Finite Group Databasebeta

← #14321212

Subgroups — #14321212

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×#6765C211yesmaximal, characteristic
2C2×C42×C5×D3C211yesmaximal, characteristic, complemented
3C2×#91338C211yesmaximal, characteristic, complemented
4#11751030C211yesmaximal, characteristic, complemented
5C3×#65228C211yesmaximal, characteristic, complemented
6#11748910C214yesmaximal, complemented
7Dic3×#685C214yesmaximal, complemented
8#11748785C211yesmaximal, characteristic, complemented
9#11748599C211yesmaximal, characteristic, complemented
10#10613801—31nomaximal, Hall
11C22×Dic30C2211yescharacteristic
12C22×C4×C5×D3C2212yes—
13C22×C4×C5×D3C2211yescharacteristic
14C22×C3×D20C2211yescharacteristic
15C22×#1201C2212yes—
16C22×D5×Dic3C2212yes—
17C2×#6765—24no—
18C2×#6765—24no—
19C2×#6765C418yescomplemented
20C2×C3×C42×C5C2211yescharacteristic
21C2×C4⋊Dic15C2211yescharacteristic
22C2×C4×C5×Dic3C2212yes—
23C2×C4×C5×Dic3C2211yescharacteristic
24C42×C5×D3—24no—
25C2×C3×#685C2214yes—
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C2×(C2×D3)⋊Dic5C2212yes—
28C2×#6665C2214yes—
29C2×Dic3×Dic5C2212yes—
30#91338—24no—
31#91115C2214yes—
32#90994C2214yes—
33#76290—51nomaximal, Hall
34C22×#709—31no—
35C22×C42×C5—31no—
36C2×#4072—31no—
37#65502—34no—
38#65228—31no—
39#65228D311yescharacteristic, complemented
40C4×#685—34no—
41#65218—31no—
42#65218—31no—
43C23×C5×D3C2311yescharacteristic
44C23×C3×D5C2312yes—
45C22×C3×C4×C5C2312yes—
46C22×C3×C4×C5C2311yescharacteristic
47C22×Dic15C2312yes—
48C22×Dic15C2312yes—
49C2×Dic30—22no—
50C2×Dic30—22no—
51C2×Dic30C2×C414yescomplemented
52C22×C5×Dic3C2312yes—
53C22×C5×Dic3C2311yescharacteristic
54C2×C4×C5×D3—24no—
55C2×C4×C5×D3—24no—
56C2×C4×C5×D3—24no—
57C2×C4×C5×D3C2×C414yes—
58C2×C4×C5×D3D414yes—
59C2×C4×C5×D3D414yes—
60C22×C3×Dic5C2312yes—
61C2×C3×D20—22no—
62C2×C3×D20—22no—
63C2×C3×D20C2×C414yescomplemented
64C2×#1201—24no—
65C2×#1201—24no—
66C2×#1201C2×C418yes—
67C2×D5×Dic3—24no—
68C2×D5×Dic3—24no—
69C2×D5×Dic3C2×C418yescomplemented
70#6765—232no—
71C3×C42×C5—22no—
72C4⋊Dic15—22no—
73C4×C5×Dic3—22no—
74C4×C5×Dic3—24no—
75C3×#685—28no—
76C3×C4⋊Dic5—22no—
77(C2×D3)⋊Dic5—24no—
78#6665—28no—
79Dic3×Dic5—24no—
80C22×#237—51no—
81C2×#892—51no—
82C2×C42×D3D511yescharacteristic, complemented
83C3×#390—51no—
84#5221—51no—
85#4973—51no—
86#4956—54no—
87Dic3×C22⋊C4—54no—
88#4929—51no—
89C23×C4×C5—31no—
90C23×C4×C5—32no—
91C22×C22⋊D5—32no—
92C2×#709—38no—
93C2×#709—64no—
94C2×#709—64no—
95C22×D20C2×D311yescharacteristic
96C22×C4×D5—32no—
97C22×Dic10—31no—
98C2×C42×C5—31no—
99C2×C42×C5—32no—
100C2×C42×C5—64no—
101C2×C42×C5C2×D311yescharacteristic
102C2×C22⋊Dic5—32no—
103C2×#685—34no—
104C2×#685C2×D314yes—
105#4072—64no—
106C2×C4⋊Dic5—31no—
107C2×C4⋊Dic5C2×D311yescharacteristic
108C2×C4×Dic5—32no—
109#3963—34no—
110#3963—34no—
111#2297—151noSylow
112C23×C3×C5C2411yescharacteristic
113C22×C5×D3—22no—
114C22×C5×D3—22no—
115C22×C5×D3C22×C412yes—
116C22×C5×D3C2×D412yes—
117C22×C5×D3C2×D412yes—
118C22×C3×D5—22no—
119C22×C3×D5—22no—
120C22×C3×D5—24no—
121C22×C3×D5—24no—
122C22×C3×D5C22×C414yes—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5C22×C412yes—
126C2×C3×C4×C5C2×D412yes—
127C2×C3×C4×C5C2×D412yes—
128C2×C3×C4×C5#4114yescomplemented
129C2×Dic15—22no—
130C2×Dic15—22no—
131C2×Dic15—22no—
132C2×Dic15—24no—
133C2×Dic15C22×C414yes—
134Dic30—28no—
135C2×C5×Dic3—22no—
136C2×C5×Dic3—22no—
137C2×C5×Dic3C22×C412yes—
138C2×C5×Dic3C2×D412yes—
139C2×C5×Dic3C2×D412yes—
140C2×C5×Dic3#4114yescomplemented
141C4×C5×D3—216no—
142C4×C5×D3C22⋊C4116yescomplemented
143C2×C3×Dic5—22no—
144C2×C3×Dic5—24no—
145C3×D20—28no—
146#1201—216no—
147D5×Dic3—216no—
148C22×C3×D4—51no—
149C22×#67—52no—
150C23×Dic3—52no—
151C2×#237—58no—
152C2×#237—104no—
153C2×#237—104no—
154C22×C4×D3C2×D512yes—
155C22×C4×D3C2×D511yescharacteristic
156C22×Dic6—51no—
157C2×C3×C4⋊C4—51no—
158C2×C3×C22⋊C4—54no—
159C2×C3×C42C2×D511yescharacteristic
160C2×#217—54no—
161C2×(C2×D3)⋊C4—52no—
162C2×C4⋊Dic3—51no—
163C2×C4×Dic3—52no—
164C2×C4×Dic3C2×D512yes—
165C2×C4×Dic3C2×D511yescharacteristic
166#892—104no—
167C42×D3—24no—
168C3⋊#95—54no—
169C3⋊#95—54no—
170C24×C5—31no—
171C23×D5C22×D312yes—
172C22×C4×C5—31no—
173C22×C4×C5—34no—
174C22×C4×C5—34no—
175C22×C4×C5—34no—
176C22×C4×C5—32no—
177C22×C4×C5—64no—
178C22×C4×C5—64no—
179C22×C4×C5—64no—
180C22×C4×C5C22×D312yes—
181C22×C4×C5C22×D311yescharacteristic
182C2×C22⋊D5—38no—
183C2×C22⋊D5—64no—
184C2×C22⋊D5—64no—
185C22×Dic5—32no—
186C22×Dic5—32no—
187C22×Dic5C22×D312yes—
188#709—632no—
189C2×D20—22no—
190C2×D20—22no—
191C2×D20C4×D314yescomplemented
192C2×C4×D5—38no—
193C2×C4×D5—64no—
194C2×C4×D5—64no—
195C2×Dic10—34no—
196C2×Dic10—62no—
197C2×Dic10—62no—
198C42×C5—22no—
199C42×C5—64no—
200C42×C5—62no—
201C22⋊Dic5—64no—
202#685—28no—
203#685—68no—
204C4⋊Dic5—22no—
205C4⋊Dic5—62no—
206C4×Dic5—64no—
207C22×#41—151no—
208C2×#117—151no—
209C22×C42—31no—
210#391—151no—
211#391—151no—
212#390—51no—
213#390—151no—
214#386—154no—
215C4×C22⋊C4—154no—
216C22×C3×C5C23×C411yescharacteristic
217C22×C3×C5C22×D411yescharacteristic
218C22×C3×C5C22×D411yescharacteristic
219C22×C3×C5C2×#4112yes—
220C22×C3×C5C2×#4112yes—
221C2×C5×D3—24no—
222C2×C5×D3—24no—
223C2×C5×D3—22no—
224C2×C5×D3—22no—
225C2×C5×D3C2×C22⋊C414yes—
226C2×C3×D5—24no—
227C2×C3×D5—28no—
228C2×C3×D5—44no—
229C2×C3×D5—42no—
230C2×C3×D5—42no—
231C3×C4×C5—24no—
232C3×C4×C5C2×C22⋊C414yes—
233Dic15—24no—
234Dic15—42no—
235C3×Dic5—42no—
236C5×Dic3—24no—
237C5×Dic3C2×C22⋊C414yes—
238C24×C3—52no—
239C23×D3C22×D511yescharacteristic
240C2×C3×D4—54no—
241C2×C3×D4—102no—
242C2×C3×D4—102no—
243C22×C3×C4—52no—
244C22×C3×C4C22×D512yes—
245C22×C3×C4C22×D511yescharacteristic
246C2×#67—58no—
247C2×#67—104no—
248C2×#67—104no—
249C22×Dic3—52no—
250C22×Dic3—58no—
251C22×Dic3—52no—
252C22×Dic3—104no—
253C22×Dic3—104no—
254C22×Dic3C22×D512yes—
255C22×Dic3C22×D511yescharacteristic
256#237—1032no—
257C2×C4×D3—24no—
258C2×C4×D3—24no—
259C2×C4×D3—24no—
260C2×C4×D3C4×D514yes—
261C2×C4×D3D2014yes—
262C2×C4×D3C22⋊D514yes—
263C2×Dic6—54no—
264C2×Dic6—102no—
265C2×Dic6—102no—
266C3×C4⋊C4—102no—
267C3×C22⋊C4—108no—
268C3×C42—22no—
269#217—108no—
270(C2×D3)⋊C4—104no—
271C4⋊Dic3—102no—
272C4×Dic3—22no—
273C4×Dic3—24no—
274C4×Dic3—104no—
275C23×C5—32no—
276C23×C5—32no—
277C23×C5—32no—
278C23×C5—62no—
279C23×C5—62no—
280C23×C5C23×D311yescharacteristic
281C22×D5—22no—
282C22×D5—22no—
283C22×D5—24no—
284C22×D5—24no—
285C22×D5C2×C4×D314yes—
286C2×C4×C5—22no—
287C2×C4×C5—22no—
288C2×C4×C5—32no—
289C2×C4×C5—32no—
290C2×C4×C5—32no—
291C2×C4×C5—316no—
292C2×C4×C5—34no—
293C2×C4×C5—616no—
294C2×C4×C5—62no—
295C2×C4×C5—62no—
296C2×C4×C5C2×C4×D312yes—
297C2×C4×C5D3×D412yes—
298C2×C4×C5D3×D412yes—
299C2×C4×C5#23714yescomplemented
300C22⋊D5—616no—
301C2×Dic5—22no—
302C2×Dic5—24no—
303C2×Dic5—34no—
304C2×Dic5—62no—
305C2×Dic5—62no—
306C2×Dic5—62no—
307C2×Dic5—64no—
308D20—28no—
309C4×D5—616no—
310Dic10—68no—
311C2×#41—158no—
312C2×#41—304no—
313C2×#41—304no—
314C22×Q8—151no—
315C22×D4—51no—
316C22×D4—152no—
317C23×C4—31no—
318C23×C4—32no—
319C23×C4—152no—
320#117—304no—
321C2×C4⋊C4—51no—
322C2×C4⋊C4—151no—
323C2×C22⋊C4—54no—
324C2×C22⋊C4—154no—
325C2×C22⋊C4—152no—
326C2×C42—32no—
327C2×C42—31no—
328C2×C42—64no—
329C2×C42—152no—
330C2×C42D3×D511yescharacteristic
331#95—154no—
332#95—154no—
333C2×C3×C5C22×C22⋊C411yescharacteristic
334C2×C3×C5C4×#4112yes—
335C2×C3×C5#52512yes—
336C2×C3×C5#53211yescharacteristic
337C2×C3×C5#53211yescharacteristic
338C3×D5—44no—
339C5×D3—24no—
340C23×C3—54no—
341C23×C3—102no—
342C23×C3—104no—
343C23×C3—104no—
344C23×C3—102no—
345C23×C3C23×D511yescharacteristic
346C22×D3—22no—
347C22×D3—22no—
348C22×D3C2×C4×D512yes—
349C22×D3C2×D2012yes—
350C22×D3C2×C22⋊D512yes—
351C3×D4—108no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—102no—
355C2×C3×C4—104no—
356C2×C3×C4C2×C4×D512yes—
357C2×C3×C4C2×D2012yes—
358C2×C3×C4#70914yescomplemented
359C2×C3×C4C2×C22⋊D512yes—
360#67—1016no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—54no—
364C2×Dic3—102no—
365C2×Dic3—102no—
366C2×Dic3—1016no—
367C2×Dic3—102no—
368C2×Dic3—104no—
369C2×Dic3C2×C4×D512yes—
370C2×Dic3C2×D2012yes—
371C2×Dic3#70914yescomplemented
372C2×Dic3C2×C22⋊D512yes—
373C4×D3—216no—
374C4×D3#685116yescomplemented
375Dic6—108no—
376C22×C5—34no—
377C22×C5—62no—
378C22×C5—62no—
379C22×C5—64no—
380C22×C5—64no—
381C22×C5C22×C4×D311yescharacteristic
382C22×C5C2×D3×D411yescharacteristic
383C22×C5C2×D3×D411yescharacteristic
384C22×C5C2×#23712yes—
385C22×C5D3×#4112yes—
386C2×D5—24no—
387C2×D5—28no—
388C2×D5—42no—
389C2×D5—42no—
390C2×D5—44no—
391C4×C5—24no—
392C4×C5—34no—
393C4×C5—64no—
394C4×C5D3×C22⋊C414yes—
395Dic5—42no—
396Dic5—64no—
397Dic5—122no—
398C24—31no—
399C24—52no—
400#41—3032no—
401C2×Q8—154no—
402C2×Q8—302no—
403C2×Q8—302no—
404C2×D4—54no—
405C2×D4—102no—
406C2×D4—102no—
407C2×D4—158no—
408C2×D4—304no—
409C2×D4—304no—
410C22×C4—31no—
411C22×C4—34no—
412C22×C4—34no—
413C22×C4—34no—
414C22×C4—32no—
415C22×C4—52no—
416C22×C4—64no—
417C22×C4—64no—
418C22×C4—64no—
419C22×C4—158no—
420C22×C4—152no—
421C22×C4—152no—
422C22×C4—304no—
423C22×C4—304no—
424C22×C4C2×D3×D512yes—
425C22×C4C2×D3×D511yescharacteristic
426C4⋊C4—102no—
427C4⋊C4—302no—
428C22⋊C4—108no—
429C22⋊C4—308no—
430C22⋊C4—304no—
431C42—22no—
432C42—64no—
433C42—62no—
434C42—304no—
435C3×C5#229711yescharacteristic, complemented, Hall
436C22×C3—108no—
437C22×C3—104no—
438C22×C3—202no—
439C22×C3—202no—
440C22×C3—204no—
441C22×C3C22×C4×D511yescharacteristic
442C22×C3C22×D2011yescharacteristic
443C22×C3C2×#70912yes—
444C22×C3C2×#70912yes—
445C22×C3C22×C22⋊D511yescharacteristic
446C2×D3—24no—
447C2×D3—22no—
448C2×D3—22no—
449C2×D3—24no—
450C2×D3C2×#68514yes—
451C3×C4—24no—
452C3×C4—202no—
453C3×C4C2×#68514yes—
454Dic3—24no—
455Dic3—104no—
456Dic3—202no—
457Dic3C2×#68514yes—
458C2×C5—64no—
459C2×C5C2×D3×C22⋊C411yescharacteristic
460C2×C5C4×#23712yes—
461C2×C5#560612yes—
462C2×C5#566811yescharacteristic
463C2×C5#571611yescharacteristic
464D5—44no—
465C23—32no—
466C23—32no—
467C23—32no—
468C23—54no—
469C23—62no—
470C23—62no—
471C23—104no—
472C23—102no—
473C23—104no—
474C23—102no—
475C23C22×D3×D511yescharacteristic
476Q8—308no—
477D4—108no—
478D4—3016no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—32no—
482C2×C4—32no—
483C2×C4—32no—
484C2×C4—316no—
485C2×C4—34no—
486C2×C4—616no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—104no—
490C2×C4—102no—
491C2×C4—154no—
492C2×C4—3016no—
493C2×C4—302no—
494C2×C4—302no—
495C2×C4—302no—
496C2×C4—304no—
497C2×C4#676514yescomplemented
498C2×C4C4×D3×D512yes—
499C2×C4D3×D2012yes—
500C2×C4D3×C22⋊D512yes—
501C2×C3—204no—
502C2×C3C4×#70912yes—
503C2×C3#6581111yescharacteristic
504C2×C3#6581111yescharacteristic
505C2×C3C22×#68511yescharacteristic
506C2×C3#6612512yes—
507D3—24no—
508C5#7629011yescharacteristic, complemented, Sylow
509C22—34no—
510C22—64no—
511C22—62no—
512C22—62no—
513C22—64no—
514C22—108no—
515C22—104no—
516C22—202no—
517C22—202no—
518C22—204no—
519C22C2×#676512yes—
520C22C2×C4×D3×D511yescharacteristic
521C22C2×D3×D2011yescharacteristic
522C22D3×#70912yes—
523C22C2×D3×C22⋊D511yescharacteristic
524C4—24no—
525C4—34no—
526C4—64no—
527C4—202no—
528C4—304no—
529C4—602no—
530C4D3×#68514yes—
531C3#1061380111yescharacteristic, complemented, Sylow
532C2—64no—
533C2—204no—
534C2C4×#676512yes—
535C2#1175254211yescharacteristic
536C2#1175254311yescharacteristic
537C2#1175434112yes—
538C2C2×D3×#68511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#6765C2
hypercenterC2×C42D3×D5
hypocenterC3×C5#2297
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C2×C42×C5×D3C2semidirect
3. C2×#91338C2semidirect
4. #11751030C2semidirect
5. C3×#65228C2semidirect
6. #11748910C2semidirect
7. Dic3×#685C2semidirect
8. #11748785C2semidirect
9. #11748599C2semidirect
19. C2×#6765C4semidirect
39. #65228D3semidirect
51. C2×Dic30C2×C4semidirect
63. C2×C3×D20C2×C4semidirect
69. C2×D5×Dic3C2×C4semidirect
82. C2×C42×D3D5semidirect
128. C2×C3×C4×C5#41semidirect
140. C2×C5×Dic3#41semidirect
142. C4×C5×D3C22⋊C4semidirect
191. C2×D20C4×D3semidirect
299. C2×C4×C5#237semidirect
358. C2×C3×C4#709semidirect
371. C2×Dic3#709semidirect
374. C4×D3#685semidirect
435. C3×C5#2297semidirect
497. C2×C4#6765semidirect
508. C5#76290semidirect
531. C3#10613801semidirect

Maximal subgroups

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

subgroupindex
C22×#67652
C2×C42×C5×D32
C2×#913382
#117510302
C3×#652282
#117489102
Dic3×#6852
#117487852
#117485992
#106138013
#762905