Finite Group Databasebeta

← #14307270

Subgroups — #14307270

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×#890C211yesmaximal, characteristic
2Dic3×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3D5⋊(Dic3⋊C8)C211yesmaximal, characteristic
4#11746745C211yesmaximal, characteristic
5C3×#64874C211yesmaximal, characteristic
6#11746530C211yesmaximal, characteristic
7#11746528C211yesmaximal, characteristic
8#10604589—31nomaximal, Hall
9C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
10C2×D5⋊(C3⋊C8)C2211yescharacteristic
11C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
12C2×C3×D5⋊C8C2211yescharacteristic
13C2×Dic3×AGL(1,𝔽5)—21no—
14#91779C411yescharacteristic, complemented
15C5×#890C411yescharacteristic, complemented
16C3×D5×C4⋊C4C2211yescharacteristic
17D5×C4⋊Dic3—21no—
18D5×C4⋊Dic3C2211yescharacteristic
19D5×Dic3⋊C4—21no—
20C4×D5×Dic3C2211yescharacteristic
21#91384C411yescharacteristic, complemented
22#91381C411yescharacteristic, complemented
23#91335—21no—
24#91334—21no—
25#91177—21no—
26Dic3⋊#170—21no—
27(C2×AGL(1,𝔽5))⋊Dic3—21no—
28#68801—51nomaximal, Hall
29D5×#125—31no—
30C4×C4⋊AGL(1,𝔽5)—31no—
31D5⋊(C4⋊C8)—31no—
32#64874—31no—
33#64874—31no—
34#64874—31no—
35#64874D311yescharacteristic
36C22×ΓL(1,𝔽16)—21no—
37C22×C3×AGL(1,𝔽5)—21no—
38C2×C3×C4×D5—21no—
39C2×C3×C4×D5C2311yescharacteristic
40C2×D5×Dic3—21no—
41C2×D5×Dic3D411yescharacteristic
42C2×D5×Dic3D411yescharacteristic
43C2×C3⋊#170—21no—
44C4⋊ΓL(1,𝔽16)—21no—
45C4⋊ΓL(1,𝔽16)Q812yes—
46D5⋊(C3⋊C8)—21no—
47D5⋊(C3⋊C8)—21no—
48C2×C3×#170—21no—
49C3×C4⋊AGL(1,𝔽5)—21no—
50C3×C4⋊AGL(1,𝔽5)Q812yes—
51C3×D5⋊C8—21no—
52C3×D5⋊C8—21no—
53Dic3×AGL(1,𝔽5)—42no—
54C3×C5×C4⋊C4C2×C411yescharacteristic
55C4⋊Dic15—21no—
56C4⋊Dic15C2×C411yescharacteristic
57#6712—21no—
58C4×Dic15C2×C411yescharacteristic
59C5×C4⋊Dic3—21no—
60C5×C4⋊Dic3C2×C411yescharacteristic
61C5×Dic3⋊C4—21no—
62C4×C5×Dic3C2×C411yescharacteristic
63C3×C4⋊Dic5C2×C411yescharacteristic
64C3×#683—21no—
65Dic3⋊Dic5—21no—
66#6669—21no—
67#6668—21no—
68#6668—21no—
69#6668—21no—
70Dic3×Dic5D411yescharacteristic
71Dic3×Dic5D411yescharacteristic
72C2×#890—51no—
73C2×Dic3⋊C8—51no—
74Dic3×C4⋊C4—51no—
75C3×#340—51no—
76#4551—51no—
77#4529—51no—
78#4529—51no—
79C2×C4⋊AGL(1,𝔽5)—31no—
80C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
81C2×C4×AGL(1,𝔽5)—61no—
82C2×D5⋊C8—31no—
83C2×D5⋊C8C2×D311yescharacteristic
84C5×#125—31no—
85D5×C4⋊C4—31no—
86D5×C4⋊C4—61no—
87D5×C4⋊C4—61no—
88D5×C4⋊C4C2×D311yescharacteristic
89#4036—31no—
90#4036—31no—
91#4035—61no—
92#4035—61no—
93C42×D5—31no—
94#4016—31no—
95(C2×AGL(1,𝔽5))⋊C4—61no—
96#4004—61no—
97C4⋊#170—61no—
98#1989—151noSylow
99C22×C3×D5C2×D411yescharacteristic
100C2×ΓL(1,𝔽16)—22no—
101C2×ΓL(1,𝔽16)—41no—
102C2×ΓL(1,𝔽16)—41no—
103C2×C3×AGL(1,𝔽5)—22no—
104C2×C3×AGL(1,𝔽5)—41no—
105C2×C3×AGL(1,𝔽5)—41no—
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5C22×C411yescharacteristic
108C2×Dic15—21no—
109C2×Dic15C22⋊C411yescharacteristic
110C2×Dic15C22⋊C411yescharacteristic
111C2×C5×Dic3—21no—
112C2×C5×Dic3C22⋊C411yescharacteristic
113C2×C5×Dic3C22⋊C411yescharacteristic
114C2×C3×Dic5—21no—
115C2×C3×Dic5C2×D411yescharacteristic
116C3×C4×D5—41no—
117C3×C4×D5—41no—
118C3×C4×D5C2×Q811yescharacteristic
119C3×C4×D5C2×Q811yescharacteristic
120C3×C4×D5#4111yescharacteristic
121C3×C4×D5#4111yescharacteristic
122D5×Dic3—21no—
123D5×Dic3—21no—
124D5×Dic3—21no—
125D5×Dic3—21no—
126D5×Dic3—41no—
127D5×Dic3—41no—
128C3⋊#170—41no—
129C3×#170—41no—
130C22×C3×C8—51no—
131C2×C3×C4⋊C4—51no—
132C2×C3×C4⋊C4—51no—
133C2×C4⋊Dic3—51no—
134C2×C4⋊Dic3—51no—
135C2×C4⋊Dic3—101no—
136C2×Dic3⋊C4—101no—
137C2×C4×Dic3—51no—
138C2×C4×Dic3—101no—
139C22×C3⋊C8—51no—
140#890—51no—
141#890—51no—
142#890—51no—
143#890—101no—
144#890—101no—
145#890AGL(1,𝔽5)11yescharacteristic, complemented
146C3⋊#95—101no—
147Dic3⋊C8—101no—
148Dic3⋊C8—101no—
149C22×AGL(1,𝔽5)—21no—
150C22×AGL(1,𝔽5)—61no—
151C2×C4×D5—21no—
152C2×C4×D5—31no—
153C2×C4×D5—31no—
154C2×C4×D5—61no—
155C2×C4×D5C22×D311yescharacteristic
156C2×#170—21no—
157C2×#170—61no—
158C4⋊AGL(1,𝔽5)—21no—
159C4⋊AGL(1,𝔽5)—32no—
160C4⋊AGL(1,𝔽5)—61no—
161C4⋊AGL(1,𝔽5)Dic612yes—
162C4×AGL(1,𝔽5)—122no—
163D5⋊C8—21no—
164D5⋊C8—21no—
165D5⋊C8—61no—
166D5⋊C8—61no—
167C5×C4⋊C4—31no—
168C5×C4⋊C4—61no—
169C5×C4⋊C4—61no—
170C5×C4⋊C4C4×D311yescharacteristic
171C42×C5—31no—
172C4⋊Dic5—31no—
173C4⋊Dic5—61no—
174C4⋊Dic5—61no—
175C4⋊Dic5C4×D311yescharacteristic
176#683—21no—
177#683—61no—
178#683—61no—
179#683—61no—
180#683—61no—
181#683—61no—
182C4×Dic5—31no—
183C4×Dic5—31no—
184C4×Dic5—31no—
185C2×#125—151no—
186C2×C4⋊C8—151no—
187C4×C4⋊C4—151no—
188#340—51no—
189#340—151no—
190#340—151no—
191#340—151no—
192C22×C3×C5C2×C22⋊C411yescharacteristic
193C2×C3×D5C22⋊Q811yescharacteristic
194C2×C3×D5C22⋊Q811yescharacteristic
195C2×C3×D5#13511yescharacteristic
196C2×C3×D5#13511yescharacteristic
197C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
198C2×C3×D5#13711yescharacteristic
199ΓL(1,𝔽16)—42no—
200C3×AGL(1,𝔽5)—42no—
201C3×C4×C5—41no—
202C3×C4×C5C4×Q811yescharacteristic
203C3×C4×C5C4×Q811yescharacteristic
204Dic15—21no—
205Dic15—21no—
206Dic15—41no—
207C3×Dic5—41no—
208C3×Dic5#12411yescharacteristic
209C3×Dic5C4⋊Q811yescharacteristic
210C5×Dic3—21no—
211C5×Dic3—21no—
212C5×Dic3—41no—
213C22×C3×C4—51no—
214C22×C3×C4—101no—
215C22×C3×C4—101no—
216C22×Dic3—51no—
217C22×Dic3—51no—
218C22×Dic3—101no—
219C22×Dic3—101no—
220C2×C3×C8—101no—
221C2×C3×C8—101no—
222C2×C3×C8—101no—
223C3×C4⋊C4—52no—
224C3×C4⋊C4—51no—
225C3×C4⋊C4—101no—
226C3×C4⋊C4—101no—
227C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
228C4⋊Dic3—21no—
229C4⋊Dic3—52no—
230C4⋊Dic3—51no—
231C4⋊Dic3—101no—
232C4⋊Dic3—101no—
233C4⋊Dic3—101no—
234C4⋊Dic3—101no—
235C4⋊Dic3—101no—
236C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
237Dic3⋊C4—21no—
238Dic3⋊C4—101no—
239Dic3⋊C4—101no—
240Dic3⋊C4—101no—
241C4×Dic3—51no—
242C4×Dic3—51no—
243C4×Dic3—51no—
244C4×Dic3—202no—
245C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
246C2×C3⋊C8—101no—
247C2×C3⋊C8—101no—
248C2×C3⋊C8—101no—
249C22×D5D3×D411yescharacteristic
250C2×AGL(1,𝔽5)—22no—
251C2×AGL(1,𝔽5)—41no—
252C2×AGL(1,𝔽5)—41no—
253C2×AGL(1,𝔽5)—62no—
254C2×AGL(1,𝔽5)—121no—
255C2×AGL(1,𝔽5)—121no—
256C2×C4×C5—21no—
257C2×C4×C5—31no—
258C2×C4×C5—31no—
259C2×C4×C5—61no—
260C2×C4×C5C2×C4×D311yescharacteristic
261C2×Dic5—21no—
262C2×Dic5—31no—
263C2×Dic5—31no—
264C2×Dic5—61no—
265C2×Dic5D3×D411yescharacteristic
266C4×D5—41no—
267C4×D5—41no—
268C4×D5—61no—
269C4×D5—61no—
270C4×D5—61no—
271C4×D5—61no—
272C4×D5—121no—
273C4×D5—121no—
274C4×D5C2×Dic611yescharacteristic
275C4×D5#23511yescharacteristic
276C4×D5#23711yescharacteristic
277C4×D5D3×Q811yescharacteristic
278#170—41no—
279#170—121no—
280C22×C8—51no—
281C22×C8—151no—
282#125—31no—
283#125—151no—
284#125—151no—
285#125—151no—
286#125—301no—
287#125—301no—
288C2×C4⋊C4—51no—
289C2×C4⋊C4—51no—
290C2×C4⋊C4—151no—
291C2×C4⋊C4—151no—
292C2×C4⋊C4—301no—
293C2×C4⋊C4—301no—
294C2×C42—151no—
295C2×C42—301no—
296C4⋊C8—301no—
297C4⋊C8—301no—
298#95—301no—
299C2×C3×C5#39111yescharacteristic
300C2×C3×C5#41611yescharacteristic
301C2×C3×C5#41711yescharacteristic
302C3×D5#47811yescharacteristic
303C3×D5#47811yescharacteristic
304C3×D5#47911yescharacteristic
305C3×D5#47911yescharacteristic
306C23×C3—51no—
307C2×C3×C4—21no—
308C2×C3×C4—51no—
309C2×C3×C4—51no—
310C2×C3×C4—51no—
311C2×C3×C4—51no—
312C2×C3×C4—51no—
313C2×C3×C4—102no—
314C2×C3×C4—101no—
315C2×C3×C4—201no—
316C2×C3×C4—201no—
317C2×C3×C4—201no—
318C2×C3×C4—201no—
319C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
320C2×Dic3—21no—
321C2×Dic3—51no—
322C2×Dic3—51no—
323C2×Dic3—101no—
324C2×Dic3—101no—
325C2×Dic3—101no—
326C2×Dic3—101no—
327C2×Dic3—102no—
328C2×Dic3—101no—
329C2×Dic3—201no—
330C2×Dic3—201no—
331C2×Dic3—201no—
332C2×Dic3—201no—
333C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
334C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
335C3×C8—201no—
336C3⋊C8—201no—
337C22×C5D3×C22⋊C411yescharacteristic
338C2×D5#87811yescharacteristic
339C2×D5#89611yescharacteristic
340C2×D5#91111yescharacteristic
341C2×D5#91511yescharacteristic
342C2×D5#91611yescharacteristic
343C2×D5#91911yescharacteristic
344AGL(1,𝔽5)—42no—
345AGL(1,𝔽5)—122no—
346C4×C5—41no—
347C4×C5—61no—
348C4×C5—61no—
349C4×C5—121no—
350C4×C5C4×Dic611yescharacteristic
351C4×C5#88711yescharacteristic
352Dic5—41no—
353Dic5—61no—
354Dic5—61no—
355Dic5—121no—
356Dic5#88511yescharacteristic
357Dic5#88811yescharacteristic
358C22×C4—51no—
359C22×C4—101no—
360C22×C4—101no—
361C22×C4—151no—
362C22×C4—151no—
363C22×C4—301no—
364C22×C4—301no—
365C2×C8—101no—
366C2×C8—101no—
367C2×C8—101no—
368C2×C8—301no—
369C2×C8—301no—
370C2×C8—301no—
371C4⋊C4—31no—
372C4⋊C4—52no—
373C4⋊C4—51no—
374C4⋊C4—61no—
375C4⋊C4—61no—
376C4⋊C4—101no—
377C4⋊C4—101no—
378C4⋊C4—152no—
379C4⋊C4—151no—
380C4⋊C4—301no—
381C4⋊C4—301no—
382C4⋊C4—301no—
383C4⋊C4—301no—
384C4⋊C4—301no—
385C4⋊C4—301no—
386C4⋊C4—301no—
387C4⋊C4—301no—
388C4⋊C4D3×AGL(1,𝔽5)11yescharacteristic
389C42—31no—
390C42—151no—
391C42—151no—
392C42—151no—
393C42—602no—
394C3×C5#198911yescharacteristic, complemented, Hall
395C22×C3—51no—
396C22×C3—51no—
397C22×C3—51no—
398C22×C3—51no—
399C22×C3—51no—
400C22×C3—51no—
401C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
402C3×C4—41no—
403C3×C4—51no—
404C3×C4—51no—
405C3×C4—202no—
406C3×C4—201no—
407C3×C4Q8×AGL(1,𝔽5)11yescharacteristic
408C3×C4Q8×AGL(1,𝔽5)11yescharacteristic
409Dic3—21no—
410Dic3—21no—
411Dic3—41no—
412Dic3—101no—
413Dic3—101no—
414Dic3—202no—
415Dic3—201no—
416C2×C5#464611yescharacteristic
417C2×C5#477111yescharacteristic
418C2×C5#480411yescharacteristic
419D5#476311yescharacteristic
420D5#479611yescharacteristic
421D5#485711yescharacteristic
422D5#489111yescharacteristic
423C23—51no—
424C2×C4—21no—
425C2×C4—31no—
426C2×C4—31no—
427C2×C4—51no—
428C2×C4—51no—
429C2×C4—51no—
430C2×C4—51no—
431C2×C4—51no—
432C2×C4—61no—
433C2×C4—102no—
434C2×C4—101no—
435C2×C4—151no—
436C2×C4—151no—
437C2×C4—201no—
438C2×C4—201no—
439C2×C4—201no—
440C2×C4—201no—
441C2×C4—301no—
442C2×C4—301no—
443C2×C4—301no—
444C2×C4—301no—
445C2×C4—302no—
446C2×C4—301no—
447C2×C4—601no—
448C2×C4—601no—
449C2×C4—601no—
450C2×C4—601no—
451C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
452C8—201no—
453C8—601no—
454C2×C3—51no—
455C2×C3—51no—
456C2×C3—51no—
457C2×C3—51no—
458C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
459C2×C3#6579311yescharacteristic
460C2×C3#6579411yescharacteristic
461C5#6880111yescharacteristic, complemented, Sylow
462C22—51no—
463C22—51no—
464C22—51no—
465C22—51no—
466C22—51no—
467C22—51no—
468C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
469C4—41no—
470C4—51no—
471C4—51no—
472C4—61no—
473C4—61no—
474C4—121no—
475C4—202no—
476C4—201no—
477C4—301no—
478C4—301no—
479C4—602no—
480C4—601no—
481C4AGL(1,𝔽5)×Dic611yescharacteristic
482C4ΓL(1,𝔽16)⋊Q811yescharacteristic
483C3#1060458911yescharacteristic, complemented, Sylow
484C2—51no—
485C2—51no—
486C2—51no—
487C2—51no—
488C2#1175184211yescharacteristic
489C2#1175193711yescharacteristic
490C2#1175197111yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C5×C4⋊C4C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC4⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1989
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
14. #91779C4semidirect
15. C5×#890C4semidirect
21. #91384C4semidirect
22. #91381C4semidirect
145. #890AGL(1,𝔽5)semidirect
394. C3×C5#1989semidirect
461. C5#68801semidirect
483. C3#10604589semidirect

Maximal subgroups

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

subgroupindex
D5×#8902
Dic3×C4⋊AGL(1,𝔽5)2
D5⋊(Dic3⋊C8)2
#117467452
C3×#648742
#117465302
#117465282
#106045893
#688015