Finite Group Databasebeta

← #14327398

Subgroups — #14327398

Order 1920.

1320 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91781C211yesmaximal, characteristic, complemented
2C2×C4×#1203C211yesmaximal, characteristic
3C2×D3×C4⋊Dic5C211yesmaximal, characteristic, complemented
4C5×#4987C211yesmaximal, characteristic, complemented
5C3×#65266C211yesmaximal, characteristic, complemented
6#11748886C212yesmaximal, complemented
7#11748883C212yesmaximal
8#11748810C212yesmaximal, complemented
9#11748799C211yesmaximal, characteristic, complemented
10Dic5×C4⋊Dic3C211yesmaximal, characteristic, complemented
11#11748668C212yesmaximal, complemented
12#10605614—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C22×C4×C5×D3C2211yescharacteristic
15C22×#1203C2211yescharacteristic
16C22×D3×Dic5C2212yes—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×#6714C2212yes—
19C2×C4⋊Dic15C2211yescharacteristic
20C2×C4×Dic15C2212yes—
21#91781—24no—
22C2×C5×(C2×D3)⋊C4C2212yes—
23C2×C5×C4⋊Dic3C2211yescharacteristic
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C2×C3×C4×Dic5C2211yescharacteristic
26C2×Dic3⋊Dic5C2212yes—
27C2×#6668C2211yescharacteristic
28C2×#6667C2212yes—
29C2×#6667C2211yescharacteristic
30C2×(C2×D3)⋊Dic5C2211yescharacteristic
31C2×Dic3×Dic5C2212yes—
32C4×#1203C418yescomplemented
33D3×C4⋊Dic5—24no—
34C5×C3⋊#95C2212yes—
35C3×#3963C2212yes—
36C3×#3963C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#74229—51nomaximal, Hall
40C2×C4×C22⋊D5—31no—
41C22×C4⋊Dic5—31no—
42C2×#4038—31no—
43C5×#380—31no—
44#65276—31no—
45#65266D311yescharacteristic, complemented
46C4⋊C4×Dic5—31no—
47#65247—32no—
48C22⋊C4×Dic5—32no—
49#64959—32no—
50#64944—32no—
51C23×D15C2311yescharacteristic
52C23×C5×D3C2311yescharacteristic
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×Dic15C2311yescharacteristic
56C22×Dic15C2312yes—
57C2×C4×D15—22no—
58C2×C4×D15—22no—
59C2×C4×D15C2×C414yescomplemented
60C22×C5×Dic3C2312yes—
61C22×C5×Dic3C2311yescharacteristic
62C2×C4×C5×D3—22no—
63C2×C4×C5×D3—22no—
64C2×C4×C5×D3C2×C414yes—
65C22×C3×Dic5C2312yes—
66C22×C3×Dic5C2311yescharacteristic
67C22×C3×Dic5C2311yescharacteristic
68C2×#1203—22no—
69C2×#1203—22no—
70C2×#1203C2×C414yes—
71C2×D3×Dic5—24no—
72C2×D3×Dic5—24no—
73C2×D3×Dic5—42no—
74C3×C5×C4⋊C4—22no—
75#6714—24no—
76C4⋊Dic15D414yescomplemented
77C4×Dic15—24no—
78C5×(C2×D3)⋊C4—24no—
79C5×C4⋊Dic3—22no—
80C3×C4⋊Dic5—22no—
81C3×C4×Dic5C2×C414yescomplemented
82Dic3⋊Dic5—24no—
83#6668C2×C414yescomplemented
84#6667—42no—
85#6667C2×C414yescomplemented
86(C2×D3)⋊Dic5C2×C414yes—
87Dic3×Dic5—42no—
88C2×#892—51no—
89C2×D3×C4⋊C4—51no—
90C2×C4×D12—51no—
91C3×#382—51no—
92#4987—51no—
93#4987D511yescharacteristic, complemented
94C4×C4⋊Dic3—51no—
95#4669—52no—
96#4666—52no—
97#4665—52no—
98#4654—52no—
99C23×C4×C5—31no—
100C22×C22⋊D5—31no—
101C23×Dic5—32no—
102C22×C4×D5—31no—
103C2×C5×C4⋊C4—31no—
104C2×C5×C4⋊C4C2×D311yescharacteristic
105C2×C5×C22⋊C4—32no—
106C2×C22⋊Dic5—31no—
107C4×C22⋊D5—38no—
108C2×#685—31no—
109C2×#685—32no—
110C2×#685—32no—
111C2×C4⋊Dic5—31no—
112C2×C4⋊Dic5—32no—
113C2×C4⋊Dic5—64no—
114C2×C4⋊Dic5C2×D311yescharacteristic
115C2×#683—31no—
116C2×C4×Dic5—32no—
117C2×C4×Dic5—32no—
118C2×C4×Dic5C2×D311yescharacteristic
119#4038—64no—
120C5×#95—32no—
121#3963—32no—
122#3963—32no—
123#3963C2×D312yes—
124#3963C2×D312yes—
125#2352—151noSylow
126C23×C3×C5C2411yescharacteristic
127C22×D15—21no—
128C22×D15—21no—
129C22×D15—21no—
130C22×D15—21no—
131C22×D15—21no—
132C22×D15—21no—
133C22×D15C22×C412yes—
134C22×C5×D3—21no—
135C22×C5×D3—21no—
136C22×C5×D3—21no—
137C22×C5×D3—21no—
138C22×C5×D3C22×C412yes—
139C22×C5×D3#4112yes—
140C22×C5×D3#4112yes—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5C22×C412yes—
145C2×C3×C4×C5C2×D412yes—
146C2×C3×C4×C5#4112yes—
147C2×Dic15—21no—
148C2×Dic15—22no—
149C2×Dic15—21no—
150C2×Dic15—22no—
151C2×Dic15C22×C412yes—
152C2×Dic15C2×D414yes—
153C4×D15—28no—
154C2×C5×Dic3—21no—
155C2×C5×Dic3—21no—
156C2×C5×Dic3—22no—
157C2×C5×Dic3—22no—
158C2×C5×Dic3—22no—
159C2×C5×Dic3C22×C412yes—
160C4×C5×D3—28no—
161C2×C3×Dic5—21no—
162C2×C3×Dic5—21no—
163C2×C3×Dic5—21no—
164C2×C3×Dic5—21no—
165C2×C3×Dic5—22no—
166C2×C3×Dic5—22no—
167C2×C3×Dic5—22no—
168C2×C3×Dic5C22×C412yes—
169C2×C3×Dic5C22×C412yes—
170#1203—28no—
171D3×Dic5—48no—
172C22×D12—51no—
173C22×C4×D3—51no—
174C22×C4×D3—52no—
175C22×C4×D3C2×D511yescharacteristic
176C2×C3×C4⋊C4—51no—
177C2×C3×C4⋊C4C2×D511yescharacteristic
178C2×C3×C42—51no—
179C2×(C2×D3)⋊C4—51no—
180C2×(C2×D3)⋊C4—51no—
181C2×(C2×D3)⋊C4—52no—
182C2×(C2×D3)⋊C4—52no—
183C2×(C2×D3)⋊C4C2×D512yes—
184C2×C4⋊Dic3—51no—
185C2×C4⋊Dic3—51no—
186C2×C4⋊Dic3C2×D511yescharacteristic
187C2×Dic3⋊C4—52no—
188C2×C4×Dic3—52no—
189C2×C4×Dic3—52no—
190#892—104no—
191D3×C4⋊C4—104no—
192C4×D12—58no—
193C3×#95—52no—
194C3×#95—52no—
195C3⋊#95—52no—
196C3⋊#95—52no—
197C3⋊#95C2×D512yes—
198C24×C5—31no—
199C23×D5—31no—
200C22×C4×C5—31no—
201C22×C4×C5—34no—
202C22×C4×C5—32no—
203C22×C4×C5—62no—
204C22×C4×C5—62no—
205C22×C4×C5C22×D312yes—
206C22×C4×C5C22×D311yescharacteristic
207C2×C22⋊D5—34no—
208C2×C22⋊D5—62no—
209C2×C22⋊D5—62no—
210C22×Dic5—31no—
211C22×Dic5—32no—
212C22×Dic5—64no—
213C22×Dic5—64no—
214C22×Dic5—122no—
215C22×Dic5C22×D312yes—
216C22×Dic5C22×D311yescharacteristic
217C22×Dic5C22×D311yescharacteristic
218C2×C4×D5—34no—
219C2×C4×D5—62no—
220C2×C4×D5—62no—
221C5×C4⋊C4—22no—
222C5×C4⋊C4—62no—
223C5×C22⋊C4—64no—
224C22⋊Dic5—34no—
225#685—34no—
226#685—64no—
227#685—122no—
228C4⋊Dic5—22no—
229C4⋊Dic5—34no—
230C4⋊Dic5—64no—
231#683—34no—
232C4×Dic5—64no—
233C4×Dic5—122no—
234C4×Dic5C4×D314yescomplemented
235C2×C4×D4—151no—
236C2×#117—151no—
237C22×C4⋊C4—151no—
238#388—152no—
239#387—152no—
240#386—152no—
241#382—51no—
242#380—31no—
243#380—151no—
244C4×C4⋊C4—151no—
245C4×C22⋊C4—152no—
246C22×C3×C5C23×C411yescharacteristic
247C22×C3×C5C22×D411yescharacteristic
248C22×C3×C5C2×#4111yescharacteristic
249C22×C3×C5C2×#4111yescharacteristic
250C22×C3×C5C2×#4111yescharacteristic
251C22×C3×C5H(5,𝔽2)11yescharacteristic
252C22×C3×C52−1+411yescharacteristic
253C2×D15—22no—
254C2×D15—22no—
255C2×D15—22no—
256C2×D15—41no—
257C2×D15—41no—
258C2×D15—41no—
259C2×D15—41no—
260C2×C5×D3—22no—
261C2×C5×D3—22no—
262C2×C5×D3—22no—
263C2×C5×D3—22no—
264C2×C5×D3—22no—
265C2×C5×D3—22no—
266C2×C5×D3#11714yescomplemented
267C3×C4×C5—42no—
268C3×C4×C5C4×D414yes—
269Dic15—22no—
270Dic15—24no—
271C3×Dic5—22no—
272C3×Dic5—22no—
273C3×Dic5—42no—
274C5×Dic3—22no—
275C5×Dic3—42no—
276C23×D3—51no—
277C23×D3C22×D511yescharacteristic
278C22×C3×C4—51no—
279C22×C3×C4—51no—
280C22×C3×C4—52no—
281C22×C3×C4C22×D512yes—
282C22×C3×C4C22×D511yescharacteristic
283C22×Dic3—51no—
284C22×Dic3—52no—
285C22×Dic3C22×D512yes—
286C22×Dic3C22×D511yescharacteristic
287C2×D12—54no—
288C2×D12—102no—
289C2×D12—102no—
290C2×C4×D3—22no—
291C2×C4×D3—22no—
292C2×C4×D3—54no—
293C2×C4×D3—102no—
294C2×C4×D3—102no—
295C2×C4×D3—104no—
296C2×C4×D3—104no—
297C2×C4×D3—202no—
298C2×C4×D3C4×D514yes—
299C3×C4⋊C4—22no—
300C3×C4⋊C4—102no—
301C3×C42—54no—
302(C2×D3)⋊C4—24no—
303(C2×D3)⋊C4—54no—
304(C2×D3)⋊C4—54no—
305(C2×D3)⋊C4—104no—
306(C2×D3)⋊C4—202no—
307C4⋊Dic3—22no—
308C4⋊Dic3—54no—
309C4⋊Dic3—54no—
310Dic3⋊C4—104no—
311C4×Dic3—104no—
312C4×Dic3—202no—
313C23×C5—32no—
314C23×C5—32no—
315C23×C5—32no—
316C23×C5—61no—
317C23×C5—61no—
318C23×C5—61no—
319C23×C5—61no—
320C23×C5C23×D311yescharacteristic
321C22×D5—32no—
322C22×D5—61no—
323C22×D5—61no—
324C22×D5—61no—
325C22×D5—61no—
326C22×D5—61no—
327C22×D5—61no—
328C2×C4×C5—22no—
329C2×C4×C5—22no—
330C2×C4×C5—22no—
331C2×C4×C5—32no—
332C2×C4×C5—61no—
333C2×C4×C5—61no—
334C2×C4×C5—68no—
335C2×C4×C5—62no—
336C2×C4×C5—62no—
337C2×C4×C5—62no—
338C2×C4×C5C2×C4×D312yes—
339C2×C4×C5D3×D412yes—
340C2×C4×C5#23912yes—
341C22⋊D5—68no—
342C2×Dic5—21no—
343C2×Dic5—21no—
344C2×Dic5—21no—
345C2×Dic5—21no—
346C2×Dic5—22no—
347C2×Dic5—22no—
348C2×Dic5—22no—
349C2×Dic5—32no—
350C2×Dic5—34no—
351C2×Dic5—61no—
352C2×Dic5—61no—
353C2×Dic5—62no—
354C2×Dic5—62no—
355C2×Dic5—128no—
356C2×Dic5C2×C4×D312yes—
357C2×Dic5C2×C4×D312yes—
358C4×D5—68no—
359C22×D4—151no—
360C23×C4—31no—
361C23×C4—152no—
362C23×C4—151no—
363C4×D4—158no—
364#117—304no—
365C2×C4⋊C4—31no—
366C2×C4⋊C4—51no—
367C2×C4⋊C4—151no—
368C2×C4⋊C4—151no—
369C2×C4⋊C4—152no—
370C2×C4⋊C4—304no—
371C2×C4⋊C4D3×D511yescharacteristic
372C2×C22⋊C4—32no—
373C2×C22⋊C4—151no—
374C2×C22⋊C4—151no—
375C2×C22⋊C4—152no—
376C2×C22⋊C4—152no—
377C2×C42—51no—
378C2×C42—152no—
379C2×C42—152no—
380#95—32no—
381#95—52no—
382#95—52no—
383#95—152no—
384#95—152no—
385C2×C3×C5C2×#11711yescharacteristic
386C2×C3×C5C2×C4×D411yescharacteristic
387C2×C3×C5#52011yescharacteristic
388C2×C3×C5#54611yescharacteristic
389C2×C3×C5#54811yescharacteristic
390C2×C3×C5#55211yescharacteristic
391C2×C3×C5#55311yescharacteristic
392D15—42no—
393C5×D3—24no—
394C23×C3C23×D511yescharacteristic
395C22×D3—21no—
396C22×D3—21no—
397C22×D3—21no—
398C22×D3—21no—
399C22×D3—52no—
400C22×D3—101no—
401C22×D3—101no—
402C22×D3—101no—
403C22×D3—101no—
404C22×D3—101no—
405C22×D3—101no—
406C22×D3C2×C4×D512yes—
407C22×D3#71112yes—
408C22×D3#71312yes—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—52no—
413C2×C3×C4—52no—
414C2×C3×C4—101no—
415C2×C3×C4—101no—
416C2×C3×C4—101no—
417C2×C3×C4—101no—
418C2×C3×C4—102no—
419C2×C3×C4—102no—
420C2×C3×C4—102no—
421C2×C3×C4C2×C4×D512yes—
422C2×C3×C4D4×D512yes—
423C2×C3×C4#71112yes—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—52no—
430C2×Dic3—54no—
431C2×Dic3—101no—
432C2×Dic3—101no—
433C2×Dic3—102no—
434C2×Dic3—102no—
435C2×Dic3C2×C4×D512yes—
436D12—108no—
437C4×D3—28no—
438C4×D3—108no—
439C4×D3—208no—
440C22×C5—34no—
441C22×C5—62no—
442C22×C5—62no—
443C22×C5—62no—
444C22×C5—62no—
445C22×C5—62no—
446C22×C5—62no—
447C22×C5C22×C4×D311yescharacteristic
448C22×C5C2×D3×D411yescharacteristic
449C22×C5C2×#23911yescharacteristic
450C22×C5D3×#4111yescharacteristic
451C22×C5D3×#4111yescharacteristic
452C22×C5#100911yescharacteristic
453C22×C5#101011yescharacteristic
454C2×D5—62no—
455C2×D5—62no—
456C2×D5—62no—
457C2×D5—121no—
458C2×D5—121no—
459C2×D5—121no—
460C2×D5—121no—
461C4×C5—42no—
462C4×C5—62no—
463C4×C5—122no—
464C4×C5#89314yes—
465Dic5—22no—
466Dic5—22no—
467Dic5—42no—
468Dic5—64no—
469Dic5—62no—
470C24—31no—
471C24—151no—
472C2×D4—154no—
473C2×D4—302no—
474C2×D4—302no—
475C22×C4—31no—
476C22×C4—34no—
477C22×C4—32no—
478C22×C4—51no—
479C22×C4—51no—
480C22×C4—52no—
481C22×C4—62no—
482C22×C4—62no—
483C22×C4—151no—
484C22×C4—152no—
485C22×C4—154no—
486C22×C4—304no—
487C22×C4—304no—
488C22×C4—302no—
489C22×C4—302no—
490C22×C4—602no—
491C22×C4C2×D3×D512yes—
492C22×C4C2×D3×D511yescharacteristic
493C4⋊C4—22no—
494C4⋊C4—62no—
495C4⋊C4—102no—
496C4⋊C4—154no—
497C4⋊C4—154no—
498C4⋊C4—304no—
499C22⋊C4—64no—
500C22⋊C4—154no—
501C22⋊C4—154no—
502C22⋊C4—304no—
503C22⋊C4—602no—
504C42—54no—
505C42—304no—
506C42—602no—
507C3×C5#235211yescharacteristic, complemented, Hall
508C22×C3C22×C4×D511yescharacteristic
509C22×C3C2×D4×D511yescharacteristic
510C22×C3C2×#71111yescharacteristic
511C22×C3C2×#71111yescharacteristic
512C22×C3C2×#71311yescharacteristic
513C22×C3#414911yescharacteristic
514C22×C3#415011yescharacteristic
515C2×D3—22no—
516C2×D3—22no—
517C2×D3—22no—
518C2×D3—22no—
519C2×D3—22no—
520C2×D3—22no—
521C2×D3—102no—
522C2×D3—102no—
523C2×D3—102no—
524C2×D3—201no—
525C2×D3—201no—
526C2×D3—201no—
527C2×D3—201no—
528C2×D3#403814yescomplemented
529C3×C4—42no—
530C3×C4—102no—
531C3×C4—102no—
532C3×C4—202no—
533C3×C4#402714yes—
534Dic3—22no—
535Dic3—42no—
536Dic3—102no—
537Dic3—104no—
538C2×C5—64no—
539C2×C5C2×#89311yescharacteristic
540C2×C5D3×#11711yescharacteristic
541C2×C5#552311yescharacteristic
542C2×C5#567611yescharacteristic
543C2×C5#567711yescharacteristic
544C2×C5#569411yescharacteristic
545C2×C5#569511yescharacteristic
546D5—122no—
547C23—32no—
548C23—32no—
549C23—32no—
550C23—61no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23—152no—
555C23—301no—
556C23—301no—
557C23—301no—
558C23—301no—
559C23—301no—
560C23—301no—
561C23C22×D3×D511yescharacteristic
562D4—308no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—32no—
567C2×C4—52no—
568C2×C4—52no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—68no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—101no—
576C2×C4—101no—
577C2×C4—101no—
578C2×C4—101no—
579C2×C4—102no—
580C2×C4—102no—
581C2×C4—102no—
582C2×C4—152no—
583C2×C4—154no—
584C2×C4—301no—
585C2×C4—302no—
586C2×C4—302no—
587C2×C4—301no—
588C2×C4—308no—
589C2×C4—608no—
590C2×C4#676812yes—
591C2×C4C4×D3×D512yes—
592C2×C4#679012yes—
593C2×C3C2×#402711yescharacteristic
594C2×C3C2×#403811yescharacteristic
595C2×C3#6585711yescharacteristic
596C2×C3#6593211yescharacteristic
597C2×C3#6595211yescharacteristic
598C2×C3#6596411yescharacteristic
599C2×C3#6597611yescharacteristic
600D3—24no—
601D3—202no—
602C5#7422911yescharacteristic, complemented, Sylow
603C22—34no—
604C22—62no—
605C22—62no—
606C22—62no—
607C22—62no—
608C22—62no—
609C22—62no—
610C22—302no—
611C22—302no—
612C22—302no—
613C22—601no—
614C22—601no—
615C22—601no—
616C22—601no—
617C22C2×#676811yescharacteristic
618C22C2×C4×D3×D511yescharacteristic
619C22#9202011yescharacteristic
620C22D3×#71111yescharacteristic
621C22D3×#71311yescharacteristic
622C22#9203511yescharacteristic
623C22C2×#679011yescharacteristic
624C4—42no—
625C4—62no—
626C4—102no—
627C4—102no—
628C4—122no—
629C4—202no—
630C4—304no—
631C4—302no—
632C4#9140614yes—
633C3#1060561411yescharacteristic, complemented, Sylow
634C2—64no—
635C2—602no—
636C2#1175358511yescharacteristic
637C2#1175359911yescharacteristic
638C2#1175367011yescharacteristic
639C2D3×#403811yescharacteristic
640C2C2×#9140611yescharacteristic
641C2#1175482711yescharacteristic
642C2#1175528411yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#1203C22
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2352
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
1. C2×#91781C2semidirect
3. C2×D3×C4⋊Dic5C2semidirect
4. C5×#4987C2semidirect
5. C3×#65266C2semidirect
6. #11748886C2semidirect
8. #11748810C2semidirect
9. #11748799C2semidirect
10. Dic5×C4⋊Dic3C2semidirect
11. #11748668C2semidirect
32. C4×#1203C4semidirect
45. #65266D3semidirect
59. C2×C4×D15C2×C4semidirect
76. C4⋊Dic15D4semidirect
81. C3×C4×Dic5C2×C4semidirect
83. #6668C2×C4semidirect
85. #6667C2×C4semidirect
93. #4987D5semidirect
234. C4×Dic5C4×D3semidirect
266. C2×C5×D3#117semidirect
507. C3×C5#2352semidirect
528. C2×D3#4038semidirect
602. C5#74229semidirect
633. C3#10605614semidirect

Maximal subgroups

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

subgroupindex
C2×#917812
C2×C4×#12032
C2×D3×C4⋊Dic52
C5×#49872
C3×#652662
#117488862
#117488832
#117488102
#117487992
Dic5×C4⋊Dic32
#117486682
#106056143
#742295