Finite Group Databasebeta

← #14331640

Subgroups — #14331640

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×#6772C211yesmaximal, characteristic, complemented
2C2×C3×#4027C211yesmaximal, characteristic, complemented
3C2×Dic5×#67C212yesmaximal, complemented
4#11751048C211yesmaximal, characteristic, complemented
5C5×#4960C211yesmaximal, characteristic, complemented
6#11749521C212yesmaximal, complemented
7#11748922C212yesmaximal, complemented
8Dic5×(C2×D3)⋊C4C212yesmaximal, complemented
9#11748846C211yesmaximal, characteristic, complemented
10#11748765C211yesmaximal, characteristic, complemented
11#11748702C211yesmaximal, characteristic, complemented
12#10608396—31nomaximal, Hall
13C22×Dic30C2211yescharacteristic
14C22×C5×#67C2212yes—
15C22×C5×D12C2211yescharacteristic
16C22×C3×C22⋊D5C2211yescharacteristic
17C23×C3×Dic5C2211yescharacteristic
18C22×C3×C4×D5C2211yescharacteristic
19C22×#1201C2212yes—
20C22×D3×Dic5C2212yes—
21C2×#6772—24no—
22C2×#6772—28no—
23C2×C3×C5×C22⋊C4C2211yescharacteristic
24C2×#6719C2212yes—
25C2×C5×(C2×D3)⋊C4C2212yes—
26C2×C3×#685C2211yescharacteristic
27C2×C3×#683C2211yescharacteristic
28C2×C3×C4×Dic5C2211yescharacteristic
29C3×#4027—24no—
30Dic5×#67—28no—
31C2×Dic3⋊Dic5C2212yes—
32C2×(C2×D3)⋊Dic5C2212yes—
33C2×(C2×D3)⋊Dic5C2212yes—
34C2×#6665C2212yes—
35C2×Dic3×Dic5C2212yes—
36#91115C2212yes—
37#91115C2211yescharacteristic
38C5×C3⋊#95C2211yescharacteristic
39#90994C2212yes—
40#82400—51nomaximal, Hall
41C22×#711—31no—
42C2×D4×Dic5—32no—
43C2×#4027D311yescharacteristic, complemented
44C5×#392—31no—
45#65512—32no—
46#65511—31no—
47#65250—32no—
48#65236—31no—
49C22⋊C4×Dic5—32no—
50#64955—31no—
51#64945—31no—
52C24×C3×C5C2311yescharacteristic
53C23×C5×D3C2312yes—
54C23×C3×D5C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×Dic15C2312yes—
58C22×Dic15C2312yes—
59C2×Dic30—22no—
60C2×Dic30—24no—
61C2×C5×#67—24no—
62C2×C5×#67—24no—
63C2×C5×#67—24no—
64C22×C5×Dic3C2312yes—
65C2×C5×D12—24no—
66C2×C5×D12—22no—
67C2×C3×C22⋊D5—22no—
68C2×C3×C22⋊D5—22no—
69C2×C3×C22⋊D5—22no—
70C22×C3×Dic5—24no—
71C22×C3×Dic5—22no—
72C22×C3×Dic5C2311yescharacteristic
73C22×C3×Dic5C2311yescharacteristic
74C22×C3×Dic5C2311yescharacteristic
75C2×C3×C4×D5—24no—
76C2×C3×C4×D5D414yescomplemented
77C2×#1201—24no—
78C2×#1201—24no—
79C2×#1201—24no—
80C2×D3×Dic5—24no—
81C2×D3×Dic5—24no—
82C2×D3×Dic5D418yescomplemented
83#6772—232no—
84C3×C5×C22⋊C4—22no—
85#6719—24no—
86C5×(C2×D3)⋊C4—24no—
87C3×#685—22no—
88C3×#683—22no—
89C3×C4×Dic5—22no—
90Dic3⋊Dic5—24no—
91(C2×D3)⋊Dic5—24no—
92(C2×D3)⋊Dic5—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#4979—51no—
102#4960D511yescharacteristic, complemented
103#4959—52no—
104#4947—51no—
105C4×(C2×D3)⋊C4—52no—
106C22×C5×D4—32no—
107C22×C5×D4—31no—
108C22×C22⋊D5—32no—
109C22×C22⋊D5C2×D311yescharacteristic
110C23×Dic5—32no—
111C23×Dic5C2×D311yescharacteristic
112C2×#711—68no—
113C2×#711—64no—
114C22×C4×D5C2×D311yescharacteristic
115C22×Dic10—31no—
116C2×C5×C22⋊C4—32no—
117C2×C5×C22⋊C4C2×D311yescharacteristic
118C2×C22⋊Dic5—32no—
119C2×C22⋊Dic5—32no—
120C2×C22⋊Dic5—32no—
121D4×Dic5—68no—
122C2×#685—32no—
123C2×#685C2×D311yescharacteristic
124C2×C4⋊Dic5—32no—
125C2×#683C2×D311yescharacteristic
126C2×C4×Dic5—32no—
127C2×C4×Dic5C2×D311yescharacteristic
128#4027—24no—
129C5×#95—31no—
130#3963—31no—
131#3963—32no—
132#3963—32no—
133#2388—151noSylow
134C23×C3×C5—22no—
135C23×C3×C5—21no—
136C23×C3×C5—21no—
137C23×C3×C5—22no—
138C23×C3×C5#4112yescomplemented
139C23×C3×C5C2411yescharacteristic
140C22×C5×D3—22no—
141C22×C5×D3—22no—
142C22×C5×D3—22no—
143C22×C5×D3—22no—
144C22×C5×D3—22no—
145C22×C5×D3—22no—
146C22×C5×D3C2×D414yes—
147C22×C3×D5—22no—
148C22×C3×D5—22no—
149C22×C3×D5—21no—
150C22×C3×D5—21no—
151C22×C3×D5C2×D412yes—
152C2×C3×C4×C5—22no—
153C2×C3×C4×C5—21no—
154C2×C3×C4×C5—22no—
155C2×C3×C4×C5C2×D412yes—
156C2×Dic15—22no—
157C2×Dic15—22no—
158C2×Dic15—22no—
159C2×Dic15—22no—
160C2×Dic15—22no—
161C2×Dic15C2×D414yes—
162Dic30—28no—
163C5×#67—216no—
164C2×C5×Dic3—22no—
165C2×C5×Dic3—22no—
166C2×C5×Dic3#4114yescomplemented
167C5×D12—28no—
168C3×C22⋊D5—28no—
169C2×C3×Dic5—21no—
170C2×C3×Dic5—21no—
171C2×C3×Dic5—22no—
172C2×C3×Dic5—21no—
173C2×C3×Dic5—44no—
174C2×C3×Dic5C2×D414yes—
175C2×C3×Dic5C2×D412yes—
176C2×C3×Dic5#4112yescomplemented
177C3×C4×D5—28no—
178#1201—216no—
179D3×Dic5—216no—
180C22×C3×D4—51no—
181C23×C3×C4—51no—
182C23×C3×C4—51no—
183C22×#67—52no—
184C22×#67C2×D512yes—
185C2×#235—108no—
186C2×#235—104no—
187C22×D12C2×D511yescharacteristic
188C22×C4×D3—52no—
189C22×Dic6—51no—
190C3×C4×D4—104no—
191C2×C3×C4⋊C4—51no—
192C2×C3×C22⋊C4—51no—
193C2×C3×C22⋊C4C2×D511yescharacteristic
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)⋊C4C2×D512yes—
201C2×Dic3⋊C4—52no—
202C2×C4×Dic3—52no—
203C3⋊#95—52no—
204C3⋊#95—51no—
205C3⋊#95—52no—
206C3⋊#95C2×D511yescharacteristic
207C24×C5—32no—
208C24×C5C22×D311yescharacteristic
209C23×D5C22×D311yescharacteristic
210C2×C5×D4—64no—
211C2×C5×D4—64no—
212C2×C5×D4—64no—
213C2×C5×D4—64no—
214C2×C5×D4—62no—
215C22×C4×C5—32no—
216C22×C4×C5C22×D311yescharacteristic
217C22×C4×C5C22×D311yescharacteristic
218C2×C22⋊D5—22no—
219C2×C22⋊D5—22no—
220C2×C22⋊D5—22no—
221C2×C22⋊D5—64no—
222C2×C22⋊D5—64no—
223C2×C22⋊D5—64no—
224C22×Dic5—24no—
225C22×Dic5—22no—
226C22×Dic5—32no—
227C22×Dic5—38no—
228C22×Dic5—32no—
229C22×Dic5—64no—
230C22×Dic5—64no—
231C22×Dic5C22×D311yescharacteristic
232C22×Dic5C22×D311yescharacteristic
233C22×Dic5C22×D311yescharacteristic
234#711—632no—
235C2×C4×D5—24no—
236C2×C4×D5#6714yescomplemented
237C2×Dic10—64no—
238C2×Dic10—62no—
239C5×C22⋊C4—22no—
240C5×C22⋊C4—64no—
241C22⋊Dic5—64no—
242C22⋊Dic5—64no—
243C22⋊Dic5—64no—
244#685—22no—
245#685—64no—
246C4⋊Dic5—64no—
247#683—22no—
248C4×Dic5—22no—
249C4×Dic5—64no—
250C22×#41—151no—
251C2×C4×D4—51no—
252C2×C4×D4—152no—
253#395—151no—
254#394—152no—
255#394—151no—
256#393—151no—
257#392—31no—
258#392—152no—
259C4×C22⋊C4—151no—
260C4×C22⋊C4—152no—
261C22×C3×C5—22no—
262C22×C3×C5—24no—
263C22×C3×C5—41no—
264C22×C3×C5—41no—
265C22×C3×C5—42no—
266C22×C3×C5C22×D412yes—
267C22×C3×C5C22×D411yescharacteristic
268C22×C3×C5C2×#4111yescharacteristic
269C22×C3×C5C2×#4112yes—
270C22×C3×C5C2×#4111yescharacteristic
271C2×C5×D3—24no—
272C2×C5×D3—24no—
273C2×C5×D3—24no—
274C2×C5×D3—42no—
275C2×C5×D3—42no—
276C2×C5×D3—42no—
277C2×C5×D3—42no—
278C2×C3×D5—22no—
279C2×C3×D5—22no—
280C2×C3×D5—22no—
281C2×C3×D5—42no—
282C2×C3×D5—42no—
283C3×C4×C5—22no—
284C3×C4×C5—41no—
285Dic15—24no—
286Dic15—42no—
287C3×Dic5—22no—
288C3×Dic5—41no—
289C3×Dic5#12014yes—
290C5×Dic3—24no—
291C24×C3—51no—
292C24×C3C22×D511yescharacteristic
293C23×D3C22×D512yes—
294C2×C3×D4—102no—
295C2×C3×D4—102no—
296C2×C3×D4—102no—
297C22×C3×C4—51no—
298C22×C3×C4—51no—
299C22×C3×C4—54no—
300C22×C3×C4—51no—
301C22×C3×C4—104no—
302C22×C3×C4—104no—
303C22×C3×C4—102no—
304C22×C3×C4C22×D511yescharacteristic
305C22×C3×C4C22×D511yescharacteristic
306C2×#67—24no—
307C2×#67—24no—
308C2×#67—24no—
309C2×#67—104no—
310C2×#67—104no—
311C2×#67—104no—
312C22×Dic3—52no—
313C22×Dic3—52no—
314C22×Dic3C22×D512yes—
315#235—1032no—
316C2×D12—24no—
317C2×D12—22no—
318C2×C4×D3—58no—
319C2×C4×D3—104no—
320C2×C4×D3—104no—
321C2×Dic6—104no—
322C2×Dic6—102no—
323C3×C4⋊C4—102no—
324C3×C22⋊C4—22no—
325C3×C22⋊C4—102no—
326C3×C42—102no—
327#217—104no—
328#217—104no—
329(C2×D3)⋊C4—24no—
330(C2×D3)⋊C4—104no—
331(C2×D3)⋊C4—104no—
332Dic3⋊C4—104no—
333C4×Dic3—104no—
334C23×C5—21no—
335C23×C5—22no—
336C23×C5—22no—
337C23×C5—21no—
338C23×C5—34no—
339C23×C5—62no—
340C23×C5—62no—
341C23×C5—62no—
342C23×C5—62no—
343C23×C5—62no—
344C23×C5—62no—
345C23×C5#23512yescomplemented
346C23×C5C23×D311yescharacteristic
347C22×D5—21no—
348C22×D5—21no—
349C22×D5—22no—
350C22×D5—22no—
351C22×D5C2×#6712yes—
352C5×D4—68no—
353C5×D4—616no—
354C2×C4×C5—22no—
355C2×C4×C5—22no—
356C2×C4×C5—21no—
357C2×C4×C5—34no—
358C2×C4×C5—62no—
359C2×C4×C5—62no—
360C2×C4×C5C2×#6712yes—
361C22⋊D5—28no—
362C22⋊D5—616no—
363C2×Dic5—21no—
364C2×Dic5—21no—
365C2×Dic5—22no—
366C2×Dic5—21no—
367C2×Dic5—34no—
368C2×Dic5—44no—
369C2×Dic5—62no—
370C2×Dic5—62no—
371C2×Dic5—616no—
372C2×Dic5—62no—
373C2×Dic5—62no—
374C2×Dic5—62no—
375C2×Dic5#23512yescomplemented
376C2×Dic5D3×D414yes—
377C2×Dic5C2×#6712yes—
378C4×D5—28no—
379Dic10—68no—
380C2×#41—304no—
381C2×#41—308no—
382C22×Q8—151no—
383C22×D4—32no—
384C22×D4—31no—
385C22×D4—51no—
386C22×D4—152no—
387C23×C4—51no—
388C23×C4—51no—
389C23×C4—152no—
390C4×D4—104no—
391C4×D4—308no—
392C2×C4⋊C4—51no—
393C2×C4⋊C4—152no—
394C2×C22⋊C4—32no—
395C2×C22⋊C4—51no—
396C2×C22⋊C4—152no—
397C2×C22⋊C4—152no—
398C2×C22⋊C4—152no—
399C2×C22⋊C4—152no—
400C2×C22⋊C4D3×D511yescharacteristic
401C2×C42—51no—
402C2×C42—152no—
403#95—31no—
404#95—152no—
405#95—152no—
406#95—151no—
407C2×C3×C5—42no—
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—44no—
415C23×C3—22no—
416C23×C3—21no—
417C23×C3—22no—
418C23×C3—21no—
419C23×C3—52no—
420C23×C3—102no—
421C23×C3—102no—
422C23×C3—101no—
423C23×C3—101no—
424C23×C3#71112yescomplemented
425C23×C3C23×D511yescharacteristic
426C22×D3—22no—
427C22×D3—22no—
428C22×D3—22no—
429C22×D3—22no—
430C22×D3—22no—
431C22×D3—22no—
432C22×D3D4×D514yes—
433C3×D4—108no—
434C2×C3×C4—22no—
435C2×C3×C4—21no—
436C2×C3×C4—22no—
437C2×C3×C4—54no—
438C2×C3×C4—52no—
439C2×C3×C4—52no—
440C2×C3×C4—108no—
441C2×C3×C4—101no—
442C2×C3×C4—101no—
443C2×C3×C4—102no—
444C2×C3×C4—101no—
445C2×C3×C4—204no—
446C2×C3×C4D4×D512yes—
447#67—216no—
448#67—1016no—
449C2×Dic3—22no—
450C2×Dic3—22no—
451C2×Dic3—54no—
452C2×Dic3—102no—
453C2×Dic3—102no—
454C2×Dic3—102no—
455C2×Dic3—102no—
456C2×Dic3—102no—
457C2×Dic3#71114yescomplemented
458D12—28no—
459C4×D3—1016no—
460Dic6—108no—
461C22×C5—22no—
462C22×C5—24no—
463C22×C5—41no—
464C22×C5—42no—
465C22×C5—41no—
466C22×C5—64no—
467C22×C5—64no—
468C22×C5—64no—
469C22×C5—122no—
470C22×C5—122no—
471C22×C5—122no—
472C22×C5—122no—
473C22×C5C2×#23511yescharacteristic
474C22×C5C2×#23511yescharacteristic
475C22×C5C2×D3×D412yes—
476C22×C5D3×#4112yes—
477C22×C5C22×#6711yescharacteristic
478C2×D5—22no—
479C2×D5—22no—
480C2×D5—22no—
481C2×D5—42no—
482C2×D5—42no—
483C4×C5—22no—
484C4×C5—41no—
485C4×C5—64no—
486Dic5—22no—
487Dic5—41no—
488Dic5—64no—
489Dic5—122no—
490Dic5#93714yes—
491C24—32no—
492C24—51no—
493C24C2×D3×D511yescharacteristic
494#41—3032no—
495C2×Q8—304no—
496C2×Q8—302no—
497C2×D4—64no—
498C2×D4—64no—
499C2×D4—64no—
500C2×D4—64no—
501C2×D4—62no—
502C2×D4—102no—
503C2×D4—102no—
504C2×D4—102no—
505C2×D4—304no—
506C2×D4—304no—
507C2×D4—304no—
508C22×C4—32no—
509C22×C4—51no—
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—152no—
519C22×C4—304no—
520C22×C4—304no—
521C22×C4C2×D3×D511yescharacteristic
522C22×C4C2×D3×D511yescharacteristic
523C4⋊C4—102no—
524C4⋊C4—304no—
525C22⋊C4—22no—
526C22⋊C4—64no—
527C22⋊C4—102no—
528C22⋊C4—304no—
529C22⋊C4—304no—
530C22⋊C4—304no—
531C22⋊C4—304no—
532C42—102no—
533C42—304no—
534C3×C5#238811yescharacteristic, complemented, Hall
535C22×C3—24no—
536C22×C3—22no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—42no—
540C22×C3—102no—
541C22×C3—102no—
542C22×C3—102no—
543C22×C3—202no—
544C22×C3—202no—
545C22×C3C2×D4×D512yes—
546C22×C3C2×D4×D511yescharacteristic
547C22×C3C2×#71111yescharacteristic
548C22×C3C2×#71112yes—
549C22×C3D5×#4111yescharacteristic
550C2×D3—24no—
551C2×D3—24no—
552C2×D3—24no—
553C2×D3—42no—
554C2×D3—42no—
555C2×D3—42no—
556C2×D3—42no—
557C3×C4—22no—
558C3×C4—41no—
559C3×C4—54no—
560C3×C4—102no—
561C3×C4—201no—
562Dic3—24no—
563Dic3—104no—
564Dic3—202no—
565C2×C5—42no—
566C2×C5—124no—
567C2×C5#554712yes—
568C2×C5#554812yes—
569C2×C5#579111yescharacteristic
570C2×C5C2×#93711yescharacteristic
571C2×C5#583211yescharacteristic
572D5—42no—
573C23—21no—
574C23—22no—
575C23—22no—
576C23—21no—
577C23—34no—
578C23—52no—
579C23—62no—
580C23—62no—
581C23—62no—
582C23—62no—
583C23—62no—
584C23—62no—
585C23—101no—
586C23—101no—
587C23—102no—
588C23—102no—
589C23#677212yescomplemented
590C23C22×D3×D511yescharacteristic
591Q8—308no—
592D4—68no—
593D4—616no—
594D4—108no—
595D4—3016no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—34no—
600C2×C4—54no—
601C2×C4—52no—
602C2×C4—52no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—108no—
606C2×C4—101no—
607C2×C4—101no—
608C2×C4—102no—
609C2×C4—101no—
610C2×C4—154no—
611C2×C4—204no—
612C2×C4—3016no—
613C2×C4—302no—
614C2×C4—302no—
615C2×C4—302no—
616C2×C4—302no—
617C2×C4—302no—
618C2×C4D5×#6712yes—
619C2×C3—42no—
620C2×C3—202no—
621C2×C3#6592012yes—
622C2×C3D5×#12011yescharacteristic
623C2×C3#6593012yes—
624C2×C3#6594011yescharacteristic
625C2×C3#6605011yescharacteristic
626D3—44no—
627C5#8240011yescharacteristic, complemented, Sylow
628C22—22no—
629C22—24no—
630C22—41no—
631C22—41no—
632C22—42no—
633C22—64no—
634C22—64no—
635C22—64no—
636C22—102no—
637C22—102no—
638C22—102no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22—122no—
643C22—202no—
644C22—202no—
645C22C2×#677211yescharacteristic
646C22D5×#23511yescharacteristic
647C22D3×D4×D512yes—
648C22D3×#71112yes—
649C22C2×D5×#6711yescharacteristic
650C4—22no—
651C4—41no—
652C4—54no—
653C4—64no—
654C4—102no—
655C4—201no—
656C4—304no—
657C4—602no—
658C3#1060839611yescharacteristic, complemented, Sylow
659C2—42no—
660C2—124no—
661C2—202no—
662C2#1175284012yes—
663C2#1175284312yes—
664C2#1175452611yescharacteristic
665C2#1175479311yescharacteristic
666C2D5×#93711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutator, socle, cosocleC23×C3×C5C24
FittingC2×C3×C5×C22⋊C4C22
hypercenterC2×C22⋊C4D3×D5
hypocenterC3×C5#2388
LCSG ▷ C23×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#6772C2semidirect
2. C2×C3×#4027C2semidirect
3. C2×Dic5×#67C2semidirect
4. #11751048C2semidirect
5. C5×#4960C2semidirect
6. #11749521C2semidirect
7. #11748922C2semidirect
8. Dic5×(C2×D3)⋊C4C2semidirect
9. #11748846C2semidirect
10. #11748765C2semidirect
11. #11748702C2semidirect
43. C2×#4027D3semidirect
76. C2×C3×C4×D5D4semidirect
82. C2×D3×Dic5D4semidirect
102. #4960D5semidirect
138. C23×C3×C5#41semidirect
166. C2×C5×Dic3#41semidirect
176. C2×C3×Dic5#41semidirect
236. C2×C4×D5#67semidirect
345. C23×C5#235semidirect
375. C2×Dic5#235semidirect
424. C23×C3#711semidirect
457. C2×Dic3#711semidirect
534. C3×C5#2388semidirect
589. C23#6772semidirect
627. C5#82400semidirect
658. C3#10608396semidirect

Maximal subgroups

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

subgroupindex
C22×#67722
C2×C3×#40272
C2×Dic5×#672
#117510482
C5×#49602
#117495212
#117489222
Dic5×(C2×D3)⋊C42
#117488462
#117487652
#117487022
#106083963
#824005