Finite Group Databasebeta

← #14322565

Subgroups — #14322565

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×#6765C211yesmaximal, characteristic
2C2×C5×#872C211yesmaximal, characteristic, complemented
3C2×D3×C4⋊Dic5C211yesmaximal, characteristic, complemented
4#11751030C211yesmaximal, characteristic, complemented
5C3×#65228C211yesmaximal, characteristic, complemented
6#11748911C214yesmaximal, complemented
7#11748878C214yesmaximal, complemented
8#11748785C211yesmaximal, characteristic, complemented
9#11748599C211yesmaximal, characteristic, complemented
10#10614001—31nomaximal, Hall
11C22×Dic30C2211yescharacteristic
12C22×C4×C5×D3C2211yescharacteristic
13C22×C3×D20C2211yescharacteristic
14C22×#1201C2212yes—
15C22×D3×Dic5C2212yes—
16C22×D5×Dic3C2212yes—
17C2×#6765—24no—
18C2×#6765—24no—
19C2×#6765C418yescomplemented
20C2×C3×C42×C5C2211yescharacteristic
21C2×C4⋊Dic15C2211yescharacteristic
22C2×C5×(C2×D3)⋊C4C2212yes—
23C2×C5×Dic3⋊C4C2212yes—
24C2×C4×C5×Dic3C2211yescharacteristic
25C5×#872—24no—
26C2×C3×#685C2214yes—
27C2×C3×C4⋊Dic5C2211yescharacteristic
28C2×Dic3⋊Dic5C2212yes—
29C2×#6665C2214yes—
30D3×C4⋊Dic5—24no—
31#91115C2214yes—
32#90994C2214yes—
33#76294—51nomaximal, Hall
34C22×#709—31no—
35C2×C5×#117—31no—
36C22×C4⋊Dic5—31no—
37#65276—34no—
38#65247—34no—
39#65228—31no—
40#65228D311yescharacteristic, complemented
41#65218—31no—
42#65218—31no—
43C23×C5×D3C2311yescharacteristic
44C23×C3×D5C2312yes—
45C22×C3×C4×C5C2312yes—
46C22×C3×C4×C5C2311yescharacteristic
47C22×Dic15C2312yes—
48C22×Dic15C2312yes—
49C2×Dic30—22no—
50C2×Dic30—22no—
51C2×Dic30C2×C414yescomplemented
52C22×C5×Dic3C2312yes—
53C22×C5×Dic3C2311yescharacteristic
54C2×C4×C5×D3C2×C414yes—
55C2×C4×C5×D3D414yes—
56C2×C4×C5×D3D414yes—
57C22×C3×Dic5C2312yes—
58C2×C3×D20—22no—
59C2×C3×D20—22no—
60C2×C3×D20C2×C414yescomplemented
61C2×#1201—24no—
62C2×#1201—24no—
63C2×#1201C2×C418yes—
64C2×D3×Dic5—24no—
65C2×D3×Dic5—44no—
66C2×D5×Dic3—24no—
67C2×D5×Dic3—24no—
68C2×D5×Dic3C2×C418yescomplemented
69#6765—232no—
70C3×C42×C5—22no—
71C4⋊Dic15—22no—
72C5×(C2×D3)⋊C4—24no—
73C5×Dic3⋊C4—24no—
74C4×C5×Dic3—22no—
75C3×#685—44no—
76C3×C4⋊Dic5—22no—
77Dic3⋊Dic5—24no—
78#6665—44no—
79C22×#237—51no—
80C2×D3×C4⋊C4—51no—
81C2×#872D511yescharacteristic, complemented
82C3×#390—51no—
83#5221—51no—
84#4973—51no—
85#4956—54no—
86#4942—54no—
87#4929—51no—
88C23×C4×C5—31no—
89C22×C22⋊D5—32no—
90C23×Dic5—32no—
91C2×#709—38no—
92C2×#709—64no—
93C2×#709—64no—
94C22×D20C2×D311yescharacteristic
95C22×C4×D5—32no—
96C22×Dic10—31no—
97C5×#117—64no—
98C2×C5×C4⋊C4—32no—
99C2×C5×C22⋊C4—32no—
100C2×C42×C5—31no—
101C2×C42×C5C2×D311yescharacteristic
102C2×#685—34no—
103C2×#685C2×D314yes—
104C2×C4⋊Dic5—31no—
105C2×C4⋊Dic5—32no—
106C2×C4⋊Dic5—64no—
107C2×C4⋊Dic5C2×D311yescharacteristic
108#3963—34no—
109#3963—34no—
110#2311—151noSylow
111C23×C3×C5C2411yescharacteristic
112C22×C5×D3—22no—
113C22×C5×D3—22no—
114C22×C5×D3C22×C412yes—
115C22×C5×D3C2×D412yes—
116C22×C5×D3C2×D412yes—
117C22×C3×D5—22no—
118C22×C3×D5—22no—
119C22×C3×D5—24no—
120C22×C3×D5—24no—
121C22×C3×D5C22×C414yes—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5C22×C412yes—
126C2×C3×C4×C5C2×D412yes—
127C2×C3×C4×C5C2×D412yes—
128C2×Dic15—22no—
129C2×Dic15—22no—
130C2×Dic15—22no—
131C2×Dic15—24no—
132C2×Dic15C22×C414yes—
133Dic30—28no—
134C2×C5×Dic3—22no—
135C2×C5×Dic3—22no—
136C2×C5×Dic3—22no—
137C2×C5×Dic3C22×C412yes—
138C2×C5×Dic3C2×D412yes—
139C2×C5×Dic3C2×D412yes—
140C4×C5×D3C22⋊C4116yescomplemented
141C2×C3×Dic5—22no—
142C2×C3×Dic5—24no—
143C3×D20—28no—
144#1201—216no—
145D3×Dic5—48no—
146D5×Dic3—216no—
147C22×C3×D4—51no—
148C22×#67—52no—
149C23×Dic3—52no—
150C2×#237—58no—
151C2×#237—104no—
152C2×#237—104no—
153C22×C4×D3—52no—
154C22×C4×D3C2×D511yescharacteristic
155C22×Dic6—51no—
156C2×C3×C4⋊C4—51no—
157C2×C3×C22⋊C4—54no—
158C2×C3×C42C2×D511yescharacteristic
159C2×#217—54no—
160C2×(C2×D3)⋊C4C2×D512yes—
161C2×C4⋊Dic3—51no—
162C2×Dic3⋊C4—52no—
163C2×Dic3⋊C4C2×D512yes—
164C2×C4×Dic3C2×D511yescharacteristic
165D3×C4⋊C4—104no—
166#872—24no—
167C3⋊#95—54no—
168C3⋊#95—54no—
169C24×C5—31no—
170C23×D5C22×D312yes—
171C22×C4×C5—31no—
172C22×C4×C5—34no—
173C22×C4×C5—34no—
174C22×C4×C5—34no—
175C22×C4×C5—32no—
176C22×C4×C5C22×D312yes—
177C22×C4×C5C22×D311yescharacteristic
178C2×C22⋊D5—38no—
179C2×C22⋊D5—64no—
180C2×C22⋊D5—64no—
181C22×Dic5—32no—
182C22×Dic5—32no—
183C22×Dic5—64no—
184C22×Dic5—124no—
185C22×Dic5C22×D312yes—
186#709—632no—
187C2×D20—22no—
188C2×D20—22no—
189C2×D20C4×D314yescomplemented
190C2×C4×D5—38no—
191C2×C4×D5—64no—
192C2×C4×D5—64no—
193C2×Dic10—34no—
194C2×Dic10—62no—
195C2×Dic10—62no—
196C5×C4⋊C4—64no—
197C5×C22⋊C4—64no—
198C42×C5—22no—
199C42×C5—62no—
200#685—44no—
201#685—124no—
202C4⋊Dic5—22no—
203C4⋊Dic5—64no—
204C4⋊Dic5—62no—
205C22×#41—151no—
206C2×#117—31no—
207C22×C4⋊C4—151no—
208#391—151no—
209#391—151no—
210#390—51no—
211#390—151no—
212#386—154no—
213#380—154no—
214C22×C3×C5C23×C411yescharacteristic
215C22×C3×C5C22×D411yescharacteristic
216C22×C3×C5C22×D411yescharacteristic
217C22×C3×C5H(5,𝔽2)12yes—
218C22×C3×C52−1+412yes—
219C2×C5×D3—24no—
220C2×C5×D3—22no—
221C2×C5×D3—22no—
222C2×C5×D3—24no—
223C2×C5×D3C2×C22⋊C414yes—
224C2×C3×D5—24no—
225C2×C3×D5—28no—
226C2×C3×D5—42no—
227C2×C3×D5—42no—
228C2×C3×D5—44no—
229C3×C4×C5—42no—
230C3×C4×C5C2×C22⋊C414yes—
231Dic15—24no—
232Dic15—42no—
233C3×Dic5—42no—
234C5×Dic3—42no—
235C5×Dic3C2×C22⋊C414yes—
236C24×C3—52no—
237C23×D3C22×D511yescharacteristic
238C2×C3×D4—54no—
239C2×C3×D4—102no—
240C2×C3×D4—102no—
241C22×C3×C4—52no—
242C22×C3×C4C22×D512yes—
243C22×C3×C4C22×D511yescharacteristic
244C2×#67—58no—
245C2×#67—104no—
246C2×#67—104no—
247C22×Dic3—52no—
248C22×Dic3—58no—
249C22×Dic3—52no—
250C22×Dic3—104no—
251C22×Dic3—104no—
252C22×Dic3C22×D512yes—
253C22×Dic3C22×D511yescharacteristic
254#237—1032no—
255C2×C4×D3—104no—
256C2×C4×D3—204no—
257C2×C4×D3C4×D514yes—
258C2×C4×D3D2014yes—
259C2×C4×D3C22⋊D514yes—
260C2×Dic6—54no—
261C2×Dic6—102no—
262C2×Dic6—102no—
263C3×C4⋊C4—102no—
264C3×C22⋊C4—204no—
265C3×C42—22no—
266#217—204no—
267(C2×D3)⋊C4—24no—
268C4⋊Dic3—102no—
269Dic3⋊C4—24no—
270Dic3⋊C4—104no—
271C4×Dic3—22no—
272C23×C5—32no—
273C23×C5—32no—
274C23×C5—32no—
275C23×C5—62no—
276C23×C5—62no—
277C23×C5C23×D311yescharacteristic
278C22×D5—22no—
279C22×D5—22no—
280C22×D5—24no—
281C22×D5—24no—
282C22×D5C2×C4×D314yes—
283C2×C4×C5—22no—
284C2×C4×C5—22no—
285C2×C4×C5—22no—
286C2×C4×C5—32no—
287C2×C4×C5—32no—
288C2×C4×C5—32no—
289C2×C4×C5—316no—
290C2×C4×C5—62no—
291C2×C4×C5—62no—
292C2×C4×C5—62no—
293C2×C4×C5C2×C4×D312yes—
294C2×C4×C5D3×D412yes—
295C2×C4×C5D3×D412yes—
296C22⋊D5—616no—
297C2×Dic5—22no—
298C2×Dic5—24no—
299C2×Dic5—34no—
300C2×Dic5—62no—
301C2×Dic5—62no—
302C2×Dic5—62no—
303C2×Dic5—64no—
304C2×Dic5—128no—
305D20—28no—
306C4×D5—616no—
307Dic10—68no—
308C2×#41—158no—
309C2×#41—304no—
310C2×#41—304no—
311C22×Q8—151no—
312C22×D4—51no—
313C22×D4—152no—
314C23×C4—31no—
315C23×C4—152no—
316C23×C4—152no—
317#117—64no—
318C2×C4⋊C4—32no—
319C2×C4⋊C4—51no—
320C2×C4⋊C4—152no—
321C2×C4⋊C4—151no—
322C2×C4⋊C4—304no—
323C2×C22⋊C4—32no—
324C2×C22⋊C4—54no—
325C2×C22⋊C4—154no—
326C2×C42—31no—
327C2×C42D3×D511yescharacteristic
328#95—154no—
329#95—154no—
330C2×C3×C5C22×C22⋊C411yescharacteristic
331C2×C3×C5#52012yes—
332C2×C3×C5#53511yescharacteristic
333C2×C3×C5#53611yescharacteristic
334C2×C3×C5#53712yes—
335C3×D5—44no—
336C5×D3—24no—
337C23×C3—54no—
338C23×C3—102no—
339C23×C3—104no—
340C23×C3—104no—
341C23×C3—102no—
342C23×C3C23×D511yescharacteristic
343C22×D3—22no—
344C22×D3—22no—
345C22×D3C2×C4×D512yes—
346C22×D3C2×D2012yes—
347C22×D3C2×C22⋊D512yes—
348C3×D4—108no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—102no—
353C2×C3×C4—104no—
354C2×C3×C4C2×C4×D512yes—
355C2×C3×C4C2×D2012yes—
356C2×C3×C4C2×C22⋊D512yes—
357#67—1016no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—54no—
362C2×Dic3—102no—
363C2×Dic3—102no—
364C2×Dic3—1016no—
365C2×Dic3—102no—
366C2×Dic3—104no—
367C2×Dic3C2×C4×D512yes—
368C2×Dic3C2×D2012yes—
369C2×Dic3C2×C22⋊D512yes—
370C4×D3—208no—
371C4×D3#685116yescomplemented
372Dic6—108no—
373C22×C5—34no—
374C22×C5—62no—
375C22×C5—62no—
376C22×C5—64no—
377C22×C5—64no—
378C22×C5C22×C4×D311yescharacteristic
379C22×C5C2×D3×D411yescharacteristic
380C22×C5C2×D3×D411yescharacteristic
381C22×C5#100412yes—
382C22×C5#101012yes—
383C2×D5—24no—
384C2×D5—28no—
385C2×D5—42no—
386C2×D5—42no—
387C2×D5—44no—
388C4×C5—34no—
389C4×C5—42no—
390C4×C5—122no—
391C4×C5D3×C22⋊C414yes—
392Dic5—42no—
393Dic5—64no—
394Dic5—122no—
395C24—31no—
396C24—52no—
397#41—3032no—
398C2×Q8—154no—
399C2×Q8—302no—
400C2×Q8—302no—
401C2×D4—54no—
402C2×D4—102no—
403C2×D4—102no—
404C2×D4—158no—
405C2×D4—304no—
406C2×D4—304no—
407C22×C4—31no—
408C22×C4—34no—
409C22×C4—34no—
410C22×C4—34no—
411C22×C4—32no—
412C22×C4—52no—
413C22×C4—158no—
414C22×C4—152no—
415C22×C4—152no—
416C22×C4—304no—
417C22×C4—304no—
418C22×C4—304no—
419C22×C4—604no—
420C22×C4C2×D3×D512yes—
421C22×C4C2×D3×D511yescharacteristic
422C4⋊C4—64no—
423C4⋊C4—102no—
424C4⋊C4—304no—
425C4⋊C4—302no—
426C22⋊C4—64no—
427C22⋊C4—204no—
428C22⋊C4—604no—
429C42—22no—
430C42—62no—
431C3×C5#231111yescharacteristic, complemented, Hall
432C22×C3—108no—
433C22×C3—104no—
434C22×C3—202no—
435C22×C3—202no—
436C22×C3—204no—
437C22×C3C22×C4×D511yescharacteristic
438C22×C3C22×D2011yescharacteristic
439C22×C3#414912yes—
440C22×C3#415012yes—
441C22×C3C22×C22⋊D511yescharacteristic
442C2×D3—24no—
443C2×D3—22no—
444C2×D3—22no—
445C2×D3—24no—
446C2×D3C2×#68514yes—
447C3×C4—42no—
448C3×C4—202no—
449C3×C4C2×#68514yes—
450Dic3—42no—
451Dic3—104no—
452Dic3—202no—
453Dic3C2×#68514yes—
454C2×C5—64no—
455C2×C5C2×D3×C22⋊C411yescharacteristic
456C2×C5#554612yes—
457C2×C5#560812yes—
458C2×C5#567511yescharacteristic
459C2×C5#571511yescharacteristic
460D5—44no—
461C23—32no—
462C23—32no—
463C23—32no—
464C23—54no—
465C23—62no—
466C23—62no—
467C23—104no—
468C23—102no—
469C23—104no—
470C23—102no—
471C23C22×D3×D511yescharacteristic
472Q8—308no—
473D4—108no—
474D4—3016no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—32no—
479C2×C4—32no—
480C2×C4—32no—
481C2×C4—316no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—102no—
486C2×C4—104no—
487C2×C4—154no—
488C2×C4—3016no—
489C2×C4—302no—
490C2×C4—302no—
491C2×C4—302no—
492C2×C4—304no—
493C2×C4—608no—
494C2×C4C4×D3×D512yes—
495C2×C4D3×D2012yes—
496C2×C4D3×C22⋊D512yes—
497C2×C3—204no—
498C2×C3#6585712yes—
499C2×C3#6585911yescharacteristic
500C2×C3#6586011yescharacteristic
501C2×C3C22×#68511yescharacteristic
502C2×C3#6616712yes—
503D3—24no—
504C5#7629411yescharacteristic, complemented, Sylow
505C22—34no—
506C22—62no—
507C22—64no—
508C22—62no—
509C22—64no—
510C22—108no—
511C22—104no—
512C22—202no—
513C22—202no—
514C22—204no—
515C22#9200112yes—
516C22C2×C4×D3×D511yescharacteristic
517C22C2×D3×D2011yescharacteristic
518C22#9201712yes—
519C22C2×D3×C22⋊D511yescharacteristic
520C4—34no—
521C4—42no—
522C4—122no—
523C4—202no—
524C4—304no—
525C4—602no—
526C4D3×#68514yes—
527C3#1061400111yescharacteristic, complemented, Sylow
528C2—64no—
529C2—204no—
530C2#1175251812yes—
531C2#1175254411yescharacteristic
532C2#1175274311yescharacteristic
533C2#1175434212yes—
534C2C2×D3×#68511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#6765C2
hypercenterC2×C42D3×D5
hypocenterC3×C5#2311
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. C2×C5×#872C2semidirect
3. C2×D3×C4⋊Dic5C2semidirect
4. #11751030C2semidirect
5. C3×#65228C2semidirect
6. #11748911C2semidirect
7. #11748878C2semidirect
8. #11748785C2semidirect
9. #11748599C2semidirect
19. C2×#6765C4semidirect
40. #65228D3semidirect
51. C2×Dic30C2×C4semidirect
60. C2×C3×D20C2×C4semidirect
68. C2×D5×Dic3C2×C4semidirect
81. C2×#872D5semidirect
140. C4×C5×D3C22⋊C4semidirect
189. C2×D20C4×D3semidirect
371. C4×D3#685semidirect
431. C3×C5#2311semidirect
504. C5#76294semidirect
527. C3#10614001semidirect

Maximal subgroups

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

subgroupindex
C22×#67652
C2×C5×#8722
C2×D3×C4⋊Dic52
#117510302
C3×#652282
#117489112
#117488782
#117487852
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