Finite Group Databasebeta

← #14329103

Subgroups — #14329103

Order 1920.

1376 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D60C211yesmaximal, characteristic, complemented
2C2×#91450C211yesmaximal, characteristic
3C2×#91358C211yesmaximal, characteristic, complemented
4C5×#4928C211yesmaximal, characteristic, complemented
5C3×C4×#685C211yesmaximal, characteristic, complemented
6#11748917C212yesmaximal, complemented
7#11748913C212yesmaximal, complemented
8#11748768C211yesmaximal, characteristic, complemented
9Dic5×C4⋊Dic3C211yesmaximal, characteristic, complemented
10#11748670C212yesmaximal, complemented
11#11748669C212yesmaximal, complemented
12#10607716—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C22×C4×D15C2212yes—
15C22×C3×C4×D5C2211yescharacteristic
16C22×#1202C2212yes—
17C2×C3×C42×C5C2211yescharacteristic
18C2×#6714C2212yes—
19C2×C4⋊Dic15C2211yescharacteristic
20C4×D60—24no—
21C2×C5×C4⋊Dic3C2211yescharacteristic
22C2×C5×C4⋊Dic3C2211yescharacteristic
23C2×C5×Dic3⋊C4C2212yes—
24C2×C3×#685C2211yescharacteristic
25C2×C3×#685C2211yescharacteristic
26C2×C3×C4×Dic5C2211yescharacteristic
27C2×#6667C2212yes—
28C2×#6667C2212yes—
29C2×#6667C2212yes—
30C2×#6665C2212yes—
31C2×Dic3×Dic5C2212yes—
32C2×Dic3×Dic5C2212yes—
33#91450C418yescomplemented
34#91358—24no—
35C5×C3⋊#95C2212yes—
36C3×#3963C2211yescharacteristic
37C3×#3963C2211yescharacteristic
38#90994C2212yes—
39#73469—51nomaximal, Hall
40C2×#4041—31no—
41C2×#4038—31no—
42C2×C4×D20—31no—
43C5×#389—31no—
44#65279—32no—
45#65277—32no—
46#65277—31no—
47C4⋊C4×Dic5—31no—
48C4×#685D311yescharacteristic, complemented
49#64955—32no—
50#64954—32no—
51C23×D15C2312yes—
52C23×C3×D5C2311yescharacteristic
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×Dic15C2312yes—
57C2×D60—22no—
58C2×D60—22no—
59C2×D60C2×C414yes—
60C2×C4×D15—24no—
61C2×C4×D15—24no—
62C2×C4×D15—42no—
63C22×C5×Dic3C2312yes—
64C22×C5×Dic3C2312yes—
65C22×C3×Dic5C2311yescharacteristic
66C22×C3×Dic5C2311yescharacteristic
67C22×C3×Dic5C2311yescharacteristic
68C2×C3×C4×D5—22no—
69C2×C3×C4×D5—22no—
70C2×C3×C4×D5C2×C414yescomplemented
71C2×#1202—42no—
72C2×#1202—42no—
73C2×#1202C2×C418yescomplemented
74C3×C42×C5—22no—
75#6714—24no—
76C4⋊Dic15D414yescomplemented
77C5×C4⋊Dic3—22no—
78C5×C4⋊Dic3C2×C414yescomplemented
79C5×Dic3⋊C4—42no—
80C3×#685—22no—
81C3×#685—22no—
82C3×C4×Dic5—22no—
83#6667—24no—
84#6667—42no—
85#6667C2×C418yescomplemented
86#6665—24no—
87Dic3×Dic5—24no—
88Dic3×Dic5—42no—
89C2×#930—51no—
90C2×#926—51no—
91C2×C4×D12—51no—
92C3×C4×C22⋊C4—51no—
93#4957—52no—
94#4956—52no—
95#4939—51no—
96C4×C4⋊Dic3—51no—
97#4928D511yescharacteristic, complemented
98#4669—52no—
99#4666—52no—
100C22×D20—31no—
101C22×D20—32no—
102C22×C4×D5—32no—
103C22×C4×D5C2×D311yescharacteristic
104C2×C5×C4⋊C4—31no—
105C2×C5×C4⋊C4—31no—
106C2×C5×C4⋊C4—32no—
107C2×C42×C5C2×D311yescharacteristic
108C2×#685—32no—
109C2×#685—32no—
110C2×#685—32no—
111C2×#685—32no—
112C2×#685—32no—
113C2×#685C2×D311yescharacteristic
114C2×#685C2×D311yescharacteristic
115C2×C4⋊Dic5—31no—
116C2×C4×Dic5—32no—
117C2×C4×Dic5—32no—
118C2×C4×Dic5C2×D311yescharacteristic
119#4041—38no—
120#4038—64no—
121C4×D20—64no—
122C5×#95—32no—
123#3963—32no—
124#3963C2×D311yescharacteristic
125#3963C2×D311yescharacteristic
126#2363—151noSylow
127C23×C3×C5C2411yescharacteristic
128C22×D15—22no—
129C22×D15—22no—
130C22×D15—22no—
131C22×D15—22no—
132C22×D15—22no—
133C22×D15—22no—
134C22×D15C22×C414yes—
135C22×C3×D5—21no—
136C22×C3×D5—21no—
137C22×C3×D5—21no—
138C22×C3×D5—21no—
139C22×C3×D5—21no—
140C22×C3×D5—21no—
141C22×C3×D5C22×C412yes—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5—21no—
144C2×C3×C4×C5—21no—
145C2×C3×C4×C5—21no—
146C2×C3×C4×C5—21no—
147C2×C3×C4×C5C22×C412yes—
148C2×C3×C4×C5C2×D412yes—
149C2×C3×C4×C5#4112yes—
150C2×C3×C4×C5#4112yes—
151C2×Dic15—22no—
152C2×Dic15—22no—
153C2×Dic15C2×D414yes—
154D60—28no—
155C4×D15—48no—
156C2×C5×Dic3—22no—
157C2×C5×Dic3—22no—
158C2×C5×Dic3—22no—
159C2×C5×Dic3—22no—
160C2×C5×Dic3—22no—
161C2×C5×Dic3C22×C414yes—
162C2×C3×Dic5—21no—
163C2×C3×Dic5—21no—
164C2×C3×Dic5—21no—
165C2×C3×Dic5—21no—
166C2×C3×Dic5—21no—
167C2×C3×Dic5—21no—
168C2×C3×Dic5—21no—
169C2×C3×Dic5C22×C412yes—
170C2×C3×Dic5#4112yes—
171C3×C4×D5—28no—
172#1202—48no—
173C23×C3×C4—51no—
174C22×#67—52no—
175C22×D12—51no—
176C22×C4×D3—52no—
177C2×C3×C22⋊C4—51no—
178C2×C3×C22⋊C4—51no—
179C2×C3×C42—51no—
180C2×C3×C42C2×D511yescharacteristic
181C2×#217—52no—
182#930—58no—
183C2×(C2×D3)⋊C4—52no—
184C2×(C2×D3)⋊C4—52no—
185C2×(C2×D3)⋊C4—52no—
186C2×(C2×D3)⋊C4—52no—
187#926—104no—
188C2×C4⋊Dic3—51no—
189C2×C4⋊Dic3C2×D511yescharacteristic
190C2×C4⋊Dic3C2×D511yescharacteristic
191C2×Dic3⋊C4C2×D512yes—
192C2×C4×Dic3—52no—
193C2×C4×Dic3—52no—
194C4×D12—104no—
195C3×#95—51no—
196C3×#95—51no—
197C3⋊#95—52no—
198C3⋊#95C2×D512yes—
199C23×D5—32no—
200C23×D5C22×D311yescharacteristic
201C22×C4×C5—32no—
202C22×C4×C5—32no—
203C22×C4×C5C22×D311yescharacteristic
204C22×C4×C5C22×D311yescharacteristic
205C22×C4×C5C22×D311yescharacteristic
206C22×Dic5—32no—
207C22×Dic5C22×D311yescharacteristic
208C22×Dic5C22×D311yescharacteristic
209C22×Dic5C22×D311yescharacteristic
210C2×D20—34no—
211C2×D20—38no—
212C2×D20—62no—
213C2×D20—62no—
214C2×D20—122no—
215C2×D20—122no—
216C2×C4×D5—22no—
217C2×C4×D5—22no—
218C2×C4×D5—64no—
219C2×C4×D5—64no—
220C2×C4×D5—122no—
221C2×C4×D5C4×D314yescomplemented
222C5×C4⋊C4—34no—
223C5×C4⋊C4—62no—
224C5×C4⋊C4—122no—
225C42×C5—22no—
226#685—22no—
227#685—22no—
228#685—38no—
229#685—64no—
230#685—64no—
231#685—64no—
232#685—122no—
233C4⋊Dic5—34no—
234C4×Dic5—22no—
235C4×Dic5—64no—
236C4×Dic5—122no—
237C2×#121—151no—
238C2×C4×D4—151no—
239C2×#117—151no—
240#390—151no—
241#390—152no—
242#389—31no—
243#388—152no—
244#386—152no—
245C4×C4⋊C4—151no—
246C4×C22⋊C4—51no—
247C4×C22⋊C4—152no—
248C22×C3×C5C23×C411yescharacteristic
249C22×C3×C5C22×D411yescharacteristic
250C22×C3×C5C2×#4111yescharacteristic
251C22×C3×C5C2×#4111yescharacteristic
252C22×C3×C5C2×#4111yescharacteristic
253C22×C3×C5H(5,𝔽2)11yescharacteristic
254C22×C3×C5H(5,𝔽2)11yescharacteristic
255C2×D15—24no—
256C2×D15—24no—
257C2×D15—24no—
258C2×D15—42no—
259C2×D15—42no—
260C2×D15—42no—
261C2×D15—42no—
262C2×C3×D5—22no—
263C2×C3×D5—22no—
264C2×C3×D5—22no—
265C2×C3×D5—41no—
266C2×C3×D5—41no—
267C2×C3×D5—41no—
268C2×C3×D5—41no—
269C3×C4×C5—22no—
270C3×C4×C5—41no—
271C3×C4×C5C4×D414yes—
272Dic15—24no—
273C3×Dic5—22no—
274C3×Dic5—22no—
275C3×Dic5—41no—
276C5×Dic3—24no—
277C5×Dic3—42no—
278C24×C3—51no—
279C23×D3—52no—
280C22×C3×C4—51no—
281C22×C3×C4—51no—
282C22×C3×C4—54no—
283C22×C3×C4—51no—
284C22×C3×C4—102no—
285C22×C3×C4—102no—
286C22×C3×C4C22×D511yescharacteristic
287C22×C3×C4C22×D511yescharacteristic
288C22×C3×C4C22×D511yescharacteristic
289C2×#67—58no—
290C2×#67—202no—
291C2×#67—202no—
292C22×Dic3—52no—
293C22×Dic3C22×D512yes—
294C22×Dic3C22×D512yes—
295C2×D12—54no—
296C2×D12—102no—
297C2×D12—102no—
298C2×C4×D3—104no—
299C2×C4×D3—104no—
300C2×C4×D3—202no—
301C3×C22⋊C4—102no—
302C3×C22⋊C4—102no—
303C3×C42—22no—
304C3×C42—102no—
305#217—104no—
306(C2×D3)⋊C4—58no—
307(C2×D3)⋊C4—104no—
308(C2×D3)⋊C4—104no—
309(C2×D3)⋊C4—202no—
310C4⋊Dic3—22no—
311C4⋊Dic3—54no—
312C4⋊Dic3C4×D514yescomplemented
313Dic3⋊C4—42no—
314C4×Dic3—104no—
315C4×Dic3—202no—
316C23×C5C23×D311yescharacteristic
317C22×D5—21no—
318C22×D5—21no—
319C22×D5—21no—
320C22×D5—21no—
321C22×D5—21no—
322C22×D5—21no—
323C22×D5—34no—
324C22×D5—62no—
325C22×D5—62no—
326C22×D5—62no—
327C22×D5—62no—
328C22×D5—62no—
329C22×D5—62no—
330C22×D5C2×C4×D312yes—
331C2×C4×C5—21no—
332C2×C4×C5—21no—
333C2×C4×C5—21no—
334C2×C4×C5—21no—
335C2×C4×C5—21no—
336C2×C4×C5—34no—
337C2×C4×C5—62no—
338C2×C4×C5—62no—
339C2×C4×C5—62no—
340C2×C4×C5—62no—
341C2×C4×C5—62no—
342C2×C4×C5C2×C4×D312yes—
343C2×C4×C5#23512yes—
344C2×C4×C5D3×D412yes—
345C2×C4×C5#23712yes—
346C2×Dic5—21no—
347C2×Dic5—21no—
348C2×Dic5—21no—
349C2×Dic5—21no—
350C2×Dic5—21no—
351C2×Dic5—21no—
352C2×Dic5—21no—
353C2×Dic5—34no—
354C2×Dic5—62no—
355C2×Dic5—62no—
356C2×Dic5C2×C4×D312yes—
357C2×Dic5#23512yes—
358D20—68no—
359D20—128no—
360C4×D5—28no—
361C4×D5—128no—
362C22×D4—152no—
363C22×D4—151no—
364C23×C4—51no—
365C23×C4—152no—
366#121—158no—
367C4×D4—304no—
368#117—304no—
369C2×C4⋊C4—31no—
370C2×C4⋊C4—31no—
371C2×C4⋊C4—32no—
372C2×C4⋊C4—151no—
373C2×C22⋊C4—51no—
374C2×C22⋊C4—51no—
375C2×C22⋊C4—152no—
376C2×C22⋊C4—152no—
377C2×C22⋊C4—152no—
378C2×C22⋊C4—152no—
379C2×C22⋊C4—152no—
380C2×C42—51no—
381C2×C42—152no—
382C2×C42—152no—
383C2×C42D3×D511yescharacteristic
384#95—32no—
385#95—51no—
386#95—51no—
387#95—152no—
388C2×C3×C5C2×C4×D411yescharacteristic
389C2×C3×C5C4×#4111yescharacteristic
390C2×C3×C5#51811yescharacteristic
391C2×C3×C5#54611yescharacteristic
392C2×C3×C5#55011yescharacteristic
393C2×C3×C5#55311yescharacteristic
394C2×C3×C5#55511yescharacteristic
395D15—44no—
396C3×D5—42no—
397C23×C3—52no—
398C23×C3—101no—
399C23×C3—101no—
400C23×C3—101no—
401C23×C3—101no—
402C23×C3—101no—
403C23×C3—101no—
404C23×C3C23×D511yescharacteristic
405C22×D3—54no—
406C22×D3—102no—
407C22×D3—102no—
408C22×D3—102no—
409C22×D3—102no—
410C22×D3—102no—
411C22×D3—102no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—21no—
417C2×C3×C4—52no—
418C2×C3×C4—52no—
419C2×C3×C4—101no—
420C2×C3×C4—101no—
421C2×C3×C4—101no—
422C2×C3×C4—101no—
423C2×C3×C4—108no—
424C2×C3×C4—101no—
425C2×C3×C4—101no—
426C2×C3×C4—101no—
427C2×C3×C4C2×C4×D512yes—
428C2×C3×C4D4×D512yes—
429C2×C3×C4#71312yes—
430C2×C3×C4#71312yes—
431#67—208no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—22no—
436C2×Dic3—22no—
437C2×Dic3—54no—
438C2×Dic3—102no—
439C2×Dic3—102no—
440C2×Dic3C2×C4×D514yes—
441D12—108no—
442C4×D3—208no—
443C22×C5C22×C4×D311yescharacteristic
444C22×C5C2×#23511yescharacteristic
445C22×C5C2×#23511yescharacteristic
446C22×C5C2×D3×D411yescharacteristic
447C22×C5C2×#23711yescharacteristic
448C22×C5#100911yescharacteristic
449C22×C5#100911yescharacteristic
450C2×D5—22no—
451C2×D5—22no—
452C2×D5—22no—
453C2×D5—41no—
454C2×D5—41no—
455C2×D5—41no—
456C2×D5—41no—
457C2×D5—64no—
458C2×D5—64no—
459C2×D5—64no—
460C2×D5—122no—
461C2×D5—122no—
462C2×D5—122no—
463C2×D5—122no—
464C4×C5—22no—
465C4×C5—41no—
466C4×C5—64no—
467C4×C5—122no—
468C4×C5#67⋊C414yes—
469Dic5—22no—
470Dic5—22no—
471Dic5—41no—
472Dic5—64no—
473C24—51no—
474C24—152no—
475C2×D4—158no—
476C2×D4—154no—
477C2×D4—302no—
478C2×D4—302no—
479C2×D4—602no—
480C2×D4—602no—
481C22×C4—32no—
482C22×C4—32no—
483C22×C4—51no—
484C22×C4—51no—
485C22×C4—54no—
486C22×C4—51no—
487C22×C4—102no—
488C22×C4—102no—
489C22×C4—152no—
490C22×C4—304no—
491C22×C4—304no—
492C22×C4—602no—
493C22×C4C2×D3×D511yescharacteristic
494C22×C4C2×D3×D511yescharacteristic
495C22×C4C2×D3×D511yescharacteristic
496C4⋊C4—34no—
497C4⋊C4—62no—
498C4⋊C4—122no—
499C4⋊C4—154no—
500C22⋊C4—102no—
501C22⋊C4—102no—
502C22⋊C4—158no—
503C22⋊C4—304no—
504C22⋊C4—304no—
505C22⋊C4—304no—
506C22⋊C4—602no—
507C42—22no—
508C42—102no—
509C42—304no—
510C42—602no—
511C3×C5#236311yescharacteristic, complemented, Hall
512C22×C3—102no—
513C22×C3—102no—
514C22×C3—102no—
515C22×C3—201no—
516C22×C3—201no—
517C22×C3—201no—
518C22×C3—201no—
519C22×C3C22×C4×D511yescharacteristic
520C22×C3C2×D4×D511yescharacteristic
521C22×C3#414411yescharacteristic
522C22×C3C2×#71311yescharacteristic
523C22×C3C2×#71311yescharacteristic
524C22×C3D5×#4111yescharacteristic
525C22×C3#414911yescharacteristic
526C2×D3—104no—
527C2×D3—104no—
528C2×D3—104no—
529C2×D3—202no—
530C2×D3—202no—
531C2×D3—202no—
532C2×D3—202no—
533C3×C4—22no—
534C3×C4—41no—
535C3×C4—102no—
536C3×C4—102no—
537C3×C4—201no—
538C3×C4#403914yes—
539Dic3—24no—
540Dic3—42no—
541Dic3—104no—
542C2×C5C4×#23511yescharacteristic
543C2×C5C2×#67⋊C411yescharacteristic
544C2×C5#552111yescharacteristic
545C2×C5#555011yescharacteristic
546C2×C5#555311yescharacteristic
547C2×C5#556311yescharacteristic
548C2×C5#556511yescharacteristic
549D5—42no—
550D5—124no—
551C23—52no—
552C23—101no—
553C23—101no—
554C23—101no—
555C23—101no—
556C23—101no—
557C23—101no—
558C23—154no—
559C23—302no—
560C23—302no—
561C23—302no—
562C23—302no—
563C23—302no—
564C23—302no—
565C23C22×D3×D511yescharacteristic
566D4—308no—
567D4—608no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—34no—
574C2×C4—52no—
575C2×C4—52no—
576C2×C4—62no—
577C2×C4—62no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—101no—
582C2×C4—101no—
583C2×C4—101no—
584C2×C4—101no—
585C2×C4—108no—
586C2×C4—101no—
587C2×C4—101no—
588C2×C4—101no—
589C2×C4—154no—
590C2×C4—302no—
591C2×C4—302no—
592C2×C4—608no—
593C2×C4#676612yes—
594C2×C4#677312yes—
595C2×C4C4×D3×D512yes—
596C2×C4#679012yes—
597C2×C3—202no—
598C2×C3C2×#403911yescharacteristic
599C2×C3#6587911yescharacteristic
600C2×C3C4×#71311yescharacteristic
601C2×C3#6593311yescharacteristic
602C2×C3#6596411yescharacteristic
603C2×C3#6602911yescharacteristic
604C2×C3#6606811yescharacteristic
605D3—204no—
606C5#7346911yescharacteristic, complemented, Sylow
607C22—102no—
608C22—102no—
609C22—102no—
610C22—201no—
611C22—201no—
612C22—201no—
613C22—201no—
614C22—304no—
615C22—304no—
616C22—304no—
617C22—602no—
618C22—602no—
619C22—602no—
620C22—602no—
621C22C2×#676611yescharacteristic
622C22C2×#677311yescharacteristic
623C22C2×C4×D3×D511yescharacteristic
624C22D5×#23511yescharacteristic
625C22#9201811yescharacteristic
626C22#9201911yescharacteristic
627C22C2×#679011yescharacteristic
628C4—22no—
629C4—41no—
630C4—64no—
631C4—102no—
632C4—102no—
633C4—122no—
634C4—201no—
635C4—304no—
636C4#9139614yes—
637C3#1060771611yescharacteristic, complemented, Sylow
638C2—202no—
639C2—604no—
640C2#1175257711yescharacteristic
641C2C4×#677311yescharacteristic
642C2#1175271511yescharacteristic
643C2#1175272511yescharacteristic
644C2C2×#9139611yescharacteristic
645C2#1175455711yescharacteristic
646C2#1175455811yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC2×#91450C2
hypercenterC2×C42D3×D5
hypocenterC3×C5#2363
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C2×C4×D60C2semidirect
3. C2×#91358C2semidirect
4. C5×#4928C2semidirect
5. C3×C4×#685C2semidirect
6. #11748917C2semidirect
7. #11748913C2semidirect
8. #11748768C2semidirect
9. Dic5×C4⋊Dic3C2semidirect
10. #11748670C2semidirect
11. #11748669C2semidirect
33. #91450C4semidirect
48. C4×#685D3semidirect
70. C2×C3×C4×D5C2×C4semidirect
73. C2×#1202C2×C4semidirect
76. C4⋊Dic15D4semidirect
78. C5×C4⋊Dic3C2×C4semidirect
85. #6667C2×C4semidirect
97. #4928D5semidirect
221. C2×C4×D5C4×D3semidirect
312. C4⋊Dic3C4×D5semidirect
511. C3×C5#2363semidirect
606. C5#73469semidirect
637. C3#10607716semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D602
C2×#914502
C2×#913582
C5×#49282
C3×C4×#6852
#117489172
#117489132
#117487682
Dic5×C4⋊Dic32
#117486702
#117486692
#106077163
#734695