Finite Group Databasebeta

← #14297382

Subgroups — #14297382

Order 1920.

1072 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4147C211yesmaximal, characteristic
2#11752443C212yesmaximal, complemented
3#11752441C212yesmaximal, complemented
4#11752420C211yesmaximal, characteristic, complemented
5#11752404C211yesmaximal, characteristic, complemented
6#10604974—31nomaximal, Hall
7C22×C3×C5×Q8C411yescharacteristic, complemented
8C3×#4147—24no—
9C3×#4147—24no—
10C2×C3×#713—21no—
11C2×C3×#713C411yescharacteristic, complemented
12C2×C3×#713C2211yescharacteristic
13C2×C3×Q8×D5—21no—
14C2×C3×Q8×D5C411yescharacteristic, complemented
15C2×C3×Q8×D5C2211yescharacteristic
16C2×C3×#709—21no—
17C2×C3×#709—21no—
18C2×C3×#709C411yescharacteristic, complemented
19C2×C3×#709C2211yescharacteristic
20#91989C2212yes—
21D5⋊#208C2212yes—
22#91242—21no—
23#91240—21no—
24#91239—24no—
25#91238—24no—
26#91234—21no—
27#91228—21no—
28#72144—51nomaximal, Hall
29C2×#4147D311yescharacteristic
30#65784—32no—
31#65782—32no—
32#65761—31no—
33#65745—31no—
34C2×C3×C5×Q8—22no—
35C2×C3×C5×Q8—22no—
36C2×C3×C5×Q8—21no—
37C2×C3×C5×Q8C2×C411yescharacteristic
38C2×C3×C5×Q8C2×C411yescharacteristic
39C22×C3×C4×C5—21no—
40C22×C3×C4×C5C2×C411yescharacteristic
41C2×C3×C22⋊D5—21no—
42C2×C3×C22⋊D5D411yescharacteristic
43C2×C3×C22⋊D5D411yescharacteristic
44C3×#713—28no—
45C3×#713—22no—
46C3×#713—22no—
47C3×#713—22no—
48C3×#713D414yes—
49C3×Q8×D5—28no—
50C3×Q8×D5—22no—
51C3×Q8×D5—22no—
52C3×Q8×D5—22no—
53C3×Q8×D5D414yes—
54C3×#709—28no—
55C3×#709—28no—
56C3×#709—22no—
57C3×#709—22no—
58C3×#709—22no—
59C3×#709—22no—
60C2×C3×D20—21no—
61C2×C3×D20—21no—
62C2×C3×D20C2×C411yescharacteristic
63C2×C3×D20D411yescharacteristic
64C2×C3×C4×D5—21no—
65C2×C3×C4×D5—21no—
66C2×C3×C4×D5—21no—
67C2×C3×C4×D5—21no—
68C2×C3×C4×D5—21no—
69C2×C3×C4×D5C2×C411yescharacteristic
70C2×C3×C4×D5C2311yescharacteristic
71C2×C3×Dic10—21no—
72C2×C3×Dic10—21no—
73C2×C3×Dic10C2×C411yescharacteristic
74C2×C3×Dic10D411yescharacteristic
75#6763—22no—
76#6762—22no—
77C4⋊ΓL(1,𝔽16)—22no—
78C4×ΓL(1,𝔽16)—22no—
79#6758—22no—
80D5⋊(C3⋊C8)—22no—
81C2×C3×2−1+4—51no—
82#5240—51no—
83#5237—51no—
84#5236—52no—
85#5225—52no—
86C22×C5×Q8Dic311yescharacteristic, complemented
87#4147—24no—
88#4147—24no—
89C2×#713—21no—
90C2×#713Dic311yescharacteristic, complemented
91C2×#713C2×D311yescharacteristic
92C2×Q8×D5—21no—
93C2×Q8×D5Dic311yescharacteristic, complemented
94C2×Q8×D5C2×D311yescharacteristic
95C2×#709—21no—
96C2×#709—21no—
97C2×#709Dic311yescharacteristic, complemented
98C2×#709C2×D311yescharacteristic
99#4130—32no—
100D5⋊M16—32no—
101#4013—61no—
102#4011—61no—
103#4010—64no—
104#4009—64no—
105#4005—61no—
106#3999—61no—
107#1796—151noSylow
108C23×C3×C5C22⋊C411yescharacteristic
109C22×C3×D5—21no—
110C22×C3×D5C22⋊C411yescharacteristic
111C22×C3×D5C2×D411yescharacteristic
112C2×ΓL(1,𝔽16)—42no—
113C3×C5×Q8—24no—
114C3×C5×Q8—21no—
115C3×C5×Q8—21no—
116C3×C5×Q8C22⋊C412yes—
117C3×C5×Q8C22⋊C412yes—
118C2×C3×C4×C5—24no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5—21no—
121C2×C3×C4×C5—21no—
122C2×C3×C4×C5—21no—
123C2×C3×C4×C5C22⋊C411yescharacteristic
124C2×C3×C4×C5C22×C411yescharacteristic
125C3×C22⋊D5—24no—
126C3×C22⋊D5—22no—
127C3×C22⋊D5—22no—
128C2×C3×Dic5—21no—
129C2×C3×Dic5C22⋊C411yescharacteristic
130C2×C3×Dic5C2×D411yescharacteristic
131C3×D20—24no—
132C3×D20—24no—
133C3×D20—22no—
134C3×D20—21no—
135C3×D20C22⋊C412yes—
136C3×C4×D5—24no—
137C3×C4×D5—24no—
138C3×C4×D5—24no—
139C3×C4×D5—24no—
140C3×C4×D5—21no—
141C3×C4×D5—21no—
142C3×C4×D5—24no—
143C3×C4×D5C2×D412yes—
144C3×C4×D5C2×D412yes—
145C3×Dic10—24no—
146C3×Dic10—21no—
147C3×Dic10—24no—
148C3×Dic10—22no—
149C3×Dic10C22⋊C412yes—
150C3⋊#170—42no—
151C3×2−1+4—104no—
152C3×2−1+4—104no—
153C2×C3×#41—51no—
154C2×C3×#41—51no—
155C2×C3×#41—51no—
156C2×C3×#41—51no—
157C2×C3×#41—101no—
158C2×C3×#41—101no—
159C2×C3×#41—101no—
160C22×C3×Q8—51no—
161C22×C3×Q8—51no—
162C22×C3×Q8—101no—
163C22×C3×Q8AGL(1,𝔽5)11yescharacteristic, complemented
164#926—52no—
165C2×#208—52no—
166#837—104no—
167#836—101no—
168#835—104no—
169#834—101no—
170#834—101no—
171#833—101no—
172C2×C5×Q8—22no—
173C2×C5×Q8—22no—
174C2×C5×Q8—21no—
175C2×C5×Q8C2×Dic311yescharacteristic
176C2×C5×Q8C2×Dic311yescharacteristic
177C22×C4×C5—21no—
178C22×C4×C5C2×Dic311yescharacteristic
179C2×C22⋊D5—21no—
180C2×C22⋊D5#6711yescharacteristic
181C2×C22⋊D5#6711yescharacteristic
182#713—28no—
183#713—22no—
184#713—22no—
185#713—22no—
186#713#6714yes—
187Q8×D5—28no—
188Q8×D5—22no—
189Q8×D5—22no—
190Q8×D5—22no—
191Q8×D5#6714yes—
192#709—28no—
193#709—28no—
194#709—22no—
195#709—22no—
196#709—22no—
197#709—22no—
198C2×D20—21no—
199C2×D20—21no—
200C2×D20C2×Dic311yescharacteristic
201C2×D20#6711yescharacteristic
202C2×C4×D5—21no—
203C2×C4×D5—21no—
204C2×C4×D5—21no—
205C2×C4×D5—21no—
206C2×C4×D5—21no—
207C2×C4×D5C2×Dic311yescharacteristic
208C2×C4×D5C22×D311yescharacteristic
209C2×Dic10—21no—
210C2×Dic10—21no—
211C2×Dic10C2×Dic311yescharacteristic
212C2×Dic10#6711yescharacteristic
213C22⋊AGL(1,𝔽5)—62no—
214#704—62no—
215C4⋊AGL(1,𝔽5)—62no—
216C4×AGL(1,𝔽5)—62no—
217#700—62no—
218D5⋊C8—62no—
219C2×2−1+4—51no—
220#421—152no—
221#419—152no—
222#413—151no—
223#410—151no—
224C22×C3×C5—21no—
225C22×C3×C5—41no—
226C22×C3×C5C2×C22⋊C411yescharacteristic
227C2×C3×D5—21no—
228C2×C3×D5—22no—
229C2×C3×D5—41no—
230C2×C3×D5—42no—
231C2×C3×D5—41no—
232C2×C3×D5#12012yes—
233ΓL(1,𝔽16)—82no—
234C3×C4×C5—22no—
235C3×C4×C5—22no—
236C3×C4×C5—21no—
237C3×C4×C5C2×C22⋊C411yescharacteristic
238C3×C4×C5C2×C22⋊C411yescharacteristic
239C3×Dic5—21no—
240C3×Dic5—22no—
241C3×Dic5#12012yes—
242C3×#41—54no—
243C3×#41—102no—
244C3×#41—102no—
245C3×#41—102no—
246C3×#41—102no—
247C3×#41—108no—
248C3×#41—108no—
249C3×#41—102no—
250C3×#41—102no—
251C3×#41—108no—
252C3×#41—102no—
253C2×C3×Q8—22no—
254C2×C3×Q8—22no—
255C2×C3×Q8—21no—
256C2×C3×Q8—51no—
257C2×C3×Q8—54no—
258C2×C3×Q8—51no—
259C2×C3×Q8—101no—
260C2×C3×Q8—101no—
261C2×C3×Q8—102no—
262C2×C3×Q8—102no—
263C2×C3×Q8—108no—
264C2×C3×Q8—102no—
265C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
266C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
267C2×C3×D4—51no—
268C2×C3×D4—51no—
269C2×C3×D4—51no—
270C2×C3×D4—51no—
271C2×C3×D4—101no—
272C2×C3×D4—101no—
273C2×C3×D4—101no—
274C22×C3×C4—21no—
275C22×C3×C4—51no—
276C22×C3×C4—51no—
277C22×C3×C4—101no—
278C22×C3×C4—101no—
279C22×C3×C4—101no—
280C22×C3×C4—101no—
281C22×C3×C4—101no—
282C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
283#217—102no—
284C4⋊Dic3—102no—
285C4×Dic3—102no—
286#208—102no—
287#208—102no—
288C2×C3⋊C8—102no—
289C23×C5#21711yescharacteristic
290C22×D5—21no—
291C22×D5#21711yescharacteristic
292C22×D5C2×#6711yescharacteristic
293C2×AGL(1,𝔽5)—122no—
294C5×Q8—24no—
295C5×Q8—21no—
296C5×Q8—21no—
297C5×Q8#21712yes—
298C5×Q8#21712yes—
299C2×C4×C5—24no—
300C2×C4×C5—21no—
301C2×C4×C5—21no—
302C2×C4×C5—21no—
303C2×C4×C5—21no—
304C2×C4×C5#21711yescharacteristic
305C2×C4×C5C22×Dic311yescharacteristic
306C22⋊D5—24no—
307C22⋊D5—22no—
308C22⋊D5—22no—
309C2×Dic5—21no—
310C2×Dic5#21711yescharacteristic
311C2×Dic5C2×#6711yescharacteristic
312D20—24no—
313D20—24no—
314D20—22no—
315D20—21no—
316D20#21712yes—
317C4×D5—24no—
318C4×D5—24no—
319C4×D5—24no—
320C4×D5—24no—
321C4×D5—21no—
322C4×D5—21no—
323C4×D5—24no—
324C4×D5C2×#6712yes—
325C4×D5C2×#6712yes—
326Dic10—24no—
327Dic10—21no—
328Dic10—24no—
329Dic10—22no—
330Dic10#21712yes—
331#170—122no—
3322−1+4—104no—
3332−1+4—104no—
334C2×#41—51no—
335C2×#41—51no—
336C2×#41—51no—
337C2×#41—51no—
338C2×#41—101no—
339C2×#41—101no—
340C2×#41—101no—
341C22×Q8—51no—
342C22×Q8—51no—
343C22×Q8—101no—
344C22×Q8ΓL(1,𝔽16)11yescharacteristic, complemented
345C2×M16—152no—
346#117—152no—
347#104—304no—
348#103—304no—
349#101—301no—
350#100—301no—
351#99—301no—
352#99—301no—
353C2×C3×C5—21no—
354C2×C3×C5—41no—
355C2×C3×C5#37911yescharacteristic
356C3×D5—22no—
357C3×D5—41no—
358C3×D5—42no—
359C23×C3—51no—
360C23×C3—51no—
361C23×C3—101no—
362C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
363C3×Q8—24no—
364C3×Q8—21no—
365C3×Q8—21no—
366C3×Q8—52no—
367C3×Q8—104no—
368C3×Q8—102no—
369C3×Q8—104no—
370C3×Q8—101no—
371C3×Q8C22⋊AGL(1,𝔽5)12yes—
372C3×Q8C22⋊AGL(1,𝔽5)12yes—
373C3×D4—52no—
374C3×D4—102no—
375C3×D4—104no—
376C3×D4—104no—
377C3×D4—102no—
378C3×D4—102no—
379C3×D4—104no—
380C3×D4—101no—
381C2×C3×C4—24no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—51no—
387C2×C3×C4—52no—
388C2×C3×C4—52no—
389C2×C3×C4—51no—
390C2×C3×C4—104no—
391C2×C3×C4—104no—
392C2×C3×C4—104no—
393C2×C3×C4—104no—
394C2×C3×C4—104no—
395C2×C3×C4—101no—
396C2×C3×C4—101no—
397C2×C3×C4—101no—
398C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
399C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
400C2×Dic3—202no—
401C3⋊C8—202no—
402C22×C5—21no—
403C22×C5—41no—
404C22×C5C2×#21711yescharacteristic
405C2×D5—21no—
406C2×D5—22no—
407C2×D5—41no—
408C2×D5—42no—
409C2×D5—41no—
410C2×D5#95312yes—
411AGL(1,𝔽5)—242no—
412C4×C5—22no—
413C4×C5—22no—
414C4×C5—21no—
415C4×C5C2×#21711yescharacteristic
416C4×C5C2×#21711yescharacteristic
417Dic5—21no—
418Dic5—22no—
419Dic5#95312yes—
420#41—54no—
421#41—102no—
422#41—102no—
423#41—102no—
424#41—102no—
425#41—108no—
426#41—108no—
427#41—102no—
428#41—102no—
429#41—108no—
430#41—102no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—51no—
435C2×Q8—54no—
436C2×Q8—51no—
437C2×Q8—101no—
438C2×Q8—101no—
439C2×Q8—102no—
440C2×Q8—102no—
441C2×Q8—108no—
442C2×Q8—102no—
443C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
444C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
445C2×D4—51no—
446C2×D4—51no—
447C2×D4—51no—
448C2×D4—51no—
449C2×D4—101no—
450C2×D4—101no—
451C2×D4—101no—
452C22×C4—21no—
453C22×C4—51no—
454C22×C4—51no—
455C22×C4—101no—
456C22×C4—101no—
457C22×C4—101no—
458C22×C4—101no—
459C22×C4—101no—
460C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
461M16—302no—
462M16—302no—
463C2×C8—302no—
464C4⋊C4—302no—
465C22⋊C4—302no—
466C42—302no—
467C3×C5#179611yescharacteristic, complemented, Hall
468C22×C3—21no—
469C22×C3—41no—
470C22×C3—52no—
471C22×C3—102no—
472C22×C3—101no—
473C22×C3—201no—
474C22×C3—201no—
475C22×C3—202no—
476C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
477C3×C4—22no—
478C3×C4—22no—
479C3×C4—21no—
480C3×C4—52no—
481C3×C4—101no—
482C3×C4—102no—
483C3×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
484C3×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
485Dic3—402no—
486C2×C5—21no—
487C2×C5—41no—
488C2×C5#524711yescharacteristic
489D5—22no—
490D5—42no—
491D5—41no—
492C23—51no—
493C23—51no—
494C23—101no—
495C23#676311yescharacteristic
496Q8—24no—
497Q8—21no—
498Q8—21no—
499Q8—52no—
500Q8—104no—
501Q8—102no—
502Q8—104no—
503Q8—101no—
504Q8#676312yes—
505Q8#676312yes—
506D4—52no—
507D4—102no—
508D4—104no—
509D4—104no—
510D4—102no—
511D4—102no—
512D4—104no—
513D4—101no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—51no—
520C2×C4—52no—
521C2×C4—52no—
522C2×C4—51no—
523C2×C4—104no—
524C2×C4—104no—
525C2×C4—104no—
526C2×C4—104no—
527C2×C4—104no—
528C2×C4—101no—
529C2×C4—101no—
530C2×C4—101no—
531C2×C4—602no—
532C2×C4#676311yescharacteristic
533C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
534C8—602no—
535C2×C3—21no—
536C2×C3—41no—
537C2×C3—102no—
538C2×C3—202no—
539C2×C3—201no—
540C2×C3#6580011yescharacteristic
541C5#7214411yescharacteristic, complemented, Sylow
542C22—21no—
543C22—41no—
544C22—52no—
545C22—102no—
546C22—101no—
547C22—201no—
548C22—201no—
549C22—202no—
550C22C2×#676311yescharacteristic
551C4—22no—
552C4—22no—
553C4—21no—
554C4—52no—
555C4—101no—
556C4—102no—
557C4—1202no—
558C4C2×#676311yescharacteristic
559C4C2×#676311yescharacteristic
560C3#1060497411yescharacteristic, complemented, Sylow
561C2—21no—
562C2—41no—
563C2—102no—
564C2—202no—
565C2—201no—
566C2#1175245911yescharacteristic
subgroupG/H
centerC2#11752459
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC22×C3×C5×Q8C4
socleC2×C3×C5#379
prime radicalC2×C3×#4147C2
cosocleC2×C3×C4×D5C23
hypercenterC22×Q8ΓL(1,𝔽16)
hypocenterC3×C5#1796
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. #11752443C2semidirect
3. #11752441C2semidirect
4. #11752420C2semidirect
5. #11752404C2semidirect
7. C22×C3×C5×Q8C4semidirect
11. C2×C3×#713C4semidirect
14. C2×C3×Q8×D5C4semidirect
18. C2×C3×#709C4semidirect
86. C22×C5×Q8Dic3semidirect
90. C2×#713Dic3semidirect
93. C2×Q8×D5Dic3semidirect
97. C2×#709Dic3semidirect
163. C22×C3×Q8AGL(1,𝔽5)semidirect
344. C22×Q8ΓL(1,𝔽16)semidirect
467. C3×C5#1796semidirect
541. C5#72144semidirect
560. C3#10604974semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#41472
#117524432
#117524412
#117524202
#117524042
#106049743
#721445