Subgroups — C₂×#562 #3527

Order 128.

576 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#562	C₂	1	16	yes	maximal, direct product factor
2	C₂×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#124	C₂	1	2	yes	maximal, complemented
4	C₂×#123	C₂	1	4	yes	maximal, complemented
5	C₂×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
6	C₂×#121	C₂	1	4	yes	maximal, complemented
7	C₂²×Q₈	C₂²	1	2	yes	—
8	C₂²×D₄	C₂²	1	4	yes	—
9	C₂²×D₄	C₂²	1	2	yes	—
10	C₂³×C₄	C₂²	1	4	yes	—
11	#125	C₂²	1	8	yes	complemented
12	#124	C₂²	1	16	yes	complemented
13	#123	C₂²	1	32	yes	complemented
14	C₂²⋊Q₈	C₂²	1	32	yes	complemented
15	#121	C₂²	1	32	yes	complemented
16	C₂×C₄⋊C₄	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄	C₂²	1	4	yes	—
18	C₂×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₄	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₄	C₂²	1	4	yes	—
22	C₂×C₄²	C₂²	1	1	yes	characteristic
23	C₂⁴	C₂³	1	4	yes	—
24	C₂×Q₈	—	4	4	no	—
25	C₂×Q₈	C₂³	1	8	yes	—
26	C₂×D₄	—	4	8	no	—
27	C₂×D₄	—	4	4	no	—
28	C₂×D₄	C₂³	1	16	yes	—
29	C₂×D₄	C₂³	1	8	yes	—
30	C₂²×C₄	—	4	8	no	—
31	C₂²×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₄	C₂³	1	2	yes	—
33	C₂²×C₄	C₂³	1	4	yes	—
34	C₂²×C₄	C₂³	1	4	yes	—
35	C₂²×C₄	C₂³	1	16	yes	—
36	C₄⋊C₄	C₂³	1	16	yes	—
37	C₄⋊C₄	C₂³	1	16	yes	—
38	C₄⋊C₄	C₂³	1	8	yes	—
39	C₂²⋊C₄	C₂³	1	16	yes	—
40	C₂²⋊C₄	C₂³	1	16	yes	—
41	C₂²⋊C₄	C₂³	1	16	yes	—
42	C₄²	C₂³	1	4	yes	—
43	C₂³	—	2	8	no	—
44	C₂³	—	2	8	no	—
45	C₂³	—	2	8	no	—
46	C₂³	C₂⁴	1	8	yes	—
47	C₂³	C₂⁴	1	1	yes	characteristic
48	Q₈	—	4	8	no	—
49	D₄	—	4	16	no	—
50	D₄	—	4	8	no	—
51	C₂×C₄	—	2	8	no	—
52	C₂×C₄	—	2	8	no	—
53	C₂×C₄	—	2	4	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	4	16	no	—
56	C₂×C₄	C₂⁴	1	2	yes	—
57	C₂×C₄	C₂⁴	1	4	yes	—
58	C₂×C₄	C₂⁴	1	8	yes	—
59	C₂×C₄	C₂⁴	1	8	yes	—
60	C₂²	—	2	8	no	—
61	C₂²	—	2	8	no	—
62	C₂²	—	2	8	no	—
63	C₂²	—	4	16	no	—
64	C₂²	H(5,𝔽₂)	1	4	yes	—
65	C₂²	2₋¹⁺⁴	1	2	yes	—
66	C₂²	C₂⁵	1	1	yes	characteristic
67	C₄	—	2	8	no	—
68	C₄	—	2	8	no	—
69	C₄	—	2	4	no	—
70	C₄	—	2	2	no	—
71	C₂	—	4	8	no	—
72	C₂	#562	1	4	yes	direct product factor
73	C₂	C₂×H(5,𝔽₂)	1	2	yes	—
74	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵
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. #562	C₂	direct (N × H)
2. C₂×#125	C₂	semidirect
3. C₂×#124	C₂	semidirect
4. C₂×#123	C₂	semidirect
5. C₂×C₂²⋊Q₈	C₂	semidirect
6. C₂×#121	C₂	semidirect
11. #125	C₂²	semidirect
12. #124	C₂²	semidirect
13. #123	C₂²	semidirect
14. C₂²⋊Q₈	C₂²	semidirect
15. #121	C₂²	semidirect
72. C₂	#562	direct (N × H)

Maximal subgroups

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

subgroup	index
#562	2
C₂×#125	2
C₂×#124	2
C₂×#123	2
C₂×C₂²⋊Q₈	2
C₂×#121	2

Subgroups — C2×#562 #3527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#562 #3527