Finite Group Databasebeta

← #14304618

Subgroups — #14304618

Order 1920.

584 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65712C211yesmaximal, characteristic, complemented
2#11751850C211yesmaximal, characteristic
3C2×#91151C211yesmaximal, characteristic
4C2×AGL(1,𝔽5)×C3⋊C8C211yesmaximal, characteristic, complemented
5D5×#809C211yesmaximal, characteristic
6#11746745C211yesmaximal, characteristic, complemented
7#11746529C211yesmaximal, characteristic, complemented
8#10603963—31nomaximal, Hall
9C2×D5×D12C2211yescharacteristic
10C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
11C2×D5⋊(C3⋊C8)C2211yescharacteristic
12C2×C3×C4⋊AGL(1,𝔽5)—21no—
13C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
14C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
15C2×(C2×D3)⋊AGL(1,𝔽5)—21no—
16C2×(C2×D3)⋊AGL(1,𝔽5)—21no—
17C3×D5×C4⋊C4C2211yescharacteristic
18C2×D5×C3⋊C8C2211yescharacteristic
19C3×(C2×AGL(1,𝔽5))⋊C4—21no—
20#91151—22no—
21AGL(1,𝔽5)×C3⋊C8—22no—
22#91093C411yescharacteristic, complemented
23C5×#809C411yescharacteristic, complemented
24#90968C411yescharacteristic, complemented
25#90966C411yescharacteristic, complemented
26#69615—51nomaximal, Hall
27#65780—31no—
28C2×#4007—31no—
29C2×C8×AGL(1,𝔽5)—31no—
30#65712D311yescharacteristic, complemented
31D5×#102—31no—
32#64893—31no—
33#64874—31no—
34C22×D3×D5—21no—
35C22×ΓL(1,𝔽16)—21no—
36C22×C3×AGL(1,𝔽5)—21no—
37C22×C3×AGL(1,𝔽5)D411yescharacteristic
38C22×C3×AGL(1,𝔽5)D411yescharacteristic
39C2×D60C2×C411yescharacteristic
40C2×C5×D12C2×C411yescharacteristic
41C2×C3×C4×D5—21no—
42C2×C3×C4×D5C2311yescharacteristic
43C2×#1203—21no—
44D5×D12—21no—
45D5×D12—21no—
46D5×D12—21no—
47D5×D12—21no—
48C2×C3⋊#170—21no—
49C4⋊ΓL(1,𝔽16)—21no—
50C4⋊ΓL(1,𝔽16)—21no—
51D5⋊(C3⋊C8)—21no—
52D5⋊(C3⋊C8)D412yes—
53C3×C4⋊AGL(1,𝔽5)—21no—
54C3×C4⋊AGL(1,𝔽5)—21no—
55C3×C4⋊AGL(1,𝔽5)—42no—
56C3×C4×AGL(1,𝔽5)—21no—
57C3×C4×AGL(1,𝔽5)—21no—
58(C2×D3)⋊AGL(1,𝔽5)—42no—
59(C2×D3)⋊AGL(1,𝔽5)—42no—
60C3×C5×C4⋊C4C2×C411yescharacteristic
61C2×C3⋊#168C2×C411yescharacteristic
62C2×C5×C3⋊C8C2×C411yescharacteristic
63C3×C4⋊Dic5C2×C411yescharacteristic
64C3×#683—21no—
65D5×C3⋊C8—21no—
66D5×C3⋊C8—21no—
67C3×#389—51no—
68C2×#809—51no—
69C2×#809—51no—
70#4939—51no—
71C2×C4×C3⋊C8—51no—
72#4529—51no—
73#4529—51no—
74C2×D4×D5—31no—
75C2×C22⋊AGL(1,𝔽5)—61no—
76C2×C22⋊AGL(1,𝔽5)—61no—
77C2×C4⋊AGL(1,𝔽5)—21no—
78C2×C4⋊AGL(1,𝔽5)—31no—
79C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
80C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
81C2×D5⋊C8—31no—
82C2×C8×D5—31no—
83D5×C4⋊C4C2×D311yescharacteristic
84#4007—62no—
85(C2×AGL(1,𝔽5))⋊C4—21no—
86C8×AGL(1,𝔽5)—62no—
87C5×#102—31no—
88#3964—31no—
89#3953—31no—
90#3941—31no—
91#1934—151noSylow
92C22×D15—21no—
93C22×C5×D3—21no—
94C22×C3×D5C2×D411yescharacteristic
95C2×D3×D5—41no—
96C2×D3×D5—41no—
97C2×D3×D5—41no—
98C2×D3×D5—41no—
99C2×D3×D5—41no—
100C2×D3×D5—41no—
101C2×ΓL(1,𝔽16)—41no—
102C2×ΓL(1,𝔽16)—41no—
103C2×ΓL(1,𝔽16)—41no—
104C2×C3×AGL(1,𝔽5)—21no—
105C2×C3×AGL(1,𝔽5)—21no—
106C2×C3×AGL(1,𝔽5)—21no—
107C2×C3×AGL(1,𝔽5)—21no—
108C2×C3×AGL(1,𝔽5)—41no—
109C2×C3×AGL(1,𝔽5)—41no—
110C2×C3×AGL(1,𝔽5)—41no—
111C2×C3×AGL(1,𝔽5)D812yes—
112C2×C3×AGL(1,𝔽5)SD1612yes—
113C2×C3×C4×C5—21no—
114C2×C3×C4×C5C22×C411yescharacteristic
115D60—21no—
116D60—21no—
117C5×D12—21no—
118C5×D12—21no—
119C2×C3×Dic5—21no—
120C2×C3×Dic5#4111yescharacteristic
121C3×C4×D5—41no—
122C3×C4×D5—41no—
123C3×C4×D5C2×D411yescharacteristic
124C3×C4×D5#4111yescharacteristic
125C3×C4×D5#4111yescharacteristic
126C3×C4×D5#4111yescharacteristic
127#1203—41no—
128#1203—41no—
129C3⋊#170—22no—
130C3⋊#168—21no—
131C5×C3⋊C8—21no—
132C22×D12—51no—
133C2×C3×C4⋊C4—51no—
134C2×C3×C4⋊C4—51no—
135C2×C3×C4⋊C4—101no—
136C2×C3×C42—51no—
137C2×(C2×D3)⋊C4—101no—
138C2×(C2×D3)⋊C4—101no—
139C2×C4⋊Dic3—51no—
140C22×C3⋊C8—51no—
141C22×C3⋊C8—51no—
142C3×#95—101no—
143#809—51no—
144#809—51no—
145#809—51no—
146#809—102no—
147#809AGL(1,𝔽5)11yescharacteristic, complemented
148C4×C3⋊C8—102no—
149C23×D5—61no—
150C22×AGL(1,𝔽5)—21no—
151C22×AGL(1,𝔽5)—61no—
152C22×AGL(1,𝔽5)#6711yescharacteristic
153C22×AGL(1,𝔽5)#6711yescharacteristic
154C2×C5×D4—31no—
155C2×C22⋊D5—61no—
156D4×D5—61no—
157D4×D5—61no—
158D4×D5—61no—
159D4×D5—61no—
160C2×D20—31no—
161C2×C4×D5—21no—
162C2×C4×D5C22×D311yescharacteristic
163C22⋊AGL(1,𝔽5)—122no—
164C22⋊AGL(1,𝔽5)—122no—
165C2×#170—61no—
166C4⋊AGL(1,𝔽5)—21no—
167C4⋊AGL(1,𝔽5)—21no—
168C4⋊AGL(1,𝔽5)—42no—
169C4⋊AGL(1,𝔽5)—61no—
170C4⋊AGL(1,𝔽5)—61no—
171C4×AGL(1,𝔽5)—21no—
172C4×AGL(1,𝔽5)—21no—
173D5⋊C8—32no—
174D5⋊C8—61no—
175C2×C5×C8—31no—
176C5×C4⋊C4C4×D311yescharacteristic
177C4⋊Dic5C4×D311yescharacteristic
178#683—21no—
179C2×#168—31no—
180C8×D5—61no—
181C8×D5—61no—
182C2×#102—151no—
183C2×#102—151no—
184C2×C4×C8—151no—
185#390—151no—
186#389—51no—
187#340—151no—
188#340—151no—
189C22×C3×C5C4×D411yescharacteristic
190C2×D15—41no—
191C2×D15—41no—
192C2×D15—41no—
193C2×C5×D3—41no—
194C2×C5×D3—41no—
195C2×C5×D3—41no—
196C2×C3×D5#12111yescharacteristic
197C2×C3×D5#12411yescharacteristic
198C2×C3×D5C2×D811yescharacteristic
199C2×C3×D5C2×SD1611yescharacteristic
200C2×C3×D5#13511yescharacteristic
201C2×C3×D5#13511yescharacteristic
202D3×D5—81no—
203D3×D5—81no—
204D3×D5—81no—
205D3×D5—81no—
206ΓL(1,𝔽16)—81no—
207C3×AGL(1,𝔽5)—22no—
208C3×AGL(1,𝔽5)—22no—
209C3×AGL(1,𝔽5)—81no—
210C3×C4×C5—41no—
211C3×C4×C5#11711yescharacteristic
212C3×C4×C5C4×D411yescharacteristic
213C3×Dic5—41no—
214C3×Dic5#12311yescharacteristic
215C3×Dic5#12611yescharacteristic
216C23×D3—101no—
217C22×C3×C4—51no—
218C22×C3×C4—51no—
219C22×C3×C4—51no—
220C22×C3×C4—101no—
221C22×C3×C4—101no—
222C22×Dic3—101no—
223C2×D12—51no—
224C2×D12—101no—
225C2×D12—101no—
226C2×D12—101no—
227C2×D12—101no—
228C2×D12—101no—
229C2×D12C2×AGL(1,𝔽5)11yescharacteristic
230C3×C4⋊C4—51no—
231C3×C4⋊C4—101no—
232C3×C4⋊C4—101no—
233C3×C4⋊C4—101no—
234C3×C4⋊C4—202no—
235C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
236C3×C42—101no—
237C3×C42—101no—
238(C2×D3)⋊C4—202no—
239(C2×D3)⋊C4—202no—
240C4⋊Dic3—101no—
241C4⋊Dic3—101no—
242C2×C3⋊C8—52no—
243C2×C3⋊C8—51no—
244C2×C3⋊C8—101no—
245C2×C3⋊C8—101no—
246C2×C3⋊C8—101no—
247C2×C3⋊C8—101no—
248C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic
249C23×C5—61no—
250C22×D5—61no—
251C22×D5—121no—
252C22×D5—121no—
253C22×D5—121no—
254C22×D5—121no—
255C22×D5—121no—
256C22×D5—121no—
257C22×D5C2×#6711yescharacteristic
258C2×AGL(1,𝔽5)—21no—
259C2×AGL(1,𝔽5)—21no—
260C2×AGL(1,𝔽5)—21no—
261C2×AGL(1,𝔽5)—21no—
262C2×AGL(1,𝔽5)—41no—
263C2×AGL(1,𝔽5)—41no—
264C2×AGL(1,𝔽5)—41no—
265C2×AGL(1,𝔽5)—121no—
266C2×AGL(1,𝔽5)—121no—
267C2×AGL(1,𝔽5)—121no—
268C2×AGL(1,𝔽5)#21312yes—
269C2×AGL(1,𝔽5)#21512yes—
270C5×D4—61no—
271C5×D4—61no—
272C2×C4×C5—21no—
273C2×C4×C5C2×C4×D311yescharacteristic
274C22⋊D5—121no—
275C22⋊D5—121no—
276C2×Dic5—21no—
277C2×Dic5#23511yescharacteristic
278D20—61no—
279D20—61no—
280C4×D5—41no—
281C4×D5—41no—
282C4×D5D3×D411yescharacteristic
283C4×D5#23711yescharacteristic
284C4×D5#23911yescharacteristic
285C4×D5#23911yescharacteristic
286#170—62no—
287C5×C8—61no—
288#168—61no—
289C22×D4—151no—
290C22×C8—151no—
291C22×C8—151no—
292C2×C4⋊C4—51no—
293C2×C4⋊C4—51no—
294C2×C4⋊C4—101no—
295C2×C4⋊C4—151no—
296C2×C22⋊C4—301no—
297C2×C22⋊C4—301no—
298C2×C42—51no—
299#102—31no—
300#102—151no—
301#102—151no—
302#102—151no—
303#102—302no—
304C4×C8—302no—
305#95—101no—
306C2×C3×C5#38811yescharacteristic
307C2×C3×C5C4×D811yescharacteristic
308C2×C3×C5C4×SD1611yescharacteristic
309D15—81no—
310C3×D5#46511yescharacteristic
311C3×D5#46611yescharacteristic
312C3×D5#48611yescharacteristic
313C3×D5#48811yescharacteristic
314C5×D3—81no—
315C23×C3—51no—
316C22×D3—21no—
317C22×D3—101no—
318C22×D3—201no—
319C22×D3—201no—
320C22×D3—201no—
321C22×D3—201no—
322C22×D3—201no—
323C22×D3—201no—
324C2×C3×C4—21no—
325C2×C3×C4—51no—
326C2×C3×C4—51no—
327C2×C3×C4—51no—
328C2×C3×C4—51no—
329C2×C3×C4—51no—
330C2×C3×C4—52no—
331C2×C3×C4—52no—
332C2×C3×C4—101no—
333C2×C3×C4—101no—
334C2×C3×C4—101no—
335C2×C3×C4—101no—
336C2×C3×C4—101no—
337C2×C3×C4—201no—
338C2×C3×C4—201no—
339C2×C3×C4—201no—
340C2×C3×C4—201no—
341C2×C3×C4—201no—
342C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
343C2×Dic3—201no—
344C2×Dic3—201no—
345C2×Dic3—201no—
346D12—21no—
347D12—21no—
348D12—101no—
349D12—101no—
350D12—201no—
351D12—201no—
352C3⋊C8—21no—
353C3⋊C8—102no—
354C3⋊C8—101no—
355C22×C5—121no—
356C22×C5—121no—
357C22×C5—121no—
358C22×C5C4×#6711yescharacteristic
359C2×D5—121no—
360C2×D5—121no—
361C2×D5—121no—
362C2×D5—241no—
363C2×D5—241no—
364C2×D5—241no—
365C2×D5—241no—
366C2×D5C2×#21311yescharacteristic
367C2×D5#93911yescharacteristic
368C2×D5C2×#21511yescharacteristic
369C2×D5#94711yescharacteristic
370C2×D5#95011yescharacteristic
371C2×D5#95011yescharacteristic
372AGL(1,𝔽5)—22no—
373AGL(1,𝔽5)—22no—
374AGL(1,𝔽5)—81no—
375AGL(1,𝔽5)—241no—
376C4×C5—41no—
377C4×C5#89211yescharacteristic
378C4×C5#89311yescharacteristic
379Dic5—41no—
380Dic5#89411yescharacteristic
381Dic5#89811yescharacteristic
382C24—301no—
383C2×D4—31no—
384C2×D4—151no—
385C2×D4—301no—
386C2×D4—301no—
387C2×D4—301no—
388C2×D4—301no—
389C2×D4—301no—
390C22×C4—51no—
391C22×C4—51no—
392C22×C4—51no—
393C22×C4—101no—
394C22×C4—101no—
395C22×C4—301no—
396C2×C8—31no—
397C2×C8—152no—
398C2×C8—151no—
399C2×C8—301no—
400C2×C8—301no—
401C2×C8—301no—
402C2×C8—301no—
403C4⋊C4—51no—
404C4⋊C4—101no—
405C4⋊C4—101no—
406C4⋊C4—101no—
407C4⋊C4—202no—
408C4⋊C4—301no—
409C4⋊C4—301no—
410C4⋊C4D3×AGL(1,𝔽5)11yescharacteristic
411C22⋊C4—602no—
412C22⋊C4—602no—
413C42—101no—
414C42—101no—
415C3×C5#193411yescharacteristic, complemented, Hall
416C22×C3—51no—
417C22×C3—51no—
418C22×C3—51no—
419C22×C3—51no—
420C22×C3—51no—
421C22×C3—51no—
422C22×C3D4×AGL(1,𝔽5)11yescharacteristic
423C2×D3—41no—
424C2×D3—41no—
425C2×D3—41no—
426C2×D3—201no—
427C2×D3—201no—
428C2×D3—201no—
429C2×D3—401no—
430C2×D3—401no—
431C2×D3—401no—
432C2×D3—401no—
433C3×C4—41no—
434C3×C4—51no—
435C3×C4—51no—
436C3×C4—102no—
437C3×C4—102no—
438C3×C4—201no—
439C3×C4—401no—
440C3×C4#413011yescharacteristic
441C3×C4D4×AGL(1,𝔽5)11yescharacteristic
442Dic3—401no—
443C2×C5—241no—
444C2×C5#499011yescharacteristic
445C2×C5C4×#21311yescharacteristic
446C2×C5C4×#21511yescharacteristic
447D5—241no—
448D5#503811yescharacteristic
449D5#504811yescharacteristic
450D5#506811yescharacteristic
451D5#508811yescharacteristic
452C23—51no—
453C23—61no—
454C23—301no—
455C23—601no—
456C23—601no—
457C23—601no—
458C23—601no—
459C23—601no—
460C23—601no—
461D4—61no—
462D4—61no—
463D4—301no—
464D4—301no—
465D4—601no—
466D4—601no—
467C2×C4—21no—
468C2×C4—51no—
469C2×C4—51no—
470C2×C4—51no—
471C2×C4—51no—
472C2×C4—51no—
473C2×C4—52no—
474C2×C4—52no—
475C2×C4—101no—
476C2×C4—101no—
477C2×C4—101no—
478C2×C4—101no—
479C2×C4—101no—
480C2×C4—201no—
481C2×C4—201no—
482C2×C4—201no—
483C2×C4—201no—
484C2×C4—201no—
485C2×C4—601no—
486C2×C4—601no—
487C2×C4—601no—
488C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
489C8—61no—
490C8—302no—
491C8—301no—
492C2×C3—51no—
493C2×C3—51no—
494C2×C3—51no—
495C2×C3—51no—
496C2×C3#6570011yescharacteristic
497C2×C3D8×AGL(1,𝔽5)11yescharacteristic
498C2×C3SD16×AGL(1,𝔽5)11yescharacteristic
499D3—81no—
500D3—401no—
501C5#6961511yescharacteristic, complemented, Sylow
502C22—51no—
503C22—51no—
504C22—51no—
505C22—51no—
506C22—51no—
507C22—51no—
508C22—121no—
509C22—121no—
510C22—121no—
511C22—601no—
512C22—601no—
513C22—601no—
514C22—1201no—
515C22—1201no—
516C22—1201no—
517C22—1201no—
518C22AGL(1,𝔽5)×#6711yescharacteristic
519C4—41no—
520C4—51no—
521C4—51no—
522C4—102no—
523C4—102no—
524C4—201no—
525C4—401no—
526C4—1201no—
527C4#9190711yescharacteristic
528C4#9192111yescharacteristic
529C3#1060396311yescharacteristic, complemented, Sylow
530C2—51no—
531C2—51no—
532C2—51no—
533C2—51no—
534C2—241no—
535C2—1201no—
536C2#1175183611yescharacteristic
537C2AGL(1,𝔽5)×#21311yescharacteristic
538C2AGL(1,𝔽5)×#21511yescharacteristic
subgroupG/H
centerC22AGL(1,𝔽5)×#67
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C5×C4⋊C4C2×C4
socleC22×C3×C5C4×D4
prime radicalC2×D5×D12C22
cosocleC2×C3×C4×D5C23
hypercenterC4⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1934
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
1. C3×#65712C2semidirect
4. C2×AGL(1,𝔽5)×C3⋊C8C2semidirect
6. #11746745C2semidirect
7. #11746529C2semidirect
22. #91093C4semidirect
23. C5×#809C4semidirect
24. #90968C4semidirect
25. #90966C4semidirect
30. #65712D3semidirect
147. #809AGL(1,𝔽5)semidirect
415. C3×C5#1934semidirect
501. C5#69615semidirect
529. C3#10603963semidirect

Maximal subgroups

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

subgroupindex
C3×#657122
#117518502
C2×#911512
C2×AGL(1,𝔽5)×C3⋊C82
D5×#8092
#117467452
#117465292
#106039633
#696155