Finite Group Databasebeta

← #14412032

Subgroups — #14412032

Order 1920.

1432 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92000C211yesmaximal, characteristic
2#11756812C212yesmaximal, complemented
3#11752404C211yesmaximal, characteristic, complemented
4C3×#65745C211yesmaximal, characteristic, complemented
5#11752030C214yesmaximal, complemented
6#11751863C214yesmaximal, complemented
7#11751817C212yesmaximal, complemented
8#10617514—31nomaximal, Hall
9C2×#6817C411yescharacteristic, complemented
10C22×C5×Dic6C411yescharacteristic, complemented
11C2×C3×#709C2211yescharacteristic
12C2×#6779C412yescomplemented
13C2×#6779C2212yes—
14C2×#6773C411yescharacteristic, complemented
15C2×#6773C2211yescharacteristic
16C2×D15⋊Q8C412yescomplemented
17#92000—24no—
18#92000—24no—
19C2×#6766C2212yes—
20C2×D5×Dic6C411yescharacteristic, complemented
21C2×D5×Dic6C2211yescharacteristic
22#91989C2212yes—
23C3×#4130C2212yes—
24#91923C2218yes—
25ΓL(1,𝔽16)⋊Q8—24no—
26AGL(1,𝔽5)×Dic6—24no—
27#91240C2212yes—
28#91228C2212yes—
29C3×#4011C2212yes—
30C3×#3999C2212yes—
31#91173C2218yes—
32#78689—51nomaximal, Hall
33C2×#4147—31no—
34#66262—32no—
35#65785—34no—
36#65785—32no—
37#65745—34no—
38#65745—31no—
39#65745D311yescharacteristic, complemented
40C22×C3×C4×C5C2×C411yescharacteristic
41C2×#1220C2×C412yes—
42#6817—22no—
43#6817—22no—
44C2×D60C2×C411yescharacteristic
45C2×C4×D15C2×C412yes—
46C2×Dic30C2×C411yescharacteristic
47C22×C5×Dic3C2×C412yes—
48C2×C5×Dic6—22no—
49C2×C5×Dic6—22no—
50C2×C5×Dic6C2×C411yescharacteristic
51C2×C5×Dic6C2×C412yes—
52C2×C5×Dic6C2×C411yescharacteristic
53C2×C3×C22⋊D5C2×C411yescharacteristic
54C2×C3×C22⋊D5C2311yescharacteristic
55C3×#709—22no—
56C3×#709—22no—
57C2×C3×D20C2311yescharacteristic
58C2×C3×C4×D5C2×C411yescharacteristic
59C2×C3×C4×D5C2311yescharacteristic
60C2×C3×Dic10C2×C411yescharacteristic
61C2×#1204C2×C412yes—
62C2×#1204C2×C412yes—
63C2×#1202C2×C412yes—
64C2×#1202C2312yes—
65C2×#1200C2×C412yes—
66C2×#1200C2×C412yes—
67#6779—28no—
68#6779—28no—
69C2×D5×Dic3C2×C412yes—
70C2×D5×Dic3C2312yes—
71#6773—22no—
72#6773—22no—
73#6773—22no—
74#6773—22no—
75D15⋊Q8—28no—
76#6766—28no—
77D5×Dic6—22no—
78D5×Dic6—22no—
79D5×Dic6D414yes—
80D5×Dic6D414yes—
81#6763C2314yes—
82C4⋊ΓL(1,𝔽16)—22no—
83C4×ΓL(1,𝔽16)—22no—
84C3×C22⋊AGL(1,𝔽5)C2314yes—
85C3×C4⋊AGL(1,𝔽5)—22no—
86C3×C4×AGL(1,𝔽5)—22no—
87Dic3⋊AGL(1,𝔽5)—28no—
88Dic3×AGL(1,𝔽5)—28no—
89C2×#1010—51no—
90#5516—52no—
91C3×#410—51no—
92#5240—51no—
93#5006—52no—
94#4740—54no—
95#4740—54no—
96C22×C5×Q8—31no—
97#4147—64no—
98#4147—64no—
99C2×#713—32no—
100C2×#713—31no—
101C2×#713—31no—
102C2×Q8×D5—31no—
103C2×Q8×D5—31no—
104C2×Q8×D5—32no—
105C2×#709—31no—
106C2×#709—32no—
107C2×#709—32no—
108C2×#709C2×D311yescharacteristic
109Q8×AGL(1,𝔽5)—64no—
110Q8×AGL(1,𝔽5)—64no—
111#4130—32no—
112#4130—38no—
113#4130C2×D312yes—
114#4011—32no—
115#4011C2×D312yes—
116#3999—32no—
117#3999—38no—
118#3999C2×D312yes—
119#2884—151noSylow
120C23×C3×C5C22×C411yescharacteristic
121C22×D15C22×C412yes—
122C22×C3×D5C22×C411yescharacteristic
123C22×C3×D5C2411yescharacteristic
124C2×ΓL(1,𝔽16)—24no—
125C2×C3×AGL(1,𝔽5)—24no—
126C2×C3×C4×C5—21no—
127C2×C3×C4×C5—21no—
128C2×C3×C4×C5C22×C411yescharacteristic
129C2×C3×C4×C5C22×C411yescharacteristic
130#1220—24no—
131C2×Dic15C22×C412yes—
132D60—21no—
133D60—21no—
134C4×D15—24no—
135Dic30C22⋊C412yes—
136Dic30C22⋊C412yes—
137C2×C5×Dic3—24no—
138C2×C5×Dic3C22×C412yes—
139C2×C5×Dic3C22×C412yes—
140C5×Dic6—21no—
141C5×Dic6—21no—
142C5×Dic6—24no—
143C5×Dic6C22⋊C412yes—
144C5×Dic6C22⋊C412yes—
145C3×C22⋊D5—22no—
146C3×C22⋊D5—22no—
147C2×C3×Dic5C22×C411yescharacteristic
148C2×C3×Dic5C22×C411yescharacteristic
149C3×D20—22no—
150C3×C4×D5—21no—
151C3×C4×D5—21no—
152C3×C4×D5C2×D412yes—
153C3×C4×D5C2×D412yes—
154C3×Dic10—22no—
155#1204—24no—
156#1204C22⋊C418yes—
157#1202—24no—
158#1202—24no—
159#1200—24no—
160#1200—24no—
161D5×Dic3—24no—
162D5×Dic3C2×D418yes—
163C2×C3×#41—51no—
164#1010—104no—
165#1010—104no—
166C2×D3×Q8—52no—
167C2×#237—52no—
168C2×#237—52no—
169C2×#237—52no—
170C2×#235—51no—
171C2×#235—51no—
172C2×#235—51no—
173C22×Dic6—51no—
174C22×Dic6—51no—
175C22×Dic6AGL(1,𝔽5)11yescharacteristic, complemented
176C3×#117—52no—
177#926—52no—
178#887—104no—
179#877—58no—
180C4×Dic6—104no—
181C3×#99—52no—
182C3×#99—52no—
183#834—52no—
184#834—52no—
185#806—58no—
186C2×C5×Q8—31no—
187C2×C5×Q8—32no—
188C2×C5×Q8—31no—
189C2×C5×Q8—62no—
190C2×C5×Q8—62no—
191C22×C4×C5—32no—
192C22×C4×C5C4×D311yescharacteristic
193C2×C22⋊D5—32no—
194C2×C22⋊D5C4×D311yescharacteristic
195C2×C22⋊D5C22×D311yescharacteristic
196#713—62no—
197#713—62no—
198#713—68no—
199#713—62no—
200#713—62no—
201Q8×D5—34no—
202Q8×D5—34no—
203Q8×D5—68no—
204Q8×D5—62no—
205Q8×D5—62no—
206#709—22no—
207#709—22no—
208#709—68no—
209#709—62no—
210#709—62no—
211#709—68no—
212C2×D20—32no—
213C2×D20—32no—
214C2×D20—31no—
215C2×D20C22×D311yescharacteristic
216C2×C4×D5—32no—
217C2×C4×D5—32no—
218C2×C4×D5—32no—
219C2×C4×D5—32no—
220C2×C4×D5—32no—
221C2×C4×D5C4×D311yescharacteristic
222C2×C4×D5C22×D311yescharacteristic
223C2×Dic10—31no—
224C2×Dic10—32no—
225C2×Dic10—32no—
226C2×Dic10C4×D311yescharacteristic
227C22⋊AGL(1,𝔽5)—34no—
228C22⋊AGL(1,𝔽5)C22×D314yes—
229C4⋊AGL(1,𝔽5)—22no—
230C4⋊AGL(1,𝔽5)—68no—
231C4⋊AGL(1,𝔽5)—62no—
232C4×AGL(1,𝔽5)—22no—
233C4×AGL(1,𝔽5)—68no—
234C4×AGL(1,𝔽5)—62no—
235C2×2−1+4—151no—
236#519—152no—
237#410—51no—
238#410—154no—
239#410—154no—
240#410—152no—
241#410—151no—
242C22×C3×C5—21no—
243C22×C3×C5—41no—
244C22×C3×C5C23×C411yescharacteristic
245C2×D15—22no—
246C2×D15—42no—
247C2×C3×D5—21no—
248C2×C3×D5—41no—
249C2×C3×D5—41no—
250C2×C3×D5C22×D412yes—
251ΓL(1,𝔽16)—44no—
252C3×AGL(1,𝔽5)—44no—
253C3×C4×C5—21no—
254C3×C4×C5C2×C22⋊C411yescharacteristic
255C3×C4×C5C2×C22⋊C411yescharacteristic
256Dic15C2×C22⋊C414yes—
257C3×Dic5—21no—
258C3×Dic5C2×C22⋊C412yes—
259C5×Dic3—22no—
260C5×Dic3C2×C22⋊C414yes—
261C3×#41—102no—
262C3×#41—102no—
263C2×C3×Q8—51no—
264C2×C3×D4—51no—
265C2×C3×D4—51no—
266C2×C3×D4—51no—
267C22×C3×C4—51no—
268C22×C3×C4—51no—
269C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
270C2×#67—52no—
271C2×#67—52no—
272C2×#67—52no—
273C22×Dic3—52no—
274C22×Dic3—52no—
275C22×Dic3C2×AGL(1,𝔽5)12yes—
276D3×Q8—108no—
277#237—108no—
278#237—108no—
279#237—108no—
280#235—102no—
281#235—102no—
282#235—102no—
283#235—102no—
284#235—102no—
285#235—102no—
286C2×D12—51no—
287C2×C4×D3—52no—
288C2×C4×D3—52no—
289C2×C4×D3—52no—
290C2×Dic6—22no—
291C2×Dic6—22no—
292C2×Dic6—51no—
293C2×Dic6—52no—
294C2×Dic6—54no—
295C2×Dic6—54no—
296C2×Dic6—52no—
297C2×Dic6—102no—
298C2×Dic6—102no—
299C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
300C2×Dic6C2×AGL(1,𝔽5)12yes—
301C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
302C3×C4⋊C4—102no—
303C3×C22⋊C4—54no—
304C3×C42—102no—
305#217—54no—
306C4⋊Dic3—102no—
307Dic3⋊C4—108no—
308C4×Dic3—108no—
309C4×Dic3—102no—
310C23×C5C2×C4×D311yescharacteristic
311C22×D5—32no—
312C22×D5C2×C4×D311yescharacteristic
313C22×D5C23×D311yescharacteristic
314C2×AGL(1,𝔽5)—24no—
315C2×AGL(1,𝔽5)—64no—
316C5×Q8—32no—
317C5×Q8—32no—
318C5×Q8—64no—
319C5×Q8—61no—
320C5×Q8—61no—
321C2×C4×C5—21no—
322C2×C4×C5—21no—
323C2×C4×C5—32no—
324C2×C4×C5—32no—
325C2×C4×C5—64no—
326C2×C4×C5C2×C4×D311yescharacteristic
327C2×C4×C5C2×C4×D311yescharacteristic
328C22⋊D5—22no—
329C22⋊D5—22no—
330C22⋊D5—64no—
331C2×Dic5—32no—
332C2×Dic5C2×C4×D311yescharacteristic
333C2×Dic5C2×C4×D311yescharacteristic
334D20—22no—
335D20—64no—
336D20—64no—
337D20—61no—
338D20—61no—
339C4×D5—21no—
340C4×D5—21no—
341C4×D5—38no—
342C4×D5—64no—
343C4×D5—64no—
344C4×D5—64no—
345C4×D5—64no—
346C4×D5D3×D412yes—
347C4×D5D3×D412yes—
348Dic10—22no—
349Dic10—32no—
350Dic10—32no—
351Dic10—38no—
352Dic10—64no—
3532−1+4—304no—
3542−1+4—304no—
355C2×#41—51no—
356C2×#41—152no—
357C2×#41—152no—
358C2×#41—151no—
359C2×#41—151no—
360C2×#41—152no—
361C2×#41—151no—
362C22×Q8—31no—
363C22×Q8—151no—
364C22×Q8—151no—
365C22×Q8—152no—
366C4×Q8—304no—
367C4×Q8—304no—
368#117—52no—
369#117—158no—
370#117—152no—
371#99—52no—
372#99—52no—
373#99—158no—
374#99—152no—
375#99—152no—
376C2×C3×C5—21no—
377C2×C3×C5—41no—
378C2×C3×C5C22×C22⋊C411yescharacteristic
379D15—42no—
380C3×D5—22no—
381C3×D5—41no—
382C23×C3—51no—
383C23×C3—51no—
384C23×C3C22×AGL(1,𝔽5)11yescharacteristic
385C22×D3—52no—
386C3×Q8—102no—
387C3×D4—102no—
388C3×D4—102no—
389C3×D4—102no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—51no—
393C2×C3×C4—52no—
394C2×C3×C4—52no—
395C2×C3×C4—51no—
396C2×C3×C4—101no—
397C2×C3×C4—101no—
398C2×C3×C4—104no—
399C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
400C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
401#67—104no—
402#67—104no—
403#67—104no—
404C2×Dic3—24no—
405C2×Dic3—52no—
406C2×Dic3—58no—
407C2×Dic3—104no—
408C2×Dic3—104no—
409C2×Dic3C22×AGL(1,𝔽5)12yes—
410C2×Dic3C22×AGL(1,𝔽5)12yes—
411D12—101no—
412D12—101no—
413C4×D3—104no—
414C4×D3—104no—
415C4×D3—104no—
416Dic6—21no—
417Dic6—21no—
418Dic6—24no—
419Dic6—52no—
420Dic6—52no—
421Dic6—58no—
422Dic6—104no—
423Dic6C22⋊AGL(1,𝔽5)12yes—
424Dic6C22⋊AGL(1,𝔽5)12yes—
425C22×C5—21no—
426C22×C5—41no—
427C22×C5C22×C4×D311yescharacteristic
428C2×D5—21no—
429C2×D5—41no—
430C2×D5—41no—
431C2×D5—62no—
432C2×D5—122no—
433C2×D5C2×D3×D412yes—
434AGL(1,𝔽5)—44no—
435AGL(1,𝔽5)—124no—
436C4×C5—21no—
437C4×C5—34no—
438C4×C5—62no—
439C4×C5D3×C22⋊C411yescharacteristic
440C4×C5D3×C22⋊C411yescharacteristic
441Dic5—21no—
442Dic5—34no—
443Dic5D3×C22⋊C412yes—
444#41—102no—
445#41—102no—
446#41—302no—
447#41—302no—
448#41—302no—
449#41—302no—
450#41—308no—
451#41—308no—
452#41—302no—
453#41—302no—
454#41—308no—
455C2×Q8—32no—
456C2×Q8—31no—
457C2×Q8—31no—
458C2×Q8—51no—
459C2×Q8—62no—
460C2×Q8—62no—
461C2×Q8—152no—
462C2×Q8—152no—
463C2×Q8—154no—
464C2×Q8—154no—
465C2×Q8—151no—
466C2×Q8—302no—
467C2×Q8—302no—
468C2×Q8—308no—
469C2×D4—51no—
470C2×D4—51no—
471C2×D4—51no—
472C2×D4—152no—
473C2×D4—152no—
474C2×D4—152no—
475C2×D4—151no—
476C22×C4—32no—
477C22×C4—51no—
478C22×C4—51no—
479C22×C4—152no—
480C22×C4—152no—
481C22×C4—152no—
482C22×C4—152no—
483C22×C4—152no—
484C22×C4D3×AGL(1,𝔽5)11yescharacteristic
485C4⋊C4—102no—
486C4⋊C4—302no—
487C4⋊C4—308no—
488C22⋊C4—54no—
489C22⋊C4—154no—
490C42—102no—
491C42—302no—
492C42—308no—
493C3×C5#288411yescharacteristic, complemented, Hall
494C22×C3—21no—
495C22×C3—41no—
496C22×C3—52no—
497C22×C3—101no—
498C22×C3—201no—
499C22×C3—201no—
500C22×C3C23×AGL(1,𝔽5)11yescharacteristic
501C2×D3—102no—
502C2×D3—202no—
503C3×C4—21no—
504C3×C4—52no—
505C3×C4—101no—
506C3×C4—204no—
507C3×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
508C3×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
509Dic3—22no—
510Dic3—54no—
511Dic3—204no—
512Dic3C2×C22⋊AGL(1,𝔽5)14yes—
513C2×C5—21no—
514C2×C5—41no—
515C2×C5C2×D3×C22⋊C411yescharacteristic
516D5—22no—
517D5—41no—
518D5—122no—
519C23—51no—
520C23—51no—
521C23—152no—
522C23C2×D3×AGL(1,𝔽5)11yescharacteristic
523Q8—32no—
524Q8—32no—
525Q8—64no—
526Q8—61no—
527Q8—61no—
528Q8—102no—
529Q8—158no—
530Q8—152no—
531Q8—152no—
532Q8—304no—
533D4—102no—
534D4—102no—
535D4—102no—
536D4—304no—
537D4—304no—
538D4—304no—
539D4—301no—
540D4—301no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—32no—
544C2×C4—32no—
545C2×C4—51no—
546C2×C4—52no—
547C2×C4—52no—
548C2×C4—51no—
549C2×C4—64no—
550C2×C4—101no—
551C2×C4—101no—
552C2×C4—104no—
553C2×C4—158no—
554C2×C4—152no—
555C2×C4—304no—
556C2×C4—304no—
557C2×C4—304no—
558C2×C4—304no—
559C2×C4—304no—
560C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
561C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
562C2×C3—21no—
563C2×C3—41no—
564C2×C3—102no—
565C2×C3—201no—
566C2×C3C22×C22⋊AGL(1,𝔽5)11yescharacteristic
567D3—202no—
568C5#7868911yescharacteristic, complemented, Sylow
569C22—21no—
570C22—41no—
571C22—52no—
572C22—101no—
573C22—201no—
574C22—201no—
575C22—302no—
576C22—602no—
577C22C22×D3×AGL(1,𝔽5)11yescharacteristic
578C4—21no—
579C4—34no—
580C4—52no—
581C4—62no—
582C4—101no—
583C4—154no—
584C4—204no—
585C4—604no—
586C4D3×C22⋊AGL(1,𝔽5)11yescharacteristic
587C4D3×C22⋊AGL(1,𝔽5)11yescharacteristic
588C3#1061751411yescharacteristic, complemented, Sylow
589C2—21no—
590C2—41no—
591C2—102no—
592C2—201no—
593C2—602no—
594C2C2×D3×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2C2×D3×C22⋊AGL(1,𝔽5)
commutatorC22×C3×C5C23×C4
FrattiniC22C22×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC2×C3×C5C22×C22⋊C4
prime radicalC2×#92000C2
cosocleC22×C3×D5C24
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2884
LCSG ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11756812C2semidirect
3. #11752404C2semidirect
4. C3×#65745C2semidirect
5. #11752030C2semidirect
6. #11751863C2semidirect
7. #11751817C2semidirect
9. C2×#6817C4semidirect
10. C22×C5×Dic6C4semidirect
12. C2×#6779C4semidirect
14. C2×#6773C4semidirect
16. C2×D15⋊Q8C4semidirect
20. C2×D5×Dic6C4semidirect
39. #65745D3semidirect
175. C22×Dic6AGL(1,𝔽5)semidirect
493. C3×C5#2884semidirect
568. C5#78689semidirect
588. C3#10617514semidirect

Maximal subgroups

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

subgroupindex
C2×#920002
#117568122
#117524042
C3×#657452
#117520302
#117518632
#117518172
#106175143
#786895