Finite Group Databasebeta

← #14417117

Subgroups — #14417117

Order 1920.

1012 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4054C211yesmaximal, characteristic
2#11754499C211yesmaximal, characteristic, complemented
3#11754488C211yesmaximal, characteristic, complemented
4#11751223C211yesmaximal, characteristic, complemented
5C5×#5121C211yesmaximal, characteristic, complemented
6#11748364C211yesmaximal, characteristic, complemented
7C2×#90964C211yesmaximal, characteristic, complemented
8#11747815C212yesmaximal, complemented
9#11747812C212yesmaximal, complemented
10#11747809C212yesmaximal, complemented
11#11747806C212yesmaximal, complemented
12#10610932—31nomaximal, Hall
13C2×C3×#709C2211yescharacteristic
14C22×C3×D20C2211yescharacteristic
15C2×C3×C5×M16C2211yescharacteristic
16#91815C2211yescharacteristic
17C22×C3⋊#168C2211yescharacteristic
18C5×#926C2211yescharacteristic
19C2×C5×C4⋊Dic3C2211yescharacteristic
20C3×#4054C418yescomplemented
21C2×C3×D40C2212yes—
22C2×C3×#677C2212yes—
23Dic3×C22⋊D5—22no—
24D5×#217—22no—
25#91434C2212yes—
26#91434C2212yes—
27#91434C2212yescomplemented
28Dic3×D20—21no—
29Dic3×D20—21no—
30D5×C4⋊Dic3—21no—
31#91358—21no—
32#91331C2212yescomplemented
33#91099C2212yescomplemented
34C8×Dic15C2212yescomplemented
35C5×#818C2212yescomplemented
36C5×#817C2212yescomplemented
37#90970C2211yescharacteristic, complemented
38#90970C2211yescharacteristic, complemented
39#90964C2212yes—
40#90964C2212yes—
41#90964C2211yescharacteristic, complemented
42#90964C2211yescharacteristic, complemented
43#83930—51nomaximal, Hall
44C2×#4054D311yescharacteristic
45#65855—31no—
46#65838—31no—
47C5×#428—31no—
48#65411—31no—
49#65291—31no—
50C2×#3941—31no—
51#65178—32no—
52#65161—32no—
53#65158—32no—
54#65141—32no—
55C23×C3×D5—21no—
56C22×C3×C4×C5C2311yescharacteristic
57C22×C5×Dic3—21no—
58C2×C3×C22⋊D5—21no—
59C3×#709—22no—
60C3×#709C2×C414yescomplemented
61C2×C3×D20—21no—
62C2×C3×D20—22no—
63C2×C3×D20C2×C414yescomplemented
64C2×C3×D20C2312yes—
65C2×C3×D20C2311yescharacteristic
66C2×C3×C4×D5—21no—
67C2×C3×Dic10C2311yescharacteristic
68C2×D5×Dic3—22no—
69C2×D5×Dic3—22no—
70C2×D5×Dic3—22no—
71C3×C5×M16C2×C414yescomplemented
72C2×C3×C5×C8C2312yes—
73#6719D412yes—
74C4⋊Dic15D411yescharacteristic
75C4⋊Dic15D411yescharacteristic
76C4×Dic15C2312yes—
77C2×C3⋊#168—22no—
78C2×C3⋊#168C2311yescharacteristic
79C2×C3⋊#168C2311yescharacteristic
80C5×#217—21no—
81C5×C4⋊Dic3—21no—
82C5×C4⋊Dic3C2311yescharacteristic
83C5×C4⋊Dic3C2311yescharacteristic
84C5×C4⋊Dic3C2312yes—
85C4×C5×Dic3—21no—
86C3×D40C2×C418yescomplemented
87C3×#677C2×C418yescomplemented
88#6668—22no—
89#6665—22no—
90#6665—22no—
91#6665—22no—
92Dic3×Dic5—22no—
93C2×C3×AGL(1,ℤ/8)—51no—
94#5827—51no—
95#5798—51no—
96#5233—51no—
97C2×#832—51no—
98#5164—52no—
99Dic3×SD16—52no—
100#5152—52no—
101Dic3×D8—52no—
102#5122—51no—
103#5121D511yescharacteristic, complemented
104C2×#709C2×D311yescharacteristic
105C22×D20C2×D311yescharacteristic
106C2×C5×M16C2×D311yescharacteristic
107C5×#117—31no—
108C2×C5×C4⋊C4—31no—
109C4×C22⋊D5—62no—
110#4072—31no—
111C22×#168—31no—
112#4054Dic318yescomplemented
113C2×D40C2×D312yes—
114C2×#677C2×D312yes—
115#4039—32no—
116#4039—32no—
117#4039—32no—
118#4038—61no—
119D5×C4⋊C4—61no—
120#4033—32no—
121D5×C22⋊C4—62no—
122C4×D20—61no—
123C4×D20—61no—
124C5×#107—32no—
125C5×#106—32no—
126#3947—32no—
127C8×Dic5—32no—
128#3942—31no—
129#3942—31no—
130#3941—31no—
131#3941—31no—
132#3941—32no—
133#3941—32no—
134#2946—151noSylow
135C23×C3×C5C2×D411yescharacteristic
136C22×C3×D5—21no—
137C22×C3×D5—22no—
138C22×C3×D5—24no—
139C22×C3×D5—81no—
140C22×C3×D5—81no—
141C2×C3×C4×C5C22×C412yes—
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5#4112yes—
144C2×C3×C4×C5C2411yescharacteristic
145C2×Dic15C2×D412yes—
146C2×Dic15C2×D412yes—
147C2×C5×Dic3—22no—
148C2×C5×Dic3—21no—
149C2×C5×Dic3—21no—
150C2×C5×Dic3—81no—
151C3×C22⋊D5—24no—
152C2×C3×Dic5—21no—
153C3×D20—24no—
154C3×D20—22no—
155C3×D20—21no—
156C3×D20C22×C414yes—
157C3×D20C22×C412yes—
158C3×C4×D5—24no—
159C3×Dic10—21no—
160C3×Dic10C22×C412yes—
161D5×Dic3—82no—
162D5×Dic3—82no—
163D5×Dic3—82no—
164C3×C5×C8C22×C414yes—
165C3⋊#168—21no—
166C3⋊#168—21no—
167C2×C3×#41—51no—
168C22×C3×D4—51no—
169C3×AGL(1,ℤ/8)—58no—
170C2×C3×SD16—52no—
171C2×C3×D8—52no—
172C2×C3×M16C2×D511yescharacteristic
173C2×#217—102no—
174Q8×Dic3—52no—
175D4×Dic3—52no—
176D4×Dic3—52no—
177D4×Dic3—52no—
178D4×Dic3—101no—
179D4×Dic3—101no—
180D4×Dic3—102no—
181#926—51no—
182#926—101no—
183#926C2×D511yescharacteristic
184C2×C4⋊Dic3—101no—
185C2×C4⋊Dic3C2×D511yescharacteristic
186C22×C3⋊C8—51no—
187#835—51no—
188#835—51no—
189#832—51no—
190#832—52no—
191#832—51no—
192#832—52no—
193#818C2×D512yescomplemented
194#817C2×D512yescomplemented
195#815—52no—
196C8×Dic3—52no—
197C23×D5—21no—
198C22×C4×C5—61no—
199C22×C4×C5C22×D311yescharacteristic
200C2×C22⋊D5—21no—
201#709—22no—
202#709C2×Dic314yescomplemented
203C2×D20—21no—
204C2×D20—22no—
205C2×D20C2×Dic314yescomplemented
206C2×D20C22×D312yes—
207C2×D20C22×D311yescharacteristic
208C2×C4×D5—21no—
209C2×C4×D5—62no—
210C2×C4×D5—62no—
211C2×C4×D5—62no—
212C2×Dic10C22×D311yescharacteristic
213C5×M16C2×Dic314yescomplemented
214C2×C5×C8C22×D312yes—
215C5×C4⋊C4—32no—
216C5×C4⋊C4—31no—
217C5×C4⋊C4—31no—
218C5×C4⋊C4—61no—
219C5×C22⋊C4—61no—
220C42×C5—61no—
221C22⋊Dic5—32no—
222#685—62no—
223#685—62no—
224#685—62no—
225C4⋊Dic5—31no—
226C4⋊Dic5—31no—
227#683—62no—
228C4×Dic5—32no—
229C4×Dic5—62no—
230C2×#168—31no—
231C2×#168—31no—
232C2×#168—62no—
233D40C2×Dic318yescomplemented
234#677C2×Dic318yescomplemented
235C2×AGL(1,ℤ/8)—51no—
236#520—151no—
237#518—151no—
238#442—152no—
239#440—152no—
240C4×SD16—152no—
241C4×D8—152no—
242#428—31no—
243#416—151no—
244C2×#102—151no—
245#405—151no—
246C22×C3×C5—21no—
247C22×C3×C5—21no—
248C22×C3×C5C4×D412yes—
249C22×C3×C5C22×D411yescharacteristic
250C2×C3×D5—22no—
251C2×C3×D5—24no—
252C2×C3×D5—41no—
253C2×C3×D5—42no—
254C2×C3×D5—42no—
255C2×C3×D5—41no—
256C2×C3×D5—84no—
257C3×C4×C5C4×D412yes—
258C3×C4×C5C23×C411yescharacteristic
259C3×C4×C5C2×#4111yescharacteristic
260Dic15—22no—
261Dic15—22no—
262C3×Dic5—22no—
263C5×Dic3—41no—
264C5×Dic3—42no—
265C5×Dic3—41no—
266C24×C3—101no—
267C3×#41—54no—
268C3×#41—102no—
269C2×C3×Q8—51no—
270C2×C3×D4—54no—
271C2×C3×D4—51no—
272C2×C3×D4—52no—
273C2×C3×D4—101no—
274C2×C3×D4—101no—
275C2×C3×D4—102no—
276C22×C3×C4—101no—
277C22×C3×C4C22×D511yescharacteristic
278C22×Dic3—21no—
279C22×Dic3—102no—
280C22×Dic3—102no—
281C22×Dic3—102no—
282C3×SD16—58no—
283C3×D8—58no—
284C3×M16C4×D514yescomplemented
285C2×C3×C8C22×D512yes—
286#217—21no—
287#217—52no—
288#217—102no—
289#217—102no—
290#217—102no—
291C4⋊Dic3—21no—
292C4⋊Dic3—51no—
293C4⋊Dic3—51no—
294C4⋊Dic3—102no—
295C4⋊Dic3C22×D511yescharacteristic
296C4⋊Dic3C22×D512yes—
297C4⋊Dic3C22×D511yescharacteristic
298C4×Dic3—21no—
299C4×Dic3—52no—
300C4×Dic3—102no—
301C2×C3⋊C8—51no—
302C2×C3⋊C8—51no—
303C2×C3⋊C8—102no—
304C23×C5D3×D411yescharacteristic
305C22×D5—21no—
306C22×D5—22no—
307C22×D5—24no—
308C22×D5—81no—
309C22×D5—81no—
310C2×C4×C5—62no—
311C2×C4×C5—61no—
312C2×C4×C5—61no—
313C2×C4×C5—241no—
314C2×C4×C5D3×D411yescharacteristic
315C2×C4×C5#23712yes—
316C2×C4×C5C22×Dic312yes—
317C2×C4×C5C23×D311yescharacteristic
318C22⋊D5—24no—
319C2×Dic5—21no—
320C2×Dic5—32no—
321C2×Dic5—32no—
322D20—24no—
323D20—22no—
324D20—21no—
325D20C22×Dic314yes—
326D20C22×Dic312yes—
327C4×D5—24no—
328C4×D5—242no—
329C4×D5—242no—
330C4×D5—242no—
331Dic10—21no—
332Dic10C22×Dic312yes—
333C5×C8C22×Dic314yes—
334#168—61no—
335#168—61no—
336C2×#41—51no—
337C22×D4—51no—
338AGL(1,ℤ/8)—58no—
339C2×SD16—52no—
340C2×D8—52no—
341C2×M16D3×D511yescharacteristic
342C22×C8—151no—
343C4×Q8—152no—
344C4×D4—152no—
345C4×D4—152no—
346C4×D4—152no—
347C4×D4—301no—
348C4×D4—301no—
349C4×D4—302no—
350#117—31no—
351#117—151no—
352#117—301no—
353C2×C4⋊C4—31no—
354C2×C4⋊C4—301no—
355C2×C22⋊C4—302no—
356#107—32no—
357#106—32no—
358#103—151no—
359#103—151no—
360#102—152no—
361#102—152no—
362#102—151no—
363#102—151no—
364#97—152no—
365C4×C8—152no—
366C2×C3×C5—22no—
367C2×C3×C5C2×C4×D411yescharacteristic
368C2×C3×C5#57611yescharacteristic
369C2×C3×C5#57711yescharacteristic
370C3×D5—42no—
371C3×D5—44no—
372C23×C3—101no—
373C23×C3—102no—
374C23×C3—104no—
375C23×C3—401no—
376C23×C3—401no—
377C23×C3D4×D511yescharacteristic
378C3×Q8—52no—
379C3×Q8—101no—
380C3×D4—52no—
381C3×D4—54no—
382C3×D4—104no—
383C3×D4—102no—
384C3×D4—101no—
385C3×D4—104no—
386C2×C3×C4—101no—
387C2×C3×C4—104no—
388C2×C3×C4C2×C4×D512yes—
389C2×C3×C4D4×D511yescharacteristic
390C2×C3×C4#71312yes—
391C2×C3×C4C23×D511yescharacteristic
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—22no—
395C2×Dic3—52no—
396C2×Dic3—52no—
397C2×Dic3—81no—
398C2×Dic3—402no—
399C2×Dic3—402no—
400C2×Dic3—402no—
401C3×C8C2×C4×D514yes—
402C3⋊C8—101no—
403C3⋊C8—101no—
404C22×C5—21no—
405C22×C5—21no—
406C22×C5D4×Dic312yes—
407C22×C5C2×D3×D411yescharacteristic
408C2×D5—22no—
409C2×D5—24no—
410C2×D5—41no—
411C2×D5—42no—
412C2×D5—42no—
413C2×D5—41no—
414C2×D5—84no—
415C4×C5—121no—
416C4×C5—122no—
417C4×C5—121no—
418C4×C5D4×Dic312yes—
419C4×C5C2×#23711yescharacteristic
420C4×C5C23×Dic311yescharacteristic
421Dic5—22no—
422Dic5—62no—
423Dic5—62no—
424C24—101no—
425#41—54no—
426#41—102no—
427C2×Q8—51no—
428C2×D4—54no—
429C2×D4—51no—
430C2×D4—52no—
431C2×D4—101no—
432C2×D4—102no—
433C2×D4—101no—
434C22×C4—61no—
435C22×C4—101no—
436C22×C4—302no—
437C22×C4—302no—
438C22×C4—302no—
439C22×C4C2×D3×D511yescharacteristic
440SD16—58no—
441D8—58no—
442M16D5×Dic314yescomplemented
443C2×C8—151no—
444C2×C8—151no—
445C2×C8—302no—
446C2×C8C2×D3×D512yes—
447C4⋊C4—32no—
448C4⋊C4—31no—
449C4⋊C4—31no—
450C4⋊C4—61no—
451C4⋊C4—151no—
452C4⋊C4—151no—
453C4⋊C4—302no—
454C22⋊C4—61no—
455C22⋊C4—152no—
456C22⋊C4—302no—
457C22⋊C4—302no—
458C22⋊C4—302no—
459C42—61no—
460C42—152no—
461C42—302no—
462C3×C5#294611yescharacteristic, complemented, Hall
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—102no—
466C22×C3—104no—
467C22×C3—201no—
468C22×C3—202no—
469C22×C3—202no—
470C22×C3—201no—
471C22×C3—404no—
472C22×C3#403912yes—
473C22×C3C2×D4×D511yescharacteristic
474C3×C4—102no—
475C3×C4#403912yes—
476C3×C4C22×C4×D511yescharacteristic
477C3×C4C2×#71311yescharacteristic
478Dic3—42no—
479Dic3—41no—
480Dic3—41no—
481Dic3—102no—
482Dic3—102no—
483C2×C5—22no—
484C2×C5#577411yescharacteristic
485C2×C5#577511yescharacteristic
486C2×C5C2×D4×Dic311yescharacteristic
487D5—44no—
488D5—42no—
489C23—101no—
490C23—102no—
491C23—104no—
492C23—401no—
493C23—401no—
494C23#677711yescharacteristic
495Q8—52no—
496Q8—101no—
497D4—52no—
498D4—54no—
499D4—104no—
500D4—104no—
501D4—102no—
502D4—101no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—62no—
506C2×C4—101no—
507C2×C4—104no—
508C2×C4—152no—
509C2×C4—152no—
510C2×C4—241no—
511C2×C4—1202no—
512C2×C4—1202no—
513C2×C4—1202no—
514C2×C4#676612yes—
515C2×C4#677711yescharacteristic
516C2×C4C2×D5×Dic312yes—
517C2×C4C22×D3×D511yescharacteristic
518C8—301no—
519C8—301no—
520C8C2×D5×Dic314yes—
521C2×C3—22no—
522C2×C3—204no—
523C2×C3—202no—
524C2×C3C2×#403911yescharacteristic
525C2×C3#6610811yescharacteristic
526C2×C3#6611211yescharacteristic
527C5#8393011yescharacteristic, complemented, Sylow
528C22—21no—
529C22—21no—
530C22—102no—
531C22—104no—
532C22—202no—
533C22—201no—
534C22—201no—
535C22—202no—
536C22—404no—
537C22#9143412yes—
538C22C2×#677711yescharacteristic
539C4—102no—
540C4—121no—
541C4—122no—
542C4—121no—
543C4—302no—
544C4—302no—
545C4#9143412yes—
546C4C2×#676611yescharacteristic
547C4C22×D5×Dic311yescharacteristic
548C3#1061093211yescharacteristic, complemented, Sylow
549C2—22no—
550C2—204no—
551C2—202no—
552C2#1175403711yescharacteristic
553C2#1175403811yescharacteristic
554C2C2×#9143411yescharacteristic
subgroupG/H
centerC22C2×#6777
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C22×D4
prime radicalC2×C3×#4054C2
cosocleC2×C3×C4×C5C24
hypercenterC2×M16D3×D5
hypocenterC3×C5#2946
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. #11754499C2semidirect
3. #11754488C2semidirect
4. #11751223C2semidirect
5. C5×#5121C2semidirect
6. #11748364C2semidirect
7. C2×#90964C2semidirect
8. #11747815C2semidirect
9. #11747812C2semidirect
10. #11747809C2semidirect
11. #11747806C2semidirect
20. C3×#4054C4semidirect
27. #91434C22semidirect
32. #91331C22semidirect
33. #91099C22semidirect
34. C8×Dic15C22semidirect
35. C5×#818C22semidirect
36. C5×#817C22semidirect
37. #90970C22semidirect
38. #90970C22semidirect
41. #90964C22semidirect
42. #90964C22semidirect
60. C3×#709C2×C4semidirect
63. C2×C3×D20C2×C4semidirect
71. C3×C5×M16C2×C4semidirect
86. C3×D40C2×C4semidirect
87. C3×#677C2×C4semidirect
103. #5121D5semidirect
112. #4054Dic3semidirect
193. #818C2×D5semidirect
194. #817C2×D5semidirect
202. #709C2×Dic3semidirect
205. C2×D20C2×Dic3semidirect
213. C5×M16C2×Dic3semidirect
233. D40C2×Dic3semidirect
234. #677C2×Dic3semidirect
284. C3×M16C4×D5semidirect
442. M16D5×Dic3semidirect
462. C3×C5#2946semidirect
527. C5#83930semidirect
548. C3#10610932semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40542
#117544992
#117544882
#117512232
C5×#51212
#117483642
C2×#909642
#117478152
#117478122
#117478092
#117478062
#106109323
#839305