Subgroups — C₂⁴×C₄×Dic₆₇ #31762380

Order 17152.

24030 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×Dic₆₇	C₂	1	2	yes	maximal
2	C₂³×C₄×Dic₆₇	C₂	1	60	yes	maximal, direct product factor
3	C₂⁵×C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×Dic₆₇	C₄	1	64	yes	direct product factor
5	C₂⁴×Dic₆₇	C₂²	1	60	yes	—
6	C₂²×C₄×Dic₆₇	C₂²	1	560	yes	direct product factor
7	C₂⁶×C₆₇	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₆₇	C₄	1	32	yes	complemented
9	C₂⁴×C₄×C₆₇	C₂²	1	30	yes	—
10	C₂³×Dic₆₇	C₂×C₄	1	960	yes	direct product factor
11	C₂³×Dic₆₇	C₂³	1	280	yes	—
12	C₂×C₄×Dic₆₇	C₂³	1	960	yes	direct product factor
13	C₂⁵×C₆₇	C₂×C₄	1	32	yes	—
14	C₂⁵×C₆₇	C₂×C₄	1	16	yes	—
15	C₂⁵×C₆₇	C₂³	1	15	yes	—
16	C₂³×C₄×C₆₇	C₂×C₄	1	480	yes	complemented
17	C₂³×C₄×C₆₇	C₂³	1	140	yes	—
18	C₂²×Dic₆₇	C₂²×C₄	1	2240	yes	direct product factor
19	C₂²×Dic₆₇	C₂⁴	1	240	yes	—
20	C₄×Dic₆₇	C₂⁴	1	256	yes	direct product factor
21	C₂⁴×C₆₇	C₄²	1	256	yes	complemented
22	C₂⁴×C₆₇	C₂²×C₄	1	240	yes	—
23	C₂⁴×C₆₇	C₂²×C₄	1	120	yes	—
24	C₂⁴×C₆₇	C₂⁴	1	35	yes	—
25	C₂²×C₄×C₆₇	C₂²×C₄	1	1120	yes	complemented
26	C₂²×C₄×C₆₇	C₂⁴	1	120	yes	—
27	C₂³×C₆₇	C₂×C₄²	1	960	yes	complemented
28	C₂³×C₆₇	C₂³×C₄	1	280	yes	—
29	C₂³×C₆₇	C₂³×C₄	1	140	yes	—
30	C₂³×C₆₇	C₂⁵	1	15	yes	—
31	C₂×C₄×C₆₇	C₂³×C₄	1	480	yes	complemented
32	C₂×C₄×C₆₇	C₂⁵	1	16	yes	—
33	C₂×Dic₆₇	C₂³×C₄	1	960	yes	direct product factor
34	C₂×Dic₆₇	C₂⁵	1	32	yes	—
35	C₂²×C₆₇	C₂²×C₄²	1	560	yes	complemented
36	C₂²×C₆₇	C₂⁴×C₄	1	60	yes	—
37	C₂²×C₆₇	C₂⁴×C₄	1	30	yes	—
38	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
39	C₄×C₆₇	C₂⁴×C₄	1	32	yes	complemented
40	Dic₆₇	C₂⁴×C₄	1	64	yes	direct product factor
41	C₂⁴×C₄²	—	67	1	no	maximal, Sylow
42	C₂×C₆₇	C₂³×C₄²	1	60	yes	complemented
43	C₂×C₆₇	C₂⁵×C₄	1	2	yes	—
44	C₂×C₆₇	C₂⁵×C₄	1	1	yes	characteristic
45	C₂⁵×C₄	—	67	2	no	—
46	C₂⁵×C₄	D₆₇	1	1	yes	characteristic
47	C₂³×C₄²	—	67	60	no	—
48	C₆₇	C₂⁴×C₄²	1	1	yes	characteristic, complemented, Sylow
49	C₂⁶	C₂×D₆₇	1	1	yes	characteristic
50	C₂⁴×C₄	—	67	64	no	—
51	C₂⁴×C₄	—	67	60	no	—
52	C₂⁴×C₄	Dic₆₇	1	32	yes	direct product factor
53	C₂⁴×C₄	C₂×D₆₇	1	30	yes	—
54	C₂²×C₄²	—	67	560	no	—
55	C₂⁵	C₄×D₆₇	1	32	yes	—
56	C₂⁵	C₂×Dic₆₇	1	16	yes	—
57	C₂⁵	C₂²×D₆₇	1	15	yes	—
58	C₂³×C₄	—	67	960	no	—
59	C₂³×C₄	—	67	280	no	—
60	C₂³×C₄	C₂×Dic₆₇	1	480	yes	direct product factor
61	C₂³×C₄	C₂²×D₆₇	1	140	yes	—
62	C₂×C₄²	—	67	960	no	—
63	C₂⁴	C₄×Dic₆₇	1	256	yes	direct product factor
64	C₂⁴	C₂×C₄×D₆₇	1	240	yes	—
65	C₂⁴	C₂²×Dic₆₇	1	120	yes	—
66	C₂⁴	C₂³×D₆₇	1	35	yes	—
67	C₂²×C₄	—	67	2240	no	—
68	C₂²×C₄	—	67	240	no	—
69	C₂²×C₄	C₂²×Dic₆₇	1	1120	yes	direct product factor
70	C₂²×C₄	C₂³×D₆₇	1	120	yes	—
71	C₄²	—	67	256	no	—
72	C₂³	C₂×C₄×Dic₆₇	1	960	yes	direct product factor
73	C₂³	C₂²×C₄×D₆₇	1	280	yes	—
74	C₂³	C₂³×Dic₆₇	1	140	yes	—
75	C₂³	C₂⁴×D₆₇	1	15	yes	—
76	C₂×C₄	—	67	960	no	—
77	C₂×C₄	—	67	32	no	—
78	C₂×C₄	C₂³×Dic₆₇	1	480	yes	direct product factor
79	C₂×C₄	C₂⁴×D₆₇	1	16	yes	—
80	C₂²	C₂²×C₄×Dic₆₇	1	560	yes	direct product factor
81	C₂²	C₂³×C₄×D₆₇	1	60	yes	—
82	C₂²	C₂⁴×Dic₆₇	1	30	yes	—
83	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
84	C₄	—	67	64	no	—
85	C₄	C₂⁴×Dic₆₇	1	32	yes	direct product factor
86	C₂	C₂³×C₄×Dic₆₇	1	60	yes	direct product factor
87	C₂	C₂⁴×C₄×D₆₇	1	2	yes	—
88	C₂	C₂⁵×Dic₆₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁵×C₄	D₆₇
commutator, hypocenter	C₆₇	C₂⁴×C₄²
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂⁵×C₄×C₆₇	C₂
socle, prime radical	C₂⁶×C₆₇	C₂²
cosocle	C₂²×C₆₇	C₂⁶
LCS	G ▷ C₆₇
UCS	C₁ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄×Dic₆₇	C₂	direct (N × H)
4. C₂⁴×Dic₆₇	C₄	direct (N × H)
6. C₂²×C₄×Dic₆₇	C₂²	direct (N × H)
8. C₂⁴×C₄×C₆₇	C₄	semidirect
10. C₂³×Dic₆₇	C₂×C₄	direct (N × H)
12. C₂×C₄×Dic₆₇	C₂³	direct (N × H)
16. C₂³×C₄×C₆₇	C₂×C₄	semidirect
18. C₂²×Dic₆₇	C₂²×C₄	direct (N × H)
20. C₄×Dic₆₇	C₂⁴	direct (N × H)
21. C₂⁴×C₆₇	C₄²	semidirect
25. C₂²×C₄×C₆₇	C₂²×C₄	semidirect
27. C₂³×C₆₇	C₂×C₄²	semidirect
31. C₂×C₄×C₆₇	C₂³×C₄	semidirect
33. C₂×Dic₆₇	C₂³×C₄	direct (N × H)
35. C₂²×C₆₇	C₂²×C₄²	semidirect
39. C₄×C₆₇	C₂⁴×C₄	semidirect
40. Dic₆₇	C₂⁴×C₄	direct (N × H)
42. C₂×C₆₇	C₂³×C₄²	semidirect
48. C₆₇	C₂⁴×C₄²	semidirect
52. C₂⁴×C₄	Dic₆₇	direct (N × H)
60. C₂³×C₄	C₂×Dic₆₇	direct (N × H)
63. C₂⁴	C₄×Dic₆₇	direct (N × H)
69. C₂²×C₄	C₂²×Dic₆₇	direct (N × H)
72. C₂³	C₂×C₄×Dic₆₇	direct (N × H)
78. C₂×C₄	C₂³×Dic₆₇	direct (N × H)
80. C₂²	C₂²×C₄×Dic₆₇	direct (N × H)
85. C₄	C₂⁴×Dic₆₇	direct (N × H)
86. C₂	C₂³×C₄×Dic₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×Dic₆₇	2
C₂³×C₄×Dic₆₇	2
C₂⁵×C₄×C₆₇	2
C₂⁴×C₄²	67

Subgroups — C24×C4×Dic67 #31762380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₄×Dic₆₇ #31762380