Finite Group Databasebeta

← #14321153

Subgroups — #14321153

Order 1920.

1872 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6819C211yesmaximal, characteristic
2C2×#91781C211yesmaximal, characteristic, complemented
3C2×C42×D15C211yesmaximal, characteristic, complemented
4C3×C5×#390C211yesmaximal, characteristic, complemented
5#11751322C211yesmaximal, characteristic, complemented
6#11751074C211yesmaximal, characteristic, complemented
7#11751057C214yesmaximal, complemented
8C22⋊C4×Dic15C214yesmaximal, complemented
9#11751030C211yesmaximal, characteristic, complemented
10#10608021—31nomaximal, Hall
11C22×C3×C5×D4C2211yescharacteristic
12C22×#1220C2212yes—
13C23×Dic15C2212yes—
14C2×#6819—24no—
15C2×#6819—24no—
16C2×#6819C418yescomplemented
17C22×C4×D15C2212yes—
18C22×C4×D15C2211yescharacteristic
19C22×Dic30C2211yescharacteristic
20C2×C3×C5×C4⋊C4C2211yescharacteristic
21C2×C3×C5×C22⋊C4C2214yes—
22C2×C3×C42×C5C2211yescharacteristic
23C2×#6719C2214yes—
24C2×#6714C2212yes—
25C2×C4⋊Dic15C2211yescharacteristic
26C2×C4×Dic15C2212yes—
27C2×C4×Dic15C2212yes—
28C2×C4×Dic15C2211yescharacteristic
29#91781—24no—
30C42×D15—24no—
31#91115C2214yes—
32#91115C2214yes—
33#76290—51nomaximal, Hall
34C22×#711—31no—
35C2×#4038—31no—
36C2×C42×D5—31no—
37C5×#390D311yescharacteristic, complemented
38#65510—31no—
39#65263—31no—
40#65245—34no—
41C22⋊C4×Dic5—34no—
42#65218—31no—
43C24×C3×C5C2312yes—
44C23×D15C2311yescharacteristic
45C2×C3×C5×D4—22no—
46C2×C3×C5×D4—22no—
47C2×C3×C5×D4C2×C414yescomplemented
48C22×C3×C4×C5C2312yes—
49C22×C3×C4×C5C2312yes—
50C22×C3×C4×C5C2311yescharacteristic
51C2×#1220—24no—
52C2×#1220—24no—
53C2×#1220C2×C418yes—
54C22×Dic15—24no—
55C22×Dic15—24no—
56C22×Dic15C2×C418yescomplemented
57C22×Dic15C2312yes—
58C22×Dic15C2312yes—
59C22×Dic15C2312yes—
60C22×Dic15C2311yescharacteristic
61#6819—232no—
62C2×C4×D15—24no—
63C2×C4×D15—24no—
64C2×C4×D15—24no—
65C2×C4×D15C2×C414yes—
66C2×C4×D15D414yes—
67C2×C4×D15D414yes—
68C2×Dic30—22no—
69C2×Dic30—22no—
70C2×Dic30C2×C414yescomplemented
71C3×C5×C4⋊C4—22no—
72C3×C5×C22⋊C4—28no—
73C3×C42×C5—22no—
74#6719—28no—
75#6714—24no—
76C4⋊Dic15—22no—
77C4×Dic15—22no—
78C4×Dic15—24no—
79C4×Dic15—24no—
80C22×#237—51no—
81C2×#892—51no—
82C2×C42×D3—51no—
83C3×#390D511yescharacteristic, complemented
84#5221—51no—
85#4973—51no—
86#4956—54no—
87Dic3×C22⋊C4—54no—
88#4929—51no—
89C22×C5×D4C2×D311yescharacteristic
90C22×C22⋊D5—32no—
91C23×Dic5—32no—
92C2×#711—38no—
93C2×#711—64no—
94C2×#711—64no—
95C22×C4×D5—31no—
96C22×C4×D5—32no—
97C22×Dic10—31no—
98C2×C5×C4⋊C4C2×D311yescharacteristic
99C2×C5×C22⋊C4C2×D314yes—
100C2×C42×C5C2×D311yescharacteristic
101C2×C22⋊Dic5—34no—
102C2×#685—32no—
103C2×C4⋊Dic5—31no—
104C2×C4×Dic5—32no—
105C2×C4×Dic5—31no—
106C2×C4×Dic5—32no—
107#4038—64no—
108C42×D5—64no—
109#3963—34no—
110#3963—34no—
111#2297—151noSylow
112C23×C3×C5—22no—
113C23×C3×C5—22no—
114C23×C3×C5—24no—
115C23×C3×C5—24no—
116C23×C3×C5C22×C414yes—
117C23×C3×C5C2411yescharacteristic
118C22×D15—22no—
119C22×D15—22no—
120C22×D15C22×C412yes—
121C22×D15C2×D412yes—
122C22×D15C2×D412yes—
123C3×C5×D4—28no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—24no—
128C2×C3×C4×C5C22×C412yes—
129C2×C3×C4×C5C2×D412yes—
130C2×C3×C4×C5C2×D412yes—
131C2×C3×C4×C5#4114yescomplemented
132#1220—216no—
133C2×Dic15—22no—
134C2×Dic15—22no—
135C2×Dic15—22no—
136C2×Dic15—22no—
137C2×Dic15—216no—
138C2×Dic15—22no—
139C2×Dic15—24no—
140C2×Dic15C22×C414yes—
141C2×Dic15C22×C412yes—
142C2×Dic15C2×D412yes—
143C2×Dic15C2×D412yes—
144C2×Dic15#4114yescomplemented
145C4×D15—216no—
146C4×D15C22⋊C4116yescomplemented
147Dic30—28no—
148C22×C3×D4C2×D511yescharacteristic
149C22×#67—52no—
150C23×Dic3—52no—
151C2×#237—58no—
152C2×#237—104no—
153C2×#237—104no—
154C22×C4×D3—51no—
155C22×C4×D3—52no—
156C22×Dic6—51no—
157C2×C3×C4⋊C4C2×D511yescharacteristic
158C2×C3×C22⋊C4C2×D514yes—
159C2×C3×C42C2×D511yescharacteristic
160C2×#217—54no—
161C2×(C2×D3)⋊C4—52no—
162C2×C4⋊Dic3—51no—
163C2×C4×Dic3—52no—
164C2×C4×Dic3—52no—
165C2×C4×Dic3—51no—
166#892—104no—
167C42×D3—104no—
168C3⋊#95—54no—
169C3⋊#95—54no—
170C24×C5C22×D312yes—
171C23×D5—31no—
172C2×C5×D4—22no—
173C2×C5×D4—22no—
174C2×C5×D4C4×D314yescomplemented
175C22×C4×C5C22×D312yes—
176C22×C4×C5C22×D312yes—
177C22×C4×C5C22×D311yescharacteristic
178C2×C22⋊D5—38no—
179C2×C22⋊D5—64no—
180C2×C22⋊D5—64no—
181C22×Dic5—31no—
182C22×Dic5—38no—
183C22×Dic5—32no—
184C22×Dic5—32no—
185C22×Dic5—32no—
186C22×Dic5—64no—
187C22×Dic5—64no—
188#711—632no—
189C2×C4×D5—34no—
190C2×C4×D5—34no—
191C2×C4×D5—34no—
192C2×C4×D5—64no—
193C2×C4×D5—64no—
194C2×C4×D5—64no—
195C2×Dic10—34no—
196C2×Dic10—62no—
197C2×Dic10—62no—
198C5×C4⋊C4—22no—
199C5×C22⋊C4—28no—
200C42×C5—22no—
201C22⋊Dic5—68no—
202#685—64no—
203C4⋊Dic5—62no—
204C4×Dic5—64no—
205C4×Dic5—64no—
206C4×Dic5—62no—
207C22×#41—151no—
208C2×#117—151no—
209C22×C42—151no—
210#391—151no—
211#391—151no—
212#390—151no—
213#390D1511yescharacteristic, complemented
214#386—154no—
215C4×C22⋊C4—154no—
216C22×C3×C5—24no—
217C22×C3×C5—28no—
218C22×C3×C5—44no—
219C22×C3×C5—42no—
220C22×C3×C5—42no—
221C22×C3×C5C23×C411yescharacteristic
222C22×C3×C5C22×D411yescharacteristic
223C22×C3×C5C22×D411yescharacteristic
224C22×C3×C5C2×#4112yes—
225C22×C3×C5C2×#4112yes—
226C2×D15—24no—
227C2×D15—24no—
228C2×D15—22no—
229C2×D15—22no—
230C2×D15C2×C22⋊C414yes—
231C3×C4×C5—24no—
232C3×C4×C5—42no—
233C3×C4×C5C2×C22⋊C414yes—
234Dic15—24no—
235Dic15—24no—
236Dic15—42no—
237Dic15C2×C22⋊C414yes—
238C24×C3C22×D512yes—
239C23×D3—51no—
240C2×C3×D4—22no—
241C2×C3×D4—22no—
242C2×C3×D4C4×D514yescomplemented
243C22×C3×C4C22×D512yes—
244C22×C3×C4C22×D512yes—
245C22×C3×C4C22×D511yescharacteristic
246C2×#67—58no—
247C2×#67—104no—
248C2×#67—104no—
249C22×Dic3—51no—
250C22×Dic3—58no—
251C22×Dic3—52no—
252C22×Dic3—52no—
253C22×Dic3—52no—
254C22×Dic3—104no—
255C22×Dic3—104no—
256#237—1032no—
257C2×C4×D3—54no—
258C2×C4×D3—54no—
259C2×C4×D3—54no—
260C2×C4×D3—104no—
261C2×C4×D3—104no—
262C2×C4×D3—104no—
263C2×Dic6—54no—
264C2×Dic6—102no—
265C2×Dic6—102no—
266C3×C4⋊C4—22no—
267C3×C22⋊C4—28no—
268C3×C42—22no—
269#217—108no—
270(C2×D3)⋊C4—104no—
271C4⋊Dic3—102no—
272C4×Dic3—104no—
273C4×Dic3—104no—
274C4×Dic3—102no—
275C23×C5—22no—
276C23×C5—22no—
277C23×C5—24no—
278C23×C5—24no—
279C23×C5C2×C4×D314yes—
280C23×C5C23×D311yescharacteristic
281C22×D5—32no—
282C22×D5—32no—
283C22×D5—32no—
284C22×D5—62no—
285C22×D5—62no—
286C5×D4—28no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—24no—
291C2×C4×C5C2×C4×D312yes—
292C2×C4×C5D3×D412yes—
293C2×C4×C5D3×D412yes—
294C2×C4×C5#23714yescomplemented
295C22⋊D5—616no—
296C2×Dic5—32no—
297C2×Dic5—32no—
298C2×Dic5—32no—
299C2×Dic5—34no—
300C2×Dic5—34no—
301C2×Dic5—616no—
302C2×Dic5—62no—
303C2×Dic5—62no—
304C2×Dic5—62no—
305C2×Dic5—62no—
306C2×Dic5—62no—
307C2×Dic5—64no—
308C4×D5—316no—
309C4×D5—616no—
310Dic10—68no—
311C2×#41—158no—
312C2×#41—304no—
313C2×#41—304no—
314C22×Q8—151no—
315C22×D4—152no—
316C22×D4C2×D1511yescharacteristic
317C23×C4—151no—
318C23×C4—152no—
319C23×C4—152no—
320#117—304no—
321C2×C4⋊C4—151no—
322C2×C4⋊C4C2×D1511yescharacteristic
323C2×C22⋊C4—154no—
324C2×C22⋊C4—152no—
325C2×C22⋊C4C2×D1514yes—
326C2×C42—152no—
327C2×C42—152no—
328C2×C42—151no—
329C2×C42—304no—
330C2×C42C2×D1511yescharacteristic
331#95—154no—
332#95—154no—
333C2×C3×C5—44no—
334C2×C3×C5C22×C22⋊C411yescharacteristic
335C2×C3×C5C4×#4112yes—
336C2×C3×C5#52512yes—
337C2×C3×C5#53211yescharacteristic
338C2×C3×C5#53211yescharacteristic
339D15—24no—
340C23×C3—22no—
341C23×C3—22no—
342C23×C3—24no—
343C23×C3—24no—
344C23×C3C2×C4×D514yes—
345C23×C3C23×D511yescharacteristic
346C22×D3—52no—
347C22×D3—52no—
348C22×D3—52no—
349C22×D3—102no—
350C22×D3—102no—
351C3×D4—28no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4C2×C4×D512yes—
357C2×C3×C4D4×D512yes—
358C2×C3×C4D4×D512yes—
359C2×C3×C4#71114yescomplemented
360#67—1016no—
361C2×Dic3—52no—
362C2×Dic3—52no—
363C2×Dic3—52no—
364C2×Dic3—54no—
365C2×Dic3—54no—
366C2×Dic3—1016no—
367C2×Dic3—102no—
368C2×Dic3—102no—
369C2×Dic3—102no—
370C2×Dic3—102no—
371C2×Dic3—102no—
372C2×Dic3—104no—
373C4×D3—516no—
374C4×D3—1016no—
375Dic6—108no—
376C22×C5—24no—
377C22×C5—28no—
378C22×C5—42no—
379C22×C5—42no—
380C22×C5—44no—
381C22×C5C22×C4×D311yescharacteristic
382C22×C5C2×D3×D411yescharacteristic
383C22×C5C2×D3×D411yescharacteristic
384C22×C5C2×#23712yes—
385C22×C5D3×#4112yes—
386C2×D5—34no—
387C2×D5—62no—
388C2×D5—62no—
389C2×D5—64no—
390C2×D5—64no—
391C4×C5—24no—
392C4×C5—42no—
393C4×C5D3×C22⋊C414yes—
394Dic5—34no—
395Dic5—64no—
396Dic5—64no—
397Dic5—122no—
398C24—151no—
399C24C22×D1512yes—
400#41—3032no—
401C2×Q8—154no—
402C2×Q8—302no—
403C2×Q8—302no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—158no—
407C2×D4—304no—
408C2×D4—304no—
409C2×D4C4×D1514yescomplemented
410C22×C4—151no—
411C22×C4—154no—
412C22×C4—154no—
413C22×C4—154no—
414C22×C4—158no—
415C22×C4—152no—
416C22×C4—152no—
417C22×C4—152no—
418C22×C4—304no—
419C22×C4—304no—
420C22×C4—304no—
421C22×C4—304no—
422C22×C4—304no—
423C22×C4C22×D1512yes—
424C22×C4C22×D1512yes—
425C22×C4C22×D1511yescharacteristic
426C4⋊C4—22no—
427C4⋊C4—302no—
428C22⋊C4—28no—
429C22⋊C4—304no—
430C22⋊C4—308no—
431C42—22no—
432C42—304no—
433C42—304no—
434C42—302no—
435C3×C5#229711yescharacteristic, complemented, Hall
436C22×C3—24no—
437C22×C3—28no—
438C22×C3—42no—
439C22×C3—44no—
440C22×C3—42no—
441C22×C3C22×C4×D511yescharacteristic
442C22×C3C2×D4×D511yescharacteristic
443C22×C3C2×D4×D511yescharacteristic
444C22×C3C2×#71112yes—
445C22×C3D5×#4112yes—
446C2×D3—54no—
447C2×D3—102no—
448C2×D3—104no—
449C2×D3—104no—
450C2×D3—102no—
451C3×C4—24no—
452C3×C4—42no—
453C3×C4D5×C22⋊C414yes—
454Dic3—54no—
455Dic3—104no—
456Dic3—104no—
457Dic3—202no—
458C2×C5—44no—
459C2×C5C2×D3×C22⋊C411yescharacteristic
460C2×C5C4×#23712yes—
461C2×C5#560612yes—
462C2×C5#566811yescharacteristic
463C2×C5#571611yescharacteristic
464D5—64no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—152no—
470C23—152no—
471C23—152no—
472C23—302no—
473C23—302no—
474C23C2×C4×D1514yes—
475C23C23×D1511yescharacteristic
476Q8—308no—
477D4—28no—
478D4—3016no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—152no—
484C2×C4—152no—
485C2×C4—152no—
486C2×C4—1516no—
487C2×C4—154no—
488C2×C4—154no—
489C2×C4—3016no—
490C2×C4—3016no—
491C2×C4—302no—
492C2×C4—302no—
493C2×C4—302no—
494C2×C4—302no—
495C2×C4—302no—
496C2×C4—304no—
497C2×C4C2×C4×D1512yes—
498C2×C4D4×D1512yes—
499C2×C4D4×D1512yes—
500C2×C4#681914yescomplemented
501C2×C3—44no—
502C2×C3C2×D5×C22⋊C411yescharacteristic
503C2×C3C4×#71112yes—
504C2×C3#6594012yes—
505C2×C3#6600211yescharacteristic
506C2×C3#6605011yescharacteristic
507D3—104no—
508C5#7629011yescharacteristic, complemented, Sylow
509C22—28no—
510C22—24no—
511C22—44no—
512C22—42no—
513C22—42no—
514C22—154no—
515C22—304no—
516C22—304no—
517C22—302no—
518C22—302no—
519C22C22×C4×D1511yescharacteristic
520C22C2×D4×D1511yescharacteristic
521C22C2×D4×D1511yescharacteristic
522C22C2×#681912yes—
523C22#41×D1512yes—
524C4—24no—
525C4—42no—
526C4—154no—
527C4—304no—
528C4—304no—
529C4—602no—
530C4C22⋊C4×D1514yes—
531C3#1060802111yescharacteristic, complemented, Sylow
532C2—44no—
533C2—304no—
534C2C2×C22⋊C4×D1511yescharacteristic
535C2C4×#681912yes—
536C2#1175636512yes—
537C2#1175642711yescharacteristic
538C2#1175647511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#390C2
socle, cosocleC23×C3×C5C24
prime radicalC22×#6819C2
hypercenter#390D15
hypocenterC3×C5#2297
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#91781C2semidirect
3. C2×C42×D15C2semidirect
4. C3×C5×#390C2semidirect
5. #11751322C2semidirect
6. #11751074C2semidirect
7. #11751057C2semidirect
8. C22⋊C4×Dic15C2semidirect
9. #11751030C2semidirect
16. C2×#6819C4semidirect
37. C5×#390D3semidirect
47. C2×C3×C5×D4C2×C4semidirect
56. C22×Dic15C2×C4semidirect
70. C2×Dic30C2×C4semidirect
83. C3×#390D5semidirect
131. C2×C3×C4×C5#41semidirect
144. C2×Dic15#41semidirect
146. C4×D15C22⋊C4semidirect
174. C2×C5×D4C4×D3semidirect
213. #390D15semidirect
242. C2×C3×D4C4×D5semidirect
294. C2×C4×C5#237semidirect
359. C2×C3×C4#711semidirect
409. C2×D4C4×D15semidirect
435. C3×C5#2297semidirect
500. C2×C4#6819semidirect
508. C5#76290semidirect
531. C3#10608021semidirect

Maximal subgroups

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

subgroupindex
C22×#68192
C2×#917812
C2×C42×D152
C3×C5×#3902
#117513222
#117510742
#117510572
C22⋊C4×Dic152
#117510302
#106080213
#762905