Finite Group Databasebeta

← #14424604

Subgroups — #14424604

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
2C22×C3×Dic20C211yesmaximal, characteristic, complemented
3#11754278C211yesmaximal, characteristic, complemented
4C2×#91309C211yesmaximal, characteristic, complemented
5#11751211C211yesmaximal, characteristic, complemented
6C5×#5113C211yesmaximal, characteristic, complemented
7#11748399C211yesmaximal, characteristic, complemented
8#11748151C214yesmaximal, complemented
9#11748126C214yesmaximal, complemented
10#10610568—31nomaximal, Hall
11C2×#6817C2211yescharacteristic
12C2×C5×#235C2211yescharacteristic
13C22×C3×Dic10C2211yescharacteristic
14C22×C3×Dic10C2211yescharacteristic
15C2×#6782—22no—
16C2×D15⋊Q8C2212yes—
17C2×#6769—21no—
18#92004—24no—
19#92004—24no—
20C2×#6768C2212yes—
21C2×D3×Dic10—21no—
22C22×C3×C5×C8C2211yescharacteristic
23C2×#6710C2211yescharacteristic
24C5×#926C2211yescharacteristic
25C2×C3×Dic20—24no—
26C2×C3×Dic20C2214yes—
27#91546—22no—
28#91390C2214yes—
29#91389—21no—
30#91374—21no—
31C2×#6661C2212yes—
32#91309—24no—
33C2×#6656C2212yes—
34#91104C2214yes—
35C5×#821C2212yes—
36C5×#816C2212yes—
37#90970C2214yes—
38#83884—51nomaximal, Hall
39C2×#4150—31no—
40C2×#4097—31no—
41C22×Dic20D311yescharacteristic, complemented
42#65860—31no—
43C5×#416—31no—
44#65399—31no—
45#65293—31no—
46#65097—34no—
47#65072—34no—
48C22×C3×C4×C5C2311yescharacteristic
49C2×#1220D412yes—
50#6817—22no—
51#6817—22no—
52C2×D60D411yescharacteristic
53C2×C4×D15C2312yes—
54C2×Dic30D411yescharacteristic
55C2×C5×#67—21no—
56C5×#235—22no—
57C5×#235D414yes—
58C2×C5×D12C2311yescharacteristic
59C2×C4×C5×D3—21no—
60C2×C5×Dic6C2311yescharacteristic
61C22×C3×Dic5—21no—
62C22×C3×Dic5—21no—
63C2×C3×Dic10—21no—
64C2×C3×Dic10—22no—
65C2×C3×Dic10—22no—
66C2×C3×Dic10—22no—
67C2×C3×Dic10—21no—
68C2×C3×Dic10D414yes—
69C2×C3×Dic10C2312yes—
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—
82C2×C3×C5×C8—22no—
83C2×C3×C5×C8C2312yes—
84#6710—22no—
85C2×C3⋊#168C2312yes—
86C5×#217—21no—
87C5×C4⋊Dic3C2312yes—
88C4×C5×Dic3—21no—
89C3×Dic20—28no—
90Dic3⋊Dic5—24no—
91#6667—24no—
92#6661—24no—
93#6656—24no—
94C2×#1007—51no—
95C22×C3×Q16—51no—
96#5816—51no—
97C2×#942—51no—
98#5225—51no—
99#5187—54no—
100#5186—54no—
101#5113D511yescharacteristic, complemented
102#5110—51no—
103C2×C5×#41—31no—
104#4150—64no—
105#4150—64no—
106C2×Q8×D5—32no—
107C2×#711—32no—
108C2×#711—62no—
109C2×#709—31no—
110C2×#709—61no—
111C22×Dic10—61no—
112C22×Dic10C2×D311yescharacteristic
113C22×Dic10C2×D311yescharacteristic
114C22×C5×C8C2×D311yescharacteristic
115C5×#117—31no—
116#4097—64no—
117C2×#689—32no—
118C2×#687—32no—
119C2×#681—31no—
120C2×Dic20—24no—
121C2×Dic20C2×D314yes—
122#4043—34no—
123#4032—62no—
124#4021—61no—
125#4015—61no—
126C5×#103—32no—
127C5×#102—32no—
128#3952—34no—
129#3942—34no—
130#3011—151noSylow
131C23×C3×C5C2×D411yescharacteristic
132C22×D15C2×D412yes—
133C22×C5×D3—21no—
134C2×C3×C4×C5C2×D411yescharacteristic
135C2×C3×C4×C5C2×D412yes—
136C2×C3×C4×C5C2×D412yes—
137C2×C3×C4×C5C2411yescharacteristic
138#1220—24no—
139C2×Dic15C2×D412yes—
140D60—22no—
141C4×D15—22no—
142C4×D15—22no—
143Dic30—22no—
144C5×#67—24no—
145C2×C5×Dic3—21no—
146C2×C5×Dic3—22no—
147C5×D12—21no—
148C5×D12C2×D412yes—
149C4×C5×D3—24no—
150C5×Dic6—21no—
151C5×Dic6C2×D412yes—
152C2×C3×Dic5—24no—
153C2×C3×Dic5—22no—
154C2×C3×Dic5—22no—
155C2×C3×Dic5—42no—
156C3×Dic10—24no—
157C3×Dic10—22no—
158C3×Dic10—22no—
159C3×Dic10—22no—
160C3×Dic10—42no—
161C3×Dic10C2×D414yes—
162#1204—28no—
163#1203—28no—
164#1200—28no—
165D3×Dic5—28no—
166C3×C5×C8—22no—
167C3⋊#168—22no—
168C22×C3×Q8—51no—
169C22×C3×Q8—51no—
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×C8C2×D511yescharacteristic
182#947—101no—
183#946—54no—
184#944—101no—
185C2×#216—52no—
186#942—104no—
187C2×#215—52no—
188#929—102no—
189#926C2×D511yescharacteristic
190C2×#208—51no—
191#835—54no—
192#821C2×D512yes—
193(C2×D3)⋊C8—54no—
194#816C2×D512yes—
195C5×#41—34no—
196C5×#41—62no—
197C2×C5×Q8—31no—
198C2×C5×D4—31no—
199C2×C5×D4—61no—
200C22×C4×C5—61no—
201C22×C4×C5C22×D311yescharacteristic
202C2×C22⋊D5—32no—
203C2×C22⋊D5—62no—
204C22×Dic5—21no—
205C22×Dic5—21no—
206C22×Dic5—62no—
207Q8×D5—64no—
208Q8×D5—64no—
209#711—64no—
210#711—616no—
211#711—64no—
212#709—68no—
213#709—62no—
214#709—62no—
215C2×D20—31no—
216C2×C4×D5—32no—
217C2×C4×D5—62no—
218C2×Dic10—21no—
219C2×Dic10—22no—
220C2×Dic10—22no—
221C2×Dic10—22no—
222C2×Dic10—21no—
223C2×Dic10—31no—
224C2×Dic10—68no—
225C2×Dic10—62no—
226C2×Dic10#6714yes—
227C2×Dic10C22×D312yes—
228C2×Dic10C22×D312yes—
229C2×C5×C8—22no—
230C2×C5×C8C22×D312yes—
231C5×C4⋊C4—32no—
232C5×C22⋊C4—61no—
233C42×C5—61no—
234#689—64no—
235#687—64no—
236#685—64no—
237C4⋊Dic5—64no—
238#681—62no—
239C2×#168—32no—
240Dic20—28no—
241C2×2−1+4—151no—
242C2×#137—151no—
243C22×Q16—51no—
244#536—151no—
245#452—154no—
246#451—154no—
247#419—151no—
248#416—31no—
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#12012yes—
258C3×C4×C5C22×D411yescharacteristic
259C3×C4×C5C22×D411yescharacteristic
260Dic15—22no—
261C3×Dic5—24no—
262C3×Dic5—42no—
263C5×Dic3—22no—
264C5×Dic3—42no—
265C2×C3×Q8—54no—
266C2×C3×Q8—52no—
267C2×C3×Q8—52no—
268C2×C3×Q8—102no—
269C2×C3×Q8—101no—
270C2×C3×Q8—101no—
271C2×C3×Q8—102no—
272C2×C3×Q8—102no—
273C22×C3×C4—101no—
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#235D2014yes—
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—22no—
301C2×C3×C8C22×D512yes—
302#217—21no—
303#216—104no—
304#215—104no—
305(C2×D3)⋊C4—104no—
306C4⋊Dic3C22×D512yes—
307Dic3⋊C4—104no—
308C4×Dic3—21no—
309#208—102no—
310C2×C3⋊C8—52no—
311C23×C5—61no—
312C23×C5D3×D411yescharacteristic
313C22×D5—32no—
314C5×Q8—32no—
315C5×Q8—61no—
316C5×D4—32no—
317C5×D4—64no—
318C5×D4—61no—
319C2×C4×C5—61no—
320C2×C4×C5—64no—
321C2×C4×C5—62no—
322C2×C4×C5D3×D412yes—
323C2×C4×C5D3×D411yescharacteristic
324C2×C4×C5C2×#6712yes—
325C2×C4×C5C23×D311yescharacteristic
326C22⋊D5—68no—
327C22⋊D5—64no—
328C2×Dic5—24no—
329C2×Dic5—22no—
330C2×Dic5—22no—
331C2×Dic5—32no—
332C2×Dic5—42no—
333C2×Dic5—68no—
334D20—62no—
335C4×D5—68no—
336C4×D5—62no—
337C4×D5—62no—
338Dic10—24no—
339Dic10—22no—
340Dic10—22no—
341Dic10—22no—
342Dic10—42no—
343Dic10—68no—
344Dic10—62no—
345Dic10C2×#6714yes—
346C5×C8—22no—
347#168—62no—
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—51no—
357C22×Q8—152no—
358C22×Q8—301no—
359#137—304no—
360C2×Q16—54no—
361C2×Q16—104no—
362C2×Q16—152no—
363C2×SD16—152no—
364C2×M16—151no—
365C22×C8D3×D511yescharacteristic
366C4⋊Q8—301no—
367#124—301no—
368#123—302no—
369C22⋊Q8—154no—
370#117—31no—
371#103—32no—
372#103—154no—
373#102—32no—
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—54no—
384C3×Q8—102no—
385C3×Q8—104no—
386C3×Q8—102no—
387C3×Q8—102no—
388C3×Q8—202no—
389C2×C3×C4—102no—
390C2×C3×C4—104no—
391C2×C3×C4—102no—
392C2×C3×C4—202no—
393C2×C3×C4C2×D2012yes—
394C2×C3×C4D4×D511yescharacteristic
395C2×C3×C4D4×D512yes—
396C2×C3×C4C23×D511yescharacteristic
397#67—24no—
398#67—104no—
399C2×Dic3—21no—
400C2×Dic3—22no—
401C2×Dic3—52no—
402D12—21no—
403D12—108no—
404D12—102no—
405D12C2×D2012yes—
406C4×D3—24no—
407C4×D3—108no—
408C4×D3—108no—
409C4×D3—102no—
410C4×D3—102no—
411Dic6—21no—
412Dic6—108no—
413Dic6—102no—
414Dic6C2×D2012yes—
415C3×C8—22no—
416C3⋊C8—102no—
417C22×C5—22no—
418C22×C5—62no—
419C22×C5—122no—
420C22×C5#93712yes—
421C22×C5C2×D3×D411yescharacteristic
422C2×D5—64no—
423C2×D5—62no—
424C4×C5—62no—
425C4×C5—122no—
426C4×C5#93712yes—
427C4×C5C2×D3×D411yescharacteristic
428C4×C5C22×#6711yescharacteristic
429Dic5—24no—
430Dic5—42no—
431Dic5—62no—
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—31no—
441C2×Q8—54no—
442C2×Q8—52no—
443C2×Q8—52no—
444C2×Q8—102no—
445C2×Q8—101no—
446C2×Q8—101no—
447C2×Q8—102no—
448C2×Q8—102no—
449C2×Q8—151no—
450C2×Q8—302no—
451C2×Q8—308no—
452C2×Q8—304no—
453C2×Q8—304no—
454C2×D4—31no—
455C2×D4—61no—
456C2×D4—152no—
457C2×D4—151no—
458C2×D4—302no—
459C22×C4—61no—
460C22×C4—101no—
461C22×C4—101no—
462C22×C4—152no—
463C22×C4—302no—
464C22×C4—302no—
465C22×C4C2×D3×D511yescharacteristic
466Q16—108no—
467Q16—304no—
468SD16—304no—
469M16—302no—
470C2×C8—22no—
471C2×C8—152no—
472C2×C8C2×D3×D512yes—
473C4⋊C4—32no—
474C4⋊C4—304no—
475C22⋊C4—61no—
476C22⋊C4—304no—
477C42—61no—
478C3×C5#301111yescharacteristic, complemented, Hall
479C22×C3—22no—
480C22×C3#402812yes—
481C22×C3C2×D4×D511yescharacteristic
482C2×D3—22no—
483C2×D3—42no—
484C2×D3—102no—
485C2×D3—104no—
486C3×C4—104no—
487C3×C4—202no—
488C3×C4#402812yes—
489C3×C4C22×D2011yescharacteristic
490C3×C4C2×D4×D511yescharacteristic
491Dic3—22no—
492Dic3—42no—
493Dic3—102no—
494C2×C5—22no—
495C2×C5—122no—
496C2×C5#576612yes—
497C2×C5C2×#93711yescharacteristic
498D5—122no—
499C23—61no—
500C23—152no—
501C23#677711yescharacteristic
502Q8—32no—
503Q8—54no—
504Q8—61no—
505Q8—104no—
506Q8—102no—
507Q8—102no—
508Q8—102no—
509Q8—202no—
510Q8—308no—
511Q8—302no—
512D4—32no—
513D4—64no—
514D4—61no—
515D4—308no—
516D4—304no—
517D4—302no—
518C2×C4—61no—
519C2×C4—64no—
520C2×C4—62no—
521C2×C4—104no—
522C2×C4—102no—
523C2×C4—102no—
524C2×C4—152no—
525C2×C4—202no—
526C2×C4—308no—
527C2×C4—308no—
528C2×C4—302no—
529C2×C4—302no—
530C2×C4D3×D2012yes—
531C2×C4#677711yescharacteristic
532C2×C4D5×#6712yes—
533C2×C4C22×D3×D511yescharacteristic
534C8—22no—
535C8—302no—
536C2×C3—22no—
537C2×C3C2×#402811yescharacteristic
538C2×C3#6610512yes—
539D3—42no—
540D3—202no—
541C5#8388411yescharacteristic, complemented, Sylow
542C22—22no—
543C22—62no—
544C22—122no—
545C22—304no—
546C22—302no—
547C22#9147412yes—
548C22C2×#677711yescharacteristic
549C4—62no—
550C4—104no—
551C4—122no—
552C4—202no—
553C4—302no—
554C4#9147412yes—
555C4C2×D3×D2011yescharacteristic
556C4C2×D5×#6711yescharacteristic
557C3#1061056811yescharacteristic, complemented, Sylow
558C2—22no—
559C2—122no—
560C2—602no—
561C2#1175401112yes—
562C2C2×#9147411yescharacteristic
subgroupG/H
centerC22C2×#6777
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×C8C22
socleC22×C3×C5C22×D4
prime radicalC2×#92004C2
hypercenterC22×C8D3×D5
hypocenterC3×C5#3011
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
2. C22×C3×Dic20C2semidirect
3. #11754278C2semidirect
4. C2×#91309C2semidirect
5. #11751211C2semidirect
6. C5×#5113C2semidirect
7. #11748399C2semidirect
8. #11748151C2semidirect
9. #11748126C2semidirect
41. C22×Dic20D3semidirect
101. #5113D5semidirect
478. C3×C5#3011semidirect
541. C5#83884semidirect
557. C3#10610568semidirect

Maximal subgroups

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

subgroupindex
C2×#920042
C22×C3×Dic202
#117542782
C2×#913092
#117512112
C5×#51132
#117483992
#117481512
#117481262
#106105683
#838845