Finite Group Databasebeta

← #14326948

Subgroups — #14326948

Order 1920.

1008 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#892C211yesmaximal, characteristic
2C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
3C3×C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
4#11748868C211yesmaximal, characteristic, complemented
5#11748842C211yesmaximal, characteristic, complemented
6C4×Dic3⋊Dic5C212yesmaximal, complemented
7#11748794C212yesmaximal
8#11748786C211yesmaximal, characteristic
9#11748747C212yesmaximal
10C4×(C2×D3)⋊Dic5C211yesmaximal, characteristic
11#11748618C212yesmaximal, complemented
12#10613971—31nomaximal, Hall
13C22×C4×C5×D3C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C4⋊Dic15C2211yescharacteristic
16C2×C4×Dic15C2211yescharacteristic
17C2×C4×Dic15C2212yes—
18C2×C5×(C2×D3)⋊C4C2212yes—
19C2×C5×C4⋊Dic3C2211yescharacteristic
20C2×C4×C5×Dic3C2212yes—
21C5×#892C418yescomplemented
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C2×C3×C4×Dic5C2211yescharacteristic
24C2×C3×C4×Dic5C2212yes—
25C2×Dic3⋊Dic5C2212yes—
26C2×Dic3⋊Dic5C2212yes—
27C2×(C2×D3)⋊Dic5C2211yescharacteristic
28C2×(C2×D3)⋊Dic5C2211yescharacteristic
29C2×(C2×D3)⋊Dic5C2212yes—
30#91115C2212yes—
31C3×#3963C2212yes—
32#90994C2212yes—
33#90994C2211yescharacteristic
34#90994C2212yes—
35#90994C2212yes—
36#90994C2211yescharacteristic
37#73634—51nomaximal, Hall
38C2×C5×#117—31no—
39#65501—31no—
40C4×C22⋊Dic5—31no—
41#65274—31no—
42#65274—31no—
43C4⋊C4×Dic5—31no—
44C4⋊C4×Dic5D311yescharacteristic, complemented
45#65240—32no—
46#65240—32no—
47#65226—32no—
48C4×C4⋊Dic5—32no—
49C23×C5×D3C2311yescharacteristic
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×Dic15C2311yescharacteristic
53C22×Dic15C2312yes—
54C22×Dic15C2311yescharacteristic
55C22×C5×Dic3C2311yescharacteristic
56C22×C5×Dic3C2312yes—
57C2×C4×C5×D3—22no—
58C2×C4×C5×D3—22no—
59C2×C4×C5×D3C2×C414yes—
60C22×C3×Dic5C2311yescharacteristic
61C22×C3×Dic5C2312yes—
62C22×C3×Dic5C2311yescharacteristic
63C3×C5×C4⋊C4C2×C414yes—
64C4⋊Dic15—22no—
65C4×Dic15—22no—
66C4×Dic15—42no—
67C5×(C2×D3)⋊C4C2×C418yes—
68C5×C4⋊Dic3C2×C414yes—
69C4×C5×Dic3C2×C418yes—
70C3×C4⋊Dic5—22no—
71C3×C4×Dic5—22no—
72C3×C4×Dic5—42no—
73Dic3⋊Dic5—24no—
74Dic3⋊Dic5—24no—
75(C2×D3)⋊Dic5—22no—
76(C2×D3)⋊Dic5—22no—
77(C2×D3)⋊Dic5—42no—
78C2×#892D511yescharacteristic
79C3×C4×C4⋊C4—51no—
80#4987—51no—
81Dic3×C4⋊C4—51no—
82#4972—51no—
83#4940—52no—
84C4×(C2×D3)⋊C4—51no—
85C4×Dic3⋊C4—52no—
86#4928—51no—
87#4669—52no—
88#4651—52no—
89C23×C4×C5—31no—
90C5×#117—38no—
91C2×C5×C4⋊C4—31no—
92C2×C5×C4⋊C4C2×D311yescharacteristic
93C2×C5×C22⋊C4—32no—
94C2×C42×C5—32no—
95C2×C22⋊Dic5—31no—
96C2×C22⋊Dic5—31no—
97C2×C22⋊Dic5—32no—
98C2×C4⋊Dic5—31no—
99C2×C4⋊Dic5—32no—
100C2×C4⋊Dic5—32no—
101C2×C4⋊Dic5C2×D311yescharacteristic
102C2×C4×Dic5—31no—
103C2×C4×Dic5—32no—
104C2×C4×Dic5C2×D311yescharacteristic
105C2×C4×Dic5C2×D312yes—
106#3963—32no—
107#3963—31no—
108#3963—32no—
109#3963—31no—
110#3963—32no—
111#3963—32no—
112#3963C2×D312yes—
113#2347—151noSylow
114C23×C3×C5C2411yescharacteristic
115C22×C5×D3—21no—
116C22×C5×D3—21no—
117C22×C5×D3—21no—
118C22×C5×D3—21no—
119C22×C5×D3—21no—
120C22×C5×D3—21no—
121C22×C5×D3C22×C412yes—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5C22×C414yes—
125C2×C3×C4×C5C22×C412yes—
126C2×C3×C4×C5#4112yes—
127C2×C3×C4×C5#4112yes—
128C2×Dic15—21no—
129C2×Dic15—21no—
130C2×Dic15—21no—
131C2×Dic15—22no—
132C2×Dic15—21no—
133C2×Dic15—22no—
134C2×Dic15—22no—
135C2×Dic15—21no—
136C2×Dic15#4112yes—
137C2×C5×Dic3—22no—
138C2×C5×Dic3—22no—
139C2×C5×Dic3—21no—
140C2×C5×Dic3—21no—
141C2×C5×Dic3C22×C412yes—
142C2×C5×Dic3C22×C414yes—
143C4×C5×D3—28no—
144C2×C3×Dic5—21no—
145C2×C3×Dic5—21no—
146C2×C3×Dic5—21no—
147C2×C3×Dic5—22no—
148C2×C3×Dic5—21no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5—21no—
152C2×C3×Dic5#4112yes—
153C22×C4×D3C2×D511yescharacteristic
154C2×C3×C4⋊C4—51no—
155C2×C3×C4⋊C4C2×D511yescharacteristic
156C2×C3×C42—51no—
157C2×C3×C42—52no—
158C2×(C2×D3)⋊C4—51no—
159C2×(C2×D3)⋊C4—51no—
160C2×(C2×D3)⋊C4—52no—
161C2×(C2×D3)⋊C4C2×D512yes—
162C2×C4⋊Dic3—51no—
163C2×C4⋊Dic3C2×D511yescharacteristic
164C2×Dic3⋊C4—52no—
165C2×Dic3⋊C4—52no—
166C2×C4×Dic3—51no—
167C2×C4×Dic3—52no—
168C2×C4×Dic3C2×D512yes—
169#892Dic518yescomplemented
170C3×#95—52no—
171C3⋊#95—52no—
172C3⋊#95—51no—
173C3⋊#95—52no—
174C3⋊#95—52no—
175C3⋊#95—52no—
176C3⋊#95—51no—
177C24×C5—31no—
178C22×C4×C5—31no—
179C22×C4×C5—32no—
180C22×C4×C5—34no—
181C22×C4×C5—62no—
182C22×C4×C5—62no—
183C22×C4×C5C22×D312yes—
184C22×C4×C5C22×D311yescharacteristic
185C22×Dic5—31no—
186C22×Dic5—31no—
187C22×Dic5—32no—
188C22×Dic5C22×D311yescharacteristic
189C22×Dic5C22×D312yes—
190C22×Dic5C22×D311yescharacteristic
191C5×C4⋊C4—34no—
192C5×C4⋊C4C4×D314yes—
193C5×C22⋊C4—38no—
194C42×C5—38no—
195C22⋊Dic5—62no—
196C22⋊Dic5—62no—
197C22⋊Dic5—122no—
198C4⋊Dic5—22no—
199C4⋊Dic5—64no—
200C4⋊Dic5—62no—
201C4⋊Dic5—64no—
202C4×Dic5—22no—
203C4×Dic5—42no—
204C4×Dic5—62no—
205C4×Dic5—122no—
206C2×#117—31no—
207#389—151no—
208#389—151no—
209#388—152no—
210#388—152no—
211#383—152no—
212#380—151no—
213C4×C4⋊C4—51no—
214C4×C4⋊C4—152no—
215C4×C4⋊C4—151no—
216C4×C22⋊C4—151no—
217C22×C3×C5C23×C411yescharacteristic
218C22×C3×C5C2×#4111yescharacteristic
219C22×C3×C5C2×#4111yescharacteristic
220C22×C3×C5C2×#4111yescharacteristic
221C22×C3×C5C2×#4111yescharacteristic
222C22×C3×C5H(5,𝔽2)11yescharacteristic
223C22×C3×C52−1+411yescharacteristic
224C2×C5×D3—22no—
225C2×C5×D3—22no—
226C2×C5×D3—22no—
227C2×C5×D3—41no—
228C2×C5×D3—41no—
229C2×C5×D3—41no—
230C2×C5×D3—41no—
231C3×C4×C5—24no—
232C3×C4×C5#11714yes—
233Dic15—22no—
234Dic15—42no—
235Dic15—41no—
236C3×Dic5—22no—
237C3×Dic5—42no—
238C3×Dic5—41no—
239C5×Dic3—22no—
240C5×Dic3—24no—
241C23×D3C22×D511yescharacteristic
242C22×C3×C4—51no—
243C22×C3×C4—51no—
244C22×C3×C4—52no—
245C22×C3×C4C22×D512yes—
246C22×C3×C4C22×D511yescharacteristic
247C22×Dic3—51no—
248C22×Dic3—51no—
249C22×Dic3—52no—
250C22×Dic3C22×D511yescharacteristic
251C22×Dic3C22×D512yes—
252C2×C4×D3—22no—
253C2×C4×D3—22no—
254C2×C4×D3C2×Dic514yes—
255C3×C4⋊C4—102no—
256C3×C4⋊C4C2×Dic514yes—
257C3×C42—102no—
258C3×C42—202no—
259(C2×D3)⋊C4—102no—
260(C2×D3)⋊C4—102no—
261(C2×D3)⋊C4—202no—
262(C2×D3)⋊C4C2×Dic518yes—
263C4⋊Dic3—102no—
264C4⋊Dic3C2×Dic514yes—
265Dic3⋊C4—104no—
266Dic3⋊C4—104no—
267C4×Dic3—102no—
268C4×Dic3—202no—
269C4×Dic3C2×Dic518yes—
270C23×C5—32no—
271C23×C5—61no—
272C23×C5—61no—
273C23×C5—61no—
274C23×C5—61no—
275C23×C5—61no—
276C23×C5—61no—
277C23×C5C23×D311yescharacteristic
278C2×C4×C5—22no—
279C2×C4×C5—22no—
280C2×C4×C5—32no—
281C2×C4×C5—34no—
282C2×C4×C5—61no—
283C2×C4×C5—61no—
284C2×C4×C5—62no—
285C2×C4×C5—62no—
286C2×C4×C5—68no—
287C2×C4×C5C2×C4×D314yes—
288C2×C4×C5C2×C4×D312yes—
289C2×C4×C5#23512yes—
290C2×C4×C5#23512yes—
291C2×Dic5—21no—
292C2×Dic5—21no—
293C2×Dic5—21no—
294C2×Dic5—22no—
295C2×Dic5—22no—
296C2×Dic5—22no—
297C2×Dic5—21no—
298C2×Dic5—21no—
299C2×Dic5—32no—
300C2×Dic5—61no—
301C2×Dic5—61no—
302C2×Dic5—61no—
303C2×Dic5—62no—
304C2×Dic5—62no—
305C2×Dic5—62no—
306C2×Dic5—61no—
307C2×Dic5—61no—
308C2×Dic5#23912yes—
309C23×C4—31no—
310#117—38no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—51no—
313C2×C4⋊C4—152no—
314C2×C4⋊C4—152no—
315C2×C4⋊C4—151no—
316C2×C4⋊C4D3×D511yescharacteristic
317C2×C22⋊C4—32no—
318C2×C22⋊C4—151no—
319C2×C22⋊C4—151no—
320C2×C22⋊C4—152no—
321C2×C42—32no—
322C2×C42—51no—
323C2×C42—52no—
324C2×C42—151no—
325C2×C42—152no—
326#95—52no—
327#95—151no—
328#95—151no—
329#95—152no—
330#95—152no—
331#95—152no—
332#95—152no—
333C2×C3×C5C2×#11711yescharacteristic
334C2×C3×C5C4×#4111yescharacteristic
335C2×C3×C5#52011yescharacteristic
336C2×C3×C5#55311yescharacteristic
337C2×C3×C5#55311yescharacteristic
338C2×C3×C5#55611yescharacteristic
339C2×C3×C5#55611yescharacteristic
340C5×D3—42no—
341C23×C3C23×D511yescharacteristic
342C22×D3—21no—
343C22×D3—21no—
344C22×D3—21no—
345C22×D3—21no—
346C22×D3—21no—
347C22×D3—21no—
348C22×D3C22×Dic512yes—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—52no—
352C2×C3×C4—101no—
353C2×C3×C4—101no—
354C2×C3×C4—101no—
355C2×C3×C4—102no—
356C2×C3×C4—102no—
357C2×C3×C4—101no—
358C2×C3×C4—102no—
359C2×C3×C4—101no—
360C2×C3×C4#70912yes—
361C2×C3×C4#70912yes—
362C2×C3×C4C22×Dic514yes—
363C2×C3×C4C22×Dic512yes—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—52no—
369C2×Dic3—101no—
370C2×Dic3—101no—
371C2×Dic3—101no—
372C2×Dic3—102no—
373C2×Dic3—102no—
374C2×Dic3—101no—
375C2×Dic3—102no—
376C2×Dic3—101no—
377C2×Dic3C22×Dic512yes—
378C2×Dic3C22×Dic514yes—
379C4×D3—28no—
380C22×C5—62no—
381C22×C5—62no—
382C22×C5—62no—
383C22×C5—121no—
384C22×C5—121no—
385C22×C5—121no—
386C22×C5—121no—
387C22×C5C22×C4×D311yescharacteristic
388C22×C5C2×#23511yescharacteristic
389C22×C5C2×#23511yescharacteristic
390C22×C5#100411yescharacteristic
391C22×C5C2×#23911yescharacteristic
392C22×C5D3×#4111yescharacteristic
393C22×C5#101011yescharacteristic
394C4×C5—24no—
395C4×C5—62no—
396C4×C5—64no—
397C4×C5#87214yes—
398Dic5—22no—
399Dic5—41no—
400Dic5—42no—
401Dic5—62no—
402Dic5—122no—
403Dic5—121no—
404C24—31no—
405C22×C4—31no—
406C22×C4—32no—
407C22×C4—34no—
408C22×C4—51no—
409C22×C4—51no—
410C22×C4—52no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—151no—
414C22×C4—151no—
415C22×C4—152no—
416C22×C4C2×D3×D512yes—
417C22×C4C2×D3×D511yescharacteristic
418C4⋊C4—34no—
419C4⋊C4—102no—
420C4⋊C4—304no—
421C4⋊C4—304no—
422C4⋊C4—302no—
423C4⋊C4D3×Dic514yes—
424C22⋊C4—38no—
425C22⋊C4—302no—
426C22⋊C4—302no—
427C22⋊C4—602no—
428C42—38no—
429C42—102no—
430C42—202no—
431C42—302no—
432C42—602no—
433C3×C5#234711yescharacteristic, complemented, Hall
434C22×C3C2×#70911yescharacteristic
435C22×C3C2×#70911yescharacteristic
436C22×C3D5×#4111yescharacteristic
437C22×C3D5×#4111yescharacteristic
438C22×C3#414911yescharacteristic
439C22×C3#415011yescharacteristic
440C22×C3C23×Dic511yescharacteristic
441C2×D3—22no—
442C2×D3—22no—
443C2×D3—22no—
444C2×D3—41no—
445C2×D3—41no—
446C2×D3—41no—
447C2×D3—41no—
448C3×C4—24no—
449C3×C4—102no—
450C3×C4—202no—
451C3×C4—201no—
452C3×C4#407214yes—
453Dic3—22no—
454Dic3—24no—
455Dic3—102no—
456Dic3—202no—
457Dic3—201no—
458C2×C5—122no—
459C2×C5C2×#87211yescharacteristic
460C2×C5#551411yescharacteristic
461C2×C5#553311yescharacteristic
462C2×C5#554611yescharacteristic
463C2×C5#555811yescharacteristic
464C2×C5C4×#23911yescharacteristic
465C2×C5#558411yescharacteristic
466C23—32no—
467C23—61no—
468C23—61no—
469C23—61no—
470C23—61no—
471C23—61no—
472C23—61no—
473C23C22×D3×D511yescharacteristic
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—32no—
477C2×C4—34no—
478C2×C4—52no—
479C2×C4—61no—
480C2×C4—61no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—68no—
484C2×C4—101no—
485C2×C4—101no—
486C2×C4—101no—
487C2×C4—102no—
488C2×C4—101no—
489C2×C4—102no—
490C2×C4—101no—
491C2×C4—102no—
492C2×C4—152no—
493C2×C4—301no—
494C2×C4—301no—
495C2×C4—301no—
496C2×C4—302no—
497C2×C4—301no—
498C2×C4—302no—
499C2×C4—301no—
500C2×C4—302no—
501C2×C4#677112yes—
502C2×C4#677112yes—
503C2×C4C2×D3×Dic514yes—
504C2×C4C2×D3×Dic512yes—
505C2×C3#6586411yescharacteristic
506C2×C3#6586411yescharacteristic
507C2×C3#6586711yescharacteristic
508C2×C3#6586711yescharacteristic
509C2×C3C2×#407211yescharacteristic
510C2×C3#41×Dic511yescharacteristic
511C2×C3#6616111yescharacteristic
512D3—42no—
513C5#7363411yescharacteristic, complemented, Sylow
514C22—62no—
515C22—62no—
516C22—62no—
517C22—121no—
518C22—121no—
519C22—121no—
520C22—121no—
521C22C2×#677111yescharacteristic
522C22C2×#677111yescharacteristic
523C22#9202011yescharacteristic
524C22#9202611yescharacteristic
525C22D5×#23911yescharacteristic
526C22#9203411yescharacteristic
527C22C22×D3×Dic511yescharacteristic
528C4—24no—
529C4—62no—
530C4—64no—
531C4—102no—
532C4—201no—
533C4—202no—
534C4—302no—
535C4—602no—
536C4—601no—
537C4#9137114yes—
538C3#1061397111yescharacteristic, complemented, Sylow
539C2—122no—
540C2#1175374011yescharacteristic
541C2#1175374411yescharacteristic
542C2#1175380311yescharacteristic
543C2#1175380811yescharacteristic
544C2C2×#9137111yescharacteristic
545C2#1175492011yescharacteristic
546C2Dic5×#23911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC23×C5×D3C23
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2347
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. C4⋊C4×Dic15C2semidirect
3. C3×C4⋊C4×Dic5C2semidirect
4. #11748868C2semidirect
5. #11748842C2semidirect
6. C4×Dic3⋊Dic5C2semidirect
11. #11748618C2semidirect
21. C5×#892C4semidirect
44. C4⋊C4×Dic5D3semidirect
169. #892Dic5semidirect
433. C3×C5#2347semidirect
513. C5#73634semidirect
538. C3#10613971semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#8922
C4⋊C4×Dic152
C3×C4⋊C4×Dic52
#117488682
#117488422
C4×Dic3⋊Dic52
#117487942
#117487862
#117487472
C4×(C2×D3)⋊Dic52
#117486182
#106139713
#736345