Finite Group Databasebeta

← #14295969

Subgroups — #14295969

Order 1920.

600 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×#907C211yesmaximal, characteristic
2D5⋊#805C211yesmaximal, characteristic
3#11751703C211yesmaximal, characteristic
4#11751695C211yesmaximal, characteristic, complemented
5C2×AGL(1,𝔽5)×C3⋊C8C211yesmaximal, characteristic, complemented
6#11746772C211yesmaximal, characteristic, complemented
7C3×#64900C211yesmaximal, characteristic, complemented
8#10604777—31nomaximal, Hall
9D5×#235C2211yescharacteristic
10D5⋊#208C2211yescharacteristic
11C2×D5⋊(C3⋊C8)C2211yescharacteristic
12C3×#4130C2211yescharacteristic
13C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
14#91796C411yescharacteristic, complemented
15C5×#907C411yescharacteristic, complemented
16C3×D5×M16C2211yescharacteristic
17#91291C411yescharacteristic, complemented
18#91288C411yescharacteristic, complemented
19C2×D5×C3⋊C8C2211yescharacteristic
20#91268—21no—
21#91267—21no—
22D5×ΓL(1,𝔽25)—21no—
23D3×C8×D5—21no—
24#91158C412yescomplemented
25#91156C412yescomplemented
26#91153C412yescomplemented
27#91151C412yescomplemented
28#91149—21no—
29AGL(1,𝔽5)×C3⋊C8—21no—
30AGL(1,𝔽5)×C3⋊C8—21no—
31AGL(1,𝔽5)×C3⋊C8—21no—
32#68739—51nomaximal, Hall
33D5×#131—31no—
34#65794—31no—
35D5⋊#104—31no—
36#65718—31no—
37C2×C8×AGL(1,𝔽5)—31no—
38#64901—31no—
39#64900D311yescharacteristic, complemented
40C22×C3×AGL(1,𝔽5)—21no—
41#6817C2×C411yescharacteristic
42C5×#235C2×C411yescharacteristic
43C2×C3×C4×D5C2311yescharacteristic
44D5×#67—21no—
45#6782—21no—
46D5×D12C2×C411yescharacteristic
47C4×D3×D5—21no—
48#6773C2×C411yescharacteristic
49#6772C2×C411yescharacteristic
50#6771—21no—
51D5×Dic6C2×C411yescharacteristic
52#6762—21no—
53C2×C3⋊#170—21no—
54#6758C2×C412yes—
55D5⋊(C3⋊C8)—21no—
56D5⋊(C3⋊C8)—21no—
57D5⋊(C3⋊C8)C2×C412yes—
58C3×C22⋊AGL(1,𝔽5)—21no—
59C3×C4⋊AGL(1,𝔽5)C2×C412yes—
60C3×C4×AGL(1,𝔽5)—21no—
61C3×C4×AGL(1,𝔽5)—21no—
62C3×C4×AGL(1,𝔽5)C2×C412yes—
63C3×C5×M16C2×C411yescharacteristic
64C2×C3⋊#168C2×C411yescharacteristic
65#6705—21no—
66C8×D15—21no—
67C2×C5×C3⋊C8C2×C411yescharacteristic
68C5×ΓL(1,𝔽25)—21no—
69C5×D3×C8—21no—
70C3×#681C2×C411yescharacteristic
71C3×#676—21no—
72C3×C8×D5—21no—
73#6651—21no—
74#6650—21no—
75#6648—21no—
76D3×#168—21no—
77D5×C3⋊C8D411yescharacteristic
78D5×C3⋊C8D411yescharacteristic
79D5×C3⋊C8D411yescharacteristic
80D5×C3⋊C8D411yescharacteristic
81C2×#907—51no—
82#5002—51no—
83#4997—51no—
84C2×#805—51no—
85C2×C4×C3⋊C8—51no—
86C3×#339—51no—
87#4558—51no—
88D5×#41—31no—
89#4130C2×D311yescharacteristic
90C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
91D5⋊M16—31no—
92C2×D5⋊C8—31no—
93C5×#131—31no—
94#4094—31no—
95#4053—31no—
96#4053—61no—
97D5×M16—61no—
98D5×M16C2×D311yescharacteristic
99#4047—31no—
100#4047—61no—
101C2×C8×D5—31no—
102C2×C8×D5—61no—
103#4010—32no—
104#4009—32no—
105#4008—32no—
106#4007—32no—
107#3992—61no—
108C8×AGL(1,𝔽5)—61no—
109C8×AGL(1,𝔽5)—61no—
110C8×AGL(1,𝔽5)—61no—
111#1764—151noSylow
112C22×C3×D5#4111yescharacteristic
113C2×D3×D5—21no—
114C2×C3×AGL(1,𝔽5)—24no—
115C2×C3×AGL(1,𝔽5)—22no—
116C2×C3×AGL(1,𝔽5)—22no—
117C2×C3×C4×C5C22×C411yescharacteristic
118#1220—21no—
119D60C4211yescharacteristic
120C4×D15—21no—
121Dic30C4211yescharacteristic
122C5×#67—21no—
123C5×D12C4211yescharacteristic
124C4×C5×D3—21no—
125C5×Dic6C4211yescharacteristic
126C2×C3×Dic5#4111yescharacteristic
127C3×C4×D5C22×C411yescharacteristic
128C3×C4×D5C22×C411yescharacteristic
129C3×C4×D5C2×D411yescharacteristic
130C3×C4×D5C2×D411yescharacteristic
131#1204—21no—
132#1203—21no—
133#1202—21no—
134#1201—21no—
135#1200—21no—
136D3×Dic5—21no—
137D5×Dic3—21no—
138C3⋊#170—22no—
139C3⋊#170—22no—
140C3×C5×C8—21no—
141C3⋊#168C22⋊C411yescharacteristic
142C3⋊#168C22⋊C411yescharacteristic
143C3×#168—21no—
144C5×C3⋊C8C22⋊C411yescharacteristic
145C5×C3⋊C8C22⋊C411yescharacteristic
146C2×#235—51no—
147C2×C3×M16—51no—
148C3×#117—51no—
149C2×C3×C42—51no—
150C2×#208—51no—
151C22×C3⋊C8—51no—
152C22×C3⋊C8—51no—
153#907—51no—
154#907—51no—
155#907—51no—
156#907—101no—
157#907—101no—
158#907AGL(1,𝔽5)11yescharacteristic, complemented
159C2×ΓL(1,𝔽25)—101no—
160C2×D3×C8—101no—
161#810—52no—
162#809—52no—
163#805—52no—
164#805—52no—
165#803—101no—
166C4×C3⋊C8—101no—
167C4×C3⋊C8—101no—
168C4×C3⋊C8—101no—
169C22×AGL(1,𝔽5)—21no—
170C5×#41—31no—
171#713—31no—
172Q8×D5—31no—
173#711—31no—
174#711—61no—
175D4×D5—31no—
176D4×D5—61no—
177#709—31no—
178#709—61no—
179C2×C4×D5—61no—
180C2×C4×D5C22×D311yescharacteristic
181C22⋊AGL(1,𝔽5)—21no—
182#704—61no—
183C2×#170—61no—
184C4⋊AGL(1,𝔽5)C4×D312yes—
185C4×AGL(1,𝔽5)—21no—
186C4×AGL(1,𝔽5)—21no—
187C4×AGL(1,𝔽5)C4×D312yes—
188#700—32no—
189D5⋊C8—32no—
190D5⋊C8—61no—
191D5⋊C8—61no—
192C5×M16—61no—
193C5×M16C4×D311yescharacteristic
194C2×C5×C8—31no—
195C2×C5×C8—61no—
196#681—61no—
197#681C4×D311yescharacteristic
198C2×#168—31no—
199C2×#168—61no—
200#676—21no—
201#676—61no—
202#676—61no—
203C8×D5—21no—
204C8×D5—31no—
205C8×D5—31no—
206C8×D5—31no—
207C8×D5—31no—
208C8×D5—61no—
209C8×D5—61no—
210C2×#131—151no—
211C2×#104—151no—
212#416—151no—
213#405—151no—
214C2×C4×C8—151no—
215#341—151no—
216#339—51no—
217C22×C3×C5#11711yescharacteristic
218C2×D15—21no—
219C2×C5×D3—21no—
220C2×C3×D5—22no—
221C2×C3×D5C4×D411yescharacteristic
222C2×C3×D5C4×D411yescharacteristic
223D3×D5—42no—
224C3×AGL(1,𝔽5)—24no—
225C3×AGL(1,𝔽5)—42no—
226C3×C4×C5C2×C4211yescharacteristic
227C3×C4×C5C2×C22⋊C411yescharacteristic
228Dic15—21no—
229C3×Dic5C4×D411yescharacteristic
230C3×Dic5C4×D411yescharacteristic
231C5×Dic3—21no—
232C22×C3×C4—51no—
233C22×C3×C4—101no—
234C2×#67—101no—
235#235—51no—
236#235—51no—
237#235—51no—
238#235—101no—
239#235—101no—
240#235C2×AGL(1,𝔽5)11yescharacteristic
241C2×D12—51no—
242C2×C4×D3—101no—
243C2×Dic6—51no—
244C3×M16—51no—
245C3×M16—101no—
246C3×M16C2×AGL(1,𝔽5)11yescharacteristic
247C2×C3×C8—101no—
248C3×C4⋊C4—52no—
249C3×C22⋊C4—101no—
250C3×C42—52no—
251C3×C42—101no—
252C3×C42—101no—
253#208—52no—
254#208—101no—
255C2×C3⋊C8—51no—
256C2×C3⋊C8—51no—
257C2×C3⋊C8—51no—
258C2×C3⋊C8—51no—
259C2×C3⋊C8—51no—
260C2×C3⋊C8—52no—
261C2×C3⋊C8—101no—
262C2×C3⋊C8—101no—
263C2×C3⋊C8—101no—
264C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic
265ΓL(1,𝔽25)—21no—
266ΓL(1,𝔽25)—101no—
267ΓL(1,𝔽25)—101no—
268ΓL(1,𝔽25)—101no—
269D3×C8—21no—
270D3×C8—101no—
271D3×C8—101no—
272D3×C8—101no—
273C22×D5—61no—
274C22×D5#23711yescharacteristic
275C2×AGL(1,𝔽5)—24no—
276C2×AGL(1,𝔽5)—22no—
277C2×AGL(1,𝔽5)—22no—
278C5×Q8—31no—
279C5×D4—31no—
280C5×D4—61no—
281C2×C4×C5—61no—
282C2×C4×C5C2×C4×D311yescharacteristic
283C22⋊D5—61no—
284C22⋊D5—61no—
285C22⋊D5—61no—
286C2×Dic5—61no—
287C2×Dic5#23911yescharacteristic
288D20—31no—
289D20—61no—
290C4×D5—61no—
291C4×D5—61no—
292C4×D5—61no—
293C4×D5C2×C4×D311yescharacteristic
294C4×D5C2×C4×D311yescharacteristic
295C4×D5D3×D411yescharacteristic
296C4×D5D3×D411yescharacteristic
297Dic10—31no—
298Dic10—61no—
299#170—62no—
300#170—62no—
301C5×C8—21no—
302C5×C8—31no—
303C5×C8—31no—
304#168—21no—
305#168—31no—
306#168—31no—
307C2×#41—151no—
308#131—31no—
309#131—151no—
310#131—151no—
311#131—151no—
312#131—301no—
313#131—301no—
314C2×M16—51no—
315C2×M16—151no—
316C2×M16—301no—
317C22×C8—151no—
318C22×C8—151no—
319C22×C8—301no—
320#117—51no—
321C2×C42—51no—
322#104—152no—
323#104—152no—
324#103—152no—
325#102—152no—
326#97—301no—
327C4×C8—301no—
328C4×C8—301no—
329C4×C8—301no—
330C2×C3×C5—21no—
331C2×C3×C5C4×C22⋊C411yescharacteristic
332D15—41no—
333C3×D5—21no—
334C3×D5#44312yes—
335C5×D3—41no—
336C23×C3—51no—
337C22×D3—101no—
338C2×C3×C4—51no—
339C2×C3×C4—51no—
340C2×C3×C4—51no—
341C2×C3×C4—51no—
342C2×C3×C4—51no—
343C2×C3×C4—102no—
344C2×C3×C4—104no—
345C2×C3×C4—102no—
346C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
347#67—21no—
348#67—101no—
349#67—101no—
350#67—101no—
351C2×Dic3—101no—
352D12—51no—
353D12—101no—
354D12C4×AGL(1,𝔽5)11yescharacteristic
355C4×D3—21no—
356C4×D3—101no—
357C4×D3—101no—
358C4×D3—101no—
359Dic6—51no—
360Dic6—101no—
361Dic6C4×AGL(1,𝔽5)11yescharacteristic
362C3×C8—21no—
363C3×C8—101no—
364C3⋊C8—51no—
365C3⋊C8—51no—
366C3⋊C8—102no—
367C3⋊C8—102no—
368C3⋊C8C22⋊AGL(1,𝔽5)11yescharacteristic
369C3⋊C8C22⋊AGL(1,𝔽5)11yescharacteristic
370C22×C5—61no—
371C22×C5#89211yescharacteristic
372C2×D5—22no—
373C2×D5—61no—
374C2×D5—122no—
375C2×D5#67⋊C411yescharacteristic
376C2×D5#67⋊C411yescharacteristic
377AGL(1,𝔽5)—24no—
378AGL(1,𝔽5)—42no—
379C4×C5—61no—
380C4×C5C42×D311yescharacteristic
381C4×C5D3×C22⋊C411yescharacteristic
382Dic5—61no—
383Dic5#89311yescharacteristic
384Dic5#89311yescharacteristic
385#41—31no—
386#41—151no—
387#41—151no—
388#41—151no—
389#41—301no—
390#41—301no—
391C2×Q8—151no—
392C2×D4—151no—
393C2×D4—301no—
394C22×C4—51no—
395C22×C4—101no—
396C22×C4—301no—
397M16—51no—
398M16—61no—
399M16—101no—
400M16—152no—
401M16—301no—
402M16—301no—
403M16—301no—
404M16—301no—
405M16D3×AGL(1,𝔽5)11yescharacteristic
406C2×C8—31no—
407C2×C8—61no—
408C2×C8—101no—
409C2×C8—151no—
410C2×C8—151no—
411C2×C8—151no—
412C2×C8—151no—
413C2×C8—151no—
414C2×C8—152no—
415C2×C8—301no—
416C2×C8—301no—
417C2×C8—301no—
418C2×C8—301no—
419C2×C8—301no—
420C2×C8—301no—
421C4⋊C4—52no—
422C22⋊C4—101no—
423C42—52no—
424C42—101no—
425C42—101no—
426C3×C5#176411yescharacteristic, complemented, Hall
427C22×C3—51no—
428C22×C3—51no—
429C22×C3—102no—
430C22×C3#413011yescharacteristic
431C2×D3—21no—
432C2×D3—101no—
433C2×D3—202no—
434C3×C4—51no—
435C3×C4—51no—
436C3×C4—104no—
437C3×C4—202no—
438C3×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
439C3×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
440Dic3—21no—
441Dic3—101no—
442C2×C5—21no—
443C2×C5—121no—
444C2×C5#466611yescharacteristic
445D5—21no—
446D5—121no—
447D5#482412yes—
448C23—51no—
449C23—301no—
450Q8—31no—
451Q8—151no—
452Q8—301no—
453D4—31no—
454D4—61no—
455D4—151no—
456D4—301no—
457D4—301no—
458D4—301no—
459D4—301no—
460C2×C4—51no—
461C2×C4—51no—
462C2×C4—51no—
463C2×C4—51no—
464C2×C4—51no—
465C2×C4—61no—
466C2×C4—102no—
467C2×C4—104no—
468C2×C4—102no—
469C2×C4—301no—
470C2×C4—301no—
471C2×C4—301no—
472C2×C4—301no—
473C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
474C8—21no—
475C8—31no—
476C8—31no—
477C8—101no—
478C8—151no—
479C8—151no—
480C8—302no—
481C8—302no—
482C2×C3—21no—
483C2×C3—52no—
484C2×C3—101no—
485C2×C3C4×C22⋊AGL(1,𝔽5)11yescharacteristic
486D3—41no—
487D3—201no—
488C5#6873911yescharacteristic, complemented, Sylow
489C22—51no—
490C22—51no—
491C22—61no—
492C22—102no—
493C22—301no—
494C22—602no—
495C22#9190711yescharacteristic
496C4—51no—
497C4—51no—
498C4—61no—
499C4—104no—
500C4—202no—
501C4—301no—
502C4C4×D3×AGL(1,𝔽5)11yescharacteristic
503C4D3×C22⋊AGL(1,𝔽5)11yescharacteristic
504C3#1060477711yescharacteristic, complemented, Sylow
505C2—21no—
506C2—52no—
507C2—101no—
508C2—121no—
509C2—601no—
510C2#1175184811yescharacteristic
subgroupG/H
centerC4D3×C22⋊AGL(1,𝔽5)
commutatorC3×C4×C5C2×C42
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C5×M16C2×C4
socleC2×C3×C5C4×C22⋊C4
prime radicalD5×#235C22
cosocleC2×C3×C4×D5C23
hypercenterM16D3×AGL(1,𝔽5)
hypocenterC3×C5#1764
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
4. #11751695C2semidirect
5. C2×AGL(1,𝔽5)×C3⋊C8C2semidirect
6. #11746772C2semidirect
7. C3×#64900C2semidirect
14. #91796C4semidirect
15. C5×#907C4semidirect
17. #91291C4semidirect
18. #91288C4semidirect
24. #91158C4semidirect
25. #91156C4semidirect
26. #91153C4semidirect
27. #91151C4semidirect
39. #64900D3semidirect
158. #907AGL(1,𝔽5)semidirect
426. C3×C5#1764semidirect
488. C5#68739semidirect
504. C3#10604777semidirect

Maximal subgroups

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

subgroupindex
D5×#9072
D5⋊#8052
#117517032
#117516952
C2×AGL(1,𝔽5)×C3⋊C82
#117467722
C3×#649002
#106047773
#687395