Finite Group Databasebeta

← #14326670

Subgroups — #14326670

Order 1920.

1096 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6712C211yesmaximal, characteristic
2C3×C5×#380C211yesmaximal, characteristic
3C4×#6719C211yesmaximal, characteristic
4#11751079C212yesmaximal, complemented
5#11751077C212yesmaximal, complemented
6#11751043C211yesmaximal, characteristic
7#11751043C211yesmaximal, characteristic
8C22⋊C4×Dic15C211yesmaximal, characteristic
9C22⋊C4×Dic15C211yesmaximal, characteristic
10#11750753C212yesmaximal, complemented
11#11750750C212yesmaximal, complemented
12#10605611—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C23×Dic15C2211yescharacteristic
15C23×Dic15C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2212yes—
17C2×C3×C5×C22⋊C4C2211yescharacteristic
18C2×C3×C5×C22⋊C4C2211yescharacteristic
19C2×#6719C2211yescharacteristic
20C2×#6719C2211yescharacteristic
21C2×#6712C418yescomplemented
22C2×#6712C2212yes—
23C2×#6712C2212yes—
24C2×#6712C2211yescharacteristic
25C2×#6712C2211yescharacteristic
26C2×#6712C2211yescharacteristic
27C2×#6712C2211yescharacteristic
28C2×C4×Dic15C2212yes—
29C2×C4×Dic15C2212yes—
30C2×C4×Dic15C2212yes—
31C3×C5×#95C2212yes—
32#91115C2212yes—
33#91115C2212yes—
34#91115C2212yes—
35#91115C2212yes—
36#91115C2212yes—
37#73880—51nomaximal, Hall
38C22×#683—31no—
39C5×#380D311yescharacteristic
40C4×C22⋊Dic5—31no—
41#65267—32no—
42#65261—32no—
43#65231—31no—
44#65231—31no—
45C22⋊C4×Dic5—31no—
46C22⋊C4×Dic5—31no—
47#64941—32no—
48#64938—32no—
49C24×C3×C5C2311yescharacteristic
50C22×C3×C4×C5—42no—
51C22×C3×C4×C5C2×C414yes—
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×Dic15—24no—
57C22×Dic15—42no—
58C22×Dic15C2×C414yes—
59C22×Dic15C2×C414yes—
60C22×Dic15C2312yes—
61C22×Dic15C2311yescharacteristic
62C22×Dic15C2311yescharacteristic
63C22×Dic15C2311yescharacteristic
64C22×Dic15C2312yes—
65C22×Dic15C2311yescharacteristic
66C3×C5×C4⋊C4—42no—
67C3×C5×C22⋊C4—22no—
68C3×C5×C22⋊C4—22no—
69#6719—22no—
70#6719Q814yes—
71#6712—24no—
72#6712—42no—
73#6712C2×C414yes—
74#6712C2×C414yes—
75#6712C2×C414yes—
76#6712C2×C414yes—
77C4×Dic15—24no—
78C4×Dic15—42no—
79C4×Dic15—42no—
80C22×Dic3⋊C4—51no—
81C3×#380D511yescharacteristic
82C4×#217—51no—
83#4978—52no—
84#4976—52no—
85#4942—51no—
86#4942—51no—
87Dic3×C22⋊C4—51no—
88Dic3×C22⋊C4—51no—
89#4652—52no—
90#4649—52no—
91C23×C4×C5C2×D311yescharacteristic
92C23×Dic5—31no—
93C23×Dic5—31no—
94C2×C5×C4⋊C4C2×D312yes—
95C2×C5×C22⋊C4C2×D311yescharacteristic
96C2×C5×C22⋊C4C2×D311yescharacteristic
97C2×C22⋊Dic5—31no—
98C2×C22⋊Dic5—31no—
99C2×#683—38no—
100C2×#683—31no—
101C2×#683—31no—
102C2×#683—31no—
103C2×#683—31no—
104C2×#683—32no—
105C2×#683—32no—
106C2×C4×Dic5—32no—
107C2×C4×Dic5—32no—
108C2×C4×Dic5—32no—
109C5×#95C2×D312yes—
110#3963—32no—
111#3963—32no—
112#3963—32no—
113#3963—32no—
114#3963—32no—
115#2345—151noSylow
116C23×C3×C5—22no—
117C23×C3×C5—22no—
118C23×C3×C5C22×C412yes—
119C23×C3×C5C2×Q812yes—
120C23×C3×C5#4112yes—
121C23×C3×C5C2411yescharacteristic
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—44no—
131C2×C3×C4×C5C22×C412yes—
132C2×C3×C4×C5C22×C412yes—
133C2×Dic15—22no—
134C2×Dic15—22no—
135C2×Dic15—22no—
136C2×Dic15—22no—
137C2×Dic15—22no—
138C2×Dic15—28no—
139C2×Dic15—22no—
140C2×Dic15—22no—
141C2×Dic15—22no—
142C2×Dic15—44no—
143C2×Dic15C22×C412yes—
144C2×Dic15C22×C412yes—
145C2×Dic15C22×C412yes—
146C2×Dic15C22×C412yes—
147C2×Dic15C2×Q814yes—
148C2×Dic15#4114yes—
149C23×C3×C4C2×D511yescharacteristic
150C23×Dic3—51no—
151C23×Dic3—51no—
152C2×C3×C4⋊C4C2×D512yes—
153C2×C3×C22⋊C4C2×D511yescharacteristic
154C2×C3×C22⋊C4C2×D511yescharacteristic
155C2×#217—51no—
156C2×#217—51no—
157C2×Dic3⋊C4—58no—
158C2×Dic3⋊C4—51no—
159C2×Dic3⋊C4—51no—
160C2×Dic3⋊C4—51no—
161C2×Dic3⋊C4—51no—
162C2×Dic3⋊C4—52no—
163C2×Dic3⋊C4—52no—
164C2×C4×Dic3—52no—
165C2×C4×Dic3—52no—
166C2×C4×Dic3—52no—
167C3×#95C2×D512yes—
168C3⋊#95—52no—
169C3⋊#95—52no—
170C3⋊#95—52no—
171C3⋊#95—52no—
172C3⋊#95—52no—
173C24×C5C22×D311yescharacteristic
174C22×C4×C5—42no—
175C22×C4×C5C4×D314yes—
176C22×C4×C5C22×D312yes—
177C22×C4×C5C22×D312yes—
178C22×C4×C5C22×D311yescharacteristic
179C22×C4×C5C22×D311yescharacteristic
180C22×Dic5—31no—
181C22×Dic5—34no—
182C22×Dic5—32no—
183C22×Dic5—31no—
184C22×Dic5—31no—
185C22×Dic5—31no—
186C22×Dic5—34no—
187C22×Dic5—32no—
188C22×Dic5—64no—
189C22×Dic5—122no—
190C5×C4⋊C4—42no—
191C5×C22⋊C4—22no—
192C5×C22⋊C4—22no—
193C22⋊Dic5—34no—
194C22⋊Dic5—62no—
195#683—34no—
196#683—34no—
197#683—34no—
198#683—34no—
199#683—64no—
200#683—122no—
201C4×Dic5—64no—
202C4×Dic5—122no—
203C4×Dic5—122no—
204C22×C4⋊C4—151no—
205#389—152no—
206#387—152no—
207#387—152no—
208#380—151no—
209#380—151no—
210#380D1511yescharacteristic
211C4×C4⋊C4—152no—
212C4×C22⋊C4—151no—
213C4×C22⋊C4—151no—
214C4×C22⋊C4—151no—
215C22×C3×C5—22no—
216C22×C3×C5—22no—
217C22×C3×C5—24no—
218C22×C3×C5—24no—
219C22×C3×C5C4×Q814yes—
220C22×C3×C5C23×C411yescharacteristic
221C22×C3×C5C22×Q811yescharacteristic
222C22×C3×C5C2×#4111yescharacteristic
223C22×C3×C5C2×#4112yes—
224C22×C3×C5H(5,𝔽2)12yes—
225C3×C4×C5—22no—
226C3×C4×C5—22no—
227C3×C4×C5—42no—
228C3×C4×C5—42no—
229Dic15—22no—
230Dic15—22no—
231Dic15—22no—
232Dic15—24no—
233Dic15—42no—
234Dic15#11714yes—
235C24×C3C22×D511yescharacteristic
236C22×C3×C4—42no—
237C22×C3×C4C4×D514yes—
238C22×C3×C4C22×D512yes—
239C22×C3×C4C22×D512yes—
240C22×C3×C4C22×D511yescharacteristic
241C22×C3×C4C22×D511yescharacteristic
242C22×Dic3—51no—
243C22×Dic3—54no—
244C22×Dic3—51no—
245C22×Dic3—52no—
246C22×Dic3—51no—
247C22×Dic3—51no—
248C22×Dic3—54no—
249C22×Dic3—52no—
250C22×Dic3—104no—
251C22×Dic3—202no—
252C3×C4⋊C4—42no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—22no—
255#217—54no—
256#217—102no—
257Dic3⋊C4—54no—
258Dic3⋊C4—54no—
259Dic3⋊C4—54no—
260Dic3⋊C4—54no—
261Dic3⋊C4—104no—
262Dic3⋊C4—202no—
263C4×Dic3—104no—
264C4×Dic3—202no—
265C4×Dic3—202no—
266C23×C5—22no—
267C23×C5—22no—
268C23×C5C2×C4×D312yes—
269C23×C5#23712yes—
270C23×C5D3×Q812yes—
271C23×C5C23×D311yescharacteristic
272C2×C4×C5—22no—
273C2×C4×C5—22no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—22no—
279C2×C4×C5—22no—
280C2×C4×C5—44no—
281C2×C4×C5C2×C4×D312yes—
282C2×C4×C5C2×C4×D312yes—
283C2×Dic5—32no—
284C2×Dic5—34no—
285C2×Dic5—32no—
286C2×Dic5—32no—
287C2×Dic5—32no—
288C2×Dic5—34no—
289C2×Dic5—68no—
290C2×Dic5—62no—
291C2×Dic5—62no—
292C2×Dic5—62no—
293C2×Dic5—62no—
294C2×Dic5—62no—
295C2×Dic5—62no—
296C2×Dic5—62no—
297C2×Dic5—62no—
298C2×Dic5—124no—
299C23×C4—151no—
300C23×C4—151no—
301C23×C4C2×D1511yescharacteristic
302C2×C4⋊C4—158no—
303C2×C4⋊C4—151no—
304C2×C4⋊C4—151no—
305C2×C4⋊C4—151no—
306C2×C4⋊C4—151no—
307C2×C4⋊C4—152no—
308C2×C4⋊C4—152no—
309C2×C4⋊C4C2×D1512yes—
310C2×C22⋊C4—151no—
311C2×C22⋊C4—151no—
312C2×C22⋊C4C2×D1511yescharacteristic
313C2×C22⋊C4C2×D1511yescharacteristic
314C2×C42—152no—
315C2×C42—152no—
316C2×C42—152no—
317#95—152no—
318#95—152no—
319#95—152no—
320#95—152no—
321#95—152no—
322#95C2×D1512yes—
323C2×C3×C5—24no—
324C2×C3×C5C2×#11711yescharacteristic
325C2×C3×C5C2×C4×Q811yescharacteristic
326C2×C3×C5#51811yescharacteristic
327C2×C3×C5#55312yes—
328C2×C3×C5#55412yes—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3C2×C4×D512yes—
332C23×C3#71112yes—
333C23×C3Q8×D512yes—
334C23×C3C23×D511yescharacteristic
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—44no—
344C2×C3×C4C2×C4×D512yes—
345C2×C3×C4C2×C4×D512yes—
346C2×Dic3—52no—
347C2×Dic3—54no—
348C2×Dic3—52no—
349C2×Dic3—52no—
350C2×Dic3—52no—
351C2×Dic3—54no—
352C2×Dic3—108no—
353C2×Dic3—102no—
354C2×Dic3—102no—
355C2×Dic3—102no—
356C2×Dic3—102no—
357C2×Dic3—102no—
358C2×Dic3—102no—
359C2×Dic3—102no—
360C2×Dic3—102no—
361C2×Dic3—204no—
362C22×C5—24no—
363C22×C5—22no—
364C22×C5—22no—
365C22×C5—24no—
366C22×C5#88714yes—
367C22×C5C22×C4×D311yescharacteristic
368C22×C5C2×#23711yescharacteristic
369C22×C5#100412yes—
370C22×C5C2×D3×Q811yescharacteristic
371C22×C5D3×#4112yes—
372C4×C5—22no—
373C4×C5—22no—
374C4×C5—42no—
375C4×C5—42no—
376Dic5—34no—
377Dic5—64no—
378Dic5—62no—
379Dic5—62no—
380Dic5—62no—
381Dic5—122no—
382C24C22×D1511yescharacteristic
383C22×C4—42no—
384C22×C4—151no—
385C22×C4—154no—
386C22×C4—151no—
387C22×C4—152no—
388C22×C4—151no—
389C22×C4—151no—
390C22×C4—154no—
391C22×C4—152no—
392C22×C4—304no—
393C22×C4—602no—
394C22×C4C4×D1514yes—
395C22×C4C22×D1512yes—
396C22×C4C22×D1512yes—
397C22×C4C22×D1511yescharacteristic
398C22×C4C22×D1511yescharacteristic
399C4⋊C4—42no—
400C4⋊C4—154no—
401C4⋊C4—154no—
402C4⋊C4—154no—
403C4⋊C4—154no—
404C4⋊C4—304no—
405C4⋊C4—602no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—154no—
409C22⋊C4—302no—
410C42—304no—
411C42—602no—
412C42—602no—
413C3×C5#234511yescharacteristic, complemented, Hall
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—24no—
417C22×C3—24no—
418C22×C3#403314yes—
419C22×C3C22×C4×D511yescharacteristic
420C22×C3C2×#71111yescharacteristic
421C22×C3#414412yes—
422C22×C3C2×Q8×D511yescharacteristic
423C22×C3D5×#4112yes—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—42no—
427C3×C4—42no—
428Dic3—54no—
429Dic3—102no—
430Dic3—102no—
431Dic3—102no—
432Dic3—104no—
433Dic3—202no—
434C2×C5—24no—
435C2×C5#548611yescharacteristic
436C2×C5C2×#88711yescharacteristic
437C2×C5D3×#11711yescharacteristic
438C2×C5#560112yes—
439C2×C5#563612yes—
440C23—22no—
441C23—22no—
442C23C2×C4×D1512yes—
443C23#681912yes—
444C23Q8×D1512yes—
445C23C23×D1511yescharacteristic
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—44no—
455C2×C4—152no—
456C2×C4—154no—
457C2×C4—152no—
458C2×C4—152no—
459C2×C4—152no—
460C2×C4—154no—
461C2×C4—308no—
462C2×C4—302no—
463C2×C4—302no—
464C2×C4—302no—
465C2×C4—302no—
466C2×C4—302no—
467C2×C4—302no—
468C2×C4—302no—
469C2×C4—302no—
470C2×C4—604no—
471C2×C4C2×C4×D1512yes—
472C2×C4C2×C4×D1512yes—
473C2×C3—24no—
474C2×C3#6582011yescharacteristic
475C2×C3C2×#403311yescharacteristic
476C2×C3D5×#11711yescharacteristic
477C2×C3#6593512yes—
478C2×C3#6597012yes—
479C5#7388011yescharacteristic, complemented, Sylow
480C22—24no—
481C22—24no—
482C22—22no—
483C22—22no—
484C22#9177614yes—
485C22C22×C4×D1511yescharacteristic
486C22C2×#681911yescharacteristic
487C22#9209512yes—
488C22C2×Q8×D1511yescharacteristic
489C22#41×D1512yes—
490C4—22no—
491C4—22no—
492C4—42no—
493C4—42no—
494C4—154no—
495C4—304no—
496C4—302no—
497C4—302no—
498C4—302no—
499C4—602no—
500C3#1060561111yescharacteristic, complemented, Sylow
501C2—24no—
502C2#1175624511yescharacteristic
503C2C2×#9177611yescharacteristic
504C2D15×#11711yescharacteristic
505C2#1175636012yes—
506C2#1175639512yes—
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#380C2
socle, cosocleC23×C3×C5C24
prime radicalC24×C3×C5C23
hypercenter#380D15
hypocenterC3×C5#2345
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
4. #11751079C2semidirect
5. #11751077C2semidirect
10. #11750753C2semidirect
11. #11750750C2semidirect
21. C2×#6712C4semidirect
413. C3×C5#2345semidirect
479. C5#73880semidirect
500. C3#10605611semidirect

Maximal subgroups

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

subgroupindex
C22×#67122
C3×C5×#3802
C4×#67192
#117510792
#117510772
#117510432
C22⋊C4×Dic152
#117507532
#117507502
#106056113
#738805