Subgroups — C₂³×#543 #10585901

Order 512.

45714 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#543	C₂	1	112	yes	maximal, direct product factor
2	C₂³×C₂²⋊Q₈	C₂	1	12	yes	maximal, complemented
3	C₂⁴×C₂²⋊C₄	C₂	1	3	yes	maximal
4	C₂⁷	C₂²	1	1	yes	characteristic
5	C₂⁴×Q₈	C₂²	1	4	yes	complemented
6	C₂⁵×C₄	C₂²	1	6	yes	—
7	C₂×#543	C₂²	1	1792	yes	direct product factor
8	C₂²×C₂²⋊Q₈	C₂²	1	672	yes	complemented
9	C₂³×C₄⋊C₄	C₂²	1	12	yes	complemented
10	C₂³×C₂²⋊C₄	C₂²	1	12	yes	—
11	C₂³×C₂²⋊C₄	C₂²	1	168	yes	—
12	C₂⁶	—	4	16	no	—
13	C₂⁶	Q₈	1	32	yes	complemented
14	C₂⁶	C₂³	1	3	yes	—
15	C₂⁶	C₂³	1	28	yes	—
16	C₂³×Q₈	—	4	32	no	—
17	C₂³×Q₈	C₂³	1	112	yes	complemented
18	C₂⁴×C₄	—	4	48	no	—
19	C₂⁴×C₄	C₂³	1	12	yes	—
20	C₂⁴×C₄	C₂³	1	168	yes	—
21	#543	C₂³	1	4096	yes	direct product factor
22	C₂×C₂²⋊Q₈	C₂³	1	5376	yes	complemented
23	C₂²×C₄⋊C₄	C₂³	1	336	yes	complemented
24	C₂²×C₂²⋊C₄	C₂³	1	336	yes	—
25	C₂²×C₂²⋊C₄	C₂³	1	1344	yes	—
26	C₂⁵	—	2	48	no	—
27	C₂⁵	—	4	256	no	—
28	C₂⁵	—	4	224	no	—
29	C₂⁵	C₂×Q₈	1	48	yes	—
30	C₂⁵	C₂×Q₈	1	448	yes	complemented
31	C₂⁵	C₂⁴	1	42	yes	—
32	C₂⁵	C₂⁴	1	112	yes	—
33	C₂⁵	C₂⁴	1	1	yes	characteristic
34	C₂²×Q₈	—	4	448	no	—
35	C₂²×Q₈	C₂⁴	1	448	yes	complemented
36	C₂³×C₄	—	2	96	no	—
37	C₂³×C₄	—	4	672	no	—
38	C₂³×C₄	C₂⁴	1	168	yes	—
39	C₂³×C₄	C₂⁴	1	672	yes	—
40	C₂²⋊Q₈	C₂⁴	1	6144	yes	complemented
41	C₂×C₄⋊C₄	C₂⁴	1	1344	yes	complemented
42	C₂×C₂²⋊C₄	C₂⁴	1	1344	yes	—
43	C₂×C₂²⋊C₄	C₂⁴	1	1536	yes	—
44	C₂⁴	—	2	384	no	—
45	C₂⁴	—	2	336	no	—
46	C₂⁴	—	4	1792	no	—
47	C₂⁴	—	4	448	no	—
48	C₂⁴	C₂²×Q₈	1	336	yes	—
49	C₂⁴	C₂²×Q₈	1	8	yes	—
50	C₂⁴	C₂²×Q₈	1	896	yes	complemented
51	C₂⁴	H(5,𝔽₂)	1	16	yes	—
52	C₂⁴	C₂⁵	1	84	yes	—
53	C₂⁴	C₂⁵	1	64	yes	—
54	C₂⁴	C₂⁵	1	7	yes	—
55	C₂×Q₈	—	4	896	no	—
56	C₂×Q₈	C₂⁵	1	256	yes	complemented
57	C₂²×C₄	—	2	672	no	—
58	C₂²×C₄	—	4	1344	no	—
59	C₂²×C₄	C₂⁵	1	336	yes	—
60	C₂²×C₄	C₂⁵	1	384	yes	—
61	C₄⋊C₄	C₂⁵	1	768	yes	complemented
62	C₂²⋊C₄	C₂⁵	1	768	yes	—
63	C₂³	—	2	336	no	—
64	C₂³	—	2	1344	no	—
65	C₂³	—	4	1792	no	—
66	C₂³	—	4	128	no	—
67	C₂³	#543	1	64	yes	direct product factor
68	C₂³	C₂³×Q₈	1	336	yes	—
69	C₂³	C₂³×Q₈	1	28	yes	—
70	C₂³	C₂³×Q₈	1	256	yes	complemented
71	C₂³	C₂×H(5,𝔽₂)	1	56	yes	—
72	C₂³	C₂⁶	1	24	yes	—
73	C₂³	C₂⁶	1	7	yes	—
74	Q₈	—	4	256	no	—
75	C₂×C₄	—	2	672	no	—
76	C₂×C₄	—	4	384	no	—
77	C₂×C₄	C₂⁶	1	96	yes	—
78	C₂²	—	2	48	no	—
79	C₂²	—	2	672	no	—
80	C₂²	—	4	256	no	—
81	C₂²	C₂×#543	1	112	yes	direct product factor
82	C₂²	C₂⁴×Q₈	1	48	yes	—
83	C₂²	C₂⁴×Q₈	1	14	yes	—
84	C₂²	C₂²×H(5,𝔽₂)	1	28	yes	—
85	C₂²	C₂⁷	1	1	yes	characteristic
86	C₄	—	2	96	no	—
87	C₂	—	2	48	no	—
88	C₂	C₂²×#543	1	28	yes	direct product factor
89	C₂	C₂⁵×Q₈	1	1	yes	characteristic
90	C₂	C₂³×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁷
prime radical	C₂⁷	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×#543	C₂	direct (N × H)
2. C₂³×C₂²⋊Q₈	C₂	semidirect
5. C₂⁴×Q₈	C₂²	semidirect
7. C₂×#543	C₂²	direct (N × H)
8. C₂²×C₂²⋊Q₈	C₂²	semidirect
9. C₂³×C₄⋊C₄	C₂²	semidirect
13. C₂⁶	Q₈	semidirect
17. C₂³×Q₈	C₂³	semidirect
21. #543	C₂³	direct (N × H)
22. C₂×C₂²⋊Q₈	C₂³	semidirect
23. C₂²×C₄⋊C₄	C₂³	semidirect
30. C₂⁵	C₂×Q₈	semidirect
35. C₂²×Q₈	C₂⁴	semidirect
40. C₂²⋊Q₈	C₂⁴	semidirect
41. C₂×C₄⋊C₄	C₂⁴	semidirect
50. C₂⁴	C₂²×Q₈	semidirect
56. C₂×Q₈	C₂⁵	semidirect
61. C₄⋊C₄	C₂⁵	semidirect
67. C₂³	#543	direct (N × H)
70. C₂³	C₂³×Q₈	semidirect
81. C₂²	C₂×#543	direct (N × H)
88. C₂	C₂²×#543	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×#543	2
C₂³×C₂²⋊Q₈	2
C₂⁴×C₂²⋊C₄	2

Subgroups — C23×#543 #10585901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#543 #10585901