Subgroups — C₂⁴×#11719910 #28330783

Order 13568.

2590 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719910	C₂	1	30	yes	maximal, direct product factor
2	C₂⁴×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11719910	C₂²	1	140	yes	direct product factor
4	C₂³×#88322	C₄	1	15	yes	—
5	C₂³×#88322	C₂²	1	15	yes	—
6	C₂⁴×C₄×C₅₃	C₄	1	1	yes	characteristic
7	C₂×#11719910	C₂³	1	120	yes	direct product factor
8	C₂²×#88322	C₂×C₄	1	105	yes	—
9	C₂²×#88322	C₂³	1	35	yes	—
10	C₂⁵×C₅₃	C₈	1	1	yes	characteristic
11	C₂³×C₄×C₅₃	C₈	1	15	yes	—
12	C₂³×C₄×C₅₃	C₂×C₄	1	15	yes	—
13	C₂⁴×C₅₃	C₁₆	1	16	yes	complemented
14	C₂⁴×C₅₃	C₂×C₈	1	15	yes	—
15	C₂²×C₄×C₅₃	C₂×C₈	1	105	yes	—
16	C₂²×C₄×C₅₃	C₂²×C₄	1	35	yes	—
17	C₂×#88322	C₂²×C₄	1	105	yes	—
18	C₂×#88322	C₂⁴	1	15	yes	—
19	#11719910	C₂⁴	1	16	yes	direct product factor
20	C₂³×C₅₃	C₂×C₁₆	1	120	yes	complemented
21	C₂³×C₅₃	C₂²×C₈	1	35	yes	—
22	C₂×C₄×C₅₃	C₂²×C₈	1	105	yes	—
23	C₂×C₄×C₅₃	C₂³×C₄	1	15	yes	—
24	#88322	C₂³×C₄	1	15	yes	—
25	#88322	C₂⁵	1	1	yes	characteristic
26	C₂⁴×C₁₆	—	53	1	no	maximal, Sylow
27	C₂²×C₅₃	C₂²×C₁₆	1	140	yes	complemented
28	C₂²×C₅₃	C₂³×C₈	1	15	yes	—
29	C₄×C₅₃	C₂³×C₈	1	15	yes	—
30	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
31	C₂⁴×C₈	—	53	1	no	—
32	C₂³×C₁₆	—	53	30	no	—
33	C₂×C₅₃	C₂³×C₁₆	1	30	yes	complemented
34	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
35	C₂⁴×C₄	#6157	1	1	yes	characteristic
36	C₂³×C₈	—	53	15	no	—
37	C₂³×C₈	—	53	15	no	—
38	C₂²×C₁₆	—	53	140	no	—
39	C₅₃	C₂⁴×C₁₆	1	1	yes	characteristic, complemented, Sylow
40	C₂⁵	#88324	1	1	yes	characteristic
41	C₂³×C₄	#88324	1	15	yes	—
42	C₂³×C₄	C₂×#6157	1	15	yes	—
43	C₂²×C₈	—	53	105	no	—
44	C₂²×C₈	—	53	35	no	—
45	C₂×C₁₆	—	53	120	no	—
46	C₂⁴	#11719910	1	16	yes	direct product factor
47	C₂⁴	C₂×#88324	1	15	yes	—
48	C₂²×C₄	C₂×#88324	1	105	yes	—
49	C₂²×C₄	C₂²×#6157	1	35	yes	—
50	C₂×C₈	—	53	15	no	—
51	C₂×C₈	—	53	105	no	—
52	C₁₆	—	53	16	no	—
53	C₂³	C₂×#11719910	1	120	yes	direct product factor
54	C₂³	C₂²×#88324	1	35	yes	—
55	C₂×C₄	C₂²×#88324	1	105	yes	—
56	C₂×C₄	C₂³×#6157	1	15	yes	—
57	C₈	—	53	15	no	—
58	C₈	—	53	1	no	—
59	C₂²	C₂²×#11719910	1	140	yes	direct product factor
60	C₂²	C₂³×#88324	1	15	yes	—
61	C₄	C₂³×#88324	1	15	yes	—
62	C₄	C₂⁴×#6157	1	1	yes	characteristic
63	C₂	C₂³×#11719910	1	30	yes	direct product factor
64	C₂	C₂⁴×#88324	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₄	#6157
commutator, hypocenter	C₅₃	C₂⁴×C₁₆
Frattini	C₄	C₂⁴×#6157
Fitting	C₂⁴×C₄×C₅₃	C₄
socle, prime radical	C₂⁵×C₅₃	C₈
cosocle	#88322	C₂⁵
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#11719910	C₂	direct (N × H)
3. C₂²×#11719910	C₂²	direct (N × H)
7. C₂×#11719910	C₂³	direct (N × H)
13. C₂⁴×C₅₃	C₁₆	semidirect
19. #11719910	C₂⁴	direct (N × H)
20. C₂³×C₅₃	C₂×C₁₆	semidirect
27. C₂²×C₅₃	C₂²×C₁₆	semidirect
33. C₂×C₅₃	C₂³×C₁₆	semidirect
39. C₅₃	C₂⁴×C₁₆	semidirect
46. C₂⁴	#11719910	direct (N × H)
53. C₂³	C₂×#11719910	direct (N × H)
59. C₂²	C₂²×#11719910	direct (N × H)
63. C₂	C₂³×#11719910	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×#11719910	2
C₂⁴×#88322	2
C₂⁴×C₁₆	53

Subgroups — C24×#11719910 #28330783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#11719910 #28330783