Finite Group Databasebeta

← #14323769

Subgroups — #14323769

Order 1920.

1112 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#924C211yesmaximal, characteristic
2C2×C3×#4072C211yesmaximal, characteristic
3C2×Dic5×Dic6C211yesmaximal, characteristic, complemented
4#11751313C211yesmaximal, characteristic
5#11749479C212yesmaximal
6#11749478C212yesmaximal
7#11748842C211yesmaximal, characteristic, complemented
8Dic5×C4⋊Dic3C211yesmaximal, characteristic, complemented
9#11748677C212yesmaximal, complemented
10#11748673C212yesmaximal, complemented
11#11748589C211yesmaximal, characteristic, complemented
12#10614475—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C2×#6719C2212yes—
16C2×C4⋊Dic15C2211yescharacteristic
17C2×C4⋊Dic15C2211yescharacteristic
18C2×C5×#217C2212yes—
19C2×C5×C4⋊Dic3C2211yescharacteristic
20C5×#924C418yescomplemented
21C2×C5×Dic3⋊C4C2212yes—
22C2×C3×C22⋊Dic5C2211yescharacteristic
23C2×C3×C22⋊Dic5C2211yescharacteristic
24C3×#4072—24no—
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C2×C3×C4×Dic5C2211yescharacteristic
28C2×C3×C4×Dic5C2211yescharacteristic
29C2×Dic3⋊Dic5C2212yes—
30C2×Dic3⋊Dic5C2212yes—
31C2×Dic3×Dic5C2212yes—
32C2×Dic3×Dic5C2212yes—
33Dic5×Dic6—24no—
34#91115C2212yes—
35#90994C2212yes—
36#90994C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#82235—51nomaximal, Hall
40C2×C5×C22⋊Q8—31no—
41C2×Q8×Dic5—31no—
42C2×#4072D311yescharacteristic
43#65519—31no—
44#65501—31no—
45#65274—32no—
46#65274—31no—
47#65270—32no—
48C4⋊C4×Dic5—31no—
49#65240—32no—
50#65240—32no—
51C24×C3×C5C2311yescharacteristic
52C22×C3×C4×C5—22no—
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5C2×C414yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×Dic15C2312yes—
58C22×Dic15C2312yes—
59C22×C5×Dic3C2312yes—
60C22×C5×Dic3C2312yes—
61C2×C5×Dic6—22no—
62C2×C5×Dic6—22no—
63C2×C5×Dic6C2×C414yes—
64C22×C3×Dic5C2311yescharacteristic
65C22×C3×Dic5C2311yescharacteristic
66C22×C3×Dic5C2311yescharacteristic
67C22×C3×Dic5C2311yescharacteristic
68#6719—42no—
69C4⋊Dic15—22no—
70C4⋊Dic15—22no—
71C5×#217C2×C418yes—
72C5×C4⋊Dic3C2×C414yes—
73C5×Dic3⋊C4C2×C418yes—
74C3×C22⋊Dic5—22no—
75C3×C22⋊Dic5—22no—
76C3×C4⋊Dic5—22no—
77C3×C4⋊Dic5—22no—
78C3×C4×Dic5—22no—
79C3×C4×Dic5D414yes—
80Dic3⋊Dic5—24no—
81Dic3⋊Dic5—42no—
82Dic3×Dic5—24no—
83Dic3×Dic5—42no—
84C2×C3×#117—51no—
85C2×#924D511yescharacteristic
86C2×C4×Dic6—51no—
87#5212—51no—
88#4978—52no—
89#4976—52no—
90#4945—52no—
91#4944—52no—
92C4×C4⋊Dic3—51no—
93#4929—51no—
94#4928—51no—
95C22×C5×Q8—31no—
96C23×C4×C5C2×D311yescharacteristic
97C5×C22⋊Q8—38no—
98C2×C5×C4⋊C4—32no—
99C2×C5×C4⋊C4—31no—
100C2×C5×C22⋊C4—32no—
101C2×C22⋊Dic5—32no—
102C2×C22⋊Dic5C2×D311yescharacteristic
103C2×C22⋊Dic5C2×D311yescharacteristic
104Q8×Dic5—64no—
105#4072—24no—
106C2×C4⋊Dic5—31no—
107C2×C4⋊Dic5—31no—
108C2×C4⋊Dic5—32no—
109C2×C4⋊Dic5—32no—
110C2×C4⋊Dic5C2×D311yescharacteristic
111C2×C4⋊Dic5C2×D311yescharacteristic
112C2×C4×Dic5—32no—
113C2×C4×Dic5—32no—
114C2×C4×Dic5C2×D311yescharacteristic
115C2×C4×Dic5C2×D311yescharacteristic
116#3963—32no—
117#3963—32no—
118#3963—32no—
119#3963—32no—
120#3963—32no—
121#2323—151noSylow
122C23×C3×C5—21no—
123C23×C3×C5—21no—
124C23×C3×C5—21no—
125C23×C3×C5—21no—
126C23×C3×C5—21no—
127C23×C3×C5—21no—
128C23×C3×C5C22×C412yes—
129C23×C3×C5C2411yescharacteristic
130C2×C3×C4×C5—21no—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5—28no—
133C2×C3×C4×C5C22×C412yes—
134C2×C3×C4×C5C22×C412yes—
135C2×C3×C4×C5C2×D412yes—
136C2×C3×C4×C5#4112yes—
137C2×Dic15—22no—
138C2×Dic15—22no—
139C2×Dic15—22no—
140C2×Dic15—22no—
141C2×Dic15—22no—
142C2×Dic15—22no—
143C2×C5×Dic3—22no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3C22×C414yes—
148C2×C5×Dic3C22×C414yes—
149C5×Dic6—28no—
150C2×C3×Dic5—21no—
151C2×C3×Dic5—21no—
152C2×C3×Dic5—21no—
153C2×C3×Dic5—21no—
154C2×C3×Dic5—21no—
155C2×C3×Dic5—21no—
156C2×C3×Dic5—21no—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—21no—
160C2×C3×Dic5C2×D412yes—
161C2×C3×Dic5C2×D412yes—
162C23×C3×C4C2×D511yescharacteristic
163C22×Dic6C2×D511yescharacteristic
164C3×#117—104no—
165C2×C3×C4⋊C4—51no—
166C2×C3×C4⋊C4—51no—
167C2×C3×C22⋊C4—51no—
168C2×C3×C22⋊C4—51no—
169C2×C3×C42—51no—
170C2×C3×C42—51no—
171C2×#217—52no—
172C2×#217C2×D512yes—
173C2×C4⋊Dic3—51no—
174C2×C4⋊Dic3—51no—
175C2×C4⋊Dic3C2×D511yescharacteristic
176#924Dic518yescomplemented
177C2×Dic3⋊C4—52no—
178C2×Dic3⋊C4—52no—
179C2×Dic3⋊C4C2×D512yes—
180C2×C4×Dic3—52no—
181C2×C4×Dic3—52no—
182C4×Dic6—104no—
183C3⋊#95—52no—
184C3⋊#95—52no—
185C3⋊#95—52no—
186C3⋊#95—52no—
187C3⋊#95—52no—
188C24×C5C22×D311yescharacteristic
189C2×C5×Q8—34no—
190C2×C5×Q8—62no—
191C2×C5×Q8—62no—
192C22×C4×C5—22no—
193C22×C4×C5—22no—
194C22×C4×C5—32no—
195C22×C4×C5—32no—
196C22×C4×C5C4×D314yes—
197C22×C4×C5C22×D311yescharacteristic
198C22×C4×C5C22×D311yescharacteristic
199C22×Dic5—32no—
200C22×Dic5—32no—
201C22×Dic5C22×D311yescharacteristic
202C22×Dic5C22×D311yescharacteristic
203C22×Dic5C22×D311yescharacteristic
204C22×Dic5C22×D311yescharacteristic
205C5×C4⋊C4—38no—
206C5×C4⋊C4—34no—
207C5×C22⋊C4—38no—
208C22⋊Dic5—22no—
209C22⋊Dic5—22no—
210C22⋊Dic5—122no—
211C4⋊Dic5—22no—
212C4⋊Dic5—22no—
213C4⋊Dic5—62no—
214C4⋊Dic5—62no—
215C4⋊Dic5—64no—
216C4⋊Dic5—122no—
217C4×Dic5—22no—
218C4×Dic5—64no—
219C4×Dic5—122no—
220C4×Dic5#6714yes—
221C2×C22⋊Q8—31no—
222C2×C4×Q8—151no—
223C2×#117—51no—
224#391—151no—
225#389—151no—
226#389—152no—
227#388—152no—
228#388—152no—
229#387—152no—
230#380—151no—
231C4×C4⋊C4—151no—
232C22×C3×C5—22no—
233C22×C3×C5—22no—
234C22×C3×C5—22no—
235C22×C3×C5—41no—
236C22×C3×C5—41no—
237C22×C3×C5—41no—
238C22×C3×C5—41no—
239C22×C3×C5C23×C411yescharacteristic
240C22×C3×C5C22×D411yescharacteristic
241C22×C3×C5C2×#4111yescharacteristic
242C22×C3×C5H(5,𝔽2)11yescharacteristic
243C22×C3×C52−1+411yescharacteristic
244C22×C3×C52−1+411yescharacteristic
245C22×C3×C52−1+411yescharacteristic
246C3×C4×C5—22no—
247C3×C4×C5C4×D414yes—
248Dic15—42no—
249Dic15—42no—
250C3×Dic5—22no—
251C3×Dic5—22no—
252C3×Dic5—41no—
253C3×Dic5—41no—
254C5×Dic3—24no—
255C5×Dic3—24no—
256C24×C3C22×D511yescharacteristic
257C22×C3×C4—22no—
258C22×C3×C4—22no—
259C22×C3×C4—51no—
260C22×C3×C4—51no—
261C22×C3×C4—51no—
262C22×C3×C4—51no—
263C22×C3×C4C2×Dic514yes—
264C22×C3×C4C22×D511yescharacteristic
265C22×C3×C4C22×D511yescharacteristic
266C22×Dic3—52no—
267C22×Dic3—52no—
268C22×Dic3C22×D512yes—
269C22×Dic3C22×D512yes—
270C2×Dic6—22no—
271C2×Dic6—22no—
272C2×Dic6C2×Dic514yes—
273C3×C4⋊C4—102no—
274C3×C4⋊C4—102no—
275C3×C22⋊C4—102no—
276C3×C22⋊C4—102no—
277C3×C42—54no—
278C3×C42—102no—
279#217—202no—
280#217C2×Dic518yes—
281C4⋊Dic3—102no—
282C4⋊Dic3—102no—
283C4⋊Dic3C2×Dic514yes—
284Dic3⋊C4—104no—
285Dic3⋊C4—202no—
286Dic3⋊C4C2×Dic518yes—
287C4×Dic3—104no—
288C4×Dic3—202no—
289C23×C5—21no—
290C23×C5—21no—
291C23×C5—21no—
292C23×C5—21no—
293C23×C5—21no—
294C23×C5—21no—
295C23×C5C2×C4×D312yes—
296C23×C5C23×D311yescharacteristic
297C5×Q8—68no—
298C2×C4×C5—21no—
299C2×C4×C5—21no—
300C2×C4×C5—28no—
301C2×C4×C5—34no—
302C2×C4×C5—34no—
303C2×C4×C5—62no—
304C2×C4×C5—62no—
305C2×C4×C5—62no—
306C2×C4×C5—62no—
307C2×C4×C5C2×C4×D312yes—
308C2×C4×C5C2×C4×D312yes—
309C2×C4×C5#23512yes—
310C2×C4×C5C2×#6712yes—
311C2×Dic5—21no—
312C2×Dic5—21no—
313C2×Dic5—21no—
314C2×Dic5—21no—
315C2×Dic5—21no—
316C2×Dic5—21no—
317C2×Dic5—21no—
318C2×Dic5—21no—
319C2×Dic5—21no—
320C2×Dic5—21no—
321C2×Dic5—62no—
322C2×Dic5—62no—
323C2×Dic5—62no—
324C2×Dic5—62no—
325C2×Dic5—62no—
326C2×Dic5—62no—
327C2×Dic5C2×#6712yes—
328C2×Dic5C2×#6712yes—
329C22×Q8—31no—
330C23×C4D3×D511yescharacteristic
331C22⋊Q8—38no—
332C4×Q8—304no—
333#117—104no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—32no—
336C2×C4⋊C4—51no—
337C2×C4⋊C4—51no—
338C2×C4⋊C4—152no—
339C2×C4⋊C4—151no—
340C2×C4⋊C4—151no—
341C2×C4⋊C4—152no—
342C2×C22⋊C4—32no—
343C2×C22⋊C4—51no—
344C2×C22⋊C4—51no—
345C2×C22⋊C4—152no—
346C2×C42—51no—
347C2×C42—51no—
348C2×C42—152no—
349C2×C42—152no—
350#95—152no—
351#95—152no—
352#95—152no—
353#95—152no—
354#95—152no—
355C2×C3×C5—42no—
356C2×C3×C5C2×C4×D411yescharacteristic
357C2×C3×C5#51911yescharacteristic
358C2×C3×C5#52011yescharacteristic
359C2×C3×C5#53611yescharacteristic
360C2×C3×C5#53711yescharacteristic
361C2×C3×C5#54111yescharacteristic
362C2×C3×C5#54211yescharacteristic
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3C22×Dic512yes—
370C23×C3C23×D511yescharacteristic
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—28no—
374C2×C3×C4—52no—
375C2×C3×C4—52no—
376C2×C3×C4—101no—
377C2×C3×C4—101no—
378C2×C3×C4—101no—
379C2×C3×C4—101no—
380C2×C3×C4—101no—
381C2×C3×C4—101no—
382C2×C3×C4—101no—
383C2×C3×C4—101no—
384C2×C3×C4—101no—
385C2×C3×C4—101no—
386C2×C3×C4D4×D512yes—
387C2×C3×C4#71112yes—
388C2×C3×C4C22×Dic512yes—
389C2×C3×C4C22×Dic512yes—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—102no—
395C2×Dic3—102no—
396C2×Dic3—102no—
397C2×Dic3—102no—
398C2×Dic3—102no—
399C2×Dic3—102no—
400C2×Dic3C22×Dic514yes—
401C2×Dic3C22×Dic514yes—
402Dic6—28no—
403C22×C5—22no—
404C22×C5—22no—
405C22×C5—22no—
406C22×C5—41no—
407C22×C5—41no—
408C22×C5—41no—
409C22×C5—41no—
410C22×C5C22×C4×D311yescharacteristic
411C22×C5C2×#23511yescharacteristic
412C22×C5#100911yescharacteristic
413C22×C5#101011yescharacteristic
414C22×C5#101011yescharacteristic
415C22×C5#101011yescharacteristic
416C22×C5C22×#6711yescharacteristic
417C4×C5—22no—
418C4×C5—64no—
419C4×C5—64no—
420C4×C5C4×#6714yes—
421Dic5—22no—
422Dic5—22no—
423Dic5—41no—
424Dic5—41no—
425Dic5—122no—
426Dic5—122no—
427C24C2×D3×D511yescharacteristic
428C2×Q8—34no—
429C2×Q8—62no—
430C2×Q8—62no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—32no—
434C22×C4—32no—
435C22×C4—51no—
436C22×C4—51no—
437C22×C4—51no—
438C22×C4—51no—
439C22×C4—152no—
440C22×C4—152no—
441C22×C4D3×Dic514yes—
442C22×C4C2×D3×D511yescharacteristic
443C22×C4C2×D3×D511yescharacteristic
444C4⋊C4—38no—
445C4⋊C4—34no—
446C4⋊C4—102no—
447C4⋊C4—102no—
448C4⋊C4—304no—
449C4⋊C4—302no—
450C4⋊C4—302no—
451C4⋊C4—602no—
452C22⋊C4—38no—
453C22⋊C4—102no—
454C22⋊C4—102no—
455C22⋊C4—602no—
456C42—54no—
457C42—102no—
458C42—304no—
459C42—602no—
460C3×C5#232311yescharacteristic, complemented, Hall
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—41no—
465C22×C3—41no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3C2×D4×D511yescharacteristic
469C22×C3C2×#71111yescharacteristic
470C22×C3#414711yescharacteristic
471C22×C3#414711yescharacteristic
472C22×C3#414911yescharacteristic
473C22×C3#415011yescharacteristic
474C22×C3C23×Dic511yescharacteristic
475C3×C4—22no—
476C3×C4—102no—
477C3×C4—102no—
478C3×C4—201no—
479C3×C4—201no—
480C3×C4D4×Dic514yes—
481Dic3—24no—
482Dic3—24no—
483Dic3—202no—
484Dic3—202no—
485C2×C5—42no—
486C2×C5#551611yescharacteristic
487C2×C5#551811yescharacteristic
488C2×C5#552311yescharacteristic
489C2×C5#553411yescharacteristic
490C2×C5C2×C4×#6711yescharacteristic
491C2×C5#582511yescharacteristic
492C2×C5#583311yescharacteristic
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23C2×D3×Dic512yes—
500C23C22×D3×D511yescharacteristic
501Q8—68no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—28no—
505C2×C4—34no—
506C2×C4—34no—
507C2×C4—52no—
508C2×C4—52no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—101no—
514C2×C4—101no—
515C2×C4—101no—
516C2×C4—101no—
517C2×C4—101no—
518C2×C4—101no—
519C2×C4—101no—
520C2×C4—101no—
521C2×C4—101no—
522C2×C4—101no—
523C2×C4—302no—
524C2×C4—302no—
525C2×C4—302no—
526C2×C4—302no—
527C2×C4—302no—
528C2×C4—302no—
529C2×C4C2×D3×Dic512yes—
530C2×C4C2×D3×Dic512yes—
531C2×C4#678212yes—
532C2×C4D5×#6712yes—
533C2×C3—42no—
534C2×C3#6595911yescharacteristic
535C2×C3#6596211yescharacteristic
536C2×C3#6596311yescharacteristic
537C2×C3#6597711yescharacteristic
538C2×C3C2×D4×Dic511yescharacteristic
539C2×C3#6614611yescharacteristic
540C2×C3#6616111yescharacteristic
541C5#8223511yescharacteristic, complemented, Sylow
542C22—22no—
543C22—22no—
544C22—22no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22#9200011yescharacteristic
550C22#9200011yescharacteristic
551C22#9200111yescharacteristic
552C22#9201911yescharacteristic
553C22C22×D3×Dic511yescharacteristic
554C22C2×#678211yescharacteristic
555C22C2×D5×#6711yescharacteristic
556C4—22no—
557C4—64no—
558C4—64no—
559C4—102no—
560C4—102no—
561C4—201no—
562C4—201no—
563C4—602no—
564C4—602no—
565C4Dic5×#6714yes—
566C3#1061447511yescharacteristic, complemented, Sylow
567C2—42no—
568C2#1175418411yescharacteristic
569C2#1175421111yescharacteristic
570C2#1175421211yescharacteristic
571C2#1175430311yescharacteristic
572C2#1175431411yescharacteristic
573C2#1175451311yescharacteristic
574C2C2×Dic5×#6711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC24×C3×C5C23
hypercenterC23×C4D3×D5
hypocenterC3×C5#2323
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
3. C2×Dic5×Dic6C2semidirect
7. #11748842C2semidirect
8. Dic5×C4⋊Dic3C2semidirect
9. #11748677C2semidirect
10. #11748673C2semidirect
11. #11748589C2semidirect
20. C5×#924C4semidirect
176. #924Dic5semidirect
460. C3×C5#2323semidirect
541. C5#82235semidirect
566. C3#10614475semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9242
C2×C3×#40722
C2×Dic5×Dic62
#117513132
#117494792
#117494782
#117488422
Dic5×C4⋊Dic32
#117486772
#117486732
#117485892
#106144753
#822355