Finite Group Databasebeta

← #14438679

Subgroups — #14438679

Order 1920.

784 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
1#11752673C211yesmaximal, characteristic
2D5×#870C211yesmaximal, characteristic, complemented
3#11750798C211yesmaximal, characteristic, complemented
4C5×#4697C211yesmaximal, characteristic, complemented
5C3×C4×#676C211yesmaximal, characteristic, complemented
6#11748025C212yesmaximal, complemented
7#11747998C212yesmaximal, complemented
8#11747606C212yesmaximal, complemented
9#11747605C212yesmaximal, complemented
10#11746913C211yesmaximal, characteristic, complemented
11#11746907C211yesmaximal, characteristic, complemented
12#10613372—31nomaximal, Hall
13C2×#6773C2211yescharacteristic
14C2×#6772C2211yescharacteristic
15#91764C2211yescharacteristic
16#91759C2211yescharacteristic
17C5×#875C2211yescharacteristic
18C5×#870C2211yescharacteristic
19C2×C3×#676C2212yes—
20C3×C42×D5C2211yescharacteristic
21#91451C2211yescharacteristic
22#91448—22no—
23D5×C4⋊Dic3C2212yes—
24#91405—22no—
25D5×Dic3⋊C4—22no—
26#91388C2212yes—
27#91378—22no—
28#91376C2211yescharacteristic
29C3×C4×C5×C8C2211yescharacteristic
30#91105C2212yes—
31#91100C2212yes—
32C5×#821C2212yes—
33C5×#816C2212yes—
34C3×#3947C2212yes—
35C3×C4×#168C2211yescharacteristic
36#90975C2212yes—
37#90973C2212yes—
38#90969C2212yes—
39#90967C2212yes—
40#78529—51nomaximal, Hall
41D5×#125—31no—
42#66002—31no—
43C5×#488—31no—
44#65357—31no—
45#65345—31no—
46#65092—32no—
47#65086—32no—
48#65060—32no—
49#65053—32no—
50#64986—31no—
51C4×#676D311yescharacteristic, complemented
52C2×D60C2311yescharacteristic
53C2×Dic30C2311yescharacteristic
54C2×C5×D12C2311yescharacteristic
55C2×C5×Dic6C2311yescharacteristic
56C2×C3×C4×D5D412yes—
57C2×C3×C4×D5C2311yescharacteristic
58C2×#1203—22no—
59C2×#1202—21no—
60C2×#1201—21no—
61C2×#1200—21no—
62C2×D3×Dic5—21no—
63C2×D5×Dic3—22no—
64#6773—24no—
65#6772—24no—
66C2×C3×C5×C8C2312yes—
67C3×C42×C5C2311yescharacteristic
68#6714—22no—
69C4⋊Dic15C2312yes—
70#6712—22no—
71C5×(C2×D3)⋊C4—22no—
72C5×C4⋊Dic3C2312yes—
73C5×Dic3⋊C4—22no—
74C3×C4×Dic5D412yes—
75C3×C4×Dic5C2311yescharacteristic
76C2×C3×#168C2312yes—
77C3×#676—24no—
78Dic3⋊Dic5—22no—
79#6669—22no—
80#6668—22no—
81#6668—22no—
82#5477—51no—
83C2×#870—51no—
84C3×C4×M16—51no—
85#5133—52no—
86#5133—52no—
87#4713—52no—
88#4703—52no—
89#4697—51no—
90#4697—51no—
91#4697—51no—
92#4697D511yescharacteristic, complemented
93C2×#713—31no—
94C2×#711—31no—
95C5×#125—31no—
96C5×#124—31no—
97#4090—31no—
98#4086—31no—
99C2×#676C2×D312yes—
100D5×C4⋊C4—32no—
101D5×C4⋊C4—62no—
102#4036—32no—
103#4035—62no—
104#4030—62no—
105#4027—62no—
106#4021—31no—
107C42×D5C2×D311yescharacteristic
108#4016—31no—
109C5×#103—32no—
110C5×#102—32no—
111C4×C5×C8C2×D311yescharacteristic
112#3967—32no—
113#3964—32no—
114#3953—32no—
115#3948—32no—
116#3947C2×D312yes—
117#3942—32no—
118#3941—32no—
119C4×#168C2×D311yescharacteristic
120#3142—151noSylow
121C22×D15—21no—
122C22×C5×D3—21no—
123C22×C3×D5C2×D411yescharacteristic
124C2×C3×C4×C5C2×D412yes—
125C2×C3×C4×C5C2411yescharacteristic
126C2×Dic15—21no—
127C2×Dic15—22no—
128D60—22no—
129Dic30—22no—
130C2×C5×Dic3—21no—
131C2×C5×Dic3—22no—
132C5×D12—22no—
133C5×Dic6—22no—
134C2×C3×Dic5C2×D412yes—
135C2×C3×Dic5C2×D411yescharacteristic
136C3×C4×D5—42no—
137C3×C4×D5#4114yes—
138#1203—44no—
139#1202—42no—
140#1201—42no—
141#1200—42no—
142D3×Dic5—42no—
143D5×Dic3—44no—
144C3×C5×C8—22no—
145C3×#168—22no—
146C2×#235—51no—
147C2×#235—51no—
148C2×C3×M16—52no—
149C2×C3×C42—51no—
150#930—102no—
151C2×C4⋊Dic3—52no—
152C2×Dic3⋊C4—102no—
153#875—51no—
154#875C2×D511yescharacteristic
155#874—51no—
156C4×D12—102no—
157#870—51no—
158#870—52no—
159#870—102no—
160#870C2×D511yescharacteristic
161#869—51no—
162C3×#97—52no—
163C3×C4×C8—51no—
164C3×C4×C8C2×D511yescharacteristic
165#821—52no—
166#821—52no—
167#821—52no—
168#821C2×D512yes—
169#816—52no—
170#816—52no—
171#816—52no—
172#816C2×D512yes—
173C2×C5×Q8—31no—
174C2×C5×D4—31no—
175C2×C22⋊D5—61no—
176C2×C22⋊D5—62no—
177C22×Dic5—61no—
178#713—64no—
179#711—64no—
180C2×D20—31no—
181C2×D20—61no—
182C2×C4×D5—62no—
183C2×C4×D5—61no—
184C2×C4×D5D1212yes—
185C2×C4×D5C22×D311yescharacteristic
186C2×Dic10—31no—
187C2×C5×C8C22×D312yes—
188C5×C4⋊C4—32no—
189C5×C4⋊C4—62no—
190C5×C22⋊C4—62no—
191C42×C5C22×D311yescharacteristic
192#685—62no—
193C4⋊Dic5—32no—
194C4⋊Dic5—62no—
195#683—62no—
196#683—62no—
197#683—62no—
198#683—62no—
199C4×Dic5D1212yes—
200C4×Dic5C22×D311yescharacteristic
201C2×#168C22×D312yes—
202#676—24no—
203#532—151no—
204C2×#125—151no—
205#491—152no—
206#490—152no—
207#488—31no—
208#488—151no—
209#488—151no—
210#488—151no—
211#417—152no—
212#417—152no—
213C4×M16—51no—
214C22×C3×C5C22×D411yescharacteristic
215C2×D15—41no—
216C2×D15—42no—
217C2×C5×D3—41no—
218C2×C5×D3—42no—
219C2×C3×D5—22no—
220C2×C3×D5#12412yes—
221C3×C4×C5—22no—
222C3×C4×C5C2×#4112yes—
223Dic15—41no—
224Dic15—42no—
225C3×Dic5—22no—
226C3×Dic5#12412yes—
227C5×Dic3—41no—
228C5×Dic3—42no—
229C22×C3×C4—51no—
230C22×C3×C4—52no—
231C2×#67—101no—
232C2×#67—101no—
233C22×Dic3—102no—
234#235—104no—
235#235—104no—
236C2×D12—51no—
237C2×D12—102no—
238C2×D12C22×D511yescharacteristic
239C2×C4×D3—101no—
240C2×C4×D3—101no—
241C2×Dic6—51no—
242C2×Dic6C22×D511yescharacteristic
243C3×M16—104no—
244C2×C3×C8—52no—
245C2×C3×C8C22×D512yes—
246C3×C42—52no—
247C3×C42—51no—
248C3×C42C22×D511yescharacteristic
249(C2×D3)⋊C4—22no—
250(C2×D3)⋊C4—102no—
251C4⋊Dic3—52no—
252C4⋊Dic3—102no—
253C4⋊Dic3—102no—
254C4⋊Dic3C22×D512yes—
255Dic3⋊C4—22no—
256Dic3⋊C4—102no—
257Dic3⋊C4—102no—
258Dic3⋊C4—102no—
259C23×C5—61no—
260C22×D5—61no—
261C22×D5C2×D1211yescharacteristic
262C5×Q8—62no—
263C5×D4—62no—
264C2×C4×C5—62no—
265C2×C4×C5—61no—
266C2×C4×C5C2×D1212yes—
267C2×C4×C5C23×D311yescharacteristic
268C22⋊D5—124no—
269C22⋊D5—122no—
270C2×Dic5—62no—
271C2×Dic5—61no—
272C2×Dic5—122no—
273C2×Dic5C2×D1212yes—
274C2×Dic5C2×D1211yescharacteristic
275D20—62no—
276D20—122no—
277C4×D5—42no—
278C4×D5—124no—
279C4×D5—122no—
280C4×D5#23514yes—
281Dic10—62no—
282C5×C8—22no—
283#168—22no—
284C2×#41—151no—
285C2×#41—151no—
286C2×M16—52no—
287C4⋊Q8—151no—
288#127—151no—
289#125—31no—
290#125—152no—
291#125—151no—
292#125—302no—
293#124—31no—
294#124—151no—
295#121—302no—
296C4×D4—302no—
297C2×C4⋊C4—152no—
298C2×C4⋊C4—302no—
299C2×C42—51no—
300#103—32no—
301#103—152no—
302#103—152no—
303#103—152no—
304#102—32no—
305#102—152no—
306#102—152no—
307#102—152no—
308#97—52no—
309C4×C8—51no—
310C4×C8D3×D511yescharacteristic
311C2×C3×C5C2×#12411yescharacteristic
312C2×C3×C5#57512yes—
313D15—81no—
314C3×D5—22no—
315C5×D3—81no—
316C23×C3—51no—
317C22×D3—21no—
318C22×D3—101no—
319C2×C3×C4—51no—
320C2×C3×C4—54no—
321C2×C3×C4—52no—
322C2×C3×C4—202no—
323C2×C3×C4D4×D512yes—
324C2×C3×C4C23×D511yescharacteristic
325#67—202no—
326#67—202no—
327C2×Dic3—22no—
328C2×Dic3—21no—
329C2×Dic3—102no—
330C2×Dic3—101no—
331C2×Dic3—204no—
332D12—22no—
333D12—102no—
334D12—204no—
335C4×D3—202no—
336C4×D3—202no—
337Dic6—22no—
338Dic6—102no—
339C3×C8—22no—
340C3×C8—102no—
341C22×C5—121no—
342C22×C5—122no—
343C22×C5C22×D1211yescharacteristic
344C2×D5—22no—
345C2×D5—121no—
346C2×D5—122no—
347C2×D5#87512yes—
348C4×C5—22no—
349C4×C5—121no—
350C4×C5—122no—
351C4×C5C2×#23512yes—
352Dic5—22no—
353Dic5—121no—
354Dic5—122no—
355Dic5#87512yes—
356#41—304no—
357#41—304no—
358C2×Q8—31no—
359C2×Q8—151no—
360C2×D4—31no—
361C2×D4—151no—
362C2×D4—302no—
363C2×D4—301no—
364C2×D4—301no—
365C22×C4—51no—
366C22×C4—52no—
367C22×C4—301no—
368C22×C4—301no—
369C22×C4—302no—
370M16—104no—
371C2×C8—52no—
372C2×C8C2×D3×D512yes—
373C4⋊C4—32no—
374C4⋊C4—62no—
375C4⋊C4—152no—
376C4⋊C4—302no—
377C4⋊C4—302no—
378C4⋊C4—302no—
379C4⋊C4—302no—
380C4⋊C4—302no—
381C22⋊C4—62no—
382C22⋊C4—302no—
383C42—52no—
384C42—51no—
385C42C2×D3×D511yescharacteristic
386C3×C5#314211yescharacteristic, complemented, Hall
387C22×C3—52no—
388C22×C3—102no—
389C22×C3C2×D4×D511yescharacteristic
390C2×D3—42no—
391C2×D3—41no—
392C2×D3—202no—
393C2×D3—201no—
394C3×C4—22no—
395C3×C4—52no—
396C3×C4—102no—
397C3×C4D5×#4112yes—
398Dic3—42no—
399Dic3—41no—
400Dic3—202no—
401Dic3—201no—
402C2×C5—241no—
403C2×C5C2×#87511yescharacteristic
404C2×C5#574812yes—
405D5—22no—
406D5—241no—
407C23—51no—
408C23—61no—
409C23—301no—
410Q8—62no—
411Q8—302no—
412D4—62no—
413D4—302no—
414D4—604no—
415D4—602no—
416D4—602no—
417C2×C4—51no—
418C2×C4—54no—
419C2×C4—52no—
420C2×C4—61no—
421C2×C4—62no—
422C2×C4—202no—
423C2×C4—301no—
424C2×C4—302no—
425C2×C4—602no—
426C2×C4—602no—
427C2×C4—604no—
428C2×C4D5×D1212yes—
429C2×C4C22×D3×D511yescharacteristic
430C8—22no—
431C8—102no—
432C2×C3—102no—
433C2×C3D5×#12411yescharacteristic
434C2×C3#6610212yes—
435D3—81no—
436D3—401no—
437C5#7852911yescharacteristic, complemented, Sylow
438C22—52no—
439C22—102no—
440C22—121no—
441C22—122no—
442C22—601no—
443C22—602no—
444C22C2×D5×D1211yescharacteristic
445C4—22no—
446C4—52no—
447C4—102no—
448C4—121no—
449C4—122no—
450C4—601no—
451C4—602no—
452C4D5×#23512yes—
453C3#1061337211yescharacteristic, complemented, Sylow
454C2—102no—
455C2—241no—
456C2—1201no—
457C2D5×#87511yescharacteristic
458C2#1175395912yes—
subgroupG/H
centerC22C2×D5×D12
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×C8C22
socleC22×C3×C5C22×D4
prime radical#11752673C2
hypercenterC4×C8D3×D5
hypocenterC3×C5#3142
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
2. D5×#870C2semidirect
3. #11750798C2semidirect
4. C5×#4697C2semidirect
5. C3×C4×#676C2semidirect
6. #11748025C2semidirect
7. #11747998C2semidirect
8. #11747606C2semidirect
9. #11747605C2semidirect
10. #11746913C2semidirect
11. #11746907C2semidirect
51. C4×#676D3semidirect
92. #4697D5semidirect
386. C3×C5#3142semidirect
437. C5#78529semidirect
453. C3#10613372semidirect

Maximal subgroups

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

subgroupindex
#117526732
D5×#8702
#117507982
C5×#46972
C3×C4×#6762
#117480252
#117479982
#117476062
#117476052
#117469132
#117469072
#106133723
#785295