Finite Group Databasebeta

← #14416626

Subgroups — #14416626

Order 1920.

984 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91273C211yesmaximal, characteristic
2#11753398C212yesmaximal, complemented
3#11750954C211yesmaximal, characteristic, complemented
4C5×#4860C211yesmaximal, characteristic, complemented
5C3×#65158C211yesmaximal, characteristic, complemented
6#11747854C211yesmaximal, characteristic, complemented
7#11747815C211yesmaximal, characteristic, complemented
8D3×#3947C211yesmaximal, characteristic, complemented
9#11747502C212yesmaximal, complemented
10#11747472C212yesmaximal, complemented
11#11747228C212yesmaximal, complemented
12#10610897—31nomaximal, Hall
13C2×#6765C2212yes—
14C2×Dic60C2211yescharacteristic
15#91776C2212yes—
16C2×C5×D3×C8C2211yescharacteristic
17C5×#892C2212yes—
18C2×C3×D40C2211yescharacteristic
19C3×#4039C2212yes—
20#91442—22no—
21#91440—22no—
22#91434C2212yes—
23C4×D3×Dic5C2211yescharacteristic
24#91393—22no—
25C4×D5×Dic3—22no—
26#91333C2212yes—
27C2×#6657C2212yes—
28C2×D3×#168C2211yescharacteristic
29#91273C418yescomplemented
30C3×C5×#106C2211yescharacteristic
31#91099C2211yescharacteristic
32#91094C2212yes—
33C5×#817C2211yescharacteristic
34C5×#807C2212yes—
35C3×#3947C2211yescharacteristic
36C3×#3941C2212yes—
37#90975C2212yes—
38#90964C2212yes—
39#90954C2211yescharacteristic
40#90952C2211yescharacteristic
41#79864—51nomaximal, Hall
42C2×#4050—31no—
43#65856—32no—
44C5×#427—31no—
45C2×#3947—31no—
46#65291—32no—
47#65178—32no—
48#65178—32no—
49#65158—31no—
50#65158D311yescharacteristic, complemented
51#65142—31no—
52#65142—31no—
53C2×Dic30C2312yes—
54C2×C4×C5×D3C2311yescharacteristic
55C2×C3×D20C2312yes—
56C2×C3×C4×D5—22no—
57C2×#1201—22no—
58C2×D3×Dic5D412yes—
59C2×D5×Dic3—22no—
60C2×D5×Dic3—22no—
61#6765—24no—
62#6765C2×C418yescomplemented
63C2×C3×C5×C8C2311yescharacteristic
64C3×C5×C4⋊C4C2312yes—
65#6712—22no—
66C4×Dic15—22no—
67C4×Dic15C2311yescharacteristic
68C2×C3⋊#168C2311yescharacteristic
69Dic60C2×C414yescomplemented
70C5×(C2×D3)⋊C4—22no—
71C5×C4⋊Dic3C2312yes—
72C4×C5×Dic3—22no—
73C2×C5×C3⋊C8C2311yescharacteristic
74C5×D3×C8C2×C414yescomplemented
75C3×#685—22no—
76C3×C4×Dic5C2311yescharacteristic
77C2×C3×#168C2311yescharacteristic
78C3×D40C2×C414yescomplemented
79#6669—22no—
80#6665—22no—
81Dic3×Dic5—22no—
82Dic3×Dic5D412yes—
83#6657C2×C418yescomplemented
84D3×#168—22no—
85C2×#912—51no—
86C4×#237—52no—
87C3×#442—51no—
88#5152—51no—
89#5018—52no—
90#4860D511yescharacteristic, complemented
91#4853—51no—
92#4772—52no—
93#4757—52no—
94#4716—51no—
95D3×#97—51no—
96C2×#709—32no—
97C22×C5×C8—31no—
98C5×#117—32no—
99C2×C4×Dic5—31no—
100C22×#168—31no—
101C2×Dic20—31no—
102#4050—38no—
103C2×D40C2×D311yescharacteristic
104C2×#677—32no—
105#4039—32no—
106#4039C2×D312yes—
107#4033—32no—
108#4033—32no—
109#4027—62no—
110#4027—62no—
111#4018—62no—
112C42×D5—62no—
113C5×#107—32no—
114C5×#106—31no—
115C5×#106C2×D311yescharacteristic
116#3947—31no—
117#3947—31no—
118#3947—31no—
119#3947C2×D311yescharacteristic
120#3942—32no—
121#3942—32no—
122#3941—32no—
123#3941C2×D312yes—
124#2943—151noSylow
125C22×C5×D3C2×D411yescharacteristic
126C22×C3×D5—22no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5C2411yescharacteristic
129C2×Dic15—22no—
130C2×Dic15C2×D412yes—
131Dic30—22no—
132Dic30C22×C414yes—
133C2×C5×Dic3—22no—
134C2×C5×Dic3C2×D411yescharacteristic
135C4×C5×D3C22×C412yes—
136C4×C5×D3#4112yes—
137C2×C3×Dic5C2×D412yes—
138C3×D20—22no—
139C3×D20C22×C414yes—
140C3×C4×D5—44no—
141#1201—28no—
142D3×Dic5—42no—
143D5×Dic3—28no—
144D5×Dic3—44no—
145C3×C5×C8C22×C412yes—
146C3⋊#168—21no—
147C3×#168—21no—
148C5×C3⋊C8C22×C412yes—
149C2×#237—52no—
150C2×C3×D8—51no—
151C3×C4×D4—52no—
152D4×Dic3—52no—
153C2×#214—52no—
154C4×#67—102no—
155C2×C4×Dic3—102no—
156#912—58no—
157C2×Dic12—51no—
158C2×D3×C8—51no—
159C2×D3×C8C2×D511yescharacteristic
160#892C2×D512yes—
161#887—52no—
162#67⋊C4—102no—
163#877—102no—
164C42×D3—51no—
165C4×Dic6—52no—
166C3×#106C2×D511yescharacteristic
167C3×#102—52no—
168C3×#97—51no—
169#832—52no—
170#817C2×D511yescharacteristic
171#816—52no—
172#815—51no—
173#815—51no—
174#810—52no—
175#807C2×D512yes—
176#803—51no—
177C22×C4×C5—31no—
178C2×C22⋊D5—62no—
179C22×Dic5—32no—
180#709—38no—
181#709—64no—
182C2×D20C22×D312yes—
183C2×C4×D5—22no—
184C2×C4×D5—62no—
185C2×C4×D5—62no—
186C2×Dic10—32no—
187C2×C5×C8—31no—
188C2×C5×C8—34no—
189C2×C5×C8C22×D311yescharacteristic
190C5×C4⋊C4—32no—
191C5×C4⋊C4C22×D312yes—
192C5×C22⋊C4—62no—
193C42×C5—62no—
194#685—22no—
195#685—62no—
196#683—62no—
197#683—62no—
198C4×Dic5—31no—
199C4×Dic5—32no—
200C4×Dic5—62no—
201C4×Dic5—62no—
202C4×Dic5C22×D311yescharacteristic
203C2×#168—31no—
204C2×#168—62no—
205C2×#168C22×D311yescharacteristic
206Dic20—34no—
207D40C4×D314yescomplemented
208#677—38no—
209C2×#135—151no—
210C4×#41—152no—
211#442—51no—
212#442—151no—
213#441—151no—
214#441—151no—
215#440—152no—
216#440—152no—
217#427—31no—
218#416—152no—
219C2×#97—151no—
220C22×C3×C5C22×D411yescharacteristic
221C2×C5×D3—22no—
222C2×C5×D3C4×D412yes—
223C2×C3×D5—24no—
224C2×C3×D5—42no—
225C2×C3×D5—42no—
226C3×C4×C5—42no—
227C3×C4×C5C23×C411yescharacteristic
228C3×C4×C5C2×#4111yescharacteristic
229Dic15—24no—
230Dic15—22no—
231C3×Dic5—22no—
232C5×Dic3—42no—
233C5×Dic3C4×D412yes—
234C2×C3×D4—52no—
235C22×C3×C4—102no—
236C2×#67—102no—
237C22×Dic3—102no—
238C22×Dic3—102no—
239#237—58no—
240#237—104no—
241C2×C4×D3—52no—
242C2×C4×D3C22×D511yescharacteristic
243C2×Dic6—52no—
244C3×D8—54no—
245C2×C3×C8—51no—
246C2×C3×C8C22×D511yescharacteristic
247C3×C4⋊C4C22×D512yes—
248C3×C22⋊C4—102no—
249C3×C42—51no—
250#217—102no—
251#214—58no—
252(C2×D3)⋊C4—22no—
253C4⋊Dic3C22×D512yes—
254Dic3⋊C4—102no—
255Dic3⋊C4—102no—
256C4×Dic3—22no—
257C4×Dic3—51no—
258C4×Dic3—52no—
259C4×Dic3—102no—
260C4×Dic3—102no—
261C2×C3⋊C8—51no—
262C2×C3⋊C8C22×D511yescharacteristic
263Dic12—54no—
264D3×C8—102no—
265D3×C8C4×D514yescomplemented
266C23×C5—31no—
267C22×D5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—31no—
270C2×C4×C5—32no—
271C2×C4×C5—32no—
272C2×C4×C5—62no—
273C2×C4×C5C23×D311yescharacteristic
274C22⋊D5—68no—
275C2×Dic5—32no—
276C2×Dic5—62no—
277C2×Dic5—122no—
278C2×Dic5D3×D412yes—
279D20—22no—
280D20C2×C4×D314yes—
281C4×D5—44no—
282C4×D5—68no—
283C4×D5—124no—
284Dic10—34no—
285Dic10—62no—
286C5×C8—32no—
287C5×C8C2×C4×D312yes—
288#168—21no—
289#168—61no—
290C2×#41—152no—
291#135—158no—
292C2×Q16—151no—
293C2×SD16—152no—
294C2×D8—51no—
295C22×C8—31no—
296C22×C8—151no—
297C4×Q8—152no—
298C4×Q8—152no—
299C4×D4—52no—
300C4×D4—152no—
301C4×D4—302no—
302C4×D4—302no—
303#117—32no—
304#117—302no—
305C2×C42—151no—
306C2×C42—302no—
307#107—32no—
308#106—31no—
309#106D3×D511yescharacteristic
310#103—152no—
311#103—152no—
312#102—52no—
313#102—152no—
314#97—51no—
315#97—151no—
316#97—151no—
317#97—151no—
318C2×C3×C5C2×C4×D411yescharacteristic
319C2×C3×C5#57512yes—
320C3×D5—44no—
321C5×D3—22no—
322C23×C3—102no—
323C22×D3D4×D511yescharacteristic
324C3×D4—54no—
325C3×D4—102no—
326C2×C3×C4—22no—
327C2×C3×C4—52no—
328C2×C3×C4—204no—
329C2×C3×C4C23×D511yescharacteristic
330#67—108no—
331C2×Dic3—22no—
332C2×Dic3—52no—
333C2×Dic3—102no—
334C2×Dic3—108no—
335C2×Dic3—204no—
336C2×Dic3D4×D511yescharacteristic
337C4×D3—202no—
338C4×D3C2×C4×D512yes—
339C4×D3#71312yes—
340Dic6—54no—
341Dic6—102no—
342C3×C8—101no—
343C3×C8C2×C4×D512yes—
344C3⋊C8—101no—
345C3⋊C8C2×C4×D512yes—
346C22×C5—32no—
347C22×C5—62no—
348C22×C5C2×D3×D411yescharacteristic
349C2×D5—24no—
350C2×D5—42no—
351C2×D5—42no—
352C4×C5—32no—
353C4×C5—42no—
354C4×C5—122no—
355C4×C5C22×C4×D311yescharacteristic
356C4×C5D3×#4111yescharacteristic
357Dic5—22no—
358Dic5—62no—
359Dic5—64no—
360#41—158no—
361#41—304no—
362C2×Q8—152no—
363C2×D4—52no—
364C2×D4—302no—
365C22×C4—31no—
366C22×C4—102no—
367C22×C4—152no—
368C22×C4—302no—
369C22×C4—302no—
370Q16—154no—
371SD16—158no—
372D8—54no—
373C2×C8—31no—
374C2×C8—34no—
375C2×C8—51no—
376C2×C8—151no—
377C2×C8—302no—
378C2×C8C2×D3×D511yescharacteristic
379C4⋊C4—32no—
380C4⋊C4—302no—
381C4⋊C4—302no—
382C4⋊C4C2×D3×D512yes—
383C22⋊C4—62no—
384C22⋊C4—102no—
385C22⋊C4—302no—
386C42—51no—
387C42—62no—
388C42—152no—
389C42—151no—
390C42—302no—
391C42—302no—
392C3×C5#294311yescharacteristic, complemented, Hall
393C22×C3—104no—
394C22×C3—202no—
395C22×C3—202no—
396C22×C3C2×D4×D511yescharacteristic
397C2×D3—22no—
398C2×D3#403912yes—
399C3×C4—42no—
400C3×C4—102no—
401C3×C4C22×C4×D511yescharacteristic
402C3×C4C2×#71311yescharacteristic
403Dic3—42no—
404Dic3—104no—
405Dic3—102no—
406Dic3#403912yes—
407C2×C5—62no—
408C2×C5C4×D3×D411yescharacteristic
409C2×C5#577312yes—
410D5—44no—
411C23—31no—
412C23—102no—
413Q8—154no—
414Q8—302no—
415D4—54no—
416D4—102no—
417D4—308no—
418C2×C4—22no—
419C2×C4—31no—
420C2×C4—32no—
421C2×C4—32no—
422C2×C4—52no—
423C2×C4—62no—
424C2×C4—152no—
425C2×C4—204no—
426C2×C4—308no—
427C2×C4—302no—
428C2×C4—602no—
429C2×C4—604no—
430C2×C4C22×D3×D511yescharacteristic
431C8—32no—
432C8—101no—
433C8—301no—
434C8C4×D3×D512yes—
435C2×C3—204no—
436C2×C3C2×#403911yescharacteristic
437C2×C3#6610212yes—
438D3—22no—
439C5#7986411yescharacteristic, complemented, Sylow
440C22—32no—
441C22—62no—
442C22—104no—
443C22—202no—
444C22—202no—
445C22D3×D4×D511yescharacteristic
446C4—32no—
447C4—42no—
448C4—102no—
449C4—122no—
450C4—302no—
451C4—304no—
452C4C2×C4×D3×D511yescharacteristic
453C4D3×#71311yescharacteristic
454C3#1061089711yescharacteristic, complemented, Sylow
455C2—62no—
456C2—204no—
457C2D3×#403911yescharacteristic
458C2#1175407312yes—
subgroupG/H
centerC22D3×D4×D5
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#106C22
socleC22×C3×C5C22×D4
prime radicalC2×#91273C2
cosocleC2×C3×C4×C5C24
hypercenter#106D3×D5
hypocenterC3×C5#2943
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
2. #11753398C2semidirect
3. #11750954C2semidirect
4. C5×#4860C2semidirect
5. C3×#65158C2semidirect
6. #11747854C2semidirect
7. #11747815C2semidirect
8. D3×#3947C2semidirect
9. #11747502C2semidirect
10. #11747472C2semidirect
11. #11747228C2semidirect
29. #91273C4semidirect
50. #65158D3semidirect
62. #6765C2×C4semidirect
69. Dic60C2×C4semidirect
74. C5×D3×C8C2×C4semidirect
78. C3×D40C2×C4semidirect
83. #6657C2×C4semidirect
90. #4860D5semidirect
207. D40C4×D3semidirect
265. D3×C8C4×D5semidirect
392. C3×C5#2943semidirect
439. C5#79864semidirect
454. C3#10610897semidirect

Maximal subgroups

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

subgroupindex
C2×#912732
#117533982
#117509542
C5×#48602
C3×#651582
#117478542
#117478152
D3×#39472
#117475022
#117474722
#117472282
#106108973
#798645