Finite Group Databasebeta

← #14417257

Subgroups — #14417257

Order 1920.

940 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5812C211yesmaximal, characteristic, complemented
2C2×C3×#4088C211yesmaximal, characteristic
3#11754509C211yesmaximal, characteristic
4C2×#91119C211yesmaximal, characteristic, complemented
5#11749256C212yesmaximal, complemented
6Dic3×#689C212yesmaximal, complemented
7#11749234C212yesmaximal, complemented
8Dic3×#688C212yesmaximal, complemented
9#11748460C211yesmaximal, characteristic, complemented
10#11748364C211yesmaximal, characteristic
11#11748327C211yesmaximal, characteristic, complemented
12#10614278—31nomaximal, Hall
13C22×C3×C5×Q8C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C2×C4⋊Dic15C2211yescharacteristic
16C22×C3⋊#168C2211yescharacteristic
17C5×Q8×Dic3—21no—
18C5×Q8×Dic3—21no—
19C5×Q8×Dic3C2212yes—
20C5×Q8×Dic3C2212yes—
21C5×#926—22no—
22C5×#926C2211yescharacteristic
23C5×#926C2211yescharacteristic
24C2×C3×#689C2212yes—
25C3×#4088C418yescomplemented
26C2×C3×#688C2212yes—
27C2×C3×#681C2211yescharacteristic
28#91559—22no—
29Dic3×D20C2212yes—
30D5×C4⋊Dic3—21no—
31#91358—21no—
32Dic3×Dic10C2212yes—
33#91119C2212yes—
34#91119C2212yes—
35#90988C2212yes—
36#90985C2212yes—
37#90970C2211yescharacteristic
38#90970C2211yescharacteristic
39#90964C2211yescharacteristic
40#90964C2211yescharacteristic
41#90953C2212yes—
42Dic3×#168C2212yes—
43#83965—51nomaximal, Hall
44C5×#519—31no—
45C2×#4088D311yescharacteristic
46#65857—31no—
47C2×#3967—31no—
48#65319—32no—
49C4×#689—32no—
50#65315—32no—
51C4×#688—32no—
52#65291—31no—
53#65286—31no—
54#65284—31no—
55C2×C3×C5×Q8—21no—
56C2×C3×C5×Q8—22no—
57C2×C3×C5×Q8C2×C414yescomplemented
58C2×C3×C5×Q8C2312yes—
59C22×C3×C4×C5—21no—
60C22×C3×C4×C5C2311yescharacteristic
61C22×Dic15—21no—
62C2×C3×C22⋊D5—21no—
63C3×#709—22no—
64C3×#709C2×C414yes—
65C2×C3×D20C2311yescharacteristic
66C2×C3×C4×D5—21no—
67C2×C3×Dic10C2311yescharacteristic
68C2×D5×Dic3—22no—
69C4⋊Dic15—21no—
70C4⋊Dic15C2312yes—
71C2×C3⋊#168—22no—
72C2×C3⋊#168C2311yescharacteristic
73C2×C3⋊#168C2311yescharacteristic
74C5×#217—21no—
75C5×#217D412yes—
76C5×C4⋊Dic3—22no—
77C5×C4⋊Dic3—22no—
78C5×C4⋊Dic3D411yescharacteristic
79C5×C4⋊Dic3D411yescharacteristic
80C5×C4⋊Dic3C2311yescharacteristic
81C5×C4⋊Dic3C2311yescharacteristic
82C4×C5×Dic3—21no—
83C4×C5×Dic3—22no—
84C4×C5×Dic3—22no—
85C4×C5×Dic3C2312yes—
86C3×#689C2×C418yescomplemented
87C3×#688C2×C418yescomplemented
88C3×#681C2×C414yescomplemented
89C2×C3×#168C2312yes—
90#6668—22no—
91#6665—22no—
92Dic3×Dic5—22no—
93C2×C3×#137—51no—
94#5827—51no—
95#5812D511yescharacteristic, complemented
96#5233—51no—
97C2×#835—51no—
98#5184—52no—
99Dic3×Q16—52no—
100#5164—52no—
101Dic3×SD16—52no—
102#5122—51no—
103#5121—51no—
104C22×C5×Q8C2×D311yescharacteristic
105C2×#709C2×D311yescharacteristic
106C4×C5×Q8—32no—
107C4×C5×Q8—32no—
108C4×C5×Q8—61no—
109C4×C5×Q8—61no—
110C5×#117—31no—
111C5×#117—31no—
112C5×#117—62no—
113C2×#689C2×D312yes—
114#4088Dic318yescomplemented
115C2×#688C2×D312yes—
116C2×C4⋊Dic5—31no—
117C2×#681C2×D311yescharacteristic
118C22×#168—31no—
119#4038—61no—
120D5×C4⋊C4—61no—
121#4027—62no—
122C4×D20—32no—
123C4×Dic10—32no—
124#3967—32no—
125#3967—32no—
126#3942—31no—
127#3942—31no—
128#3941—31no—
129#3941—31no—
130#3940—32no—
131#3939—32no—
132#3935—32no—
133C4×#168—32no—
134#2947—151noSylow
135C23×C3×C5C2×D411yescharacteristic
136C22×C3×D5—21no—
137C3×C5×Q8—24no—
138C3×C5×Q8—22no—
139C3×C5×Q8C22×C414yes—
140C2×C3×C4×C5—24no—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5C22×C412yes—
143C2×C3×C4×C5C2×D411yescharacteristic
144C2×C3×C4×C5#4112yes—
145C2×C3×C4×C5C2411yescharacteristic
146C2×Dic15—22no—
147C2×Dic15—81no—
148C2×C5×Dic3—21no—
149C2×C5×Dic3—21no—
150C2×C5×Dic3C2×D412yes—
151C2×C5×Dic3C2×D412yes—
152C3×C22⋊D5—24no—
153C2×C3×Dic5—21no—
154C3×D20—21no—
155C3×D20C22×C412yes—
156C3×C4×D5—24no—
157C3×Dic10—21no—
158C3×Dic10C22×C412yes—
159D5×Dic3—82no—
160C3⋊#168—21no—
161C3⋊#168—21no—
162C3×#168C22×C414yes—
163C2×C3×#41—51no—
164C22×C3×Q8C2×D511yescharacteristic
165C3×#137—58no—
166C2×C3×Q16—52no—
167C2×C3×SD16—52no—
168C2×C3×M16—51no—
169Q8×Dic3—21no—
170Q8×Dic3—21no—
171Q8×Dic3—52no—
172Q8×Dic3C2×D512yes—
173Q8×Dic3C2×D512yes—
174D4×Dic3—52no—
175D4×Dic3—102no—
176#926—22no—
177#926—101no—
178#926C2×D511yescharacteristic
179#926C2×D511yescharacteristic
180C2×C4⋊Dic3—51no—
181C2×C4⋊Dic3—101no—
182C22×C3⋊C8—51no—
183#835—52no—
184#835—52no—
185#835—51no—
186#835—51no—
187#832—51no—
188#832—51no—
189#818—52no—
190#817—52no—
191#815—52no—
192C8×Dic3—52no—
193C2×C5×Q8—21no—
194C2×C5×Q8—22no—
195C2×C5×Q8C2×Dic314yescomplemented
196C2×C5×Q8C22×D312yes—
197C22×C4×C5—21no—
198C22×C4×C5C22×D311yescharacteristic
199C2×C22⋊D5—21no—
200C22×Dic5—61no—
201#709—22no—
202#709C2×Dic314yes—
203C2×D20C22×D311yescharacteristic
204C2×C4×D5—21no—
205C2×C4×D5—62no—
206C2×Dic10C22×D311yescharacteristic
207C5×C4⋊C4—31no—
208C5×C4⋊C4—31no—
209C5×C4⋊C4—31no—
210C5×C4⋊C4—31no—
211C5×C4⋊C4—62no—
212C5×C4⋊C4—62no—
213C5×C22⋊C4—32no—
214C5×C22⋊C4—61no—
215C42×C5—32no—
216C42×C5—62no—
217C42×C5—62no—
218C42×C5—61no—
219#689C2×Dic318yescomplemented
220#688C2×Dic318yescomplemented
221#685—62no—
222C4⋊Dic5—32no—
223C4⋊Dic5—61no—
224#683—62no—
225C4×Dic5—62no—
226#681C2×Dic314yescomplemented
227C2×#168—31no—
228C2×#168—31no—
229C2×#168—62no—
230C2×#168C22×D312yes—
231C2×#137—51no—
232#520—151no—
233#519—31no—
234#441—152no—
235#440—152no—
236C4×Q16—152no—
237C4×SD16—152no—
238#428—151no—
239#416—151no—
240C2×#103—151no—
241#405—151no—
242C22×C3×C5—21no—
243C22×C3×C5—21no—
244C22×C3×C5C4×D412yes—
245C22×C3×C5C22×D411yescharacteristic
246C2×C3×D5—22no—
247C2×C3×D5—41no—
248C2×C3×D5—41no—
249C3×C4×C5—24no—
250C3×C4×C5C4×D412yes—
251C3×C4×C5C23×C411yescharacteristic
252C3×C4×C5C2×#4111yescharacteristic
253Dic15—42no—
254C3×Dic5—22no—
255C5×Dic3—22no—
256C5×Dic3—22no—
257C5×Dic3—41no—
258C5×Dic3—41no—
259C3×#41—54no—
260C3×#41—102no—
261C2×C3×Q8—22no—
262C2×C3×Q8—21no—
263C2×C3×Q8—51no—
264C2×C3×Q8C4×D514yescomplemented
265C2×C3×Q8C22×D512yes—
266C2×C3×D4—51no—
267C2×C3×D4—101no—
268C22×C3×C4—21no—
269C22×C3×C4—101no—
270C22×C3×C4C22×D511yescharacteristic
271C22×Dic3—101no—
272C22×Dic3—102no—
273C3×Q16—58no—
274C3×SD16—58no—
275C3×M16—54no—
276C2×C3×C8—52no—
277#217—21no—
278#217—102no—
279#217C22⋊D512yes—
280C4⋊Dic3—22no—
281C4⋊Dic3—22no—
282C4⋊Dic3—52no—
283C4⋊Dic3—101no—
284C4⋊Dic3—102no—
285C4⋊Dic3C22⋊D511yescharacteristic
286C4⋊Dic3C22⋊D511yescharacteristic
287C4⋊Dic3C22×D511yescharacteristic
288C4⋊Dic3C22×D511yescharacteristic
289C4×Dic3—21no—
290C4×Dic3—22no—
291C4×Dic3—22no—
292C4×Dic3—102no—
293C4×Dic3C22×D512yes—
294C2×C3⋊C8—51no—
295C2×C3⋊C8—51no—
296C2×C3⋊C8—102no—
297C23×C5D3×D411yescharacteristic
298C22×D5—21no—
299C5×Q8—24no—
300C5×Q8—22no—
301C5×Q8C22×Dic314yes—
302C2×C4×C5—24no—
303C2×C4×C5—22no—
304C2×C4×C5—32no—
305C2×C4×C5—32no—
306C2×C4×C5—61no—
307C2×C4×C5—61no—
308C2×C4×C5D3×D411yescharacteristic
309C2×C4×C5#23712yes—
310C2×C4×C5C22×Dic312yes—
311C2×C4×C5C23×D311yescharacteristic
312C22⋊D5—24no—
313C2×Dic5—21no—
314C2×Dic5—62no—
315C2×Dic5—241no—
316D20—21no—
317D20C22×Dic312yes—
318C4×D5—24no—
319C4×D5—242no—
320Dic10—21no—
321Dic10C22×Dic312yes—
322#168—61no—
323#168—61no—
324#168C22×Dic314yes—
325C2×#41—51no—
326C22×Q8D3×D511yescharacteristic
327#137—58no—
328C2×Q16—52no—
329C2×SD16—52no—
330C2×M16—51no—
331C22×C8—151no—
332C4×Q8—32no—
333C4×Q8—32no—
334C4×Q8—61no—
335C4×Q8—61no—
336C4×Q8—152no—
337C4×D4—152no—
338C4×D4—302no—
339#117—31no—
340#117—31no—
341#117—62no—
342#117—301no—
343C2×C4⋊C4—151no—
344C2×C4⋊C4—301no—
345#107—152no—
346#106—152no—
347#103—151no—
348#103—151no—
349#103—152no—
350#103—152no—
351#102—151no—
352#102—151no—
353#97—152no—
354C4×C8—152no—
355C2×C3×C5—22no—
356C2×C3×C5C2×C4×D411yescharacteristic
357C2×C3×C5#57711yescharacteristic
358C2×C3×C5#57811yescharacteristic
359C3×D5—42no—
360C23×C3—101no—
361C23×C3C2×C22⋊D511yescharacteristic
362C3×Q8—22no—
363C3×Q8—24no—
364C3×Q8—52no—
365C3×Q8—101no—
366C3×Q8C2×C4×D514yes—
367C3×D4—52no—
368C3×D4—104no—
369C3×D4—101no—
370C2×C3×C4—22no—
371C2×C3×C4—24no—
372C2×C3×C4—101no—
373C2×C3×C4—104no—
374C2×C3×C4C2×C4×D512yes—
375C2×C3×C4#70912yes—
376C2×C3×C4C2×C22⋊D511yescharacteristic
377C2×C3×C4C23×D511yescharacteristic
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—102no—
381C2×Dic3—402no—
382C2×Dic3—401no—
383C2×Dic3C2×C22⋊D512yes—
384C2×Dic3C2×C22⋊D512yes—
385C3×C8—54no—
386C3⋊C8—101no—
387C3⋊C8—101no—
388C22×C5—21no—
389C22×C5—21no—
390C22×C5D4×Dic312yes—
391C22×C5C2×D3×D411yescharacteristic
392C2×D5—22no—
393C2×D5—41no—
394C2×D5—41no—
395C4×C5—24no—
396C4×C5—62no—
397C4×C5—62no—
398C4×C5—121no—
399C4×C5—121no—
400C4×C5D4×Dic312yes—
401C4×C5C2×#23711yescharacteristic
402C4×C5C23×Dic311yescharacteristic
403Dic5—22no—
404Dic5—122no—
405#41—54no—
406#41—102no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—51no—
410C2×Q8D5×Dic314yescomplemented
411C2×Q8C2×D3×D512yes—
412C2×D4—51no—
413C2×D4—101no—
414C22×C4—21no—
415C22×C4—101no—
416C22×C4—302no—
417C22×C4—301no—
418C22×C4C2×D3×D511yescharacteristic
419Q16—58no—
420SD16—58no—
421M16—54no—
422C2×C8—52no—
423C2×C8—151no—
424C2×C8—151no—
425C2×C8—302no—
426C4⋊C4—31no—
427C4⋊C4—31no—
428C4⋊C4—31no—
429C4⋊C4—31no—
430C4⋊C4—62no—
431C4⋊C4—62no—
432C4⋊C4—152no—
433C4⋊C4—302no—
434C4⋊C4—301no—
435C22⋊C4—32no—
436C22⋊C4—61no—
437C22⋊C4—302no—
438C42—32no—
439C42—62no—
440C42—62no—
441C42—61no—
442C42—302no—
443C3×C5#294711yescharacteristic, complemented, Hall
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—102no—
447C22×C3—201no—
448C22×C3—201no—
449C22×C3C4×C22⋊D512yes—
450C22×C3C22×C22⋊D511yescharacteristic
451C3×C4—24no—
452C3×C4—102no—
453C3×C4C4×C22⋊D512yes—
454C3×C4C22×C4×D511yescharacteristic
455C3×C4C2×#70911yescharacteristic
456Dic3—22no—
457Dic3—22no—
458Dic3—41no—
459Dic3—41no—
460Dic3—202no—
461C2×C5—22no—
462C2×C5#577811yescharacteristic
463C2×C5#577911yescharacteristic
464C2×C5C2×D4×Dic311yescharacteristic
465D5—42no—
466C23—101no—
467C23D3×C22⋊D511yescharacteristic
468Q8—22no—
469Q8—24no—
470Q8—52no—
471Q8—101no—
472Q8C2×D5×Dic314yes—
473D4—52no—
474D4—104no—
475D4—101no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—32no—
479C2×C4—32no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—101no—
483C2×C4—104no—
484C2×C4—302no—
485C2×C4—1202no—
486C2×C4—1201no—
487C2×C4C2×D5×Dic312yes—
488C2×C4#677912yes—
489C2×C4D3×C22⋊D511yescharacteristic
490C2×C4C22×D3×D511yescharacteristic
491C8—54no—
492C8—301no—
493C8—301no—
494C2×C3—22no—
495C2×C3—202no—
496C2×C3C2×C4×C22⋊D511yescharacteristic
497C2×C3#6617111yescharacteristic
498C2×C3#6617211yescharacteristic
499C5#8396511yescharacteristic, complemented, Sylow
500C22—21no—
501C22—21no—
502C22—102no—
503C22—201no—
504C22—201no—
505C22Dic3×C22⋊D512yes—
506C22C2×D3×C22⋊D511yescharacteristic
507C4—24no—
508C4—62no—
509C4—62no—
510C4—102no—
511C4—121no—
512C4—121no—
513C4—602no—
514C4Dic3×C22⋊D512yes—
515C4C22×D5×Dic311yescharacteristic
516C4C2×#677911yescharacteristic
517C3#1061427811yescharacteristic, complemented, Sylow
518C2—22no—
519C2—202no—
520C2#1175509211yescharacteristic
521C2#1175511411yescharacteristic
522C2C2×Dic3×C22⋊D511yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×Q8C22
socleC22×C3×C5C22×D4
prime radicalC2×C3×#709C22
cosocleC2×C3×C4×C5C24
hypercenterC22×Q8D3×D5
hypocenterC3×C5#2947
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C5×#5812C2semidirect
4. C2×#91119C2semidirect
5. #11749256C2semidirect
6. Dic3×#689C2semidirect
7. #11749234C2semidirect
8. Dic3×#688C2semidirect
9. #11748460C2semidirect
11. #11748327C2semidirect
25. C3×#4088C4semidirect
57. C2×C3×C5×Q8C2×C4semidirect
86. C3×#689C2×C4semidirect
87. C3×#688C2×C4semidirect
88. C3×#681C2×C4semidirect
95. #5812D5semidirect
114. #4088Dic3semidirect
195. C2×C5×Q8C2×Dic3semidirect
219. #689C2×Dic3semidirect
220. #688C2×Dic3semidirect
226. #681C2×Dic3semidirect
264. C2×C3×Q8C4×D5semidirect
410. C2×Q8D5×Dic3semidirect
443. C3×C5#2947semidirect
499. C5#83965semidirect
517. C3#10614278semidirect

Maximal subgroups

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

subgroupindex
C5×#58122
C2×C3×#40882
#117545092
C2×#911192
#117492562
Dic3×#6892
#117492342
Dic3×#6882
#117484602
#117483642
#117483272
#106142783
#839655