Finite Group Databasebeta

← #14305764

Subgroups — #14305764

Order 1920.

768 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91552C211yesmaximal, characteristic
2C2×#91241C211yesmaximal, characteristic, complemented
3C2×C3×C22⋊#170C211yesmaximal, characteristic, complemented
4C2×Dic3⋊#170C211yesmaximal, characteristic, complemented
5C2×#91171C211yesmaximal, characteristic, complemented
6C2×#91171C211yesmaximal, characteristic, complemented
7C2×#91169C211yesmaximal, characteristic, complemented
8#10604242—31nomaximal, Hall
9C22×#1220C411yescharacteristic
10C23×C3×Dic5C2211yescharacteristic
11C22×#1200C2211yescharacteristic
12C22×C3⋊#170C2211yescharacteristic
13C22×C3⋊#170C2211yescharacteristic
14C22×C3×#170C2211yescharacteristic
15C22×C3×#170C2211yescharacteristic
16C2×C5×#217C411yescharacteristic
17#91552—22no—
18#91552—22no—
19C2×#6669C411yescharacteristic
20C2×#6667C411yescharacteristic
21C2×Dic3×Dic5C2211yescharacteristic
22#91241—22no—
23C3×C22⋊#170—22no—
24Dic3⋊#170—22no—
25#91171—22no—
26#91171—22no—
27#91169—22no—
28#69917—51nomaximal, Hall
29C2×#4027—31no—
30C2×C22⋊#170—31no—
31C2×C22⋊#170D311yescharacteristic, complemented
32C2×#4003—31no—
33C2×#4003—31no—
34C2×#4002—31no—
35C2×C4⋊#170—31no—
36C24×C3×C5C2×C411yescharacteristic
37C23×D15C2×C411yescharacteristic
38C2×#1220—22no—
39C2×#1220—22no—
40C2×#1220—22no—
41C22×Dic15C2×C411yescharacteristic
42C22×C5×Dic3C2×C411yescharacteristic
43C22×C5×Dic3C2×C411yescharacteristic
44C22×C3×Dic5—21no—
45C22×C3×Dic5—21no—
46C22×C3×Dic5—21no—
47C22×C3×Dic5—21no—
48C22×C3×Dic5—21no—
49C22×C3×Dic5—21no—
50C22×C3×Dic5C2×C411yescharacteristic
51C22×C3×Dic5C2311yescharacteristic
52C2×#1200—21no—
53C2×#1200—21no—
54C2×#1200—21no—
55C2×#1200—21no—
56C2×#1200—21no—
57C2×#1200—21no—
58C2×C3⋊#170—21no—
59C2×C3⋊#170—21no—
60C2×C3⋊#170—21no—
61C2×C3⋊#170—21no—
62C2×C3⋊#170—21no—
63C2×C3⋊#170D412yescomplemented
64C2×C3×#170—21no—
65C2×C3×#170—21no—
66C2×C3×#170—21no—
67C2×C3×#170—21no—
68C2×C3×#170—21no—
69C2×C3×#170D412yescomplemented
70C5×#217—22no—
71#6669—22no—
72#6667—22no—
73Dic3×Dic5D412yes—
74Dic3×Dic5D412yes—
75C2×C4×#67—51no—
76C2×C3×C22⋊C8—51no—
77C2×#830—51no—
78C2×(C2×D3)⋊C8—51no—
79C2×(C2×D3)⋊C8—51no—
80C2×#815—51no—
81C2×Dic3⋊C8—51no—
82C22×C22⋊D5—31no—
83C23×Dic5C2×D311yescharacteristic
84C22×C4×D5—31no—
85C22×#170—31no—
86C22×#170—31no—
87C22×#170C2×D311yescharacteristic
88C22×#170C2×D311yescharacteristic
89C2×C5×C22⋊C4—31no—
90C2×#685—31no—
91C2×#683—31no—
92C2×C4×Dic5—31no—
93#4027—62no—
94#4027—62no—
95C22⋊#170—22no—
96C22⋊#170—62no—
97#4003—62no—
98#4003—62no—
99#4002—62no—
100C4⋊#170—62no—
101#1955—151noSylow
102C23×C3×C5—21no—
103C23×C3×C5—21no—
104C23×C3×C5—21no—
105C23×C3×C5—21no—
106C23×C3×C5—21no—
107C23×C3×C5—21no—
108C23×C3×C5M1612yes—
109C23×C3×C5C22×C411yescharacteristic
110C22×D15—21no—
111C22×D15—21no—
112C22×D15—21no—
113C22×D15—21no—
114C22×D15—21no—
115C22×D15—21no—
116C22×D15M1612yes—
117#1220—28no—
118C2×Dic15—21no—
119C2×Dic15—21no—
120C2×Dic15C22⋊C412yes—
121C2×C5×Dic3—21no—
122C2×C5×Dic3—21no—
123C2×C5×Dic3—21no—
124C2×C5×Dic3—21no—
125C2×C5×Dic3—21no—
126C2×C5×Dic3C22⋊C412yes—
127C2×C3×Dic5—21no—
128C2×C3×Dic5—21no—
129C2×C3×Dic5—21no—
130C2×C3×Dic5—41no—
131C2×C3×Dic5—41no—
132C2×C3×Dic5—41no—
133C2×C3×Dic5—41no—
134C2×C3×Dic5C2×D411yescharacteristic
135C2×C3×Dic5C2×D411yescharacteristic
136C2×C3×Dic5C2×D411yescharacteristic
137C2×C3×Dic5C2×D411yescharacteristic
138C2×C3×Dic5#4111yescharacteristic
139C2×C3×Dic5#4111yescharacteristic
140#1200—41no—
141#1200—41no—
142#1200—41no—
143#1200—41no—
144C3⋊#170—22no—
145C3⋊#170—41no—
146C3×#170—22no—
147C3×#170—41no—
148C23×C3×C4—51no—
149C22×#67—51no—
150C22×C4×D3—51no—
151C22×C3×C8—51no—
152C22×C3×C8—51no—
153C2×#217AGL(1,𝔽5)11yescharacteristic
154C4×#67—102no—
155C4×#67—102no—
156C2×(C2×D3)⋊C4—51no—
157C2×Dic3⋊C4—51no—
158C2×C4×Dic3—51no—
159C22×C3⋊C8—51no—
160C22×C3⋊C8—51no—
161C3×C22⋊C8—102no—
162#830—102no—
163(C2×D3)⋊C8—102no—
164(C2×D3)⋊C8—102no—
165#815—102no—
166Dic3⋊C8—102no—
167C24×C5C4×D311yescharacteristic
168C23×D5—31no—
169C22×C4×C5—31no—
170C22×C4×C5—31no—
171C2×C22⋊D5—62no—
172C2×C22⋊D5—62no—
173C2×C22⋊D5—62no—
174C22×Dic5—21no—
175C22×Dic5—21no—
176C22×Dic5—21no—
177C22×Dic5—21no—
178C22×Dic5—21no—
179C22×Dic5—21no—
180C22×Dic5—31no—
181C22×Dic5C4×D311yescharacteristic
182C22×Dic5C22×D311yescharacteristic
183C2×C4×D5—61no—
184C2×C4×D5—61no—
185C2×C4×D5—61no—
186C2×C4×D5—61no—
187C2×C4×D5—61no—
188C2×C4×D5—61no—
189C2×#170—21no—
190C2×#170—21no—
191C2×#170—21no—
192C2×#170—21no—
193C2×#170—21no—
194C2×#170—32no—
195C2×#170—61no—
196C2×#170—61no—
197C2×#170—61no—
198C2×#170—61no—
199C2×#170—61no—
200C2×#170#6712yescomplemented
201C5×C22⋊C4—62no—
202#685—62no—
203#683—62no—
204C4×Dic5—32no—
205C4×Dic5—32no—
206C2×C4×D4—151no—
207C2×C4⋊C8—151no—
208C2×C22⋊C8—51no—
209C2×C22⋊C8—151no—
210C2×C22⋊C8—151no—
211C2×C22⋊C8—151no—
212C2×#97—151no—
213C22×C3×C5—22no—
214C22×C3×C5—22no—
215C22×C3×C5—22no—
216C22×C3×C5—41no—
217C22×C3×C5—41no—
218C22×C3×C5—41no—
219C22×C3×C5—41no—
220C22×C3×C5C2×C22⋊C411yescharacteristic
221C22×C3×C5C4×D411yescharacteristic
222C22×C3×C5C4×D411yescharacteristic
223C22×C3×C5C2×M1611yescharacteristic
224C22×C3×C5C2×M1611yescharacteristic
225C22×C3×C5#13111yescharacteristic
226C22×C3×C5#13111yescharacteristic
227C2×D15—22no—
228C2×D15—22no—
229C2×D15—22no—
230C2×D15—41no—
231C2×D15—41no—
232C2×D15—41no—
233C2×D15—41no—
234Dic15—22no—
235C3×Dic5—41no—
236C3×Dic5#12111yescharacteristic
237C3×Dic5#12111yescharacteristic
238C3×Dic5#12411yescharacteristic
239C3×Dic5#12711yescharacteristic
240C5×Dic3—22no—
241C5×Dic3—41no—
242C24×C3C2×AGL(1,𝔽5)11yescharacteristic
243C23×D3—51no—
244C22×C3×C4—51no—
245C22×C3×C4—51no—
246C22×C3×C4—101no—
247C22×C3×C4—101no—
248C22×C3×C4—101no—
249C22×C3×C4—101no—
250C22×C3×C4—101no—
251C22×C3×C4—101no—
252C2×#67—102no—
253C2×#67—102no—
254C2×#67—102no—
255C22×Dic3—51no—
256C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
257C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
258C2×C4×D3—101no—
259C2×C4×D3—101no—
260C2×C4×D3—101no—
261C2×C4×D3—101no—
262C2×C4×D3—101no—
263C2×C4×D3—101no—
264C2×C3×C8—52no—
265C2×C3×C8—101no—
266C2×C3×C8—101no—
267C2×C3×C8—101no—
268C2×C3×C8—101no—
269C2×C3×C8—101no—
270#217—22no—
271(C2×D3)⋊C4—102no—
272Dic3⋊C4—102no—
273C4×Dic3—52no—
274C4×Dic3—52no—
275C2×C3⋊C8—52no—
276C2×C3⋊C8—101no—
277C2×C3⋊C8—101no—
278C2×C3⋊C8—101no—
279C2×C3⋊C8—101no—
280C2×C3⋊C8—101no—
281C23×C5—21no—
282C23×C5—21no—
283C23×C5—21no—
284C23×C5—21no—
285C23×C5—21no—
286C23×C5—21no—
287C23×C5ΓL(1,𝔽25)12yes—
288C23×C5C2×C4×D311yescharacteristic
289C22×D5—32no—
290C22×D5—61no—
291C22×D5—61no—
292C22×D5—61no—
293C22×D5—61no—
294C22×D5—61no—
295C22×D5—61no—
296C2×C4×C5—32no—
297C2×C4×C5—61no—
298C2×C4×C5—61no—
299C2×C4×C5—61no—
300C2×C4×C5—61no—
301C2×C4×C5—61no—
302C22⋊D5—68no—
303C2×Dic5—21no—
304C2×Dic5—21no—
305C2×Dic5—21no—
306C2×Dic5—32no—
307C2×Dic5—41no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310C2×Dic5—41no—
311C2×Dic5—61no—
312C2×Dic5—61no—
313C2×Dic5#23511yescharacteristic
314C2×Dic5D3×D411yescharacteristic
315C2×Dic5D3×D411yescharacteristic
316C2×Dic5D3×D411yescharacteristic
317C2×Dic5#23711yescharacteristic
318C2×Dic5C2×#6711yescharacteristic
319C4×D5—121no—
320C4×D5—121no—
321C4×D5—121no—
322C4×D5—121no—
323#170—22no—
324#170—41no—
325#170—62no—
326#170—121no—
327C22×D4—151no—
328C23×C4—51no—
329C23×C4—151no—
330C22×C8—51no—
331C22×C8—51no—
332C22×C8—151no—
333C22×C8—151no—
334C4×D4—302no—
335C4×D4—302no—
336C2×C4⋊C4—151no—
337C2×C22⋊C4—31no—
338C2×C22⋊C4—151no—
339C2×C42—151no—
340C4⋊C8—302no—
341C22⋊C8—102no—
342C22⋊C8—302no—
343C22⋊C8—302no—
344C22⋊C8—302no—
345#97—302no—
346C2×C3×C5—42no—
347C2×C3×C5#39011yescharacteristic
348C2×C3×C5#40711yescharacteristic
349C2×C3×C5#40811yescharacteristic
350C2×C3×C5#43511yescharacteristic
351C2×C3×C5#43511yescharacteristic
352C2×C3×C5#43511yescharacteristic
353C2×C3×C5#43511yescharacteristic
354D15—42no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3#70012yes—
362C23×C3C22×AGL(1,𝔽5)11yescharacteristic
363C22×D3—52no—
364C22×D3—101no—
365C22×D3—101no—
366C22×D3—101no—
367C22×D3—101no—
368C22×D3—101no—
369C22×D3—101no—
370C2×C3×C4—51no—
371C2×C3×C4—51no—
372C2×C3×C4—51no—
373C2×C3×C4—51no—
374C2×C3×C4—51no—
375C2×C3×C4—51no—
376C2×C3×C4—101no—
377C2×C3×C4—101no—
378C2×C3×C4—101no—
379C2×C3×C4—201no—
380C2×C3×C4—201no—
381C2×C3×C4—201no—
382C2×C3×C4—201no—
383#67—108no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—52no—
390C2×Dic3—101no—
391C2×Dic3—101no—
392C2×Dic3C22⋊AGL(1,𝔽5)12yes—
393C4×D3—201no—
394C4×D3—201no—
395C4×D3—201no—
396C4×D3—201no—
397C3×C8—102no—
398C3×C8—201no—
399C3⋊C8—102no—
400C3⋊C8—201no—
401C22×C5—22no—
402C22×C5—22no—
403C22×C5—22no—
404C22×C5—41no—
405C22×C5—41no—
406C22×C5—41no—
407C22×C5—41no—
408C22×C5D3×C22⋊C411yescharacteristic
409C22×C5#67⋊C411yescharacteristic
410C22×C5C2×ΓL(1,𝔽25)11yescharacteristic
411C22×C5#90111yescharacteristic
412C22×C5D3×M1611yescharacteristic
413C22×C5#90711yescharacteristic
414C22×C5C4×#6711yescharacteristic
415C2×D5—62no—
416C2×D5—62no—
417C2×D5—62no—
418C2×D5—121no—
419C2×D5—121no—
420C2×D5—121no—
421C2×D5—121no—
422C4×C5—62no—
423C4×C5—121no—
424Dic5—41no—
425Dic5—62no—
426Dic5#88411yescharacteristic
427Dic5#88511yescharacteristic
428Dic5#93911yescharacteristic
429Dic5#94011yescharacteristic
430C24—151no—
431C24D3×AGL(1,𝔽5)11yescharacteristic
432C2×D4—302no—
433C2×D4—302no—
434C2×D4—302no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—51no—
438C22×C4—51no—
439C22×C4—101no—
440C22×C4—101no—
441C22×C4—101no—
442C22×C4—101no—
443C22×C4—101no—
444C22×C4—101no—
445C22×C4—151no—
446C22×C4—301no—
447C22×C4—301no—
448C22×C4—301no—
449C22×C4—301no—
450C22×C4—301no—
451C22×C4—301no—
452C2×C8—52no—
453C2×C8—101no—
454C2×C8—101no—
455C2×C8—101no—
456C2×C8—101no—
457C2×C8—101no—
458C2×C8—152no—
459C2×C8—301no—
460C2×C8—301no—
461C2×C8—301no—
462C2×C8—301no—
463C2×C8—301no—
464C4⋊C4—302no—
465C22⋊C4—62no—
466C22⋊C4—302no—
467C42—152no—
468C42—152no—
469C3×C5#195511yescharacteristic, complemented, Hall
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3C2×#70011yescharacteristic
478C22×C3D5⋊M1611yescharacteristic
479C22×C3#413111yescharacteristic
480C22×C3#413111yescharacteristic
481C22×C3D4×AGL(1,𝔽5)11yescharacteristic
482C22×C3D4×AGL(1,𝔽5)11yescharacteristic
483C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
484C2×D3—102no—
485C2×D3—102no—
486C2×D3—102no—
487C2×D3—201no—
488C2×D3—201no—
489C2×D3—201no—
490C2×D3—201no—
491C3×C4—51no—
492C3×C4—51no—
493C3×C4—51no—
494C3×C4—51no—
495C3×C4—201no—
496Dic3—22no—
497Dic3—41no—
498Dic3—102no—
499C2×C5—42no—
500C2×C5#472611yescharacteristic
501C2×C5#472711yescharacteristic
502C2×C5#472811yescharacteristic
503C2×C5#473011yescharacteristic
504C2×C5#495711yescharacteristic
505C2×C5#511111yescharacteristic
506C2×C5#512711yescharacteristic
507D5—122no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—152no—
515C23—301no—
516C23—301no—
517C23—301no—
518C23—301no—
519C23—301no—
520C23—301no—
521C23#674012yes—
522C23C2×D3×AGL(1,𝔽5)11yescharacteristic
523D4—308no—
524C2×C4—32no—
525C2×C4—51no—
526C2×C4—51no—
527C2×C4—51no—
528C2×C4—51no—
529C2×C4—51no—
530C2×C4—51no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—101no—
537C2×C4—101no—
538C2×C4—101no—
539C2×C4—152no—
540C2×C4—201no—
541C2×C4—201no—
542C2×C4—201no—
543C2×C4—201no—
544C2×C4—301no—
545C2×C4—301no—
546C2×C4—601no—
547C2×C4—601no—
548C2×C4—601no—
549C2×C4—601no—
550C8—102no—
551C8—201no—
552C8—302no—
553C8—601no—
554C2×C3—42no—
555C2×C3#6569311yescharacteristic
556C2×C3#6569311yescharacteristic
557C2×C3#6569411yescharacteristic
558C2×C3#6569411yescharacteristic
559C2×C3#6575611yescharacteristic
560C2×C3#6577511yescharacteristic
561C2×C3#6578011yescharacteristic
562D3—202no—
563C5#6991711yescharacteristic, complemented, Sylow
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—302no—
572C22—302no—
573C22—302no—
574C22—601no—
575C22—601no—
576C22—601no—
577C22—601no—
578C22#9192711yescharacteristic
579C22#9192911yescharacteristic
580C22#9193111yescharacteristic
581C22C2×#674011yescharacteristic
582C22AGL(1,𝔽5)×#6711yescharacteristic
583C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
584C22#67⋊AGL(1,𝔽5)11yescharacteristic
585C4—51no—
586C4—51no—
587C4—51no—
588C4—51no—
589C4—62no—
590C4—121no—
591C4—201no—
592C4—302no—
593C3#1060424211yescharacteristic, complemented, Sylow
594C2—42no—
595C2—602no—
596C2#1175201911yescharacteristic
597C2#1175203711yescharacteristic
598C2#1175203811yescharacteristic
599C2#1175204111yescharacteristic
600C2#1175204611yescharacteristic
601C2#1175205211yescharacteristic
602C2#1175205411yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×AGL(1,𝔽5)
commutator, socleC23×C3×C5C22×C4
FittingC24×C3×C5C2×C4
prime radicalC22×#1220C4
cosocleC22×C3×Dic5C23
hypercenterC24D3×AGL(1,𝔽5)
hypocenterC3×C5#1955
LCSG ▷ C23×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×#91241C2semidirect
3. C2×C3×C22⋊#170C2semidirect
4. C2×Dic3⋊#170C2semidirect
5. C2×#91171C2semidirect
6. C2×#91171C2semidirect
7. C2×#91169C2semidirect
31. C2×C22⋊#170D3semidirect
63. C2×C3⋊#170D4semidirect
69. C2×C3×#170D4semidirect
200. C2×#170#67semidirect
469. C3×C5#1955semidirect
563. C5#69917semidirect
593. C3#10604242semidirect

Maximal subgroups

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

subgroupindex
C2×#915522
C2×#912412
C2×C3×C22⋊#1702
C2×Dic3⋊#1702
C2×#911712
C2×#911692
#106042423
#699175