Finite Group Databasebeta

← #14331627

Subgroups — #14331627

Order 1920.

1888 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×#237C211yesmaximal, characteristic, complemented
2C2×Dic3×C22⋊D5C212yesmaximal, complemented
3C2×#91441C211yesmaximal, characteristic, complemented
4#11751323C211yesmaximal, characteristic, complemented
5C3×#65512C211yesmaximal, characteristic, complemented
6#11749521C212yesmaximal, complemented
7#11749493C212yesmaximal, complemented
8Dic3×C22⋊Dic5C212yesmaximal, complemented
9#11748833C211yesmaximal, characteristic, complemented
10#11748773C211yesmaximal, characteristic, complemented
11#11748719C211yesmaximal, characteristic, complemented
12#10615826—31nomaximal, Hall
13C22×C3×C5×D4C2211yescharacteristic
14C22×C5×#67C2212yes—
15C23×C5×Dic3C2212yes—
16C2×C5×#237—28no—
17C2×C5×#237—24no—
18C22×C4×C5×D3C2211yescharacteristic
19C22×C5×Dic6C2211yescharacteristic
20C22×C3×C22⋊D5C2212yes—
21C22×#1201C2211yescharacteristic
22C22×D5×Dic3C2211yescharacteristic
23C2×#6719C2212yes—
24C2×#6719C2212yes—
25C2×#6712C2211yescharacteristic
26C2×C3×C22⋊Dic5C2212yes—
27C2×C3×#685C2211yescharacteristic
28Dic3×C22⋊D5—28no—
29C2×#6668C2212yes—
30C2×(C2×D3)⋊Dic5C2211yescharacteristic
31C2×(C2×D3)⋊Dic5C2212yes—
32C2×#6665C2212yes—
33C2×Dic3×Dic5C2211yescharacteristic
34C2×Dic3×Dic5C2212yes—
35#91441—24no—
36#91115C2211yescharacteristic
37C3×#3963C2211yescharacteristic
38#90994C2212yes—
39#90994C2212yes—
40#76665—51nomaximal, Hall
41C22×C5×#41—31no—
42C2×C4×C22⋊D5—31no—
43C2×C4×C22⋊D5—32no—
44#65520—31no—
45#65518—31no—
46#65512—32no—
47#65512D311yescharacteristic, complemented
48#65511—31no—
49#65511—32no—
50C4×C22⋊Dic5—31no—
51C4×C22⋊Dic5—32no—
52C24×C3×C5C2312yes—
53C23×C5×D3C2311yescharacteristic
54C23×C3×D5C2311yescharacteristic
55C2×C3×C5×D4—24no—
56C2×C3×C5×D4—22no—
57C22×C3×C4×C5C2311yescharacteristic
58C22×Dic15C2312yes—
59C22×Dic15C2311yescharacteristic
60C22×Dic15C2311yescharacteristic
61C2×C5×#67—24no—
62C2×C5×#67—24no—
63C2×C5×#67—24no—
64C22×C5×Dic3—24no—
65C22×C5×Dic3—24no—
66C22×C5×Dic3D418yescomplemented
67C22×C5×Dic3C2312yes—
68C22×C5×Dic3C2311yescharacteristic
69C5×#237—232no—
70C2×C4×C5×D3—24no—
71C2×C4×C5×D3D414yescomplemented
72C2×C5×Dic6—22no—
73C2×C5×Dic6—24no—
74C2×C3×C22⋊D5—24no—
75C2×C3×C22⋊D5—24no—
76C2×C3×C22⋊D5—24no—
77C22×C3×Dic5C2311yescharacteristic
78C22×C3×Dic5C2312yes—
79C2×#1201—22no—
80C2×#1201—22no—
81C2×#1201—22no—
82C2×D5×Dic3—22no—
83C2×D5×Dic3—24no—
84#6719—24no—
85#6719—24no—
86#6712—22no—
87C3×C22⋊Dic5—24no—
88C3×#685—22no—
89#6668—24no—
90(C2×D3)⋊Dic5—22no—
91(C2×D3)⋊Dic5—24no—
92#6665—24no—
93Dic3×Dic5—22no—
94Dic3×Dic5—24no—
95C22×#237D511yescharacteristic, complemented
96C2×D4×Dic3—52no—
97C2×#67⋊C4—51no—
98C3×#392—51no—
99#5223—52no—
100#5222—51no—
101#4961—52no—
102#4947—51no—
103Dic3×C22⋊C4—52no—
104#4666—51no—
105#4656—51no—
106C2×C5×#41—68no—
107C2×C5×#41—64no—
108C22×C5×Q8—31no—
109C22×C5×D4—32no—
110C22×C5×D4C2×D311yescharacteristic
111C23×C4×C5—32no—
112C23×C4×C5—31no—
113C22×C22⋊D5—31no—
114C22×C22⋊D5C2×D312yes—
115C22×C4×D5—31no—
116C2×C22⋊Dic5—32no—
117C2×C22⋊Dic5—32no—
118C2×C22⋊Dic5—31no—
119C2×C22⋊Dic5—32no—
120C2×C22⋊Dic5C2×D312yes—
121C4×C22⋊D5—64no—
122C4×C22⋊D5—68no—
123C2×#685—32no—
124C2×#685C2×D311yescharacteristic
125C2×#683—31no—
126C2×#683—32no—
127C2×C4×Dic5—31no—
128C2×C4×Dic5—32no—
129#3963—31no—
130#3963—32no—
131#3963—32no—
132#3963C2×D311yescharacteristic
133#2388—151noSylow
134C23×C3×C5—22no—
135C23×C3×C5—22no—
136C23×C3×C5—22no—
137C23×C3×C5—22no—
138C23×C3×C5—22no—
139C23×C3×C5—22no—
140C23×C3×C5C2×D414yes—
141C23×C3×C5C2411yescharacteristic
142C22×C5×D3—22no—
143C22×C5×D3—22no—
144C22×C5×D3—21no—
145C22×C5×D3—21no—
146C22×C5×D3C2×D412yes—
147C22×C3×D5—22no—
148C22×C3×D5—21no—
149C22×C3×D5—21no—
150C22×C3×D5—22no—
151C22×C3×D5#4112yescomplemented
152C3×C5×D4—28no—
153C2×C3×C4×C5—22no—
154C2×C3×C4×C5C2×D412yes—
155C2×Dic15—21no—
156C2×Dic15—21no—
157C2×Dic15—22no—
158C2×Dic15—21no—
159C2×Dic15—22no—
160C2×Dic15—22no—
161C2×Dic15—22no—
162C2×Dic15#4112yescomplemented
163C5×#67—216no—
164C2×C5×Dic3—22no—
165C2×C5×Dic3—22no—
166C2×C5×Dic3—216no—
167C2×C5×Dic3C2×D414yes—
168C2×C5×Dic3C2×D414yes—
169C2×C5×Dic3C2×D412yes—
170C4×C5×D3—28no—
171C5×Dic6—28no—
172C3×C22⋊D5—216no—
173C2×C3×Dic5—22no—
174C2×C3×Dic5—22no—
175C2×C3×Dic5—21no—
176C2×C3×Dic5—22no—
177C2×C3×Dic5#4114yescomplemented
178#1201—28no—
179D5×Dic3—44no—
180C22×C3×D4—52no—
181C22×C3×D4C2×D511yescharacteristic
182C22×#67—51no—
183C22×#67C2×D512yes—
184C23×Dic3—51no—
185C23×Dic3C2×D512yes—
186C2×#237—28no—
187C2×#237—24no—
188C22×C4×D3C2×D511yescharacteristic
189C22×Dic6C2×D511yescharacteristic
190C2×C3×C22⋊C4—52no—
191C2×C3×C22⋊C4—51no—
192C2×#217—52no—
193C2×#217—52no—
194C2×#217—52no—
195D4×Dic3—108no—
196C2×(C2×D3)⋊C4—52no—
197C2×(C2×D3)⋊C4—51no—
198C2×C4⋊Dic3—52no—
199C2×Dic3⋊C4—51no—
200C2×C4×Dic3—52no—
201C2×C4×Dic3—51no—
202#67⋊C4—104no—
203C3×#95—51no—
204C3⋊#95—51no—
205C3⋊#95—52no—
206C3⋊#95—52no—
207C24×C5—31no—
208C24×C5C22×D312yes—
209C23×D5C22×D311yescharacteristic
210C5×#41—632no—
211C2×C5×Q8—64no—
212C2×C5×Q8—62no—
213C2×C5×D4—24no—
214C2×C5×D4—22no—
215C2×C5×D4—64no—
216C2×C5×D4—64no—
217C2×C5×D4—64no—
218C22×C4×C5—31no—
219C22×C4×C5—38no—
220C22×C4×C5—32no—
221C22×C4×C5—34no—
222C22×C4×C5—64no—
223C22×C4×C5—64no—
224C22×C4×C5—64no—
225C22×C4×C5C22×D311yescharacteristic
226C2×C22⋊D5—24no—
227C2×C22⋊D5—24no—
228C2×C22⋊D5—24no—
229C2×C22⋊D5—62no—
230C2×C22⋊D5—62no—
231C2×C22⋊D5—62no—
232C22×Dic5—31no—
233C22×Dic5—31no—
234C22×Dic5—32no—
235C22×Dic5C22×D311yescharacteristic
236C22×Dic5C22×D312yes—
237C2×C4×D5—62no—
238C2×C4×D5—64no—
239C22⋊Dic5—24no—
240C22⋊Dic5—62no—
241C22⋊Dic5—64no—
242C22⋊Dic5—64no—
243C22⋊Dic5—64no—
244#685—22no—
245#685—64no—
246#683—64no—
247#683—62no—
248C4×Dic5—64no—
249C4×Dic5—62no—
250C22×#41—31no—
251C2×C4×D4—151no—
252C2×C4×D4—152no—
253#395—151no—
254#394—152no—
255#394—151no—
256#393—151no—
257#392—51no—
258#392—152no—
259C4×C22⋊C4—151no—
260C4×C22⋊C4—152no—
261C22×C3×C5—24no—
262C22×C3×C5—24no—
263C22×C3×C5—24no—
264C22×C3×C5—42no—
265C22×C3×C5—42no—
266C22×C3×C5—42no—
267C22×C3×C5—42no—
268C22×C3×C5C22×D411yescharacteristic
269C22×C3×C5C22×D412yes—
270C22×C3×C5C2×#4111yescharacteristic
271C22×C3×C5C2×#4112yes—
272C22×C3×C5C2×#4111yescharacteristic
273C2×C5×D3—22no—
274C2×C5×D3—22no—
275C2×C5×D3—22no—
276C2×C5×D3—42no—
277C2×C5×D3—42no—
278C2×C3×D5—22no—
279C2×C3×D5—24no—
280C2×C3×D5—41no—
281C2×C3×D5—41no—
282C2×C3×D5—42no—
283C3×C4×C5—22no—
284Dic15—22no—
285Dic15—41no—
286Dic15—42no—
287C3×Dic5—24no—
288C3×Dic5—41no—
289C5×Dic3—24no—
290C5×Dic3#12014yes—
291C24×C3—51no—
292C24×C3C22×D512yes—
293C23×D3C22×D511yescharacteristic
294C2×C3×D4—24no—
295C2×C3×D4—22no—
296C2×C3×D4—104no—
297C2×C3×D4—104no—
298C2×C3×D4—104no—
299C22×C3×C4—52no—
300C22×C3×C4—51no—
301C22×C3×C4C22×D511yescharacteristic
302C2×#67—24no—
303C2×#67—24no—
304C2×#67—24no—
305C2×#67—102no—
306C2×#67—102no—
307C2×#67—102no—
308C22×Dic3—24no—
309C22×Dic3—24no—
310C22×Dic3—51no—
311C22×Dic3—52no—
312C22×Dic3—51no—
313C22×Dic3—102no—
314C22×Dic3—104no—
315C22×Dic3C22⋊D518yescomplemented
316C22×Dic3C22×D512yes—
317C22×Dic3C22×D511yescharacteristic
318#237—232no—
319C2×C4×D3—24no—
320C2×C4×D3C22⋊D514yescomplemented
321C2×Dic6—22no—
322C2×Dic6—24no—
323C3×C22⋊C4—104no—
324C3×C22⋊C4—102no—
325#217—104no—
326#217—104no—
327#217—104no—
328(C2×D3)⋊C4—104no—
329(C2×D3)⋊C4—102no—
330C4⋊Dic3—104no—
331Dic3⋊C4—102no—
332C4×Dic3—102no—
333C4×Dic3—104no—
334C23×C5—22no—
335C23×C5—22no—
336C23×C5—22no—
337C23×C5—22no—
338C23×C5—22no—
339C23×C5—22no—
340C23×C5—32no—
341C23×C5—62no—
342C23×C5—62no—
343C23×C5—61no—
344C23×C5—61no—
345C23×C5D3×D414yes—
346C23×C5C23×D311yescharacteristic
347C22×D5—21no—
348C22×D5—22no—
349C22×D5—22no—
350C22×D5—21no—
351C22×D5#23712yescomplemented
352C5×Q8—68no—
353C5×D4—28no—
354C5×D4—616no—
355C2×C4×C5—22no—
356C2×C4×C5—32no—
357C2×C4×C5—34no—
358C2×C4×C5—34no—
359C2×C4×C5—68no—
360C2×C4×C5—616no—
361C2×C4×C5—62no—
362C2×C4×C5—62no—
363C2×C4×C5D3×D412yes—
364C22⋊D5—216no—
365C22⋊D5—68no—
366C2×Dic5—22no—
367C2×Dic5—22no—
368C2×Dic5—21no—
369C2×Dic5—22no—
370C2×Dic5—32no—
371C2×Dic5—61no—
372C2×Dic5—61no—
373C2×Dic5—62no—
374C2×Dic5—61no—
375C2×Dic5—62no—
376C2×Dic5—62no—
377C2×Dic5—62no—
378C2×Dic5#23714yescomplemented
379C4×D5—124no—
380C2×#41—68no—
381C2×#41—64no—
382C22×Q8—31no—
383C22×D4—32no—
384C22×D4—52no—
385C22×D4—151no—
386C22×D4D3×D511yescharacteristic
387C23×C4—32no—
388C23×C4—31no—
389C23×C4—151no—
390C4×D4—304no—
391C4×D4—308no—
392C2×C4⋊C4—152no—
393C2×C4⋊C4—151no—
394C2×C22⋊C4—52no—
395C2×C22⋊C4—51no—
396C2×C22⋊C4—152no—
397C2×C22⋊C4—151no—
398C2×C22⋊C4—152no—
399C2×C22⋊C4—152no—
400C2×C22⋊C4—152no—
401C2×C42—152no—
402C2×C42—151no—
403#95—51no—
404#95—152no—
405#95—152no—
406#95—151no—
407C2×C3×C5—44no—
408C2×C3×C5C2×#12011yescharacteristic
409C2×C3×C5#52512yes—
410C2×C3×C5#52511yescharacteristic
411C2×C3×C5#53211yescharacteristic
412C2×C3×C5#53212yes—
413C3×D5—42no—
414C5×D3—42no—
415C23×C3—22no—
416C23×C3—22no—
417C23×C3—22no—
418C23×C3—22no—
419C23×C3—22no—
420C23×C3—22no—
421C23×C3—52no—
422C23×C3—101no—
423C23×C3—102no—
424C23×C3—102no—
425C23×C3—101no—
426C23×C3C2×C22⋊D514yes—
427C23×C3C23×D511yescharacteristic
428C22×D3—22no—
429C22×D3—21no—
430C22×D3—21no—
431C22×D3—22no—
432C22×D3C2×C22⋊D512yes—
433C3×D4—28no—
434C3×D4—1016no—
435C2×C3×C4—22no—
436C2×C3×C4—54no—
437C2×C3×C4—102no—
438C2×C3×C4—102no—
439C2×C3×C4—102no—
440C2×C3×C4—101no—
441C2×C3×C4C2×C22⋊D512yes—
442#67—216no—
443#67—108no—
444C2×Dic3—22no—
445C2×Dic3—22no—
446C2×Dic3—216no—
447C2×Dic3—52no—
448C2×Dic3—101no—
449C2×Dic3—101no—
450C2×Dic3—102no—
451C2×Dic3—102no—
452C2×Dic3—102no—
453C2×Dic3—101no—
454C2×Dic3—102no—
455C2×Dic3—204no—
456C2×Dic3C2×C22⋊D514yes—
457C2×Dic3C2×C22⋊D514yes—
458C2×Dic3C2×C22⋊D512yes—
459C4×D3—28no—
460Dic6—28no—
461C22×C5—24no—
462C22×C5—24no—
463C22×C5—24no—
464C22×C5—42no—
465C22×C5—42no—
466C22×C5—42no—
467C22×C5—42no—
468C22×C5—62no—
469C22×C5—62no—
470C22×C5—62no—
471C22×C5—122no—
472C22×C5—122no—
473C22×C5C2×D3×D411yescharacteristic
474C22×C5C2×D3×D412yes—
475C22×C5C2×#23712yes—
476C22×C5C2×#23711yescharacteristic
477C22×C5D3×#4111yescharacteristic
478C2×D5—24no—
479C2×D5—22no—
480C2×D5—41no—
481C2×D5—41no—
482C2×D5—42no—
483C4×C5—22no—
484C4×C5—34no—
485C4×C5—64no—
486Dic5—24no—
487Dic5—41no—
488Dic5—62no—
489Dic5—121no—
490Dic5—122no—
491C24—31no—
492C24—51no—
493C24C2×D3×D512yes—
494#41—632no—
495C2×Q8—64no—
496C2×Q8—62no—
497C2×D4—24no—
498C2×D4—22no—
499C2×D4—64no—
500C2×D4—64no—
501C2×D4—64no—
502C2×D4—104no—
503C2×D4—104no—
504C2×D4—104no—
505C2×D4—302no—
506C2×D4—302no—
507C2×D4—302no—
508C22×C4—31no—
509C22×C4—38no—
510C22×C4—32no—
511C22×C4—34no—
512C22×C4—52no—
513C22×C4—51no—
514C22×C4—64no—
515C22×C4—64no—
516C22×C4—64no—
517C22×C4—151no—
518C22×C4—152no—
519C22×C4—151no—
520C22×C4—302no—
521C22×C4—304no—
522C22×C4C2×D3×D511yescharacteristic
523C4⋊C4—304no—
524C4⋊C4—302no—
525C22⋊C4—104no—
526C22⋊C4—102no—
527C22⋊C4—304no—
528C22⋊C4—304no—
529C22⋊C4—302no—
530C22⋊C4—304no—
531C22⋊C4—304no—
532C42—304no—
533C42—302no—
534C3×C5#238811yescharacteristic, complemented, Hall
535C22×C3—24no—
536C22×C3—24no—
537C22×C3—24no—
538C22×C3—42no—
539C22×C3—42no—
540C22×C3—42no—
541C22×C3—42no—
542C22×C3—102no—
543C22×C3—104no—
544C22×C3—202no—
545C22×C3—201no—
546C22×C3—201no—
547C22×C3D5×#4111yescharacteristic
548C22×C3D5×#4112yes—
549C22×C3D5×#4111yescharacteristic
550C22×C3C22×C22⋊D511yescharacteristic
551C22×C3C22×C22⋊D512yes—
552C2×D3—22no—
553C2×D3—22no—
554C2×D3—22no—
555C2×D3—42no—
556C2×D3—42no—
557C3×C4—22no—
558C3×C4—104no—
559C3×C4—201no—
560Dic3—24no—
561Dic3—102no—
562Dic3—202no—
563Dic3—201no—
564Dic3#409914yes—
565C2×C5—44no—
566C2×C5—122no—
567C2×C5#558612yes—
568C2×C5D3×#12011yescharacteristic
569C2×C5#559612yes—
570C2×C5#560611yescharacteristic
571C2×C5#571611yescharacteristic
572D5—42no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—32no—
580C23—52no—
581C23—62no—
582C23—61no—
583C23—61no—
584C23—62no—
585C23—102no—
586C23—101no—
587C23—101no—
588C23—102no—
589C23D3×C22⋊D514yes—
590C23C22×D3×D511yescharacteristic
591Q8—68no—
592D4—28no—
593D4—616no—
594D4—1016no—
595D4—308no—
596C2×C4—22no—
597C2×C4—34no—
598C2×C4—32no—
599C2×C4—34no—
600C2×C4—54no—
601C2×C4—68no—
602C2×C4—616no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—102no—
606C2×C4—102no—
607C2×C4—102no—
608C2×C4—101no—
609C2×C4—152no—
610C2×C4—301no—
611C2×C4—301no—
612C2×C4—302no—
613C2×C4—302no—
614C2×C4—302no—
615C2×C4—302no—
616C2×C4—301no—
617C2×C4—604no—
618C2×C4D3×C22⋊D512yes—
619C2×C3—44no—
620C2×C3—202no—
621C2×C3#6616212yes—
622C2×C3#6616211yescharacteristic
623C2×C3#6616612yes—
624C2×C3#6616611yescharacteristic
625C2×C3C2×#409911yescharacteristic
626D3—42no—
627C5#7666511yescharacteristic, complemented, Sylow
628C22—24no—
629C22—24no—
630C22—24no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—62no—
636C22—62no—
637C22—62no—
638C22—104no—
639C22—102no—
640C22—122no—
641C22—122no—
642C22—202no—
643C22—201no—
644C22—201no—
645C22D5×#23711yescharacteristic
646C22D5×#23712yes—
647C22#9202411yescharacteristic
648C22C2×D3×C22⋊D512yes—
649C22C2×D3×C22⋊D511yescharacteristic
650C4—22no—
651C4—34no—
652C4—64no—
653C4—104no—
654C4—201no—
655C4—302no—
656C4—602no—
657C4—601no—
658C3#1061582611yescharacteristic, complemented, Sylow
659C2—44no—
660C2—122no—
661C2—202no—
662C2#1175480112yes—
663C2#1175480312yes—
664C2#1175480511yescharacteristic
665C2#1175506511yescharacteristic
666C2D3×#409911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutator, socle, cosocleC23×C3×C5C24
FittingC22×C3×C5×D4C22
hypercenterC22×D4D3×D5
hypocenterC3×C5#2388
LCSG ▷ C23×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×C5×#237C2semidirect
2. C2×Dic3×C22⋊D5C2semidirect
3. C2×#91441C2semidirect
4. #11751323C2semidirect
5. C3×#65512C2semidirect
6. #11749521C2semidirect
7. #11749493C2semidirect
8. Dic3×C22⋊Dic5C2semidirect
9. #11748833C2semidirect
10. #11748773C2semidirect
11. #11748719C2semidirect
47. #65512D3semidirect
66. C22×C5×Dic3D4semidirect
71. C2×C4×C5×D3D4semidirect
95. C22×#237D5semidirect
151. C22×C3×D5#41semidirect
162. C2×Dic15#41semidirect
177. C2×C3×Dic5#41semidirect
315. C22×Dic3C22⋊D5semidirect
320. C2×C4×D3C22⋊D5semidirect
351. C22×D5#237semidirect
378. C2×Dic5#237semidirect
534. C3×C5#2388semidirect
627. C5#76665semidirect
658. C3#10615826semidirect

Maximal subgroups

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

subgroupindex
C22×C5×#2372
C2×Dic3×C22⋊D52
C2×#914412
#117513232
C3×#655122
#117495212
#117494932
Dic3×C22⋊Dic52
#117488332
#117487732
#117487192
#106158263
#766655