Finite Group Databasebeta

← #14300208

Subgroups — #14300208

Order 1920.

596 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
1C2×ΓL(1,𝔽16)⋊Q8C211yesmaximal, characteristic
2C2×C3×#3993C211yesmaximal, characteristic
3#11751861C211yesmaximal, characteristic
4C2×#91153C211yesmaximal, characteristic
5D5×#816C211yesmaximal, characteristic
6#11746763C211yesmaximal, characteristic
7#11746528C211yesmaximal, characteristic
8#10603990—31nomaximal, Hall
9C2×D5×Dic6C2211yescharacteristic
10C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
11C2×D5⋊(C3⋊C8)C2211yescharacteristic
12C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
13C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
14C2×Dic3⋊AGL(1,𝔽5)—21no—
15C2×Dic3⋊AGL(1,𝔽5)—21no—
16C2×Dic3×AGL(1,𝔽5)—21no—
17ΓL(1,𝔽16)⋊Q8—22no—
18ΓL(1,𝔽16)⋊Q8—22no—
19C2×C3×C8×D5C2211yescharacteristic
20D5×C4⋊Dic3C2211yescharacteristic
21C3×#3993—22no—
22(C2×AGL(1,𝔽5))⋊Dic3—21no—
23#91153—22no—
24#91100C411yescharacteristic, complemented
25C5×#816C411yescharacteristic, complemented
26#90975C411yescharacteristic, complemented
27#90973C411yescharacteristic, complemented
28#70588—51nomaximal, Hall
29C2×Q8×AGL(1,𝔽5)—31no—
30C2×#4009—31no—
31C2×#3993D311yescharacteristic
32#65712—31no—
33D5×#103—31no—
34(C2×AGL(1,𝔽5))⋊C8—31no—
35#64874—31no—
36C22×ΓL(1,𝔽16)D411yescharacteristic
37C22×ΓL(1,𝔽16)D411yescharacteristic
38C22×C3×AGL(1,𝔽5)—21no—
39C22×C3×AGL(1,𝔽5)—21no—
40C2×Dic30C2×C411yescharacteristic
41C2×C5×Dic6C2×C411yescharacteristic
42C2×C3×C4×D5C2311yescharacteristic
43C2×#1204—21no—
44C2×D5×Dic3—21no—
45C2×D5×Dic3—21no—
46D5×Dic6—21no—
47D5×Dic6—21no—
48D5×Dic6D412yes—
49D5×Dic6Q812yes—
50C2×C3⋊#170—21no—
51C4×ΓL(1,𝔽16)—21no—
52C4×ΓL(1,𝔽16)—21no—
53D5⋊(C3⋊C8)—21no—
54D5⋊(C3⋊C8)—21no—
55C3×C4⋊AGL(1,𝔽5)—21no—
56C3×C4⋊AGL(1,𝔽5)—21no—
57C3×C4⋊AGL(1,𝔽5)—21no—
58C3×C4⋊AGL(1,𝔽5)D412yes—
59Dic3⋊AGL(1,𝔽5)—24no—
60Dic3⋊AGL(1,𝔽5)—42no—
61Dic3×AGL(1,𝔽5)—24no—
62C2×C3×C5×C8C2×C411yescharacteristic
63C4⋊Dic15C2×C411yescharacteristic
64C5×C4⋊Dic3C2×C411yescharacteristic
65C2×C3×#168C2×C411yescharacteristic
66C3×C8×D5—21no—
67C3×C8×D5—21no—
68#6668—21no—
69C2×#887—51no—
70C2×C3×#106—51no—
71C2×#816—51no—
72#4972—51no—
73C2×#810—51no—
74#4550—51no—
75#4529—51no—
76C2×Q8×D5—31no—
77Q8×AGL(1,𝔽5)—62no—
78Q8×AGL(1,𝔽5)—62no—
79C2×C4⋊AGL(1,𝔽5)—61no—
80C2×C4⋊AGL(1,𝔽5)—61no—
81C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
82C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
83C2×C4×AGL(1,𝔽5)—31no—
84C2×C4×AGL(1,𝔽5)—61no—
85C2×D5⋊C8—31no—
86C2×C8×D5C2×D311yescharacteristic
87D5×C4⋊C4—31no—
88#4009—62no—
89(C2×AGL(1,𝔽5))⋊C4—61no—
90#3993—22no—
91C5×#103—31no—
92#3967—31no—
93#3948—31no—
94#3942—31no—
95#1865—151noSylow
96C22×C3×D5C2×D411yescharacteristic
97C2×ΓL(1,𝔽16)—21no—
98C2×ΓL(1,𝔽16)—21no—
99C2×ΓL(1,𝔽16)—21no—
100C2×ΓL(1,𝔽16)—21no—
101C2×ΓL(1,𝔽16)—21no—
102C2×ΓL(1,𝔽16)SD1612yes—
103C2×C3×AGL(1,𝔽5)—22no—
104C2×C3×AGL(1,𝔽5)—41no—
105C2×C3×AGL(1,𝔽5)—41no—
106C2×C3×AGL(1,𝔽5)—41no—
107C2×C3×AGL(1,𝔽5)—41no—
108C2×C3×AGL(1,𝔽5)—41no—
109C2×C3×C4×C5C22×C411yescharacteristic
110C2×Dic15—21no—
111C2×Dic15—21no—
112Dic30—21no—
113Dic30C4⋊C412yescomplemented
114C2×C5×Dic3—21no—
115C2×C5×Dic3—21no—
116C5×Dic6—21no—
117C5×Dic6C4⋊C412yescomplemented
118C2×C3×Dic5#4111yescharacteristic
119C3×C4×D5C2×D411yescharacteristic
120C3×C4×D5C2×D411yescharacteristic
121C3×C4×D5C2×Q811yescharacteristic
122C3×C4×D5#4111yescharacteristic
123#1204—22no—
124#1204—22no—
125D5×Dic3—22no—
126D5×Dic3—22no—
127D5×Dic3—41no—
128D5×Dic3—41no—
129C3⋊#170—41no—
130C3×C5×C8—21no—
131C3×#168—21no—
132C22×Dic6—51no—
133C22×C3×C8—51no—
134C2×C3×C4⋊C4—51no—
135C2×C3×C4⋊C4—51no—
136C2×C4⋊Dic3—51no—
137C2×Dic3⋊C4—101no—
138C2×Dic3⋊C4—101no—
139C2×C4×Dic3—51no—
140C2×C4×Dic3—101no—
141C22×C3⋊C8—51no—
142#887—102no—
143#887—102no—
144C3×#106—102no—
145C3⋊#95—101no—
146#816—51no—
147#816—51no—
148#816—51no—
149#816AGL(1,𝔽5)11yescharacteristic, complemented
150#810—102no—
151C22×AGL(1,𝔽5)—21no—
152C22×AGL(1,𝔽5)—21no—
153C22×AGL(1,𝔽5)—31no—
154C22×AGL(1,𝔽5)—31no—
155C2×C5×Q8—31no—
156Q8×D5—32no—
157Q8×D5—32no—
158Q8×D5—61no—
159Q8×D5—61no—
160C2×C4×D5—61no—
161C2×C4×D5—61no—
162C2×C4×D5C22×D311yescharacteristic
163C2×Dic10—31no—
164C2×Dic10—61no—
165C2×#170—61no—
166C4⋊AGL(1,𝔽5)—21no—
167C4⋊AGL(1,𝔽5)—21no—
168C4⋊AGL(1,𝔽5)—21no—
169C4⋊AGL(1,𝔽5)—64no—
170C4⋊AGL(1,𝔽5)—122no—
171C4⋊AGL(1,𝔽5)#6712yes—
172C4×AGL(1,𝔽5)—64no—
173C4×AGL(1,𝔽5)—61no—
174C4×AGL(1,𝔽5)—61no—
175D5⋊C8—61no—
176D5⋊C8—61no—
177C2×C5×C8C4×D311yescharacteristic
178C5×C4⋊C4—31no—
179C4⋊Dic5—31no—
180#683—61no—
181C2×#168C4×D311yescharacteristic
182C8×D5—21no—
183C8×D5—21no—
184C2×C4×Q8—151no—
185C2×#106—51no—
186C2×#103—151no—
187C2×#103—151no—
188#389—151no—
189#340—151no—
190#336—151no—
191C22×C3×C5C4×D411yescharacteristic
192C2×C3×D5#12011yescharacteristic
193C2×C3×D5C22⋊Q811yescharacteristic
194C2×C3×D5C2×SD1611yescharacteristic
195C2×C3×D5#13511yescharacteristic
196C2×C3×D5#13711yescharacteristic
197C2×C3×D5#13711yescharacteristic
198ΓL(1,𝔽16)—22no—
199ΓL(1,𝔽16)—41no—
200C3×AGL(1,𝔽5)—42no—
201C3×AGL(1,𝔽5)—81no—
202C3×C4×C5C2×C4⋊C411yescharacteristic
203C3×C4×C5C4×D411yescharacteristic
204Dic15—22no—
205Dic15—41no—
206C3×Dic5C22⋊Q811yescharacteristic
207C3×Dic5#12311yescharacteristic
208C5×Dic3—22no—
209C5×Dic3—41no—
210C22×C3×C4—51no—
211C22×C3×C4—101no—
212C22×C3×C4—101no—
213C22×Dic3—51no—
214C22×Dic3—51no—
215C22×Dic3—101no—
216C22×Dic3—101no—
217C2×Dic6—51no—
218C2×Dic6—52no—
219C2×Dic6—52no—
220C2×Dic6—101no—
221C2×Dic6—101no—
222C2×Dic6—101no—
223C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
224C2×C3×C8—51no—
225C2×C3×C8—101no—
226C2×C3×C8—101no—
227C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
228C3×C4⋊C4—52no—
229C3×C4⋊C4—101no—
230C3×C4⋊C4—101no—
231C3×C4⋊C4—101no—
232C4⋊Dic3—51no—
233C4⋊Dic3—101no—
234C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
235Dic3⋊C4—104no—
236Dic3⋊C4—202no—
237C4×Dic3—101no—
238C4×Dic3—101no—
239C4×Dic3—104no—
240C2×C3⋊C8—101no—
241C2×C3⋊C8—101no—
242C2×C3⋊C8—101no—
243C22×D5D3×D411yescharacteristic
244C2×AGL(1,𝔽5)—22no—
245C2×AGL(1,𝔽5)—32no—
246C2×AGL(1,𝔽5)—41no—
247C2×AGL(1,𝔽5)—41no—
248C2×AGL(1,𝔽5)—41no—
249C2×AGL(1,𝔽5)—41no—
250C2×AGL(1,𝔽5)—41no—
251C2×AGL(1,𝔽5)—61no—
252C2×AGL(1,𝔽5)—61no—
253C2×AGL(1,𝔽5)—61no—
254C2×AGL(1,𝔽5)—61no—
255C2×AGL(1,𝔽5)—61no—
256C5×Q8—32no—
257C5×Q8—61no—
258C2×C4×C5—61no—
259C2×C4×C5—61no—
260C2×C4×C5C2×C4×D311yescharacteristic
261C2×Dic5—61no—
262C2×Dic5—61no—
263C2×Dic5#23911yescharacteristic
264C4×D5—62no—
265C4×D5—62no—
266C4×D5—121no—
267C4×D5—121no—
268C4×D5#23511yescharacteristic
269C4×D5D3×D411yescharacteristic
270C4×D5D3×Q811yescharacteristic
271C4×D5C2×#6711yescharacteristic
272Dic10—32no—
273Dic10—61no—
274Dic10—62no—
275Dic10—62no—
276#170—121no—
277C5×C8—21no—
278#168—21no—
279C22×Q8—151no—
280C22×C8—51no—
281C22×C8—151no—
282C4×Q8—302no—
283C4×Q8—302no—
284C2×C4⋊C4—51no—
285C2×C4⋊C4—51no—
286C2×C4⋊C4—151no—
287C2×C4⋊C4—301no—
288C2×C4⋊C4—301no—
289C2×C42—151no—
290C2×C42—301no—
291#106—102no—
292#103—31no—
293#103—151no—
294#103—151no—
295#103—151no—
296#103—302no—
297#95—301no—
298C2×C3×C5#38511yescharacteristic
299C2×C3×C5C4×SD1611yescharacteristic
300C2×C3×C5#44111yescharacteristic
301C3×D5#44811yescharacteristic
302C3×D5#45211yescharacteristic
303C3×D5#47511yescharacteristic
304C3×D5#47911yescharacteristic
305C23×C3—51no—
306C2×C3×C4—51no—
307C2×C3×C4—51no—
308C2×C3×C4—51no—
309C2×C3×C4—51no—
310C2×C3×C4—51no—
311C2×C3×C4—102no—
312C2×C3×C4—201no—
313C2×C3×C4—201no—
314C2×C3×C4—201no—
315C2×C3×C4—201no—
316C2×C3×C4—201no—
317C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—52no—
321C2×Dic3—102no—
322C2×Dic3—102no—
323C2×Dic3—101no—
324C2×Dic3—101no—
325C2×Dic3—101no—
326C2×Dic3—101no—
327C2×Dic3—101no—
328C2×Dic3—101no—
329C2×Dic3—101no—
330C2×Dic3—201no—
331C2×Dic3—201no—
332Dic6—21no—
333Dic6—52no—
334Dic6—102no—
335Dic6—102no—
336Dic6—101no—
337Dic6C4⋊AGL(1,𝔽5)12yescomplemented
338C3×C8—21no—
339C3×C8—101no—
340C3⋊C8—201no—
341C22×C5#89311yescharacteristic
342C2×D5#89611yescharacteristic
343C2×D5D3×SD1611yescharacteristic
344C2×D5#91511yescharacteristic
345C2×D5#91511yescharacteristic
346C2×D5#91611yescharacteristic
347C2×D5#93711yescharacteristic
348AGL(1,𝔽5)—42no—
349AGL(1,𝔽5)—62no—
350AGL(1,𝔽5)—81no—
351AGL(1,𝔽5)—121no—
352C4×C5—62no—
353C4×C5—121no—
354C4×C5D3×C4⋊C411yescharacteristic
355C4×C5C4×#6711yescharacteristic
356Dic5—62no—
357Dic5—121no—
358Dic5#89411yescharacteristic
359Dic5#94611yescharacteristic
360C2×Q8—31no—
361C2×Q8—152no—
362C2×Q8—152no—
363C2×Q8—151no—
364C2×Q8—301no—
365C2×Q8—301no—
366C2×Q8—301no—
367C22×C4—51no—
368C22×C4—101no—
369C22×C4—101no—
370C22×C4—151no—
371C22×C4—151no—
372C22×C4—301no—
373C22×C4—301no—
374C2×C8—51no—
375C2×C8—101no—
376C2×C8—101no—
377C2×C8—301no—
378C2×C8—301no—
379C2×C8—301no—
380C2×C8D3×AGL(1,𝔽5)11yescharacteristic
381C4⋊C4—31no—
382C4⋊C4—52no—
383C4⋊C4—101no—
384C4⋊C4—101no—
385C4⋊C4—101no—
386C4⋊C4—151no—
387C4⋊C4—301no—
388C4⋊C4—304no—
389C4⋊C4—602no—
390C42—301no—
391C42—301no—
392C42—304no—
393C3×C5#186511yescharacteristic, complemented, Hall
394C22×C3—51no—
395C22×C3—51no—
396C22×C3—51no—
397C22×C3—51no—
398C22×C3—51no—
399C22×C3—51no—
400C22×C3D4×AGL(1,𝔽5)11yescharacteristic
401C3×C4—51no—
402C3×C4—51no—
403C3×C4—202no—
404C3×C4—401no—
405C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
406C3×C4D4×AGL(1,𝔽5)11yescharacteristic
407Dic3—22no—
408Dic3—41no—
409Dic3—102no—
410Dic3—102no—
411Dic3—201no—
412Dic3—201no—
413C2×C5#485211yescharacteristic
414C2×C5#485311yescharacteristic
415C2×C5#498911yescharacteristic
416D5#485411yescharacteristic
417D5#485711yescharacteristic
418D5#517011yescharacteristic
419D5#517311yescharacteristic
420C23—51no—
421Q8—32no—
422Q8—61no—
423Q8—152no—
424Q8—302no—
425Q8—302no—
426Q8—301no—
427C2×C4—51no—
428C2×C4—51no—
429C2×C4—51no—
430C2×C4—51no—
431C2×C4—51no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—102no—
435C2×C4—152no—
436C2×C4—201no—
437C2×C4—201no—
438C2×C4—201no—
439C2×C4—201no—
440C2×C4—201no—
441C2×C4—302no—
442C2×C4—302no—
443C2×C4—301no—
444C2×C4—301no—
445C2×C4—301no—
446C2×C4—301no—
447C2×C4—301no—
448C2×C4—301no—
449C2×C4—301no—
450C2×C4—601no—
451C2×C4—601no—
452C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
453C8—21no—
454C8—101no—
455C8—601no—
456C2×C3—51no—
457C2×C3—51no—
458C2×C3—51no—
459C2×C3—51no—
460C2×C3#6569911yescharacteristic
461C2×C3SD16×AGL(1,𝔽5)11yescharacteristic
462C2×C3#6573911yescharacteristic
463C5#7058811yescharacteristic, complemented, Sylow
464C22—51no—
465C22—51no—
466C22—51no—
467C22—51no—
468C22—51no—
469C22—51no—
470C22#9192111yescharacteristic
471C4—51no—
472C4—51no—
473C4—62no—
474C4—121no—
475C4—202no—
476C4—302no—
477C4—302no—
478C4—401no—
479C4—601no—
480C4—601no—
481C4D3×C4⋊AGL(1,𝔽5)11yescharacteristic
482C4AGL(1,𝔽5)×#6711yescharacteristic
483C3#1060399011yescharacteristic, complemented, Sylow
484C2—51no—
485C2—51no—
486C2—51no—
487C2—51no—
488C2#1175166311yescharacteristic
489C2#1175167011yescharacteristic
490C2#1175193211yescharacteristic
subgroupG/H
centerC22#91921
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×C8C2×C4
socleC22×C3×C5C4×D4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC2×C8D3×AGL(1,𝔽5)
hypocenterC3×C5#1865
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
24. #91100C4semidirect
25. C5×#816C4semidirect
26. #90975C4semidirect
27. #90973C4semidirect
113. Dic30C4⋊C4semidirect
117. C5×Dic6C4⋊C4semidirect
149. #816AGL(1,𝔽5)semidirect
337. Dic6C4⋊AGL(1,𝔽5)semidirect
393. C3×C5#1865semidirect
463. C5#70588semidirect
483. C3#10603990semidirect

Maximal subgroups

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

subgroupindex
C2×ΓL(1,𝔽16)⋊Q82
C2×C3×#39932
#117518612
C2×#911532
D5×#8162
#117467632
#117465282
#106039903
#705885