Subgroups — C₂×#10600110 #12252560

Order 1280.

806 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈₀	C₂	1	1	yes	maximal, characteristic
2	C₂×#65161	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#366	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#64711	C₂	1	2	yes	maximal, complemented
6	#10600110	C₂	1	8	yes	maximal, direct product factor
7	C₂²×D₄₀	C₂²	1	2	yes	—
8	C₂×#4039	—	2	2	no	—
9	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₅×#107	C₂²	1	2	yes	—
11	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
12	C₂²×#672	C₂²	1	1	yes	characteristic
13	C₂×#3941	—	2	2	no	—
14	C₂×D₈₀	C₄	1	8	yes	complemented
15	C₂×D₈₀	C₂²	1	4	yes	—
16	#65161	—	2	4	no	—
17	#65161	C₂²	1	8	yes	complemented
18	C₅×#366	C₂²	1	4	yes	complemented
19	C₁₆×Dic₅	C₂²	1	4	yes	complemented
20	#64711	C₂²	1	8	yes	complemented
21	C₂×C₄×D₁₆	—	5	1	no	maximal, Sylow
22	C₂²×D₂₀	—	2	2	no	—
23	C₂²×C₄×D₅	—	4	2	no	—
24	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₅×C₄⋊C₄	—	2	2	no	—
26	C₂×#685	—	4	2	no	—
27	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
28	C₂²×#168	D₄	1	1	yes	characteristic
29	C₂×D₄₀	—	2	2	no	—
30	C₂×D₄₀	—	2	4	no	—
31	C₂×D₄₀	C₂×C₄	1	8	yes	complemented
32	C₂×D₄₀	C₂³	1	4	yes	—
33	#4039	—	2	8	no	—
34	#4039	—	4	4	no	—
35	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
36	C₂×C₅×C₁₆	C₂³	1	2	yes	—
37	C₅×#107	—	2	2	no	—
38	C₅×#107	C₂³	1	4	yes	—
39	C₈×Dic₅	D₄	1	2	yes	—
40	C₈×Dic₅	C₂³	1	2	yes	—
41	C₂×#672	—	2	2	no	—
42	C₂×#672	C₂³	1	2	yes	—
43	#3941	—	2	8	no	—
44	D₈₀	C₂×C₄	1	16	yes	complemented
45	C₂²×D₁₆	—	5	1	no	—
46	C₂×C₄×D₈	—	5	2	no	—
47	C₄×D₁₆	—	5	8	no	—
48	C₂×#366	D₅	1	1	yes	characteristic, complemented
49	C₂×#357	—	5	2	no	—
50	C₂×C₄×C₁₆	—	5	1	no	—
51	C₂³×D₅	—	4	2	no	—
52	C₂²×C₄×C₅	—	4	2	no	—
53	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₅	D₈	1	1	yes	characteristic
55	C₂²×Dic₅	D₈	1	1	yes	characteristic
56	C₂×D₂₀	—	2	8	no	—
57	C₂×D₂₀	—	2	4	no	—
58	C₂×D₂₀	—	4	2	no	—
59	C₂×D₂₀	—	4	4	no	—
60	C₂×C₄×D₅	—	4	8	no	—
61	C₂×C₄×D₅	—	8	8	no	—
62	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
63	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₅×C₈	#41	1	2	yes	—
65	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
66	C₅×C₄⋊C₄	—	2	4	no	—
67	C₅×C₄⋊C₄	—	4	2	no	—
68	#685	—	4	8	no	—
69	C₄×Dic₅	D₈	1	2	yes	—
70	C₄×Dic₅	C₂×D₄	1	2	yes	—
71	C₂×#168	—	2	2	no	—
72	C₂×#168	C₂×D₄	1	2	yes	—
73	D₄₀	—	2	8	no	—
74	D₄₀	—	2	4	no	—
75	D₄₀	C₂²×C₄	1	8	yes	complemented
76	C₅×C₁₆	C₂²×C₄	1	4	yes	complemented
77	#672	—	2	2	no	—
78	C₂²×D₈	—	5	2	no	—
79	C₂×C₄×D₄	—	10	2	no	—
80	C₂×D₁₆	—	5	8	no	—
81	C₂×D₁₆	—	5	4	no	—
82	C₂²×C₁₆	—	5	1	no	—
83	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
84	C₄×D₈	—	5	8	no	—
85	C₄×D₈	—	10	4	no	—
86	C₂×#107	C₂×D₅	1	2	yes	—
87	C₂×#102	—	10	2	no	—
88	C₂×C₄×C₈	—	5	1	no	—
89	#366	C₂×D₅	1	4	yes	complemented
90	#357	—	5	8	no	—
91	C₄×C₁₆	—	5	4	no	—
92	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
93	C₂²×D₅	—	4	4	no	—
94	C₂²×D₅	—	4	8	no	—
95	C₂²×D₅	—	8	4	no	—
96	C₂²×D₅	—	8	4	no	—
97	C₂×C₄×C₅	—	4	4	no	—
98	C₂×C₄×C₅	—	8	4	no	—
99	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₅	#135	1	2	yes	—
102	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
103	C₂×Dic₅	—	2	2	no	—
104	C₂×Dic₅	D₁₆	1	4	yes	complemented
105	C₂×Dic₅	C₂×D₈	1	2	yes	—
106	C₂×Dic₅	C₂×D₈	1	2	yes	—
107	D₂₀	—	2	8	no	—
108	D₂₀	—	4	4	no	—
109	D₂₀	—	4	8	no	—
110	C₄×D₅	—	8	16	no	—
111	C₅×C₈	C₄×D₄	1	2	yes	—
112	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
113	C₅×C₈	C₂×#41	1	1	yes	characteristic
114	#168	—	2	2	no	—
115	C₂²×D₄	—	10	2	no	—
116	C₂³×C₄	—	20	2	no	—
117	C₂×D₈	—	5	8	no	—
118	C₂×D₈	—	5	4	no	—
119	C₂×D₈	—	10	2	no	—
120	C₂×D₈	—	10	4	no	—
121	C₂²×C₈	—	5	1	no	—
122	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
123	C₄×D₄	—	10	8	no	—
124	C₄×D₄	—	20	4	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₂²⋊C₄	—	20	2	no	—
127	C₂×C₄²	—	5	1	no	—
128	D₁₆	—	5	16	no	—
129	C₂×C₁₆	—	5	2	no	—
130	C₂×C₁₆	—	10	2	no	—
131	C₂×C₁₆	C₄×D₅	1	4	yes	complemented
132	C₂×C₁₆	C₂²×D₅	1	2	yes	—
133	#107	—	2	2	no	—
134	#107	C₂²×D₅	1	4	yes	—
135	#102	—	10	8	no	—
136	C₄×C₈	—	5	2	no	—
137	C₄×C₈	—	5	2	no	—
138	C₂²×C₅	C₄×D₈	1	2	yes	—
139	C₂²×C₅	C₂×D₁₆	1	2	yes	—
140	C₂²×C₅	#508	1	2	yes	—
141	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
142	C₂×D₅	—	4	8	no	—
143	C₂×D₅	—	8	4	no	—
144	C₂×D₅	—	8	4	no	—
145	C₂×D₅	—	8	16	no	—
146	C₄×C₅	—	8	4	no	—
147	C₄×C₅	C₄×D₈	1	2	yes	—
148	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
149	C₄×C₅	C₂×#135	1	1	yes	characteristic
150	Dic₅	—	2	2	no	—
151	Dic₅	C₂×D₁₆	1	4	yes	complemented
152	C₂⁴	—	20	2	no	—
153	C₂×D₄	—	10	4	no	—
154	C₂×D₄	—	10	8	no	—
155	C₂×D₄	—	20	2	no	—
156	C₂×D₄	—	20	4	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	—	5	1	no	—
159	C₂²×C₄	—	5	1	no	—
160	C₂²×C₄	—	20	8	no	—
161	C₂²×C₄	—	40	8	no	—
162	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
163	D₈	—	5	8	no	—
164	D₈	—	10	4	no	—
165	D₈	—	10	8	no	—
166	C₂×C₈	—	5	2	no	—
167	C₂×C₈	—	10	2	no	—
168	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
169	C₂×C₈	D₄×D₅	1	1	yes	characteristic
170	C₂×C₈	#713	1	2	yes	—
171	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₂²⋊C₄	—	20	8	no	—
175	C₄²	—	5	2	no	—
176	C₄²	—	5	2	no	—
177	C₁₆	—	10	2	no	—
178	C₁₆	C₂×C₄×D₅	1	4	yes	complemented
179	C₂×C₅	C₄×D₁₆	1	4	yes	complemented
180	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
181	C₂×C₅	C₂²×D₁₆	1	1	yes	characteristic
182	C₂×C₅	C₂×#508	1	1	yes	characteristic
183	D₅	—	8	8	no	—
184	C₂³	—	20	4	no	—
185	C₂³	—	20	8	no	—
186	C₂³	—	40	4	no	—
187	C₂³	—	40	4	no	—
188	C₂³	D₅×D₈	1	1	yes	characteristic
189	D₄	—	10	8	no	—
190	D₄	—	20	4	no	—
191	D₄	—	20	8	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	—	5	2	no	—
194	C₂×C₄	—	5	4	no	—
195	C₂×C₄	—	5	2	no	—
196	C₂×C₄	—	8	4	no	—
197	C₂×C₄	—	10	2	no	—
198	C₂×C₄	—	40	16	no	—
199	C₂×C₄	#4039	1	2	yes	—
200	C₂×C₄	D₅×D₈	1	1	yes	characteristic
201	C₂×C₄	#4065	1	2	yes	—
202	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
203	C₈	—	10	2	no	—
204	C₈	#4039	1	2	yes	—
205	C₈	C₂²×C₄×D₅	1	1	yes	characteristic
206	C₈	C₂×#713	1	1	yes	characteristic
207	C₅	C₂×C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
208	C₂²	—	20	8	no	—
209	C₂²	—	40	16	no	—
210	C₂²	—	40	4	no	—
211	C₂²	—	40	4	no	—
212	C₂²	#65161	1	2	yes	—
213	C₂²	D₅×D₁₆	1	2	yes	—
214	C₂²	#65208	1	2	yes	—
215	C₂²	C₂×D₅×D₈	1	1	yes	characteristic
216	C₄	—	5	4	no	—
217	C₄	—	8	4	no	—
218	C₄	—	10	2	no	—
219	C₄	#65161	1	2	yes	—
220	C₄	C₂×#4039	1	1	yes	characteristic
221	C₄	C₂×#4065	1	1	yes	characteristic
222	C₂	—	40	8	no	—
223	C₂	#10600110	1	4	yes	direct product factor
224	C₂	C₂×#65161	1	1	yes	characteristic
225	C₂	C₂×D₅×D₁₆	1	1	yes	characteristic
226	C₂	C₂×#65208	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₅×D₈
commutator	C₅×C₈	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#366	C₂
socle	C₂³×C₅	C₂×D₈
prime radical	C₂²×D₈₀	C₂
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×#366	D₅
hypocenter	C₅	C₂×C₄×D₁₆
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
2. C₂×#65161	C₂	semidirect
3. C₂×C₅×#366	C₂	semidirect
4. C₂×C₁₆×Dic₅	C₂	semidirect
5. C₂×#64711	C₂	semidirect
6. #10600110	C₂	direct (N × H)
14. C₂×D₈₀	C₄	semidirect
17. #65161	C₂²	semidirect
18. C₅×#366	C₂²	semidirect
19. C₁₆×Dic₅	C₂²	semidirect
20. #64711	C₂²	semidirect
31. C₂×D₄₀	C₂×C₄	semidirect
35. C₂×C₅×C₁₆	C₂×C₄	semidirect
44. D₈₀	C₂×C₄	semidirect
48. C₂×#366	D₅	semidirect
75. D₄₀	C₂²×C₄	semidirect
76. C₅×C₁₆	C₂²×C₄	semidirect
89. #366	C₂×D₅	semidirect
104. C₂×Dic₅	D₁₆	semidirect
131. C₂×C₁₆	C₄×D₅	semidirect
151. Dic₅	C₂×D₁₆	semidirect
178. C₁₆	C₂×C₄×D₅	semidirect
179. C₂×C₅	C₄×D₁₆	semidirect
207. C₅	C₂×C₄×D₁₆	semidirect
223. C₂	#10600110	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×D₈₀	2
C₂×#65161	2
C₂×C₅×#366	2
C₂×C₁₆×Dic₅	2
C₂×#64711	2
#10600110	2
C₂×C₄×D₁₆	5

Subgroups — C2×#10600110 #12252560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10600110 #12252560