Subgroups — C₂²×#499 #60653

Order 256.

983 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125	C₂	1	2	yes	maximal
2	C₂×#499	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#107	C₂	1	2	yes	maximal
4	C₂²×#106	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈	C₂²	1	1	yes	characteristic
7	C₂³×C₈	C₂²	1	1	yes	characteristic
8	C₂×#125	—	2	12	no	—
9	C₂×#125	C₂²	1	24	yes	—
10	C₂²×C₄⋊C₄	—	2	4	no	—
11	C₂²×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂²×C₄²	C₂²	1	1	yes	characteristic
13	#499	C₂²	1	64	yes	direct product factor
14	C₂×#107	C₂²	1	24	yes	—
15	C₂×#106	C₂²	1	24	yes	—
16	C₂×C₄×C₈	C₂²	1	12	yes	—
17	C₂³×C₄	—	2	4	no	—
18	C₂³×C₄	D₄	1	2	yes	—
19	C₂³×C₄	C₂³	1	1	yes	characteristic
20	C₂²×C₈	Q₈	1	8	yes	—
21	C₂²×C₈	Q₈	1	8	yes	—
22	C₂²×C₈	C₂³	1	6	yes	—
23	C₂²×C₈	C₂³	1	6	yes	—
24	#125	—	2	48	no	—
25	#125	C₂³	1	32	yes	—
26	C₂×C₄⋊C₄	—	2	48	no	—
27	C₂×C₄⋊C₄	—	2	12	no	—
28	C₂×C₄⋊C₄	C₂³	1	24	yes	—
29	C₂×C₄²	D₄	1	6	yes	—
30	C₂×C₄²	C₂³	1	6	yes	—
31	#107	C₂³	1	32	yes	—
32	#106	C₂³	1	32	yes	—
33	C₄×C₈	C₂³	1	16	yes	—
34	C₂⁴	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄	—	2	8	no	—
36	C₂²×C₄	—	2	24	no	—
37	C₂²×C₄	—	4	16	no	—
38	C₂²×C₄	C₂×D₄	1	12	yes	—
39	C₂²×C₄	C₂×D₄	1	3	yes	—
40	C₂²×C₄	C₂×Q₈	1	4	yes	—
41	C₂²×C₄	C₂×Q₈	1	4	yes	—
42	C₂²×C₄	C₂⁴	1	3	yes	—
43	C₂×C₈	C₂×Q₈	1	24	yes	—
44	C₂×C₈	C₂×Q₈	1	24	yes	—
45	C₂×C₈	C₂⁴	1	4	yes	—
46	C₂×C₈	C₂⁴	1	4	yes	—
47	C₄⋊C₄	—	2	64	no	—
48	C₄⋊C₄	—	2	24	no	—
49	C₄⋊C₄	C₂⁴	1	16	yes	—
50	C₄²	C₂×D₄	1	12	yes	—
51	C₄²	C₂⁴	1	4	yes	—
52	C₂³	C₄⋊Q₈	1	4	yes	—
53	C₂³	#135	1	8	yes	—
54	C₂³	C₂²×D₄	1	3	yes	—
55	C₂×C₄	—	2	24	no	—
56	C₂×C₄	—	2	16	no	—
57	C₂×C₄	—	4	48	no	—
58	C₂×C₄	C₄⋊Q₈	1	12	yes	—
59	C₂×C₄	C₂²×D₄	1	8	yes	—
60	C₂×C₄	C₂²×D₄	1	3	yes	—
61	C₂×C₄	C₂²×Q₈	1	6	yes	—
62	C₂×C₄	C₂²×Q₈	1	6	yes	—
63	C₂×C₄	C₂⁵	1	1	yes	characteristic
64	C₈	C₂²×Q₈	1	8	yes	—
65	C₈	C₂²×Q₈	1	8	yes	—
66	C₂²	#499	1	16	yes	direct product factor
67	C₂²	C₂×C₄⋊Q₈	1	6	yes	—
68	C₂²	C₂×#135	1	12	yes	—
69	C₂²	C₂³×D₄	1	1	yes	characteristic
70	C₄	—	2	8	no	—
71	C₄	—	4	16	no	—
72	C₄	C₂×C₄⋊Q₈	1	6	yes	—
73	C₄	C₂³×Q₈	1	1	yes	characteristic
74	C₄	C₂³×Q₈	1	1	yes	characteristic
75	C₂	C₂×#499	1	12	yes	direct product factor
76	C₂	C₂²×C₄⋊Q₈	1	1	yes	characteristic
77	C₂	C₂²×#135	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#499	C₂	direct (N × H)
13. #499	C₂²	direct (N × H)
66. C₂²	#499	direct (N × H)
75. C₂	C₂×#499	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#125	2
C₂×#499	2
C₂²×#107	2
C₂²×#106	2
C₂²×C₄×C₈	2

Subgroups — C22×#499 #60653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#499 #60653