Finite Group Databasebeta

← #14327706

Subgroups — #14327706

Order 1920.

1264 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#926C211yesmaximal, characteristic, complemented
2C2×C3×C4×C22⋊D5C211yesmaximal, characteristic
3C2×#91358C211yesmaximal, characteristic, complemented
4#11751313C211yesmaximal, characteristic, complemented
5#11749489C212yesmaximal
6#11749485C212yesmaximal, complemented
7#11748834C211yesmaximal, characteristic, complemented
8#11748770C212yesmaximal, complemented
9#11748759C211yesmaximal, characteristic, complemented
10Dic5×C4⋊Dic3C211yesmaximal, characteristic, complemented
11Dic3×#683C212yesmaximal, complemented
12#10613972—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×C3×C22⋊D5C2211yescharacteristic
15C22×C3×C4×D5C2211yescharacteristic
16C2×#6719C2212yes—
17C2×C4⋊Dic15C2211yescharacteristic
18C2×C4⋊Dic15C2211yescharacteristic
19C2×C5×#217C2212yes—
20C5×#926—24no—
21C2×C5×C4⋊Dic3C2211yescharacteristic
22C2×C5×C4⋊Dic3C2211yescharacteristic
23C2×C4×C5×Dic3C2212yes—
24C2×C3×C22⋊Dic5C2211yescharacteristic
25C3×C4×C22⋊D5C418yescomplemented
26C2×C3×#685C2211yescharacteristic
27C2×C3×#683C2211yescharacteristic
28C2×C3×C4×Dic5C2211yescharacteristic
29C2×#6668C2212yes—
30C2×#6668C2212yes—
31C2×#6665C2212yes—
32C2×#6665C2212yes—
33C2×Dic3×Dic5C2212yes—
34C2×Dic3×Dic5C2212yes—
35#91358—24no—
36#91115C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#82337—51nomaximal, Hall
40C2×C5×#117—31no—
41C2×C4×C22⋊D5D311yescharacteristic
42C2×#4038—31no—
43#65501—31no—
44#65276—31no—
45#65265—31no—
46C4⋊C4×Dic5—31no—
47#65246—32no—
48#65233—32no—
49#65229—32no—
50C4×#683—32no—
51C24×C3×C5C2311yescharacteristic
52C23×C3×D5C2311yescharacteristic
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5—22no—
55C22×C3×C4×C5C2×C414yes—
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C22×Dic15C2312yes—
59C22×Dic15C2312yes—
60C22×C5×Dic3C2312yes—
61C22×C5×Dic3C2312yes—
62C2×C3×C22⋊D5—22no—
63C2×C3×C22⋊D5—22no—
64C2×C3×C22⋊D5C2×C414yes—
65C22×C3×Dic5C2311yescharacteristic
66C22×C3×Dic5C2311yescharacteristic
67C22×C3×Dic5C2311yescharacteristic
68C2×C3×C4×D5—22no—
69C2×C3×C4×D5—22no—
70C2×C3×C4×D5C2×C414yes—
71#6719—42no—
72C4⋊Dic15—22no—
73C4⋊Dic15—22no—
74C5×#217—24no—
75C5×C4⋊Dic3—22no—
76C5×C4⋊Dic3D414yescomplemented
77C4×C5×Dic3—24no—
78C3×C22⋊Dic5C2×C414yes—
79C3×#685C2×C414yes—
80C3×#683C2×C414yes—
81C3×C4×Dic5C2×C414yes—
82#6668—24no—
83#6668—42no—
84#6665—24no—
85#6665—42no—
86Dic3×Dic5—24no—
87Dic3×Dic5—24no—
88C2×C3×C4×D4—51no—
89C2×#926—51no—
90C2×#926D511yescharacteristic, complemented
91#5221—52no—
92#5212—51no—
93#5212—51no—
94Dic3×C4⋊C4—52no—
95#4951—52no—
96#4951—52no—
97#4935—51no—
98C4×C4⋊Dic3—51no—
99C23×C4×C5C2×D311yescharacteristic
100C22×C22⋊D5C2×D311yescharacteristic
101C22×C4×D5C2×D311yescharacteristic
102C5×#117—64no—
103C2×C5×C4⋊C4—31no—
104C2×C5×C4⋊C4—31no—
105C2×C5×C22⋊C4—32no—
106C2×C42×C5—32no—
107C2×C22⋊Dic5—32no—
108C2×C22⋊Dic5C2×D311yescharacteristic
109C4×C22⋊D5Dic318yescomplemented
110C2×#685—32no—
111C2×#685—32no—
112C2×#685C2×D311yescharacteristic
113C2×C4⋊Dic5—31no—
114C2×C4⋊Dic5—31no—
115C2×#683—32no—
116C2×#683—32no—
117C2×#683C2×D311yescharacteristic
118C2×C4×Dic5—32no—
119C2×C4×Dic5—32no—
120C2×C4×Dic5C2×D311yescharacteristic
121#4038—64no—
122#3963—32no—
123#3963—32no—
124#3963—32no—
125#2354—151noSylow
126C23×C3×C5—21no—
127C23×C3×C5—21no—
128C23×C3×C5—21no—
129C23×C3×C5—21no—
130C23×C3×C5—21no—
131C23×C3×C5—21no—
132C23×C3×C5C22×C412yes—
133C23×C3×C5C2411yescharacteristic
134C22×C3×D5—21no—
135C22×C3×D5—21no—
136C22×C3×D5—21no—
137C22×C3×D5—21no—
138C22×C3×D5—21no—
139C22×C3×D5—21no—
140C22×C3×D5C22×C412yes—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5—28no—
144C2×C3×C4×C5C22×C412yes—
145C2×C3×C4×C5C22×C412yes—
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5#4112yes—
148C2×Dic15—22no—
149C2×Dic15—22no—
150C2×Dic15—22no—
151C2×Dic15—22no—
152C2×Dic15—22no—
153C2×Dic15—22no—
154C2×C5×Dic3—22no—
155C2×C5×Dic3—22no—
156C2×C5×Dic3—22no—
157C2×C5×Dic3—22no—
158C2×C5×Dic3—22no—
159C2×C5×Dic3C2×D414yes—
160C3×C22⋊D5—28no—
161C2×C3×Dic5—21no—
162C2×C3×Dic5—21no—
163C2×C3×Dic5—21no—
164C2×C3×Dic5—21no—
165C2×C3×Dic5C22×C412yes—
166C2×C3×Dic5C22×C412yes—
167C2×C3×Dic5C22×C412yes—
168C2×C3×Dic5#4112yes—
169C2×C3×Dic5#4112yes—
170C3×C4×D5—28no—
171C22×C3×D4—51no—
172C23×C3×C4—51no—
173C23×C3×C4C2×D511yescharacteristic
174C3×C4×D4—58no—
175C2×C3×C4⋊C4—51no—
176C2×C3×C22⋊C4—51no—
177C2×C3×C22⋊C4—51no—
178C2×C3×C42—51no—
179C2×#217—52no—
180C2×#217—52no—
181C2×#217—52no—
182C2×#217C2×D512yes—
183#926—24no—
184#926—104no—
185C2×C4⋊Dic3—51no—
186C2×C4⋊Dic3—51no—
187C2×C4⋊Dic3—52no—
188C2×C4⋊Dic3—52no—
189C2×C4⋊Dic3C2×D511yescharacteristic
190C2×C4⋊Dic3C2×D511yescharacteristic
191C2×C4×Dic3—52no—
192C2×C4×Dic3—52no—
193C2×C4×Dic3C2×D512yes—
194C3⋊#95—52no—
195C3⋊#95—52no—
196C3⋊#95—52no—
197C24×C5C22×D311yescharacteristic
198C23×D5C22×D311yescharacteristic
199C22×C4×C5—22no—
200C22×C4×C5—22no—
201C22×C4×C5—32no—
202C22×C4×C5—32no—
203C22×C4×C5C2×Dic314yes—
204C22×C4×C5C22×D311yescharacteristic
205C22×C4×C5C22×D311yescharacteristic
206C2×C22⋊D5—22no—
207C2×C22⋊D5—22no—
208C2×C22⋊D5C2×Dic314yes—
209C22×Dic5—32no—
210C22×Dic5—32no—
211C22×Dic5C22×D311yescharacteristic
212C22×Dic5C22×D311yescharacteristic
213C22×Dic5C22×D311yescharacteristic
214C2×C4×D5—22no—
215C2×C4×D5—22no—
216C2×C4×D5C2×Dic314yes—
217C5×C4⋊C4—34no—
218C5×C4⋊C4—62no—
219C5×C22⋊C4—64no—
220C42×C5—64no—
221C22⋊Dic5—122no—
222C22⋊Dic5C2×Dic314yes—
223#685—64no—
224#685—122no—
225#685C2×Dic314yes—
226C4⋊Dic5—62no—
227C4⋊Dic5—62no—
228#683—64no—
229#683—122no—
230#683C2×Dic314yes—
231C4×Dic5—64no—
232C4×Dic5—64no—
233C4×Dic5C2×Dic314yes—
234C2×C4×D4—51no—
235C2×#117—31no—
236C2×#117—151no—
237#390—152no—
238#388—152no—
239#388—152no—
240#384—151no—
241#380—151no—
242#380—151no—
243C4×C4⋊C4—152no—
244C4×C4⋊C4—151no—
245C22×C3×C5—22no—
246C22×C3×C5—22no—
247C22×C3×C5—22no—
248C22×C3×C5—41no—
249C22×C3×C5—41no—
250C22×C3×C5—41no—
251C22×C3×C5—41no—
252C22×C3×C5C23×C411yescharacteristic
253C22×C3×C5C22×D411yescharacteristic
254C22×C3×C5C2×#4111yescharacteristic
255C22×C3×C5C2×#4111yescharacteristic
256C22×C3×C5C2×#4111yescharacteristic
257C22×C3×C5H(5,𝔽2)11yescharacteristic
258C22×C3×C52−1+411yescharacteristic
259C2×C3×D5—22no—
260C2×C3×D5—22no—
261C2×C3×D5—22no—
262C2×C3×D5—41no—
263C2×C3×D5—41no—
264C2×C3×D5—41no—
265C2×C3×D5—41no—
266C3×C4×C5—22no—
267C3×C4×C5C4×D414yes—
268Dic15—42no—
269Dic15—42no—
270C3×Dic5—22no—
271C3×Dic5—22no—
272C3×Dic5#11714yescomplemented
273C5×Dic3—24no—
274C5×Dic3—42no—
275C24×C3—51no—
276C24×C3C22×D511yescharacteristic
277C2×C3×D4—54no—
278C2×C3×D4—102no—
279C2×C3×D4—102no—
280C22×C3×C4—22no—
281C22×C3×C4—22no—
282C22×C3×C4—51no—
283C22×C3×C4—51no—
284C22×C3×C4—51no—
285C22×C3×C4—54no—
286C22×C3×C4—102no—
287C22×C3×C4—102no—
288C22×C3×C4C4×D514yes—
289C22×C3×C4C22×D511yescharacteristic
290C22×C3×C4C22×D511yescharacteristic
291C22×Dic3—52no—
292C22×Dic3—52no—
293C22×Dic3C22×D512yes—
294C22×Dic3C22×D512yes—
295C3×C4⋊C4—54no—
296C3×C22⋊C4—54no—
297C3×C22⋊C4—54no—
298C3×C42—54no—
299#217—24no—
300#217—104no—
301#217—202no—
302#217—202no—
303C4⋊Dic3—22no—
304C4⋊Dic3—104no—
305C4⋊Dic3—102no—
306C4⋊Dic3—102no—
307C4⋊Dic3—202no—
308C4⋊Dic3C22⋊D514yescomplemented
309C4×Dic3—24no—
310C4×Dic3—104no—
311C4×Dic3—104no—
312C23×C5—21no—
313C23×C5—21no—
314C23×C5—21no—
315C23×C5—21no—
316C23×C5—21no—
317C23×C5—21no—
318C23×C5C22×Dic312yes—
319C23×C5C23×D311yescharacteristic
320C22×D5—21no—
321C22×D5—21no—
322C22×D5—21no—
323C22×D5—21no—
324C22×D5—21no—
325C22×D5—21no—
326C22×D5C22×Dic312yes—
327C2×C4×C5—21no—
328C2×C4×C5—21no—
329C2×C4×C5—28no—
330C2×C4×C5—34no—
331C2×C4×C5—62no—
332C2×C4×C5—62no—
333C2×C4×C5—62no—
334C2×C4×C5—62no—
335C2×C4×C5—62no—
336C2×C4×C5D3×D412yes—
337C2×C4×C5#23712yes—
338C2×C4×C5C22×Dic312yes—
339C2×C4×C5C22×Dic312yes—
340C22⋊D5—28no—
341C2×Dic5—21no—
342C2×Dic5—21no—
343C2×Dic5—21no—
344C2×Dic5—21no—
345C2×Dic5—62no—
346C2×Dic5—62no—
347C2×Dic5—62no—
348C2×Dic5—62no—
349C2×Dic5—62no—
350C2×Dic5—62no—
351C2×Dic5#23512yes—
352C2×Dic5#23512yes—
353C2×Dic5C22×Dic312yes—
354C2×Dic5C22×Dic312yes—
355C2×Dic5C22×Dic312yes—
356C4×D5—28no—
357C22×D4—51no—
358C23×C4—51no—
359C23×C4D3×D511yescharacteristic
360C4×D4—58no—
361#117—64no—
362#117—304no—
363C2×C4⋊C4—31no—
364C2×C4⋊C4—31no—
365C2×C4⋊C4—51no—
366C2×C4⋊C4—152no—
367C2×C4⋊C4—151no—
368C2×C4⋊C4—151no—
369C2×C4⋊C4—152no—
370C2×C22⋊C4—32no—
371C2×C22⋊C4—51no—
372C2×C22⋊C4—51no—
373C2×C22⋊C4—152no—
374C2×C22⋊C4—152no—
375C2×C22⋊C4—152no—
376C2×C42—32no—
377C2×C42—51no—
378C2×C42—152no—
379C2×C42—152no—
380#95—152no—
381#95—152no—
382#95—152no—
383C2×C3×C5—42no—
384C2×C3×C5C2×#11711yescharacteristic
385C2×C3×C5C2×C4×D411yescharacteristic
386C2×C3×C5#52011yescharacteristic
387C2×C3×C5#54711yescharacteristic
388C2×C3×C5#55011yescharacteristic
389C2×C3×C5#55511yescharacteristic
390C2×C3×C5#55611yescharacteristic
391C3×D5—42no—
392C23×C3—21no—
393C23×C3—21no—
394C23×C3—21no—
395C23×C3—21no—
396C23×C3—21no—
397C23×C3—21no—
398C23×C3—52no—
399C23×C3—101no—
400C23×C3—101no—
401C23×C3—101no—
402C23×C3—101no—
403C23×C3—101no—
404C23×C3—101no—
405C23×C3C2×C4×D512yes—
406C23×C3C23×D511yescharacteristic
407C3×D4—108no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—28no—
411C2×C3×C4—52no—
412C2×C3×C4—52no—
413C2×C3×C4—52no—
414C2×C3×C4—52no—
415C2×C3×C4—52no—
416C2×C3×C4—101no—
417C2×C3×C4—101no—
418C2×C3×C4—101no—
419C2×C3×C4—101no—
420C2×C3×C4—108no—
421C2×C3×C4C2×C4×D512yes—
422C2×C3×C4C2×C4×D512yes—
423C2×C3×C4#70912yes—
424C2×C3×C4C2×C22⋊D512yes—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—102no—
431C2×Dic3—102no—
432C2×Dic3—102no—
433C2×Dic3—102no—
434C2×Dic3—102no—
435C2×Dic3—102no—
436C2×Dic3C2×C22⋊D514yes—
437C22×C5—22no—
438C22×C5—22no—
439C22×C5—22no—
440C22×C5—41no—
441C22×C5—41no—
442C22×C5—41no—
443C22×C5—41no—
444C22×C5C2×#23511yescharacteristic
445C22×C5C2×#23511yescharacteristic
446C22×C5C2×D3×D411yescharacteristic
447C22×C5C2×#23711yescharacteristic
448C22×C5#100911yescharacteristic
449C22×C5#101011yescharacteristic
450C22×C5C23×Dic311yescharacteristic
451C2×D5—22no—
452C2×D5—22no—
453C2×D5—22no—
454C2×D5—41no—
455C2×D5—41no—
456C2×D5—41no—
457C2×D5—41no—
458C4×C5—22no—
459C4×C5—64no—
460C4×C5—122no—
461C4×C5D4×Dic314yes—
462Dic5—22no—
463Dic5—22no—
464Dic5—122no—
465Dic5—122no—
466Dic5#92614yescomplemented
467C24—51no—
468C24C2×D3×D511yescharacteristic
469C2×D4—54no—
470C2×D4—102no—
471C2×D4—102no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—32no—
475C22×C4—32no—
476C22×C4—51no—
477C22×C4—51no—
478C22×C4—51no—
479C22×C4—54no—
480C22×C4—102no—
481C22×C4—102no—
482C22×C4—152no—
483C22×C4—152no—
484C22×C4D5×Dic314yes—
485C22×C4C2×D3×D511yescharacteristic
486C22×C4C2×D3×D511yescharacteristic
487C4⋊C4—34no—
488C4⋊C4—54no—
489C4⋊C4—62no—
490C4⋊C4—304no—
491C4⋊C4—302no—
492C4⋊C4—302no—
493C4⋊C4—602no—
494C22⋊C4—54no—
495C22⋊C4—54no—
496C22⋊C4—64no—
497C22⋊C4—304no—
498C22⋊C4—602no—
499C22⋊C4—602no—
500C42—54no—
501C42—64no—
502C42—304no—
503C42—304no—
504C3×C5#235411yescharacteristic, complemented, Hall
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—102no—
513C22×C3—102no—
514C22×C3—102no—
515C22×C3—201no—
516C22×C3—201no—
517C22×C3—201no—
518C22×C3—201no—
519C22×C3C22×C4×D511yescharacteristic
520C22×C3C2×#70911yescharacteristic
521C22×C3D5×#4111yescharacteristic
522C22×C3D5×#4111yescharacteristic
523C22×C3#414911yescharacteristic
524C22×C3#415011yescharacteristic
525C22×C3C22×C22⋊D511yescharacteristic
526C3×C4—22no—
527C3×C4—54no—
528C3×C4—102no—
529C3×C4—102no—
530C3×C4C4×C22⋊D514yes—
531Dic3—24no—
532Dic3—42no—
533Dic3—202no—
534Dic3—202no—
535C2×C5—42no—
536C2×C5#554211yescharacteristic
537C2×C5#555111yescharacteristic
538C2×C5#555311yescharacteristic
539C2×C5#556311yescharacteristic
540C2×C5C2×#92611yescharacteristic
541C2×C5C2×D4×Dic311yescharacteristic
542C2×C5#582711yescharacteristic
543D5—42no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—52no—
551C23—101no—
552C23—101no—
553C23—101no—
554C23—101no—
555C23—101no—
556C23—101no—
557C23C2×D5×Dic312yes—
558C23C22×D3×D511yescharacteristic
559D4—108no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—28no—
563C2×C4—34no—
564C2×C4—52no—
565C2×C4—52no—
566C2×C4—52no—
567C2×C4—52no—
568C2×C4—52no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—101no—
575C2×C4—101no—
576C2×C4—101no—
577C2×C4—101no—
578C2×C4—108no—
579C2×C4—302no—
580C2×C4—302no—
581C2×C4—302no—
582C2×C4—302no—
583C2×C4—302no—
584C2×C4—302no—
585C2×C4C2×D5×Dic312yes—
586C2×C4C2×D5×Dic312yes—
587C2×C4#677912yes—
588C2×C4D3×C22⋊D512yes—
589C2×C3—42no—
590C2×C3—202no—
591C2×C3D5×#11711yescharacteristic
592C2×C3#6585711yescharacteristic
593C2×C3#6586611yescharacteristic
594C2×C3#6586711yescharacteristic
595C2×C3C2×C4×C22⋊D511yescharacteristic
596C2×C3#6615811yescharacteristic
597C2×C3#6616811yescharacteristic
598C5#8233711yescharacteristic, complemented, Sylow
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—102no—
607C22—102no—
608C22—102no—
609C22—201no—
610C22—201no—
611C22—201no—
612C22—201no—
613C22#9200111yescharacteristic
614C22D5×#23511yescharacteristic
615C22D5×#23511yescharacteristic
616C22#9201911yescharacteristic
617C22C22×D5×Dic311yescharacteristic
618C22C2×#677911yescharacteristic
619C22C2×D3×C22⋊D511yescharacteristic
620C4—22no—
621C4—54no—
622C4—64no—
623C4—102no—
624C4—102no—
625C4—122no—
626C4—602no—
627C4—602no—
628C4Dic3×C22⋊D514yes—
629C3#1061397211yescharacteristic, complemented, Sylow
630C2—42no—
631C2—202no—
632C2#1175422811yescharacteristic
633C2#1175423011yescharacteristic
634C2D5×#92611yescharacteristic
635C2#1175450911yescharacteristic
636C2#1175454111yescharacteristic
637C2#1175454511yescharacteristic
638C2C2×Dic3×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×C3×C22⋊D5C22
hypercenterC23×C4D3×D5
hypocenterC3×C5#2354
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C2×C5×#926C2semidirect
3. C2×#91358C2semidirect
4. #11751313C2semidirect
6. #11749485C2semidirect
7. #11748834C2semidirect
8. #11748770C2semidirect
9. #11748759C2semidirect
10. Dic5×C4⋊Dic3C2semidirect
11. Dic3×#683C2semidirect
25. C3×C4×C22⋊D5C4semidirect
76. C5×C4⋊Dic3D4semidirect
90. C2×#926D5semidirect
109. C4×C22⋊D5Dic3semidirect
272. C3×Dic5#117semidirect
308. C4⋊Dic3C22⋊D5semidirect
466. Dic5#926semidirect
504. C3×C5#2354semidirect
598. C5#82337semidirect
629. C3#10613972semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9262
C2×C3×C4×C22⋊D52
C2×#913582
#117513132
#117494892
#117494852
#117488342
#117487702
#117487592
Dic5×C4⋊Dic32
Dic3×#6832
#106139723
#823375