Finite Group Databasebeta

← #14426089

Subgroups — #14426089

Order 1920.

960 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×Q8×D15C211yesmaximal, characteristic
2C2×#91511C212yesmaximal
3#11752562C211yesmaximal, characteristic
4C5×#5031C211yesmaximal, characteristic, complemented
5C3×#65320C211yesmaximal, characteristic, complemented
6#11749364C212yesmaximal, complemented
7#11747490C212yesmaximal
8#11747427C212yesmaximal, complemented
9#11747105C211yesmaximal, characteristic, complemented
10#11747104C211yesmaximal, characteristic, complemented
11#11747000C211yesmaximal, characteristic
12D5×#462—31nomaximal, Hall
13C2×Q8×D15C2211yescharacteristic
14C2×D15⋊Q8C2212yes—
15C3×C4×C5×Q8C2211yescharacteristic
16Q8×Dic15C2211yescharacteristic
17C4⋊C4×D15—21no—
18C4⋊C4×D15C2211yescharacteristic
19#91776—21no—
20#91776C2211yescharacteristic
21C42×D15—21no—
22C42×D15C2211yescharacteristic
23C4×Dic30—21no—
24C4×Dic30C2211yescharacteristic
25C2×C5×#216C2212yes—
26C5×#869C2211yescharacteristic
27C2×C3×#689C2212yes—
28C3×#4015C2211yescharacteristic
29#91511—28no—
30#91470—21no—
31#91470—21no—
32#91437—21no—
33#91437—21no—
34#91385C2212yes—
35C2×#6663C2212yes—
36C2×#6662C2212yes—
37C2×#6648C2212yes—
38C5×#810C2212yes—
39C5×C4⋊(C3⋊C8)C2211yescharacteristic
40C3×#3942C2212yes—
41C3×C4⋊#168C2211yescharacteristic
42#90983C2212yes—
43#90973C2212yes—
44#90971C2212yes—
45D3×#462—51nomaximal, Hall
46C2×D5×Q16—32no—
47D5×C4⋊Q8—31no—
48C4×Q8×D5—31no—
49C5×#462—31no—
50#65471—32no—
51#65379—31no—
52#65320D311yescharacteristic, complemented
53#65138—31no—
54D5×C4⋊C8—31no—
55D5×#103—32no—
56#65082—32no—
57C2×C3×C5×Q8C2311yescharacteristic
58Q8×D15—22no—
59Q8×D15D414yes—
60C2×C4×D15—21no—
61C2×C4×D15—21no—
62C2×C4×D15D411yescharacteristic
63C2×C4×D15D411yescharacteristic
64C2×C4×D15C2311yescharacteristic
65C2×Dic30—21no—
66C2×Dic30C2311yescharacteristic
67C2×C5×Dic6C2312yes—
68C2×C3×Dic10C2312yes—
69C2×#1204—22no—
70C2×#1200—22no—
71D15⋊Q8—24no—
72D15⋊Q8—24no—
73C3×C5×C4⋊C4—21no—
74C3×C5×C4⋊C4C2311yescharacteristic
75C3×C42×C5—21no—
76C3×C42×C5C2311yescharacteristic
77C4⋊Dic15—21no—
78C4⋊Dic15C2311yescharacteristic
79#6712—21no—
80#6712—21no—
81#6712—21no—
82C4×Dic15—21no—
83C4×Dic15—21no—
84C4×Dic15—21no—
85C4×Dic15D411yescharacteristic
86C4×Dic15D411yescharacteristic
87C4×Dic15C2311yescharacteristic
88C5×#216—24no—
89C5×C4⋊Dic3—21no—
90C5×C4⋊Dic3—21no—
91C2×C5×C3⋊C8C2312yes—
92C3×#689—24no—
93C3×C4⋊Dic5—21no—
94C3×C4⋊Dic5—21no—
95C2×C3×#168C2312yes—
96#6669—22no—
97#6669—22no—
98#6663—24no—
99#6662—24no—
100#6648—24no—
101C2×D3×Q16—52no—
102D3×C4⋊Q8—51no—
103C4×D3×Q8—51no—
104C3×#462—51no—
105#5182—52no—
106#5090—51no—
107#5031D511yescharacteristic, complemented
108#4849—51no—
109D3×C4⋊C8—51no—
110D3×#103—52no—
111#4795—52no—
112C2×Q8×D5—31no—
113C2×Q8×D5—32no—
114C2×C5×Q16—32no—
115C5×C4⋊Q8—31no—
116C4×C5×Q8C2×D311yescharacteristic
117Q8×Dic5—31no—
118#4090—32no—
119C2×#689—32no—
120C2×#689C2×D312yes—
121D5×Q16—68no—
122C2×Dic20—32no—
123C2×C8×D5—32no—
124D5×C4⋊C4—31no—
125D5×C4⋊C4—61no—
126D5×C4⋊C4—61no—
127D5×C4⋊C4—61no—
128#4034—61no—
129#4034—61no—
130#4033—31no—
131#4033—61no—
132C42×D5—31no—
133C42×D5—61no—
134#4015C2×D311yescharacteristic
135C4×Dic10—31no—
136C4×Dic10—61no—
137C5×C4⋊C8—31no—
138C5×#103—32no—
139#3967—32no—
140#3948—32no—
141#3946—32no—
142#3942C2×D312yes—
143C4⋊#168C2×D311yescharacteristic
144C2×#462—151noSylow
145C22×D15C2×D411yescharacteristic
146C3×C5×Q8—21no—
147C3×C5×Q8C2×D412yes—
148C2×C3×C4×C5—21no—
149C2×C3×C4×C5—21no—
150C2×C3×C4×C5C2×D411yescharacteristic
151C2×C3×C4×C5C2×D411yescharacteristic
152C2×C3×C4×C5C2411yescharacteristic
153C2×Dic15—21no—
154C2×Dic15—21no—
155C2×Dic15C2×D411yescharacteristic
156C2×Dic15C2×D411yescharacteristic
157C2×Dic15C2×D411yescharacteristic
158C4×D15—24no—
159C4×D15—22no—
160C4×D15—42no—
161C4×D15Q1614yescomplemented
162C4×D15C2×D412yes—
163C4×D15#4112yes—
164Dic30—21no—
165Dic30—24no—
166Dic30C2×D412yes—
167C2×C5×Dic3—22no—
168C5×Dic6—22no—
169C5×Dic6—22no—
170C2×C3×Dic5—22no—
171C3×Dic10—22no—
172C3×Dic10—22no—
173#1204—44no—
174#1200—44no—
175C3×#168—22no—
176C5×C3⋊C8—22no—
177C2×D3×Q8—51no—
178C2×D3×Q8—52no—
179C2×C3×Q16—52no—
180C3×C4⋊Q8—51no—
181C3×C4×Q8C2×D511yescharacteristic
182Q8×Dic3—51no—
183#944—52no—
184C2×#216—52no—
185C2×#216C2×D512yes—
186D3×Q16—108no—
187C2×Dic12—52no—
188C2×D3×C8—52no—
189D3×C4⋊C4—51no—
190D3×C4⋊C4—101no—
191D3×C4⋊C4—101no—
192D3×C4⋊C4—101no—
193#888—101no—
194#888—101no—
195#887—51no—
196#887—101no—
197C42×D3—51no—
198C42×D3—101no—
199#869C2×D511yescharacteristic
200C4×Dic6—51no—
201C4×Dic6—101no—
202C3×C4⋊C8—51no—
203C3×#103—52no—
204#835—52no—
205#816—52no—
206Dic3⋊C8—52no—
207#810C2×D512yes—
208C4⋊(C3⋊C8)C2×D511yescharacteristic
209C2×C5×Q8—32no—
210C2×C5×Q8C22×D311yescharacteristic
211Q8×D5—34no—
212Q8×D5—64no—
213Q8×D5—64no—
214Q8×D5—62no—
215C2×C4×D5—31no—
216C2×C4×D5—31no—
217C2×C4×D5—31no—
218C2×C4×D5—61no—
219C2×C4×D5—62no—
220C2×C4×D5—61no—
221C2×Dic10—31no—
222C2×Dic10—61no—
223C2×Dic10—62no—
224C2×Dic10C22×D312yes—
225C5×Q16—64no—
226C2×C5×C8—32no—
227C5×C4⋊C4—21no—
228C5×C4⋊C4—61no—
229C5×C4⋊C4—61no—
230C5×C4⋊C4C22×D311yescharacteristic
231C42×C5—21no—
232C42×C5C22×D311yescharacteristic
233#689—24no—
234#689—64no—
235C4⋊Dic5—21no—
236C4⋊Dic5—21no—
237C4⋊Dic5—31no—
238C4⋊Dic5—61no—
239#683—62no—
240#683—62no—
241#683—61no—
242#683—61no—
243#683—61no—
244C4×Dic5—31no—
245C4×Dic5—31no—
246C4×Dic5—31no—
247C4×Dic5—61no—
248C4×Dic5—61no—
249C4×Dic5—61no—
250C2×#168C22×D312yes—
251Dic20—64no—
252C8×D5—64no—
253C22×Q16—152no—
254C2×C4⋊Q8—151no—
255C2×C4×Q8—151no—
256#462—31no—
257#462—51no—
258#462—152no—
259#462—151no—
260#462—151no—
261#462—152no—
262C2×C4⋊C8—151no—
263C2×#103—152no—
264C22×C3×C5C22×D411yescharacteristic
265C2×D15#12112yes—
266C2×D15C2×Q1612yes—
267C2×D15#13712yes—
268C3×C4×C5—22no—
269C3×C4×C5—21no—
270C3×C4×C5—41no—
271C3×C4×C5C2×Q1612yescomplemented
272C3×C4×C5C22×D411yescharacteristic
273C3×C4×C5C2×#4111yescharacteristic
274Dic15—22no—
275Dic15—21no—
276Dic15—41no—
277Dic15#12112yes—
278Dic15C2×Q1612yescomplemented
279C3×Dic5—42no—
280C5×Dic3—42no—
281C2×C3×Q8—52no—
282C2×C3×Q8C22×D511yescharacteristic
283D3×Q8—54no—
284D3×Q8—104no—
285D3×Q8—104no—
286D3×Q8—102no—
287C2×C4×D3—51no—
288C2×C4×D3—51no—
289C2×C4×D3—51no—
290C2×C4×D3—101no—
291C2×C4×D3—102no—
292C2×C4×D3—101no—
293C2×Dic6—51no—
294C2×Dic6—101no—
295C2×Dic6—102no—
296C2×Dic6C22×D512yes—
297C3×Q16—104no—
298C2×C3×C8—52no—
299C3×C4⋊C4—21no—
300C3×C4⋊C4—101no—
301C3×C4⋊C4—101no—
302C3×C4⋊C4C22×D511yescharacteristic
303C3×C42—21no—
304C3×C42C22×D511yescharacteristic
305#216—24no—
306#216—104no—
307C4⋊Dic3—21no—
308C4⋊Dic3—21no—
309C4⋊Dic3—51no—
310C4⋊Dic3—101no—
311Dic3⋊C4—102no—
312Dic3⋊C4—102no—
313Dic3⋊C4—101no—
314Dic3⋊C4—101no—
315Dic3⋊C4—101no—
316C4×Dic3—51no—
317C4×Dic3—51no—
318C4×Dic3—51no—
319C4×Dic3—101no—
320C4×Dic3—101no—
321C4×Dic3—101no—
322C2×C3⋊C8C22×D512yes—
323Dic12—104no—
324D3×C8—104no—
325C22×D5—31no—
326C5×Q8—21no—
327C5×Q8—62no—
328C5×Q8—62no—
329C5×Q8D3×D412yes—
330C2×C4×C5—21no—
331C2×C4×C5—21no—
332C2×C4×C5—62no—
333C2×C4×C5D3×D411yescharacteristic
334C2×C4×C5D3×D411yescharacteristic
335C2×C4×C5C23×D311yescharacteristic
336C2×Dic5—22no—
337C2×Dic5—31no—
338C2×Dic5—31no—
339C2×Dic5—31no—
340C2×Dic5—61no—
341C2×Dic5—61no—
342C4×D5—34no—
343C4×D5—32no—
344C4×D5—32no—
345C4×D5—62no—
346C4×D5—64no—
347C4×D5—124no—
348C4×D5—122no—
349Dic10—22no—
350Dic10—22no—
351Dic10—32no—
352Dic10—64no—
353Dic10—61no—
354Dic10—124no—
355C5×C8—62no—
356#168—22no—
357C22×Q8—152no—
358C22×Q8—151no—
359C2×Q16—32no—
360C2×Q16—52no—
361C2×Q16—152no—
362C2×Q16—152no—
363C2×Q16—308no—
364C22×C8—152no—
365C4⋊Q8—31no—
366C4⋊Q8—51no—
367C4⋊Q8—152no—
368C4⋊Q8—301no—
369C4⋊Q8—301no—
370C4×Q8—151no—
371C4×Q8—151no—
372C4×Q8—151no—
373C4×Q8—301no—
374C4×Q8—301no—
375C4×Q8D3×D511yescharacteristic
376C2×C4⋊C4—151no—
377C2×C4⋊C4—301no—
378C2×C4⋊C4—301no—
379C2×C4⋊C4—301no—
380C2×C42—151no—
381C2×C42—301no—
382C4⋊C8—31no—
383C4⋊C8—51no—
384C4⋊C8—152no—
385#103—32no—
386#103—52no—
387#103—152no—
388#103—152no—
389C2×C3×C5C2×#12111yescharacteristic
390C2×C3×C5C22×Q1611yescharacteristic
391C2×C3×C5C2×#13711yescharacteristic
392D15#46214yescomplemented
393C22×D3—51no—
394C3×Q8—21no—
395C3×Q8—102no—
396C3×Q8—102no—
397C3×Q8D4×D512yes—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—102no—
401C2×C3×C4D4×D511yescharacteristic
402C2×C3×C4D4×D511yescharacteristic
403C2×C3×C4C23×D511yescharacteristic
404C2×Dic3—22no—
405C2×Dic3—51no—
406C2×Dic3—51no—
407C2×Dic3—51no—
408C2×Dic3—101no—
409C2×Dic3—101no—
410C4×D3—52no—
411C4×D3—52no—
412C4×D3—54no—
413C4×D3—102no—
414C4×D3—104no—
415C4×D3—204no—
416C4×D3—202no—
417Dic6—22no—
418Dic6—22no—
419Dic6—52no—
420Dic6—104no—
421Dic6—101no—
422Dic6—204no—
423C3×C8—102no—
424C3⋊C8—22no—
425C22×C5C2×D3×D411yescharacteristic
426C2×D5—32no—
427C2×D5—32no—
428C2×D5—32no—
429C4×C5—21no—
430C4×C5—22no—
431C4×C5—41no—
432C4×C5—122no—
433C4×C5D3×Q1612yescomplemented
434C4×C5C2×D3×D411yescharacteristic
435C4×C5D3×#4111yescharacteristic
436Dic5—32no—
437Dic5—32no—
438Dic5—42no—
439Dic5—61no—
440Dic5—62no—
441Dic5—121no—
442C2×Q8—32no—
443C2×Q8—52no—
444C2×Q8—154no—
445C2×Q8—151no—
446C2×Q8—301no—
447C2×Q8—302no—
448C2×Q8—302no—
449C2×Q8—304no—
450C2×Q8—304no—
451C2×Q8C2×D3×D511yescharacteristic
452C22×C4—151no—
453C22×C4—151no—
454C22×C4—151no—
455C22×C4—301no—
456C22×C4—301no—
457C22×C4—302no—
458Q16—64no—
459Q16—104no—
460Q16—304no—
461Q16—304no—
462C2×C8—32no—
463C2×C8—52no—
464C2×C8—304no—
465C4⋊C4—21no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—101no—
469C4⋊C4—101no—
470C4⋊C4—151no—
471C4⋊C4—301no—
472C4⋊C4—301no—
473C4⋊C4—301no—
474C4⋊C4—302no—
475C4⋊C4—302no—
476C4⋊C4—301no—
477C4⋊C4C2×D3×D511yescharacteristic
478C42—21no—
479C42—151no—
480C42—151no—
481C42—151no—
482C42—301no—
483C42—301no—
484C42—301no—
485C42C2×D3×D511yescharacteristic
486C3×C5C2×#46211yescharacteristic, complemented, Hall
487C22×C3C2×D4×D511yescharacteristic
488C2×D3—52no—
489C2×D3—52no—
490C2×D3—52no—
491C3×C4—21no—
492C3×C4—22no—
493C3×C4—41no—
494C3×C4—202no—
495C3×C4D5×Q1612yescomplemented
496C3×C4C2×D4×D511yescharacteristic
497C3×C4D5×#4111yescharacteristic
498Dic3—42no—
499Dic3—52no—
500Dic3—52no—
501Dic3—101no—
502Dic3—102no—
503Dic3—201no—
504C2×C5D3×#12111yescharacteristic
505C2×C5C2×D3×Q1611yescharacteristic
506C2×C5D3×#13711yescharacteristic
507D5—34no—
508C23—151no—
509Q8—21no—
510Q8—62no—
511Q8—62no—
512Q8—102no—
513Q8—102no—
514Q8—152no—
515Q8—304no—
516Q8—301no—
517Q8—604no—
518Q8#677712yes—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—62no—
522C2×C4—102no—
523C2×C4—151no—
524C2×C4—151no—
525C2×C4—152no—
526C2×C4—152no—
527C2×C4—154no—
528C2×C4—151no—
529C2×C4—302no—
530C2×C4—304no—
531C2×C4—301no—
532C2×C4—301no—
533C2×C4—602no—
534C2×C4—604no—
535C2×C4#679011yescharacteristic
536C2×C4#679011yescharacteristic
537C2×C4C22×D3×D511yescharacteristic
538C8—62no—
539C8—102no—
540C2×C3D5×#12111yescharacteristic
541C2×C3C2×D5×Q1611yescharacteristic
542C2×C3D5×#13711yescharacteristic
543D3—54no—
544C5D3×#46211yescharacteristic, complemented, Sylow
545C22—152no—
546C22—152no—
547C22—152no—
548C22C2×#679011yescharacteristic
549C4—21no—
550C4—22no—
551C4—41no—
552C4—122no—
553C4—152no—
554C4—152no—
555C4—202no—
556C4—301no—
557C4—302no—
558C4—601no—
559C4#9151112yescomplemented
560C4C2×#677711yescharacteristic
561C4#9201611yescharacteristic
562C3D5×#46211yescharacteristic, complemented, Sylow
563C2—154no—
564C2#1175467011yescharacteristic
565C2C2×#9151111yescharacteristic
566C2#1175534211yescharacteristic
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×Q8C22
socleC22×C3×C5C22×D4
prime radicalC22×D15C2×D4
hypercenterC4×Q8D3×D5
hypocenterC3×C5C2×#462
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
4. C5×#5031C2semidirect
5. C3×#65320C2semidirect
6. #11749364C2semidirect
8. #11747427C2semidirect
9. #11747105C2semidirect
10. #11747104C2semidirect
52. #65320D3semidirect
107. #5031D5semidirect
161. C4×D15Q16semidirect
271. C3×C4×C5C2×Q16semidirect
278. Dic15C2×Q16semidirect
392. D15#462semidirect
433. C4×C5D3×Q16semidirect
486. C3×C5C2×#462semidirect
495. C3×C4D5×Q16semidirect
544. C5D3×#462semidirect
559. C4#91511semidirect
562. C3D5×#462semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×D152
C2×#915112
#117525622
C5×#50312
C3×#653202
#117493642
#117474902
#117474272
#117471052
#117471042
#117470002
D5×#4623
D3×#4625