Finite Group Databasebeta

← #14326150

Subgroups — #14326150

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×C3×Q8×D5C211yesmaximal, characteristic
2C2×#91331C211yesmaximal, characteristic, complemented
3C2×Dic3×Dic10C211yesmaximal, characteristic, complemented
4#11751330C211yesmaximal, characteristic, complemented
5C5×#5229C211yesmaximal, characteristic, complemented
6#11748759C211yesmaximal, characteristic
7#11748738C212yesmaximal
8#11748702C212yesmaximal
9C4×#6665C211yesmaximal, characteristic
10#11748672C212yesmaximal, complemented
11#11748672C212yesmaximal, complemented
12#10608345—31nomaximal, Hall
13C22×C3×C5×Q8C2211yescharacteristic
14C2×C3×Q8×D5—24no—
15C2×C3×Q8×D5—24no—
16C2×C3×Q8×D5C418yescomplemented
17C22×C3×C4×D5C2211yescharacteristic
18C22×C3×C4×D5C2212yes—
19C22×C3×Dic10C2212yes—
20C22×C3×Dic10C2211yescharacteristic
21C2×C4⋊Dic15C2211yescharacteristic
22C2×C4×Dic15C2211yescharacteristic
23C2×C5×C4⋊Dic3C2211yescharacteristic
24C2×C4×C5×Dic3C2211yescharacteristic
25C2×#6668C2212yes—
26C2×#6668C2212yes—
27C2×#6665C2211yescharacteristic
28C2×#6665C2212yes—
29C2×#6665C2211yescharacteristic
30C2×Dic3×Dic5C2212yes—
31C2×Dic3×Dic5C2212yes—
32#91331—24no—
33Dic3×Dic10—24no—
34#91115C2212yes—
35#91115C2212yes—
36C5×C3⋊#95C2212yes—
37C5×C3⋊#95C2212yes—
38#90994C2211yescharacteristic
39#90994C2211yescharacteristic
40#90994C2212yes—
41Q8×#217—51nomaximal, Hall
42C22×Q8×D5D311yescharacteristic
43C2×#4033—31no—
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×C3×D5C2311yescharacteristic
54C2×C3×C5×Q8—22no—
55C2×C3×C5×Q8—22no—
56C2×C3×C5×Q8C2×C414yes—
57C22×C3×C4×C5C2312yes—
58C22×C3×C4×C5C2311yescharacteristic
59C22×Dic15C2312yes—
60C22×Dic15C2311yescharacteristic
61C22×Dic15C2311yescharacteristic
62C22×C5×Dic3C2311yescharacteristic
63C22×C5×Dic3C2312yes—
64C22×C5×Dic3C2311yescharacteristic
65C22×C3×Dic5C2311yescharacteristic
66C22×C3×Dic5C2312yes—
67C3×Q8×D5—232no—
68C2×C3×C4×D5—22no—
69C2×C3×C4×D5—22no—
70C2×C3×C4×D5—24no—
71C2×C3×C4×D5—24no—
72C2×C3×C4×D5C2×C414yes—
73C2×C3×C4×D5C2×C418yes—
74C2×C3×Dic10—24no—
75C2×C3×Dic10—24no—
76C2×C3×Dic10C2×C418yes—
77C2×C3×Dic10C2×C414yes—
78C2×C3×Dic10D414yes—
79C2×C3×Dic10D414yes—
80C4⋊Dic15—22no—
81C4×Dic15—22no—
82C5×C4⋊Dic3—22no—
83C4×C5×Dic3—22no—
84#6668—24no—
85#6668—24no—
86#6665—22no—
87#6665—24no—
88#6665Q814yescomplemented
89Dic3×Dic5—24no—
90Dic3×Dic5—24no—
91C23×C3×Q8—51no—
92C2×Q8×Dic3—51no—
93C2×Q8×Dic3—51no—
94#5229—51no—
95#5229—52no—
96#5229—52no—
97#5229D511yescharacteristic, complemented
98#5212—51no—
99#5212—52no—
100C4×#217—51no—
101C4×#217—52no—
102C22×C5×Q8C2×D311yescharacteristic
103C2×Q8×D5—24no—
104C2×Q8×D5—24no—
105C2×Q8×D5Dic318yescomplemented
106C22×C4×D5C2×D312yes—
107C22×C4×D5C2×D311yescharacteristic
108C22×Dic10C2×D312yes—
109C22×Dic10C2×D311yescharacteristic
110C2×C5×C4⋊C4—31no—
111C2×C42×C5—31no—
112C2×#685—31no—
113C2×#685—32no—
114C2×#685—31no—
115C2×C4⋊Dic5—31no—
116C2×#683—32no—
117C2×#683—32no—
118C2×C4×Dic5—31no—
119C2×C4×Dic5—32no—
120C2×C4×Dic5—32no—
121#4033—64no—
122C4×Dic10—64no—
123C5×#95—32no—
124C5×#95—32no—
125#3963—32no—
126#3963—32no—
127#3963—31no—
128#3963—31no—
129#3963—32no—
130Q8×C22⋊C4—151noSylow
131C23×C3×C5C2411yescharacteristic
132C22×C3×D5—21no—
133C22×C3×D5—21no—
134C22×C3×D5—21no—
135C22×C3×D5—21no—
136C22×C3×D5C22×C412yes—
137C22×C3×D5C2×Q812yes—
138C22×C3×D5#4112yes—
139C3×C5×Q8—28no—
140C2×C3×C4×C5—22no—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5C22×C414yes—
143C2×C3×C4×C5C22×C412yes—
144C2×C3×C4×C5C2×D412yes—
145C2×C3×C4×C5C2×D412yes—
146C2×Dic15—21no—
147C2×Dic15—21no—
148C2×Dic15—22no—
149C2×Dic15—21no—
150C2×Dic15—22no—
151C2×Dic15—21no—
152C2×Dic15—22no—
153C2×Dic15—21no—
154C2×Dic15C2×Q812yescomplemented
155C2×C5×Dic3—21no—
156C2×C5×Dic3—21no—
157C2×C5×Dic3—22no—
158C2×C5×Dic3—21no—
159C2×C5×Dic3—21no—
160C2×C5×Dic3—22no—
161C2×C5×Dic3—21no—
162C2×C5×Dic3—22no—
163C2×C5×Dic3C2×Q812yescomplemented
164C2×C3×Dic5—21no—
165C2×C3×Dic5—21no—
166C2×C3×Dic5C22×C412yes—
167C2×C3×Dic5C22×C414yes—
168C2×C3×Dic5C2×D414yes—
169C2×C3×Dic5C2×D414yes—
170C3×C4×D5—28no—
171C3×C4×D5—216no—
172C3×Dic10—216no—
173C3×Dic10C22⋊C4116yescomplemented
174C22×C3×Q8—58no—
175C22×C3×Q8—51no—
176C22×C3×Q8—52no—
177C22×C3×Q8—104no—
178C22×C3×Q8—104no—
179C22×C3×Q8C2×D511yescharacteristic
180C23×C3×C4—52no—
181C23×C3×C4—51no—
182C2×#217—51no—
183C2×#217—52no—
184C2×#217—51no—
185Q8×Dic3—104no—
186Q8×Dic3—104no—
187C2×C4⋊Dic3—51no—
188C2×C4⋊Dic3—52no—
189C2×C4⋊Dic3—52no—
190C2×C4⋊Dic3C2×D511yescharacteristic
191C2×C4×Dic3—51no—
192C2×C4×Dic3—52no—
193C2×C4×Dic3—52no—
194C2×C4×Dic3C2×D511yescharacteristic
195C3⋊#95—52no—
196C3⋊#95—52no—
197C3⋊#95—51no—
198C3⋊#95—51no—
199C3⋊#95—52no—
200C3⋊#95C2×D512yes—
201C3⋊#95C2×D512yes—
202C23×D5C22×D311yescharacteristic
203C2×C5×Q8—22no—
204C2×C5×Q8—22no—
205C2×C5×Q8C2×Dic314yes—
206C22×C4×C5—31no—
207C22×C4×C5—31no—
208C22×C4×C5—32no—
209C22×C4×C5C22×D312yes—
210C22×C4×C5C22×D311yescharacteristic
211C22×Dic5—31no—
212C22×Dic5—31no—
213C22×Dic5—32no—
214C22×Dic5C22×D311yescharacteristic
215C22×Dic5C22×D312yes—
216Q8×D5—232no—
217C2×C4×D5—22no—
218C2×C4×D5—22no—
219C2×C4×D5—24no—
220C2×C4×D5—24no—
221C2×C4×D5C2×Dic314yes—
222C2×C4×D5C2×Dic318yes—
223C2×Dic10—24no—
224C2×Dic10—24no—
225C2×Dic10C2×Dic318yes—
226C2×Dic10C2×Dic314yes—
227C2×Dic10#6714yes—
228C2×Dic10#6714yes—
229C5×C4⋊C4—62no—
230C42×C5—62no—
231#685—34no—
232#685—64no—
233#685—62no—
234C4⋊Dic5—62no—
235#683—64no—
236#683—64no—
237C4×Dic5—62no—
238C4×Dic5—64no—
239C4×Dic5—64no—
240C23×Q8—51no—
241C2×C4×Q8—151no—
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×C3×D5—22no—
259C2×C3×D5—22no—
260C2×C3×D5—22no—
261C2×C3×D5—22no—
262C2×C3×D5—22no—
263C2×C3×D5—22no—
264C2×C3×D5C4×Q814yescomplemented
265C3×C4×C5—24no—
266C3×C4×C5C2×C22⋊C414yes—
267Dic15—22no—
268Dic15—42no—
269Dic15—41no—
270C3×Dic5—22no—
271C3×Dic5C2×C22⋊C418yes—
272C5×Dic3—22no—
273C5×Dic3—41no—
274C5×Dic3—42no—
275C24×C3—51no—
276C2×C3×Q8—22no—
277C2×C3×Q8—22no—
278C2×C3×Q8—54no—
279C2×C3×Q8—58no—
280C2×C3×Q8—54no—
281C2×C3×Q8—54no—
282C2×C3×Q8—104no—
283C2×C3×Q8—104no—
284C2×C3×Q8—1032no—
285C2×C3×Q8C4×D514yes—
286C22×C3×C4—52no—
287C22×C3×C4—51no—
288C22×C3×C4—58no—
289C22×C3×C4—54no—
290C22×C3×C4—104no—
291C22×C3×C4—104no—
292C22×C3×C4—102no—
293C22×C3×C4—102no—
294C22×C3×C4C22×D512yes—
295C22×C3×C4C22×D511yescharacteristic
296C22×Dic3—51no—
297C22×Dic3—52no—
298C22×Dic3—51no—
299C22×Dic3C22×D512yes—
300C22×Dic3C22×D511yescharacteristic
301C22×Dic3C22×D511yescharacteristic
302#217—54no—
303#217—102no—
304#217—104no—
305C4⋊Dic3—22no—
306C4⋊Dic3—104no—
307C4⋊Dic3—104no—
308C4⋊Dic3—102no—
309C4×Dic3—22no—
310C4×Dic3—104no—
311C4×Dic3—104no—
312C4×Dic3—102no—
313C23×C5C23×D311yescharacteristic
314C22×D5—21no—
315C22×D5—21no—
316C22×D5—21no—
317C22×D5—21no—
318C22×D5D3×Q812yes—
319C22×D5#23912yes—
320C22×D5C22×Dic312yes—
321C5×Q8—28no—
322C2×C4×C5—22no—
323C2×C4×C5—22no—
324C2×C4×C5—32no—
325C2×C4×C5—61no—
326C2×C4×C5—61no—
327C2×C4×C5—61no—
328C2×C4×C5—62no—
329C2×C4×C5—61no—
330C2×C4×C5—62no—
331C2×C4×C5—62no—
332C2×C4×C5—61no—
333C2×C4×C5C22×Dic314yes—
334C2×C4×C5C22×Dic312yes—
335C2×C4×C5C2×#6712yes—
336C2×C4×C5C2×#6712yes—
337C2×Dic5—21no—
338C2×Dic5—21no—
339C2×Dic5—32no—
340C2×Dic5—61no—
341C2×Dic5—61no—
342C2×Dic5—61no—
343C2×Dic5—62no—
344C2×Dic5—62no—
345C2×Dic5—61no—
346C2×Dic5—61no—
347C2×Dic5—62no—
348C2×Dic5C22×Dic312yes—
349C2×Dic5C22×Dic314yes—
350C2×Dic5C2×#6714yes—
351C2×Dic5C2×#6714yes—
352C4×D5—28no—
353C4×D5—216no—
354Dic10—216no—
355Dic10#217116yescomplemented
356C22×Q8—58no—
357C22×Q8—51no—
358C22×Q8—52no—
359C22×Q8—104no—
360C22×Q8—104no—
361C22×Q8D3×D511yescharacteristic
362C23×C4—52no—
363C23×C4—51no—
364C4×Q8—304no—
365C4×Q8—304no—
366C2×C4⋊C4—31no—
367C2×C4⋊C4—152no—
368C2×C4⋊C4—152no—
369C2×C4⋊C4—151no—
370C2×C22⋊C4—151no—
371C2×C22⋊C4—151no—
372C2×C22⋊C4—152no—
373C2×C42—31no—
374C2×C42—152no—
375C2×C42—152no—
376C2×C42—151no—
377#95—32no—
378#95—32no—
379#95—151no—
380#95—151no—
381#95—152no—
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
391C3×D5—24no—
392C23×C3—52no—
393C23×C3—52no—
394C23×C3—52no—
395C23×C3—101no—
396C23×C3—101no—
397C23×C3—101no—
398C23×C3—101no—
399C23×C3C23×D511yescharacteristic
400C3×Q8—28no—
401C3×Q8—516no—
402C3×Q8—1016no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—54no—
406C2×C3×C4—54no—
407C2×C3×C4—54no—
408C2×C3×C4—52no—
409C2×C3×C4—1016no—
410C2×C3×C4—101no—
411C2×C3×C4—101no—
412C2×C3×C4—108no—
413C2×C3×C4C2×C4×D514yes—
414C2×C3×C4C2×C4×D512yes—
415C2×C3×C4D4×D512yes—
416C2×C3×C4D4×D512yes—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—22no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—52no—
426C2×Dic3—101no—
427C2×Dic3—101no—
428C2×Dic3—102no—
429C2×Dic3—101no—
430C2×Dic3—102no—
431C2×Dic3—102no—
432C2×Dic3—101no—
433C2×Dic3—101no—
434C2×Dic3Q8×D512yescomplemented
435C22×C5C2×D3×Q811yescharacteristic
436C22×C5C2×#23911yescharacteristic
437C22×C5#100711yescharacteristic
438C22×C5#100711yescharacteristic
439C22×C5C23×Dic311yescharacteristic
440C22×C5C22×#6711yescharacteristic
441C22×C5C22×#6711yescharacteristic
442C2×D5—22no—
443C2×D5—22no—
444C2×D5—22no—
445C2×D5—22no—
446C2×D5—22no—
447C2×D5—22no—
448C2×D5Q8×Dic314yescomplemented
449C4×C5—24no—
450C4×C5—62no—
451C4×C5—122no—
452C4×C5—121no—
453C4×C5C2×#21714yes—
454Dic5—22no—
455Dic5—62no—
456Dic5—121no—
457Dic5—122no—
458Dic5C2×#21718yes—
459C24—51no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—54no—
463C2×Q8—58no—
464C2×Q8—54no—
465C2×Q8—54no—
466C2×Q8—104no—
467C2×Q8—104no—
468C2×Q8—1032no—
469C2×Q8D5×Dic314yes—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—32no—
473C22×C4—52no—
474C22×C4—51no—
475C22×C4—58no—
476C22×C4—54no—
477C22×C4—104no—
478C22×C4—104no—
479C22×C4—102no—
480C22×C4—102no—
481C22×C4—151no—
482C22×C4—152no—
483C22×C4—151no—
484C22×C4C2×D3×D512yes—
485C22×C4C2×D3×D511yescharacteristic
486C4⋊C4—62no—
487C4⋊C4—304no—
488C4⋊C4—304no—
489C4⋊C4—302no—
490C22⋊C4—154no—
491C22⋊C4—304no—
492C22⋊C4—302no—
493C42—62no—
494C42—304no—
495C42—304no—
496C42—302no—
497C3×C5Q8×C22⋊C411yescharacteristic, complemented, Hall
498C22×C3—54no—
499C22×C3—102no—
500C22×C3—102no—
501C22×C3—102no—
502C22×C3—102no—
503C22×C3—102no—
504C22×C3—102no—
505C22×C3C22×C4×D511yescharacteristic
506C22×C3C2×D4×D511yescharacteristic
507C22×C3C2×D4×D511yescharacteristic
508C22×C3C2×Q8×D511yescharacteristic
509C22×C3#414711yescharacteristic
510C22×C3D5×#4111yescharacteristic
511C22×C3#415011yescharacteristic
512C3×C4—24no—
513C3×C4—58no—
514C3×C4—102no—
515C3×C4D5×C22⋊C414yes—
516Dic3—22no—
517Dic3—41no—
518Dic3—42no—
519Dic3—102no—
520Dic3—202no—
521Dic3—201no—
522C2×C5C2×Q8×Dic311yescharacteristic
523C2×C5#581211yescharacteristic
524C2×C5Q8×#6711yescharacteristic
525C2×C5Q8×#6711yescharacteristic
526C2×C5#581711yescharacteristic
527C2×C5#581711yescharacteristic
528C2×C5C22×#21711yescharacteristic
529D5—24no—
530C23—52no—
531C23—52no—
532C23—52no—
533C23—101no—
534C23—101no—
535C23—101no—
536C23—101no—
537C23C22×D3×D511yescharacteristic
538Q8—28no—
539Q8—516no—
540Q8—1016no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—32no—
544C2×C4—54no—
545C2×C4—54no—
546C2×C4—54no—
547C2×C4—52no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—62no—
556C2×C4—1016no—
557C2×C4—101no—
558C2×C4—101no—
559C2×C4—108no—
560C2×C4—152no—
561C2×C4—301no—
562C2×C4—301no—
563C2×C4—302no—
564C2×C4—301no—
565C2×C4—301no—
566C2×C4—302no—
567C2×C4—301no—
568C2×C4—302no—
569C2×C4C2×D5×Dic314yes—
570C2×C4C2×D5×Dic312yes—
571C2×C4D5×#6712yes—
572C2×C4D5×#6712yes—
573C2×C3—104no—
574C2×C3C2×D5×C22⋊C411yescharacteristic
575C2×C3C4×Q8×D511yescharacteristic
576C2×C3#6590611yescharacteristic
577C2×C3#6596611yescharacteristic
578C2×C3#6596711yescharacteristic
579C2×C3#6601111yescharacteristic
580C2×C3#6606111yescharacteristic
581C5Q8×#21711yescharacteristic, complemented, Sylow
582C22—54no—
583C22—102no—
584C22—102no—
585C22—102no—
586C22—102no—
587C22—102no—
588C22—102no—
589C22#9202911yescharacteristic
590C22#9203011yescharacteristic
591C22D3×Q8×D511yescharacteristic
592C22D5×#23911yescharacteristic
593C22C22×D5×Dic311yescharacteristic
594C22C2×D5×#6711yescharacteristic
595C22C2×D5×#6711yescharacteristic
596C4—24no—
597C4—58no—
598C4—62no—
599C4—102no—
600C4—122no—
601C4—121no—
602C4—302no—
603C4—601no—
604C4—602no—
605C4D5×#21714yes—
606C3#1060834511yescharacteristic, complemented, Sylow
607C2—104no—
608C2#1175514011yescharacteristic
609C2Q8×D5×Dic311yescharacteristic
610C2#1175522511yescharacteristic
611C2#1175522611yescharacteristic
612C2#1175522711yescharacteristic
613C2#1175522811yescharacteristic
614C2C2×D5×#21711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC22×C3×C5×Q8C22
socle, cosocleC23×C3×C5C24
prime radicalC23×C3×D5C23
hypercenterC22×Q8D3×D5
hypocenterC3×C5Q8×C22⋊C4
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×#91331C2semidirect
3. C2×Dic3×Dic10C2semidirect
4. #11751330C2semidirect
5. C5×#5229C2semidirect
10. #11748672C2semidirect
11. #11748672C2semidirect
16. C2×C3×Q8×D5C4semidirect
88. #6665Q8semidirect
97. #5229D5semidirect
105. C2×Q8×D5Dic3semidirect
154. C2×Dic15C2×Q8semidirect
163. C2×C5×Dic3C2×Q8semidirect
173. C3×Dic10C22⋊C4semidirect
264. C2×C3×D5C4×Q8semidirect
355. Dic10#217semidirect
434. C2×Dic3Q8×D5semidirect
448. C2×D5Q8×Dic3semidirect
497. C3×C5Q8×C22⋊C4semidirect
581. C5Q8×#217semidirect
606. C3#10608345semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Q8×D52
C2×#913312
C2×Dic3×Dic102
#117513302
C5×#52292
#117487592
#117487382
#117487022
C4×#66652
#117486722
#106083453
Q8×#2175