Finite Group Databasebeta

← #14512993

Subgroups — #14512993

Order 1920.

598 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊(C5⋊S4)C211yesmaximal, characteristic, complemented
2C5×#5912C211yesmaximal, characteristic, complemented
3A4×#681C211yesmaximal, characteristic, complemented
4#11751575C212yesmaximal, complemented
5#11751574C212yesmaximal, complemented
6#10614655—31nomaximal, Hall
7C22×C5⋊S4—21no—
8C2×C4×C5×A4C2211yescharacteristic
9C4⋊(C5⋊S4)—21no—
10C4⋊(C5⋊S4)C2212yes—
11C5×GL(2,ℤ/4)—21no—
12C5×C4⋊S4C2211yescharacteristic, complemented
13C4×C5×S4—21no—
14C5×A4⋊Q8C2211yescharacteristic, complemented
15#91305—41nomaximal
16A4×#168C2212yescomplemented
17#85443—51nomaximal, Hall
18C5×#525—31no—
19C2×#4095—31no—
20C2×#4076—31no—
21C22×#681D311yescharacteristic, complemented
22#65521—31no—
23#65338—32no—
24#65337—32no—
25#65328—32no—
26#65327—32no—
27#65290—31no—
28#65284—31no—
29C22×C5×A4D411yescharacteristic
30C2×C5⋊S4—22no—
31C2×C5⋊S4—42no—
32C2×C5×S4—21no—
33C2×D60—41no—
34C5×#235—41no—
35C4×C5×A4D411yescharacteristic
36C4×C5×A4C2311yescharacteristic
37C5×SL(2,ℤ/4)—21no—
38C3×#681—41no—
39#6660—42no—
40#6654—42no—
41#5912D511yescharacteristic, complemented
42C2×C4⋊S4—51no—
43A4×M16—51no—
44#5402—52no—
45#5401—52no—
46C2×C5×#41—31no—
47C23×C4×C5C2×D311yescharacteristic
48C22×C22⋊D5—61no—
49C22×D20—31no—
50C5×#123—61no—
51C5×C22⋊Q8—31no—
52C5×C22⋊Q8—31no—
53C5×#121—31no—
54C5×#121—31no—
55C5×#120—61no—
56C4×C5×D4—61no—
57C4×C5×D4—61no—
58C5×#117—31no—
59#4095—61no—
60#4095—62no—
61#4095—61no—
62C2×#688—32no—
63C2×#686—32no—
64#4076—32no—
65#4076—32no—
66#4076—61no—
67#4076—61no—
68C2×#685—61no—
69C2×C4⋊Dic5—31no—
70C2×#681—31no—
71C2×#681—31no—
72C2×#681—32no—
73C22×#168C2×D312yescomplemented
74#3967—32no—
75#3964—32no—
76#3941—32no—
77#3941—32no—
78#3940—32no—
79#3939—32no—
80#3055—151noSylow
81C22×D15—81no—
82C2×C5×A4—21no—
83C2×C5×A4C2×D411yescharacteristic
84C5⋊S4—42no—
85C5×S4—41no—
86C2×C3×C4×C5—41no—
87D60—42no—
88D60—81no—
89C5×#67—81no—
90C5×D12—41no—
91C4×C5×D3—81no—
92C5×Dic6—41no—
93C3×#168—42no—
94C22×S4—101no—
95C2×C4×A4C2×D511yescharacteristic
96GL(2,ℤ/4)—21no—
97C4⋊S4—52no—
98C4⋊S4—101no—
99C4⋊S4C2×D511yescharacteristic, complemented
100C4×S4—21no—
101A4⋊Q8C2×D511yescharacteristic, complemented
102#908—201no—
103C8×A4—52no—
104C24×C5D1211yescharacteristic
105C23×D5—61no—
106C5×#41—61no—
107C5×#41—61no—
108C5×#41—61no—
109C5×#41—61no—
110C2×C5×Q8—31no—
111C2×C5×D4—31no—
112C2×C5×D4—61no—
113C2×C5×D4—61no—
114C2×C5×D4—61no—
115C22×C4×C5—31no—
116C22×C4×C5—31no—
117C22×C4×C5—31no—
118C22×C4×C5—31no—
119C22×C4×C5—61no—
120C22×C4×C5D1211yescharacteristic
121C22×C4×C5C22×D311yescharacteristic
122C2×C22⋊D5—62no—
123C2×C22⋊D5—62no—
124C2×C22⋊D5—122no—
125C2×C22⋊D5—122no—
126C22×Dic5—61no—
127C2×D20—32no—
128C2×D20—61no—
129C2×D20—61no—
130C2×D20—61no—
131C2×D20—61no—
132C2×D20—61no—
133C5×C4⋊C4—31no—
134C5×C4⋊C4—31no—
135C5×C4⋊C4—61no—
136C5×C4⋊C4—61no—
137C5×C22⋊C4—61no—
138C5×C22⋊C4—61no—
139C5×C22⋊C4—61no—
140C5×C22⋊C4—61no—
141C5×C22⋊C4—61no—
142C42×C5—61no—
143#688—62no—
144#688—62no—
145#686—62no—
146#686—62no—
147#685—62no—
148#685—62no—
149C4⋊Dic5—32no—
150C4⋊Dic5—61no—
151#681—31no—
152#681—32no—
153#681—61no—
154#681S411yescharacteristic, complemented
155C2×#168—32no—
156C2×#168—32no—
157C2×AGL(1,ℤ/8)—151no—
158C22×M16—51no—
159#525—31no—
160C2×#121—151no—
161#469—152no—
162#468—152no—
163#466—152no—
164#465—152no—
165#428—151no—
166#418—151no—
167#417—151no—
168C22×C3×C5—41no—
169C2×D15—82no—
170C2×D15—162no—
171C2×C5×D3—81no—
172C5×A4AGL(1,ℤ/8)11yescharacteristic, complemented
173C3×C4×C5—41no—
174C3×C4×C5—41no—
175C5×Dic3—81no—
176C22×A4C22⋊D511yescharacteristic
177C2×S4—21no—
178C2×S4—102no—
179C2×S4—202no—
180#235—41no—
181C2×D12—201no—
182C4×A4C22⋊D511yescharacteristic
183C4×A4C22×D511yescharacteristic
184SL(2,ℤ/4)—21no—
185C3×M16—201no—
186D24—202no—
187#204—202no—
188C23×C5—21no—
189C23×C5—31no—
190C23×C5—31no—
191C23×C5—61no—
192C23×C5—61no—
193C23×C5C2×D1211yescharacteristic
194C22×D5—62no—
195C22×D5—121no—
196C22×D5—122no—
197C22×D5—122no—
198C22×D5—121no—
199C5×Q8—61no—
200C5×Q8—61no—
201C5×D4—61no—
202C5×D4—61no—
203C5×D4—121no—
204C5×D4—121no—
205C5×D4—122no—
206C5×D4—121no—
207C5×D4—121no—
208C2×C4×C5—31no—
209C2×C4×C5—31no—
210C2×C4×C5—31no—
211C2×C4×C5—31no—
212C2×C4×C5—31no—
213C2×C4×C5—31no—
214C2×C4×C5—31no—
215C2×C4×C5—61no—
216C2×C4×C5—61no—
217C2×C4×C5—61no—
218C2×C4×C5—61no—
219C2×C4×C5—121no—
220C2×C4×C5—121no—
221C2×C4×C5C2×S411yescharacteristic
222C22⋊D5—122no—
223C22⋊D5—124no—
224C22⋊D5—122no—
225C2×Dic5—62no—
226C2×Dic5—122no—
227D20—62no—
228D20—62no—
229D20—121no—
230D20—121no—
231#168—32no—
232#168C2×S412yescomplemented
233C2×#41—31no—
234C22×D4—151no—
235C22×D4—301no—
236C23×C4D3×D511yescharacteristic
237AGL(1,ℤ/8)—301no—
238AGL(1,ℤ/8)—302no—
239AGL(1,ℤ/8)—301no—
240C2×SD16—152no—
241C2×D8—152no—
242C2×M16—151no—
243C2×M16—151no—
244C2×M16—152no—
245C22×C8—52no—
246#123—61no—
247C22⋊Q8—31no—
248C22⋊Q8—31no—
249#121—31no—
250#121—31no—
251#121—152no—
252#121—152no—
253#121—301no—
254#121—301no—
255#120—61no—
256C4×D4—61no—
257C4×D4—61no—
258#117—31no—
259C2×C4⋊C4—151no—
260C2×C22⋊C4—301no—
261#107—152no—
262#106—152no—
263#103—152no—
264#102—152no—
265#102—152no—
266#102—152no—
267C2×C3×C5—41no—
268C2×C3×C5—81no—
269D15—162no—
270C5×D3—161no—
271C22×D3—401no—
272C2×A4—21no—
273C2×A4C2×C22⋊D511yescharacteristic
274S4—41no—
275S4—202no—
276C2×C3×C4—41no—
277#67—81no—
278D12—41no—
279D12—202no—
280D12—401no—
281C4×D3—81no—
282Dic6—41no—
283C3×C8—202no—
284C22×C5—31no—
285C22×C5—31no—
286C22×C5—31no—
287C22×C5—61no—
288C22×C5—61no—
289C22×C5—61no—
290C22×C5—61no—
291C22×C5—121no—
292C22×C5—121no—
293C22×C5—121no—
294C22×C5#90811yescharacteristic, complemented
295C22×C5C4⋊S411yescharacteristic
296C2×D5—122no—
297C2×D5—122no—
298C2×D5—122no—
299C2×D5—242no—
300C2×D5—242no—
301C4×C5—31no—
302C4×C5—31no—
303C4×C5—121no—
304C4×C5—121no—
305C4×C5—121no—
306C4×C5C4⋊S411yescharacteristic
307C4×C5C22×S411yescharacteristic
308Dic5—122no—
309C24—301no—
310C24#120311yescharacteristic
311#41—61no—
312#41—61no—
313#41—61no—
314#41—61no—
315C2×Q8—31no—
316C2×D4—31no—
317C2×D4—61no—
318C2×D4—61no—
319C2×D4—61no—
320C2×D4—152no—
321C2×D4—301no—
322C2×D4—301no—
323C2×D4—301no—
324C2×D4—301no—
325C2×D4—302no—
326C2×D4—302no—
327C2×D4—301no—
328C2×D4—602no—
329C2×D4—602no—
330C22×C4—31no—
331C22×C4—31no—
332C22×C4—31no—
333C22×C4—31no—
334C22×C4—61no—
335C22×C4—301no—
336C22×C4#120311yescharacteristic
337C22×C4C2×D3×D511yescharacteristic
338SD16—302no—
339SD16—302no—
340D8—302no—
341D8—302no—
342M16—51no—
343M16—152no—
344M16—151no—
345M16—301no—
346C2×C8—152no—
347C2×C8—152no—
348C4⋊C4—31no—
349C4⋊C4—31no—
350C4⋊C4—61no—
351C4⋊C4—61no—
352C4⋊C4—152no—
353C4⋊C4—301no—
354C22⋊C4—61no—
355C22⋊C4—61no—
356C22⋊C4—61no—
357C22⋊C4—61no—
358C22⋊C4—61no—
359C22⋊C4—302no—
360C22⋊C4—302no—
361C42—61no—
362C3×C5—41noHall
363C22×C3—41no—
364C2×D3—81no—
365C2×D3—402no—
366C2×D3—802no—
367A4#409511yescharacteristic, complemented
368C3×C4—41no—
369C3×C4—41no—
370Dic3—81no—
371C2×C5—21no—
372C2×C5—31no—
373C2×C5—31no—
374C2×C5—61no—
375C2×C5—241no—
376C2×C5C2×C4⋊S411yescharacteristic
377D5—242no—
378C23—21no—
379C23—31no—
380C23—31no—
381C23—61no—
382C23—61no—
383C23—302no—
384C23—601no—
385C23—602no—
386C23—602no—
387C23—601no—
388C23C2×#120311yescharacteristic
389Q8—61no—
390Q8—61no—
391D4—61no—
392D4—61no—
393D4—121no—
394D4—121no—
395D4—122no—
396D4—121no—
397D4—121no—
398D4—302no—
399D4—302no—
400D4—601no—
401D4—602no—
402D4—604no—
403D4—602no—
404D4—601no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—31no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—61no—
415C2×C4—61no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—302no—
419C2×C4—602no—
420C2×C4D5×S411yescharacteristic
421C8—52no—
422C8—152no—
423C2×C3—41no—
424C2×C3—81no—
425D3—161no—
426D3—802no—
427C5#8544311yescharacteristic, complemented, Sylow
428C22—31no—
429C22—31no—
430C22—31no—
431C22—61no—
432C22—61no—
433C22—61no—
434C22—61no—
435C22—121no—
436C22—121no—
437C22—121no—
438C22—602no—
439C22—602no—
440C22—602no—
441C22—1202no—
442C22—1202no—
443C22#9130511yescharacteristic, complemented
444C22#9190211yescharacteristic
445C4—31no—
446C4—31no—
447C4—121no—
448C4—121no—
449C4—121no—
450C4—602no—
451C4#9190211yescharacteristic
452C4C2×D5×S411yescharacteristic
453C3—41noSylow
454C2—21no—
455C2—31no—
456C2—31no—
457C2—61no—
458C2—241no—
459C2—1202no—
460C2C2×#9190211yescharacteristic
subgroupG/H
centerC2C2×#91902
commutator, cosocleC4×C5×A4C23
FrattiniC4C2×D5×S4
FittingC23×C4×C5C2×D3
socleC23×C5C2×D12
hypercenterC2×C4D5×S4
hypocenterC5×A4AGL(1,ℤ/8)
LCSG ▷ C4×C5×A4 ▷ C2×C5×A4 ▷ C5×A4
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊(C5⋊S4)C2semidirect
2. C5×#5912C2semidirect
3. A4×#681C2semidirect
4. #11751575C2semidirect
5. #11751574C2semidirect
12. C5×C4⋊S4C22semidirect
14. C5×A4⋊Q8C22semidirect
16. A4×#168C22semidirect
21. C22×#681D3semidirect
41. #5912D5semidirect
73. C22×#168C2×D3semidirect
99. C4⋊S4C2×D5semidirect
101. A4⋊Q8C2×D5semidirect
154. #681S4semidirect
172. C5×A4AGL(1,ℤ/8)semidirect
232. #168C2×S4semidirect
294. C22×C5#908semidirect
367. A4#4095semidirect
427. C5#85443semidirect
443. C22#91305semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C5⋊S4)2
C5×#59122
A4×#6812
#117515752
#117515742
#106146553
#913054
#854435