Finite Group Databasebeta

← #14472164

Subgroups — #14472164

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756480C211yesmaximal, characteristic, complemented
2#11752480C211yesmaximal, characteristic
3C5×#5087C211yesmaximal, characteristic
4C3×#65382C211yesmaximal, characteristic, complemented
5#11747387C212yesmaximal
6#11747385C212yesmaximal
7#11747301C212yesmaximal, complemented
8#11747271C212yesmaximal, complemented
9#11747104C211yesmaximal, characteristic, complemented
10C4×#6659C211yesmaximal, characteristic
11#11746851C211yesmaximal, characteristic, complemented
12#10612725—31nomaximal, Hall
13C3×C5×C4⋊Q8C2211yescharacteristic
14Q8×Dic15—21no—
15#91833—21no—
16#91779C2212yes—
17#91778—22no—
18#91776C2212yes—
19C4×Dic30C2211yescharacteristic
20C4×Dic30C2211yescharacteristic
21C4×C5×D12C2211yescharacteristic
22C3×#4015C2211yescharacteristic
23#91370C2212yes—
24#91365—21no—
25#91341—22no—
26#91338—21no—
27#91331C2212yes—
28C2×#6662C2212yes—
29C2×#6659C2211yescharacteristic
30C5×#809C2212yes—
31C5×#807C2212yes—
32C5×C4⋊(C3⋊C8)C2211yescharacteristic
33C3×#3942C2212yes—
34C3×#3942C2212yes—
35C3×C4×#168C2211yescharacteristic
36#90985C2211yescharacteristic
37#90983C2212yes—
38#90975C2211yescharacteristic
39#90971C2212yes—
40#90967C2211yescharacteristic
41#90951C2212yes—
42#81625—51nomaximal, Hall
43#66055—31no—
44#65876—31no—
45C5×#474—31no—
46#65382D311yescharacteristic, complemented
47C4×#687—31no—
48#65166—32no—
49#65162—32no—
50#65138—31no—
51#65119—31no—
52#65057—32no—
53#65056—32no—
54C2×C3×C5×Q8—21no—
55C2×Dic30D411yescharacteristic
56C2×Dic30C2311yescharacteristic
57C2×C5×D12C2311yescharacteristic
58C2×C4×C5×D3—21no—
59C2×C3×Dic10C2312yes—
60C3×C5×C4⋊C4—21no—
61C3×C5×C4⋊C4C2312yes—
62C3×C42×C5C2311yescharacteristic
63C4⋊Dic15—22no—
64C4⋊Dic15D411yescharacteristic
65C4⋊Dic15C2311yescharacteristic
66#6712—21no—
67#6712—22no—
68#6712—22no—
69#6712D412yes—
70C4×Dic15—21no—
71C4×Dic15—22no—
72C4×Dic15C2312yes—
73C5×(C2×D3)⋊C4—21no—
74C5×C4⋊Dic3C2311yescharacteristic
75C2×C5×C3⋊C8C2312yes—
76C3×C4⋊Dic5—21no—
77C3×C4⋊Dic5—21no—
78C2×C3×#168C2311yescharacteristic
79C2×C3×#168C2311yescharacteristic
80#6668—22no—
81(C2×D3)⋊Dic5—22no—
82(C2×D3)⋊Dic5—22no—
83Dic3×Dic5—22no—
84#6662—24no—
85#6659—22no—
86#5732—51no—
87#5721—51no—
88C3×#487—51no—
89#5090—51no—
90#5087D511yescharacteristic
91#4800—52no—
92#4799—52no—
93#4798—52no—
94#4790—52no—
95C4×#204—51no—
96C8×Dic6—51no—
97C5×C4⋊Q8C2×D311yescharacteristic
98C4×C5×D4—31no—
99Q8×Dic5—61no—
100#4090—61no—
101#4082—32no—
102C2×#687—31no—
103#4072—61no—
104#4070—61no—
105C2×Dic20—32no—
106#4036—32no—
107#4035—62no—
108#4033—32no—
109#4033—32no—
110#4025—62no—
111#4015C2×D311yescharacteristic
112C4×Dic10—31no—
113C4×Dic10—31no—
114C5×#107—32no—
115C5×C4⋊C8—31no—
116C5×#102—32no—
117#3964—31no—
118#3948—32no—
119#3946—32no—
120C8×Dic5—32no—
121#3942—31no—
122#3942C2×D312yes—
123#3942C2×D312yes—
124#3940—31no—
125C4×#168C2×D311yescharacteristic
126#3364—151noSylow
127C22×C5×D3—21no—
128C3×C5×Q8—42no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5C2×D411yescharacteristic
132C2×C3×C4×C5C2411yescharacteristic
133C2×Dic15—21no—
134C2×Dic15—21no—
135C2×Dic15C2×D412yes—
136C2×Dic15C2×D412yes—
137Dic30—22no—
138Dic30—21no—
139Dic30#4112yes—
140C2×C5×Dic3—21no—
141C5×D12—21no—
142C5×D12—21no—
143C4×C5×D3—42no—
144C2×C3×Dic5—22no—
145C3×Dic10—22no—
146C3×Dic10—22no—
147C3×#168—21no—
148C3×#168—21no—
149C5×C3⋊C8—22no—
150C3×C4⋊Q8—51no—
151C3×C4⋊Q8C2×D511yescharacteristic
152#947—52no—
153Q8×Dic3—52no—
154Q8×Dic3—101no—
155#944—101no—
156C2×#216—52no—
157C2×#204—51no—
158#898—102no—
159#897—101no—
160#892—101no—
161#890—52no—
162#889—102no—
163#887—52no—
164C4×D12C2×D511yescharacteristic
165C4×Dic6—51no—
166C4×Dic6—51no—
167C3×#103—52no—
168C3×#103—52no—
169C3×C4×C8—51no—
170#835—52no—
171#821—51no—
172#817—51no—
173#816—51no—
174Dic3⋊C8—52no—
175C8×Dic3—52no—
176#809C2×D512yes—
177#807C2×D512yes—
178C4⋊(C3⋊C8)C2×D511yescharacteristic
179C2×C5×Q8—21no—
180C2×C5×D4—31no—
181C22×C4×C5—61no—
182C2×Dic10—31no—
183C2×Dic10—31no—
184C2×Dic10C22×D312yes—
185C2×C5×C8—32no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—31no—
188C5×C4⋊C4C22×D312yes—
189C5×C22⋊C4—61no—
190C42×C5C22×D311yescharacteristic
191C22⋊Dic5—62no—
192C22⋊Dic5—62no—
193#687—62no—
194C4⋊Dic5—21no—
195C4⋊Dic5—21no—
196C4⋊Dic5—31no—
197C4⋊Dic5—31no—
198C4⋊Dic5—62no—
199#683—32no—
200#683—62no—
201#683—62no—
202#683—62no—
203#683—61no—
204C4×Dic5—32no—
205C4×Dic5—62no—
206C4×Dic5—62no—
207C4×Dic5—61no—
208C2×#168C22×D311yescharacteristic
209C2×#168C22×D311yescharacteristic
210Dic20—64no—
211#557—151no—
212#556—151no—
213#488—152no—
214#487—51no—
215#479—152no—
216#474—31no—
217#464—152no—
218#462—151no—
219C8×Q8—151no—
220C4×Q16—152no—
221C4×SD16—151no—
222C22×C3×C5C22×D411yescharacteristic
223C2×C5×D3—41no—
224C2×C5×D3—41no—
225C2×C5×D3—41no—
226C3×C4×C5—21no—
227C3×C4×C5—42no—
228C3×C4×C5#13512yes—
229C3×C4×C5C2×#4111yescharacteristic
230C3×C4×C52−1+411yescharacteristic
231Dic15—22no—
232Dic15—22no—
233Dic15—22no—
234Dic15—41no—
235C3×Dic5—42no—
236C5×Dic3—41no—
237C2×C3×Q8—21no—
238C2×C3×Q8—52no—
239C2×D12C22×D511yescharacteristic
240C2×C4×D3—21no—
241C2×Dic6—51no—
242C2×Dic6—51no—
243C2×C3×C8—51no—
244C2×C3×C8—51no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—101no—
247C3×C4⋊C4—101no—
248C3×C4⋊C4C22×D512yes—
249C3×C42C22×D511yescharacteristic
250#216—104no—
251(C2×D3)⋊C4—21no—
252(C2×D3)⋊C4—102no—
253(C2×D3)⋊C4—102no—
254C4⋊Dic3—51no—
255C4⋊Dic3—51no—
256C4⋊Dic3—102no—
257C4⋊Dic3—102no—
258C4⋊Dic3C22×D511yescharacteristic
259Dic3⋊C4—52no—
260Dic3⋊C4—102no—
261Dic3⋊C4—102no—
262Dic3⋊C4—101no—
263C4×Dic3—52no—
264C4×Dic3—102no—
265C4×Dic3—102no—
266C4×Dic3—101no—
267C2×C3⋊C8C22×D512yes—
268#204—102no—
269C23×C5—61no—
270C5×Q8—42no—
271C5×D4—61no—
272C5×D4—61no—
273C2×C4×C5—22no—
274C2×C4×C5—61no—
275C2×C4×C5—122no—
276C2×C4×C5D3×D411yescharacteristic
277C2×C4×C5D3×D411yescharacteristic
278C2×C4×C5C23×D311yescharacteristic
279C2×Dic5—22no—
280C2×Dic5—32no—
281C2×Dic5—32no—
282C2×Dic5—61no—
283C2×Dic5—61no—
284Dic10—22no—
285Dic10—22no—
286Dic10—32no—
287Dic10—62no—
288Dic10—61no—
289C5×C8—62no—
290#168—21no—
291#168—21no—
292C2×Q16—152no—
293C2×SD16—151no—
294C4⋊Q8—51no—
295C4⋊Q8—301no—
296C4⋊Q8D3×D511yescharacteristic
297#126—302no—
298#125—152no—
299#125—302no—
300#124—152no—
301C22⋊Q8—301no—
302C4×Q8—151no—
303C4×Q8—152no—
304C4×Q8—152no—
305C4×Q8—151no—
306C4×Q8—301no—
307C4×D4—31no—
308#117—301no—
309#107—32no—
310#106—151no—
311C4⋊C8—31no—
312C4⋊C8—152no—
313#103—52no—
314#103—52no—
315#103—151no—
316#103—152no—
317#102—32no—
318#102—151no—
319C4×C8—51no—
320C4×C8—152no—
321C2×C3×C5#54811yescharacteristic
322C2×C3×C5C2×#13511yescharacteristic
323C2×C3×C5#57811yescharacteristic
324C5×D3—81no—
325C22×D3—21no—
326C3×Q8—42no—
327C3×Q8—102no—
328C3×Q8—102no—
329C2×C3×C4—22no—
330C2×C3×C4—102no—
331C2×C3×C4D4×D511yescharacteristic
332C2×C3×C4D4×D511yescharacteristic
333C2×C3×C4C23×D511yescharacteristic
334C2×Dic3—21no—
335C2×Dic3—52no—
336C2×Dic3—52no—
337C2×Dic3—101no—
338C2×Dic3—101no—
339D12—21no—
340D12—21no—
341C4×D3—42no—
342Dic6—52no—
343Dic6—101no—
344Dic6—102no—
345C3×C8—101no—
346C3×C8—101no—
347C3⋊C8—22no—
348C22×C5—121no—
349C22×C5—121no—
350C22×C5—121no—
351C22×C5C2×D3×D411yescharacteristic
352C4×C5—21no—
353C4×C5—42no—
354C4×C5—121no—
355C4×C5#91612yes—
356C4×C5D3×#4111yescharacteristic
357C4×C5#101011yescharacteristic
358Dic5—42no—
359Dic5—62no—
360Dic5—62no—
361Dic5—62no—
362Dic5—121no—
363C2×Q8—21no—
364C2×Q8—52no—
365C2×Q8—151no—
366C2×Q8—151no—
367C2×D4—31no—
368C22×C4—61no—
369Q16—304no—
370SD16—302no—
371C2×C8—32no—
372C2×C8—51no—
373C2×C8—51no—
374C4⋊C4—21no—
375C4⋊C4—31no—
376C4⋊C4—101no—
377C4⋊C4—101no—
378C4⋊C4—151no—
379C4⋊C4—152no—
380C4⋊C4—151no—
381C4⋊C4—302no—
382C4⋊C4—302no—
383C4⋊C4—302no—
384C4⋊C4—302no—
385C4⋊C4—301no—
386C4⋊C4C2×D3×D512yes—
387C22⋊C4—61no—
388C22⋊C4—302no—
389C22⋊C4—302no—
390C42—152no—
391C42—302no—
392C42—302no—
393C42—301no—
394C42C2×D3×D511yescharacteristic
395C3×C5#336411yescharacteristic, complemented, Hall
396C22×C3C2×D4×D511yescharacteristic
397C2×D3—41no—
398C2×D3—41no—
399C2×D3—41no—
400C3×C4—21no—
401C3×C4—42no—
402C3×C4—202no—
403C3×C4#405812yes—
404C3×C4#414711yescharacteristic
405C3×C4D5×#4111yescharacteristic
406Dic3—41no—
407Dic3—102no—
408Dic3—102no—
409Dic3—102no—
410Dic3—201no—
411C2×C5—241no—
412C2×C5#567711yescharacteristic
413C2×C5C2×#91611yescharacteristic
414C2×C5#576611yescharacteristic
415C23—61no—
416Q8—42no—
417Q8—102no—
418Q8—102no—
419Q8—152no—
420Q8—302no—
421Q8—301no—
422D4—61no—
423D4—61no—
424C2×C4—22no—
425C2×C4—61no—
426C2×C4—102no—
427C2×C4—122no—
428C2×C4—152no—
429C2×C4—152no—
430C2×C4—301no—
431C2×C4—301no—
432C2×C4#679011yescharacteristic
433C2×C4#679011yescharacteristic
434C2×C4C22×D3×D511yescharacteristic
435C8—62no—
436C8—101no—
437C8—101no—
438C2×C3#6596711yescharacteristic
439C2×C3C2×#405811yescharacteristic
440C2×C3#6611311yescharacteristic
441D3—81no—
442C5#8162511yescharacteristic, complemented, Sylow
443C22—121no—
444C22—121no—
445C22—121no—
446C22C2×#679011yescharacteristic
447C4—21no—
448C4—42no—
449C4—121no—
450C4—202no—
451C4—302no—
452C4—302no—
453C4—302no—
454C4—601no—
455C4#9149912yes—
456C4#9202811yescharacteristic
457C4#9203411yescharacteristic
458C3#1061272511yescharacteristic, complemented, Sylow
459C2—241no—
460C2C2×#9149911yescharacteristic
461C2#1175512211yescharacteristic
462C2#1175516611yescharacteristic
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5C22×D4
prime radicalC2×C5×D12C23
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#3364
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #11756480C2semidirect
4. C3×#65382C2semidirect
7. #11747301C2semidirect
8. #11747271C2semidirect
9. #11747104C2semidirect
11. #11746851C2semidirect
46. #65382D3semidirect
395. C3×C5#3364semidirect
442. C5#81625semidirect
458. C3#10612725semidirect

Maximal subgroups

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

subgroupindex
#117564802
#117524802
C5×#50872
C3×#653822
#117473872
#117473852
#117473012
#117472712
#117471042
C4×#66592
#117468512
#106127253
#816255