Finite Group Databasebeta

← #14424586

Subgroups — #14424586

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×#92004C211yesmaximal, characteristic
2#11756575C211yesmaximal, characteristic, complemented
3C2×C5×#942C211yesmaximal, characteristic, complemented
4C22×C3×#689C211yesmaximal, characteristic, complemented
5#11749407C214yesmaximal, complemented
6#11749402C214yesmaximal, complemented
7#11748400C211yesmaximal, characteristic, complemented
8#11748388C211yesmaximal, characteristic, complemented
9#11748349C211yesmaximal, characteristic, complemented
10#10615714—31nomaximal, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C2×#6817C2211yescharacteristic
13C2×C5×#235C2211yescharacteristic
14C22×C3×Dic10C2211yescharacteristic
15C2×#6782—22no—
16C2×D15⋊Q8C2212yes—
17C2×#6769—21no—
18#92004—24no—
19#92004—24no—
20C2×#6768C2212yes—
21C2×D3×Dic10—21no—
22#91836—21no—
23#91835C2214yes—
24#91833—21no—
25#91818—22no—
26#91815C2211yescharacteristic
27C2×C5×#216C2212yes—
28C5×#942—24no—
29C2×C5×#215C2212yes—
30C2×C5×#208C2211yescharacteristic
31C2×C3×#689—24no—
32C2×C3×#689C2214yes—
33C22×C3×#168C2211yescharacteristic
34#90973C2212yes—
35#90971C2214yes—
36#90967C2212yes—
37#90957C2214yes—
38#83884—51nomaximal, Hall
39C2×#4150—31no—
40C2×C5×#137—31no—
41#66150—31no—
42C22×#689D311yescharacteristic, complemented
43#65522—31no—
44#65475—34no—
45#65462—34no—
46#65418—31no—
47#65409—31no—
48C2×C3×C5×Q8—21no—
49C2×C3×C5×Q8—22no—
50C2×C3×C5×Q8—22no—
51C2×C3×C5×Q8C2312yes—
52C22×C3×C4×C5—21no—
53C22×C3×C4×C5C2311yescharacteristic
54C2×#1220D412yes—
55#6817—22no—
56#6817—22no—
57C2×D60D411yescharacteristic
58C2×C4×D15C2312yes—
59C2×Dic30D411yescharacteristic
60C2×C5×#67—21no—
61C5×#235—22no—
62C5×#235D414yes—
63C2×C5×D12C2311yescharacteristic
64C2×C4×C5×D3—21no—
65C2×C5×Dic6C2311yescharacteristic
66C22×C3×Dic5—21no—
67C2×C3×Dic10—21no—
68C2×C3×Dic10—22no—
69C2×C3×Dic10D414yes—
70C2×C3×Dic10C2312yes—
71C2×#1204—22no—
72C2×#1203—22no—
73C2×#1200—22no—
74#6782—216no—
75C2×D3×Dic5—22no—
76D15⋊Q8—24no—
77D15⋊Q8—24no—
78#6769—28no—
79#6768—24no—
80#6768—24no—
81D3×Dic10—28no—
82#6719—21no—
83#6714—24no—
84C4⋊Dic15C2312yes—
85#6712—24no—
86C4×Dic15—21no—
87C5×#216—24no—
88C5×#215—24no—
89C5×#208—22no—
90C2×C5×C3⋊C8C2312yes—
91C3×#689—28no—
92C2×C3×#168—22no—
93C2×C3×#168C2312yes—
94C2×#1007—51no—
95C22×C3×Q16—51no—
96#5816—51no—
97C2×#942D511yescharacteristic, complemented
98#5225—51no—
99#5187—54no—
100#5186—54no—
101#5113—51no—
102#5110—51no—
103C2×C5×#41—31no—
104C22×C5×Q8C2×D311yescharacteristic
105#4150—64no—
106#4150—64no—
107C2×Q8×D5—32no—
108C2×#711—32no—
109C2×#711—62no—
110C2×#709—31no—
111C2×#709—61no—
112C22×Dic10—61no—
113C22×Dic10C2×D311yescharacteristic
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—24no—
122C2×#689C2×D314yes—
123#4075—62no—
124#4072—31no—
125C22×#168C2×D311yescharacteristic
126#3967—32no—
127#3964—32no—
128#3952—34no—
129#3948—34no—
130#3011—151noSylow
131C23×C3×C5C2×D411yescharacteristic
132C22×D15C2×D412yes—
133C22×C5×D3—21no—
134C3×C5×Q8—22no—
135C3×C5×Q8—22no—
136C3×C5×Q8—42no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—42no—
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2×D412yes—
141C2×C3×C4×C5C2×D412yes—
142C2×C3×C4×C5C2411yescharacteristic
143#1220—24no—
144C2×Dic15—22no—
145C2×Dic15C2×D412yes—
146D60—22no—
147C4×D15—22no—
148C4×D15—22no—
149Dic30—22no—
150C5×#67—24no—
151C2×C5×Dic3—21no—
152C5×D12—21no—
153C5×D12C2×D412yes—
154C4×C5×D3—24no—
155C5×Dic6—21no—
156C5×Dic6C2×D412yes—
157C2×C3×Dic5—24no—
158C2×C3×Dic5—22no—
159C3×Dic10—24no—
160C3×Dic10—22no—
161C3×Dic10C2×D414yes—
162#1204—28no—
163#1203—28no—
164#1200—28no—
165D3×Dic5—28no—
166C3×#168—22no—
167C5×C3⋊C8—22no—
168C22×C3×Q8—51no—
169C22×C3×Q8C2×D511yescharacteristic
170#1007—104no—
171#1007—104no—
172C2×#239—52no—
173C2×#239—101no—
174C2×D3×Q8—52no—
175C2×D3×Q8—101no—
176C2×#235—51no—
177C2×#235—102no—
178C2×#235C2×D511yescharacteristic
179C2×C3×Q16—54no—
180C2×C3×Q16—104no—
181C22×C3×C8—51no—
182#947—101no—
183#946—54no—
184#944—101no—
185C2×#216C2×D512yes—
186#942—24no—
187C2×#215C2×D512yes—
188#929—102no—
189#926—51no—
190C2×#208C2×D511yescharacteristic
191#835—54no—
192#821—52no—
193(C2×D3)⋊C8—54no—
194#816—52no—
195C5×#41—34no—
196C5×#41—62no—
197C2×C5×Q8—21no—
198C2×C5×Q8—22no—
199C2×C5×Q8—22no—
200C2×C5×Q8—31no—
201C2×C5×Q8C22×D312yes—
202C2×C5×D4—31no—
203C2×C5×D4—61no—
204C22×C4×C5—21no—
205C22×C4×C5—61no—
206C22×C4×C5C22×D311yescharacteristic
207C2×C22⋊D5—32no—
208C2×C22⋊D5—62no—
209C22×Dic5—21no—
210C22×Dic5—62no—
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—21no—
223C2×Dic10—22no—
224C2×Dic10—31no—
225C2×Dic10—68no—
226C2×Dic10—62no—
227C2×Dic10#6714yes—
228C2×Dic10C22×D312yes—
229C5×Q16—64no—
230C5×SD16—64no—
231C5×M16—62no—
232C2×C5×C8—32no—
233C22⋊Dic5—61no—
234#689—28no—
235#685—64no—
236C4⋊Dic5—32no—
237#683—64no—
238C4×Dic5—61no—
239C2×#168—22no—
240C2×#168C22×D312yes—
241C2×2−1+4—151no—
242C2×#137—31no—
243C22×Q16—51no—
244#536—151no—
245#452—154no—
246#451—154no—
247#419—151no—
248#416—151no—
249#408—151no—
250C22×C3×C5—22no—
251C22×C3×C5#12012yes—
252C22×C3×C5C22×D411yescharacteristic
253C2×D15—22no—
254C2×D15—24no—
255C2×C5×D3—22no—
256C2×C5×D3—42no—
257C3×C4×C5—42no—
258C3×C4×C5#12012yes—
259C3×C4×C5C22×D411yescharacteristic
260C3×C4×C5C22×D411yescharacteristic
261Dic15—22no—
262Dic15—42no—
263C3×Dic5—24no—
264C5×Dic3—22no—
265C2×C3×Q8—21no—
266C2×C3×Q8—22no—
267C2×C3×Q8—22no—
268C2×C3×Q8—54no—
269C2×C3×Q8—52no—
270C2×C3×Q8—101no—
271C2×C3×Q8—102no—
272C2×C3×Q8C22×D512yes—
273C22×C3×C4—21no—
274C22×C3×C4—101no—
275C22×C3×C4C22×D511yescharacteristic
276C2×#67—21no—
277C2×#67—52no—
278#239—108no—
279#239—104no—
280#239—104no—
281D3×Q8—104no—
282D3×Q8—108no—
283D3×Q8—104no—
284#235—22no—
285#235—1016no—
286#235—102no—
287#235—102no—
288#235C22⋊D514yes—
289C2×D12—51no—
290C2×D12—102no—
291C2×D12C22×D511yescharacteristic
292C2×C4×D3—21no—
293C2×C4×D3—52no—
294C2×C4×D3—102no—
295C2×C4×D3—102no—
296C2×Dic6—51no—
297C2×Dic6—102no—
298C2×Dic6C22×D511yescharacteristic
299C3×Q16—108no—
300C2×C3×C8—52no—
301C2×C3×C8—102no—
302#217—101no—
303#216—24no—
304#215—24no—
305(C2×D3)⋊C4—104no—
306C4⋊Dic3—52no—
307Dic3⋊C4—104no—
308C4×Dic3—101no—
309#208—22no—
310C2×C3⋊C8C22×D512yes—
311C23×C5—61no—
312C23×C5D3×D411yescharacteristic
313C22×D5—32no—
314C5×Q8—22no—
315C5×Q8—22no—
316C5×Q8—32no—
317C5×Q8—42no—
318C5×Q8—61no—
319C5×D4—32no—
320C5×D4—64no—
321C5×D4—61no—
322C2×C4×C5—22no—
323C2×C4×C5—42no—
324C2×C4×C5—61no—
325C2×C4×C5—64no—
326C2×C4×C5D3×D412yes—
327C2×C4×C5D3×D411yescharacteristic
328C2×C4×C5C2×#6712yes—
329C2×C4×C5C23×D311yescharacteristic
330C22⋊D5—68no—
331C22⋊D5—64no—
332C2×Dic5—24no—
333C2×Dic5—22no—
334C2×Dic5—32no—
335C2×Dic5—68no—
336C2×Dic5—62no—
337D20—62no—
338C4×D5—68no—
339C4×D5—62no—
340C4×D5—62no—
341Dic10—24no—
342Dic10—22no—
343Dic10—68no—
344Dic10—62no—
345Dic10C2×#6714yes—
346C5×C8—62no—
347#168—22no—
3482−1+4—304no—
3492−1+4—304no—
350C2×#41—31no—
351C2×#41—151no—
352C2×#41—152no—
353C2×#41—301no—
354C2×#41—302no—
355C22×Q8—51no—
356C22×Q8—152no—
357C22×Q8—301no—
358C22×Q8D3×D511yescharacteristic
359#137—64no—
360C2×Q16—32no—
361C2×Q16—54no—
362C2×Q16—104no—
363C2×SD16—32no—
364C2×M16—31no—
365C22×C8—51no—
366C4⋊Q8—301no—
367#124—301no—
368#123—302no—
369C22⋊Q8—154no—
370#117—151no—
371#103—154no—
372#103—152no—
373#102—152no—
374C22⋊C8—154no—
375C2×C3×C5—22no—
376C2×C3×C5C2×#12011yescharacteristic
377C2×C3×C5#57812yes—
378D15—42no—
379C5×D3—42no—
380C23×C3D4×D511yescharacteristic
381C22×D3—21no—
382C22×D3—52no—
383C3×Q8—22no—
384C3×Q8—22no—
385C3×Q8—42no—
386C3×Q8—54no—
387C3×Q8—102no—
388C3×Q8—104no—
389C2×C3×C4—22no—
390C2×C3×C4—42no—
391C2×C3×C4—102no—
392C2×C3×C4—104no—
393C2×C3×C4D4×D512yes—
394C2×C3×C4D4×D511yescharacteristic
395C2×C3×C4C2×C22⋊D512yes—
396C2×C3×C4C23×D511yescharacteristic
397#67—24no—
398#67—104no—
399C2×Dic3—21no—
400C2×Dic3—52no—
401C2×Dic3—102no—
402D12—21no—
403D12—108no—
404D12—102no—
405D12C2×C22⋊D512yes—
406C4×D3—24no—
407C4×D3—108no—
408C4×D3—108no—
409C4×D3—102no—
410C4×D3—102no—
411Dic6—21no—
412Dic6—108no—
413Dic6—102no—
414Dic6C2×C22⋊D512yes—
415C3×C8—102no—
416C3⋊C8—22no—
417C22×C5—22no—
418C22×C5—62no—
419C22×C5—122no—
420C22×C5#93712yes—
421C22×C5C2×D3×D411yescharacteristic
422C2×D5—64no—
423C2×D5—62no—
424C4×C5—42no—
425C4×C5—62no—
426C4×C5#93712yes—
427C4×C5C2×D3×D411yescharacteristic
428C4×C5C22×#6711yescharacteristic
429Dic5—24no—
430Dic5—62no—
431Dic5—122no—
432#41—34no—
433#41—62no—
434#41—308no—
435#41—3016no—
436#41—304no—
437#41—304no—
438#41—302no—
439#41—302no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—31no—
444C2×Q8—54no—
445C2×Q8—52no—
446C2×Q8—102no—
447C2×Q8—101no—
448C2×Q8—151no—
449C2×Q8—302no—
450C2×Q8—308no—
451C2×Q8—304no—
452C2×Q8—304no—
453C2×Q8C2×D3×D512yes—
454C2×D4—31no—
455C2×D4—61no—
456C2×D4—152no—
457C2×D4—151no—
458C2×D4—302no—
459C22×C4—21no—
460C22×C4—61no—
461C22×C4—101no—
462C22×C4—152no—
463C22×C4—302no—
464C22×C4—302no—
465C22×C4C2×D3×D511yescharacteristic
466Q16—64no—
467Q16—108no—
468SD16—64no—
469M16—62no—
470C2×C8—32no—
471C2×C8—52no—
472C2×C8—102no—
473C4⋊C4—152no—
474C4⋊C4—304no—
475C22⋊C4—304no—
476C22⋊C4—301no—
477C42—301no—
478C3×C5#301111yescharacteristic, complemented, Hall
479C22×C3—22no—
480C22×C3#408312yes—
481C22×C3C2×D4×D511yescharacteristic
482C2×D3—22no—
483C2×D3—42no—
484C2×D3—104no—
485C2×D3—102no—
486C3×C4—42no—
487C3×C4—104no—
488C3×C4#408312yes—
489C3×C4C2×D4×D511yescharacteristic
490C3×C4C22×C22⋊D511yescharacteristic
491Dic3—22no—
492Dic3—102no—
493Dic3—202no—
494C2×C5—22no—
495C2×C5—122no—
496C2×C5#576612yes—
497C2×C5C2×#93711yescharacteristic
498D5—122no—
499C23—61no—
500C23—152no—
501C23#679011yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—32no—
505Q8—42no—
506Q8—54no—
507Q8—61no—
508Q8—102no—
509Q8—104no—
510Q8—308no—
511Q8—302no—
512D4—32no—
513D4—64no—
514D4—61no—
515D4—308no—
516D4—304no—
517D4—302no—
518C2×C4—22no—
519C2×C4—42no—
520C2×C4—61no—
521C2×C4—64no—
522C2×C4—102no—
523C2×C4—104no—
524C2×C4—152no—
525C2×C4—308no—
526C2×C4—308no—
527C2×C4—302no—
528C2×C4—302no—
529C2×C4—302no—
530C2×C4D5×#6712yes—
531C2×C4D3×C22⋊D512yes—
532C2×C4#679011yescharacteristic
533C2×C4C22×D3×D511yescharacteristic
534C8—62no—
535C8—102no—
536C2×C3—22no—
537C2×C3#6611312yes—
538C2×C3C2×#408311yescharacteristic
539D3—42no—
540D3—202no—
541C5#8388411yescharacteristic, complemented, Sylow
542C22—22no—
543C22—62no—
544C22—122no—
545C22—304no—
546C22—302no—
547C22#9157712yes—
548C22C2×#679011yescharacteristic
549C4—42no—
550C4—62no—
551C4—104no—
552C4—302no—
553C4—602no—
554C4#9157712yes—
555C4C2×D5×#6711yescharacteristic
556C4C2×D3×C22⋊D511yescharacteristic
557C3#1061571411yescharacteristic, complemented, Sylow
558C2—22no—
559C2—122no—
560C2—602no—
561C2#1175512212yes—
562C2C2×#9157711yescharacteristic
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×Q8C22
socleC22×C3×C5C22×D4
prime radicalC2×#92004C2
hypercenterC22×Q8D3×D5
hypocenterC3×C5#3011
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. #11756575C2semidirect
3. C2×C5×#942C2semidirect
4. C22×C3×#689C2semidirect
5. #11749407C2semidirect
6. #11749402C2semidirect
7. #11748400C2semidirect
8. #11748388C2semidirect
9. #11748349C2semidirect
42. C22×#689D3semidirect
97. C2×#942D5semidirect
478. C3×C5#3011semidirect
541. C5#83884semidirect
557. C3#10615714semidirect

Maximal subgroups

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

subgroupindex
C2×#920042
#117565752
C2×C5×#9422
C22×C3×#6892
#117494072
#117494022
#117484002
#117483882
#117483492
#106157143
#838845