Finite Group Databasebeta

← #14323698

Subgroups — #14323698

Order 1920.

1112 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91815C211yesmaximal, characteristic
2C2×#91813C211yesmaximal, characteristic
3C2×#91776C211yesmaximal, characteristic, complemented
4C3×C5×#380C211yesmaximal, characteristic
5C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
6#11751074C211yesmaximal, characteristic, complemented
7#11751073C211yesmaximal, characteristic, complemented
8#11751046C212yesmaximal
9#11751045C212yesmaximal
10#11750756C212yesmaximal, complemented
11#11750748C212yesmaximal, complemented
12#10605602—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×Dic30C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C5×C22⋊C4C2212yes—
18C2×#6719C2211yescharacteristic
19C2×#6719C2211yescharacteristic
20C2×#6719C2212yes—
21#91815—24no—
22C2×C4⋊Dic15C2211yescharacteristic
23C2×C4⋊Dic15C2211yescharacteristic
24C2×C4⋊Dic15C2211yescharacteristic
25#91813C418yescomplemented
26C2×#6712C2212yes—
27C2×#6712C2212yes—
28C2×#6712C2212yes—
29C2×C4×Dic15C2211yescharacteristic
30C2×C4×Dic15C2212yes—
31C2×C4×Dic15C2212yes—
32C2×C4×Dic15C2211yescharacteristic
33#91776—24no—
34C3×C5×#95C2212yes—
35#91115C2212yes—
36#91115C2212yes—
37#91115C2212yes—
38#91115C2212yes—
39#73871—51nomaximal, Hall
40C2×#4072—31no—
41C2×#4070—31no—
42C2×#4033—31no—
43C5×#380D311yescharacteristic
44C4⋊C4×Dic5—31no—
45#65263—31no—
46#65262—31no—
47#65234—32no—
48#65233—32no—
49#64944—32no—
50#64936—32no—
51C24×C3×C5C2311yescharacteristic
52C22×C3×C4×C5—22no—
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5C2×C414yes—
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2312yes—
57C22×C3×C4×C5C2311yescharacteristic
58C22×C3×C4×C5C2311yescharacteristic
59C22×Dic15C2311yescharacteristic
60C22×Dic15C2311yescharacteristic
61C22×Dic15C2312yes—
62C22×Dic15C2312yes—
63C22×Dic15C2311yescharacteristic
64C22×Dic15C2311yescharacteristic
65C2×Dic30—22no—
66C2×Dic30—22no—
67C2×Dic30C2×C414yes—
68C3×C5×C4⋊C4—22no—
69C3×C5×C4⋊C4—22no—
70C3×C5×C22⋊C4—42no—
71#6719—22no—
72#6719—22no—
73#6719C2×C418yes—
74C4⋊Dic15—22no—
75C4⋊Dic15—22no—
76C4⋊Dic15C2×C414yes—
77#6712—24no—
78#6712—42no—
79#6712C2×C418yes—
80C4×Dic15—22no—
81C4×Dic15—24no—
82C4×Dic15—42no—
83C4×Dic15D414yes—
84C2×#926—51no—
85C2×#924—51no—
86C2×#887—51no—
87C3×#380D511yescharacteristic
88Dic3×C4⋊C4—51no—
89#4973—51no—
90#4972—51no—
91#4945—52no—
92#4944—52no—
93#4655—52no—
94#4647—52no—
95C23×C4×C5C2×D311yescharacteristic
96C22×Dic10—31no—
97C2×C5×C4⋊C4C2×D311yescharacteristic
98C2×C5×C4⋊C4C2×D311yescharacteristic
99C2×C5×C22⋊C4C2×D312yes—
100C2×C22⋊Dic5—32no—
101C2×C22⋊Dic5—31no—
102C2×C22⋊Dic5—31no—
103#4072—64no—
104C2×C4⋊Dic5—31no—
105C2×C4⋊Dic5—31no—
106C2×C4⋊Dic5—31no—
107#4070—38no—
108C2×#683—32no—
109C2×#683—32no—
110C2×#683—32no—
111C2×C4×Dic5—31no—
112C2×C4×Dic5—32no—
113C2×C4×Dic5—31no—
114C2×C4×Dic5—32no—
115#4033—64no—
116C5×#95C2×D312yes—
117#3963—32no—
118#3963—32no—
119#3963—32no—
120#3963—32no—
121#2323—151noSylow
122C23×C3×C5—21no—
123C23×C3×C5—21no—
124C23×C3×C5—21no—
125C23×C3×C5—21no—
126C23×C3×C5—21no—
127C23×C3×C5—21no—
128C23×C3×C5C22×C412yes—
129C23×C3×C5C2411yescharacteristic
130C2×C3×C4×C5—21no—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5—28no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—22no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5C22×C412yes—
140C2×C3×C4×C5C22×C412yes—
141C2×C3×C4×C5C2×D412yes—
142C2×C3×C4×C5#4112yes—
143C2×Dic15—22no—
144C2×Dic15—22no—
145C2×Dic15—21no—
146C2×Dic15—22no—
147C2×Dic15—22no—
148C2×Dic15—21no—
149C2×Dic15—21no—
150C2×Dic15—21no—
151C2×Dic15—21no—
152C2×Dic15—21no—
153C2×Dic15—21no—
154C2×Dic15—21no—
155C2×Dic15—21no—
156C2×Dic15—21no—
157C2×Dic15C22×C414yes—
158C2×Dic15C22×C414yes—
159C2×Dic15C2×D412yes—
160C2×Dic15C2×D412yes—
161Dic30—28no—
162C23×C3×C4C2×D511yescharacteristic
163C22×Dic6—51no—
164C2×C3×C4⋊C4C2×D511yescharacteristic
165C2×C3×C4⋊C4C2×D511yescharacteristic
166C2×C3×C22⋊C4C2×D512yes—
167C2×#217—52no—
168C2×#217—51no—
169C2×#217—51no—
170#926—104no—
171C2×C4⋊Dic3—51no—
172C2×C4⋊Dic3—51no—
173C2×C4⋊Dic3—51no—
174#924—58no—
175C2×Dic3⋊C4—52no—
176C2×Dic3⋊C4—52no—
177C2×Dic3⋊C4—52no—
178C2×C4×Dic3—51no—
179C2×C4×Dic3—52no—
180C2×C4×Dic3—51no—
181C2×C4×Dic3—52no—
182#887—104no—
183C3×#95C2×D512yes—
184C3⋊#95—52no—
185C3⋊#95—52no—
186C3⋊#95—52no—
187C3⋊#95—52no—
188C24×C5C22×D311yescharacteristic
189C22×C4×C5—22no—
190C22×C4×C5—22no—
191C22×C4×C5C4×D314yes—
192C22×C4×C5C22×D312yes—
193C22×C4×C5C22×D312yes—
194C22×C4×C5C22×D311yescharacteristic
195C22×C4×C5C22×D311yescharacteristic
196C22×Dic5—32no—
197C22×Dic5—31no—
198C22×Dic5—32no—
199C22×Dic5—31no—
200C22×Dic5—31no—
201C22×Dic5—31no—
202C2×Dic10—34no—
203C2×Dic10—62no—
204C2×Dic10—62no—
205C5×C4⋊C4—22no—
206C5×C4⋊C4—22no—
207C5×C22⋊C4—42no—
208C22⋊Dic5—38no—
209C22⋊Dic5—62no—
210C22⋊Dic5—62no—
211C4⋊Dic5—34no—
212C4⋊Dic5—62no—
213C4⋊Dic5—62no—
214#683—38no—
215#683—64no—
216#683—122no—
217C4×Dic5—34no—
218C4×Dic5—64no—
219C4×Dic5—62no—
220C4×Dic5—122no—
221C2×C22⋊Q8—151no—
222C2×C4×Q8—151no—
223C2×#117—151no—
224#391—151no—
225#389—151no—
226#389—152no—
227#388—152no—
228#388—152no—
229#387—152no—
230#380D1511yescharacteristic
231C4×C4⋊C4—151no—
232C22×C3×C5—22no—
233C22×C3×C5—22no—
234C22×C3×C5—22no—
235C22×C3×C5—41no—
236C22×C3×C5—41no—
237C22×C3×C5—41no—
238C22×C3×C5—41no—
239C22×C3×C5C23×C411yescharacteristic
240C22×C3×C5C22×D411yescharacteristic
241C22×C3×C5C2×#4111yescharacteristic
242C22×C3×C5H(5,𝔽2)11yescharacteristic
243C22×C3×C52−1+411yescharacteristic
244C22×C3×C52−1+411yescharacteristic
245C22×C3×C52−1+411yescharacteristic
246C3×C4×C5—22no—
247C3×C4×C5—42no—
248C3×C4×C5—42no—
249C3×C4×C5C4×D414yes—
250Dic15—24no—
251Dic15—24no—
252Dic15—22no—
253Dic15—22no—
254Dic15—41no—
255Dic15—41no—
256C24×C3C22×D511yescharacteristic
257C22×C3×C4—22no—
258C22×C3×C4—22no—
259C22×C3×C4C4×D514yes—
260C22×C3×C4C22×D512yes—
261C22×C3×C4C22×D512yes—
262C22×C3×C4C22×D511yescharacteristic
263C22×C3×C4C22×D511yescharacteristic
264C22×Dic3—51no—
265C22×Dic3—52no—
266C22×Dic3—52no—
267C22×Dic3—51no—
268C22×Dic3—51no—
269C22×Dic3—51no—
270C2×Dic6—54no—
271C2×Dic6—102no—
272C2×Dic6—102no—
273C3×C4⋊C4—22no—
274C3×C4⋊C4—22no—
275C3×C22⋊C4—42no—
276#217—58no—
277#217—102no—
278#217—102no—
279C4⋊Dic3—54no—
280C4⋊Dic3—102no—
281C4⋊Dic3—102no—
282Dic3⋊C4—58no—
283Dic3⋊C4—104no—
284Dic3⋊C4—202no—
285C4×Dic3—54no—
286C4×Dic3—104no—
287C4×Dic3—102no—
288C4×Dic3—202no—
289C23×C5—21no—
290C23×C5—21no—
291C23×C5—21no—
292C23×C5—21no—
293C23×C5—21no—
294C23×C5—21no—
295C23×C5C2×C4×D312yes—
296C23×C5C23×D311yescharacteristic
297C2×C4×C5—21no—
298C2×C4×C5—21no—
299C2×C4×C5—28no—
300C2×C4×C5—22no—
301C2×C4×C5—22no—
302C2×C4×C5—22no—
303C2×C4×C5—22no—
304C2×C4×C5—22no—
305C2×C4×C5—22no—
306C2×C4×C5C2×C4×D312yes—
307C2×C4×C5C2×C4×D312yes—
308C2×C4×C5D3×D412yes—
309C2×C4×C5#23712yes—
310C2×Dic5—34no—
311C2×Dic5—34no—
312C2×Dic5—32no—
313C2×Dic5—32no—
314C2×Dic5—62no—
315C2×Dic5—62no—
316C2×Dic5—61no—
317C2×Dic5—62no—
318C2×Dic5—62no—
319C2×Dic5—61no—
320C2×Dic5—61no—
321C2×Dic5—61no—
322C2×Dic5—61no—
323C2×Dic5—61no—
324C2×Dic5—61no—
325C2×Dic5—61no—
326C2×Dic5—61no—
327C2×Dic5—61no—
328Dic10—68no—
329C22×Q8—151no—
330C23×C4C2×D1511yescharacteristic
331C22⋊Q8—158no—
332C4×Q8—304no—
333#117—304no—
334C2×C4⋊C4—151no—
335C2×C4⋊C4—152no—
336C2×C4⋊C4—151no—
337C2×C4⋊C4—151no—
338C2×C4⋊C4—152no—
339C2×C4⋊C4—152no—
340C2×C4⋊C4C2×D1511yescharacteristic
341C2×C4⋊C4C2×D1511yescharacteristic
342C2×C22⋊C4—152no—
343C2×C22⋊C4—151no—
344C2×C22⋊C4—151no—
345C2×C22⋊C4C2×D1512yes—
346C2×C42—151no—
347C2×C42—151no—
348C2×C42—152no—
349C2×C42—152no—
350#95—152no—
351#95—152no—
352#95—152no—
353#95—152no—
354#95C2×D1512yes—
355C2×C3×C5—42no—
356C2×C3×C5C2×C4×D411yescharacteristic
357C2×C3×C5#51911yescharacteristic
358C2×C3×C5#52011yescharacteristic
359C2×C3×C5#53611yescharacteristic
360C2×C3×C5#53711yescharacteristic
361C2×C3×C5#54111yescharacteristic
362C2×C3×C5#54211yescharacteristic
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3C2×C4×D512yes—
370C23×C3C23×D511yescharacteristic
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—28no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4C2×C4×D512yes—
381C2×C3×C4C2×C4×D512yes—
382C2×C3×C4D4×D512yes—
383C2×C3×C4#71112yes—
384C2×Dic3—54no—
385C2×Dic3—54no—
386C2×Dic3—52no—
387C2×Dic3—52no—
388C2×Dic3—102no—
389C2×Dic3—102no—
390C2×Dic3—101no—
391C2×Dic3—102no—
392C2×Dic3—102no—
393C2×Dic3—101no—
394C2×Dic3—101no—
395C2×Dic3—101no—
396C2×Dic3—101no—
397C2×Dic3—101no—
398C2×Dic3—101no—
399C2×Dic3—101no—
400C2×Dic3—101no—
401C2×Dic3—101no—
402Dic6—108no—
403C22×C5—22no—
404C22×C5—22no—
405C22×C5—22no—
406C22×C5—41no—
407C22×C5—41no—
408C22×C5—41no—
409C22×C5—41no—
410C22×C5C22×C4×D311yescharacteristic
411C22×C5C2×D3×D411yescharacteristic
412C22×C5C2×#23711yescharacteristic
413C22×C5#100411yescharacteristic
414C22×C5#100711yescharacteristic
415C22×C5#101011yescharacteristic
416C22×C5#101011yescharacteristic
417C4×C5—22no—
418C4×C5—42no—
419C4×C5—42no—
420C4×C5#67⋊C414yes—
421Dic5—64no—
422Dic5—64no—
423Dic5—62no—
424Dic5—62no—
425Dic5—121no—
426Dic5—121no—
427C24C22×D1511yescharacteristic
428C2×Q8—154no—
429C2×Q8—302no—
430C2×Q8—302no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—151no—
434C22×C4—152no—
435C22×C4—152no—
436C22×C4—151no—
437C22×C4—151no—
438C22×C4—151no—
439C22×C4C4×D1514yes—
440C22×C4C22×D1512yes—
441C22×C4C22×D1512yes—
442C22×C4C22×D1511yescharacteristic
443C22×C4C22×D1511yescharacteristic
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—158no—
447C4⋊C4—154no—
448C4⋊C4—302no—
449C4⋊C4—302no—
450C4⋊C4—304no—
451C4⋊C4—602no—
452C22⋊C4—42no—
453C22⋊C4—158no—
454C22⋊C4—302no—
455C22⋊C4—302no—
456C42—154no—
457C42—304no—
458C42—302no—
459C42—602no—
460C3×C5#232311yescharacteristic, complemented, Hall
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—41no—
465C22×C3—41no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3C22×C4×D511yescharacteristic
469C22×C3C2×D4×D511yescharacteristic
470C22×C3C2×#71111yescharacteristic
471C22×C3#414411yescharacteristic
472C22×C3#414711yescharacteristic
473C22×C3#415011yescharacteristic
474C22×C3#415011yescharacteristic
475C3×C4—22no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4#402714yes—
479Dic3—104no—
480Dic3—104no—
481Dic3—102no—
482Dic3—102no—
483Dic3—201no—
484Dic3—201no—
485C2×C5—42no—
486C2×C5C2×#67⋊C411yescharacteristic
487C2×C5#550411yescharacteristic
488C2×C5#551611yescharacteristic
489C2×C5#560311yescharacteristic
490C2×C5#560811yescharacteristic
491C2×C5#562511yescharacteristic
492C2×C5#562811yescharacteristic
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23C2×C4×D1512yes—
500C23C23×D1511yescharacteristic
501Q8—308no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—28no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—154no—
512C2×C4—154no—
513C2×C4—152no—
514C2×C4—152no—
515C2×C4—302no—
516C2×C4—302no—
517C2×C4—301no—
518C2×C4—302no—
519C2×C4—301no—
520C2×C4—302no—
521C2×C4—301no—
522C2×C4—301no—
523C2×C4—301no—
524C2×C4—301no—
525C2×C4—301no—
526C2×C4—301no—
527C2×C4—301no—
528C2×C4—301no—
529C2×C4C2×C4×D1512yes—
530C2×C4C2×C4×D1512yes—
531C2×C4D4×D1512yes—
532C2×C4#681912yes—
533C2×C3—42no—
534C2×C3C2×#402711yescharacteristic
535C2×C3#6583811yescharacteristic
536C2×C3#6585011yescharacteristic
537C2×C3#6593711yescharacteristic
538C2×C3#6594211yescharacteristic
539C2×C3#6595911yescharacteristic
540C2×C3#6596211yescharacteristic
541C5#7387111yescharacteristic, complemented, Sylow
542C22—22no—
543C22—22no—
544C22—22no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22C22×C4×D1511yescharacteristic
550C22C2×D4×D1511yescharacteristic
551C22C2×#681911yescharacteristic
552C22#9209511yescharacteristic
553C22#9209811yescharacteristic
554C22#9210111yescharacteristic
555C22#9210111yescharacteristic
556C4—22no—
557C4—42no—
558C4—42no—
559C4—302no—
560C4—302no—
561C4—304no—
562C4—304no—
563C4—601no—
564C4—601no—
565C4#9177014yes—
566C3#1060560211yescharacteristic, complemented, Sylow
567C2—42no—
568C2C2×#9177011yescharacteristic
569C2#1175626311yescharacteristic
570C2#1175627511yescharacteristic
571C2#1175636211yescharacteristic
572C2#1175636711yescharacteristic
573C2#1175638411yescharacteristic
574C2#1175638711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#380C2
socle, cosocleC23×C3×C5C24
prime radicalC24×C3×C5C23
hypercenter#380D15
hypocenterC3×C5#2323
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
3. C2×#91776C2semidirect
5. C4⋊C4×Dic15C2semidirect
6. #11751074C2semidirect
7. #11751073C2semidirect
10. #11750756C2semidirect
11. #11750748C2semidirect
25. #91813C4semidirect
460. C3×C5#2323semidirect
541. C5#73871semidirect
566. C3#10605602semidirect

Maximal subgroups

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

subgroupindex
C2×#918152
C2×#918132
C2×#917762
C3×C5×#3802
C4⋊C4×Dic152
#117510742
#117510732
#117510462
#117510452
#117507562
#117507482
#106056023
#738715