Finite Group Databasebeta

← #14326880

Subgroups — #14326880

Order 1920.

1008 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×#926C211yesmaximal, characteristic
2#11752424C211yesmaximal, characteristic
3C3×C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic
4#11752017C214yesmaximal
5#11751859C212yesmaximal, complemented
6AGL(1,𝔽5)×C4⋊Dic3C212yesmaximal, complemented
7#11751832C212yesmaximal, complemented
8#11751746C212yesmaximal, complemented
9#10617133—31nomaximal, Hall
10C22×C3×C4×D5C2211yescharacteristic
11C2×#6763C2212yes—
12C2×C4⋊ΓL(1,𝔽16)C2212yes—
13C2×C3×C22⋊AGL(1,𝔽5)C2212yes—
14C2×C3×C4×AGL(1,𝔽5)C2212yes—
15C2×Dic3⋊AGL(1,𝔽5)C2214yes—
16C2×Dic3×AGL(1,𝔽5)C2214yes—
17#91815C411yescharacteristic, complemented
18C5×#926C411yescharacteristic, complemented
19D5×#217C2212yes—
20#91528C412yescomplemented
21D5×C4⋊Dic3C2211yescharacteristic
22D5×C4⋊Dic3C2211yescharacteristic
23C4×D5×Dic3C2212yes—
24#91358C411yescharacteristic, complemented
25#91358C411yescharacteristic, complemented
26#91357C412yescomplemented
27#91235C2212yes—
28C3×(C2×AGL(1,𝔽5))⋊C4C2211yescharacteristic
29C3×(C2×AGL(1,𝔽5))⋊C4C2211yescharacteristic
30(C2×AGL(1,𝔽5))⋊Dic3C2214yes—
31(C2×AGL(1,𝔽5))⋊Dic3C2214yes—
32#73459—51nomaximal, Hall
33D5×#117—31no—
34#65765—31no—
35C4×C22⋊AGL(1,𝔽5)D311yescharacteristic
36#65712—32no—
37C4⋊C4×AGL(1,𝔽5)—32no—
38#65700—34no—
39#65689—32no—
40#65686—32no—
41C23×C3×D5C2311yescharacteristic
42C22×ΓL(1,𝔽16)C2314yes—
43C22×C3×AGL(1,𝔽5)C2312yes—
44C22×C3×AGL(1,𝔽5)C2312yes—
45C22×C3×C4×C5C2×C411yescharacteristic
46C22×C3×Dic5C2×C411yescharacteristic
47C2×C3×C4×D5—24no—
48C2×C3×C4×D5C2×C411yescharacteristic
49C2×C3×C4×D5C2×C411yescharacteristic
50C2×C3×C4×D5C2311yescharacteristic
51C2×C3×C4×D5C2311yescharacteristic
52C2×D5×Dic3C2312yes—
53C2×D5×Dic3C2312yes—
54#6763—42no—
55C4⋊ΓL(1,𝔽16)—24no—
56C3×C22⋊AGL(1,𝔽5)—24no—
57C3×C4×AGL(1,𝔽5)—24no—
58Dic3⋊AGL(1,𝔽5)—44no—
59Dic3×AGL(1,𝔽5)—28no—
60#6719C2×C412yes—
61C4⋊Dic15C2×C411yescharacteristic
62C4⋊Dic15C2×C411yescharacteristic
63C4×Dic15C2×C412yes—
64C5×#217C2×C412yes—
65C5×C4⋊Dic3C2×C411yescharacteristic
66C5×C4⋊Dic3C2×C411yescharacteristic
67C4×C5×Dic3C2×C412yes—
68#6668C2×C412yes—
69#6668C2×C412yes—
70#6665C2×C412yes—
71#6665C2×C412yes—
72Dic3×Dic5C2×C412yes—
73Dic3×Dic5C2×C412yes—
74C2×#926—51no—
75C3×C4×C22⋊C4—51no—
76#5212—51no—
77#4944—54no—
78C4×C4⋊Dic3—52no—
79#4928—52no—
80#4650—52no—
81#4648—52no—
82C22×C4×D5C2×D311yescharacteristic
83C2×C22⋊AGL(1,𝔽5)—32no—
84C2×C22⋊AGL(1,𝔽5)C2×D312yes—
85C2×C4⋊AGL(1,𝔽5)—32no—
86C2×C4⋊AGL(1,𝔽5)—34no—
87C2×C4×AGL(1,𝔽5)—34no—
88C2×C4×AGL(1,𝔽5)C2×D312yes—
89C5×#117—31no—
90#4072—31no—
91#4038—31no—
92#4038—31no—
93D5×C4⋊C4—31no—
94D5×C4⋊C4—31no—
95D5×C22⋊C4—32no—
96#4023—32no—
97#4018—32no—
98C42×D5—32no—
99(C2×AGL(1,𝔽5))⋊C4—34no—
100(C2×AGL(1,𝔽5))⋊C4—34no—
101(C2×AGL(1,𝔽5))⋊C4—32no—
102(C2×AGL(1,𝔽5))⋊C4C2×D311yescharacteristic
103(C2×AGL(1,𝔽5))⋊C4C2×D311yescharacteristic
104#2346—151noSylow
105C23×C3×C5C22×C411yescharacteristic
106C22×C3×D5—22no—
107C22×C3×D5—22no—
108C22×C3×D5—21no—
109C22×C3×D5—21no—
110C22×C3×D5C22×C411yescharacteristic
111C22×C3×D5C2411yescharacteristic
112C2×ΓL(1,𝔽16)—24no—
113C2×ΓL(1,𝔽16)—24no—
114C2×ΓL(1,𝔽16)—24no—
115C2×C3×AGL(1,𝔽5)—22no—
116C2×C3×AGL(1,𝔽5)—22no—
117C2×C3×AGL(1,𝔽5)—22no—
118C2×C3×AGL(1,𝔽5)—22no—
119C2×C3×AGL(1,𝔽5)—22no—
120C2×C3×AGL(1,𝔽5)#4114yes—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5C22×C411yescharacteristic
123C2×C3×C4×C5C22×C411yescharacteristic
124C2×Dic15C22×C412yes—
125C2×Dic15C22×C412yes—
126C2×C5×Dic3C22×C412yes—
127C2×C5×Dic3C22×C412yes—
128C2×C3×Dic5—22no—
129C2×C3×Dic5C22×C411yescharacteristic
130C2×C3×Dic5C22×C411yescharacteristic
131C3×C4×D5—22no—
132C3×C4×D5—22no—
133C3×C4×D5—22no—
134C3×C4×D5#4114yes—
135D5×Dic3—24no—
136D5×Dic3—22no—
137D5×Dic3—22no—
138C23×C3×C4—51no—
139C2×C3×C22⋊C4—52no—
140C2×C3×C42—52no—
141C2×#217—52no—
142C2×#217—52no—
143#926—51no—
144#926—52no—
145#926—51no—
146#926—51no—
147#926—52no—
148#926AGL(1,𝔽5)11yescharacteristic, complemented
149C2×C4⋊Dic3—51no—
150C2×C4⋊Dic3—51no—
151C2×C4⋊Dic3—52no—
152C2×Dic3⋊C4—54no—
153C2×C4×Dic3—52no—
154C2×C4×Dic3—54no—
155C3×#95—51no—
156C3×#95—51no—
157C3⋊#95—54no—
158C3⋊#95—54no—
159C3⋊#95—52no—
160C23×D5C22×D311yescharacteristic
161C22×AGL(1,𝔽5)—34no—
162C22×AGL(1,𝔽5)C22×D312yes—
163C22×AGL(1,𝔽5)C22×D312yes—
164C22×C4×C5C4×D311yescharacteristic
165C22×Dic5C4×D311yescharacteristic
166C2×C4×D5—24no—
167C2×C4×D5—32no—
168C2×C4×D5—32no—
169C2×C4×D5C4×D311yescharacteristic
170C2×C4×D5C4×D311yescharacteristic
171C2×C4×D5C22×D311yescharacteristic
172C2×C4×D5C22×D311yescharacteristic
173C22⋊AGL(1,𝔽5)—24no—
174C22⋊AGL(1,𝔽5)—122no—
175C4⋊AGL(1,𝔽5)—64no—
176C4⋊AGL(1,𝔽5)—124no—
177C4×AGL(1,𝔽5)—24no—
178C4×AGL(1,𝔽5)—68no—
179C5×C4⋊C4—31no—
180C5×C4⋊C4—31no—
181C5×C22⋊C4—32no—
182C42×C5—32no—
183C22⋊Dic5—32no—
184#685—32no—
185#685—32no—
186C4⋊Dic5—31no—
187C4⋊Dic5—31no—
188#683—32no—
189#683—32no—
190C4×Dic5—32no—
191C4×Dic5—32no—
192C4×Dic5—32no—
193C2×#117—151no—
194#389—152no—
195#388—154no—
196#382—152no—
197#380—151no—
198C4×C4⋊C4—152no—
199C4×C22⋊C4—51no—
200#376—152no—
201C22×C3×C5—21no—
202C22×C3×C5—21no—
203C22×C3×C5—21no—
204C22×C3×C5C23×C411yescharacteristic
205C2×C3×D5—21no—
206C2×C3×D5—21no—
207C2×C3×D5—21no—
208C2×C3×D5—42no—
209C2×C3×D5—42no—
210C2×C3×D5C2×#4112yes—
211C2×C3×D5C2×#4112yes—
212C2×C3×D5H(5,𝔽2)11yescharacteristic
213C2×C3×D52−1+411yescharacteristic
214ΓL(1,𝔽16)—44no—
215C3×AGL(1,𝔽5)—24no—
216C3×AGL(1,𝔽5)—42no—
217C3×C4×C5—21no—
218C3×C4×C5#11712yes—
219Dic15—22no—
220Dic15—22no—
221C3×Dic5—21no—
222C3×Dic5#11712yes—
223C5×Dic3—22no—
224C5×Dic3—22no—
225C24×C3—51no—
226C22×C3×C4—51no—
227C22×C3×C4—52no—
228C22×C3×C4—51no—
229C22×C3×C4—51no—
230C22×C3×C4—51no—
231C22×C3×C4—51no—
232C22×C3×C4—52no—
233C22×C3×C4—104no—
234C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
235C22×Dic3—52no—
236C22×Dic3—52no—
237C22×Dic3—54no—
238C3×C22⋊C4—104no—
239C3×C42—104no—
240#217—52no—
241#217—52no—
242#217—52no—
243#217—202no—
244#217C2×AGL(1,𝔽5)12yes—
245C4⋊Dic3—51no—
246C4⋊Dic3—51no—
247C4⋊Dic3—52no—
248C4⋊Dic3—52no—
249C4⋊Dic3—104no—
250C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
251C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
252Dic3⋊C4—204no—
253C4×Dic3—52no—
254C4×Dic3—52no—
255C4×Dic3—52no—
256C4×Dic3—108no—
257C4×Dic3C2×AGL(1,𝔽5)12yes—
258C23×C5C2×C4×D311yescharacteristic
259C22×D5—22no—
260C22×D5—22no—
261C22×D5—21no—
262C22×D5—21no—
263C22×D5C2×C4×D311yescharacteristic
264C22×D5C23×D311yescharacteristic
265C2×AGL(1,𝔽5)—22no—
266C2×AGL(1,𝔽5)—22no—
267C2×AGL(1,𝔽5)—22no—
268C2×AGL(1,𝔽5)—22no—
269C2×AGL(1,𝔽5)—22no—
270C2×AGL(1,𝔽5)—64no—
271C2×AGL(1,𝔽5)—64no—
272C2×AGL(1,𝔽5)—64no—
273C2×AGL(1,𝔽5)#23514yes—
274C2×C4×C5—22no—
275C2×C4×C5—32no—
276C2×C4×C5—32no—
277C2×C4×C5C2×C4×D311yescharacteristic
278C2×C4×C5C2×C4×D311yescharacteristic
279C2×Dic5—22no—
280C2×Dic5—32no—
281C2×Dic5—32no—
282C2×Dic5C2×C4×D311yescharacteristic
283C2×Dic5C2×C4×D311yescharacteristic
284C4×D5—22no—
285C4×D5—22no—
286C4×D5—22no—
287C4×D5—64no—
288C4×D5—62no—
289C4×D5—62no—
290C4×D5#23714yes—
291C23×C4—51no—
292#117—31no—
293#117—152no—
294#117—152no—
295#117—151no—
296#117—151no—
297#117—151no—
298C2×C4⋊C4—151no—
299C2×C4⋊C4—151no—
300C2×C4⋊C4—152no—
301C2×C4⋊C4—154no—
302C2×C22⋊C4—52no—
303C2×C22⋊C4—152no—
304C2×C22⋊C4—152no—
305C2×C42—52no—
306C2×C42—152no—
307C2×C42—154no—
308#95—51no—
309#95—51no—
310#95—152no—
311#95—154no—
312#95—154no—
313C2×C3×C5—41no—
314C2×C3×C5C2×#11711yescharacteristic
315C2×C3×C5C4×#4111yescharacteristic
316C2×C3×C5#52011yescharacteristic
317C3×D5—41no—
318C3×D5#55212yes—
319C3×D5#55512yes—
320C23×C3—51no—
321C23×C3—51no—
322C23×C3—101no—
323C23×C3—102no—
324C23×C3—102no—
325C23×C3—101no—
326C23×C3C22×AGL(1,𝔽5)11yescharacteristic
327C2×C3×C4—22no—
328C2×C3×C4—51no—
329C2×C3×C4—54no—
330C2×C3×C4—54no—
331C2×C3×C4—51no—
332C2×C3×C4—102no—
333C2×C3×C4—102no—
334C2×C3×C4—102no—
335C2×C3×C4—102no—
336C2×C3×C4—102no—
337C2×C3×C4—102no—
338C2×C3×C4—102no—
339C2×C3×C4—102no—
340C2×C3×C4—102no—
341C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
342C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
343C2×Dic3—52no—
344C2×Dic3—52no—
345C2×Dic3—104no—
346C2×Dic3—102no—
347C2×Dic3—102no—
348C2×Dic3—104no—
349C2×Dic3—104no—
350C2×Dic3—104no—
351C2×Dic3C22×AGL(1,𝔽5)12yes—
352C2×Dic3C22×AGL(1,𝔽5)12yes—
353C22×C5—21no—
354C22×C5—21no—
355C22×C5—21no—
356C22×C5C22×C4×D311yescharacteristic
357C2×D5—21no—
358C2×D5—21no—
359C2×D5—21no—
360C2×D5—42no—
361C2×D5—42no—
362C2×D5C2×#23512yes—
363C2×D5C2×#23712yes—
364C2×D5#100911yescharacteristic
365C2×D5#101011yescharacteristic
366AGL(1,𝔽5)—24no—
367AGL(1,𝔽5)—42no—
368AGL(1,𝔽5)—124no—
369C4×C5—21no—
370C4×C5—62no—
371C4×C5—62no—
372C4×C5#87712yes—
373Dic5—21no—
374Dic5—62no—
375Dic5—62no—
376Dic5#87712yes—
377C24—51no—
378C22×C4—51no—
379C22×C4—52no—
380C22×C4—51no—
381C22×C4—51no—
382C22×C4—51no—
383C22×C4—51no—
384C22×C4—52no—
385C22×C4—104no—
386C22×C4—152no—
387C22×C4—152no—
388C22×C4—154no—
389C22×C4D3×AGL(1,𝔽5)11yescharacteristic
390C4⋊C4—31no—
391C4⋊C4—31no—
392C4⋊C4—152no—
393C4⋊C4—152no—
394C4⋊C4—151no—
395C4⋊C4—151no—
396C4⋊C4—304no—
397C4⋊C4—604no—
398C22⋊C4—32no—
399C22⋊C4—104no—
400C22⋊C4—152no—
401C22⋊C4—152no—
402C22⋊C4—152no—
403C22⋊C4—602no—
404C42—32no—
405C42—104no—
406C42—152no—
407C42—152no—
408C42—152no—
409C42—308no—
410C3×C5#234611yescharacteristic, complemented, Hall
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—52no—
415C22×C3—52no—
416C22×C3—51no—
417C22×C3—51no—
418C22×C3—101no—
419C22×C3—101no—
420C22×C3—101no—
421C22×C3—202no—
422C22×C3—202no—
423C22×C3C23×AGL(1,𝔽5)11yescharacteristic
424C3×C4—21no—
425C3×C4—52no—
426C3×C4—104no—
427C3×C4—101no—
428C3×C4—202no—
429C3×C4#413012yes—
430Dic3—22no—
431Dic3—22no—
432Dic3—102no—
433Dic3—102no—
434Dic3—204no—
435C2×C5—41no—
436C2×C5C4×#23511yescharacteristic
437C2×C5C2×#87711yescharacteristic
438C2×C5#552311yescharacteristic
439D5—41no—
440D5#554112yes—
441D5#556312yes—
442C23—51no—
443C23—51no—
444C23—102no—
445C23—101no—
446C23—102no—
447C23—101no—
448C23C2×D3×AGL(1,𝔽5)11yescharacteristic
449C2×C4—22no—
450C2×C4—32no—
451C2×C4—32no—
452C2×C4—51no—
453C2×C4—54no—
454C2×C4—54no—
455C2×C4—51no—
456C2×C4—102no—
457C2×C4—102no—
458C2×C4—102no—
459C2×C4—102no—
460C2×C4—102no—
461C2×C4—102no—
462C2×C4—102no—
463C2×C4—102no—
464C2×C4—102no—
465C2×C4—152no—
466C2×C4—152no—
467C2×C4—304no—
468C2×C4—302no—
469C2×C4—302no—
470C2×C4—304no—
471C2×C4—304no—
472C2×C4—304no—
473C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
474C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
475C2×C3—41no—
476C2×C3—52no—
477C2×C3—52no—
478C2×C3—201no—
479C2×C3C2×#413011yescharacteristic
480C2×C3#41×AGL(1,𝔽5)11yescharacteristic
481C2×C3#6626611yescharacteristic
482C5#7345911yescharacteristic, complemented, Sylow
483C22—21no—
484C22—21no—
485C22—21no—
486C22—52no—
487C22—51no—
488C22—51no—
489C22—52no—
490C22—101no—
491C22—101no—
492C22—101no—
493C22—202no—
494C22—202no—
495C22C22×D3×AGL(1,𝔽5)11yescharacteristic
496C4—21no—
497C4—52no—
498C4—62no—
499C4—62no—
500C4—104no—
501C4—101no—
502C4—202no—
503C4—302no—
504C4—302no—
505C4—604no—
506C4#9192312yes—
507C3#1061713311yescharacteristic, complemented, Sylow
508C2—41no—
509C2—52no—
510C2—52no—
511C2—201no—
512C2AGL(1,𝔽5)×#23511yescharacteristic
513C2#1175683811yescharacteristic
514C2C2×#9192311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC22×C3×C4×C5C2×C4
prime radicalC23×C3×D5C23
cosocleC22×C3×D5C24
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2346
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
5. #11751859C2semidirect
6. AGL(1,𝔽5)×C4⋊Dic3C2semidirect
7. #11751832C2semidirect
8. #11751746C2semidirect
17. #91815C4semidirect
18. C5×#926C4semidirect
20. #91528C4semidirect
24. #91358C4semidirect
25. #91358C4semidirect
26. #91357C4semidirect
148. #926AGL(1,𝔽5)semidirect
410. C3×C5#2346semidirect
482. C5#73459semidirect
507. C3#10617133semidirect

Maximal subgroups

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

subgroupindex
D5×#9262
#117524242
C3×C4×C22⋊AGL(1,𝔽5)2
#117520172
#117518592
AGL(1,𝔽5)×C4⋊Dic32
#117518322
#117517462
#106171333
#734595