Subgroups — C₂⁴×#1088 #13180100

Order 1728.

3539 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#272	C₂	1	1	yes	maximal, characteristic
2	C₂³×#1088	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×H(3,𝔽₃)	C₄	1	1	yes	characteristic, complemented
4	C₂³×#272	C₄	1	15	yes	complemented
5	C₂³×#272	C₂²	1	15	yes	—
6	C₂²×#1088	C₂²	1	140	yes	direct product factor
7	C₂⁴×C₃×D₃	—	6	2	no	—
8	C₂³×H(3,𝔽₃)	C₂×C₄	1	15	yes	complemented
9	C₂²×#272	C₂×C₄	1	105	yes	complemented
10	C₂²×#272	C₂³	1	35	yes	—
11	C₂×#1088	C₂³	1	120	yes	direct product factor
12	C₂⁴×C₃×C₄	—	9	1	no	maximal
13	C₂⁴×C₃²	—	2	2	no	—
14	C₂³×C₃×D₃	—	6	30	no	—
15	C₂³×C₃×D₃	—	6	30	no	—
16	C₂²×H(3,𝔽₃)	C₂²×C₄	1	35	yes	complemented
17	C₂×#272	C₂²×C₄	1	105	yes	complemented
18	C₂×#272	C₂⁴	1	15	yes	—
19	#1088	C₂⁴	1	16	yes	direct product factor
20	C₂⁵×C₃	—	9	1	no	—
21	C₂⁴×D₃	—	6	2	no	—
22	C₂³×C₃×C₄	—	9	30	no	—
23	C₂³×C₃²	—	2	30	no	—
24	C₂²×C₃×D₃	—	6	210	no	—
25	C₂²×C₃×D₃	—	6	70	no	—
26	C₂⁴×C₄	—	9	1	no	Sylow
27	C₂×H(3,𝔽₃)	C₂³×C₄	1	15	yes	complemented
28	#272	C₂³×C₄	1	15	yes	complemented
29	#272	C₂⁵	1	1	yes	characteristic
30	C₂⁴×C₃	—	6	2	no	—
31	C₂⁴×C₃	—	9	15	no	—
32	C₂⁴×C₃	—	9	15	no	—
33	C₂⁴×C₃	#157	1	1	yes	characteristic
34	C₂³×D₃	—	6	30	no	—
35	C₂³×D₃	—	6	30	no	—
36	C₂²×C₃×C₄	—	9	140	no	—
37	C₂²×C₃²	—	2	70	no	—
38	C₂×C₃×D₃	—	6	210	no	—
39	C₂×C₃×D₃	—	6	30	no	—
40	C₂⁵	—	9	1	no	—
41	C₂³×C₄	—	9	30	no	—
42	H(3,𝔽₃)	C₂⁴×C₄	1	1	yes	characteristic, complemented, Sylow
43	C₂³×C₃	—	6	30	no	—
44	C₂³×C₃	—	9	35	no	—
45	C₂³×C₃	—	9	105	no	—
46	C₂³×C₃	C₂×#157	1	15	yes	—
47	C₂²×D₃	—	6	70	no	—
48	C₂²×D₃	—	6	210	no	—
49	C₂×C₃×C₄	—	9	120	no	—
50	C₂×C₃²	—	2	30	no	—
51	C₃×D₃	—	6	30	no	—
52	C₃×D₃	—	6	2	no	—
53	C₂⁴	—	9	15	no	—
54	C₂⁴	—	9	15	no	—
55	C₂⁴	#1088	1	1	yes	characteristic, direct product factor
56	C₂²×C₄	—	9	140	no	—
57	C₂²×C₃	—	6	70	no	—
58	C₂²×C₃	—	9	15	no	—
59	C₂²×C₃	—	9	105	no	—
60	C₂²×C₃	C₂²×#157	1	35	yes	—
61	C₂×D₃	—	6	30	no	—
62	C₂×D₃	—	6	210	no	—
63	C₃×C₄	—	9	16	no	—
64	C₃²	—	2	2	no	—
65	C₂³	—	9	35	no	—
66	C₂³	—	9	105	no	—
67	C₂³	C₂×#1088	1	15	yes	direct product factor
68	C₂×C₄	—	9	120	no	—
69	C₂×C₃	—	6	30	no	—
70	C₂×C₃	—	9	1	no	—
71	C₂×C₃	—	9	15	no	—
72	C₂×C₃	C₂³×#157	1	15	yes	—
73	D₃	—	6	2	no	—
74	D₃	—	6	30	no	—
75	C₂²	—	9	15	no	—
76	C₂²	—	9	105	no	—
77	C₂²	C₂²×#1088	1	35	yes	direct product factor
78	C₄	—	9	16	no	—
79	C₃	—	6	2	no	—
80	C₃	C₂⁴×#157	1	1	yes	characteristic
81	C₂	—	9	1	no	—
82	C₂	—	9	15	no	—
83	C₂	C₂³×#1088	1	15	yes	direct product factor

	subgroup	G/H
center, socle, hypercenter	C₂⁴×C₃	#157
commutator, hypocenter	H(3,𝔽₃)	C₂⁴×C₄
Frattini	C₃	C₂⁴×#157
Fitting	C₂⁴×H(3,𝔽₃)	C₄
prime radical	C₂⁴×#272	C₂
cosocle	#272	C₂⁵
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂⁴×C₃

Complements

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

N	H ≅ G/N	type
2. C₂³×#1088	C₂	direct (N × H)
3. C₂⁴×H(3,𝔽₃)	C₄	semidirect
4. C₂³×#272	C₄	semidirect
6. C₂²×#1088	C₂²	direct (N × H)
8. C₂³×H(3,𝔽₃)	C₂×C₄	semidirect
9. C₂²×#272	C₂×C₄	semidirect
11. C₂×#1088	C₂³	direct (N × H)
16. C₂²×H(3,𝔽₃)	C₂²×C₄	semidirect
17. C₂×#272	C₂²×C₄	semidirect
19. #1088	C₂⁴	direct (N × H)
27. C₂×H(3,𝔽₃)	C₂³×C₄	semidirect
28. #272	C₂³×C₄	semidirect
42. H(3,𝔽₃)	C₂⁴×C₄	semidirect
55. C₂⁴	#1088	direct (N × H)
67. C₂³	C₂×#1088	direct (N × H)
77. C₂²	C₂²×#1088	direct (N × H)
83. C₂	C₂³×#1088	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×#272	2
C₂³×#1088	2
C₂⁴×C₃×C₄	9

Subgroups — C24×#1088 #13180100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#1088 #13180100