Finite Group Databasebeta

← #14300522

Subgroups — #14300522

Order 1920.

816 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6740C211yesmaximal, characteristic, complemented
2C2×#91418C211yesmaximal, characteristic
3C2×#91171C211yesmaximal, characteristic, complemented
4C2×#91171C211yesmaximal, characteristic, complemented
5C2×#91171C211yesmaximal, characteristic, complemented
6#11746789C211yesmaximal, characteristic, complemented
7C3×#64918C211yesmaximal, characteristic, complemented
8#10604320—31nomaximal, Hall
9C22×D60C411yescharacteristic
10C22×#1200C2211yescharacteristic
11C22×#1200C2211yescharacteristic
12C22×C3⋊#170C2211yescharacteristic
13C22×C3⋊#170C2211yescharacteristic
14C22×C3×#170C2211yescharacteristic
15C22×C3×#170C2211yescharacteristic
16C2×#6740—22no—
17C2×#6740—22no—
18C2×#6740—22no—
19C2×C5×C4⋊Dic3C411yescharacteristic
20C2×C3×C4×Dic5C2211yescharacteristic
21C2×#6667C411yescharacteristic
22C2×#6667C411yescharacteristic
23#91418—22no—
24#91418—22no—
25#91171—22no—
26#91171—22no—
27#91171—22no—
28#68669—51nomaximal, Hall
29C22×#700—31no—
30C2×#4039—31no—
31C2×#4003—31no—
32C2×#4003—31no—
33C2×#4003—31no—
34#64918—31no—
35#64918D311yescharacteristic, complemented
36C23×D15C2×C411yescharacteristic
37C23×D15C2×C411yescharacteristic
38C22×C3×C4×C5C2×C411yescharacteristic
39C2×D60—22no—
40C2×D60—22no—
41C2×D60—22no—
42C22×C5×Dic3C2×C411yescharacteristic
43C22×C5×Dic3C2×C411yescharacteristic
44C22×C3×Dic5C2×C411yescharacteristic
45C22×C3×Dic5C2311yescharacteristic
46C2×#1200—21no—
47C2×#1200—21no—
48C2×#1200—21no—
49C2×#1200—21no—
50C2×#1200—21no—
51C2×#1200—21no—
52C2×#1200—21no—
53C2×#1200—21no—
54C2×#1200—21no—
55C2×#1200—21no—
56C2×#1200D412yes—
57C2×#1200D412yes—
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
70#6740—28no—
71C5×C4⋊Dic3—22no—
72C3×C4×Dic5—21no—
73C3×C4×Dic5—21no—
74#6667—22no—
75#6667—22no—
76C22×ΓL(1,𝔽25)—51no—
77C2×C4×D12—51no—
78C2×(C2×D3)⋊C8—51no—
79C2×(C2×D3)⋊C8—51no—
80C2×(C2×D3)⋊C8—51no—
81C3×#336—51no—
82#4524—51no—
83C22×D20—31no—
84C22×C4×D5—31no—
85C22×C4×D5—31no—
86C22×#170—31no—
87C22×#170—31no—
88C22×#170C2×D311yescharacteristic
89C22×#170C2×D311yescharacteristic
90C2×#700—62no—
91C2×#700—62no—
92C2×#700—62no—
93C2×C5×C4⋊C4—31no—
94C2×#685—31no—
95C2×#685—31no—
96C2×C4×Dic5C2×D311yescharacteristic
97#4039—62no—
98#4039—62no—
99#4003—62no—
100#4003—62no—
101#4003—62no—
102#1870—151noSylow
103C23×C3×C5C22×C411yescharacteristic
104C22×D15—21no—
105C22×D15—21no—
106C22×D15—21no—
107C22×D15—21no—
108C22×D15—21no—
109C22×D15—21no—
110C22×D15—21no—
111C22×D15—21no—
112C22×D15—21no—
113C22×D15—21no—
114C22×D15—21no—
115C22×D15—21no—
116C22×D15C22⋊C412yes—
117C22×D15M1612yescomplemented
118C2×C3×C4×C5—21no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5M1612yescomplemented
121D60—28no—
122C2×C5×Dic3—21no—
123C2×C5×Dic3—21no—
124C2×C5×Dic3—21no—
125C2×C5×Dic3—21no—
126C2×C5×Dic3—21no—
127C2×C5×Dic3C22⋊C412yes—
128C2×C3×Dic5—21no—
129C2×C3×Dic5—21no—
130C2×C3×Dic5—21no—
131C2×C3×Dic5C2×D411yescharacteristic
132C2×C3×Dic5C2×D411yescharacteristic
133C2×C3×Dic5C2×D411yescharacteristic
134C2×C3×Dic5C2×D411yescharacteristic
135C2×C3×Dic5#4111yescharacteristic
136C2×C3×Dic5#4111yescharacteristic
137#1200—22no—
138#1200—22no—
139#1200—22no—
140#1200—22no—
141#1200—41no—
142#1200—41no—
143#1200—41no—
144#1200—41no—
145C3⋊#170—22no—
146C3⋊#170—41no—
147C3×#170—22no—
148C3×#170—41no—
149C22×D12—51no—
150C22×C4×D3—51no—
151C22×C4×D3—51no—
152C22×C3×C8—51no—
153C22×C3×C8—51no—
154C2×C3×C42—51no—
155C2×(C2×D3)⋊C4—51no—
156C2×(C2×D3)⋊C4—51no—
157C2×C4⋊Dic3AGL(1,𝔽5)11yescharacteristic
158C22×C3⋊C8—51no—
159C22×C3⋊C8—51no—
160C2×ΓL(1,𝔽25)—102no—
161C2×ΓL(1,𝔽25)—102no—
162C2×ΓL(1,𝔽25)—102no—
163C4×D12—102no—
164C4×D12—102no—
165(C2×D3)⋊C8—102no—
166(C2×D3)⋊C8—102no—
167(C2×D3)⋊C8—102no—
168C23×D5—31no—
169C23×D5—31no—
170C22×C4×C5—31no—
171C22×C4×C5—31no—
172C22×C4×C5C4×D311yescharacteristic
173C22×Dic5C4×D311yescharacteristic
174C22×Dic5C22×D311yescharacteristic
175C2×D20—62no—
176C2×D20—62no—
177C2×D20—62no—
178C2×C4×D5—32no—
179C2×C4×D5—32no—
180C2×C4×D5—61no—
181C2×C4×D5—61no—
182C2×C4×D5—61no—
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×C4×D5—61no—
190C2×#170—21no—
191C2×#170—21no—
192C2×#170—21no—
193C2×#170—21no—
194C2×#170—21no—
195C2×#170—32no—
196C2×#170—61no—
197C2×#170—61no—
198C2×#170—61no—
199C2×#170—61no—
200C2×#170—61no—
201C2×#170D1212yescomplemented
202#700—68no—
203C5×C4⋊C4—62no—
204#685—62no—
205#685—62no—
206C4×Dic5—21no—
207C4×Dic5—21no—
208C22×M16—151no—
209C2×C4×D4—151no—
210C2×C22⋊C8—151no—
211C2×C22⋊C8—151no—
212C2×C22⋊C8—151no—
213#336—51no—
214#336—151no—
215C22×C3×C5C2×C22⋊C411yescharacteristic
216C22×C3×C5C4×D411yescharacteristic
217C22×C3×C5C4×D411yescharacteristic
218C22×C3×C5C2×M1611yescharacteristic
219C22×C3×C5C2×M1611yescharacteristic
220C22×C3×C5#13111yescharacteristic
221C22×C3×C5#13111yescharacteristic
222C2×D15—22no—
223C2×D15—22no—
224C2×D15—22no—
225C2×D15—22no—
226C2×D15—22no—
227C2×D15—22no—
228C2×D15—41no—
229C2×D15—41no—
230C2×D15—41no—
231C2×D15—41no—
232C2×D15—41no—
233C2×D15—41no—
234C2×D15—41no—
235C2×D15—41no—
236C3×C4×C5—22no—
237C3×Dic5—41no—
238C3×Dic5#12011yescharacteristic
239C3×Dic5#12111yescharacteristic
240C3×Dic5#12111yescharacteristic
241C3×Dic5#12311yescharacteristic
242C5×Dic3—22no—
243C5×Dic3—41no—
244C23×D3—51no—
245C23×D3—51no—
246C22×C3×C4—51no—
247C22×C3×C4—51no—
248C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
249C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
250C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
251C2×D12—102no—
252C2×D12—102no—
253C2×D12—102no—
254C2×C4×D3—52no—
255C2×C4×D3—52no—
256C2×C4×D3—101no—
257C2×C4×D3—101no—
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×C4×D3—101no—
265C2×C4×D3—101no—
266C2×C3×C8—52no—
267C2×C3×C8—101no—
268C2×C3×C8—101no—
269C2×C3×C8—101no—
270C2×C3×C8—101no—
271C2×C3×C8—101no—
272C3×C42—101no—
273C3×C42—101no—
274(C2×D3)⋊C4—102no—
275(C2×D3)⋊C4—102no—
276C4⋊Dic3—22no—
277C2×C3⋊C8—52no—
278C2×C3⋊C8—101no—
279C2×C3⋊C8—101no—
280C2×C3⋊C8—101no—
281C2×C3⋊C8—101no—
282C2×C3⋊C8—101no—
283ΓL(1,𝔽25)—108no—
284C23×C5C2×C4×D311yescharacteristic
285C22×D5—32no—
286C22×D5—32no—
287C22×D5—61no—
288C22×D5—61no—
289C22×D5—61no—
290C22×D5—61no—
291C22×D5—61no—
292C22×D5—61no—
293C22×D5—61no—
294C22×D5—61no—
295C22×D5—61no—
296C22×D5—61no—
297C22×D5—61no—
298C22×D5—61no—
299C2×C4×C5—21no—
300C2×C4×C5—21no—
301C2×C4×C5—32no—
302C2×C4×C5—61no—
303C2×C4×C5—61no—
304C2×C4×C5—61no—
305C2×C4×C5—61no—
306C2×C4×C5—61no—
307C2×C4×C5ΓL(1,𝔽25)12yescomplemented
308C2×Dic5—21no—
309C2×Dic5—21no—
310C2×Dic5—21no—
311C2×Dic5C2×D1211yescharacteristic
312C2×Dic5#23511yescharacteristic
313C2×Dic5D3×D411yescharacteristic
314C2×Dic5D3×D411yescharacteristic
315C2×Dic5D3×D411yescharacteristic
316C2×Dic5#23911yescharacteristic
317D20—68no—
318C4×D5—62no—
319C4×D5—62no—
320C4×D5—62no—
321C4×D5—62no—
322C4×D5—121no—
323C4×D5—121no—
324C4×D5—121no—
325C4×D5—121no—
326#170—22no—
327#170—41no—
328#170—62no—
329#170—121no—
330C22×D4—151no—
331C23×C4—151no—
332C23×C4—151no—
333C2×M16—302no—
334C2×M16—302no—
335C2×M16—302no—
336C22×C8—51no—
337C22×C8—51no—
338C22×C8—151no—
339C22×C8—151no—
340C4×D4—302no—
341C4×D4—302no—
342C2×C4⋊C4—31no—
343C2×C22⋊C4—151no—
344C2×C22⋊C4—151no—
345C2×C42—51no—
346C22⋊C8—302no—
347C22⋊C8—302no—
348C22⋊C8—302no—
349C2×C3×C5#38611yescharacteristic
350C2×C3×C5#40711yescharacteristic
351C2×C3×C5#40811yescharacteristic
352C2×C3×C5#43511yescharacteristic
353C2×C3×C5#43511yescharacteristic
354C2×C3×C5#43611yescharacteristic
355C2×C3×C5#43611yescharacteristic
356D15—42no—
357D15—42no—
358C23×C3C22×AGL(1,𝔽5)11yescharacteristic
359C22×D3—52no—
360C22×D3—52no—
361C22×D3—101no—
362C22×D3—101no—
363C22×D3—101no—
364C22×D3—101no—
365C22×D3—101no—
366C22×D3—101no—
367C22×D3—101no—
368C22×D3—101no—
369C22×D3—101no—
370C22×D3—101no—
371C22×D3—101no—
372C22×D3—101no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—51no—
376C2×C3×C4—51no—
377C2×C3×C4—51no—
378C2×C3×C4—51no—
379C2×C3×C4—51no—
380C2×C3×C4—51no—
381C2×C3×C4—101no—
382C2×C3×C4—101no—
383C2×C3×C4—101no—
384C2×C3×C4#70012yescomplemented
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3C22⋊AGL(1,𝔽5)12yes—
391D12—108no—
392C4×D3—102no—
393C4×D3—102no—
394C4×D3—102no—
395C4×D3—102no—
396C4×D3—201no—
397C4×D3—201no—
398C4×D3—201no—
399C4×D3—201no—
400C3×C8—102no—
401C3×C8—201no—
402C3⋊C8—102no—
403C3⋊C8—201no—
404C22×C5C4×D1211yescharacteristic
405C22×C5D3×C22⋊C411yescharacteristic
406C22×C5#89311yescharacteristic
407C22×C5C2×ΓL(1,𝔽25)11yescharacteristic
408C22×C5#90111yescharacteristic
409C22×C5D3×M1611yescharacteristic
410C22×C5#90711yescharacteristic
411C2×D5—62no—
412C2×D5—62no—
413C2×D5—62no—
414C2×D5—62no—
415C2×D5—62no—
416C2×D5—62no—
417C2×D5—121no—
418C2×D5—121no—
419C2×D5—121no—
420C2×D5—121no—
421C2×D5—121no—
422C2×D5—121no—
423C2×D5—121no—
424C2×D5—121no—
425C4×C5—22no—
426C4×C5—62no—
427C4×C5—121no—
428Dic5—41no—
429Dic5#88211yescharacteristic
430Dic5#88411yescharacteristic
431Dic5#89411yescharacteristic
432Dic5#89511yescharacteristic
433C24—151no—
434C24—151no—
435C2×D4—302no—
436C2×D4—302no—
437C2×D4—302no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—51no—
441C22×C4—51no—
442C22×C4—152no—
443C22×C4—152no—
444C22×C4—301no—
445C22×C4—301no—
446C22×C4—301no—
447C22×C4—301no—
448C22×C4—301no—
449C22×C4—301no—
450C22×C4—301no—
451C22×C4—301no—
452C22×C4—301no—
453C22×C4—301no—
454C22×C4D3×AGL(1,𝔽5)11yescharacteristic
455M16—308no—
456C2×C8—52no—
457C2×C8—101no—
458C2×C8—101no—
459C2×C8—101no—
460C2×C8—101no—
461C2×C8—101no—
462C2×C8—152no—
463C2×C8—301no—
464C2×C8—301no—
465C2×C8—301no—
466C2×C8—301no—
467C2×C8—301no—
468C4⋊C4—62no—
469C22⋊C4—302no—
470C22⋊C4—302no—
471C42—101no—
472C42—101no—
473C3×C5#187011yescharacteristic, complemented, Hall
474C22×C3C2×#70011yescharacteristic
475C22×C3D5⋊M1611yescharacteristic
476C22×C3D4×AGL(1,𝔽5)11yescharacteristic
477C22×C3D4×AGL(1,𝔽5)11yescharacteristic
478C22×C3#413311yescharacteristic
479C22×C3#413311yescharacteristic
480C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
481C2×D3—102no—
482C2×D3—102no—
483C2×D3—102no—
484C2×D3—102no—
485C2×D3—102no—
486C2×D3—102no—
487C2×D3—201no—
488C2×D3—201no—
489C2×D3—201no—
490C2×D3—201no—
491C2×D3—201no—
492C2×D3—201no—
493C2×D3—201no—
494C2×D3—201no—
495C3×C4—22no—
496C3×C4—51no—
497C3×C4—51no—
498C3×C4—51no—
499C3×C4—51no—
500C3×C4—201no—
501Dic3—22no—
502Dic3—41no—
503C2×C5#466511yescharacteristic
504C2×C5#468811yescharacteristic
505C2×C5#470611yescharacteristic
506C2×C5#472611yescharacteristic
507C2×C5#472711yescharacteristic
508C2×C5#483611yescharacteristic
509C2×C5#483811yescharacteristic
510D5—122no—
511D5—122no—
512C23—152no—
513C23—152no—
514C23—301no—
515C23—301no—
516C23—301no—
517C23—301no—
518C23—301no—
519C23—301no—
520C23—301no—
521C23—301no—
522C23—301no—
523C23—301no—
524C23—301no—
525C23—301no—
526C23C2×D3×AGL(1,𝔽5)11yescharacteristic
527D4—308no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—32no—
531C2×C4—51no—
532C2×C4—51no—
533C2×C4—51no—
534C2×C4—51no—
535C2×C4—51no—
536C2×C4—51no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—101no—
543C2×C4—101no—
544C2×C4—101no—
545C2×C4—302no—
546C2×C4—302no—
547C2×C4—302no—
548C2×C4—302no—
549C2×C4—601no—
550C2×C4—601no—
551C2×C4—601no—
552C2×C4—601no—
553C2×C4#674012yescomplemented
554C8—102no—
555C8—201no—
556C8—302no—
557C8—601no—
558C2×C3#6570411yescharacteristic
559C2×C3#6570411yescharacteristic
560C2×C3#6570511yescharacteristic
561C2×C3#6570511yescharacteristic
562C2×C3#6575611yescharacteristic
563C2×C3#6577911yescharacteristic
564C2×C3#6578711yescharacteristic
565D3—202no—
566D3—202no—
567C5#6866911yescharacteristic, complemented, Sylow
568C22—302no—
569C22—302no—
570C22—302no—
571C22—302no—
572C22—302no—
573C22—302no—
574C22—601no—
575C22—601no—
576C22—601no—
577C22—601no—
578C22—601no—
579C22—601no—
580C22—601no—
581C22—601no—
582C22#9191411yescharacteristic
583C22#9191611yescharacteristic
584C22AGL(1,𝔽5)×D1211yescharacteristic
585C22#9192111yescharacteristic
586C22#9193111yescharacteristic
587C22C2×#674011yescharacteristic
588C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
589C4—22no—
590C4—51no—
591C4—51no—
592C4—51no—
593C4—51no—
594C4—62no—
595C4—121no—
596C4—201no—
597C3#1060432011yescharacteristic, complemented, Sylow
598C2—602no—
599C2—602no—
600C2#1175189411yescharacteristic
601C2#1175190011yescharacteristic
602C2#1175190511yescharacteristic
603C2#1175191411yescharacteristic
604C2#1175192811yescharacteristic
605C2#1175193311yescharacteristic
606C2#1175205211yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×AGL(1,𝔽5)
commutator, socleC23×C3×C5C22×C4
FittingC22×C3×C4×C5C2×C4
prime radicalC22×D60C4
cosocleC22×C3×Dic5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1870
LCSG ▷ C23×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#6740C2semidirect
3. C2×#91171C2semidirect
4. C2×#91171C2semidirect
5. C2×#91171C2semidirect
6. #11746789C2semidirect
7. C3×#64918C2semidirect
35. #64918D3semidirect
63. C2×C3⋊#170D4semidirect
69. C2×C3×#170D4semidirect
117. C22×D15M16semidirect
120. C2×C3×C4×C5M16semidirect
201. C2×#170D12semidirect
307. C2×C4×C5ΓL(1,𝔽25)semidirect
384. C2×C3×C4#700semidirect
473. C3×C5#1870semidirect
553. C2×C4#6740semidirect
567. C5#68669semidirect
597. C3#10604320semidirect

Maximal subgroups

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

subgroupindex
C22×#67402
C2×#914182
C2×#911712
#117467892
C3×#649182
#106043203
#686695