Subgroups — #12069833

Order 1280.

614 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610206	C₂	1	1	yes	maximal, characteristic
2	#10610176	C₂	1	2	yes	maximal
3	C₂×C₅×C₈²	C₂	1	1	yes	maximal, characteristic
4	#10600531	C₂	1	2	yes	maximal
5	#10600524	C₂	1	1	yes	maximal, characteristic
6	#10598849	C₂	1	4	yes	maximal, complemented
7	C₈×Dic₂₀	C₂	1	4	yes	maximal, complemented
8	#65804	C₂²	1	2	yes	—
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	#65405	C₄	1	4	yes	—
13	#65391	—	2	2	no	—
14	#65390	—	2	2	no	—
15	#65213	C₂²	1	2	yes	—
16	#64988	C₄	1	1	yes	characteristic
17	C₄×Dic₂₀	C₂²	1	4	yes	—
18	#64971	C₄	1	1	yes	characteristic
19	#64970	C₄	1	2	yes	—
20	#64968	—	2	2	no	—
21	#64968	C₂²	1	4	yes	—
22	C₈×Dic₁₀	—	2	2	no	—
23	C₈×Dic₁₀	C₂²	1	4	yes	—
24	C₅×C₈²	C₂²	1	2	yes	—
25	C₅×C₈²	C₂²	1	2	yes	—
26	#64678	C₂²	1	4	yes	—
27	#64676	C₂²	1	8	yes	—
28	#17958	—	5	1	no	maximal, Sylow
29	C₂²×C₅×C₈	—	2	1	no	—
30	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
34	#4072	—	2	2	no	—
35	#4070	—	2	2	no	—
36	#4070	C₂×C₄	1	4	yes	—
37	C₂×#681	—	2	2	no	—
38	C₂×Dic₂₀	C₂×C₄	1	4	yes	—
39	#4016	—	2	2	no	—
40	#4015	C₂×C₄	1	2	yes	—
41	#4015	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₀	—	2	2	no	—
43	C₄×Dic₁₀	C₂³	1	4	yes	—
44	C₄×C₅×C₈	—	2	1	no	—
45	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₅×C₈	D₄	1	1	yes	characteristic
48	C₄×C₅×C₈	D₄	1	1	yes	characteristic
49	C₄×C₅×C₈	C₂³	1	2	yes	—
50	C₄×C₅×C₈	C₂³	1	2	yes	—
51	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
53	#3950	C₂×C₄	1	4	yes	—
54	#3948	C₂×C₄	1	8	yes	—
55	#3947	—	2	4	no	—
56	#3946	—	2	4	no	—
57	#3946	—	2	4	no	—
58	C₈×Dic₅	—	2	4	no	—
59	C₄⋊#168	—	2	2	no	—
60	C₄⋊#168	C₂³	1	4	yes	—
61	#3170	—	5	1	no	—
62	#2960	—	5	2	no	—
63	#1591	—	5	4	no	—
64	C₈×Q₁₆	—	5	4	no	—
65	#1567	—	5	1	no	—
66	#1493	—	5	2	no	—
67	C₂×C₈²	D₅	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	#41	1	1	yes	characteristic
71	C₂×Dic₁₀	—	2	2	no	—
72	C₂×Dic₁₀	C₂²×C₄	1	4	yes	—
73	C₂×C₅×C₈	—	2	4	no	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	2	4	no	—
77	C₂×C₅×C₈	Q₁₆	1	4	yes	complemented
78	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
79	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅	#41	1	1	yes	characteristic
87	C₄²×C₅	C₂⁴	1	1	yes	characteristic
88	C₂²⋊Dic₅	—	2	4	no	—
89	C₄⋊Dic₅	—	2	2	no	—
90	C₄⋊Dic₅	—	2	2	no	—
91	C₄⋊Dic₅	—	2	2	no	—
92	C₄⋊Dic₅	C₂²×C₄	1	4	yes	—
93	#683	—	2	4	no	—
94	#683	—	2	4	no	—
95	C₄×Dic₅	—	2	4	no	—
96	#681	—	4	4	no	—
97	C₂×#168	—	2	4	no	—
98	Dic₂₀	—	2	8	no	—
99	#533	—	5	2	no	—
100	#500	—	5	1	no	—
101	#494	—	5	1	no	—
102	#487	—	5	2	no	—
103	#467	—	5	4	no	—
104	#446	—	5	4	no	—
105	#446	—	10	2	no	—
106	C₈×Q₈	—	5	4	no	—
107	C₈×Q₈	—	10	2	no	—
108	C₄×Q₁₆	—	5	4	no	—
109	#424	—	10	2	no	—
110	#423	—	5	2	no	—
111	#405	—	10	2	no	—
112	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
115	#335	—	5	4	no	—
116	Q₈⋊C₈	—	5	8	no	—
117	C₈²	C₂×D₅	1	2	yes	—
118	C₈²	C₂×D₅	1	2	yes	—
119	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
126	C₂×C₄×C₅	#135	1	2	yes	—
127	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
130	C₂×Dic₅	—	2	4	no	—
131	C₂×Dic₅	—	2	4	no	—
132	Dic₁₀	—	2	8	no	—
133	Dic₁₀	—	4	4	no	—
134	C₅×C₈	—	2	4	no	—
135	C₅×C₈	—	2	2	no	—
136	C₅×C₈	—	2	1	no	—
137	C₅×C₈	—	2	1	no	—
138	C₅×C₈	#131	1	4	yes	—
139	C₅×C₈	C₂×Q₁₆	1	2	yes	—
140	C₅×C₈	C₂×Q₁₆	1	2	yes	—
141	#168	—	4	4	no	—
142	C₂×Q₁₆	—	5	4	no	—
143	C₂×M₁₆	—	10	2	no	—
144	C₂²×C₈	—	2	1	no	—
145	C₂²×C₈	C₄×D₅	1	2	yes	—
146	C₂²×C₈	D₂₀	1	1	yes	characteristic
147	C₂²×C₈	D₂₀	1	1	yes	characteristic
148	C₄⋊Q₈	—	5	2	no	—
149	C₄⋊Q₈	—	5	2	no	—
150	#125	—	10	2	no	—
151	C₂²⋊Q₈	—	5	4	no	—
152	C₂²⋊Q₈	—	10	2	no	—
153	C₄×Q₈	—	5	4	no	—
154	C₄×Q₈	—	10	2	no	—
155	#117	—	10	2	no	—
156	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
157	#107	—	5	4	no	—
158	C₄⋊C₈	—	5	4	no	—
159	C₄⋊C₈	—	10	4	no	—
160	C₄⋊C₈	—	10	4	no	—
161	C₄⋊C₈	—	10	2	no	—
162	#103	—	5	8	no	—
163	#97	—	10	4	no	—
164	C₄×C₈	—	2	1	no	—
165	C₄×C₈	—	10	4	no	—
166	C₄×C₈	C₄×D₅	1	1	yes	characteristic
167	C₄×C₈	C₄×D₅	1	1	yes	characteristic
168	C₄×C₈	D₂₀	1	1	yes	characteristic
169	C₄×C₈	D₂₀	1	1	yes	characteristic
170	C₄×C₈	C₂²×D₅	1	2	yes	—
171	C₄×C₈	C₂²×D₅	1	2	yes	—
172	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
173	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
174	C₂²×C₅	—	2	1	no	—
175	C₂²×C₅	—	2	1	no	—
176	C₂²×C₅	C₄×Q₁₆	1	2	yes	—
177	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
178	C₄×C₅	—	2	1	no	—
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	C₄×Q₁₆	1	2	yes	—
181	C₄×C₅	#443	1	2	yes	—
182	C₄×C₅	C₂×#131	1	2	yes	—
183	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
184	C₄×C₅	C₂×#135	1	1	yes	characteristic
185	Dic₅	—	4	4	no	—
186	Dic₅	—	4	4	no	—
187	C₂×Q₈	—	5	4	no	—
188	C₂×Q₈	—	10	2	no	—
189	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
190	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
191	C₂²×C₄	#709	1	1	yes	characteristic
192	Q₁₆	—	10	8	no	—
193	M₁₆	—	20	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	10	4	no	—
199	C₂×C₈	Dic₂₀	1	4	yes	complemented
200	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
201	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
202	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
203	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
204	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
205	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
206	C₄⋊C₄	—	5	4	no	—
207	C₄⋊C₄	—	10	4	no	—
208	C₄⋊C₄	—	10	2	no	—
209	C₄⋊C₄	—	10	2	no	—
210	C₄⋊C₄	—	10	4	no	—
211	C₄⋊C₄	—	10	2	no	—
212	C₂²⋊C₄	—	10	4	no	—
213	C₄²	—	10	4	no	—
214	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
215	C₄²	C₂×D₂₀	1	1	yes	characteristic
216	C₄²	#709	1	1	yes	characteristic
217	C₄²	C₂³×D₅	1	1	yes	characteristic
218	C₂×C₅	—	2	2	no	—
219	C₂×C₅	#2931	1	1	yes	characteristic
220	C₂×C₅	C₂×C₄×Q₁₆	1	1	yes	characteristic
221	C₂×C₅	C₂×#443	1	1	yes	characteristic
222	C₂³	C₄×D₂₀	1	1	yes	characteristic
223	Q₈	—	10	8	no	—
224	Q₈	—	20	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	10	4	no	—
228	C₂×C₄	—	10	4	no	—
229	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
230	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
231	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
232	C₂×C₄	#4050	1	2	yes	—
233	C₂×C₄	C₂×Dic₂₀	1	2	yes	—
234	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
235	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
236	C₂×C₄	C₂×#709	1	1	yes	characteristic
237	C₈	—	2	4	no	—
238	C₈	—	2	2	no	—
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	20	4	no	—
242	C₈	#4047	1	4	yes	—
243	C₈	C₂×Dic₂₀	1	2	yes	—
244	C₈	C₂×Dic₂₀	1	2	yes	—
245	C₅	#17958	1	1	yes	characteristic, complemented, Sylow
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	C₄×Dic₂₀	1	2	yes	—
249	C₂²	C₂×C₄×D₂₀	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	20	4	no	—
253	C₄	—	20	4	no	—
254	C₄	C₄×Dic₂₀	1	2	yes	—
255	C₄	#64989	1	2	yes	—
256	C₄	C₂×#4047	1	2	yes	—
257	C₄	C₂×#4050	1	1	yes	characteristic
258	C₄	C₂²×Dic₂₀	1	1	yes	characteristic
259	C₂	—	2	2	no	—
260	C₂	#10610189	1	1	yes	characteristic
261	C₂	C₂×C₄×Dic₂₀	1	1	yes	characteristic
262	C₂	C₂×#64989	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₅×C₈²	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂³×C₅	C₄×D₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂×C₈²	D₅
hypocenter	C₅	#17958
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
6. #10598849	C₂	semidirect
7. C₈×Dic₂₀	C₂	semidirect
77. C₂×C₅×C₈	Q₁₆	semidirect
199. C₂×C₈	Dic₂₀	semidirect
245. C₅	#17958	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#10610206	2
#10610176	2
C₂×C₅×C₈²	2
#10600531	2
#10600524	2
#10598849	2
C₈×Dic₂₀	2
#17958	5