Finite Group Databasebeta

← #14471587

Subgroups — #14471587

Order 1920.

700 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#556C211yesmaximal, characteristic, complemented
2#11756338C211yesmaximal, characteristic, complemented
3#11751184C211yesmaximal, characteristic, complemented
4#11751180C211yesmaximal, characteristic, complemented
5#11751158C212yesmaximal
6#11751157C212yesmaximal, complemented
7C4×#6717C212yesmaximal, complemented
8Q8×C3⋊#168C211yesmaximal, characteristic, complemented
9#11751123C212yesmaximal, complemented
10C4×#6715C211yesmaximal, characteristic
11#11751109C211yesmaximal, characteristic, complemented
12#10615398—31nomaximal, Hall
13C3×C5×C4⋊Q8C2211yescharacteristic
14C3×C5×#126—22no—
15C3×C5×#124C2212yes—
16C3×C5×C22⋊Q8—21no—
17C3×C4×C5×Q8C2212yes—
18C3×C4×C5×Q8C2211yescharacteristic
19C3×C4×C5×D4C2211yescharacteristic
20C3×C5×#117—21no—
21#91835—21no—
22C2×#6717C2212yes—
23C2×#6715C2211yescharacteristic
24#91785—22no—
25C4⋊C4×D15—21no—
26C4×D60C2211yescharacteristic
27C4×D60C2212yes—
28#91759C2212yes—
29#91758C2211yescharacteristic
30#91119C2212yes—
31#91119C2212yes—
32#91119C2212yes—
33#91116C2212yes—
34#91116C2211yescharacteristic
35#91093C2211yescharacteristic
36#91092C2212yes—
37#91091C2211yescharacteristic
38C4⋊(C3⋊#168)C2211yescharacteristic
39C4⋊(C3⋊#168)C2212yes—
40C4×C3⋊#168C2211yescharacteristic
41C4×C3⋊#168C2212yes—
42#83667—51nomaximal, Hall
43C5×#556D311yescharacteristic, complemented
44#65913—31no—
45#65372—31no—
46#65370—31no—
47#65346—32no—
48#65345—32no—
49C4×#688—32no—
50Q8×#168—31no—
51#65311—32no—
52C4×#686—31no—
53#65297—31no—
54C2×C3×C5×Q8D411yescharacteristic
55C2×C3×C5×Q8C2312yes—
56C2×C3×C5×D4C2311yescharacteristic
57C22×C3×C4×C5—21no—
58C2×D60C2311yescharacteristic
59C2×C4×D15—21no—
60C2×C4×D15—22no—
61C2×Dic30—21no—
62C3×C5×C4⋊C4—21no—
63C3×C5×C4⋊C4—22no—
64C3×C5×C4⋊C4—21no—
65C3×C5×C4⋊C4D411yescharacteristic
66C3×C5×C4⋊C4D412yes—
67C3×C5×C4⋊C4C2311yescharacteristic
68C3×C5×C22⋊C4—22no—
69C3×C5×C22⋊C4—21no—
70C3×C5×C22⋊C4—22no—
71C3×C42×C5—22no—
72C3×C42×C5C2311yescharacteristic
73C3×C42×C5C2312yes—
74#6717—24no—
75#6715—22no—
76#6714—21no—
77#6714—22no—
78C4⋊Dic15—21no—
79C4⋊Dic15C2312yes—
80C4⋊Dic15C2311yescharacteristic
81#6712—22no—
82#6712—22no—
83C2×C3⋊#168C2311yescharacteristic
84C2×C3⋊#168C2312yes—
85C2×C3⋊#168C2311yescharacteristic
86C3×#556D511yescharacteristic, complemented
87#5579—51no—
88#5083—51no—
89#5079—51no—
90#5057—52no—
91#5056—52no—
92C4×#215—52no—
93Q8×C3⋊C8—51no—
94#5022—52no—
95C4×#213—51no—
96#5008—51no—
97C5×C4⋊Q8C2×D311yescharacteristic
98C5×#126—22no—
99C5×#124C2×D312yes—
100C5×C22⋊Q8—21no—
101C4×C5×Q8C2×D312yes—
102C4×C5×Q8C2×D311yescharacteristic
103C4×C5×D4C2×D311yescharacteristic
104C5×#117—21no—
105#4092—61no—
106C2×#688—32no—
107C2×#686—31no—
108#4042—62no—
109D5×C4⋊C4—61no—
110C4×D20—31no—
111C4×D20—32no—
112#4016—32no—
113#4015—31no—
114#3967—32no—
115#3967—32no—
116#3967—32no—
117#3964—32no—
118#3964—31no—
119#3941—31no—
120#3940—32no—
121#3939—31no—
122C4⋊#168—31no—
123C4⋊#168—32no—
124C4×#168—31no—
125C4×#168—32no—
126#3360—151noSylow
127C23×C3×C5—21no—
128C22×D15—21no—
129C3×C5×Q8—22no—
130C3×C5×Q8—22no—
131C3×C5×Q8—22no—
132C3×C5×D4—21no—
133C3×C5×D4—21no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—42no—
137C2×C3×C4×C5C2×D412yes—
138C2×C3×C4×C5C2×D412yes—
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2411yescharacteristic
142C2×Dic15—21no—
143C2×Dic15—22no—
144D60—21no—
145D60#4112yes—
146C4×D15—42no—
147C4×D15—82no—
148Dic30—81no—
149C3⋊#168—21no—
150C3⋊#168—22no—
151C3⋊#168—21no—
152C3×C4⋊Q8C2×D511yescharacteristic
153C3×#126—22no—
154C3×#124C2×D512yes—
155C3×C22⋊Q8—21no—
156C3×C4×Q8C2×D512yes—
157C3×C4×Q8C2×D511yescharacteristic
158C3×C4×D4C2×D511yescharacteristic
159C3×#117—21no—
160#946—101no—
161C2×#215—52no—
162C2×#213—51no—
163#896—102no—
164D3×C4⋊C4—101no—
165C4×D12—51no—
166C4×D12—52no—
167#870—52no—
168#869—51no—
169#835—52no—
170#835—52no—
171#835—52no—
172#832—52no—
173#832—51no—
174#809—51no—
175#808—52no—
176#807—51no—
177C4⋊(C3⋊C8)—51no—
178C4⋊(C3⋊C8)—52no—
179C4×C3⋊C8—51no—
180C4×C3⋊C8—52no—
181C2×C5×Q8#6711yescharacteristic
182C2×C5×Q8C22×D312yes—
183C2×C5×D4C22×D311yescharacteristic
184C22×C4×C5—21no—
185C2×D20—31no—
186C2×C4×D5—61no—
187C2×C4×D5—62no—
188C2×Dic10—61no—
189C5×C4⋊C4—21no—
190C5×C4⋊C4—22no—
191C5×C4⋊C4—21no—
192C5×C4⋊C4#6711yescharacteristic
193C5×C4⋊C4#6712yes—
194C5×C4⋊C4C22×D311yescharacteristic
195C5×C22⋊C4—22no—
196C5×C22⋊C4—21no—
197C5×C22⋊C4—22no—
198C42×C5—22no—
199C42×C5C22×D311yescharacteristic
200C42×C5C22×D312yes—
201#688—64no—
202#686—62no—
203#685—61no—
204#685—62no—
205C4⋊Dic5—31no—
206C4⋊Dic5—32no—
207C4⋊Dic5—61no—
208#683—62no—
209#683—62no—
210C2×#168—31no—
211C2×#168—31no—
212C2×#168—32no—
213#556D1511yescharacteristic, complemented
214#554—151no—
215#488—152no—
216#487—151no—
217#479—152no—
218#476—151no—
219#463—152no—
220#461—151no—
221C8×Q8—151no—
222C4×SD16—152no—
223C4×D8—151no—
224C22×C3×C5—41no—
225C22×C3×C5—41no—
226C22×C3×C5—41no—
227C22×C3×C5C22×D411yescharacteristic
228C2×D15—22no—
229C2×D15—41no—
230C2×D15—41no—
231C3×C4×C5—22no—
232C3×C4×C5—21no—
233C3×C4×C5—22no—
234C3×C4×C5—42no—
235C3×C4×C5—41no—
236C3×C4×C5#13512yes—
237C3×C4×C5C2×#4111yescharacteristic
238C3×C4×C52−1+411yescharacteristic
239Dic15—41no—
240Dic15—42no—
241C2×C3×Q8C22⋊D511yescharacteristic
242C2×C3×Q8C22×D512yes—
243C2×C3×D4C22×D511yescharacteristic
244C22×C3×C4—21no—
245C2×D12—51no—
246C2×C4×D3—101no—
247C2×C4×D3—102no—
248C2×Dic6—101no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4C22⋊D511yescharacteristic
253C3×C4⋊C4C22⋊D512yes—
254C3×C4⋊C4C22×D511yescharacteristic
255C3×C22⋊C4—22no—
256C3×C22⋊C4—21no—
257C3×C22⋊C4—22no—
258C3×C42—22no—
259C3×C42C22×D511yescharacteristic
260C3×C42C22×D512yes—
261#215—104no—
262#213—102no—
263(C2×D3)⋊C4—101no—
264(C2×D3)⋊C4—102no—
265C4⋊Dic3—51no—
266C4⋊Dic3—52no—
267C4⋊Dic3—101no—
268Dic3⋊C4—102no—
269Dic3⋊C4—102no—
270C2×C3⋊C8—51no—
271C2×C3⋊C8—51no—
272C2×C3⋊C8—52no—
273C23×C5—21no—
274C22×D5—61no—
275C5×Q8—22no—
276C5×Q8—22no—
277C5×Q8—22no—
278C5×D4—21no—
279C5×D4—21no—
280C2×C4×C5—21no—
281C2×C4×C5—22no—
282C2×C4×C5—42no—
283C2×C4×C5C2×#6712yes—
284C2×C4×C5C2×#6712yes—
285C2×C4×C5C2×#6711yescharacteristic
286C2×C4×C5C2×#6711yescharacteristic
287C2×C4×C5C23×D311yescharacteristic
288C2×Dic5—61no—
289C2×Dic5—62no—
290D20—32no—
291D20—61no—
292C4×D5—122no—
293C4×D5—242no—
294Dic10—241no—
295#168—61no—
296#168—62no—
297#168—61no—
298C2×SD16—152no—
299C2×D8—151no—
300C4⋊Q8—151no—
301C4⋊Q8C2×D1511yescharacteristic
302#126—22no—
303#125—152no—
304#124C2×D1512yes—
305C22⋊Q8—21no—
306C22⋊Q8—301no—
307C22⋊Q8—302no—
308C4×Q8C2×D1512yes—
309C4×Q8C2×D1511yescharacteristic
310C4×D4—151no—
311C4×D4—152no—
312C4×D4C2×D1511yescharacteristic
313#117—21no—
314C2×C4⋊C4—301no—
315#107—151no—
316#106—152no—
317C4⋊C8—151no—
318C4⋊C8—152no—
319#103—152no—
320#103—152no—
321#103—152no—
322#102—151no—
323#102—152no—
324#102—151no—
325C4×C8—151no—
326C4×C8—152no—
327C2×C3×C5—81no—
328C2×C3×C5#54811yescharacteristic
329C2×C3×C5C2×#13511yescharacteristic
330C2×C3×C5#57711yescharacteristic
331D15—42no—
332C23×C3—21no—
333C22×D3—101no—
334C3×Q8—22no—
335C3×Q8—22no—
336C3×Q8—22no—
337C3×D4—21no—
338C3×D4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—42no—
342C2×C3×C4C2×C22⋊D512yes—
343C2×C3×C4C2×C22⋊D512yes—
344C2×C3×C4C2×C22⋊D511yescharacteristic
345C2×C3×C4C2×C22⋊D511yescharacteristic
346C2×C3×C4C23×D511yescharacteristic
347C2×Dic3—101no—
348C2×Dic3—102no—
349D12—52no—
350D12—101no—
351C4×D3—202no—
352C4×D3—402no—
353Dic6—401no—
354C3⋊C8—101no—
355C3⋊C8—102no—
356C3⋊C8—101no—
357C22×C5—41no—
358C22×C5—41no—
359C22×C5—41no—
360C22×C5C22×#6711yescharacteristic
361C2×D5—62no—
362C2×D5—121no—
363C2×D5—121no—
364C4×C5—22no—
365C4×C5—21no—
366C4×C5—22no—
367C4×C5—41no—
368C4×C5—42no—
369C4×C5#95012yes—
370C4×C5D3×#4111yescharacteristic
371C4×C5#101011yescharacteristic
372Dic5—121no—
373Dic5—122no—
374C2×Q8—301no—
375C2×Q8#122011yescharacteristic
376C2×Q8C22×D1512yes—
377C2×D4—151no—
378C2×D4C22×D1511yescharacteristic
379C22×C4—21no—
380C22×C4—301no—
381C22×C4—302no—
382SD16—304no—
383D8—302no—
384C2×C8—151no—
385C2×C8—152no—
386C2×C8—151no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—152no—
391C4⋊C4—151no—
392C4⋊C4—302no—
393C4⋊C4—301no—
394C4⋊C4—302no—
395C4⋊C4#122011yescharacteristic
396C4⋊C4#122012yes—
397C4⋊C4C22×D1511yescharacteristic
398C22⋊C4—22no—
399C22⋊C4—21no—
400C22⋊C4—22no—
401C22⋊C4—301no—
402C22⋊C4—302no—
403C42—22no—
404C42C22×D1511yescharacteristic
405C42C22×D1512yes—
406C3×C5#336011yescharacteristic, complemented, Hall
407C22×C3—41no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3C22×C22⋊D511yescharacteristic
411C2×D3—102no—
412C2×D3—201no—
413C2×D3—201no—
414C3×C4—22no—
415C3×C4—21no—
416C3×C4—22no—
417C3×C4—41no—
418C3×C4—42no—
419C3×C4#409612yes—
420C3×C4D5×#4111yescharacteristic
421C3×C4#415011yescharacteristic
422Dic3—201no—
423Dic3—202no—
424C2×C5—81no—
425C2×C5C2×#95011yescharacteristic
426C2×C5#583111yescharacteristic
427C2×C5#583711yescharacteristic
428D5—122no—
429C23—21no—
430C23—301no—
431Q8—22no—
432Q8—22no—
433Q8—22no—
434Q8—1201no—
435D4—21no—
436D4—21no—
437D4—152no—
438D4—301no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—42no—
442C2×C4—301no—
443C2×C4—302no—
444C2×C4—602no—
445C2×C4—1202no—
446C2×C4C2×#122012yes—
447C2×C4C2×#122012yes—
448C2×C4C2×#122011yescharacteristic
449C2×C4C2×#122011yescharacteristic
450C2×C4C23×D1511yescharacteristic
451C8—301no—
452C8—302no—
453C8—301no—
454C2×C3—81no—
455C2×C3C2×#409611yescharacteristic
456C2×C3#6616511yescharacteristic
457C2×C3#6617111yescharacteristic
458D3—202no—
459C5#8366711yescharacteristic, complemented, Sylow
460C22—41no—
461C22—41no—
462C22—41no—
463C22—302no—
464C22—601no—
465C22—601no—
466C22C22×#122011yescharacteristic
467C4—22no—
468C4—22no—
469C4—21no—
470C4—41no—
471C4—42no—
472C4—601no—
473C4—602no—
474C4#9183912yes—
475C4#41×D1511yescharacteristic
476C4#9210111yescharacteristic
477C3#1061539811yescharacteristic, complemented, Sylow
478C2—81no—
479C2—602no—
480C2C2×#9183911yescharacteristic
481C2#1175659011yescharacteristic
482C2#1175659611yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#556C2
socleC22×C3×C5C22×D4
prime radicalC2×#6715C22
hypercenter#556D15
hypocenterC3×C5#3360
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. C3×C5×#556C2semidirect
2. #11756338C2semidirect
3. #11751184C2semidirect
4. #11751180C2semidirect
6. #11751157C2semidirect
7. C4×#6717C2semidirect
8. Q8×C3⋊#168C2semidirect
9. #11751123C2semidirect
11. #11751109C2semidirect
43. C5×#556D3semidirect
86. C3×#556D5semidirect
213. #556D15semidirect
406. C3×C5#3360semidirect
459. C5#83667semidirect
477. C3#10615398semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5562
#117563382
#117511842
#117511802
#117511582
#117511572
C4×#67172
Q8×C3⋊#1682
#117511232
C4×#67152
#117511092
#106153983
#836675