Subgroups — C₃×C₅×#423 #11751398

Order 960.

252 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₃×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×C₄⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₅×#423	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₄×C₅	C₄	1	2	yes	—
6	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₃×C₅×M₁₆	—	2	4	no	—
8	C₂×C₃×C₅×C₈	C₂²	1	4	yes	—
9	C₃×C₄²×C₅	C₄	1	1	yes	characteristic
10	C₃×C₄²×C₅	C₄	1	1	yes	characteristic
11	C₃×C₄²×C₅	C₂²	1	2	yes	—
12	C₃×#423	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₅×M₁₆	C₂×C₃	1	2	yes	—
14	C₂×C₄²×C₅	C₂×C₃	1	1	yes	characteristic
15	C₅×C₄⋊C₈	C₂×C₃	1	4	yes	complemented
16	C₂³×C₃×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃×C₄×C₅	—	2	4	no	—
18	C₂×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
19	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
21	C₂×C₃×C₄×C₅	D₄	1	2	yes	—
22	C₂×C₃×C₄×C₅	Q₈	1	2	yes	—
23	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₃×C₅×C₈	—	2	4	no	—
25	C₂×C₃×M₁₆	C₂×C₅	1	2	yes	—
26	C₂×C₃×C₄²	C₂×C₅	1	1	yes	characteristic
27	C₃×C₄⋊C₈	C₂×C₅	1	4	yes	complemented
28	C₂²×C₄×C₅	C₃×C₄	1	2	yes	—
29	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
30	C₅×M₁₆	—	2	4	no	—
31	C₂×C₅×C₈	C₂²×C₃	1	4	yes	—
32	C₄²×C₅	C₃×C₄	1	1	yes	characteristic
33	C₄²×C₅	C₃×C₄	1	1	yes	characteristic
34	C₄²×C₅	C₂²×C₃	1	2	yes	—
35	#423	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
36	C₂²×C₃×C₅	—	2	2	no	—
37	C₂²×C₃×C₅	C₄⋊C₄	1	2	yes	—
38	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
39	C₃×C₄×C₅	—	2	2	no	—
40	C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
41	C₃×C₄×C₅	M₁₆	1	4	yes	complemented
42	C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₃×C₄	C₄×C₅	1	2	yes	—
45	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
46	C₃×M₁₆	—	2	4	no	—
47	C₂×C₃×C₈	C₂²×C₅	1	4	yes	—
48	C₃×C₄²	C₄×C₅	1	1	yes	characteristic
49	C₃×C₄²	C₄×C₅	1	1	yes	characteristic
50	C₃×C₄²	C₂²×C₅	1	2	yes	—
51	C₂³×C₅	C₂×C₃×C₄	1	1	yes	characteristic
52	C₂×C₄×C₅	—	2	4	no	—
53	C₂×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
54	C₂×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
55	C₂×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
56	C₂×C₄×C₅	C₃×D₄	1	2	yes	—
57	C₂×C₄×C₅	C₃×Q₈	1	2	yes	—
58	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
59	C₅×C₈	—	2	4	no	—
60	C₂×M₁₆	C₂×C₃×C₅	1	2	yes	—
61	C₂×C₄²	C₂×C₃×C₅	1	1	yes	characteristic
62	C₄⋊C₈	C₂×C₃×C₅	1	4	yes	complemented
63	C₂×C₃×C₅	—	2	2	no	—
64	C₂×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₃×C₅	C₂×M₁₆	1	2	yes	—
66	C₂³×C₃	C₂×C₄×C₅	1	1	yes	characteristic
67	C₂×C₃×C₄	—	2	4	no	—
68	C₂×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
69	C₂×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
70	C₂×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
71	C₂×C₃×C₄	C₅×D₄	1	2	yes	—
72	C₂×C₃×C₄	C₅×Q₈	1	2	yes	—
73	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
74	C₃×C₈	—	2	4	no	—
75	C₂²×C₅	—	2	2	no	—
76	C₂²×C₅	C₃×C₄⋊C₄	1	2	yes	—
77	C₂²×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
78	C₄×C₅	—	2	2	no	—
79	C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
80	C₄×C₅	C₃×M₁₆	1	4	yes	complemented
81	C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
82	C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
83	C₂²×C₄	C₃×C₄×C₅	1	2	yes	—
84	C₂²×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
85	M₁₆	—	2	4	no	—
86	C₂×C₈	C₂²×C₃×C₅	1	4	yes	—
87	C₄²	C₃×C₄×C₅	1	1	yes	characteristic
88	C₄²	C₃×C₄×C₅	1	1	yes	characteristic
89	C₄²	C₂²×C₃×C₅	1	2	yes	—
90	C₃×C₅	#423	1	1	yes	characteristic, direct product factor, Hall
91	C₂²×C₃	—	2	2	no	—
92	C₂²×C₃	C₅×C₄⋊C₄	1	2	yes	—
93	C₂²×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
94	C₃×C₄	—	2	2	no	—
95	C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
96	C₃×C₄	C₅×M₁₆	1	4	yes	complemented
97	C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
98	C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
99	C₂×C₅	—	2	2	no	—
100	C₂×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₅	C₂×C₃×M₁₆	1	2	yes	—
102	C₂³	C₂×C₃×C₄×C₅	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
105	C₂×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
106	C₂×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
107	C₂×C₄	C₃×C₅×D₄	1	2	yes	—
108	C₂×C₄	C₃×C₅×Q₈	1	2	yes	—
109	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
110	C₈	—	2	4	no	—
111	C₂×C₃	—	2	2	no	—
112	C₂×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₃	C₂×C₅×M₁₆	1	2	yes	—
114	C₅	C₃×#423	1	1	yes	characteristic, direct product factor, Sylow
115	C₂²	—	2	2	no	—
116	C₂²	C₃×C₅×C₄⋊C₄	1	2	yes	—
117	C₂²	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	C₃×C₅×C₄⋊C₄	1	2	yes	—
120	C₄	C₃×C₅×M₁₆	1	4	yes	complemented
121	C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
122	C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
123	C₃	C₅×#423	1	1	yes	characteristic, direct product factor, Sylow
124	C₂	—	2	2	no	—
125	C₂	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
126	C₂	C₂×C₃×C₅×M₁₆	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂³
commutator	C₂²	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂²×C₄
prime radical	C₂³×C₃×C₅	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×C₄⋊C₈	C₂	semidirect
4. C₅×#423	C₃	direct (N × H)
12. C₃×#423	C₅	direct (N × H)
15. C₅×C₄⋊C₈	C₂×C₃	semidirect
27. C₃×C₄⋊C₈	C₂×C₅	semidirect
35. #423	C₃×C₅	direct (N × H)
41. C₃×C₄×C₅	M₁₆	semidirect
62. C₄⋊C₈	C₂×C₃×C₅	semidirect
80. C₄×C₅	C₃×M₁₆	semidirect
90. C₃×C₅	#423	direct (N × H)
96. C₃×C₄	C₅×M₁₆	semidirect
114. C₅	C₃×#423	direct (N × H)
120. C₄	C₃×C₅×M₁₆	semidirect
123. C₃	C₅×#423	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×M₁₆	2
C₂×C₃×C₄²×C₅	2
C₃×C₅×C₄⋊C₈	2
C₅×#423	3
C₃×#423	5

Subgroups — C3×C5×#423 #11751398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#423 #11751398