Subgroups — C₂⁴×#634 #11919176

Order 1152.

18792 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃²×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₃⋊Dic₃	C₂	1	2	yes	maximal
3	C₂³×#634	C₂	1	60	yes	maximal, direct product factor
4	C₂⁴×Dic₆	—	3	4	no	maximal
5	C₂⁵×C₃²	C₂²	1	1	yes	characteristic
6	C₂³×C₃²×C₄	C₂²	1	30	yes	—
7	C₂³×C₃⋊Dic₃	C₂²	1	60	yes	—
8	C₂²×#634	C₂²	1	560	yes	direct product factor
9	C₂⁴×C₃×C₄	D₃	1	4	yes	—
10	C₂⁴×Dic₃	—	3	8	no	—
11	C₂³×Dic₆	—	3	240	no	—
12	C₂⁴×C₃²	Q₈	1	16	yes	complemented
13	C₂⁴×C₃²	C₂³	1	15	yes	—
14	C₂²×C₃²×C₄	C₂³	1	140	yes	—
15	C₂²×C₃⋊Dic₃	C₂³	1	280	yes	—
16	C₂×#634	C₂³	1	960	yes	direct product factor
17	C₂⁴×Q₈	—	9	1	no	Sylow
18	C₂⁵×C₃	C₂×D₃	1	4	yes	—
19	C₂³×C₃×C₄	C₂×D₃	1	120	yes	—
20	C₂³×Dic₃	—	3	240	no	—
21	C₂²×Dic₆	—	3	2240	no	—
22	C₂³×C₃²	C₂×Q₈	1	120	yes	complemented
23	C₂³×C₃²	C₂⁴	1	35	yes	—
24	C₂×C₃²×C₄	C₂⁴	1	120	yes	—
25	C₂×C₃⋊Dic₃	C₂⁴	1	240	yes	—
26	#634	C₂⁴	1	256	yes	direct product factor
27	C₂³×Q₈	—	9	60	no	—
28	C₂⁴×C₄	—	9	2	no	—
29	C₂⁴×C₄	C₃⋊D₃	1	1	yes	characteristic
30	C₂⁴×C₃	Dic₆	1	64	yes	complemented
31	C₂⁴×C₃	C₂²×D₃	1	60	yes	—
32	C₂²×C₃×C₄	C₂²×D₃	1	560	yes	—
33	C₂²×Dic₃	—	3	1120	no	—
34	C₂×Dic₆	—	3	3840	no	—
35	C₂²×C₃²	C₂²×Q₈	1	140	yes	complemented
36	C₂²×C₃²	C₂⁵	1	15	yes	—
37	C₃²×C₄	C₂⁵	1	16	yes	—
38	C₃⋊Dic₃	C₂⁵	1	32	yes	—
39	C₂⁵	C₂×C₃⋊D₃	1	1	yes	characteristic
40	C₂²×Q₈	—	9	560	no	—
41	C₂³×C₄	—	9	60	no	—
42	C₂³×C₄	C₂×C₃⋊D₃	1	30	yes	—
43	C₂³×C₃	C₂×Dic₆	1	480	yes	complemented
44	C₂³×C₃	C₂³×D₃	1	140	yes	—
45	C₂×C₃×C₄	C₂³×D₃	1	480	yes	—
46	C₂×Dic₃	—	3	960	no	—
47	Dic₆	—	3	1024	no	—
48	C₂×C₃²	C₂³×Q₈	1	30	yes	complemented
49	C₂×C₃²	C₂⁶	1	1	yes	characteristic
50	C₂⁴	#634	1	16	yes	direct product factor
51	C₂⁴	C₂²×C₃⋊D₃	1	15	yes	—
52	C₂×Q₈	—	9	960	no	—
53	C₂²×C₄	—	9	280	no	—
54	C₂²×C₄	C₂²×C₃⋊D₃	1	140	yes	—
55	C₂²×C₃	C₂²×Dic₆	1	560	yes	complemented
56	C₂²×C₃	C₂⁴×D₃	1	60	yes	—
57	C₃×C₄	C₂⁴×D₃	1	64	yes	—
58	Dic₃	—	3	128	no	—
59	C₃²	C₂⁴×Q₈	1	1	yes	characteristic, complemented, Sylow
60	C₂³	C₂×#634	1	120	yes	direct product factor
61	C₂³	C₂³×C₃⋊D₃	1	35	yes	—
62	Q₈	—	9	256	no	—
63	C₂×C₄	—	9	240	no	—
64	C₂×C₄	C₂³×C₃⋊D₃	1	120	yes	—
65	C₂×C₃	C₂³×Dic₆	1	120	yes	complemented
66	C₂×C₃	C₂⁵×D₃	1	4	yes	—
67	C₂²	C₂²×#634	1	140	yes	direct product factor
68	C₂²	C₂⁴×C₃⋊D₃	1	15	yes	—
69	C₄	—	9	32	no	—
70	C₄	C₂⁴×C₃⋊D₃	1	16	yes	—
71	C₃	C₂⁴×Dic₆	1	4	yes	complemented
72	C₂	C₂³×#634	1	30	yes	direct product factor
73	C₂	C₂⁵×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂×C₃⋊D₃
commutator, cosocle	C₂×C₃²	C₂⁶
Frattini	C₂	C₂⁵×C₃⋊D₃
Fitting	C₂⁴×C₃²×C₄	C₂
socle, prime radical	C₂⁵×C₃²	C₂²
hypercenter	C₂⁴×C₄	C₃⋊D₃
hypocenter	C₃²	C₂⁴×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
3. C₂³×#634	C₂	direct (N × H)
8. C₂²×#634	C₂²	direct (N × H)
12. C₂⁴×C₃²	Q₈	semidirect
16. C₂×#634	C₂³	direct (N × H)
22. C₂³×C₃²	C₂×Q₈	semidirect
26. #634	C₂⁴	direct (N × H)
30. C₂⁴×C₃	Dic₆	semidirect
35. C₂²×C₃²	C₂²×Q₈	semidirect
43. C₂³×C₃	C₂×Dic₆	semidirect
48. C₂×C₃²	C₂³×Q₈	semidirect
50. C₂⁴	#634	direct (N × H)
55. C₂²×C₃	C₂²×Dic₆	semidirect
59. C₃²	C₂⁴×Q₈	semidirect
60. C₂³	C₂×#634	direct (N × H)
65. C₂×C₃	C₂³×Dic₆	semidirect
67. C₂²	C₂²×#634	direct (N × H)
71. C₃	C₂⁴×Dic₆	semidirect
72. C₂	C₂³×#634	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×C₃²×C₄	2
C₂⁴×C₃⋊Dic₃	2
C₂³×#634	2
C₂⁴×Dic₆	3

Subgroups — C24×#634 #11919176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#634 #11919176