Finite Group Databasebeta

← #14320070

Subgroups — #14320070

Order 1920.

1136 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#1200C211yesmaximal, characteristic
2C4×#6714C211yesmaximal, characteristic
3C5×#4932C211yesmaximal, characteristic, complemented
4C3×C4×#683C211yesmaximal, characteristic, complemented
5#11748889C212yesmaximal
6#11748876C212yesmaximal
7C4×#6668C211yesmaximal, characteristic, complemented
8C4×#6667C211yesmaximal, characteristic
9#11748647C212yesmaximal, complemented
10#11748642C212yesmaximal, complemented
11#11748611C211yesmaximal, characteristic, complemented
12#10605144—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C22×#1200C2211yescharacteristic
15C22×#1200C2211yescharacteristic
16C2×C3×C42×C5C2211yescharacteristic
17C2×#6714C2212yes—
18C2×C4×Dic15C2211yescharacteristic
19C2×C4×C5×Dic3C2211yescharacteristic
20C2×C4×C5×Dic3C2211yescharacteristic
21C2×C3×#683C2212yes—
22C2×C3×C4×Dic5C2211yescharacteristic
23C2×C3×C4×Dic5C2211yescharacteristic
24C2×#6668C2212yes—
25C2×#6667C2212yes—
26C2×Dic3×Dic5C2211yescharacteristic
27C2×Dic3×Dic5C2211yescharacteristic
28C4×#1200C418yescomplemented
29#91115C2212yes—
30C5×C3⋊#95C2212yes—
31C5×C3⋊#95C2212yes—
32C3×#3963C2212yes—
33#90994C2212yes—
34#90994C2212yes—
35#90994C2212yes—
36#90994C2212yes—
37#73752—51nomaximal, Hall
38C2×C42×D5—31no—
39C5×#376—31no—
40C4×#685—31no—
41C4×#685—31no—
42#65221—31no—
43C4×#683—31no—
44C4×#683D311yescharacteristic, complemented
45#64956—32no—
46#64953—32no—
47#64940—32no—
48#64939—32no—
49C23×D15C2311yescharacteristic
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×Dic15C2312yes—
53C22×Dic15C2311yescharacteristic
54C2×C4×D15—22no—
55C2×C4×D15—22no—
56C2×C4×D15C2×C414yes—
57C22×C5×Dic3C2312yes—
58C22×C5×Dic3C2311yescharacteristic
59C22×C5×Dic3C2311yescharacteristic
60C22×C3×Dic5C2312yes—
61C22×C3×Dic5C2311yescharacteristic
62C22×C3×Dic5C2311yescharacteristic
63C2×#1200—24no—
64C2×#1200—24no—
65C2×#1200C2×C414yes—
66C2×#1200C2×C414yes—
67C3×C42×C5—22no—
68#6714—24no—
69C4×Dic15—22no—
70C4×C5×Dic3—22no—
71C4×C5×Dic3C2×C414yes—
72C3×#683—24no—
73C3×C4×Dic5—22no—
74C3×C4×Dic5C2×C414yes—
75#6668—24no—
76#6667—24no—
77Dic3×Dic5C2×C414yes—
78Dic3×Dic5C2×C414yes—
79C2×C42×D3—51no—
80C3×C4×C4⋊C4—51no—
81#4987—52no—
82#4977—52no—
83C4×(C2×D3)⋊C4—51no—
84C4×(C2×D3)⋊C4—51no—
85C4×C4⋊Dic3—51no—
86#4932—51no—
87#4932D511yescharacteristic, complemented
88#4664—52no—
89#4651—52no—
90C22×C4×D5—31no—
91C22×C4×D5—31no—
92C22×C4×D5—31no—
93C2×C42×C5—31no—
94C2×C42×C5—31no—
95C2×C42×C5C2×D311yescharacteristic
96C2×#685—32no—
97C2×#685—32no—
98C2×#683—32no—
99C2×#683C2×D312yes—
100C2×C4×Dic5—31no—
101C2×C4×Dic5—31no—
102C2×C4×Dic5—31no—
103C2×C4×Dic5C2×D311yescharacteristic
104C2×C4×Dic5C2×D311yescharacteristic
105C42×D5—38no—
106C5×#95—32no—
107C5×#95—32no—
108#3963—32no—
109#3963—32no—
110#3963—32no—
111#3963—32no—
112#3963—32no—
113#3963C2×D312yes—
114#2283—151noSylow
115C23×C3×C5C2411yescharacteristic
116C22×D15—22no—
117C22×D15—21no—
118C22×D15—21no—
119C22×D15C22×C412yes—
120C22×D15#4114yes—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5C22×C412yes—
125C2×C3×C4×C5#4112yes—
126C2×C3×C4×C5#4112yes—
127C2×Dic15—21no—
128C2×Dic15—21no—
129C2×Dic15—22no—
130C2×Dic15—22no—
131C2×Dic15—22no—
132C2×Dic15C22×C412yes—
133C4×D15—28no—
134C2×C5×Dic3—22no—
135C2×C5×Dic3—22no—
136C2×C5×Dic3—22no—
137C2×C5×Dic3—22no—
138C2×C5×Dic3—22no—
139C2×C5×Dic3C22×C412yes—
140C2×C5×Dic3C22×C412yes—
141C2×C3×Dic5—22no—
142C2×C3×Dic5—22no—
143C2×C3×Dic5—22no—
144C2×C3×Dic5—22no—
145C2×C3×Dic5C22×C412yes—
146C2×C3×Dic5C22×C412yes—
147C2×C3×Dic5#4114yes—
148#1200—28no—
149#1200—28no—
150C22×C4×D3—51no—
151C22×C4×D3—51no—
152C22×C4×D3—51no—
153C2×C3×C4⋊C4—52no—
154C2×C3×C42—51no—
155C2×C3×C42—51no—
156C2×C3×C42C2×D511yescharacteristic
157C2×(C2×D3)⋊C4—52no—
158C2×(C2×D3)⋊C4—52no—
159C2×C4⋊Dic3—52no—
160C2×C4×Dic3—51no—
161C2×C4×Dic3—51no—
162C2×C4×Dic3—51no—
163C2×C4×Dic3C2×D511yescharacteristic
164C2×C4×Dic3C2×D511yescharacteristic
165C42×D3—58no—
166C3×#95—52no—
167C3⋊#95—52no—
168C3⋊#95—52no—
169C3⋊#95—52no—
170C3⋊#95—52no—
171C3⋊#95—52no—
172C3⋊#95C2×D512yes—
173C3⋊#95C2×D512yes—
174C23×D5—31no—
175C22×C4×C5—31no—
176C22×C4×C5—31no—
177C22×C4×C5—32no—
178C22×C4×C5C22×D312yes—
179C22×C4×C5C22×D311yescharacteristic
180C22×Dic5—31no—
181C22×Dic5—32no—
182C22×Dic5C22×D312yes—
183C22×Dic5C22×D311yescharacteristic
184C22×Dic5C22×D311yescharacteristic
185C2×C4×D5—34no—
186C2×C4×D5—34no—
187C2×C4×D5—34no—
188C2×C4×D5—64no—
189C2×C4×D5—62no—
190C2×C4×D5—62no—
191C2×C4×D5—64no—
192C42×C5—22no—
193C42×C5—34no—
194C42×C5—62no—
195#685—64no—
196#685—64no—
197#683—24no—
198#683—64no—
199C4×Dic5—22no—
200C4×Dic5—34no—
201C4×Dic5—34no—
202C4×Dic5—62no—
203C4×Dic5C4×D314yes—
204C22×C42—151no—
205#383—152no—
206#382—152no—
207#381—152no—
208#380—152no—
209C4×C4⋊C4—51no—
210C4×C4⋊C4—151no—
211C4×C22⋊C4—151no—
212C4×C22⋊C4—151no—
213#376—31no—
214#376—151no—
215C22×C3×C5C23×C411yescharacteristic
216C22×C3×C5C2×#4111yescharacteristic
217C22×C3×C5C2×#4111yescharacteristic
218C22×C3×C5C2×#4112yes—
219C22×C3×C5C2×#4112yes—
220C2×D15—22no—
221C2×D15—22no—
222C2×D15—24no—
223C2×D15—24no—
224C2×D15#11714yes—
225C3×C4×C5—42no—
226C3×C4×C5#11714yes—
227Dic15—22no—
228Dic15—42no—
229C3×Dic5—22no—
230C3×Dic5—42no—
231C3×Dic5#11714yes—
232C5×Dic3—22no—
233C5×Dic3—22no—
234C5×Dic3—42no—
235C23×D3—51no—
236C22×C3×C4—51no—
237C22×C3×C4—51no—
238C22×C3×C4—52no—
239C22×C3×C4C22×D512yes—
240C22×C3×C4C22×D511yescharacteristic
241C22×Dic3—51no—
242C22×Dic3—52no—
243C22×Dic3C22×D512yes—
244C22×Dic3C22×D511yescharacteristic
245C22×Dic3C22×D511yescharacteristic
246C2×C4×D3—54no—
247C2×C4×D3—54no—
248C2×C4×D3—54no—
249C2×C4×D3—104no—
250C2×C4×D3—104no—
251C2×C4×D3—102no—
252C2×C4×D3—102no—
253C3×C4⋊C4—104no—
254C3×C42—22no—
255C3×C42—54no—
256C3×C42—102no—
257(C2×D3)⋊C4—104no—
258(C2×D3)⋊C4—104no—
259C4⋊Dic3—104no—
260C4×Dic3—22no—
261C4×Dic3—54no—
262C4×Dic3—54no—
263C4×Dic3—102no—
264C4×Dic3C4×D514yes—
265C23×C5C23×D311yescharacteristic
266C22×D5—32no—
267C22×D5—34no—
268C22×D5—62no—
269C22×D5—61no—
270C22×D5—61no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—22no—
274C2×C4×C5—32no—
275C2×C4×C5—32no—
276C2×C4×C5—62no—
277C2×C4×C5—62no—
278C2×C4×C5—62no—
279C2×C4×C5—62no—
280C2×C4×C5—62no—
281C2×C4×C5C2×C4×D312yes—
282C2×C4×C5#23712yes—
283C2×C4×C5#23912yes—
284C2×Dic5—22no—
285C2×Dic5—22no—
286C2×Dic5—22no—
287C2×Dic5—22no—
288C2×Dic5—32no—
289C2×Dic5—61no—
290C2×Dic5—61no—
291C2×Dic5—62no—
292C2×Dic5—62no—
293C2×Dic5—62no—
294C2×Dic5C2×C4×D312yes—
295C2×Dic5C2×C4×D312yes—
296C2×Dic5#23514yes—
297C4×D5—68no—
298C4×D5—68no—
299C4×D5—68no—
300C23×C4—151no—
301C23×C4—151no—
302C23×C4—151no—
303C2×C4⋊C4—52no—
304C2×C4⋊C4—152no—
305C2×C22⋊C4—152no—
306C2×C22⋊C4—152no—
307C2×C42—31no—
308C2×C42—31no—
309C2×C42—51no—
310C2×C42—51no—
311C2×C42—158no—
312C2×C42—151no—
313C2×C42—151no—
314C2×C42—151no—
315C2×C42D3×D511yescharacteristic
316#95—32no—
317#95—32no—
318#95—52no—
319#95—152no—
320#95—152no—
321#95—152no—
322#95—152no—
323#95—152no—
324C2×C3×C5C2×#11711yescharacteristic
325C2×C3×C5C2×#11711yescharacteristic
326C2×C3×C5C2×#11711yescharacteristic
327C2×C3×C5#52912yes—
328C2×C3×C5#52912yes—
329D15—24no—
330C23×C3C23×D511yescharacteristic
331C22×D3—52no—
332C22×D3—54no—
333C22×D3—101no—
334C22×D3—101no—
335C22×D3—102no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—52no—
340C2×C3×C4—54no—
341C2×C3×C4—52no—
342C2×C3×C4—102no—
343C2×C3×C4—102no—
344C2×C3×C4—102no—
345C2×C3×C4—102no—
346C2×C3×C4C2×C4×D512yes—
347C2×C3×C4#70912yes—
348C2×C3×C4#70912yes—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—52no—
355C2×Dic3—101no—
356C2×Dic3—101no—
357C2×Dic3—102no—
358C2×Dic3—102no—
359C2×Dic3—102no—
360C2×Dic3C2×C4×D512yes—
361C2×Dic3C2×C4×D512yes—
362C4×D3—108no—
363C4×D3—108no—
364C4×D3—108no—
365C22×C5C22×C4×D311yescharacteristic
366C22×C5C2×#23512yes—
367C22×C5C2×#23711yescharacteristic
368C22×C5C2×#23911yescharacteristic
369C22×C5D3×#4112yes—
370C2×D5—34no—
371C2×D5—62no—
372C2×D5—62no—
373C2×D5—64no—
374C2×D5—64no—
375C4×C5—42no—
376C4×C5—62no—
377C4×C5—62no—
378C4×C5—122no—
379C4×C5#89214yes—
380Dic5—22no—
381Dic5—42no—
382Dic5—62no—
383Dic5—122no—
384Dic5#87214yes—
385C24—151no—
386C22×C4—31no—
387C22×C4—31no—
388C22×C4—32no—
389C22×C4—51no—
390C22×C4—51no—
391C22×C4—52no—
392C22×C4—154no—
393C22×C4—154no—
394C22×C4—151no—
395C22×C4—154no—
396C22×C4—152no—
397C22×C4—304no—
398C22×C4—304no—
399C22×C4—302no—
400C22×C4—302no—
401C22×C4C2×D3×D512yes—
402C22×C4C2×D3×D511yescharacteristic
403C4⋊C4—104no—
404C4⋊C4—304no—
405C22⋊C4—304no—
406C22⋊C4—304no—
407C42—22no—
408C42—34no—
409C42—54no—
410C42—62no—
411C42—102no—
412C42—154no—
413C42—154no—
414C42—302no—
415C3×C5#228311yescharacteristic, complemented, Hall
416C22×C3C22×C4×D511yescharacteristic
417C22×C3C2×#70911yescharacteristic
418C22×C3C2×#70911yescharacteristic
419C22×C3D5×#4112yes—
420C22×C3D5×#4112yes—
421C2×D3—54no—
422C2×D3—102no—
423C2×D3—102no—
424C2×D3—104no—
425C2×D3—104no—
426C3×C4—42no—
427C3×C4—54no—
428C3×C4—102no—
429C3×C4—202no—
430C3×C4#401814yes—
431Dic3—22no—
432Dic3—22no—
433Dic3—42no—
434Dic3—102no—
435Dic3—202no—
436C2×C5C2×#87211yescharacteristic
437C2×C5C2×#89211yescharacteristic
438C2×C5D3×#11711yescharacteristic
439C2×C5#555712yes—
440C2×C5#558012yes—
441D5—64no—
442C23—152no—
443C23—154no—
444C23—301no—
445C23—302no—
446C23—301no—
447C23C22×D3×D511yescharacteristic
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—32no—
452C2×C4—32no—
453C2×C4—52no—
454C2×C4—54no—
455C2×C4—52no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—102no—
462C2×C4—102no—
463C2×C4—102no—
464C2×C4—102no—
465C2×C4—152no—
466C2×C4—308no—
467C2×C4—308no—
468C2×C4—301no—
469C2×C4—301no—
470C2×C4—308no—
471C2×C4—302no—
472C2×C4—302no—
473C2×C4—302no—
474C2×C4#676912yes—
475C2×C4C4×D3×D512yes—
476C2×C4#677912yes—
477C2×C3C2×#401811yescharacteristic
478C2×C3D5×#11711yescharacteristic
479C2×C3D5×#11711yescharacteristic
480C2×C3#6586212yes—
481C2×C3#6586212yes—
482D3—104no—
483C5#7375211yescharacteristic, complemented, Sylow
484C22—154no—
485C22—302no—
486C22—302no—
487C22—304no—
488C22—304no—
489C22C2×#676911yescharacteristic
490C22C2×C4×D3×D511yescharacteristic
491C22D5×#23512yes—
492C22#9201612yes—
493C22C2×#677911yescharacteristic
494C4—42no—
495C4—54no—
496C4—62no—
497C4—62no—
498C4—102no—
499C4—122no—
500C4—202no—
501C4—302no—
502C4—602no—
503C4#9140314yes—
504C3#1060514411yescharacteristic, complemented, Sylow
505C2—304no—
506C2#1175258712yes—
507C2D5×#87211yescharacteristic
508C2#1175272311yescharacteristic
509C2#1175422612yes—
510C2C2×#9140311yescharacteristic
subgroupG/H
centerC22×C4C2×D3×D5
commutatorC22×C3×C5C23×C4
FrattiniC23C22×D3×D5
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC23×D15C23
hypercenterC2×C42D3×D5
hypocenterC3×C5#2283
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
3. C5×#4932C2semidirect
4. C3×C4×#683C2semidirect
7. C4×#6668C2semidirect
9. #11748647C2semidirect
10. #11748642C2semidirect
11. #11748611C2semidirect
28. C4×#1200C4semidirect
44. C4×#683D3semidirect
87. #4932D5semidirect
415. C3×C5#2283semidirect
483. C5#73752semidirect
504. C3#10605144semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#12002
C4×#67142
C5×#49322
C3×C4×#6832
#117488892
#117488762
C4×#66682
C4×#66672
#117486472
#117486422
#117486112
#106051443
#737525