Finite Group Databasebeta

← #14302612

Subgroups — #14302612

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#709C211yesmaximal, characteristic, complemented
2#11754138C211yesmaximal, characteristic
3C2×C5×#805C211yesmaximal, characteristic, complemented
4#11748625C211yesmaximal, characteristic, complemented
5#11748420C211yesmaximal, characteristic, complemented
6#11746344C211yesmaximal, characteristic, complemented
7#11745992C211yesmaximal, characteristic, complemented
8#10601184—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11C2×C3×C42×C5C2211yescharacteristic
12#91815C2211yescharacteristic
13C2×#6710C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C3×C4×C22⋊D5—22no—
16C3×#4072C2211yescharacteristic
17C3×C4×D20—21no—
18C3×#4018—21no—
19C3×C42×D5—21no—
20C3×C4×Dic10—21no—
21#91309—21no—
22#91307C412yescomplemented
23#91299C412yescomplemented
24#91298—21no—
25#91297—22no—
26C5×#805—21no—
27C5×#805—21no—
28#90996—22no—
29#90976—22no—
30#71645—51nomaximal, Hall
31#66156—31no—
32C4×#709D311yescharacteristic, complemented
33C2×C5×#104—31no—
34#65529—31no—
35#65428—31no—
36#64776—31no—
37#64743—31no—
38C22×C3×C4×C5—21no—
39C22×C3×C4×C5C2311yescharacteristic
40C2×C5×#67—21no—
41C5×#235—21no—
42C5×#235—21no—
43C5×#235—21no—
44C5×#235—21no—
45C2×C5×D12C2×C411yescharacteristic, complemented
46C2×C4×C5×D3—21no—
47C2×C5×Dic6C2×C411yescharacteristic, complemented
48C2×C3×C22⋊D5—21no—
49C3×#709—21no—
50C3×#709—21no—
51C3×#709C2×C412yes—
52C3×#709D412yes—
53C2×C3×D20D411yescharacteristic
54C2×C3×C4×D5—21no—
55C2×C3×C4×D5—22no—
56C2×C3×Dic10D411yescharacteristic
57C3×C42×C5—21no—
58C3×C42×C5D411yescharacteristic
59C3×C42×C5D411yescharacteristic
60#6719—21no—
61C4⋊Dic15—21no—
62C4×Dic15—21no—
63#6710—21no—
64#6710C2×C412yescomplemented
65C2×C3⋊#168—21no—
66C5×#208—21no—
67C5×#208—21no—
68C2×C5×C3⋊C8—21no—
69C3×C22⋊Dic5D412yes—
70C3×#685—22no—
71C3×C4⋊Dic5—21no—
72C3×#683—22no—
73C3×C4×Dic5—22no—
74C3×C4×Dic5—21no—
75#6661—22no—
76#6656—22no—
77#6655—22no—
78#6652—22no—
79C3×C4×#41—51no—
80#5822—51no—
81#5236—51no—
82#5006—51no—
83C2×#805D511yescharacteristic, complemented
84#4533—51no—
85#4523—51no—
86C2×C5×#41—31no—
87C2×#709C2×D311yescharacteristic
88C2×C5×M16—31no—
89C2×C42×C5C2×D311yescharacteristic
90#4097—61no—
91#4096—62no—
92#4095—61no—
93#4088—32no—
94#4078—32no—
95C4×C22⋊D5—22no—
96#4072—31no—
97#4072C2×D311yescharacteristic
98C2×#681—31no—
99C4×D20—21no—
100#4018—21no—
101C42×D5—21no—
102C4×Dic10—21no—
103C5×#104—61no—
104C5×#104—61no—
105#3966—62no—
106#3957—62no—
107#1906—151noSylow
108C23×C3×C5C2×D411yescharacteristic
109C22×C5×D3—21no—
110C22×C3×D5—21no—
111C2×C3×C4×C5—21no—
112C2×C3×C4×C5—21no—
113C2×C3×C4×C5—42no—
114C2×C3×C4×C5C22×C411yescharacteristic
115C2×C3×C4×C5C2×D411yescharacteristic
116C2×C3×C4×C5C2×D411yescharacteristic
117C2×C3×C4×C5C2×D411yescharacteristic
118C2×C3×C4×C5#4111yescharacteristic
119C2×C3×C4×C5#4111yescharacteristic
120C2×Dic15—41no—
121C5×#67—41no—
122C5×#67—41no—
123C2×C5×Dic3—21no—
124C5×D12—21no—
125C5×D12—21no—
126C4×C5×D3—41no—
127C4×C5×D3—41no—
128C5×Dic6—21no—
129C5×Dic6—21no—
130C3×C22⋊D5—22no—
131C3×C22⋊D5—22no—
132C2×C3×Dic5—21no—
133C2×C3×Dic5—22no—
134C3×D20—21no—
135C3×D20C22⋊C412yes—
136C3×C4×D5—22no—
137C3×C4×D5—22no—
138C3×C4×D5—44no—
139C3×Dic10—21no—
140C3×Dic10C22⋊C412yes—
141C3⋊#168—22no—
142C5×C3⋊C8—41no—
143C2×C3×#41—51no—
144C2×#235C2×D511yescharacteristic
145C3×C4×Q8—101no—
146C3×C4×D4—101no—
147C3×C4×D4—102no—
148C3×#117—51no—
149C3×#117—101no—
150C2×C3×C42—101no—
151C2×C3×C42C2×D511yescharacteristic
152#951—52no—
153#950—102no—
154#949—52no—
155#942—101no—
156#932—101no—
157#926—51no—
158C2×#208—51no—
159C2×#208C2×D511yescharacteristic
160#837—102no—
161#806—102no—
162#805—21no—
163#805—21no—
164C5×#41—61no—
165C5×#41—61no—
166C5×#41—61no—
167C5×#41—61no—
168C2×C5×Q8—31no—
169C2×C5×D4—31no—
170C2×C5×D4—61no—
171C22×C4×C5—21no—
172C22×C4×C5—61no—
173C22×C4×C5C22×D311yescharacteristic
174C2×C22⋊D5—21no—
175#709—21no—
176#709—21no—
177#709C4×D312yes—
178#709D1212yes—
179C2×D20#6711yescharacteristic
180C2×C4×D5—21no—
181C2×C4×D5—22no—
182C2×Dic10#6711yescharacteristic
183C5×M16—61no—
184C5×M16—61no—
185C2×C5×C8—61no—
186C42×C5—21no—
187C42×C5#6711yescharacteristic
188C42×C5#6711yescharacteristic
189C22⋊Dic5—61no—
190C22⋊Dic5#6712yes—
191#689—62no—
192#688—62no—
193#687—62no—
194#686—62no—
195#685—22no—
196C4⋊Dic5—21no—
197C4⋊Dic5—61no—
198#683—22no—
199C4×Dic5—22no—
200C4×Dic5—21no—
201C4×Dic5—61no—
202#681—32no—
203#681—61no—
204C2×#168—61no—
205#575—151no—
206C4×#41—51no—
207#421—151no—
208C2×#104—31no—
209#410—151no—
210#344—151no—
211#339—151no—
212C22×C3×C5—41no—
213C22×C3×C5C4×D411yescharacteristic
214C22×C3×C5#12011yescharacteristic
215C22×C3×C5#12111yescharacteristic
216C2×C5×D3—41no—
217C2×C5×D3—81no—
218C2×C3×D5—22no—
219C2×C3×D5—42no—
220C3×C4×C5—22no—
221C3×C4×C5—41no—
222C3×C4×C5C2×C22⋊C411yescharacteristic
223C3×C4×C5C4×D411yescharacteristic
224C3×C4×C5#12111yescharacteristic
225C3×C4×C5#12311yescharacteristic
226Dic15—81no—
227C3×Dic5—22no—
228C3×Dic5—42no—
229C5×Dic3—41no—
230C3×#41—52no—
231C3×#41—52no—
232C3×#41—101no—
233C3×#41—101no—
234C2×C3×Q8—51no—
235C2×C3×D4—51no—
236C2×C3×D4—101no—
237C22×C3×C4—21no—
238C22×C3×C4—102no—
239C22×C3×C4—101no—
240C22×C3×C4C22×D511yescharacteristic
241C2×#67—21no—
242#235—21no—
243#235—21no—
244#235—21no—
245#235—21no—
246C2×D12C4×D511yescharacteristic, complemented
247C2×C4×D3—21no—
248C2×Dic6C4×D511yescharacteristic, complemented
249C3×C4⋊C4—102no—
250C3×C4⋊C4—101no—
251C3×C22⋊C4—52no—
252C3×C22⋊C4—102no—
253C3×C42—21no—
254C3×C42—102no—
255C3×C42—101no—
256C3×C42C22⋊D511yescharacteristic
257C3×C42C22⋊D511yescharacteristic
258#217—101no—
259#216—102no—
260#215—102no—
261#214—102no—
262#213—102no—
263C4⋊Dic3—101no—
264C4×Dic3—101no—
265#208—21no—
266#208—21no—
267#208—52no—
268#208—101no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—101no—
271C23×C5—61no—
272C23×C5C2×#6711yescharacteristic
273C22×D5—21no—
274C5×Q8—61no—
275C5×Q8—61no—
276C5×D4—61no—
277C5×D4—61no—
278C5×D4—121no—
279C5×D4—121no—
280C2×C4×C5—21no—
281C2×C4×C5—21no—
282C2×C4×C5—42no—
283C2×C4×C5—61no—
284C2×C4×C5—121no—
285C2×C4×C5—121no—
286C2×C4×C5C2×C4×D311yescharacteristic
287C2×C4×C5C2×D1211yescharacteristic
288C2×C4×C5#23511yescharacteristic
289C2×C4×C5#23511yescharacteristic
290C2×C4×C5C2×#6711yescharacteristic
291C2×C4×C5C2×#6711yescharacteristic
292C22⋊D5—22no—
293C22⋊D5—22no—
294C2×Dic5—21no—
295C2×Dic5—22no—
296C2×Dic5—121no—
297D20—21no—
298D20(C2×D3)⋊C412yes—
299C4×D5—22no—
300C4×D5—22no—
301C4×D5—44no—
302Dic10—21no—
303Dic10(C2×D3)⋊C412yes—
304C5×C8—121no—
305#168—62no—
306C2×#41—31no—
307C2×#41—51no—
308#137—152no—
309#137—301no—
310AGL(1,ℤ/8)—152no—
311AGL(1,ℤ/8)—301no—
312#135—302no—
313C2×M16—31no—
314C2×M16—151no—
315C4×Q8—101no—
316C4×D4—101no—
317C4×D4—102no—
318#117—51no—
319#117—101no—
320#117—151no—
321C2×C42—101no—
322C2×C42D3×D511yescharacteristic
323#104—61no—
324#104—61no—
325#104—302no—
326#99—302no—
327C2×C3×C5—21no—
328C2×C3×C5—21no—
329C2×C3×C5—21no—
330C2×C3×C5#38611yescharacteristic
331C3×D5—42no—
332C5×D3—81no—
333C23×C3—101no—
334C23×C3D4×D511yescharacteristic
335C22×D3—21no—
336C3×Q8—52no—
337C3×Q8—101no—
338C3×D4—52no—
339C3×D4—102no—
340C3×D4—102no—
341C3×D4—101no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—42no—
345C2×C3×C4—101no—
346C2×C3×C4—102no—
347C2×C3×C4—102no—
348C2×C3×C4—102no—
349C2×C3×C4—204no—
350C2×C3×C4C2×C4×D511yescharacteristic
351C2×C3×C4#70911yescharacteristic
352C2×C3×C4D4×D511yescharacteristic
353C2×C3×C4D4×D511yescharacteristic
354C2×C3×C4#71111yescharacteristic
355C2×C3×C4C2×C22⋊D511yescharacteristic
356#67—41no—
357#67—41no—
358C2×Dic3—21no—
359C2×Dic3—201no—
360D12—21no—
361D12—21no—
362C4×D3—41no—
363C4×D3—41no—
364Dic6—21no—
365Dic6—21no—
366C3⋊C8—41no—
367C3⋊C8—102no—
368C22×C5—41no—
369C22×C5—121no—
370C22×C5—241no—
371C22×C5C4×#6711yescharacteristic
372C22×C5#93011yescharacteristic
373C22×C5#95311yescharacteristic
374C2×D5—22no—
375C2×D5—42no—
376C4×C5—22no—
377C4×C5—41no—
378C4×C5—121no—
379C4×C5C2×(C2×D3)⋊C411yescharacteristic
380C4×C5C4×#6711yescharacteristic
381C4×C5#92911yescharacteristic
382C4×C5#93011yescharacteristic
383Dic5—22no—
384Dic5—42no—
385Dic5—241no—
386#41—52no—
387#41—52no—
388#41—61no—
389#41—61no—
390#41—61no—
391#41—61no—
392#41—101no—
393#41—101no—
394C2×Q8—31no—
395C2×Q8—51no—
396C2×D4—31no—
397C2×D4—51no—
398C2×D4—61no—
399C2×D4—101no—
400C22×C4—21no—
401C22×C4—61no—
402C22×C4—102no—
403C22×C4—101no—
404C22×C4C2×D3×D511yescharacteristic
405Q16—302no—
406SD16—302no—
407SD16—302no—
408D8—302no—
409M16—61no—
410M16—61no—
411M16—152no—
412M16—301no—
413C2×C8—61no—
414C2×C8—301no—
415C4⋊C4—102no—
416C4⋊C4—101no—
417C4⋊C4—301no—
418C22⋊C4—52no—
419C22⋊C4—102no—
420C22⋊C4—301no—
421C42—21no—
422C42—102no—
423C42—101no—
424C42—301no—
425C42#120111yescharacteristic
426C42#120111yescharacteristic
427C3×C5#190611yescharacteristic, complemented, Hall
428C22×C3—41no—
429C22×C3—102no—
430C22×C3—202no—
431C22×C3#402711yescharacteristic
432C22×C3#403011yescharacteristic
433C22×C3#408311yescharacteristic
434C2×D3—41no—
435C2×D3—81no—
436C3×C4—22no—
437C3×C4—41no—
438C3×C4—102no—
439C3×C4—202no—
440C3×C4D5×C22⋊C411yescharacteristic
441C3×C4#402911yescharacteristic
442C3×C4C4×C22⋊D511yescharacteristic
443C3×C4#408411yescharacteristic
444Dic3—41no—
445Dic3—401no—
446C2×C5—21no—
447C2×C5—21no—
448C2×C5—21no—
449C2×C5—241no—
450C2×C5#521311yescharacteristic
451D5—42no—
452C23—61no—
453C23—101no—
454C23D5×#6711yescharacteristic
455Q8—52no—
456Q8—61no—
457Q8—61no—
458Q8—101no—
459D4—52no—
460D4—61no—
461D4—61no—
462D4—102no—
463D4—102no—
464D4—101no—
465D4—121no—
466D4—121no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—42no—
470C2×C4—61no—
471C2×C4—101no—
472C2×C4—102no—
473C2×C4—102no—
474C2×C4—102no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—204no—
478C2×C4—601no—
479C2×C4#677111yescharacteristic
480C2×C4#677211yescharacteristic
481C2×C4C4×D3×D511yescharacteristic
482C2×C4D5×D1211yescharacteristic
483C2×C4C2×#120111yescharacteristic
484C2×C4D5×#6711yescharacteristic
485C8—121no—
486C8—302no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—202no—
491C2×C3#6524511yescharacteristic
492D3—81no—
493C5#7164511yescharacteristic, complemented, Sylow
494C22—41no—
495C22—102no—
496C22—121no—
497C22—202no—
498C22—241no—
499C22#9144011yescharacteristic
500C22#9144811yescharacteristic
501C22#9156311yescharacteristic
502C4—22no—
503C4—41no—
504C4—102no—
505C4—121no—
506C4—202no—
507C4—1201no—
508C4C4×#120111yescharacteristic
509C4#9144711yescharacteristic
510C4#9144911yescharacteristic
511C4D5×(C2×D3)⋊C411yescharacteristic
512C3#1060118411yescharacteristic, complemented, Sylow
513C2—21no—
514C2—21no—
515C2—21no—
516C2—202no—
517C2—241no—
518C2#1174890511yescharacteristic
subgroupG/H
centerC4#91449
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C42×C5C22
socleC2×C3×C5#386
prime radical#11754138C2
cosocleC22×C3×C4×C5C23
hypercenterC2×C42D3×D5
hypocenterC3×C5#1906
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C3×C4×#709C2semidirect
3. C2×C5×#805C2semidirect
4. #11748625C2semidirect
5. #11748420C2semidirect
6. #11746344C2semidirect
7. #11745992C2semidirect
22. #91307C4semidirect
23. #91299C4semidirect
32. C4×#709D3semidirect
45. C2×C5×D12C2×C4semidirect
47. C2×C5×Dic6C2×C4semidirect
64. #6710C2×C4semidirect
83. C2×#805D5semidirect
246. C2×D12C4×D5semidirect
248. C2×Dic6C4×D5semidirect
427. C3×C5#1906semidirect
493. C5#71645semidirect
512. C3#10601184semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#7092
#117541382
C2×C5×#8052
#117486252
#117484202
#117463442
#117459922
#106011843
#716455