Finite Group Databasebeta

← #14472136

Subgroups — #14472136

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#11756301C211yesmaximal, characteristic
2#11752482C211yesmaximal, characteristic, complemented
3C4×C5×#214C211yesmaximal, characteristic
4C3×#65296C211yesmaximal, characteristic
5#11749015C212yesmaximal
6#11748944C212yesmaximal
7#11747278C212yesmaximal, complemented
8#11747184C212yesmaximal, complemented
9#11747101C211yesmaximal, characteristic, complemented
10#11746921C211yesmaximal, characteristic, complemented
11#11746855C211yesmaximal, characteristic, complemented
12#10611354—31nomaximal, Hall
13C3×C4×C5×D4C2211yescharacteristic
14#91825C2212yes—
15#91815—21no—
16#91813—21no—
17#91776C2212yes—
18#91768—22no—
19#91758C2211yescharacteristic
20C4×Dic30C2211yescharacteristic
21C2×C5×#214C2211yescharacteristic
22C4×C5×Dic6C2211yescharacteristic
23C3×#4015C2211yescharacteristic
24#91389—21no—
25#91384C2212yes—
26#91346—22no—
27#91344C2212yes—
28Dic3×Dic10—21no—
29C2×#6663C2212yes—
30C5×#832C2211yescharacteristic
31C5×#810C2211yescharacteristic
32C5×#808C2211yescharacteristic
33C4×C5×C3⋊C8C2211yescharacteristic
34C3×#3964C2212yes—
35C3×#3939C2212yes—
36C3×C4⋊#168C2211yescharacteristic
37#90983C2212yes—
38#90975C2212yes—
39#90974C2212yes—
40#90974C2212yes—
41#90951C2212yes—
42#80994—51nomaximal, Hall
43#65904—31no—
44#65876—31no—
45C4×C5×SD16—31no—
46#65455—32no—
47#65320—31no—
48#65296D311yescharacteristic
49#65146—32no—
50#65079—32no—
51#65057—32no—
52#64970—31no—
53C8×Dic10—31no—
54C2×C3×C5×D4C2311yescharacteristic
55C22×C3×C4×C5—21no—
56C2×Dic30D411yescharacteristic
57C2×Dic30C2312yes—
58C2×C5×Dic6C2311yescharacteristic
59C2×C3×Dic10—21no—
60C3×C5×C4⋊C4C2311yescharacteristic
61C3×C5×C22⋊C4—21no—
62C3×C42×C5C2311yescharacteristic
63#6719—22no—
64#6719—22no—
65C4⋊Dic15—21no—
66C4⋊Dic15—21no—
67C4⋊Dic15D411yescharacteristic
68#6712—22no—
69#6712D412yes—
70C4×Dic15—22no—
71C4×Dic15C2312yes—
72C5×#214—22no—
73C5×C4⋊Dic3C2311yescharacteristic
74C5×Dic3⋊C4—21no—
75C4×C5×Dic3—21no—
76C2×C5×C3⋊C8C2311yescharacteristic
77C2×C5×C3⋊C8C2311yescharacteristic
78C3×C4⋊Dic5—21no—
79C3×C4⋊Dic5C2312yes—
80C2×C3×#168C2312yes—
81Dic3⋊Dic5—22no—
82Dic3⋊Dic5—22no—
83#6668—22no—
84Dic3×Dic5—22no—
85#6663—24no—
86#5721—51no—
87#5542—51no—
88C3×#474—51no—
89#5093—51no—
90C4×#214D511yescharacteristic
91#4877—52no—
92#4873—52no—
93#4849—51no—
94#4830—51no—
95#4768—52no—
96#4767—52no—
97C2×C5×SD16—31no—
98C4×C5×Q8—31no—
99C4×C5×D4C2×D311yescharacteristic
100Q8×Dic5—32no—
101C2×#689—32no—
102#4082—32no—
103#4072—61no—
104#4070—61no—
105#4036—32no—
106#4036—62no—
107#4033—32no—
108#4025—62no—
109#4015—31no—
110#4015—61no—
111#4015C2×D311yescharacteristic
112C4×Dic10—31no—
113C4×Dic10—61no—
114C5×#106—31no—
115C5×#103—31no—
116C5×#102—31no—
117C4×C5×C8—31no—
118#3964C2×D312yes—
119#3948—32no—
120#3948—32no—
121#3946—32no—
122C8×Dic5—32no—
123#3942—32no—
124#3939C2×D312yes—
125C4⋊#168C2×D311yescharacteristic
126#3364—151noSylow
127C23×C3×C5—21no—
128C3×C5×D4—21no—
129C3×C5×D4—21no—
130C2×C3×C4×C5—21no—
131C2×C3×C4×C5—42no—
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5C2×D411yescharacteristic
134C2×C3×C4×C5C2411yescharacteristic
135C2×Dic15—22no—
136C2×Dic15C2×D412yes—
137C2×Dic15C2×D412yes—
138Dic30—22no—
139Dic30—22no—
140Dic30—22no—
141C2×C5×Dic3—21no—
142C2×C5×Dic3—21no—
143C5×Dic6—21no—
144C5×Dic6#4112yes—
145C2×C3×Dic5—22no—
146C3×Dic10—42no—
147C3×#168—22no—
148C5×C3⋊C8—21no—
149C5×C3⋊C8—21no—
150C3×C4⋊Q8—51no—
151C3×C4×D4C2×D511yescharacteristic
152Q8×Dic3—101no—
153#944—101no—
154#936—52no—
155C2×#214C2×D511yescharacteristic
156#926—101no—
157#924—101no—
158C2×Dic12—52no—
159#890—52no—
160#889—102no—
161#887—52no—
162#887—52no—
163#879—102no—
164#869—51no—
165C4×Dic6—51no—
166C4×Dic6C2×D511yescharacteristic
167C3×#107—52no—
168C3×C4⋊C8—51no—
169C3×#102—52no—
170#832C2×D511yescharacteristic
171#816—52no—
172Dic3⋊C8—52no—
173C8×Dic3—52no—
174#810—52no—
175#810—52no—
176#810C2×D511yescharacteristic
177#808C2×D511yescharacteristic
178C4×C3⋊C8C2×D511yescharacteristic
179C2×C5×Q8—31no—
180C2×C5×D4C22×D311yescharacteristic
181C22×C4×C5—21no—
182C2×Dic10—21no—
183C2×Dic10—31no—
184C2×Dic10—32no—
185C5×SD16—62no—
186C2×C5×C8—31no—
187C2×C5×C8—31no—
188C5×C4⋊C4—31no—
189C5×C4⋊C4—61no—
190C5×C4⋊C4C22×D311yescharacteristic
191C5×C22⋊C4—21no—
192C42×C5—61no—
193C42×C5C22×D311yescharacteristic
194C22⋊Dic5—62no—
195C22⋊Dic5—62no—
196#689—64no—
197C4⋊Dic5—21no—
198C4⋊Dic5—31no—
199C4⋊Dic5—62no—
200C4⋊Dic5—62no—
201C4⋊Dic5—61no—
202C4⋊Dic5—61no—
203C4⋊Dic5C22×D312yes—
204#683—32no—
205#683—62no—
206#683—62no—
207C4×Dic5—32no—
208C4×Dic5—62no—
209C4×Dic5—62no—
210C2×#168C22×D312yes—
211#557—151no—
212#556—151no—
213#488—152no—
214#487—151no—
215#479—152no—
216#474—51no—
217#464—152no—
218#462—151no—
219C8×Q8—151no—
220C4×Q16—152no—
221C4×SD16—31no—
222C22×C3×C5—41no—
223C22×C3×C5—41no—
224C22×C3×C5—41no—
225C22×C3×C5C22×D411yescharacteristic
226C3×C4×C5—21no—
227C3×C4×C5—41no—
228C3×C4×C5#13512yes—
229C3×C4×C5C2×#4111yescharacteristic
230C3×C4×C52−1+411yescharacteristic
231Dic15—22no—
232Dic15—22no—
233Dic15—42no—
234C3×Dic5—42no—
235C5×Dic3—22no—
236C5×Dic3—41no—
237C2×C3×Q8—101no—
238C2×C3×D4C22×D511yescharacteristic
239C22×C3×C4—21no—
240C2×Dic6—51no—
241C2×Dic6—52no—
242C2×Dic6C22×D511yescharacteristic
243C2×C3×C8—52no—
244C3×C4⋊C4—52no—
245C3×C4⋊C4—101no—
246C3×C4⋊C4C22×D511yescharacteristic
247C3×C22⋊C4—21no—
248C3×C42C22×D511yescharacteristic
249#217—102no—
250#217—102no—
251#214—22no—
252C4⋊Dic3—51no—
253C4⋊Dic3—102no—
254C4⋊Dic3—101no—
255C4⋊Dic3—101no—
256C4⋊Dic3C22×D511yescharacteristic
257Dic3⋊C4—21no—
258Dic3⋊C4—52no—
259Dic3⋊C4—102no—
260Dic3⋊C4—102no—
261Dic3⋊C4—102no—
262C4×Dic3—21no—
263C4×Dic3—52no—
264C4×Dic3—102no—
265C4×Dic3—102no—
266C2×C3⋊C8C22×D511yescharacteristic
267C2×C3⋊C8C22×D511yescharacteristic
268Dic12—104no—
269C23×C5—21no—
270C5×Q8—32no—
271C5×Q8—61no—
272C5×D4—21no—
273C5×D4—21no—
274C2×C4×C5—21no—
275C2×C4×C5—42no—
276C2×C4×C5—61no—
277C2×C4×C5—61no—
278C2×C4×C5D3×D411yescharacteristic
279C2×C4×C5D3×D411yescharacteristic
280C2×C4×C5C23×D311yescharacteristic
281C2×Dic5—22no—
282C2×Dic5—32no—
283C2×Dic5—32no—
284C2×Dic5—62no—
285Dic10—42no—
286Dic10—62no—
287Dic10—62no—
288Dic10—62no—
289C5×C8—61no—
290C5×C8—61no—
291#168—22no—
292C2×Q16—152no—
293C2×SD16—31no—
294C4⋊Q8—51no—
295C4⋊Q8—151no—
296C4⋊Q8—301no—
297#126—302no—
298#125—152no—
299#125—302no—
300#124—152no—
301C22⋊Q8—301no—
302C4×Q8—31no—
303C4×Q8—151no—
304C4×Q8—152no—
305C4×Q8—152no—
306C4×Q8—301no—
307C4×D4D3×D511yescharacteristic
308#117—301no—
309#107—52no—
310#106—31no—
311C4⋊C8—51no—
312C4⋊C8—152no—
313#103—31no—
314#103—152no—
315#103—152no—
316#103—152no—
317#102—31no—
318#102—52no—
319C4×C8—31no—
320C4×C8—152no—
321C2×C3×C5—81no—
322C2×C3×C5#54811yescharacteristic
323C2×C3×C5C2×#13511yescharacteristic
324C2×C3×C5#57811yescharacteristic
325C23×C3—21no—
326C3×Q8—202no—
327C3×D4—21no—
328C3×D4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—42no—
331C2×C3×C4—102no—
332C2×C3×C4D4×D511yescharacteristic
333C2×C3×C4D4×D511yescharacteristic
334C2×C3×C4C23×D511yescharacteristic
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—52no—
338C2×Dic3—52no—
339C2×Dic3—102no—
340Dic6—21no—
341Dic6—102no—
342Dic6—102no—
343Dic6—102no—
344Dic6#70912yes—
345C3×C8—102no—
346C3⋊C8—21no—
347C3⋊C8—21no—
348C22×C5—41no—
349C22×C5—41no—
350C22×C5—41no—
351C22×C5C2×D3×D411yescharacteristic
352C4×C5—21no—
353C4×C5—41no—
354C4×C5—62no—
355C4×C5—121no—
356C4×C5#91212yes—
357C4×C5#100711yescharacteristic
358C4×C5D3×#4111yescharacteristic
359Dic5—42no—
360Dic5—62no—
361Dic5—62no—
362Dic5—122no—
363C2×Q8—31no—
364C2×Q8—101no—
365C2×Q8—151no—
366C2×Q8—152no—
367C2×D4C2×D3×D511yescharacteristic
368C22×C4—21no—
369Q16—304no—
370SD16—62no—
371C2×C8—31no—
372C2×C8—31no—
373C2×C8—52no—
374C4⋊C4—31no—
375C4⋊C4—52no—
376C4⋊C4—61no—
377C4⋊C4—101no—
378C4⋊C4—151no—
379C4⋊C4—152no—
380C4⋊C4—302no—
381C4⋊C4—302no—
382C4⋊C4—302no—
383C4⋊C4—302no—
384C4⋊C4—301no—
385C4⋊C4—301no—
386C4⋊C4C2×D3×D511yescharacteristic
387C22⋊C4—21no—
388C22⋊C4—302no—
389C22⋊C4—302no—
390C42—61no—
391C42—152no—
392C42—302no—
393C42—302no—
394C42C2×D3×D511yescharacteristic
395C3×C5#336411yescharacteristic, complemented, Hall
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3C2×D4×D511yescharacteristic
400C3×C4—21no—
401C3×C4—41no—
402C3×C4—202no—
403C3×C4#406212yes—
404C3×C4C2×#70911yescharacteristic
405C3×C4#415011yescharacteristic
406Dic3—22no—
407Dic3—41no—
408Dic3—102no—
409Dic3—102no—
410Dic3—202no—
411C2×C5—81no—
412C2×C5#563311yescharacteristic
413C2×C5C2×#91211yescharacteristic
414C2×C5#577911yescharacteristic
415C23—21no—
416Q8—32no—
417Q8—61no—
418Q8—202no—
419Q8—302no—
420Q8—302no—
421Q8—302no—
422D4—21no—
423D4—21no—
424C2×C4—21no—
425C2×C4—42no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—102no—
429C2×C4—152no—
430C2×C4—152no—
431C2×C4—302no—
432C2×C4#679011yescharacteristic
433C2×C4#679011yescharacteristic
434C2×C4C22×D3×D511yescharacteristic
435C8—61no—
436C8—61no—
437C8—102no—
438C2×C3—81no—
439C2×C3#6588611yescharacteristic
440C2×C3C2×#406211yescharacteristic
441C2×C3#6610511yescharacteristic
442C5#8099411yescharacteristic, complemented, Sylow
443C22—41no—
444C22—41no—
445C22—41no—
446C22C2×#679011yescharacteristic
447C4—21no—
448C4—41no—
449C4—62no—
450C4—121no—
451C4—202no—
452C4—302no—
453C4—302no—
454C4—602no—
455C4#9149712yes—
456C4#9200411yescharacteristic
457C4D3×#70911yescharacteristic
458C3#1061135411yescharacteristic, complemented, Sylow
459C2—81no—
460C2#1175437811yescharacteristic
461C2C2×#9149711yescharacteristic
462C2#1175535311yescharacteristic
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×D4C22
socleC22×C3×C5C22×D4
prime radicalC2×C3×C5×D4C23
hypercenterC4×D4D3×D5
hypocenterC3×C5#3364
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. #11752482C2semidirect
7. #11747278C2semidirect
8. #11747184C2semidirect
9. #11747101C2semidirect
10. #11746921C2semidirect
11. #11746855C2semidirect
395. C3×C5#3364semidirect
442. C5#80994semidirect
458. C3#10611354semidirect

Maximal subgroups

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

subgroupindex
#117563012
#117524822
C4×C5×#2142
C3×#652962
#117490152
#117489442
#117472782
#117471842
#117471012
#117469212
#117468552
#106113543
#809945