Finite Group Databasebeta

← #14460006

Subgroups — #14460006

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
1C5×#5555C211yesmaximal, characteristic
2#11752491C211yesmaximal, characteristic, complemented
3#11751108C211yesmaximal, characteristic, complemented
4C3×C4×#687C211yesmaximal, characteristic, complemented
5#11749059C212yesmaximal, complemented
6#11748965C212yesmaximal, complemented
7#11747353C212yesmaximal, complemented
8#11747211C212yesmaximal, complemented
9#11747094C211yesmaximal, characteristic, complemented
10D12×#168C211yesmaximal, characteristic, complemented
11#11746929C211yesmaximal, characteristic, complemented
12#10615241—31nomaximal, Hall
13C2×C5×#237C2212yes—
14C3×C4×C5×D4C2211yescharacteristic
15#91758C2211yescharacteristic
16C5×#930—21no—
17C2×C5×C4⋊Dic3—21no—
18C5×#897C2212yes—
19C5×#886—22no—
20C4×C5×D12C2211yescharacteristic
21C5×#869C2211yescharacteristic
22C2×C3×#687C2211yescharacteristic
23C3×C4×Dic10C2211yescharacteristic
24#91373—21no—
25#91369C2212yes—
26#91342—22no—
27#91338C2212yes—
28#91333—21no—
29C2×#6661C2212yes—
30C2×D3×#168C2212yes—
31#91116C2212yes—
32#91091C2212yes—
33C4⋊(C3⋊#168)C2211yescharacteristic
34C3×#3964C2211yescharacteristic
35C3×#3942C2211yescharacteristic
36C3×#3940C2211yescharacteristic
37C3×C4×#168C2211yescharacteristic
38#90973C2212yes—
39#90973C2212yes—
40#90972C2212yes—
41(C2×D3)⋊#168C2212yes—
42#79680—51nomaximal, Hall
43C5×#547—31no—
44#65876—31no—
45#65528—32no—
46#65522—32no—
47#65379—31no—
48#65370—31no—
49#65340—32no—
50#65334—32no—
51C4×#687D311yescharacteristic, complemented
52D4×#168—31no—
53#65296—31no—
54C2×C3×C5×D4C2311yescharacteristic
55C22×C3×C4×C5—21no—
56C2×Dic30—21no—
57C2×C5×#67—22no—
58C22×C5×Dic3—22no—
59C5×#237—24no—
60C5×#237—42no—
61C2×C5×D12D411yescharacteristic
62C2×C4×C5×D3C2312yes—
63C2×C5×Dic6C2312yes—
64C2×C3×Dic10C2311yescharacteristic
65C3×C5×C4⋊C4C2311yescharacteristic
66C3×C5×C22⋊C4—21no—
67C3×C42×C5C2311yescharacteristic
68C4⋊Dic15—21no—
69C4⋊Dic15C2312yes—
70C2×C3⋊#168C2312yes—
71C5×(C2×D3)⋊C4—22no—
72C5×(C2×D3)⋊C4D412yes—
73C5×C4⋊Dic3—21no—
74C5×C4⋊Dic3—21no—
75C5×C4⋊Dic3—22no—
76C5×C4⋊Dic3D411yescharacteristic
77C3×#687—22no—
78C3×C4⋊Dic5C2311yescharacteristic
79C3×#683—21no—
80C3×C4×Dic5—21no—
81C2×C3×#168C2311yescharacteristic
82C2×C3×#168C2311yescharacteristic
83#6669—22no—
84(C2×D3)⋊Dic5—22no—
85(C2×D3)⋊Dic5—22no—
86Dic3×Dic5—22no—
87#6661—42no—
88D3×#168—42no—
89#5584—51no—
90#5555D511yescharacteristic
91C3×C4×SD16—51no—
92#5173—52no—
93#5031—51no—
94#5007—51no—
95#4868—52no—
96#4813—52no—
97#4772—52no—
98C8×D12—51no—
99#4681—51no—
100C2×C5×#41—32no—
101C5×C4⋊Q8—31no—
102C5×#123—62no—
103C5×C22⋊Q8—32no—
104C5×#121—61no—
105C4×C5×D4—31no—
106C4×C5×D4C2×D311yescharacteristic
107C2×C5×C4⋊C4—61no—
108C2×#689—32no—
109#4082—61no—
110C2×#687C2×D311yescharacteristic
111#4072—32no—
112#4070—32no—
113C22×#168—32no—
114#4033—61no—
115#4025—62no—
116#4015—31no—
117C4×Dic10C2×D311yescharacteristic
118#3967—32no—
119#3967—32no—
120#3967—32no—
121#3964—32no—
122#3964C2×D311yescharacteristic
123C22⋊#168—32no—
124#3942C2×D311yescharacteristic
125#3940C2×D311yescharacteristic
126#3939—32no—
127C4⋊#168—31no—
128C4×#168C2×D311yescharacteristic
129#3302—151noSylow
130C23×C3×C5—21no—
131C22×C5×D3C2×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
139C2×Dic15—22no—
140Dic30—42no—
141C5×#67—44no—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—44no—
144C2×C5×Dic3C2×D412yes—
145C5×D12—22no—
146C4×C5×D3—22no—
147C4×C5×D3—22no—
148C5×Dic6—22no—
149C5×Dic6—22no—
150C2×C3×Dic5—21no—
151C2×C3×Dic5—21no—
152C3×Dic10—21no—
153C3×Dic10—21no—
154C3⋊#168—22no—
155C3×#168—21no—
156C3×#168—21no—
157C2×#237C2×D512yes—
158C2×C3×SD16—51no—
159C3×C4×Q8—51no—
160C3×C4×D4C2×D511yescharacteristic
161#946—52no—
162C2×#216—52no—
163#930—21no—
164C2×C4⋊Dic3—21no—
165C2×D3×C8—52no—
166#898—102no—
167#897C2×D512yes—
168#892—52no—
169#886—22no—
170#875—101no—
171C4×D12C2×D511yescharacteristic
172#869—51no—
173#869C2×D511yescharacteristic
174C4×Dic6—101no—
175C3×#106—51no—
176C3×#103—51no—
177C3×#102—51no—
178C3×C4×C8—51no—
179#832—52no—
180(C2×D3)⋊C8—52no—
181#816—52no—
182#816—52no—
183#810—52no—
184#807—52no—
185C4⋊(C3⋊C8)—51no—
186C5×#41—64no—
187C5×#41—122no—
188C2×C5×Q8—32no—
189C2×C5×D4—31no—
190C2×C5×D4—62no—
191C2×C5×D4C22×D311yescharacteristic
192C22×C4×C5—21no—
193C22×C4×C5—32no—
194C22×C4×C5—62no—
195C2×Dic10—61no—
196C2×Dic10C22×D311yescharacteristic
197C5×C4⋊C4—31no—
198C5×C4⋊C4—62no—
199C5×C4⋊C4—61no—
200C5×C4⋊C4—61no—
201C5×C4⋊C4C22×D311yescharacteristic
202C5×C22⋊C4—21no—
203C5×C22⋊C4—32no—
204C5×C22⋊C4—62no—
205C42×C5C22×D311yescharacteristic
206C22⋊Dic5—62no—
207C22⋊Dic5—62no—
208#689—122no—
209#687—22no—
210C4⋊Dic5—32no—
211C4⋊Dic5—61no—
212C4⋊Dic5C22×D311yescharacteristic
213#683—21no—
214#683—62no—
215C4×Dic5—21no—
216C4×Dic5—62no—
217C2×#168—32no—
218C2×#168—122no—
219C2×#168C22×D311yescharacteristic
220C2×#168C22×D311yescharacteristic
221#556—151no—
222#547—31no—
223#487—151no—
224#485—152no—
225#474—151no—
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×C5×D3—22no—
237C2×C5×D3—22no—
238C2×C5×D3—22no—
239C3×C4×C5—21no—
240C3×C4×C5—41no—
241C3×C4×C5#13512yes—
242C3×C4×C5C2×#4111yescharacteristic
243C3×C4×C5H(5,𝔽2)11yescharacteristic
244Dic15—42no—
245C3×Dic5—41no—
246C3×Dic5—41no—
247C5×Dic3—22no—
248C5×Dic3—42no—
249C2×C3×Q8—51no—
250C2×C3×D4C22×D511yescharacteristic
251C22×C3×C4—21no—
252C2×#67—22no—
253C22×Dic3—22no—
254#237—24no—
255#237—42no—
256C2×D12C22⋊D511yescharacteristic
257C2×C4×D3C22×D512yes—
258C2×Dic6—101no—
259C2×Dic6C22×D512yes—
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—22no—
271(C2×D3)⋊C4—102no—
272(C2×D3)⋊C4—102no—
273(C2×D3)⋊C4C22⋊D512yes—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—22no—
277C4⋊Dic3—52no—
278C4⋊Dic3—101no—
279C4⋊Dic3C22⋊D511yescharacteristic
280Dic3⋊C4—102no—
281C4×Dic3—102no—
282C2×C3⋊C8—52no—
283D3×C8—202no—
284C23×C5—21no—
285C23×C5—32no—
286C5×Q8—62no—
287C5×Q8—62no—
288C5×D4—21no—
289C5×D4—62no—
290C5×D4—124no—
291C5×D4#23512yes—
292C2×C4×C5—21no—
293C2×C4×C5—32no—
294C2×C4×C5—42no—
295C2×C4×C5—62no—
296C2×C4×C5—62no—
297C2×C4×C5—62no—
298C2×C4×C5—124no—
299C2×C4×C5D3×D411yescharacteristic
300C2×C4×C5D3×D411yescharacteristic
301C2×C4×C5C23×D311yescharacteristic
302C2×Dic5—21no—
303C2×Dic5—21no—
304C2×Dic5—62no—
305Dic10—21no—
306Dic10—21no—
307Dic10—122no—
308#168—21no—
309#168—21no—
310#168—62no—
311C2×#41—32no—
312C2×Q16—152no—
313C2×SD16—51no—
314C22×C8—152no—
315C4⋊Q8—31no—
316C4⋊Q8—151no—
317#126—302no—
318#124—301no—
319#123—62no—
320C22⋊Q8—32no—
321C22⋊Q8—152no—
322#121—61no—
323C4×Q8—51no—
324C4×Q8—301no—
325C4×D4—31no—
326C4×D4D3×D511yescharacteristic
327#117—152no—
328C2×C4⋊C4—61no—
329#107—152no—
330#106—51no—
331C4⋊C8—151no—
332#103—51no—
333#103—152no—
334#103—152no—
335#103—152no—
336#102—51no—
337#102—152no—
338C22⋊C8—152no—
339C4×C8—51no—
340C2×C3×C5—42no—
341C2×C3×C5#54611yescharacteristic
342C2×C3×C5C2×#13511yescharacteristic
343C2×C3×C5#57811yescharacteristic
344C5×D3—42no—
345C23×C3—21no—
346C22×D3C2×C22⋊D512yes—
347C3×Q8—101no—
348C3×Q8—101no—
349C3×D4—21no—
350C3×D4#71112yes—
351C2×C3×C4—21no—
352C2×C3×C4—42no—
353C2×C3×C4—101no—
354C2×C3×C4—101no—
355C2×C3×C4C2×C22⋊D511yescharacteristic
356C2×C3×C4C2×C22⋊D511yescharacteristic
357C2×C3×C4C23×D511yescharacteristic
358#67—44no—
359C2×Dic3—22no—
360C2×Dic3—44no—
361C2×Dic3—102no—
362C2×Dic3C2×C22⋊D512yes—
363D12—22no—
364C4×D3—22no—
365C4×D3—22no—
366Dic6—22no—
367Dic6—22no—
368Dic6—202no—
369C3×C8—101no—
370C3×C8—101no—
371C3⋊C8—102no—
372C22×C5—22no—
373C22×C5—41no—
374C22×C5—41no—
375C22×C5—62no—
376C22×C5—62no—
377C22×C5—62no—
378C22×C5C2×D3×D411yescharacteristic
379C4×C5—21no—
380C4×C5—41no—
381C4×C5—62no—
382C4×C5—122no—
383C4×C5#91612yes—
384C4×C5C2×#23511yescharacteristic
385C4×C5#100911yescharacteristic
386Dic5—41no—
387Dic5—41no—
388Dic5—122no—
389#41—64no—
390#41—122no—
391C2×Q8—32no—
392C2×Q8—51no—
393C2×Q8—301no—
394C2×D4—31no—
395C2×D4—62no—
396C2×D4C2×D3×D511yescharacteristic
397C22×C4—21no—
398C22×C4—32no—
399C22×C4—62no—
400Q16—602no—
401SD16—102no—
402C2×C8—51no—
403C2×C8—51no—
404C2×C8—152no—
405C2×C8—602no—
406C4⋊C4—31no—
407C4⋊C4—51no—
408C4⋊C4—62no—
409C4⋊C4—61no—
410C4⋊C4—61no—
411C4⋊C4—101no—
412C4⋊C4—152no—
413C4⋊C4—302no—
414C4⋊C4—301no—
415C4⋊C4C2×D3×D511yescharacteristic
416C22⋊C4—21no—
417C22⋊C4—32no—
418C22⋊C4—62no—
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×C3C22×C22⋊D511yescharacteristic
429C2×D3—22no—
430C2×D3—22no—
431C2×D3—22no—
432C3×C4—21no—
433C3×C4—41no—
434C3×C4—201no—
435C3×C4—201no—
436C3×C4#409612yes—
437C3×C4C2×#71111yescharacteristic
438C3×C4#414411yescharacteristic
439Dic3—22no—
440Dic3—42no—
441Dic3—202no—
442C2×C5—42no—
443C2×C5—122no—
444C2×C5#555011yescharacteristic
445C2×C5C2×#91611yescharacteristic
446C2×C5#577111yescharacteristic
447C23—21no—
448C23—32no—
449Q8—62no—
450Q8—62no—
451Q8—101no—
452Q8—101no—
453Q8—602no—
454D4—21no—
455D4—62no—
456D4—124no—
457D4#677212yes—
458C2×C4—21no—
459C2×C4—32no—
460C2×C4—42no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—101no—
465C2×C4—101no—
466C2×C4—124no—
467C2×C4—302no—
468C2×C4D3×C22⋊D511yescharacteristic
469C2×C4D3×C22⋊D511yescharacteristic
470C2×C4C22×D3×D511yescharacteristic
471C8—101no—
472C8—101no—
473C8—302no—
474C2×C3—42no—
475C2×C3#6614011yescharacteristic
476C2×C3C2×#409611yescharacteristic
477C2×C3#6617211yescharacteristic
478D3—42no—
479C5#7968011yescharacteristic, complemented, Sylow
480C22—22no—
481C22—41no—
482C22—41no—
483C22—62no—
484C22—62no—
485C22—62no—
486C22C2×D3×C22⋊D511yescharacteristic
487C4—21no—
488C4—41no—
489C4—62no—
490C4—122no—
491C4—201no—
492C4—201no—
493C4—602no—
494C4#9149512yes—
495C4C2×#677211yescharacteristic
496C4#9201811yescharacteristic
497C3#1061524111yescharacteristic, complemented, Sylow
498C2—42no—
499C2—122no—
500C2#1175454011yescharacteristic
501C2C2×#9149511yescharacteristic
502C2#1175534611yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×D4C22
socleC22×C3×C5C22×D4
prime radicalC5×#5555C2
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. #11752491C2semidirect
3. #11751108C2semidirect
4. C3×C4×#687C2semidirect
5. #11749059C2semidirect
6. #11748965C2semidirect
7. #11747353C2semidirect
8. #11747211C2semidirect
9. #11747094C2semidirect
10. D12×#168C2semidirect
11. #11746929C2semidirect
51. C4×#687D3semidirect
424. C3×C5#3302semidirect
479. C5#79680semidirect
497. C3#10615241semidirect

Maximal subgroups

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

subgroupindex
C5×#55552
#117524912
#117511082
C3×C4×#6872
#117490592
#117489652
#117473532
#117472112
#117470942
D12×#1682
#117469292
#106152413
#796805