Finite Group Databasebeta

← #14320113

Subgroups — #14320113

Order 1920.

1136 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
1C2×C4×#1200C211yesmaximal, characteristic
2#11751088C211yesmaximal, characteristic
3C5×#4977C211yesmaximal, characteristic, complemented
4C3×C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
5#11748889C212yesmaximal
6#11748875C212yesmaximal
7#11748799C211yesmaximal, characteristic
8Dic5×C4⋊Dic3C211yesmaximal, characteristic, complemented
9#11748646C212yesmaximal, complemented
10#11748635C212yesmaximal, complemented
11#11748619C211yesmaximal, characteristic, complemented
12#10605878—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C22×#1200C2211yescharacteristic
15C22×#1200C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×#6714C2212yes—
18C2×C4⋊Dic15C2211yescharacteristic
19C2×C5×C4⋊Dic3C2211yescharacteristic
20C2×C4×C5×Dic3C2211yescharacteristic
21C2×C3×C4⋊Dic5C2211yescharacteristic
22C2×C3×C4×Dic5C2212yes—
23C2×C3×C4×Dic5C2211yescharacteristic
24C2×#6667C2212yes—
25C2×Dic3×Dic5C2212yes—
26C2×Dic3×Dic5C2211yescharacteristic
27C2×Dic3×Dic5C2211yescharacteristic
28C4×#1200C418yescomplemented
29#91115C2212yes—
30C5×C3⋊#95C2212yes—
31C5×C3⋊#95C2212yes—
32C3×#3963C2212yes—
33#90994C2212yes—
34#90994C2212yes—
35#90994C2212yes—
36#90994C2212yes—
37#73752—51nomaximal, Hall
38C2×C42×D5—31no—
39C5×#382—31no—
40#65276—31no—
41#65276—31no—
42C4⋊C4×Dic5—31no—
43C4⋊C4×Dic5D311yescharacteristic, complemented
44#65225—31no—
45#64955—32no—
46#64953—32no—
47#64940—32no—
48#64937—32no—
49C23×D15C2311yescharacteristic
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×Dic15C2312yes—
53C22×Dic15C2311yescharacteristic
54C2×C4×D15—22no—
55C2×C4×D15—22no—
56C2×C4×D15C2×C414yes—
57C22×C5×Dic3C2312yes—
58C22×C5×Dic3C2311yescharacteristic
59C22×C5×Dic3C2311yescharacteristic
60C22×C3×Dic5C2312yes—
61C22×C3×Dic5C2311yescharacteristic
62C22×C3×Dic5C2311yescharacteristic
63C2×#1200—22no—
64C2×#1200—22no—
65C2×#1200—22no—
66C2×#1200—22no—
67C2×#1200C2×C414yes—
68C2×#1200C2×C414yes—
69C3×C5×C4⋊C4—22no—
70#6714—24no—
71C4⋊Dic15—22no—
72C5×C4⋊Dic3—22no—
73C4×C5×Dic3C2×C414yes—
74C3×C4⋊Dic5—22no—
75C3×C4×Dic5—24no—
76C3×C4×Dic5C2×C414yes—
77#6667—24no—
78Dic3×Dic5—24no—
79Dic3×Dic5C2×C414yes—
80Dic3×Dic5C2×C414yes—
81C2×C42×D3—51no—
82C3×C4×C4⋊C4—51no—
83#4987—51no—
84#4987—51no—
85#4977—51no—
86#4977D511yescharacteristic, complemented
87C4×(C2×D3)⋊C4—52no—
88C4×C4⋊Dic3—51no—
89#4932—52no—
90#4664—52no—
91#4651—52no—
92C22×C4×D5—31no—
93C22×C4×D5—31no—
94C22×C4×D5—31no—
95C2×C5×C4⋊C4—31no—
96C2×C5×C4⋊C4C2×D311yescharacteristic
97C2×C42×C5—31no—
98C2×#685—32no—
99C2×#685—32no—
100C2×C4⋊Dic5—31no—
101C2×C4⋊Dic5C2×D311yescharacteristic
102C2×C4×Dic5—31no—
103C2×C4×Dic5—31no—
104C2×C4×Dic5—32no—
105C2×C4×Dic5C2×D312yes—
106C2×C4×Dic5C2×D311yescharacteristic
107C42×D5—38no—
108C5×#95—32no—
109C5×#95—32no—
110#3963—32no—
111#3963—32no—
112#3963—32no—
113#3963—32no—
114#3963—32no—
115#3963C2×D312yes—
116#2283—151noSylow
117C23×C3×C5C2411yescharacteristic
118C22×D15—21no—
119C22×D15—21no—
120C22×D15—21no—
121C22×D15—21no—
122C22×D15C22×C412yes—
123C22×D15#4112yes—
124C22×D15#4112yes—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5C22×C412yes—
129C2×C3×C4×C5#4112yes—
130C2×C3×C4×C5#4112yes—
131C2×Dic15—21no—
132C2×Dic15—21no—
133C2×Dic15—22no—
134C2×Dic15—22no—
135C2×Dic15—22no—
136C2×Dic15C22×C412yes—
137C4×D15—28no—
138C2×C5×Dic3—21no—
139C2×C5×Dic3—21no—
140C2×C5×Dic3—21no—
141C2×C5×Dic3—21no—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—22no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3C22×C412yes—
146C2×C5×Dic3C22×C412yes—
147C2×C3×Dic5—21no—
148C2×C3×Dic5—21no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5—22no—
152C2×C3×Dic5C22×C412yes—
153C2×C3×Dic5C22×C412yes—
154C2×C3×Dic5#4112yes—
155C2×C3×Dic5#4112yes—
156#1200—28no—
157#1200—28no—
158C22×C4×D3—51no—
159C22×C4×D3—51no—
160C22×C4×D3—51no—
161C2×C3×C4⋊C4—51no—
162C2×C3×C4⋊C4C2×D511yescharacteristic
163C2×C3×C42—51no—
164C2×C3×C42—52no—
165C2×(C2×D3)⋊C4—52no—
166C2×(C2×D3)⋊C4—52no—
167C2×C4⋊Dic3—51no—
168C2×C4⋊Dic3C2×D511yescharacteristic
169C2×C4×Dic3—51no—
170C2×C4×Dic3—51no—
171C2×C4×Dic3—52no—
172C2×C4×Dic3C2×D511yescharacteristic
173C42×D3—58no—
174C3×#95—52no—
175C3⋊#95—52no—
176C3⋊#95—52no—
177C3⋊#95—52no—
178C3⋊#95—52no—
179C3⋊#95—52no—
180C3⋊#95C2×D512yes—
181C3⋊#95C2×D512yes—
182C23×D5—31no—
183C22×C4×C5—31no—
184C22×C4×C5—31no—
185C22×C4×C5—32no—
186C22×C4×C5C22×D312yes—
187C22×C4×C5C22×D311yescharacteristic
188C22×Dic5—31no—
189C22×Dic5—32no—
190C22×Dic5C22×D312yes—
191C22×Dic5C22×D311yescharacteristic
192C22×Dic5C22×D311yescharacteristic
193C2×C4×D5—34no—
194C2×C4×D5—34no—
195C2×C4×D5—34no—
196C2×C4×D5—62no—
197C2×C4×D5—62no—
198C2×C4×D5—62no—
199C2×C4×D5—62no—
200C2×C4×D5—62no—
201C2×C4×D5—62no—
202C5×C4⋊C4—22no—
203C5×C4⋊C4—62no—
204C42×C5—34no—
205#685—64no—
206#685—64no—
207C4⋊Dic5—22no—
208C4⋊Dic5—62no—
209C4×Dic5—24no—
210C4×Dic5—34no—
211C4×Dic5—34no—
212C4×Dic5—64no—
213C4×Dic5C4×D314yes—
214C22×C42—151no—
215#383—152no—
216#382—31no—
217#382—151no—
218#381—152no—
219#380—151no—
220#380—151no—
221C4×C4⋊C4—51no—
222C4×C4⋊C4—151no—
223C4×C22⋊C4—152no—
224#376—152no—
225C22×C3×C5C23×C411yescharacteristic
226C22×C3×C5C2×#4111yescharacteristic
227C22×C3×C5C2×#4111yescharacteristic
228C22×C3×C5C2×#4111yescharacteristic
229C22×C3×C5C2×#4111yescharacteristic
230C22×C3×C5C2×#4111yescharacteristic
231C22×C3×C5C2×#4111yescharacteristic
232C2×D15—22no—
233C2×D15—22no—
234C2×D15—22no—
235C2×D15—22no—
236C2×D15—22no—
237C2×D15—22no—
238C2×D15#11714yes—
239C3×C4×C5—42no—
240C3×C4×C5#11714yes—
241Dic15—22no—
242Dic15—42no—
243C3×Dic5—22no—
244C3×Dic5—42no—
245C3×Dic5#11714yes—
246C5×Dic3—22no—
247C5×Dic3—22no—
248C5×Dic3—42no—
249C23×D3—51no—
250C22×C3×C4—51no—
251C22×C3×C4—51no—
252C22×C3×C4—52no—
253C22×C3×C4C22×D512yes—
254C22×C3×C4C22×D511yescharacteristic
255C22×Dic3—51no—
256C22×Dic3—52no—
257C22×Dic3C22×D512yes—
258C22×Dic3C22×D511yescharacteristic
259C22×Dic3C22×D511yescharacteristic
260C2×C4×D3—54no—
261C2×C4×D3—54no—
262C2×C4×D3—54no—
263C2×C4×D3—102no—
264C2×C4×D3—102no—
265C2×C4×D3—102no—
266C2×C4×D3—102no—
267C2×C4×D3—102no—
268C2×C4×D3—102no—
269C3×C4⋊C4—22no—
270C3×C4⋊C4—102no—
271C3×C42—54no—
272C3×C42—104no—
273(C2×D3)⋊C4—104no—
274(C2×D3)⋊C4—104no—
275C4⋊Dic3—22no—
276C4⋊Dic3—102no—
277C4×Dic3—54no—
278C4×Dic3—54no—
279C4×Dic3—104no—
280C4×Dic3C4×D514yes—
281C23×C5C23×D311yescharacteristic
282C22×D5—32no—
283C22×D5—32no—
284C22×D5—32no—
285C22×D5—61no—
286C22×D5—61no—
287C22×D5—61no—
288C22×D5—61no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—22no—
292C2×C4×C5—32no—
293C2×C4×C5—32no—
294C2×C4×C5—61no—
295C2×C4×C5—61no—
296C2×C4×C5—61no—
297C2×C4×C5—61no—
298C2×C4×C5—62no—
299C2×C4×C5—62no—
300C2×C4×C5—62no—
301C2×C4×C5C2×C4×D312yes—
302C2×C4×C5#23512yes—
303C2×C4×C5#23512yes—
304C2×Dic5—21no—
305C2×Dic5—21no—
306C2×Dic5—22no—
307C2×Dic5—22no—
308C2×Dic5—22no—
309C2×Dic5—32no—
310C2×Dic5—61no—
311C2×Dic5—61no—
312C2×Dic5—62no—
313C2×Dic5—62no—
314C2×Dic5—62no—
315C2×Dic5C2×C4×D312yes—
316C2×Dic5C2×C4×D312yes—
317C2×Dic5#23712yes—
318C2×Dic5#23912yes—
319C4×D5—68no—
320C4×D5—68no—
321C4×D5—68no—
322C23×C4—151no—
323C23×C4—151no—
324C23×C4—151no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—51no—
327C2×C4⋊C4—151no—
328C2×C4⋊C4D3×D511yescharacteristic
329C2×C22⋊C4—152no—
330C2×C22⋊C4—152no—
331C2×C42—31no—
332C2×C42—51no—
333C2×C42—52no—
334C2×C42—158no—
335C2×C42—151no—
336C2×C42—151no—
337C2×C42—152no—
338#95—32no—
339#95—32no—
340#95—52no—
341#95—152no—
342#95—152no—
343#95—152no—
344#95—152no—
345#95—152no—
346C2×C3×C5C2×#11711yescharacteristic
347C2×C3×C5C2×#11711yescharacteristic
348C2×C3×C5C2×#11711yescharacteristic
349C2×C3×C5#52911yescharacteristic
350C2×C3×C5#52911yescharacteristic
351C2×C3×C5#52911yescharacteristic
352C2×C3×C5#52911yescharacteristic
353D15—24no—
354C23×C3C23×D511yescharacteristic
355C22×D3—52no—
356C22×D3—52no—
357C22×D3—52no—
358C22×D3—101no—
359C22×D3—101no—
360C22×D3—101no—
361C22×D3—101no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—52no—
366C2×C3×C4—52no—
367C2×C3×C4—52no—
368C2×C3×C4—52no—
369C2×C3×C4—101no—
370C2×C3×C4—101no—
371C2×C3×C4—102no—
372C2×C3×C4—102no—
373C2×C3×C4—102no—
374C2×C3×C4C2×C4×D512yes—
375C2×C3×C4#71112yes—
376C2×C3×C4#71312yes—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—52no—
385C2×Dic3—101no—
386C2×Dic3—101no—
387C2×Dic3—102no—
388C2×Dic3—102no—
389C2×Dic3—102no—
390C2×Dic3C2×C4×D512yes—
391C2×Dic3C2×C4×D512yes—
392C4×D3—108no—
393C4×D3—108no—
394C4×D3—108no—
395C22×C5C22×C4×D311yescharacteristic
396C22×C5C2×#23511yescharacteristic
397C22×C5C2×#23511yescharacteristic
398C22×C5C2×#23711yescharacteristic
399C22×C5C2×#23911yescharacteristic
400C22×C5D3×#4111yescharacteristic
401C22×C5D3×#4111yescharacteristic
402C2×D5—34no—
403C2×D5—62no—
404C2×D5—62no—
405C2×D5—62no—
406C2×D5—62no—
407C2×D5—62no—
408C2×D5—62no—
409C4×C5—42no—
410C4×C5—62no—
411C4×C5—62no—
412C4×C5—122no—
413C4×C5#87214yes—
414Dic5—22no—
415Dic5—42no—
416Dic5—62no—
417Dic5—122no—
418Dic5#89214yes—
419C24—151no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—32no—
423C22×C4—51no—
424C22×C4—51no—
425C22×C4—52no—
426C22×C4—154no—
427C22×C4—154no—
428C22×C4—151no—
429C22×C4—154no—
430C22×C4—152no—
431C22×C4—302no—
432C22×C4—302no—
433C22×C4—302no—
434C22×C4—302no—
435C22×C4—302no—
436C22×C4—302no—
437C22×C4C2×D3×D512yes—
438C22×C4C2×D3×D511yescharacteristic
439C4⋊C4—22no—
440C4⋊C4—62no—
441C4⋊C4—102no—
442C4⋊C4—302no—
443C22⋊C4—304no—
444C22⋊C4—304no—
445C42—34no—
446C42—54no—
447C42—104no—
448C42—154no—
449C42—154no—
450C42—304no—
451C3×C5#228311yescharacteristic, complemented, Hall
452C22×C3C22×C4×D511yescharacteristic
453C22×C3C2×#71111yescharacteristic
454C22×C3C2×#71311yescharacteristic
455C22×C3D5×#4111yescharacteristic
456C22×C3D5×#4111yescharacteristic
457C22×C3D5×#4111yescharacteristic
458C22×C3D5×#4111yescharacteristic
459C2×D3—54no—
460C2×D3—102no—
461C2×D3—102no—
462C2×D3—102no—
463C2×D3—102no—
464C2×D3—102no—
465C2×D3—102no—
466C3×C4—42no—
467C3×C4—54no—
468C3×C4—102no—
469C3×C4—202no—
470C3×C4#403814yes—
471Dic3—22no—
472Dic3—22no—
473Dic3—42no—
474Dic3—102no—
475Dic3—202no—
476C2×C5C2×#87211yescharacteristic
477C2×C5C2×#89211yescharacteristic
478C2×C5D3×#11711yescharacteristic
479C2×C5#555711yescharacteristic
480C2×C5#555711yescharacteristic
481C2×C5#558011yescharacteristic
482C2×C5#558011yescharacteristic
483D5—64no—
484C23—152no—
485C23—152no—
486C23—152no—
487C23—301no—
488C23—301no—
489C23—301no—
490C23—301no—
491C23C22×D3×D511yescharacteristic
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—32no—
496C2×C4—32no—
497C2×C4—52no—
498C2×C4—52no—
499C2×C4—52no—
500C2×C4—52no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—101no—
509C2×C4—101no—
510C2×C4—102no—
511C2×C4—102no—
512C2×C4—102no—
513C2×C4—152no—
514C2×C4—308no—
515C2×C4—308no—
516C2×C4—301no—
517C2×C4—301no—
518C2×C4—308no—
519C2×C4—302no—
520C2×C4—302no—
521C2×C4—302no—
522C2×C4#677312yes—
523C2×C4C4×D3×D512yes—
524C2×C4#678212yes—
525C2×C3C2×#403811yescharacteristic
526C2×C3D5×#11711yescharacteristic
527C2×C3D5×#11711yescharacteristic
528C2×C3#6601411yescharacteristic
529C2×C3#6601411yescharacteristic
530C2×C3#6602511yescharacteristic
531C2×C3#6602511yescharacteristic
532D3—104no—
533C5#7375211yescharacteristic, complemented, Sylow
534C22—154no—
535C22—302no—
536C22—302no—
537C22—302no—
538C22—302no—
539C22—302no—
540C22—302no—
541C22C2×#677311yescharacteristic
542C22C2×C4×D3×D511yescharacteristic
543C22D5×#23711yescharacteristic
544C22#9202411yescharacteristic
545C22D5×#23911yescharacteristic
546C22#9203411yescharacteristic
547C22C2×#678211yescharacteristic
548C4—42no—
549C4—54no—
550C4—62no—
551C4—62no—
552C4—102no—
553C4—122no—
554C4—202no—
555C4—302no—
556C4—602no—
557C4#9140214yes—
558C3#1060587811yescharacteristic, complemented, Sylow
559C2—304no—
560C2#1175352111yescharacteristic
561C2#1175354911yescharacteristic
562C2D5×#89211yescharacteristic
563C2#1175383511yescharacteristic
564C2C2×#9140211yescharacteristic
565C2#1175485411yescharacteristic
566C2#1175519611yescharacteristic
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#2283
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. C5×#4977C2semidirect
4. C3×C4⋊C4×Dic5C2semidirect
8. Dic5×C4⋊Dic3C2semidirect
9. #11748646C2semidirect
10. #11748635C2semidirect
11. #11748619C2semidirect
28. C4×#1200C4semidirect
43. C4⋊C4×Dic5D3semidirect
86. #4977D5semidirect
451. C3×C5#2283semidirect
533. C5#73752semidirect
558. C3#10605878semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#12002
#117510882
C5×#49772
C3×C4⋊C4×Dic52
#117488892
#117488752
#117487992
Dic5×C4⋊Dic32
#117486462
#117486352
#117486192
#106058783
#737525