Finite Group Databasebeta

← #14326928

Subgroups — #14326928

Order 1920.

1008 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×C5×#872C211yesmaximal, characteristic
2#11751038C211yesmaximal, characteristic, complemented
3C3×C4×C4⋊Dic5C211yesmaximal, characteristic, complemented
4#11748782C211yesmaximal, characteristic
5#11748747C212yesmaximal
6#11748719C211yesmaximal, characteristic
7#11748677C212yesmaximal, complemented
8#11748675C212yesmaximal
9#11748667C211yesmaximal, characteristic, complemented
10Dic5×Dic3⋊C4C212yesmaximal, complemented
11#11748618C211yesmaximal, characteristic, complemented
12#10613971—31nomaximal, Hall
13C22×C4×C5×D3C2211yescharacteristic
14C2×C3×C42×C5C2211yescharacteristic
15C2×C5×(C2×D3)⋊C4C2211yescharacteristic
16C2×C5×(C2×D3)⋊C4C2211yescharacteristic
17C2×C5×Dic3⋊C4C2211yescharacteristic
18C2×C5×Dic3⋊C4C2211yescharacteristic
19C2×C4×C5×Dic3C2211yescharacteristic
20C5×#872C418yescomplemented
21C2×C3×C4⋊Dic5C2212yes—
22C2×C3×C4×Dic5C2212yes—
23C2×Dic3⋊Dic5C2212yes—
24C2×Dic3⋊Dic5C2212yes—
25C2×(C2×D3)⋊Dic5C2212yes—
26C2×(C2×D3)⋊Dic5C2212yes—
27C2×Dic3×Dic5C2212yes—
28C2×Dic3×Dic5C2212yes—
29#91115C2211yescharacteristic
30#91115C2212yes—
31#91115C2211yescharacteristic
32#91115C2212yes—
33C3×#3963C2211yescharacteristic
34C3×#3963C2211yescharacteristic
35#90994C2212yes—
36#90994C2212yes—
37#73635—51nomaximal, Hall
38C2×C5×#117—31no—
39#65501—31no—
40C4×C22⋊Dic5—31no—
41#65274—32no—
42C4⋊C4×Dic5—32no—
43#65240—32no—
44#65240—32no—
45#65226—31no—
46#65226—31no—
47C4×C4⋊Dic5—31no—
48C4×C4⋊Dic5D311yescharacteristic, complemented
49C23×C5×D3C2311yescharacteristic
50C22×C3×C4×C5C2311yescharacteristic
51C22×C3×C4×C5C2311yescharacteristic
52C22×C3×C4×C5C2311yescharacteristic
53C22×Dic15C2312yes—
54C22×Dic15C2312yes—
55C22×C5×Dic3C2311yescharacteristic
56C22×C5×Dic3C2311yescharacteristic
57C22×C5×Dic3C2311yescharacteristic
58C2×C4×C5×D3—24no—
59C2×C4×C5×D3C2×C414yes—
60C22×C3×Dic5C2312yes—
61C22×C3×Dic5C2312yes—
62C3×C42×C5C2×C414yes—
63C5×(C2×D3)⋊C4C2×C414yes—
64C5×(C2×D3)⋊C4C2×C414yes—
65C5×Dic3⋊C4C2×C414yes—
66C5×Dic3⋊C4C2×C414yes—
67C4×C5×Dic3C2×C414yes—
68C3×C4⋊Dic5—24no—
69C3×C4×Dic5—42no—
70Dic3⋊Dic5—24no—
71Dic3⋊Dic5—24no—
72(C2×D3)⋊Dic5—24no—
73(C2×D3)⋊Dic5—42no—
74Dic3×Dic5—24no—
75Dic3×Dic5—42no—
76C2×#872D511yescharacteristic
77C3×C4×C4⋊C4—51no—
78#4990—52no—
79#4976—52no—
80#4940—52no—
81#4937—51no—
82C4×Dic3⋊C4—52no—
83#4667—51no—
84#4666—51no—
85#4651—51no—
86#4649—51no—
87C23×C4×C5—31no—
88C5×#117—38no—
89C2×C5×C4⋊C4—31no—
90C2×C5×C4⋊C4—31no—
91C2×C5×C22⋊C4—31no—
92C2×C5×C22⋊C4—31no—
93C2×C42×C5—31no—
94C2×C42×C5C2×D311yescharacteristic
95C2×C22⋊Dic5—32no—
96C2×C22⋊Dic5—32no—
97C2×C4⋊Dic5—32no—
98C2×C4⋊Dic5—32no—
99C2×C4⋊Dic5C2×D312yes—
100C2×C4×Dic5—32no—
101C2×C4×Dic5—32no—
102C2×C4×Dic5C2×D312yes—
103#3963—31no—
104#3963—31no—
105#3963—32no—
106#3963—32no—
107#3963—32no—
108#3963—32no—
109#3963C2×D311yescharacteristic
110#3963C2×D311yescharacteristic
111#2347—151noSylow
112C23×C3×C5C2411yescharacteristic
113C22×C5×D3—22no—
114C22×C5×D3—22no—
115C22×C5×D3—21no—
116C22×C5×D3—21no—
117C22×C5×D3C22×C412yes—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—21no—
121C2×C3×C4×C5—21no—
122C2×C3×C4×C5C22×C412yes—
123C2×C3×C4×C5C22×C412yes—
124C2×C3×C4×C5C22×C412yes—
125C2×Dic15—22no—
126C2×Dic15—22no—
127C2×Dic15—22no—
128C2×Dic15—22no—
129C2×Dic15—22no—
130C2×Dic15—22no—
131C2×C5×Dic3—22no—
132C2×C5×Dic3—21no—
133C2×C5×Dic3—21no—
134C2×C5×Dic3C22×C412yes—
135C2×C5×Dic3C22×C412yes—
136C2×C5×Dic3C22×C412yes—
137C2×C5×Dic3#4114yes—
138C4×C5×D3—28no—
139C2×C3×Dic5—22no—
140C2×C3×Dic5—22no—
141C2×C3×Dic5—22no—
142C2×C3×Dic5—22no—
143C2×C3×Dic5—22no—
144C2×C3×Dic5#4114yes—
145C22×C4×D3C2×D511yescharacteristic
146C2×C3×C4⋊C4—52no—
147C2×C3×C42—52no—
148C2×C3×C42C2×D511yescharacteristic
149C2×(C2×D3)⋊C4—52no—
150C2×(C2×D3)⋊C4—52no—
151C2×(C2×D3)⋊C4C2×D511yescharacteristic
152C2×(C2×D3)⋊C4C2×D511yescharacteristic
153C2×Dic3⋊C4—52no—
154C2×Dic3⋊C4—52no—
155C2×Dic3⋊C4C2×D511yescharacteristic
156C2×Dic3⋊C4C2×D511yescharacteristic
157C2×C4×Dic3—52no—
158C2×C4×Dic3—52no—
159C2×C4×Dic3C2×D511yescharacteristic
160#872Dic518yescomplemented
161C3×#95—51no—
162C3×#95—51no—
163C3⋊#95—51no—
164C3⋊#95—51no—
165C3⋊#95—52no—
166C3⋊#95—52no—
167C3⋊#95—52no—
168C3⋊#95—52no—
169C24×C5—31no—
170C22×C4×C5—31no—
171C22×C4×C5—31no—
172C22×C4×C5—31no—
173C22×C4×C5—34no—
174C22×C4×C5—64no—
175C22×C4×C5C22×D311yescharacteristic
176C22×C4×C5C22×D311yescharacteristic
177C22×C4×C5C22×D311yescharacteristic
178C22×Dic5—32no—
179C22×Dic5—32no—
180C22×Dic5C22×D312yes—
181C22×Dic5C22×D312yes—
182C5×C4⋊C4—34no—
183C5×C4⋊C4—34no—
184C5×C22⋊C4—34no—
185C5×C22⋊C4—34no—
186C42×C5—34no—
187C42×C5C4×D314yes—
188C22⋊Dic5—64no—
189C22⋊Dic5—122no—
190C4⋊Dic5—24no—
191C4⋊Dic5—64no—
192C4⋊Dic5—64no—
193C4×Dic5—42no—
194C4×Dic5—64no—
195C4×Dic5—122no—
196C2×#117—31no—
197#389—152no—
198#388—152no—
199#388—152no—
200#383—151no—
201#383—151no—
202#380—151no—
203C4×C4⋊C4—51no—
204C4×C4⋊C4—151no—
205C4×C4⋊C4—152no—
206C4×C22⋊C4—151no—
207C22×C3×C5C23×C411yescharacteristic
208C22×C3×C5C2×#4112yes—
209C22×C3×C5C2×#4112yes—
210C22×C3×C5H(5,𝔽2)11yescharacteristic
211C22×C3×C52−1+411yescharacteristic
212C2×C5×D3—22no—
213C2×C5×D3—22no—
214C2×C5×D3—22no—
215C2×C5×D3—42no—
216C2×C5×D3—42no—
217C3×C4×C5—22no—
218C3×C4×C5—22no—
219C3×C4×C5—22no—
220Dic15—42no—
221Dic15—42no—
222C3×Dic5—24no—
223C3×Dic5—42no—
224C5×Dic3—22no—
225C5×Dic3—22no—
226C5×Dic3#11714yes—
227C23×D3C22×D511yescharacteristic
228C22×C3×C4—52no—
229C22×C3×C4—52no—
230C22×C3×C4C22×D511yescharacteristic
231C22×C3×C4C22×D511yescharacteristic
232C22×C3×C4C22×D511yescharacteristic
233C22×Dic3—52no—
234C22×Dic3—52no—
235C22×Dic3C22×D511yescharacteristic
236C22×Dic3C22×D511yescharacteristic
237C22×Dic3C22×D511yescharacteristic
238C2×C4×D3—24no—
239C2×C4×D3C2×Dic514yes—
240C3×C4⋊C4—104no—
241C3×C42—202no—
242C3×C42C2×Dic514yes—
243(C2×D3)⋊C4—104no—
244(C2×D3)⋊C4—202no—
245(C2×D3)⋊C4C2×Dic514yes—
246(C2×D3)⋊C4C2×Dic514yes—
247Dic3⋊C4—104no—
248Dic3⋊C4—104no—
249Dic3⋊C4C2×Dic514yes—
250Dic3⋊C4C2×Dic514yes—
251C4×Dic3—104no—
252C4×Dic3—202no—
253C4×Dic3C2×Dic514yes—
254C23×C5—32no—
255C23×C5—62no—
256C23×C5—61no—
257C23×C5—61no—
258C23×C5—62no—
259C23×C5C23×D311yescharacteristic
260C2×C4×C5—22no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—22no—
264C2×C4×C5—32no—
265C2×C4×C5—34no—
266C2×C4×C5—32no—
267C2×C4×C5—32no—
268C2×C4×C5—68no—
269C2×C4×C5—62no—
270C2×C4×C5—61no—
271C2×C4×C5—61no—
272C2×C4×C5C2×C4×D312yes—
273C2×C4×C5C2×C4×D312yes—
274C2×C4×C5C2×C4×D312yes—
275C2×Dic5—22no—
276C2×Dic5—22no—
277C2×Dic5—22no—
278C2×Dic5—22no—
279C2×Dic5—22no—
280C2×Dic5—62no—
281C2×Dic5—62no—
282C2×Dic5—62no—
283C2×Dic5—62no—
284C2×Dic5—62no—
285C2×Dic5—62no—
286C2×Dic5#23514yes—
287C23×C4—31no—
288#117—38no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—52no—
292C2×C4⋊C4—152no—
293C2×C4⋊C4—152no—
294C2×C22⋊C4—31no—
295C2×C22⋊C4—31no—
296C2×C22⋊C4—152no—
297C2×C22⋊C4—152no—
298C2×C42—31no—
299C2×C42—52no—
300C2×C42—152no—
301C2×C42—152no—
302C2×C42D3×D511yescharacteristic
303#95—51no—
304#95—51no—
305#95—152no—
306#95—152no—
307#95—151no—
308#95—151no—
309#95—152no—
310#95—152no—
311C2×C3×C5C2×#11711yescharacteristic
312C2×C3×C5C4×#4111yescharacteristic
313C2×C3×C5#52011yescharacteristic
314C2×C3×C5#55312yes—
315C2×C3×C5#55612yes—
316C5×D3—42no—
317C23×C3C23×D511yescharacteristic
318C22×D3—22no—
319C22×D3—22no—
320C22×D3—21no—
321C22×D3—21no—
322C22×D3C22×Dic512yes—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—54no—
328C2×C3×C4—102no—
329C2×C3×C4—102no—
330C2×C3×C4—102no—
331C2×C3×C4—102no—
332C2×C3×C4—102no—
333C2×C3×C4C22×Dic512yes—
334C2×C3×C4C22×Dic512yes—
335C2×C3×C4C22×Dic512yes—
336C2×Dic3—22no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—102no—
340C2×Dic3—102no—
341C2×Dic3—102no—
342C2×Dic3—102no—
343C2×Dic3—102no—
344C2×Dic3—102no—
345C2×Dic3#70914yes—
346C2×Dic3C22×Dic512yes—
347C2×Dic3C22×Dic512yes—
348C2×Dic3C22×Dic512yes—
349C4×D3—28no—
350C22×C5—62no—
351C22×C5—62no—
352C22×C5—62no—
353C22×C5—122no—
354C22×C5—122no—
355C22×C5C22×C4×D311yescharacteristic
356C22×C5C2×#23512yes—
357C22×C5#100411yescharacteristic
358C22×C5#100711yescharacteristic
359C22×C5D3×#4112yes—
360C4×C5—22no—
361C4×C5—22no—
362C4×C5—22no—
363C4×C5—34no—
364C4×C5—62no—
365C4×C5—62no—
366Dic5—24no—
367Dic5—42no—
368Dic5—122no—
369Dic5—122no—
370C24—31no—
371C22×C4—31no—
372C22×C4—31no—
373C22×C4—31no—
374C22×C4—34no—
375C22×C4—52no—
376C22×C4—52no—
377C22×C4—64no—
378C22×C4—152no—
379C22×C4—152no—
380C22×C4C2×D3×D511yescharacteristic
381C22×C4C2×D3×D511yescharacteristic
382C22×C4C2×D3×D511yescharacteristic
383C4⋊C4—34no—
384C4⋊C4—34no—
385C4⋊C4—104no—
386C4⋊C4—304no—
387C4⋊C4—304no—
388C22⋊C4—34no—
389C22⋊C4—34no—
390C22⋊C4—304no—
391C22⋊C4—602no—
392C42—34no—
393C42—202no—
394C42—304no—
395C42—602no—
396C42D3×Dic514yes—
397C3×C5#234711yescharacteristic, complemented, Hall
398C22×C3C2×#70912yes—
399C22×C3D5×#4112yes—
400C22×C3#414911yescharacteristic
401C22×C3#415011yescharacteristic
402C22×C3C23×Dic511yescharacteristic
403C2×D3—22no—
404C2×D3—22no—
405C2×D3—22no—
406C2×D3—42no—
407C2×D3—42no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—104no—
412C3×C4—202no—
413Dic3—22no—
414Dic3—22no—
415Dic3—202no—
416Dic3—202no—
417Dic3#407214yes—
418C2×C5—122no—
419C2×C5C4×#23511yescharacteristic
420C2×C5#550411yescharacteristic
421C2×C5D3×#11711yescharacteristic
422C2×C5#554012yes—
423C2×C5#556812yes—
424C23—32no—
425C23—61no—
426C23—62no—
427C23—62no—
428C23—61no—
429C23C22×D3×D511yescharacteristic
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—34no—
435C2×C4—32no—
436C2×C4—32no—
437C2×C4—32no—
438C2×C4—54no—
439C2×C4—68no—
440C2×C4—61no—
441C2×C4—62no—
442C2×C4—61no—
443C2×C4—102no—
444C2×C4—102no—
445C2×C4—102no—
446C2×C4—102no—
447C2×C4—102no—
448C2×C4—302no—
449C2×C4—302no—
450C2×C4—302no—
451C2×C4—302no—
452C2×C4—302no—
453C2×C4—302no—
454C2×C4C2×D3×Dic512yes—
455C2×C4C2×D3×Dic512yes—
456C2×C4C2×D3×Dic512yes—
457C2×C3#6586412yes—
458C2×C3#6586712yes—
459C2×C3C2×#407211yescharacteristic
460C2×C3#41×Dic511yescharacteristic
461C2×C3#6616111yescharacteristic
462D3—42no—
463C5#7363511yescharacteristic, complemented, Sylow
464C22—62no—
465C22—62no—
466C22—62no—
467C22—122no—
468C22—122no—
469C22#9200411yescharacteristic
470C22D5×#23512yes—
471C22D3×#70912yes—
472C22#9201711yescharacteristic
473C22C22×D3×Dic511yescharacteristic
474C4—22no—
475C4—22no—
476C4—22no—
477C4—34no—
478C4—62no—
479C4—62no—
480C4—104no—
481C4—202no—
482C4—602no—
483C4—602no—
484C3#1061397111yescharacteristic, complemented, Sylow
485C2—122no—
486C2#1175257412yes—
487C2#1175259212yes—
488C2Dic5×#23511yescharacteristic
489C2D3×#407211yescharacteristic
490C2#1175449211yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC23×C5×D3C23
hypercenterC2×C42D3×D5
hypocenterC3×C5#2347
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. #11751038C2semidirect
3. C3×C4×C4⋊Dic5C2semidirect
7. #11748677C2semidirect
9. #11748667C2semidirect
10. Dic5×Dic3⋊C4C2semidirect
11. #11748618C2semidirect
20. C5×#872C4semidirect
48. C4×C4⋊Dic5D3semidirect
160. #872Dic5semidirect
397. C3×C5#2347semidirect
463. C5#73635semidirect
484. C3#10613971semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#8722
#117510382
C3×C4×C4⋊Dic52
#117487822
#117487472
#117487192
#117486772
#117486752
#117486672
Dic5×Dic3⋊C42
#117486182
#106139713
#736355