Subgroups — #12068527

Order 1280.

610 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610731	C₂	1	2	yes	maximal
2	C₈×#709	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601215	C₂	1	2	yes	maximal, complemented
5	#10600523	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599326	C₂	1	4	yes	maximal, complemented
7	#10599323	C₂	1	2	yes	maximal, complemented
8	#10599320	C₂	1	2	yes	maximal, complemented
9	C₄×#709	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	#65399	—	2	2	no	—
13	C₈×C₂²⋊D₅	C₄	1	2	yes	—
14	C₈×C₂²⋊D₅	C₄	1	2	yes	—
15	#65390	—	2	2	no	—
16	#65214	C₂²	1	1	yes	characteristic
17	#65213	C₂²	1	2	yes	—
18	#65129	—	2	4	no	—
19	#65124	C₂²	1	4	yes	—
20	#65122	C₂²	1	4	yes	—
21	C₈×D₂₀	C₄	1	1	yes	characteristic
22	C₈×D₂₀	C₄	1	1	yes	characteristic
23	#64974	C₂²	1	2	yes	—
24	C₄×C₈×D₅	C₂²	1	2	yes	—
25	C₈×Dic₁₀	C₄	1	1	yes	characteristic
26	C₈×Dic₁₀	C₄	1	1	yes	characteristic
27	C₅×#334	C₂²	1	1	yes	characteristic
28	C₅×#334	C₂²	1	1	yes	characteristic
29	C₅×#334	C₂²	1	2	yes	—
30	#64699	C₂²	1	4	yes	—
31	#64699	C₂²	1	4	yes	—
32	#64677	C₂²	1	2	yes	—
33	#64677	C₂²	1	2	yes	—
34	#17827	—	5	1	no	maximal, Sylow
35	C₂×#709	D₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	—	2	2	no	—
37	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₄×C₂²⋊D₅	C₂×C₄	1	2	yes	—
41	C₄×C₂²⋊D₅	Q₈	1	2	yes	—
42	#4072	D₄	1	1	yes	characteristic
43	C₂×#681	—	2	2	no	—
44	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
46	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
47	C₄×D₂₀	Q₈	1	1	yes	characteristic
48	#4018	D₄	1	2	yes	—
49	C₄²×D₅	C₂³	1	2	yes	—
50	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
51	C₄×Dic₁₀	Q₈	1	1	yes	characteristic
52	C₅×C₄⋊C₈	—	2	2	no	—
53	C₅×C₄⋊C₈	C₂³	1	2	yes	—
54	C₅×C₄⋊C₈	C₂³	1	2	yes	—
55	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
56	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
57	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
58	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
59	C₂²⋊#168	C₂×C₄	1	2	yes	—
60	#3952	—	2	4	no	—
61	#3952	—	2	4	no	—
62	#3952	C₂×C₄	1	2	yes	—
63	#3952	C₂×C₄	1	2	yes	—
64	#3947	—	2	4	no	—
65	#3946	C₂×C₄	1	2	yes	—
66	#3946	C₂×C₄	1	2	yes	—
67	C₈×Dic₅	—	2	4	no	—
68	C₈×Dic₅	C₂×C₄	1	2	yes	—
69	C₈×Dic₅	C₂×C₄	1	2	yes	—
70	C₄⋊#168	—	2	2	no	—
71	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
72	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
73	C₄⋊#168	C₂³	1	4	yes	—
74	C₄×#168	C₂³	1	2	yes	—
75	C₈×#41	—	5	1	no	—
76	#2925	—	5	2	no	—
77	#1572	—	5	4	no	—
78	#1571	—	5	2	no	—
79	#1568	—	5	2	no	—
80	#1566	—	5	1	no	—
81	#1564	—	5	2	no	—
82	C₂×#334	D₅	1	1	yes	characteristic, complemented
83	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₂²⋊D₅	C₄⋊C₄	1	2	yes	—
87	#709	—	2	4	no	—
88	C₂×D₂₀	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
90	C₂×C₄×D₅	C₂×D₄	1	2	yes	—
91	C₂×C₄×D₅	C₂×Q₈	1	2	yes	—
92	C₂×Dic₁₀	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	4	1	no	—
97	C₂×C₅×C₈	—	4	4	no	—
98	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
105	C₄²×C₅	C₂⁴	1	1	yes	characteristic
106	C₂²⋊Dic₅	C₄⋊C₄	1	2	yes	—
107	#685	C₄⋊C₄	1	2	yes	—
108	#685	C₄⋊C₄	1	2	yes	—
109	C₄⋊Dic₅	C₄⋊C₄	1	1	yes	characteristic
110	C₄⋊Dic₅	C₄⋊C₄	1	1	yes	characteristic
111	#683	C₄⋊C₄	1	2	yes	—
112	#683	C₄⋊C₄	1	2	yes	—
113	C₄×Dic₅	C₂²×C₄	1	2	yes	—
114	C₄×Dic₅	C₂×D₄	1	2	yes	—
115	C₄×Dic₅	C₂×Q₈	1	2	yes	—
116	#681	—	4	4	no	—
117	C₂×#168	—	2	4	no	—
118	C₂×#168	C₂²×C₄	1	2	yes	—
119	C₂×#168	C₂²×C₄	1	2	yes	—
120	C₈×D₅	—	2	4	no	—
121	C₈×D₅	—	4	2	no	—
122	C₄×#41	—	5	1	no	—
123	C₈×Q₈	—	5	1	no	—
124	C₈×Q₈	—	5	1	no	—
125	C₈×D₄	—	5	2	no	—
126	C₈×D₄	—	5	2	no	—
127	C₈×D₄	—	5	1	no	—
128	C₈×D₄	—	5	1	no	—
129	#433	—	10	4	no	—
130	#432	—	5	4	no	—
131	#431	—	5	1	no	—
132	#431	—	5	2	no	—
133	#431	—	5	4	no	—
134	#423	—	5	2	no	—
135	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
136	#408	—	10	2	no	—
137	#405	—	10	2	no	—
138	C₂×C₄×C₈	—	5	2	no	—
139	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
140	#335	—	5	4	no	—
141	#335	—	5	4	no	—
142	#334	—	5	2	no	—
143	#334	—	5	2	no	—
144	#334	C₂×D₅	1	1	yes	characteristic
145	#334	C₂×D₅	1	1	yes	characteristic
146	#334	C₂×D₅	1	2	yes	—
147	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂²×D₅	C₂×C₄⋊C₄	1	2	yes	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	4	1	no	—
152	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₅	#135	1	4	yes	—
156	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
159	C₂²⋊D₅	—	2	4	no	—
160	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
161	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
162	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
163	D₂₀	—	2	2	no	—
164	C₄×D₅	—	2	2	no	—
165	C₄×D₅	—	2	2	no	—
166	C₄×D₅	—	2	4	no	—
167	C₄×D₅	—	4	2	no	—
168	Dic₁₀	—	2	2	no	—
169	C₅×C₈	—	2	1	no	—
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	—	2	1	no	—
172	C₅×C₈	—	4	2	no	—
173	C₅×C₈	—	4	2	no	—
174	C₅×C₈	#131	1	2	yes	—
175	#168	—	2	2	no	—
176	#168	—	2	2	no	—
177	#168	—	4	4	no	—
178	C₂×#41	—	5	1	no	—
179	C₂×M₁₆	—	10	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	5	2	no	—
182	C₂²×C₈	—	5	2	no	—
183	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
184	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
185	C₄×Q₈	—	5	1	no	—
186	C₄×Q₈	—	5	1	no	—
187	C₄×D₄	—	5	2	no	—
188	C₄×D₄	—	5	2	no	—
189	C₄×D₄	—	5	1	no	—
190	C₄×D₄	—	5	1	no	—
191	#117	—	5	1	no	—
192	#117	—	5	2	no	—
193	C₂×C₄²	—	5	2	no	—
194	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	5	2	no	—
197	C₄⋊C₈	—	5	2	no	—
198	C₄⋊C₈	—	5	1	no	—
199	C₄⋊C₈	—	5	1	no	—
200	C₄⋊C₈	—	5	4	no	—
201	C₄⋊C₈	—	10	2	no	—
202	C₄⋊C₈	C₂²×D₅	1	2	yes	—
203	C₄⋊C₈	C₂²×D₅	1	2	yes	—
204	C₂²⋊C₈	—	5	2	no	—
205	C₂²⋊C₈	—	5	2	no	—
206	C₂²⋊C₈	—	5	2	no	—
207	C₂²⋊C₈	—	10	4	no	—
208	C₂²⋊C₈	—	10	4	no	—
209	#97	—	10	4	no	—
210	C₄×C₈	—	5	2	no	—
211	C₄×C₈	—	5	2	no	—
212	C₄×C₈	—	5	2	no	—
213	C₄×C₈	—	10	4	no	—
214	C₄×C₈	C₄×D₅	1	1	yes	characteristic
215	C₄×C₈	C₄×D₅	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
218	C₂²×C₅	—	2	1	no	—
219	C₂²×C₅	—	2	1	no	—
220	C₂²×C₅	#427	1	2	yes	—
221	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
222	C₂×D₅	—	2	2	no	—
223	C₂×D₅	—	2	2	no	—
224	C₂×D₅	—	2	2	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	#427	1	2	yes	—
229	C₄×C₅	C₂×#131	1	1	yes	characteristic
230	C₄×C₅	#568	1	1	yes	characteristic
231	C₄×C₅	C₂×#135	1	2	yes	—
232	Dic₅	—	2	2	no	—
233	Dic₅	—	2	2	no	—
234	Dic₅	—	2	2	no	—
235	#41	—	10	4	no	—
236	C₂×Q₈	—	5	1	no	—
237	C₂×D₄	—	5	2	no	—
238	C₂×D₄	—	5	1	no	—
239	C₂²×C₄	—	5	2	no	—
240	C₂²×C₄	—	5	2	no	—
241	C₂²×C₄	—	5	2	no	—
242	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
243	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
244	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
245	M₁₆	—	20	4	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	4	no	—
251	C₂×C₈	—	5	2	no	—
252	C₂×C₈	—	5	2	no	—
253	C₂×C₈	—	10	4	no	—
254	C₂×C₈	—	10	4	no	—
255	C₂×C₈	—	20	2	no	—
256	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
257	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
258	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
259	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
260	C₄⋊C₄	—	5	2	no	—
261	C₄⋊C₄	—	5	2	no	—
262	C₄⋊C₄	—	5	1	no	—
263	C₄⋊C₄	—	5	1	no	—
264	C₂²⋊C₄	—	5	2	no	—
265	C₂²⋊C₄	—	5	2	no	—
266	C₂²⋊C₄	—	5	2	no	—
267	C₄²	—	5	2	no	—
268	C₄²	—	5	2	no	—
269	C₄²	—	5	2	no	—
270	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
271	C₄²	D₄×D₅	1	1	yes	characteristic
272	C₄²	Q₈×D₅	1	1	yes	characteristic
273	C₄²	C₂³×D₅	1	1	yes	characteristic
274	C₂×C₅	—	2	2	no	—
275	C₂×C₅	#2906	1	1	yes	characteristic
276	C₂×C₅	C₂×#427	1	1	yes	characteristic
277	C₂×C₅	#2916	1	1	yes	characteristic
278	D₅	—	4	2	no	—
279	C₂³	—	5	2	no	—
280	C₂³	D₅×C₄⋊C₄	1	1	yes	characteristic
281	Q₈	—	10	2	no	—
282	D₄	—	10	4	no	—
283	D₄	—	10	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	5	2	no	—
288	C₂×C₄	—	5	2	no	—
289	C₂×C₄	—	5	2	no	—
290	C₂×C₄	—	10	2	no	—
291	C₂×C₄	—	10	2	no	—
292	C₂×C₄	—	10	4	no	—
293	C₂×C₄	—	20	2	no	—
294	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
295	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
296	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
297	C₂×C₄	#4062	1	4	yes	—
298	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
299	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
300	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	4	2	no	—
305	C₈	—	4	2	no	—
306	C₈	—	10	2	no	—
307	C₈	—	10	2	no	—
308	C₈	—	20	4	no	—
309	C₈	#4047	1	2	yes	—
310	C₅	#17827	1	1	yes	characteristic, complemented, Sylow
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	10	2	no	—
314	C₂²	—	10	2	no	—
315	C₂²	—	10	2	no	—
316	C₂²	#65149	1	2	yes	—
317	C₂²	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	10	2	no	—
322	C₄	—	10	2	no	—
323	C₄	—	10	2	no	—
324	C₄	#65149	1	2	yes	—
325	C₄	C₂×#4047	1	1	yes	characteristic
326	C₄	#66079	1	1	yes	characteristic
327	C₄	C₂×#4062	1	2	yes	—
328	C₂	—	2	2	no	—
329	C₂	—	20	2	no	—
330	C₂	#10610743	1	1	yes	characteristic
331	C₂	C₂×#65149	1	1	yes	characteristic
332	C₂	#10610983	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×#334	C₂
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂×#709	D₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂×#334	D₅
hypocenter	C₅	#17827
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#334	C₂	semidirect
4. #10601215	C₂	semidirect
5. #10600523	C₂	semidirect
6. #10599326	C₂	semidirect
7. #10599323	C₂	semidirect
8. #10599320	C₂	semidirect
82. C₂×#334	D₅	semidirect
310. C₅	#17827	semidirect

Maximal subgroups

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

subgroup	index
#10610731	2
C₈×#709	2
C₂×C₅×#334	2
#10601215	2
#10600523	2
#10599326	2
#10599323	2
#10599320	2
#17827	5