Subgroups — C₂²×#64972 #12968894

Order 1280.

1966 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#4016	C₂	1	2	yes	maximal
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3950	C₂	1	2	yes	maximal
4	C₂²×#3949	C₂	1	2	yes	maximal
5	C₂×#64972	C₂	1	24	yes	maximal, direct product factor
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₅	C₂²	1	4	yes	—
10	C₂²×#683	—	2	4	no	—
11	C₂×#4016	—	2	12	no	—
12	C₂×#4016	C₂²	1	24	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	12	yes	—
14	C₂×#3950	C₂²	1	24	yes	—
15	C₂×#3949	C₂²	1	24	yes	—
16	#64972	C₂²	1	64	yes	direct product factor
17	C₂²×#499	—	5	1	no	maximal, Sylow
18	C₂³×C₄×C₅	D₄	1	2	yes	—
19	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂³×Dic₅	—	2	4	no	—
21	C₂²×C₅×C₈	Q₈	1	8	yes	—
22	C₂²×C₅×C₈	Q₈	1	8	yes	—
23	C₂²×C₅×C₈	C₂³	1	6	yes	—
24	C₂²×C₅×C₈	C₂³	1	6	yes	—
25	C₂×C₄²×C₅	D₄	1	6	yes	—
26	C₂×C₄²×C₅	C₂³	1	6	yes	—
27	C₂×C₄⋊Dic₅	—	2	12	no	—
28	C₂×C₄⋊Dic₅	C₂³	1	24	yes	—
29	C₂×#683	—	2	48	no	—
30	#4016	—	2	48	no	—
31	#4016	C₂³	1	32	yes	—
32	C₄×C₅×C₈	C₂³	1	16	yes	—
33	#3950	C₂³	1	32	yes	—
34	#3949	C₂³	1	32	yes	—
35	C₂²×#125	—	5	2	no	—
36	C₂×#499	—	5	24	no	—
37	C₂²×#107	—	5	2	no	—
38	C₂²×#106	—	5	2	no	—
39	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
40	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅	—	2	8	no	—
42	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
43	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
44	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
45	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
46	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
47	C₂²×Dic₅	—	2	24	no	—
48	C₂²×Dic₅	—	4	16	no	—
49	C₂×C₅×C₈	C₂×Q₈	1	24	yes	—
50	C₂×C₅×C₈	C₂×Q₈	1	24	yes	—
51	C₂×C₅×C₈	C₂⁴	1	4	yes	—
52	C₂×C₅×C₈	C₂⁴	1	4	yes	—
53	C₄²×C₅	C₂×D₄	1	12	yes	—
54	C₄²×C₅	C₂⁴	1	4	yes	—
55	C₄⋊Dic₅	—	2	24	no	—
56	C₄⋊Dic₅	C₂⁴	1	16	yes	—
57	#683	—	2	64	no	—
58	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
59	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
60	C₂×#125	—	5	24	no	—
61	C₂×#125	—	10	12	no	—
62	C₂²×C₄⋊C₄	—	5	4	no	—
63	C₂²×C₄⋊C₄	—	10	4	no	—
64	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
65	#499	—	5	64	no	—
66	C₂×#107	—	5	24	no	—
67	C₂×#106	—	5	24	no	—
68	C₂×C₄×C₈	C₂×D₅	1	12	yes	—
69	C₂³×C₅	C₄⋊Q₈	1	4	yes	—
70	C₂³×C₅	#135	1	8	yes	—
71	C₂³×C₅	C₂²×D₄	1	3	yes	—
72	C₂×C₄×C₅	—	2	24	no	—
73	C₂×C₄×C₅	C₄⋊Q₈	1	12	yes	—
74	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
75	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
76	C₂×C₄×C₅	C₂²×Q₈	1	6	yes	—
77	C₂×C₄×C₅	C₂²×Q₈	1	6	yes	—
78	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
79	C₂×Dic₅	—	2	16	no	—
80	C₂×Dic₅	—	4	48	no	—
81	C₅×C₈	C₂²×Q₈	1	8	yes	—
82	C₅×C₈	C₂²×Q₈	1	8	yes	—
83	C₂³×C₄	—	10	4	no	—
84	C₂³×C₄	D₂₀	1	2	yes	—
85	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
86	C₂²×C₈	Dic₁₀	1	8	yes	—
87	C₂²×C₈	Dic₁₀	1	8	yes	—
88	C₂²×C₈	C₂²×D₅	1	6	yes	—
89	C₂²×C₈	C₂²×D₅	1	6	yes	—
90	#125	—	5	32	no	—
91	#125	—	10	48	no	—
92	C₂×C₄⋊C₄	—	5	24	no	—
93	C₂×C₄⋊C₄	—	10	12	no	—
94	C₂×C₄⋊C₄	—	10	48	no	—
95	C₂×C₄²	D₂₀	1	6	yes	—
96	C₂×C₄²	C₂²×D₅	1	6	yes	—
97	#107	—	5	32	no	—
98	#106	—	5	32	no	—
99	C₄×C₈	C₂²×D₅	1	16	yes	—
100	C₂²×C₅	#499	1	16	yes	complemented
101	C₂²×C₅	C₂×C₄⋊Q₈	1	6	yes	—
102	C₂²×C₅	C₂×#135	1	12	yes	—
103	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
104	C₄×C₅	—	2	8	no	—
105	C₄×C₅	C₂×C₄⋊Q₈	1	6	yes	—
106	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
107	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
108	Dic₅	—	4	16	no	—
109	C₂⁴	C₂×D₂₀	1	1	yes	characteristic
110	C₂²×C₄	—	2	8	no	—
111	C₂²×C₄	—	10	24	no	—
112	C₂²×C₄	—	20	16	no	—
113	C₂²×C₄	C₂×Dic₁₀	1	4	yes	—
114	C₂²×C₄	C₂×Dic₁₀	1	4	yes	—
115	C₂²×C₄	C₂×D₂₀	1	12	yes	—
116	C₂²×C₄	C₂×D₂₀	1	3	yes	—
117	C₂²×C₄	C₂³×D₅	1	3	yes	—
118	C₂×C₈	C₂×Dic₁₀	1	24	yes	—
119	C₂×C₈	C₂×Dic₁₀	1	24	yes	—
120	C₂×C₈	C₂³×D₅	1	4	yes	—
121	C₂×C₈	C₂³×D₅	1	4	yes	—
122	C₄⋊C₄	—	5	16	no	—
123	C₄⋊C₄	—	10	24	no	—
124	C₄⋊C₄	—	10	64	no	—
125	C₄²	C₂×D₂₀	1	12	yes	—
126	C₄²	C₂³×D₅	1	4	yes	—
127	C₂×C₅	C₂×#499	1	12	yes	complemented
128	C₂×C₅	C₂²×C₄⋊Q₈	1	1	yes	characteristic
129	C₂×C₅	C₂²×#135	1	2	yes	—
130	C₂³	#4015	1	4	yes	—
131	C₂³	#4050	1	8	yes	—
132	C₂³	C₂²×D₂₀	1	3	yes	—
133	C₂×C₄	—	2	24	no	—
134	C₂×C₄	—	10	16	no	—
135	C₂×C₄	—	20	48	no	—
136	C₂×C₄	#4015	1	12	yes	—
137	C₂×C₄	C₂²×Dic₁₀	1	6	yes	—
138	C₂×C₄	C₂²×Dic₁₀	1	6	yes	—
139	C₂×C₄	C₂²×D₂₀	1	8	yes	—
140	C₂×C₄	C₂²×D₂₀	1	3	yes	—
141	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
142	C₈	C₂²×Dic₁₀	1	8	yes	—
143	C₈	C₂²×Dic₁₀	1	8	yes	—
144	C₅	C₂²×#499	1	1	yes	characteristic, complemented, Sylow
145	C₂²	#64972	1	16	yes	direct product factor
146	C₂²	C₂×#4015	1	6	yes	—
147	C₂²	C₂×#4050	1	12	yes	—
148	C₂²	C₂³×D₂₀	1	1	yes	characteristic
149	C₄	—	2	8	no	—
150	C₄	—	20	16	no	—
151	C₄	C₂×#4015	1	6	yes	—
152	C₄	C₂³×Dic₁₀	1	1	yes	characteristic
153	C₄	C₂³×Dic₁₀	1	1	yes	characteristic
154	C₂	C₂×#64972	1	12	yes	direct product factor
155	C₂	C₂²×#4015	1	1	yes	characteristic
156	C₂	C₂²×#4050	1	2	yes	—

	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×#499
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
5. C₂×#64972	C₂	direct (N × H)
16. #64972	C₂²	direct (N × H)
100. C₂²×C₅	#499	semidirect
127. C₂×C₅	C₂×#499	semidirect
144. C₅	C₂²×#499	semidirect
145. C₂²	#64972	direct (N × H)
154. C₂	C₂×#64972	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×#4016	2
C₂²×C₄×C₅×C₈	2
C₂²×#3950	2
C₂²×#3949	2
C₂×#64972	2
C₂²×#499	5

Subgroups — C22×#64972 #12968894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#64972 #12968894