Finite Group Databasebeta

← #14295976

Subgroups — #14295976

Order 1920.

600 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5⋊#901C211yesmaximal, characteristic
2C3×C4×D5⋊C8C211yesmaximal, characteristic, complemented
3#11751704C211yesmaximal, characteristic
4#11751694C211yesmaximal, characteristic, complemented
5D5×#805C211yesmaximal, characteristic
6#11746733C211yesmaximal, characteristic, complemented
7#11746533C211yesmaximal, characteristic, complemented
8#10604005—31nomaximal, Hall
9D5×#235C2211yescharacteristic
10#91989C2211yescharacteristic
11D5⋊#208C2211yescharacteristic
12C2×C3×D5⋊C8C2211yescharacteristic
13C2×C3×D5⋊C8C2211yescharacteristic
14#91927—22no—
15D5⋊ΓL(1,𝔽25)—21no—
16#91914C412yescomplemented
17#91911C412yescomplemented
18D3×D5⋊C8—21no—
19C3×C42×D5C2211yescharacteristic
20D5×#208C2211yescharacteristic
21C3×C4×#170—22no—
22#91154C412yescomplemented
23#91152C412yescomplemented
24#91150—21no—
25ΓL(1,𝔽16)⋊C8—21no—
26#91089C411yescharacteristic, complemented
27C5×#805C411yescharacteristic, complemented
28#90981C411yescharacteristic, complemented
29#90979C411yescharacteristic, complemented
30#69789—51nomaximal, Hall
31D5⋊#131—31no—
32#65794—31no—
33#65728—31no—
34C4×D5⋊C8D311yescharacteristic, complemented
35D5×#104—31no—
36#64901—31no—
37#64862—31no—
38#6817C2×C411yescharacteristic
39C5×#235C2×C411yescharacteristic
40C2×C3×C4×D5—21no—
41C2×C3×C4×D5C2311yescharacteristic
42D5×#67—21no—
43#6782—21no—
44D5×D12C2×C411yescharacteristic
45C4×D3×D5—21no—
46#6773C2×C411yescharacteristic
47#6772C2×C411yescharacteristic
48#6771—21no—
49D5×Dic6C2×C411yescharacteristic
50#6763—21no—
51#6762—21no—
52C4⋊ΓL(1,𝔽16)C2×C412yes—
53C4×ΓL(1,𝔽16)—21no—
54#6758C2×C412yes—
55D5⋊(C3⋊C8)—21no—
56C2×C3×#170—21no—
57C2×C3×#170D412yes—
58C3×D5⋊C8—21no—
59C3×D5⋊C8C2×C412yes—
60C3×D5⋊C8C2×C412yes—
61C3×D5⋊C8D411yescharacteristic
62C3×D5⋊C8D411yescharacteristic
63#6740—22no—
64#6739—22no—
65#6738—22no—
66D3×#170—22no—
67C3×C42×C5C2×C411yescharacteristic
68#6710C2×C411yescharacteristic
69C5×#208C2×C411yescharacteristic
70C3×C4×Dic5—21no—
71C3×C4×Dic5C2×C411yescharacteristic
72#6649—21no—
73D5×C3⋊C8—21no—
74C2×#901—51no—
75C2×C3×C4×C8—51no—
76#5113—51no—
77#5101—51no—
78C2×#805—51no—
79#4552—51no—
80#4523—51no—
81D5×#41—31no—
82#4133—32no—
83#4131—32no—
84#4131—62no—
85#4130—31no—
86D5⋊M16—31no—
87D5⋊M16—61no—
88C2×D5⋊C8—61no—
89C2×D5⋊C8C2×D311yescharacteristic
90C2×D5⋊C8C2×D311yescharacteristic
91D5×M16—31no—
92C42×D5C2×D311yescharacteristic
93#4009—32no—
94#4007—32no—
95C4×#170—22no—
96#3992—61no—
97C8×AGL(1,𝔽5)—61no—
98C5×#104—31no—
99#3969—31no—
100#3957—31no—
101#3937—31no—
102#1764—151noSylow
103C22×C3×D5#4111yescharacteristic
104C2×D3×D5—21no—
105C2×ΓL(1,𝔽16)—22no—
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5C22×C411yescharacteristic
108#1220—21no—
109D60C4211yescharacteristic
110C4×D15—21no—
111Dic30C4211yescharacteristic
112C5×#67—21no—
113C5×D12C4211yescharacteristic
114C4×C5×D3—21no—
115C5×Dic6C4211yescharacteristic
116C2×C3×Dic5—21no—
117C2×C3×Dic5C2×D411yescharacteristic
118C3×C4×D5—24no—
119C3×C4×D5C22×C411yescharacteristic
120C3×C4×D5C22×C411yescharacteristic
121C3×C4×D5C2×D411yescharacteristic
122C3×C4×D5#4111yescharacteristic
123#1204—21no—
124#1203—21no—
125#1202—21no—
126#1201—21no—
127#1200—21no—
128D3×Dic5—21no—
129D5×Dic3—21no—
130C3⋊#170—22no—
131C3×#170—22no—
132C3×#170C22⋊C412yes—
133C3×#170C22⋊C412yes—
134C3⋊#168—21no—
135C5×C3⋊C8—21no—
136C2×#235—51no—
137C22×C3×C8—51no—
138C22×C3×C8—51no—
139C2×C3×C42—51no—
140#926—51no—
141C2×#208—51no—
142C2×#208—51no—
143#901—52no—
144#901—52no—
145#901—102no—
146C2×ΓL(1,𝔽25)—101no—
147C2×D3×C8—101no—
148C3×C4×C8—102no—
149#821—52no—
150#816—52no—
151#815—101no—
152C8×Dic3—101no—
153#805—51no—
154#805—51no—
155#805—51no—
156#805AGL(1,𝔽5)11yescharacteristic, complemented
157C5×#41—31no—
158#713—31no—
159Q8×D5—31no—
160#711—31no—
161#711—61no—
162D4×D5—31no—
163D4×D5—61no—
164#709—31no—
165#709—61no—
166C2×C4×D5—21no—
167C2×C4×D5—61no—
168C2×C4×D5C22×D311yescharacteristic
169C22⋊AGL(1,𝔽5)—61no—
170#704—61no—
171#704—62no—
172C2×#170—21no—
173C2×#170—62no—
174C2×#170#6712yes—
175C4⋊AGL(1,𝔽5)—32no—
176C4×AGL(1,𝔽5)—61no—
177#700—32no—
178#700—62no—
179D5⋊C8—21no—
180D5⋊C8—61no—
181D5⋊C8—62no—
182D5⋊C8C4×D312yes—
183D5⋊C8C4×D312yes—
184D5⋊C8D1211yescharacteristic
185D5⋊C8D1211yescharacteristic
186C5×M16—31no—
187C42×C5C4×D311yescharacteristic
188C4×Dic5—21no—
189C4×Dic5C4×D311yescharacteristic
190#681—31no—
191#676—61no—
192C8×D5—61no—
193C2×#131—151no—
194C2×#104—151no—
195#416—151no—
196#405—151no—
197C2×C4×C8—51no—
198#341—151no—
199#339—151no—
200C22×C3×C5C4×D411yescharacteristic
201C2×D15—21no—
202C2×C5×D3—21no—
203C2×C3×D5—22no—
204C2×C3×D5#11711yescharacteristic
205C2×C3×D5C4×D411yescharacteristic
206D3×D5—42no—
207ΓL(1,𝔽16)—42no—
208C3×C4×C5—22no—
209C3×C4×C5C2×C4211yescharacteristic
210C3×C4×C5C4×D411yescharacteristic
211Dic15—21no—
212C3×Dic5—22no—
213C3×Dic5C2×C22⋊C411yescharacteristic
214C3×Dic5C4×D411yescharacteristic
215C5×Dic3—21no—
216C22×C3×C4—51no—
217C22×C3×C4—101no—
218C2×#67—101no—
219#235—51no—
220#235—51no—
221#235—51no—
222#235—101no—
223#235—101no—
224#235C2×AGL(1,𝔽5)11yescharacteristic
225C2×D12—51no—
226C2×C4×D3—101no—
227C2×Dic6—51no—
228C2×C3×C8—51no—
229C2×C3×C8—51no—
230C2×C3×C8—52no—
231C2×C3×C8—52no—
232C2×C3×C8—52no—
233C2×C3×C8—101no—
234C2×C3×C8—101no—
235C3×C42—51no—
236C3×C42—101no—
237C3×C42C2×AGL(1,𝔽5)11yescharacteristic
238#217—101no—
239C4⋊Dic3—52no—
240C4×Dic3—101no—
241#208—52no—
242#208—51no—
243#208—101no—
244#208—101no—
245#208C2×AGL(1,𝔽5)11yescharacteristic
246C2×C3⋊C8—101no—
247C2×C3⋊C8—101no—
248ΓL(1,𝔽25)—102no—
249ΓL(1,𝔽25)—102no—
250D3×C8—102no—
251D3×C8—102no—
252C22×D5—61no—
253C22×D5#23511yescharacteristic
254C2×AGL(1,𝔽5)—62no—
255C5×Q8—31no—
256C5×D4—31no—
257C5×D4—61no—
258C2×C4×C5—21no—
259C2×C4×C5—61no—
260C2×C4×C5C2×C4×D311yescharacteristic
261C22⋊D5—61no—
262C22⋊D5—61no—
263C22⋊D5—61no—
264C2×Dic5—21no—
265C2×Dic5—61no—
266C2×Dic5C2×#6711yescharacteristic
267D20—31no—
268D20—61no—
269C4×D5—24no—
270C4×D5—61no—
271C4×D5—61no—
272C4×D5—61no—
273C4×D5C2×C4×D311yescharacteristic
274C4×D5C2×C4×D311yescharacteristic
275C4×D5C2×D1211yescharacteristic
276C4×D5#23511yescharacteristic
277Dic10—31no—
278Dic10—61no—
279#170—22no—
280#170—62no—
281#170(C2×D3)⋊C412yes—
282#170(C2×D3)⋊C412yes—
283C5×C8—61no—
284#168—61no—
285C2×#41—151no—
286#131—152no—
287#131—152no—
288#131—302no—
289C2×M16—151no—
290C2×M16—151no—
291C2×M16—301no—
292C22×C8—51no—
293C22×C8—51no—
294C22×C8—301no—
295#117—151no—
296C2×C42—51no—
297#104—31no—
298#104—151no—
299#104—151no—
300#104—151no—
301#103—152no—
302#102—152no—
303#97—301no—
304C4×C8—102no—
305C4×C8—301no—
306C2×C3×C5—21no—
307C2×C3×C5C4×C22⋊C411yescharacteristic
308D15—41no—
309C3×D5—21no—
310C3×D5#44312yes—
311C5×D3—41no—
312C23×C3—51no—
313C22×D3—101no—
314C2×C3×C4—21no—
315C2×C3×C4—51no—
316C2×C3×C4—51no—
317C2×C3×C4—51no—
318C2×C3×C4—51no—
319C2×C3×C4—51no—
320C2×C3×C4—104no—
321C2×C3×C4—101no—
322C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
323#67—21no—
324#67—101no—
325#67—101no—
326#67—101no—
327C2×Dic3—101no—
328C2×Dic3—102no—
329D12—51no—
330D12—101no—
331D12C4×AGL(1,𝔽5)11yescharacteristic
332C4×D3—21no—
333C4×D3—101no—
334C4×D3—101no—
335C4×D3—101no—
336Dic6—51no—
337Dic6—101no—
338Dic6C4×AGL(1,𝔽5)11yescharacteristic
339C3×C8—52no—
340C3×C8—52no—
341C3×C8—102no—
342C3⋊C8—21no—
343C3⋊C8—102no—
344C3⋊C8—101no—
345C22×C5—61no—
346C22×C5C4×#6711yescharacteristic
347C2×D5—22no—
348C2×D5—61no—
349C2×D5—122no—
350C2×D5#87211yescharacteristic
351C2×D5C4×D1211yescharacteristic
352AGL(1,𝔽5)—122no—
353C4×C5—22no—
354C4×C5—61no—
355C4×C5C42×D311yescharacteristic
356C4×C5C4×D1211yescharacteristic
357Dic5—22no—
358Dic5—61no—
359Dic5C2×(C2×D3)⋊C411yescharacteristic
360Dic5C4×#6711yescharacteristic
361#41—31no—
362#41—151no—
363#41—151no—
364#41—151no—
365#41—301no—
366#41—301no—
367C2×Q8—151no—
368C2×D4—151no—
369C2×D4—301no—
370C22×C4—51no—
371C22×C4—101no—
372C22×C4—301no—
373M16—31no—
374M16—152no—
375M16—151no—
376M16—302no—
377M16—302no—
378M16—301no—
379M16—301no—
380C2×C8—52no—
381C2×C8—51no—
382C2×C8—51no—
383C2×C8—52no—
384C2×C8—52no—
385C2×C8—101no—
386C2×C8—101no—
387C2×C8—302no—
388C2×C8—302no—
389C2×C8—301no—
390C2×C8—301no—
391C4⋊C4—152no—
392C22⋊C4—301no—
393C42—51no—
394C42—101no—
395C42—301no—
396C42D3×AGL(1,𝔽5)11yescharacteristic
397C3×C5#176411yescharacteristic, complemented, Hall
398C22×C3—51no—
399C22×C3—51no—
400C22×C3—102no—
401C22×C3D4×AGL(1,𝔽5)11yescharacteristic
402C2×D3—21no—
403C2×D3—101no—
404C2×D3—202no—
405C3×C4—22no—
406C3×C4—51no—
407C3×C4—51no—
408C3×C4—102no—
409C3×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
410C3×C4D4×AGL(1,𝔽5)11yescharacteristic
411Dic3—21no—
412Dic3—101no—
413Dic3—202no—
414C2×C5—21no—
415C2×C5—121no—
416C2×C5C4×(C2×D3)⋊C411yescharacteristic
417D5—21no—
418D5—121no—
419D5#470012yes—
420C23—51no—
421C23—301no—
422Q8—31no—
423Q8—151no—
424Q8—301no—
425D4—31no—
426D4—61no—
427D4—151no—
428D4—301no—
429D4—301no—
430D4—301no—
431D4—301no—
432C2×C4—21no—
433C2×C4—51no—
434C2×C4—51no—
435C2×C4—51no—
436C2×C4—51no—
437C2×C4—51no—
438C2×C4—61no—
439C2×C4—104no—
440C2×C4—101no—
441C2×C4—301no—
442C2×C4—301no—
443C2×C4—301no—
444C2×C4—301no—
445C2×C4—302no—
446C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
447C8—52no—
448C8—52no—
449C8—61no—
450C8—102no—
451C8—302no—
452C8—301no—
453C2×C3—21no—
454C2×C3—52no—
455C2×C3—101no—
456C2×C3C22⋊C4×AGL(1,𝔽5)11yescharacteristic
457D3—41no—
458D3—201no—
459C5#6978911yescharacteristic, complemented, Sylow
460C22—51no—
461C22—51no—
462C22—61no—
463C22—102no—
464C22—301no—
465C22—602no—
466C22AGL(1,𝔽5)×#6711yescharacteristic
467C4—22no—
468C4—51no—
469C4—51no—
470C4—61no—
471C4—102no—
472C4—301no—
473C4—602no—
474C4C4×D3×AGL(1,𝔽5)11yescharacteristic
475C4AGL(1,𝔽5)×D1211yescharacteristic
476C3#1060400511yescharacteristic, complemented, Sylow
477C2—21no—
478C2—52no—
479C2—101no—
480C2—121no—
481C2—601no—
482C2AGL(1,𝔽5)×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC4AGL(1,𝔽5)×D12
commutatorC3×C4×C5C2×C42
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C42×C5C2×C4
socleC2×C3×C5C4×C22⋊C4
prime radicalD5×#235C22
cosocleC2×C3×C4×D5C23
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#1764
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×C4×D5⋊C8C2semidirect
4. #11751694C2semidirect
6. #11746733C2semidirect
7. #11746533C2semidirect
16. #91914C4semidirect
17. #91911C4semidirect
22. #91154C4semidirect
23. #91152C4semidirect
26. #91089C4semidirect
27. C5×#805C4semidirect
28. #90981C4semidirect
29. #90979C4semidirect
34. C4×D5⋊C8D3semidirect
156. #805AGL(1,𝔽5)semidirect
397. C3×C5#1764semidirect
459. C5#69789semidirect
476. C3#10604005semidirect

Maximal subgroups

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

subgroupindex
D5⋊#9012
C3×C4×D5⋊C82
#117517042
#117516942
D5×#8052
#117467332
#117465332
#106040053
#697895