Subgroups — #12071076

Order 1280.

586 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610739	C₂	1	2	yes	maximal
2	#10610206	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic
4	#10601222	C₂	1	2	yes	maximal
5	C₈×#681	C₂	1	1	yes	maximal, characteristic
6	#10599357	C₂	1	2	yes	maximal, complemented
7	#10599353	C₂	1	4	yes	maximal, complemented
8	#10599350	C₂	1	2	yes	maximal, complemented
9	#65804	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	#65405	C₄	1	2	yes	—
13	#65405	C₄	1	2	yes	—
14	#65391	—	2	2	no	—
15	#65390	—	2	2	no	—
16	#65214	C₂²	1	1	yes	characteristic
17	C₄×#681	C₂²	1	1	yes	characteristic
18	#65120	C₂²	1	4	yes	—
19	#65119	C₂²	1	4	yes	—
20	#65117	—	2	4	no	—
21	#64988	C₄	1	1	yes	characteristic
22	C₄×Dic₂₀	C₂²	1	2	yes	—
23	C₄×Dic₂₀	C₂²	1	2	yes	—
24	#64971	C₄	1	1	yes	characteristic
25	#64970	C₄	1	2	yes	—
26	C₅×#335	C₂²	1	1	yes	characteristic
27	C₅×#335	C₂²	1	2	yes	—
28	C₅×#335	C₂²	1	1	yes	characteristic
29	#64704	C₂²	1	4	yes	—
30	#64704	C₂²	1	4	yes	—
31	#64674	C₂²	1	2	yes	—
32	C₈×#168	C₂²	1	2	yes	—
33	#18068	—	5	1	no	maximal, Sylow
34	C₂²×C₅×C₈	—	2	2	no	—
35	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	#4072	—	2	2	no	—
39	#4070	—	2	2	no	—
40	#4070	C₂×C₄	1	4	yes	—
41	C₂×#681	D₄	1	1	yes	characteristic
42	C₂×#681	D₄	1	1	yes	characteristic
43	C₂×Dic₂₀	C₂×C₄	1	2	yes	—
44	C₂×Dic₂₀	C₂×C₄	1	2	yes	—
45	#4016	—	2	2	no	—
46	#4015	C₂×C₄	1	2	yes	—
47	#4015	C₂×C₄	1	2	yes	—
48	C₄×Dic₁₀	—	2	2	no	—
49	C₄×Dic₁₀	C₂³	1	4	yes	—
50	C₅×C₄⋊C₈	—	2	2	no	—
51	C₅×C₄⋊C₈	C₂³	1	2	yes	—
52	C₅×C₄⋊C₈	C₂³	1	2	yes	—
53	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
56	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
57	C₂²⋊#168	—	2	1	no	—
58	#3950	C₂×C₄	1	2	yes	—
59	#3950	C₂×C₄	1	2	yes	—
60	#3948	C₂×C₄	1	4	yes	—
61	#3948	C₂×C₄	1	4	yes	—
62	#3947	—	2	4	no	—
63	#3946	—	2	4	no	—
64	#3946	—	2	4	no	—
65	C₈×Dic₅	—	2	4	no	—
66	C₄⋊#168	—	2	1	no	—
67	#3935	D₄	1	2	yes	—
68	C₄×#168	C₂³	1	2	yes	—
69	C₄×#168	C₂³	1	2	yes	—
70	#3170	—	5	1	no	—
71	#2962	—	5	2	no	—
72	#1591	—	5	2	no	—
73	#1584	—	5	4	no	—
74	C₈×Q₁₆	—	5	2	no	—
75	C₂×#335	D₅	1	1	yes	characteristic
76	#1481	—	5	2	no	—
77	C₈×M₁₆	—	5	1	no	—
78	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	#41	1	1	yes	characteristic
81	C₂×Dic₁₀	—	2	2	no	—
82	C₂×Dic₁₀	C₂²×C₄	1	4	yes	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	—	4	1	no	—
87	C₂×C₅×C₈	—	4	4	no	—
88	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅	#41	1	1	yes	characteristic
95	C₄²×C₅	C₂⁴	1	1	yes	characteristic
96	C₂²⋊Dic₅	—	2	4	no	—
97	C₄⋊Dic₅	—	2	2	no	—
98	C₄⋊Dic₅	—	2	2	no	—
99	C₄⋊Dic₅	—	2	2	no	—
100	C₄⋊Dic₅	C₂²×C₄	1	4	yes	—
101	#683	—	2	4	no	—
102	#683	—	2	4	no	—
103	C₄×Dic₅	—	2	4	no	—
104	#681	—	2	2	no	—
105	#681	Q₁₆	1	4	yes	complemented
106	C₂×#168	—	2	2	no	—
107	C₂×#168	C₂×D₄	1	2	yes	—
108	C₂×#168	C₂×D₄	1	2	yes	—
109	Dic₂₀	—	2	4	no	—
110	Dic₂₀	—	4	2	no	—
111	#533	—	5	2	no	—
112	#500	—	5	1	no	—
113	#494	—	5	1	no	—
114	#487	—	5	2	no	—
115	#467	—	5	2	no	—
116	#467	—	5	2	no	—
117	#446	—	5	4	no	—
118	#446	—	10	4	no	—
119	C₈×Q₈	—	5	4	no	—
120	C₄×Q₁₆	—	5	2	no	—
121	C₄×Q₁₆	—	5	2	no	—
122	#431	—	5	1	no	—
123	#424	—	10	2	no	—
124	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
125	#405	—	10	2	no	—
126	C₄×M₁₆	—	5	1	no	—
127	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
128	#335	C₂×D₅	1	1	yes	characteristic
129	#335	C₂×D₅	1	2	yes	—
130	#335	C₂×D₅	1	1	yes	characteristic
131	Q₈⋊C₈	—	5	4	no	—
132	Q₈⋊C₈	—	5	4	no	—
133	#322	—	5	2	no	—
134	C₈²	—	5	2	no	—
135	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
143	C₂×C₄×C₅	#135	1	2	yes	—
144	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
147	C₂×Dic₅	—	2	4	no	—
148	C₂×Dic₅	—	2	4	no	—
149	Dic₁₀	—	2	4	no	—
150	Dic₁₀	—	2	4	no	—
151	Dic₁₀	—	8	2	no	—
152	C₅×C₈	—	2	1	no	—
153	C₅×C₈	—	2	1	no	—
154	C₅×C₈	—	2	1	no	—
155	C₅×C₈	—	4	2	no	—
156	C₅×C₈	—	4	2	no	—
157	C₅×C₈	#131	1	2	yes	—
158	#168	—	2	2	no	—
159	#168	—	4	2	no	—
160	#168	C₂×Q₁₆	1	4	yes	—
161	C₂×Q₁₆	—	5	2	no	—
162	C₂×Q₁₆	—	5	2	no	—
163	C₂×M₁₆	—	5	1	no	—
164	C₂×M₁₆	—	5	1	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
167	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
168	C₄⋊Q₈	—	5	2	no	—
169	C₄⋊Q₈	—	5	2	no	—
170	#125	—	10	2	no	—
171	C₂²⋊Q₈	—	5	4	no	—
172	C₂²⋊Q₈	—	10	2	no	—
173	C₄×Q₈	—	5	4	no	—
174	C₄×Q₈	—	10	2	no	—
175	#117	—	10	2	no	—
176	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
177	#107	—	5	2	no	—
178	#107	—	5	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	10	4	no	—
181	C₄⋊C₈	—	10	4	no	—
182	C₄⋊C₈	—	10	1	no	—
183	C₄⋊C₈	C₂²×D₅	1	2	yes	—
184	C₄⋊C₈	C₂²×D₅	1	2	yes	—
185	#103	—	5	4	no	—
186	#103	—	5	4	no	—
187	C₂²⋊C₈	—	10	1	no	—
188	#97	—	5	2	no	—
189	#97	—	10	4	no	—
190	C₄×C₈	—	5	2	no	—
191	C₄×C₈	—	5	2	no	—
192	C₄×C₈	—	10	4	no	—
193	C₄×C₈	C₄×D₅	1	1	yes	characteristic
194	C₄×C₈	C₄×D₅	1	1	yes	characteristic
195	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
196	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
197	C₂²×C₅	—	2	1	no	—
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	C₄×Q₁₆	1	2	yes	—
200	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
201	C₄×C₅	—	2	1	no	—
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	2	1	no	—
204	C₄×C₅	C₄×Q₁₆	1	2	yes	—
205	C₄×C₅	C₂×#131	1	1	yes	characteristic
206	C₄×C₅	#568	1	1	yes	characteristic
207	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
208	C₄×C₅	C₂×#135	1	1	yes	characteristic
209	Dic₅	—	4	4	no	—
210	Dic₅	—	4	4	no	—
211	C₂×Q₈	—	5	4	no	—
212	C₂×Q₈	—	10	2	no	—
213	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
214	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
215	C₂²×C₄	#713	1	1	yes	characteristic
216	Q₁₆	—	10	4	no	—
217	Q₁₆	—	20	2	no	—
218	M₁₆	—	5	4	no	—
219	M₁₆	—	10	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	4	no	—
225	C₂×C₈	—	5	2	no	—
226	C₂×C₈	—	5	2	no	—
227	C₂×C₈	—	10	2	no	—
228	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
231	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
232	C₄⋊C₄	—	5	4	no	—
233	C₄⋊C₄	—	10	4	no	—
234	C₄⋊C₄	—	10	2	no	—
235	C₄⋊C₄	—	10	2	no	—
236	C₄⋊C₄	—	10	4	no	—
237	C₄⋊C₄	—	10	2	no	—
238	C₂²⋊C₄	—	10	4	no	—
239	C₄²	—	10	4	no	—
240	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
241	C₄²	D₄×D₅	1	1	yes	characteristic
242	C₄²	#713	1	1	yes	characteristic
243	C₄²	C₂³×D₅	1	1	yes	characteristic
244	C₂×C₅	—	2	2	no	—
245	C₂×C₅	#2935	1	1	yes	characteristic
246	C₂×C₅	C₂×C₄×Q₁₆	1	1	yes	characteristic
247	C₂×C₅	#2957	1	1	yes	characteristic
248	C₂³	#4039	1	1	yes	characteristic
249	Q₈	—	10	4	no	—
250	Q₈	—	10	4	no	—
251	Q₈	—	40	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	10	4	no	—
256	C₂×C₄	—	10	4	no	—
257	C₂×C₄	#4039	1	1	yes	characteristic
258	C₂×C₄	#4039	1	1	yes	characteristic
259	C₂×C₄	#4039	1	1	yes	characteristic
260	C₂×C₄	#4058	1	2	yes	—
261	C₂×C₄	D₅×Q₁₆	1	2	yes	—
262	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
263	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
264	C₂×C₄	C₂×#713	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	4	2	no	—
269	C₈	—	4	2	no	—
270	C₈	—	5	4	no	—
271	C₈	—	10	2	no	—
272	C₈	—	20	2	no	—
273	C₈	#4047	1	2	yes	—
274	C₅	#18068	1	1	yes	characteristic, complemented, Sylow
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	#65162	1	2	yes	—
278	C₂²	C₂×#4039	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	20	4	no	—
283	C₄	—	20	4	no	—
284	C₄	#65162	1	2	yes	—
285	C₄	C₂×#4047	1	1	yes	characteristic
286	C₄	#66079	1	1	yes	characteristic
287	C₄	C₂×#4058	1	1	yes	characteristic
288	C₄	C₂×D₅×Q₁₆	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	#10610735	1	1	yes	characteristic
291	C₂	C₂×#65162	1	1	yes	characteristic
292	C₂	#10610980	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₅×#335	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂³×C₅	C₄×D₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂×#335	D₅
hypocenter	C₅	#18068
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
6. #10599357	C₂	semidirect
7. #10599353	C₂	semidirect
8. #10599350	C₂	semidirect
105. #681	Q₁₆	semidirect
274. C₅	#18068	semidirect

Maximal subgroups

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

subgroup	index
#10610739	2
#10610206	2
C₂×C₅×#335	2
#10601222	2
C₈×#681	2
#10599357	2
#10599353	2
#10599350	2
#18068	5