Subgroups — #12015737

Order 1280.

374 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#64725	C₂	1	4	yes	maximal
3	#10600558	C₂	1	2	yes	maximal
4	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
5	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂×#3950	—	2	2	no	—
7	C₂×#3950	C₂²	1	4	yes	—
8	#65224	—	2	2	no	—
9	C₄×C₅×C₁₆	C₄	1	4	yes	complemented
10	#64725	—	2	8	no	—
11	#12906	—	5	1	no	maximal, Sylow
12	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
13	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
15	C₂×C₄⋊Dic₅	—	2	4	no	—
16	C₂×C₄⋊Dic₅	—	4	4	no	—
17	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
18	C₂×C₅×C₁₆	D₄	1	4	yes	—
19	C₂×C₅×C₁₆	Q₈	1	4	yes	—
20	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
21	C₄×C₅×C₈	Q₈	1	2	yes	—
22	#3950	—	2	8	no	—
23	#3950	—	4	4	no	—
24	C₂×#367	—	5	4	no	—
25	C₂×C₄×C₁₆	D₅	1	1	yes	characteristic
26	#1845	—	5	2	no	—
27	C₂²×C₄×C₅	D₈	1	2	yes	—
28	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
29	C₂²×Dic₅	—	4	4	no	—
30	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
31	C₂×C₅×C₈	D₈	1	2	yes	—
32	C₂×C₅×C₈	Q₁₆	1	2	yes	—
33	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
35	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
36	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₅	C₄⋊C₄	1	2	yes	—
38	C₄²×C₅	Q₁₆	1	2	yes	—
39	C₄⋊Dic₅	—	4	4	no	—
40	C₄⋊Dic₅	—	4	4	no	—
41	C₄⋊Dic₅	—	8	8	no	—
42	C₅×C₁₆	C₄⋊C₄	1	8	yes	complemented
43	C₂²×C₁₆	C₂×D₅	1	2	yes	—
44	C₂×#107	—	5	4	no	—
45	C₂×#107	—	10	2	no	—
46	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
47	#384	—	10	2	no	—
48	#367	—	10	8	no	—
49	C₄×C₁₆	Dic₅	1	4	yes	complemented
50	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₅	#107	1	4	yes	—
52	C₂×C₄×C₅	SD₃₂	1	8	yes	—
53	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅	#127	1	1	yes	characteristic
55	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
56	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
58	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₅	—	8	4	no	—
60	C₂×Dic₅	—	8	8	no	—
61	C₅×C₈	#107	1	4	yes	—
62	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
63	C₅×C₈	C₄⋊Q₈	1	2	yes	—
64	C₂²×C₈	D₂₀	1	1	yes	characteristic
65	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
66	C₂×C₄⋊C₄	—	10	4	no	—
67	C₂×C₄⋊C₄	—	20	4	no	—
68	C₂×C₄²	D₂₀	1	1	yes	characteristic
69	C₂×C₁₆	Dic₁₀	1	4	yes	—
70	C₂×C₁₆	D₂₀	1	4	yes	—
71	C₂×C₁₆	C₂×Dic₅	1	4	yes	—
72	#107	—	10	8	no	—
73	#107	—	20	4	no	—
74	C₄×C₈	Dic₁₀	1	2	yes	—
75	C₄×C₈	C₂×Dic₅	1	2	yes	—
76	C₂²×C₅	C₂×#107	1	1	yes	characteristic
77	C₂²×C₅	#493	1	1	yes	characteristic
78	C₂²×C₅	#500	1	1	yes	characteristic
79	C₂²×C₅	C₂×SD₃₂	1	4	yes	—
80	C₄×C₅	#367	1	8	yes	complemented
81	C₄×C₅	#384	1	1	yes	characteristic
82	C₄×C₅	C₂×#107	1	1	yes	characteristic
83	C₄×C₅	#494	1	1	yes	characteristic
84	C₄×C₅	#500	1	1	yes	characteristic
85	Dic₅	—	16	4	no	—
86	C₂²×C₄	—	20	4	no	—
87	C₂²×C₄	D₄₀	1	2	yes	—
88	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
89	C₂×C₈	D₄₀	1	2	yes	—
90	C₂×C₈	Dic₂₀	1	2	yes	—
91	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
92	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
93	C₂×C₈	C₂×Dic₁₀	1	2	yes	—
94	C₂×C₈	C₂×D₂₀	1	2	yes	—
95	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
96	C₄⋊C₄	—	20	4	no	—
97	C₄⋊C₄	—	20	4	no	—
98	C₄⋊C₄	—	40	8	no	—
99	C₄²	Dic₂₀	1	2	yes	—
100	C₄²	C₄⋊Dic₅	1	2	yes	—
101	C₁₆	C₄⋊Dic₅	1	8	yes	complemented
102	C₂×C₅	#1845	1	1	yes	characteristic
103	C₂×C₅	C₂×#367	1	2	yes	—
104	C₂×C₅	#2248	1	2	yes	—
105	C₂×C₅	#2254	1	2	yes	—
106	C₂³	C₂×D₄₀	1	1	yes	characteristic
107	C₂×C₄	—	40	8	no	—
108	C₂×C₄	—	40	4	no	—
109	C₂×C₄	#3932	1	8	yes	—
110	C₂×C₄	#3950	1	4	yes	—
111	C₂×C₄	#4015	1	1	yes	characteristic
112	C₂×C₄	#4020	1	1	yes	characteristic
113	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
114	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
115	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
116	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
117	C₈	#3950	1	4	yes	—
118	C₈	#4015	1	2	yes	—
119	C₈	C₂×C₄⋊Dic₅	1	2	yes	—
120	C₅	#12906	1	1	yes	characteristic, complemented, Sylow
121	C₂²	#64971	1	1	yes	characteristic
122	C₂²	#64984	1	1	yes	characteristic
123	C₂²	C₂×#3932	1	4	yes	—
124	C₂²	C₂×#3950	1	1	yes	characteristic
125	C₄	—	80	4	no	—
126	C₄	#64725	1	8	yes	complemented
127	C₄	#64971	1	1	yes	characteristic
128	C₄	#64988	1	1	yes	characteristic
129	C₄	#65224	1	1	yes	characteristic
130	C₄	C₂×#3950	1	1	yes	characteristic
131	C₂	#10599839	1	2	yes	—
132	C₂	#10599854	1	2	yes	—
133	C₂	#10600558	1	1	yes	characteristic
134	C₂	C₂×#64725	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₀
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₄×C₅×C₁₆	C₂
socle, prime radical	C₂³×C₅	C₂×D₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×C₄×C₁₆	D₅
hypocenter	C₅	#12906
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
9. C₄×C₅×C₁₆	C₄	semidirect
42. C₅×C₁₆	C₄⋊C₄	semidirect
49. C₄×C₁₆	Dic₅	semidirect
80. C₄×C₅	#367	semidirect
101. C₁₆	C₄⋊Dic₅	semidirect
120. C₅	#12906	semidirect
126. C₄	#64725	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×C₁₆	2
C₂×#64725	2
#10600558	2
#12906	5