Subgroups — C₂⁴×#11764047 #31797772

Order 17536.

1976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11764047	C₂	1	30	yes	maximal, direct product factor
2	C₂⁴×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11764047	C₂²	1	140	yes	direct product factor
4	C₂³×Dic₁₃₇	C₄	1	15	yes	—
5	C₂³×Dic₁₃₇	C₂²	1	15	yes	—
6	C₂⁵×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂×#11764047	C₂³	1	120	yes	direct product factor
8	C₂²×Dic₁₃₇	C₂×C₄	1	105	yes	—
9	C₂²×Dic₁₃₇	C₂³	1	35	yes	—
10	C₂⁴×C₁₃₇	C₈	1	16	yes	complemented
11	C₂⁴×C₁₃₇	C₂×C₄	1	15	yes	—
12	#11764047	C₂⁴	1	16	yes	direct product factor
13	C₂×Dic₁₃₇	C₂²×C₄	1	105	yes	—
14	C₂×Dic₁₃₇	C₂⁴	1	15	yes	—
15	C₂³×C₁₃₇	C₂×C₈	1	120	yes	complemented
16	C₂³×C₁₃₇	C₂²×C₄	1	35	yes	—
17	C₂²×C₁₃₇	C₂²×C₈	1	140	yes	complemented
18	C₂²×C₁₃₇	C₂³×C₄	1	15	yes	—
19	Dic₁₃₇	C₂³×C₄	1	15	yes	—
20	Dic₁₃₇	C₂⁵	1	1	yes	characteristic
21	C₂×C₁₃₇	C₂³×C₈	1	30	yes	complemented
22	C₂×C₁₃₇	C₂⁴×C₄	1	1	yes	characteristic
23	C₁₃₇	C₂⁴×C₈	1	1	yes	characteristic, complemented, Sylow
24	C₂⁴×C₈	—	137	1	no	maximal, Sylow
25	C₂⁴×C₄	—	137	1	no	—
26	C₂³×C₈	—	137	30	no	—
27	C₂⁵	#10587739	1	1	yes	characteristic
28	C₂³×C₄	—	137	15	no	—
29	C₂³×C₄	—	137	15	no	—
30	C₂²×C₈	—	137	140	no	—
31	C₂⁴	#11764047	1	16	yes	direct product factor
32	C₂⁴	C₂×#10587739	1	15	yes	—
33	C₂²×C₄	—	137	105	no	—
34	C₂²×C₄	—	137	35	no	—
35	C₂×C₈	—	137	120	no	—
36	C₂³	C₂×#11764047	1	120	yes	direct product factor
37	C₂³	C₂²×#10587739	1	35	yes	—
38	C₂×C₄	—	137	15	no	—
39	C₂×C₄	—	137	105	no	—
40	C₈	—	137	16	no	—
41	C₂²	C₂²×#11764047	1	140	yes	direct product factor
42	C₂²	C₂³×#10587739	1	15	yes	—
43	C₄	—	137	1	no	—
44	C₄	—	137	15	no	—
45	C₂	C₂³×#11764047	1	30	yes	direct product factor
46	C₂	C₂⁴×#10587739	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁵	#10587739
commutator, hypocenter	C₁₃₇	C₂⁴×C₈
Frattini	C₂	C₂⁴×#10587739
Fitting, socle, prime radical	C₂⁵×C₁₃₇	C₄
cosocle	Dic₁₃₇	C₂⁵
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
1. C₂³×#11764047	C₂	direct (N × H)
3. C₂²×#11764047	C₂²	direct (N × H)
7. C₂×#11764047	C₂³	direct (N × H)
10. C₂⁴×C₁₃₇	C₈	semidirect
12. #11764047	C₂⁴	direct (N × H)
15. C₂³×C₁₃₇	C₂×C₈	semidirect
17. C₂²×C₁₃₇	C₂²×C₈	semidirect
21. C₂×C₁₃₇	C₂³×C₈	semidirect
23. C₁₃₇	C₂⁴×C₈	semidirect
31. C₂⁴	#11764047	direct (N × H)
36. C₂³	C₂×#11764047	direct (N × H)
41. C₂²	C₂²×#11764047	direct (N × H)
45. C₂	C₂³×#11764047	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×#11764047	2
C₂⁴×Dic₁₃₇	2
C₂⁴×C₈	137

Subgroups — C24×#11764047 #31797772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#11764047 #31797772