Subgroups — C₂×C₃²×C₅×C₄⋊C₄ #13088225

Order 1440.

552 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄×C₅	C₂	1	2	yes	maximal
2	C₂²×C₃²×C₄×C₅	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₅×C₄⋊C₄	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×C₅×C₄⋊C₄	C₃	1	4	yes	maximal, direct product factor
5	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic
6	C₂×C₃²×C₄×C₅	—	2	4	no	—
7	C₂×C₃²×C₄×C₅	C₄	1	4	yes	complemented
8	C₂×C₃²×C₄×C₅	C₂²	1	4	yes	—
9	C₂×C₃²×C₄×C₅	C₂²	1	2	yes	—
10	C₂×C₃²×C₄⋊C₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂²×C₃×C₄×C₅	C₂×C₃	1	8	yes	—
12	C₂²×C₃×C₄×C₅	C₂×C₃	1	4	yes	—
13	C₃×C₅×C₄⋊C₄	C₂×C₃	1	16	yes	direct product factor
14	C₂²×C₃²×C₅	C₂×C₄	1	2	yes	—
15	C₂²×C₃²×C₅	D₄	1	2	yes	—
16	C₂²×C₃²×C₅	Q₈	1	2	yes	—
17	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
18	C₃²×C₄×C₅	—	2	4	no	—
19	C₃²×C₄×C₅	C₂×C₄	1	4	yes	complemented
20	C₂×C₅×C₄⋊C₄	C₃²	1	1	yes	characteristic, direct product factor, Hall
21	C₂²×C₃²×C₄	C₂×C₅	1	2	yes	—
22	C₂²×C₃²×C₄	C₂×C₅	1	1	yes	characteristic
23	C₃²×C₄⋊C₄	C₂×C₅	1	4	yes	direct product factor
24	C₂³×C₃×C₅	C₂²×C₃	1	4	yes	—
25	C₂×C₃×C₄×C₅	—	2	16	no	—
26	C₂×C₃×C₄×C₅	C₃×C₄	1	16	yes	complemented
27	C₂×C₃×C₄×C₅	C₂²×C₃	1	16	yes	—
28	C₂×C₃×C₄×C₅	C₂²×C₃	1	8	yes	—
29	C₂×C₃×C₄⋊C₄	C₃×C₅	1	4	yes	direct product factor
30	C₂×C₃²×C₅	C₄⋊C₄	1	4	yes	direct product factor
31	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
33	C₂×C₃²×C₅	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₅	C₂×C₃²	1	2	yes	—
35	C₂²×C₄×C₅	C₂×C₃²	1	1	yes	characteristic
36	C₅×C₄⋊C₄	C₂×C₃²	1	4	yes	direct product factor
37	C₂³×C₃²	C₂²×C₅	1	1	yes	characteristic
38	C₂×C₃²×C₄	—	2	4	no	—
39	C₂×C₃²×C₄	C₄×C₅	1	4	yes	complemented
40	C₂×C₃²×C₄	C₂²×C₅	1	4	yes	—
41	C₂×C₃²×C₄	C₂²×C₅	1	2	yes	—
42	C₂²×C₃×C₅	C₂×C₃×C₄	1	8	yes	—
43	C₂²×C₃×C₅	C₃×D₄	1	8	yes	—
44	C₂²×C₃×C₅	C₃×Q₈	1	8	yes	—
45	C₂²×C₃×C₅	C₂³×C₃	1	4	yes	—
46	C₃×C₄×C₅	—	2	16	no	—
47	C₃×C₄×C₅	C₂×C₃×C₄	1	16	yes	complemented
48	C₂²×C₃×C₄	C₂×C₃×C₅	1	8	yes	—
49	C₂²×C₃×C₄	C₂×C₃×C₅	1	4	yes	—
50	C₃×C₄⋊C₄	C₂×C₃×C₅	1	16	yes	direct product factor
51	C₃²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Hall
52	C₂³×C₅	C₂²×C₃²	1	1	yes	characteristic
53	C₂×C₄×C₅	—	2	4	no	—
54	C₂×C₄×C₅	C₃²×C₄	1	4	yes	complemented
55	C₂×C₄×C₅	C₂²×C₃²	1	4	yes	—
56	C₂×C₄×C₅	C₂²×C₃²	1	2	yes	—
57	C₂²×C₃²	C₂×C₄×C₅	1	2	yes	—
58	C₂²×C₃²	C₅×D₄	1	2	yes	—
59	C₂²×C₃²	C₅×Q₈	1	2	yes	—
60	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
61	C₃²×C₄	—	2	4	no	—
62	C₃²×C₄	C₂×C₄×C₅	1	4	yes	complemented
63	C₂×C₄⋊C₄	C₃²×C₅	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×C₃×C₅	C₃×C₄⋊C₄	1	16	yes	direct product factor
65	C₂×C₃×C₅	C₂²×C₃×C₄	1	4	yes	—
66	C₂×C₃×C₅	C₂×C₃×D₄	1	4	yes	—
67	C₂×C₃×C₅	C₂×C₃×Q₈	1	4	yes	—
68	C₂³×C₃	C₂²×C₃×C₅	1	4	yes	—
69	C₂×C₃×C₄	—	2	16	no	—
70	C₂×C₃×C₄	C₃×C₄×C₅	1	16	yes	complemented
71	C₂×C₃×C₄	C₂²×C₃×C₅	1	16	yes	—
72	C₂×C₃×C₄	C₂²×C₃×C₅	1	8	yes	—
73	C₂²×C₅	C₂×C₃²×C₄	1	2	yes	—
74	C₂²×C₅	C₃²×D₄	1	2	yes	—
75	C₂²×C₅	C₃²×Q₈	1	2	yes	—
76	C₂²×C₅	C₂³×C₃²	1	1	yes	characteristic
77	C₄×C₅	—	2	4	no	—
78	C₄×C₅	C₂×C₃²×C₄	1	4	yes	complemented
79	C₂×C₃²	C₅×C₄⋊C₄	1	4	yes	direct product factor
80	C₂×C₃²	C₂²×C₄×C₅	1	1	yes	characteristic
81	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
82	C₂×C₃²	C₂×C₅×Q₈	1	1	yes	characteristic
83	C₂²×C₄	C₂×C₃²×C₅	1	2	yes	—
84	C₂²×C₄	C₂×C₃²×C₅	1	1	yes	characteristic
85	C₄⋊C₄	C₂×C₃²×C₅	1	4	yes	direct product factor
86	C₃×C₅	C₂×C₃×C₄⋊C₄	1	4	yes	direct product factor
87	C₂²×C₃	C₂×C₃×C₄×C₅	1	8	yes	—
88	C₂²×C₃	C₃×C₅×D₄	1	8	yes	—
89	C₂²×C₃	C₃×C₅×Q₈	1	8	yes	—
90	C₂²×C₃	C₂³×C₃×C₅	1	4	yes	—
91	C₃×C₄	—	2	16	no	—
92	C₃×C₄	C₂×C₃×C₄×C₅	1	16	yes	complemented
93	C₂×C₅	C₃²×C₄⋊C₄	1	4	yes	direct product factor
94	C₂×C₅	C₂²×C₃²×C₄	1	1	yes	characteristic
95	C₂×C₅	C₂×C₃²×D₄	1	1	yes	characteristic
96	C₂×C₅	C₂×C₃²×Q₈	1	1	yes	characteristic
97	C₃²	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
98	C₂³	C₂²×C₃²×C₅	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	C₃²×C₄×C₅	1	4	yes	complemented
101	C₂×C₄	C₂²×C₃²×C₅	1	4	yes	—
102	C₂×C₄	C₂²×C₃²×C₅	1	2	yes	—
103	C₂×C₃	C₃×C₅×C₄⋊C₄	1	16	yes	direct product factor
104	C₂×C₃	C₂²×C₃×C₄×C₅	1	4	yes	—
105	C₂×C₃	C₂×C₃×C₅×D₄	1	4	yes	—
106	C₂×C₃	C₂×C₃×C₅×Q₈	1	4	yes	—
107	C₅	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
108	C₂²	C₂×C₃²×C₄×C₅	1	2	yes	—
109	C₂²	C₃²×C₅×D₄	1	2	yes	—
110	C₂²	C₃²×C₅×Q₈	1	2	yes	—
111	C₂²	C₂³×C₃²×C₅	1	1	yes	characteristic
112	C₄	—	2	4	no	—
113	C₄	C₂×C₃²×C₄×C₅	1	4	yes	complemented
114	C₃	C₂×C₃×C₅×C₄⋊C₄	1	4	yes	direct product factor
115	C₂	C₃²×C₅×C₄⋊C₄	1	4	yes	direct product factor
116	C₂	C₂²×C₃²×C₄×C₅	1	1	yes	characteristic
117	C₂	C₂×C₃²×C₅×D₄	1	1	yes	characteristic
118	C₂	C₂×C₃²×C₅×Q₈	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃²×C₅	C₂²
commutator	C₂	C₂²×C₃²×C₄×C₅
Frattini, cosocle	C₂²	C₂³×C₃²×C₅
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×C₅×C₄⋊C₄	C₂	direct (N × H)
4. C₂×C₃×C₅×C₄⋊C₄	C₃	direct (N × H)
7. C₂×C₃²×C₄×C₅	C₄	semidirect
10. C₂×C₃²×C₄⋊C₄	C₅	direct (N × H)
13. C₃×C₅×C₄⋊C₄	C₂×C₃	direct (N × H)
19. C₃²×C₄×C₅	C₂×C₄	semidirect
20. C₂×C₅×C₄⋊C₄	C₃²	direct (N × H)
23. C₃²×C₄⋊C₄	C₂×C₅	direct (N × H)
26. C₂×C₃×C₄×C₅	C₃×C₄	semidirect
29. C₂×C₃×C₄⋊C₄	C₃×C₅	direct (N × H)
30. C₂×C₃²×C₅	C₄⋊C₄	direct (N × H)
36. C₅×C₄⋊C₄	C₂×C₃²	direct (N × H)
39. C₂×C₃²×C₄	C₄×C₅	semidirect
47. C₃×C₄×C₅	C₂×C₃×C₄	semidirect
50. C₃×C₄⋊C₄	C₂×C₃×C₅	direct (N × H)
51. C₃²×C₅	C₂×C₄⋊C₄	direct (N × H)
54. C₂×C₄×C₅	C₃²×C₄	semidirect
62. C₃²×C₄	C₂×C₄×C₅	semidirect
63. C₂×C₄⋊C₄	C₃²×C₅	direct (N × H)
64. C₂×C₃×C₅	C₃×C₄⋊C₄	direct (N × H)
70. C₂×C₃×C₄	C₃×C₄×C₅	semidirect
78. C₄×C₅	C₂×C₃²×C₄	semidirect
79. C₂×C₃²	C₅×C₄⋊C₄	direct (N × H)
85. C₄⋊C₄	C₂×C₃²×C₅	direct (N × H)
86. C₃×C₅	C₂×C₃×C₄⋊C₄	direct (N × H)
92. C₃×C₄	C₂×C₃×C₄×C₅	semidirect
93. C₂×C₅	C₃²×C₄⋊C₄	direct (N × H)
97. C₃²	C₂×C₅×C₄⋊C₄	direct (N × H)
100. C₂×C₄	C₃²×C₄×C₅	semidirect
103. C₂×C₃	C₃×C₅×C₄⋊C₄	direct (N × H)
107. C₅	C₂×C₃²×C₄⋊C₄	direct (N × H)
113. C₄	C₂×C₃²×C₄×C₅	semidirect
114. C₃	C₂×C₃×C₅×C₄⋊C₄	direct (N × H)
115. C₂	C₃²×C₅×C₄⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₄×C₅	2
C₃²×C₅×C₄⋊C₄	2
C₂×C₃×C₅×C₄⋊C₄	3
C₂×C₃²×C₄⋊C₄	5

Subgroups — C2×C32×C5×C4⋊C4 #13088225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₅×C₄⋊C₄ #13088225