Finite Group Databasebeta

← #14322540

Subgroups — #14322540

Order 1920.

1760 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6819C211yesmaximal, characteristic
2C2×#91437C211yesmaximal, characteristic, complemented
3C2×#91386C211yesmaximal, characteristic, complemented
4C5×#5221C211yesmaximal, characteristic, complemented
5C3×#65510C211yesmaximal, characteristic, complemented
6#11749520C214yesmaximal, complemented
7#11749516C214yesmaximal, complemented
8#11748599C211yesmaximal, characteristic, complemented
9#11748587C211yesmaximal, characteristic, complemented
10#10608025—31nomaximal, Hall
11C22×C3×C5×D4C2211yescharacteristic
12C22×#1220C2212yes—
13C23×Dic15C2212yes—
14C2×#6819—24no—
15C2×#6819—24no—
16C2×#6819C418yescomplemented
17C22×C4×D15C2211yescharacteristic
18C22×Dic30C2211yescharacteristic
19C22×#1200C2212yes—
20C2×C5×#217C2214yes—
21C2×C5×C4⋊Dic3C2211yescharacteristic
22C2×C4×C5×Dic3C2211yescharacteristic
23C2×C3×C22⋊Dic5C2214yes—
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C2×C3×C4×Dic5C2211yescharacteristic
26C2×#6669C2212yes—
27C2×#6669C2212yes—
28C2×#6667C2212yes—
29#91437—24no—
30#91386—24no—
31#90994C2214yes—
32#90994C2214yes—
33#76294—51nomaximal, Hall
34C22×#711—31no—
35C2×D5×C4⋊C4—31no—
36C2×#4018—31no—
37C5×#390—31no—
38#65510D311yescharacteristic, complemented
39#65263—31no—
40#65245—34no—
41#65231—34no—
42#65218—31no—
43C24×C3×C5C2312yes—
44C23×D15C2311yescharacteristic
45C2×C3×C5×D4—22no—
46C2×C3×C5×D4—22no—
47C2×C3×C5×D4C2×C414yescomplemented
48C22×C3×C4×C5C2311yescharacteristic
49C2×#1220—24no—
50C2×#1220—24no—
51C2×#1220C2×C418yes—
52C22×Dic15—24no—
53C22×Dic15—24no—
54C22×Dic15C2×C418yescomplemented
55C22×Dic15C2312yes—
56C22×Dic15C2311yescharacteristic
57#6819—232no—
58C2×C4×D15C2×C414yes—
59C2×C4×D15D414yes—
60C2×C4×D15D414yes—
61C2×Dic30—22no—
62C2×Dic30—22no—
63C2×Dic30C2×C414yescomplemented
64C22×C5×Dic3C2312yes—
65C22×C5×Dic3C2312yes—
66C22×C3×Dic5C2312yes—
67C22×C3×Dic5C2312yes—
68C2×#1200—24no—
69C2×#1200—44no—
70C5×#217—44no—
71C5×C4⋊Dic3—22no—
72C4×C5×Dic3—22no—
73C3×C22⋊Dic5—44no—
74C3×C4⋊Dic5—22no—
75C3×C4×Dic5—22no—
76#6669—24no—
77#6669—24no—
78#6667—24no—
79C22×#237—51no—
80C2×D3×C4⋊C4—51no—
81C2×#872—51no—
82C3×#390—51no—
83#5221D511yescharacteristic, complemented
84#4973—51no—
85#4956—54no—
86#4942—54no—
87#4929—51no—
88C22×C5×D4C2×D311yescharacteristic
89C22×C22⋊D5—32no—
90C23×Dic5—32no—
91C2×#711—38no—
92C2×#711—64no—
93C2×#711—64no—
94C22×C4×D5—31no—
95C22×C4×D5—32no—
96C22×Dic10—31no—
97C2×C5×C4⋊C4—31no—
98C2×C5×C22⋊C4—34no—
99C2×C42×C5—31no—
100C2×C22⋊Dic5C2×D314yes—
101C2×#685—32no—
102C2×C4⋊Dic5C2×D311yescharacteristic
103C2×#683—32no—
104C2×#683—32no—
105C2×C4×Dic5C2×D311yescharacteristic
106D5×C4⋊C4—64no—
107#4018—64no—
108#3963—34no—
109#3963—34no—
110#2311—151noSylow
111C23×C3×C5—22no—
112C23×C3×C5—22no—
113C23×C3×C5—24no—
114C23×C3×C5—24no—
115C23×C3×C5C22×C414yes—
116C23×C3×C5C2411yescharacteristic
117C22×D15—22no—
118C22×D15—22no—
119C22×D15C22×C412yes—
120C22×D15C2×D412yes—
121C22×D15C2×D412yes—
122C3×C5×D4—28no—
123C2×C3×C4×C5C22×C412yes—
124C2×C3×C4×C5C2×D412yes—
125C2×C3×C4×C5C2×D412yes—
126#1220—216no—
127C2×Dic15—22no—
128C2×Dic15—22no—
129C2×Dic15—216no—
130C2×Dic15C22×C414yes—
131C2×Dic15C22×C412yes—
132C2×Dic15C2×D412yes—
133C2×Dic15C2×D412yes—
134C4×D15C22⋊C4116yescomplemented
135Dic30—28no—
136C2×C5×Dic3—22no—
137C2×C5×Dic3—22no—
138C2×C5×Dic3—22no—
139C2×C5×Dic3—24no—
140C2×C5×Dic3—22no—
141C2×C3×Dic5—22no—
142C2×C3×Dic5—22no—
143C2×C3×Dic5—22no—
144C2×C3×Dic5—24no—
145C2×C3×Dic5—22no—
146#1200—48no—
147C22×C3×D4C2×D511yescharacteristic
148C22×#67—52no—
149C23×Dic3—52no—
150C2×#237—58no—
151C2×#237—104no—
152C2×#237—104no—
153C22×C4×D3—51no—
154C22×C4×D3—52no—
155C22×Dic6—51no—
156C2×C3×C4⋊C4—51no—
157C2×C3×C22⋊C4—54no—
158C2×C3×C42—51no—
159C2×#217C2×D514yes—
160C2×(C2×D3)⋊C4—52no—
161C2×C4⋊Dic3C2×D511yescharacteristic
162C2×Dic3⋊C4—52no—
163C2×Dic3⋊C4—52no—
164C2×C4×Dic3C2×D511yescharacteristic
165D3×C4⋊C4—104no—
166#872—104no—
167C3⋊#95—54no—
168C3⋊#95—54no—
169C24×C5C22×D312yes—
170C23×D5—31no—
171C2×C5×D4—22no—
172C2×C5×D4—22no—
173C2×C5×D4C4×D314yescomplemented
174C22×C4×C5—32no—
175C22×C4×C5—32no—
176C22×C4×C5C22×D311yescharacteristic
177C2×C22⋊D5—38no—
178C2×C22⋊D5—64no—
179C2×C22⋊D5—64no—
180C22×Dic5—31no—
181C22×Dic5—38no—
182C22×Dic5—32no—
183C22×Dic5—64no—
184C22×Dic5—64no—
185C22×Dic5C22×D312yes—
186C22×Dic5C22×D312yes—
187#711—632no—
188C2×C4×D5—34no—
189C2×C4×D5—34no—
190C2×C4×D5—34no—
191C2×C4×D5—64no—
192C2×C4×D5—124no—
193C2×Dic10—34no—
194C2×Dic10—62no—
195C2×Dic10—62no—
196C5×C4⋊C4—62no—
197C5×C22⋊C4—124no—
198C42×C5—62no—
199C22⋊Dic5—44no—
200#685—64no—
201C4⋊Dic5—22no—
202#683—64no—
203#683—64no—
204C4×Dic5—22no—
205C22×#41—151no—
206C2×#117—151no—
207C22×C4⋊C4—151no—
208#391—151no—
209#391—151no—
210#390—31no—
211#390—51no—
212#386—154no—
213#380—154no—
214C22×C3×C5—24no—
215C22×C3×C5—28no—
216C22×C3×C5—42no—
217C22×C3×C5—42no—
218C22×C3×C5—44no—
219C22×C3×C5C23×C411yescharacteristic
220C22×C3×C5C22×D411yescharacteristic
221C22×C3×C5C22×D411yescharacteristic
222C22×C3×C5H(5,𝔽2)12yes—
223C22×C3×C52−1+412yes—
224C2×D15—24no—
225C2×D15—22no—
226C2×D15—22no—
227C2×D15—24no—
228C2×D15C2×C22⋊C414yes—
229C3×C4×C5C2×C22⋊C414yes—
230Dic15—24no—
231Dic15C2×C22⋊C414yes—
232C3×Dic5—42no—
233C3×Dic5—42no—
234C5×Dic3—42no—
235C5×Dic3—42no—
236C24×C3C22×D512yes—
237C23×D3—51no—
238C2×C3×D4—22no—
239C2×C3×D4—22no—
240C2×C3×D4C4×D514yescomplemented
241C22×C3×C4—52no—
242C22×C3×C4—52no—
243C22×C3×C4C22×D511yescharacteristic
244C2×#67—58no—
245C2×#67—104no—
246C2×#67—104no—
247C22×Dic3—51no—
248C22×Dic3—58no—
249C22×Dic3—52no—
250C22×Dic3—104no—
251C22×Dic3—104no—
252C22×Dic3C22×D512yes—
253C22×Dic3C22×D512yes—
254#237—1032no—
255C2×C4×D3—54no—
256C2×C4×D3—54no—
257C2×C4×D3—54no—
258C2×C4×D3—104no—
259C2×C4×D3—204no—
260C2×Dic6—54no—
261C2×Dic6—102no—
262C2×Dic6—102no—
263C3×C4⋊C4—102no—
264C3×C22⋊C4—204no—
265C3×C42—102no—
266#217—44no—
267(C2×D3)⋊C4—104no—
268C4⋊Dic3—22no—
269Dic3⋊C4—104no—
270Dic3⋊C4—104no—
271C4×Dic3—22no—
272C23×C5—22no—
273C23×C5—22no—
274C23×C5—24no—
275C23×C5—24no—
276C23×C5C2×C4×D314yes—
277C23×C5C23×D311yescharacteristic
278C22×D5—32no—
279C22×D5—32no—
280C22×D5—32no—
281C22×D5—62no—
282C22×D5—62no—
283C5×D4—28no—
284C2×C4×C5—62no—
285C2×C4×C5—62no—
286C2×C4×C5—62no—
287C2×C4×C5—64no—
288C2×C4×C5—62no—
289C2×C4×C5C2×C4×D312yes—
290C2×C4×C5D3×D412yes—
291C2×C4×C5D3×D412yes—
292C22⋊D5—616no—
293C2×Dic5—22no—
294C2×Dic5—22no—
295C2×Dic5—22no—
296C2×Dic5—24no—
297C2×Dic5—22no—
298C2×Dic5—32no—
299C2×Dic5—32no—
300C2×Dic5—32no—
301C2×Dic5—34no—
302C2×Dic5—616no—
303C2×Dic5—62no—
304C2×Dic5—62no—
305C4×D5—316no—
306C4×D5—128no—
307Dic10—68no—
308C2×#41—158no—
309C2×#41—304no—
310C2×#41—304no—
311C22×Q8—151no—
312C22×D4—152no—
313C22×D4D3×D511yescharacteristic
314C23×C4—152no—
315C23×C4—151no—
316C23×C4—152no—
317#117—304no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—51no—
320C2×C4⋊C4—152no—
321C2×C4⋊C4—152no—
322C2×C4⋊C4—304no—
323C2×C22⋊C4—34no—
324C2×C22⋊C4—54no—
325C2×C22⋊C4—152no—
326C2×C42—31no—
327C2×C42—51no—
328#95—154no—
329#95—154no—
330C2×C3×C5—44no—
331C2×C3×C5C22×C22⋊C411yescharacteristic
332C2×C3×C5#52012yes—
333C2×C3×C5#53511yescharacteristic
334C2×C3×C5#53611yescharacteristic
335C2×C3×C5#53712yes—
336D15—24no—
337C23×C3—22no—
338C23×C3—22no—
339C23×C3—24no—
340C23×C3—24no—
341C23×C3C2×C4×D514yes—
342C23×C3C23×D511yescharacteristic
343C22×D3—52no—
344C22×D3—52no—
345C22×D3—52no—
346C22×D3—102no—
347C22×D3—102no—
348C3×D4—28no—
349C2×C3×C4—102no—
350C2×C3×C4—102no—
351C2×C3×C4—102no—
352C2×C3×C4—104no—
353C2×C3×C4—102no—
354C2×C3×C4C2×C4×D512yes—
355C2×C3×C4D4×D512yes—
356C2×C3×C4D4×D512yes—
357#67—1016no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—24no—
362C2×Dic3—22no—
363C2×Dic3—52no—
364C2×Dic3—52no—
365C2×Dic3—52no—
366C2×Dic3—54no—
367C2×Dic3—1016no—
368C2×Dic3—102no—
369C2×Dic3—102no—
370C4×D3—516no—
371C4×D3—208no—
372Dic6—108no—
373C22×C5—24no—
374C22×C5—28no—
375C22×C5—42no—
376C22×C5—42no—
377C22×C5—44no—
378C22×C5C22×C4×D311yescharacteristic
379C22×C5C2×D3×D411yescharacteristic
380C22×C5C2×D3×D411yescharacteristic
381C22×C5#100412yes—
382C22×C5#101012yes—
383C2×D5—34no—
384C2×D5—62no—
385C2×D5—62no—
386C2×D5—64no—
387C2×D5—64no—
388C4×C5—122no—
389C4×C5—122no—
390C4×C5D3×C22⋊C414yes—
391Dic5—34no—
392Dic5—42no—
393Dic5—42no—
394Dic5—64no—
395C24—151no—
396C24C2×D3×D512yes—
397#41—3032no—
398C2×Q8—154no—
399C2×Q8—302no—
400C2×Q8—302no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—158no—
404C2×D4—304no—
405C2×D4—304no—
406C2×D4#120014yescomplemented
407C22×C4—32no—
408C22×C4—32no—
409C22×C4—52no—
410C22×C4—52no—
411C22×C4—151no—
412C22×C4—154no—
413C22×C4—154no—
414C22×C4—154no—
415C22×C4—158no—
416C22×C4—152no—
417C22×C4—304no—
418C22×C4—304no—
419C22×C4—304no—
420C22×C4—604no—
421C22×C4C2×D3×D511yescharacteristic
422C4⋊C4—62no—
423C4⋊C4—102no—
424C4⋊C4—304no—
425C4⋊C4—304no—
426C22⋊C4—124no—
427C22⋊C4—204no—
428C22⋊C4—304no—
429C42—62no—
430C42—102no—
431C3×C5#231111yescharacteristic, complemented, Hall
432C22×C3—24no—
433C22×C3—28no—
434C22×C3—42no—
435C22×C3—44no—
436C22×C3—42no—
437C22×C3C22×C4×D511yescharacteristic
438C22×C3C2×D4×D511yescharacteristic
439C22×C3C2×D4×D511yescharacteristic
440C22×C3#414412yes—
441C22×C3#415012yes—
442C2×D3—54no—
443C2×D3—102no—
444C2×D3—104no—
445C2×D3—104no—
446C2×D3—102no—
447C3×C4—202no—
448C3×C4—202no—
449C3×C4D5×C22⋊C414yes—
450Dic3—42no—
451Dic3—42no—
452Dic3—54no—
453Dic3—104no—
454C2×C5—44no—
455C2×C5C2×D3×C22⋊C411yescharacteristic
456C2×C5#554612yes—
457C2×C5#560812yes—
458C2×C5#567511yescharacteristic
459C2×C5#571511yescharacteristic
460D5—64no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—152no—
466C23—152no—
467C23—152no—
468C23—302no—
469C23—302no—
470C23C2×#120014yes—
471C23C22×D3×D511yescharacteristic
472Q8—308no—
473D4—28no—
474D4—3016no—
475C2×C4—62no—
476C2×C4—62no—
477C2×C4—64no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—102no—
481C2×C4—102no—
482C2×C4—104no—
483C2×C4—102no—
484C2×C4—102no—
485C2×C4—152no—
486C2×C4—152no—
487C2×C4—152no—
488C2×C4—1516no—
489C2×C4—154no—
490C2×C4—3016no—
491C2×C4—302no—
492C2×C4—302no—
493C2×C4—608no—
494C2×C4C2×#120012yes—
495C2×C4#679012yes—
496C2×C4#679012yes—
497C2×C3—44no—
498C2×C3C2×D5×C22⋊C411yescharacteristic
499C2×C3#6588012yes—
500C2×C3#6594212yes—
501C2×C3#6600911yescharacteristic
502C2×C3#6604911yescharacteristic
503D3—104no—
504C5#7629411yescharacteristic, complemented, Sylow
505C22—24no—
506C22—28no—
507C22—42no—
508C22—42no—
509C22—44no—
510C22—154no—
511C22—302no—
512C22—302no—
513C22—304no—
514C22—304no—
515C22#9202012yes—
516C22#9202512yes—
517C22C22×#120011yescharacteristic
518C22C2×#679011yescharacteristic
519C22C2×#679011yescharacteristic
520C4—122no—
521C4—122no—
522C4—154no—
523C4—202no—
524C4—202no—
525C4—304no—
526C4#9156114yes—
527C3#1060802511yescharacteristic, complemented, Sylow
528C2—44no—
529C2—304no—
530C2#1175483211yescharacteristic
531C2#1175491612yes—
532C2#1175495412yes—
533C2#1175507411yescharacteristic
534C2C2×#9156111yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC22×C3×C5×D4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#6819C2
hypercenterC22×D4D3×D5
hypocenterC3×C5#2311
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C2×#91437C2semidirect
3. C2×#91386C2semidirect
4. C5×#5221C2semidirect
5. C3×#65510C2semidirect
6. #11749520C2semidirect
7. #11749516C2semidirect
8. #11748599C2semidirect
9. #11748587C2semidirect
16. C2×#6819C4semidirect
38. #65510D3semidirect
47. C2×C3×C5×D4C2×C4semidirect
54. C22×Dic15C2×C4semidirect
63. C2×Dic30C2×C4semidirect
83. #5221D5semidirect
134. C4×D15C22⋊C4semidirect
173. C2×C5×D4C4×D3semidirect
240. C2×C3×D4C4×D5semidirect
406. C2×D4#1200semidirect
431. C3×C5#2311semidirect
504. C5#76294semidirect
527. C3#10608025semidirect

Maximal subgroups

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

subgroupindex
C22×#68192
C2×#914372
C2×#913862
C5×#52212
C3×#655102
#117495202
#117495162
#117485992
#117485872
#106080253
#762945