Subgroups — C₂⁵×Dic₁₃₄ #31769711

Order 17152.

63326 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₃₄	C₂	1	124	yes	maximal, direct product factor
2	C₂⁵×Dic₆₇	C₂	1	2	yes	maximal
3	C₂⁵×C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₃₄	C₂²	1	2480	yes	direct product factor
5	C₂⁴×Dic₆₇	C₂²	1	124	yes	—
6	C₂⁶×C₆₇	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄×C₆₇	C₂²	1	62	yes	—
8	C₂²×Dic₁₃₄	C₂³	1	9920	yes	direct product factor
9	C₂³×Dic₆₇	C₂³	1	1240	yes	—
10	C₂⁵×C₆₇	Q₈	1	32	yes	complemented
11	C₂⁵×C₆₇	C₂³	1	31	yes	—
12	C₂³×C₄×C₆₇	C₂³	1	620	yes	—
13	C₂×Dic₁₃₄	C₂⁴	1	7936	yes	direct product factor
14	C₂²×Dic₆₇	C₂⁴	1	2480	yes	—
15	C₂⁴×C₆₇	C₂×Q₈	1	496	yes	complemented
16	C₂⁴×C₆₇	C₂⁴	1	155	yes	—
17	C₂²×C₄×C₆₇	C₂⁴	1	1240	yes	—
18	C₂³×C₆₇	C₂²×Q₈	1	1240	yes	complemented
19	C₂³×C₆₇	C₂⁵	1	155	yes	—
20	C₂×C₄×C₆₇	C₂⁵	1	496	yes	—
21	C₂×Dic₆₇	C₂⁵	1	992	yes	—
22	Dic₁₃₄	C₂⁵	1	1024	yes	direct product factor
23	C₂²×C₆₇	C₂³×Q₈	1	620	yes	complemented
24	C₂²×C₆₇	C₂⁶	1	31	yes	—
25	C₄×C₆₇	C₂⁶	1	32	yes	—
26	Dic₆₇	C₂⁶	1	64	yes	—
27	C₂⁵×Q₈	—	67	1	no	maximal, Sylow
28	C₂×C₆₇	C₂⁴×Q₈	1	62	yes	complemented
29	C₂×C₆₇	C₂⁷	1	1	yes	characteristic
30	C₂⁴×Q₈	—	67	124	no	—
31	C₂⁵×C₄	—	67	2	no	—
32	C₂⁵×C₄	D₆₇	1	1	yes	characteristic
33	C₆₇	C₂⁵×Q₈	1	1	yes	characteristic, complemented, Sylow
34	C₂⁶	C₂×D₆₇	1	1	yes	characteristic
35	C₂³×Q₈	—	67	2480	no	—
36	C₂⁴×C₄	—	67	124	no	—
37	C₂⁴×C₄	C₂×D₆₇	1	62	yes	—
38	C₂⁵	Dic₁₃₄	1	32	yes	direct product factor
39	C₂⁵	C₂²×D₆₇	1	31	yes	—
40	C₂²×Q₈	—	67	9920	no	—
41	C₂³×C₄	—	67	1240	no	—
42	C₂³×C₄	C₂²×D₆₇	1	620	yes	—
43	C₂⁴	C₂×Dic₁₃₄	1	496	yes	direct product factor
44	C₂⁴	C₂³×D₆₇	1	155	yes	—
45	C₂×Q₈	—	67	7936	no	—
46	C₂²×C₄	—	67	2480	no	—
47	C₂²×C₄	C₂³×D₆₇	1	1240	yes	—
48	C₂³	C₂²×Dic₁₃₄	1	1240	yes	direct product factor
49	C₂³	C₂⁴×D₆₇	1	155	yes	—
50	Q₈	—	67	1024	no	—
51	C₂×C₄	—	67	992	no	—
52	C₂×C₄	C₂⁴×D₆₇	1	496	yes	—
53	C₂²	C₂³×Dic₁₃₄	1	620	yes	direct product factor
54	C₂²	C₂⁵×D₆₇	1	31	yes	—
55	C₄	—	67	64	no	—
56	C₄	C₂⁵×D₆₇	1	32	yes	—
57	C₂	C₂⁴×Dic₁₃₄	1	62	yes	direct product factor
58	C₂	C₂⁶×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁶	C₂×D₆₇
commutator, cosocle	C₂×C₆₇	C₂⁷
Frattini	C₂	C₂⁶×D₆₇
Fitting	C₂⁵×C₄×C₆₇	C₂
socle, prime radical	C₂⁶×C₆₇	C₂²
hypercenter	C₂⁵×C₄	D₆₇
hypocenter	C₆₇	C₂⁵×Q₈
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁶ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Dic₁₃₄	C₂	direct (N × H)
4. C₂³×Dic₁₃₄	C₂²	direct (N × H)
8. C₂²×Dic₁₃₄	C₂³	direct (N × H)
10. C₂⁵×C₆₇	Q₈	semidirect
13. C₂×Dic₁₃₄	C₂⁴	direct (N × H)
15. C₂⁴×C₆₇	C₂×Q₈	semidirect
18. C₂³×C₆₇	C₂²×Q₈	semidirect
22. Dic₁₃₄	C₂⁵	direct (N × H)
23. C₂²×C₆₇	C₂³×Q₈	semidirect
28. C₂×C₆₇	C₂⁴×Q₈	semidirect
33. C₆₇	C₂⁵×Q₈	semidirect
38. C₂⁵	Dic₁₃₄	direct (N × H)
43. C₂⁴	C₂×Dic₁₃₄	direct (N × H)
48. C₂³	C₂²×Dic₁₃₄	direct (N × H)
53. C₂²	C₂³×Dic₁₃₄	direct (N × H)
57. C₂	C₂⁴×Dic₁₃₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₁₃₄	2
C₂⁵×Dic₆₇	2
C₂⁵×C₄×C₆₇	2
C₂⁵×Q₈	67

Subgroups — C25×Dic134 #31769711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×Dic₁₃₄ #31769711