Finite Group Databasebeta

← #14326130

Subgroups — #14326130

Order 1920.

1640 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D15⋊Q8C211yesmaximal, characteristic
2C2×C3×#4033C211yesmaximal, characteristic, complemented
3C2×Dic3×Dic10C211yesmaximal, characteristic, complemented
4#11751088C211yesmaximal, characteristic
5C5×#4973C211yesmaximal, characteristic, complemented
6#11748933C212yesmaximal, complemented
7#11748876C212yesmaximal
8#11748875C212yesmaximal
9C4×#6667C211yesmaximal, characteristic
10#11748628C212yesmaximal, complemented
11#11748589C211yesmaximal, characteristic, complemented
12#10608345—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C22×C3×Dic10C2211yescharacteristic
16C22×#1204C2212yes—
17C22×#1200C2212yes—
18C2×D15⋊Q8—24no—
19C2×D15⋊Q8—24no—
20C2×D15⋊Q8C418yescomplemented
21C2×C3×C5×C4⋊C4C2211yescharacteristic
22C2×#6714C2212yes—
23C2×C4⋊Dic15C2211yescharacteristic
24C2×C4×C5×Dic3C2211yescharacteristic
25C2×C3×#683C2212yes—
26C2×C3×C4×Dic5C2212yes—
27C2×C3×C4×Dic5C2211yescharacteristic
28C3×#4033—24no—
29C2×#6668C2212yes—
30C2×#6667C2211yescharacteristic
31C2×#6667C2211yescharacteristic
32C2×Dic3×Dic5C2212yes—
33Dic3×Dic10—24no—
34#91115C2212yes—
35C5×C3⋊#95C2212yes—
36C5×C3⋊#95C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#90994C2211yescharacteristic
40#90994C2211yescharacteristic
41Q8×(C2×D3)⋊C4—51nomaximal, Hall
42C22×Q8×D5—31no—
43C2×#4033D311yescharacteristic, complemented
44C2×C4×Dic10—31no—
45C5×#391—31no—
46#65519—31no—
47#65276—31no—
48C4×#685—31no—
49#64956—32no—
50#64955—32no—
51#64935—32no—
52#64935—32no—
53C23×D15C2311yescharacteristic
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×Dic15C2312yes—
57C22×Dic15C2311yescharacteristic
58C2×C4×D15—22no—
59C2×C4×D15—22no—
60C2×C4×D15C2×C414yes—
61C22×C5×Dic3C2311yescharacteristic
62C22×C5×Dic3C2312yes—
63C22×C5×Dic3C2311yescharacteristic
64C2×C5×Dic6—22no—
65C2×C5×Dic6—22no—
66C2×C5×Dic6C2×C414yes—
67C22×C3×Dic5C2311yescharacteristic
68C22×C3×Dic5C2311yescharacteristic
69C22×C3×Dic5C2312yes—
70C2×C3×Dic10C2×C414yes—
71C2×C3×Dic10D414yes—
72C2×C3×Dic10D414yes—
73C2×#1204—24no—
74C2×#1204—24no—
75C2×#1204C2×C418yes—
76C2×#1200—24no—
77C2×#1200—24no—
78C2×#1200C2×C418yes—
79D15⋊Q8—232no—
80C3×C5×C4⋊C4—22no—
81#6714—24no—
82C4⋊Dic15—22no—
83C4×C5×Dic3—22no—
84C3×#683—24no—
85C3×C4×Dic5—22no—
86C3×C4×Dic5—24no—
87#6668—24no—
88#6667—22no—
89#6667Q814yescomplemented
90Dic3×Dic5—24no—
91C22×D3×Q8—51no—
92C2×C3×C4×Q8—51no—
93C2×Q8×Dic3—51no—
94#4987—51no—
95#4987—52no—
96#4973—52no—
97#4973D511yescharacteristic, complemented
98C4×(C2×D3)⋊C4—51no—
99C4×(C2×D3)⋊C4—52no—
100#4929—51no—
101#4929—52no—
102C22×C5×Q8—31no—
103C2×Q8×D5—38no—
104C2×Q8×D5—64no—
105C2×Q8×D5—64no—
106C22×C4×D5—31no—
107C22×C4×D5—32no—
108C22×Dic10—32no—
109C22×Dic10C2×D311yescharacteristic
110C2×C5×C4⋊C4C2×D311yescharacteristic
111C2×C42×C5—31no—
112C2×#685—31no—
113C2×#685—32no—
114C2×#685—31no—
115C2×C4⋊Dic5—31no—
116C2×#683—32no—
117C2×#683C2×D312yes—
118C2×C4×Dic5—32no—
119C2×C4×Dic5C2×D312yes—
120C2×C4×Dic5C2×D311yescharacteristic
121#4033—24no—
122C4×Dic10—64no—
123C5×#95—32no—
124C5×#95—32no—
125#3963—32no—
126#3963—31no—
127#3963—31no—
128#3963—32no—
129#3963—32no—
130Q8×C22⋊C4—151noSylow
131C23×C3×C5C2411yescharacteristic
132C22×D15—21no—
133C22×D15—21no—
134C22×D15—21no—
135C22×D15—21no—
136C22×D15C22×C412yes—
137C22×D15C2×Q812yes—
138C22×D15#4112yes—
139C2×C3×C4×C5—22no—
140C2×C3×C4×C5—22no—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5C22×C412yes—
143C2×C3×C4×C5C2×D412yes—
144C2×C3×C4×C5C2×D412yes—
145C2×Dic15—21no—
146C2×Dic15—21no—
147C2×Dic15—22no—
148C2×Dic15—22no—
149C2×Dic15—22no—
150C2×Dic15C22×C412yes—
151C4×D15—28no—
152C2×C5×Dic3—21no—
153C2×C5×Dic3—22no—
154C2×C5×Dic3—22no—
155C2×C5×Dic3—21no—
156C2×C5×Dic3—21no—
157C2×C5×Dic3—21no—
158C2×C5×Dic3—21no—
159C2×C5×Dic3C22×C414yes—
160C2×C5×Dic3C2×Q812yescomplemented
161C5×Dic6—28no—
162C2×C3×Dic5—21no—
163C2×C3×Dic5—21no—
164C2×C3×Dic5—21no—
165C2×C3×Dic5—21no—
166C2×C3×Dic5—21no—
167C2×C3×Dic5C22×C414yes—
168C2×C3×Dic5C2×D414yes—
169C2×C3×Dic5C2×D414yes—
170C2×C3×Dic5C2×Q812yescomplemented
171C3×Dic10C22⋊C4116yescomplemented
172#1204—216no—
173#1200—216no—
174C22×C3×Q8—51no—
175C2×D3×Q8—58no—
176C2×D3×Q8—104no—
177C2×D3×Q8—104no—
178C22×C4×D3—51no—
179C22×C4×D3—52no—
180C22×Dic6—52no—
181C22×Dic6C2×D511yescharacteristic
182C3×C4×Q8—104no—
183C2×C3×C4⋊C4—52no—
184C2×C3×C4⋊C4C2×D511yescharacteristic
185C2×C3×C42—52no—
186C2×C3×C42—51no—
187Q8×Dic3—104no—
188C2×(C2×D3)⋊C4—51no—
189C2×(C2×D3)⋊C4—52no—
190C2×(C2×D3)⋊C4—51no—
191C2×C4⋊Dic3—51no—
192C2×C4⋊Dic3—52no—
193C2×C4×Dic3—52no—
194C2×C4×Dic3C2×D511yescharacteristic
195C3⋊#95—52no—
196C3⋊#95—51no—
197C3⋊#95—52no—
198C3⋊#95—52no—
199C3⋊#95—51no—
200C3⋊#95C2×D512yes—
201C3⋊#95C2×D512yes—
202C23×D5—31no—
203C2×C5×Q8—34no—
204C2×C5×Q8—62no—
205C2×C5×Q8—62no—
206C22×C4×C5—31no—
207C22×C4×C5—32no—
208C22×C4×C5—31no—
209C22×C4×C5C22×D312yes—
210C22×C4×C5C22×D311yescharacteristic
211C22×Dic5—31no—
212C22×Dic5—32no—
213C22×Dic5C22×D311yescharacteristic
214C22×Dic5C22×D311yescharacteristic
215C22×Dic5C22×D312yes—
216Q8×D5—632no—
217C2×C4×D5—34no—
218C2×C4×D5—38no—
219C2×C4×D5—62no—
220C2×C4×D5—62no—
221C2×C4×D5—64no—
222C2×C4×D5—64no—
223C2×Dic10—38no—
224C2×Dic10—64no—
225C2×Dic10—64no—
226C2×Dic10C4×D314yes—
227C2×Dic10D1214yes—
228C2×Dic10#6714yes—
229C5×C4⋊C4—22no—
230C42×C5—62no—
231#685—34no—
232#685—62no—
233#685—64no—
234C4⋊Dic5—62no—
235#683—24no—
236#683—64no—
237C4×Dic5—22no—
238C4×Dic5—24no—
239C4×Dic5—64no—
240C23×Q8—151no—
241C2×C4×Q8—51no—
242C2×C4×Q8—151no—
243#391—31no—
244#391—152no—
245#391—152no—
246#391—151no—
247#380—152no—
248#380—151no—
249C4×C22⋊C4—152no—
250C4×C22⋊C4—151no—
251C22×C3×C5C23×C411yescharacteristic
252C22×C3×C5C22×D411yescharacteristic
253C22×C3×C5C22×D411yescharacteristic
254C22×C3×C5C22×Q811yescharacteristic
255C22×C3×C5C2×#4111yescharacteristic
256C22×C3×C52−1+411yescharacteristic
257C22×C3×C52−1+411yescharacteristic
258C2×D15—22no—
259C2×D15—22no—
260C2×D15—22no—
261C2×D15—22no—
262C2×D15—22no—
263C2×D15—22no—
264C2×D15C4×Q814yescomplemented
265C3×C4×C5—42no—
266C3×C4×C5C2×C22⋊C414yes—
267Dic15—22no—
268Dic15—42no—
269C3×Dic5—22no—
270C3×Dic5—41no—
271C3×Dic5C2×C22⋊C418yes—
272C5×Dic3—24no—
273C5×Dic3—22no—
274C5×Dic3—41no—
275C23×D3—51no—
276C2×C3×Q8—54no—
277C2×C3×Q8—54no—
278C2×C3×Q8—54no—
279C22×C3×C4—52no—
280C22×C3×C4—51no—
281C22×C3×C4—51no—
282C22×C3×C4C22×D512yes—
283C22×C3×C4C22×D511yescharacteristic
284C22×Dic3—51no—
285C22×Dic3—52no—
286C22×Dic3C22×D511yescharacteristic
287C22×Dic3C22×D512yes—
288C22×Dic3C22×D511yescharacteristic
289D3×Q8—1032no—
290C2×C4×D3—54no—
291C2×C4×D3—58no—
292C2×C4×D3—104no—
293C2×C4×D3—104no—
294C2×C4×D3—102no—
295C2×C4×D3—102no—
296C2×Dic6—22no—
297C2×Dic6—22no—
298C2×Dic6—58no—
299C2×Dic6—104no—
300C2×Dic6—104no—
301C2×Dic6C4×D514yes—
302C3×C4⋊C4—22no—
303C3×C4⋊C4—104no—
304C3×C42—104no—
305C3×C42—102no—
306(C2×D3)⋊C4—54no—
307(C2×D3)⋊C4—102no—
308(C2×D3)⋊C4—104no—
309C4⋊Dic3—104no—
310C4⋊Dic3—102no—
311C4×Dic3—22no—
312C4×Dic3—104no—
313C23×C5C23×D311yescharacteristic
314C22×D5—32no—
315C22×D5—32no—
316C22×D5—32no—
317C22×D5—61no—
318C22×D5—61no—
319C22×D5—61no—
320C22×D5—61no—
321C5×Q8—68no—
322C2×C4×C5—22no—
323C2×C4×C5—22no—
324C2×C4×C5—22no—
325C2×C4×C5—34no—
326C2×C4×C5—32no—
327C2×C4×C5—61no—
328C2×C4×C5—62no—
329C2×C4×C5—61no—
330C2×C4×C5—62no—
331C2×C4×C5—61no—
332C2×C4×C5—61no—
333C2×C4×C5—61no—
334C2×C4×C5C2×C4×D312yes—
335C2×C4×C5C2×D1212yes—
336C2×C4×C5C2×#6712yes—
337C2×Dic5—21no—
338C2×Dic5—21no—
339C2×Dic5—21no—
340C2×Dic5—21no—
341C2×Dic5—21no—
342C2×Dic5—32no—
343C2×Dic5—61no—
344C2×Dic5—61no—
345C2×Dic5—62no—
346C2×Dic5—62no—
347C2×Dic5—62no—
348C2×Dic5C2×C4×D314yes—
349C2×Dic5C2×D1214yes—
350C2×Dic5D3×Q812yescomplemented
351C2×Dic5C2×#6714yes—
352C4×D5—68no—
353C4×D5—616no—
354Dic10—616no—
355Dic10(C2×D3)⋊C4116yescomplemented
356C22×Q8—31no—
357C22×Q8—51no—
358C22×Q8—152no—
359C22×Q8—158no—
360C22×Q8—304no—
361C22×Q8—304no—
362C23×C4—152no—
363C23×C4—151no—
364C4×Q8—104no—
365C4×Q8—304no—
366C2×C4⋊C4—52no—
367C2×C4⋊C4—152no—
368C2×C4⋊C4—151no—
369C2×C4⋊C4D3×D511yescharacteristic
370C2×C22⋊C4—151no—
371C2×C22⋊C4—151no—
372C2×C22⋊C4—152no—
373C2×C42—31no—
374C2×C42—52no—
375C2×C42—51no—
376C2×C42—152no—
377#95—32no—
378#95—32no—
379#95—152no—
380#95—151no—
381#95—151no—
382#95—152no—
383#95—152no—
384C2×C3×C5C22×C22⋊C411yescharacteristic
385C2×C3×C5C2×C4×Q811yescharacteristic
386C2×C3×C5#51911yescharacteristic
387C2×C3×C5#54811yescharacteristic
388C2×C3×C5#54811yescharacteristic
389C2×C3×C5D4×Q811yescharacteristic
390C2×C3×C5D4×Q811yescharacteristic
391D15—24no—
392C23×C3C23×D511yescharacteristic
393C22×D3—52no—
394C22×D3—52no—
395C22×D3—52no—
396C22×D3—101no—
397C22×D3—101no—
398C22×D3—101no—
399C22×D3—101no—
400C3×Q8—516no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—54no—
405C2×C3×C4—54no—
406C2×C3×C4—54no—
407C2×C3×C4—52no—
408C2×C3×C4—101no—
409C2×C3×C4—101no—
410C2×C3×C4—101no—
411C2×C3×C4—101no—
412C2×C3×C4—101no—
413C2×C3×C4C2×C4×D512yes—
414C2×C3×C4D4×D512yes—
415C2×C3×C4D4×D512yes—
416C2×Dic3—21no—
417C2×Dic3—22no—
418C2×Dic3—22no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—52no—
424C2×Dic3—101no—
425C2×Dic3—101no—
426C2×Dic3—102no—
427C2×Dic3—102no—
428C2×Dic3—102no—
429C2×Dic3C2×C4×D514yes—
430C2×Dic3Q8×D512yescomplemented
431C4×D3—1016no—
432C4×D3—108no—
433Dic6—28no—
434Dic6—1016no—
435C22×C5C22×C4×D311yescharacteristic
436C22×C5C22×D1211yescharacteristic
437C22×C5C2×D3×Q811yescharacteristic
438C22×C5#100711yescharacteristic
439C22×C5D3×#4111yescharacteristic
440C22×C5#101011yescharacteristic
441C22×C5C22×#6711yescharacteristic
442C2×D5—34no—
443C2×D5—62no—
444C2×D5—62no—
445C2×D5—62no—
446C2×D5—62no—
447C2×D5—62no—
448C2×D5—62no—
449C4×C5—42no—
450C4×C5—64no—
451C4×C5—62no—
452C4×C5—121no—
453C4×C5C2×(C2×D3)⋊C414yes—
454Dic5—22no—
455Dic5—41no—
456Dic5—62no—
457Dic5—122no—
458Dic5C2×(C2×D3)⋊C418yes—
459C24—151no—
460C2×Q8—34no—
461C2×Q8—54no—
462C2×Q8—54no—
463C2×Q8—54no—
464C2×Q8—62no—
465C2×Q8—62no—
466C2×Q8—158no—
467C2×Q8—304no—
468C2×Q8—304no—
469C2×Q8—3032no—
470C22×C4—32no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—52no—
474C22×C4—51no—
475C22×C4—51no—
476C22×C4—158no—
477C22×C4—151no—
478C22×C4—154no—
479C22×C4—152no—
480C22×C4—304no—
481C22×C4—304no—
482C22×C4—302no—
483C22×C4—302no—
484C22×C4C2×D3×D512yes—
485C22×C4C2×D3×D511yescharacteristic
486C4⋊C4—22no—
487C4⋊C4—104no—
488C4⋊C4—304no—
489C4⋊C4—302no—
490C22⋊C4—154no—
491C22⋊C4—304no—
492C22⋊C4—302no—
493C42—62no—
494C42—104no—
495C42—102no—
496C42—304no—
497C3×C5Q8×C22⋊C411yescharacteristic, complemented, Hall
498C22×C3C22×C4×D511yescharacteristic
499C22×C3C2×D4×D511yescharacteristic
500C22×C3C2×D4×D511yescharacteristic
501C22×C3C2×Q8×D511yescharacteristic
502C22×C3#414711yescharacteristic
503C22×C3D5×#4111yescharacteristic
504C22×C3#415011yescharacteristic
505C2×D3—54no—
506C2×D3—102no—
507C2×D3—102no—
508C2×D3—102no—
509C2×D3—102no—
510C2×D3—102no—
511C2×D3—102no—
512C3×C4—42no—
513C3×C4—58no—
514C3×C4—102no—
515C3×C4—201no—
516C3×C4D5×C22⋊C414yes—
517Dic3—24no—
518Dic3—22no—
519Dic3—41no—
520Dic3—102no—
521Dic3—202no—
522C2×C5C4×D3×Q811yescharacteristic
523C2×C5#557211yescharacteristic
524C2×C5Q8×D1211yescharacteristic
525C2×C5#557611yescharacteristic
526C2×C5C22×(C2×D3)⋊C411yescharacteristic
527C2×C5Q8×#6711yescharacteristic
528C2×C5#583111yescharacteristic
529D5—64no—
530C23—152no—
531C23—152no—
532C23—152no—
533C23—301no—
534C23—301no—
535C23—301no—
536C23—301no—
537C23C22×D3×D511yescharacteristic
538Q8—516no—
539Q8—68no—
540Q8—3016no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—34no—
545C2×C4—32no—
546C2×C4—54no—
547C2×C4—54no—
548C2×C4—54no—
549C2×C4—52no—
550C2×C4—61no—
551C2×C4—62no—
552C2×C4—61no—
553C2×C4—62no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—101no—
558C2×C4—101no—
559C2×C4—101no—
560C2×C4—101no—
561C2×C4—101no—
562C2×C4—152no—
563C2×C4—3016no—
564C2×C4—301no—
565C2×C4—301no—
566C2×C4—308no—
567C2×C4—302no—
568C2×C4—302no—
569C2×C4—302no—
570C2×C4C4×D3×D512yes—
571C2×C4D5×D1212yes—
572C2×C4D5×#6712yes—
573C2×C3C2×D5×C22⋊C411yescharacteristic
574C2×C3C4×Q8×D511yescharacteristic
575C2×C3#6590611yescharacteristic
576C2×C3#6596611yescharacteristic
577C2×C3#6596711yescharacteristic
578C2×C3#6601111yescharacteristic
579C2×C3#6606111yescharacteristic
580D3—104no—
581C5Q8×(C2×D3)⋊C411yescharacteristic, complemented, Sylow
582C22—154no—
583C22—302no—
584C22—302no—
585C22—302no—
586C22—302no—
587C22—302no—
588C22—302no—
589C22C2×C4×D3×D511yescharacteristic
590C22C2×D5×D1211yescharacteristic
591C22#9202011yescharacteristic
592C22#9202411yescharacteristic
593C22#9202911yescharacteristic
594C22D3×Q8×D511yescharacteristic
595C22C2×D5×#6711yescharacteristic
596C4—42no—
597C4—58no—
598C4—64no—
599C4—62no—
600C4—102no—
601C4—121no—
602C4—201no—
603C4—302no—
604C4—602no—
605C4D5×(C2×D3)⋊C414yes—
606C3#1060834511yescharacteristic, complemented, Sylow
607C2—304no—
608C2#1175334111yescharacteristic
609C2#1175338111yescharacteristic
610C2#1175340911yescharacteristic
611C2#1175343811yescharacteristic
612C2C2×D5×(C2×D3)⋊C411yescharacteristic
613C2#1175493911yescharacteristic
614C2#1175522811yescharacteristic
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×C5Q8×C22⋊C4
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
2. C2×C3×#4033C2semidirect
3. C2×Dic3×Dic10C2semidirect
5. C5×#4973C2semidirect
6. #11748933C2semidirect
10. #11748628C2semidirect
11. #11748589C2semidirect
20. C2×D15⋊Q8C4semidirect
43. C2×#4033D3semidirect
89. #6667Q8semidirect
97. #4973D5semidirect
160. C2×C5×Dic3C2×Q8semidirect
170. C2×C3×Dic5C2×Q8semidirect
171. C3×Dic10C22⋊C4semidirect
264. C2×D15C4×Q8semidirect
350. C2×Dic5D3×Q8semidirect
355. Dic10(C2×D3)⋊C4semidirect
430. C2×Dic3Q8×D5semidirect
497. C3×C5Q8×C22⋊C4semidirect
581. C5Q8×(C2×D3)⋊C4semidirect
606. C3#10608345semidirect

Maximal subgroups

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

subgroupindex
C22×D15⋊Q82
C2×C3×#40332
C2×Dic3×Dic102
#117510882
C5×#49732
#117489332
#117488762
#117488752
C4×#66672
#117486282
#117485892
#106083453
Q8×(C2×D3)⋊C45