Subgroups — C₂×#1710 #19598

Order 256.

375 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500	C₂	1	2	yes	maximal
2	C₂×#487	C₂	1	4	yes	maximal
3	#1710	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₈²	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈	C₂²	1	4	yes	—
6	#500	—	2	4	no	—
7	#500	C₂²	1	8	yes	—
8	#487	—	2	8	no	—
9	#487	C₂²	1	16	yes	—
10	C₂×#107	—	2	4	no	—
11	C₂×#103	—	2	8	no	—
12	C₂×C₄×C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₈²	C₂²	1	4	yes	—
15	C₂²×Q₈	—	2	4	no	—
16	C₂²×C₈	D₄	1	4	yes	—
17	C₂²×C₈	D₄	1	2	yes	—
18	C₄⋊Q₈	—	2	4	no	—
19	C₄⋊Q₈	—	2	4	no	—
20	C₄⋊Q₈	—	2	4	no	—
21	C₄⋊Q₈	C₂³	1	8	yes	—
22	C₂×C₄⋊C₄	—	2	4	no	—
23	C₂×C₄⋊C₄	—	2	4	no	—
24	C₂×C₄²	C₂³	1	1	yes	characteristic
25	#107	—	2	16	no	—
26	#103	—	2	32	no	—
27	C₄×C₈	D₄	1	4	yes	—
28	C₄×C₈	D₄	1	2	yes	—
29	C₄×C₈	C₂³	1	4	yes	—
30	C₄×C₈	C₂³	1	2	yes	—
31	C₂×Q₈	—	2	8	no	—
32	C₂×Q₈	—	4	4	no	—
33	C₂×Q₈	—	4	16	no	—
34	C₂²×C₄	—	4	4	no	—
35	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄	C₂×D₄	1	2	yes	—
37	C₂×C₈	—	2	8	no	—
38	C₂×C₈	Q₁₆	1	8	yes	—
39	C₂×C₈	C₂×D₄	1	8	yes	—
40	C₂×C₈	C₂×D₄	1	4	yes	—
41	C₄⋊C₄	—	2	8	no	—
42	C₄⋊C₄	—	2	8	no	—
43	C₄⋊C₄	—	4	4	no	—
44	C₄⋊C₄	—	4	4	no	—
45	C₄²	C₂×D₄	1	2	yes	—
46	C₄²	C₂×D₄	1	1	yes	characteristic
47	C₄²	C₂⁴	1	1	yes	characteristic
48	C₂³	#127	1	1	yes	characteristic
49	Q₈	—	4	16	no	—
50	Q₈	—	8	8	no	—
51	C₂×C₄	—	4	8	no	—
52	C₂×C₄	—	8	8	no	—
53	C₂×C₄	#127	1	1	yes	characteristic
54	C₂×C₄	#127	1	2	yes	—
55	C₂×C₄	C₂×Q₁₆	1	4	yes	—
56	C₂×C₄	#135	1	8	yes	—
57	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
58	C₂×C₄	C₂²×D₄	1	2	yes	—
59	C₈	—	2	8	no	—
60	C₈	C₂×Q₁₆	1	8	yes	—
61	C₂²	#494	1	2	yes	—
62	C₂²	#495	1	4	yes	—
63	C₂²	C₂×#127	1	1	yes	characteristic
64	C₄	—	8	8	no	—
65	C₄	#494	1	2	yes	—
66	C₄	#495	1	4	yes	—
67	C₄	C₂²×Q₁₆	1	2	yes	—
68	C₄	C₂×#135	1	4	yes	—
69	C₂	#1710	1	4	yes	direct product factor
70	C₂	C₂×#494	1	1	yes	characteristic
71	C₂	C₂×#495	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	#127
commutator, Frattini, cosocle	C₄²	C₂⁴
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
3. #1710	C₂	direct (N × H)
69. C₂	#1710	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×#500	2
C₂×#487	2
#1710	2
C₂×C₈²	2

Subgroups — C2×#1710 #19598

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1710 #19598