Finite Group Databasebeta

← #14307491

Subgroups — #14307491

Order 1920.

648 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4042C211yesmaximal, characteristic
2#11752424C211yesmaximal, characteristic
3C4×#6763C211yesmaximal, characteristic
4#11746788C212yesmaximal
5#11746788C212yesmaximal
6#10604311—31nomaximal, Hall
7C22×C3×C4×D5C2211yescharacteristic
8C22×C3×Dic10C411yescharacteristic, complemented
9C2×#6763C2212yes—
10C2×#6763C2212yes—
11C2×C4⋊ΓL(1,𝔽16)—21no—
12C2×C4×ΓL(1,𝔽16)—21no—
13C2×C3×C5×C4⋊C4C411yescharacteristic, complemented
14C2×C3×#685C2211yescharacteristic
15C2×C3×#685C2211yescharacteristic
16C2×C3×#683C411yescharacteristic, complemented
17C2×C3×#683C411yescharacteristic, complemented
18C3×#4042—21no—
19C3×#4042—21no—
20C3×#4042—21no—
21C3×#4042—21no—
22#91235—21no—
23#91235—21no—
24#71701—51nomaximal, Hall
25C2×#4042D311yescharacteristic
26#65765—31no—
27C4×C22⋊AGL(1,𝔽5)—31no—
28#64917—32no—
29#64917—32no—
30C23×C3×D5C2311yescharacteristic
31C22×ΓL(1,𝔽16)—22no—
32C22×ΓL(1,𝔽16)—22no—
33C22×C3×C4×C5C2×C411yescharacteristic
34C22×C3×C4×C5C2×C411yescharacteristic
35C22×C3×C4×C5C2×C411yescharacteristic
36C22×C3×Dic5C2×C411yescharacteristic
37C22×C3×Dic5C2×C411yescharacteristic
38C22×C3×Dic5C2×C411yescharacteristic
39C2×C3×C4×D5—42no—
40C2×C3×C4×D5D411yescharacteristic
41C2×C3×C4×D5D411yescharacteristic
42C2×C3×C4×D5D411yescharacteristic
43C2×C3×C4×D5D411yescharacteristic
44C2×C3×Dic10—21no—
45C2×C3×Dic10—21no—
46C2×C3×Dic10—42no—
47#6763—22no—
48#6763—24no—
49#6763Q814yescomplemented
50C4⋊ΓL(1,𝔽16)—42no—
51C4×ΓL(1,𝔽16)—42no—
52C3×C5×C4⋊C4—21no—
53C3×C5×C4⋊C4—21no—
54C3×#685—21no—
55C3×#685—21no—
56C3×#685—21no—
57C3×#685—21no—
58C3×#683—21no—
59C3×#683—21no—
60C3×#683—21no—
61C3×#683—21no—
62C2×C3×C22⋊Q8—51no—
63#5212—51no—
64C4×#217—51no—
65#4526—52no—
66#4526—52no—
67C22×C4×D5C2×D311yescharacteristic
68C22×Dic10Dic311yescharacteristic, complemented
69C2×C22⋊AGL(1,𝔽5)—32no—
70C2×C22⋊AGL(1,𝔽5)—32no—
71C2×C4⋊AGL(1,𝔽5)—61no—
72C2×C4×AGL(1,𝔽5)—61no—
73C2×C5×C4⋊C4Dic311yescharacteristic, complemented
74C2×#685C2×D311yescharacteristic
75C2×#685C2×D311yescharacteristic
76C2×#683Dic311yescharacteristic, complemented
77C2×#683Dic311yescharacteristic, complemented
78#4042—21no—
79#4042—21no—
80#4042—21no—
81#4042—21no—
82(C2×AGL(1,𝔽5))⋊C4—61no—
83(C2×AGL(1,𝔽5))⋊C4—61no—
84#1996—151noSylow
85C23×C3×C5C22×C411yescharacteristic
86C22×C3×D5—41no—
87C22×C3×D5—41no—
88C22×C3×D5C2×D411yescharacteristic
89C22×C3×D5C2×D411yescharacteristic
90C22×C3×D5C2×Q812yes—
91C22×C3×D5#4112yes—
92C2×ΓL(1,𝔽16)—24no—
93C2×ΓL(1,𝔽16)—42no—
94C2×ΓL(1,𝔽16)—42no—
95C2×ΓL(1,𝔽16)—42no—
96C2×ΓL(1,𝔽16)—42no—
97C2×ΓL(1,𝔽16)—42no—
98C2×C3×C4×C5—21no—
99C2×C3×C4×C5—21no—
100C2×C3×C4×C5—41no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5—41no—
103C2×C3×C4×C5C22⋊C411yescharacteristic
104C2×C3×C4×C5C22⋊C411yescharacteristic
105C2×C3×Dic5—22no—
106C2×C3×Dic5—21no—
107C2×C3×Dic5—21no—
108C2×C3×Dic5—41no—
109C2×C3×Dic5—41no—
110C2×C3×Dic5C22⋊C411yescharacteristic
111C2×C3×Dic5C22⋊C411yescharacteristic
112C3×C4×D5—22no—
113C3×C4×D5—22no—
114C3×C4×D5—41no—
115C3×C4×D5—41no—
116C3×Dic10—42no—
117C3×Dic10—42no—
118C22×C3×Q8—51no—
119C23×C3×C4—51no—
120C3×C22⋊Q8—101no—
121C3×C22⋊Q8—101no—
122C3×C22⋊Q8—101no—
123C3×C22⋊Q8—101no—
124C2×C3×C4⋊C4—51no—
125C2×C3×C4⋊C4—51no—
126C2×C3×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
127C2×C3×C22⋊C4—51no—
128C2×C3×C22⋊C4—51no—
129C2×#217—52no—
130C2×#217—52no—
131C2×C4⋊Dic3—101no—
132C2×C4×Dic3—101no—
133C3⋊#95—101no—
134C3⋊#95—101no—
135C23×D5C22×D311yescharacteristic
136C22×AGL(1,𝔽5)—62no—
137C22×AGL(1,𝔽5)—62no—
138C22×C4×C5C2×Dic311yescharacteristic
139C22×C4×C5C2×Dic311yescharacteristic
140C22×C4×C5C2×Dic311yescharacteristic
141C22×Dic5C2×Dic311yescharacteristic
142C22×Dic5C2×Dic311yescharacteristic
143C22×Dic5C2×Dic311yescharacteristic
144C2×C4×D5—42no—
145C2×C4×D5#6711yescharacteristic
146C2×C4×D5#6711yescharacteristic
147C2×C4×D5#6711yescharacteristic
148C2×C4×D5#6711yescharacteristic
149C2×Dic10—21no—
150C2×Dic10—21no—
151C2×Dic10—42no—
152C22⋊AGL(1,𝔽5)—34no—
153C22⋊AGL(1,𝔽5)—62no—
154C22⋊AGL(1,𝔽5)—64no—
155C4⋊AGL(1,𝔽5)—122no—
156C4×AGL(1,𝔽5)—122no—
157C5×C4⋊C4—21no—
158C5×C4⋊C4—21no—
159#685—21no—
160#685—21no—
161#685—21no—
162#685—21no—
163#683—21no—
164#683—21no—
165#683—21no—
166#683—21no—
167C2×C22⋊Q8—51no—
168#380—151no—
169C4×C22⋊C4—151no—
170#342—152no—
171#342—152no—
172C22×C3×C5—21no—
173C22×C3×C5—21no—
174C22×C3×C5C2×C22⋊C411yescharacteristic
175C22×C3×C5C4×Q812yes—
176C2×C3×D5—21no—
177C2×C3×D5—21no—
178C2×C3×D5—21no—
179C2×C3×D5—21no—
180C2×C3×D5—42no—
181C2×C3×D5—42no—
182C2×C3×D5C22⋊Q812yes—
183C2×C3×D5#12411yescharacteristic
184C2×C3×D5C4⋊Q811yescharacteristic
185ΓL(1,𝔽16)—44no—
186ΓL(1,𝔽16)—82no—
187C3×C4×C5—21no—
188C3×C4×C5—21no—
189C3×C4×C5—41no—
190C3×C4×C5—41no—
191C3×Dic5—21no—
192C3×Dic5—41no—
193C3×Dic5—41no—
194C3×Dic5#9912yes—
195C24×C3—51no—
196C2×C3×Q8—101no—
197C2×C3×Q8—101no—
198C2×C3×Q8—202no—
199C22×C3×C4—51no—
200C22×C3×C4—51no—
201C22×C3×C4—51no—
202C22×C3×C4—51no—
203C22×C3×C4—51no—
204C22×C3×C4—51no—
205C22×C3×C4—51no—
206C22×C3×C4—202no—
207C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
208C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
209C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
210C22×Dic3—102no—
211C22×Dic3—102no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—101no—
215C3×C4⋊C4—101no—
216C3×C4⋊C4—101no—
217C3×C4⋊C4—101no—
218C3×C22⋊C4—101no—
219C3×C22⋊C4—101no—
220C3×C22⋊C4—101no—
221C3×C22⋊C4—101no—
222#217—54no—
223#217—102no—
224#217—104no—
225C4⋊Dic3—202no—
226C4×Dic3—202no—
227C23×C5C22×Dic311yescharacteristic
228C22×D5—41no—
229C22×D5—41no—
230C22×D5D3×Q812yes—
231C22×D5#23912yes—
232C22×D5C2×#6711yescharacteristic
233C22×D5C2×#6711yescharacteristic
234C2×AGL(1,𝔽5)—64no—
235C2×AGL(1,𝔽5)—122no—
236C2×AGL(1,𝔽5)—122no—
237C2×AGL(1,𝔽5)—122no—
238C2×AGL(1,𝔽5)—122no—
239C2×AGL(1,𝔽5)—122no—
240C2×C4×C5—21no—
241C2×C4×C5—21no—
242C2×C4×C5—41no—
243C2×C4×C5—41no—
244C2×C4×C5—41no—
245C2×C4×C5#21711yescharacteristic
246C2×C4×C5#21711yescharacteristic
247C2×Dic5—22no—
248C2×Dic5—21no—
249C2×Dic5—21no—
250C2×Dic5—41no—
251C2×Dic5—41no—
252C2×Dic5#21711yescharacteristic
253C2×Dic5#21711yescharacteristic
254C4×D5—22no—
255C4×D5—22no—
256C4×D5—41no—
257C4×D5—41no—
258Dic10—42no—
259Dic10—42no—
260C22×Q8—51no—
261C23×C4—51no—
262C22⋊Q8—101no—
263C22⋊Q8—101no—
264C22⋊Q8—101no—
265C22⋊Q8—101no—
266C2×C4⋊C4—51no—
267C2×C4⋊C4—51no—
268C2×C4⋊C4—301no—
269C2×C4⋊C4ΓL(1,𝔽16)11yescharacteristic, complemented
270C2×C22⋊C4—51no—
271C2×C22⋊C4—51no—
272C2×C22⋊C4—152no—
273C2×C22⋊C4—152no—
274C2×C42—301no—
275#95—301no—
276#95—301no—
277C2×C3×C5—22no—
278C2×C3×C5#39111yescharacteristic
279C2×C3×C5C2×#9911yescharacteristic
280C2×C3×C5#41011yescharacteristic
281C3×D5—22no—
282C3×D5—21no—
283C3×D5—21no—
284C23×C3—51no—
285C23×C3—51no—
286C23×C3—52no—
287C23×C3—52no—
288C23×C3—201no—
289C23×C3—201no—
290C23×C3C22×AGL(1,𝔽5)11yescharacteristic
291C3×Q8—202no—
292C3×Q8—202no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—51no—
299C2×C3×C4—51no—
300C2×C3×C4—102no—
301C2×C3×C4—102no—
302C2×C3×C4—102no—
303C2×C3×C4—101no—
304C2×C3×C4—101no—
305C2×C3×C4—201no—
306C2×C3×C4—201no—
307C2×C3×C4—201no—
308C2×C3×C4—201no—
309C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
310C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
311C2×Dic3—104no—
312C2×Dic3—202no—
313C2×Dic3—202no—
314C2×Dic3—202no—
315C2×Dic3—202no—
316C2×Dic3—202no—
317C22×C5—21no—
318C22×C5—21no—
319C22×C5Q8×Dic312yes—
320C22×C5C2×#21711yescharacteristic
321C2×D5—21no—
322C2×D5—21no—
323C2×D5—21no—
324C2×D5—21no—
325C2×D5—42no—
326C2×D5—42no—
327C2×D5#94411yescharacteristic
328C2×D5#94612yes—
329C2×D5#94711yescharacteristic
330AGL(1,𝔽5)—124no—
331AGL(1,𝔽5)—242no—
332C4×C5—21no—
333C4×C5—21no—
334C4×C5—41no—
335C4×C5—41no—
336Dic5—21no—
337Dic5—41no—
338Dic5—41no—
339Dic5#83412yes—
340C24—51no—
341C2×Q8—101no—
342C2×Q8—101no—
343C2×Q8—202no—
344C22×C4—51no—
345C22×C4—51no—
346C22×C4—51no—
347C22×C4—51no—
348C22×C4—51no—
349C22×C4—51no—
350C22×C4—51no—
351C22×C4—202no—
352C22×C4—302no—
353C22×C4—302no—
354C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
355C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
356C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—101no—
360C4⋊C4—101no—
361C4⋊C4—101no—
362C4⋊C4—101no—
363C4⋊C4—602no—
364C22⋊C4—101no—
365C22⋊C4—101no—
366C22⋊C4—101no—
367C22⋊C4—101no—
368C22⋊C4—154no—
369C22⋊C4—302no—
370C22⋊C4—304no—
371C42—602no—
372C3×C5#199611yescharacteristic, complemented, Hall
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—52no—
376C22×C3—51no—
377C22×C3—51no—
378C22×C3—101no—
379C22×C3—101no—
380C22×C3—101no—
381C22×C3—101no—
382C22×C3—202no—
383C22×C3—202no—
384C22×C3Q8×AGL(1,𝔽5)12yes—
385C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—41no—
389C3×C4—41no—
390C3×C4—52no—
391C3×C4—101no—
392C3×C4—201no—
393C3×C4—201no—
394Dic3—204no—
395Dic3—402no—
396C2×C5—22no—
397C2×C5C2×#83411yescharacteristic
398C2×C5#522911yescharacteristic
399C2×C5#524011yescharacteristic
400D5—22no—
401D5—21no—
402D5—21no—
403C23—51no—
404C23—51no—
405C23—52no—
406C23—52no—
407C23—201no—
408C23—201no—
409C23C22×ΓL(1,𝔽16)11yescharacteristic
410Q8—202no—
411Q8—202no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—51no—
418C2×C4—51no—
419C2×C4—102no—
420C2×C4—102no—
421C2×C4—102no—
422C2×C4—101no—
423C2×C4—101no—
424C2×C4—201no—
425C2×C4—201no—
426C2×C4—201no—
427C2×C4—201no—
428C2×C4—304no—
429C2×C4—602no—
430C2×C4—602no—
431C2×C4—602no—
432C2×C4—602no—
433C2×C4—602no—
434C2×C4#676311yescharacteristic
435C2×C4#676311yescharacteristic
436C2×C3—22no—
437C2×C3—102no—
438C2×C3—101no—
439C2×C3—101no—
440C2×C3#6574511yescharacteristic
441C2×C3D5⋊#9911yescharacteristic
442C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
443C5#7170111yescharacteristic, complemented, Sylow
444C22—21no—
445C22—21no—
446C22—52no—
447C22—51no—
448C22—51no—
449C22—101no—
450C22—101no—
451C22—101no—
452C22—101no—
453C22—202no—
454C22—202no—
455C22Q8×ΓL(1,𝔽16)12yes—
456C22C2×#676311yescharacteristic
457C4—21no—
458C4—21no—
459C4—41no—
460C4—41no—
461C4—52no—
462C4—101no—
463C4—201no—
464C4—201no—
465C4—604no—
466C4—1202no—
467C3#1060431111yescharacteristic, complemented, Sylow
468C2—22no—
469C2—102no—
470C2—101no—
471C2—101no—
472C2#1175240411yescharacteristic
473C2D5⋊#83411yescharacteristic
474C2#1175244911yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC23×C3×C5C22×C4
FrattiniC23C22×ΓL(1,𝔽16)
FittingC2×C3×C5×C4⋊C4C4
socleC22×C3×C5C2×C22⋊C4
prime radical, cosocleC23×C3×D5C23
hypercenterC2×C4⋊C4ΓL(1,𝔽16)
hypocenterC3×C5#1996
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
8. C22×C3×Dic10C4semidirect
13. C2×C3×C5×C4⋊C4C4semidirect
16. C2×C3×#683C4semidirect
17. C2×C3×#683C4semidirect
49. #6763Q8semidirect
68. C22×Dic10Dic3semidirect
73. C2×C5×C4⋊C4Dic3semidirect
76. C2×#683Dic3semidirect
77. C2×#683Dic3semidirect
126. C2×C3×C4⋊C4AGL(1,𝔽5)semidirect
269. C2×C4⋊C4ΓL(1,𝔽16)semidirect
372. C3×C5#1996semidirect
443. C5#71701semidirect
467. C3#10604311semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40422
#117524242
C4×#67632
#117467882
#106043113
#717015