Finite Group Databasebeta

← #14322617

Subgroups — #14322617

Order 1920.

1344 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6667C211yesmaximal, characteristic
2#11751056C211yesmaximal, characteristic
3C5×Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
4C3×#65231C211yesmaximal, characteristic, complemented
5#11749506C211yesmaximal, characteristic, complemented
6#11749502C211yesmaximal, characteristic
7Dic3×C22⋊Dic5C211yesmaximal, characteristic, complemented
8#11748730C212yesmaximal, complemented
9#11748728C212yesmaximal, complemented
10#11748671C212yesmaximal, complemented
11#11748666C212yesmaximal, complemented
12#10605298—31nomaximal, Hall
13C24×D15C2211yescharacteristic
14C23×C5×Dic3C2211yescharacteristic
15C23×C3×Dic5C2211yescharacteristic
16C2×C3×C5×C22⋊C4C2211yescharacteristic
17C2×#6719C2211yescharacteristic
18C2×#6714C2212yes—
19C2×#6714C2212yes—
20C2×C5×#217C2211yescharacteristic
21C2×C4×C5×Dic3C2212yescomplemented
22C2×C3×C22⋊Dic5C2211yescharacteristic
23C2×C3×#683C2212yescomplemented
24C2×#6668C2212yescomplemented
25C2×#6667C418yescomplemented
26C2×#6667C2212yes—
27C2×#6667C2212yes—
28C2×#6667C2211yescharacteristic
29C2×#6667C2211yescharacteristic
30C2×#6667C2211yescharacteristic
31C2×#6667C2211yescharacteristic
32C2×Dic3×Dic5C2212yescomplemented
33C5×C3⋊#95C2212yescomplemented
34C3×#3963C2212yescomplemented
35#90994C2212yescomplemented
36#90994C2212yescomplemented
37#74004—51nomaximal, Hall
38C22×#685—31no—
39C4×C5×C22⋊C4—31no—
40C4×C22⋊Dic5—31no—
41#65244—31no—
42#65244—31no—
43#65231—31no—
44#65231D311yescharacteristic, complemented
45#65229—32no—
46#65229—32no—
47#64958—32no—
48#64958—32no—
49C24×C3×C5C2311yescharacteristic
50C23×D15—22no—
51C23×D15—22no—
52C23×D15—42no—
53C23×D15—42no—
54C23×D15C2×C414yes—
55C23×D15C2311yescharacteristic
56C23×D15C2311yescharacteristic
57C22×C3×C4×C5C2312yes—
58C22×Dic15C2312yes—
59C22×C5×Dic3—42no—
60C22×C5×Dic3C2×C414yes—
61C22×C5×Dic3C2312yes—
62C22×C5×Dic3C2311yescharacteristic
63C22×C5×Dic3C2311yescharacteristic
64C22×C3×Dic5—42no—
65C22×C3×Dic5C2×C414yes—
66C22×C3×Dic5C2312yes—
67C22×C3×Dic5C2311yescharacteristic
68C22×C3×Dic5C2311yescharacteristic
69C3×C5×C22⋊C4—22no—
70#6719—22no—
71#6714—42no—
72#6714—42no—
73C5×#217—22no—
74C4×C5×Dic3—42no—
75C3×C22⋊Dic5—22no—
76C3×#683—42no—
77#6668—42no—
78#6667—42no—
79#6667—42no—
80#6667C2×C414yes—
81#6667C2×C414yes—
82#6667C2×C414yes—
83#6667C2×C414yes—
84Dic3×Dic5—42no—
85C22×(C2×D3)⋊C4—51no—
86C3×#380—51no—
87#5212—51no—
88#4990—52no—
89#4990—52no—
90#4955—51no—
91#4955—51no—
92Dic3×C22⋊C4—51no—
93Dic3×C22⋊C4D511yescharacteristic, complemented
94#4670—52no—
95#4670—52no—
96C24×D5—31no—
97C23×C4×C5—31no—
98C23×Dic5C2×D311yescharacteristic
99C2×C5×C22⋊C4—31no—
100C2×C5×C22⋊C4C2×D311yescharacteristic
101C2×C42×C5—32no—
102C2×C22⋊Dic5—31no—
103C2×C22⋊Dic5C2×D311yescharacteristic
104C2×#685—38no—
105C2×#685—31no—
106C2×#685—31no—
107C2×#685—31no—
108C2×#685—31no—
109C2×#685—32no—
110C2×#685—32no—
111C2×#685—32no—
112C2×#685—32no—
113C2×#683—32no—
114C2×#683C2×D312yescomplemented
115C2×C4×Dic5—32no—
116C5×#95—32no—
117#3963—32no—
118#3963—32no—
119#3963C2×D312yescomplemented
120#2312—151noSylow
121C23×C3×C5—22no—
122C23×C3×C5—22no—
123C23×C3×C5C22×C412yes—
124C23×C3×C5#4112yes—
125C23×C3×C5#4112yes—
126C23×C3×C5C2411yescharacteristic
127C22×D15—28no—
128C22×D15—21no—
129C22×D15—21no—
130C22×D15—21no—
131C22×D15—21no—
132C22×D15—22no—
133C22×D15—22no—
134C22×D15—22no—
135C22×D15—22no—
136C22×D15—44no—
137C22×D15—44no—
138C22×D15—48no—
139C22×D15—48no—
140C22×D15C22×C412yes—
141C22×D15C22×C412yes—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5—22no—
145C2×Dic15—22no—
146C2×Dic15—22no—
147C2×Dic15—22no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—22no—
150C2×C5×Dic3—22no—
151C2×C5×Dic3—22no—
152C2×C5×Dic3—22no—
153C2×C5×Dic3—44no—
154C2×C5×Dic3C22×C412yes—
155C2×C5×Dic3C22×C412yes—
156C2×C3×Dic5—22no—
157C2×C3×Dic5—22no—
158C2×C3×Dic5—22no—
159C2×C3×Dic5—22no—
160C2×C3×Dic5—22no—
161C2×C3×Dic5—44no—
162C2×C3×Dic5C22×C412yes—
163C2×C3×Dic5C22×C412yes—
164C24×D3—51no—
165C23×C3×C4—51no—
166C23×Dic3C2×D511yescharacteristic
167C2×C3×C4⋊C4—52no—
168C2×C3×C22⋊C4—51no—
169C2×C3×C22⋊C4C2×D511yescharacteristic
170C2×#217—51no—
171C2×#217C2×D511yescharacteristic
172C2×(C2×D3)⋊C4—58no—
173C2×(C2×D3)⋊C4—51no—
174C2×(C2×D3)⋊C4—51no—
175C2×(C2×D3)⋊C4—51no—
176C2×(C2×D3)⋊C4—51no—
177C2×(C2×D3)⋊C4—52no—
178C2×(C2×D3)⋊C4—52no—
179C2×(C2×D3)⋊C4—52no—
180C2×(C2×D3)⋊C4—52no—
181C2×C4⋊Dic3—52no—
182C2×C4×Dic3—52no—
183C2×C4×Dic3C2×D512yescomplemented
184C3×#95—52no—
185C3⋊#95—52no—
186C3⋊#95—52no—
187C3⋊#95C2×D512yescomplemented
188C24×C5C22×D311yescharacteristic
189C23×D5—31no—
190C23×D5—31no—
191C23×D5—34no—
192C23×D5—62no—
193C23×D5—62no—
194C23×D5—122no—
195C23×D5—122no—
196C22×C4×C5—31no—
197C22×C4×C5—31no—
198C22×C4×C5—34no—
199C22×C4×C5—32no—
200C22×C4×C5—122no—
201C22×C4×C5C22×D312yes—
202C22×Dic5—32no—
203C22×Dic5—42no—
204C22×Dic5C4×D314yes—
205C22×Dic5C22×D312yes—
206C22×Dic5C22×D311yescharacteristic
207C22×Dic5C22×D311yescharacteristic
208C5×C22⋊C4—22no—
209C5×C22⋊C4—62no—
210C42×C5—122no—
211C22⋊Dic5—22no—
212C22⋊Dic5—62no—
213#685—34no—
214#685—34no—
215#685—34no—
216#685—34no—
217#685—122no—
218#685—122no—
219#685—122no—
220#685—122no—
221#683—42no—
222#683—122no—
223C4×Dic5—122no—
224C22×C22⋊C4—151no—
225#388—152no—
226#388—152no—
227#388—152no—
228#388—152no—
229#380—51no—
230#380—151no—
231#379—151no—
232#379—151no—
233C4×C22⋊C4—31no—
234C4×C22⋊C4—151no—
235C22×C3×C5—22no—
236C22×C3×C5—22no—
237C22×C3×C5—24no—
238C22×C3×C5—24no—
239C22×C3×C5#11714yes—
240C22×C3×C5C23×C411yescharacteristic
241C22×C3×C5C2×#4111yescharacteristic
242C22×C3×C5C2×#4111yescharacteristic
243C22×C3×C5H(5,𝔽2)12yes—
244C22×C3×C5H(5,𝔽2)12yes—
245C2×D15—22no—
246C2×D15—22no—
247C2×D15—22no—
248C2×D15—22no—
249C2×D15—22no—
250C2×D15—22no—
251C2×D15—416no—
252C2×D15—42no—
253C2×D15—42no—
254C2×D15—42no—
255C2×D15—42no—
256C3×C4×C5—42no—
257Dic15—42no—
258C3×Dic5—22no—
259C3×Dic5—22no—
260C3×Dic5—42no—
261C5×Dic3—22no—
262C5×Dic3—22no—
263C5×Dic3—42no—
264C24×C3C22×D511yescharacteristic
265C23×D3—51no—
266C23×D3—51no—
267C23×D3—54no—
268C23×D3—102no—
269C23×D3—102no—
270C23×D3—202no—
271C23×D3—202no—
272C22×C3×C4—51no—
273C22×C3×C4—51no—
274C22×C3×C4—54no—
275C22×C3×C4—52no—
276C22×C3×C4—202no—
277C22×C3×C4C22×D512yes—
278C22×Dic3—42no—
279C22×Dic3—52no—
280C22×Dic3C4×D514yes—
281C22×Dic3C22×D512yes—
282C22×Dic3C22×D511yescharacteristic
283C22×Dic3C22×D511yescharacteristic
284C3×C4⋊C4—202no—
285C3×C22⋊C4—22no—
286C3×C22⋊C4—102no—
287#217—22no—
288#217—102no—
289(C2×D3)⋊C4—54no—
290(C2×D3)⋊C4—54no—
291(C2×D3)⋊C4—54no—
292(C2×D3)⋊C4—54no—
293(C2×D3)⋊C4—202no—
294(C2×D3)⋊C4—202no—
295(C2×D3)⋊C4—202no—
296(C2×D3)⋊C4—202no—
297C4⋊Dic3—202no—
298C4×Dic3—42no—
299C4×Dic3—202no—
300C23×C5—22no—
301C23×C5—22no—
302C23×C5C2×C4×D312yes—
303C23×C5#23712yes—
304C23×C5#23912yes—
305C23×C5C23×D311yescharacteristic
306C22×D5—32no—
307C22×D5—32no—
308C22×D5—68no—
309C22×D5—61no—
310C22×D5—61no—
311C22×D5—62no—
312C22×D5—62no—
313C22×D5—61no—
314C22×D5—61no—
315C22×D5—62no—
316C22×D5—62no—
317C22×D5—124no—
318C22×D5—124no—
319C22×D5—128no—
320C22×D5—128no—
321C2×C4×C5—22no—
322C2×C4×C5—22no—
323C2×C4×C5—22no—
324C2×C4×C5—32no—
325C2×C4×C5—32no—
326C2×C4×C5—62no—
327C2×C4×C5—62no—
328C2×C4×C5—62no—
329C2×C4×C5—62no—
330C2×C4×C5—62no—
331C2×C4×C5—124no—
332C2×Dic5—22no—
333C2×Dic5—22no—
334C2×Dic5—22no—
335C2×Dic5—22no—
336C2×Dic5—22no—
337C2×Dic5—44no—
338C2×Dic5—62no—
339C2×Dic5—62no—
340C2×Dic5—62no—
341C2×Dic5C2×C4×D312yes—
342C2×Dic5C2×C4×D312yes—
343C25—151no—
344C23×C4—31no—
345C23×C4—51no—
346C2×C4⋊C4—52no—
347C2×C4⋊C4—152no—
348C2×C22⋊C4—31no—
349C2×C22⋊C4—51no—
350C2×C22⋊C4—151no—
351C2×C22⋊C4—151no—
352C2×C22⋊C4—151no—
353C2×C22⋊C4—151no—
354C2×C22⋊C4—158no—
355C2×C22⋊C4—151no—
356C2×C22⋊C4—152no—
357C2×C22⋊C4—152no—
358C2×C22⋊C4—152no—
359C2×C22⋊C4—152no—
360C2×C22⋊C4D3×D511yescharacteristic
361C2×C42—32no—
362C2×C42—152no—
363#95—32no—
364#95—52no—
365#95—152no—
366#95—152no—
367C2×C3×C5—24no—
368C2×C3×C5C2×#11711yescharacteristic
369C2×C3×C5#51811yescharacteristic
370C2×C3×C5#51811yescharacteristic
371C2×C3×C5#53812yes—
372C2×C3×C5#53812yes—
373D15—42no—
374D15—42no—
375C23×C3—22no—
376C23×C3—22no—
377C23×C3C2×C4×D512yes—
378C23×C3#70912yes—
379C23×C3#70912yes—
380C23×C3C23×D511yescharacteristic
381C22×D3—52no—
382C22×D3—52no—
383C22×D3—101no—
384C22×D3—101no—
385C22×D3—101no—
386C22×D3—101no—
387C22×D3—102no—
388C22×D3—102no—
389C22×D3—102no—
390C22×D3—102no—
391C22×D3—108no—
392C22×D3—204no—
393C22×D3—204no—
394C22×D3—208no—
395C22×D3—208no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—52no—
400C2×C3×C4—52no—
401C2×C3×C4—102no—
402C2×C3×C4—102no—
403C2×C3×C4—102no—
404C2×C3×C4—102no—
405C2×C3×C4—102no—
406C2×C3×C4—204no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—44no—
413C2×Dic3—102no—
414C2×Dic3—102no—
415C2×Dic3—102no—
416C2×Dic3C2×C4×D512yes—
417C2×Dic3C2×C4×D512yes—
418C22×C5—24no—
419C22×C5—22no—
420C22×C5—22no—
421C22×C5—24no—
422C22×C5#89214yes—
423C22×C5C22×C4×D311yescharacteristic
424C22×C5C2×#23711yescharacteristic
425C22×C5#100412yes—
426C22×C5C2×#23911yescharacteristic
427C22×C5#100912yes—
428C2×D5—62no—
429C2×D5—62no—
430C2×D5—62no—
431C2×D5—62no—
432C2×D5—62no—
433C2×D5—62no—
434C2×D5—1216no—
435C2×D5—122no—
436C2×D5—122no—
437C2×D5—122no—
438C2×D5—122no—
439C4×C5—42no—
440C4×C5—62no—
441C4×C5—62no—
442C4×C5—122no—
443Dic5—22no—
444Dic5—22no—
445Dic5—42no—
446Dic5—122no—
447C24—151no—
448C24—151no—
449C24—154no—
450C24—302no—
451C24—302no—
452C24—602no—
453C24—602no—
454C24C2×D3×D511yescharacteristic
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—34no—
458C22×C4—32no—
459C22×C4—51no—
460C22×C4—51no—
461C22×C4—54no—
462C22×C4—52no—
463C22×C4—122no—
464C22×C4—152no—
465C22×C4—202no—
466C22×C4C2×D3×D512yes—
467C4⋊C4—202no—
468C4⋊C4—602no—
469C22⋊C4—22no—
470C22⋊C4—62no—
471C22⋊C4—102no—
472C22⋊C4—154no—
473C22⋊C4—154no—
474C22⋊C4—154no—
475C22⋊C4—154no—
476C22⋊C4—302no—
477C22⋊C4—602no—
478C22⋊C4—602no—
479C22⋊C4—602no—
480C22⋊C4—602no—
481C42—122no—
482C42—602no—
483C3×C5#231211yescharacteristic, complemented, Hall
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—24no—
487C22×C3—24no—
488C22×C3#401814yes—
489C22×C3C22×C4×D511yescharacteristic
490C22×C3C2×#70911yescharacteristic
491C22×C3C2×#70911yescharacteristic
492C22×C3#414412yes—
493C22×C3#414912yes—
494C2×D3—102no—
495C2×D3—102no—
496C2×D3—102no—
497C2×D3—102no—
498C2×D3—102no—
499C2×D3—102no—
500C2×D3—2016no—
501C2×D3—202no—
502C2×D3—202no—
503C2×D3—202no—
504C2×D3—202no—
505C3×C4—42no—
506C3×C4—102no—
507C3×C4—102no—
508C3×C4—202no—
509Dic3—22no—
510Dic3—22no—
511Dic3—42no—
512Dic3—202no—
513C2×C5—24no—
514C2×C5#548611yescharacteristic
515C2×C5C2×#89211yescharacteristic
516C2×C5#552111yescharacteristic
517C2×C5#561712yes—
518C2×C5#564412yes—
519D5—122no—
520D5—122no—
521C23—22no—
522C23—22no—
523C23—152no—
524C23—152no—
525C23—308no—
526C23—301no—
527C23—301no—
528C23—302no—
529C23—302no—
530C23—302no—
531C23—302no—
532C23—301no—
533C23—301no—
534C23—604no—
535C23—604no—
536C23—608no—
537C23—608no—
538C23#676912yes—
539C23C4×D3×D512yes—
540C23#677912yes—
541C23C22×D3×D511yescharacteristic
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—32no—
546C2×C4—32no—
547C2×C4—52no—
548C2×C4—52no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—102no—
555C2×C4—102no—
556C2×C4—102no—
557C2×C4—102no—
558C2×C4—102no—
559C2×C4—124no—
560C2×C4—204no—
561C2×C4—302no—
562C2×C4—302no—
563C2×C4—302no—
564C2×C3—24no—
565C2×C3C2×#401811yescharacteristic
566C2×C3#6582011yescharacteristic
567C2×C3#6585511yescharacteristic
568C2×C3#6589412yes—
569C2×C3#6589412yes—
570D3—202no—
571D3—202no—
572C5#7400411yescharacteristic, complemented, Sylow
573C22—22no—
574C22—22no—
575C22—24no—
576C22—24no—
577C22—302no—
578C22—302no—
579C22—302no—
580C22—302no—
581C22—302no—
582C22—302no—
583C22—6016no—
584C22—602no—
585C22—602no—
586C22—602no—
587C22—602no—
588C22#9140314yes—
589C22C2×#676911yescharacteristic
590C22C2×C4×D3×D511yescharacteristic
591C22#9202612yes—
592C22#9202712yes—
593C22C2×#677911yescharacteristic
594C4—42no—
595C4—62no—
596C4—62no—
597C4—102no—
598C4—102no—
599C4—122no—
600C4—202no—
601C4—602no—
602C3#1060529811yescharacteristic, complemented, Sylow
603C2—24no—
604C2—602no—
605C2—602no—
606C2#1175306611yescharacteristic
607C2#1175306711yescharacteristic
608C2#1175318312yes—
609C2C2×#9140311yescharacteristic
610C2#1175491912yes—
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C22⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC24×D15C22
hypercenterC2×C22⋊C4D3×D5
hypocenterC3×C5#2312
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C5×Dic3×C22⋊C4C2semidirect
4. C3×#65231C2semidirect
5. #11749506C2semidirect
7. Dic3×C22⋊Dic5C2semidirect
8. #11748730C2semidirect
9. #11748728C2semidirect
10. #11748671C2semidirect
11. #11748666C2semidirect
21. C2×C4×C5×Dic3C22semidirect
23. C2×C3×#683C22semidirect
24. C2×#6668C22semidirect
25. C2×#6667C4semidirect
32. C2×Dic3×Dic5C22semidirect
33. C5×C3⋊#95C22semidirect
34. C3×#3963C22semidirect
35. #90994C22semidirect
36. #90994C22semidirect
44. #65231D3semidirect
93. Dic3×C22⋊C4D5semidirect
114. C2×#683C2×D3semidirect
119. #3963C2×D3semidirect
183. C2×C4×Dic3C2×D5semidirect
187. C3⋊#95C2×D5semidirect
483. C3×C5#2312semidirect
572. C5#74004semidirect
602. C3#10605298semidirect

Maximal subgroups

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

subgroupindex
C22×#66672
#117510562
C5×Dic3×C22⋊C42
C3×#652312
#117495062
#117495022
Dic3×C22⋊Dic52
#117487302
#117487282
#117486712
#117486662
#106052983
#740045