Subgroups — C₂×#1705 #19592

Order 256.

497 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#498	C₂	1	3	yes	maximal, complemented
2	C₂×#492	C₂	1	3	yes	maximal
3	#1705	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×SD₁₆	—	2	6	no	—
6	C₂×C₄⋊Q₈	C₂²	1	3	yes	—
7	C₂×#127	C₂²	1	1	yes	characteristic
8	#498	—	2	6	no	—
9	#498	C₂²	1	12	yes	complemented
10	#492	—	2	6	no	—
11	#492	C₂²	1	12	yes	—
12	C₂×#106	—	2	6	no	—
13	C₂×C₄×C₈	C₂²	1	3	yes	—
14	C₈²	C₂²	1	4	yes	complemented
15	C₂²×Q₈	—	2	3	no	—
16	C₂²×D₄	—	2	3	no	—
17	C₂×SD₁₆	—	2	24	no	—
18	C₂×SD₁₆	—	4	24	no	—
19	C₂²×C₈	D₄	1	6	yes	—
20	C₄⋊Q₈	—	2	6	no	—
21	C₄⋊Q₈	—	2	3	no	—
22	C₄⋊Q₈	C₂³	1	6	yes	—
23	#127	—	2	3	no	—
24	#127	C₂³	1	2	yes	—
25	C₂×C₄⋊C₄	—	2	6	no	—
26	C₂×C₄²	C₂³	1	1	yes	characteristic
27	#106	—	2	24	no	—
28	C₄×C₈	D₄	1	6	yes	—
29	C₄×C₈	C₂³	1	6	yes	—
30	C₂⁴	—	4	1	no	—
31	C₂×Q₈	—	2	6	no	—
32	C₂×Q₈	—	4	3	no	—
33	C₂×Q₈	—	4	12	no	—
34	C₂×D₄	—	2	6	no	—
35	C₂×D₄	—	4	3	no	—
36	C₂×D₄	—	4	12	no	—
37	C₂²×C₄	—	4	3	no	—
38	C₂²×C₄	C₂×D₄	1	3	yes	—
39	SD₁₆	—	4	48	no	—
40	C₂×C₈	SD₁₆	1	24	yes	complemented
41	C₂×C₈	C₂×D₄	1	12	yes	—
42	C₄⋊C₄	—	2	12	no	—
43	C₄⋊C₄	—	4	6	no	—
44	C₄²	C₂×D₄	1	3	yes	—
45	C₄²	C₂⁴	1	1	yes	characteristic
46	C₂³	—	4	2	no	—
47	C₂³	—	8	6	no	—
48	C₂³	#127	1	1	yes	characteristic
49	Q₈	—	4	12	no	—
50	Q₈	—	8	6	no	—
51	D₄	—	4	12	no	—
52	D₄	—	8	6	no	—
53	C₂×C₄	—	4	6	no	—
54	C₂×C₄	—	8	6	no	—
55	C₂×C₄	#127	1	3	yes	—
56	C₂×C₄	C₂×SD₁₆	1	12	yes	—
57	C₂×C₄	C₂²×D₄	1	3	yes	—
58	C₈	C₂×SD₁₆	1	24	yes	complemented
59	C₂²	—	8	6	no	—
60	C₂²	—	16	4	no	—
61	C₂²	#492	1	6	yes	—
62	C₂²	C₂×#127	1	1	yes	characteristic
63	C₄	—	8	6	no	—
64	C₄	#492	1	6	yes	—
65	C₄	C₂²×SD₁₆	1	6	yes	—
66	C₂	—	16	2	no	—
67	C₂	#1705	1	4	yes	direct product factor
68	C₂	C₂×#492	1	3	yes	—

	subgroup	G/H
center, socle	C₂³	#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#498	C₂	semidirect
3. #1705	C₂	direct (N × H)
4. C₂×C₈²	C₂	semidirect
9. #498	C₂²	semidirect
14. C₈²	C₂²	semidirect
40. C₂×C₈	SD₁₆	semidirect
58. C₈	C₂×SD₁₆	semidirect
67. C₂	#1705	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×#498	2
C₂×#492	2
#1705	2
C₂×C₈²	2

Subgroups — C2×#1705 #19592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1705 #19592