Subgroups — C₂×#578 #3583

Order 128.

704 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×2₋¹⁺⁴	C₂	1	2	yes	maximal, complemented
2	#578	C₂	1	16	yes	maximal, direct product factor
3	C₂×#137	C₂	1	6	yes	maximal, complemented
4	C₂×#135	C₂	1	3	yes	maximal, complemented
5	C₂²×Q₁₆	C₂	1	3	yes	maximal, complemented
6	C₂×#131	C₂	1	1	yes	maximal, characteristic, complemented
7	2₋¹⁺⁴	—	2	8	no	—
8	2₋¹⁺⁴	C₂²	1	16	yes	complemented
9	C₂×#41	—	2	6	no	—
10	C₂×#41	C₂²	1	6	yes	—
11	C₂×#41	C₂²	1	1	yes	characteristic
12	C₂²×Q₈	—	2	2	no	—
13	C₂²×Q₈	C₂²	1	6	yes	—
14	#137	C₂²	1	48	yes	complemented
15	#135	C₂²	1	24	yes	complemented
16	C₂×Q₁₆	C₂²	1	3	yes	—
17	C₂×Q₁₆	C₂²	1	6	yes	—
18	C₂×Q₁₆	C₂²	1	24	yes	complemented
19	C₂×SD₁₆	C₂²	1	6	yes	—
20	C₂×D₈	C₂²	1	1	yes	characteristic
21	#131	C₂²	1	8	yes	complemented
22	C₂×M₁₆	C₂²	1	3	yes	—
23	C₂²×C₈	C₂²	1	3	yes	—
24	#41	—	2	48	no	—
25	#41	—	2	12	no	—
26	#41	D₄	1	4	yes	—
27	#41	C₂³	1	24	yes	—
28	#41	C₂³	1	4	yes	—
29	C₂×Q₈	—	2	16	no	—
30	C₂×Q₈	—	2	6	no	—
31	C₂×Q₈	—	2	12	no	—
32	C₂×Q₈	D₄	1	1	yes	characteristic
33	C₂×Q₈	C₂³	1	6	yes	—
34	C₂×Q₈	C₂³	1	24	yes	—
35	C₂×D₄	—	2	6	no	—
36	C₂×D₄	D₄	1	3	yes	—
37	C₂×D₄	C₂³	1	2	yes	—
38	C₂²×C₄	—	2	6	no	—
39	C₂²×C₄	—	2	6	no	—
40	C₂²×C₄	C₂³	1	3	yes	—
41	Q₁₆	C₂³	1	12	yes	—
42	Q₁₆	C₂³	1	24	yes	—
43	SD₁₆	C₂³	1	24	yes	—
44	D₈	C₂³	1	4	yes	—
45	M₁₆	C₂³	1	12	yes	—
46	C₂×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₈	C₂³	1	3	yes	—
48	C₂×C₈	C₂³	1	12	yes	—
49	C₂³	—	2	2	no	—
50	C₂³	C₂×D₄	1	3	yes	—
51	Q₈	—	2	24	no	—
52	Q₈	—	2	6	no	—
53	Q₈	C₂×D₄	1	4	yes	—
54	Q₈	C₂⁴	1	12	yes	—
55	D₄	—	2	24	no	—
56	D₄	—	2	2	no	—
57	D₄	C₂×D₄	1	12	yes	—
58	D₄	C₂⁴	1	4	yes	—
59	C₂×C₄	—	2	24	no	—
60	C₂×C₄	—	2	6	no	—
61	C₂×C₄	—	2	24	no	—
62	C₂×C₄	C₂×D₄	1	6	yes	—
63	C₂×C₄	C₂×D₄	1	3	yes	—
64	C₂×C₄	C₂⁴	1	6	yes	—
65	C₂×C₄	C₂⁴	1	1	yes	characteristic
66	C₈	C₂⁴	1	2	yes	—
67	C₈	C₂⁴	1	6	yes	—
68	C₂²	—	2	6	no	—
69	C₂²	—	2	4	no	—
70	C₂²	—	4	4	no	—
71	C₂²	C₂²×D₄	1	6	yes	—
72	C₂²	C₂²×D₄	1	1	yes	characteristic
73	C₄	—	2	12	no	—
74	C₄	C₂²×D₄	1	6	yes	—
75	C₄	C₂²×D₄	1	1	yes	characteristic
76	C₄	C₂⁵	1	1	yes	characteristic
77	C₂	—	2	6	no	—
78	C₂	—	4	4	no	—
79	C₂	#578	1	2	yes	direct product factor
80	C₂	C₂³×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁵
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×2₋¹⁺⁴	C₂	semidirect
2. #578	C₂	direct (N × H)
3. C₂×#137	C₂	semidirect
4. C₂×#135	C₂	semidirect
5. C₂²×Q₁₆	C₂	semidirect
6. C₂×#131	C₂	semidirect
8. 2₋¹⁺⁴	C₂²	semidirect
14. #137	C₂²	semidirect
15. #135	C₂²	semidirect
18. C₂×Q₁₆	C₂²	semidirect
21. #131	C₂²	semidirect
79. C₂	#578	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×2₋¹⁺⁴	2
#578	2
C₂×#137	2
C₂×#135	2
C₂²×Q₁₆	2
C₂×#131	2

Subgroups — C2×#578 #3583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#578 #3583