Finite Group Databasebeta

← #14321723

Subgroups — #14321723

Order 1920.

1376 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Dic5C211yesmaximal, characteristic, complemented
2C2×Dic5×#67C211yesmaximal, characteristic
3C2×#91539C211yesmaximal, characteristic
4#11751311C211yesmaximal, characteristic, complemented
5C5×#5213C211yesmaximal, characteristic
6Dic5×#217C211yesmaximal, characteristic, complemented
7#11749471C211yesmaximal, characteristic, complemented
8Dic5×(C2×D3)⋊C4C212yesmaximal, complemented
9#11748815C212yesmaximal, complemented
10#11748715C212yesmaximal, complemented
11#11748640C212yesmaximal, complemented
12#10606392—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×C5×#67C2211yescharacteristic
15C23×C3×Dic5C2211yescharacteristic
16C23×C3×Dic5C2211yescharacteristic
17C22×D3×Dic5C2211yescharacteristic
18C2×#6719C2211yescharacteristic
19C2×#6719C2211yescharacteristic
20C2×#6712C2212yes—
21C2×C5×#217C2211yescharacteristic
22C2×C5×(C2×D3)⋊C4C2212yes—
23C2×C3×C4×Dic5—24no—
24C2×C3×C4×Dic5C2212yes—
25C2×C3×C4×Dic5C2212yes—
26Dic5×#67—24no—
27C2×Dic3⋊Dic5C2211yescharacteristic
28C2×#6669C2212yes—
29#91539C418yescomplemented
30C2×(C2×D3)⋊Dic5C2211yescharacteristic
31C2×(C2×D3)⋊Dic5C2212yes—
32C2×Dic3×Dic5C2212yes—
33C2×Dic3×Dic5C2211yescharacteristic
34#91115C2212yes—
35C5×C3⋊#95C2212yes—
36#90994C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39C4×#929—51nomaximal, Hall
40C2×#4081—31no—
41C2×D4×Dic5—31no—
42C22×C4×Dic5D311yescharacteristic, complemented
43C5×#386—31no—
44#65499—31no—
45#65234—32no—
46#65232—31no—
47C22⋊C4×Dic5—32no—
48C22⋊C4×Dic5—31no—
49#64944—32no—
50#64938—32no—
51C24×C3×C5C2311yescharacteristic
52C23×C5×D3C2311yescharacteristic
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5—24no—
55C22×C3×C4×C5C2312yes—
56C22×Dic15C2311yescharacteristic
57C22×Dic15C2311yescharacteristic
58C22×Dic15C2312yes—
59C2×C5×#67—22no—
60C2×C5×#67—22no—
61C2×C5×#67C2×C414yes—
62C22×C5×Dic3C2312yes—
63C22×C5×Dic3C2311yescharacteristic
64C22×C3×Dic5—22no—
65C22×C3×Dic5—22no—
66C22×C3×Dic5—24no—
67C22×C3×Dic5C2×C414yescomplemented
68C22×C3×Dic5D414yes—
69C22×C3×Dic5C2312yes—
70C22×C3×Dic5C2311yescharacteristic
71C22×C3×Dic5C2311yescharacteristic
72C2×D3×Dic5—22no—
73C2×D3×Dic5—24no—
74#6719—22no—
75#6719C2×C414yescomplemented
76#6712—24no—
77C5×#217—22no—
78C5×(C2×D3)⋊C4—24no—
79C3×C4×Dic5—24no—
80C3×C4×Dic5—24no—
81Dic3⋊Dic5—22no—
82#6669C2×C418yescomplemented
83(C2×D3)⋊Dic5—22no—
84(C2×D3)⋊Dic5C2×C418yescomplemented
85Dic3×Dic5—22no—
86Dic3×Dic5—24no—
87C22×C3×C42—51no—
88C2×#929—51no—
89C2×C4×#67—51no—
90#5213D511yescharacteristic
91#5211—51no—
92#5210—51no—
93C4×#217—51no—
94#4940—52no—
95C4×(C2×D3)⋊C4—52no—
96#4933—52no—
97C4×Dic3⋊C4—52no—
98C22×C5×D4—31no—
99C23×C4×C5C2×D311yescharacteristic
100C23×Dic5—31no—
101C23×Dic5C2×D311yescharacteristic
102C23×Dic5C2×D311yescharacteristic
103C2×C5×C22⋊C4—31no—
104C2×C5×C22⋊C4—32no—
105C2×C22⋊Dic5—31no—
106C2×C22⋊Dic5—32no—
107C2×C22⋊Dic5—31no—
108C2×C22⋊Dic5—31no—
109#4081—38no—
110D4×Dic5—64no—
111C2×C4⋊Dic5—31no—
112C2×#683—32no—
113C2×#683—32no—
114C2×C4×Dic5—24no—
115C2×C4×Dic5—31no—
116C2×C4×Dic5—32no—
117C2×C4×Dic5C2×D312yes—
118C2×C4×Dic5C2×D312yes—
119C5×#95—32no—
120#3963—32no—
121#3963—32no—
122#3963—32no—
123#3963—32no—
124C4×#123—151noSylow
125C23×C3×C5—22no—
126C23×C3×C5—22no—
127C23×C3×C5C22×C412yes—
128C23×C3×C5C2×D412yes—
129C23×C3×C5#4112yes—
130C23×C3×C5C2411yescharacteristic
131C22×C5×D3—21no—
132C22×C5×D3—22no—
133C22×C5×D3—22no—
134C22×C5×D3—21no—
135C22×C5×D3C22×C412yes—
136C2×C3×C4×C5—28no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5#4114yes—
140C2×Dic15—22no—
141C2×Dic15—22no—
142C2×Dic15—21no—
143C2×Dic15—21no—
144C2×Dic15—22no—
145C2×Dic15—22no—
146C2×Dic15C22×C414yes—
147C5×#67—28no—
148C2×C5×Dic3—21no—
149C2×C5×Dic3—21no—
150C2×C5×Dic3—22no—
151C2×C5×Dic3—22no—
152C2×C5×Dic3—22no—
153C2×C5×Dic3C22×C412yes—
154C2×C3×Dic5—28no—
155C2×C3×Dic5—22no—
156C2×C3×Dic5—22no—
157C2×C3×Dic5—22no—
158C2×C3×Dic5—28no—
159C2×C3×Dic5C22×C414yes—
160C2×C3×Dic5C2×D412yes—
161C2×C3×Dic5C2×D412yes—
162C2×C3×Dic5#4114yes—
163D3×Dic5—44no—
164C23×C3×C4—51no—
165C23×C3×C4—51no—
166C23×C3×C4C2×D511yescharacteristic
167C22×#67C2×D511yescharacteristic
168C22×C4×D3—51no—
169C2×C3×C42—52no—
170C2×C3×C42—52no—
171C2×C3×C42—104no—
172C2×#217—51no—
173C2×#217—51no—
174C2×#217C2×D511yescharacteristic
175#929—58no—
176C4×#67—104no—
177C2×(C2×D3)⋊C4—52no—
178C2×(C2×D3)⋊C4—51no—
179C2×(C2×D3)⋊C4C2×D512yes—
180C2×Dic3⋊C4—52no—
181C2×Dic3⋊C4—52no—
182C2×Dic3⋊C4—51no—
183C2×C4×Dic3—51no—
184C2×C4×Dic3—52no—
185C3⋊#95—52no—
186C3⋊#95—52no—
187C3⋊#95—52no—
188C3⋊#95—52no—
189C3⋊#95C2×D512yes—
190C24×C5—31no—
191C24×C5C22×D311yescharacteristic
192C2×C5×D4—34no—
193C2×C5×D4—62no—
194C2×C5×D4—62no—
195C22×C4×C5—22no—
196C22×C4×C5—24no—
197C22×C4×C5—31no—
198C22×C4×C5—32no—
199C22×C4×C5C22×D312yes—
200C22×Dic5—22no—
201C22×Dic5—22no—
202C22×Dic5—24no—
203C22×Dic5—32no—
204C22×Dic5—31no—
205C22×Dic5—31no—
206C22×Dic5—64no—
207C22×Dic5—62no—
208C22×Dic5C4×D314yescomplemented
209C22×Dic5#6714yes—
210C22×Dic5C22×D312yes—
211C22×Dic5C22×D311yescharacteristic
212C22×Dic5C22×D311yescharacteristic
213C5×C22⋊C4—62no—
214C5×C22⋊C4—64no—
215C22⋊Dic5—34no—
216C22⋊Dic5—38no—
217C22⋊Dic5—62no—
218C22⋊Dic5—62no—
219C4⋊Dic5—62no—
220#683—38no—
221#683—64no—
222C4×Dic5—24no—
223C4×Dic5—24no—
224C4×Dic5—64no—
225C4×Dic5—62no—
226C2×#123—151no—
227C2×C4×D4—151no—
228C22×C42—51no—
229#388—152no—
230#387—152no—
231#386—31no—
232#385—151no—
233#381—151no—
234C4×C4⋊C4—152no—
235C4×C22⋊C4—152no—
236C4×C22⋊C4—151no—
237C22×C3×C5—24no—
238C22×C3×C5—22no—
239C22×C3×C5—22no—
240C22×C3×C5—24no—
241C22×C3×C5C4×D414yes—
242C22×C3×C5C23×C411yescharacteristic
243C22×C3×C5C22×D411yescharacteristic
244C22×C3×C5C2×#4111yescharacteristic
245C22×C3×C5C2×#4112yes—
246C22×C3×C5C2×#4112yes—
247C2×C5×D3—22no—
248C2×C5×D3—24no—
249C2×C5×D3—41no—
250C2×C5×D3—41no—
251C2×C5×D3—42no—
252C3×C4×C5—24no—
253Dic15—24no—
254Dic15—41no—
255Dic15—41no—
256C3×Dic5—24no—
257C3×Dic5—22no—
258C3×Dic5#12314yescomplemented
259C5×Dic3—22no—
260C5×Dic3—42no—
261C24×C3C22×D511yescharacteristic
262C23×D3C22×D511yescharacteristic
263C22×C3×C4—22no—
264C22×C3×C4—24no—
265C22×C3×C4—51no—
266C22×C3×C4—54no—
267C22×C3×C4—54no—
268C22×C3×C4—52no—
269C22×C3×C4—51no—
270C22×C3×C4—104no—
271C22×C3×C4—102no—
272C22×C3×C4—102no—
273C22×C3×C4C22×D512yes—
274C2×#67—22no—
275C2×#67—22no—
276C2×#67C4×D514yes—
277C22×Dic3—52no—
278C22×Dic3—51no—
279C22×Dic3—51no—
280C22×Dic3C22×D512yes—
281C22×Dic3C22×D511yescharacteristic
282C2×C4×D3—102no—
283C2×C4×D3—104no—
284C3×C42—104no—
285C3×C42—104no—
286#217—22no—
287#217—54no—
288#217—102no—
289(C2×D3)⋊C4—24no—
290(C2×D3)⋊C4—58no—
291(C2×D3)⋊C4—102no—
292Dic3⋊C4—58no—
293Dic3⋊C4—102no—
294Dic3⋊C4—104no—
295C4×Dic3—104no—
296C4×Dic3—102no—
297C23×C5—22no—
298C23×C5—22no—
299C23×C5—32no—
300C23×C5—62no—
301C23×C5—62no—
302C23×C5—61no—
303C23×C5—61no—
304C23×C5C2×C4×D312yes—
305C23×C5#23512yes—
306C23×C5C2×#6712yes—
307C23×C5C23×D311yescharacteristic
308C5×D4—68no—
309C2×C4×C5—28no—
310C2×C4×C5—22no—
311C2×C4×C5—22no—
312C2×C4×C5—32no—
313C2×C4×C5—61no—
314C2×C4×C5—61no—
315C2×C4×C5—62no—
316C2×C4×C5—62no—
317C2×C4×C5—62no—
318C2×C4×C5#23514yes—
319C2×Dic5—28no—
320C2×Dic5—22no—
321C2×Dic5—22no—
322C2×Dic5—22no—
323C2×Dic5—28no—
324C2×Dic5—34no—
325C2×Dic5—62no—
326C2×Dic5—62no—
327C2×Dic5—62no—
328C2×Dic5—62no—
329C2×Dic5—61no—
330C2×Dic5—61no—
331C2×Dic5—124no—
332C2×Dic5C2×C4×D314yes—
333C2×Dic5#23514yes—
334C2×Dic5C2×#6712yes—
335C2×Dic5C2×#6712yes—
336C22×D4—31no—
337C23×C4—51no—
338C23×C4—51no—
339C23×C4—151no—
340C23×C4D3×D511yescharacteristic
341#123—158no—
342C4×D4—304no—
343C2×C4⋊C4—152no—
344C2×C4⋊C4—151no—
345C2×C4⋊C4—152no—
346C2×C22⋊C4—31no—
347C2×C22⋊C4—32no—
348C2×C22⋊C4—152no—
349C2×C22⋊C4—151no—
350C2×C22⋊C4—151no—
351C2×C22⋊C4—151no—
352C2×C42—52no—
353C2×C42—52no—
354C2×C42—104no—
355C2×C42—151no—
356C2×C42—152no—
357#95—32no—
358#95—152no—
359#95—152no—
360#95—152no—
361#95—152no—
362C2×C3×C5—24no—
363C2×C3×C5C2×C4×D411yescharacteristic
364C2×C3×C5C4×#4112yes—
365C2×C3×C5C2×#12311yescharacteristic
366C2×C3×C5#52511yescharacteristic
367C2×C3×C5#52912yes—
368C5×D3—42no—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3C2×C4×D512yes—
372C23×C3D4×D512yes—
373C23×C3#71112yes—
374C23×C3C23×D511yescharacteristic
375C22×D3—21no—
376C22×D3—22no—
377C22×D3—21no—
378C22×D3—22no—
379C22×D3C2×C4×D512yes—
380C2×C3×C4—28no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—52no—
384C2×C3×C4—54no—
385C2×C3×C4—54no—
386C2×C3×C4—52no—
387C2×C3×C4—108no—
388C2×C3×C4—102no—
389C2×C3×C4—102no—
390C2×C3×C4—102no—
391C2×C3×C4—108no—
392C2×C3×C4#71114yes—
393#67—28no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—54no—
400C2×Dic3—102no—
401C2×Dic3—102no—
402C2×Dic3—101no—
403C2×Dic3—101no—
404C2×Dic3—102no—
405C2×Dic3—102no—
406C2×Dic3C2×C4×D512yes—
407C4×D3—204no—
408C22×C5—24no—
409C22×C5—24no—
410C22×C5—22no—
411C22×C5—22no—
412C22×C5—64no—
413C22×C5—62no—
414C22×C5—122no—
415C22×C5—121no—
416C22×C5—121no—
417C22×C5C4×#6714yes—
418C22×C5C22×C4×D311yescharacteristic
419C22×C5C2×#23511yescharacteristic
420C22×C5C2×#23512yes—
421C22×C5C2×#23512yes—
422C22×C5C22×#6711yescharacteristic
423C4×C5—24no—
424C4×C5—62no—
425C4×C5—122no—
426Dic5—24no—
427Dic5—22no—
428Dic5—64no—
429Dic5—121no—
430Dic5—121no—
431Dic5#92914yescomplemented
432C24—31no—
433C24C2×D3×D511yescharacteristic
434C2×D4—34no—
435C2×D4—62no—
436C2×D4—62no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—31no—
440C22×C4—32no—
441C22×C4—51no—
442C22×C4—54no—
443C22×C4—54no—
444C22×C4—52no—
445C22×C4—51no—
446C22×C4—104no—
447C22×C4—102no—
448C22×C4—102no—
449C22×C4—152no—
450C22×C4—151no—
451C22×C4—151no—
452C22×C4—302no—
453C22×C4—304no—
454C22×C4C2×D3×D512yes—
455C4⋊C4—158no—
456C4⋊C4—302no—
457C4⋊C4—304no—
458C22⋊C4—62no—
459C22⋊C4—64no—
460C22⋊C4—158no—
461C22⋊C4—154no—
462C22⋊C4—302no—
463C22⋊C4—302no—
464C42—104no—
465C42—104no—
466C42—302no—
467C42—304no—
468C3×C5C4×#12311yescharacteristic, complemented, Hall
469C22×C3—24no—
470C22×C3—22no—
471C22×C3—24no—
472C22×C3—22no—
473C22×C3#402714yes—
474C22×C3C22×C4×D511yescharacteristic
475C22×C3C2×D4×D511yescharacteristic
476C22×C3C2×#71111yescharacteristic
477C22×C3C2×#71112yes—
478C22×C3D5×#4112yes—
479C2×D3—22no—
480C2×D3—24no—
481C2×D3—41no—
482C2×D3—42no—
483C2×D3—41no—
484C3×C4—24no—
485C3×C4—54no—
486C3×C4—104no—
487C3×C4—102no—
488Dic3—22no—
489Dic3—42no—
490Dic3—104no—
491Dic3—201no—
492Dic3—201no—
493C2×C5—24no—
494C2×C5—122no—
495C2×C5C4×#23512yes—
496C2×C5#547912yes—
497C2×C5C2×C4×#6711yescharacteristic
498C2×C5C2×#92911yescharacteristic
499C2×C5#579111yescharacteristic
500C23—22no—
501C23—22no—
502C23—32no—
503C23—61no—
504C23—62no—
505C23—62no—
506C23—61no—
507C23#677212yes—
508C23C4×D3×D512yes—
509C23D5×#6712yes—
510C23C22×D3×D511yescharacteristic
511D4—68no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—32no—
516C2×C4—54no—
517C2×C4—52no—
518C2×C4—54no—
519C2×C4—52no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—108no—
526C2×C4—108no—
527C2×C4—102no—
528C2×C4—102no—
529C2×C4—102no—
530C2×C4—154no—
531C2×C4—302no—
532C2×C4—302no—
533C2×C4—302no—
534C2×C4—302no—
535C2×C4—301no—
536C2×C4—301no—
537C2×C4—604no—
538C2×C4#678214yes—
539C2×C3—24no—
540C2×C3C2×#402711yescharacteristic
541C2×C3C4×#71112yes—
542C2×C3#6592011yescharacteristic
543C2×C3#6593412yes—
544C2×C3D5×#12311yescharacteristic
545D3—42no—
546C5C4×#92911yescharacteristic, complemented, Sylow
547C22—24no—
548C22—22no—
549C22—22no—
550C22—24no—
551C22—62no—
552C22—64no—
553C22—121no—
554C22—121no—
555C22—122no—
556C22#9144014yes—
557C22C2×#677211yescharacteristic
558C22C2×C4×D3×D511yescharacteristic
559C22D5×#23512yes—
560C22C2×#678212yes—
561C22C2×D5×#6711yescharacteristic
562C4—24no—
563C4—54no—
564C4—62no—
565C4—104no—
566C4—102no—
567C4—122no—
568C4—304no—
569C4—601no—
570C4—601no—
571C3#1060639211yescharacteristic, complemented, Sylow
572C2—24no—
573C2—122no—
574C2C4×#678212yes—
575C2#1175441012yes—
576C2C2×#9144011yescharacteristic
577C2D5×#92911yescharacteristic
578C2#1175538511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×C5×#67C22
hypercenterC23×C4D3×D5
hypocenterC3×C5C4×#123
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. C22×C3×C4×Dic5C2semidirect
4. #11751311C2semidirect
6. Dic5×#217C2semidirect
7. #11749471C2semidirect
8. Dic5×(C2×D3)⋊C4C2semidirect
9. #11748815C2semidirect
10. #11748715C2semidirect
11. #11748640C2semidirect
29. #91539C4semidirect
42. C22×C4×Dic5D3semidirect
67. C22×C3×Dic5C2×C4semidirect
75. #6719C2×C4semidirect
82. #6669C2×C4semidirect
84. (C2×D3)⋊Dic5C2×C4semidirect
208. C22×Dic5C4×D3semidirect
258. C3×Dic5#123semidirect
431. Dic5#929semidirect
468. C3×C5C4×#123semidirect
546. C5C4×#929semidirect
571. C3#10606392semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×Dic52
C2×Dic5×#672
C2×#915392
#117513112
C5×#52132
Dic5×#2172
#117494712
Dic5×(C2×D3)⋊C42
#117488152
#117487152
#117486402
#106063923
C4×#9295