Finite Group Databasebeta

← D5×#4702 #14439915

Subgroups — D5×#4702 #14439915

Order 1920.

768 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×#870C211yesmaximal, characteristic
2D5×#869C211yesmaximal, characteristic
3#11750803C211yesmaximal, characteristic, complemented
4C5×#4702C211yesmaximal, characteristic, complemented
5C3×D5×#97C211yesmaximal, characteristic
6#11748021C212yesmaximal, complemented
7#11748020C212yesmaximal, complemented
8D5×#818C212yesmaximal
9D5×#817C212yesmaximal
10#11746958C211yesmaximal, characteristic, complemented
11#11746957C211yesmaximal, characteristic, complemented
12D5×#501—31nomaximal, Hall
13C2×D5×Dic6—21no—
14#91759C2211yescharacteristic
15#91758C2211yescharacteristic
16C5×#870C2211yescharacteristic
17C5×#869C2211yescharacteristic
18C2×C3×C8×D5C2212yes—
19C3×C42×D5C2211yescharacteristic
20#91468—21no—
21#91468C2212yes—
22D5×C4⋊Dic3—21no—
23D5×C4⋊Dic3C2211yescharacteristic
24D5×C4⋊Dic3C2212yes—
25D5×C4⋊Dic3C2211yescharacteristic
26D5×Dic3⋊C4—21no—
27D5×Dic3⋊C4—21no—
28#91388C2211yescharacteristic
29#91388C2211yescharacteristic
30#91378—21no—
31#91378—21no—
32#91376—21no—
33C3×C5×#97C2211yescharacteristic
34#91102C2212yes—
35#91101C2212yes—
36C5×#818C2212yes—
37C5×#817C2212yes—
38C3×#3947C2212yes—
39C3×#3935C2211yescharacteristic
40#90988C2212yes—
41#90988C2212yes—
42#90985C2212yes—
43#90985C2212yes—
44C2×#4702—51nomaximal, Hall
45D5×C4⋊Q8—31no—
46D5×#125—31no—
47C5×#501—31no—
48#65369—31no—
49#65354—31no—
50D5×#107—32no—
51#65165—32no—
52D5×#106—32no—
53#65145—32no—
54D5×#97D311yescharacteristic
55#64991—31no—
56C2×Dic30—21no—
57C2×C5×Dic6—21no—
58C2×C3×C4×D5D411yescharacteristic
59C2×C3×C4×D5D411yescharacteristic
60C2×C3×C4×D5C2311yescharacteristic
61C2×#1204—22no—
62C2×D5×Dic3—21no—
63C2×D5×Dic3—21no—
64C2×D5×Dic3—22no—
65D5×Dic6—44no—
66C2×C3×C5×C8C2312yes—
67C3×C42×C5C2311yescharacteristic
68C4⋊Dic15—21no—
69C4⋊Dic15C2312yes—
70C4⋊Dic15C2311yescharacteristic
71C4⋊Dic15C2311yescharacteristic
72#6712—21no—
73#6712—21no—
74C5×C4⋊Dic3—21no—
75C5×C4⋊Dic3C2312yes—
76C5×C4⋊Dic3C2311yescharacteristic
77C5×C4⋊Dic3C2311yescharacteristic
78C5×Dic3⋊C4—21no—
79C5×Dic3⋊C4—21no—
80C3×C4×Dic5D411yescharacteristic
81C3×C4×Dic5D411yescharacteristic
82C3×C4×Dic5C2311yescharacteristic
83C2×C3×#168C2312yes—
84C3×C8×D5Q818yescomplemented
85Dic3⋊Dic5—21no—
86Dic3⋊Dic5—21no—
87#6669—21no—
88#6669—21no—
89#6668—22no—
90#6668—22no—
91#6668—21no—
92#6668—21no—
93C2×#870—51no—
94C2×#869—51no—
95C2×C3×#97—51no—
96C2×#818—52no—
97C2×#817—52no—
98#4702—51no—
99#4702—51no—
100#4702—51no—
101#4702—52no—
102#4702—52no—
103#4702D511yescharacteristic, direct product factor
104C2×Q8×D5—61no—
105C5×C4⋊Q8—31no—
106C5×#125—31no—
107#4090—61no—
108C2×C8×D5C2×D312yes—
109D5×C4⋊C4—32no—
110D5×C4⋊C4—31no—
111D5×C4⋊C4—31no—
112D5×C4⋊C4—61no—
113D5×C4⋊C4—61no—
114D5×C4⋊C4—61no—
115#4036—31no—
116#4036—31no—
117#4035—61no—
118#4035—61no—
119#4034—32no—
120#4034—61no—
121C42×D5C2×D311yescharacteristic
122#4016—31no—
123#4015—31no—
124C5×#107—32no—
125C5×#106—32no—
126C5×#97C2×D311yescharacteristic
127#3950—32no—
128#3949—32no—
129#3947C2×D312yes—
130#3940—32no—
131#3940—32no—
132#3939—32no—
133#3939—32no—
134#3935C2×D311yescharacteristic
135C2×#501—151noSylow
136C22×C3×D5C2×D411yescharacteristic
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2×D411yescharacteristic
139C2×C3×C4×C5C2411yescharacteristic
140C2×Dic15—22no—
141C2×Dic15—21no—
142C2×Dic15—21no—
143Dic30—42no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—21no—
146C2×C5×Dic3—21no—
147C5×Dic6—42no—
148C2×C3×Dic5C2×D411yescharacteristic
149C2×C3×Dic5C2×D411yescharacteristic
150C2×C3×Dic5C2×D411yescharacteristic
151C3×C4×D5—22no—
152C3×C4×D5—22no—
153C3×C4×D5C2×Q814yes—
154#1204—44no—
155D5×Dic3—44no—
156D5×Dic3—42no—
157D5×Dic3—42no—
158C3×C5×C8C2×Q814yescomplemented
159C3×#168C2×Q814yescomplemented
160C22×Dic6—101no—
161C22×C3×C8—52no—
162C2×C3×C42—51no—
163C2×C4⋊Dic3—52no—
164C2×C4⋊Dic3—51no—
165C2×C4⋊Dic3—51no—
166C2×C4⋊Dic3—101no—
167C2×Dic3⋊C4—101no—
168C2×Dic3⋊C4—101no—
169#870—51no—
170#870—51no—
171#870—51no—
172#870—101no—
173#870—101no—
174#870C2×D511yescharacteristic
175#869—51no—
176#869—52no—
177#869—101no—
178#869—101no—
179#869C2×D511yescharacteristic
180C3×#97—52no—
181C3×#97—51no—
182C3×#97C2×D511yescharacteristic
183#818—52no—
184#818—52no—
185#818—52no—
186#818C2×D512yes—
187#817—52no—
188#817—52no—
189#817—52no—
190#817C2×D512yes—
191C2×C5×Q8—61no—
192Q8×D5—124no—
193C2×C4×D5—62no—
194C2×C4×D5—61no—
195C2×C4×D5—61no—
196C2×C4×D5D1211yescharacteristic
197C2×C4×D5D1211yescharacteristic
198C2×C4×D5C22×D311yescharacteristic
199C2×Dic10—61no—
200C2×Dic10—62no—
201C2×C5×C8C22×D312yes—
202C5×C4⋊C4—31no—
203C5×C4⋊C4—31no—
204C5×C4⋊C4—32no—
205C5×C4⋊C4—61no—
206C5×C4⋊C4—61no—
207C5×C4⋊C4—61no—
208C42×C5C22×D311yescharacteristic
209C4⋊Dic5—31no—
210C4⋊Dic5—32no—
211C4⋊Dic5—31no—
212C4⋊Dic5—61no—
213C4⋊Dic5—61no—
214C4⋊Dic5—61no—
215#683—61no—
216#683—61no—
217#683—62no—
218#683—62no—
219#683—61no—
220#683—61no—
221#683—61no—
222#683—61no—
223C4×Dic5D1211yescharacteristic
224C4×Dic5D1211yescharacteristic
225C4×Dic5C22×D311yescharacteristic
226C2×#168C22×D312yes—
227C8×D5Dic618yescomplemented
228C2×C4⋊Q8—151no—
229C2×#125—151no—
230#501—31no—
231#501—152no—
232#501—152no—
233#501—151no—
234#501—151no—
235#501—151no—
236C2×#107—152no—
237C2×#106—152no—
238C2×#97—51no—
239C22×C3×C5C22×D411yescharacteristic
240C2×C3×D5C4⋊Q812yes—
241C2×C3×D5AGL(1,ℤ/8)12yes—
242C2×C3×D5#13712yes—
243C3×C4×C5—21no—
244C3×C4×C5—21no—
245C3×C4×C5C22×Q812yes—
246Dic15—42no—
247Dic15—41no—
248Dic15—41no—
249C3×Dic5—21no—
250C3×Dic5—21no—
251C3×Dic5C4⋊Q812yes—
252C5×Dic3—42no—
253C5×Dic3—41no—
254C5×Dic3—41no—
255C22×C3×C4—51no—
256C22×C3×C4—51no—
257C22×C3×C4—51no—
258C22×Dic3—101no—
259C22×Dic3—101no—
260C22×Dic3—102no—
261C2×Dic6—21no—
262C2×Dic6—102no—
263C2×Dic6—101no—
264C2×Dic6—204no—
265C2×C3×C8—52no—
266C2×C3×C8—58no—
267C2×C3×C8C22×D512yes—
268C3×C42—51no—
269C3×C42—51no—
270C3×C42—51no—
271C3×C42C22×D511yescharacteristic
272C4⋊Dic3—21no—
273C4⋊Dic3—51no—
274C4⋊Dic3—52no—
275C4⋊Dic3—51no—
276C4⋊Dic3—102no—
277C4⋊Dic3—102no—
278C4⋊Dic3—101no—
279C4⋊Dic3—101no—
280C4⋊Dic3—101no—
281C4⋊Dic3C22×D511yescharacteristic
282C4⋊Dic3C22×D512yes—
283C4⋊Dic3C22×D511yescharacteristic
284Dic3⋊C4—21no—
285Dic3⋊C4—21no—
286Dic3⋊C4—101no—
287Dic3⋊C4—101no—
288Dic3⋊C4—101no—
289Dic3⋊C4—101no—
290Dic3⋊C4—101no—
291Dic3⋊C4—101no—
292C22×D5C2×D1211yescharacteristic
293C5×Q8—122no—
294C2×C4×C5—61no—
295C2×C4×C5—61no—
296C2×C4×C5—62no—
297C2×C4×C5C2×D1211yescharacteristic
298C2×C4×C5C2×D1211yescharacteristic
299C2×C4×C5C23×D311yescharacteristic
300C2×Dic5—61no—
301C2×Dic5—61no—
302C2×Dic5—62no—
303C2×Dic5C2×D1211yescharacteristic
304C2×Dic5C2×D1211yescharacteristic
305C2×Dic5C2×D1211yescharacteristic
306C4×D5—22no—
307C4×D5—22no—
308C4×D5—122no—
309C4×D5—124no—
310C4×D5—122no—
311C4×D5C2×Dic614yes—
312Dic10—124no—
313Dic10—122no—
314C5×C8C2×Dic614yescomplemented
315#168C2×Dic614yescomplemented
316C22×Q8—301no—
317C22×C8—52no—
318C4⋊Q8—31no—
319C4⋊Q8—152no—
320C4⋊Q8—151no—
321C4⋊Q8—301no—
322C4⋊Q8—301no—
323#125—31no—
324#125—151no—
325#125—151no—
326#125—151no—
327#125—301no—
328#125—301no—
329C2×C4⋊C4—151no—
330C2×C4⋊C4—151no—
331C2×C4⋊C4—152no—
332C2×C4⋊C4—301no—
333C2×C4⋊C4—301no—
334C2×C4⋊C4—301no—
335C2×C42—51no—
336#107—32no—
337#107—152no—
338#107—152no—
339#107—152no—
340#106—32no—
341#106—152no—
342#106—152no—
343#106—152no—
344#97—52no—
345#97—51no—
346#97D3×D511yescharacteristic
347C2×C3×C5C2×C4⋊Q811yescharacteristic
348C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
349C2×C3×C5C2×#13711yescharacteristic
350C3×D5#50114yescomplemented
351C23×C3—51no—
352C2×C3×C4—51no—
353C2×C3×C4—54no—
354C2×C3×C4—51no—
355C2×C3×C4—51no—
356C2×C3×C4—102no—
357C2×C3×C4—102no—
358C2×C3×C4D4×D511yescharacteristic
359C2×C3×C4D4×D511yescharacteristic
360C2×C3×C4C23×D511yescharacteristic
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—22no—
364C2×Dic3—101no—
365C2×Dic3—101no—
366C2×Dic3—102no—
367C2×Dic3—204no—
368C2×Dic3—202no—
369C2×Dic3—202no—
370Dic6—42no—
371Dic6—204no—
372Dic6—202no—
373C3×C8—54no—
374C3×C8Q8×D514yescomplemented
375C22×C5C22×D1211yescharacteristic
376C2×D5#86912yes—
377C2×D5#90812yes—
378C2×D5#90912yes—
379C4×C5—21no—
380C4×C5—21no—
381C4×C5—122no—
382C4×C5—121no—
383C4×C5—121no—
384C4×C5C22×Dic612yes—
385Dic5—21no—
386Dic5—21no—
387Dic5—122no—
388Dic5—121no—
389Dic5—121no—
390Dic5#86912yes—
391C2×Q8—61no—
392C2×Q8—302no—
393C2×Q8—301no—
394C2×Q8—604no—
395C22×C4—51no—
396C22×C4—51no—
397C22×C4—51no—
398C22×C4—301no—
399C22×C4—301no—
400C22×C4—302no—
401C2×C8—52no—
402C2×C8—58no—
403C2×C8C2×D3×D512yes—
404C4⋊C4—31no—
405C4⋊C4—32no—
406C4⋊C4—31no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—61no—
410C4⋊C4—151no—
411C4⋊C4—152no—
412C4⋊C4—151no—
413C4⋊C4—301no—
414C4⋊C4—301no—
415C4⋊C4—302no—
416C4⋊C4—301no—
417C4⋊C4—301no—
418C4⋊C4—301no—
419C4⋊C4—301no—
420C4⋊C4—302no—
421C4⋊C4—301no—
422C4⋊C4—301no—
423C4⋊C4—301no—
424C42—51no—
425C42—51no—
426C42—51no—
427C42C2×D3×D511yescharacteristic
428C3×C5C2×#50111yescharacteristic, complemented, Hall
429C22×C3—52no—
430C22×C3—52no—
431C22×C3—52no—
432C22×C3C2×D4×D511yescharacteristic
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—52no—
436C3×C4—101no—
437C3×C4—101no—
438C3×C4C2×Q8×D512yes—
439Dic3—42no—
440Dic3—41no—
441Dic3—41no—
442Dic3—202no—
443Dic3—201no—
444Dic3—201no—
445C2×C5C2×#86911yescharacteristic
446C2×C5C2×#90811yescharacteristic
447C2×C5C2×#90911yescharacteristic
448D5#470214yesdirect product factor
449C23—51no—
450Q8—122no—
451Q8—602no—
452Q8—604no—
453C2×C4—51no—
454C2×C4—54no—
455C2×C4—51no—
456C2×C4—51no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4—62no—
460C2×C4—102no—
461C2×C4—102no—
462C2×C4—301no—
463C2×C4—301no—
464C2×C4—302no—
465C2×C4—604no—
466C2×C4—602no—
467C2×C4—602no—
468C2×C4D5×D1211yescharacteristic
469C2×C4D5×D1211yescharacteristic
470C2×C4C22×D3×D511yescharacteristic
471C8—54no—
472C8D5×Dic614yescomplemented
473C2×C3—54no—
474C2×C3D5×C4⋊Q811yescharacteristic
475C2×C3D5×AGL(1,ℤ/8)11yescharacteristic
476C2×C3D5×#13711yescharacteristic
477C5C2×#470211yescharacteristic, complemented, Sylow
478C22—52no—
479C22—52no—
480C22—52no—
481C22C2×D5×D1211yescharacteristic
482C4—21no—
483C4—21no—
484C4—52no—
485C4—101no—
486C4—101no—
487C4—122no—
488C4—121no—
489C4—121no—
490C4—602no—
491C4—601no—
492C4—601no—
493C4C2×D5×Dic612yes—
494C3D5×#50111yescharacteristic, complemented, Sylow
495C2—54no—
496C2D5×#86911yescharacteristic
497C2D5×#90811yescharacteristic
498C2D5×#90911yescharacteristic
subgroupG/H
centerC22C2×D5×D12
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#97C22
socleC22×C3×C5C22×D4
prime radicalC22×C3×D5C2×D4
hypercenter#97D3×D5
hypocenterC3×C5C2×#501
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
3. #11750803C2semidirect
4. C5×#4702C2semidirect
6. #11748021C2semidirect
7. #11748020C2semidirect
10. #11746958C2semidirect
11. #11746957C2semidirect
84. C3×C8×D5Q8semidirect
103. #4702D5direct (N × H)
158. C3×C5×C8C2×Q8semidirect
159. C3×#168C2×Q8semidirect
227. C8×D5Dic6semidirect
314. C5×C8C2×Dic6semidirect
315. #168C2×Dic6semidirect
350. C3×D5#501semidirect
374. C3×C8Q8×D5semidirect
428. C3×C5C2×#501semidirect
448. D5#4702direct (N × H)
472. C8D5×Dic6semidirect
477. C5C2×#4702semidirect
494. C3D5×#501semidirect

Maximal subgroups

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

subgroupindex
D5×#8702
D5×#8692
#117508032
C5×#47022
C3×D5×#972
#117480212
#117480202
D5×#8182
D5×#8172
#117469582
#117469572
D5×#5013
C2×#47025