Finite Group Databasebeta

← #14323637

Subgroups — #14323637

Order 1920.

1264 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#886C211yesmaximal, characteristic
2C2×Dic5×#67C211yesmaximal, characteristic
3C22×Dic3⋊Dic5C211yesmaximal, characteristic, complemented
4#11751051C211yesmaximal, characteristic, complemented
5C3×C22⋊C4×Dic5C211yesmaximal, characteristic, complemented
6#11749501C211yesmaximal, characteristic
7#11749472C211yesmaximal, characteristic, complemented
8#11748903C212yesmaximal, complemented
9#11748842C212yesmaximal, complemented
10#11748747C212yesmaximal, complemented
11#11748682C212yesmaximal, complemented
12#10614567—31nomaximal, Hall
13C23×Dic15C2211yescharacteristic
14C22×C5×#67C2211yescharacteristic
15C23×C5×Dic3C2211yescharacteristic
16C23×C3×Dic5C2211yescharacteristic
17C22×D3×Dic5C2211yescharacteristic
18C2×C3×C5×C22⋊C4C2211yescharacteristic
19C2×#6719C2211yescharacteristic
20C2×C4⋊Dic15C2212yes—
21C2×C5×(C2×D3)⋊C4C2212yes—
22C2×C5×C4⋊Dic3C2212yes—
23C5×#886C418yescomplemented
24C2×C3×C22⋊Dic5C2211yescharacteristic
25C2×C3×C4×Dic5C2212yes—
26Dic5×#67—24no—
27C2×Dic3⋊Dic5—24no—
28C2×Dic3⋊Dic5C2212yes—
29C2×Dic3⋊Dic5C2212yes—
30C2×Dic3⋊Dic5C2211yescharacteristic
31C2×(C2×D3)⋊Dic5C2211yescharacteristic
32C2×(C2×D3)⋊Dic5C2212yes—
33C2×Dic3×Dic5C2211yescharacteristic
34#91115C2212yes—
35C3×#3963C2212yes—
36#90994C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#73446—51nomaximal, Hall
40C2×C5×#123—31no—
41C2×D4×Dic5—31no—
42C22×C4⋊Dic5—31no—
43#65509—31no—
44#65500—31no—
45#65274—32no—
46#65270—32no—
47#65240—32no—
48#65239—32no—
49#65239—31no—
50C22⋊C4×Dic5D311yescharacteristic, complemented
51C24×C3×C5C2311yescharacteristic
52C23×C5×D3C2311yescharacteristic
53C22×C3×C4×C5C2312yes—
54C22×Dic15—22no—
55C22×Dic15—42no—
56C22×Dic15C2311yescharacteristic
57C22×Dic15C2311yescharacteristic
58C22×Dic15C2312yes—
59C2×C5×#67—22no—
60C2×C5×#67—22no—
61C2×C5×#67C2×C414yes—
62C22×C5×Dic3—22no—
63C22×C5×Dic3—22no—
64C22×C5×Dic3C2×C414yescomplemented
65C22×C5×Dic3C2311yescharacteristic
66C22×C5×Dic3C2312yes—
67C22×C3×Dic5—42no—
68C22×C3×Dic5D414yes—
69C22×C3×Dic5C2312yes—
70C22×C3×Dic5C2311yescharacteristic
71C22×C3×Dic5C2311yescharacteristic
72C2×D3×Dic5—22no—
73C2×D3×Dic5—42no—
74C3×C5×C22⋊C4C2×C414yescomplemented
75#6719—22no—
76C4⋊Dic15—42no—
77C5×(C2×D3)⋊C4C2×C418yescomplemented
78C5×C4⋊Dic3C2×C418yescomplemented
79C3×C22⋊Dic5—22no—
80C3×C4×Dic5—42no—
81Dic3⋊Dic5—24no—
82Dic3⋊Dic5—24no—
83Dic3⋊Dic5—22no—
84(C2×D3)⋊Dic5—22no—
85(C2×D3)⋊Dic5—42no—
86Dic3×Dic5—22no—
87C2×C4×#67—51no—
88C22×Dic3⋊C4—51no—
89C2×#886D511yescharacteristic
90C3×C4×C22⋊C4—51no—
91#4958—51no—
92#4950—51no—
93#4943—51no—
94#4940—52no—
95#4928—52no—
96#4668—52no—
97#4653—52no—
98C22×C5×D4—31no—
99C23×C4×C5—31no—
100C23×Dic5—31no—
101C23×Dic5—31no—
102C23×Dic5C2×D311yescharacteristic
103C5×#123—38no—
104C2×C5×C4⋊C4—32no—
105C2×C5×C22⋊C4—32no—
106C2×C5×C22⋊C4C2×D311yescharacteristic
107C2×C22⋊Dic5—31no—
108C2×C22⋊Dic5—31no—
109C2×C22⋊Dic5—32no—
110C2×C22⋊Dic5C2×D311yescharacteristic
111D4×Dic5—64no—
112C2×C4⋊Dic5—32no—
113C2×C4⋊Dic5—32no—
114C2×C4⋊Dic5—31no—
115C2×C4⋊Dic5—32no—
116C2×C4⋊Dic5—64no—
117C2×C4×Dic5—31no—
118C2×C4×Dic5C2×D312yes—
119#3963—32no—
120#3963—32no—
121#3963—32no—
122#3963—32no—
123#3963C2×D312yes—
124#2322—151noSylow
125C23×C3×C5—22no—
126C23×C3×C5—22no—
127C23×C3×C5C22×C412yes—
128C23×C3×C5C2×D412yes—
129C23×C3×C5#4112yes—
130C23×C3×C5C2411yescharacteristic
131C22×C5×D3—21no—
132C22×C5×D3—21no—
133C22×C5×D3—22no—
134C22×C5×D3—22no—
135C22×C5×D3C22×C412yes—
136C2×C3×C4×C5—22no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5C22×C414yes—
139C2×Dic15—21no—
140C2×Dic15—22no—
141C2×Dic15—21no—
142C2×Dic15—22no—
143C2×Dic15—22no—
144C2×Dic15—22no—
145C2×Dic15—22no—
146C2×Dic15—44no—
147C5×#67—28no—
148C2×C5×Dic3—28no—
149C2×C5×Dic3—21no—
150C2×C5×Dic3—22no—
151C2×C5×Dic3—22no—
152C2×C5×Dic3—21no—
153C2×C5×Dic3C22×C412yes—
154C2×C5×Dic3C22×C414yes—
155C2×C3×Dic5—22no—
156C2×C3×Dic5—22no—
157C2×C3×Dic5—22no—
158C2×C3×Dic5—22no—
159C2×C3×Dic5—22no—
160C2×C3×Dic5—44no—
161C2×C3×Dic5C2×D412yes—
162C2×C3×Dic5C2×D412yes—
163D3×Dic5—44no—
164C23×C3×C4—51no—
165C22×#67C2×D511yescharacteristic
166C23×Dic3—51no—
167C23×Dic3C2×D511yescharacteristic
168C22×C4×D3—51no—
169C2×C3×C22⋊C4—51no—
170C2×C3×C22⋊C4C2×D511yescharacteristic
171C2×C3×C42—52no—
172C2×#217—51no—
173C4×#67—104no—
174C2×(C2×D3)⋊C4—51no—
175C2×(C2×D3)⋊C4—52no—
176C2×(C2×D3)⋊C4C2×D512yes—
177C2×C4⋊Dic3—52no—
178C2×C4⋊Dic3C2×D512yes—
179C2×Dic3⋊C4—52no—
180C2×Dic3⋊C4—51no—
181C2×Dic3⋊C4—52no—
182C2×Dic3⋊C4—104no—
183C2×C4×Dic3—51no—
184#886Dic518yescomplemented
185C3×#95—52no—
186C3⋊#95—52no—
187C3⋊#95—52no—
188C3⋊#95—52no—
189C3⋊#95—52no—
190C24×C5—31no—
191C24×C5C22×D311yescharacteristic
192C2×C5×D4—34no—
193C2×C5×D4—62no—
194C2×C5×D4—62no—
195C22×C4×C5—34no—
196C22×C4×C5—32no—
197C22×C4×C5—31no—
198C22×C4×C5—62no—
199C22×C4×C5—62no—
200C22×C4×C5C22×D312yes—
201C22×Dic5—31no—
202C22×Dic5—32no—
203C22×Dic5—31no—
204C22×Dic5—42no—
205C22×Dic5—62no—
206C22×Dic5—62no—
207C22×Dic5—122no—
208C22×Dic5—122no—
209C22×Dic5D1214yes—
210C22×Dic5C22×D312yes—
211C22×Dic5C22×D311yescharacteristic
212C22×Dic5C22×D311yescharacteristic
213C5×C4⋊C4—38no—
214C5×C22⋊C4—38no—
215C5×C22⋊C4C4×D314yescomplemented
216C22⋊Dic5—22no—
217C22⋊Dic5—62no—
218C22⋊Dic5—62no—
219C22⋊Dic5—122no—
220C4⋊Dic5—64no—
221C4⋊Dic5—64no—
222C4⋊Dic5—62no—
223C4⋊Dic5—122no—
224C4×Dic5—42no—
225C4×Dic5—62no—
226C2×#123—31no—
227C2×C4×D4—151no—
228C22×C4⋊C4—151no—
229#389—152no—
230#388—152no—
231#387—152no—
232#386—151no—
233#385—152no—
234#385—151no—
235#381—151no—
236C4×C22⋊C4—51no—
237C22×C3×C5—24no—
238C22×C3×C5—22no—
239C22×C3×C5—22no—
240C22×C3×C5—24no—
241C22×C3×C5C4×D414yes—
242C22×C3×C5C23×C411yescharacteristic
243C22×C3×C5C22×D411yescharacteristic
244C22×C3×C5C2×#4111yescharacteristic
245C22×C3×C5H(5,𝔽2)12yes—
246C22×C3×C52−1+412yes—
247C2×C5×D3—22no—
248C2×C5×D3—24no—
249C2×C5×D3—41no—
250C2×C5×D3—41no—
251C2×C5×D3—42no—
252C3×C4×C5—24no—
253Dic15—41no—
254Dic15—42no—
255Dic15—41no—
256C3×Dic5—22no—
257C3×Dic5—22no—
258C3×Dic5—42no—
259C5×Dic3—24no—
260C5×Dic3—22no—
261C24×C3C22×D511yescharacteristic
262C23×D3C22×D511yescharacteristic
263C22×C3×C4—51no—
264C22×C3×C4—51no—
265C22×C3×C4—54no—
266C22×C3×C4—52no—
267C22×C3×C4—202no—
268C22×C3×C4C22×D512yes—
269C2×#67—22no—
270C2×#67—22no—
271C2×#67C2×Dic514yes—
272C22×Dic3—22no—
273C22×Dic3—22no—
274C22×Dic3—51no—
275C22×Dic3—52no—
276C22×Dic3—51no—
277C22×Dic3—102no—
278C22×Dic3—202no—
279C22×Dic3C2×Dic514yescomplemented
280C22×Dic3C22×D511yescharacteristic
281C22×Dic3C22×D512yes—
282C2×C4×D3—102no—
283C2×C4×D3—202no—
284C3×C22⋊C4—102no—
285C3×C22⋊C4C2×Dic514yescomplemented
286C3×C42—202no—
287#217—102no—
288(C2×D3)⋊C4—102no—
289(C2×D3)⋊C4—202no—
290(C2×D3)⋊C4C2×Dic518yescomplemented
291C4⋊Dic3—202no—
292C4⋊Dic3C2×Dic518yescomplemented
293Dic3⋊C4—104no—
294Dic3⋊C4—104no—
295Dic3⋊C4—102no—
296C4×Dic3—102no—
297C23×C5—22no—
298C23×C5—22no—
299C23×C5—32no—
300C23×C5—61no—
301C23×C5—61no—
302C23×C5—62no—
303C23×C5—62no—
304C23×C5C2×C4×D312yes—
305C23×C5C2×D1212yes—
306C23×C5#23512yes—
307C23×C5C23×D311yescharacteristic
308C5×D4—68no—
309C2×C4×C5—22no—
310C2×C4×C5—22no—
311C2×C4×C5—34no—
312C2×C4×C5—32no—
313C2×C4×C5—68no—
314C2×C4×C5—61no—
315C2×C4×C5—62no—
316C2×C4×C5—62no—
317C2×C4×C5—61no—
318C2×C4×C5C2×C4×D314yes—
319C2×Dic5—22no—
320C2×Dic5—22no—
321C2×Dic5—22no—
322C2×Dic5—22no—
323C2×Dic5—22no—
324C2×Dic5—44no—
325C2×Dic5—61no—
326C2×Dic5—62no—
327C2×Dic5—61no—
328C2×Dic5—62no—
329C2×Dic5—62no—
330C2×Dic5—62no—
331C2×Dic5—62no—
332C2×Dic5—124no—
333C2×Dic5—124no—
334C2×Dic5C2×D1212yes—
335C2×Dic5C2×D1212yes—
336C22×D4—31no—
337C23×C4—31no—
338C23×C4—51no—
339C23×C4—151no—
340C23×C4—151no—
341#123—38no—
342C4×D4—304no—
343C2×C4⋊C4—32no—
344C2×C4⋊C4—152no—
345C2×C4⋊C4—152no—
346C2×C4⋊C4—151no—
347C2×C4⋊C4—152no—
348C2×C4⋊C4—304no—
349C2×C22⋊C4—32no—
350C2×C22⋊C4—51no—
351C2×C22⋊C4—151no—
352C2×C22⋊C4—151no—
353C2×C22⋊C4—152no—
354C2×C22⋊C4D3×D511yescharacteristic
355C2×C42—52no—
356C2×C42—151no—
357#95—52no—
358#95—152no—
359#95—152no—
360#95—152no—
361#95—152no—
362C2×C3×C5—24no—
363C2×C3×C5C2×C4×D411yescharacteristic
364C2×C3×C5#52012yes—
365C2×C3×C5#53411yescharacteristic
366C2×C3×C5#53611yescharacteristic
367C2×C3×C5#53912yes—
368C5×D3—42no—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3D4×D512yes—
372C23×C3#71112yes—
373C23×C3C22×Dic512yes—
374C23×C3C23×D511yescharacteristic
375C22×D3—21no—
376C22×D3—21no—
377C22×D3—22no—
378C22×D3—22no—
379C22×D3C22×Dic512yes—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—52no—
383C2×C3×C4—52no—
384C2×C3×C4—102no—
385C2×C3×C4—102no—
386C2×C3×C4—102no—
387C2×C3×C4—102no—
388C2×C3×C4—102no—
389C2×C3×C4—204no—
390C2×C3×C4C22×Dic514yes—
391#67—28no—
392C2×Dic3—21no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—21no—
396C2×Dic3—28no—
397C2×Dic3—101no—
398C2×Dic3—102no—
399C2×Dic3—101no—
400C2×Dic3—102no—
401C2×Dic3—102no—
402C2×Dic3—102no—
403C2×Dic3—102no—
404C2×Dic3—204no—
405C2×Dic3C22×Dic512yes—
406C2×Dic3C22×Dic514yes—
407C4×D3—204no—
408C22×C5—22no—
409C22×C5—22no—
410C22×C5—24no—
411C22×C5—24no—
412C22×C5—62no—
413C22×C5—64no—
414C22×C5—121no—
415C22×C5—121no—
416C22×C5—122no—
417C22×C5C4×D1214yes—
418C22×C5C22×C4×D311yescharacteristic
419C22×C5C22×D1211yescharacteristic
420C22×C5C2×#23511yescharacteristic
421C22×C5#100412yes—
422C22×C5#101012yes—
423C4×C5—24no—
424C4×C5—64no—
425C4×C5—62no—
426Dic5—22no—
427Dic5—22no—
428Dic5—42no—
429Dic5—121no—
430Dic5—122no—
431Dic5—121no—
432C24—31no—
433C24C2×D3×D511yescharacteristic
434C2×D4—34no—
435C2×D4—62no—
436C2×D4—62no—
437C22×C4—34no—
438C22×C4—31no—
439C22×C4—32no—
440C22×C4—51no—
441C22×C4—51no—
442C22×C4—54no—
443C22×C4—52no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—151no—
447C22×C4—151no—
448C22×C4—152no—
449C22×C4—202no—
450C22×C4—302no—
451C22×C4—302no—
452C22×C4—602no—
453C22×C4—602no—
454C22×C4C2×D3×D512yes—
455C4⋊C4—38no—
456C4⋊C4—304no—
457C4⋊C4—304no—
458C4⋊C4—302no—
459C4⋊C4—602no—
460C22⋊C4—38no—
461C22⋊C4—102no—
462C22⋊C4—302no—
463C22⋊C4—302no—
464C22⋊C4—602no—
465C22⋊C4D3×Dic514yescomplemented
466C42—202no—
467C42—302no—
468C3×C5#232211yescharacteristic, complemented, Hall
469C22×C3—24no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—24no—
473C22×C3D4×Dic514yes—
474C22×C3C2×D4×D511yescharacteristic
475C22×C3C2×#71111yescharacteristic
476C22×C3#414912yes—
477C22×C3#415012yes—
478C22×C3C23×Dic511yescharacteristic
479C2×D3—22no—
480C2×D3—24no—
481C2×D3—41no—
482C2×D3—41no—
483C2×D3—42no—
484C3×C4—24no—
485C3×C4—102no—
486C3×C4—102no—
487C3×C4—202no—
488Dic3—24no—
489Dic3—22no—
490Dic3—201no—
491Dic3—201no—
492Dic3—202no—
493C2×C5—24no—
494C2×C5—122no—
495C2×C5C2×C4×D1211yescharacteristic
496C2×C5#549311yescharacteristic
497C2×C5#552611yescharacteristic
498C2×C5#554612yes—
499C2×C5#556412yes—
500C23—22no—
501C23—22no—
502C23—32no—
503C23—61no—
504C23—61no—
505C23—62no—
506C23—62no—
507C23#677212yes—
508C23D5×D1212yes—
509C23C2×D3×Dic512yes—
510C23C22×D3×D511yescharacteristic
511D4—68no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—34no—
515C2×C4—32no—
516C2×C4—52no—
517C2×C4—52no—
518C2×C4—68no—
519C2×C4—62no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—62no—
523C2×C4—102no—
524C2×C4—102no—
525C2×C4—102no—
526C2×C4—102no—
527C2×C4—102no—
528C2×C4—204no—
529C2×C4—301no—
530C2×C4—301no—
531C2×C4—302no—
532C2×C4—302no—
533C2×C4—302no—
534C2×C4—302no—
535C2×C4—302no—
536C2×C4—604no—
537C2×C4—604no—
538C2×C4C2×D3×Dic514yes—
539C2×C3—24no—
540C2×C3#6598211yescharacteristic
541C2×C3#6598711yescharacteristic
542C2×C3#6599912yes—
543C2×C3C2×D4×Dic511yescharacteristic
544C2×C3#6616112yes—
545D3—42no—
546C5#7344611yescharacteristic, complemented, Sylow
547C22—22no—
548C22—24no—
549C22—22no—
550C22—24no—
551C22—62no—
552C22—64no—
553C22—121no—
554C22—121no—
555C22—122no—
556C22Dic5×D1214yes—
557C22C2×#677211yescharacteristic
558C22C2×D5×D1211yescharacteristic
559C22#9202012yes—
560C22#9202612yes—
561C22C22×D3×Dic511yescharacteristic
562C4—24no—
563C4—64no—
564C4—62no—
565C4—102no—
566C4—102no—
567C4—202no—
568C4—601no—
569C4—602no—
570C4—601no—
571C3#1061456711yescharacteristic, complemented, Sylow
572C2—24no—
573C2—122no—
574C2#1175287411yescharacteristic
575C2#1175317812yes—
576C2#1175326411yescharacteristic
577C2C2×Dic5×D1211yescharacteristic
578C2#1175492012yes—
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C22⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×C5×#67C22
hypercenterC2×C22⋊C4D3×D5
hypocenterC3×C5#2322
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C22×Dic3⋊Dic5C2semidirect
4. #11751051C2semidirect
5. C3×C22⋊C4×Dic5C2semidirect
7. #11749472C2semidirect
8. #11748903C2semidirect
9. #11748842C2semidirect
10. #11748747C2semidirect
11. #11748682C2semidirect
23. C5×#886C4semidirect
50. C22⋊C4×Dic5D3semidirect
64. C22×C5×Dic3C2×C4semidirect
74. C3×C5×C22⋊C4C2×C4semidirect
77. C5×(C2×D3)⋊C4C2×C4semidirect
78. C5×C4⋊Dic3C2×C4semidirect
184. #886Dic5semidirect
215. C5×C22⋊C4C4×D3semidirect
279. C22×Dic3C2×Dic5semidirect
285. C3×C22⋊C4C2×Dic5semidirect
290. (C2×D3)⋊C4C2×Dic5semidirect
292. C4⋊Dic3C2×Dic5semidirect
465. C22⋊C4D3×Dic5semidirect
468. C3×C5#2322semidirect
546. C5#73446semidirect
571. C3#10614567semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#8862
C2×Dic5×#672
C22×Dic3⋊Dic52
#117510512
C3×C22⋊C4×Dic52
#117495012
#117494722
#117489032
#117488422
#117487472
#117486822
#106145673
#734465