Subgroups — C₂×#10604025 #13043630

Order 1280.

542 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65119	C₂	1	1	yes	maximal, characteristic
2	C₂×#64888	C₂	1	2	yes	maximal
3	C₂×#64886	C₂	1	2	yes	maximal
4	C₂×C₄⋊#674	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#674	C₂	1	1	yes	maximal, characteristic
6	#10604025	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₄×Dic₁₀	C₄	1	1	yes	characteristic
8	C₂²×#674	C₂²	1	1	yes	characteristic
9	C₂²×#674	C₂²	1	2	yes	—
10	C₂²×#674	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×#3946	C₄	1	2	yes	—
13	C₂×C₈×Dic₅	C₂²	1	2	yes	—
14	C₂×C₄×#168	C₂²	1	1	yes	characteristic
15	#65119	C₄	1	4	yes	—
16	#65119	C₂²	1	4	yes	—
17	#64888	C₂²	1	8	yes	—
18	#64886	C₂²	1	8	yes	—
19	C₄⋊#674	C₂²	1	4	yes	—
20	C₄×#674	C₂²	1	4	yes	—
21	C₂×#2183	—	5	1	no	maximal, Sylow
22	C₂²×Dic₁₀	C₈	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄⋊Dic₅	C₈	1	1	yes	characteristic
26	C₂×#683	C₈	1	2	yes	—
27	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
28	C₂²×#168	C₂×C₄	1	1	yes	characteristic
29	C₂²×#168	C₂³	1	1	yes	characteristic
30	C₄×Dic₁₀	C₈	1	4	yes	—
31	C₄×Dic₁₀	C₂×C₄	1	4	yes	—
32	C₂×#674	—	2	2	no	—
33	C₂×#674	—	2	4	no	—
34	C₂×#674	Q₈	1	4	yes	complemented
35	C₂×#674	C₂³	1	2	yes	—
36	C₂×#674	C₂³	1	4	yes	—
37	C₂×#674	C₂³	1	2	yes	—
38	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
39	#3946	C₂×C₄	1	8	yes	—
40	C₈×Dic₅	C₂×C₄	1	4	yes	—
41	C₈×Dic₅	C₂³	1	4	yes	—
42	C₄×#168	C₂×C₄	1	2	yes	—
43	C₄×#168	C₂³	1	2	yes	—
44	C₂×C₈×Q₈	—	5	1	no	—
45	#2183	—	5	8	no	—
46	C₂×C₄⋊C₁₆	—	5	1	no	—
47	C₂×C₄⋊C₁₆	—	5	2	no	—
48	C₂×#346	—	5	2	no	—
49	C₂×C₄×C₁₆	—	5	1	no	—
50	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₅	C₂×C₈	1	2	yes	—
54	C₂²×Dic₅	C₂×C₈	1	2	yes	—
55	C₂×Dic₁₀	—	2	4	no	—
56	C₂×Dic₁₀	C₂×C₈	1	4	yes	—
57	C₂×C₅×C₈	—	2	4	no	—
58	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
59	C₄²×C₅	C₂×C₈	1	2	yes	—
60	C₄²×C₅	C₂²×C₄	1	2	yes	—
61	C₄⋊Dic₅	C₂×C₈	1	4	yes	—
62	#683	C₂×C₈	1	8	yes	—
63	C₄×Dic₅	C₂×C₈	1	4	yes	—
64	C₄×Dic₅	C₂²×C₄	1	4	yes	—
65	C₂×#168	—	2	2	no	—
66	C₂×#168	C₂²×C₄	1	2	yes	—
67	C₂×#168	C₂²×C₄	1	1	yes	characteristic
68	C₂×#168	C₂×Q₈	1	2	yes	—
69	C₂×#168	#41	1	2	yes	—
70	C₂×#168	C₂⁴	1	1	yes	characteristic
71	#674	—	2	2	no	—
72	#674	—	2	4	no	—
73	#674	C₂×Q₈	1	4	yes	complemented
74	C₂×C₄×Q₈	—	5	1	no	—
75	C₂²×C₁₆	—	5	1	no	—
76	C₂²×C₁₆	—	5	1	no	—
77	C₂²×C₁₆	—	5	2	no	—
78	C₈×Q₈	—	5	4	no	—
79	C₈×Q₈	—	5	4	no	—
80	C₂×C₄⋊C₈	—	5	2	no	—
81	C₂×C₄⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
82	C₂×C₄×C₈	—	5	2	no	—
83	C₂×C₄×C₈	—	5	1	no	—
84	C₄⋊C₁₆	—	5	4	no	—
85	C₄⋊C₁₆	—	5	8	no	—
86	#346	—	5	8	no	—
87	C₄×C₁₆	—	5	4	no	—
88	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	M₃₂	1	4	yes	—
91	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
92	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
93	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₅	#131	1	2	yes	—
96	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
97	C₂×Dic₅	—	2	4	no	—
98	C₂×Dic₅	—	2	4	no	—
99	C₂×Dic₅	C₂²×C₈	1	4	yes	—
100	C₂×Dic₅	C₂²×C₈	1	4	yes	—
101	Dic₁₀	—	2	8	no	—
102	C₅×C₈	—	2	4	no	—
103	#168	—	2	2	no	—
104	#168	C₄×Q₈	1	2	yes	—
105	#168	C₂²×Q₈	1	1	yes	characteristic
106	#168	C₂×#41	1	1	yes	characteristic
107	C₂²×Q₈	—	5	1	no	—
108	C₂²×C₈	—	5	1	no	—
109	C₂²×C₈	—	5	1	no	—
110	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
111	C₄×Q₈	—	5	4	no	—
112	C₄×Q₈	—	5	4	no	—
113	C₂×C₄⋊C₄	—	5	2	no	—
114	C₂×C₄⋊C₄	—	5	1	no	—
115	C₂×C₄²	—	5	2	no	—
116	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₂×C₁₆	—	5	2	no	—
118	C₂×C₁₆	—	5	4	no	—
119	C₂×C₁₆	—	5	2	no	—
120	C₂×C₁₆	—	5	4	no	—
121	C₂×C₁₆	—	10	2	no	—
122	C₂×C₁₆	—	10	4	no	—
123	C₄⋊C₈	—	5	8	no	—
124	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	4	yes	—
125	C₄×C₈	—	5	4	no	—
126	C₄×C₈	—	5	4	no	—
127	C₄×C₈	—	5	2	no	—
128	C₄×C₈	—	5	2	no	—
129	C₂²×C₅	C₈×Q₈	1	2	yes	—
130	C₂²×C₅	C₂×M₃₂	1	2	yes	—
131	C₂²×C₅	#504	1	2	yes	—
132	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
133	C₄×C₅	—	2	2	no	—
134	C₄×C₅	C₈×Q₈	1	2	yes	—
135	C₄×C₅	C₂×M₃₂	1	4	yes	—
136	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
137	C₄×C₅	C₂×#131	1	1	yes	characteristic
138	Dic₅	—	2	4	no	—
139	Dic₅	—	2	4	no	—
140	C₂×Q₈	—	5	4	no	—
141	C₂×Q₈	—	10	4	no	—
142	C₂²×C₄	—	5	2	no	—
143	C₂²×C₄	—	5	2	no	—
144	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
145	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
146	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	5	2	no	—
149	C₂×C₈	—	5	2	no	—
150	C₂×C₈	—	5	1	no	—
151	C₂×C₈	—	5	1	no	—
152	C₂×C₈	—	5	2	no	—
153	C₂×C₈	—	10	2	no	—
154	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	4	yes	—
155	C₄⋊C₄	—	5	8	no	—
156	C₄⋊C₄	—	5	4	no	—
157	C₄²	—	5	4	no	—
158	C₄²	—	5	4	no	—
159	C₄²	D₅⋊C₈	1	2	yes	—
160	C₄²	C₂²×AGL(1,𝔽₅)	1	2	yes	—
161	C₁₆	—	5	4	no	—
162	C₁₆	—	10	2	no	—
163	C₁₆	—	10	4	no	—
164	C₂×C₅	#2183	1	4	yes	complemented
165	C₂×C₅	C₂×C₈×Q₈	1	1	yes	characteristic
166	C₂×C₅	C₂²×M₃₂	1	1	yes	characteristic
167	C₂×C₅	C₂×#504	1	1	yes	characteristic
168	C₂³	C₂×D₅⋊C₈	1	1	yes	characteristic
169	Q₈	—	10	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	5	4	no	—
172	C₂×C₄	—	5	4	no	—
173	C₂×C₄	—	10	4	no	—
174	C₂×C₄	—	10	4	no	—
175	C₂×C₄	#3990	1	4	yes	—
176	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
177	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
178	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
179	C₂×C₄	#4131	1	2	yes	—
180	C₂×C₄	Q₈×AGL(1,𝔽₅)	1	2	yes	—
181	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
182	C₈	—	2	4	no	—
183	C₈	—	5	1	no	—
184	C₈	—	5	1	no	—
185	C₈	—	5	2	no	—
186	C₈	—	10	2	no	—
187	C₅	C₂×#2183	1	1	yes	characteristic, complemented, Sylow
188	C₂²	#65703	1	2	yes	—
189	C₂²	C₂×#3990	1	2	yes	—
190	C₂²	#65725	1	2	yes	—
191	C₂²	C₂²×D₅⋊C₈	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	10	4	no	—
194	C₄	—	10	4	no	—
195	C₄	#65703	1	2	yes	—
196	C₄	C₂×#3990	1	4	yes	—
197	C₄	C₂×#4131	1	1	yes	characteristic
198	C₄	C₂×Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
199	C₂	#10604025	1	4	yes	direct product factor
200	C₂	C₂×#65703	1	1	yes	characteristic
201	C₂	C₂²×#3990	1	1	yes	characteristic
202	C₂	C₂×#65725	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂³×C₈
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₄⋊C₈	C₄
socle, prime radical	C₂³×C₅	C₂²×C₈
cosocle	C₂×#168	C₂⁴
hypercenter	C₂×C₄⋊C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#2183
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
6. #10604025	C₂	direct (N × H)
34. C₂×#674	Q₈	semidirect
73. #674	C₂×Q₈	semidirect
164. C₂×C₅	#2183	semidirect
187. C₅	C₂×#2183	semidirect
199. C₂	#10604025	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×#65119	2
C₂×#64888	2
C₂×#64886	2
C₂×C₄⋊#674	2
C₂×C₄×#674	2
#10604025	2
C₂×#2183	5

Subgroups — C2×#10604025 #13043630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10604025 #13043630