Finite Group Databasebeta

← #14437833

Subgroups — #14437833

Order 1920.

800 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66062C211yesmaximal, characteristic, complemented
2#11752672C211yesmaximal, characteristic
3#11751194C211yesmaximal, characteristic, complemented
4C5×#5088C211yesmaximal, characteristic, complemented
5#11747499C214yesmaximal, complemented
6#11747142C214yesmaximal, complemented
7C4×D5×C3⋊C8C211yesmaximal, characteristic, complemented
8#11746921C211yesmaximal, characteristic, complemented
9#11746901C211yesmaximal, characteristic, complemented
10#10613293—31nomaximal, Hall
11C2×#6772C2212yes—
12C3×C5×C4⋊Q8C2211yescharacteristic
13#91758C2211yescharacteristic
14C5×#874C2211yescharacteristic
15C3×#4092—21no—
16C3×Q8×Dic5—21no—
17C3×#4043—21no—
18C3×#4038C2212yes—
19C3×#4036C2212yes—
20C3×#4033—21no—
21C3×C42×D5C2211yescharacteristic
22C3×#4015C2211yescharacteristic
23#91449—21no—
24#91449—21no—
25Dic5×D12—21no—
26Dic5×D12—21no—
27#91373C2212yes—
28C2×D5×C3⋊C8C2212yes—
29#91094C2214yes—
30C4×C3⋊#168C2211yescharacteristic
31C5×#809C2214yes—
32C4×C5×C3⋊C8C2211yescharacteristic
33#90974C2214yes—
34#90966C2214yes—
35Dic5×C3⋊C8C2212yes—
36#83334—51nomaximal, Hall
37#66062D311yescharacteristic, complemented
38#66050—31no—
39C5×#486—31no—
40#65382—31no—
41#65359—31no—
42#65061—34no—
43#65057—34no—
44C4×C8×D5—31no—
45#64970—31no—
46C2×C3×C5×Q8—21no—
47C2×Dic30C2312yes—
48C2×C5×D12—21no—
49C2×C5×D12—21no—
50C2×C5×D12C2312yes—
51C2×C3×C4×D5D411yescharacteristic
52C2×C3×C4×D5D411yescharacteristic
53C2×C3×C4×D5C2311yescharacteristic
54C2×C3×Dic10—21no—
55C2×#1201—22no—
56C2×D3×Dic5—22no—
57#6772—28no—
58C3×C5×C4⋊C4—21no—
59C3×C5×C4⋊C4C2312yes—
60C3×C42×C5C2311yescharacteristic
61C4⋊Dic15—21no—
62C4⋊Dic15—21no—
63C2×C3⋊#168C2312yes—
64C2×C5×C3⋊C8C2312yes—
65C3×#685—22no—
66C3×C4⋊Dic5—21no—
67C3×C4⋊Dic5—22no—
68C3×C4⋊Dic5C2312yes—
69C3×#683—22no—
70C3×C4×Dic5—22no—
71C3×C4×Dic5D411yescharacteristic
72C3×C4×Dic5D411yescharacteristic
73C3×C4×Dic5C2311yescharacteristic
74(C2×D3)⋊Dic5—22no—
75(C2×D3)⋊Dic5—22no—
76D5×C3⋊C8—24no—
77C3×#533—51no—
78#5477—51no—
79#5093—51no—
80#5093—51no—
81#5088—51no—
82#5088D511yescharacteristic, complemented
83#5068—54no—
84#5002—54no—
85C2×C4×C3⋊C8—51no—
86C2×#711—32no—
87C5×C4⋊Q8C2×D311yescharacteristic
88C5×#127—31no—
89#4092—21no—
90Q8×Dic5—21no—
91#4084—61no—
92#4084—61no—
93#4082—32no—
94D4×Dic5—61no—
95D4×Dic5—61no—
96C2×C8×D5—32no—
97#4043—21no—
98#4038C2×D312yes—
99#4036C2×D312yes—
100#4033—21no—
101C42×D5C2×D311yescharacteristic
102#4015—31no—
103#4015C2×D311yescharacteristic
104C5×#102—34no—
105C4×C5×C8—31no—
106#3964—34no—
107#3948—34no—
108C8×Dic5—32no—
109#3942—34no—
110C4×#168—31no—
111#3131—151noSylow
112C22×C5×D3—22no—
113C22×C3×D5C2×D411yescharacteristic
114C3×C5×Q8—42no—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5C2×D411yescharacteristic
117C2×C3×C4×C5C2×D411yescharacteristic
118C2×C3×C4×C5C2411yescharacteristic
119C2×Dic15—22no—
120Dic30—24no—
121C5×D12—24no—
122C5×D12—42no—
123C5×D12—42no—
124C2×C3×Dic5—22no—
125C2×C3×Dic5C2×D411yescharacteristic
126C2×C3×Dic5C2×D411yescharacteristic
127C2×C3×Dic5C2×D411yescharacteristic
128C3×C4×D5—22no—
129C3×C4×D5—22no—
130C3×C4×D5#4114yes—
131C3×Dic10—42no—
132#1201—44no—
133D3×Dic5—44no—
134C3⋊#168—22no—
135C5×C3⋊C8—22no—
136C2×#235—52no—
137C3×C4⋊Q8—51no—
138C3×C4⋊Q8C2×D511yescharacteristic
139C3×#125—52no—
140C3×C22⋊Q8—101no—
141C3×C22⋊Q8—101no—
142C3×C4×Q8—101no—
143C3×C4×Q8—101no—
144C3×#117—52no—
145C2×C3×C42—51no—
146#930—101no—
147#930—101no—
148C22×C3⋊C8—52no—
149#875—52no—
150#874C2×D511yescharacteristic
151C4×D12—101no—
152C4×D12—101no—
153#869—51no—
154#810—54no—
155#810—54no—
156#809—54no—
157#809C2×D514yes—
158C4×C3⋊C8—51no—
159C4×C3⋊C8—52no—
160C4×C3⋊C8C2×D511yescharacteristic
161C2×C5×Q8—21no—
162C2×C5×D4—32no—
163C2×C5×D4—61no—
164C2×C5×D4—61no—
165C2×C22⋊D5—62no—
166C22×Dic5—62no—
167#711—68no—
168C2×C4×D5#6711yescharacteristic
169C2×C4×D5#6711yescharacteristic
170C2×C4×D5C22×D311yescharacteristic
171C2×Dic10—21no—
172C2×Dic10—32no—
173C2×C5×C8—32no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4C22×D312yes—
176C42×C5C22×D311yescharacteristic
177C22⋊Dic5—62no—
178C22⋊Dic5—62no—
179#685—22no—
180C4⋊Dic5—21no—
181C4⋊Dic5—22no—
182C4⋊Dic5—61no—
183C4⋊Dic5—61no—
184C4⋊Dic5C22×D312yes—
185#683—22no—
186C4×Dic5—22no—
187C4×Dic5#6711yescharacteristic
188C4×Dic5#6711yescharacteristic
189C4×Dic5C22×D311yescharacteristic
190C2×#168—32no—
191C8×D5—64no—
192#533—51no—
193#532—151no—
194#488—154no—
195#487—151no—
196#487—151no—
197#486—31no—
198#486—151no—
199#416—154no—
200C2×C4×C8—151no—
201C22×C3×C5C22×D411yescharacteristic
202C2×C5×D3—42no—
203C2×C5×D3—42no—
204C2×C5×D3—42no—
205C2×C3×D5—21no—
206C2×C3×D5—21no—
207C2×C3×D5#12412yes—
208C3×C4×C5—42no—
209C3×C4×C5#13512yes—
210C3×C4×C5#13512yes—
211C3×C4×C5C2×#4112yes—
212Dic15—42no—
213C3×Dic5—21no—
214C3×Dic5—21no—
215C3×Dic5—42no—
216C3×Dic5#12412yes—
217C2×C3×Q8—21no—
218C2×C3×Q8—101no—
219C22×C3×C4—51no—
220C22×C3×C4—51no—
221C22×C3×C4—51no—
222C2×#67—102no—
223#235—108no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12C22×D512yes—
227C2×C4×D3—102no—
228C2×Dic6—52no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—52no—
231C3×C4⋊C4—102no—
232C3×C4⋊C4—102no—
233C3×C4⋊C4—101no—
234C3×C4⋊C4C22×D512yes—
235C3×C22⋊C4—102no—
236C3×C42—51no—
237C3×C42—51no—
238C3×C42—51no—
239C3×C42—102no—
240C3×C42C22×D511yescharacteristic
241(C2×D3)⋊C4—102no—
242(C2×D3)⋊C4—102no—
243C4⋊Dic3—101no—
244C4⋊Dic3—101no—
245C2×C3⋊C8—52no—
246C2×C3⋊C8—104no—
247C2×C3⋊C8C22×D512yes—
248C23×C5—62no—
249C22×D5C2×#6711yescharacteristic
250C5×Q8—42no—
251C5×D4—64no—
252C5×D4—122no—
253C5×D4—122no—
254C2×C4×C5—22no—
255C2×C4×C5C2×#6711yescharacteristic
256C2×C4×C5C2×#6711yescharacteristic
257C2×C4×C5C23×D311yescharacteristic
258C22⋊D5—124no—
259C2×Dic5—22no—
260C2×Dic5—62no—
261C2×Dic5—124no—
262C2×Dic5C2×#6711yescharacteristic
263C2×Dic5C2×#6711yescharacteristic
264C2×Dic5C2×#6711yescharacteristic
265C4×D5—22no—
266C4×D5—22no—
267C4×D5#23914yes—
268Dic10—42no—
269Dic10—64no—
270C5×C8—62no—
271#168—62no—
272C2×#41—152no—
273C22×C8—152no—
274C4⋊Q8—51no—
275C4⋊Q8—151no—
276C4⋊Q8D3×D511yescharacteristic
277#127—31no—
278#125—52no—
279#124—152no—
280C22⋊Q8—101no—
281C22⋊Q8—101no—
282#121—301no—
283#121—301no—
284C4×Q8—101no—
285C4×Q8—101no—
286C4×D4—301no—
287C4×D4—301no—
288#117—52no—
289C2×C42—51no—
290#103—154no—
291#103—154no—
292#102—34no—
293#102—154no—
294C4×C8—31no—
295C4×C8—152no—
296C4×C8—151no—
297C2×C3×C5C2×#12411yescharacteristic
298C2×C3×C5C2×#13511yescharacteristic
299C2×C3×C5C2×#13511yescharacteristic
300C3×D5—22no—
301C5×D3—82no—
302C23×C3—51no—
303C22×D3—22no—
304C3×Q8—42no—
305C3×Q8—202no—
306C2×C3×C4—22no—
307C2×C3×C4—51no—
308C2×C3×C4—54no—
309C2×C3×C4—51no—
310C2×C3×C4—51no—
311C2×C3×C4—102no—
312C2×C3×C4—102no—
313C2×C3×C4—102no—
314C2×C3×C4D4×D511yescharacteristic
315C2×C3×C4D4×D511yescharacteristic
316C2×C3×C4C23×D511yescharacteristic
317#67—204no—
318C2×Dic3—102no—
319D12—24no—
320D12—42no—
321D12—42no—
322C4×D3—204no—
323Dic6—104no—
324C3⋊C8—22no—
325C3⋊C8—102no—
326C22×C5—122no—
327C22×C5—122no—
328C22×C5—122no—
329C22×C5C22×#6711yescharacteristic
330C2×D5—21no—
331C2×D5—21no—
332C2×D5#94712yes—
333C4×C5—42no—
334C4×C5#95012yes—
335C4×C5#95012yes—
336C4×C5C2×#23912yes—
337Dic5—21no—
338Dic5—21no—
339Dic5—42no—
340Dic5—122no—
341Dic5#94712yes—
342#41—308no—
343C2×Q8—21no—
344C2×Q8—101no—
345C2×Q8—152no—
346C2×D4—32no—
347C2×D4—61no—
348C2×D4—61no—
349C2×D4—302no—
350C22×C4—51no—
351C22×C4—51no—
352C22×C4—51no—
353C22×C4—302no—
354C2×C8—32no—
355C2×C8—152no—
356C2×C8—304no—
357C4⋊C4—21no—
358C4⋊C4—52no—
359C4⋊C4—102no—
360C4⋊C4—102no—
361C4⋊C4—101no—
362C4⋊C4—301no—
363C4⋊C4—301no—
364C4⋊C4C2×D3×D512yes—
365C22⋊C4—102no—
366C22⋊C4—302no—
367C22⋊C4—302no—
368C42—51no—
369C42—51no—
370C42—51no—
371C42—102no—
372C42C2×D3×D511yescharacteristic
373C3×C5#313111yescharacteristic, complemented, Hall
374C22×C3—52no—
375C22×C3—101no—
376C22×C3—101no—
377C22×C3C2×D4×D511yescharacteristic
378C2×D3—42no—
379C2×D3—42no—
380C2×D3—42no—
381C3×C4—42no—
382C3×C4—52no—
383C3×C4—101no—
384C3×C4—101no—
385C3×C4—202no—
386C3×C4#405812yes—
387C3×C4#406212yes—
388C3×C4D5×#4112yes—
389Dic3—202no—
390C2×C5—242no—
391C2×C5C2×#94711yescharacteristic
392C2×C5C2×#95011yescharacteristic
393C2×C5C2×#95011yescharacteristic
394D5—22no—
395C23—51no—
396C23—62no—
397Q8—42no—
398Q8—202no—
399Q8—304no—
400D4—64no—
401D4—122no—
402D4—122no—
403D4—604no—
404C2×C4—22no—
405C2×C4—51no—
406C2×C4—54no—
407C2×C4—51no—
408C2×C4—51no—
409C2×C4—102no—
410C2×C4—102no—
411C2×C4—102no—
412C2×C4—302no—
413C2×C4—604no—
414C2×C4D5×#6711yescharacteristic
415C2×C4D5×#6711yescharacteristic
416C2×C4C22×D3×D511yescharacteristic
417C8—62no—
418C8—302no—
419C2×C3—102no—
420C2×C3D5×#12411yescharacteristic
421C2×C3C2×#405811yescharacteristic
422C2×C3C2×#406211yescharacteristic
423D3—82no—
424C5#8333411yescharacteristic, complemented, Sylow
425C22—52no—
426C22—101no—
427C22—101no—
428C22—122no—
429C22—122no—
430C22—122no—
431C22C2×D5×#6711yescharacteristic
432C4—42no—
433C4—52no—
434C4—101no—
435C4—101no—
436C4—202no—
437C4—602no—
438C4#9149012yes—
439C4#9151412yes—
440C4D5×#23912yes—
441C3#1061329311yescharacteristic, complemented, Sylow
442C2—102no—
443C2—242no—
444C2C2×#9149011yescharacteristic
445C2C2×#9151411yescharacteristic
446C2D5×#94711yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5C22×D4
prime radical#11752672C2
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#3131
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#66062C2semidirect
3. #11751194C2semidirect
4. C5×#5088C2semidirect
5. #11747499C2semidirect
6. #11747142C2semidirect
7. C4×D5×C3⋊C8C2semidirect
8. #11746921C2semidirect
9. #11746901C2semidirect
37. #66062D3semidirect
82. #5088D5semidirect
373. C3×C5#3131semidirect
424. C5#83334semidirect
441. C3#10613293semidirect

Maximal subgroups

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

subgroupindex
C3×#660622
#117526722
#117511942
C5×#50882
#117474992
#117471422
C4×D5×C3⋊C82
#117469212
#117469012
#106132933
#833345