Finite Group Databasebeta

← #14424613

Subgroups — #14424613

Order 1920.

1400 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92101C211yesmaximal, characteristic
2C2×C5×#951C211yesmaximal, characteristic, complemented
3#11754270C211yesmaximal, characteristic, complemented
4C22×#6663C211yesmaximal, characteristic, complemented
5C3×#65522C211yesmaximal, characteristic, complemented
6#11749308C214yesmaximal, complemented
7#11749061C214yesmaximal, complemented
8#11748415C211yesmaximal, characteristic, complemented
9#11748349C211yesmaximal, characteristic, complemented
10#10615714—31nomaximal, Hall
11C2×C3×C5×#41C2211yescharacteristic
12#92101—24no—
13#92101—24no—
14C2×Q8×D15C2212yes—
15C2×#6819—22no—
16C2×#6819C2212yes—
17C2×#6817—21no—
18C2×#6817C2211yescharacteristic
19C22×Dic30—21no—
20C22×Dic30C2211yescharacteristic
21C22×C5×Dic6C2211yescharacteristic
22C5×#951—24no—
23C2×C5×#216C2212yes—
24C2×C5×#214C2212yes—
25C2×C5×#208C2211yescharacteristic
26C3×#4072C2211yescharacteristic
27C22×C3×#168C2211yescharacteristic
28#91542—22no—
29#91383C2214yes—
30#91376—21no—
31#91375—21no—
32C2×#6663—24no—
33C2×#6663C2214yes—
34C3×#3967C2212yes—
35C3×#3964C2212yes—
36#90974C2214yes—
37#90957C2214yes—
38#78837—51nomaximal, Hall
39C2×#4150—31no—
40C2×C5×#137—31no—
41#66150—31no—
42C22×#689—31no—
43#65522D311yescharacteristic, complemented
44#65475—34no—
45#65462—34no—
46#65418—31no—
47#65409—31no—
48C3×C5×#41—22no—
49C3×C5×#41D414yes—
50C2×C3×C5×Q8C2311yescharacteristic
51C2×C3×C5×D4—21no—
52C2×C3×C5×D4C2311yescharacteristic
53C22×C3×C4×C5—21no—
54C22×C3×C4×C5C2311yescharacteristic
55C2×#1220—22no—
56C2×#1220D412yes—
57C22×Dic15—21no—
58C22×Dic15—22no—
59Q8×D15—24no—
60Q8×D15—24no—
61#6819—216no—
62#6819—24no—
63#6819—24no—
64#6817—28no—
65#6817—22no—
66#6817—22no—
67C2×D60D411yescharacteristic
68C2×C4×D15—22no—
69C2×C4×D15C2312yes—
70C2×Dic30—28no—
71C2×Dic30—21no—
72C2×Dic30—22no—
73C2×Dic30—22no—
74C2×Dic30D411yescharacteristic
75C2×Dic30D414yes—
76C2×Dic30C2312yes—
77C22×C5×Dic3—21no—
78C2×C5×Dic6—22no—
79C2×C5×Dic6—22no—
80C2×C5×Dic6—21no—
81C2×C5×Dic6C2312yes—
82C5×#216—24no—
83C5×#214—24no—
84C5×#208—22no—
85C2×C5×C3⋊C8C2312yes—
86C3×C22⋊Dic5—21no—
87C3×C4⋊Dic5C2312yes—
88C3×C4×Dic5—21no—
89C2×C3×#168—22no—
90C2×C3×#168C2312yes—
91#6668—24no—
92#6667—24no—
93#6663—28no—
94C2×#1010—51no—
95C2×#951D511yescharacteristic, complemented
96C22×Dic12—51no—
97#5526—51no—
98C3×#416—51no—
99#5110—51no—
100#5004—51no—
101#4809—54no—
102#4783—54no—
103C2×C5×#41C2×D311yescharacteristic
104C22×C5×Q8—31no—
105#4150—64no—
106#4150—64no—
107C2×Q8×D5—32no—
108C2×#711—32no—
109C2×#711—62no—
110C2×#709—31no—
111C2×#709—61no—
112C22×Dic10—31no—
113C22×Dic10—61no—
114C5×#137—64no—
115C2×C5×Q16—32no—
116C2×C5×SD16—32no—
117C2×C5×M16—31no—
118#4093—61no—
119#4092—34no—
120#4090—61no—
121C2×#689—34no—
122C2×#689—64no—
123#4075—62no—
124#4072C2×D311yescharacteristic
125C22×#168C2×D311yescharacteristic
126#3967C2×D312yes—
127#3964C2×D312yes—
128#3952—34no—
129#3948—34no—
130#3011—151noSylow
131C23×C3×C5—21no—
132C23×C3×C5C2×D411yescharacteristic
133C22×D15C2×D412yes—
134C3×C5×Q8—21no—
135C3×C5×Q8C2×D412yes—
136C3×C5×D4—24no—
137C3×C5×D4—21no—
138C3×C5×D4C2×D412yes—
139C2×C3×C4×C5—21no—
140C2×C3×C4×C5—24no—
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5C2×D412yes—
143C2×C3×C4×C5C2×D412yes—
144C2×C3×C4×C5C2411yescharacteristic
145#1220—28no—
146#1220—24no—
147C2×Dic15—28no—
148C2×Dic15—24no—
149C2×Dic15—22no—
150C2×Dic15C2×D412yes—
151D60—22no—
152C4×D15—28no—
153C4×D15—22no—
154C4×D15—22no—
155Dic30—28no—
156Dic30—22no—
157Dic30—24no—
158Dic30—22no—
159Dic30C2×D414yes—
160C2×C5×Dic3—22no—
161C2×C5×Dic3—42no—
162C5×Dic6—22no—
163C5×Dic6—22no—
164C5×Dic6—42no—
165C2×C3×Dic5—22no—
166C3×#168—22no—
167C5×C3⋊C8—22no—
168C2×C3×#41C2×D511yescharacteristic
169#1010—104no—
170#1010—104no—
171C2×D3×Q8—52no—
172C2×#237—52no—
173C2×#237—102no—
174C2×#235—51no—
175C2×#235—101no—
176C22×Dic6—51no—
177C22×Dic6—101no—
178C22×Dic6C2×D511yescharacteristic
179C22×C3×C8—51no—
180C3×#117—51no—
181#951—24no—
182C2×#216C2×D512yes—
183C2×#214C2×D512yes—
184C2×#208C2×D511yescharacteristic
185C2×Dic12—54no—
186C2×Dic12—104no—
187#897—54no—
188#886—102no—
189#875—101no—
190#869—101no—
191C3×#103—52no—
192C3×#102—52no—
193(C2×D3)⋊C8—54no—
194#810—54no—
195C5×#41—22no—
196C5×#41D1214yes—
197C2×C5×Q8—32no—
198C2×C5×Q8—61no—
199C2×C5×Q8—62no—
200C2×C5×Q8—62no—
201C2×C5×Q8C22×D311yescharacteristic
202C2×C5×D4—21no—
203C2×C5×D4C22×D311yescharacteristic
204C22×C4×C5—21no—
205C22×C4×C5—61no—
206C22×C4×C5C22×D311yescharacteristic
207C2×C22⋊D5—32no—
208C2×C22⋊D5—62no—
209C22×Dic5—62no—
210C22×Dic5—61no—
211Q8×D5—64no—
212Q8×D5—64no—
213#711—64no—
214#711—616no—
215#711—64no—
216#709—68no—
217#709—62no—
218#709—62no—
219C2×D20—31no—
220C2×C4×D5—32no—
221C2×C4×D5—62no—
222C2×Dic10—34no—
223C2×Dic10—32no—
224C2×Dic10—31no—
225C2×Dic10—62no—
226C2×Dic10—68no—
227C2×Dic10—62no—
228C2×Dic10—61no—
229C5×Q16—64no—
230C5×SD16—64no—
231C5×M16—62no—
232C2×C5×C8—32no—
233C22⋊Dic5—21no—
234#689—68no—
235#685—64no—
236C4⋊Dic5C22×D312yes—
237#683—64no—
238C4×Dic5—21no—
239C2×#168—22no—
240C2×#168C22×D312yes—
241C2×2−1+4—151no—
242C2×#137—31no—
243C22×Q16—151no—
244#536—151no—
245#452—154no—
246#451—154no—
247#419—151no—
248#416—51no—
249#408—151no—
250C22×C3×C5—22no—
251C22×C3×C5—22no—
252C22×C3×C5—42no—
253C22×C3×C5#12012yes—
254C22×C3×C5C22×D411yescharacteristic
255C2×D15—22no—
256C2×D15—24no—
257C3×C4×C5—22no—
258C3×C4×C5#12012yes—
259C3×C4×C5C22×D411yescharacteristic
260C3×C4×C5C22×D411yescharacteristic
261Dic15—22no—
262Dic15—24no—
263C3×Dic5—42no—
264C5×Dic3—42no—
265C3×#41—22no—
266C3×#41C22⋊D514yes—
267C2×C3×Q8C22×D511yescharacteristic
268C2×C3×D4—21no—
269C2×C3×D4C22×D511yescharacteristic
270C22×C3×C4—21no—
271C22×C3×C4C22×D511yescharacteristic
272C2×#67—52no—
273C2×#67—102no—
274C22×Dic3—21no—
275C22×Dic3—102no—
276C22×Dic3—101no—
277D3×Q8—104no—
278D3×Q8—104no—
279#237—104no—
280#237—1016no—
281#237—104no—
282#235—108no—
283#235—102no—
284#235—102no—
285C2×D12—51no—
286C2×C4×D3—52no—
287C2×C4×D3—102no—
288C2×Dic6—22no—
289C2×Dic6—22no—
290C2×Dic6—21no—
291C2×Dic6—54no—
292C2×Dic6—52no—
293C2×Dic6—51no—
294C2×Dic6—102no—
295C2×Dic6—108no—
296C2×Dic6—101no—
297C2×Dic6—102no—
298C2×Dic6C22×D512yes—
299C2×C3×C8—52no—
300C2×C3×C8—102no—
301C3×C4⋊C4—52no—
302C3×C22⋊C4—101no—
303C3×C42—101no—
304#216—24no—
305#214—24no—
306(C2×D3)⋊C4—104no—
307C4⋊Dic3—104no—
308#208—22no—
309C2×C3⋊C8C22×D512yes—
310Dic12—108no—
311C23×C5—21no—
312C23×C5D3×D411yescharacteristic
313C22×D5—32no—
314C5×Q8—21no—
315C5×Q8—62no—
316C5×Q8—62no—
317C5×Q8—122no—
318C5×Q8C2×D1212yes—
319C5×D4—24no—
320C5×D4—21no—
321C5×D4C2×D1212yes—
322C2×C4×C5—21no—
323C2×C4×C5—24no—
324C2×C4×C5—62no—
325C2×C4×C5—122no—
326C2×C4×C5C2×D1212yes—
327C2×C4×C5D3×D411yescharacteristic
328C2×C4×C5D3×D412yes—
329C2×C4×C5C23×D311yescharacteristic
330C22⋊D5—68no—
331C22⋊D5—64no—
332C2×Dic5—22no—
333C2×Dic5—32no—
334C2×Dic5—68no—
335C2×Dic5—64no—
336C2×Dic5—62no—
337D20—62no—
338C4×D5—68no—
339C4×D5—62no—
340C4×D5—62no—
341Dic10—34no—
342Dic10—68no—
343Dic10—64no—
344Dic10—62no—
345Dic10—62no—
346C5×C8—62no—
347#168—22no—
3482−1+4—304no—
3492−1+4—304no—
350C2×#41—151no—
351C2×#41—152no—
352C2×#41—301no—
353C2×#41—302no—
354C2×#41D3×D511yescharacteristic
355C22×Q8—31no—
356C22×Q8—152no—
357C22×Q8—151no—
358C22×Q8—301no—
359#137—64no—
360C2×Q16—32no—
361C2×Q16—154no—
362C2×Q16—304no—
363C2×SD16—32no—
364C2×M16—31no—
365C22×C8—51no—
366C4⋊Q8—301no—
367#124—301no—
368#123—302no—
369C22⋊Q8—154no—
370#117—51no—
371#103—52no—
372#103—154no—
373#102—52no—
374C22⋊C8—154no—
375C2×C3×C5—22no—
376C2×C3×C5—42no—
377C2×C3×C5C2×#12011yescharacteristic
378C2×C3×C5#57812yes—
379D15—42no—
380C23×C3—21no—
381C23×C3D4×D511yescharacteristic
382C22×D3—52no—
383C3×Q8—21no—
384C3×Q8C2×C22⋊D512yes—
385C3×D4—24no—
386C3×D4—21no—
387C3×D4C2×C22⋊D512yes—
388C2×C3×C4—21no—
389C2×C3×C4—24no—
390C2×C3×C4—102no—
391C2×C3×C4D4×D512yes—
392C2×C3×C4D4×D511yescharacteristic
393C2×C3×C4C2×C22⋊D512yes—
394C2×C3×C4C23×D511yescharacteristic
395#67—108no—
396#67—104no—
397C2×Dic3—22no—
398C2×Dic3—42no—
399C2×Dic3—52no—
400C2×Dic3—108no—
401C2×Dic3—102no—
402C2×Dic3—104no—
403D12—102no—
404C4×D3—108no—
405C4×D3—102no—
406C4×D3—102no—
407Dic6—22no—
408Dic6—22no—
409Dic6—42no—
410Dic6—54no—
411Dic6—108no—
412Dic6—102no—
413Dic6—104no—
414Dic6—102no—
415C3×C8—102no—
416C3⋊C8—22no—
417C22×C5—22no—
418C22×C5—22no—
419C22×C5—42no—
420C22×C5#88212yes—
421C22×C5C2×D3×D411yescharacteristic
422C2×D5—62no—
423C2×D5—64no—
424C4×C5—22no—
425C4×C5—122no—
426C4×C5#88212yes—
427C4×C5C22×D1211yescharacteristic
428C4×C5C2×D3×D411yescharacteristic
429Dic5—42no—
430Dic5—64no—
431Dic5—62no—
432#41—22no—
433#41—308no—
434#41—304no—
435#41—3016no—
436#41—304no—
437#41—302no—
438#41—302no—
439#41#120314yes—
440C2×Q8—32no—
441C2×Q8—61no—
442C2×Q8—62no—
443C2×Q8—62no—
444C2×Q8—151no—
445C2×Q8—154no—
446C2×Q8—152no—
447C2×Q8—302no—
448C2×Q8—308no—
449C2×Q8—304no—
450C2×Q8—304no—
451C2×Q8—302no—
452C2×Q8—301no—
453C2×Q8C2×D3×D511yescharacteristic
454C2×D4—21no—
455C2×D4—152no—
456C2×D4—151no—
457C2×D4—302no—
458C2×D4C2×D3×D511yescharacteristic
459C22×C4—21no—
460C22×C4—61no—
461C22×C4—152no—
462C22×C4—302no—
463C22×C4—302no—
464C22×C4—301no—
465C22×C4C2×D3×D511yescharacteristic
466Q16—64no—
467Q16—308no—
468SD16—64no—
469M16—62no—
470C2×C8—32no—
471C2×C8—52no—
472C2×C8—102no—
473C4⋊C4—52no—
474C4⋊C4—304no—
475C22⋊C4—101no—
476C22⋊C4—304no—
477C42—101no—
478C3×C5#301111yescharacteristic, complemented, Hall
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—42no—
482C22×C3#408312yes—
483C22×C3C2×D4×D511yescharacteristic
484C2×D3—104no—
485C2×D3—102no—
486C3×C4—22no—
487C3×C4—202no—
488C3×C4#408312yes—
489C3×C4C2×D4×D511yescharacteristic
490C3×C4C22×C22⋊D511yescharacteristic
491Dic3—42no—
492Dic3—104no—
493Dic3—102no—
494C2×C5—22no—
495C2×C5—42no—
496C2×C5C2×#88211yescharacteristic
497C2×C5#577112yes—
498D5—122no—
499C23—21no—
500C23—152no—
501C23#679011yescharacteristic
502Q8—21no—
503Q8—62no—
504Q8—62no—
505Q8—122no—
506Q8—154no—
507Q8—308no—
508Q8—302no—
509Q8—302no—
510Q8—304no—
511Q8C2×#120312yes—
512D4—24no—
513D4—21no—
514D4—308no—
515D4—304no—
516D4—302no—
517D4C2×#120312yes—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—62no—
521C2×C4—102no—
522C2×C4—122no—
523C2×C4—152no—
524C2×C4—308no—
525C2×C4—308no—
526C2×C4—302no—
527C2×C4—304no—
528C2×C4—302no—
529C2×C4—302no—
530C2×C4C2×#120312yes—
531C2×C4#679012yes—
532C2×C4#679011yescharacteristic
533C2×C4C22×D3×D511yescharacteristic
534C8—62no—
535C8—102no—
536C2×C3—22no—
537C2×C3—42no—
538C2×C3#6611312yes—
539C2×C3C2×#408311yescharacteristic
540D3—202no—
541C5#7883711yescharacteristic, complemented, Sylow
542C22—22no—
543C22—22no—
544C22—42no—
545C22—304no—
546C22—302no—
547C22#9157312yes—
548C22C2×#679011yescharacteristic
549C4—22no—
550C4—122no—
551C4—202no—
552C4—302no—
553C4—304no—
554C4#9157312yes—
555C4C22×#120311yescharacteristic
556C4C2×#679011yescharacteristic
557C3#1061571411yescharacteristic, complemented, Sylow
558C2—22no—
559C2—42no—
560C2—602no—
561C2#1175535012yes—
562C2C2×#9157311yescharacteristic
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×#41C22
socleC22×C3×C5C22×D4
prime radicalC2×#92101C2
hypercenterC2×#41D3×D5
hypocenterC3×C5#3011
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C2×C5×#951C2semidirect
3. #11754270C2semidirect
4. C22×#6663C2semidirect
5. C3×#65522C2semidirect
6. #11749308C2semidirect
7. #11749061C2semidirect
8. #11748415C2semidirect
9. #11748349C2semidirect
43. #65522D3semidirect
95. C2×#951D5semidirect
478. C3×C5#3011semidirect
541. C5#78837semidirect
557. C3#10615714semidirect

Maximal subgroups

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

subgroupindex
C2×#921012
C2×C5×#9512
#117542702
C22×#66632
C3×#655222
#117493082
#117490612
#117484152
#117483492
#106157143
#788375