Finite Group Databasebeta

← #14331682

Subgroups — #14331682

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×#6773C211yesmaximal, characteristic, complemented
2C2×C3×#4039C211yesmaximal, characteristic, complemented
3C2×#91444C212yesmaximal, complemented
4#11751092C211yesmaximal, characteristic, complemented
5C5×#4979C211yesmaximal, characteristic, complemented
6#11748931C211yesmaximal, characteristic, complemented
7#11748923C212yesmaximal, complemented
8#11748875C212yesmaximal, complemented
9#11748773C211yesmaximal, characteristic, complemented
10#11748732C212yesmaximal, complemented
11#11748702C211yesmaximal, characteristic, complemented
12#10608900—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C22×C3×D20C2211yescharacteristic
16C22×C3×C4×D5C2211yescharacteristic
17C22×C3×C4×D5C2211yescharacteristic
18C22×#1202C2212yes—
19C22×#1202C2212yes—
20C22×#1200C2212yes—
21C2×#6773—24no—
22C2×#6773—28no—
23C2×C3×C5×C4⋊C4C2211yescharacteristic
24C2×#6714C2212yes—
25C2×#6714C2212yes—
26C2×C5×Dic3⋊C4C2212yes—
27C2×C3×#685C2211yescharacteristic
28C2×C3×#685C2211yescharacteristic
29C2×C3×C4×Dic5C2211yescharacteristic
30C3×#4039—24no—
31C2×#6667C2212yes—
32C2×#6665C2212yes—
33C2×#6665C2212yes—
34C2×Dic3×Dic5C2212yes—
35#91444—28no—
36#91115C2211yescharacteristic
37C5×C3⋊#95C2212yes—
38C5×C3⋊#95C2211yescharacteristic
39#90994C2212yes—
40#82400—51nomaximal, Hall
41C22×#713—31no—
42C2×#4039—32no—
43C2×#4039D311yescharacteristic, complemented
44C5×#395—31no—
45#65520—31no—
46#65280—32no—
47#65280—31no—
48#64960—31no—
49#64960—32no—
50#64955—32no—
51#64955—31no—
52C23×D15C2312yes—
53C23×C3×D5C2311yescharacteristic
54C23×C3×D5C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C22×Dic15C2312yes—
59C2×D60—24no—
60C2×D60—22no—
61C22×C5×Dic3C2312yes—
62C22×C5×Dic3C2312yes—
63C2×C5×Dic6—22no—
64C2×C5×Dic6—24no—
65C22×C3×Dic5C2311yescharacteristic
66C22×C3×Dic5C2311yescharacteristic
67C2×C3×D20—22no—
68C2×C3×D20—22no—
69C2×C3×D20—22no—
70C2×C3×C4×D5—24no—
71C2×C3×C4×D5—24no—
72C2×C3×C4×D5—22no—
73C2×C3×C4×D5D414yescomplemented
74C2×#1202—24no—
75C2×#1202—24no—
76C2×#1202—24no—
77C2×#1202—24no—
78C2×#1202—24no—
79C2×#1202—24no—
80C2×#1200—24no—
81C2×#1200—24no—
82C2×#1200D418yescomplemented
83#6773—232no—
84C3×C5×C4⋊C4—22no—
85#6714—24no—
86#6714—24no—
87C5×Dic3⋊C4—24no—
88C3×#685—22no—
89C3×#685—22no—
90C3×C4×Dic5—22no—
91#6667—24no—
92#6665—24no—
93#6665—24no—
94Dic3×Dic5—24no—
95C22×#235—51no—
96C2×C3×C4×D4—51no—
97C2×C4×#67—52no—
98#5223—52no—
99C4×#217—51no—
100#4991—51no—
101#4979D511yescharacteristic, complemented
102#4960—51no—
103#4959—52no—
104#4947—51no—
105C4×(C2×D3)⋊C4—52no—
106C22×C5×Q8—31no—
107C2×#713—68no—
108C2×#713—64no—
109C22×D20—32no—
110C22×D20—32no—
111C22×D20—31no—
112C22×D20C2×D311yescharacteristic
113C22×C4×D5—32no—
114C22×C4×D5C2×D311yescharacteristic
115C22×C4×D5C2×D311yescharacteristic
116C2×C5×C4⋊C4—32no—
117C2×C5×C4⋊C4C2×D311yescharacteristic
118C2×#685—32no—
119C2×#685—32no—
120C2×#685—32no—
121C2×#685—32no—
122C2×#685—32no—
123C2×#685C2×D311yescharacteristic
124C2×#685C2×D311yescharacteristic
125C2×C4×Dic5—32no—
126C2×C4×Dic5C2×D311yescharacteristic
127#4039—24no—
128#4039—68no—
129C5×#95—32no—
130C5×#95—31no—
131#3963—31no—
132#3963—32no—
133#2388—151noSylow
134C23×C3×C5C2411yescharacteristic
135C22×D15—22no—
136C22×D15—22no—
137C22×D15—22no—
138C22×D15—22no—
139C22×D15—22no—
140C22×D15—22no—
141C22×D15C2×D414yes—
142C22×C3×D5—22no—
143C22×C3×D5—21no—
144C22×C3×D5—22no—
145C22×C3×D5—22no—
146C22×C3×D5—22no—
147C22×C3×D5—21no—
148C22×C3×D5—21no—
149C22×C3×D5—21no—
150C22×C3×D5C2×D412yes—
151C22×C3×D5#4112yescomplemented
152C2×C3×C4×C5—21no—
153C2×C3×C4×C5—22no—
154C2×C3×C4×C5—21no—
155C2×C3×C4×C5—21no—
156C2×C3×C4×C5—22no—
157C2×C3×C4×C5C2×D412yes—
158C2×C3×C4×C5#4112yescomplemented
159C2×Dic15—22no—
160C2×Dic15—22no—
161C2×Dic15—22no—
162D60—28no—
163C2×C5×Dic3—22no—
164C2×C5×Dic3—22no—
165C2×C5×Dic3—22no—
166C2×C5×Dic3—22no—
167C2×C5×Dic3C2×D414yes—
168C2×C5×Dic3#4114yescomplemented
169C5×Dic6—28no—
170C2×C3×Dic5—21no—
171C2×C3×Dic5—22no—
172C2×C3×Dic5C2×D414yes—
173C2×C3×Dic5C2×D412yes—
174C3×D20—28no—
175C3×C4×D5—28no—
176C3×C4×D5—44no—
177#1202—216no—
178#1202—216no—
179#1200—216no—
180C22×C3×D4—51no—
181C23×C3×C4—51no—
182C23×C3×C4—51no—
183C22×#67—52no—
184C22×#67—52no—
185C2×#235—108no—
186C2×#235—104no—
187C22×D12—51no—
188C22×C4×D3—52no—
189C22×Dic6C2×D511yescharacteristic
190C3×C4×D4—104no—
191C2×C3×C4⋊C4C2×D511yescharacteristic
192C2×C3×C22⋊C4—51no—
193C2×C3×C22⋊C4—51no—
194C2×C3×C42—51no—
195C2×#217—52no—
196C2×#217—52no—
197C4×#67—108no—
198C2×(C2×D3)⋊C4—52no—
199C2×(C2×D3)⋊C4—52no—
200C2×(C2×D3)⋊C4—52no—
201C2×Dic3⋊C4C2×D512yes—
202C2×C4×Dic3—52no—
203C3⋊#95—51no—
204C3⋊#95—52no—
205C3⋊#95C2×D512yes—
206C3⋊#95C2×D511yescharacteristic
207C23×D5—32no—
208C23×D5C22×D311yescharacteristic
209C23×D5C22×D311yescharacteristic
210C2×C5×Q8—64no—
211C2×C5×Q8—62no—
212C22×C4×C5—32no—
213C22×C4×C5—32no—
214C22×C4×C5C22×D311yescharacteristic
215C22×C4×C5C22×D311yescharacteristic
216C22×C4×C5C22×D311yescharacteristic
217C22×Dic5—32no—
218C22×Dic5C22×D311yescharacteristic
219C22×Dic5C22×D311yescharacteristic
220#713—632no—
221C2×D20—22no—
222C2×D20—22no—
223C2×D20—22no—
224C2×D20—64no—
225C2×D20—64no—
226C2×D20—64no—
227C2×D20—64no—
228C2×D20—64no—
229C2×D20—64no—
230C2×D20—64no—
231C2×D20—62no—
232C2×C4×D5—24no—
233C2×C4×D5—24no—
234C2×C4×D5—22no—
235C2×C4×D5—38no—
236C2×C4×D5—64no—
237C2×C4×D5—64no—
238C2×C4×D5#6714yescomplemented
239C5×C4⋊C4—22no—
240C5×C4⋊C4—64no—
241#685—22no—
242#685—22no—
243#685—64no—
244#685—64no—
245#685—64no—
246#685—64no—
247#685—64no—
248C4×Dic5—22no—
249C4×Dic5—64no—
250C22×#41—151no—
251C2×C4×D4—51no—
252C2×C4×D4—152no—
253#395—31no—
254#394—152no—
255#394—151no—
256#393—151no—
257#392—151no—
258#392—152no—
259C4×C22⋊C4—152no—
260C4×C22⋊C4—151no—
261C22×C3×C5C22×D412yes—
262C22×C3×C5C22×D411yescharacteristic
263C22×C3×C5C2×#4111yescharacteristic
264C22×C3×C5C2×#4112yes—
265C22×C3×C5C2×#4111yescharacteristic
266C2×D15—24no—
267C2×D15—24no—
268C2×D15—24no—
269C2×D15—42no—
270C2×D15—42no—
271C2×D15—42no—
272C2×D15—42no—
273C2×C3×D5—22no—
274C2×C3×D5—24no—
275C2×C3×D5—22no—
276C2×C3×D5—22no—
277C2×C3×D5—22no—
278C2×C3×D5—41no—
279C2×C3×D5—42no—
280C2×C3×D5—42no—
281C2×C3×D5—42no—
282C2×C3×D5—41no—
283C3×C4×C5—22no—
284C3×C4×C5—22no—
285C3×C4×C5—41no—
286Dic15—42no—
287C3×Dic5—41no—
288C3×Dic5#12014yes—
289C5×Dic3—24no—
290C5×Dic3—24no—
291C24×C3—51no—
292C24×C3—51no—
293C23×D3—52no—
294C2×C3×D4—102no—
295C2×C3×D4—102no—
296C2×C3×D4—102no—
297C22×C3×C4—51no—
298C22×C3×C4—54no—
299C22×C3×C4—51no—
300C22×C3×C4—104no—
301C22×C3×C4—104no—
302C22×C3×C4—102no—
303C22×C3×C4C22×D511yescharacteristic
304C22×C3×C4C22×D511yescharacteristic
305C22×C3×C4C22×D511yescharacteristic
306C2×#67—104no—
307C2×#67—104no—
308C2×#67—104no—
309C2×#67—104no—
310C2×#67—104no—
311C2×#67—104no—
312C22×Dic3—52no—
313C22×Dic3C22×D512yes—
314C22×Dic3C22×D512yes—
315#235—1032no—
316C2×D12—104no—
317C2×D12—102no—
318C2×C4×D3—58no—
319C2×C4×D3—104no—
320C2×C4×D3—104no—
321C2×Dic6—22no—
322C2×Dic6—24no—
323C3×C4⋊C4—22no—
324C3×C22⋊C4—102no—
325C3×C22⋊C4—102no—
326C3×C42—102no—
327#217—104no—
328#217—104no—
329(C2×D3)⋊C4—104no—
330(C2×D3)⋊C4—104no—
331(C2×D3)⋊C4—104no—
332Dic3⋊C4—24no—
333C4×Dic3—104no—
334C23×C5C23×D311yescharacteristic
335C22×D5—21no—
336C22×D5—21no—
337C22×D5—21no—
338C22×D5—22no—
339C22×D5—22no—
340C22×D5—22no—
341C22×D5—22no—
342C22×D5—21no—
343C22×D5—34no—
344C22×D5—62no—
345C22×D5—62no—
346C22×D5—62no—
347C22×D5—62no—
348C22×D5—62no—
349C22×D5—62no—
350C22×D5#23512yescomplemented
351C22×D5C2×#6712yes—
352C5×Q8—68no—
353C2×C4×C5—21no—
354C2×C4×C5—22no—
355C2×C4×C5—21no—
356C2×C4×C5—22no—
357C2×C4×C5—21no—
358C2×C4×C5—34no—
359C2×C4×C5—34no—
360C2×C4×C5—62no—
361C2×C4×C5—62no—
362C2×C4×C5—62no—
363C2×C4×C5—62no—
364C2×C4×C5#23512yescomplemented
365C2×C4×C5C2×#6712yes—
366C2×Dic5—22no—
367C2×Dic5—21no—
368C2×Dic5—62no—
369C2×Dic5—62no—
370C2×Dic5—62no—
371C2×Dic5D3×D414yes—
372C2×Dic5C2×#6712yes—
373D20—28no—
374D20—68no—
375D20—616no—
376D20—616no—
377C4×D5—28no—
378C4×D5—44no—
379C4×D5—616no—
380C2×#41—304no—
381C2×#41—308no—
382C22×Q8—31no—
383C22×D4—51no—
384C22×D4—152no—
385C22×D4—152no—
386C22×D4—151no—
387C23×C4—51no—
388C23×C4—51no—
389C23×C4—152no—
390C4×D4—104no—
391C4×D4—308no—
392C2×C4⋊C4—32no—
393C2×C4⋊C4D3×D511yescharacteristic
394C2×C22⋊C4—51no—
395C2×C22⋊C4—51no—
396C2×C22⋊C4—152no—
397C2×C22⋊C4—152no—
398C2×C22⋊C4—152no—
399C2×C22⋊C4—152no—
400C2×C22⋊C4—152no—
401C2×C42—51no—
402C2×C42—152no—
403#95—31no—
404#95—32no—
405#95—152no—
406#95—151no—
407C2×C3×C5C2×#12011yescharacteristic
408C2×C3×C5#52511yescharacteristic
409C2×C3×C5#52512yes—
410C2×C3×C5#53212yes—
411C2×C3×C5#53211yescharacteristic
412D15—44no—
413C3×D5—42no—
414C3×D5—42no—
415C23×C3—52no—
416C23×C3—52no—
417C23×C3—102no—
418C23×C3—101no—
419C23×C3—102no—
420C23×C3—102no—
421C23×C3—102no—
422C23×C3—101no—
423C23×C3—101no—
424C23×C3—101no—
425C23×C3C23×D511yescharacteristic
426C22×D3—54no—
427C22×D3—102no—
428C22×D3—102no—
429C22×D3—102no—
430C22×D3—102no—
431C22×D3—102no—
432C22×D3—102no—
433C3×D4—108no—
434C2×C3×C4—21no—
435C2×C3×C4—22no—
436C2×C3×C4—21no—
437C2×C3×C4—22no—
438C2×C3×C4—21no—
439C2×C3×C4—54no—
440C2×C3×C4—52no—
441C2×C3×C4—108no—
442C2×C3×C4—101no—
443C2×C3×C4—102no—
444C2×C3×C4—204no—
445C2×C3×C4D4×D512yes—
446C2×C3×C4#71312yescomplemented
447#67—1016no—
448#67—1016no—
449C2×Dic3—22no—
450C2×Dic3—22no—
451C2×Dic3—22no—
452C2×Dic3—22no—
453C2×Dic3—102no—
454C2×Dic3—102no—
455C2×Dic3—102no—
456C2×Dic3D4×D514yes—
457C2×Dic3#71314yescomplemented
458D12—108no—
459C4×D3—1016no—
460Dic6—28no—
461C22×C5C2×#23511yescharacteristic
462C22×C5C2×#23511yescharacteristic
463C22×C5C2×D3×D412yes—
464C22×C5D3×#4112yes—
465C22×C5C22×#6711yescharacteristic
466C2×D5—22no—
467C2×D5—22no—
468C2×D5—22no—
469C2×D5—24no—
470C2×D5—22no—
471C2×D5—41no—
472C2×D5—42no—
473C2×D5—42no—
474C2×D5—42no—
475C2×D5—41no—
476C2×D5—64no—
477C2×D5—64no—
478C2×D5—64no—
479C2×D5—122no—
480C2×D5—122no—
481C2×D5—122no—
482C2×D5—122no—
483C4×C5—22no—
484C4×C5—22no—
485C4×C5—41no—
486C4×C5—64no—
487C4×C5—64no—
488Dic5—41no—
489Dic5—122no—
490Dic5#93714yes—
491C24—51no—
492C24—51no—
493C24—152no—
494#41—3032no—
495C2×Q8—64no—
496C2×Q8—62no—
497C2×D4—102no—
498C2×D4—102no—
499C2×D4—102no—
500C2×D4—304no—
501C2×D4—304no—
502C2×D4—304no—
503C2×D4—304no—
504C2×D4—304no—
505C2×D4—304no—
506C2×D4—304no—
507C2×D4—302no—
508C22×C4—32no—
509C22×C4—32no—
510C22×C4—51no—
511C22×C4—54no—
512C22×C4—51no—
513C22×C4—104no—
514C22×C4—102no—
515C22×C4—104no—
516C22×C4—158no—
517C22×C4—152no—
518C22×C4—304no—
519C22×C4—304no—
520C22×C4C2×D3×D511yescharacteristic
521C22×C4C2×D3×D511yescharacteristic
522C22×C4C2×D3×D511yescharacteristic
523C4⋊C4—22no—
524C4⋊C4—64no—
525C22⋊C4—102no—
526C22⋊C4—102no—
527C22⋊C4—304no—
528C22⋊C4—304no—
529C22⋊C4—304no—
530C22⋊C4—304no—
531C22⋊C4—304no—
532C42—102no—
533C42—304no—
534C3×C5#238811yescharacteristic, complemented, Hall
535C22×C3—102no—
536C22×C3—102no—
537C22×C3—104no—
538C22×C3—102no—
539C22×C3—102no—
540C22×C3—201no—
541C22×C3—202no—
542C22×C3—202no—
543C22×C3—201no—
544C22×C3—202no—
545C22×C3C2×D4×D512yes—
546C22×C3C2×D4×D511yescharacteristic
547C22×C3C2×#71311yescharacteristic
548C22×C3C2×#71312yes—
549C22×C3D5×#4111yescharacteristic
550C2×D3—104no—
551C2×D3—104no—
552C2×D3—104no—
553C2×D3—202no—
554C2×D3—202no—
555C2×D3—202no—
556C2×D3—202no—
557C3×C4—22no—
558C3×C4—22no—
559C3×C4—41no—
560C3×C4—54no—
561C3×C4—201no—
562Dic3—24no—
563Dic3—24no—
564Dic3—202no—
565C2×C5#554712yes—
566C2×C5#554812yes—
567C2×C5#579111yescharacteristic
568C2×C5C2×#93711yescharacteristic
569C2×C5#583211yescharacteristic
570D5—42no—
571D5—42no—
572D5—124no—
573C23—52no—
574C23—52no—
575C23—101no—
576C23—101no—
577C23—101no—
578C23—102no—
579C23—102no—
580C23—102no—
581C23—102no—
582C23—101no—
583C23—154no—
584C23—302no—
585C23—302no—
586C23—302no—
587C23—302no—
588C23—302no—
589C23—302no—
590C23C22×D3×D511yescharacteristic
591Q8—68no—
592D4—108no—
593D4—308no—
594D4—3016no—
595D4—3016no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—34no—
602C2×C4—34no—
603C2×C4—54no—
604C2×C4—52no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—108no—
610C2×C4—102no—
611C2×C4—101no—
612C2×C4—204no—
613C2×C4—3016no—
614C2×C4—302no—
615C2×C4—302no—
616C2×C4—302no—
617C2×C4#677312yescomplemented
618C2×C4D5×#6712yes—
619C2×C3—202no—
620C2×C3—202no—
621C2×C3D5×#12011yescharacteristic
622C2×C3#6596111yescharacteristic
623C2×C3#6596112yes—
624C2×C3#6605911yescharacteristic
625C2×C3#6605912yes—
626D3—204no—
627C5#8240011yescharacteristic, complemented, Sylow
628C22—102no—
629C22—102no—
630C22—102no—
631C22—104no—
632C22—102no—
633C22—201no—
634C22—202no—
635C22—201no—
636C22—202no—
637C22—202no—
638C22—304no—
639C22—304no—
640C22—304no—
641C22—602no—
642C22—602no—
643C22—602no—
644C22—602no—
645C22C2×#677311yescharacteristic
646C22D5×#23511yescharacteristic
647C22D3×D4×D512yes—
648C22D3×#71312yes—
649C22C2×D5×#6711yescharacteristic
650C4—22no—
651C4—22no—
652C4—41no—
653C4—54no—
654C4—64no—
655C4—64no—
656C4—201no—
657C4—602no—
658C3#1060890011yescharacteristic, complemented, Sylow
659C2—202no—
660C2—202no—
661C2—604no—
662C2#1175368212yes—
663C2#1175386512yes—
664C2#1175452711yescharacteristic
665C2D5×#93711yescharacteristic
666C2#1175515911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutator, socle, cosocleC23×C3×C5C24
FittingC2×C3×C5×C4⋊C4C22
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2388
LCSG ▷ C23×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#6773C2semidirect
2. C2×C3×#4039C2semidirect
3. C2×#91444C2semidirect
4. #11751092C2semidirect
5. C5×#4979C2semidirect
6. #11748931C2semidirect
7. #11748923C2semidirect
8. #11748875C2semidirect
9. #11748773C2semidirect
10. #11748732C2semidirect
11. #11748702C2semidirect
43. C2×#4039D3semidirect
73. C2×C3×C4×D5D4semidirect
82. C2×#1200D4semidirect
101. #4979D5semidirect
151. C22×C3×D5#41semidirect
158. C2×C3×C4×C5#41semidirect
168. C2×C5×Dic3#41semidirect
238. C2×C4×D5#67semidirect
350. C22×D5#235semidirect
364. C2×C4×C5#235semidirect
446. C2×C3×C4#713semidirect
457. C2×Dic3#713semidirect
534. C3×C5#2388semidirect
617. C2×C4#6773semidirect
627. C5#82400semidirect
658. C3#10608900semidirect

Maximal subgroups

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

subgroupindex
C22×#67732
C2×C3×#40392
C2×#914442
#117510922
C5×#49792
#117489312
#117489232
#117488752
#117487732
#117487322
#117487022
#106089003
#824005