Finite Group Databasebeta

← #14322568

Subgroups — #14322568

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×#6773C211yesmaximal, characteristic
2C2×C3×#4018C211yesmaximal, characteristic, complemented
3C2×D5×C4⋊Dic3C211yesmaximal, characteristic, complemented
4#11751040C211yesmaximal, characteristic, complemented
5C5×#4929C211yesmaximal, characteristic, complemented
6#11748876C214yesmaximal, complemented
7#11748872C214yesmaximal, complemented
8#11748768C211yesmaximal, characteristic, complemented
9#11748589C211yesmaximal, characteristic, complemented
10#10608213—31nomaximal, Hall
11C22×D60C2211yescharacteristic
12C22×C5×Dic6C2211yescharacteristic
13C22×C3×C4×D5C2211yescharacteristic
14C22×#1202C2212yes—
15C22×#1200C2212yes—
16C22×D5×Dic3C2212yes—
17C2×#6773—24no—
18C2×#6773—24no—
19C2×#6773C418yescomplemented
20C2×C3×C42×C5C2211yescharacteristic
21C2×#6714C2214yes—
22C2×C4⋊Dic15C2211yescharacteristic
23C2×C5×C4⋊Dic3C2211yescharacteristic
24C2×C3×#685C2212yes—
25C2×C3×#683C2212yes—
26C2×C3×C4×Dic5C2211yescharacteristic
27C3×#4018—24no—
28C2×#6668C2212yes—
29C2×#6667C2214yes—
30D5×C4⋊Dic3—24no—
31C5×C3⋊#95C2214yes—
32#90994C2214yes—
33#82270—51nomaximal, Hall
34C22×#713—31no—
35C2×D5×C4⋊C4—31no—
36C2×#4018D311yescharacteristic, complemented
37C5×#391—31no—
38#65519—31no—
39#65277—31no—
40#65228—31no—
41#64956—34no—
42#64954—34no—
43C23×D15C2312yes—
44C23×C3×D5C2311yescharacteristic
45C22×C3×C4×C5C2312yes—
46C22×C3×C4×C5C2311yescharacteristic
47C22×Dic15C2312yes—
48C2×D60—22no—
49C2×D60—22no—
50C2×D60C2×C414yescomplemented
51C22×C5×Dic3C2312yes—
52C22×C5×Dic3C2312yes—
53C2×C5×Dic6—22no—
54C2×C5×Dic6—22no—
55C2×C5×Dic6C2×C414yescomplemented
56C22×C3×Dic5C2312yes—
57C22×C3×Dic5C2311yescharacteristic
58C2×C3×C4×D5C2×C414yes—
59C2×C3×C4×D5D414yes—
60C2×C3×C4×D5D414yes—
61C2×#1202—24no—
62C2×#1202—24no—
63C2×#1202C2×C418yes—
64C2×#1200—24no—
65C2×#1200—24no—
66C2×#1200C2×C418yescomplemented
67C2×D5×Dic3—24no—
68C2×D5×Dic3—44no—
69#6773—232no—
70C3×C42×C5—22no—
71#6714—44no—
72C4⋊Dic15—22no—
73C5×C4⋊Dic3—22no—
74C3×#685—24no—
75C3×#683—24no—
76C3×C4×Dic5—22no—
77#6668—24no—
78#6667—44no—
79C22×#235—51no—
80C2×C3×#117—51no—
81C22×C4⋊Dic3—51no—
82#4987—54no—
83#4958—54no—
84#4939—51no—
85#4939—51no—
86#4929—51no—
87#4929D511yescharacteristic, complemented
88C22×C5×Q8—31no—
89C2×#713—38no—
90C2×#713—64no—
91C2×#713—64no—
92C22×D20—32no—
93C22×D20—31no—
94C22×C4×D5—32no—
95C22×C4×D5—32no—
96C22×C4×D5C2×D311yescharacteristic
97C2×C5×C4⋊C4—31no—
98C2×C42×C5C2×D311yescharacteristic
99C2×#685—34no—
100C2×#685—34no—
101C2×#685C2×D312yes—
102C2×C4⋊Dic5—31no—
103C2×#683—32no—
104C2×#683C2×D312yes—
105C2×C4×Dic5C2×D311yescharacteristic
106D5×C4⋊C4—64no—
107#4018—24no—
108C5×#95—34no—
109#3963—34no—
110#2311—151noSylow
111C23×C3×C5C2411yescharacteristic
112C22×D15—22no—
113C22×D15—22no—
114C22×D15—24no—
115C22×D15—24no—
116C22×D15C22×C414yes—
117C22×C3×D5—22no—
118C22×C3×D5—22no—
119C22×C3×D5C22×C412yes—
120C22×C3×D5C2×D412yes—
121C22×C3×D5C2×D412yes—
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—24no—
130D60—28no—
131C2×C5×Dic3—22no—
132C2×C5×Dic3—22no—
133C2×C5×Dic3—22no—
134C2×C5×Dic3—24no—
135C2×C5×Dic3C22×C414yes—
136C5×Dic6—28no—
137C2×C3×Dic5—22no—
138C2×C3×Dic5—22no—
139C2×C3×Dic5—22no—
140C2×C3×Dic5C22×C412yes—
141C2×C3×Dic5C2×D412yes—
142C2×C3×Dic5C2×D412yes—
143C3×C4×D5C22⋊C4116yescomplemented
144#1202—216no—
145#1200—216no—
146D5×Dic3—48no—
147C23×C3×C4—51no—
148C22×#67—52no—
149C23×Dic3—52no—
150C2×#235—58no—
151C2×#235—104no—
152C2×#235—104no—
153C22×D12—51no—
154C22×C4×D3—52no—
155C22×Dic6C2×D511yescharacteristic
156C3×#117—104no—
157C2×C3×C4⋊C4—52no—
158C2×C3×C22⋊C4—52no—
159C2×C3×C42—51no—
160C2×C3×C42C2×D511yescharacteristic
161C2×(C2×D3)⋊C4—54no—
162C2×(C2×D3)⋊C4—54no—
163C2×C4⋊Dic3—51no—
164C2×C4⋊Dic3—52no—
165C2×C4⋊Dic3—104no—
166C2×C4⋊Dic3C2×D511yescharacteristic
167C3⋊#95—54no—
168C3⋊#95C2×D514yes—
169C23×D5—32no—
170C23×D5C22×D311yescharacteristic
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—32no—
179C22×Dic5C22×D312yes—
180C22×Dic5C22×D311yescharacteristic
181#713—632no—
182C2×D20—38no—
183C2×D20—34no—
184C2×D20—64no—
185C2×D20—64no—
186C2×D20—62no—
187C2×D20—62no—
188C2×C4×D5—38no—
189C2×C4×D5—64no—
190C2×C4×D5—64no—
191C2×C4×D5—64no—
192C2×C4×D5—124no—
193C2×C4×D5C4×D314yes—
194C2×C4×D5D1214yes—
195C2×C4×D5#6714yes—
196C5×C4⋊C4—62no—
197C42×C5—22no—
198#685—24no—
199#685—124no—
200#685—124no—
201C4⋊Dic5—62no—
202#683—24no—
203#683—64no—
204C4×Dic5—22no—
205C22×#41—151no—
206C2×#117—51no—
207C22×C4⋊C4—151no—
208#391—31no—
209#391—151no—
210#390—151no—
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—28no—
221C2×D15—42no—
222C2×D15—42no—
223C2×D15—44no—
224C2×C3×D5—24no—
225C2×C3×D5—22no—
226C2×C3×D5—22no—
227C2×C3×D5—24no—
228C2×C3×D5C2×C22⋊C414yes—
229C3×C4×C5—42no—
230C3×C4×C5C2×C22⋊C414yes—
231Dic15—42no—
232C3×Dic5—42no—
233C3×Dic5C2×C22⋊C414yes—
234C5×Dic3—24no—
235C5×Dic3—42no—
236C24×C3—51no—
237C23×D3—52no—
238C22×C3×C4—51no—
239C22×C3×C4—54no—
240C22×C3×C4—54no—
241C22×C3×C4—54no—
242C22×C3×C4—52no—
243C22×C3×C4C22×D512yes—
244C22×C3×C4C22×D511yescharacteristic
245C2×#67—58no—
246C2×#67—104no—
247C2×#67—104no—
248C22×Dic3—52no—
249C22×Dic3—104no—
250C22×Dic3—204no—
251C22×Dic3C22×D512yes—
252C22×Dic3C22×D512yes—
253#235—1032no—
254C2×D12—54no—
255C2×D12—102no—
256C2×D12—102no—
257C2×C4×D3—58no—
258C2×C4×D3—104no—
259C2×C4×D3—104no—
260C2×Dic6—22no—
261C2×Dic6—22no—
262C2×Dic6C4×D514yescomplemented
263C3×C4⋊C4—104no—
264C3×C22⋊C4—104no—
265C3×C42—22no—
266C3×C42—102no—
267(C2×D3)⋊C4—204no—
268(C2×D3)⋊C4—204no—
269C4⋊Dic3—22no—
270C4⋊Dic3—104no—
271C4⋊Dic3—102no—
272C23×C5C23×D311yescharacteristic
273C22×D5—22no—
274C22×D5—22no—
275C22×D5—34no—
276C22×D5—62no—
277C22×D5—64no—
278C22×D5—64no—
279C22×D5—62no—
280C22×D5C2×C4×D312yes—
281C22×D5C2×D1212yes—
282C22×D5C2×#6712yes—
283C5×Q8—68no—
284C2×C4×C5—22no—
285C2×C4×C5—22no—
286C2×C4×C5—22no—
287C2×C4×C5—34no—
288C2×C4×C5—62no—
289C2×C4×C5—62no—
290C2×C4×C5—62no—
291C2×C4×C5—64no—
292C2×C4×C5C2×C4×D312yes—
293C2×C4×C5C2×D1212yes—
294C2×C4×C5C2×#6712yes—
295C2×Dic5—22no—
296C2×Dic5—22no—
297C2×Dic5—22no—
298C2×Dic5—62no—
299C2×Dic5—64no—
300C2×Dic5C2×C4×D312yes—
301C2×Dic5C2×D1212yes—
302C2×Dic5C2×#6712yes—
303D20—616no—
304D20—68no—
305C4×D5—616no—
306C4×D5—128no—
307C4×D5(C2×D3)⋊C4116yescomplemented
308C2×#41—158no—
309C2×#41—304no—
310C2×#41—304no—
311C22×Q8—31no—
312C22×D4—152no—
313C22×D4—151no—
314C23×C4—51no—
315C23×C4—152no—
316C23×C4—152no—
317#117—104no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—52no—
320C2×C4⋊C4—152no—
321C2×C4⋊C4—151no—
322C2×C4⋊C4—304no—
323C2×C22⋊C4—52no—
324C2×C22⋊C4—154no—
325C2×C22⋊C4—154no—
326C2×C42—51no—
327C2×C42D3×D511yescharacteristic
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—44no—
336C3×D5—24no—
337C23×C3—52no—
338C23×C3—52no—
339C23×C3—52no—
340C23×C3—102no—
341C23×C3—102no—
342C23×C3C23×D511yescharacteristic
343C22×D3—54no—
344C22×D3—102no—
345C22×D3—102no—
346C22×D3—104no—
347C22×D3—104no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—52no—
352C2×C3×C4—52no—
353C2×C3×C4—52no—
354C2×C3×C4—516no—
355C2×C3×C4—102no—
356C2×C3×C4—102no—
357C2×C3×C4—102no—
358C2×C3×C4C2×C4×D512yes—
359C2×C3×C4D4×D512yes—
360C2×C3×C4D4×D512yes—
361#67—1016no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—24no—
366C2×Dic3—102no—
367C2×Dic3—104no—
368C2×Dic3—208no—
369C2×Dic3C2×C4×D514yes—
370D12—108no—
371C4×D3—1016no—
372Dic6—28no—
373C22×C5C22×C4×D311yescharacteristic
374C22×C5C22×D1211yescharacteristic
375C22×C5#100912yes—
376C22×C5#101012yes—
377C22×C5C22×#6711yescharacteristic
378C2×D5—24no—
379C2×D5—22no—
380C2×D5—22no—
381C2×D5—24no—
382C2×D5—68no—
383C2×D5—64no—
384C2×D5—122no—
385C2×D5—122no—
386C2×D5—124no—
387C2×D5C2×(C2×D3)⋊C414yes—
388C4×C5—42no—
389C4×C5—64no—
390C4×C5—122no—
391C4×C5C2×(C2×D3)⋊C414yes—
392Dic5—42no—
393Dic5—122no—
394Dic5C2×(C2×D3)⋊C414yes—
395C24—51no—
396C24—152no—
397#41—3032no—
398C2×Q8—34no—
399C2×Q8—62no—
400C2×Q8—62no—
401C2×D4—158no—
402C2×D4—154no—
403C2×D4—304no—
404C2×D4—304no—
405C2×D4—302no—
406C2×D4—302no—
407C22×C4—32no—
408C22×C4—32no—
409C22×C4—51no—
410C22×C4—54no—
411C22×C4—54no—
412C22×C4—54no—
413C22×C4—52no—
414C22×C4—158no—
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—62no—
423C4⋊C4—104no—
424C4⋊C4—304no—
425C4⋊C4—302no—
426C22⋊C4—104no—
427C22⋊C4—604no—
428C22⋊C4—604no—
429C42—22no—
430C42—102no—
431C3×C5#231111yescharacteristic, complemented, Hall
432C22×C3—54no—
433C22×C3—102no—
434C22×C3—104no—
435C22×C3—104no—
436C22×C3—102no—
437C22×C3C22×C4×D511yescharacteristic
438C22×C3C2×D4×D511yescharacteristic
439C22×C3C2×D4×D511yescharacteristic
440C22×C3#414712yes—
441C22×C3#414912yes—
442C2×D3—104no—
443C2×D3—108no—
444C2×D3—202no—
445C2×D3—204no—
446C2×D3—202no—
447C3×C4—42no—
448C3×C4—54no—
449C3×C4—202no—
450C3×C4D5×C22⋊C414yes—
451Dic3—24no—
452Dic3—42no—
453Dic3—202no—
454C2×C5#552312yes—
455C2×C5#552511yescharacteristic
456C2×C5#552611yescharacteristic
457C2×C5C22×(C2×D3)⋊C411yescharacteristic
458C2×C5#583312yes—
459D5—24no—
460D5—124no—
461C23—52no—
462C23—52no—
463C23—52no—
464C23—102no—
465C23—102no—
466C23—154no—
467C23—304no—
468C23—302no—
469C23—304no—
470C23—302no—
471C23C22×D3×D511yescharacteristic
472Q8—68no—
473D4—3016no—
474D4—308no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—34no—
479C2×C4—52no—
480C2×C4—52no—
481C2×C4—52no—
482C2×C4—516no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—64no—
487C2×C4—102no—
488C2×C4—102no—
489C2×C4—102no—
490C2×C4—3016no—
491C2×C4—302no—
492C2×C4—304no—
493C2×C4—608no—
494C2×C4C4×D3×D512yes—
495C2×C4D5×D1212yes—
496C2×C4D5×#6712yes—
497C2×C3—104no—
498C2×C3C2×D5×C22⋊C411yescharacteristic
499C2×C3#6590812yes—
500C2×C3#6596312yes—
501C2×C3#6600811yescharacteristic
502C2×C3#6605811yescharacteristic
503D3—204no—
504C5#8227011yescharacteristic, complemented, Sylow
505C22—54no—
506C22—102no—
507C22—104no—
508C22—104no—
509C22—102no—
510C22—308no—
511C22—304no—
512C22—602no—
513C22—604no—
514C22—602no—
515C22#9200012yes—
516C22C2×C4×D3×D511yescharacteristic
517C22C2×D5×D1211yescharacteristic
518C22#9201912yes—
519C22C2×D5×#6711yescharacteristic
520C4—42no—
521C4—54no—
522C4—64no—
523C4—122no—
524C4—202no—
525C4—602no—
526C4D5×(C2×D3)⋊C414yes—
527C3#1060821311yescharacteristic, complemented, Sylow
528C2—104no—
529C2—604no—
530C2#1175252112yes—
531C2#1175253011yescharacteristic
532C2#1175273411yescharacteristic
533C2#1175431412yes—
534C2C2×D5×(C2×D3)⋊C411yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#6773C2
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×C3×#4018C2semidirect
3. C2×D5×C4⋊Dic3C2semidirect
4. #11751040C2semidirect
5. C5×#4929C2semidirect
6. #11748876C2semidirect
7. #11748872C2semidirect
8. #11748768C2semidirect
9. #11748589C2semidirect
19. C2×#6773C4semidirect
36. C2×#4018D3semidirect
50. C2×D60C2×C4semidirect
55. C2×C5×Dic6C2×C4semidirect
66. C2×#1200C2×C4semidirect
87. #4929D5semidirect
143. C3×C4×D5C22⋊C4semidirect
262. C2×Dic6C4×D5semidirect
307. C4×D5(C2×D3)⋊C4semidirect
431. C3×C5#2311semidirect
504. C5#82270semidirect
527. C3#10608213semidirect

Maximal subgroups

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

subgroupindex
C22×#67732
C2×C3×#40182
C2×D5×C4⋊Dic32
#117510402
C5×#49292
#117488762
#117488722
#117487682
#117485892
#106082133
#822705