Finite Group Databasebeta

← #14322573

Subgroups — #14322573

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×#6766C211yesmaximal, characteristic
2C2×C4⋊C4×D15C211yesmaximal, characteristic, complemented
3C2×#91386C211yesmaximal, characteristic, complemented
4C5×#4973C211yesmaximal, characteristic, complemented
5C3×#65277C211yesmaximal, characteristic, complemented
6#11748917C214yesmaximal, complemented
7#11748885C214yesmaximal, complemented
8#11748785C211yesmaximal, characteristic, complemented
9#11748589C211yesmaximal, characteristic, complemented
10#10608213—31nomaximal, Hall
11C22×C4×D15C2212yes—
12C22×C4×D15C2211yescharacteristic
13C22×C5×Dic6C2211yescharacteristic
14C22×C3×D20C2211yescharacteristic
15C22×#1202C2212yes—
16C22×D5×Dic3C2212yes—
17C2×#6766—24no—
18C2×#6766—24no—
19C2×#6766C418yescomplemented
20C2×C3×C5×C4⋊C4C2211yescharacteristic
21C2×C4⋊Dic15C2211yescharacteristic
22C2×#6712C2212yes—
23C4⋊C4×D15—24no—
24C2×C4×C5×Dic3C2211yescharacteristic
25C2×C3×#685C2214yes—
26C2×C3×C4×Dic5C2211yescharacteristic
27C2×#6669C2212yes—
28C2×#6667C2212yes—
29C2×#6665C2214yes—
30#91386—24no—
31C5×C3⋊#95C2214yes—
32#90994C2214yes—
33#76294—51nomaximal, Hall
34C22×#713—31no—
35C2×D5×C4⋊C4—31no—
36C2×#4018—31no—
37C5×#391—31no—
38#65519—31no—
39#65277D311yescharacteristic, complemented
40#65228—31no—
41#64956—34no—
42#64954—34no—
43C23×D15C2311yescharacteristic
44C23×C3×D5C2312yes—
45C22×C3×C4×C5C2312yes—
46C22×C3×C4×C5C2311yescharacteristic
47C22×Dic15C2312yes—
48C22×Dic15C2311yescharacteristic
49C2×C4×D15—24no—
50C2×C4×D15—44no—
51C2×C4×D15C2×C414yes—
52C2×C4×D15D414yes—
53C2×C4×D15D414yes—
54C22×C5×Dic3C2312yes—
55C22×C5×Dic3C2312yes—
56C2×C5×Dic6—22no—
57C2×C5×Dic6—22no—
58C2×C5×Dic6C2×C414yescomplemented
59C22×C3×Dic5C2312yes—
60C2×C3×D20—22no—
61C2×C3×D20—22no—
62C2×C3×D20C2×C414yescomplemented
63C2×#1202—24no—
64C2×#1202—24no—
65C2×#1202C2×C418yes—
66C2×D5×Dic3—24no—
67C2×D5×Dic3—24no—
68C2×D5×Dic3C2×C418yescomplemented
69#6766—232no—
70C3×C5×C4⋊C4—22no—
71C4⋊Dic15—22no—
72#6712—24no—
73C4×C5×Dic3—22no—
74C3×#685—44no—
75C3×C4×Dic5—22no—
76#6669—24no—
77#6667—24no—
78#6665—44no—
79C22×#237—51no—
80C2×D3×C4⋊C4—51no—
81C2×#872—51no—
82C3×#390—51no—
83#5221—51no—
84#4973D511yescharacteristic, complemented
85#4956—54no—
86#4942—54no—
87#4929—51no—
88C22×C5×Q8—31no—
89C2×#713—38no—
90C2×#713—64no—
91C2×#713—64no—
92C22×D20—32no—
93C22×D20C2×D311yescharacteristic
94C22×C4×D5—31no—
95C22×C4×D5—32no—
96C22×C4×D5—32no—
97C2×C5×C4⋊C4C2×D311yescharacteristic
98C2×C42×C5—31no—
99C2×#685—32no—
100C2×#685—34no—
101C2×#685C2×D314yes—
102C2×C4⋊Dic5—31no—
103C2×#683—32no—
104C2×#683—32no—
105C2×C4×Dic5C2×D311yescharacteristic
106D5×C4⋊C4—64no—
107#4018—64no—
108C5×#95—34no—
109#3963—34no—
110#2311—151noSylow
111C23×C3×C5C2411yescharacteristic
112C22×D15—22no—
113C22×D15—22no—
114C22×D15C22×C412yes—
115C22×D15C2×D412yes—
116C22×D15C2×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—24no—
124C2×C3×C4×C5C22×C412yes—
125C2×C3×C4×C5C2×D412yes—
126C2×C3×C4×C5C2×D412yes—
127C2×Dic15—22no—
128C2×Dic15—24no—
129C2×Dic15C22×C412yes—
130C2×Dic15C2×D412yes—
131C2×Dic15C2×D412yes—
132C4×D15—48no—
133C4×D15C22⋊C4116yescomplemented
134C2×C5×Dic3—22no—
135C2×C5×Dic3—22no—
136C2×C5×Dic3—22no—
137C2×C5×Dic3—22no—
138C2×C5×Dic3—22no—
139C2×C5×Dic3C22×C414yes—
140C5×Dic6—28no—
141C2×C3×Dic5—22no—
142C2×C3×Dic5—22no—
143C2×C3×Dic5—22no—
144C3×D20—28no—
145#1202—216no—
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—51no—
154C22×C4×D3—52no—
155C22×Dic6C2×D511yescharacteristic
156C2×C3×C4⋊C4C2×D511yescharacteristic
157C2×C3×C22⋊C4—54no—
158C2×C3×C42—51no—
159C2×#217—54no—
160C2×(C2×D3)⋊C4—52no—
161C2×C4⋊Dic3—51no—
162C2×Dic3⋊C4—52no—
163C2×Dic3⋊C4—52no—
164C2×C4×Dic3C2×D511yescharacteristic
165D3×C4⋊C4—104no—
166#872—104no—
167C3⋊#95—54no—
168C3⋊#95C2×D514yes—
169C23×D5—31no—
170C23×D5C22×D312yes—
171C2×C5×Q8—34no—
172C2×C5×Q8—62no—
173C2×C5×Q8—62no—
174C22×C4×C5—32no—
175C22×C4×C5—32no—
176C22×C4×C5C22×D312yes—
177C22×C4×C5C22×D311yescharacteristic
178C22×Dic5—31no—
179C22×Dic5—32no—
180C22×Dic5C22×D312yes—
181#713—632no—
182C2×D20—22no—
183C2×D20—22no—
184C2×D20—38no—
185C2×D20—64no—
186C2×D20—64no—
187C2×D20C4×D314yescomplemented
188C2×C4×D5—34no—
189C2×C4×D5—34no—
190C2×C4×D5—34no—
191C2×C4×D5—38no—
192C2×C4×D5—64no—
193C2×C4×D5—64no—
194C2×C4×D5—64no—
195C2×C4×D5—124no—
196C5×C4⋊C4—22no—
197C42×C5—62no—
198#685—44no—
199#685—64no—
200#685—124no—
201C4⋊Dic5—62no—
202#683—64no—
203#683—64no—
204C4×Dic5—22no—
205C22×#41—151no—
206C2×#117—151no—
207C22×C4⋊C4—151no—
208#391—31no—
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×D15—24no—
220C2×D15—22no—
221C2×D15—22no—
222C2×D15—24no—
223C2×D15C2×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—42no—
232Dic15C2×C22⋊C414yes—
233C3×Dic5—42no—
234C5×Dic3—24no—
235C5×Dic3—42no—
236C24×C3—52no—
237C23×D3—51no—
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—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—22no—
261C2×Dic6—22no—
262C2×Dic6C4×D514yescomplemented
263C3×C4⋊C4—22no—
264C3×C22⋊C4—204no—
265C3×C42—102no—
266#217—204no—
267(C2×D3)⋊C4—104no—
268C4⋊Dic3—102no—
269Dic3⋊C4—104no—
270Dic3⋊C4—104no—
271C4×Dic3—22no—
272C23×C5C23×D311yescharacteristic
273C22×D5—22no—
274C22×D5—22no—
275C22×D5—24no—
276C22×D5—24no—
277C22×D5—32no—
278C22×D5—32no—
279C22×D5—32no—
280C22×D5—62no—
281C22×D5—62no—
282C22×D5C2×C4×D314yes—
283C5×Q8—68no—
284C2×C4×C5—22no—
285C2×C4×C5—24no—
286C2×C4×C5—34no—
287C2×C4×C5—62no—
288C2×C4×C5—62no—
289C2×C4×C5—62no—
290C2×C4×C5—62no—
291C2×C4×C5—62no—
292C2×C4×C5C2×C4×D312yes—
293C2×C4×C5D3×D412yes—
294C2×C4×C5D3×D412yes—
295C2×Dic5—22no—
296C2×Dic5—22no—
297C2×Dic5—22no—
298C2×Dic5—32no—
299C2×Dic5—32no—
300C2×Dic5—32no—
301C2×Dic5—62no—
302C2×Dic5—64no—
303D20—28no—
304D20—616no—
305C4×D5—316no—
306C4×D5—616no—
307C4×D5—128no—
308C2×#41—158no—
309C2×#41—304no—
310C2×#41—304no—
311C22×Q8—31no—
312C22×D4—51no—
313C22×D4—152no—
314C23×C4—152no—
315C23×C4—151no—
316C23×C4—152no—
317#117—304no—
318C2×C4⋊C4—152no—
319C2×C4⋊C4—152no—
320C2×C4⋊C4—151no—
321C2×C4⋊C4—304no—
322C2×C4⋊C4D3×D511yescharacteristic
323C2×C22⋊C4—54no—
324C2×C22⋊C4—152no—
325C2×C22⋊C4—154no—
326C2×C42—31no—
327C2×C42—51no—
328#95—34no—
329#95—154no—
330C2×C3×C5C22×C22⋊C411yescharacteristic
331C2×C3×C5#52012yes—
332C2×C3×C5#53511yescharacteristic
333C2×C3×C5#53611yescharacteristic
334C2×C3×C5#53712yes—
335D15—24no—
336C3×D5—44no—
337C23×C3—54no—
338C23×C3—102no—
339C23×C3—102no—
340C23×C3—104no—
341C23×C3—104no—
342C23×C3C23×D511yescharacteristic
343C22×D3—52no—
344C22×D3—52no—
345C22×D3—52no—
346C22×D3—102no—
347C22×D3—102no—
348C3×D4—108no—
349C2×C3×C4—22no—
350C2×C3×C4—24no—
351C2×C3×C4—102no—
352C2×C3×C4—102no—
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—22no—
362C2×Dic3—22no—
363C2×Dic3—52no—
364C2×Dic3—52no—
365C2×Dic3—52no—
366C2×Dic3—1016no—
367C2×Dic3—102no—
368C2×Dic3—104no—
369C2×Dic3C2×C4×D514yes—
370C4×D3—516no—
371C4×D3—208no—
372Dic6—28no—
373C22×C5C22×C4×D311yescharacteristic
374C22×C5C2×D3×D411yescharacteristic
375C22×C5C2×D3×D411yescharacteristic
376C22×C5#100412yes—
377C22×C5#101012yes—
378C2×D5—24no—
379C2×D5—28no—
380C2×D5—34no—
381C2×D5—42no—
382C2×D5—42no—
383C2×D5—44no—
384C2×D5—62no—
385C2×D5—62no—
386C2×D5—64no—
387C2×D5—64no—
388C4×C5—42no—
389C4×C5—64no—
390C4×C5—122no—
391C4×C5D3×C22⋊C414yes—
392Dic5—34no—
393Dic5—42no—
394Dic5—122no—
395C24—52no—
396C24—151no—
397#41—3032no—
398C2×Q8—34no—
399C2×Q8—62no—
400C2×Q8—62no—
401C2×D4—54no—
402C2×D4—102no—
403C2×D4—102no—
404C2×D4—158no—
405C2×D4—304no—
406C2×D4—304no—
407C22×C4—32no—
408C22×C4—32no—
409C22×C4—52no—
410C22×C4—158no—
411C22×C4—151no—
412C22×C4—154no—
413C22×C4—154no—
414C22×C4—154no—
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—22no—
423C4⋊C4—304no—
424C4⋊C4—304no—
425C4⋊C4—302no—
426C22⋊C4—204no—
427C22⋊C4—304no—
428C22⋊C4—604no—
429C42—62no—
430C42—102no—
431C3×C5#231111yescharacteristic, complemented, Hall
432C22×C3—104no—
433C22×C3—108no—
434C22×C3—202no—
435C22×C3—204no—
436C22×C3—202no—
437C22×C3C22×C4×D511yescharacteristic
438C22×C3C2×D4×D511yescharacteristic
439C22×C3C2×D4×D511yescharacteristic
440C22×C3#414712yes—
441C22×C3#414912yes—
442C2×D3—54no—
443C2×D3—102no—
444C2×D3—104no—
445C2×D3—104no—
446C2×D3—102no—
447C3×C4—42no—
448C3×C4—202no—
449C3×C4D5×C22⋊C414yes—
450Dic3—24no—
451Dic3—42no—
452Dic3—54no—
453Dic3—202no—
454C2×C5C2×D3×C22⋊C411yescharacteristic
455C2×C5#554612yes—
456C2×C5#560812yes—
457C2×C5#567511yescharacteristic
458C2×C5#571511yescharacteristic
459D5—44no—
460D5—64no—
461C23—54no—
462C23—102no—
463C23—104no—
464C23—104no—
465C23—102no—
466C23—152no—
467C23—152no—
468C23—152no—
469C23—302no—
470C23—302no—
471C23C22×D3×D511yescharacteristic
472Q8—68no—
473D4—108no—
474D4—3016no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—34no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—102no—
484C2×C4—102no—
485C2×C4—102no—
486C2×C4—152no—
487C2×C4—152no—
488C2×C4—152no—
489C2×C4—1516no—
490C2×C4—3016no—
491C2×C4—304no—
492C2×C4—302no—
493C2×C4—608no—
494C2×C4C4×D3×D512yes—
495C2×C4#677712yes—
496C2×C4#679012yes—
497C2×C3—204no—
498C2×C3C2×D5×C22⋊C411yescharacteristic
499C2×C3#6590812yes—
500C2×C3#6596312yes—
501C2×C3#6600811yescharacteristic
502C2×C3#6605811yescharacteristic
503D3—104no—
504C5#7629411yescharacteristic, complemented, Sylow
505C22—104no—
506C22—108no—
507C22—154no—
508C22—202no—
509C22—202no—
510C22—204no—
511C22—302no—
512C22—302no—
513C22—304no—
514C22—304no—
515C22C2×C4×D3×D511yescharacteristic
516C22C2×#677711yescharacteristic
517C22#9202612yes—
518C22#9202812yes—
519C22C2×#679011yescharacteristic
520C4—42no—
521C4—64no—
522C4—122no—
523C4—154no—
524C4—202no—
525C4—602no—
526C4#9147314yes—
527C3#1060821311yescharacteristic, complemented, Sylow
528C2—204no—
529C2—304no—
530C2#1175340112yes—
531C2#1175370112yes—
532C2C2×#9147311yescharacteristic
533C2#1175482311yescharacteristic
534C2#1175516811yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#6766C2
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2311
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×C4⋊C4×D15C2semidirect
3. C2×#91386C2semidirect
4. C5×#4973C2semidirect
5. C3×#65277C2semidirect
6. #11748917C2semidirect
7. #11748885C2semidirect
8. #11748785C2semidirect
9. #11748589C2semidirect
19. C2×#6766C4semidirect
39. #65277D3semidirect
58. C2×C5×Dic6C2×C4semidirect
62. C2×C3×D20C2×C4semidirect
68. C2×D5×Dic3C2×C4semidirect
84. #4973D5semidirect
133. C4×D15C22⋊C4semidirect
187. C2×D20C4×D3semidirect
262. C2×Dic6C4×D5semidirect
431. C3×C5#2311semidirect
504. C5#76294semidirect
527. C3#10608213semidirect

Maximal subgroups

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

subgroupindex
C22×#67662
C2×C4⋊C4×D152
C2×#913862
C5×#49732
C3×#652772
#117489172
#117488852
#117487852
#117485892
#106082133
#762945