Finite Group Databasebeta

← #14296875

Subgroups — #14296875

Order 1920.

1224 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×C5×#834C211yesmaximal, characteristic, complemented
3C2×C3×#3966C211yesmaximal, characteristic, complemented
4C2×#90997C212yesmaximal, complemented
5#11749473C212yesmaximal, complemented
6#10600820—31nomaximal, Hall
7C22×C3×C5×D4C2211yescharacteristic
8C22×#1220C411yescharacteristic, complemented
9C22×#1220C2211yescharacteristic
10C23×Dic15C411yescharacteristic, complemented
11C23×Dic15C2211yescharacteristic
12C2×#6819—22no—
13C2×#6819—22no—
14C2×#6819—24no—
15C2×#6819—24no—
16C22×C4×D15C411yescharacteristic, complemented
17C22×Dic30C411yescharacteristic, complemented
18C2×C5×#217C2212yes—
19C2×C3×C22⋊Dic5C2212yes—
20C5×#834—22no—
21C3×#3966—22no—
22#90997—24no—
23#90994—24no—
24#70027—51nomaximal, Hall
25C22×#711—31no—
26C2×C5×#99—31no—
27C2×#3966D311yescharacteristic, complemented
28C2×#3938—32no—
29#65232—32no—
30C24×C3×C5C2×C411yescharacteristic
31C24×C3×C5C2311yescharacteristic
32C23×D15C2×C411yescharacteristic
33C2×C3×C5×D4—22no—
34C2×C3×C5×D4—22no—
35C2×C3×C5×D4D412yes—
36C2×C3×C5×D4D412yes—
37C22×C3×C4×C5C2×C411yescharacteristic
38C2×#1220—22no—
39C2×#1220—21no—
40C2×#1220—22no—
41C2×#1220—21no—
42C2×#1220—22no—
43C2×#1220—22no—
44C2×#1220D414yes—
45C22×Dic15—22no—
46C22×Dic15—21no—
47C22×Dic15—22no—
48C22×Dic15—21no—
49C22×Dic15—22no—
50C22×Dic15—22no—
51C22×Dic15C2×C411yescharacteristic
52C22×Dic15C2×C411yescharacteristic
53C22×Dic15C2×C411yescharacteristic
54C22×Dic15D414yes—
55#6819—216no—
56#6819—216no—
57C2×C4×D15—22no—
58C2×C4×D15—22no—
59C2×C4×D15—22no—
60C2×Dic30—22no—
61C2×Dic30—22no—
62C2×Dic30—22no—
63C22×C5×Dic3—22no—
64C22×C3×Dic5—22no—
65C5×#217—22no—
66C5×#217—22no—
67C3×C22⋊Dic5—22no—
68C3×C22⋊Dic5—22no—
69C22×#237—51no—
70C2×C3×#99—51no—
71C2×#834D511yescharacteristic, complemented
72C2×#806—52no—
73#4943—52no—
74C22×C5×D4C2×D311yescharacteristic
75C22×C22⋊D5—31no—
76C22×C22⋊D5—31no—
77C23×Dic5—31no—
78C23×Dic5—31no—
79C2×#711—64no—
80C2×#711—64no—
81C2×#711—62no—
82C2×#711—62no—
83C22×C4×D5—31no—
84C22×Dic10—31no—
85C2×C5×C22⋊C4—32no—
86C2×C22⋊Dic5C2×D312yes—
87C5×#99—62no—
88#3966—22no—
89#3963—64no—
90#3938—64no—
91#1782—151noSylow
92C23×C3×C5—21no—
93C23×C3×C5—21no—
94C23×C3×C5—42no—
95C23×C3×C5—42no—
96C23×C3×C5C22⋊C412yes—
97C23×C3×C5C22×C411yescharacteristic
98C23×C3×C5C2×D412yes—
99C23×C3×C5C2×D411yescharacteristic
100C23×C3×C5C2×D412yes—
101C23×C3×C5C2×D411yescharacteristic
102C22×D15—21no—
103C22×D15—21no—
104C22×D15—42no—
105C22×D15C22⋊C412yes—
106C3×C5×D4—24no—
107C3×C5×D4—24no—
108C2×C3×C4×C5—21no—
109C2×C3×C4×C5—21no—
110C2×C3×C4×C5C22⋊C412yes—
111#1220—28no—
112#1220—24no—
113#1220—24no—
114C2×Dic15—28no—
115C2×Dic15—21no—
116C2×Dic15—21no—
117C2×Dic15—21no—
118C2×Dic15—21no—
119C2×Dic15—21no—
120C2×Dic15—21no—
121C2×Dic15—24no—
122C2×Dic15—24no—
123C2×Dic15C22⋊C412yes—
124C2×Dic15C22⋊C412yes—
125C2×Dic15C22⋊C412yes—
126C4×D15—28no—
127Dic30—28no—
128C2×C5×Dic3—44no—
129C2×C5×Dic3—42no—
130C2×C3×Dic5—44no—
131C2×C3×Dic5—42no—
132C22×C3×D4C2×D511yescharacteristic
133C22×#67—51no—
134C22×#67—51no—
135C23×Dic3—51no—
136C23×Dic3—51no—
137C2×#237—104no—
138C2×#237—104no—
139C2×#237—102no—
140C2×#237—102no—
141C22×C4×D3—51no—
142C22×Dic6—51no—
143C2×C3×C22⋊C4—52no—
144C2×#217C2×D512yes—
145C3×#99—102no—
146#834—22no—
147C3⋊#95—104no—
148#806—104no—
149C24×C5C4×D311yescharacteristic
150C24×C5C22×D311yescharacteristic
151C23×D5—31no—
152C2×C5×D4—22no—
153C2×C5×D4—22no—
154C2×C5×D4D1212yes—
155C2×C5×D4#6712yes—
156C22×C4×C5—62no—
157C22×C4×C5C4×D311yescharacteristic
158C2×C22⋊D5—34no—
159C2×C22⋊D5—62no—
160C2×C22⋊D5—62no—
161C2×C22⋊D5—62no—
162C2×C22⋊D5—61no—
163C2×C22⋊D5—62no—
164C2×C22⋊D5—61no—
165C22×Dic5—22no—
166C22×Dic5—31no—
167C22×Dic5—31no—
168C22×Dic5—31no—
169C22×Dic5—34no—
170C22×Dic5—62no—
171C22×Dic5—62no—
172C22×Dic5—62no—
173C22×Dic5—61no—
174C22×Dic5—62no—
175C22×Dic5—61no—
176#711—616no—
177#711—616no—
178C2×C4×D5—62no—
179C2×C4×D5—62no—
180C2×C4×D5—62no—
181C2×Dic10—62no—
182C2×Dic10—62no—
183C2×Dic10—62no—
184C5×C22⋊C4—62no—
185C5×C22⋊C4—62no—
186C22⋊Dic5—22no—
187C22⋊Dic5—22no—
188C22×#41—151no—
189C2×#99—31no—
190C2×#99—51no—
191C2×#99—152no—
192#381—152no—
193C22×C3×C5—24no—
194C22×C3×C5—22no—
195C22×C3×C5—22no—
196C22×C3×C5—41no—
197C22×C3×C5—42no—
198C22×C3×C5—41no—
199C22×C3×C5—42no—
200C22×C3×C5—42no—
201C22×C3×C5—42no—
202C22×C3×C5C2×C22⋊C411yescharacteristic
203C22×C3×C5C2×C22⋊C411yescharacteristic
204C22×C3×C5C2×C22⋊C411yescharacteristic
205C22×C3×C5#12012yes—
206C22×C3×C5#12012yes—
207C2×D15—22no—
208C2×D15—42no—
209C2×D15—42no—
210C2×D15—42no—
211C3×C4×C5—22no—
212Dic15—22no—
213Dic15—22no—
214Dic15—22no—
215C3×Dic5—82no—
216C5×Dic3—82no—
217C24×C3C4×D511yescharacteristic
218C24×C3C22×D511yescharacteristic
219C23×D3—51no—
220C2×C3×D4—22no—
221C2×C3×D4—22no—
222C2×C3×D4D2012yes—
223C2×C3×D4C22⋊D512yes—
224C22×C3×C4—102no—
225C22×C3×C4C4×D511yescharacteristic
226C2×#67—54no—
227C2×#67—102no—
228C2×#67—102no—
229C2×#67—102no—
230C2×#67—101no—
231C2×#67—102no—
232C2×#67—101no—
233C22×Dic3—22no—
234C22×Dic3—51no—
235C22×Dic3—51no—
236C22×Dic3—51no—
237C22×Dic3—54no—
238C22×Dic3—102no—
239C22×Dic3—102no—
240C22×Dic3—102no—
241C22×Dic3—101no—
242C22×Dic3—102no—
243C22×Dic3—101no—
244#237—1016no—
245#237—1016no—
246C2×C4×D3—102no—
247C2×C4×D3—102no—
248C2×C4×D3—102no—
249C2×Dic6—102no—
250C2×Dic6—102no—
251C2×Dic6—102no—
252C3×C22⋊C4—102no—
253C3×C22⋊C4—102no—
254#217—22no—
255#217—22no—
256C23×C5—21no—
257C23×C5—21no—
258C23×C5—42no—
259C23×C5—42no—
260C23×C5(C2×D3)⋊C412yes—
261C23×C5C2×C4×D311yescharacteristic
262C23×C5C2×D1211yescharacteristic
263C23×C5D3×D412yes—
264C23×C5D3×D412yes—
265C23×C5C2×#6711yescharacteristic
266C22×D5—32no—
267C22×D5—61no—
268C22×D5—61no—
269C22×D5—122no—
270C5×D4—24no—
271C5×D4—24no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—124no—
275C2×C4×C5—122no—
276C2×C4×C5(C2×D3)⋊C412yes—
277C22⋊D5—68no—
278C22⋊D5—64no—
279C22⋊D5—64no—
280C2×Dic5—32no—
281C2×Dic5—32no—
282C2×Dic5—32no—
283C2×Dic5—44no—
284C2×Dic5—42no—
285C2×Dic5—68no—
286C2×Dic5—61no—
287C2×Dic5—61no—
288C2×Dic5—61no—
289C2×Dic5—64no—
290C2×Dic5—61no—
291C2×Dic5—61no—
292C2×Dic5—61no—
293C2×Dic5—64no—
294C4×D5—68no—
295Dic10—68no—
296C2×#41—304no—
297C2×#41—304no—
298C2×#41—302no—
299C2×#41—302no—
300C22×Q8—151no—
301C22×D4—151no—
302C22×D4—151no—
303C22×D4D3×D511yescharacteristic
304C23×C4—151no—
305C23×C4—151no—
306C23×C4—151no—
307C2×C22⋊C4—32no—
308C2×C22⋊C4—52no—
309#99—62no—
310#99—102no—
311#99—304no—
312#95—304no—
313C2×C3×C5—22no—
314C2×C3×C5—22no—
315C2×C3×C5—21no—
316C2×C3×C5—21no—
317C2×C3×C5—42no—
318C2×C3×C5#37911yescharacteristic
319C2×C3×C5#41012yes—
320D15—42no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3#68512yes—
326C23×C3C2×C4×D511yescharacteristic
327C23×C3C2×D2011yescharacteristic
328C23×C3D4×D512yes—
329C23×C3D4×D512yes—
330C23×C3C2×C22⋊D511yescharacteristic
331C22×D3—52no—
332C22×D3—101no—
333C22×D3—101no—
334C22×D3—202no—
335C3×D4—24no—
336C3×D4—24no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—204no—
340C2×C3×C4—202no—
341C2×C3×C4#68512yes—
342#67—108no—
343#67—104no—
344#67—104no—
345C2×Dic3—44no—
346C2×Dic3—42no—
347C2×Dic3—52no—
348C2×Dic3—52no—
349C2×Dic3—52no—
350C2×Dic3—108no—
351C2×Dic3—101no—
352C2×Dic3—101no—
353C2×Dic3—101no—
354C2×Dic3—104no—
355C2×Dic3—101no—
356C2×Dic3—101no—
357C2×Dic3—101no—
358C2×Dic3—104no—
359C4×D3—108no—
360Dic6—108no—
361C22×C5—24no—
362C22×C5—22no—
363C22×C5—22no—
364C22×C5—41no—
365C22×C5—42no—
366C22×C5—41no—
367C22×C5—42no—
368C22×C5—42no—
369C22×C5—42no—
370C22×C5D3×C22⋊C411yescharacteristic
371C22×C5D3×C22⋊C411yescharacteristic
372C22×C5#88212yes—
373C22×C5C2×(C2×D3)⋊C411yescharacteristic
374C22×C5#93712yes—
375C2×D5—62no—
376C2×D5—122no—
377C2×D5—122no—
378C2×D5—122no—
379C4×C5—22no—
380C4×C5—242no—
381Dic5—62no—
382Dic5—62no—
383Dic5—62no—
384Dic5—82no—
385C24—151no—
386C24#120011yescharacteristic
387C24C2×D3×D511yescharacteristic
388#41—3016no—
389#41—3016no—
390C2×Q8—302no—
391C2×Q8—302no—
392C2×Q8—302no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—154no—
396C2×D4—302no—
397C2×D4—302no—
398C2×D4—302no—
399C2×D4—301no—
400C2×D4—301no—
401C2×D4—302no—
402C2×D4#120212yes—
403C2×D4#120312yes—
404C22×C4—62no—
405C22×C4—102no—
406C22×C4—151no—
407C22×C4—151no—
408C22×C4—151no—
409C22×C4—154no—
410C22×C4—302no—
411C22×C4—302no—
412C22×C4—302no—
413C22×C4—302no—
414C22×C4—302no—
415C22×C4—301no—
416C22×C4—301no—
417C22×C4—302no—
418C22×C4—302no—
419C22×C4#120011yescharacteristic
420C22⋊C4—62no—
421C22⋊C4—62no—
422C22⋊C4—102no—
423C22⋊C4—102no—
424C3×C5#178211yescharacteristic, complemented, Hall
425C22×C3—24no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—41no—
429C22×C3—42no—
430C22×C3—41no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3D5×C22⋊C411yescharacteristic
435C22×C3D5×C22⋊C411yescharacteristic
436C22×C3#402812yes—
437C22×C3C2×#68511yescharacteristic
438C22×C3#408312yes—
439C2×D3—102no—
440C2×D3—202no—
441C2×D3—202no—
442C2×D3—202no—
443C3×C4—22no—
444C3×C4—402no—
445Dic3—82no—
446Dic3—102no—
447Dic3—102no—
448Dic3—102no—
449C2×C5—22no—
450C2×C5—22no—
451C2×C5—21no—
452C2×C5—21no—
453C2×C5—42no—
454C2×C5#474012yes—
455C2×C5#495511yescharacteristic
456D5—122no—
457C23—21no—
458C23—21no—
459C23—42no—
460C23—42no—
461C23—152no—
462C23—301no—
463C23—301no—
464C23—602no—
465C23#666712yes—
466C23C2×#120011yescharacteristic
467C23C2×#120211yescharacteristic
468C23C2×#120311yescharacteristic
469C23#679012yes—
470C23#679012yes—
471Q8—308no—
472D4—24no—
473D4—24no—
474D4—308no—
475D4—304no—
476D4—304no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—124no—
480C2×C4—122no—
481C2×C4—152no—
482C2×C4—152no—
483C2×C4—152no—
484C2×C4—204no—
485C2×C4—202no—
486C2×C4—308no—
487C2×C4—308no—
488C2×C4—301no—
489C2×C4—301no—
490C2×C4—301no—
491C2×C4—301no—
492C2×C4—301no—
493C2×C4—301no—
494C2×C4—304no—
495C2×C4—304no—
496C2×C4#666712yes—
497C2×C3—22no—
498C2×C3—22no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—42no—
502C2×C3#6502912yes—
503C2×C3#6524411yescharacteristic
504D3—202no—
505C5#7002711yescharacteristic, complemented, Sylow
506C22—24no—
507C22—22no—
508C22—22no—
509C22—41no—
510C22—41no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—302no—
516C22—602no—
517C22—602no—
518C22—602no—
519C22C2×#666711yescharacteristic
520C22#9156111yescharacteristic
521C22#9156111yescharacteristic
522C22#9157212yes—
523C22#9157312yes—
524C4—22no—
525C4—242no—
526C4—302no—
527C4—302no—
528C4—302no—
529C4—402no—
530C3#1060082011yescharacteristic, complemented, Sylow
531C2—22no—
532C2—22no—
533C2—21no—
534C2—21no—
535C2—42no—
536C2—602no—
537C2#1174917412yes—
538C2#1174950211yescharacteristic
subgroupG/H
centerC22#91561
commutatorC23×C3×C5C22×C4
FrattiniC24C2×D3×D5
FittingC22×C3×C5×D4C22
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×#6819C2
cosocleC24×C3×C5C23
hypercenterC22×D4D3×D5
hypocenterC3×C5#1782
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C2×C5×#834C2semidirect
3. C2×C3×#3966C2semidirect
4. C2×#90997C2semidirect
5. #11749473C2semidirect
8. C22×#1220C4semidirect
10. C23×Dic15C4semidirect
16. C22×C4×D15C4semidirect
17. C22×Dic30C4semidirect
27. C2×#3966D3semidirect
71. C2×#834D5semidirect
424. C3×C5#1782semidirect
505. C5#70027semidirect
530. C3#10600820semidirect

Maximal subgroups

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

subgroupindex
C22×#68192
C2×C5×#8342
C2×C3×#39662
C2×#909972
#117494732
#106008203
#700275