Finite Group Databasebeta

← #14323776

Subgroups — #14323776

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×#91360C211yesmaximal, characteristic
2C2×#91357C211yesmaximal, characteristic
3C2×#91340C211yesmaximal, characteristic, complemented
4#11751074C211yesmaximal, characteristic, complemented
5C5×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
6C3×#65276C211yesmaximal, characteristic
7#11748868C211yesmaximal, characteristic, complemented
8#11748803C212yesmaximal, complemented
9#11748712C212yesmaximal
10#11748705C212yesmaximal
11#11748656C212yesmaximal, complemented
12#10605493—31nomaximal, Hall
13C22×Dic30C2211yescharacteristic
14C22×C3×C4×D5C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C4×Dic15C2211yescharacteristic
17C2×C5×C4⋊Dic3C2211yescharacteristic
18C2×C4×C5×Dic3C2212yes—
19C2×C4×C5×Dic3C2211yescharacteristic
20C2×C3×#685C2212yes—
21C2×C3×C4⋊Dic5C2211yescharacteristic
22C2×Dic3⋊Dic5C2212yes—
23C2×Dic3⋊Dic5C2212yes—
24C2×#6669C2212yes—
25C2×#6668C2211yescharacteristic
26C2×#6668C2211yescharacteristic
27C2×#6665C2211yescharacteristic
28C2×#6665C2211yescharacteristic
29C2×#6665C2212yes—
30C2×Dic3×Dic5C2212yes—
31#91360C418yescomplemented
32#91357—24no—
33#91340—24no—
34#91115C2212yes—
35#91115C2212yes—
36C5×C3⋊#95C2212yes—
37C3×#3963C2212yes—
38#90994C2212yes—
39#73871—51nomaximal, Hall
40C2×#4043—31no—
41C2×#4018—31no—
42C2×C4×Dic10—31no—
43C4×C5×C4⋊C4—31no—
44#65276D311yescharacteristic
45#65274—31no—
46#65263—31no—
47#65229—32no—
48#65217—32no—
49#64959—32no—
50#64942—32no—
51C23×C3×D5C2311yescharacteristic
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×Dic15C2311yescharacteristic
55C22×Dic15C2311yescharacteristic
56C22×Dic15C2312yes—
57C2×Dic30—22no—
58C2×Dic30—22no—
59C2×Dic30C2×C414yes—
60C22×C5×Dic3C2312yes—
61C22×C5×Dic3C2311yescharacteristic
62C22×C5×Dic3C2311yescharacteristic
63C22×C3×Dic5C2312yes—
64C22×C3×Dic5C2311yescharacteristic
65C2×C3×C4×D5—22no—
66C2×C3×C4×D5—22no—
67C2×C3×C4×D5C2×C414yes—
68C3×C5×C4⋊C4—22no—
69C4×Dic15—22no—
70C5×C4⋊Dic3C2×C414yes—
71C4×C5×Dic3—42no—
72C4×C5×Dic3D414yes—
73C3×#685—42no—
74C3×C4⋊Dic5—22no—
75Dic3⋊Dic5—42no—
76Dic3⋊Dic5C2×C418yes—
77#6669—24no—
78#6668—22no—
79#6668—22no—
80#6665—22no—
81#6665—22no—
82#6665C2×C418yes—
83Dic3×Dic5—24no—
84C2×#926—51no—
85C2×#924—51no—
86C2×#887—51no—
87C3×#380—51no—
88Dic3×C4⋊C4D511yescharacteristic, complemented
89#4973—51no—
90#4972—51no—
91#4945—52no—
92#4944—52no—
93#4655—52no—
94#4647—52no—
95C22×C4×D5C2×D311yescharacteristic
96C22×Dic10—31no—
97C2×C5×C4⋊C4—31no—
98C2×C5×C4⋊C4C2×D311yescharacteristic
99C2×C42×C5—31no—
100C2×C42×C5—32no—
101C2×#685—32no—
102C2×#685—31no—
103C2×#685—31no—
104C2×#685C2×D312yes—
105C2×C4⋊Dic5—32no—
106C2×C4⋊Dic5—32no—
107C2×C4⋊Dic5C2×D311yescharacteristic
108C2×#683—32no—
109C2×#683—31no—
110C2×#683—31no—
111C2×C4×Dic5—31no—
112C2×C4×Dic5—32no—
113#4043—38no—
114#4018—64no—
115C4×Dic10—64no—
116C5×#95—32no—
117#3963—32no—
118#3963—32no—
119#3963—32no—
120#3963C2×D312yes—
121#2323—151noSylow
122C23×C3×C5C2411yescharacteristic
123C22×C3×D5—21no—
124C22×C3×D5—21no—
125C22×C3×D5—21no—
126C22×C3×D5—21no—
127C22×C3×D5—21no—
128C22×C3×D5—21no—
129C22×C3×D5C22×C412yes—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5C22×C412yes—
134C2×C3×C4×C5C2×D412yes—
135C2×C3×C4×C5#4112yes—
136C2×Dic15—22no—
137C2×Dic15—22no—
138C2×Dic15—21no—
139C2×Dic15—21no—
140C2×Dic15—21no—
141C2×Dic15—21no—
142C2×Dic15—21no—
143C2×Dic15—21no—
144C2×Dic15C22×C414yes—
145Dic30—28no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3—21no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—21no—
150C2×C5×Dic3—21no—
151C2×C5×Dic3—21no—
152C2×C5×Dic3C22×C414yes—
153C2×C5×Dic3C2×D412yes—
154C2×C5×Dic3C2×D412yes—
155C2×C3×Dic5—21no—
156C2×C3×Dic5—21no—
157C2×C3×Dic5—22no—
158C2×C3×Dic5—22no—
159C2×C3×Dic5—22no—
160C2×C3×Dic5C22×C412yes—
161C3×C4×D5—28no—
162C23×C3×C4—51no—
163C22×Dic6—51no—
164C2×C3×C4⋊C4—51no—
165C2×C3×C4⋊C4C2×D511yescharacteristic
166C2×C3×C22⋊C4—52no—
167C2×#217—52no—
168C2×#217—51no—
169C2×#217—51no—
170#926—104no—
171C2×C4⋊Dic3—51no—
172C2×C4⋊Dic3—51no—
173C2×C4⋊Dic3C2×D511yescharacteristic
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×Dic3C2×D512yes—
181C2×C4×Dic3C2×D511yescharacteristic
182#887—104no—
183C3×#95—52no—
184C3⋊#95—52no—
185C3⋊#95—52no—
186C3⋊#95—52no—
187C3⋊#95C2×D512yes—
188C23×D5C22×D311yescharacteristic
189C22×C4×C5—31no—
190C22×C4×C5—32no—
191C22×C4×C5—31no—
192C22×C4×C5C22×D312yes—
193C22×C4×C5C22×D311yescharacteristic
194C22×Dic5—32no—
195C22×Dic5—31no—
196C22×Dic5—31no—
197C22×Dic5C22×D312yes—
198C22×Dic5C22×D311yescharacteristic
199C2×C4×D5—22no—
200C2×C4×D5—22no—
201C2×C4×D5C4×D314yes—
202C2×Dic10—34no—
203C2×Dic10—62no—
204C2×Dic10—62no—
205C5×C4⋊C4—22no—
206C5×C4⋊C4—34no—
207C42×C5—34no—
208C42×C5—122no—
209#685—38no—
210#685—42no—
211#685—62no—
212#685—62no—
213C4⋊Dic5—22no—
214C4⋊Dic5—38no—
215C4⋊Dic5—122no—
216#683—62no—
217#683—62no—
218#683—64no—
219C4×Dic5—64no—
220C4×Dic5—62no—
221C2×C22⋊Q8—151no—
222C2×C4×Q8—151no—
223C2×#117—151no—
224#391—151no—
225#389—152no—
226#389—151no—
227#388—152no—
228#388—152no—
229#387—152no—
230#380—51no—
231C4×C4⋊C4—31no—
232C22×C3×C5C23×C411yescharacteristic
233C22×C3×C5C22×D411yescharacteristic
234C22×C3×C5C2×#4111yescharacteristic
235C22×C3×C5H(5,𝔽2)11yescharacteristic
236C22×C3×C52−1+411yescharacteristic
237C22×C3×C52−1+411yescharacteristic
238C22×C3×C52−1+411yescharacteristic
239C2×C3×D5—22no—
240C2×C3×D5—22no—
241C2×C3×D5—22no—
242C2×C3×D5—41no—
243C2×C3×D5—41no—
244C2×C3×D5—41no—
245C2×C3×D5—41no—
246C3×C4×C5—42no—
247C3×C4×C5C4×D414yes—
248Dic15—24no—
249Dic15—41no—
250Dic15—41no—
251C3×Dic5—22no—
252C3×Dic5—42no—
253C5×Dic3—24no—
254C5×Dic3—22no—
255C5×Dic3—22no—
256C24×C3—51no—
257C22×C3×C4—51no—
258C22×C3×C4—54no—
259C22×C3×C4—52no—
260C22×C3×C4—102no—
261C22×C3×C4—102no—
262C22×C3×C4C22×D512yes—
263C22×C3×C4C22×D511yescharacteristic
264C22×Dic3—52no—
265C22×Dic3—51no—
266C22×Dic3—51no—
267C22×Dic3C22×D512yes—
268C22×Dic3C22×D511yescharacteristic
269C22×Dic3C22×D511yescharacteristic
270C2×Dic6—54no—
271C2×Dic6—102no—
272C2×Dic6—102no—
273C3×C4⋊C4—22no—
274C3×C4⋊C4—102no—
275C3×C22⋊C4—202no—
276#217—58no—
277#217—102no—
278#217—102no—
279C4⋊Dic3—102no—
280C4⋊Dic3—102no—
281C4⋊Dic3C4×D514yes—
282Dic3⋊C4—58no—
283Dic3⋊C4—104no—
284Dic3⋊C4—202no—
285C4×Dic3—42no—
286C4×Dic3—102no—
287C4×Dic3—104no—
288C4×Dic3D2014yes—
289C23×C5C23×D311yescharacteristic
290C22×D5—21no—
291C22×D5—21no—
292C22×D5—21no—
293C22×D5—21no—
294C22×D5—21no—
295C22×D5—21no—
296C22×D5C2×C4×D312yes—
297C2×C4×C5—22no—
298C2×C4×C5—22no—
299C2×C4×C5—22no—
300C2×C4×C5—34no—
301C2×C4×C5—32no—
302C2×C4×C5—32no—
303C2×C4×C5—62no—
304C2×C4×C5—61no—
305C2×C4×C5—62no—
306C2×C4×C5—61no—
307C2×C4×C5—61no—
308C2×C4×C5—61no—
309C2×C4×C5C2×C4×D312yes—
310C2×C4×C5D3×D412yes—
311C2×C4×C5#23712yes—
312C2×Dic5—21no—
313C2×Dic5—21no—
314C2×Dic5—22no—
315C2×Dic5—22no—
316C2×Dic5—22no—
317C2×Dic5—34no—
318C2×Dic5—62no—
319C2×Dic5—62no—
320C2×Dic5—61no—
321C2×Dic5—61no—
322C2×Dic5—61no—
323C2×Dic5—61no—
324C2×Dic5—61no—
325C2×Dic5—61no—
326C2×Dic5C2×C4×D312yes—
327C4×D5—28no—
328Dic10—68no—
329C22×Q8—151no—
330C23×C4—51no—
331C22⋊Q8—158no—
332C4×Q8—304no—
333#117—304no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—51no—
336C2×C4⋊C4—152no—
337C2×C4⋊C4—151no—
338C2×C4⋊C4—151no—
339C2×C4⋊C4—152no—
340C2×C4⋊C4—152no—
341C2×C4⋊C4D3×D511yescharacteristic
342C2×C22⋊C4—52no—
343C2×C22⋊C4—152no—
344C2×C22⋊C4—151no—
345C2×C22⋊C4—151no—
346C2×C42—31no—
347C2×C42—32no—
348C2×C42—152no—
349C2×C42—151no—
350#95—32no—
351#95—52no—
352#95—152no—
353#95—152no—
354#95—152no—
355C2×C3×C5C2×C4×D411yescharacteristic
356C2×C3×C5#51911yescharacteristic
357C2×C3×C5#52011yescharacteristic
358C2×C3×C5#53611yescharacteristic
359C2×C3×C5#53711yescharacteristic
360C2×C3×C5#54111yescharacteristic
361C2×C3×C5#54211yescharacteristic
362C3×D5—42no—
363C23×C3—52no—
364C23×C3—101no—
365C23×C3—101no—
366C23×C3—101no—
367C23×C3—101no—
368C23×C3—101no—
369C23×C3—101no—
370C23×C3C23×D511yescharacteristic
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—52no—
375C2×C3×C4—101no—
376C2×C3×C4—101no—
377C2×C3×C4—108no—
378C2×C3×C4—102no—
379C2×C3×C4—102no—
380C2×C3×C4—102no—
381C2×C3×C4C2×C4×D512yes—
382C2×C3×C4C2×D2012yes—
383C2×C3×C4#70912yes—
384C2×Dic3—21no—
385C2×Dic3—22no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—22no—
390C2×Dic3—54no—
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×Dic3C2×C4×D514yes—
400C2×Dic3C2×D2012yes—
401C2×Dic3C2×D2012yes—
402Dic6—108no—
403C22×C5C22×C4×D311yescharacteristic
404C22×C5C2×D3×D411yescharacteristic
405C22×C5C2×#23711yescharacteristic
406C22×C5#100411yescharacteristic
407C22×C5#100711yescharacteristic
408C22×C5#101011yescharacteristic
409C22×C5#101011yescharacteristic
410C2×D5—22no—
411C2×D5—22no—
412C2×D5—22no—
413C2×D5—41no—
414C2×D5—41no—
415C2×D5—41no—
416C2×D5—41no—
417C4×C5—42no—
418C4×C5—64no—
419C4×C5—62no—
420C4×C5—62no—
421C4×C5#67⋊C414yes—
422Dic5—22no—
423Dic5—42no—
424Dic5—64no—
425Dic5—121no—
426Dic5—121no—
427C24—51no—
428C2×Q8—154no—
429C2×Q8—302no—
430C2×Q8—302no—
431C22×C4—31no—
432C22×C4—32no—
433C22×C4—31no—
434C22×C4—51no—
435C22×C4—54no—
436C22×C4—52no—
437C22×C4—102no—
438C22×C4—102no—
439C22×C4—152no—
440C22×C4—151no—
441C22×C4—151no—
442C22×C4C2×D3×D512yes—
443C22×C4C2×D3×D511yescharacteristic
444C4⋊C4—22no—
445C4⋊C4—34no—
446C4⋊C4—102no—
447C4⋊C4—158no—
448C4⋊C4—302no—
449C4⋊C4—302no—
450C4⋊C4—304no—
451C4⋊C4—602no—
452C22⋊C4—158no—
453C22⋊C4—202no—
454C22⋊C4—302no—
455C22⋊C4—302no—
456C42—34no—
457C42—122no—
458C42—304no—
459C42—302no—
460C3×C5#232311yescharacteristic, complemented, Hall
461C22×C3—102no—
462C22×C3—102no—
463C22×C3—102no—
464C22×C3—201no—
465C22×C3—201no—
466C22×C3—201no—
467C22×C3—201no—
468C22×C3C22×C4×D511yescharacteristic
469C22×C3C22×D2011yescharacteristic
470C22×C3C2×#70911yescharacteristic
471C22×C3#414411yescharacteristic
472C22×C3#414711yescharacteristic
473C22×C3#415011yescharacteristic
474C22×C3#415011yescharacteristic
475C3×C4—42no—
476C3×C4—102no—
477C3×C4—202no—
478C3×C4C4×D2014yes—
479Dic3—22no—
480Dic3—22no—
481Dic3—24no—
482Dic3—104no—
483Dic3—201no—
484Dic3—201no—
485C2×C5C2×#67⋊C411yescharacteristic
486C2×C5#550411yescharacteristic
487C2×C5#551611yescharacteristic
488C2×C5#560311yescharacteristic
489C2×C5#560811yescharacteristic
490C2×C5#562511yescharacteristic
491C2×C5#562811yescharacteristic
492D5—42no—
493C23—52no—
494C23—101no—
495C23—101no—
496C23—101no—
497C23—101no—
498C23—101no—
499C23—101no—
500C23C22×D3×D511yescharacteristic
501Q8—308no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—34no—
506C2×C4—32no—
507C2×C4—32no—
508C2×C4—52no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—62no—
512C2×C4—61no—
513C2×C4—62no—
514C2×C4—61no—
515C2×C4—101no—
516C2×C4—101no—
517C2×C4—108no—
518C2×C4—102no—
519C2×C4—102no—
520C2×C4—102no—
521C2×C4—154no—
522C2×C4—302no—
523C2×C4—302no—
524C2×C4—301no—
525C2×C4—301no—
526C2×C4—301no—
527C2×C4—301no—
528C2×C4—301no—
529C2×C4—301no—
530C2×C4C4×D3×D512yes—
531C2×C4D3×D2012yes—
532C2×C4#677912yes—
533C2×C3—202no—
534C2×C3C2×C4×D2011yescharacteristic
535C2×C3#6584411yescharacteristic
536C2×C3#6586011yescharacteristic
537C2×C3#6588011yescharacteristic
538C2×C3#6589511yescharacteristic
539C2×C3#6590611yescharacteristic
540C2×C3#6591711yescharacteristic
541C5#7387111yescharacteristic, complemented, Sylow
542C22—102no—
543C22—102no—
544C22—102no—
545C22—201no—
546C22—201no—
547C22—201no—
548C22—201no—
549C22C2×C4×D3×D511yescharacteristic
550C22C2×D3×D2011yescharacteristic
551C22#9202011yescharacteristic
552C22#9202511yescharacteristic
553C22#9202811yescharacteristic
554C22#9203011yescharacteristic
555C22C2×#677911yescharacteristic
556C4—42no—
557C4—64no—
558C4—62no—
559C4—62no—
560C4—102no—
561C4—202no—
562C4—304no—
563C4—601no—
564C4—601no—
565C4#9139514yes—
566C3#1060549311yescharacteristic, complemented, Sylow
567C2—202no—
568C2#1175339711yescharacteristic
569C2#1175344111yescharacteristic
570C2#1175367111yescharacteristic
571C2#1175369511yescharacteristic
572C2C2×#9139511yescharacteristic
573C2#1175475011yescharacteristic
574C2#1175514111yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC23×C3×D5C23
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2323
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#91340C2semidirect
4. #11751074C2semidirect
5. C5×Dic3×C4⋊C4C2semidirect
7. #11748868C2semidirect
8. #11748803C2semidirect
11. #11748656C2semidirect
31. #91360C4semidirect
88. Dic3×C4⋊C4D5semidirect
460. C3×C5#2323semidirect
541. C5#73871semidirect
566. C3#10605493semidirect

Maximal subgroups

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

subgroupindex
C2×#913602
C2×#913572
C2×#913402
#117510742
C5×Dic3×C4⋊C42
C3×#652762
#117488682
#117488032
#117487122
#117487052
#117486562
#106054933
#738715