Subgroups — C₂×#510 #3413

Order 128.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135	C₂	1	1	yes	maximal, characteristic
2	C₂²×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#510	C₂	1	8	yes	maximal, direct product factor
4	C₂×Q₃₂	C₂	1	2	yes	maximal, complemented
5	C₂×SD₃₂	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41	—	2	1	no	—
8	C₂²×Q₈	—	2	1	no	—
9	#135	—	2	2	no	—
10	#135	C₂²	1	4	yes	—
11	C₂×Q₁₆	—	2	1	no	—
12	C₂×Q₁₆	—	2	2	no	—
13	C₂×Q₁₆	C₂²	1	2	yes	—
14	C₂×Q₁₆	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆	C₂²	1	4	yes	complemented
16	C₂×SD₁₆	—	2	1	no	—
17	C₂×D₈	C₂²	1	1	yes	characteristic
18	C₂²×C₈	C₂²	1	1	yes	characteristic
19	Q₃₂	C₂²	1	8	yes	complemented
20	SD₃₂	C₂²	1	8	yes	complemented
21	M₃₂	C₂²	1	4	yes	complemented
22	C₂×C₁₆	C₂²	1	2	yes	—
23	#41	—	2	4	no	—
24	#41	—	4	2	no	—
25	C₂×Q₈	—	2	1	no	—
26	C₂×Q₈	—	2	2	no	—
27	C₂×Q₈	—	2	4	no	—
28	C₂×Q₈	—	4	2	no	—
29	C₂×Q₈	—	4	1	no	—
30	C₂×D₄	—	2	1	no	—
31	C₂×D₄	—	4	1	no	—
32	C₂²×C₄	—	4	1	no	—
33	C₂²×C₄	—	4	1	no	—
34	C₂²×C₄	D₄	1	1	yes	characteristic
35	Q₁₆	—	2	2	no	—
36	Q₁₆	—	2	4	no	—
37	Q₁₆	—	2	1	no	—
38	Q₁₆	C₂³	1	4	yes	—
39	Q₁₆	C₂³	1	2	yes	—
40	SD₁₆	—	2	4	no	—
41	D₈	—	2	1	no	—
42	D₈	C₂³	1	2	yes	—
43	C₂×C₈	D₄	1	1	yes	characteristic
44	C₂×C₈	D₄	1	2	yes	—
45	C₂×C₈	C₂³	1	1	yes	characteristic
46	C₂×C₈	C₂³	1	2	yes	—
47	C₁₆	C₂³	1	4	yes	—
48	C₂³	—	4	1	no	—
49	C₂³	D₈	1	1	yes	characteristic
50	Q₈	—	2	2	no	—
51	Q₈	—	2	4	no	—
52	Q₈	—	4	4	no	—
53	Q₈	—	4	2	no	—
54	Q₈	—	4	1	no	—
55	D₄	—	2	2	no	—
56	D₄	—	4	4	no	—
57	D₄	—	4	1	no	—
58	C₂×C₄	—	4	4	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	4	4	no	—
62	C₂×C₄	D₈	1	2	yes	—
63	C₂×C₄	D₈	1	1	yes	characteristic
64	C₂×C₄	C₂×D₄	1	2	yes	—
65	C₂×C₄	C₂×D₄	1	1	yes	characteristic
66	C₈	C₂×D₄	1	1	yes	characteristic
67	C₈	C₂×D₄	1	2	yes	—
68	C₈	C₂⁴	1	1	yes	characteristic
69	C₂²	—	2	2	no	—
70	C₂²	—	4	2	no	—
71	C₂²	—	8	2	no	—
72	C₂²	C₂×D₈	1	2	yes	—
73	C₂²	C₂×D₈	1	1	yes	characteristic
74	C₄	—	4	2	no	—
75	C₄	—	4	4	no	—
76	C₄	C₂×D₈	1	2	yes	—
77	C₄	C₂×D₈	1	1	yes	characteristic
78	C₄	C₂²×D₄	1	1	yes	characteristic
79	C₂	—	2	2	no	—
80	C₂	—	8	2	no	—
81	C₂	#510	1	2	yes	direct product factor
82	C₂	C₂²×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂⁴
prime radical	C₂×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²×Q₁₆	C₂	semidirect
3. #510	C₂	direct (N × H)
4. C₂×Q₃₂	C₂	semidirect
5. C₂×SD₃₂	C₂	semidirect
6. C₂×M₃₂	C₂	semidirect
15. C₂×Q₁₆	C₂²	semidirect
19. Q₃₂	C₂²	semidirect
20. SD₃₂	C₂²	semidirect
21. M₃₂	C₂²	semidirect
81. C₂	#510	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×#135	2
C₂²×Q₁₆	2
#510	2
C₂×Q₃₂	2
C₂×SD₃₂	2
C₂×M₃₂	2

Subgroups — C2×#510 #3413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#510 #3413