Finite Group Databasebeta

← #14323233

Subgroups — #14323233

Order 1920.

1080 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D5×D12C211yesmaximal, characteristic
2C4×C4⋊ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
3C3×#65686C211yesmaximal, characteristic, complemented
4#11751859C212yesmaximal, complemented
5#11751856C214yesmaximal, complemented
6#11751854C212yesmaximal, complemented
7#11751853C212yesmaximal, complemented
8#11751850C212yesmaximal
9#10617176—31nomaximal, Hall
10C2×D5×D12C2211yescharacteristic
11C2×C4×D3×D5C2212yes—
12C2×C4⋊ΓL(1,𝔽16)C2212yes—
13C2×C4×ΓL(1,𝔽16)C2212yes—
14C2×C3×C4×AGL(1,𝔽5)C2212yes—
15C2×Dic3⋊AGL(1,𝔽5)C2214yes—
16C2×(C2×D3)⋊AGL(1,𝔽5)C2214yes—
17C2×(C2×D3)⋊AGL(1,𝔽5)C2214yes—
18C4×D60C411yescharacteristic, complemented
19C4×C5×D12C411yescharacteristic, complemented
20C3×C42×D5C2211yescharacteristic
21D5×(C2×D3)⋊C4C2212yes—
22C4×#1203C412yescomplemented
23#91418C411yescharacteristic, complemented
24Dic5×D12C411yescharacteristic, complemented
25D5×C4⋊Dic3C2211yescharacteristic
26#91405C412yescomplemented
27#91235C2212yes—
28C3×(C2×AGL(1,𝔽5))⋊C4C2211yescharacteristic
29C3×(C2×AGL(1,𝔽5))⋊C4C2211yescharacteristic
30C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
31(C2×AGL(1,𝔽5))⋊Dic3C2214yes—
32#73415—51nomaximal, Hall
33C4×D4×D5—31no—
34#65780—32no—
35#65765—32no—
36#65764—32no—
37#65712—32no—
38#65700—34no—
39C4×C4⋊AGL(1,𝔽5)—31no—
40#65686D311yescharacteristic, complemented
41C22×D3×D5C2312yes—
42C22×ΓL(1,𝔽16)C2314yes—
43C22×C3×AGL(1,𝔽5)C2312yes—
44C22×C3×AGL(1,𝔽5)C2312yes—
45C2×D60C2×C411yescharacteristic
46C2×C4×D15C2×C412yes—
47C2×C5×D12C2×C411yescharacteristic
48C2×C4×C5×D3C2×C412yes—
49C2×C3×C4×D5C2311yescharacteristic
50C2×C3×C4×D5C2311yescharacteristic
51C2×C3×C4×D5C2311yescharacteristic
52C2×#1203C2×C412yes—
53C2×#1200C2×C412yes—
54C2×D3×Dic5C2×C412yes—
55C2×D5×Dic3C2312yes—
56D5×D12—24no—
57C4×D3×D5—44no—
58C4⋊ΓL(1,𝔽16)—24no—
59C4×ΓL(1,𝔽16)—42no—
60C3×C4×AGL(1,𝔽5)—24no—
61Dic3⋊AGL(1,𝔽5)—44no—
62(C2×D3)⋊AGL(1,𝔽5)—44no—
63(C2×D3)⋊AGL(1,𝔽5)—44no—
64C3×C42×C5C2×C411yescharacteristic
65#6714C2×C412yes—
66C4⋊Dic15C2×C411yescharacteristic
67C5×(C2×D3)⋊C4C2×C412yes—
68C5×C4⋊Dic3C2×C411yescharacteristic
69C3×C4×Dic5C2×C411yescharacteristic
70C3×C4×Dic5C2×C411yescharacteristic
71C3×C4×Dic5C2×C411yescharacteristic
72#6668C2×C412yes—
73#6667C2×C412yes—
74(C2×D3)⋊Dic5C2×C412yes—
75C2×C4×D12—51no—
76C3×#376—51no—
77#4939—52no—
78C4×C4⋊Dic3—51no—
79#4928—52no—
80#4669—54no—
81#4667—52no—
82#4664—52no—
83C2×D4×D5—31no—
84C22×C4×D5—32no—
85C2×C22⋊AGL(1,𝔽5)—34no—
86C2×C22⋊AGL(1,𝔽5)—34no—
87C2×C4⋊AGL(1,𝔽5)—32no—
88C2×C4⋊AGL(1,𝔽5)—34no—
89C2×C4×AGL(1,𝔽5)—32no—
90C2×C4×AGL(1,𝔽5)C2×D312yes—
91C4×C5×D4—31no—
92D4×Dic5—31no—
93C4×C22⋊D5—32no—
94#4039—31no—
95D5×C4⋊C4—31no—
96#4027—32no—
97D5×C22⋊C4—32no—
98C4×D20—31no—
99C42×D5C2×D311yescharacteristic
100(C2×AGL(1,𝔽5))⋊C4—34no—
101(C2×AGL(1,𝔽5))⋊C4—32no—
102(C2×AGL(1,𝔽5))⋊C4C2×D312yes—
103(C2×AGL(1,𝔽5))⋊C4C2×D311yescharacteristic
104(C2×AGL(1,𝔽5))⋊C4C2×D311yescharacteristic
105#2319—151noSylow
106C22×D15C22×C412yes—
107C22×C5×D3C22×C412yes—
108C22×C3×D5C2411yescharacteristic
109C2×D3×D5—24no—
110C2×D3×D5—22no—
111C2×D3×D5—24no—
112C2×D3×D5—22no—
113C2×ΓL(1,𝔽16)—24no—
114C2×ΓL(1,𝔽16)—24no—
115C2×ΓL(1,𝔽16)—24no—
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)—22no—
121C2×C3×AGL(1,𝔽5)—22no—
122C2×C3×C4×C5C22×C411yescharacteristic
123C2×C3×C4×C5C22×C411yescharacteristic
124C2×C3×C4×C5C22×C411yescharacteristic
125C2×Dic15C22×C412yes—
126D60—22no—
127C4×D15—42no—
128C2×C5×Dic3C22×C412yes—
129C5×D12—22no—
130C4×C5×D3—42no—
131C2×C3×Dic5C22×C411yescharacteristic
132C2×C3×Dic5C22×C411yescharacteristic
133C2×C3×Dic5C22×C411yescharacteristic
134C3×C4×D5—21no—
135C3×C4×D5—22no—
136C3×C4×D5—21no—
137C3×C4×D5#4114yes—
138#1203—24no—
139#1200—42no—
140D3×Dic5—42no—
141D5×Dic3—24no—
142C22×D12—51no—
143C22×C4×D3—52no—
144C2×C3×C42—51no—
145C2×C3×C42—52no—
146C2×(C2×D3)⋊C4—52no—
147C2×(C2×D3)⋊C4—54no—
148C2×(C2×D3)⋊C4—54no—
149C2×C4⋊Dic3—51no—
150C2×C4⋊Dic3—52no—
151C2×Dic3⋊C4—54no—
152C2×C4×Dic3—52no—
153C4×D12—51no—
154C4×D12—52no—
155C4×D12—51no—
156C4×D12—52no—
157C4×D12—51no—
158C4×D12AGL(1,𝔽5)11yescharacteristic, complemented
159C3×#95—51no—
160C3×#95—51no—
161C3×#95—52no—
162C3⋊#95—54no—
163C3⋊#95—52no—
164C23×D5—32no—
165C22×AGL(1,𝔽5)—34no—
166C22×AGL(1,𝔽5)C22×D312yes—
167C22×AGL(1,𝔽5)C22×D312yes—
168C2×C5×D4—31no—
169C22×C4×C5—32no—
170C2×C22⋊D5—32no—
171C22×Dic5—32no—
172D4×D5—64no—
173C2×D20—31no—
174C2×C4×D5—32no—
175C2×C4×D5—32no—
176C2×C4×D5—32no—
177C2×C4×D5—124no—
178C2×C4×D5C22×D311yescharacteristic
179C2×C4×D5C22×D311yescharacteristic
180C2×C4×D5C22×D311yescharacteristic
181C22⋊AGL(1,𝔽5)—124no—
182C22⋊AGL(1,𝔽5)—124no—
183C4⋊AGL(1,𝔽5)—64no—
184C4⋊AGL(1,𝔽5)—124no—
185C4×AGL(1,𝔽5)—24no—
186C4×AGL(1,𝔽5)—122no—
187C5×C4⋊C4—31no—
188C5×C22⋊C4—32no—
189C42×C5C4×D311yescharacteristic
190C22⋊Dic5—32no—
191#685—32no—
192#685—32no—
193C4⋊Dic5—31no—
194#683—32no—
195C4×Dic5C4×D311yescharacteristic
196C4×Dic5C4×D311yescharacteristic
197C4×Dic5C4×D311yescharacteristic
198C2×C4×D4—151no—
199#390—152no—
200#389—152no—
201#388—154no—
202#381—152no—
203#380—152no—
204C4×C4⋊C4—151no—
205#376—51no—
206C22×C3×C5C23×C411yescharacteristic
207C2×D15—22no—
208C2×D15—22no—
209C2×D15—22no—
210C2×C5×D3—22no—
211C2×C5×D3—22no—
212C2×C5×D3—22no—
213C2×C3×D5C2×#4112yes—
214C2×C3×D5H(5,𝔽2)11yescharacteristic
215C2×C3×D5H(5,𝔽2)12yes—
216C2×C3×D52−1+411yescharacteristic
217D3×D5—44no—
218D3×D5—44no—
219ΓL(1,𝔽16)—44no—
220C3×AGL(1,𝔽5)—42no—
221C3×AGL(1,𝔽5)—42no—
222C3×C4×C5—21no—
223C3×C4×C5—21no—
224C3×C4×C5#11712yes—
225Dic15—22no—
226C3×Dic5—21no—
227C3×Dic5—21no—
228C3×Dic5#11712yes—
229C5×Dic3—22no—
230C23×D3—52no—
231C22×C3×C4—51no—
232C22×C3×C4—51no—
233C22×C3×C4—51no—
234C22×C3×C4—52no—
235C22×C3×C4—52no—
236C22×Dic3—52no—
237C22×Dic3—54no—
238C2×D12—52no—
239C2×D12—51no—
240C2×D12—104no—
241C2×D12C2×AGL(1,𝔽5)11yescharacteristic
242C2×C4×D3—52no—
243C2×C4×D3—52no—
244C2×C4×D3—52no—
245C2×C4×D3—204no—
246C2×C4×D3C2×AGL(1,𝔽5)12yes—
247C3×C42—51no—
248C3×C42—51no—
249C3×C42—51no—
250C3×C42—104no—
251C3×C42C2×AGL(1,𝔽5)11yescharacteristic
252(C2×D3)⋊C4—52no—
253(C2×D3)⋊C4—52no—
254(C2×D3)⋊C4—52no—
255(C2×D3)⋊C4—204no—
256(C2×D3)⋊C4—204no—
257(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
258C4⋊Dic3—51no—
259C4⋊Dic3—52no—
260C4⋊Dic3—104no—
261C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
262Dic3⋊C4—204no—
263C4×Dic3—202no—
264C23×C5—32no—
265C22×D5—32no—
266C22×D5—64no—
267C22×D5—62no—
268C22×D5—64no—
269C22×D5—62no—
270C22×D5C23×D311yescharacteristic
271C2×AGL(1,𝔽5)—22no—
272C2×AGL(1,𝔽5)—22no—
273C2×AGL(1,𝔽5)—22no—
274C2×AGL(1,𝔽5)—22no—
275C2×AGL(1,𝔽5)—22no—
276C2×AGL(1,𝔽5)—22no—
277C2×AGL(1,𝔽5)—64no—
278C2×AGL(1,𝔽5)—64no—
279C2×AGL(1,𝔽5)—64no—
280C5×D4—62no—
281C2×C4×C5—32no—
282C2×C4×C5—122no—
283C2×C4×C5C2×C4×D311yescharacteristic
284C2×C4×C5C2×C4×D311yescharacteristic
285C2×C4×C5C2×C4×D311yescharacteristic
286C22⋊D5—64no—
287C2×Dic5—32no—
288C2×Dic5—122no—
289C2×Dic5C2×C4×D311yescharacteristic
290C2×Dic5C2×C4×D311yescharacteristic
291C2×Dic5C2×C4×D311yescharacteristic
292D20—62no—
293C4×D5—21no—
294C4×D5—22no—
295C4×D5—21no—
296C4×D5—64no—
297C4×D5—122no—
298C4×D5—122no—
299C4×D5#23514yes—
300C22×D4—151no—
301C23×C4—152no—
302C4×D4—31no—
303C4×D4—152no—
304C4×D4—151no—
305C4×D4—152no—
306C4×D4—151no—
307C4×D4—151no—
308C2×C4⋊C4—151no—
309C2×C4⋊C4—152no—
310C2×C4⋊C4—154no—
311C2×C22⋊C4—152no—
312C2×C22⋊C4—154no—
313C2×C22⋊C4—154no—
314C2×C42—51no—
315C2×C42—52no—
316C2×C42—152no—
317#95—51no—
318#95—51no—
319#95—52no—
320#95—152no—
321#95—154no—
322C2×C3×C5C2×#11711yescharacteristic
323C2×C3×C5#51811yescharacteristic
324C2×C3×C5#52011yescharacteristic
325D15—42no—
326C3×D5#54012yes—
327C3×D5#54212yes—
328C5×D3—42no—
329C23×C3—51no—
330C22×D3—52no—
331C22×D3—102no—
332C22×D3—104no—
333C22×D3—104no—
334C22×D3—102no—
335C22×D3C22×AGL(1,𝔽5)12yes—
336C2×C3×C4—51no—
337C2×C3×C4—54no—
338C2×C3×C4—51no—
339C2×C3×C4—51no—
340C2×C3×C4—101no—
341C2×C3×C4—102no—
342C2×C3×C4—101no—
343C2×C3×C4—102no—
344C2×C3×C4—102no—
345C2×C3×C4—102no—
346C2×C3×C4—102no—
347C2×C3×C4—102no—
348C2×C3×C4—102no—
349C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
350C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
351C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
352C2×Dic3—52no—
353C2×Dic3—104no—
354C2×Dic3—104no—
355C2×Dic3—104no—
356C2×Dic3—104no—
357C2×Dic3C22×AGL(1,𝔽5)12yes—
358D12—22no—
359D12—104no—
360D12—102no—
361C4×D3—42no—
362C4×D3—202no—
363C4×D3—202no—
364C4×D3—202no—
365C22×C5—62no—
366C22×C5—62no—
367C22×C5—62no—
368C22×C5C22×C4×D311yescharacteristic
369C2×D5—62no—
370C2×D5—62no—
371C2×D5—62no—
372C2×D5—124no—
373C2×D5—124no—
374C2×D5C2×#23512yes—
375C2×D5#100912yes—
376C2×D5#100911yescharacteristic
377C2×D5#101011yescharacteristic
378AGL(1,𝔽5)—42no—
379AGL(1,𝔽5)—42no—
380AGL(1,𝔽5)—124no—
381C4×C5—21no—
382C4×C5—21no—
383C4×C5—62no—
384C4×C5#87212yes—
385Dic5—21no—
386Dic5—21no—
387Dic5—62no—
388Dic5#87212yes—
389C24—152no—
390C2×D4—31no—
391C2×D4—152no—
392C2×D4—151no—
393C2×D4—304no—
394C22×C4—32no—
395C22×C4—51no—
396C22×C4—51no—
397C22×C4—51no—
398C22×C4—52no—
399C22×C4—52no—
400C22×C4—152no—
401C22×C4—152no—
402C22×C4—152no—
403C22×C4—152no—
404C22×C4—154no—
405C22×C4—604no—
406C4⋊C4—31no—
407C4⋊C4—152no—
408C4⋊C4—151no—
409C4⋊C4—304no—
410C4⋊C4—604no—
411C22⋊C4—32no—
412C22⋊C4—152no—
413C22⋊C4—152no—
414C22⋊C4—152no—
415C22⋊C4—604no—
416C22⋊C4—604no—
417C42—51no—
418C42—51no—
419C42—51no—
420C42—104no—
421C42—602no—
422C42D3×AGL(1,𝔽5)11yescharacteristic
423C3×C5#231911yescharacteristic, complemented, Hall
424C22×C3—52no—
425C22×C3—52no—
426C22×C3—51no—
427C22×C3—51no—
428C22×C3C23×AGL(1,𝔽5)11yescharacteristic
429C2×D3—22no—
430C2×D3—22no—
431C2×D3—22no—
432C2×D3—102no—
433C2×D3—102no—
434C2×D3—102no—
435C2×D3—204no—
436C2×D3—204no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—52no—
440C3×C4—101no—
441C3×C4—101no—
442C3×C4—202no—
443C3×C4—202no—
444C3×C4#413012yes—
445Dic3—22no—
446Dic3—102no—
447Dic3—204no—
448C2×C5—122no—
449C2×C5C2×#87211yescharacteristic
450C2×C5#552111yescharacteristic
451C2×C5#552311yescharacteristic
452D5—122no—
453D5#553412yes—
454D5#553512yes—
455C23—32no—
456C23—51no—
457C23—152no—
458C23—302no—
459C23—304no—
460C23—304no—
461C23—302no—
462D4—62no—
463D4—304no—
464D4—302no—
465C2×C4—32no—
466C2×C4—51no—
467C2×C4—54no—
468C2×C4—51no—
469C2×C4—51no—
470C2×C4—101no—
471C2×C4—102no—
472C2×C4—101no—
473C2×C4—102no—
474C2×C4—102no—
475C2×C4—102no—
476C2×C4—102no—
477C2×C4—102no—
478C2×C4—102no—
479C2×C4—122no—
480C2×C4—152no—
481C2×C4—304no—
482C2×C4—304no—
483C2×C4—304no—
484C2×C4—304no—
485C2×C4—602no—
486C2×C4—602no—
487C2×C4—602no—
488C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
489C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
490C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
491C2×C3—52no—
492C2×C3—52no—
493C2×C3C2×#413011yescharacteristic
494C2×C3#6625811yescharacteristic
495C2×C3#6626611yescharacteristic
496D3—42no—
497D3—202no—
498C5#7341511yescharacteristic, complemented, Sylow
499C22—52no—
500C22—52no—
501C22—51no—
502C22—51no—
503C22—62no—
504C22—62no—
505C22—62no—
506C22—302no—
507C22—302no—
508C22—302no—
509C22—604no—
510C22—604no—
511C22C22×D3×AGL(1,𝔽5)11yescharacteristic
512C4—21no—
513C4—21no—
514C4—52no—
515C4—62no—
516C4—101no—
517C4—101no—
518C4—202no—
519C4—202no—
520C4—302no—
521C4—604no—
522C4#9190912yes—
523C3#1061717611yescharacteristic, complemented, Sylow
524C2—52no—
525C2—52no—
526C2—122no—
527C2—602no—
528C2C2×#9190911yescharacteristic
529C2#1175683711yescharacteristic
530C2#1175683811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC3×C42×C5C2×C4
prime radicalC2×D5×D12C22
cosocleC22×C3×D5C24
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#2319
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊ΓL(1,𝔽16)C2semidirect
3. C3×#65686C2semidirect
4. #11751859C2semidirect
5. #11751856C2semidirect
6. #11751854C2semidirect
7. #11751853C2semidirect
18. C4×D60C4semidirect
19. C4×C5×D12C4semidirect
22. C4×#1203C4semidirect
23. #91418C4semidirect
24. Dic5×D12C4semidirect
26. #91405C4semidirect
40. #65686D3semidirect
158. C4×D12AGL(1,𝔽5)semidirect
423. C3×C5#2319semidirect
498. C5#73415semidirect
523. C3#10617176semidirect

Maximal subgroups

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

subgroupindex
C4×D5×D122
C4×C4⋊ΓL(1,𝔽16)2
C3×#656862
#117518592
#117518562
#117518542
#117518532
#117518502
#106171763
#734155