Finite Group Databasebeta

← #14321804

Subgroups — #14321804

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91835C211yesmaximal, characteristic
2C2×C3×Q8×Dic5C211yesmaximal, characteristic, complemented
3C2×C4×#1200C211yesmaximal, characteristic
4C5×#5229C211yesmaximal, characteristic, complemented
5#11748898C212yesmaximal
6#11748875C212yesmaximal
7#11748858C211yesmaximal, characteristic, complemented
8#11748799C211yesmaximal, characteristic
9Dic5×C4⋊Dic3C211yesmaximal, characteristic, complemented
10#11748674C212yesmaximal, complemented
11#11748640C212yesmaximal, complemented
12#10608210—31nomaximal, Hall
13C22×C3×C5×Q8C2211yescharacteristic
14C22×C4×D15C2211yescharacteristic
15C22×#1200C2211yescharacteristic
16C22×#1200C2211yescharacteristic
17#91835C418yescomplemented
18C2×#6714C2212yes—
19C2×C4⋊Dic15C2211yescharacteristic
20C2×#6712C2212yes—
21C2×C5×C4⋊Dic3C2211yescharacteristic
22C2×C4×C5×Dic3C2211yescharacteristic
23C3×Q8×Dic5—24no—
24C2×C3×C4⋊Dic5C2212yes—
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×C4×Dic5C2212yes—
27C2×C3×C4×Dic5C2211yescharacteristic
28C2×#6669C2212yes—
29C2×#6667C2212yes—
30C2×Dic3×Dic5C2212yes—
31C2×Dic3×Dic5C2211yescharacteristic
32C2×Dic3×Dic5C2211yescharacteristic
33C4×#1200—24no—
34C5×C3⋊#95C2212yes—
35C5×C3⋊#95C2212yes—
36#90994C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39C4×#946—51nomaximal, Hall
40C2×#4092—31no—
41C2×Q8×Dic5D311yescharacteristic, complemented
42C2×C42×D5—31no—
43C5×#391—31no—
44#65276—31no—
45C4⋊C4×Dic5—31no—
46#65217—31no—
47#64957—32no—
48#64955—32no—
49#64944—32no—
50#64938—32no—
51C23×D15C2311yescharacteristic
52C2×C3×C5×Q8—22no—
53C2×C3×C5×Q8—22no—
54C2×C3×C5×Q8C2×C414yescomplemented
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2311yescharacteristic
57C22×Dic15C2311yescharacteristic
58C22×Dic15C2312yes—
59C2×C4×D15—22no—
60C2×C4×D15—22no—
61C2×C4×D15C2×C414yes—
62C22×C5×Dic3C2312yes—
63C22×C5×Dic3C2311yescharacteristic
64C22×C5×Dic3C2311yescharacteristic
65C22×C3×Dic5C2312yes—
66C22×C3×Dic5C2311yescharacteristic
67C22×C3×Dic5C2311yescharacteristic
68C2×#1200—22no—
69C2×#1200—22no—
70C2×#1200—22no—
71C2×#1200—22no—
72C2×#1200—22no—
73C2×#1200Q814yescomplemented
74#6714C2×C418yes—
75C4⋊Dic15C2×C414yescomplemented
76#6712C2×C418yescomplemented
77C5×C4⋊Dic3—22no—
78C4×C5×Dic3—22no—
79C3×C4⋊Dic5—22no—
80C3×C4⋊Dic5—24no—
81C3×C4×Dic5—24no—
82C3×C4×Dic5D414yes—
83#6669—24no—
84#6667—24no—
85Dic3×Dic5—22no—
86Dic3×Dic5—22no—
87Dic3×Dic5—24no—
88C2×C3×C4×Q8—51no—
89C2×#946—51no—
90C2×C42×D3—51no—
91#5229D511yescharacteristic, complemented
92#4989—52no—
93#4987—51no—
94#4978—52no—
95#4972—51no—
96C4×(C2×D3)⋊C4—52no—
97C4×C4⋊Dic3—51no—
98C4×Dic3⋊C4—52no—
99C22×C5×Q8C2×D311yescharacteristic
100C22×C4×D5—31no—
101C22×C4×D5—31no—
102C22×C4×D5—31no—
103C2×C5×C4⋊C4—31no—
104C2×C42×C5—31no—
105#4092—38no—
106Q8×Dic5—24no—
107C2×#685—32no—
108C2×#685—32no—
109C2×C4⋊Dic5—31no—
110C2×C4⋊Dic5C2×D312yes—
111C2×C4⋊Dic5C2×D311yescharacteristic
112C2×#683—32no—
113C2×#683—32no—
114C2×C4×Dic5—31no—
115C2×C4×Dic5—31no—
116C2×C4×Dic5—32no—
117C2×C4×Dic5C2×D312yes—
118C2×C4×Dic5C2×D311yescharacteristic
119C42×D5—64no—
120C5×#95—32no—
121C5×#95—32no—
122#3963—32no—
123#3963—32no—
124#3963—32no—
125C4×C22⋊Q8—151noSylow
126C23×C3×C5C2411yescharacteristic
127C22×D15—21no—
128C22×D15—21no—
129C22×D15—21no—
130C22×D15—21no—
131C22×D15C22×C412yes—
132C22×D15C2×Q812yes—
133C22×D15#4112yes—
134C3×C5×Q8—28no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—22no—
137C2×C3×C4×C5C22×C414yes—
138C2×C3×C4×C5C22×C412yes—
139C2×C3×C4×C5C2×D412yes—
140C2×C3×C4×C5#4112yes—
141C2×Dic15—21no—
142C2×Dic15—22no—
143C2×Dic15—22no—
144C2×Dic15—21no—
145C2×Dic15C22×C412yes—
146C2×Dic15C22×C414yes—
147C4×D15—28no—
148C2×C5×Dic3—21no—
149C2×C5×Dic3—21no—
150C2×C5×Dic3—21no—
151C2×C5×Dic3—21no—
152C2×C5×Dic3—22no—
153C2×C5×Dic3—22no—
154C2×C5×Dic3—22no—
155C2×C5×Dic3C2×Q812yes—
156C2×C5×Dic3#4112yes—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—22no—
160C2×C3×Dic5—22no—
161C2×C3×Dic5—22no—
162C2×C3×Dic5C2×D412yes—
163C2×C3×Dic5C2×D412yes—
164C2×C3×Dic5C2×Q812yes—
165C2×C3×Dic5#4112yes—
166#1200—28no—
167#1200—28no—
168C22×C3×Q8C2×D511yescharacteristic
169C22×C4×D3—51no—
170C22×C4×D3—51no—
171C22×C4×D3—51no—
172C3×C4×Q8—104no—
173C2×C3×C4⋊C4—51no—
174C2×C3×C4⋊C4—52no—
175C2×C3×C42—51no—
176C2×C3×C42—52no—
177#946—58no—
178C2×(C2×D3)⋊C4—52no—
179C2×(C2×D3)⋊C4—52no—
180C2×C4⋊Dic3—51no—
181C2×C4⋊Dic3C2×D511yescharacteristic
182C2×Dic3⋊C4—52no—
183C2×Dic3⋊C4—52no—
184C2×C4×Dic3—51no—
185C2×C4×Dic3—52no—
186C2×C4×Dic3—51no—
187C2×C4×Dic3C2×D511yescharacteristic
188C42×D3—104no—
189C3⋊#95—52no—
190C3⋊#95—52no—
191C3⋊#95—52no—
192C3⋊#95C2×D512yes—
193C3⋊#95C2×D512yes—
194C23×D5—31no—
195C2×C5×Q8—22no—
196C2×C5×Q8—22no—
197C2×C5×Q8C4×D314yescomplemented
198C22×C4×C5—31no—
199C22×C4×C5—31no—
200C22×C4×C5—32no—
201C22×C4×C5C22×D312yes—
202C22×C4×C5C22×D311yescharacteristic
203C22×Dic5—31no—
204C22×Dic5—32no—
205C22×Dic5C22×D312yes—
206C22×Dic5C22×D311yescharacteristic
207C22×Dic5C22×D311yescharacteristic
208C2×C4×D5—34no—
209C2×C4×D5—34no—
210C2×C4×D5—62no—
211C2×C4×D5—62no—
212C2×C4×D5—62no—
213C2×C4×D5—62no—
214C2×C4×D5—62no—
215C2×C4×D5—62no—
216C2×C4×D5—62no—
217C5×C4⋊C4—62no—
218C42×C5—62no—
219#685—38no—
220#685—64no—
221C4⋊Dic5—22no—
222C4⋊Dic5—24no—
223C4⋊Dic5—34no—
224#683—38no—
225#683—64no—
226C4×Dic5—24no—
227C4×Dic5—64no—
228C4×Dic5—62no—
229C4×Dic5—62no—
230C4×Dic5#6714yes—
231C2×C22⋊Q8—151no—
232C2×C4×Q8—51no—
233C22×C42—151no—
234#391—31no—
235#389—151no—
236#387—152no—
237#385—152no—
238#380—151no—
239C4×C4⋊C4—152no—
240C4×C4⋊C4—151no—
241C4×C22⋊C4—152no—
242C22×C3×C5C23×C411yescharacteristic
243C22×C3×C5C22×D411yescharacteristic
244C22×C3×C5C22×Q811yescharacteristic
245C22×C3×C5C2×#4111yescharacteristic
246C22×C3×C5C2×#4111yescharacteristic
247C22×C3×C5C2×#4111yescharacteristic
248C22×C3×C5C2×#4111yescharacteristic
249C2×D15—22no—
250C2×D15—22no—
251C2×D15—22no—
252C2×D15—22no—
253C2×D15—22no—
254C2×D15—22no—
255C2×D15C4×Q814yescomplemented
256C3×C4×C5—24no—
257C3×C4×C5C4×D414yes—
258Dic15—22no—
259Dic15—24no—
260C3×Dic5—22no—
261C3×Dic5—42no—
262C3×Dic5C22⋊Q814yescomplemented
263C5×Dic3—22no—
264C5×Dic3—22no—
265C5×Dic3—42no—
266C23×D3—51no—
267C2×C3×Q8—22no—
268C2×C3×Q8—22no—
269C2×C3×Q8C4×D514yescomplemented
270C22×C3×C4—51no—
271C22×C3×C4—51no—
272C22×C3×C4—52no—
273C22×C3×C4C22×D512yes—
274C22×C3×C4C22×D511yescharacteristic
275C22×Dic3—51no—
276C22×Dic3—52no—
277C22×Dic3C22×D512yes—
278C22×Dic3C22×D511yescharacteristic
279C22×Dic3C22×D511yescharacteristic
280C2×C4×D3—54no—
281C2×C4×D3—54no—
282C2×C4×D3—102no—
283C2×C4×D3—102no—
284C2×C4×D3—102no—
285C2×C4×D3—102no—
286C2×C4×D3—102no—
287C2×C4×D3—102no—
288C2×C4×D3—102no—
289C3×C4⋊C4—104no—
290C3×C4⋊C4—102no—
291C3×C42—54no—
292C3×C42—104no—
293(C2×D3)⋊C4—58no—
294(C2×D3)⋊C4—104no—
295C4⋊Dic3—22no—
296C4⋊Dic3—54no—
297Dic3⋊C4—58no—
298Dic3⋊C4—104no—
299C4×Dic3—22no—
300C4×Dic3—102no—
301C4×Dic3—102no—
302C4×Dic3—104no—
303C23×C5C23×D311yescharacteristic
304C22×D5—32no—
305C22×D5—32no—
306C22×D5—32no—
307C22×D5—61no—
308C22×D5—61no—
309C22×D5—61no—
310C22×D5—61no—
311C5×Q8—28no—
312C2×C4×C5—22no—
313C2×C4×C5—22no—
314C2×C4×C5—32no—
315C2×C4×C5—32no—
316C2×C4×C5—61no—
317C2×C4×C5—61no—
318C2×C4×C5—61no—
319C2×C4×C5—61no—
320C2×C4×C5—62no—
321C2×C4×C5—62no—
322C2×C4×C5—62no—
323C2×C4×C5C2×C4×D314yes—
324C2×C4×C5C2×C4×D312yes—
325C2×C4×C5#23512yes—
326C2×C4×C5C2×#6712yes—
327C2×Dic5—21no—
328C2×Dic5—21no—
329C2×Dic5—22no—
330C2×Dic5—22no—
331C2×Dic5—22no—
332C2×Dic5—32no—
333C2×Dic5—34no—
334C2×Dic5—61no—
335C2×Dic5—62no—
336C2×Dic5—62no—
337C2×Dic5—61no—
338C2×Dic5D3×Q812yes—
339C2×Dic5#23912yes—
340C2×Dic5C2×#6712yes—
341C2×Dic5C2×#6712yes—
342C4×D5—68no—
343C4×D5—68no—
344C4×D5—68no—
345C22×Q8D3×D511yescharacteristic
346C23×C4—151no—
347C23×C4—151no—
348C23×C4—151no—
349C22⋊Q8—158no—
350C4×Q8—104no—
351C2×C4⋊C4—31no—
352C2×C4⋊C4—51no—
353C2×C4⋊C4—52no—
354C2×C4⋊C4—152no—
355C2×C4⋊C4—151no—
356C2×C4⋊C4—152no—
357C2×C22⋊C4—152no—
358C2×C22⋊C4—152no—
359C2×C42—31no—
360C2×C42—51no—
361C2×C42—52no—
362C2×C42—152no—
363C2×C42—151no—
364C2×C42—151no—
365C2×C42—304no—
366#95—32no—
367#95—32no—
368#95—152no—
369#95—152no—
370#95—152no—
371C2×C3×C5C2×C4×D411yescharacteristic
372C2×C3×C5C2×C4×Q811yescharacteristic
373C2×C3×C5C4×#4111yescharacteristic
374C2×C3×C5C2×C22⋊Q811yescharacteristic
375C2×C3×C5#52511yescharacteristic
376C2×C3×C5#52911yescharacteristic
377C2×C3×C5#53311yescharacteristic
378D15—24no—
379C23×C3C23×D511yescharacteristic
380C22×D3—52no—
381C22×D3—52no—
382C22×D3—52no—
383C22×D3—101no—
384C22×D3—101no—
385C22×D3—101no—
386C22×D3—101no—
387C3×Q8—28no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—52no—
391C2×C3×C4—52no—
392C2×C3×C4—52no—
393C2×C3×C4—52no—
394C2×C3×C4—101no—
395C2×C3×C4—101no—
396C2×C3×C4—102no—
397C2×C3×C4—102no—
398C2×C3×C4—102no—
399C2×C3×C4C2×C4×D514yes—
400C2×C3×C4C2×C4×D512yes—
401C2×C3×C4D4×D512yes—
402C2×C3×C4#71112yes—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—52no—
411C2×Dic3—54no—
412C2×Dic3—101no—
413C2×Dic3—101no—
414C2×Dic3—102no—
415C2×Dic3—102no—
416C2×Dic3Q8×D512yes—
417C2×Dic3#71312yes—
418C4×D3—108no—
419C4×D3—108no—
420C4×D3—108no—
421C22×C5C22×C4×D311yescharacteristic
422C22×C5C2×#23511yescharacteristic
423C22×C5C2×D3×Q811yescharacteristic
424C22×C5C2×#23911yescharacteristic
425C22×C5D3×#4111yescharacteristic
426C22×C5D3×#4111yescharacteristic
427C22×C5C22×#6711yescharacteristic
428C2×D5—34no—
429C2×D5—62no—
430C2×D5—62no—
431C2×D5—62no—
432C2×D5—62no—
433C2×D5—62no—
434C2×D5—62no—
435C4×C5—24no—
436C4×C5—62no—
437C4×C5—62no—
438C4×C5—122no—
439C4×C5C4×#6714yes—
440Dic5—22no—
441Dic5—42no—
442Dic5—62no—
443Dic5—64no—
444Dic5#94614yescomplemented
445C24—151no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8#120014yescomplemented
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—32no—
452C22×C4—51no—
453C22×C4—51no—
454C22×C4—52no—
455C22×C4—154no—
456C22×C4—151no—
457C22×C4—152no—
458C22×C4—154no—
459C22×C4—302no—
460C22×C4—302no—
461C22×C4—302no—
462C22×C4—302no—
463C22×C4—302no—
464C22×C4—302no—
465C22×C4—302no—
466C22×C4C2×D3×D512yes—
467C22×C4C2×D3×D511yescharacteristic
468C4⋊C4—62no—
469C4⋊C4—104no—
470C4⋊C4—102no—
471C4⋊C4—158no—
472C4⋊C4—154no—
473C4⋊C4—304no—
474C22⋊C4—158no—
475C22⋊C4—304no—
476C42—54no—
477C42—62no—
478C42—104no—
479C42—302no—
480C42—302no—
481C42—304no—
482C3×C5C4×C22⋊Q811yescharacteristic, complemented, Hall
483C22×C3C22×C4×D511yescharacteristic
484C22×C3C2×D4×D511yescharacteristic
485C22×C3C2×#71111yescharacteristic
486C22×C3C2×Q8×D511yescharacteristic
487C22×C3C2×#71311yescharacteristic
488C22×C3D5×#4111yescharacteristic
489C22×C3D5×#4111yescharacteristic
490C2×D3—54no—
491C2×D3—102no—
492C2×D3—102no—
493C2×D3—102no—
494C2×D3—102no—
495C2×D3—102no—
496C2×D3—102no—
497C3×C4—24no—
498C3×C4—54no—
499C3×C4—102no—
500C3×C4—202no—
501C3×C4#402714yes—
502Dic3—22no—
503Dic3—22no—
504Dic3—42no—
505Dic3—102no—
506Dic3—104no—
507C2×C5C4×D3×Q811yescharacteristic
508C2×C5C4×#23911yescharacteristic
509C2×C5#557811yescharacteristic
510C2×C5#558011yescharacteristic
511C2×C5C2×C4×#6711yescharacteristic
512C2×C5C2×#94611yescharacteristic
513C2×C5#582811yescharacteristic
514D5—64no—
515C23—152no—
516C23—152no—
517C23—152no—
518C23—301no—
519C23—301no—
520C23—301no—
521C23—301no—
522C23C22×D3×D511yescharacteristic
523Q8—28no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—32no—
527C2×C4—32no—
528C2×C4—52no—
529C2×C4—52no—
530C2×C4—52no—
531C2×C4—52no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—101no—
540C2×C4—101no—
541C2×C4—102no—
542C2×C4—102no—
543C2×C4—102no—
544C2×C4—154no—
545C2×C4—152no—
546C2×C4—308no—
547C2×C4—308no—
548C2×C4—302no—
549C2×C4—302no—
550C2×C4—301no—
551C2×C4—301no—
552C2×C4—308no—
553C2×C4#678212yes—
554C2×C4C2×#120014yes—
555C2×C4C2×#120012yes—
556C2×C4D5×#6712yes—
557C2×C3C2×#402711yescharacteristic
558C2×C3C4×Q8×D511yescharacteristic
559C2×C3C4×#71311yescharacteristic
560C2×C3D5×C22⋊Q811yescharacteristic
561C2×C3#6596111yescharacteristic
562C2×C3#6601411yescharacteristic
563C2×C3#6606211yescharacteristic
564D3—104no—
565C5C4×#94611yescharacteristic, complemented, Sylow
566C22—154no—
567C22—302no—
568C22—302no—
569C22—302no—
570C22—302no—
571C22—302no—
572C22—302no—
573C22D3×Q8×D511yescharacteristic
574C22D5×#23911yescharacteristic
575C22D3×#71311yescharacteristic
576C22#9203411yescharacteristic
577C22C2×#678211yescharacteristic
578C22C22×#120011yescharacteristic
579C22C2×D5×#6711yescharacteristic
580C4—24no—
581C4—54no—
582C4—62no—
583C4—62no—
584C4—102no—
585C4—122no—
586C4—202no—
587C4—302no—
588C4—304no—
589C4#9155214yes—
590C3#1060821011yescharacteristic, complemented, Sylow
591C2—304no—
592C2Q8×#120011yescharacteristic
593C2#1175518411yescharacteristic
594C2#1175519611yescharacteristic
595C2#1175520011yescharacteristic
596C2D5×#94611yescharacteristic
597C2#1175527011yescharacteristic
598C2C2×#9155211yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC22×C3×C5×Q8C22
socle, cosocleC23×C3×C5C24
prime radicalC23×D15C23
hypercenterC22×Q8D3×D5
hypocenterC3×C5C4×C22⋊Q8
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×Q8×Dic5C2semidirect
4. C5×#5229C2semidirect
7. #11748858C2semidirect
9. Dic5×C4⋊Dic3C2semidirect
10. #11748674C2semidirect
11. #11748640C2semidirect
17. #91835C4semidirect
41. C2×Q8×Dic5D3semidirect
54. C2×C3×C5×Q8C2×C4semidirect
73. C2×#1200Q8semidirect
75. C4⋊Dic15C2×C4semidirect
76. #6712C2×C4semidirect
91. #5229D5semidirect
197. C2×C5×Q8C4×D3semidirect
255. C2×D15C4×Q8semidirect
262. C3×Dic5C22⋊Q8semidirect
269. C2×C3×Q8C4×D5semidirect
444. Dic5#946semidirect
448. C2×Q8#1200semidirect
482. C3×C5C4×C22⋊Q8semidirect
565. C5C4×#946semidirect
590. C3#10608210semidirect

Maximal subgroups

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

subgroupindex
C2×#918352
C2×C3×Q8×Dic52
C2×C4×#12002
C5×#52292
#117488982
#117488752
#117488582
#117487992
Dic5×C4⋊Dic32
#117486742
#117486402
#106082103
C4×#9465