Finite Group Databasebeta

← #14417277

Subgroups — #14417277

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
1C2×#91840C211yesmaximal, characteristic
2C3×#66161C211yesmaximal, characteristic
3#11754213C211yesmaximal, characteristic, complemented
4C5×#5233C211yesmaximal, characteristic
5#11749250C212yesmaximal, complemented
6#11749244C212yesmaximal, complemented
7#11749004C212yesmaximal, complemented
8#11748998C212yesmaximal, complemented
9#11748470C211yesmaximal, characteristic, complemented
10C2×#90974C211yesmaximal, characteristic, complemented
11#11748318C211yesmaximal, characteristic, complemented
12#10610552—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C22×Dic30C2211yescharacteristic
15#91840C418yescomplemented
16C2×#6718C2212yes—
17C2×#6716C2212yes—
18C2×#6710C2211yescharacteristic
19C5×#926C2211yescharacteristic
20C22×C5×C3⋊C8C2211yescharacteristic
21C3×Q8×Dic5C2212yes—
22C3×D4×Dic5—22no—
23C3×D4×Dic5C2212yes—
24C3×#4072—21no—
25C3×#4072C2211yescharacteristic
26C2×C3×C4⋊Dic5—21no—
27C2×C3×C4⋊Dic5C2211yescharacteristic
28#91532—22no—
29#91340—21no—
30#91340—21no—
31#91333C2212yes—
32#91333C2212yes—
33C5×#835C2211yescharacteristic
34C5×#835C2211yescharacteristic
35C5×#832C2211yescharacteristic
36C5×#832C2211yescharacteristic
37#90989C2212yes—
38#90986C2212yes—
39#90974C2212yes—
40#90974C2212yes—
41#90952C2212yes—
42Dic5×C3⋊C8C2212yes—
43#82571—51nomaximal, Hall
44#66161D311yescharacteristic
45C2×#4097—31no—
46#65850—31no—
47C5×#416—31no—
48C2×#3948—31no—
49#65390—31no—
50#65284—31no—
51#65079—32no—
52#65078—32no—
53#65047—32no—
54#65046—32no—
55C3×C5×#41—22no—
56C3×C5×#41C2×C414yes—
57C2×C3×C5×Q8C2311yescharacteristic
58C2×C3×C5×D4—21no—
59C2×C3×C5×D4C2311yescharacteristic
60C22×C3×C4×C5—21no—
61C22×C3×C4×C5C2311yescharacteristic
62C22×Dic15—21no—
63C2×Dic30—21no—
64C2×Dic30—22no—
65C2×Dic30C2×C414yescomplemented
66C2×Dic30C2312yes—
67C22×C3×Dic5—21no—
68C22×C3×Dic5—22no—
69#6718C2×C418yescomplemented
70#6716C2×C418yescomplemented
71#6710C2×C414yescomplemented
72C2×C3⋊#168C2312yes—
73C5×#217—21no—
74C5×C4⋊Dic3C2311yescharacteristic
75C5×C4⋊Dic3C2311yescharacteristic
76C4×C5×Dic3—21no—
77C2×C5×C3⋊C8—22no—
78C2×C5×C3⋊C8C2311yescharacteristic
79C2×C5×C3⋊C8C2311yescharacteristic
80C3×C22⋊Dic5—22no—
81C3×C22⋊Dic5D412yes—
82C3×C4⋊Dic5—21no—
83C3×C4⋊Dic5—22no—
84C3×C4⋊Dic5D411yescharacteristic
85C3×C4⋊Dic5D411yescharacteristic
86C3×C4⋊Dic5C2312yes—
87C3×C4×Dic5—22no—
88C3×C4×Dic5C2312yes—
89#6669—22no—
90#6669—22no—
91Dic3×Dic5—22no—
92Dic3×Dic5—22no—
93C3×#520—51no—
94C2×#951—51no—
95#5516—51no—
96#5233D511yescharacteristic
97#5030—52no—
98C4×#216—52no—
99#5018—52no—
100C4×#214—52no—
101#4997—51no—
102C2×#810—51no—
103#4964—51no—
104C2×C5×#41C2×D311yescharacteristic
105C22×Dic10—31no—
106C22×C5×C8—31no—
107C5×#117—31no—
108#4097—38no—
109Q8×Dic5C2×D312yes—
110C2×#689—32no—
111D4×Dic5—22no—
112D4×Dic5C2×D312yes—
113C2×#687—32no—
114#4072—21no—
115#4072C2×D311yescharacteristic
116C2×C4⋊Dic5—21no—
117C2×C4⋊Dic5C2×D311yescharacteristic
118C2×#681—31no—
119#4033—32no—
120#4033—32no—
121#4023—62no—
122C4×Dic10—61no—
123C4×Dic10—61no—
124C5×#103—31no—
125C5×#103—31no—
126C5×#102—31no—
127C5×#102—31no—
128#3948—32no—
129#3948—32no—
130#3947—32no—
131C8×Dic5—32no—
132#3940—32no—
133#3939—32no—
134#2947—151noSylow
135C23×C3×C5—21no—
136C23×C3×C5C2×D411yescharacteristic
137C3×C5×Q8—21no—
138C3×C5×Q8C22×C412yes—
139C3×C5×D4—24no—
140C3×C5×D4—21no—
141C3×C5×D4C22×C412yes—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5—24no—
144C2×C3×C4×C5C22×C412yes—
145C2×C3×C4×C5C2×D411yescharacteristic
146C2×C3×C4×C5#4112yes—
147C2×C3×C4×C5C2411yescharacteristic
148C2×Dic15—24no—
149C2×Dic15—22no—
150Dic30—24no—
151Dic30—22no—
152Dic30C22×C414yes—
153C2×C5×Dic3—21no—
154C2×C5×Dic3—21no—
155C2×C3×Dic5—22no—
156C2×C3×Dic5—82no—
157C2×C3×Dic5—81no—
158C2×C3×Dic5C2×D412yes—
159C2×C3×Dic5C2×D412yes—
160C3⋊#168C22×C414yes—
161C5×C3⋊C8—21no—
162C5×C3⋊C8—21no—
163C2×C3×#41C2×D511yescharacteristic
164C22×Dic6—51no—
165C3×C4×Q8—52no—
166C3×C4×D4—52no—
167C3×C4×D4—102no—
168C3×#117—51no—
169C3×#117—101no—
170C2×C3×C4⋊C4—51no—
171C2×C3×C4⋊C4—101no—
172#951—58no—
173C2×#216—52no—
174C2×#214—52no—
175#926C2×D511yescharacteristic
176C2×#208—51no—
177C22×C3⋊C8C2×D511yescharacteristic
178#887—101no—
179#887—101no—
180#877—102no—
181C4×Dic6—52no—
182C4×Dic6—52no—
183#835C2×D511yescharacteristic
184#835C2×D511yescharacteristic
185#832C2×D511yescharacteristic
186#832C2×D511yescharacteristic
187#810—52no—
188#810—52no—
189#808—52no—
190#807—52no—
191#803—52no—
192C4×C3⋊C8—52no—
193C5×#41—22no—
194C5×#41C4×D314yes—
195C2×C5×Q8C22×D311yescharacteristic
196C2×C5×D4—21no—
197C2×C5×D4C22×D311yescharacteristic
198C22×C4×C5—21no—
199C22×C4×C5C22×D311yescharacteristic
200C22×Dic5—21no—
201C22×Dic5—22no—
202C22×Dic5—61no—
203C2×Dic10—34no—
204C2×Dic10—32no—
205C2×Dic10—62no—
206C2×Dic10—61no—
207C2×C5×C8—31no—
208C2×C5×C8—31no—
209C2×C5×C8—62no—
210C5×C4⋊C4—31no—
211C5×C4⋊C4—31no—
212C5×C22⋊C4—61no—
213C42×C5—61no—
214C22⋊Dic5—22no—
215C22⋊Dic5#6712yes—
216#689—38no—
217#687—38no—
218C4⋊Dic5—21no—
219C4⋊Dic5—22no—
220C4⋊Dic5#6711yescharacteristic
221C4⋊Dic5#6711yescharacteristic
222C4⋊Dic5C22×D312yes—
223#683—62no—
224#683—62no—
225C4×Dic5—22no—
226C4×Dic5—62no—
227C4×Dic5—62no—
228C4×Dic5C22×D312yes—
229#681—34no—
230C2×#168—32no—
231C2×#137—151no—
232#520—51no—
233#519—151no—
234#441—152no—
235#440—152no—
236C4×Q16—152no—
237C4×SD16—152no—
238#428—151no—
239#416—31no—
240C2×#103—151no—
241#405—151no—
242C22×C3×C5—21no—
243C22×C3×C5—21no—
244C22×C3×C5—22no—
245C22×C3×C5—41no—
246C22×C3×C5—41no—
247C22×C3×C5C4×D412yes—
248C22×C3×C5C22×D411yescharacteristic
249C3×C4×C5—22no—
250C3×C4×C5C4×D412yes—
251C3×C4×C5C23×C411yescharacteristic
252C3×C4×C5C2×#4111yescharacteristic
253Dic15—24no—
254C3×Dic5—22no—
255C3×Dic5—22no—
256C3×Dic5—42no—
257C5×Dic3—41no—
258C5×Dic3—41no—
259C3×#41—22no—
260C3×#41C4×D514yes—
261C2×C3×Q8C22×D511yescharacteristic
262C2×C3×D4—21no—
263C2×C3×D4C22×D511yescharacteristic
264C22×C3×C4—21no—
265C22×C3×C4—102no—
266C22×C3×C4—101no—
267C22×C3×C4C22×D511yescharacteristic
268C22×Dic3—101no—
269C2×Dic6—54no—
270C2×Dic6—52no—
271C2×Dic6—101no—
272C2×Dic6—102no—
273C3×C4⋊C4—51no—
274C3×C4⋊C4—51no—
275C3×C4⋊C4—52no—
276C3×C4⋊C4—102no—
277C3×C4⋊C4—101no—
278C3×C22⋊C4—52no—
279C3×C22⋊C4—102no—
280C3×C42—52no—
281C3×C42—102no—
282#217—21no—
283#216—58no—
284#214—58no—
285C4⋊Dic3C22×D511yescharacteristic
286C4⋊Dic3C22×D511yescharacteristic
287Dic3⋊C4—102no—
288Dic3⋊C4—102no—
289C4×Dic3—21no—
290C4×Dic3—102no—
291C4×Dic3—102no—
292#208—54no—
293C2×C3⋊C8—22no—
294C2×C3⋊C8—52no—
295C2×C3⋊C8C22×D511yescharacteristic
296C2×C3⋊C8C22×D511yescharacteristic
297C23×C5—21no—
298C23×C5C2×#6711yescharacteristic
299C5×Q8—21no—
300C5×Q8C2×C4×D312yes—
301C5×D4—24no—
302C5×D4—21no—
303C5×D4C2×C4×D312yes—
304C2×C4×C5—21no—
305C2×C4×C5—24no—
306C2×C4×C5—61no—
307C2×C4×C5—61no—
308C2×C4×C5C2×C4×D312yes—
309C2×C4×C5#23512yes—
310C2×C4×C5C2×#6711yescharacteristic
311C2×C4×C5C23×D311yescharacteristic
312C2×Dic5—22no—
313C2×Dic5—64no—
314C2×Dic5—62no—
315C2×Dic5—82no—
316C2×Dic5—81no—
317C2×Dic5C2×#6712yes—
318C2×Dic5C2×#6712yes—
319Dic10—34no—
320Dic10—64no—
321Dic10—62no—
322C5×C8—61no—
323C5×C8—61no—
324#168—34no—
325C2×#41D3×D511yescharacteristic
326C22×Q8—151no—
327#137—158no—
328C2×Q16—152no—
329C2×SD16—152no—
330C2×M16—151no—
331C22×C8—31no—
332C4×Q8—52no—
333C4×Q8—152no—
334C4×Q8—152no—
335C4×Q8—301no—
336C4×Q8—301no—
337C4×D4—52no—
338C4×D4—102no—
339#117—31no—
340#117—51no—
341#117—101no—
342#117—302no—
343C2×C4⋊C4—51no—
344C2×C4⋊C4—101no—
345#107—152no—
346#106—152no—
347#103—31no—
348#103—31no—
349#103—152no—
350#103—152no—
351#102—31no—
352#102—31no—
353#97—152no—
354C4×C8—152no—
355C2×C3×C5—22no—
356C2×C3×C5—42no—
357C2×C3×C5C2×C4×D411yescharacteristic
358C2×C3×C5#57711yescharacteristic
359C2×C3×C5#57811yescharacteristic
360C23×C3—21no—
361C23×C3D4×D511yescharacteristic
362C3×Q8—21no—
363C3×Q8C2×C4×D512yes—
364C3×D4—24no—
365C3×D4—21no—
366C3×D4C2×C4×D512yes—
367C2×C3×C4—21no—
368C2×C3×C4—24no—
369C2×C3×C4—52no—
370C2×C3×C4—52no—
371C2×C3×C4—102no—
372C2×C3×C4—402no—
373C2×C3×C4—401no—
374C2×C3×C4C2×C4×D512yes—
375C2×C3×C4D4×D511yescharacteristic
376C2×C3×C4#71112yes—
377C2×C3×C4C23×D511yescharacteristic
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—102no—
381C2×Dic3—104no—
382Dic6—54no—
383Dic6—102no—
384Dic6—104no—
385C3⋊C8—21no—
386C3⋊C8—21no—
387C3⋊C8—54no—
388C22×C5—21no—
389C22×C5—21no—
390C22×C5—22no—
391C22×C5—41no—
392C22×C5—41no—
393C22×C5C4×#6712yes—
394C22×C5C22×#6711yescharacteristic
395C4×C5—22no—
396C4×C5—121no—
397C4×C5—121no—
398C4×C5C4×#6712yes—
399C4×C5C22×C4×D311yescharacteristic
400C4×C5C2×#23511yescharacteristic
401Dic5—22no—
402Dic5—22no—
403Dic5—42no—
404Dic5—64no—
405#41—22no—
406#41#120014yes—
407C2×Q8—154no—
408C2×Q8—152no—
409C2×Q8—302no—
410C2×Q8—301no—
411C2×Q8C2×D3×D511yescharacteristic
412C2×D4—21no—
413C2×D4C2×D3×D511yescharacteristic
414C22×C4—21no—
415C22×C4—102no—
416C22×C4—101no—
417C22×C4—301no—
418C22×C4C2×D3×D511yescharacteristic
419Q16—158no—
420SD16—158no—
421M16—154no—
422C2×C8—31no—
423C2×C8—31no—
424C2×C8—62no—
425C2×C8—152no—
426C4⋊C4—31no—
427C4⋊C4—31no—
428C4⋊C4—51no—
429C4⋊C4—51no—
430C4⋊C4—52no—
431C4⋊C4—102no—
432C4⋊C4—101no—
433C4⋊C4—302no—
434C4⋊C4—302no—
435C22⋊C4—52no—
436C22⋊C4—61no—
437C22⋊C4—102no—
438C42—52no—
439C42—61no—
440C42—102no—
441C42—302no—
442C42—302no—
443C3×C5#294711yescharacteristic, complemented, Hall
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—22no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3#402712yes—
450C22×C3C2×D4×D511yescharacteristic
451C3×C4—22no—
452C3×C4—102no—
453C3×C4—102no—
454C3×C4—202no—
455C3×C4#402712yes—
456C3×C4C22×C4×D511yescharacteristic
457C3×C4C2×#71111yescharacteristic
458Dic3—41no—
459Dic3—41no—
460Dic3—104no—
461C2×C5—22no—
462C2×C5—42no—
463C2×C5C2×C4×#6711yescharacteristic
464C2×C5#583611yescharacteristic
465C2×C5#583811yescharacteristic
466C23—21no—
467C23D5×#6711yescharacteristic
468Q8—21no—
469Q8—154no—
470Q8—302no—
471Q8—304no—
472Q8C2×#120012yes—
473D4—24no—
474D4—21no—
475D4C2×#120012yes—
476C2×C4—21no—
477C2×C4—24no—
478C2×C4—52no—
479C2×C4—52no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—102no—
483C2×C4—302no—
484C2×C4—304no—
485C2×C4—402no—
486C2×C4—401no—
487C2×C4#678212yes—
488C2×C4C2×#120012yes—
489C2×C4D5×#6711yescharacteristic
490C2×C4C22×D3×D511yescharacteristic
491C8—61no—
492C8—61no—
493C8—154no—
494C2×C3—22no—
495C2×C3—42no—
496C2×C3C2×#402711yescharacteristic
497C2×C3#6610411yescharacteristic
498C2×C3#6610511yescharacteristic
499C5#8257111yescharacteristic, complemented, Sylow
500C22—21no—
501C22—21no—
502C22—22no—
503C22—41no—
504C22—41no—
505C22#9155212yes—
506C22C2×D5×#6711yescharacteristic
507C4—22no—
508C4—102no—
509C4—102no—
510C4—121no—
511C4—121no—
512C4—202no—
513C4—304no—
514C4#9155212yes—
515C4C2×#678211yescharacteristic
516C4C22×#120011yescharacteristic
517C3#1061055211yescharacteristic, complemented, Sylow
518C2—22no—
519C2—42no—
520C2#1175533711yescharacteristic
521C2#1175533811yescharacteristic
522C2C2×#9155211yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
Fitting, prime radicalC2×C3×C5×#41C22
socleC22×C3×C5C22×D4
cosocleC2×C3×C4×C5C24
hypercenterC2×#41D3×D5
hypocenterC3×C5#2947
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
3. #11754213C2semidirect
5. #11749250C2semidirect
6. #11749244C2semidirect
7. #11749004C2semidirect
8. #11748998C2semidirect
9. #11748470C2semidirect
10. C2×#90974C2semidirect
11. #11748318C2semidirect
15. #91840C4semidirect
65. C2×Dic30C2×C4semidirect
69. #6718C2×C4semidirect
70. #6716C2×C4semidirect
71. #6710C2×C4semidirect
443. C3×C5#2947semidirect
499. C5#82571semidirect
517. C3#10610552semidirect

Maximal subgroups

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

subgroupindex
C2×#918402
C3×#661612
#117542132
C5×#52332
#117492502
#117492442
#117490042
#117489982
#117484702
C2×#909742
#117483182
#106105523
#825715