Subgroups — C₂×#563 #3528

Order 128.

512 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#563	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
3	C₂×#126	C₂	1	4	yes	maximal, complemented
4	C₂×#125	C₂	1	2	yes	maximal, complemented
5	C₂×#124	C₂	1	1	yes	maximal, characteristic
6	C₂×#123	C₂	1	2	yes	maximal
7	C₂×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
8	C₂²×Q₈	C₂²	1	2	yes	—
9	C₂²×Q₈	C₂²	1	1	yes	characteristic
10	C₂²×D₄	C₂²	1	1	yes	characteristic
11	C₂³×C₄	C₂²	1	2	yes	—
12	C₄⋊Q₈	C₂²	1	16	yes	complemented
13	#126	C₂²	1	32	yes	complemented
14	#125	C₂²	1	16	yes	complemented
15	#124	C₂²	1	8	yes	—
16	#123	C₂²	1	16	yes	—
17	C₂²⋊Q₈	C₂²	1	32	yes	complemented
18	C₂×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₄	C₂²	1	4	yes	—
25	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₄²	C₂²	1	1	yes	characteristic
27	C₂×C₄²	C₂²	1	2	yes	—
28	C₂⁴	C₂³	1	2	yes	—
29	C₂×Q₈	—	4	4	no	—
30	C₂×Q₈	—	4	2	no	—
31	C₂×Q₈	C₂³	1	8	yes	—
32	C₂×Q₈	C₂³	1	4	yes	—
33	C₂×D₄	—	4	2	no	—
34	C₂×D₄	C₂³	1	4	yes	—
35	C₂²×C₄	—	4	4	no	—
36	C₂²×C₄	C₂³	1	2	yes	—
37	C₂²×C₄	C₂³	1	4	yes	—
38	C₂²×C₄	C₂³	1	2	yes	—
39	C₂²×C₄	C₂³	1	1	yes	characteristic
40	C₂²×C₄	C₂³	1	4	yes	—
41	C₂²×C₄	C₂³	1	8	yes	—
42	C₄⋊C₄	C₂³	1	8	yes	—
43	C₄⋊C₄	C₂³	1	16	yes	—
44	C₄⋊C₄	C₂³	1	16	yes	—
45	C₄⋊C₄	C₂³	1	16	yes	—
46	C₄⋊C₄	C₂³	1	8	yes	—
47	C₂²⋊C₄	C₂³	1	16	yes	—
48	C₂²⋊C₄	C₂³	1	16	yes	—
49	C₂²⋊C₄	C₂³	1	8	yes	—
50	C₄²	C₂³	1	4	yes	—
51	C₄²	C₂³	1	8	yes	—
52	C₂³	—	2	4	no	—
53	C₂³	—	2	8	no	—
54	C₂³	C₂⁴	1	4	yes	—
55	C₂³	C₂⁴	1	1	yes	characteristic
56	Q₈	—	4	8	no	—
57	Q₈	—	4	4	no	—
58	D₄	—	4	4	no	—
59	C₂×C₄	—	2	8	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	—	2	4	no	—
63	C₂×C₄	—	2	8	no	—
64	C₂×C₄	—	4	8	no	—
65	C₂×C₄	C₂⁴	1	4	yes	—
66	C₂×C₄	C₂⁴	1	8	yes	—
67	C₂×C₄	C₂⁴	1	4	yes	—
68	C₂×C₄	C₂⁴	1	2	yes	—
69	C₂×C₄	C₂⁴	1	8	yes	—
70	C₂²	—	2	4	no	—
71	C₂²	—	2	8	no	—
72	C₂²	—	4	8	no	—
73	C₂²	H(5,𝔽₂)	1	2	yes	—
74	C₂²	2₋¹⁺⁴	1	4	yes	—
75	C₂²	C₂⁵	1	1	yes	characteristic
76	C₄	—	2	8	no	—
77	C₄	—	2	2	no	—
78	C₄	—	2	4	no	—
79	C₄	—	2	4	no	—
80	C₄	—	2	8	no	—
81	C₂	—	4	4	no	—
82	C₂	#563	1	4	yes	direct product factor
83	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
84	C₂	C₂×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵
prime radical	C₂²×D₄	C₂²
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. #563	C₂	direct (N × H)
2. C₂×C₄⋊Q₈	C₂	semidirect
3. C₂×#126	C₂	semidirect
4. C₂×#125	C₂	semidirect
7. C₂×C₂²⋊Q₈	C₂	semidirect
12. C₄⋊Q₈	C₂²	semidirect
13. #126	C₂²	semidirect
14. #125	C₂²	semidirect
17. C₂²⋊Q₈	C₂²	semidirect
82. C₂	#563	direct (N × H)

Maximal subgroups

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

subgroup	index
#563	2
C₂×C₄⋊Q₈	2
C₂×#126	2
C₂×#125	2
C₂×#124	2
C₂×#123	2
C₂×C₂²⋊Q₈	2

Subgroups — C2×#563 #3528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#563 #3528