Subgroups — C₂×#10599837 #12254777

Order 1280.

558 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64971	C₂	1	2	yes	maximal
2	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#64724	C₂	1	4	yes	maximal
4	#10599837	C₂	1	8	yes	maximal, direct product factor
5	C₂×#4015	—	2	2	no	—
6	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#3950	—	2	2	no	—
9	C₂×#3950	C₂²	1	4	yes	—
10	#64971	—	2	4	no	—
11	#64971	C₂²	1	8	yes	—
12	C₄×C₅×C₁₆	C₂²	1	4	yes	—
13	#64724	C₂²	1	16	yes	—
14	C₂×#2252	—	5	1	no	maximal, Sylow
15	C₂²×Dic₁₀	—	4	2	no	—
16	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
19	C₂×C₄⋊Dic₅	—	2	4	no	—
20	C₂×C₄⋊Dic₅	—	4	2	no	—
21	#4015	—	2	8	no	—
22	#4015	—	4	4	no	—
23	C₂×C₅×C₁₆	Q₈	1	8	yes	complemented
24	C₂×C₅×C₁₆	C₂³	1	4	yes	—
25	C₄×C₅×C₈	D₄	1	2	yes	—
26	C₄×C₅×C₈	C₂³	1	2	yes	—
27	#3950	—	2	8	no	—
28	#3950	—	2	4	no	—
29	#3950	C₂³	1	8	yes	—
30	C₂×#500	—	5	2	no	—
31	#2252	—	5	8	no	—
32	C₂×#366	—	5	4	no	—
33	C₂×C₄×C₁₆	D₅	1	1	yes	characteristic
34	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₅	—	4	4	no	—
38	C₂×Dic₁₀	—	4	8	no	—
39	C₂×Dic₁₀	—	8	8	no	—
40	C₂×C₅×C₈	Q₁₆	1	4	yes	—
41	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
42	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
43	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
44	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
45	C₄²×C₅	D₈	1	2	yes	—
46	C₄²×C₅	C₂×D₄	1	2	yes	—
47	C₄⋊Dic₅	—	2	8	no	—
48	C₄⋊Dic₅	—	4	8	no	—
49	C₄⋊Dic₅	—	4	4	no	—
50	C₅×C₁₆	C₂×Q₈	1	8	yes	complemented
51	C₂×C₄⋊Q₈	—	10	2	no	—
52	C₂²×C₁₆	C₂×D₅	1	2	yes	—
53	#500	—	5	8	no	—
54	#500	—	10	4	no	—
55	C₂×#107	—	5	4	no	—
56	C₂×#107	—	10	2	no	—
57	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
58	#366	—	5	16	no	—
59	C₄×C₁₆	C₂×D₅	1	4	yes	—
60	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₅	D₁₆	1	4	yes	—
62	C₂×C₄×C₅	Q₃₂	1	4	yes	—
63	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
64	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
66	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
67	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
68	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
69	C₂×Dic₅	—	4	8	no	—
70	C₂×Dic₅	—	8	8	no	—
71	Dic₁₀	—	8	16	no	—
72	C₅×C₈	C₄⋊Q₈	1	2	yes	—
73	C₅×C₈	C₂×Q₁₆	1	4	yes	—
74	C₅×C₈	C₂²×Q₈	1	2	yes	—
75	C₂²×Q₈	—	20	2	no	—
76	C₂²×C₈	D₂₀	1	1	yes	characteristic
77	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
78	C₄⋊Q₈	—	10	8	no	—
79	C₄⋊Q₈	—	20	4	no	—
80	C₂×C₄⋊C₄	—	10	4	no	—
81	C₂×C₄⋊C₄	—	20	2	no	—
82	C₂×C₄²	D₂₀	1	1	yes	characteristic
83	C₂×C₁₆	Dic₁₀	1	8	yes	complemented
84	C₂×C₁₆	C₂²×D₅	1	4	yes	—
85	#107	—	5	8	no	—
86	#107	—	10	4	no	—
87	#107	—	10	8	no	—
88	C₄×C₈	D₂₀	1	2	yes	—
89	C₄×C₈	C₂²×D₅	1	2	yes	—
90	C₂²×C₅	#500	1	2	yes	—
91	C₂²×C₅	C₂×D₁₆	1	2	yes	—
92	C₂²×C₅	C₂×Q₃₂	1	2	yes	—
93	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₅	#500	1	2	yes	—
95	C₄×C₅	C₂×D₁₆	1	4	yes	—
96	C₄×C₅	C₂×Q₃₂	1	4	yes	—
97	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
98	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
99	Dic₅	—	8	8	no	—
100	C₂×Q₈	—	20	8	no	—
101	C₂×Q₈	—	40	8	no	—
102	C₂²×C₄	—	20	4	no	—
103	C₂²×C₄	D₄₀	1	1	yes	characteristic
104	C₂²×C₄	D₄₀	1	1	yes	characteristic
105	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
106	C₂×C₈	Dic₂₀	1	4	yes	—
107	C₂×C₈	C₂×Dic₁₀	1	4	yes	—
108	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
109	C₂×C₈	C₂×D₂₀	1	2	yes	—
110	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
111	C₄⋊C₄	—	10	8	no	—
112	C₄⋊C₄	—	20	4	no	—
113	C₄⋊C₄	—	20	8	no	—
114	C₄²	D₄₀	1	2	yes	—
115	C₄²	C₂×D₂₀	1	2	yes	—
116	C₁₆	C₂×Dic₁₀	1	8	yes	complemented
117	C₂×C₅	#2252	1	4	yes	complemented
118	C₂×C₅	C₂×#500	1	1	yes	characteristic
119	C₂×C₅	C₂²×D₁₆	1	1	yes	characteristic
120	C₂×C₅	C₂²×Q₃₂	1	1	yes	characteristic
121	C₂³	C₂×D₄₀	1	1	yes	characteristic
122	Q₈	—	40	16	no	—
123	C₂×C₄	—	20	8	no	—
124	C₂×C₄	—	40	8	no	—
125	C₂×C₄	D₈₀	1	4	yes	—
126	C₂×C₄	Dic₄₀	1	4	yes	—
127	C₂×C₄	#4015	1	2	yes	—
128	C₂×C₄	C₂×D₄₀	1	2	yes	—
129	C₂×C₄	C₂×D₄₀	1	2	yes	—
130	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
131	C₂×C₄	C₂×Dic₂₀	1	2	yes	—
132	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
133	C₈	#4015	1	2	yes	—
134	C₈	C₂×Dic₂₀	1	4	yes	—
135	C₈	C₂²×Dic₁₀	1	2	yes	—
136	C₅	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
137	C₂²	#64971	1	2	yes	—
138	C₂²	C₂×D₈₀	1	2	yes	—
139	C₂²	C₂×Dic₄₀	1	2	yes	—
140	C₂²	C₂²×D₄₀	1	1	yes	characteristic
141	C₄	—	40	8	no	—
142	C₄	#64971	1	2	yes	—
143	C₄	C₂×D₈₀	1	4	yes	—
144	C₄	C₂×Dic₄₀	1	4	yes	—
145	C₄	C₂×#4015	1	1	yes	characteristic
146	C₄	C₂²×Dic₂₀	1	1	yes	characteristic
147	C₂	#10599837	1	4	yes	direct product factor
148	C₂	C₂×#64971	1	1	yes	characteristic
149	C₂	C₂²×D₈₀	1	1	yes	characteristic
150	C₂	C₂²×Dic₄₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₀
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₄×C₅×C₁₆	C₂
socle, prime radical	C₂³×C₅	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₅
hypocenter	C₅	C₂×#2252
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
4. #10599837	C₂	direct (N × H)
23. C₂×C₅×C₁₆	Q₈	semidirect
50. C₅×C₁₆	C₂×Q₈	semidirect
83. C₂×C₁₆	Dic₁₀	semidirect
116. C₁₆	C₂×Dic₁₀	semidirect
117. C₂×C₅	#2252	semidirect
136. C₅	C₂×#2252	semidirect
147. C₂	#10599837	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×#64971	2
C₂×C₄×C₅×C₁₆	2
C₂×#64724	2
#10599837	2
C₂×#2252	5

Subgroups — C2×#10599837 #12254777

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10599837 #12254777