Subgroups — C₂⁴×#11764049 #31798045

Order 17536.

1976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11764049	C₂	1	30	yes	maximal, direct product factor
2	C₂⁴×#10587739	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11764049	C₂²	1	140	yes	direct product factor
4	C₂³×#10587739	C₄	1	15	yes	—
5	C₂³×#10587739	C₂²	1	15	yes	—
6	C₂⁴×D₁₃₇	C₄	1	1	yes	characteristic
7	C₂×#11764049	C₂³	1	120	yes	direct product factor
8	C₂²×#10587739	C₂×C₄	1	105	yes	—
9	C₂²×#10587739	C₂³	1	35	yes	—
10	C₂³×D₁₃₇	C₈	1	15	yes	complemented
11	C₂³×D₁₃₇	C₂×C₄	1	15	yes	—
12	C₂⁴×C₁₃₇	C₈	1	1	yes	characteristic, complemented
13	#11764049	C₂⁴	1	16	yes	direct product factor
14	C₂×#10587739	C₂²×C₄	1	105	yes	—
15	C₂×#10587739	C₂⁴	1	15	yes	—
16	C₂²×D₁₃₇	C₂×C₈	1	105	yes	complemented
17	C₂²×D₁₃₇	C₂²×C₄	1	35	yes	—
18	C₂³×C₁₃₇	C₂×C₈	1	15	yes	complemented
19	C₂²×C₁₃₇	C₂²×C₈	1	35	yes	complemented
20	C₂×D₁₃₇	C₂²×C₈	1	105	yes	complemented
21	C₂×D₁₃₇	C₂³×C₄	1	15	yes	—
22	#10587739	C₂³×C₄	1	15	yes	—
23	#10587739	C₂⁵	1	1	yes	characteristic
24	C₂×C₁₃₇	C₂³×C₈	1	15	yes	complemented
25	D₁₃₇	C₂³×C₈	1	15	yes	complemented
26	D₁₃₇	C₂⁴×C₄	1	1	yes	characteristic
27	C₁₃₇	C₂⁴×C₈	1	1	yes	characteristic, complemented, Sylow
28	C₂⁴×C₈	—	137	1	no	maximal, Sylow
29	C₂⁴×C₄	—	137	1	no	—
30	C₂³×C₈	—	137	30	no	—
31	C₂⁵	—	137	1	no	—
32	C₂³×C₄	—	137	15	no	—
33	C₂³×C₄	—	137	15	no	—
34	C₂²×C₈	—	137	140	no	—
35	C₂⁴	—	137	15	no	—
36	C₂⁴	—	137	15	no	—
37	C₂⁴	#11764049	1	1	yes	characteristic, direct product factor
38	C₂²×C₄	—	137	35	no	—
39	C₂²×C₄	—	137	105	no	—
40	C₂×C₈	—	137	120	no	—
41	C₂³	—	137	35	no	—
42	C₂³	—	137	105	no	—
43	C₂³	C₂×#11764049	1	15	yes	direct product factor
44	C₂×C₄	—	137	15	no	—
45	C₂×C₄	—	137	105	no	—
46	C₈	—	137	16	no	—
47	C₂²	—	137	15	no	—
48	C₂²	—	137	105	no	—
49	C₂²	C₂²×#11764049	1	35	yes	direct product factor
50	C₄	—	137	1	no	—
51	C₄	—	137	15	no	—
52	C₂	—	137	1	no	—
53	C₂	—	137	15	no	—
54	C₂	C₂³×#11764049	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴	#11764049
commutator, hypocenter	C₁₃₇	C₂⁴×C₈
Frattini	C₁	G
Fitting, socle	C₂⁴×C₁₃₇	C₈
prime radical	C₂⁴×D₁₃₇	C₄
cosocle	#10587739	C₂⁵
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂³×#11764049	C₂	direct (N × H)
3. C₂²×#11764049	C₂²	direct (N × H)
7. C₂×#11764049	C₂³	direct (N × H)
10. C₂³×D₁₃₇	C₈	semidirect
12. C₂⁴×C₁₃₇	C₈	semidirect
13. #11764049	C₂⁴	direct (N × H)
16. C₂²×D₁₃₇	C₂×C₈	semidirect
18. C₂³×C₁₃₇	C₂×C₈	semidirect
19. C₂²×C₁₃₇	C₂²×C₈	semidirect
20. C₂×D₁₃₇	C₂²×C₈	semidirect
24. C₂×C₁₃₇	C₂³×C₈	semidirect
25. D₁₃₇	C₂³×C₈	semidirect
27. C₁₃₇	C₂⁴×C₈	semidirect
37. C₂⁴	#11764049	direct (N × H)
43. C₂³	C₂×#11764049	direct (N × H)
49. C₂²	C₂²×#11764049	direct (N × H)
54. C₂	C₂³×#11764049	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×#11764049	2
C₂⁴×#10587739	2
C₂⁴×C₈	137

Subgroups — C24×#11764049 #31798045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#11764049 #31798045