Subgroups — C₂³×#117 #62641

Order 256.

7167 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×#117	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₂³×C₂²⋊C₄	C₂	1	2	yes	maximal
5	C₂³×C₄²	C₂	1	2	yes	maximal, complemented
6	C₂⁶	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄	C₄	1	32	yes	complemented
8	C₂⁴×C₄	C₂²	1	4	yes	—
9	C₂⁴×C₄	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄	C₂²	1	28	yes	—
11	C₂⁴×C₄	C₂²	1	1	yes	characteristic
12	C₂×#117	C₂²	1	448	yes	direct product factor
13	C₂²×C₄⋊C₄	C₂²	1	56	yes	complemented
14	C₂²×C₂²⋊C₄	C₂²	1	56	yes	—
15	C₂²×C₄²	C₂²	1	56	yes	complemented
16	C₂⁵	—	2	16	no	—
17	C₂⁵	C₂×C₄	1	16	yes	—
18	C₂⁵	C₂³	1	14	yes	—
19	C₂⁵	C₂³	1	1	yes	characteristic
20	C₂³×C₄	—	2	32	no	—
21	C₂³×C₄	C₂×C₄	1	16	yes	—
22	C₂³×C₄	C₂×C₄	1	16	yes	—
23	C₂³×C₄	C₂×C₄	1	448	yes	complemented
24	C₂³×C₄	C₂³	1	56	yes	—
25	C₂³×C₄	C₂³	1	14	yes	—
26	C₂³×C₄	C₂³	1	14	yes	—
27	C₂³×C₄	C₂³	1	112	yes	—
28	#117	C₂³	1	512	yes	direct product factor
29	C₂×C₄⋊C₄	C₂³	1	224	yes	complemented
30	C₂×C₂²⋊C₄	C₂³	1	224	yes	—
31	C₂×C₄²	C₂³	1	224	yes	complemented
32	C₂⁴	—	2	128	no	—
33	C₂⁴	—	2	112	no	—
34	C₂⁴	C₂²×C₄	1	112	yes	—
35	C₂⁴	C₂²×C₄	1	8	yes	—
36	C₂⁴	#41	1	16	yes	—
37	C₂⁴	C₂⁴	1	28	yes	—
38	C₂⁴	C₂⁴	1	7	yes	—
39	C₂²×C₄	—	2	224	no	—
40	C₂²×C₄	C₂²×C₄	1	112	yes	—
41	C₂²×C₄	C₂²×C₄	1	112	yes	—
42	C₂²×C₄	C₂²×C₄	1	896	yes	complemented
43	C₂²×C₄	C₂⁴	1	112	yes	—
44	C₂²×C₄	C₂⁴	1	28	yes	—
45	C₂²×C₄	C₂⁴	1	28	yes	—
46	C₂²×C₄	C₂⁴	1	64	yes	—
47	C₄⋊C₄	C₂⁴	1	128	yes	complemented
48	C₂²⋊C₄	C₂⁴	1	128	yes	—
49	C₄²	C₂⁴	1	128	yes	complemented
50	C₂³	—	2	448	no	—
51	C₂³	—	2	112	no	—
52	C₂³	#117	1	64	yes	direct product factor
53	C₂³	C₂³×C₄	1	112	yes	—
54	C₂³	C₂³×C₄	1	28	yes	—
55	C₂³	C₂×#41	1	56	yes	—
56	C₂³	C₂⁵	1	8	yes	—
57	C₂³	C₂⁵	1	7	yes	—
58	C₂×C₄	—	2	224	no	—
59	C₂×C₄	C₂³×C₄	1	112	yes	—
60	C₂×C₄	C₂³×C₄	1	112	yes	—
61	C₂×C₄	C₂³×C₄	1	256	yes	complemented
62	C₂×C₄	C₂⁵	1	32	yes	—
63	C₂×C₄	C₂⁵	1	8	yes	—
64	C₂×C₄	C₂⁵	1	8	yes	—
65	C₂²	—	2	16	no	—
66	C₂²	—	2	224	no	—
67	C₂²	C₂×#117	1	112	yes	direct product factor
68	C₂²	C₂⁴×C₄	1	16	yes	—
69	C₂²	C₂⁴×C₄	1	14	yes	—
70	C₂²	C₂²×#41	1	28	yes	—
71	C₂²	C₂⁶	1	1	yes	characteristic
72	C₄	—	2	32	no	—
73	C₄	C₂⁴×C₄	1	16	yes	—
74	C₄	C₂⁴×C₄	1	16	yes	—
75	C₂	—	2	16	no	—
76	C₂	C₂²×#117	1	28	yes	direct product factor
77	C₂	C₂⁵×C₄	1	1	yes	characteristic
78	C₂	C₂³×#41	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂²
commutator	C₂	C₂⁵×C₄
Frattini, cosocle	C₂²	C₂⁶
socle	C₂⁵	C₂³
prime radical	C₂⁶	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²×#117	C₂	direct (N × H)
3. C₂³×C₄⋊C₄	C₂	semidirect
5. C₂³×C₄²	C₂	semidirect
7. C₂⁴×C₄	C₄	semidirect
12. C₂×#117	C₂²	direct (N × H)
13. C₂²×C₄⋊C₄	C₂²	semidirect
15. C₂²×C₄²	C₂²	semidirect
23. C₂³×C₄	C₂×C₄	semidirect
28. #117	C₂³	direct (N × H)
29. C₂×C₄⋊C₄	C₂³	semidirect
31. C₂×C₄²	C₂³	semidirect
42. C₂²×C₄	C₂²×C₄	semidirect
47. C₄⋊C₄	C₂⁴	semidirect
49. C₄²	C₂⁴	semidirect
52. C₂³	#117	direct (N × H)
61. C₂×C₄	C₂³×C₄	semidirect
67. C₂²	C₂×#117	direct (N × H)
76. C₂	C₂²×#117	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄	2
C₂²×#117	2
C₂³×C₄⋊C₄	2
C₂³×C₂²⋊C₄	2
C₂³×C₄²	2

Subgroups — C23×#117 #62641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#117 #62641