Finite Group Databasebeta

← #14295613

Subgroups — #14295613

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11753402C211yesmaximal, characteristic
2C4×Dic3×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
3Dic3×#700C211yesmaximal, characteristic, complemented
4C2×#91156C211yesmaximal, characteristic
5C2×#91155C211yesmaximal, characteristic
6#11746746C211yesmaximal, characteristic, complemented
7C3×#64875C211yesmaximal, characteristic, complemented
8C4×#4008—31nomaximal, Hall
9C2×#6772C2211yescharacteristic
10C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
11C2×#6758C2211yescharacteristic
12C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
13C2×C3×#700C2211yescharacteristic
14C2×Dic3×AGL(1,𝔽5)—21no—
15C2×Dic3×AGL(1,𝔽5)—21no—
16#91776C411yescharacteristic, complemented
17C5×#893C411yescharacteristic, complemented
18C3×#4038C2211yescharacteristic
19#91442—21no—
20#91441—21no—
21#91435C411yescharacteristic, complemented
22#91398—21no—
23C4×D3×Dic5—21no—
24C4×D5×Dic3C2211yescharacteristic
25#91340C411yescharacteristic, complemented
26#91169—21no—
27Dic3×#170—21no—
28#91156C414yescomplemented
29#91155C414yescomplemented
30#68728—51nomaximal, Hall
31C4×#711—31no—
32C2×#4008—31no—
33C2×#4008—31no—
34C42×AGL(1,𝔽5)—31no—
35C4×#700—31no—
36#64875—31no—
37#64875D311yescharacteristic, complemented
38C22×ΓL(1,𝔽16)—21no—
39C22×C3×AGL(1,𝔽5)—21no—
40C2×Dic30C2×C411yescharacteristic
41C2×C5×D12C2×C411yescharacteristic
42C2×C4×C5×D3—21no—
43C2×C3×C4×D5C2311yescharacteristic
44C2×#1201—21no—
45C2×D3×Dic5—21no—
46C2×D3×Dic5—21no—
47C2×D5×Dic3D411yescharacteristic
48C2×D5×Dic3D411yescharacteristic
49#6772—21no—
50#6772—21no—
51#6772C2×C412yes—
52#6772C2×C412yes—
53C2×C3⋊#170—21no—
54C4×ΓL(1,𝔽16)—21no—
55C4×ΓL(1,𝔽16)C2×C412yescomplemented
56#6758—21no—
57#6758C2×C412yescomplemented
58C2×C3×#170—21no—
59C3×C4×AGL(1,𝔽5)—21no—
60C3×C4×AGL(1,𝔽5)C2×C412yescomplemented
61C3×#700—21no—
62C3×#700C2×C412yescomplemented
63Dic3×AGL(1,𝔽5)—24no—
64Dic3×AGL(1,𝔽5)—24no—
65C3×C5×C4⋊C4C2×C411yescharacteristic
66#6712—21no—
67C4×Dic15—21no—
68C4×Dic15C2×C411yescharacteristic
69C5×(C2×D3)⋊C4—21no—
70C4×C5×Dic3C2×C411yescharacteristic
71C3×#685—21no—
72C3×C4⋊Dic5C2×C411yescharacteristic
73C3×C4×Dic5—21no—
74#6669—21no—
75(C2×D3)⋊Dic5—21no—
76Dic3×Dic5—21no—
77Dic3×Dic5D411yescharacteristic
78Dic3×Dic5D411yescharacteristic
79#5522—51no—
80C2×#825—51no—
81Dic3×M16—51no—
82C42×Dic3—51no—
83C2×#805—51no—
84C3×#339—51no—
85#4532—51no—
86C2×#711—31no—
87C2×C4×AGL(1,𝔽5)—31no—
88C2×C4×AGL(1,𝔽5)—61no—
89C2×C4×AGL(1,𝔽5)—61no—
90C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
91C2×#700—31no—
92C2×#700C2×D311yescharacteristic
93C4×C5×D4—31no—
94D4×Dic5—31no—
95C4×C22⋊D5—61no—
96C2×C4×Dic5—61no—
97#4038C2×D311yescharacteristic
98#4033—31no—
99#4027—61no—
100#4023—61no—
101C42×D5—31no—
102C4×Dic10—31no—
103#4008—34no—
104#4008—34no—
105#4002—61no—
106C4×#170—61no—
107C4×#104—151noSylow
108C22×C5×D3—21no—
109C22×C3×D5C2×D411yescharacteristic
110C2×ΓL(1,𝔽16)—22no—
111C2×ΓL(1,𝔽16)—22no—
112C2×ΓL(1,𝔽16)—22no—
113C2×C3×AGL(1,𝔽5)—22no—
114C2×C3×AGL(1,𝔽5)—22no—
115C2×C3×AGL(1,𝔽5)—22no—
116C2×C3×C4×C5—21no—
117C2×C3×C4×C5C22×C411yescharacteristic
118C2×Dic15—21no—
119C2×Dic15C22⋊C411yescharacteristic
120C2×Dic15C22⋊C411yescharacteristic
121Dic30—21no—
122Dic30C4212yescomplemented
123C2×C5×Dic3C22⋊C411yescharacteristic
124C2×C5×Dic3C22⋊C411yescharacteristic
125C5×D12—21no—
126C5×D12C4212yescomplemented
127C4×C5×D3—42no—
128C2×C3×Dic5—21no—
129C2×C3×Dic5C2×D411yescharacteristic
130C3×C4×D5C22×C411yescharacteristic
131C3×C4×D5C22×C411yescharacteristic
132C3×C4×D5#4111yescharacteristic
133C3×C4×D5#4111yescharacteristic
134#1201—22no—
135#1201—22no—
136D3×Dic5—22no—
137D3×Dic5—22no—
138D3×Dic5—42no—
139D5×Dic3—22no—
140D5×Dic3—22no—
141C3⋊#170—22no—
142C3×#170—22no—
143C2×#235—51no—
144C2×C3×M16—51no—
145C3×#117—51no—
146C2×C3×C42—51no—
147C2×C4×Dic3—51no—
148C2×C4×Dic3—51no—
149C2×C4×Dic3—101no—
150C2×C4×Dic3—101no—
151C2×#208—51no—
152#893—51no—
153#893AGL(1,𝔽5)11yescharacteristic, complemented
154#887—51no—
155#887—51no—
156#67⋊C4—101no—
157#67⋊C4—101no—
158#872—101no—
159C42×D3—101no—
160#825—54no—
161#815—101no—
162C8×Dic3—101no—
163#805—54no—
164C22×AGL(1,𝔽5)—21no—
165C22×AGL(1,𝔽5)—61no—
166C2×C5×D4—31no—
167C22×C4×C5—61no—
168C2×C22⋊D5—61no—
169C22×Dic5—61no—
170C22×Dic5—61no—
171#711—32no—
172#711—32no—
173#711—61no—
174#711—61no—
175C2×C4×D5—31no—
176C2×C4×D5—31no—
177C2×C4×D5C22×D311yescharacteristic
178C2×Dic10—31no—
179C2×#170—21no—
180C2×#170—61no—
181C4×AGL(1,𝔽5)—21no—
182C4×AGL(1,𝔽5)—32no—
183C4×AGL(1,𝔽5)—64no—
184C4×AGL(1,𝔽5)—64no—
185C4×AGL(1,𝔽5)—61no—
186C4×AGL(1,𝔽5)C4×D312yescomplemented
187#700—21no—
188#700—32no—
189#700—61no—
190#700C4×D312yescomplemented
191C5×C4⋊C4C4×D311yescharacteristic
192C5×C22⋊C4—61no—
193C42×C5—31no—
194C22⋊Dic5—61no—
195#685—21no—
196C4⋊Dic5C4×D311yescharacteristic
197#683—61no—
198#683—61no—
199C4×Dic5—21no—
200C4×Dic5—31no—
201C4×Dic5—31no—
202C4×Dic5—31no—
203C4×Dic5—61no—
204C4×Dic5—61no—
205C4×#41—151no—
206C2×#104—151no—
207C2×#104—151no—
208C4×M16—151no—
209C43—151no—
210#339—51no—
211#339—151no—
212C22×C3×C5C2×C22⋊C411yescharacteristic
213C2×C5×D3—22no—
214C2×C5×D3—41no—
215C2×C5×D3—41no—
216C2×C3×D5—21no—
217C2×C3×D5—21no—
218C2×C3×D5C4×D411yescharacteristic
219C2×C3×D5C4×D411yescharacteristic
220ΓL(1,𝔽16)—24no—
221C3×AGL(1,𝔽5)—24no—
222C3×C4×C5—41no—
223C3×C4×C5C2×C4211yescharacteristic
224C3×C4×C5#11711yescharacteristic
225Dic15—22no—
226Dic15—21no—
227Dic15#10412yescomplemented
228C3×Dic5—41no—
229C3×Dic5C4×D411yescharacteristic
230C3×Dic5C4×D411yescharacteristic
231C5×Dic3—21no—
232C5×Dic3#10412yescomplemented
233C22×C3×C4—51no—
234C22×C3×C4—101no—
235C2×#67—101no—
236C22×Dic3—51no—
237C22×Dic3—51no—
238C22×Dic3—101no—
239#235—52no—
240#235—52no—
241#235—101no—
242#235—101no—
243C2×D12C2×AGL(1,𝔽5)11yescharacteristic
244C2×C4×D3—21no—
245C2×C4×D3—101no—
246C2×C4×D3—101no—
247C2×Dic6—51no—
248C3×M16—52no—
249C3×M16—101no—
250C2×C3×C8—101no—
251C3×C4⋊C4—51no—
252C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
253C3×C22⋊C4—101no—
254C3×C42—52no—
255C3×C42—101no—
256C3×C42—101no—
257(C2×D3)⋊C4—21no—
258(C2×D3)⋊C4—101no—
259Dic3⋊C4—101no—
260Dic3⋊C4—101no—
261C4×Dic3—51no—
262C4×Dic3—51no—
263C4×Dic3—51no—
264C4×Dic3—52no—
265C4×Dic3—101no—
266C4×Dic3—101no—
267C4×Dic3—104no—
268C4×Dic3—104no—
269C4×Dic3—101no—
270C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
271#208—52no—
272#208—101no—
273C2×C3⋊C8—101no—
274C23×C5—61no—
275C22×D5D3×D411yescharacteristic
276C2×AGL(1,𝔽5)—22no—
277C2×AGL(1,𝔽5)—22no—
278C2×AGL(1,𝔽5)—22no—
279C2×AGL(1,𝔽5)—62no—
280C2×AGL(1,𝔽5)—62no—
281C2×AGL(1,𝔽5)—62no—
282C5×D4—32no—
283C5×D4—61no—
284C2×C4×C5—21no—
285C2×C4×C5—31no—
286C2×C4×C5—31no—
287C2×C4×C5—122no—
288C2×C4×C5C2×C4×D311yescharacteristic
289C22⋊D5—62no—
290C22⋊D5—62no—
291C2×Dic5—21no—
292C2×Dic5—31no—
293C2×Dic5—31no—
294C2×Dic5—61no—
295C2×Dic5—62no—
296C2×Dic5—62no—
297C2×Dic5—122no—
298C2×Dic5D3×D411yescharacteristic
299C4×D5—62no—
300C4×D5—62no—
301C4×D5C2×C4×D311yescharacteristic
302C4×D5C2×C4×D311yescharacteristic
303C4×D5#23711yescharacteristic
304C4×D5#23911yescharacteristic
305Dic10—32no—
306Dic10—61no—
307#170—22no—
308#170—62no—
309C2×#41—151no—
310C2×M16—51no—
311C2×M16—151no—
312C4×Q8—151no—
313C4×Q8—151no—
314C4×D4—31no—
315C4×D4—151no—
316C4×D4—301no—
317C4×D4—301no—
318#117—51no—
319#117—301no—
320C2×C42—51no—
321C2×C42—151no—
322C2×C42—151no—
323C2×C42—301no—
324C2×C42—301no—
325C2×C42—301no—
326#104—154no—
327#104—154no—
328#97—301no—
329C4×C8—301no—
330C2×C3×C5C4×C22⋊C411yescharacteristic
331C2×C3×C5C2×#10411yescharacteristic
332C2×C3×C5C2×#10411yescharacteristic
333C3×D5—21no—
334C3×D5—21no—
335C5×D3—42no—
336C23×C3—51no—
337C22×D3—21no—
338C2×C3×C4—21no—
339C2×C3×C4—51no—
340C2×C3×C4—51no—
341C2×C3×C4—51no—
342C2×C3×C4—51no—
343C2×C3×C4—51no—
344C2×C3×C4—102no—
345C2×C3×C4—102no—
346C2×C3×C4—102no—
347C2×C3×C4—101no—
348C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
349#67—102no—
350#67—102no—
351C2×Dic3—51no—
352C2×Dic3—51no—
353C2×Dic3—102no—
354C2×Dic3—102no—
355C2×Dic3—102no—
356C2×Dic3—102no—
357C2×Dic3—101no—
358C2×Dic3—102no—
359C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
360C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
361D12—21no—
362D12C4×AGL(1,𝔽5)12yescomplemented
363C4×D3—42no—
364C4×D3—102no—
365C4×D3—102no—
366C4×D3—202no—
367Dic6—52no—
368Dic6—101no—
369C3×C8—102no—
370C3⋊C8—102no—
371C22×C5—62no—
372C22×C5—121no—
373C22×C5—121no—
374C22×C5D3×C22⋊C411yescharacteristic
375C2×D5—21no—
376C2×D5—21no—
377C2×D5#67⋊C411yescharacteristic
378C2×D5#89311yescharacteristic
379AGL(1,𝔽5)—24no—
380AGL(1,𝔽5)—64no—
381C4×C5—32no—
382C4×C5—41no—
383C4×C5—61no—
384C4×C5C42×D311yescharacteristic
385C4×C5#89211yescharacteristic
386Dic5—32no—
387Dic5—41no—
388Dic5—62no—
389Dic5—61no—
390Dic5#67⋊C411yescharacteristic
391Dic5#89311yescharacteristic
392#41—152no—
393#41—152no—
394#41—301no—
395#41—301no—
396C2×Q8—151no—
397C2×D4—31no—
398C2×D4—301no—
399C22×C4—51no—
400C22×C4—61no—
401C22×C4—101no—
402C22×C4—151no—
403C22×C4—151no—
404C22×C4—301no—
405C22×C4—301no—
406C22×C4—301no—
407M16—52no—
408M16—101no—
409M16—152no—
410M16—301no—
411C2×C8—101no—
412C2×C8—301no—
413C4⋊C4—51no—
414C4⋊C4—301no—
415C4⋊C4—301no—
416C4⋊C4D3×AGL(1,𝔽5)11yescharacteristic
417C22⋊C4—61no—
418C22⋊C4—101no—
419C22⋊C4—301no—
420C42—31no—
421C42—52no—
422C42—101no—
423C42—101no—
424C42—151no—
425C42—151no—
426C42—152no—
427C42—151no—
428C42—301no—
429C42—301no—
430C42—301no—
431C42—304no—
432C42—304no—
433C3×C5C4×#10411yescharacteristic, complemented, Hall
434C22×C3—51no—
435C22×C3—51no—
436C22×C3—101no—
437C22×C3—101no—
438C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
439C2×D3—22no—
440C2×D3—41no—
441C2×D3—41no—
442C3×C4—41no—
443C3×C4—51no—
444C3×C4—51no—
445C3×C4—104no—
446C3×C4—201no—
447C3×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
448C3×C4#413011yescharacteristic
449Dic3—21no—
450Dic3—52no—
451Dic3—104no—
452Dic3—102no—
453Dic3—101no—
454Dic3#400812yescomplemented
455C2×C5—122no—
456C2×C5#466611yescharacteristic
457C2×C5D3×#10411yescharacteristic
458C2×C5D3×#10411yescharacteristic
459D5—21no—
460D5—21no—
461C23—51no—
462C23—61no—
463Q8—152no—
464Q8—301no—
465D4—32no—
466D4—61no—
467D4—302no—
468D4—302no—
469C2×C4—21no—
470C2×C4—31no—
471C2×C4—31no—
472C2×C4—51no—
473C2×C4—51no—
474C2×C4—51no—
475C2×C4—51no—
476C2×C4—51no—
477C2×C4—102no—
478C2×C4—102no—
479C2×C4—102no—
480C2×C4—101no—
481C2×C4—122no—
482C2×C4—151no—
483C2×C4—151no—
484C2×C4—302no—
485C2×C4—302no—
486C2×C4—302no—
487C2×C4—302no—
488C2×C4—302no—
489C2×C4—302no—
490C2×C4—302no—
491C2×C4—301no—
492C2×C4—602no—
493C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
494C8—102no—
495C8—302no—
496C2×C3—101no—
497C2×C3—101no—
498C2×C3C4×C22⋊AGL(1,𝔽5)11yescharacteristic
499C2×C3C2×#400811yescharacteristic
500C2×C3D5⋊#10411yescharacteristic
501D3—42no—
502C5#6872811yescharacteristic, complemented, Sylow
503C22—51no—
504C22—51no—
505C22—62no—
506C22—101no—
507C22—101no—
508C22—121no—
509C22—121no—
510C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
511C4—32no—
512C4—41no—
513C4—51no—
514C4—51no—
515C4—61no—
516C4—104no—
517C4—152no—
518C4—201no—
519C4—304no—
520C4—301no—
521C4—302no—
522C4#9190711yescharacteristic
523C4C4×D3×AGL(1,𝔽5)11yescharacteristic
524C3C4×#400811yescharacteristic, complemented, Sylow
525C2—101no—
526C2—101no—
527C2—122no—
528C2#1175184811yescharacteristic
529C2D3×#400811yescharacteristic
530C2#1175195911yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC3×C4×C5C2×C42
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C5×C4⋊C4C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×#6772C22
cosocleC2×C3×C4×D5C23
hypercenterC4⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5C4×#104
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4×Dic3×AGL(1,𝔽5)C2semidirect
3. Dic3×#700C2semidirect
6. #11746746C2semidirect
7. C3×#64875C2semidirect
16. #91776C4semidirect
17. C5×#893C4semidirect
21. #91435C4semidirect
25. #91340C4semidirect
28. #91156C4semidirect
29. #91155C4semidirect
37. #64875D3semidirect
55. C4×ΓL(1,𝔽16)C2×C4semidirect
57. #6758C2×C4semidirect
60. C3×C4×AGL(1,𝔽5)C2×C4semidirect
62. C3×#700C2×C4semidirect
122. Dic30C42semidirect
126. C5×D12C42semidirect
153. #893AGL(1,𝔽5)semidirect
186. C4×AGL(1,𝔽5)C4×D3semidirect
190. #700C4×D3semidirect
227. Dic15#104semidirect
232. C5×Dic3#104semidirect
362. D12C4×AGL(1,𝔽5)semidirect
433. C3×C5C4×#104semidirect
454. Dic3#4008semidirect
502. C5#68728semidirect
524. C3C4×#4008semidirect

Maximal subgroups

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

subgroupindex
#117534022
C4×Dic3×AGL(1,𝔽5)2
Dic3×#7002
C2×#911562
C2×#911552
#117467462
C3×#648752
C4×#40083
#687285