Subgroups — C₂²×#64971 #12968898

Order 1280.

2398 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#4015	C₂	1	2	yes	maximal
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3950	C₂	1	4	yes	maximal
4	C₂×#64971	C₂	1	24	yes	maximal, direct product factor
5	C₂³×Dic₁₀	—	2	2	no	—
6	C₂³×C₅×C₈	C₂²	1	2	yes	—
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₅	—	2	2	no	—
9	C₂²×C₄⋊Dic₅	C₂²	1	4	yes	—
10	C₂×#4015	—	2	12	no	—
11	C₂×#4015	C₂²	1	24	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	12	yes	—
13	C₂×#3950	C₂²	1	48	yes	—
14	#64971	C₂²	1	64	yes	direct product factor
15	C₂²×#500	—	5	1	no	maximal, Sylow
16	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
18	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₂³×Dic₅	—	2	4	no	—
20	C₂²×Dic₁₀	—	2	24	no	—
21	C₂²×Dic₁₀	—	4	16	no	—
22	C₂²×C₅×C₈	Q₈	1	16	yes	complemented
23	C₂²×C₅×C₈	C₂³	1	12	yes	—
24	C₂×C₄²×C₅	D₄	1	6	yes	—
25	C₂×C₄²×C₅	C₂³	1	6	yes	—
26	C₂×C₄⋊Dic₅	—	2	12	no	—
27	C₂×C₄⋊Dic₅	—	2	24	no	—
28	C₂×C₄⋊Dic₅	C₂³	1	24	yes	—
29	#4015	—	2	48	no	—
30	#4015	C₂³	1	32	yes	—
31	C₄×C₅×C₈	C₂³	1	16	yes	—
32	#3950	C₂³	1	64	yes	—
33	C₂²×C₄⋊Q₈	—	5	2	no	—
34	C₂×#500	—	5	24	no	—
35	C₂²×#107	—	5	4	no	—
36	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
37	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅	D₈	1	8	yes	—
39	C₂²×C₄×C₅	Q₁₆	1	8	yes	—
40	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
41	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
42	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
43	C₂²×C₄×C₅	C₂×Q₈	1	8	yes	—
44	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
45	C₂²×Dic₅	—	2	24	no	—
46	C₂²×Dic₅	—	4	16	no	—
47	C₂×Dic₁₀	—	2	32	no	—
48	C₂×Dic₁₀	—	4	96	no	—
49	C₂×C₅×C₈	C₂×Q₈	1	48	yes	complemented
50	C₂×C₅×C₈	C₂⁴	1	8	yes	—
51	C₄²×C₅	C₂×D₄	1	12	yes	—
52	C₄²×C₅	C₂⁴	1	4	yes	—
53	C₄⋊Dic₅	—	2	24	no	—
54	C₄⋊Dic₅	—	2	32	no	—
55	C₄⋊Dic₅	C₂⁴	1	16	yes	—
56	C₂³×Q₈	—	10	2	no	—
57	C₂³×C₈	C₂×D₅	1	2	yes	—
58	C₂×C₄⋊Q₈	—	5	24	no	—
59	C₂×C₄⋊Q₈	—	10	12	no	—
60	C₂²×C₄⋊C₄	—	5	4	no	—
61	C₂²×C₄⋊C₄	—	10	2	no	—
62	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
63	#500	—	5	64	no	—
64	C₂×#107	—	5	48	no	—
65	C₂×C₄×C₈	C₂×D₅	1	12	yes	—
66	C₂³×C₅	C₄⋊Q₈	1	4	yes	—
67	C₂³×C₅	C₂×D₈	1	4	yes	—
68	C₂³×C₅	C₂×Q₁₆	1	4	yes	—
69	C₂³×C₅	C₂²×D₄	1	3	yes	—
70	C₂×C₄×C₅	C₄⋊Q₈	1	12	yes	—
71	C₂×C₄×C₅	C₂×D₈	1	24	yes	—
72	C₂×C₄×C₅	C₂×Q₁₆	1	24	yes	—
73	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
74	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
75	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
76	C₂×C₄×C₅	C₂²×Q₈	1	12	yes	—
77	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
78	C₂×Dic₅	—	2	16	no	—
79	C₂×Dic₅	—	4	48	no	—
80	Dic₁₀	—	4	64	no	—
81	C₅×C₈	C₂²×Q₈	1	16	yes	complemented
82	C₂²×Q₈	—	10	24	no	—
83	C₂²×Q₈	—	20	16	no	—
84	C₂³×C₄	—	10	4	no	—
85	C₂³×C₄	D₂₀	1	1	yes	characteristic
86	C₂³×C₄	D₂₀	1	1	yes	characteristic
87	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
88	C₂²×C₈	Dic₁₀	1	16	yes	complemented
89	C₂²×C₈	C₂²×D₅	1	12	yes	—
90	C₄⋊Q₈	—	5	32	no	—
91	C₄⋊Q₈	—	10	48	no	—
92	C₂×C₄⋊C₄	—	5	24	no	—
93	C₂×C₄⋊C₄	—	10	12	no	—
94	C₂×C₄⋊C₄	—	10	24	no	—
95	C₂×C₄²	D₂₀	1	6	yes	—
96	C₂×C₄²	C₂²×D₅	1	6	yes	—
97	#107	—	5	64	no	—
98	C₄×C₈	C₂²×D₅	1	16	yes	—
99	C₂²×C₅	#500	1	16	yes	complemented
100	C₂²×C₅	C₂×C₄⋊Q₈	1	6	yes	—
101	C₂²×C₅	C₂²×D₈	1	6	yes	—
102	C₂²×C₅	C₂²×Q₁₆	1	6	yes	—
103	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
104	C₄×C₅	C₂×C₄⋊Q₈	1	6	yes	—
105	C₄×C₅	C₂²×D₈	1	8	yes	—
106	C₄×C₅	C₂²×Q₁₆	1	8	yes	—
107	C₄×C₅	C₂³×Q₈	1	2	yes	—
108	Dic₅	—	4	16	no	—
109	C₂⁴	C₂×D₂₀	1	1	yes	characteristic
110	C₂×Q₈	—	10	32	no	—
111	C₂×Q₈	—	20	96	no	—
112	C₂²×C₄	—	10	24	no	—
113	C₂²×C₄	—	20	16	no	—
114	C₂²×C₄	D₄₀	1	8	yes	—
115	C₂²×C₄	Dic₂₀	1	8	yes	—
116	C₂²×C₄	C₂×Dic₁₀	1	8	yes	—
117	C₂²×C₄	C₂×D₂₀	1	3	yes	—
118	C₂²×C₄	C₂×D₂₀	1	6	yes	—
119	C₂²×C₄	C₂×D₂₀	1	6	yes	—
120	C₂²×C₄	C₂³×D₅	1	3	yes	—
121	C₂×C₈	C₂×Dic₁₀	1	48	yes	complemented
122	C₂×C₈	C₂³×D₅	1	8	yes	—
123	C₄⋊C₄	—	5	16	no	—
124	C₄⋊C₄	—	10	24	no	—
125	C₄⋊C₄	—	10	32	no	—
126	C₄²	C₂×D₂₀	1	12	yes	—
127	C₄²	C₂³×D₅	1	4	yes	—
128	C₂×C₅	C₂×#500	1	12	yes	complemented
129	C₂×C₅	C₂²×C₄⋊Q₈	1	1	yes	characteristic
130	C₂×C₅	C₂³×D₈	1	1	yes	characteristic
131	C₂×C₅	C₂³×Q₁₆	1	1	yes	characteristic
132	C₂³	#4015	1	4	yes	—
133	C₂³	C₂×D₄₀	1	4	yes	—
134	C₂³	C₂×Dic₂₀	1	4	yes	—
135	C₂³	C₂²×D₂₀	1	3	yes	—
136	Q₈	—	20	64	no	—
137	C₂×C₄	—	10	16	no	—
138	C₂×C₄	—	20	48	no	—
139	C₂×C₄	#4015	1	12	yes	—
140	C₂×C₄	C₂×D₄₀	1	24	yes	—
141	C₂×C₄	C₂×Dic₂₀	1	24	yes	—
142	C₂×C₄	C₂²×Dic₁₀	1	12	yes	—
143	C₂×C₄	C₂²×D₂₀	1	3	yes	—
144	C₂×C₄	C₂²×D₂₀	1	4	yes	—
145	C₂×C₄	C₂²×D₂₀	1	4	yes	—
146	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
147	C₈	C₂²×Dic₁₀	1	16	yes	complemented
148	C₅	C₂²×#500	1	1	yes	characteristic, complemented, Sylow
149	C₂²	#64971	1	16	yes	direct product factor
150	C₂²	C₂×#4015	1	6	yes	—
151	C₂²	C₂²×D₄₀	1	6	yes	—
152	C₂²	C₂²×Dic₂₀	1	6	yes	—
153	C₂²	C₂³×D₂₀	1	1	yes	characteristic
154	C₄	—	20	16	no	—
155	C₄	C₂×#4015	1	6	yes	—
156	C₄	C₂²×D₄₀	1	8	yes	—
157	C₄	C₂²×Dic₂₀	1	8	yes	—
158	C₄	C₂³×Dic₁₀	1	2	yes	—
159	C₂	C₂×#64971	1	12	yes	direct product factor
160	C₂	C₂²×#4015	1	1	yes	characteristic
161	C₂	C₂³×D₄₀	1	1	yes	characteristic
162	C₂	C₂³×Dic₂₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle, prime radical	C₂⁴×C₅	C₂×D₄
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	C₂²×#500
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₂×#64971	C₂	direct (N × H)
14. #64971	C₂²	direct (N × H)
22. C₂²×C₅×C₈	Q₈	semidirect
49. C₂×C₅×C₈	C₂×Q₈	semidirect
81. C₅×C₈	C₂²×Q₈	semidirect
88. C₂²×C₈	Dic₁₀	semidirect
99. C₂²×C₅	#500	semidirect
121. C₂×C₈	C₂×Dic₁₀	semidirect
128. C₂×C₅	C₂×#500	semidirect
147. C₈	C₂²×Dic₁₀	semidirect
148. C₅	C₂²×#500	semidirect
149. C₂²	#64971	direct (N × H)
159. C₂	C₂×#64971	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×#4015	2
C₂²×C₄×C₅×C₈	2
C₂²×#3950	2
C₂×#64971	2
C₂²×#500	5

Subgroups — C22×#64971 #12968898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#64971 #12968898