Finite Group Databasebeta

← #14331651

Subgroups — #14331651

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×#6817C211yesmaximal, characteristic, complemented
2C2×C3×C4×C22⋊D5C211yesmaximal, characteristic, complemented
3C2×C4×#1202C212yesmaximal, complemented
4C4×C5×#217C211yesmaximal, characteristic, complemented
5#11749522C212yesmaximal, complemented
6#11749499C212yesmaximal, complemented
7#11748931C211yesmaximal, characteristic, complemented
8#11748846C211yesmaximal, characteristic, complemented
9#11748793C211yesmaximal, characteristic, complemented
10#11748769C211yesmaximal, characteristic, complemented
11C4×#6667C212yesmaximal, complemented
12#10605311—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×#1220C2212yes—
15C2×#6817—24no—
16C2×#6817—28no—
17C22×D60C2211yescharacteristic
18C22×C4×D15C2212yes—
19C22×Dic30C2211yescharacteristic
20C22×C3×C22⋊D5C2211yescharacteristic
21C22×C3×C4×D5C2211yescharacteristic
22C22×#1202C2212yes—
23C2×C5×#217C2212yes—
24C2×C4×C5×Dic3C2212yes—
25C2×C3×C22⋊Dic5C2211yescharacteristic
26C3×C4×C22⋊D5—24no—
27C2×C3×#685C2211yescharacteristic
28C2×C3×#683C2211yescharacteristic
29C2×C3×C4×Dic5C2211yescharacteristic
30C2×Dic3⋊Dic5C2212yes—
31C2×#6667C2212yes—
32C2×#6667C2212yes—
33C2×#6667C2212yes—
34C2×#6665C2212yes—
35C4×#1202—28no—
36C5×C3⋊#95C2211yescharacteristic
37C5×C3⋊#95C2211yescharacteristic
38#90994C2212yes—
39#90994C2212yes—
40#82400—51nomaximal, Hall
41C22×#709—31no—
42C2×C4×C22⋊D5D311yescharacteristic, complemented
43C2×C4×D20—32no—
44C4×C5×C22⋊C4—31no—
45#65250—32no—
46#65249—32no—
47C4×#685—32no—
48#64962—31no—
49#64961—31no—
50#64960—31no—
51#64945—31no—
52C24×C3×C5C2311yescharacteristic
53C23×D15C2312yes—
54C23×C3×D5C2311yescharacteristic
55C22×C3×C4×C5—24no—
56C22×C3×C4×C5D414yescomplemented
57C22×C3×C4×C5C2311yescharacteristic
58C22×C3×C4×C5C2311yescharacteristic
59C2×#1220—24no—
60C2×#1220—24no—
61C2×#1220—24no—
62C22×Dic15C2312yes—
63#6817—232no—
64C2×D60—24no—
65C2×D60—22no—
66C2×C4×D15—24no—
67C2×C4×D15—24no—
68C2×C4×D15D418yescomplemented
69C2×Dic30—22no—
70C2×Dic30—24no—
71C22×C5×Dic3C2312yes—
72C22×C5×Dic3C2312yes—
73C2×C3×C22⋊D5—22no—
74C2×C3×C22⋊D5—22no—
75C2×C3×C22⋊D5—22no—
76C22×C3×Dic5C2311yescharacteristic
77C22×C3×Dic5C2311yescharacteristic
78C22×C3×Dic5C2311yescharacteristic
79C2×C3×C4×D5—24no—
80C2×C3×C4×D5—22no—
81C2×#1202—24no—
82C2×#1202—24no—
83C2×#1202—24no—
84C5×#217—24no—
85C4×C5×Dic3—24no—
86C3×C22⋊Dic5—22no—
87C3×#685—22no—
88C3×#683—22no—
89C3×C4×Dic5—22no—
90Dic3⋊Dic5—24no—
91#6667—24no—
92#6667—24no—
93#6667—24no—
94#6665—24no—
95C22×#235—51no—
96C2×C3×C4×D4—51no—
97C2×C4×#67—52no—
98#5223—52no—
99C4×#217D511yescharacteristic, complemented
100#4991—51no—
101#4979—51no—
102#4960—51no—
103#4959—52no—
104#4947—51no—
105C4×(C2×D3)⋊C4—52no—
106C23×C4×C5C2×D311yescharacteristic
107C22×C22⋊D5—32no—
108C22×C22⋊D5C2×D311yescharacteristic
109C2×#709—68no—
110C2×#709—64no—
111C22×D20—32no—
112C22×D20—31no—
113C22×C4×D5—32no—
114C22×C4×D5C2×D311yescharacteristic
115C22×Dic10—31no—
116C2×C5×C22⋊C4—32no—
117C2×C42×C5—32no—
118C2×C22⋊Dic5C2×D311yescharacteristic
119C4×C22⋊D5—24no—
120C2×#685—32no—
121C2×#685—32no—
122C2×#685—32no—
123C2×#685—32no—
124C2×#685C2×D311yescharacteristic
125C2×C4⋊Dic5—32no—
126C2×#683C2×D311yescharacteristic
127C2×C4×Dic5C2×D311yescharacteristic
128C4×D20—68no—
129C5×#95—31no—
130C5×#95—31no—
131#3963—32no—
132#3963—32no—
133#2388—151noSylow
134C23×C3×C5—22no—
135C23×C3×C5—22no—
136C23×C3×C5—21no—
137C23×C3×C5—21no—
138C23×C3×C5C2×D412yes—
139C23×C3×C5C2411yescharacteristic
140C22×D15—22no—
141C22×D15—22no—
142C22×D15—22no—
143C22×D15—22no—
144C22×D15—22no—
145C22×D15—22no—
146C22×D15C2×D414yes—
147C22×C3×D5—22no—
148C22×C3×D5—21no—
149C22×C3×D5—22no—
150C22×C3×D5—21no—
151C22×C3×D5#4112yescomplemented
152C2×C3×C4×C5—22no—
153C2×C3×C4×C5—28no—
154C2×C3×C4×C5C2×D412yes—
155C2×C3×C4×C5C2×D412yes—
156C2×C3×C4×C5C2×D414yes—
157#1220—216no—
158C2×Dic15—22no—
159C2×Dic15—22no—
160C2×Dic15C2×D414yes—
161D60—28no—
162C4×D15—216no—
163Dic30—28no—
164C2×C5×Dic3—22no—
165C2×C5×Dic3—22no—
166C2×C5×Dic3—22no—
167C2×C5×Dic3—22no—
168C2×C5×Dic3—22no—
169C2×C5×Dic3#4114yescomplemented
170C3×C22⋊D5—28no—
171C2×C3×Dic5—21no—
172C2×C3×Dic5—21no—
173C2×C3×Dic5—22no—
174C2×C3×Dic5—21no—
175C2×C3×Dic5—21no—
176C2×C3×Dic5—22no—
177C2×C3×Dic5#4112yescomplemented
178C3×C4×D5—44no—
179#1202—216no—
180C22×C3×D4—51no—
181C23×C3×C4—51no—
182C23×C3×C4C2×D511yescharacteristic
183C22×#67—52no—
184C22×#67—52no—
185C2×#235—108no—
186C2×#235—104no—
187C22×D12—51no—
188C22×C4×D3—52no—
189C22×Dic6—51no—
190C3×C4×D4—104no—
191C2×C3×C4⋊C4—51no—
192C2×C3×C22⋊C4—51no—
193C2×C3×C22⋊C4—51no—
194C2×C3×C42—51no—
195C2×#217—52no—
196C2×#217C2×D512yes—
197C4×#67—108no—
198C2×(C2×D3)⋊C4—52no—
199C2×(C2×D3)⋊C4—52no—
200C2×(C2×D3)⋊C4—52no—
201C2×Dic3⋊C4—52no—
202C2×C4×Dic3C2×D512yes—
203C3⋊#95—52no—
204C3⋊#95—52no—
205C3⋊#95C2×D511yescharacteristic
206C3⋊#95C2×D511yescharacteristic
207C24×C5C22×D311yescharacteristic
208C23×D5—32no—
209C23×D5C22×D311yescharacteristic
210C22×C4×C5—24no—
211C22×C4×C5—32no—
212C22×C4×C5—32no—
213C22×C4×C5#6714yescomplemented
214C22×C4×C5C22×D311yescharacteristic
215C22×C4×C5C22×D311yescharacteristic
216C2×C22⋊D5—22no—
217C2×C22⋊D5—22no—
218C2×C22⋊D5—22no—
219C2×C22⋊D5—64no—
220C2×C22⋊D5—64no—
221C2×C22⋊D5—64no—
222C22×Dic5—32no—
223C22×Dic5C22×D311yescharacteristic
224C22×Dic5C22×D311yescharacteristic
225C22×Dic5C22×D311yescharacteristic
226#709—632no—
227C2×D20—64no—
228C2×D20—64no—
229C2×D20—64no—
230C2×D20—64no—
231C2×D20—62no—
232C2×C4×D5—24no—
233C2×C4×D5—22no—
234C2×C4×D5—38no—
235C2×C4×D5—64no—
236C2×C4×D5—64no—
237C2×Dic10—64no—
238C2×Dic10—62no—
239C5×C22⋊C4—64no—
240C42×C5—64no—
241C22⋊Dic5—22no—
242#685—22no—
243#685—64no—
244#685—64no—
245#685—64no—
246#685—64no—
247C4⋊Dic5—64no—
248#683—22no—
249C4×Dic5—22no—
250C22×#41—151no—
251C2×C4×D4—51no—
252C2×C4×D4—152no—
253#395—151no—
254#394—151no—
255#394—152no—
256#393—151no—
257#392—151no—
258#392—152no—
259C4×C22⋊C4—31no—
260C4×C22⋊C4—152no—
261C22×C3×C5—22no—
262C22×C3×C5—22no—
263C22×C3×C5—22no—
264C22×C3×C5—42no—
265C22×C3×C5—42no—
266C22×C3×C5C22×D411yescharacteristic
267C22×C3×C5C22×D412yes—
268C22×C3×C5C2×#4112yes—
269C22×C3×C5C2×#4111yescharacteristic
270C22×C3×C5C2×#4111yescharacteristic
271C2×D15—24no—
272C2×D15—24no—
273C2×D15—24no—
274C2×D15—42no—
275C2×D15—42no—
276C2×D15—42no—
277C2×D15—42no—
278C2×C3×D5—24no—
279C2×C3×D5—22no—
280C2×C3×D5—41no—
281C2×C3×D5—41no—
282C2×C3×D5—42no—
283C3×C4×C5—22no—
284C3×C4×C5#12014yes—
285Dic15—24no—
286C3×Dic5—22no—
287C3×Dic5—41no—
288C3×Dic5—41no—
289C5×Dic3—24no—
290C5×Dic3—42no—
291C24×C3—51no—
292C24×C3C22×D511yescharacteristic
293C23×D3—52no—
294C2×C3×D4—102no—
295C2×C3×D4—102no—
296C2×C3×D4—102no—
297C22×C3×C4—24no—
298C22×C3×C4—51no—
299C22×C3×C4—51no—
300C22×C3×C4—51no—
301C22×C3×C4—102no—
302C22×C3×C4—104no—
303C22×C3×C4D2014yescomplemented
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—104no—
322C2×Dic6—102no—
323C3×C4⋊C4—102no—
324C3×C22⋊C4—102no—
325C3×C22⋊C4—102no—
326C3×C42—102no—
327#217—24no—
328#217—104no—
329(C2×D3)⋊C4—104no—
330(C2×D3)⋊C4—104no—
331(C2×D3)⋊C4—104no—
332Dic3⋊C4—104no—
333C4×Dic3—24no—
334C23×C5—21no—
335C23×C5—21no—
336C23×C5—22no—
337C23×C5—22no—
338C23×C5C2×#6712yes—
339C23×C5C23×D311yescharacteristic
340C22×D5—21no—
341C22×D5—22no—
342C22×D5—22no—
343C22×D5—21no—
344C22×D5—34no—
345C22×D5—62no—
346C22×D5—62no—
347C22×D5—62no—
348C22×D5—62no—
349C22×D5—62no—
350C22×D5—62no—
351C22×D5#23512yescomplemented
352C2×C4×C5—22no—
353C2×C4×C5—28no—
354C2×C4×C5—34no—
355C2×C4×C5—62no—
356C2×C4×C5—62no—
357C2×C4×C5—62no—
358C2×C4×C5—62no—
359C2×C4×C5—62no—
360C2×C4×C5D3×D414yes—
361C2×C4×C5C2×#6712yes—
362C2×C4×C5C2×#6712yes—
363C22⋊D5—28no—
364C22⋊D5—616no—
365C2×Dic5—21no—
366C2×Dic5—21no—
367C2×Dic5—22no—
368C2×Dic5—22no—
369C2×Dic5—21no—
370C2×Dic5—21no—
371C2×Dic5—34no—
372C2×Dic5—62no—
373C2×Dic5—62no—
374C2×Dic5#23512yescomplemented
375D20—68no—
376D20—616no—
377C4×D5—44no—
378C4×D5—616no—
379Dic10—68no—
380C2×#41—308no—
381C2×#41—304no—
382C22×Q8—151no—
383C22×D4—51no—
384C22×D4—152no—
385C22×D4—152no—
386C22×D4—151no—
387C23×C4—51no—
388C23×C4—152no—
389C23×C4D3×D511yescharacteristic
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—51no—
397C2×C22⋊C4—152no—
398C2×C22⋊C4—152no—
399C2×C22⋊C4—152no—
400C2×C22⋊C4—152no—
401C2×C42—32no—
402C2×C42—51no—
403#95—31no—
404#95—31no—
405#95—152no—
406#95—152no—
407C2×C3×C5—42no—
408C2×C3×C5C2×#12011yescharacteristic
409C2×C3×C5#52511yescharacteristic
410C2×C3×C5#52512yes—
411C2×C3×C5#53211yescharacteristic
412C2×C3×C5#53212yes—
413D15—44no—
414C3×D5—42no—
415C23×C3—22no—
416C23×C3—22no—
417C23×C3—21no—
418C23×C3—21no—
419C23×C3—52no—
420C23×C3—101no—
421C23×C3—102no—
422C23×C3—102no—
423C23×C3—101no—
424C23×C3C2×D2012yes—
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—22no—
435C2×C3×C4—28no—
436C2×C3×C4—52no—
437C2×C3×C4—101no—
438C2×C3×C4—101no—
439C2×C3×C4—101no—
440C2×C3×C4—101no—
441C2×C3×C4—102no—
442C2×C3×C4—102no—
443C2×C3×C4—204no—
444C2×C3×C4C2×D2012yes—
445C2×C3×C4C2×D2012yes—
446C2×C3×C4D4×D514yes—
447#67—1016no—
448#67—1016no—
449C2×Dic3—22no—
450C2×Dic3—22no—
451C2×Dic3—22no—
452C2×Dic3—22no—
453C2×Dic3—22no—
454C2×Dic3—54no—
455C2×Dic3—102no—
456C2×Dic3—102no—
457C2×Dic3#70914yescomplemented
458D12—108no—
459C4×D3—1016no—
460Dic6—108no—
461C22×C5—22no—
462C22×C5—22no—
463C22×C5—22no—
464C22×C5—42no—
465C22×C5—42no—
466C22×C5C2×#23511yescharacteristic
467C22×C5C2×#23511yescharacteristic
468C22×C5C2×D3×D412yes—
469C22×C5D3×#4112yes—
470C22×C5C22×#6711yescharacteristic
471C2×D5—22no—
472C2×D5—24no—
473C2×D5—41no—
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—64no—
485C4×C5—122no—
486C4×C5#93714yes—
487Dic5—22no—
488Dic5—41no—
489Dic5—41no—
490Dic5—64no—
491C24—51no—
492C24—152no—
493C24C2×D3×D511yescharacteristic
494#41—3032no—
495C2×Q8—304no—
496C2×Q8—302no—
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—24no—
509C22×C4—32no—
510C22×C4—32no—
511C22×C4—51no—
512C22×C4—51no—
513C22×C4—51no—
514C22×C4—102no—
515C22×C4—104no—
516C22×C4—152no—
517C22×C4—158no—
518C22×C4—304no—
519C22×C4—304no—
520C22×C4#120214yescomplemented
521C22×C4C2×D3×D511yescharacteristic
522C22×C4C2×D3×D511yescharacteristic
523C4⋊C4—102no—
524C4⋊C4—304no—
525C22⋊C4—64no—
526C22⋊C4—102no—
527C22⋊C4—102no—
528C22⋊C4—304no—
529C22⋊C4—304no—
530C22⋊C4—304no—
531C22⋊C4—304no—
532C42—64no—
533C42—102no—
534C3×C5#238811yescharacteristic, complemented, Hall
535C22×C3—22no—
536C22×C3—22no—
537C22×C3—22no—
538C22×C3—42no—
539C22×C3—42no—
540C22×C3—102no—
541C22×C3—104no—
542C22×C3—201no—
543C22×C3—201no—
544C22×C3—202no—
545C22×C3C22×D2011yescharacteristic
546C22×C3C2×#70912yes—
547C22×C3C2×D4×D512yes—
548C22×C3D5×#4111yescharacteristic
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—102no—
559C3×C4—201no—
560C3×C4—201no—
561C3×C4#402814yes—
562Dic3—24no—
563Dic3—42no—
564Dic3—104no—
565C2×C5—42no—
566C2×C5#554712yes—
567C2×C5#554812yes—
568C2×C5#579111yescharacteristic
569C2×C5C2×#93711yescharacteristic
570C2×C5#583211yescharacteristic
571D5—42no—
572D5—124no—
573C23—21no—
574C23—21no—
575C23—22no—
576C23—22no—
577C23—52no—
578C23—101no—
579C23—102no—
580C23—101no—
581C23—102no—
582C23—154no—
583C23—302no—
584C23—302no—
585C23—302no—
586C23—302no—
587C23—302no—
588C23—302no—
589C23C2×#120212yes—
590C23C22×D3×D511yescharacteristic
591Q8—308no—
592D4—108no—
593D4—308no—
594D4—3016no—
595D4—3016no—
596C2×C4—22no—
597C2×C4—28no—
598C2×C4—34no—
599C2×C4—52no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—101no—
606C2×C4—101no—
607C2×C4—102no—
608C2×C4—101no—
609C2×C4—101no—
610C2×C4—102no—
611C2×C4—154no—
612C2×C4—204no—
613C2×C4—302no—
614C2×C4—302no—
615C2×C4—3016no—
616C2×C4C2×#120212yes—
617C2×C4C2×#120212yes—
618C2×C4#679014yes—
619C2×C3—42no—
620C2×C3—202no—
621C2×C3#6581111yescharacteristic
622C2×C3C2×#402811yescharacteristic
623C2×C3#6585811yescharacteristic
624C2×C3#6588112yes—
625C2×C3#6588212yes—
626D3—204no—
627C5#8240011yescharacteristic, complemented, Sylow
628C22—22no—
629C22—22no—
630C22—22no—
631C22—42no—
632C22—42no—
633C22—102no—
634C22—104no—
635C22—202no—
636C22—201no—
637C22—201no—
638C22—304no—
639C22—304no—
640C22—304no—
641C22—602no—
642C22—602no—
643C22—602no—
644C22—602no—
645C22D5×#23511yescharacteristic
646C22D5×#23511yescharacteristic
647C22D3×#70912yes—
648C22C22×#120211yescharacteristic
649C22C2×#679012yes—
650C4—22no—
651C4—64no—
652C4—102no—
653C4—122no—
654C4—201no—
655C4—201no—
656C4—304no—
657C4#9157214yes—
658C3#1060531111yescharacteristic, complemented, Sylow
659C2—42no—
660C2—202no—
661C2—604no—
662C2#1175455512yes—
663C2#1175456411yescharacteristic
664C2#1175458111yescharacteristic
665C2#1175466312yes—
666C2C2×#9157211yescharacteristic
subgroupG/H
centerC22×C4C2×D3×D5
commutator, socle, cosocleC23×C3×C5C24
FrattiniC23C22×D3×D5
FittingC23×C3×C4×C5C22
hypercenterC23×C4D3×D5
hypocenterC3×C5#2388
LCSG ▷ C23×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C22×#6817C2semidirect
2. C2×C3×C4×C22⋊D5C2semidirect
3. C2×C4×#1202C2semidirect
4. C4×C5×#217C2semidirect
5. #11749522C2semidirect
6. #11749499C2semidirect
7. #11748931C2semidirect
8. #11748846C2semidirect
9. #11748793C2semidirect
10. #11748769C2semidirect
11. C4×#6667C2semidirect
42. C2×C4×C22⋊D5D3semidirect
56. C22×C3×C4×C5D4semidirect
68. C2×C4×D15D4semidirect
99. C4×#217D5semidirect
151. C22×C3×D5#41semidirect
169. C2×C5×Dic3#41semidirect
177. C2×C3×Dic5#41semidirect
213. C22×C4×C5#67semidirect
303. C22×C3×C4D20semidirect
351. C22×D5#235semidirect
374. C2×Dic5#235semidirect
457. C2×Dic3#709semidirect
520. C22×C4#1202semidirect
534. C3×C5#2388semidirect
627. C5#82400semidirect
658. C3#10605311semidirect

Maximal subgroups

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

subgroupindex
C22×#68172
C2×C3×C4×C22⋊D52
C2×C4×#12022
C4×C5×#2172
#117495222
#117494992
#117489312
#117488462
#117487932
#117487692
C4×#66672
#106053113
#824005