Finite Group Databasebeta

← #14323780

Subgroups — #14323780

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×#91386C211yesmaximal, characteristic
2C2×#91383C211yesmaximal, characteristic
3C2×Dic5×Dic6C211yesmaximal, characteristic, complemented
4#11751088C211yesmaximal, characteristic
5C5×#4973C211yesmaximal, characteristic, complemented
6C3×C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
7#11748857C211yesmaximal, characteristic, complemented
8#11748802C212yesmaximal, complemented
9#11748726C212yesmaximal
10#11748671C212yesmaximal
11#11748648C212yesmaximal, complemented
12#10608342—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×#6714C2212yes—
17C2×C4⋊Dic15C2211yescharacteristic
18C2×C4×C5×Dic3C2211yescharacteristic
19C2×C3×C4⋊Dic5C2211yescharacteristic
20C2×C3×C4×Dic5C2212yes—
21C2×C3×C4×Dic5C2211yescharacteristic
22C2×Dic3⋊Dic5C2212yes—
23C2×#6669C2211yescharacteristic
24C2×#6669C2211yescharacteristic
25C2×#6668C2212yes—
26C2×#6668C2212yes—
27C2×#6667C2211yescharacteristic
28C2×#6667C2211yescharacteristic
29C2×#6667C2212yes—
30C2×Dic3×Dic5C2212yes—
31#91386—24no—
32#91383C418yescomplemented
33Dic5×Dic6—24no—
34#91115C2212yes—
35C5×C3⋊#95C2212yes—
36C5×C3⋊#95C2212yes—
37C3×#3963C2212yes—
38#90994C2212yes—
39#73762—51nomaximal, Hall
40C2×#4092—31no—
41C2×Q8×Dic5—31no—
42C2×#4018—31no—
43C5×#391—31no—
44#65276—31no—
45C4⋊C4×Dic5D311yescharacteristic, complemented
46#65217—31no—
47#64959—32no—
48#64958—32no—
49#64941—32no—
50#64936—32no—
51C23×D15C2311yescharacteristic
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×Dic15C2312yes—
55C22×Dic15C2311yescharacteristic
56C2×C4×D15—22no—
57C2×C4×D15—22no—
58C2×C4×D15C2×C414yes—
59C22×C5×Dic3C2311yescharacteristic
60C22×C5×Dic3C2311yescharacteristic
61C22×C5×Dic3C2312yes—
62C2×C5×Dic6—22no—
63C2×C5×Dic6—22no—
64C2×C5×Dic6C2×C414yes—
65C22×C3×Dic5C2312yes—
66C22×C3×Dic5C2311yescharacteristic
67C22×C3×Dic5C2311yescharacteristic
68C3×C5×C4⋊C4—22no—
69#6714—42no—
70C4⋊Dic15—22no—
71C4×C5×Dic3—22no—
72C3×C4⋊Dic5C2×C414yes—
73C3×C4×Dic5—42no—
74C3×C4×Dic5D414yes—
75Dic3⋊Dic5—24no—
76#6669—22no—
77#6669—22no—
78#6668—42no—
79#6668C2×C418yes—
80#6667—22no—
81#6667—22no—
82#6667C2×C418yes—
83Dic3×Dic5—24no—
84C2×#897—51no—
85C2×#872—51no—
86C2×C4×Dic6—51no—
87C3×C4×C4⋊C4—51no—
88#4987—51no—
89#4984—51no—
90#4973D511yescharacteristic, complemented
91#4940—52no—
92#4928—52no—
93#4670—52no—
94#4653—52no—
95C22×C5×Q8—31no—
96C22×C4×D5—31no—
97C2×C5×C4⋊C4C2×D311yescharacteristic
98C2×C42×C5—31no—
99#4092—38no—
100Q8×Dic5—64no—
101C2×#685—32no—
102C2×#685—31no—
103C2×#685—31no—
104C2×#685—32no—
105C2×C4⋊Dic5—31no—
106C2×C4⋊Dic5—32no—
107C2×C4⋊Dic5C2×D311yescharacteristic
108C2×#683—32no—
109C2×#683—31no—
110C2×#683—31no—
111C2×#683—32no—
112C2×C4×Dic5—32no—
113C2×C4×Dic5C2×D312yes—
114C2×C4×Dic5C2×D311yescharacteristic
115#4018—64no—
116C5×#95—32no—
117C5×#95—32no—
118#3963—32no—
119#3963—32no—
120#3963C2×D312yes—
121#2323—151noSylow
122C23×C3×C5C2411yescharacteristic
123C22×D15—21no—
124C22×D15—21no—
125C22×D15—21no—
126C22×D15—21no—
127C22×D15—21no—
128C22×D15—21no—
129C22×D15C22×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—21no—
137C2×Dic15—21no—
138C2×Dic15—22no—
139C2×Dic15—22no—
140C2×Dic15—22no—
141C2×Dic15C22×C412yes—
142C4×D15—28no—
143C2×C5×Dic3—22no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—21no—
146C2×C5×Dic3—21no—
147C2×C5×Dic3—21no—
148C2×C5×Dic3—21no—
149C2×C5×Dic3—21no—
150C2×C5×Dic3—21no—
151C2×C5×Dic3C22×C414yes—
152C5×Dic6—28no—
153C2×C3×Dic5—22no—
154C2×C3×Dic5—21no—
155C2×C3×Dic5—22no—
156C2×C3×Dic5—21no—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5C22×C414yes—
160C2×C3×Dic5C2×D412yes—
161C2×C3×Dic5C2×D412yes—
162C22×C4×D3—51no—
163C22×Dic6C2×D511yescharacteristic
164C2×C3×C4⋊C4—51no—
165C2×C3×C4⋊C4C2×D511yescharacteristic
166C2×C3×C42—51no—
167C2×C3×C42—52no—
168C2×(C2×D3)⋊C4—52no—
169C2×(C2×D3)⋊C4—51no—
170C2×(C2×D3)⋊C4—51no—
171C2×(C2×D3)⋊C4—52no—
172C2×C4⋊Dic3—52no—
173C2×C4⋊Dic3—51no—
174C2×C4⋊Dic3—52no—
175C2×Dic3⋊C4—51no—
176C2×Dic3⋊C4—51no—
177C2×Dic3⋊C4—52no—
178C2×C4×Dic3—52no—
179C2×C4×Dic3C2×D511yescharacteristic
180#897—58no—
181#872—104no—
182C4×Dic6—104no—
183C3×#95—52no—
184C3⋊#95—52no—
185C3⋊#95—52no—
186C3⋊#95C2×D512yes—
187C3⋊#95C2×D512yes—
188C23×D5—31no—
189C2×C5×Q8—34no—
190C2×C5×Q8—62no—
191C2×C5×Q8—62no—
192C22×C4×C5—32no—
193C22×C4×C5—31no—
194C22×C4×C5—31no—
195C22×C4×C5C22×D312yes—
196C22×C4×C5C22×D311yescharacteristic
197C22×Dic5—31no—
198C22×Dic5—32no—
199C22×Dic5C22×D312yes—
200C22×Dic5C22×D311yescharacteristic
201C22×Dic5C22×D311yescharacteristic
202C2×C4×D5—34no—
203C2×C4×D5—62no—
204C2×C4×D5—62no—
205C5×C4⋊C4—22no—
206C42×C5—62no—
207#685—38no—
208#685—62no—
209#685—62no—
210#685—122no—
211C4⋊Dic5—62no—
212C4⋊Dic5—64no—
213C4⋊Dic5C4×D314yes—
214#683—38no—
215#683—62no—
216#683—62no—
217#683—122no—
218C4×Dic5—42no—
219C4×Dic5—64no—
220C4×Dic5D1214yes—
221C2×C22⋊Q8—151no—
222C2×C4×Q8—151no—
223C2×#117—151no—
224#391—31no—
225#389—152no—
226#389—151no—
227#388—152no—
228#388—152no—
229#387—152no—
230#380—151no—
231C4×C4⋊C4—51no—
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×D15—22no—
240C2×D15—22no—
241C2×D15—22no—
242C2×D15—41no—
243C2×D15—41no—
244C2×D15—41no—
245C2×D15—41no—
246C3×C4×C5—42no—
247C3×C4×C5C4×D414yes—
248Dic15—22no—
249Dic15—42no—
250C3×Dic5—24no—
251C3×Dic5—22no—
252C3×Dic5—22no—
253C5×Dic3—24no—
254C5×Dic3—41no—
255C5×Dic3—41no—
256C23×D3—51no—
257C22×C3×C4—52no—
258C22×C3×C4—51no—
259C22×C3×C4—51no—
260C22×C3×C4C22×D512yes—
261C22×C3×C4C22×D511yescharacteristic
262C22×Dic3—51no—
263C22×Dic3—52no—
264C22×Dic3C22×D511yescharacteristic
265C22×Dic3C22×D511yescharacteristic
266C22×Dic3C22×D512yes—
267C2×C4×D3—54no—
268C2×C4×D3—102no—
269C2×C4×D3—102no—
270C2×Dic6—22no—
271C2×Dic6—22no—
272C2×Dic6C4×D514yes—
273C3×C4⋊C4—22no—
274C3×C4⋊C4—54no—
275C3×C42—54no—
276C3×C42—202no—
277(C2×D3)⋊C4—58no—
278(C2×D3)⋊C4—102no—
279(C2×D3)⋊C4—102no—
280(C2×D3)⋊C4—202no—
281C4⋊Dic3—58no—
282C4⋊Dic3—102no—
283C4⋊Dic3—202no—
284Dic3⋊C4—102no—
285Dic3⋊C4—102no—
286Dic3⋊C4—104no—
287C4×Dic3—22no—
288C4×Dic3—104no—
289C23×C5C23×D311yescharacteristic
290C22×D5—32no—
291C22×D5—61no—
292C22×D5—61no—
293C22×D5—61no—
294C22×D5—61no—
295C22×D5—61no—
296C22×D5—61no—
297C5×Q8—68no—
298C2×C4×C5—22no—
299C2×C4×C5—22no—
300C2×C4×C5—22no—
301C2×C4×C5—34no—
302C2×C4×C5—62no—
303C2×C4×C5—62no—
304C2×C4×C5—61no—
305C2×C4×C5—61no—
306C2×C4×C5—61no—
307C2×C4×C5—61no—
308C2×C4×C5—61no—
309C2×C4×C5—61no—
310C2×C4×C5C2×C4×D312yes—
311C2×C4×C5C2×D1212yes—
312C2×C4×C5#23512yes—
313C2×Dic5—21no—
314C2×Dic5—22no—
315C2×Dic5—21no—
316C2×Dic5—21no—
317C2×Dic5—21no—
318C2×Dic5—22no—
319C2×Dic5—32no—
320C2×Dic5—61no—
321C2×Dic5—61no—
322C2×Dic5—62no—
323C2×Dic5—62no—
324C2×Dic5—62no—
325C2×Dic5C2×C4×D314yes—
326C2×Dic5C2×D1212yes—
327C2×Dic5C2×D1212yes—
328C4×D5—68no—
329C22×Q8—31no—
330C23×C4—151no—
331C22⋊Q8—158no—
332C4×Q8—304no—
333#117—304no—
334C2×C4⋊C4—51no—
335C2×C4⋊C4—152no—
336C2×C4⋊C4—151no—
337C2×C4⋊C4—151no—
338C2×C4⋊C4—152no—
339C2×C4⋊C4—151no—
340C2×C4⋊C4—152no—
341C2×C4⋊C4D3×D511yescharacteristic
342C2×C22⋊C4—152no—
343C2×C22⋊C4—151no—
344C2×C22⋊C4—151no—
345C2×C22⋊C4—152no—
346C2×C42—31no—
347C2×C42—51no—
348C2×C42—52no—
349C2×C42—152no—
350#95—32no—
351#95—32no—
352#95—52no—
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
362D15—42no—
363C23×C3C23×D511yescharacteristic
364C22×D3—52no—
365C22×D3—101no—
366C22×D3—101no—
367C22×D3—101no—
368C22×D3—101no—
369C22×D3—101no—
370C22×D3—101no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—54no—
375C2×C3×C4—52no—
376C2×C3×C4—52no—
377C2×C3×C4—102no—
378C2×C3×C4—101no—
379C2×C3×C4—101no—
380C2×C3×C4—102no—
381C2×C3×C4—101no—
382C2×C3×C4—101no—
383C2×C3×C4C2×C4×D512yes—
384C2×C3×C4D4×D512yes—
385C2×C3×C4#71112yes—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—52no—
395C2×Dic3—101no—
396C2×Dic3—101no—
397C2×Dic3—102no—
398C2×Dic3—102no—
399C2×Dic3—102no—
400C2×Dic3C2×C4×D514yes—
401C4×D3—108no—
402Dic6—28no—
403C22×C5C22×C4×D311yescharacteristic
404C22×C5C22×D1211yescharacteristic
405C22×C5C2×#23511yescharacteristic
406C22×C5#100411yescharacteristic
407C22×C5#100711yescharacteristic
408C22×C5#101011yescharacteristic
409C22×C5#101011yescharacteristic
410C2×D5—62no—
411C2×D5—62no—
412C2×D5—62no—
413C2×D5—121no—
414C2×D5—121no—
415C2×D5—121no—
416C2×D5—121no—
417C4×C5—42no—
418C4×C5—64no—
419C4×C5—121no—
420C4×C5—121no—
421C4×C5C4×D1214yes—
422Dic5—22no—
423Dic5—22no—
424Dic5—24no—
425Dic5—62no—
426Dic5—122no—
427C24—151no—
428C2×Q8—34no—
429C2×Q8—62no—
430C2×Q8—62no—
431C22×C4—32no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—51no—
435C22×C4—52no—
436C22×C4—51no—
437C22×C4—151no—
438C22×C4—154no—
439C22×C4—152no—
440C22×C4—302no—
441C22×C4—302no—
442C22×C4C2×D3×D512yes—
443C22×C4C2×D3×D511yescharacteristic
444C4⋊C4—22no—
445C4⋊C4—54no—
446C4⋊C4—158no—
447C4⋊C4—302no—
448C4⋊C4—302no—
449C4⋊C4—304no—
450C4⋊C4—302no—
451C4⋊C4—602no—
452C22⋊C4—158no—
453C22⋊C4—302no—
454C22⋊C4—302no—
455C22⋊C4—602no—
456C42—54no—
457C42—62no—
458C42—202no—
459C42—304no—
460C3×C5#232311yescharacteristic, complemented, Hall
461C22×C3C22×C4×D511yescharacteristic
462C22×C3C2×D4×D511yescharacteristic
463C22×C3C2×#71111yescharacteristic
464C22×C3#414711yescharacteristic
465C22×C3#414711yescharacteristic
466C22×C3#414911yescharacteristic
467C22×C3#415011yescharacteristic
468C2×D3—102no—
469C2×D3—102no—
470C2×D3—102no—
471C2×D3—201no—
472C2×D3—201no—
473C2×D3—201no—
474C2×D3—201no—
475C3×C4—42no—
476C3×C4—104no—
477C3×C4—102no—
478C3×C4—102no—
479C3×C4#402714yes—
480Dic3—24no—
481Dic3—41no—
482Dic3—41no—
483Dic3—102no—
484Dic3—202no—
485C2×C5C2×C4×D1211yescharacteristic
486C2×C5#551011yescharacteristic
487C2×C5#552611yescharacteristic
488C2×C5#554611yescharacteristic
489C2×C5#556111yescharacteristic
490C2×C5#557211yescharacteristic
491C2×C5#558311yescharacteristic
492D5—122no—
493C23—152no—
494C23—301no—
495C23—301no—
496C23—301no—
497C23—301no—
498C23—301no—
499C23—301no—
500C23C22×D3×D511yescharacteristic
501Q8—68no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—34no—
506C2×C4—54no—
507C2×C4—52no—
508C2×C4—52no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—101no—
518C2×C4—101no—
519C2×C4—102no—
520C2×C4—102no—
521C2×C4—101no—
522C2×C4—101no—
523C2×C4—152no—
524C2×C4—301no—
525C2×C4—301no—
526C2×C4—308no—
527C2×C4—302no—
528C2×C4—302no—
529C2×C4—302no—
530C2×C4C4×D3×D512yes—
531C2×C4D5×D1212yes—
532C2×C4#678212yes—
533C2×C3C2×#402711yescharacteristic
534C2×C3#6590611yescharacteristic
535C2×C3#6590811yescharacteristic
536C2×C3#6596211yescharacteristic
537C2×C3#6596311yescharacteristic
538C2×C3#6601611yescharacteristic
539C2×C3#6606511yescharacteristic
540D3—202no—
541C5#7376211yescharacteristic, complemented, Sylow
542C22—302no—
543C22—302no—
544C22—302no—
545C22—601no—
546C22—601no—
547C22—601no—
548C22—601no—
549C22C2×C4×D3×D511yescharacteristic
550C22C2×D5×D1211yescharacteristic
551C22#9202011yescharacteristic
552C22#9202611yescharacteristic
553C22#9202811yescharacteristic
554C22#9202911yescharacteristic
555C22C2×#678211yescharacteristic
556C4—42no—
557C4—64no—
558C4—104no—
559C4—102no—
560C4—102no—
561C4—121no—
562C4—121no—
563C4—302no—
564C4—602no—
565C4#9140514yes—
566C3#1060834211yescharacteristic, complemented, Sylow
567C2—602no—
568C2#1175338111yescharacteristic
569C2#1175340111yescharacteristic
570C2#1175343511yescharacteristic
571C2#1175369111yescharacteristic
572C2C2×#9140511yescharacteristic
573C2#1175474111yescharacteristic
574C2#1175515111yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC23×D15C23
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×Dic5×Dic6C2semidirect
5. C5×#4973C2semidirect
6. C3×C4⋊C4×Dic5C2semidirect
7. #11748857C2semidirect
8. #11748802C2semidirect
11. #11748648C2semidirect
32. #91383C4semidirect
45. C4⋊C4×Dic5D3semidirect
90. #4973D5semidirect
460. C3×C5#2323semidirect
541. C5#73762semidirect
566. C3#10608342semidirect

Maximal subgroups

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

subgroupindex
C2×#913862
C2×#913832
C2×Dic5×Dic62
#117510882
C5×#49732
C3×C4⋊C4×Dic52
#117488572
#117488022
#117487262
#117486712
#117486482
#106083423
#737625