirect
69. C₂	#544	direct (N × H)

Maximal subgroups

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

subgroup	index
#544	2
C₂×C₄⋊Q₈	2
C₂×#125	2
C₂×C₂²⋊Q₈	2
C₂×#117	2
C₂²×C₄⋊C₄	2

Subgroups — C2×#544 #3457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#544 #3457