Finite Group Databasebeta

← #14319925

Subgroups — #14319925

Order 1920.

2176 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#1200C211yesmaximal, characteristic
2C2×C5×#877C211yesmaximal, characteristic, complemented
3C2×#91529C211yesmaximal, characteristic, complemented
4#11751056C211yesmaximal, characteristic
5C3×#65232C211yesmaximal, characteristic, complemented
6#11749502C211yesmaximal, characteristic
7#11749471C211yesmaximal, characteristic, complemented
8#11748882C214yesmaximal, complemented
9#11748881C214yesmaximal, complemented
10#10613998—31nomaximal, Hall
11C24×D15C2211yescharacteristic
12C23×C5×Dic3C2211yescharacteristic
13C23×C3×Dic5C2211yescharacteristic
14C22×#1200—24no—
15C22×#1200—24no—
16C22×#1200C418yescomplemented
17C22×#1200C2212yes—
18C22×#1200C2212yes—
19C2×C3×C5×C22⋊C4C2211yescharacteristic
20C2×#6719C2211yescharacteristic
21C2×#6714C2214yes—
22C2×C5×#217C2211yescharacteristic
23C2×C5×Dic3⋊C4C2212yescomplemented
24C2×C4×C5×Dic3C2212yescomplemented
25C5×#877—24no—
26C2×C3×C22⋊Dic5C2211yescharacteristic
27C2×Dic3⋊Dic5C2212yescomplemented
28C2×#6667C2214yes—
29#91529—24no—
30C2×Dic3×Dic5C2212yescomplemented
31C3×#3963C2214yescomplemented
32#90994C2214yescomplemented
33#74050—51nomaximal, Hall
34C23×C4×D5—31no—
35C2×C5×#117—31no—
36C2×#4072—31no—
37#65276—34no—
38#65244—31no—
39#65244—31no—
40#65232—31no—
41#65232D311yescharacteristic, complemented
42C4×#685—34no—
43C24×C3×C5C2311yescharacteristic
44C23×D15—22no—
45C23×D15—22no—
46C23×D15—28no—
47C23×D15C2×C414yes—
48C23×D15C2312yes—
49C22×C3×C4×C5C2312yes—
50C22×Dic15C2312yes—
51C22×C5×Dic3C2×C414yes—
52C22×C5×Dic3D414yes—
53C22×C5×Dic3D414yes—
54C22×C5×Dic3C2312yes—
55C22×C5×Dic3C2311yescharacteristic
56C22×C5×Dic3C2311yescharacteristic
57C22×C3×Dic5—22no—
58C22×C3×Dic5—22no—
59C22×C3×Dic5C2×C414yes—
60C22×C3×Dic5C2312yes—
61C22×C3×Dic5C2312yes—
62C2×#1200—232no—
63C2×#1200—24no—
64C2×#1200—24no—
65C2×#1200—24no—
66C2×#1200—24no—
67C2×#1200C2×C418yes—
68C2×#1200C2×C418yes—
69C3×C5×C22⋊C4—22no—
70#6719—22no—
71#6714—28no—
72C5×#217—22no—
73C5×Dic3⋊C4—24no—
74C4×C5×Dic3—24no—
75C3×C22⋊Dic5—22no—
76Dic3⋊Dic5—24no—
77#6667—28no—
78Dic3×Dic5—24no—
79C23×C4×D3—51no—
80C2×#877—51no—
81C2×#877D511yescharacteristic, complemented
82C3×#381—51no—
83#5211—51no—
84#4955—51no—
85#4955—51no—
86#4667—54no—
87#4666—54no—
88C24×D5—31no—
89C23×C4×C5—31no—
90C23×Dic5C2×D311yescharacteristic
91C22×C4×D5—38no—
92C22×C4×D5—32no—
93C22×C4×D5—32no—
94C22×C4×D5—64no—
95C22×C4×D5—64no—
96C5×#117—64no—
97C2×C5×C4⋊C4—32no—
98C2×C5×C22⋊C4—31no—
99C2×C5×C22⋊C4C2×D311yescharacteristic
100C2×C42×C5—32no—
101C2×C22⋊Dic5—31no—
102C2×C22⋊Dic5C2×D311yescharacteristic
103C2×#685—34no—
104C2×#685—34no—
105#4072—64no—
106C2×C4⋊Dic5—32no—
107C2×C4×Dic5—32no—
108#3963—34no—
109#3963C2×D314yescomplemented
110#2280—151noSylow
111C23×C3×C5—24no—
112C23×C3×C5C22×C412yes—
113C23×C3×C5C2×D412yes—
114C23×C3×C5C2×D412yes—
115C23×C3×C5C2411yescharacteristic
116C22×D15—216no—
117C22×D15—216no—
118C22×D15—28no—
119C22×D15—22no—
120C22×D15—22no—
121C22×D15—24no—
122C22×D15—216no—
123C22×D15C22×C414yes—
124C22×D15#4118yes—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—24no—
127C2×Dic15—22no—
128C2×Dic15—24no—
129C2×C5×Dic3—22no—
130C2×C5×Dic3—24no—
131C2×C5×Dic3C22⋊C4116yescomplemented
132C2×C5×Dic3C22×C412yes—
133C2×C5×Dic3C22×C412yes—
134C2×C5×Dic3C2×D412yes—
135C2×C5×Dic3C2×D412yes—
136C2×C5×Dic3C2×D412yes—
137C2×C5×Dic3C2×D412yes—
138C2×C3×Dic5—28no—
139C2×C3×Dic5—22no—
140C2×C3×Dic5—22no—
141C2×C3×Dic5—22no—
142C2×C3×Dic5—24no—
143C2×C3×Dic5C22×C414yes—
144#1200—216no—
145#1200—216no—
146C24×D3—51no—
147C23×C3×C4—51no—
148C23×Dic3C2×D511yescharacteristic
149C22×C4×D3—58no—
150C22×C4×D3—52no—
151C22×C4×D3—52no—
152C22×C4×D3—104no—
153C22×C4×D3—104no—
154C2×C3×C22⋊C4—51no—
155C2×C3×C22⋊C4C2×D511yescharacteristic
156C2×#217—51no—
157C2×#217C2×D511yescharacteristic
158C2×(C2×D3)⋊C4—54no—
159C2×(C2×D3)⋊C4—54no—
160C2×Dic3⋊C4—52no—
161C2×Dic3⋊C4C2×D512yescomplemented
162C2×C4×Dic3—52no—
163C2×C4×Dic3C2×D512yescomplemented
164#877—24no—
165#877—104no—
166C3×#95—54no—
167C3⋊#95—54no—
168C24×C5C22×D311yescharacteristic
169C23×D5—32no—
170C23×D5—34no—
171C23×D5—68no—
172C23×D5—62no—
173C23×D5—62no—
174C22×C4×C5—34no—
175C22×C4×C5—34no—
176C22×C4×C5—34no—
177C22×C4×C5—31no—
178C22×C4×C5—31no—
179C22×C4×C5—32no—
180C22×C4×C5C22×D312yes—
181C22×Dic5—22no—
182C22×Dic5—22no—
183C22×Dic5—32no—
184C22×Dic5C4×D314yes—
185C22×Dic5C22×D312yes—
186C22×Dic5C22×D312yes—
187C2×C4×D5—38no—
188C2×C4×D5—38no—
189C2×C4×D5—632no—
190C2×C4×D5—64no—
191C2×C4×D5—64no—
192C2×C4×D5—64no—
193C2×C4×D5—64no—
194C5×C4⋊C4—64no—
195C5×C22⋊C4—22no—
196C5×C22⋊C4—62no—
197C42×C5—64no—
198C22⋊Dic5—22no—
199C22⋊Dic5—62no—
200#685—68no—
201#685—68no—
202C4⋊Dic5—64no—
203C4×Dic5—64no—
204C24×C4—151no—
205C2×#117—31no—
206C2×#117—151no—
207#381—51no—
208#381—151no—
209#380—154no—
210#379—151no—
211#379—151no—
212C4×C22⋊C4—154no—
213C22×C3×C5—24no—
214C22×C3×C5—24no—
215C22×C3×C5—24no—
216C22×C3×C5C2×C22⋊C414yes—
217C22×C3×C5C23×C411yescharacteristic
218C22×C3×C5C22×D411yescharacteristic
219C22×C3×C5C22×D411yescharacteristic
220C22×C3×C5C2×#4114yes—
221C2×D15—232no—
222C2×D15—24no—
223C2×D15—24no—
224C2×D15—28no—
225C2×D15—28no—
226C2×D15#11718yescomplemented
227C3×C4×C5—42no—
228Dic15—42no—
229C3×Dic5—24no—
230C3×Dic5—42no—
231C5×Dic3—42no—
232C5×Dic3C2×C22⋊C414yes—
233C5×Dic3C2×C22⋊C414yes—
234C24×C3C22×D511yescharacteristic
235C23×D3—52no—
236C23×D3—54no—
237C23×D3—108no—
238C23×D3—102no—
239C23×D3—102no—
240C22×C3×C4—54no—
241C22×C3×C4—52no—
242C22×C3×C4—52no—
243C22×C3×C4—102no—
244C22×C3×C4—102no—
245C22×C3×C4C22×D512yes—
246C22×Dic3—52no—
247C22×Dic3C4×D514yes—
248C22×Dic3D2014yes—
249C22×Dic3C22⋊D514yes—
250C22×Dic3C22×D512yes—
251C22×Dic3C22×D511yescharacteristic
252C22×Dic3C22×D511yescharacteristic
253C2×C4×D3—58no—
254C2×C4×D3—58no—
255C2×C4×D3—1032no—
256C2×C4×D3—104no—
257C2×C4×D3—104no—
258C2×C4×D3—104no—
259C2×C4×D3—104no—
260C3×C22⋊C4—22no—
261C3×C22⋊C4—102no—
262#217—22no—
263#217—102no—
264(C2×D3)⋊C4—108no—
265(C2×D3)⋊C4—108no—
266Dic3⋊C4—24no—
267Dic3⋊C4—104no—
268C4×Dic3—24no—
269C4×Dic3—104no—
270C23×C5—24no—
271C23×C5C2×C4×D312yes—
272C23×C5D3×D412yes—
273C23×C5D3×D412yes—
274C23×C5C23×D311yescharacteristic
275C22×D5—34no—
276C22×D5—38no—
277C22×D5—616no—
278C22×D5—616no—
279C22×D5—616no—
280C22×D5—68no—
281C22×D5—62no—
282C22×D5—62no—
283C22×D5—64no—
284C2×C4×C5—22no—
285C2×C4×C5—24no—
286C2×C4×C5—316no—
287C2×C4×C5—32no—
288C2×C4×C5—32no—
289C2×C4×C5—32no—
290C2×C4×C5—32no—
291C2×C4×C5—32no—
292C2×C4×C5—32no—
293C2×C4×C5—62no—
294C2×C4×C5—64no—
295C2×Dic5—28no—
296C2×Dic5—22no—
297C2×Dic5—22no—
298C2×Dic5—22no—
299C2×Dic5—24no—
300C2×Dic5—62no—
301C2×Dic5—64no—
302C2×Dic5C2×C4×D314yes—
303C4×D5—616no—
304C4×D5—616no—
305C25—151no—
306C23×C4—31no—
307C23×C4—51no—
308C23×C4—152no—
309C23×C4—152no—
310C23×C4—158no—
311C23×C4—304no—
312C23×C4—304no—
313#117—64no—
314#117—304no—
315C2×C4⋊C4—32no—
316C2×C4⋊C4—152no—
317C2×C22⋊C4—31no—
318C2×C22⋊C4—51no—
319C2×C22⋊C4—154no—
320C2×C22⋊C4—151no—
321C2×C22⋊C4—154no—
322C2×C22⋊C4D3×D511yescharacteristic
323C2×C42—32no—
324C2×C42—152no—
325#95—54no—
326#95—154no—
327C2×C3×C5—24no—
328C2×C3×C5C22×C22⋊C411yescharacteristic
329C2×C3×C5C2×#11712yes—
330C2×C3×C5#52512yes—
331C2×C3×C5#52512yes—
332D15—28no—
333C23×C3—24no—
334C23×C3C2×C4×D512yes—
335C23×C3C2×D2012yes—
336C23×C3C2×C22⋊D512yes—
337C23×C3C23×D511yescharacteristic
338C22×D3—54no—
339C22×D3—58no—
340C22×D3—1016no—
341C22×D3—1016no—
342C22×D3—1016no—
343C22×D3—102no—
344C22×D3—104no—
345C22×D3—102no—
346C22×D3—108no—
347C2×C3×C4—24no—
348C2×C3×C4—22no—
349C2×C3×C4—54no—
350C2×C3×C4—102no—
351C2×C3×C4—102no—
352C2×C3×C4—108no—
353C2×C3×C4—102no—
354C2×C3×C4—104no—
355C2×Dic3—24no—
356C2×Dic3—22no—
357C2×Dic3—102no—
358C2×Dic3—104no—
359C2×Dic3#685116yescomplemented
360C2×Dic3C2×C4×D512yes—
361C2×Dic3C2×C4×D512yes—
362C2×Dic3C2×D2012yes—
363C2×Dic3C2×D2012yes—
364C2×Dic3C2×C22⋊D512yes—
365C2×Dic3C2×C22⋊D512yes—
366C4×D3—1016no—
367C4×D3—1016no—
368C22×C5—24no—
369C22×C5—24no—
370C22×C5—24no—
371C22×C5D3×C22⋊C414yes—
372C22×C5C22×C4×D311yescharacteristic
373C22×C5C2×D3×D411yescharacteristic
374C22×C5C2×D3×D411yescharacteristic
375C22×C5D3×#4114yes—
376C2×D5—38no—
377C2×D5—632no—
378C2×D5—68no—
379C2×D5—68no—
380C2×D5—64no—
381C2×D5—64no—
382C4×C5—34no—
383C4×C5—34no—
384C4×C5—42no—
385C4×C5—122no—
386Dic5—24no—
387Dic5—42no—
388Dic5—122no—
389C24—152no—
390C24—154no—
391C24—308no—
392C24—302no—
393C24—302no—
394C24C2×D3×D511yescharacteristic
395C22×C4—34no—
396C22×C4—34no—
397C22×C4—34no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—32no—
401C22×C4—54no—
402C22×C4—52no—
403C22×C4—52no—
404C22×C4—102no—
405C22×C4—102no—
406C22×C4—158no—
407C22×C4—158no—
408C22×C4—152no—
409C22×C4—3032no—
410C22×C4—304no—
411C22×C4—304no—
412C22×C4—304no—
413C22×C4—304no—
414C22×C4C2×D3×D512yes—
415C4⋊C4—64no—
416C4⋊C4—304no—
417C22⋊C4—22no—
418C22⋊C4—62no—
419C22⋊C4—102no—
420C22⋊C4—308no—
421C22⋊C4—308no—
422C22⋊C4—302no—
423C42—64no—
424C42—304no—
425C3×C5#228011yescharacteristic, complemented, Hall
426C22×C3—24no—
427C22×C3—24no—
428C22×C3—24no—
429C22×C3C2×#68514yes—
430C22×C3C22×C4×D511yescharacteristic
431C22×C3C22×D2011yescharacteristic
432C22×C3D5×#4114yes—
433C22×C3C22×C22⋊D511yescharacteristic
434C2×D3—58no—
435C2×D3—1032no—
436C2×D3—104no—
437C2×D3—108no—
438C2×D3—104no—
439C2×D3—108no—
440C3×C4—42no—
441C3×C4—104no—
442C3×C4—202no—
443Dic3—42no—
444Dic3—202no—
445Dic3C2×#68514yes—
446Dic3C2×#68514yes—
447C2×C5—24no—
448C2×C5C2×D3×C22⋊C411yescharacteristic
449C2×C5D3×#11712yes—
450C2×C5#565612yes—
451C2×C5#565612yes—
452D5—68no—
453C23—24no—
454C23—154no—
455C23—158no—
456C23—3016no—
457C23—3016no—
458C23—3016no—
459C23—308no—
460C23—302no—
461C23—304no—
462C23—302no—
463C23C4×D3×D512yes—
464C23D3×D2012yes—
465C23D3×C22⋊D512yes—
466C23C22×D3×D511yescharacteristic
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—316no—
470C2×C4—32no—
471C2×C4—32no—
472C2×C4—32no—
473C2×C4—32no—
474C2×C4—32no—
475C2×C4—32no—
476C2×C4—54no—
477C2×C4—64no—
478C2×C4—62no—
479C2×C4—108no—
480C2×C4—102no—
481C2×C4—102no—
482C2×C4—102no—
483C2×C4—104no—
484C2×C4—3016no—
485C2×C4—3016no—
486C2×C4—302no—
487C2×C4—304no—
488C2×C3—24no—
489C2×C3D5×#11712yes—
490C2×C3#6585812yes—
491C2×C3C22×#68511yescharacteristic
492C2×C3#6616212yes—
493D3—108no—
494C5#7405011yescharacteristic, complemented, Sylow
495C22—24no—
496C22—24no—
497C22—24no—
498C22—158no—
499C22—3032no—
500C22—308no—
501C22—304no—
502C22—304no—
503C22—308no—
504C22D3×#68514yes—
505C22C2×C4×D3×D511yescharacteristic
506C22C2×D3×D2011yescharacteristic
507C22#9202414yes—
508C22C2×D3×C22⋊D511yescharacteristic
509C4—34no—
510C4—34no—
511C4—42no—
512C4—104no—
513C4—122no—
514C4—202no—
515C4—602no—
516C3#1061399811yescharacteristic, complemented, Sylow
517C2—24no—
518C2—308no—
519C2#1175282512yes—
520C2#1175287612yes—
521C2C2×D3×#68511yescharacteristic
522C2#1175481312yes—
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C22⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC24×D15C22
hypercenterC2×C22⋊C4D3×D5
hypocenterC3×C5#2280
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C5×#877C2semidirect
3. C2×#91529C2semidirect
5. C3×#65232C2semidirect
7. #11749471C2semidirect
8. #11748882C2semidirect
9. #11748881C2semidirect
16. C22×#1200C4semidirect
23. C2×C5×Dic3⋊C4C22semidirect
24. C2×C4×C5×Dic3C22semidirect
27. C2×Dic3⋊Dic5C22semidirect
30. C2×Dic3×Dic5C22semidirect
31. C3×#3963C22semidirect
32. #90994C22semidirect
41. #65232D3semidirect
81. C2×#877D5semidirect
109. #3963C2×D3semidirect
131. C2×C5×Dic3C22⋊C4semidirect
161. C2×Dic3⋊C4C2×D5semidirect
163. C2×C4×Dic3C2×D5semidirect
226. C2×D15#117semidirect
359. C2×Dic3#685semidirect
425. C3×C5#2280semidirect
494. C5#74050semidirect
516. C3#10613998semidirect

Maximal subgroups

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

subgroupindex
C23×#12002
C2×C5×#8772
C2×#915292
#117510562
C3×#652322
#117495022
#117494712
#117488822
#117488812
#106139983
#740505