Subgroups — #12968989

Order 1280.

1762 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10617730	C₂	1	2	yes	maximal
2	C₄×C₅×#131	C₂	1	1	yes	maximal, characteristic
3	#10615516	C₂	1	2	yes	maximal
4	#10615515	C₂	1	2	yes	maximal
5	#10610277	C₂	1	6	yes	maximal, complemented
6	#10610276	C₂	1	6	yes	maximal, complemented
7	#10610238	C₂	1	6	yes	maximal, complemented
8	#10610173	C₂	1	3	yes	maximal, complemented
9	C₂×#64972	C₂	1	3	yes	maximal, complemented
10	C₂×C₅×#131	C₂²	1	1	yes	characteristic
11	C₂×C₅×#131	C₂²	1	1	yes	characteristic
12	C₄×C₅×#41	C₂²	1	1	yes	characteristic
13	#66161	C₂²	1	4	yes	—
14	#65903	—	2	4	no	—
15	#65873	—	2	12	no	—
16	#65853	C₂²	1	12	yes	—
17	#65804	C₂²	1	6	yes	—
18	C₂×#4016	C₂²	1	6	yes	—
19	C₅×#405	C₂²	1	3	yes	—
20	C₅×#405	C₂²	1	3	yes	—
21	C₄×C₅×M₁₆	C₂²	1	3	yes	—
22	C₂×C₄×C₅×C₈	C₂²	1	3	yes	—
23	#65410	C₂²	1	12	yes	—
24	#65410	C₂²	1	12	yes	—
25	#65395	C₂²	1	6	yes	—
26	#65395	C₂²	1	6	yes	—
27	C₂×#3950	C₂²	1	6	yes	—
28	C₂×#3949	C₂²	1	6	yes	—
29	#64991	C₂²	1	24	yes	—
30	#64972	C₂²	1	6	yes	—
31	#64972	C₂²	1	6	yes	—
32	#64972	C₂²	1	6	yes	—
33	#64972	C₂²	1	6	yes	—
34	#64972	C₂²	1	3	yes	—
35	#64972	C₂²	1	1	yes	characteristic
36	#64971	C₂²	1	6	yes	—
37	#64969	C₂²	1	6	yes	—
38	#60666	—	5	1	no	maximal, Sylow
39	C₂×C₅×#41	C₂³	1	1	yes	characteristic
40	C₅×#131	Q₈	1	8	yes	complemented
41	C₅×#131	Q₈	1	8	yes	complemented
42	C₂×C₅×M₁₆	C₂³	1	3	yes	—
43	C₂×C₅×M₁₆	C₂³	1	3	yes	—
44	C₂²×C₅×C₈	C₂³	1	3	yes	—
45	C₂²×C₅×C₈	C₂³	1	3	yes	—
46	C₄×C₅×Q₈	D₄	1	2	yes	—
47	C₄×C₅×D₄	D₄	1	6	yes	—
48	C₅×#117	C₂³	1	3	yes	—
49	C₂×C₄²×C₅	C₂³	1	3	yes	—
50	Q₈×Dic₅	—	2	4	no	—
51	D₄×Dic₅	—	2	12	no	—
52	#4072	C₂³	1	12	yes	—
53	C₂×C₄⋊Dic₅	C₂³	1	12	yes	—
54	#4070	—	2	12	no	—
55	C₂×#683	—	2	12	no	—
56	#4036	—	2	12	no	—
57	#4034	—	2	12	no	—
58	#4024	—	2	24	no	—
59	#4016	—	2	12	no	—
60	#4016	C₂³	1	6	yes	—
61	#4016	C₂³	1	12	yes	—
62	#4016	C₂³	1	2	yes	—
63	#4015	C₂³	1	12	yes	—
64	C₄×Dic₁₀	—	2	12	no	—
65	C₅×#97	C₂³	1	6	yes	—
66	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₅×C₈	C₂³	1	3	yes	—
68	C₄×C₅×C₈	C₂³	1	3	yes	—
69	C₄×C₅×C₈	C₂³	1	3	yes	—
70	#3950	C₂³	1	6	yes	—
71	#3950	C₂³	1	12	yes	—
72	#3950	C₂³	1	2	yes	—
73	#3950	C₂³	1	12	yes	—
74	#3949	C₂³	1	6	yes	—
75	#3949	C₂³	1	12	yes	—
76	#3949	C₂³	1	2	yes	—
77	#3949	C₂³	1	12	yes	—
78	#3504	—	5	2	no	—
79	#3166	—	5	6	no	—
80	#3164	—	5	6	no	—
81	#3163	—	5	6	no	—
82	#3160	—	5	3	no	—
83	C₂×#499	—	5	3	no	—
84	#2913	—	5	2	no	—
85	#2912	—	5	2	no	—
86	C₄×#131	D₅	1	1	yes	characteristic
87	C₅×#41	C₂×Q₈	1	4	yes	—
88	C₅×#41	C₂×Q₈	1	4	yes	—
89	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₅×D₄	C₂×D₄	1	3	yes	—
91	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
92	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
93	C₂²×Dic₅	—	2	12	no	—
94	C₂×Dic₁₀	—	2	12	no	—
95	C₅×M₁₆	C₂×Q₈	1	12	yes	—
96	C₅×M₁₆	C₂×Q₈	1	12	yes	—
97	C₂×C₅×C₈	C₂×Q₈	1	12	yes	—
98	C₂×C₅×C₈	C₂×Q₈	1	12	yes	—
99	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂⁴	1	3	yes	—
102	C₂×C₅×C₈	C₂⁴	1	3	yes	—
103	C₅×C₄⋊C₄	C₂×D₄	1	6	yes	—
104	C₅×C₂²⋊C₄	C₂×D₄	1	6	yes	—
105	C₄²×C₅	C₂×D₄	1	6	yes	—
106	C₄²×C₅	C₂⁴	1	1	yes	characteristic
107	C₄²×C₅	C₂⁴	1	3	yes	—
108	C₂²⋊Dic₅	—	2	12	no	—
109	C₄⋊Dic₅	—	2	12	no	—
110	C₄⋊Dic₅	—	2	12	no	—
111	C₄⋊Dic₅	C₂⁴	1	4	yes	—
112	C₄⋊Dic₅	C₂⁴	1	12	yes	—
113	#683	—	2	4	no	—
114	#683	—	2	12	no	—
115	#683	—	2	24	no	—
116	C₄×Dic₅	—	2	12	no	—
117	C₂×#131	C₂×D₅	1	1	yes	characteristic
118	C₂×#131	C₂×D₅	1	1	yes	characteristic
119	#557	—	10	4	no	—
120	#554	—	10	12	no	—
121	#544	—	5	12	no	—
122	#533	—	5	6	no	—
123	C₂×#125	—	5	6	no	—
124	#520	—	5	4	no	—
125	C₄×#41	C₂×D₅	1	1	yes	characteristic
126	#501	—	5	24	no	—
127	#500	—	5	6	no	—
128	#499	—	5	3	no	—
129	#499	—	5	6	no	—
130	#499	—	5	6	no	—
131	#499	—	5	6	no	—
132	#499	—	5	1	no	—
133	#499	—	5	6	no	—
134	#498	—	5	6	no	—
135	#428	—	5	12	no	—
136	#428	—	5	12	no	—
137	#427	—	5	6	no	—
138	#427	—	5	6	no	—
139	C₂×#107	—	5	6	no	—
140	C₂×#106	—	5	6	no	—
141	#405	C₂×D₅	1	3	yes	—
142	#405	C₂×D₅	1	3	yes	—
143	C₄×M₁₆	C₂×D₅	1	3	yes	—
144	C₂×C₄×C₈	C₂×D₅	1	3	yes	—
145	C₂³×C₅	C₂²×D₄	1	3	yes	—
146	C₅×Q₈	C₄⋊Q₈	1	4	yes	—
147	C₅×D₄	C₄⋊Q₈	1	12	yes	—
148	C₂×C₄×C₅	—	4	6	no	—
149	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
150	C₂×C₄×C₅	C₂²×D₄	1	6	yes	—
151	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
152	C₂×C₄×C₅	C₂²×Q₈	1	6	yes	—
153	C₂×C₄×C₅	C₂²×Q₈	1	6	yes	—
154	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
155	C₂×Dic₅	—	2	4	no	—
156	C₂×Dic₅	—	2	12	no	—
157	C₂×Dic₅	—	8	12	no	—
158	Dic₁₀	—	8	12	no	—
159	C₅×C₈	C₂²×Q₈	1	2	yes	—
160	C₅×C₈	C₂²×Q₈	1	2	yes	—
161	C₅×C₈	C₂²×Q₈	1	6	yes	—
162	C₅×C₈	C₂²×Q₈	1	6	yes	—
163	C₂×#41	C₂²×D₅	1	1	yes	characteristic
164	#131	Dic₁₀	1	8	yes	complemented
165	#131	Dic₁₀	1	8	yes	complemented
166	C₂×M₁₆	C₂²×D₅	1	3	yes	—
167	C₂×M₁₆	C₂²×D₅	1	3	yes	—
168	C₂²×C₈	C₂²×D₅	1	3	yes	—
169	C₂²×C₈	C₂²×D₅	1	3	yes	—
170	C₄⋊Q₈	—	5	12	no	—
171	C₄⋊Q₈	—	10	12	no	—
172	#125	—	5	6	no	—
173	#125	—	5	12	no	—
174	#125	—	5	2	no	—
175	#125	—	10	12	no	—
176	#125	—	10	12	no	—
177	C₂²⋊Q₈	—	10	12	no	—
178	C₂²⋊Q₈	—	10	24	no	—
179	C₄×Q₈	—	10	4	no	—
180	C₄×Q₈	—	10	12	no	—
181	C₄×Q₈	D₂₀	1	2	yes	—
182	C₄×D₄	—	10	12	no	—
183	C₄×D₄	D₂₀	1	6	yes	—
184	#117	—	5	12	no	—
185	#117	C₂²×D₅	1	3	yes	—
186	C₂×C₄⋊C₄	—	5	12	no	—
187	C₂×C₄⋊C₄	—	10	12	no	—
188	C₂×C₄²	C₂²×D₅	1	3	yes	—
189	#107	—	5	6	no	—
190	#107	—	5	12	no	—
191	#107	—	5	2	no	—
192	#107	—	5	12	no	—
193	#106	—	5	6	no	—
194	#106	—	5	12	no	—
195	#106	—	5	2	no	—
196	#106	—	5	12	no	—
197	#97	C₂²×D₅	1	6	yes	—
198	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
199	C₄×C₈	C₂²×D₅	1	3	yes	—
200	C₄×C₈	C₂²×D₅	1	3	yes	—
201	C₄×C₈	C₂²×D₅	1	3	yes	—
202	C₂²×C₅	—	2	6	no	—
203	C₂²×C₅	C₂×C₄⋊Q₈	1	6	yes	—
204	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
205	C₄×C₅	—	2	6	no	—
206	C₄×C₅	—	2	2	no	—
207	C₄×C₅	C₂×C₄⋊Q₈	1	6	yes	—
208	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
209	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
210	Dic₅	—	4	4	no	—
211	Dic₅	—	4	12	no	—
212	#41	C₂×Dic₁₀	1	4	yes	—
213	#41	C₂×Dic₁₀	1	4	yes	—
214	C₂×Q₈	—	10	12	no	—
215	C₂×Q₈	C₂×D₂₀	1	1	yes	characteristic
216	C₂×D₄	C₂×D₂₀	1	3	yes	—
217	C₂²×C₄	—	10	12	no	—
218	C₂²×C₄	C₂×D₂₀	1	6	yes	—
219	C₂²×C₄	C₂³×D₅	1	3	yes	—
220	M₁₆	C₂×Dic₁₀	1	12	yes	—
221	M₁₆	C₂×Dic₁₀	1	12	yes	—
222	C₂×C₈	C₂×Dic₁₀	1	12	yes	—
223	C₂×C₈	C₂×Dic₁₀	1	12	yes	—
224	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
225	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
226	C₂×C₈	C₂³×D₅	1	3	yes	—
227	C₂×C₈	C₂³×D₅	1	3	yes	—
228	C₄⋊C₄	—	5	12	no	—
229	C₄⋊C₄	—	5	4	no	—
230	C₄⋊C₄	—	10	12	no	—
231	C₄⋊C₄	—	10	12	no	—
232	C₄⋊C₄	—	10	24	no	—
233	C₄⋊C₄	—	10	4	no	—
234	C₄⋊C₄	—	10	12	no	—
235	C₄⋊C₄	C₂×D₂₀	1	6	yes	—
236	C₂²⋊C₄	—	10	12	no	—
237	C₂²⋊C₄	C₂×D₂₀	1	6	yes	—
238	C₄²	—	10	12	no	—
239	C₄²	C₂×D₂₀	1	6	yes	—
240	C₄²	C₂³×D₅	1	1	yes	characteristic
241	C₄²	C₂³×D₅	1	3	yes	—
242	C₂×C₅	—	2	6	no	—
243	C₂×C₅	C₂²×C₄⋊Q₈	1	1	yes	characteristic
244	C₂×C₅	#3584	1	2	yes	—
245	C₂³	C₂²×D₂₀	1	3	yes	—
246	Q₈	—	40	12	no	—
247	Q₈	#4015	1	4	yes	—
248	D₄	#4015	1	12	yes	—
249	C₂×C₄	—	4	6	no	—
250	C₂×C₄	—	10	4	no	—
251	C₂×C₄	—	10	12	no	—
252	C₂×C₄	—	40	12	no	—
253	C₂×C₄	C₂²×Dic₁₀	1	6	yes	—
254	C₂×C₄	C₂²×Dic₁₀	1	6	yes	—
255	C₂×C₄	C₂²×D₂₀	1	2	yes	—
256	C₂×C₄	C₂²×D₂₀	1	6	yes	—
257	C₂×C₄	C₂²×D₂₀	1	3	yes	—
258	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
259	C₈	C₂²×Dic₁₀	1	2	yes	—
260	C₈	C₂²×Dic₁₀	1	2	yes	—
261	C₈	C₂²×Dic₁₀	1	6	yes	—
262	C₈	C₂²×Dic₁₀	1	6	yes	—
263	C₅	#60666	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	2	6	no	—
265	C₂²	C₂×#4015	1	6	yes	—
266	C₂²	C₂³×D₂₀	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	6	no	—
269	C₄	—	20	12	no	—
270	C₄	—	20	4	no	—
271	C₄	C₂×#4015	1	6	yes	—
272	C₄	C₂³×Dic₁₀	1	1	yes	characteristic
273	C₄	C₂³×Dic₁₀	1	1	yes	characteristic
274	C₂	—	2	6	no	—
275	C₂	C₂²×#4015	1	1	yes	characteristic
276	C₂	#10619064	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₄×C₅×#131	C₂
socle	C₂²×C₅	C₂³×D₄
prime radical	C₂×C₅×#41	C₂³
hypercenter	C₄×#131	D₅
hypocenter	C₅	#60666
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
5. #10610277	C₂	semidirect
6. #10610276	C₂	semidirect
7. #10610238	C₂	semidirect
8. #10610173	C₂	semidirect
9. C₂×#64972	C₂	semidirect
40. C₅×#131	Q₈	semidirect
41. C₅×#131	Q₈	semidirect
164. #131	Dic₁₀	semidirect
165. #131	Dic₁₀	semidirect
263. C₅	#60666	semidirect

Maximal subgroups

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

subgroup	index
#10617730	2
C₄×C₅×#131	2
#10615516	2
#10615515	2
#10610277	2
#10610276	2
#10610238	2
#10610173	2
C₂×#64972	2
#60666	5