Finite Group Databasebeta

← #14460269

Subgroups — #14460269

Order 1920.

792 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#550C211yesmaximal, characteristic
2#11756301C211yesmaximal, characteristic, complemented
3#11751340C212yesmaximal, complemented
4#11751334C212yesmaximal, complemented
5#11751176C211yesmaximal, characteristic, complemented
6#11751171C211yesmaximal, characteristic, complemented
7#11751142C212yesmaximal, complemented
8#11751136C212yesmaximal, complemented
9C4×#6718C211yesmaximal, characteristic, complemented
10#11751121C211yesmaximal, characteristic, complemented
11D4×C3⋊#168C211yesmaximal, characteristic, complemented
12#10615301—31nomaximal, Hall
13C2×C3×C5×#41C2212yes—
14C3×C5×#127—21no—
15C3×C5×#127C2211yescharacteristic
16C3×C5×#121—22no—
17C3×C5×#121C2212yes—
18C3×C4×C5×Q8C2211yescharacteristic
19C3×C4×C5×D4—21no—
20C3×C4×C5×D4C2211yescharacteristic
21C2×#6718C2211yescharacteristic
22#91825—21no—
23C2×#6716C2212yes—
24#91815C2212yes—
25#91813C2212yes—
26C22×C3⋊#168C2212yes—
27#91776—21no—
28#91768—22no—
29#91758C2211yescharacteristic
30C4×Dic30C2211yescharacteristic
31#91119C2211yescharacteristic
32#91119C2212yes—
33#91116C2212yes—
34#91116C2212yes—
35#91116C2212yes—
36#91114C2212yes—
37#91094C2211yescharacteristic
38#91092C2212yes—
39#91091C2211yescharacteristic
40C4⋊(C3⋊#168)C2211yescharacteristic
41C4×C3⋊#168C2211yescharacteristic
42#83570—51nomaximal, Hall
43C5×#550D311yescharacteristic
44#65876—31no—
45#65528—32no—
46#65522—32no—
47#65364—31no—
48#65359—31no—
49#65330—32no—
50#65324—32no—
51C4×#689—31no—
52#65309—31no—
53D4×#168—31no—
54C3×C5×#41—24no—
55C3×C5×#41—42no—
56C2×C3×C5×Q8C2311yescharacteristic
57C2×C3×C5×D4—22no—
58C2×C3×C5×D4—22no—
59C2×C3×C5×D4—21no—
60C2×C3×C5×D4—21no—
61C2×C3×C5×D4D411yescharacteristic
62C2×C3×C5×D4C2312yes—
63C22×C3×C4×C5—22no—
64C22×C3×C4×C5C2312yes—
65C2×Dic30—21no—
66C2×Dic30C2311yescharacteristic
67C3×C5×C4⋊C4—21no—
68C3×C5×C4⋊C4D411yescharacteristic
69C3×C5×C4⋊C4C2311yescharacteristic
70C3×C5×C22⋊C4—22no—
71C3×C5×C22⋊C4D412yes—
72C3×C42×C5—21no—
73C3×C42×C5C2311yescharacteristic
74#6719—22no—
75#6719—22no—
76#6718—22no—
77#6716—42no—
78C4⋊Dic15—21no—
79C4⋊Dic15C2311yescharacteristic
80C4⋊Dic15C2312yes—
81#6712—21no—
82#6712—22no—
83C4×Dic15—21no—
84C4×Dic15—22no—
85C2×C3⋊#168—42no—
86C2×C3⋊#168C2311yescharacteristic
87C2×C3⋊#168C2312yes—
88C2×C3⋊#168C2311yescharacteristic
89C3×#550D511yescharacteristic
90#5542—51no—
91#5239—52no—
92#5233—52no—
93#5075—51no—
94#5070—51no—
95#5041—52no—
96#5035—52no—
97C4×#216—51no—
98#5020—51no—
99D4×C3⋊C8—51no—
100C2×C5×#41C2×D312yes—
101C5×#127—21no—
102C5×#127C2×D311yescharacteristic
103C5×#121—22no—
104C5×#121C2×D312yes—
105C4×C5×Q8C2×D311yescharacteristic
106C4×C5×D4—21no—
107C4×C5×D4C2×D311yescharacteristic
108C2×#689—31no—
109#4082—61no—
110C2×#687—32no—
111#4072—32no—
112#4070—32no—
113C22×#168—32no—
114#4033—61no—
115#4025—62no—
116#4015—31no—
117C4×Dic10—31no—
118#3967—32no—
119#3967—31no—
120#3964—32no—
121#3964—32no—
122#3964—32no—
123C22⋊#168—32no—
124#3942—31no—
125#3940—32no—
126#3939—31no—
127C4⋊#168—31no—
128C4×#168—31no—
129#3304—151noSylow
130C23×C3×C5—22no—
131C23×C3×C5C2×D412yes—
132C3×C5×Q8—21no—
133C3×C5×Q8#4112yes—
134C3×C5×D4—22no—
135C3×C5×D4—22no—
136C3×C5×D4—22no—
137C3×C5×D4—44no—
138C3×C5×D4—42no—
139C3×C5×D4—81no—
140C3×C5×D4—82no—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5—21no—
145C2×C3×C4×C5—44no—
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5C2×D411yescharacteristic
148C2×C3×C4×C5C2×D411yescharacteristic
149C2×C3×C4×C5C2411yescharacteristic
150C2×Dic15—21no—
151C2×Dic15—21no—
152C2×Dic15—22no—
153Dic30—21no—
154Dic30—21no—
155Dic30—42no—
156C3⋊#168—21no—
157C3⋊#168—21no—
158C3⋊#168—22no—
159C2×C3×#41C2×D512yes—
160C3×#127—21no—
161C3×#127C2×D511yescharacteristic
162C3×#121—22no—
163C3×#121C2×D512yes—
164C3×C4×Q8C2×D511yescharacteristic
165C3×C4×D4—21no—
166C3×C4×D4C2×D511yescharacteristic
167C2×#216—51no—
168#936—101no—
169C2×#214—52no—
170#926—52no—
171#924—52no—
172C22×C3⋊C8—52no—
173#887—101no—
174#879—102no—
175#869—51no—
176C4×Dic6—51no—
177#835—51no—
178#835—52no—
179#832—52no—
180#832—52no—
181#832—52no—
182#830—52no—
183#810—51no—
184#808—52no—
185#807—51no—
186C4⋊(C3⋊C8)—51no—
187C4×C3⋊C8—51no—
188C5×#41—24no—
189C5×#41—42no—
190C2×C5×Q8C22×D311yescharacteristic
191C2×C5×D4—22no—
192C2×C5×D4—22no—
193C2×C5×D4—21no—
194C2×C5×D4—21no—
195C2×C5×D4#6711yescharacteristic
196C2×C5×D4C22×D312yes—
197C22×C4×C5—22no—
198C22×C4×C5C22×D312yes—
199C2×Dic10—31no—
200C2×Dic10—61no—
201C5×C4⋊C4—21no—
202C5×C4⋊C4#6711yescharacteristic
203C5×C4⋊C4C22×D311yescharacteristic
204C5×C22⋊C4—22no—
205C5×C22⋊C4#6712yes—
206C42×C5—21no—
207C42×C5C22×D311yescharacteristic
208C22⋊Dic5—62no—
209C22⋊Dic5—62no—
210#689—62no—
211#687—122no—
212C4⋊Dic5—31no—
213C4⋊Dic5—32no—
214C4⋊Dic5—61no—
215#683—61no—
216#683—62no—
217C4×Dic5—61no—
218C4×Dic5—62no—
219C2×#168—31no—
220C2×#168—31no—
221C2×#168—32no—
222C2×#168—122no—
223#556—151no—
224#550D1511yescharacteristic
225#486—151no—
226#482—152no—
227#475—151no—
228#465—152no—
229#460—151no—
230#452—152no—
231C4×Q16—151no—
232C8×D4—151no—
233#416—152no—
234C22×C3×C5—22no—
235C22×C3×C5—22no—
236C22×C3×C5—22no—
237C22×C3×C5—42no—
238C22×C3×C5—42no—
239C22×C3×C5—42no—
240C22×C3×C5C22×D411yescharacteristic
241C3×C4×C5—22no—
242C3×C4×C5—22no—
243C3×C4×C5—21no—
244C3×C4×C5—41no—
245C3×C4×C5#13512yes—
246C3×C4×C5C2×#4111yescharacteristic
247C3×C4×C5H(5,𝔽2)11yescharacteristic
248Dic15—41no—
249Dic15—42no—
250Dic15—41no—
251C3×#41—24no—
252C3×#41—42no—
253C2×C3×Q8C22×D511yescharacteristic
254C2×C3×D4—22no—
255C2×C3×D4—22no—
256C2×C3×D4—21no—
257C2×C3×D4—21no—
258C2×C3×D4C22⋊D511yescharacteristic
259C2×C3×D4C22×D512yes—
260C22×C3×C4—22no—
261C22×C3×C4C22×D512yes—
262C2×Dic6—51no—
263C2×Dic6—101no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4C22⋊D511yescharacteristic
266C3×C4⋊C4C22×D511yescharacteristic
267C3×C22⋊C4—22no—
268C3×C22⋊C4C22⋊D512yes—
269C3×C42—21no—
270C3×C42C22×D511yescharacteristic
271#217—102no—
272#217—102no—
273#216—102no—
274#214—202no—
275C4⋊Dic3—51no—
276C4⋊Dic3—52no—
277C4⋊Dic3—101no—
278Dic3⋊C4—101no—
279Dic3⋊C4—102no—
280C4×Dic3—101no—
281C4×Dic3—102no—
282C2×C3⋊C8—51no—
283C2×C3⋊C8—51no—
284C2×C3⋊C8—52no—
285C2×C3⋊C8—202no—
286C23×C5—22no—
287C23×C5C2×#6712yes—
288C5×Q8—21no—
289C5×Q8#23712yes—
290C5×D4—22no—
291C5×D4—22no—
292C5×D4—22no—
293C5×D4—44no—
294C5×D4—42no—
295C5×D4—81no—
296C5×D4—82no—
297C2×C4×C5—21no—
298C2×C4×C5—22no—
299C2×C4×C5—22no—
300C2×C4×C5—21no—
301C2×C4×C5—44no—
302C2×C4×C5C2×#6712yes—
303C2×C4×C5C2×#6711yescharacteristic
304C2×C4×C5C2×#6711yescharacteristic
305C2×C4×C5C23×D311yescharacteristic
306C2×Dic5—61no—
307C2×Dic5—61no—
308C2×Dic5—62no—
309Dic10—61no—
310Dic10—61no—
311Dic10—122no—
312#168—61no—
313#168—61no—
314#168—62no—
315C2×#41C2×D1512yes—
316C2×Q16—151no—
317C2×SD16—152no—
318C22×C8—152no—
319C4⋊Q8—151no—
320#127—21no—
321#127C2×D1511yescharacteristic
322#126—302no—
323#124—301no—
324C22⋊Q8—152no—
325#121—22no—
326#121C2×D1512yes—
327C4×Q8—151no—
328C4×Q8—301no—
329C4×Q8C2×D1511yescharacteristic
330C4×D4—21no—
331C4×D4C2×D1511yescharacteristic
332#117—152no—
333#107—151no—
334#106—152no—
335C4⋊C8—151no—
336#103—151no—
337#103—152no—
338#103—151no—
339#102—152no—
340#102—152no—
341#102—152no—
342C22⋊C8—152no—
343C4×C8—151no—
344C2×C3×C5—42no—
345C2×C3×C5—82no—
346C2×C3×C5#54611yescharacteristic
347C2×C3×C5C2×#13511yescharacteristic
348C2×C3×C5#57711yescharacteristic
349C23×C3—22no—
350C23×C3C2×C22⋊D512yes—
351C3×Q8—21no—
352C3×Q8#71112yes—
353C3×D4—22no—
354C3×D4—22no—
355C3×D4—22no—
356C3×D4—44no—
357C3×D4—42no—
358C3×D4—81no—
359C3×D4—82no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—44no—
365C2×C3×C4C2×C22⋊D512yes—
366C2×C3×C4C2×C22⋊D511yescharacteristic
367C2×C3×C4C2×C22⋊D511yescharacteristic
368C2×C3×C4C23×D511yescharacteristic
369C2×Dic3—101no—
370C2×Dic3—101no—
371C2×Dic3—102no—
372Dic6—101no—
373Dic6—101no—
374Dic6—202no—
375C3⋊C8—101no—
376C3⋊C8—101no—
377C3⋊C8—102no—
378C22×C5—22no—
379C22×C5—22no—
380C22×C5—22no—
381C22×C5—42no—
382C22×C5—42no—
383C22×C5—42no—
384C22×C5C22×#6711yescharacteristic
385C4×C5—22no—
386C4×C5—22no—
387C4×C5—21no—
388C4×C5—41no—
389C4×C5#95012yes—
390C4×C5C2×#23711yescharacteristic
391C4×C5#100411yescharacteristic
392Dic5—121no—
393Dic5—122no—
394Dic5—121no—
395#41—24no—
396#41—42no—
397C2×Q8—151no—
398C2×Q8—301no—
399C2×Q8C22×D1511yescharacteristic
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4#122011yescharacteristic
405C2×D4C22×D1512yes—
406C22×C4—22no—
407C22×C4C22×D1512yes—
408Q16—302no—
409SD16—602no—
410C2×C8—151no—
411C2×C8—151no—
412C2×C8—152no—
413C2×C8—602no—
414C4⋊C4—21no—
415C4⋊C4—152no—
416C4⋊C4—151no—
417C4⋊C4—301no—
418C4⋊C4—302no—
419C4⋊C4—301no—
420C4⋊C4#122011yescharacteristic
421C4⋊C4C22×D1511yescharacteristic
422C22⋊C4—22no—
423C22⋊C4—302no—
424C22⋊C4—302no—
425C22⋊C4#122012yes—
426C42—21no—
427C42—302no—
428C42—301no—
429C42C22×D1511yescharacteristic
430C3×C5#330411yescharacteristic, complemented, Hall
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3C22×C22⋊D511yescharacteristic
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—21no—
441C3×C4—41no—
442C3×C4#409612yes—
443C3×C4C2×#71111yescharacteristic
444C3×C4#414411yescharacteristic
445Dic3—201no—
446Dic3—202no—
447Dic3—201no—
448C2×C5—42no—
449C2×C5—82no—
450C2×C5#580611yescharacteristic
451C2×C5C2×#95011yescharacteristic
452C2×C5#583611yescharacteristic
453C23—22no—
454C23C2×#122012yes—
455Q8—21no—
456Q8—301no—
457Q8—301no—
458Q8—602no—
459Q8#681912yes—
460D4—22no—
461D4—22no—
462D4—22no—
463D4—44no—
464D4—42no—
465D4—81no—
466D4—82no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—44no—
472C2×C4—301no—
473C2×C4—302no—
474C2×C4—301no—
475C2×C4C2×#122012yes—
476C2×C4C2×#122011yescharacteristic
477C2×C4C2×#122011yescharacteristic
478C2×C4C23×D1511yescharacteristic
479C8—301no—
480C8—301no—
481C8—302no—
482C2×C3—42no—
483C2×C3—82no—
484C2×C3#6614011yescharacteristic
485C2×C3C2×#409611yescharacteristic
486C2×C3#6617011yescharacteristic
487C5#8357011yescharacteristic, complemented, Sylow
488C22—22no—
489C22—22no—
490C22—22no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22C22×#122011yescharacteristic
495C4—22no—
496C4—22no—
497C4—21no—
498C4—41no—
499C4—601no—
500C4—602no—
501C4—601no—
502C4#9183912yes—
503C4C2×#681911yescharacteristic
504C4#9209511yescharacteristic
505C3#1061530111yescharacteristic, complemented, Sylow
506C2—42no—
507C2—82no—
508C2#1175656511yescharacteristic
509C2C2×#9183911yescharacteristic
510C2#1175659511yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
Fitting, prime radicalC3×C5×#550C2
socleC22×C3×C5C22×D4
hypercenter#550D15
hypocenterC3×C5#3304
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
2. #11756301C2semidirect
3. #11751340C2semidirect
4. #11751334C2semidirect
5. #11751176C2semidirect
6. #11751171C2semidirect
7. #11751142C2semidirect
8. #11751136C2semidirect
9. C4×#6718C2semidirect
10. #11751121C2semidirect
11. D4×C3⋊#168C2semidirect
430. C3×C5#3304semidirect
487. C5#83570semidirect
505. C3#10615301semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5502
#117563012
#117513402
#117513342
#117511762
#117511712
#117511422
#117511362
C4×#67182
#117511212
D4×C3⋊#1682
#106153013
#835705