Subgroups — C₂×#10600070 #12252449

Order 1280.

806 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₅×C₁₆	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₂₀	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#64727	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#672	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600070	C₂	1	8	yes	maximal, direct product factor
7	C₂²×C₈×D₅	C₂²	1	2	yes	—
8	C₂×C₄×D₂₀	C₄	1	1	yes	characteristic
9	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3952	C₄	1	2	yes	—
12	C₂²×#672	C₂²	1	1	yes	characteristic
13	C₂²×#672	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#168	C₄	1	1	yes	characteristic
15	C₂×D₅×C₁₆	—	2	8	no	—
16	C₂×D₅×C₁₆	C₂²	1	8	yes	complemented
17	C₈×D₂₀	C₄	1	4	yes	—
18	C₈×D₂₀	C₂²	1	4	yes	—
19	C₅×C₄⋊C₁₆	C₂²	1	4	yes	complemented
20	#64727	C₂²	1	8	yes	complemented
21	C₄×#672	C₂²	1	4	yes	complemented
22	C₂×D₄×C₁₆	—	5	1	no	maximal, Sylow
23	C₂²×D₂₀	C₈	1	1	yes	characteristic
24	C₂²×C₄×D₅	C₂×C₄	1	2	yes	—
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂×#685	C₈	1	2	yes	—
29	C₂×C₄⋊Dic₅	C₈	1	1	yes	characteristic
30	C₂²×#168	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₅	—	2	4	no	—
32	C₂×C₈×D₅	—	2	4	no	—
33	C₂×C₈×D₅	C₂×C₄	1	4	yes	—
34	C₂×C₈×D₅	C₂³	1	4	yes	—
35	C₄×D₂₀	C₈	1	4	yes	—
36	C₄×D₂₀	C₂×C₄	1	4	yes	—
37	C₂×C₅×C₁₆	—	2	4	no	—
38	C₂×C₅×C₁₆	C₂³	1	4	yes	—
39	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
40	C₄×C₅×C₈	C₂³	1	2	yes	—
41	#3952	C₂×C₄	1	8	yes	—
42	C₂×#672	—	2	2	no	—
43	C₂×#672	D₄	1	4	yes	complemented
44	C₂×#672	C₂³	1	2	yes	—
45	C₂×#672	C₂³	1	2	yes	—
46	C₄⋊#168	C₂×C₄	1	4	yes	—
47	D₅×C₁₆	—	2	16	no	—
48	C₂³×C₁₆	—	5	2	no	—
49	C₂×C₈×D₄	—	5	1	no	—
50	D₄×C₁₆	—	5	8	no	—
51	C₂×C₄⋊C₁₆	D₅	1	1	yes	characteristic, complemented
52	C₂×C₂²⋊C₁₆	—	5	2	no	—
53	C₂×C₄×C₁₆	—	5	1	no	—
54	C₂³×D₅	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₅	C₂×C₈	1	2	yes	—
59	C₂×D₂₀	C₁₆	1	8	yes	complemented
60	C₂×D₂₀	C₂×C₈	1	4	yes	—
61	C₂×C₄×D₅	—	2	4	no	—
62	C₂×C₄×D₅	—	2	4	no	—
63	C₂×C₄×D₅	C₂×C₈	1	4	yes	—
64	C₂×C₄×D₅	C₂²×C₄	1	4	yes	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
67	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
69	C₂×C₅×C₈	#41	1	2	yes	—
70	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
71	C₄²×C₅	C₂×C₈	1	2	yes	—
72	C₄²×C₅	C₂²×C₄	1	2	yes	—
73	#685	C₂×C₈	1	8	yes	—
74	C₄⋊Dic₅	C₂×C₈	1	4	yes	—
75	C₂×#168	—	2	4	no	—
76	C₂×#168	C₂²×C₄	1	4	yes	—
77	C₈×D₅	—	2	8	no	—
78	C₈×D₅	—	2	4	no	—
79	C₈×D₅	—	2	4	no	—
80	C₅×C₁₆	—	2	4	no	—
81	#672	—	2	2	no	—
82	#672	C₂×D₄	1	4	yes	complemented
83	C₂³×C₈	—	5	2	no	—
84	C₂×C₄×D₄	—	5	1	no	—
85	C₂²×C₁₆	—	5	1	no	—
86	C₂²×C₁₆	—	5	1	no	—
87	C₂²×C₁₆	—	5	8	no	—
88	C₂²×C₁₆	—	10	8	no	—
89	C₂²×C₁₆	C₂×D₅	1	2	yes	—
90	C₈×D₄	—	5	4	no	—
91	C₈×D₄	—	5	4	no	—
92	C₂×C₄⋊C₈	—	5	1	no	—
93	C₂×C₂²⋊C₈	—	5	2	no	—
94	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
95	C₄⋊C₁₆	C₂×D₅	1	4	yes	complemented
96	C₂²⋊C₁₆	—	5	8	no	—
97	C₄×C₁₆	—	5	4	no	—
98	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
99	C₂²×D₅	—	2	4	no	—
100	C₂²×D₅	—	2	4	no	—
101	C₂²×D₅	C₂×C₁₆	1	8	yes	complemented
102	C₂²×D₅	C₂²×C₈	1	4	yes	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	C₂×C₁₆	1	4	yes	complemented
105	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₅	#131	1	2	yes	—
110	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
111	C₂×Dic₅	—	2	4	no	—
112	C₂×Dic₅	C₂²×C₈	1	4	yes	—
113	D₂₀	C₂×C₁₆	1	16	yes	complemented
114	C₄×D₅	—	2	4	no	—
115	C₄×D₅	—	2	4	no	—
116	C₄×D₅	—	2	8	no	—
117	C₅×C₈	—	2	2	no	—
118	C₅×C₈	C₄×D₄	1	2	yes	—
119	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
120	C₅×C₈	C₂×#41	1	1	yes	characteristic
121	#168	—	2	4	no	—
122	C₂²×D₄	—	5	1	no	—
123	C₂³×C₄	—	5	2	no	—
124	C₂²×C₈	—	5	2	no	—
125	C₂²×C₈	—	5	4	no	—
126	C₂²×C₈	—	5	4	no	—
127	C₂²×C₈	—	10	4	no	—
128	C₂²×C₈	—	10	4	no	—
129	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
131	C₄×D₄	—	5	4	no	—
132	C₄×D₄	—	5	4	no	—
133	C₂×C₄⋊C₄	—	5	1	no	—
134	C₂×C₂²⋊C₄	—	5	2	no	—
135	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
136	C₂×C₁₆	—	2	4	no	—
137	C₂×C₁₆	—	5	4	no	—
138	C₂×C₁₆	—	5	2	no	—
139	C₂×C₁₆	—	5	2	no	—
140	C₂×C₁₆	—	10	16	no	—
141	C₂×C₁₆	—	10	2	no	—
142	C₂×C₁₆	C₂²×D₅	1	4	yes	—
143	C₄⋊C₈	—	5	4	no	—
144	C₂²⋊C₈	—	5	8	no	—
145	C₄×C₈	C₄×D₅	1	2	yes	—
146	C₄×C₈	C₂²×D₅	1	2	yes	—
147	C₂²×C₅	C₈×D₄	1	2	yes	—
148	C₂²×C₅	C₂²×C₁₆	1	2	yes	—
149	C₂²×C₅	#504	1	2	yes	—
150	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
151	C₂×D₅	—	2	4	no	—
152	C₂×D₅	—	2	4	no	—
153	C₂×D₅	—	2	16	no	—
154	C₂×D₅	C₂²×C₁₆	1	8	yes	complemented
155	C₄×C₅	—	2	2	no	—
156	C₄×C₅	C₈×D₄	1	2	yes	—
157	C₄×C₅	C₂²×C₁₆	1	4	yes	complemented
158	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
159	C₄×C₅	C₂×#131	1	1	yes	characteristic
160	Dic₅	—	2	4	no	—
161	C₂⁴	—	5	2	no	—
162	C₂×D₄	—	5	8	no	—
163	C₂×D₄	—	5	4	no	—
164	C₂²×C₄	—	5	2	no	—
165	C₂²×C₄	—	5	4	no	—
166	C₂²×C₄	—	5	4	no	—
167	C₂²×C₄	—	10	4	no	—
168	C₂²×C₄	—	10	4	no	—
169	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
170	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	5	4	no	—
174	C₂×C₈	—	10	4	no	—
175	C₂×C₈	—	10	4	no	—
176	C₂×C₈	—	10	4	no	—
177	C₂×C₈	—	10	8	no	—
178	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
179	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
180	C₂×C₈	D₄×D₅	1	2	yes	—
181	C₂×C₈	#713	1	2	yes	—
182	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
183	C₄⋊C₄	—	5	4	no	—
184	C₂²⋊C₄	—	5	8	no	—
185	C₄²	C₈×D₅	1	2	yes	—
186	C₄²	C₂×C₄×D₅	1	2	yes	—
187	C₁₆	—	2	4	no	—
188	C₁₆	—	5	4	no	—
189	C₁₆	—	10	2	no	—
190	C₂×C₅	D₄×C₁₆	1	4	yes	complemented
191	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
192	C₂×C₅	C₂³×C₁₆	1	1	yes	characteristic
193	C₂×C₅	C₂×#504	1	1	yes	characteristic
194	D₅	—	2	8	no	—
195	C₂³	—	5	4	no	—
196	C₂³	—	5	8	no	—
197	C₂³	—	10	4	no	—
198	C₂³	—	10	4	no	—
199	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
200	D₄	—	5	16	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	5	4	no	—
203	C₂×C₄	—	10	4	no	—
204	C₂×C₄	—	10	4	no	—
205	C₂×C₄	—	10	4	no	—
206	C₂×C₄	—	10	8	no	—
207	C₂×C₄	D₅×C₁₆	1	4	yes	complemented
208	C₂×C₄	#4039	1	2	yes	—
209	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
210	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
211	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
212	C₂×C₄	#4053	1	2	yes	—
213	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	10	4	no	—
216	C₈	#4039	1	2	yes	—
217	C₈	C₂×D₄×D₅	1	1	yes	characteristic
218	C₈	C₂×#713	1	1	yes	characteristic
219	C₅	C₂×D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
220	C₂²	—	5	8	no	—
221	C₂²	—	10	4	no	—
222	C₂²	—	10	16	no	—
223	C₂²	—	10	4	no	—
224	C₂²	#65123	1	2	yes	—
225	C₂²	C₂×D₅×C₁₆	1	2	yes	—
226	C₂²	#65196	1	2	yes	—
227	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	10	4	no	—
230	C₄	#65123	1	2	yes	—
231	C₄	C₂×D₅×C₁₆	1	4	yes	complemented
232	C₄	C₂×#4039	1	1	yes	characteristic
233	C₄	C₂×#4053	1	1	yes	characteristic
234	C₂	—	10	8	no	—
235	C₂	#10600070	1	4	yes	direct product factor
236	C₂	C₂×#65123	1	1	yes	characteristic
237	C₂	C₂²×D₅×C₁₆	1	1	yes	characteristic
238	C₂	C₂×#65196	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅
commutator	C₂×C₅	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×C₄⋊C₁₆	C₂
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂²×D₂₀	C₈
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₅
hypocenter	C₅	C₂×D₄×C₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×D₅×C₁₆	C₂	semidirect
3. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
4. C₂×#64727	C₂	semidirect
5. C₂×C₄×#672	C₂	semidirect
6. #10600070	C₂	direct (N × H)
16. C₂×D₅×C₁₆	C₂²	semidirect
19. C₅×C₄⋊C₁₆	C₂²	semidirect
20. #64727	C₂²	semidirect
21. C₄×#672	C₂²	semidirect
43. C₂×#672	D₄	semidirect
51. C₂×C₄⋊C₁₆	D₅	semidirect
59. C₂×D₂₀	C₁₆	semidirect
82. #672	C₂×D₄	semidirect
95. C₄⋊C₁₆	C₂×D₅	semidirect
101. C₂²×D₅	C₂×C₁₆	semidirect
104. C₂×C₄×C₅	C₂×C₁₆	semidirect
113. D₂₀	C₂×C₁₆	semidirect
154. C₂×D₅	C₂²×C₁₆	semidirect
157. C₄×C₅	C₂²×C₁₆	semidirect
190. C₂×C₅	D₄×C₁₆	semidirect
207. C₂×C₄	D₅×C₁₆	semidirect
219. C₅	C₂×D₄×C₁₆	semidirect
231. C₄	C₂×D₅×C₁₆	semidirect
235. C₂	#10600070	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×D₅×C₁₆	2
C₂×C₈×D₂₀	2
C₂×C₅×C₄⋊C₁₆	2
C₂×#64727	2
C₂×C₄×#672	2
#10600070	2
C₂×D₄×C₁₆	5

Subgroups — C2×#10600070 #12252449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10600070 #12252449