Finite Group Databasebeta

← #14321111

Subgroups — #14321111

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic15C211yesmaximal, characteristic
2C2×#91532C211yesmaximal, characteristic
3C2×#91532C211yesmaximal, characteristic
4C5×#5210C211yesmaximal, characteristic
5C3×#65499C211yesmaximal, characteristic
6#11749517C211yesmaximal, characteristic
7#11749515C211yesmaximal, characteristic
8#11748818C212yesmaximal, complemented
9#11748804C212yesmaximal, complemented
10#11748651C212yesmaximal, complemented
11#11748637C212yesmaximal, complemented
12#10606109—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C23×Dic15C2211yescharacteristic
15C23×Dic15C2211yescharacteristic
16C2×C4×Dic15—24no—
17C2×C4×Dic15C2212yes—
18C2×C4×Dic15C2212yes—
19C2×C5×#217C2211yescharacteristic
20C2×C5×#217C2211yescharacteristic
21C2×C5×Dic3⋊C4C2212yes—
22C2×C3×C22⋊Dic5C2211yescharacteristic
23C2×C3×C22⋊Dic5C2211yescharacteristic
24C2×C3×#683C2212yes—
25C2×Dic3⋊Dic5C2212yes—
26C2×#6669C2211yescharacteristic
27C2×#6669C2211yescharacteristic
28C2×#6669C2212yes—
29C2×#6668C2212yes—
30#91532—24no—
31#91532C418yescomplemented
32C2×Dic3×Dic5C2211yescharacteristic
33C2×Dic3×Dic5C2211yescharacteristic
34C2×Dic3×Dic5C2212yes—
35C5×C3⋊#95C2212yes—
36C3×#3963C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#74378—51nomaximal, Hall
40C22×C4×Dic5—31no—
41C2×#4023—31no—
42C2×#4023—31no—
43C5×#385—31no—
44#65499D311yescharacteristic
45C4⋊C4×Dic5—32no—
46#65261—32no—
47#65239—31no—
48#65235—31no—
49#64938—32no—
50#64936—32no—
51C24×C3×C5C2311yescharacteristic
52C22×C3×C4×C5—22no—
53C22×C3×C4×C5—24no—
54C22×C3×C4×C5C2312yes—
55C22×Dic15—22no—
56C22×Dic15—24no—
57C22×Dic15C2×C414yes—
58C22×Dic15D414yes—
59C22×Dic15Q814yes—
60C22×Dic15C2312yes—
61C22×Dic15C2311yescharacteristic
62C22×Dic15C2311yescharacteristic
63C22×C5×Dic3C2311yescharacteristic
64C22×C5×Dic3C2311yescharacteristic
65C22×C5×Dic3C2312yes—
66C22×C3×Dic5C2311yescharacteristic
67C22×C3×Dic5C2311yescharacteristic
68C22×C3×Dic5C2312yes—
69C4×Dic15—24no—
70C4×Dic15—24no—
71C5×#217—22no—
72C5×#217C2×C414yes—
73C5×Dic3⋊C4—24no—
74C3×C22⋊Dic5—22no—
75C3×C22⋊Dic5C2×C414yes—
76C3×#683—24no—
77Dic3⋊Dic5—24no—
78#6669—22no—
79#6669—22no—
80#6669C2×C418yescomplemented
81#6668—24no—
82Dic3×Dic5—22no—
83Dic3×Dic5—22no—
84Dic3×Dic5C2×C418yescomplemented
85C22×C4×Dic3—51no—
86C2×#877—51no—
87C2×#877—51no—
88C3×#385—51no—
89#5210D511yescharacteristic
90#4976—52no—
91Dic3×C4⋊C4—52no—
92#4950—51no—
93#4946—51no—
94#4649—52no—
95#4647—52no—
96C23×C4×C5C2×D311yescharacteristic
97C23×Dic5—31no—
98C23×Dic5—31no—
99C2×C5×C4⋊C4—32no—
100C2×C5×C22⋊C4—31no—
101C2×C5×C22⋊C4—31no—
102C2×C22⋊Dic5C2×D311yescharacteristic
103C2×C22⋊Dic5C2×D311yescharacteristic
104C2×C4⋊Dic5—32no—
105C2×#683—32no—
106C2×#683—31no—
107C2×#683—31no—
108C2×#683—32no—
109C2×#683C2×D312yes—
110C2×C4×Dic5—32no—
111C2×C4×Dic5—32no—
112C2×C4×Dic5—32no—
113C2×C4×Dic5—31no—
114C2×C4×Dic5—31no—
115C2×C4×Dic5—64no—
116#4023—38no—
117#4023—64no—
118C5×#95—32no—
119#3963—32no—
120#3963—32no—
121#3963C2×D312yes—
122#2296—151noSylow
123C23×C3×C5—22no—
124C23×C3×C5—22no—
125C23×C3×C5C22×C412yes—
126C23×C3×C5C2×D412yes—
127C23×C3×C5C2×Q812yes—
128C23×C3×C5C2411yescharacteristic
129C2×C3×C4×C5—28no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5#4114yes—
133C2×Dic15—28no—
134C2×Dic15—22no—
135C2×Dic15—22no—
136C2×Dic15C4⋊C4116yescomplemented
137C2×Dic15C22×C412yes—
138C2×Dic15C22×C412yes—
139C2×Dic15C2×D412yes—
140C2×Dic15C2×D412yes—
141C2×Dic15C2×Q812yes—
142C2×Dic15C2×Q812yes—
143C2×Dic15#4114yes—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—21no—
147C2×C5×Dic3—21no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—22no—
150C2×C5×Dic3C22×C414yes—
151C2×C3×Dic5—22no—
152C2×C3×Dic5—22no—
153C2×C3×Dic5—21no—
154C2×C3×Dic5—21no—
155C2×C3×Dic5—22no—
156C2×C3×Dic5—22no—
157C2×C3×Dic5C22×C414yes—
158C23×C3×C4C2×D511yescharacteristic
159C23×Dic3—51no—
160C23×Dic3—51no—
161C2×C3×C4⋊C4—52no—
162C2×C3×C22⋊C4—51no—
163C2×C3×C22⋊C4—51no—
164C2×#217C2×D511yescharacteristic
165C2×#217C2×D511yescharacteristic
166C2×C4⋊Dic3—52no—
167C2×Dic3⋊C4—52no—
168C2×Dic3⋊C4—51no—
169C2×Dic3⋊C4—52no—
170C2×Dic3⋊C4—51no—
171C2×Dic3⋊C4C2×D512yes—
172C2×C4×Dic3—52no—
173C2×C4×Dic3—52no—
174C2×C4×Dic3—52no—
175C2×C4×Dic3—51no—
176C2×C4×Dic3—51no—
177C2×C4×Dic3—104no—
178#877—58no—
179#877—104no—
180C3×#95—52no—
181C3⋊#95—52no—
182C3⋊#95—52no—
183C3⋊#95C2×D512yes—
184C24×C5C22×D311yescharacteristic
185C22×C4×C5—22no—
186C22×C4×C5—24no—
187C22×C4×C5—32no—
188C22×C4×C5—31no—
189C22×C4×C5—31no—
190C22×C4×C5C22×D312yes—
191C22×Dic5—34no—
192C22×Dic5—34no—
193C22×Dic5—34no—
194C22×Dic5—31no—
195C22×Dic5—31no—
196C22×Dic5—32no—
197C22×Dic5—64no—
198C22×Dic5—62no—
199C22×Dic5C22×D311yescharacteristic
200C22×Dic5C22×D311yescharacteristic
201C22×Dic5C22×D312yes—
202C5×C4⋊C4—64no—
203C5×C22⋊C4—34no—
204C5×C22⋊C4—62no—
205C22⋊Dic5—22no—
206C22⋊Dic5C4×D314yes—
207C4⋊Dic5—64no—
208#683—24no—
209#683—38no—
210#683—64no—
211#683—62no—
212#683—62no—
213C4×Dic5—38no—
214C4×Dic5—62no—
215C4×Dic5—62no—
216C4×Dic5—64no—
217C4×Dic5—64no—
218C2×#117—151no—
219C2×#117—151no—
220C22×C42—151no—
221#389—152no—
222#389—152no—
223#385—31no—
224#385—51no—
225#385—151no—
226#385—151no—
227C4×C4⋊C4—152no—
228C4×C4⋊C4—152no—
229C22×C3×C5—24no—
230C22×C3×C5—22no—
231C22×C3×C5—22no—
232C22×C3×C5—24no—
233C22×C3×C5C2×C4⋊C414yes—
234C22×C3×C5C23×C411yescharacteristic
235C22×C3×C5C22×D411yescharacteristic
236C22×C3×C5C22×Q811yescharacteristic
237C22×C3×C5C2×#4112yes—
238C22×C3×C5C2×#4112yes—
239C3×C4×C5—24no—
240Dic15—24no—
241Dic15C2×C4⋊C414yes—
242Dic15C2×C4⋊C414yes—
243C3×Dic5—24no—
244C3×Dic5—41no—
245C3×Dic5—41no—
246C5×Dic3—24no—
247C5×Dic3—41no—
248C5×Dic3—41no—
249C24×C3C22×D511yescharacteristic
250C22×C3×C4—22no—
251C22×C3×C4—24no—
252C22×C3×C4—52no—
253C22×C3×C4—51no—
254C22×C3×C4—51no—
255C22×C3×C4C22×D512yes—
256C22×Dic3—54no—
257C22×Dic3—54no—
258C22×Dic3—54no—
259C22×Dic3—51no—
260C22×Dic3—51no—
261C22×Dic3—52no—
262C22×Dic3—104no—
263C22×Dic3—102no—
264C22×Dic3C22×D511yescharacteristic
265C22×Dic3C22×D511yescharacteristic
266C22×Dic3C22×D512yes—
267C3×C4⋊C4—104no—
268C3×C22⋊C4—54no—
269C3×C22⋊C4—102no—
270#217—22no—
271#217C4×D514yes—
272C4⋊Dic3—104no—
273Dic3⋊C4—24no—
274Dic3⋊C4—58no—
275Dic3⋊C4—102no—
276Dic3⋊C4—102no—
277Dic3⋊C4—104no—
278C4×Dic3—58no—
279C4×Dic3—102no—
280C4×Dic3—102no—
281C4×Dic3—104no—
282C4×Dic3—104no—
283C23×C5—22no—
284C23×C5—22no—
285C23×C5C2×C4×D312yes—
286C23×C5D3×D412yes—
287C23×C5D3×Q812yes—
288C23×C5C23×D311yescharacteristic
289C2×C4×C5—28no—
290C2×C4×C5—22no—
291C2×C4×C5—22no—
292C2×C4×C5—34no—
293C2×C4×C5—62no—
294C2×C4×C5—62no—
295C2×C4×C5—61no—
296C2×C4×C5—61no—
297C2×C4×C5—62no—
298C2×C4×C5—62no—
299C2×C4×C5#23714yes—
300C2×Dic5—22no—
301C2×Dic5—22no—
302C2×Dic5—21no—
303C2×Dic5—21no—
304C2×Dic5—22no—
305C2×Dic5—22no—
306C2×Dic5—316no—
307C2×Dic5—32no—
308C2×Dic5—32no—
309C2×Dic5—32no—
310C2×Dic5—32no—
311C2×Dic5—32no—
312C2×Dic5—32no—
313C2×Dic5—34no—
314C2×Dic5—68no—
315C2×Dic5—62no—
316C2×Dic5—62no—
317C2×Dic5C2×C4×D314yes—
318C23×C4—151no—
319C23×C4—151no—
320C23×C4D3×D511yescharacteristic
321#117—158no—
322#117—304no—
323C2×C4⋊C4—32no—
324C2×C4⋊C4—52no—
325C2×C4⋊C4—152no—
326C2×C4⋊C4—151no—
327C2×C4⋊C4—151no—
328C2×C4⋊C4—152no—
329C2×C4⋊C4—152no—
330C2×C22⋊C4—31no—
331C2×C22⋊C4—31no—
332C2×C22⋊C4—51no—
333C2×C22⋊C4—51no—
334C2×C42—152no—
335C2×C42—151no—
336C2×C42—151no—
337C2×C42—152no—
338C2×C42—152no—
339C2×C42—304no—
340#95—32no—
341#95—52no—
342#95—152no—
343#95—152no—
344C2×C3×C5—24no—
345C2×C3×C5C22×C4⋊C411yescharacteristic
346C2×C3×C5C4×#4112yes—
347C2×C3×C5#52511yescharacteristic
348C2×C3×C5#52511yescharacteristic
349C2×C3×C5#53312yes—
350C23×C3—22no—
351C23×C3—22no—
352C23×C3C2×C4×D512yes—
353C23×C3D4×D512yes—
354C23×C3Q8×D512yes—
355C23×C3C23×D511yescharacteristic
356C2×C3×C4—28no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—54no—
360C2×C3×C4—102no—
361C2×C3×C4—102no—
362C2×C3×C4—101no—
363C2×C3×C4—101no—
364C2×C3×C4—102no—
365C2×C3×C4—102no—
366C2×C3×C4#71114yes—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—52no—
374C2×Dic3—52no—
375C2×Dic3—52no—
376C2×Dic3—52no—
377C2×Dic3—52no—
378C2×Dic3—52no—
379C2×Dic3—516no—
380C2×Dic3—54no—
381C2×Dic3—108no—
382C2×Dic3—102no—
383C2×Dic3—102no—
384C2×Dic3C2×C4×D514yes—
385C22×C5—22no—
386C22×C5—24no—
387C22×C5—22no—
388C22×C5—24no—
389C22×C5D3×C4⋊C414yes—
390C22×C5C22×C4×D311yescharacteristic
391C22×C5C2×D3×D411yescharacteristic
392C22×C5C2×#23712yes—
393C22×C5C2×D3×Q811yescharacteristic
394C22×C5D3×#4112yes—
395C4×C5—24no—
396C4×C5—64no—
397C4×C5—121no—
398C4×C5—121no—
399Dic5—24no—
400Dic5—34no—
401Dic5—34no—
402Dic5—41no—
403Dic5—41no—
404Dic5—64no—
405C24C2×D3×D511yescharacteristic
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—32no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—52no—
412C22×C4—51no—
413C22×C4—51no—
414C22×C4—154no—
415C22×C4—154no—
416C22×C4—154no—
417C22×C4—151no—
418C22×C4—151no—
419C22×C4—152no—
420C22×C4—304no—
421C22×C4—302no—
422C22×C4C2×D3×D512yes—
423C4⋊C4—64no—
424C4⋊C4—104no—
425C4⋊C4—158no—
426C4⋊C4—302no—
427C4⋊C4—302no—
428C4⋊C4—304no—
429C4⋊C4—304no—
430C22⋊C4—34no—
431C22⋊C4—54no—
432C22⋊C4—62no—
433C22⋊C4—102no—
434C42—158no—
435C42—304no—
436C42—304no—
437C42—302no—
438C42—302no—
439C3×C5#229611yescharacteristic, complemented, Hall
440C22×C3—24no—
441C22×C3—22no—
442C22×C3—24no—
443C22×C3—22no—
444C22×C3D5×C4⋊C414yes—
445C22×C3C22×C4×D511yescharacteristic
446C22×C3C2×D4×D511yescharacteristic
447C22×C3C2×#71112yes—
448C22×C3C2×Q8×D511yescharacteristic
449C22×C3D5×#4112yes—
450C3×C4—24no—
451C3×C4—104no—
452C3×C4—201no—
453C3×C4—201no—
454Dic3—24no—
455Dic3—41no—
456Dic3—41no—
457Dic3—54no—
458Dic3—54no—
459Dic3—104no—
460C2×C5—24no—
461C2×C5C2×D3×C4⋊C411yescharacteristic
462C2×C5C4×#23712yes—
463C2×C5#558611yescharacteristic
464C2×C5#562212yes—
465C2×C5#565611yescharacteristic
466C23—22no—
467C23—22no—
468C23D15⋊Q812yes—
469C23C4×D3×D512yes—
470C23#679012yes—
471C23C22×D3×D511yescharacteristic
472C2×C4—28no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—34no—
476C2×C4—54no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—102no—
484C2×C4—102no—
485C2×C4—102no—
486C2×C4—102no—
487C2×C4—101no—
488C2×C4—101no—
489C2×C4—1516no—
490C2×C4—152no—
491C2×C4—152no—
492C2×C4—152no—
493C2×C4—152no—
494C2×C4—152no—
495C2×C4—152no—
496C2×C4—154no—
497C2×C4—308no—
498C2×C4—302no—
499C2×C4—302no—
500C2×C4#678014yes—
501C2×C3—24no—
502C2×C3C2×D5×C4⋊C411yescharacteristic
503C2×C3C4×#71112yes—
504C2×C3#6592011yescharacteristic
505C2×C3#6595612yes—
506C2×C3#6599011yescharacteristic
507C5#7437811yescharacteristic, complemented, Sylow
508C22—24no—
509C22—22no—
510C22—22no—
511C22—24no—
512C22#9140414yes—
513C22C2×D15⋊Q811yescharacteristic
514C22C2×C4×D3×D511yescharacteristic
515C22#9201612yes—
516C22C2×#678012yes—
517C22C2×#679011yescharacteristic
518C4—24no—
519C4—64no—
520C4—104no—
521C4—121no—
522C4—121no—
523C4—154no—
524C4—154no—
525C4—201no—
526C4—201no—
527C4—304no—
528C3#1060610911yescharacteristic, complemented, Sylow
529C2—24no—
530C2#1175417912yes—
531C2C4×#678012yes—
532C2C2×#9140411yescharacteristic
533C2#1175437111yescharacteristic
534C2#1175539911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC24×C3×C5C23
hypercenterC23×C4D3×D5
hypocenterC3×C5#2296
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
8. #11748818C2semidirect
9. #11748804C2semidirect
10. #11748651C2semidirect
11. #11748637C2semidirect
31. #91532C4semidirect
80. #6669C2×C4semidirect
84. Dic3×Dic5C2×C4semidirect
136. C2×Dic15C4⋊C4semidirect
439. C3×C5#2296semidirect
507. C5#74378semidirect
528. C3#10606109semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic152
C2×#915322
C5×#52102
C3×#654992
#117495172
#117495152
#117488182
#117488042
#117486512
#117486372
#106061093
#743785