Finite Group Databasebeta

← #14306047

Subgroups — #14306047

Order 1920.

640 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754581C211yesmaximal, characteristic
2C2×#91109C211yesmaximal, characteristic, complemented
3C5×#5118C211yesmaximal, characteristic, complemented
4C2×C3×#3957C211yesmaximal, characteristic, complemented
5#11748611C211yesmaximal, characteristic, complemented
6C2×#90981C211yesmaximal, characteristic, complemented
7C2×#90976C211yesmaximal, characteristic, complemented
8#10601306—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C2×C4×Dic15C2211yescharacteristic
13C2×C4×C5×Dic3C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C2×C3×C4×Dic5C2211yescharacteristic
16#91569—22no—
17#91458C411yescharacteristic, complemented
18C4×#1202—22no—
19#91389C411yescharacteristic, complemented
20#91374C411yescharacteristic, complemented
21#91364C411yescharacteristic, complemented
22#91109—22no—
23C5×Dic3⋊C8—22no—
24C3×#3957—22no—
25#90994—22no—
26#90981—22no—
27#90976—22no—
28#70357—51nomaximal, Hall
29#65811—31no—
30C5×#423—31no—
31C2×#3957—31no—
32C2×#3957—31no—
33C2×#3957—31no—
34C2×#3957D311yescharacteristic, complemented
35#65221—31no—
36C22×C3×C4×C5C2311yescharacteristic
37C2×#1220—21no—
38C22×Dic15—21no—
39#6817—21no—
40#6817—21no—
41#6817—21no—
42#6817—21no—
43C2×D60C2×C411yescharacteristic
44C2×C4×D15—21no—
45C2×Dic30C2×C411yescharacteristic
46C22×C5×Dic3D412yes—
47C2×C3×C22⋊D5—21no—
48C22×C3×Dic5—21no—
49C3×#709—21no—
50C3×#709—21no—
51C3×#709—21no—
52C3×#709—21no—
53C2×C3×D20C2×C411yescharacteristic
54C2×C3×C4×D5—21no—
55C2×C3×Dic10C2×C411yescharacteristic
56C2×#1202—22no—
57C3×C5×M16—21no—
58C3×C5×M16—21no—
59C2×C3×C5×C8—21no—
60C4×Dic15—21no—
61C4×Dic15D412yescomplemented
62C4×C5×Dic3C2×C411yescharacteristic
63C4×C5×Dic3C2×C411yescharacteristic
64C4×C5×Dic3D412yes—
65C5×#208—21no—
66C5×#208—21no—
67C2×C5×C3⋊C8—21no—
68C3×C4×Dic5—21no—
69C3×C4×Dic5D412yescomplemented
70Dic3⋊Dic5—22no—
71#6667—22no—
72#6665—22no—
73#5832—51no—
74C2×C3×#104—51no—
75C2×#837—51no—
76C2×#825—51no—
77#5118D511yescharacteristic, complemented
78#4932—51no—
79C2×#805—51no—
80C2×#709—31no—
81C2×#709C2×D311yescharacteristic
82C2×C5×M16—31no—
83C2×C5×M16C2×D311yescharacteristic
84C2×C42×C5—31no—
85#4076—62no—
86C2×C4×Dic5—31no—
87C2×C4×Dic5C2×D311yescharacteristic
88#4021—31no—
89#4021—31no—
90#4020—31no—
91C4×D20—62no—
92#4015—31no—
93C5×C4⋊C8—62no—
94#3963—62no—
95#3957—22no—
96#3957—62no—
97#3957—62no—
98#3957—62no—
99#1963—151noSylow
100C23×C3×C5C2×D411yescharacteristic
101C22×D15—21no—
102C22×C3×D5—21no—
103C2×C3×C4×C5C22×C411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5#4111yescharacteristic
108C2×C3×C4×C5#4111yescharacteristic
109#1220—41no—
110#1220—41no—
111C2×Dic15—21no—
112C2×Dic15—22no—
113C2×Dic15—42no—
114D60—21no—
115D60—21no—
116C4×D15—41no—
117C4×D15—41no—
118Dic30—21no—
119Dic30—21no—
120C2×C5×Dic3—42no—
121C2×C5×Dic3C22⋊C412yes—
122C2×C5×Dic3C22⋊C412yes—
123C3×C22⋊D5—41no—
124C3×C22⋊D5—41no—
125C2×C3×Dic5—21no—
126C2×C3×Dic5—22no—
127C2×C3×Dic5—42no—
128C3×D20—21no—
129C3×D20—21no—
130C3×C4×D5—41no—
131C3×C4×D5—41no—
132C3×Dic10—21no—
133C3×Dic10—21no—
134#1202—44no—
135C3×C5×C8—41no—
136C5×C3⋊C8—41no—
137C2×C3×#41—51no—
138C2×#235—51no—
139C2×C3×M16C2×D511yescharacteristic
140C2×C3×C42—51no—
141#947—51no—
142#944—51no—
143#940—51no—
144#939—102no—
145#936—51no—
146C4×#67—102no—
147C2×C4×Dic3—51no—
148C2×C4×Dic3C2×D511yescharacteristic
149C2×#208C2×D511yescharacteristic
150C3×#104—102no—
151#837—102no—
152C3⋊#95—102no—
153#825—102no—
154Dic3⋊C8—22no—
155#805—102no—
156C22×C4×C5—32no—
157C22×C4×C5C22×D311yescharacteristic
158C2×C22⋊D5—21no—
159C2×C22⋊D5—61no—
160C22×Dic5—21no—
161C22×Dic5—61no—
162#709—21no—
163#709—21no—
164#709—21no—
165#709—21no—
166#709—61no—
167#709—61no—
168#709—61no—
169#709—61no—
170C2×D20—31no—
171C2×D20—62no—
172C2×D20C4×D311yescharacteristic
173C2×C4×D5—21no—
174C2×C4×D5—61no—
175C2×Dic10—31no—
176C2×Dic10C4×D311yescharacteristic
177C5×M16—21no—
178C5×M16—21no—
179C5×M16—61no—
180C5×M16—61no—
181C2×C5×C8—21no—
182C2×C5×C8—61no—
183C42×C5—31no—
184C42×C5—31no—
185C42×C5—32no—
186#685—62no—
187#685—62no—
188C4⋊Dic5—62no—
189C4×Dic5—21no—
190C4×Dic5—32no—
191C4×Dic5—61no—
192C4×Dic5#6712yescomplemented
193#532—151no—
194#423—31no—
195C2×#104—51no—
196C2×#104—151no—
197C2×#104—151no—
198C2×#104—151no—
199#376—151no—
200C22×C3×C5—22no—
201C22×C3×C5C2×C22⋊C411yescharacteristic
202C22×C3×C5#12111yescharacteristic
203C22×C3×C5#12111yescharacteristic
204C2×D15—41no—
205C2×D15—42no—
206C2×C3×D5—41no—
207C2×C3×D5—42no—
208C3×C4×C5C4×D411yescharacteristic
209C3×C4×C5C4×D411yescharacteristic
210C3×C4×C5#12411yescharacteristic
211C3×C4×C5#12711yescharacteristic
212Dic15—42no—
213Dic15—41no—
214C3×Dic5—42no—
215C3×Dic5—41no—
216C5×Dic3—22no—
217C5×Dic3—22no—
218C3×#41—101no—
219C3×#41—101no—
220C3×#41—101no—
221C3×#41—101no—
222C2×C3×Q8—51no—
223C2×C3×D4—51no—
224C2×C3×D4—101no—
225C22×C3×C4—101no—
226C22×C3×C4—101no—
227C22×C3×C4C22×D511yescharacteristic
228C2×#67—101no—
229C2×#67—102no—
230C22×Dic3—101no—
231C22×Dic3D2012yes—
232#235—101no—
233#235—101no—
234#235—101no—
235#235—101no—
236C2×D12—51no—
237C2×C4×D3—101no—
238C2×Dic6—51no—
239C3×M16—21no—
240C3×M16—21no—
241C2×C3×C8—21no—
242C3×C42—52no—
243C3×C42—101no—
244#217—102no—
245(C2×D3)⋊C4—102no—
246Dic3⋊C4—102no—
247C4×Dic3—52no—
248C4×Dic3—101no—
249C4×Dic3C4×D511yescharacteristic
250C4×Dic3C4×D511yescharacteristic
251C4×Dic3C22⋊D512yes—
252#208—21no—
253#208—21no—
254C2×C3⋊C8—21no—
255C23×C5D3×D411yescharacteristic
256C22×D5—21no—
257C22×D5—61no—
258C2×C4×C5—32no—
259C2×C4×C5—32no—
260C2×C4×C5—122no—
261C2×C4×C5C2×C4×D311yescharacteristic
262C2×C4×C5#23511yescharacteristic
263C2×C4×C5D3×D411yescharacteristic
264C2×C4×C5D3×D411yescharacteristic
265C2×C4×C5#23711yescharacteristic
266C2×C4×C5C2×#6711yescharacteristic
267C22⋊D5—41no—
268C22⋊D5—41no—
269C22⋊D5—121no—
270C22⋊D5—121no—
271C2×Dic5—21no—
272C2×Dic5—22no—
273C2×Dic5—42no—
274C2×Dic5—61no—
275C2×Dic5—62no—
276C2×Dic5—122no—
277D20—21no—
278D20—21no—
279D20—61no—
280D20—61no—
281D20—124no—
282C4×D5—41no—
283C4×D5—41no—
284C4×D5—121no—
285C4×D5—121no—
286Dic10—21no—
287Dic10—21no—
288Dic10—61no—
289Dic10—61no—
290C5×C8—41no—
291C5×C8—121no—
292C2×#41—51no—
293C2×#41—151no—
294C2×M16—31no—
295C2×M16D3×D511yescharacteristic
296C4⋊Q8—151no—
297#127—151no—
298#124—151no—
299#124—151no—
300#121—302no—
301C4×D4—302no—
302C2×C42—31no—
303C2×C42—51no—
304C2×C42—151no—
305C4⋊C8—62no—
306#104—102no—
307#104—302no—
308#104—302no—
309#104—302no—
310#95—302no—
311C2×C3×C5—21no—
312C2×C3×C5—21no—
313C2×C3×C5#39011yescharacteristic
314C2×C3×C5#42112yes—
315D15—81no—
316C3×D5—81no—
317C23×C3—101no—
318C23×C3C2×D2011yescharacteristic
319C22×D3—101no—
320C3×Q8—101no—
321C3×Q8—101no—
322C3×D4—101no—
323C3×D4—101no—
324C3×D4—201no—
325C3×D4—201no—
326C2×C3×C4—102no—
327C2×C3×C4—101no—
328C2×C3×C4—202no—
329C2×C3×C4—201no—
330C2×C3×C4—201no—
331C2×C3×C4C2×C4×D511yescharacteristic
332C2×C3×C4D4×D511yescharacteristic
333C2×C3×C4D4×D511yescharacteristic
334C2×C3×C4#71311yescharacteristic
335C2×C3×C4#71311yescharacteristic
336C2×C3×C4C2×C22⋊D511yescharacteristic
337#67—204no—
338#67—201no—
339#67—201no—
340C2×Dic3—42no—
341C2×Dic3—102no—
342C2×Dic3—101no—
343C2×Dic3—202no—
344C2×Dic3#68512yes—
345C2×Dic3#68512yes—
346D12—101no—
347D12—101no—
348C4×D3—201no—
349C4×D3—201no—
350Dic6—101no—
351Dic6—101no—
352C3×C8—41no—
353C3⋊C8—41no—
354C22×C5—22no—
355C22×C5D3×C22⋊C411yescharacteristic
356C22×C5#88411yescharacteristic
357C22×C5#93911yescharacteristic
358C2×D5—41no—
359C2×D5—42no—
360C2×D5—121no—
361C2×D5—122no—
362C4×C5—62no—
363C4×C5—62no—
364C4×C5#67⋊C411yescharacteristic
365C4×C5#88511yescharacteristic
366C4×C5C4×#6711yescharacteristic
367C4×C5#94011yescharacteristic
368Dic5—42no—
369Dic5—41no—
370Dic5—122no—
371Dic5—121no—
372#41—101no—
373#41—101no—
374#41—101no—
375#41—101no—
376#41—301no—
377#41—301no—
378#41—301no—
379#41—301no—
380C2×Q8—51no—
381C2×Q8—151no—
382C2×D4—51no—
383C2×D4—101no—
384C2×D4—151no—
385C2×D4—302no—
386C2×D4—301no—
387C22×C4—32no—
388C22×C4—101no—
389C22×C4—101no—
390C22×C4—301no—
391C22×C4—301no—
392C22×C4C2×D3×D511yescharacteristic
393M16—21no—
394M16—21no—
395M16—61no—
396M16—61no—
397C2×C8—21no—
398C2×C8—61no—
399C4⋊C4—302no—
400C22⋊C4—302no—
401C22⋊C4—302no—
402C42—31no—
403C42—31no—
404C42—32no—
405C42—52no—
406C42—101no—
407C42—152no—
408C42—301no—
409C3×C5#196311yescharacteristic, complemented, Hall
410C22×C3—22no—
411C22×C3—202no—
412C22×C3—201no—
413C22×C3#404111yescharacteristic
414C22×C3#404111yescharacteristic
415C22×C3C2×#68511yescharacteristic
416C2×D3—202no—
417C2×D3—201no—
418C3×C4—202no—
419C3×C4—201no—
420C3×C4#403911yescharacteristic
421C3×C4#403911yescharacteristic
422C3×C4#408611yescharacteristic
423C3×C4#409311yescharacteristic
424Dic3—22no—
425Dic3—22no—
426Dic3—202no—
427Dic3—201no—
428C2×C5—21no—
429C2×C5—21no—
430C2×C5#482112yes—
431C2×C5#495711yescharacteristic
432D5—81no—
433D5—241no—
434C23—101no—
435C23—301no—
436C23D3×D2011yescharacteristic
437Q8—101no—
438Q8—101no—
439Q8—301no—
440Q8—301no—
441D4—101no—
442D4—101no—
443D4—201no—
444D4—201no—
445D4—301no—
446D4—301no—
447D4—604no—
448D4—601no—
449D4—601no—
450C2×C4—32no—
451C2×C4—32no—
452C2×C4—102no—
453C2×C4—101no—
454C2×C4—122no—
455C2×C4—202no—
456C2×C4—201no—
457C2×C4—201no—
458C2×C4—302no—
459C2×C4—301no—
460C2×C4—602no—
461C2×C4—601no—
462C2×C4—601no—
463C2×C4#676611yescharacteristic
464C2×C4#677311yescharacteristic
465C2×C4C4×D3×D511yescharacteristic
466C2×C4D5×#6711yescharacteristic
467C2×C4D3×C22⋊D511yescharacteristic
468C2×C4#679011yescharacteristic
469C8—41no—
470C8—121no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—401no—
474C2×C3#6527711yescharacteristic
475C2×C3#6542812yes—
476D3—401no—
477C5#7035711yescharacteristic, complemented, Sylow
478C22—22no—
479C22—201no—
480C22—202no—
481C22—601no—
482C22—602no—
483C22#9140911yescharacteristic
484C22#9145911yescharacteristic
485C22D3×#68511yescharacteristic
486C4—62no—
487C4—62no—
488C4—202no—
489C4—201no—
490C4—602no—
491C4—601no—
492C4#9139611yescharacteristic
493C4#9144411yescharacteristic
494C4#9145511yescharacteristic
495C4#9156711yescharacteristic
496C3#1060130611yescharacteristic, complemented, Sylow
497C2—21no—
498C2—21no—
499C2—401no—
500C2—1201no—
501C2#1174823312yes—
502C2#1174891311yescharacteristic
subgroupG/H
centerC2×C4D3×C22⋊D5
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C2×C22⋊C4
prime radical#11754581C2
cosocleC22×C3×C4×C5C23
hypercenterC2×M16D3×D5
hypocenterC3×C5#1963
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×#91109C2semidirect
3. C5×#5118C2semidirect
4. C2×C3×#3957C2semidirect
5. #11748611C2semidirect
6. C2×#90981C2semidirect
7. C2×#90976C2semidirect
17. #91458C4semidirect
19. #91389C4semidirect
20. #91374C4semidirect
21. #91364C4semidirect
34. C2×#3957D3semidirect
61. C4×Dic15D4semidirect
69. C3×C4×Dic5D4semidirect
77. #5118D5semidirect
192. C4×Dic5#67semidirect
409. C3×C5#1963semidirect
477. C5#70357semidirect
496. C3#10601306semidirect

Maximal subgroups

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

subgroupindex
#117545812
C2×#911092
C5×#51182
C2×C3×#39572
#117486112
C2×#909812
C2×#909762
#106013063
#703575