Subgroups — C₂⁴×Dic₂₇₄ #31795584

Order 17536.

6264 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₇₄	C₂	1	60	yes	maximal, direct product factor
2	C₂⁴×Dic₁₃₇	C₂	1	2	yes	maximal
3	C₂⁴×C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₇₄	C₂²	1	560	yes	direct product factor
5	C₂³×Dic₁₃₇	C₂²	1	60	yes	—
6	C₂⁵×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₁₃₇	C₂²	1	30	yes	—
8	C₂×Dic₂₇₄	C₂³	1	960	yes	direct product factor
9	C₂²×Dic₁₃₇	C₂³	1	280	yes	—
10	C₂⁴×C₁₃₇	Q₈	1	16	yes	complemented
11	C₂⁴×C₁₃₇	C₂³	1	15	yes	—
12	C₂²×C₄×C₁₃₇	C₂³	1	140	yes	—
13	Dic₂₇₄	C₂⁴	1	256	yes	direct product factor
14	C₂×Dic₁₃₇	C₂⁴	1	240	yes	—
15	C₂³×C₁₃₇	C₂×Q₈	1	120	yes	complemented
16	C₂³×C₁₃₇	C₂⁴	1	35	yes	—
17	C₂×C₄×C₁₃₇	C₂⁴	1	120	yes	—
18	C₂²×C₁₃₇	C₂²×Q₈	1	140	yes	complemented
19	C₂²×C₁₃₇	C₂⁵	1	15	yes	—
20	C₄×C₁₃₇	C₂⁵	1	16	yes	—
21	Dic₁₃₇	C₂⁵	1	32	yes	—
22	C₂×C₁₃₇	C₂³×Q₈	1	30	yes	complemented
23	C₂×C₁₃₇	C₂⁶	1	1	yes	characteristic
24	C₁₃₇	C₂⁴×Q₈	1	1	yes	characteristic, complemented, Sylow
25	C₂⁴×Q₈	—	137	1	no	maximal, Sylow
26	C₂³×Q₈	—	137	60	no	—
27	C₂⁴×C₄	—	137	2	no	—
28	C₂⁴×C₄	D₁₃₇	1	1	yes	characteristic
29	C₂⁵	C₂×D₁₃₇	1	1	yes	characteristic
30	C₂²×Q₈	—	137	560	no	—
31	C₂³×C₄	—	137	60	no	—
32	C₂³×C₄	C₂×D₁₃₇	1	30	yes	—
33	C₂⁴	Dic₂₇₄	1	16	yes	direct product factor
34	C₂⁴	C₂²×D₁₃₇	1	15	yes	—
35	C₂×Q₈	—	137	960	no	—
36	C₂²×C₄	—	137	280	no	—
37	C₂²×C₄	C₂²×D₁₃₇	1	140	yes	—
38	C₂³	C₂×Dic₂₇₄	1	120	yes	direct product factor
39	C₂³	C₂³×D₁₃₇	1	35	yes	—
40	Q₈	—	137	256	no	—
41	C₂×C₄	—	137	240	no	—
42	C₂×C₄	C₂³×D₁₃₇	1	120	yes	—
43	C₂²	C₂²×Dic₂₇₄	1	140	yes	direct product factor
44	C₂²	C₂⁴×D₁₃₇	1	15	yes	—
45	C₄	—	137	32	no	—
46	C₄	C₂⁴×D₁₃₇	1	16	yes	—
47	C₂	C₂³×Dic₂₇₄	1	30	yes	direct product factor
48	C₂	C₂⁵×D₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂×D₁₃₇
commutator, cosocle	C₂×C₁₃₇	C₂⁶
Frattini	C₂	C₂⁵×D₁₃₇
Fitting	C₂⁴×C₄×C₁₃₇	C₂
socle, prime radical	C₂⁵×C₁₃₇	C₂²
hypercenter	C₂⁴×C₄	D₁₃₇
hypocenter	C₁₃₇	C₂⁴×Q₈
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₂₇₄	C₂	direct (N × H)
4. C₂²×Dic₂₇₄	C₂²	direct (N × H)
8. C₂×Dic₂₇₄	C₂³	direct (N × H)
10. C₂⁴×C₁₃₇	Q₈	semidirect
13. Dic₂₇₄	C₂⁴	direct (N × H)
15. C₂³×C₁₃₇	C₂×Q₈	semidirect
18. C₂²×C₁₃₇	C₂²×Q₈	semidirect
22. C₂×C₁₃₇	C₂³×Q₈	semidirect
24. C₁₃₇	C₂⁴×Q₈	semidirect
33. C₂⁴	Dic₂₇₄	direct (N × H)
38. C₂³	C₂×Dic₂₇₄	direct (N × H)
43. C₂²	C₂²×Dic₂₇₄	direct (N × H)
47. C₂	C₂³×Dic₂₇₄	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×Dic₂₇₄	2
C₂⁴×Dic₁₃₇	2
C₂⁴×C₄×C₁₃₇	2
C₂⁴×Q₈	137

Subgroups — C24×Dic274 #31795584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×Dic₂₇₄ #31795584