Subgroups — C₂⁴×#11758187 #30626704

Order 15616.

2590 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11758187	C₂	1	30	yes	maximal, direct product factor
2	C₂⁴×#92418	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11758187	C₂²	1	140	yes	direct product factor
4	C₂³×#92418	C₄	1	15	yes	—
5	C₂³×#92418	C₂²	1	15	yes	—
6	C₂⁴×C₄×C₆₁	C₄	1	1	yes	characteristic
7	C₂×#11758187	C₂³	1	120	yes	direct product factor
8	C₂²×#92418	C₂×C₄	1	105	yes	—
9	C₂²×#92418	C₂³	1	35	yes	—
10	C₂⁵×C₆₁	C₈	1	1	yes	characteristic
11	C₂³×C₄×C₆₁	C₈	1	15	yes	—
12	C₂³×C₄×C₆₁	C₂×C₄	1	15	yes	—
13	C₂⁴×C₆₁	C₁₆	1	16	yes	complemented
14	C₂⁴×C₆₁	C₂×C₈	1	15	yes	—
15	C₂²×C₄×C₆₁	C₂×C₈	1	105	yes	—
16	C₂²×C₄×C₆₁	C₂²×C₄	1	35	yes	—
17	C₂×#92418	C₂²×C₄	1	105	yes	—
18	C₂×#92418	C₂⁴	1	15	yes	—
19	#11758187	C₂⁴	1	16	yes	direct product factor
20	C₂³×C₆₁	C₂×C₁₆	1	120	yes	complemented
21	C₂³×C₆₁	C₂²×C₈	1	35	yes	—
22	C₂×C₄×C₆₁	C₂²×C₈	1	105	yes	—
23	C₂×C₄×C₆₁	C₂³×C₄	1	15	yes	—
24	#92418	C₂³×C₄	1	15	yes	—
25	#92418	C₂⁵	1	1	yes	characteristic
26	C₂⁴×C₁₆	—	61	1	no	maximal, Sylow
27	C₂²×C₆₁	C₂²×C₁₆	1	140	yes	complemented
28	C₂²×C₆₁	C₂³×C₈	1	15	yes	—
29	C₄×C₆₁	C₂³×C₈	1	15	yes	—
30	C₄×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
31	C₂⁴×C₈	—	61	1	no	—
32	C₂³×C₁₆	—	61	30	no	—
33	C₂×C₆₁	C₂³×C₁₆	1	30	yes	complemented
34	C₂×C₆₁	C₂⁴×C₈	1	1	yes	characteristic
35	C₂⁴×C₄	#6923	1	1	yes	characteristic
36	C₂³×C₈	—	61	15	no	—
37	C₂³×C₈	—	61	15	no	—
38	C₂²×C₁₆	—	61	140	no	—
39	C₆₁	C₂⁴×C₁₆	1	1	yes	characteristic, complemented, Sylow
40	C₂⁵	#92420	1	1	yes	characteristic
41	C₂³×C₄	#92420	1	15	yes	—
42	C₂³×C₄	C₂×#6923	1	15	yes	—
43	C₂²×C₈	—	61	105	no	—
44	C₂²×C₈	—	61	35	no	—
45	C₂×C₁₆	—	61	120	no	—
46	C₂⁴	#11758187	1	16	yes	direct product factor
47	C₂⁴	C₂×#92420	1	15	yes	—
48	C₂²×C₄	C₂×#92420	1	105	yes	—
49	C₂²×C₄	C₂²×#6923	1	35	yes	—
50	C₂×C₈	—	61	15	no	—
51	C₂×C₈	—	61	105	no	—
52	C₁₆	—	61	16	no	—
53	C₂³	C₂×#11758187	1	120	yes	direct product factor
54	C₂³	C₂²×#92420	1	35	yes	—
55	C₂×C₄	C₂²×#92420	1	105	yes	—
56	C₂×C₄	C₂³×#6923	1	15	yes	—
57	C₈	—	61	15	no	—
58	C₈	—	61	1	no	—
59	C₂²	C₂²×#11758187	1	140	yes	direct product factor
60	C₂²	C₂³×#92420	1	15	yes	—
61	C₄	C₂³×#92420	1	15	yes	—
62	C₄	C₂⁴×#6923	1	1	yes	characteristic
63	C₂	C₂³×#11758187	1	30	yes	direct product factor
64	C₂	C₂⁴×#92420	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₄	#6923
commutator, hypocenter	C₆₁	C₂⁴×C₁₆
Frattini	C₄	C₂⁴×#6923
Fitting	C₂⁴×C₄×C₆₁	C₄
socle, prime radical	C₂⁵×C₆₁	C₈
cosocle	#92418	C₂⁵
LCS	G ▷ C₆₁
UCS	C₁ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#11758187	C₂	direct (N × H)
3. C₂²×#11758187	C₂²	direct (N × H)
7. C₂×#11758187	C₂³	direct (N × H)
13. C₂⁴×C₆₁	C₁₆	semidirect
19. #11758187	C₂⁴	direct (N × H)
20. C₂³×C₆₁	C₂×C₁₆	semidirect
27. C₂²×C₆₁	C₂²×C₁₆	semidirect
33. C₂×C₆₁	C₂³×C₁₆	semidirect
39. C₆₁	C₂⁴×C₁₆	semidirect
46. C₂⁴	#11758187	direct (N × H)
53. C₂³	C₂×#11758187	direct (N × H)
59. C₂²	C₂²×#11758187	direct (N × H)
63. C₂	C₂³×#11758187	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11758187	2
C₂⁴×#92418	2
C₂⁴×C₁₆	61

Subgroups — C24×#11758187 #30626704

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#11758187 #30626704