Finite Group Databasebeta

← #14460004

Subgroups — #14460004

Order 1920.

772 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#11756314C211yesmaximal, characteristic
2#11752486C211yesmaximal, characteristic, complemented
3C5×#5007C211yesmaximal, characteristic, complemented
4C3×C4×#687C211yesmaximal, characteristic, complemented
5#11749063C212yesmaximal, complemented
6#11748969C212yesmaximal, complemented
7#11747359C212yesmaximal, complemented
8#11747215C212yesmaximal, complemented
9#11747095C211yesmaximal, characteristic, complemented
10#11746988C211yesmaximal, characteristic, complemented
11#11746930C211yesmaximal, characteristic, complemented
12#10612712—31nomaximal, Hall
13C2×#6819C2212yes—
14C3×C4×C5×D4C2211yescharacteristic
15#91819—21no—
16C2×C4⋊Dic15—21no—
17#91786C2212yes—
18#91775—22no—
19C4×D60C2211yescharacteristic
20#91758C2211yescharacteristic
21C5×#869C2211yescharacteristic
22C2×C3×#687C2211yescharacteristic
23C3×C4×Dic10C2211yescharacteristic
24#91390C2212yes—
25#91375—21no—
26#91347C2212yes—
27#91337—22no—
28Dic5×Dic6—21no—
29C2×#6662C2212yes—
30C2×#6648C2212yes—
31C5×#832C2212yes—
32C5×#807C2212yes—
33C5×C4⋊(C3⋊C8)C2211yescharacteristic
34C3×#3964C2211yescharacteristic
35C3×#3942C2211yescharacteristic
36C3×#3940C2211yescharacteristic
37C3×C4×#168C2211yescharacteristic
38#90975C2212yes—
39#90975C2212yes—
40#90974C2212yes—
41#90957C2212yes—
42#79680—51nomaximal, Hall
43#66066—31no—
44#65889—31no—
45C5×#474—31no—
46#65382—31no—
47C4×#687D311yescharacteristic, complemented
48#65177—32no—
49#65176—32no—
50#65138—31no—
51#65123—31no—
52#65072—32no—
53#65061—32no—
54C2×C3×C5×D4C2311yescharacteristic
55C22×C3×C4×C5—21no—
56C2×#1220—22no—
57C22×Dic15—22no—
58#6819—24no—
59#6819—42no—
60C2×D60D411yescharacteristic
61C2×C4×D15C2312yes—
62C2×Dic30C2312yes—
63C2×C5×Dic6—21no—
64C2×C3×Dic10C2311yescharacteristic
65C3×C5×C4⋊C4C2311yescharacteristic
66C3×C5×C22⋊C4—21no—
67C3×C42×C5C2311yescharacteristic
68#6714—22no—
69#6714D412yes—
70C4⋊Dic15—21no—
71C4⋊Dic15—21no—
72C4⋊Dic15—22no—
73C4⋊Dic15D411yescharacteristic
74C5×C4⋊Dic3—21no—
75C5×C4⋊Dic3C2312yes—
76C2×C5×C3⋊C8C2312yes—
77C3×#687—22no—
78C3×C4⋊Dic5C2311yescharacteristic
79C3×#683—21no—
80C3×C4×Dic5—21no—
81C2×C3×#168C2311yescharacteristic
82C2×C3×#168C2311yescharacteristic
83Dic3⋊Dic5—22no—
84#6667—22no—
85#6667—22no—
86Dic3×Dic5—22no—
87#6662—42no—
88#6648—42no—
89#5584—51no—
90#5555—51no—
91C3×C4×SD16—51no—
92#5173—52no—
93#5031—51no—
94#5007D511yescharacteristic, complemented
95#4868—52no—
96#4813—52no—
97#4772—52no—
98C8×D12—51no—
99#4681—51no—
100C2×#711—32no—
101C5×C4⋊Q8—31no—
102C4×C5×D4C2×D311yescharacteristic
103#4093—61no—
104Q8×Dic5—61no—
105C2×#687C2×D311yescharacteristic
106#4076—61no—
107C2×C4⋊Dic5—61no—
108C2×Dic20—32no—
109C2×C8×D5—32no—
110#4044—62no—
111#4043—32no—
112#4043—32no—
113#4038—32no—
114#4032—62no—
115C4×D20—31no—
116#4015—31no—
117C4×Dic10C2×D311yescharacteristic
118C5×#107—32no—
119C5×C4⋊C8—31no—
120C5×#102—32no—
121#3964C2×D311yescharacteristic
122#3952—32no—
123#3948—32no—
124#3942—32no—
125#3942—32no—
126#3942C2×D311yescharacteristic
127#3940C2×D311yescharacteristic
128C4×#168C2×D311yescharacteristic
129#3302—151noSylow
130C23×C3×C5—21no—
131C22×D15C2×D412yes—
132C3×C5×D4—21no—
133C3×C5×D4#4112yes—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—42no—
136C2×C3×C4×C5C2×D411yescharacteristic
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2411yescharacteristic
139#1220—44no—
140C2×Dic15—22no—
141C2×Dic15—44no—
142C2×Dic15C2×D412yes—
143D60—22no—
144C4×D15—22no—
145C4×D15—22no—
146Dic30—22no—
147Dic30—22no—
148C2×C5×Dic3—22no—
149C5×Dic6—42no—
150C2×C3×Dic5—21no—
151C2×C3×Dic5—21no—
152C3×Dic10—21no—
153C3×Dic10—21no—
154C3×#168—21no—
155C3×#168—21no—
156C5×C3⋊C8—22no—
157C2×#237—52no—
158C2×C3×SD16—51no—
159C3×C4×Q8—51no—
160C3×C4×D4C2×D511yescharacteristic
161#946—52no—
162C2×#216—52no—
163#930—101no—
164C2×C4⋊Dic3—101no—
165C2×D3×C8—52no—
166#898—102no—
167#897—52no—
168#892—52no—
169#886—102no—
170#875—101no—
171C4×D12—51no—
172#869—51no—
173#869C2×D511yescharacteristic
174C4×Dic6—101no—
175C3×#106—51no—
176C3×#103—51no—
177C3×#102—51no—
178C3×C4×C8—51no—
179#832C2×D512yes—
180(C2×D3)⋊C8—52no—
181#816—52no—
182#816—52no—
183#810—52no—
184#807C2×D512yes—
185C4⋊(C3⋊C8)C2×D511yescharacteristic
186C2×C5×Q8—61no—
187C2×C5×D4C22×D311yescharacteristic
188C22×C4×C5—21no—
189C2×C22⋊D5—62no—
190C22×Dic5—62no—
191#711—64no—
192#711—122no—
193C2×D20—31no—
194C2×C4×D5—32no—
195C2×Dic10—32no—
196C2×Dic10C22×D311yescharacteristic
197C2×C5×C8—32no—
198C5×C4⋊C4—32no—
199C5×C4⋊C4—61no—
200C5×C4⋊C4C22×D311yescharacteristic
201C5×C22⋊C4—21no—
202C42×C5C22×D311yescharacteristic
203#687—22no—
204#685—32no—
205#685—62no—
206#685—62no—
207#685—62no—
208C4⋊Dic5—31no—
209C4⋊Dic5—62no—
210C4⋊Dic5—62no—
211C4⋊Dic5—61no—
212C4⋊Dic5—61no—
213C4⋊Dic5C22×D311yescharacteristic
214#683—21no—
215C4×Dic5—21no—
216C4×Dic5—62no—
217C2×#168C22×D311yescharacteristic
218C2×#168C22×D311yescharacteristic
219Dic20—122no—
220C8×D5—122no—
221#556—151no—
222#547—151no—
223#487—151no—
224#485—152no—
225#474—31no—
226#467—152no—
227#462—151no—
228#452—152no—
229C4×SD16—51no—
230C8×D4—151no—
231#416—152no—
232C22×C3×C5—22no—
233C22×C3×C5—41no—
234C22×C3×C5—41no—
235C22×C3×C5C22×D411yescharacteristic
236C2×D15—22no—
237C2×D15—22no—
238C2×D15—22no—
239C3×C4×C5—21no—
240C3×C4×C5—41no—
241C3×C4×C5#13512yes—
242C3×C4×C5C2×#4111yescharacteristic
243C3×C4×C5H(5,𝔽2)11yescharacteristic
244Dic15—22no—
245Dic15—42no—
246C3×Dic5—41no—
247C3×Dic5—41no—
248C5×Dic3—42no—
249C2×C3×Q8—51no—
250C2×C3×D4C22×D511yescharacteristic
251C22×C3×C4—21no—
252C2×#67—102no—
253C22×Dic3—102no—
254#237—104no—
255#237—202no—
256C2×D12—51no—
257C2×C4×D3—52no—
258C2×Dic6—21no—
259C2×Dic6—52no—
260C3×SD16—102no—
261C2×C3×C8—51no—
262C2×C3×C8—51no—
263C3×C4⋊C4—51no—
264C3×C4⋊C4—101no—
265C3×C4⋊C4C22×D511yescharacteristic
266C3×C22⋊C4—21no—
267C3×C42—101no—
268C3×C42C22×D511yescharacteristic
269#216—202no—
270(C2×D3)⋊C4—52no—
271(C2×D3)⋊C4—102no—
272(C2×D3)⋊C4—102no—
273(C2×D3)⋊C4—102no—
274C4⋊Dic3—21no—
275C4⋊Dic3—51no—
276C4⋊Dic3—102no—
277C4⋊Dic3—101no—
278C4⋊Dic3—101no—
279C4⋊Dic3C22×D512yes—
280Dic3⋊C4—102no—
281C4×Dic3—102no—
282C2×C3⋊C8C22×D512yes—
283D3×C8—202no—
284C23×C5—21no—
285C22×D5—32no—
286C5×Q8—122no—
287C5×D4—21no—
288C5×D4#23512yes—
289C2×C4×C5—21no—
290C2×C4×C5—42no—
291C2×C4×C5—62no—
292C2×C4×C5D3×D411yescharacteristic
293C2×C4×C5D3×D411yescharacteristic
294C2×C4×C5C23×D311yescharacteristic
295C22⋊D5—124no—
296C2×Dic5—21no—
297C2×Dic5—21no—
298C2×Dic5—32no—
299C2×Dic5—62no—
300C2×Dic5—124no—
301D20—62no—
302C4×D5—62no—
303C4×D5—62no—
304Dic10—21no—
305Dic10—21no—
306Dic10—62no—
307Dic10—62no—
308C5×C8—62no—
309#168—21no—
310#168—21no—
311C2×#41—152no—
312C2×Q16—152no—
313C2×SD16—51no—
314C22×C8—152no—
315C4⋊Q8—31no—
316C4⋊Q8—151no—
317#126—302no—
318#124—301no—
319#123—302no—
320C22⋊Q8—152no—
321C22⋊Q8—152no—
322#121—301no—
323C4×Q8—51no—
324C4×Q8—301no—
325C4×D4—151no—
326C4×D4D3×D511yescharacteristic
327#117—152no—
328C2×C4⋊C4—301no—
329#107—32no—
330#106—51no—
331C4⋊C8—31no—
332#103—51no—
333#103—152no—
334#103—152no—
335#103—152no—
336#102—32no—
337#102—51no—
338C22⋊C8—152no—
339C4×C8—51no—
340C2×C3×C5—42no—
341C2×C3×C5#54611yescharacteristic
342C2×C3×C5C2×#13511yescharacteristic
343C2×C3×C5#57811yescharacteristic
344D15—42no—
345C23×C3—21no—
346C22×D3—52no—
347C3×Q8—101no—
348C3×Q8—101no—
349C3×D4—21no—
350C3×D4#71312yes—
351C2×C3×C4—21no—
352C2×C3×C4—42no—
353C2×C3×C4—101no—
354C2×C3×C4—101no—
355C2×C3×C4D4×D511yescharacteristic
356C2×C3×C4D4×D511yescharacteristic
357C2×C3×C4C23×D511yescharacteristic
358#67—204no—
359C2×Dic3—22no—
360C2×Dic3—52no—
361C2×Dic3—102no—
362C2×Dic3—204no—
363D12—102no—
364C4×D3—102no—
365C4×D3—102no—
366Dic6—42no—
367Dic6—102no—
368Dic6—102no—
369C3×C8—101no—
370C3×C8—101no—
371C3⋊C8—22no—
372C22×C5—22no—
373C22×C5—41no—
374C22×C5—41no—
375C22×C5C2×D3×D411yescharacteristic
376C2×D5—62no—
377C2×D5—62no—
378C2×D5—62no—
379C4×C5—21no—
380C4×C5—41no—
381C4×C5—122no—
382C4×C5#91612yes—
383C4×C5C2×#23511yescharacteristic
384C4×C5#100911yescharacteristic
385Dic5—41no—
386Dic5—41no—
387Dic5—62no—
388Dic5—122no—
389#41—304no—
390#41—602no—
391C2×Q8—51no—
392C2×Q8—61no—
393C2×Q8—152no—
394C2×D4—151no—
395C2×D4—302no—
396C2×D4C2×D3×D511yescharacteristic
397C22×C4—21no—
398C22×C4—152no—
399C22×C4—302no—
400Q16—602no—
401SD16—102no—
402C2×C8—32no—
403C2×C8—51no—
404C2×C8—51no—
405C2×C8—602no—
406C4⋊C4—32no—
407C4⋊C4—51no—
408C4⋊C4—61no—
409C4⋊C4—101no—
410C4⋊C4—151no—
411C4⋊C4—302no—
412C4⋊C4—302no—
413C4⋊C4—301no—
414C4⋊C4—301no—
415C4⋊C4C2×D3×D511yescharacteristic
416C22⋊C4—21no—
417C22⋊C4—152no—
418C22⋊C4—302no—
419C22⋊C4—302no—
420C22⋊C4—302no—
421C42—101no—
422C42—302no—
423C42C2×D3×D511yescharacteristic
424C3×C5#330211yescharacteristic, complemented, Hall
425C22×C3—22no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3C2×D4×D511yescharacteristic
429C2×D3—102no—
430C2×D3—102no—
431C2×D3—102no—
432C3×C4—21no—
433C3×C4—41no—
434C3×C4—201no—
435C3×C4—201no—
436C3×C4#405812yes—
437C3×C4C2×#71311yescharacteristic
438C3×C4#414911yescharacteristic
439Dic3—42no—
440Dic3—102no—
441Dic3—202no—
442C2×C5—42no—
443C2×C5#555011yescharacteristic
444C2×C5C2×#91611yescharacteristic
445C2×C5#577111yescharacteristic
446D5—122no—
447C23—21no—
448C23—152no—
449Q8—101no—
450Q8—101no—
451Q8—122no—
452Q8—302no—
453Q8—302no—
454D4—21no—
455D4—302no—
456D4—604no—
457D4#677312yes—
458C2×C4—21no—
459C2×C4—42no—
460C2×C4—62no—
461C2×C4—101no—
462C2×C4—101no—
463C2×C4—152no—
464C2×C4—302no—
465C2×C4—302no—
466C2×C4—302no—
467C2×C4—604no—
468C2×C4#679011yescharacteristic
469C2×C4#679011yescharacteristic
470C2×C4C22×D3×D511yescharacteristic
471C8—62no—
472C8—101no—
473C8—101no—
474C2×C3—42no—
475C2×C3#6596411yescharacteristic
476C2×C3C2×#405811yescharacteristic
477C2×C3#6611311yescharacteristic
478D3—202no—
479C5#7968011yescharacteristic, complemented, Sylow
480C22—22no—
481C22—41no—
482C22—41no—
483C22—302no—
484C22—302no—
485C22—302no—
486C22C2×#679011yescharacteristic
487C4—21no—
488C4—41no—
489C4—122no—
490C4—201no—
491C4—201no—
492C4—302no—
493C4—602no—
494C4#9149912yes—
495C4C2×#677311yescharacteristic
496C4#9201911yescharacteristic
497C3#1061271211yescharacteristic, complemented, Sylow
498C2—42no—
499C2—602no—
500C2#1175455811yescharacteristic
501C2C2×#9149911yescharacteristic
502C2#1175535011yescharacteristic
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×D4C22
socleC22×C3×C5C22×D4
prime radical#11756314C2
hypercenterC4×D4D3×D5
hypocenterC3×C5#3302
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. #11752486C2semidirect
3. C5×#5007C2semidirect
4. C3×C4×#687C2semidirect
5. #11749063C2semidirect
6. #11748969C2semidirect
7. #11747359C2semidirect
8. #11747215C2semidirect
9. #11747095C2semidirect
10. #11746988C2semidirect
11. #11746930C2semidirect
47. C4×#687D3semidirect
94. #5007D5semidirect
424. C3×C5#3302semidirect
479. C5#79680semidirect
497. C3#10612712semidirect

Maximal subgroups

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

subgroupindex
#117563142
#117524862
C5×#50072
C3×C4×#6872
#117490632
#117489692
#117473592
#117472152
#117470952
#117469882
#117469302
#106127123
#796805