Finite Group Databasebeta

← #14297722

Subgroups — #14297722

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91782C211yesmaximal, characteristic
2C2×#91105C211yesmaximal, characteristic
3C2×#91098C211yesmaximal, characteristic, complemented
4#11751078C211yesmaximal, characteristic, complemented
5C2×#91093C211yesmaximal, characteristic
6C3×C5×#340C211yesmaximal, characteristic, complemented
7#11746350C211yesmaximal, characteristic, complemented
8#10599686—31nomaximal, Hall
9C22×D60C2211yescharacteristic
10C22×C4×D15—21no—
11C22×C3×C5×C8C2211yescharacteristic
12C2×C3×C5×C4⋊C4C2211yescharacteristic
13C2×C3×C5×C4⋊C4C2211yescharacteristic
14C2×#6714—21no—
15C2×C4⋊Dic15C2211yescharacteristic
16C2×C4×Dic15C2211yescharacteristic
17C22×C3⋊#168C2211yescharacteristic
18#91782—22no—
19#91782C414yescomplemented
20#91115—21no—
21#91115—21no—
22#91105—22no—
23#91098—22no—
24#91093—22no—
25#68893—51nomaximal, Hall
26C2×#4039—31no—
27C2×#3953—31no—
28C2×#3946—31no—
29#65266—31no—
30C2×#3941—31no—
31C5×#340D311yescharacteristic, complemented
32#64749—31no—
33C23×D15—21no—
34C22×C3×C4×C5—21no—
35C22×C3×C4×C5—21no—
36C22×C3×C4×C5C2311yescharacteristic
37C22×Dic15—21no—
38C22×Dic15D411yescharacteristic
39C22×Dic15D411yescharacteristic
40C2×D60—21no—
41C2×D60—21no—
42C2×D60—21no—
43C2×D60C2×C412yes—
44C2×D60D412yes—
45C2×D60Q812yes—
46C2×C4×D15—24no—
47C2×C4×D15—42no—
48C2×C4×D15—42no—
49C2×C3×C5×C8—21no—
50C2×C3×C5×C8—21no—
51C2×C3×C5×C8—21no—
52C3×C5×C4⋊C4—21no—
53C3×C5×C4⋊C4—21no—
54C3×C5×C4⋊C4—21no—
55C3×C5×C4⋊C4C2×C412yes—
56#6714—24no—
57C4⋊Dic15—21no—
58C4⋊Dic15—21no—
59C4×Dic15C2×C412yes—
60C4×Dic15D412yes—
61C2×C3⋊#168—21no—
62C2×C3⋊#168—21no—
63C2×C3⋊#168—21no—
64C2×#893—51no—
65C2×#821—51no—
66C2×Dic3⋊C8—51no—
67#4977—51no—
68C2×#809—51no—
69C3×#340D511yescharacteristic, complemented
70#4529—51no—
71C22×D20—31no—
72C22×C4×D5—61no—
73C22×C5×C8C2×D311yescharacteristic
74C2×C5×C4⋊C4C2×D311yescharacteristic
75C2×C5×C4⋊C4C2×D311yescharacteristic
76C2×#685—61no—
77C2×C4⋊Dic5—31no—
78C2×C4×Dic5—31no—
79C22×#168—31no—
80#4039—34no—
81#4039—62no—
82#3963—61no—
83#3963—61no—
84#3953—62no—
85#3946—62no—
86#3941—62no—
87#1804—151noSylow
88C23×C3×C5C2×D411yescharacteristic
89C22×D15—22no—
90C22×D15—22no—
91C22×D15—22no—
92C22×D15—41no—
93C22×D15—41no—
94C22×D15—41no—
95C22×D15—41no—
96C2×C3×C4×C5—22no—
97C2×C3×C4×C5—41no—
98C2×C3×C4×C5—41no—
99C2×C3×C4×C5—41no—
100C2×C3×C4×C5—41no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5C22×C411yescharacteristic
103C2×C3×C4×C5C2×D411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5C2×Q811yescharacteristic
106C2×C3×C4×C5#4111yescharacteristic
107C2×C3×C4×C5#4111yescharacteristic
108C2×Dic15—21no—
109C2×Dic15—21no—
110C2×Dic15—21no—
111C2×Dic15—21no—
112C2×Dic15—41no—
113C2×Dic15—41no—
114C2×Dic15—41no—
115C2×Dic15C22⋊C412yes—
116C2×Dic15C22⋊C412yes—
117D60—22no—
118D60—22no—
119D60—22no—
120D60C4⋊C414yes—
121C4×D15—48no—
122C3×C5×C8—41no—
123C3⋊#168—41no—
124C22×D12—51no—
125C22×C4×D3—101no—
126C22×C3×C8C2×D511yescharacteristic
127C2×C3×C4⋊C4C2×D511yescharacteristic
128C2×C3×C4⋊C4C2×D511yescharacteristic
129C2×(C2×D3)⋊C4—101no—
130C2×C4⋊Dic3—51no—
131C2×C4×Dic3—51no—
132C22×C3⋊C8—51no—
133#893—54no—
134#893—102no—
135C3⋊#95—101no—
136C3⋊#95—101no—
137#821—102no—
138Dic3⋊C8—102no—
139#809—102no—
140C23×D5—61no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5C22×D311yescharacteristic
144C22×Dic5—31no—
145C22×Dic5—31no—
146C22×Dic5—61no—
147C2×D20—32no—
148C2×D20—32no—
149C2×D20—32no—
150C2×D20—61no—
151C2×D20—61no—
152C2×D20—61no—
153C2×C4×D5—64no—
154C2×C4×D5—122no—
155C2×C4×D5—122no—
156C2×C5×C8—21no—
157C2×C5×C8—21no—
158C2×C5×C8—21no—
159C5×C4⋊C4—21no—
160C5×C4⋊C4—21no—
161C5×C4⋊C4—21no—
162C5×C4⋊C4C4×D312yes—
163#685—64no—
164C4⋊Dic5—61no—
165C4⋊Dic5—61no—
166C4×Dic5—32no—
167C4×Dic5—32no—
168C2×#168—61no—
169C2×#168—61no—
170C2×#168—61no—
171C2×C4×D4—151no—
172C2×C4⋊C8—151no—
173C2×#102—151no—
174C2×#102—151no—
175#382—151no—
176#340—151no—
177#340D1511yescharacteristic, complemented
178C22×C3×C5C2×C22⋊C411yescharacteristic
179C22×C3×C5#12111yescharacteristic
180C22×C3×C5C22⋊Q811yescharacteristic
181C22×C3×C5#13511yescharacteristic
182C22×C3×C5#13511yescharacteristic
183C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
184C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
185C2×D15—24no—
186C2×D15—42no—
187C2×D15—42no—
188C2×D15—42no—
189C2×D15—42no—
190C2×D15—42no—
191C2×D15—42no—
192C3×C4×C5—42no—
193C3×C4×C5—81no—
194C3×C4×C5C2×C4⋊C411yescharacteristic
195C3×C4×C5#11711yescharacteristic
196C3×C4×C5#12111yescharacteristic
197C3×C4×C5C22⋊Q811yescharacteristic
198Dic15—22no—
199Dic15—22no—
200Dic15—81no—
201C23×D3—101no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C22×D511yescharacteristic
205C22×Dic3—51no—
206C22×Dic3—51no—
207C22×Dic3—101no—
208C2×D12—52no—
209C2×D12—52no—
210C2×D12—52no—
211C2×D12—101no—
212C2×D12—101no—
213C2×D12—101no—
214C2×C4×D3—104no—
215C2×C4×D3—202no—
216C2×C4×D3—202no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4C4×D512yes—
224(C2×D3)⋊C4—104no—
225C4⋊Dic3—101no—
226C4⋊Dic3—101no—
227C4×Dic3—52no—
228C4×Dic3—52no—
229C2×C3⋊C8—101no—
230C2×C3⋊C8—101no—
231C2×C3⋊C8—101no—
232C23×C5D3×D411yescharacteristic
233C22×D5—62no—
234C22×D5—62no—
235C22×D5—62no—
236C22×D5—121no—
237C22×D5—121no—
238C22×D5—121no—
239C22×D5—121no—
240C2×C4×C5—22no—
241C2×C4×C5—41no—
242C2×C4×C5—41no—
243C2×C4×C5—41no—
244C2×C4×C5—41no—
245C2×C4×C5—41no—
246C2×C4×C5C2×C4×D311yescharacteristic
247C2×C4×C5#23511yescharacteristic
248C2×C4×C5#23511yescharacteristic
249C2×C4×C5D3×D411yescharacteristic
250C2×C4×C5D3×D411yescharacteristic
251C2×C4×C5D3×Q811yescharacteristic
252C2×Dic5—32no—
253C2×Dic5—32no—
254C2×Dic5—61no—
255C2×Dic5—61no—
256C2×Dic5—61no—
257C2×Dic5—61no—
258C2×Dic5—121no—
259C2×Dic5—121no—
260C2×Dic5—121no—
261D20—34no—
262D20—62no—
263D20—62no—
264D20—62no—
265C4×D5—128no—
266C5×C8—41no—
267#168—121no—
268C22×D4—151no—
269C23×C4—301no—
270C22×C8—151no—
271C22×C8C2×D1511yescharacteristic
272C4×D4—154no—
273C4×D4—302no—
274C2×C4⋊C4—151no—
275C2×C4⋊C4C2×D1511yescharacteristic
276C2×C4⋊C4C2×D1511yescharacteristic
277C2×C22⋊C4—301no—
278C2×C42—151no—
279C4⋊C8—302no—
280#102—302no—
281#102—302no—
282#95—301no—
283#95—301no—
284C2×C3×C5#38011yescharacteristic
285C2×C3×C5#41611yescharacteristic
286C2×C3×C5#41811yescharacteristic
287C2×C3×C5#46311yescharacteristic
288C2×C3×C5#46311yescharacteristic
289C2×C3×C5#47811yescharacteristic
290C2×C3×C5#47811yescharacteristic
291D15—44no—
292C23×C3D4×D511yescharacteristic
293C22×D3—102no—
294C22×D3—102no—
295C22×D3—102no—
296C22×D3—201no—
297C22×D3—201no—
298C22×D3—201no—
299C22×D3—201no—
300C2×C3×C4—22no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4C2×C4×D511yescharacteristic
307C2×C3×C4#70911yescharacteristic
308C2×C3×C4#70911yescharacteristic
309C2×C3×C4D4×D511yescharacteristic
310C2×C3×C4D4×D511yescharacteristic
311C2×C3×C4Q8×D511yescharacteristic
312C2×Dic3—52no—
313C2×Dic3—52no—
314C2×Dic3—101no—
315C2×Dic3—101no—
316C2×Dic3—101no—
317C2×Dic3—101no—
318C2×Dic3—201no—
319C2×Dic3—201no—
320C2×Dic3—201no—
321D12—54no—
322D12—102no—
323D12—102no—
324D12—102no—
325C4×D3—208no—
326C3×C8—41no—
327C3⋊C8—201no—
328C22×C5D3×C22⋊C411yescharacteristic
329C22×C5#88411yescharacteristic
330C22×C5#89611yescharacteristic
331C22×C5#91111yescharacteristic
332C22×C5#91411yescharacteristic
333C22×C5#91611yescharacteristic
334C22×C5#91911yescharacteristic
335C2×D5—64no—
336C2×D5—122no—
337C2×D5—122no—
338C2×D5—122no—
339C2×D5—122no—
340C2×D5—122no—
341C2×D5—122no—
342C4×C5—42no—
343C4×C5—81no—
344C4×C5#87211yescharacteristic
345C4×C5#88411yescharacteristic
346C4×C5D3×C4⋊C411yescharacteristic
347C4×C5#89611yescharacteristic
348Dic5—62no—
349Dic5—62no—
350Dic5—241no—
351C24—301no—
352C2×D4—152no—
353C2×D4—152no—
354C2×D4—152no—
355C2×D4—301no—
356C2×D4—301no—
357C2×D4—301no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—151no—
361C22×C4—151no—
362C22×C4—304no—
363C22×C4—301no—
364C22×C4—602no—
365C22×C4—602no—
366C22×C4C22×D1511yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—301no—
371C2×C8—301no—
372C2×C8—301no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—301no—
377C4⋊C4—301no—
378C4⋊C4C4×D1512yes—
379C22⋊C4—304no—
380C42—152no—
381C42—152no—
382C3×C5#180411yescharacteristic, complemented, Hall
383C22×C3D5×C22⋊C411yescharacteristic
384C22×C3#403011yescharacteristic
385C22×C3#404211yescharacteristic
386C22×C3#405711yescharacteristic
387C22×C3#406011yescharacteristic
388C22×C3#406211yescharacteristic
389C22×C3#406511yescharacteristic
390C2×D3—104no—
391C2×D3—202no—
392C2×D3—202no—
393C2×D3—202no—
394C2×D3—202no—
395C2×D3—202no—
396C2×D3—202no—
397C3×C4—42no—
398C3×C4—81no—
399C3×C4#401811yescharacteristic
400C3×C4#403011yescharacteristic
401C3×C4D5×C4⋊C411yescharacteristic
402C3×C4#404211yescharacteristic
403Dic3—102no—
404Dic3—102no—
405Dic3—401no—
406C2×C5#466711yescharacteristic
407C2×C5#477011yescharacteristic
408C2×C5#478711yescharacteristic
409C2×C5#480411yescharacteristic
410C2×C5#481911yescharacteristic
411C2×C5#487111yescharacteristic
412C2×C5#489111yescharacteristic
413D5—124no—
414C23—302no—
415C23—302no—
416C23—302no—
417C23—601no—
418C23—601no—
419C23—601no—
420C23—601no—
421C23D4×D1511yescharacteristic
422D4—154no—
423D4—302no—
424D4—302no—
425D4—302no—
426C2×C4—22no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—152no—
433C2×C4—152no—
434C2×C4—301no—
435C2×C4—301no—
436C2×C4—301no—
437C2×C4—301no—
438C2×C4—608no—
439C2×C4—601no—
440C2×C4—601no—
441C2×C4—601no—
442C2×C4C2×C4×D1511yescharacteristic
443C2×C4#681711yescharacteristic
444C2×C4#681711yescharacteristic
445C2×C4D4×D1511yescharacteristic
446C2×C4D4×D1511yescharacteristic
447C2×C4Q8×D1511yescharacteristic
448C8—41no—
449C8—601no—
450C2×C3#6495611yescharacteristic
451C2×C3#6505911yescharacteristic
452C2×C3#6507611yescharacteristic
453C2×C3#6509311yescharacteristic
454C2×C3#6510811yescharacteristic
455C2×C3#6516011yescharacteristic
456C2×C3#6518011yescharacteristic
457D3—204no—
458C5#6889311yescharacteristic, complemented, Sylow
459C22—304no—
460C22—602no—
461C22—602no—
462C22—602no—
463C22—602no—
464C22—602no—
465C22—602no—
466C22C22⋊C4×D1511yescharacteristic
467C22#9177311yescharacteristic
468C22#9178511yescharacteristic
469C22#9180011yescharacteristic
470C22#9180311yescharacteristic
471C22#9180511yescharacteristic
472C22#9180811yescharacteristic
473C4—42no—
474C4—81no—
475C4—302no—
476C4—302no—
477C4—1201no—
478C4#9176111yescharacteristic
479C4#9177311yescharacteristic
480C4C4⋊C4×D1511yescharacteristic
481C4#9178511yescharacteristic
482C3#1059968611yescharacteristic, complemented, Sylow
483C2—604no—
484C2#1175076811yescharacteristic
485C2#1175087111yescharacteristic
486C2#1175088811yescharacteristic
487C2#1175090511yescharacteristic
488C2#1175092011yescharacteristic
489C2#1175097211yescharacteristic
490C2#1175099211yescharacteristic
subgroupG/H
centerC23D4×D15
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C22×D15
FittingC3×C5×#340C2
socleC23×C3×C5C2×D4
prime radicalC22×D60C22
cosocleC22×C3×C4×C5C23
hypercenter#340D15
hypocenterC3×C5#1804
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
3. C2×#91098C2semidirect
4. #11751078C2semidirect
6. C3×C5×#340C2semidirect
7. #11746350C2semidirect
19. #91782C4semidirect
31. C5×#340D3semidirect
69. C3×#340D5semidirect
177. #340D15semidirect
382. C3×C5#1804semidirect
458. C5#68893semidirect
482. C3#10599686semidirect

Maximal subgroups

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

subgroupindex
C2×#917822
C2×#911052
C2×#910982
#117510782
C2×#910932
C3×C5×#3402
#117463502
#105996863
#688935