Subgroups — C₂×#10610173 #12968905

Order 1280.

1942 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65804	C₂	1	2	yes	maximal
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#65395	C₂	1	4	yes	maximal
4	#10610173	C₂	1	16	yes	maximal, direct product factor
5	C₂×#64971	C₂	1	2	yes	maximal, complemented
6	C₂×#64972	C₂	1	4	yes	maximal, complemented
7	C₂×#64969	C₂	1	2	yes	maximal, complemented
8	C₂³×C₅×C₈	C₂²	1	2	yes	—
9	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×#4072	C₂²	1	4	yes	—
11	C₂×#4070	—	2	4	no	—
12	#65804	—	2	8	no	—
13	#65804	C₂²	1	16	yes	—
14	C₂×#4016	C₂²	1	4	yes	—
15	C₂×#4015	C₂²	1	4	yes	—
16	C₂×C₄×Dic₁₀	—	2	4	no	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	C₂²	1	8	yes	—
21	#65395	C₂²	1	32	yes	—
22	C₂×#3950	C₂²	1	8	yes	—
23	C₂×#3949	C₂²	1	8	yes	—
24	#64972	C₂²	1	32	yes	complemented
25	#64971	C₂²	1	16	yes	complemented
26	#64969	C₂²	1	16	yes	complemented
27	C₂×#3160	—	5	1	no	maximal, Sylow
28	C₂³×C₄×C₅	D₄	1	2	yes	—
29	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂²×Dic₁₀	—	2	4	no	—
31	C₂²×C₅×C₈	Q₈	1	16	yes	complemented
32	C₂²×C₅×C₈	C₂³	1	2	yes	—
33	C₂²×C₅×C₈	C₂³	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	8	yes	—
35	C₂×C₄²×C₅	D₄	1	2	yes	—
36	C₂×C₄²×C₅	D₄	1	4	yes	—
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂³	1	4	yes	—
40	C₂×C₂²⋊Dic₅	—	2	4	no	—
41	#4072	—	2	8	no	—
42	#4072	C₂³	1	16	yes	—
43	C₂×C₄⋊Dic₅	—	2	4	no	—
44	C₂×C₄⋊Dic₅	C₂³	1	8	yes	—
45	#4070	—	2	32	no	—
46	C₂×#683	—	2	8	no	—
47	C₂×C₄×Dic₅	—	2	4	no	—
48	#4016	—	2	8	no	—
49	#4016	C₂³	1	16	yes	—
50	#4015	—	2	8	no	—
51	#4015	C₂³	1	16	yes	—
52	C₄×Dic₁₀	—	2	32	no	—
53	C₄×C₅×C₈	C₂³	1	4	yes	—
54	C₄×C₅×C₈	C₂³	1	4	yes	—
55	C₄×C₅×C₈	C₂³	1	8	yes	—
56	#3950	C₂³	1	32	yes	—
57	#3949	C₂³	1	32	yes	—
58	C₂×#533	—	5	2	no	—
59	#3160	—	5	16	no	—
60	C₂×#500	—	5	2	no	—
61	C₂×#499	—	5	4	no	—
62	C₂×#498	—	5	2	no	—
63	C₂×#427	—	5	4	no	—
64	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
65	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	—	2	8	no	—
67	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
69	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×Q₈	1	8	yes	—
73	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
75	C₂²×Dic₅	—	2	8	no	—
76	C₂×Dic₁₀	—	2	16	no	—
77	C₂×Dic₁₀	—	4	16	no	—
78	C₂×C₅×C₈	C₂×Q₈	1	8	yes	—
79	C₂×C₅×C₈	C₂×Q₈	1	8	yes	—
80	C₂×C₅×C₈	C₂×Q₈	1	32	yes	complemented
81	C₂×C₅×C₈	C₂⁴	1	4	yes	—
82	C₂×C₅×C₈	C₂⁴	1	4	yes	—
83	C₄²×C₅	C₂×D₄	1	8	yes	—
84	C₄²×C₅	C₂×D₄	1	2	yes	—
85	C₄²×C₅	C₂×D₄	1	2	yes	—
86	C₄²×C₅	C₂⁴	1	2	yes	—
87	C₄²×C₅	C₂⁴	1	2	yes	—
88	C₂²⋊Dic₅	—	2	16	no	—
89	C₄⋊Dic₅	—	2	16	no	—
90	C₄⋊Dic₅	—	2	8	no	—
91	C₄⋊Dic₅	C₂⁴	1	16	yes	—
92	#683	—	2	32	no	—
93	C₄×Dic₅	—	2	16	no	—
94	C₂³×C₈	C₂×D₅	1	2	yes	—
95	#533	—	5	16	no	—
96	#533	—	10	8	no	—
97	C₂×C₄⋊Q₈	—	5	4	no	—
98	C₂×#125	—	5	4	no	—
99	C₂×C₂²⋊Q₈	—	10	4	no	—
100	C₂×C₄×Q₈	—	10	4	no	—
101	C₂×#117	—	5	4	no	—
102	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
103	#500	—	5	16	no	—
104	#499	—	5	32	no	—
105	#498	—	5	16	no	—
106	#427	—	5	32	no	—
107	C₂×#107	—	5	8	no	—
108	C₂×#106	—	5	8	no	—
109	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
111	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂×D₅	1	8	yes	—
113	C₂³×C₅	—	2	4	no	—
114	C₂³×C₅	C₄⋊Q₈	1	4	yes	—
115	C₂³×C₅	C₂²×D₄	1	2	yes	—
116	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	16	no	—
119	C₂×C₄×C₅	C₄⋊Q₈	1	8	yes	—
120	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
121	C₂×C₄×C₅	#135	1	8	yes	—
122	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
123	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
124	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
126	C₂×C₄×C₅	C₂²×Q₈	1	4	yes	—
127	C₂×C₄×C₅	C₂²×Q₈	1	8	yes	—
128	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
129	C₂×Dic₅	—	2	16	no	—
130	C₂×Dic₅	—	4	16	no	—
131	Dic₁₀	—	4	32	no	—
132	C₅×C₈	C₂²×Q₈	1	8	yes	—
133	C₅×C₈	C₂²×Q₈	1	8	yes	—
134	C₂²×Q₈	—	10	4	no	—
135	C₂³×C₄	D₂₀	1	2	yes	—
136	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
137	C₂²×C₈	Dic₁₀	1	16	yes	complemented
138	C₂²×C₈	C₂²×D₅	1	2	yes	—
139	C₂²×C₈	C₂²×D₅	1	2	yes	—
140	C₂²×C₈	C₂²×D₅	1	8	yes	—
141	C₄⋊Q₈	—	5	16	no	—
142	C₄⋊Q₈	—	10	8	no	—
143	#125	—	5	16	no	—
144	#125	—	10	8	no	—
145	C₂²⋊Q₈	—	10	32	no	—
146	C₄×Q₈	—	10	32	no	—
147	#117	—	5	16	no	—
148	#117	—	10	8	no	—
149	C₂×C₄⋊C₄	—	5	8	no	—
150	C₂×C₄⋊C₄	—	10	8	no	—
151	C₂×C₄⋊C₄	—	10	4	no	—
152	C₂×C₂²⋊C₄	—	10	4	no	—
153	C₂×C₄²	—	10	4	no	—
154	C₂×C₄²	D₂₀	1	4	yes	—
155	C₂×C₄²	D₂₀	1	2	yes	—
156	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
157	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
158	C₂×C₄²	C₂²×D₅	1	4	yes	—
159	#107	—	5	32	no	—
160	#106	—	5	32	no	—
161	C₄×C₈	C₂²×D₅	1	4	yes	—
162	C₄×C₈	C₂²×D₅	1	8	yes	—
163	C₄×C₈	C₂²×D₅	1	4	yes	—
164	C₂²×C₅	—	2	8	no	—
165	C₂²×C₅	—	2	4	no	—
166	C₂²×C₅	C₂×C₄⋊Q₈	1	4	yes	—
167	C₂²×C₅	C₂×C₄⋊Q₈	1	2	yes	—
168	C₂²×C₅	C₂×#135	1	4	yes	—
169	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
170	C₄×C₅	—	2	4	no	—
171	C₄×C₅	C₂×C₄⋊Q₈	1	4	yes	—
172	C₄×C₅	C₂×C₄⋊Q₈	1	2	yes	—
173	C₄×C₅	C₂×#135	1	8	yes	—
174	C₄×C₅	C₂³×Q₈	1	2	yes	—
175	Dic₅	—	4	16	no	—
176	C₂⁴	C₂×D₂₀	1	1	yes	characteristic
177	C₂×Q₈	—	10	16	no	—
178	C₂×Q₈	—	20	16	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	10	8	no	—
181	C₂²×C₄	C₂×Dic₁₀	1	8	yes	—
182	C₂²×C₄	C₂×D₂₀	1	2	yes	—
183	C₂²×C₄	C₂×D₂₀	1	8	yes	—
184	C₂²×C₄	C₂×D₂₀	1	2	yes	—
185	C₂²×C₄	C₂×D₂₀	1	2	yes	—
186	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
187	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
188	C₂²×C₄	C₂³×D₅	1	2	yes	—
189	C₂×C₈	C₂×Dic₁₀	1	8	yes	—
190	C₂×C₈	C₂×Dic₁₀	1	8	yes	—
191	C₂×C₈	C₂×Dic₁₀	1	32	yes	complemented
192	C₂×C₈	C₂³×D₅	1	4	yes	—
193	C₂×C₈	C₂³×D₅	1	4	yes	—
194	C₄⋊C₄	—	5	16	no	—
195	C₄⋊C₄	—	10	8	no	—
196	C₄⋊C₄	—	10	32	no	—
197	C₄⋊C₄	—	10	16	no	—
198	C₂²⋊C₄	—	10	16	no	—
199	C₄²	—	10	16	no	—
200	C₄²	C₂×D₂₀	1	8	yes	—
201	C₄²	C₂×D₂₀	1	2	yes	—
202	C₄²	C₂×D₂₀	1	2	yes	—
203	C₄²	C₂³×D₅	1	2	yes	—
204	C₄²	C₂³×D₅	1	2	yes	—
205	C₂×C₅	—	2	4	no	—
206	C₂×C₅	#3160	1	4	yes	complemented
207	C₂×C₅	C₂²×C₄⋊Q₈	1	1	yes	characteristic
208	C₂×C₅	C₂²×#135	1	2	yes	—
209	C₂³	—	2	4	no	—
210	C₂³	#4015	1	4	yes	—
211	C₂³	C₂²×D₂₀	1	2	yes	—
212	C₂³	C₂²×D₂₀	1	1	yes	characteristic
213	Q₈	—	20	32	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	16	no	—
216	C₂×C₄	—	10	16	no	—
217	C₂×C₄	—	20	16	no	—
218	C₂×C₄	#4015	1	8	yes	—
219	C₂×C₄	#4015	1	4	yes	—
220	C₂×C₄	#4050	1	8	yes	—
221	C₂×C₄	C₂²×Dic₁₀	1	4	yes	—
222	C₂×C₄	C₂²×Dic₁₀	1	8	yes	—
223	C₂×C₄	C₂²×D₂₀	1	4	yes	—
224	C₂×C₄	C₂²×D₂₀	1	2	yes	—
225	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
226	C₂×C₄	C₂²×D₂₀	1	4	yes	—
227	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
228	C₈	C₂²×Dic₁₀	1	8	yes	—
229	C₈	C₂²×Dic₁₀	1	8	yes	—
230	C₅	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
231	C₂²	—	2	4	no	—
232	C₂²	—	2	8	no	—
233	C₂²	C₂×#4015	1	4	yes	—
234	C₂²	C₂×#4015	1	2	yes	—
235	C₂²	C₂×#4050	1	4	yes	—
236	C₂²	C₂³×D₂₀	1	1	yes	characteristic
237	C₄	—	2	4	no	—
238	C₄	—	20	16	no	—
239	C₄	C₂×#4015	1	2	yes	—
240	C₄	C₂×#4015	1	4	yes	—
241	C₄	C₂×#4050	1	8	yes	—
242	C₄	C₂³×Dic₁₀	1	2	yes	—
243	C₂	—	2	4	no	—
244	C₂	#10610173	1	4	yes	direct product factor
245	C₂	C₂²×#4015	1	1	yes	characteristic
246	C₂	C₂²×#4050	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle	C₂³×C₅	C₂²×D₄
prime radical	C₂⁴×C₅	C₂×D₄
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	C₂×#3160
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #10610173	C₂	direct (N × H)
5. C₂×#64971	C₂	semidirect
6. C₂×#64972	C₂	semidirect
7. C₂×#64969	C₂	semidirect
24. #64972	C₂²	semidirect
25. #64971	C₂²	semidirect
26. #64969	C₂²	semidirect
31. C₂²×C₅×C₈	Q₈	semidirect
80. C₂×C₅×C₈	C₂×Q₈	semidirect
137. C₂²×C₈	Dic₁₀	semidirect
191. C₂×C₈	C₂×Dic₁₀	semidirect
206. C₂×C₅	#3160	semidirect
230. C₅	C₂×#3160	semidirect
244. C₂	#10610173	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×#65804	2
C₂²×C₄×C₅×C₈	2
C₂×#65395	2
#10610173	2
C₂×#64971	2
C₂×#64972	2
C₂×#64969	2
C₂×#3160	5

Subgroups — C2×#10610173 #12968905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10610173 #12968905