Finite Group Databasebeta

← #14321205

Subgroups — #14321205

Order 1920.

1872 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6768C211yesmaximal, characteristic
2C2×#91781C211yesmaximal, characteristic, complemented
3C2×C4×#1200C211yesmaximal, characteristic, complemented
4C5×#4988C211yesmaximal, characteristic, complemented
5C3×#65263C211yesmaximal, characteristic, complemented
6#11748919C214yesmaximal, complemented
7#11748886C214yesmaximal, complemented
8#11748764C211yesmaximal, characteristic, complemented
9#11748609C211yesmaximal, characteristic, complemented
10#10608021—31nomaximal, Hall
11C22×C4×D15C2211yescharacteristic
12C22×C5×D12C2211yescharacteristic
13C22×C3×Dic10C2211yescharacteristic
14C22×#1203C2212yes—
15C22×#1200C2212yes—
16C22×D3×Dic5C2212yes—
17C2×#6768—24no—
18C2×#6768—24no—
19C2×#6768C418yescomplemented
20C2×C3×C5×C4⋊C4C2211yescharacteristic
21C2×#6714C2212yes—
22C2×C4⋊Dic15C2211yescharacteristic
23C2×C4×Dic15C2212yes—
24#91781—24no—
25C2×C5×(C2×D3)⋊C4C2214yes—
26C2×C4×C5×Dic3C2211yescharacteristic
27C2×C3×C4×Dic5C2211yescharacteristic
28C2×(C2×D3)⋊Dic5C2214yes—
29C2×Dic3×Dic5C2212yes—
30C4×#1200—24no—
31C3×#3963C2214yes—
32#90994C2214yes—
33#76478—51nomaximal, Hall
34C22×#711—31no—
35C2×#4038—31no—
36C2×C42×D5—31no—
37C5×#390—31no—
38#65510—31no—
39#65263D311yescharacteristic, complemented
40#65245—34no—
41C22⋊C4×Dic5—34no—
42#65218—31no—
43C23×D15C2311yescharacteristic
44C23×C5×D3C2312yes—
45C22×C3×C4×C5C2312yes—
46C22×C3×C4×C5C2311yescharacteristic
47C22×Dic15C2312yes—
48C22×Dic15C2311yescharacteristic
49C2×C4×D15C2×C414yes—
50C2×C4×D15D414yes—
51C2×C4×D15D414yes—
52C22×C5×Dic3C2312yes—
53C2×C5×D12—22no—
54C2×C5×D12—22no—
55C2×C5×D12C2×C414yescomplemented
56C22×C3×Dic5C2312yes—
57C22×C3×Dic5C2312yes—
58C2×C3×Dic10—22no—
59C2×C3×Dic10—22no—
60C2×C3×Dic10C2×C414yescomplemented
61C2×#1203—24no—
62C2×#1203—24no—
63C2×#1203C2×C418yes—
64C2×#1200—24no—
65C2×#1200—24no—
66C2×#1200—24no—
67C2×D3×Dic5—24no—
68C2×D3×Dic5—24no—
69C2×D3×Dic5C2×C418yescomplemented
70#6768—232no—
71C3×C5×C4⋊C4—22no—
72#6714—24no—
73C4⋊Dic15—22no—
74C4×Dic15—24no—
75C5×(C2×D3)⋊C4—28no—
76C4×C5×Dic3—22no—
77C3×C4×Dic5—22no—
78(C2×D3)⋊Dic5—28no—
79Dic3×Dic5—24no—
80C22×#239—51no—
81C2×#892—51no—
82C2×C42×D3—51no—
83C3×#391—51no—
84#5229—51no—
85#4988D511yescharacteristic, complemented
86#4939—51no—
87#4666—54no—
88#4665—54no—
89C22×C5×D4—31no—
90C22×C22⋊D5—32no—
91C23×Dic5—32no—
92C2×#711—38no—
93C2×#711—64no—
94C2×#711—64no—
95C22×C4×D5—31no—
96C22×C4×D5—32no—
97C22×Dic10C2×D311yescharacteristic
98C2×C5×C4⋊C4C2×D311yescharacteristic
99C2×C5×C22⋊C4—34no—
100C2×C42×C5—31no—
101C2×C22⋊Dic5—34no—
102C2×#685—32no—
103C2×C4⋊Dic5—31no—
104C2×C4×Dic5—32no—
105C2×C4×Dic5—32no—
106C2×C4×Dic5C2×D311yescharacteristic
107#4038—64no—
108C42×D5—64no—
109#3963—34no—
110#3963C2×D314yes—
111#2297—151noSylow
112C23×C3×C5C2411yescharacteristic
113C22×D15—22no—
114C22×D15—22no—
115C22×D15C22×C412yes—
116C22×D15C2×D412yes—
117C22×D15C2×D412yes—
118C22×C5×D3—22no—
119C22×C5×D3—22no—
120C22×C5×D3—24no—
121C22×C5×D3—24no—
122C22×C5×D3C22×C414yes—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—24no—
125C2×C3×C4×C5C22×C412yes—
126C2×C3×C4×C5C2×D412yes—
127C2×C3×C4×C5C2×D412yes—
128C2×Dic15—22no—
129C2×Dic15—24no—
130C2×Dic15C22×C412yes—
131C2×Dic15C2×D412yes—
132C2×Dic15C2×D412yes—
133C4×D15C22⋊C4116yescomplemented
134C2×C5×Dic3—22no—
135C2×C5×Dic3—22no—
136C2×C5×Dic3#4114yescomplemented
137C5×D12—28no—
138C2×C3×Dic5—22no—
139C2×C3×Dic5—22no—
140C2×C3×Dic5—22no—
141C2×C3×Dic5—22no—
142C2×C3×Dic5C22×C414yes—
143C2×C3×Dic5#4114yescomplemented
144C3×Dic10—28no—
145#1203—216no—
146#1200—216no—
147D3×Dic5—216no—
148C22×C3×Q8—51no—
149C2×#239—58no—
150C2×#239—104no—
151C2×#239—104no—
152C22×D12—52no—
153C22×D12C2×D511yescharacteristic
154C22×C4×D3—51no—
155C22×C4×D3—52no—
156C22×C4×D3—52no—
157C2×C3×C4⋊C4C2×D511yescharacteristic
158C2×C3×C42—51no—
159C2×(C2×D3)⋊C4—54no—
160C2×(C2×D3)⋊C4—52no—
161C2×(C2×D3)⋊C4C2×D514yes—
162C2×C4⋊Dic3—51no—
163C2×C4×Dic3—52no—
164C2×C4×Dic3—52no—
165C2×C4×Dic3C2×D511yescharacteristic
166#892—104no—
167C42×D3—104no—
168C3×#95—54no—
169C3⋊#95—54no—
170C24×C5—32no—
171C23×D5—31no—
172C2×C5×D4—34no—
173C2×C5×D4—62no—
174C2×C5×D4—62no—
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—31no—
182C22×Dic5—38no—
183C22×Dic5—32no—
184C22×Dic5—64no—
185C22×Dic5—64no—
186C22×Dic5C22×D312yes—
187C22×Dic5C22×D312yes—
188#711—632no—
189C2×C4×D5—34no—
190C2×C4×D5—34no—
191C2×C4×D5—34no—
192C2×C4×D5—64no—
193C2×C4×D5—64no—
194C2×C4×D5—64no—
195C2×Dic10—22no—
196C2×Dic10—22no—
197C2×Dic10C4×D314yescomplemented
198C5×C4⋊C4—22no—
199C5×C22⋊C4—68no—
200C42×C5—62no—
201C22⋊Dic5—68no—
202#685—64no—
203C4⋊Dic5—62no—
204C4×Dic5—22no—
205C4×Dic5—64no—
206C4×Dic5—64no—
207C22×#41—151no—
208C2×#117—151no—
209C22×C42—151no—
210#391—51no—
211#391—151no—
212#390—31no—
213#390—151no—
214#386—154no—
215C4×C22⋊C4—154no—
216C22×C3×C5C23×C411yescharacteristic
217C22×C3×C5C22×D411yescharacteristic
218C22×C3×C5C22×D411yescharacteristic
219C22×C3×C5C2×#4112yes—
220C22×C3×C5C2×#4112yes—
221C2×D15—24no—
222C2×D15—24no—
223C2×D15—22no—
224C2×D15—22no—
225C2×D15C2×C22⋊C414yes—
226C2×C5×D3—24no—
227C2×C5×D3—28no—
228C2×C5×D3—44no—
229C2×C5×D3—42no—
230C2×C5×D3—42no—
231C3×C4×C5—42no—
232C3×C4×C5C2×C22⋊C414yes—
233Dic15—42no—
234Dic15C2×C22⋊C414yes—
235C3×Dic5—24no—
236C3×Dic5—24no—
237C5×Dic3—24no—
238C23×D3—51no—
239C23×D3C22×D512yes—
240C2×C3×Q8—54no—
241C2×C3×Q8—102no—
242C2×C3×Q8—102no—
243C22×C3×C4—52no—
244C22×C3×C4—52no—
245C22×C3×C4C22×D512yes—
246C22×C3×C4C22×D511yescharacteristic
247C22×Dic3—51no—
248C22×Dic3—52no—
249C22×Dic3C22×D512yes—
250#239—1032no—
251C2×D12—22no—
252C2×D12—22no—
253C2×D12—58no—
254C2×D12—104no—
255C2×D12—104no—
256C2×D12C4×D514yescomplemented
257C2×C4×D3—54no—
258C2×C4×D3—54no—
259C2×C4×D3—54no—
260C2×C4×D3—58no—
261C2×C4×D3—104no—
262C2×C4×D3—104no—
263C2×C4×D3—104no—
264C2×C4×D3—104no—
265C2×C4×D3—104no—
266C3×C4⋊C4—22no—
267C3×C42—102no—
268(C2×D3)⋊C4—28no—
269(C2×D3)⋊C4—104no—
270(C2×D3)⋊C4—108no—
271C4⋊Dic3—102no—
272C4×Dic3—22no—
273C4×Dic3—104no—
274C4×Dic3—104no—
275C23×C5—34no—
276C23×C5—62no—
277C23×C5—62no—
278C23×C5—64no—
279C23×C5—64no—
280C23×C5C23×D311yescharacteristic
281C22×D5—32no—
282C22×D5—32no—
283C22×D5—32no—
284C22×D5—62no—
285C22×D5—62no—
286C5×D4—68no—
287C2×C4×C5—22no—
288C2×C4×C5—24no—
289C2×C4×C5—34no—
290C2×C4×C5—62no—
291C2×C4×C5—62no—
292C2×C4×C5C2×C4×D312yes—
293C2×C4×C5D3×D412yes—
294C2×C4×C5D3×D412yes—
295C22⋊D5—616no—
296C2×Dic5—22no—
297C2×Dic5—22no—
298C2×Dic5—22no—
299C2×Dic5—22no—
300C2×Dic5—32no—
301C2×Dic5—32no—
302C2×Dic5—32no—
303C2×Dic5—616no—
304C2×Dic5—62no—
305C2×Dic5—64no—
306C2×Dic5C2×C4×D314yes—
307C2×Dic5#23914yescomplemented
308C4×D5—316no—
309C4×D5—616no—
310Dic10—28no—
311C2×#41—158no—
312C2×#41—304no—
313C2×#41—304no—
314C22×Q8—51no—
315C22×D4—31no—
316C22×D4—152no—
317C23×C4—152no—
318C23×C4—152no—
319C23×C4—151no—
320#117—304no—
321C2×C4⋊C4—151no—
322C2×C4⋊C4D3×D511yescharacteristic
323C2×C22⋊C4—34no—
324C2×C22⋊C4—154no—
325C2×C22⋊C4—152no—
326C2×C42—31no—
327C2×C42—51no—
328C2×C42—152no—
329C2×C42—152no—
330C2×C42—304no—
331#95—54no—
332#95—154no—
333C2×C3×C5C22×C22⋊C411yescharacteristic
334C2×C3×C5C4×#4112yes—
335C2×C3×C5#52512yes—
336C2×C3×C5#53211yescharacteristic
337C2×C3×C5#53211yescharacteristic
338D15—24no—
339C5×D3—44no—
340C23×C3C23×D511yescharacteristic
341C22×D3—22no—
342C22×D3—22no—
343C22×D3—24no—
344C22×D3—24no—
345C22×D3—52no—
346C22×D3—52no—
347C22×D3—52no—
348C22×D3—102no—
349C22×D3—102no—
350C22×D3C2×C4×D514yes—
351C3×Q8—108no—
352C2×C3×C4—22no—
353C2×C3×C4—24no—
354C2×C3×C4—54no—
355C2×C3×C4—54no—
356C2×C3×C4—102no—
357C2×C3×C4—102no—
358C2×C3×C4—102no—
359C2×C3×C4—102no—
360C2×C3×C4C2×C4×D512yes—
361C2×C3×C4D4×D512yes—
362C2×C3×C4D4×D512yes—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—52no—
366C2×Dic3—52no—
367C2×Dic3—52no—
368C2×Dic3—102no—
369C2×Dic3—104no—
370C2×Dic3#71114yescomplemented
371D12—28no—
372D12—1016no—
373C4×D3—516no—
374C4×D3—1016no—
375C4×D3—1016no—
376C22×C5—64no—
377C22×C5—68no—
378C22×C5—122no—
379C22×C5—122no—
380C22×C5—124no—
381C22×C5C22×C4×D311yescharacteristic
382C22×C5C2×D3×D411yescharacteristic
383C22×C5C2×D3×D411yescharacteristic
384C22×C5C2×#23912yes—
385C22×C5D3×#4112yes—
386C2×D5—34no—
387C2×D5—62no—
388C2×D5—62no—
389C2×D5—64no—
390C2×D5—64no—
391C4×C5—42no—
392C4×C5—64no—
393C4×C5D3×C22⋊C414yes—
394Dic5—24no—
395Dic5—24no—
396Dic5—34no—
397Dic5—122no—
398C24—32no—
399C24—151no—
400#41—3032no—
401C2×Q8—54no—
402C2×Q8—102no—
403C2×Q8—102no—
404C2×D4—34no—
405C2×D4—62no—
406C2×D4—62no—
407C2×D4—158no—
408C2×D4—304no—
409C2×D4—304no—
410C22×C4—32no—
411C22×C4—52no—
412C22×C4—52no—
413C22×C4—158no—
414C22×C4—151no—
415C22×C4—154no—
416C22×C4—154no—
417C22×C4—154no—
418C22×C4—152no—
419C22×C4—304no—
420C22×C4—304no—
421C22×C4—304no—
422C22×C4—304no—
423C22×C4—304no—
424C22×C4C2×D3×D512yes—
425C22×C4C2×D3×D511yescharacteristic
426C4⋊C4—22no—
427C4⋊C4—302no—
428C22⋊C4—68no—
429C22⋊C4—304no—
430C22⋊C4—308no—
431C42—62no—
432C42—102no—
433C42—304no—
434C42—304no—
435C3×C5#229711yescharacteristic, complemented, Hall
436C22×C3C22×C4×D511yescharacteristic
437C22×C3C2×D4×D511yescharacteristic
438C22×C3C2×D4×D511yescharacteristic
439C22×C3C2×#71112yes—
440C22×C3D5×#4112yes—
441C2×D3—24no—
442C2×D3—28no—
443C2×D3—42no—
444C2×D3—44no—
445C2×D3—42no—
446C2×D3—54no—
447C2×D3—102no—
448C2×D3—104no—
449C2×D3—104no—
450C2×D3—102no—
451C3×C4—42no—
452C3×C4—104no—
453C3×C4—104no—
454C3×C4D5×C22⋊C414yes—
455Dic3—24no—
456Dic3—54no—
457Dic3—202no—
458C2×C5—124no—
459C2×C5C2×D3×C22⋊C411yescharacteristic
460C2×C5C4×#23912yes—
461C2×C5#562712yes—
462C2×C5#566811yescharacteristic
463C2×C5#572511yescharacteristic
464D5—64no—
465C23—34no—
466C23—62no—
467C23—64no—
468C23—64no—
469C23—62no—
470C23—152no—
471C23—152no—
472C23—152no—
473C23—302no—
474C23—302no—
475C23C22×D3×D511yescharacteristic
476Q8—108no—
477D4—68no—
478D4—3016no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—34no—
482C2×C4—54no—
483C2×C4—54no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—102no—
487C2×C4—102no—
488C2×C4—102no—
489C2×C4—102no—
490C2×C4—152no—
491C2×C4—152no—
492C2×C4—152no—
493C2×C4—1516no—
494C2×C4—3016no—
495C2×C4—3016no—
496C2×C4—304no—
497C2×C4—302no—
498C2×C4C4×D3×D512yes—
499C2×C4#677712yes—
500C2×C4#679012yes—
501C2×C3C2×D5×C22⋊C411yescharacteristic
502C2×C3C4×#71112yes—
503C2×C3#6594012yes—
504C2×C3#6600211yescharacteristic
505C2×C3#6605011yescharacteristic
506D3—44no—
507D3—104no—
508C5#7647811yescharacteristic, complemented, Sylow
509C22—64no—
510C22—68no—
511C22—122no—
512C22—124no—
513C22—122no—
514C22—154no—
515C22—302no—
516C22—304no—
517C22—302no—
518C22—304no—
519C22C2×C4×D3×D511yescharacteristic
520C22C2×#677711yescharacteristic
521C22D3×#71112yes—
522C22D5×#23912yes—
523C22C2×#679011yescharacteristic
524C4—42no—
525C4—64no—
526C4—104no—
527C4—104no—
528C4—154no—
529C4—602no—
530C4#9147314yes—
531C3#1060802111yescharacteristic, complemented, Sylow
532C2—124no—
533C2—304no—
534C2#1175340612yes—
535C2#1175388612yes—
536C2C2×#9147311yescharacteristic
537C2#1175482811yescharacteristic
538C2#1175517011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#6768C2
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2297
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#91781C2semidirect
3. C2×C4×#1200C2semidirect
4. C5×#4988C2semidirect
5. C3×#65263C2semidirect
6. #11748919C2semidirect
7. #11748886C2semidirect
8. #11748764C2semidirect
9. #11748609C2semidirect
19. C2×#6768C4semidirect
39. #65263D3semidirect
55. C2×C5×D12C2×C4semidirect
60. C2×C3×Dic10C2×C4semidirect
69. C2×D3×Dic5C2×C4semidirect
85. #4988D5semidirect
133. C4×D15C22⋊C4semidirect
136. C2×C5×Dic3#41semidirect
143. C2×C3×Dic5#41semidirect
197. C2×Dic10C4×D3semidirect
256. C2×D12C4×D5semidirect
307. C2×Dic5#239semidirect
370. C2×Dic3#711semidirect
435. C3×C5#2297semidirect
508. C5#76478semidirect
531. C3#10608021semidirect

Maximal subgroups

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

subgroupindex
C22×#67682
C2×#917812
C2×C4×#12002
C5×#49882
C3×#652632
#117489192
#117488862
#117487642
#117486092
#106080213
#764785