Finite Group Databasebeta

← #14297863

Subgroups — #14297863

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×#713C211yesmaximal, characteristic
2C2×#91239C211yesmaximal, characteristic
3C2×C3×#4010C211yesmaximal, characteristic
4C4×Dic3⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
5#11751824C211yesmaximal, characteristic, complemented
6#11746526C211yesmaximal, characteristic, complemented
7#11746524C211yesmaximal, characteristic, complemented
8#10604455—31nomaximal, Hall
9C2×C3×#713C2211yescharacteristic
10C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
11C2×#6758C2211yescharacteristic
12C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
13C2×C3×#700C2211yescharacteristic
14C2×Dic3⋊AGL(1,𝔽5)—21no—
15Q8×Dic15C411yescharacteristic, complemented
16C5×Q8×Dic3C411yescharacteristic, complemented
17#91434—21no—
18#91434C411yescharacteristic, complemented
19Dic3×D20—21no—
20Dic3×D20C411yescharacteristic, complemented
21C4×D5×Dic3—21no—
22C4×D5×Dic3C2211yescharacteristic
23#91358—21no—
24#91358C2211yescharacteristic
25#91239—22no—
26C3×#4010—22no—
27Dic3⋊#170—21no—
28Dic3⋊#170—21no—
29(C2×AGL(1,𝔽5))⋊Dic3—21no—
30#70355—51nomaximal, Hall
31C4×#713—31no—
32C2×#4010—31no—
33C2×#4010D311yescharacteristic
34C4×C4⋊AGL(1,𝔽5)—31no—
35#65681—31no—
36#64875—31no—
37#64875—31no—
38C22×ΓL(1,𝔽16)—21no—
39C22×C3×AGL(1,𝔽5)—21no—
40C2×C3×C5×Q8C2×C411yescharacteristic
41C3×#713—21no—
42C3×#713—21no—
43C3×#713D412yes—
44C3×#713Q812yes—
45C2×C3×D20—21no—
46C2×C3×D20C2×C411yescharacteristic
47C2×C3×C4×D5—21no—
48C2×C3×C4×D5C2311yescharacteristic
49C2×D5×Dic3—21no—
50C2×D5×Dic3—21no—
51C2×D5×Dic3D411yescharacteristic
52C2×D5×Dic3D411yescharacteristic
53C2×C3⋊#170—21no—
54C4×ΓL(1,𝔽16)—21no—
55C4×ΓL(1,𝔽16)—21no—
56#6758—21no—
57#6758—21no—
58C2×C3×#170—21no—
59C3×C4×AGL(1,𝔽5)—21no—
60C3×C4×AGL(1,𝔽5)—21no—
61C3×#700—21no—
62C3×#700—21no—
63Dic3⋊AGL(1,𝔽5)—24no—
64C4⋊Dic15—21no—
65C4⋊Dic15C2×C411yescharacteristic
66C4×Dic15—21no—
67C4×Dic15C2×C411yescharacteristic
68C5×C4⋊Dic3—21no—
69C5×C4⋊Dic3C2×C411yescharacteristic
70C4×C5×Dic3—21no—
71C4×C5×Dic3C2×C411yescharacteristic
72#6665—21no—
73#6665—21no—
74#6665—21no—
75Dic3×Dic5—21no—
76Dic3×Dic5D411yescharacteristic
77Dic3×Dic5D411yescharacteristic
78Dic3×#41—51no—
79C2×C3×#104—51no—
80C2×#837—51no—
81#5118—51no—
82C4×Dic3⋊C4—51no—
83#4555—51no—
84#4523—51no—
85C2×#713C2×D311yescharacteristic
86C2×C4⋊AGL(1,𝔽5)—61no—
87C2×C4×AGL(1,𝔽5)—31no—
88C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
89C2×#700—31no—
90C2×#700C2×D311yescharacteristic
91C4×C5×Q8—31no—
92Q8×Dic5—31no—
93#4039—31no—
94#4039—61no—
95#4038—31no—
96#4038—61no—
97C4×D20—31no—
98C4×D20—61no—
99C42×D5—31no—
100C42×D5—61no—
101#4010—22no—
102#4010—62no—
103(C2×AGL(1,𝔽5))⋊C4—61no—
104C4⋊#170—61no—
105C4⋊#170—61no—
106#1806—151noSylow
107C22×C3×D5—21no—
108C22×C3×D5C2×D411yescharacteristic
109C2×ΓL(1,𝔽16)—22no—
110C2×ΓL(1,𝔽16)—41no—
111C2×ΓL(1,𝔽16)—41no—
112C2×C3×AGL(1,𝔽5)—22no—
113C2×C3×AGL(1,𝔽5)—41no—
114C2×C3×AGL(1,𝔽5)—41no—
115C3×C5×Q8—21no—
116C3×C5×Q8C4⋊C412yescomplemented
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5C22×C411yescharacteristic
119C2×Dic15—21no—
120C2×Dic15C22⋊C411yescharacteristic
121C2×Dic15C22⋊C411yescharacteristic
122C2×C5×Dic3—21no—
123C2×C5×Dic3C22⋊C411yescharacteristic
124C2×C5×Dic3C22⋊C411yescharacteristic
125C2×C3×Dic5C2×D411yescharacteristic
126C3×D20—22no—
127C3×D20—22no—
128C3×D20—21no—
129C3×D20C4⋊C412yescomplemented
130C3×C4×D5—22no—
131C3×C4×D5—22no—
132C3×C4×D5C2×D411yescharacteristic
133C3×C4×D5C2×Q811yescharacteristic
134C3×C4×D5#4111yescharacteristic
135C3×C4×D5#4111yescharacteristic
136D5×Dic3—22no—
137D5×Dic3—41no—
138D5×Dic3—42no—
139D5×Dic3—42no—
140C3⋊#170—41no—
141C3×#170—41no—
142C2×C3×#41—51no—
143C2×C3×M16—51no—
144C2×C3×C42—51no—
145Q8×Dic3—51no—
146Q8×Dic3AGL(1,𝔽5)11yescharacteristic, complemented
147D4×Dic3—51no—
148D4×Dic3—51no—
149D4×Dic3—101no—
150D4×Dic3—101no—
151#926—51no—
152#926—101no—
153C2×Dic3⋊C4—101no—
154C2×C4×Dic3—51no—
155C2×C4×Dic3—51no—
156C2×C4×Dic3—101no—
157C2×#208—51no—
158C3×#104—102no—
159#837—102no—
160C3⋊#95—101no—
161Dic3⋊C8—101no—
162Dic3⋊C8—101no—
163C22×AGL(1,𝔽5)—21no—
164C22×AGL(1,𝔽5)—61no—
165C2×C5×Q8C4×D311yescharacteristic
166#713—21no—
167#713—21no—
168#713Dic612yes—
169#713#6712yes—
170C2×D20—21no—
171C2×D20C4×D311yescharacteristic
172C2×C4×D5—21no—
173C2×C4×D5—31no—
174C2×C4×D5—31no—
175C2×C4×D5—61no—
176C2×C4×D5—61no—
177C2×C4×D5C22×D311yescharacteristic
178C2×#170—21no—
179C2×#170—61no—
180C4⋊AGL(1,𝔽5)—64no—
181C4×AGL(1,𝔽5)—21no—
182C4×AGL(1,𝔽5)—21no—
183C4×AGL(1,𝔽5)—61no—
184C4×AGL(1,𝔽5)—61no—
185#700—21no—
186#700—21no—
187#700—61no—
188#700—61no—
189C5×C4⋊C4—31no—
190C5×C4⋊C4—61no—
191C42×C5—31no—
192C42×C5—61no—
193#685—61no—
194#685—61no—
195#685—61no—
196C4⋊Dic5—31no—
197C4⋊Dic5—61no—
198C4×Dic5—31no—
199C4×Dic5—31no—
200C4×Dic5—31no—
201C4×Dic5—61no—
202C4×Dic5—61no—
203C4×#41—151no—
204#423—151no—
205C2×#104—51no—
206C2×#104—151no—
207C4×C4⋊C4—151no—
208#339—151no—
209#339—151no—
210C22×C3×C5C2×C22⋊C411yescharacteristic
211C2×C3×D5—21no—
212C2×C3×D5—21no—
213C2×C3×D5—22no—
214C2×C3×D5—41no—
215C2×C3×D5—41no—
216C2×C3×D5#12111yescharacteristic
217C2×C3×D5C22⋊Q811yescharacteristic
218ΓL(1,𝔽16)—42no—
219C3×AGL(1,𝔽5)—42no—
220C3×C4×C5—22no—
221C3×C4×C5C2×C4⋊C411yescharacteristic
222C3×C4×C5#11711yescharacteristic
223Dic15—21no—
224Dic15—21no—
225Dic15—41no—
226C3×Dic5#12111yescharacteristic
227C3×Dic5C22⋊Q811yescharacteristic
228C5×Dic3—21no—
229C5×Dic3—41no—
230C5×Dic3#10412yescomplemented
231C3×#41—52no—
232C3×#41—52no—
233C3×#41—101no—
234C3×#41—101no—
235C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
236C2×C3×D4—51no—
237C2×C3×D4—101no—
238C22×C3×C4—51no—
239C22×C3×C4—101no—
240C22×C3×C4—101no—
241C22×Dic3—51no—
242C22×Dic3—51no—
243C22×Dic3—101no—
244C22×Dic3—101no—
245C22×Dic3—101no—
246C3×M16—101no—
247C3×M16—101no—
248C2×C3×C8—101no—
249C3×C42—101no—
250C3×C42—101no—
251#217—101no—
252#217—101no—
253#217—101no—
254C4⋊Dic3—21no—
255C4⋊Dic3—51no—
256C4⋊Dic3—101no—
257C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
258Dic3⋊C4—104no—
259C4×Dic3—21no—
260C4×Dic3—51no—
261C4×Dic3—51no—
262C4×Dic3—51no—
263C4×Dic3—101no—
264C4×Dic3—101no—
265C4×Dic3—101no—
266C4×Dic3—101no—
267C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
268#208—101no—
269#208—101no—
270C2×C3⋊C8—101no—
271C22×D5—21no—
272C22×D5D3×D411yescharacteristic
273C2×AGL(1,𝔽5)—22no—
274C2×AGL(1,𝔽5)—41no—
275C2×AGL(1,𝔽5)—41no—
276C2×AGL(1,𝔽5)—62no—
277C2×AGL(1,𝔽5)—121no—
278C2×AGL(1,𝔽5)—121no—
279C5×Q8—21no—
280C5×Q8Dic3⋊C412yescomplemented
281C2×C4×C5—21no—
282C2×C4×C5—31no—
283C2×C4×C5—31no—
284C2×C4×C5—61no—
285C2×C4×C5C2×C4×D311yescharacteristic
286C2×Dic5—31no—
287C2×Dic5—31no—
288C2×Dic5—61no—
289C2×Dic5D3×D411yescharacteristic
290D20—22no—
291D20—22no—
292D20—21no—
293D20Dic3⋊C412yescomplemented
294C4×D5—22no—
295C4×D5—22no—
296C4×D5—62no—
297C4×D5—121no—
298C4×D5—122no—
299C4×D5—122no—
300C4×D5C2×Dic611yescharacteristic
301C4×D5#23711yescharacteristic
302C4×D5#23711yescharacteristic
303C4×D5C2×#6711yescharacteristic
304#170—41no—
305#170—121no—
306C2×#41—51no—
307C2×M16—51no—
308C2×M16—151no—
309C4×Q8—31no—
310C4×Q8—151no—
311C4×D4—151no—
312C4×D4—151no—
313C4×D4—301no—
314C4×D4—301no—
315#117—151no—
316#117—301no—
317C2×C4⋊C4—301no—
318C2×C42—51no—
319C2×C42—151no—
320C2×C42—151no—
321C2×C42—301no—
322C4⋊C8—301no—
323C4⋊C8—301no—
324#104—102no—
325#104—302no—
326#95—301no—
327C2×C3×C5#38011yescharacteristic
328C2×C3×C5C2×#10411yescharacteristic
329C2×C3×C5#42111yescharacteristic
330C3×D5—21no—
331C3×D5—21no—
332C3×D5—42no—
333C23×C3—51no—
334C23×C3—101no—
335C3×Q8—21no—
336C3×Q8C4⋊AGL(1,𝔽5)12yescomplemented
337C3×D4—52no—
338C3×D4—102no—
339C3×D4—102no—
340C3×D4—101no—
341C2×C3×C4—21no—
342C2×C3×C4—51no—
343C2×C3×C4—51no—
344C2×C3×C4—51no—
345C2×C3×C4—51no—
346C2×C3×C4—51no—
347C2×C3×C4—102no—
348C2×C3×C4—102no—
349C2×C3×C4—102no—
350C2×C3×C4—201no—
351C2×C3×C4—201no—
352C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
353C2×Dic3—21no—
354C2×Dic3—51no—
355C2×Dic3—51no—
356C2×Dic3—102no—
357C2×Dic3—102no—
358C2×Dic3—101no—
359C2×Dic3—201no—
360C2×Dic3—202no—
361C2×Dic3—201no—
362C2×Dic3—201no—
363C2×Dic3—202no—
364C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
365C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
366C3×C8—201no—
367C3⋊C8—201no—
368C22×C5D3×C22⋊C411yescharacteristic
369C2×D5—21no—
370C2×D5—21no—
371C2×D5—22no—
372C2×D5—41no—
373C2×D5—41no—
374C2×D5#87811yescharacteristic
375C2×D5#93911yescharacteristic
376AGL(1,𝔽5)—42no—
377AGL(1,𝔽5)—122no—
378C4×C5—22no—
379C4×C5—32no—
380C4×C5—61no—
381C4×C5—121no—
382C4×C5#87711yescharacteristic
383C4×C5C2×Dic3⋊C411yescharacteristic
384Dic5—61no—
385Dic5—61no—
386Dic5—121no—
387Dic5#87811yescharacteristic
388Dic5#93911yescharacteristic
389#41—52no—
390#41—52no—
391#41—101no—
392#41—101no—
393C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
394C2×D4—51no—
395C2×D4—101no—
396C22×C4—51no—
397C22×C4—101no—
398C22×C4—101no—
399C22×C4—151no—
400C22×C4—151no—
401C22×C4—301no—
402C22×C4—301no—
403C22×C4—301no—
404M16—101no—
405M16—101no—
406M16—301no—
407M16—301no—
408C2×C8—101no—
409C2×C8—301no—
410C4⋊C4—31no—
411C4⋊C4—61no—
412C4⋊C4—151no—
413C4⋊C4—301no—
414C4⋊C4—304no—
415C22⋊C4—301no—
416C22⋊C4—301no—
417C22⋊C4—301no—
418C42—31no—
419C42—61no—
420C42—101no—
421C42—101no—
422C42—151no—
423C42—151no—
424C42—151no—
425C42—301no—
426C42—301no—
427C42—301no—
428C42—301no—
429C3×C5#180611yescharacteristic, complemented, Hall
430C22×C3—51no—
431C22×C3—51no—
432C22×C3—101no—
433C22×C3—101no—
434C22×C3—102no—
435C22×C3—201no—
436C22×C3—201no—
437C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
438C3×C4—22no—
439C3×C4—51no—
440C3×C4—51no—
441C3×C4—202no—
442C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
443C3×C4#413011yescharacteristic
444Dic3—21no—
445Dic3—41no—
446Dic3—101no—
447Dic3—101no—
448Dic3—202no—
449Dic3—201no—
450Dic3#401012yescomplemented
451C2×C5D3×#10411yescharacteristic
452C2×C5#482111yescharacteristic
453C2×C5#494211yescharacteristic
454D5—21no—
455D5—21no—
456D5—42no—
457C23—51no—
458C23—101no—
459Q8—21no—
460Q8Dic3⋊AGL(1,𝔽5)12yescomplemented
461D4—52no—
462D4—102no—
463D4—102no—
464D4—101no—
465C2×C4—21no—
466C2×C4—31no—
467C2×C4—31no—
468C2×C4—51no—
469C2×C4—51no—
470C2×C4—51no—
471C2×C4—51no—
472C2×C4—51no—
473C2×C4—61no—
474C2×C4—102no—
475C2×C4—102no—
476C2×C4—102no—
477C2×C4—151no—
478C2×C4—151no—
479C2×C4—201no—
480C2×C4—201no—
481C2×C4—302no—
482C2×C4—302no—
483C2×C4—301no—
484C2×C4—602no—
485C2×C4—601no—
486C2×C4—602no—
487C2×C4—601no—
488C2×C4—601no—
489C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
490C8—201no—
491C8—601no—
492C2×C3—101no—
493C2×C3—101no—
494C2×C3—202no—
495C2×C3#6576511yescharacteristic
496C2×C3C2×#401011yescharacteristic
497C2×C3#6579511yescharacteristic
498C5#7035511yescharacteristic, complemented, Sylow
499C22—51no—
500C22—51no—
501C22—101no—
502C22—101no—
503C22—102no—
504C22—201no—
505C22—201no—
506C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
507C4—22no—
508C4—32no—
509C4—51no—
510C4—51no—
511C4—61no—
512C4—121no—
513C4—202no—
514C4—301no—
515C4—301no—
516C4—602no—
517C4—601no—
518C4#9192311yescharacteristic
519C4C2×Dic3⋊AGL(1,𝔽5)11yescharacteristic
520C3#1060445511yescharacteristic, complemented, Sylow
521C2—101no—
522C2—101no—
523C2—202no—
524C2D3×#401011yescharacteristic
525C2#1175197311yescharacteristic
526C2#1175202811yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C3×#713C22
cosocleC2×C3×C4×D5C23
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#1806
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
4. C4×Dic3⋊AGL(1,𝔽5)C2semidirect
5. #11751824C2semidirect
6. #11746526C2semidirect
7. #11746524C2semidirect
15. Q8×Dic15C4semidirect
16. C5×Q8×Dic3C4semidirect
18. #91434C4semidirect
20. Dic3×D20C4semidirect
116. C3×C5×Q8C4⋊C4semidirect
129. C3×D20C4⋊C4semidirect
146. Q8×Dic3AGL(1,𝔽5)semidirect
230. C5×Dic3#104semidirect
280. C5×Q8Dic3⋊C4semidirect
293. D20Dic3⋊C4semidirect
336. C3×Q8C4⋊AGL(1,𝔽5)semidirect
429. C3×C5#1806semidirect
450. Dic3#4010semidirect
460. Q8Dic3⋊AGL(1,𝔽5)semidirect
498. C5#70355semidirect
520. C3#10604455semidirect

Maximal subgroups

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

subgroupindex
Dic3×#7132
C2×#912392
C2×C3×#40102
C4×Dic3⋊AGL(1,𝔽5)2
#117518242
#117465262
#117465242
#106044553
#703555