Finite Group Databasebeta

← #14322859

Subgroups — #14322859

Order 1920.

1200 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×D3×C4⋊C4C211yesmaximal, characteristic
2C2×#91371C211yesmaximal, characteristic
3C2×#91338C211yesmaximal, characteristic
4#11751085C211yesmaximal, characteristic, complemented
5C3×#65274C211yesmaximal, characteristic, complemented
6#11748868C211yesmaximal, characteristic, complemented
7#11748842C211yesmaximal, characteristic, complemented
8#11748786C212yesmaximal
9#11748782C212yesmaximal
10#11748693C212yesmaximal, complemented
11#11748677C212yesmaximal, complemented
12#10613887—31nomaximal, Hall
13C22×C4×C5×D3C2212yes—
14C22×C4×C5×D3C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C4⋊Dic15C2211yescharacteristic
17C2×C4⋊Dic15C2212yes—
18C2×C4×Dic15C2211yescharacteristic
19C2×C5×C4⋊Dic3C2211yescharacteristic
20C2×C5×Dic3⋊C4C2212yes—
21C5×D3×C4⋊C4C418yescomplemented
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C2×C3×C4⋊Dic5C2212yes—
24C2×C3×C4×Dic5C2211yescharacteristic
25C2×Dic3⋊Dic5C2211yescharacteristic
26C2×Dic3⋊Dic5C2211yescharacteristic
27C2×Dic3⋊Dic5C2212yes—
28C2×Dic3⋊Dic5C2212yes—
29C2×(C2×D3)⋊Dic5C2211yescharacteristic
30C2×(C2×D3)⋊Dic5C2211yescharacteristic
31C2×(C2×D3)⋊Dic5C2212yes—
32C2×Dic3×Dic5C2212yes—
33#91371—24no—
34#91338—24no—
35#91115C2212yes—
36C3×#3963C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#76195—51nomaximal, Hall
40C22×C5×C4⋊C4—31no—
41C2×#4072—31no—
42C2×#4072—31no—
43#65501—32no—
44#65501—32no—
45#65274—31no—
46#65274—32no—
47#65274—31no—
48#65274—32no—
49#65274—31no—
50#65274D311yescharacteristic, complemented
51C23×C5×D3C2311yescharacteristic
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×Dic15C2311yescharacteristic
55C22×Dic15C2312yes—
56C22×Dic15C2311yescharacteristic
57C22×C5×Dic3C2312yes—
58C22×C5×Dic3C2311yescharacteristic
59C2×C4×C5×D3—42no—
60C2×C4×C5×D3—42no—
61C2×C4×C5×D3C2×C418yes—
62C2×C4×C5×D3C2×C414yes—
63C2×C4×C5×D3D414yes—
64C2×C4×C5×D3Q814yes—
65C22×C3×Dic5C2311yescharacteristic
66C22×C3×Dic5C2312yes—
67C22×C3×Dic5C2311yescharacteristic
68C3×C5×C4⋊C4C2×C414yes—
69C4⋊Dic15—22no—
70C4⋊Dic15—42no—
71C4×Dic15—22no—
72C5×C4⋊Dic3C2×C414yes—
73C5×Dic3⋊C4C2×C418yes—
74C3×C4⋊Dic5—22no—
75C3×C4⋊Dic5—42no—
76C3×C4×Dic5—22no—
77Dic3⋊Dic5—22no—
78Dic3⋊Dic5—22no—
79Dic3⋊Dic5—42no—
80Dic3⋊Dic5—42no—
81(C2×D3)⋊Dic5—24no—
82(C2×D3)⋊Dic5—22no—
83(C2×D3)⋊Dic5—22no—
84Dic3×Dic5—24no—
85C2×#892—51no—
86C2×D3×C4⋊C4D511yescharacteristic
87C2×#872—51no—
88C3×#389—51no—
89#4987—52no—
90#4984—51no—
91#4976—52no—
92#4972—51no—
93#4928—51no—
94#4667—52no—
95#4647—52no—
96C23×C4×C5—31no—
97C23×C4×C5—32no—
98C2×C5×C4⋊C4—32no—
99C2×C5×C4⋊C4—31no—
100C2×C5×C4⋊C4—38no—
101C2×C5×C4⋊C4C2×D311yescharacteristic
102C2×C22⋊Dic5—31no—
103C2×C22⋊Dic5—32no—
104C2×C22⋊Dic5—31no—
105#4072—64no—
106#4072—64no—
107C2×C4⋊Dic5—31no—
108C2×C4⋊Dic5—31no—
109C2×C4⋊Dic5—31no—
110C2×C4⋊Dic5—32no—
111C2×C4⋊Dic5—32no—
112C2×C4⋊Dic5—32no—
113C2×C4⋊Dic5C2×D311yescharacteristic
114C2×C4⋊Dic5C2×D312yes—
115C2×C4×Dic5—31no—
116C2×C4×Dic5—32no—
117C2×C4×Dic5C2×D311yescharacteristic
118#3963—32no—
119#3963—32no—
120#3963—32no—
121#3963C2×D312yes—
122#2316—151noSylow
123C23×C3×C5C2411yescharacteristic
124C22×C5×D3—21no—
125C22×C5×D3—21no—
126C22×C5×D3—21no—
127C22×C5×D3—21no—
128C22×C5×D3C22×C412yes—
129C22×C5×D3C2×D412yes—
130C22×C5×D3C2×Q812yes—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5C22×C414yes—
134C2×C3×C4×C5C22×C412yes—
135C2×C3×C4×C5C2×D412yes—
136C2×C3×C4×C5C2×Q812yes—
137C2×Dic15—22no—
138C2×Dic15—21no—
139C2×Dic15—21no—
140C2×Dic15—21no—
141C2×Dic15—22no—
142C2×Dic15—21no—
143C2×Dic15—21no—
144C2×Dic15—22no—
145C2×Dic15—21no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3—22no—
148C2×C5×Dic3C22×C414yes—
149C2×C5×Dic3C22×C412yes—
150C2×C5×Dic3C2×D412yes—
151C2×C5×Dic3C2×Q812yes—
152C4×C5×D3—48no—
153C4×C5×D3C4⋊C4116yescomplemented
154C2×C3×Dic5—22no—
155C2×C3×Dic5—21no—
156C2×C3×Dic5—21no—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—22no—
159C2×C3×Dic5—21no—
160C2×C3×Dic5—21no—
161C2×C3×Dic5—21no—
162C2×C3×Dic5—22no—
163C22×C4×D3C2×D512yes—
164C22×C4×D3C2×D511yescharacteristic
165C2×C3×C4⋊C4—51no—
166C2×C3×C4⋊C4—52no—
167C2×C3×C4⋊C4C2×D511yescharacteristic
168C2×C3×C42—51no—
169C2×(C2×D3)⋊C4—51no—
170C2×(C2×D3)⋊C4—52no—
171C2×(C2×D3)⋊C4—51no—
172C2×C4⋊Dic3—51no—
173C2×C4⋊Dic3—52no—
174C2×C4⋊Dic3C2×D511yescharacteristic
175C2×Dic3⋊C4—51no—
176C2×Dic3⋊C4—51no—
177C2×Dic3⋊C4—52no—
178C2×Dic3⋊C4—52no—
179C2×Dic3⋊C4C2×D512yes—
180C2×C4×Dic3—51no—
181C2×C4×Dic3—52no—
182#892—104no—
183D3×C4⋊C4Dic518yescomplemented
184#872—104no—
185C3×#95—52no—
186C3⋊#95—52no—
187C3⋊#95—52no—
188C3⋊#95—52no—
189C24×C5—31no—
190C22×C4×C5—31no—
191C22×C4×C5—34no—
192C22×C4×C5—34no—
193C22×C4×C5—34no—
194C22×C4×C5—32no—
195C22×C4×C5—38no—
196C22×C4×C5—122no—
197C22×C4×C5—122no—
198C22×C4×C5C22×D312yes—
199C22×C4×C5C22×D311yescharacteristic
200C22×Dic5—31no—
201C22×Dic5—32no—
202C22×Dic5—31no—
203C22×Dic5C22×D311yescharacteristic
204C22×Dic5C22×D312yes—
205C22×Dic5C22×D311yescharacteristic
206C5×C4⋊C4—38no—
207C5×C4⋊C4—34no—
208C5×C4⋊C4C4×D314yes—
209C22⋊Dic5—62no—
210C22⋊Dic5—62no—
211C22⋊Dic5—64no—
212C4⋊Dic5—22no—
213C4⋊Dic5—42no—
214C4⋊Dic5—62no—
215C4⋊Dic5—62no—
216C4⋊Dic5—62no—
217C4⋊Dic5—122no—
218C4⋊Dic5—122no—
219C4⋊Dic5—122no—
220C4×Dic5—22no—
221C4×Dic5—64no—
222C4×Dic5—62no—
223C2×#117—151no—
224C2×#117—151no—
225C22×C4⋊C4—31no—
226#389—51no—
227#389—152no—
228#389—151no—
229#389—151no—
230#389—152no—
231#389—151no—
232#380—152no—
233#380—152no—
234C22×C3×C5C23×C411yescharacteristic
235C22×C3×C5C22×D411yescharacteristic
236C22×C3×C5C22×Q811yescharacteristic
237C22×C3×C5H(5,𝔽2)11yescharacteristic
238C22×C3×C5H(5,𝔽2)11yescharacteristic
239C22×C3×C52−1+411yescharacteristic
240C22×C3×C52−1+411yescharacteristic
241C2×C5×D3—22no—
242C2×C5×D3—22no—
243C2×C5×D3—22no—
244C2×C5×D3—22no—
245C2×C5×D3—22no—
246C2×C5×D3—22no—
247C2×C5×D3C2×C4⋊C414yes—
248C3×C4×C5—24no—
249C3×C4×C5C2×C4⋊C414yes—
250Dic15—41no—
251Dic15—42no—
252Dic15—41no—
253C3×Dic5—42no—
254C3×Dic5—41no—
255C3×Dic5—41no—
256C5×Dic3—24no—
257C5×Dic3C2×C4⋊C414yes—
258C23×D3C22×D511yescharacteristic
259C22×C3×C4—51no—
260C22×C3×C4—52no—
261C22×C3×C4—51no—
262C22×C3×C4C22×D512yes—
263C22×C3×C4C22×D511yescharacteristic
264C22×Dic3—51no—
265C22×Dic3—51no—
266C22×Dic3—52no—
267C22×Dic3C22×D512yes—
268C22×Dic3C22×D511yescharacteristic
269C2×C4×D3—42no—
270C2×C4×D3—42no—
271C2×C4×D3Dic1014yes—
272C2×C4×D3D2014yes—
273C2×C4×D3C2×Dic518yes—
274C2×C4×D3C2×Dic514yes—
275C3×C4⋊C4—102no—
276C3×C4⋊C4—202no—
277C3×C4⋊C4C2×Dic514yes—
278C3×C42—102no—
279(C2×D3)⋊C4—102no—
280(C2×D3)⋊C4—102no—
281(C2×D3)⋊C4—104no—
282C4⋊Dic3—102no—
283C4⋊Dic3—202no—
284C4⋊Dic3C2×Dic514yes—
285Dic3⋊C4—102no—
286Dic3⋊C4—102no—
287Dic3⋊C4—202no—
288Dic3⋊C4—202no—
289Dic3⋊C4C2×Dic518yes—
290C4×Dic3—102no—
291C4×Dic3—104no—
292C23×C5—32no—
293C23×C5—32no—
294C23×C5—32no—
295C23×C5—61no—
296C23×C5—61no—
297C23×C5—61no—
298C23×C5—61no—
299C23×C5C23×D311yescharacteristic
300C2×C4×C5—22no—
301C2×C4×C5—22no—
302C2×C4×C5—32no—
303C2×C4×C5—32no—
304C2×C4×C5—32no—
305C2×C4×C5—316no—
306C2×C4×C5—34no—
307C2×C4×C5—62no—
308C2×C4×C5—62no—
309C2×C4×C5—128no—
310C2×C4×C5C2×C4×D314yes—
311C2×C4×C5C2×C4×D312yes—
312C2×C4×C5D3×D412yes—
313C2×C4×C5D3×Q812yes—
314C2×Dic5—21no—
315C2×Dic5—21no—
316C2×Dic5—22no—
317C2×Dic5—21no—
318C2×Dic5—22no—
319C2×Dic5—21no—
320C2×Dic5—21no—
321C2×Dic5—21no—
322C2×Dic5—22no—
323C2×Dic5—61no—
324C2×Dic5—62no—
325C2×Dic5—61no—
326C2×Dic5—61no—
327C2×Dic5—61no—
328C2×Dic5—62no—
329C2×Dic5—62no—
330C2×Dic5—61no—
331C2×Dic5—61no—
332C23×C4—31no—
333C23×C4—32no—
334#117—304no—
335#117—304no—
336C2×C4⋊C4—32no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—38no—
339C2×C4⋊C4—51no—
340C2×C4⋊C4—52no—
341C2×C4⋊C4—151no—
342C2×C4⋊C4—151no—
343C2×C4⋊C4—151no—
344C2×C4⋊C4—152no—
345C2×C4⋊C4—152no—
346C2×C4⋊C4—152no—
347C2×C4⋊C4D3×D511yescharacteristic
348C2×C22⋊C4—151no—
349C2×C22⋊C4—152no—
350C2×C22⋊C4—151no—
351C2×C42—51no—
352C2×C42—152no—
353C2×C42—151no—
354#95—52no—
355#95—152no—
356#95—152no—
357#95—152no—
358C2×C3×C5C22×C4⋊C411yescharacteristic
359C2×C3×C5#51811yescharacteristic
360C2×C3×C5#51911yescharacteristic
361C2×C3×C5#53711yescharacteristic
362C2×C3×C5#53711yescharacteristic
363C2×C3×C5#54411yescharacteristic
364C2×C3×C5#54411yescharacteristic
365C5×D3—24no—
366C23×C3C23×D511yescharacteristic
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—21no—
370C22×D3—21no—
371C22×D3C2×Dic1012yes—
372C22×D3C2×D2012yes—
373C22×D3C22×Dic512yes—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—102no—
377C2×C3×C4—101no—
378C2×C3×C4—101no—
379C2×C3×C4—101no—
380C2×C3×C4—102no—
381C2×C3×C4—101no—
382C2×C3×C4—101no—
383C2×C3×C4—102no—
384C2×C3×C4—101no—
385C2×C3×C4C2×Dic1012yes—
386C2×C3×C4C2×D2012yes—
387C2×C3×C4C22×Dic514yes—
388C2×C3×C4C22×Dic512yes—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—101no—
392C2×Dic3—101no—
393C2×Dic3—102no—
394C2×Dic3—101no—
395C2×Dic3—102no—
396C2×Dic3—101no—
397C2×Dic3—101no—
398C2×Dic3—101no—
399C2×Dic3—102no—
400C2×Dic3C2×Dic1012yes—
401C2×Dic3C2×D2012yes—
402C2×Dic3C22×Dic514yes—
403C2×Dic3C22×Dic512yes—
404C4×D3—48no—
405C4×D3C4⋊Dic5116yescomplemented
406C22×C5—34no—
407C22×C5—62no—
408C22×C5—62no—
409C22×C5—62no—
410C22×C5—62no—
411C22×C5—62no—
412C22×C5—62no—
413C22×C5C22×C4×D311yescharacteristic
414C22×C5C2×D3×D411yescharacteristic
415C22×C5#100411yescharacteristic
416C22×C5C2×D3×Q811yescharacteristic
417C22×C5#100711yescharacteristic
418C22×C5#100911yescharacteristic
419C22×C5#101011yescharacteristic
420C4×C5—24no—
421C4×C5—34no—
422C4×C5—64no—
423C4×C5D3×C4⋊C414yes—
424Dic5—41no—
425Dic5—42no—
426Dic5—41no—
427Dic5—121no—
428Dic5—122no—
429Dic5—121no—
430C24—31no—
431C22×C4—31no—
432C22×C4—34no—
433C22×C4—34no—
434C22×C4—34no—
435C22×C4—32no—
436C22×C4—38no—
437C22×C4—51no—
438C22×C4—52no—
439C22×C4—51no—
440C22×C4—122no—
441C22×C4—122no—
442C22×C4—151no—
443C22×C4—151no—
444C22×C4—152no—
445C22×C4C2×D3×D512yes—
446C22×C4C2×D3×D511yescharacteristic
447C4⋊C4—38no—
448C4⋊C4—34no—
449C4⋊C4—102no—
450C4⋊C4—202no—
451C4⋊C4—302no—
452C4⋊C4—302no—
453C4⋊C4—302no—
454C4⋊C4—602no—
455C4⋊C4—602no—
456C4⋊C4—602no—
457C4⋊C4D3×Dic514yes—
458C22⋊C4—302no—
459C22⋊C4—302no—
460C22⋊C4—304no—
461C42—102no—
462C42—304no—
463C42—302no—
464C3×C5#231611yescharacteristic, complemented, Hall
465C22×C3C22×Dic1011yescharacteristic
466C22×C3C22×D2011yescharacteristic
467C22×C3#414411yescharacteristic
468C22×C3#414411yescharacteristic
469C22×C3#414711yescharacteristic
470C22×C3#414711yescharacteristic
471C22×C3C23×Dic511yescharacteristic
472C2×D3—22no—
473C2×D3—22no—
474C2×D3—22no—
475C2×D3—22no—
476C2×D3—22no—
477C2×D3—22no—
478C2×D3C2×C4⋊Dic514yes—
479C3×C4—24no—
480C3×C4—201no—
481C3×C4—202no—
482C3×C4—201no—
483C3×C4C2×C4⋊Dic514yes—
484Dic3—24no—
485Dic3—201no—
486Dic3—202no—
487Dic3—201no—
488Dic3C2×C4⋊Dic514yes—
489C2×C5—64no—
490C2×C5C2×D3×C4⋊C411yescharacteristic
491C2×C5#554511yescharacteristic
492C2×C5#557211yescharacteristic
493C2×C5#560811yescharacteristic
494C2×C5#562911yescharacteristic
495C2×C5#568911yescharacteristic
496C2×C5#572911yescharacteristic
497C23—32no—
498C23—32no—
499C23—32no—
500C23—61no—
501C23—61no—
502C23—61no—
503C23—61no—
504C23C22×D3×D511yescharacteristic
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—32no—
508C2×C4—32no—
509C2×C4—32no—
510C2×C4—316no—
511C2×C4—34no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—101no—
515C2×C4—101no—
516C2×C4—102no—
517C2×C4—101no—
518C2×C4—102no—
519C2×C4—101no—
520C2×C4—102no—
521C2×C4—101no—
522C2×C4—101no—
523C2×C4—128no—
524C2×C4—302no—
525C2×C4—301no—
526C2×C4—301no—
527C2×C4—301no—
528C2×C4—301no—
529C2×C4—302no—
530C2×C4—301no—
531C2×C4—301no—
532C2×C4—302no—
533C2×C4D3×Dic1012yes—
534C2×C4D3×D2012yes—
535C2×C4C2×D3×Dic514yes—
536C2×C4C2×D3×Dic512yes—
537C2×C3#6583411yescharacteristic
538C2×C3#6583411yescharacteristic
539C2×C3#6584411yescharacteristic
540C2×C3#6584411yescharacteristic
541C2×C3C22×C4⋊Dic511yescharacteristic
542C2×C3#6613211yescharacteristic
543C2×C3#6614611yescharacteristic
544D3—24no—
545C5#7619511yescharacteristic, complemented, Sylow
546C22—34no—
547C22—62no—
548C22—62no—
549C22—62no—
550C22—62no—
551C22—62no—
552C22—62no—
553C22C2×D3×Dic1011yescharacteristic
554C22C2×D3×D2011yescharacteristic
555C22#9202511yescharacteristic
556C22#9202711yescharacteristic
557C22#9202811yescharacteristic
558C22#9202911yescharacteristic
559C22C22×D3×Dic511yescharacteristic
560C4—24no—
561C4—34no—
562C4—64no—
563C4—201no—
564C4—201no—
565C4—202no—
566C4—602no—
567C4—601no—
568C4—601no—
569C4D3×C4⋊Dic514yes—
570C3#1061388711yescharacteristic, complemented, Sylow
571C2—64no—
572C2#1175333111yescharacteristic
573C2#1175337011yescharacteristic
574C2#1175344111yescharacteristic
575C2#1175369411yescharacteristic
576C2C2×D3×C4⋊Dic511yescharacteristic
577C2#1175491411yescharacteristic
578C2#1175514411yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC23×C5×D3C23
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2316
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #11751085C2semidirect
5. C3×#65274C2semidirect
6. #11748868C2semidirect
7. #11748842C2semidirect
10. #11748693C2semidirect
11. #11748677C2semidirect
21. C5×D3×C4⋊C4C4semidirect
50. #65274D3semidirect
153. C4×C5×D3C4⋊C4semidirect
183. D3×C4⋊C4Dic5semidirect
405. C4×D3C4⋊Dic5semidirect
464. C3×C5#2316semidirect
545. C5#76195semidirect
570. C3#10613887semidirect

Maximal subgroups

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

subgroupindex
C2×C5×D3×C4⋊C42
C2×#913712
C2×#913382
#117510852
C3×#652742
#117488682
#117488422
#117487862
#117487822
#117486932
#117486772
#106138873
#761955