Finite Group Databasebeta

← #14415027

Subgroups — #14415027

Order 1920.

1040 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×#170C211yesmaximal, characteristic, complemented
2C2×#91927C211yesmaximal, characteristic, complemented
3#11755385C211yesmaximal, characteristic
4#11752457C211yesmaximal, characteristic, complemented
5#11752046C212yesmaximal, complemented
6#11752044C212yesmaximal, complemented
7#11752036C212yesmaximal, complemented
8#67×#170C212yesmaximal, complemented
9#11752011C211yesmaximal, characteristic, complemented
10#11751999C211yesmaximal, characteristic, complemented
11#11751988C211yesmaximal, characteristic, complemented
12#10617215—31nomaximal, Hall
13C23×C3×Dic5C2211yescharacteristic
14C2×#6782C2211yescharacteristic
15C2×#6762C2211yescharacteristic
16C2×#6762C2211yescharacteristic
17C22×C3×#170—24no—
18C22×C3×#170C2211yescharacteristic
19C22×C3×#170C2212yes—
20C22×C3×#170C2212yes—
21C22×C3×#170C2211yescharacteristic
22C2×#6740C2211yescharacteristic, complemented
23C2×#6739C2211yescharacteristic, complemented
24C2×#6738C2211yescharacteristic, complemented
25#91927—24no—
26C2×D3×#170C2211yescharacteristic, complemented
27#91842C411yescharacteristic
28C5×#953C411yescharacteristic
29#91574C412yes—
30#91566C412yes—
31#91552C2212yes—
32Dic5×#67C2212yes—
33#91546C411yescharacteristic
34#91539C411yescharacteristic
35#91528C2211yescharacteristic
36#91241C2212yes—
37#91241C2212yes—
38#91177C2212yescomplemented
39Dic3⋊#170C2212yescomplemented
40#91171C2212yescomplemented
41(C2×D3)⋊#170C2212yescomplemented
42#91169C2212yescomplemented
43Dic3×#170C2212yescomplemented
44#83734—51nomaximal, Hall
45C23×#170D311yescharacteristic, complemented
46C2×#4131—31no—
47#65920—31no—
48#65798—31no—
49#65775—31no—
50#65773—32no—
51D4×#170—32no—
52#65696—31no—
53#65693—32no—
54#65692—32no—
55#65691—31no—
56C24×C3×C5C2×C411yescharacteristic
57C2×#1220C2×C411yescharacteristic
58C2×#1220C2×C412yes—
59C2×C5×#67C2×C411yescharacteristic
60C2×C5×#67C2×C412yes—
61C22×C3×Dic5—22no—
62C22×C3×Dic5—22no—
63C22×C3×Dic5C2×C411yescharacteristic
64C22×C3×Dic5C2×C412yes—
65C22×C3×Dic5C2312yes—
66C22×C3×Dic5C2311yescharacteristic
67C2×#1204C2×C411yescharacteristic
68C2×#1203C2×C411yescharacteristic
69C2×#1200C2311yescharacteristic
70#6782—22no—
71#6782—22no—
72C2×D3×Dic5C2311yescharacteristic
73#6762—22no—
74#6762—22no—
75C2×C3⋊#170C2311yescharacteristic
76C2×C3⋊#170C2311yescharacteristic
77C2×C3⋊#170C2312yes—
78C2×C3×#170—22no—
79C2×C3×#170—24no—
80C2×C3×#170—24no—
81C2×C3×#170D414yescomplemented
82C2×C3×#170C2312yes—
83C2×C3×#170C2311yescharacteristic
84C2×C3×#170C2311yescharacteristic
85#6740—22no—
86#6739—22no—
87#6738—22no—
88D3×#170—22no—
89#6719C2×C411yescharacteristic
90#6719C2×C412yes—
91C5×#217C2×C411yescharacteristic
92C5×#217C2×C412yes—
93Dic3⋊Dic5C2×C412yes—
94#6669C2×C412yes—
95#6668C2×C412yes—
96#6667C2×C412yes—
97(C2×D3)⋊Dic5C2×C412yes—
98Dic3×Dic5C2312yes—
99C23×C3×C8—51no—
100#5791—51no—
101C2×#901—51no—
102#5207—51no—
103#5111—52no—
104#5111—52no—
105#5110—51no—
106C8×#67—52no—
107C8×#67—52no—
108#5102—51no—
109#5101—51no—
110C23×Dic5C2×D311yescharacteristic
111C2×#711—31no—
112C2×#704—31no—
113C2×#704—31no—
114C2×#704—31no—
115C22×#170—24no—
116C22×#170—31no—
117C22×#170C2×D312yes—
118C22×#170C2×D312yes—
119C22×#170C2×D311yescharacteristic
120C22×#170C2×D311yescharacteristic
121#4131—64no—
122C2×#700—31no—
123C2×D5⋊C8—31no—
124C5×#120—31no—
125#4099—31no—
126#4085—32no—
127#4081—31no—
128D4×Dic5—32no—
129#4032—31no—
130#4027—32no—
131#4024—32no—
132#4023—31no—
133C22⋊#170—32no—
134C22⋊#170—32no—
135C22⋊#170—32no—
136#4004—32no—
137#4003—32no—
138#4002—32no—
139C4⋊#170—32no—
140C4×#170—32no—
141#2929—151noSylow
142C23×C3×C5—22no—
143C23×C3×C5—24no—
144C23×C3×C5C22×C411yescharacteristic
145C23×C3×C5C22×C412yes—
146C22×D15C22×C411yescharacteristic
147C22×C5×D3C22×C411yescharacteristic
148#1220—22no—
149#1220—24no—
150C2×Dic15C22×C412yes—
151C2×Dic15C22×C411yescharacteristic
152C5×#67—22no—
153C5×#67—24no—
154C2×C5×Dic3C22×C412yes—
155C2×C5×Dic3C22×C411yescharacteristic
156C2×C3×Dic5—22no—
157C2×C3×Dic5—24no—
158C2×C3×Dic5—22no—
159C2×C3×Dic5—22no—
160C2×C3×Dic5C22×C412yes—
161C2×C3×Dic5C22×C411yescharacteristic
162C2×C3×Dic5C2×D412yes—
163C2×C3×Dic5#4112yes—
164C2×C3×Dic5C2411yescharacteristic
165#1204—22no—
166#1203—22no—
167#1200—21no—
168#1200—21no—
169D3×Dic5—21no—
170D3×Dic5—21no—
171C3⋊#170—21no—
172C3⋊#170—21no—
173C3⋊#170—22no—
174C3×#170—22no—
175C3×#170C2×D412yes—
176C3×#170C2×D412yes—
177C23×C3×C4—51no—
178C2×#235—51no—
179C22×C3×C8—51no—
180C22×C3×C8—51no—
181C22×C3×C8—52no—
182C22×C3×C8—52no—
183C22×C3×C8—104no—
184#953—51no—
185#953AGL(1,𝔽5)11yescharacteristic
186#930—52no—
187#929—51no—
188#929—51no—
189C4×#67—52no—
190C4×#67—52no—
191#926—51no—
192#924—52no—
193C2×#208—51no—
194C2×#208—51no—
195#901—104no—
196C2×ΓL(1,𝔽25)—51no—
197C2×ΓL(1,𝔽25)—51no—
198C2×D3×C8—51no—
199C2×D3×C8—51no—
200#830—52no—
201#830—52no—
202(C2×D3)⋊C8—52no—
203(C2×D3)⋊C8—52no—
204#815—52no—
205Dic3⋊C8—52no—
206Dic3⋊C8—52no—
207C8×Dic3—52no—
208C24×C5C4×D311yescharacteristic
209C2×C5×D4—31no—
210C2×C5×D4—32no—
211C2×C22⋊D5—31no—
212C2×C22⋊D5—32no—
213C2×C22⋊D5—31no—
214C22×Dic5—22no—
215C22×Dic5—22no—
216C22×Dic5—31no—
217C22×Dic5C4×D312yes—
218C22×Dic5C4×D311yescharacteristic
219C22×Dic5C22×D312yes—
220C22×Dic5C22×D311yescharacteristic
221#711—62no—
222#711—62no—
223C2×C4×D5—31no—
224C2×Dic10—31no—
225#704—62no—
226#704—62no—
227#704—62no—
228C2×#170—22no—
229C2×#170—24no—
230C2×#170—24no—
231C2×#170—31no—
232C2×#170—31no—
233C2×#170—32no—
234C2×#170—62no—
235C2×#170#6714yescomplemented
236C2×#170C22×D312yes—
237C2×#170C22×D311yescharacteristic
238C2×#170C22×D311yescharacteristic
239#700—62no—
240D5⋊C8—62no—
241C5×C22⋊C4—31no—
242C5×C22⋊C4—32no—
243C22⋊Dic5—31no—
244C22⋊Dic5—32no—
245C22⋊Dic5—32no—
246#685—32no—
247C4⋊Dic5—32no—
248#683—32no—
249#683—32no—
250C4×Dic5—32no—
251C2×#131—151no—
252C23×C8—51no—
253#525—151no—
254#435—152no—
255#435—152no—
256C8×D4—152no—
257C8×D4—152no—
258#424—151no—
259#408—151no—
260#407—151no—
261#405—151no—
262C22×C3×C5—28no—
263C22×C3×C5—22no—
264C22×C3×C5—22no—
265C22×C3×C5—22no—
266C22×C3×C5C4×D412yes—
267C22×C3×C5#13114yescomplemented
268C22×C3×C5C23×C411yescharacteristic
269C2×D15—21no—
270C2×D15—22no—
271C2×C5×D3—21no—
272C2×C5×D3—22no—
273Dic15—21no—
274Dic15—22no—
275C3×Dic5—22no—
276C3×Dic5C4×D412yes—
277C3×Dic5C22×D411yescharacteristic
278C3×Dic5C2×#4111yescharacteristic
279C5×Dic3—21no—
280C5×Dic3—22no—
281C24×C3C2×AGL(1,𝔽5)11yescharacteristic
282C22×C3×C4—51no—
283C22×C3×C4—52no—
284C22×C3×C4—52no—
285C22×C3×C4—51no—
286C22×C3×C4—102no—
287C22×C3×C4—102no—
288C2×#67—52no—
289C2×#67—51no—
290C2×#67C2×AGL(1,𝔽5)12yes—
291C2×#67C2×AGL(1,𝔽5)11yescharacteristic
292#235—102no—
293#235—102no—
294C2×D12—51no—
295C2×C4×D3—51no—
296C2×C4×D3—51no—
297C2×Dic6—51no—
298C2×C3×C8—51no—
299C2×C3×C8—51no—
300C2×C3×C8—54no—
301C2×C3×C8—52no—
302C2×C3×C8—104no—
303C2×C3×C8—104no—
304C2×C3×C8—102no—
305#217—52no—
306#217—51no—
307#217C2×AGL(1,𝔽5)12yes—
308#217C2×AGL(1,𝔽5)11yescharacteristic
309(C2×D3)⋊C4—52no—
310(C2×D3)⋊C4—52no—
311C4⋊Dic3—52no—
312Dic3⋊C4—52no—
313Dic3⋊C4—52no—
314C4×Dic3—52no—
315#208—102no—
316#208—102no—
317C2×C3⋊C8—52no—
318C2×C3⋊C8—51no—
319C2×C3⋊C8—51no—
320ΓL(1,𝔽25)—102no—
321ΓL(1,𝔽25)—102no—
322D3×C8—102no—
323D3×C8—102no—
324C23×C5—24no—
325C23×C5—22no—
326C23×C5—31no—
327C23×C5C2×C4×D312yes—
328C23×C5C2×C4×D311yescharacteristic
329C22×D5—31no—
330C5×D4—64no—
331C5×D4—62no—
332C2×C4×C5—32no—
333C2×C4×C5—31no—
334C22⋊D5—64no—
335C22⋊D5—62no—
336C22⋊D5—62no—
337C2×Dic5—22no—
338C2×Dic5—24no—
339C2×Dic5—22no—
340C2×Dic5—22no—
341C2×Dic5—32no—
342C2×Dic5—31no—
343C2×Dic5—61no—
344C2×Dic5—61no—
345C2×Dic5C2×C4×D312yes—
346C2×Dic5C2×C4×D311yescharacteristic
347C2×Dic5#23512yes—
348C2×Dic5C2×#6712yes—
349C2×Dic5C23×D311yescharacteristic
350C4×D5—61no—
351C4×D5—61no—
352Dic10—62no—
353#170—22no—
354#170—62no—
355#170—61no—
356#170—61no—
357#170C2×#6712yes—
358#170C2×#6712yes—
359C2×#41—151no—
360C23×C4—51no—
361#131—304no—
362C2×M16—151no—
363C2×M16—151no—
364C2×M16—151no—
365C2×M16—151no—
366C22×C8—51no—
367C22×C8—51no—
368C22×C8—52no—
369C22×C8—52no—
370C22×C8—104no—
371C22×C8—151no—
372C22×C8—151no—
373#123—151no—
374#123—151no—
375C22⋊Q8—152no—
376#121—152no—
377#120—31no—
378#120—151no—
379C4×D4—152no—
380C4×D4—152no—
381#117—151no—
382C4⋊C8—152no—
383C4⋊C8—152no—
384C22⋊C8—152no—
385C22⋊C8—152no—
386C22⋊C8—152no—
387C22⋊C8—152no—
388#97—152no—
389C4×C8—152no—
390C2×C3×C5—22no—
391C2×C3×C5—24no—
392C2×C3×C5C2×C4×D411yescharacteristic
393C2×C3×C5C2×#13112yes—
394D15—41no—
395C5×D3—41no—
396C23×C3—24no—
397C23×C3—22no—
398C23×C3C22×AGL(1,𝔽5)12yes—
399C23×C3C22×AGL(1,𝔽5)11yescharacteristic
400C22×D3—51no—
401C22×D3C22×AGL(1,𝔽5)11yescharacteristic
402C2×C3×C4—52no—
403C2×C3×C4—52no—
404C2×C3×C4—51no—
405C2×C3×C4—51no—
406C2×C3×C4—52no—
407C2×C3×C4—102no—
408C2×C3×C4—104no—
409C2×C3×C4—102no—
410C2×C3×C4—102no—
411#67—24no—
412#67—22no—
413#67—104no—
414#67—102no—
415C2×Dic3—52no—
416C2×Dic3—51no—
417C2×Dic3C22×AGL(1,𝔽5)12yes—
418C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
419D12—102no—
420C4×D3—101no—
421C4×D3—101no—
422C4×D3—101no—
423C4×D3—101no—
424Dic6—102no—
425C3×C8—52no—
426C3×C8—52no—
427C3×C8—102no—
428C3⋊C8—101no—
429C3⋊C8—101no—
430C3⋊C8—102no—
431C22×C5—28no—
432C22×C5—22no—
433C22×C5—22no—
434C22×C5—22no—
435C22×C5—62no—
436C22×C5—61no—
437C22×C5#90114yescomplemented
438C22×C5C4×#6712yes—
439C22×C5C22×C4×D311yescharacteristic
440C2×D5—62no—
441C2×D5—61no—
442C4×C5—62no—
443C4×C5—61no—
444Dic5—22no—
445Dic5—62no—
446Dic5—61no—
447Dic5C4×#6712yes—
448Dic5C2×#23511yescharacteristic
449Dic5C22×#6711yescharacteristic
450C24D3×AGL(1,𝔽5)11yescharacteristic
451#41—302no—
452#41—302no—
453C2×Q8—151no—
454C2×D4—32no—
455C2×D4—31no—
456C2×D4—151no—
457C2×D4—152no—
458C2×D4—151no—
459C22×C4—51no—
460C22×C4—52no—
461C22×C4—52no—
462C22×C4—51no—
463C22×C4—102no—
464C22×C4—102no—
465C22×C4—151no—
466C22×C4—151no—
467M16—302no—
468M16—302no—
469M16—302no—
470M16—302no—
471C2×C8—51no—
472C2×C8—51no—
473C2×C8—54no—
474C2×C8—52no—
475C2×C8—104no—
476C2×C8—104no—
477C2×C8—102no—
478C2×C8—152no—
479C2×C8—151no—
480C2×C8—151no—
481C2×C8—302no—
482C2×C8—302no—
483C4⋊C4—152no—
484C4⋊C4—152no—
485C4⋊C4—152no—
486C22⋊C4—31no—
487C22⋊C4—32no—
488C22⋊C4—152no—
489C22⋊C4—152no—
490C22⋊C4—151no—
491C22⋊C4—152no—
492C42—152no—
493C3×C5#292911yescharacteristic, complemented, Hall
494C22×C3—28no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—22no—
498C22×C3#413114yescomplemented
499C22×C3D4×AGL(1,𝔽5)12yes—
500C22×C3C23×AGL(1,𝔽5)11yescharacteristic
501C2×D3—22no—
502C2×D3—21no—
503C2×D3—102no—
504C2×D3—101no—
505C3×C4—52no—
506C3×C4—51no—
507C3×C4—51no—
508C3×C4—102no—
509Dic3—22no—
510Dic3—21no—
511Dic3—102no—
512Dic3—101no—
513C2×C5—24no—
514C2×C5—22no—
515C2×C5—121no—
516C2×C5C2×#90112yes—
517C2×C5C2×C4×#6711yescharacteristic
518D5—121no—
519C23—24no—
520C23—22no—
521C23—31no—
522C23—151no—
523C23C2×D3×AGL(1,𝔽5)12yes—
524C23C2×D3×AGL(1,𝔽5)11yescharacteristic
525Q8—302no—
526D4—64no—
527D4—62no—
528D4—302no—
529D4—304no—
530D4—302no—
531C2×C4—32no—
532C2×C4—31no—
533C2×C4—52no—
534C2×C4—52no—
535C2×C4—51no—
536C2×C4—51no—
537C2×C4—52no—
538C2×C4—102no—
539C2×C4—102no—
540C2×C4—104no—
541C2×C4—102no—
542C2×C4—152no—
543C2×C4—151no—
544C2×C4—301no—
545C2×C4—301no—
546C2×C4—301no—
547C2×C4—301no—
548C8—52no—
549C8—52no—
550C8—102no—
551C8—301no—
552C8—301no—
553C8—302no—
554C2×C3—24no—
555C2×C3—22no—
556C2×C3C2×#413112yes—
557C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
558D3—41no—
559D3—201no—
560C5#8373411yescharacteristic, complemented, Sylow
561C22—28no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—62no—
566C22—61no—
567C22—302no—
568C22—301no—
569C22#9192714yescomplemented
570C22AGL(1,𝔽5)×#6712yes—
571C22C22×D3×AGL(1,𝔽5)11yescharacteristic
572C4—51no—
573C4—51no—
574C4—52no—
575C4—62no—
576C4—61no—
577C4—102no—
578C4—302no—
579C4—301no—
580C3#1061721511yescharacteristic, complemented, Sylow
581C2—24no—
582C2—22no—
583C2—121no—
584C2—601no—
585C2C2×#9192712yes—
586C2C2×AGL(1,𝔽5)×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC24×C3×C5C2×C4
prime radical#11755385C2
cosocleC2×C3×Dic5C24
hypercenterC24D3×AGL(1,𝔽5)
hypocenterC3×C5#2929
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×C3×#170C2semidirect
2. C2×#91927C2semidirect
4. #11752457C2semidirect
5. #11752046C2semidirect
6. #11752044C2semidirect
7. #11752036C2semidirect
8. #67×#170C2semidirect
9. #11752011C2semidirect
10. #11751999C2semidirect
11. #11751988C2semidirect
22. C2×#6740C22semidirect
23. C2×#6739C22semidirect
24. C2×#6738C22semidirect
26. C2×D3×#170C22semidirect
38. #91177C22semidirect
39. Dic3⋊#170C22semidirect
40. #91171C22semidirect
41. (C2×D3)⋊#170C22semidirect
42. #91169C22semidirect
43. Dic3×#170C22semidirect
45. C23×#170D3semidirect
81. C2×C3×#170D4semidirect
235. C2×#170#67semidirect
267. C22×C3×C5#131semidirect
437. C22×C5#901semidirect
493. C3×C5#2929semidirect
498. C22×C3#4131semidirect
560. C5#83734semidirect
569. C22#91927semidirect
580. C3#10617215semidirect

Maximal subgroups

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

subgroupindex
C23×C3×#1702
C2×#919272
#117553852
#117524572
#117520462
#117520442
#117520362
#67×#1702
#117520112
#117519992
#117519882
#106172153
#837345