Finite Group Databasebeta

← #14323791

Subgroups — #14323791

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×#897C211yesmaximal, characteristic
2C2×#91371C211yesmaximal, characteristic
3C2×#91344C211yesmaximal, characteristic, complemented
4C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
5C3×#65274C211yesmaximal, characteristic, complemented
6#11748868C212yesmaximal, complemented
7#11748794C212yesmaximal
8#11748786C211yesmaximal, characteristic
9#11748683C212yesmaximal, complemented
10#11748675C212yesmaximal
11#11748589C211yesmaximal, characteristic, complemented
12#10614475—31nomaximal, Hall
13C22×C4×C5×D3C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C4⋊Dic15C2211yescharacteristic
17C2×C4×Dic15C2212yes—
18C2×C4×Dic15C2211yescharacteristic
19C2×C5×(C2×D3)⋊C4C2212yes—
20C2×C5×C4⋊Dic3C2212yes—
21C5×#897C418yescomplemented
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C2×C3×C4⋊Dic5C2212yes—
24C2×C3×C4×Dic5C2211yescharacteristic
25C2×Dic3⋊Dic5C2211yescharacteristic
26C2×Dic3⋊Dic5C2212yes—
27C2×Dic3⋊Dic5C2211yescharacteristic
28C2×(C2×D3)⋊Dic5C2211yescharacteristic
29C2×(C2×D3)⋊Dic5C2211yescharacteristic
30C2×(C2×D3)⋊Dic5C2212yes—
31C2×Dic3×Dic5C2212yes—
32#91371—24no—
33#91344—24no—
34#91115C2212yes—
35C3×#3963C2212yes—
36#90994C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#76197—51nomaximal, Hall
40C2×C5×C22⋊Q8—31no—
41C2×Q8×Dic5—31no—
42C2×#4072—31no—
43#65519—31no—
44#65501—31no—
45#65274—32no—
46#65274D311yescharacteristic, complemented
47#65270—32no—
48C4⋊C4×Dic5—31no—
49#65240—32no—
50#65240—32no—
51C23×C5×D3C2311yescharacteristic
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×Dic15C2312yes—
55C22×Dic15C2311yescharacteristic
56C22×Dic15C2311yescharacteristic
57C22×C5×Dic3C2311yescharacteristic
58C22×C5×Dic3C2312yes—
59C2×C4×C5×D3—22no—
60C2×C4×C5×D3—22no—
61C2×C4×C5×D3C2×C414yes—
62C2×C5×Dic6—22no—
63C2×C5×Dic6—22no—
64C2×C5×Dic6C2×C414yes—
65C22×C3×Dic5C2311yescharacteristic
66C22×C3×Dic5C2311yescharacteristic
67C22×C3×Dic5C2312yes—
68C3×C5×C4⋊C4C2×C414yes—
69C4⋊Dic15—22no—
70C4×Dic15—42no—
71C4×Dic15D414yes—
72C5×(C2×D3)⋊C4C2×C418yes—
73C5×C4⋊Dic3C2×C418yes—
74C3×C4⋊Dic5—22no—
75C3×C4⋊Dic5—42no—
76C3×C4×Dic5—22no—
77Dic3⋊Dic5—24no—
78Dic3⋊Dic5—22no—
79Dic3⋊Dic5—22no—
80(C2×D3)⋊Dic5—22no—
81(C2×D3)⋊Dic5—22no—
82(C2×D3)⋊Dic5—42no—
83Dic3×Dic5—24no—
84C2×#897D511yescharacteristic
85C2×#887—51no—
86C2×#872—51no—
87C3×#389—51no—
88#4990—52no—
89#4987—51no—
90Dic3×C4⋊C4—51no—
91#4972—52no—
92#4929—51no—
93#4669—52no—
94#4654—52no—
95C22×C5×Q8—31no—
96C23×C4×C5—31no—
97C5×C22⋊Q8—38no—
98C2×C5×C4⋊C4—32no—
99C2×C5×C4⋊C4C2×D311yescharacteristic
100C2×C5×C22⋊C4—32no—
101C2×C22⋊Dic5—31no—
102C2×C22⋊Dic5—31no—
103C2×C22⋊Dic5—32no—
104Q8×Dic5—64no—
105#4072—64no—
106C2×C4⋊Dic5—31no—
107C2×C4⋊Dic5—31no—
108C2×C4⋊Dic5—31no—
109C2×C4⋊Dic5—32no—
110C2×C4⋊Dic5C2×D311yescharacteristic
111C2×C4⋊Dic5C2×D312yes—
112C2×C4×Dic5—31no—
113C2×C4×Dic5—32no—
114C2×C4×Dic5—32no—
115C2×C4×Dic5C2×D311yescharacteristic
116#3963—32no—
117#3963—32no—
118#3963—32no—
119#3963—32no—
120#3963C2×D312yes—
121#2323—151noSylow
122C23×C3×C5C2411yescharacteristic
123C22×C5×D3—21no—
124C22×C5×D3—21no—
125C22×C5×D3—21no—
126C22×C5×D3—21no—
127C22×C5×D3—21no—
128C22×C5×D3—21no—
129C22×C5×D3C22×C412yes—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5C22×C414yes—
133C2×C3×C4×C5C22×C412yes—
134C2×C3×C4×C5C2×D412yes—
135C2×C3×C4×C5#4112yes—
136C2×Dic15—21no—
137C2×Dic15—21no—
138C2×Dic15—21no—
139C2×Dic15—21no—
140C2×Dic15—22no—
141C2×Dic15—22no—
142C2×Dic15—22no—
143C2×Dic15C2×D412yes—
144C2×Dic15C2×D412yes—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—21no—
147C2×C5×Dic3—22no—
148C2×C5×Dic3—21no—
149C2×C5×Dic3C22×C412yes—
150C2×C5×Dic3C22×C414yes—
151C4×C5×D3—28no—
152C5×Dic6—28no—
153C2×C3×Dic5—22no—
154C2×C3×Dic5—21no—
155C2×C3×Dic5—21no—
156C2×C3×Dic5—22no—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—21no—
160C2×C3×Dic5—21no—
161C2×C3×Dic5—22no—
162C22×C4×D3C2×D511yescharacteristic
163C22×Dic6C2×D511yescharacteristic
164C2×C3×C4⋊C4—51no—
165C2×C3×C4⋊C4—52no—
166C2×C3×C4⋊C4C2×D511yescharacteristic
167C2×C3×C42—51no—
168C2×(C2×D3)⋊C4—51no—
169C2×(C2×D3)⋊C4—51no—
170C2×(C2×D3)⋊C4—52no—
171C2×(C2×D3)⋊C4C2×D512yes—
172C2×C4⋊Dic3—51no—
173C2×C4⋊Dic3C2×D512yes—
174C2×Dic3⋊C4—51no—
175C2×Dic3⋊C4—52no—
176C2×Dic3⋊C4—51no—
177C2×C4×Dic3—51no—
178C2×C4×Dic3—52no—
179C2×C4×Dic3—52no—
180#897Dic518yescomplemented
181#887—104no—
182#872—104no—
183C3×#95—52no—
184C3⋊#95—52no—
185C3⋊#95—52no—
186C3⋊#95—52no—
187C3⋊#95—52no—
188C24×C5—31no—
189C2×C5×Q8—34no—
190C2×C5×Q8—62no—
191C2×C5×Q8—62no—
192C22×C4×C5—32no—
193C22×C4×C5—31no—
194C22×C4×C5—34no—
195C22×C4×C5—62no—
196C22×C4×C5—62no—
197C22×C4×C5C22×D312yes—
198C22×C4×C5C22×D311yescharacteristic
199C22×Dic5—31no—
200C22×Dic5—31no—
201C22×Dic5—32no—
202C22×Dic5C22×D311yescharacteristic
203C22×Dic5C22×D311yescharacteristic
204C22×Dic5C22×D312yes—
205C5×C4⋊C4—38no—
206C5×C4⋊C4C4×D314yes—
207C5×C22⋊C4—38no—
208C22⋊Dic5—62no—
209C22⋊Dic5—62no—
210C22⋊Dic5—122no—
211C4⋊Dic5—22no—
212C4⋊Dic5—42no—
213C4⋊Dic5—62no—
214C4⋊Dic5—62no—
215C4⋊Dic5—64no—
216C4⋊Dic5—62no—
217C4×Dic5—22no—
218C4×Dic5—34no—
219C4×Dic5—64no—
220C4×Dic5—122no—
221C2×C22⋊Q8—31no—
222C2×C4×Q8—151no—
223C2×#117—151no—
224#391—151no—
225#389—51no—
226#389—152no—
227#388—152no—
228#388—152no—
229#387—152no—
230#380—151no—
231C4×C4⋊C4—151no—
232C22×C3×C5C23×C411yescharacteristic
233C22×C3×C5C22×D411yescharacteristic
234C22×C3×C5C2×#4111yescharacteristic
235C22×C3×C5H(5,𝔽2)11yescharacteristic
236C22×C3×C52−1+411yescharacteristic
237C22×C3×C52−1+411yescharacteristic
238C22×C3×C52−1+411yescharacteristic
239C2×C5×D3—22no—
240C2×C5×D3—22no—
241C2×C5×D3—22no—
242C2×C5×D3—41no—
243C2×C5×D3—41no—
244C2×C5×D3—41no—
245C2×C5×D3—41no—
246C3×C4×C5—24no—
247C3×C4×C5C4×D414yes—
248Dic15—22no—
249Dic15—22no—
250Dic15—42no—
251C3×Dic5—41no—
252C3×Dic5—42no—
253C3×Dic5—41no—
254C5×Dic3—24no—
255C5×Dic3—22no—
256C23×D3C22×D511yescharacteristic
257C22×C3×C4—51no—
258C22×C3×C4—51no—
259C22×C3×C4—52no—
260C22×C3×C4C22×D512yes—
261C22×C3×C4C22×D511yescharacteristic
262C22×Dic3—51no—
263C22×Dic3—51no—
264C22×Dic3—52no—
265C22×Dic3C22×D512yes—
266C22×Dic3C22×D511yescharacteristic
267C2×C4×D3—22no—
268C2×C4×D3—22no—
269C2×C4×D3C2×Dic514yes—
270C2×Dic6—22no—
271C2×Dic6—22no—
272C2×Dic6C2×Dic514yes—
273C3×C4⋊C4—102no—
274C3×C4⋊C4—202no—
275C3×C4⋊C4C2×Dic514yes—
276C3×C42—102no—
277(C2×D3)⋊C4—102no—
278(C2×D3)⋊C4—102no—
279(C2×D3)⋊C4—202no—
280(C2×D3)⋊C4C2×Dic518yes—
281C4⋊Dic3—102no—
282C4⋊Dic3C2×Dic518yes—
283Dic3⋊C4—102no—
284Dic3⋊C4—102no—
285Dic3⋊C4—104no—
286C4×Dic3—54no—
287C4×Dic3—104no—
288C4×Dic3—202no—
289C23×C5—32no—
290C23×C5—61no—
291C23×C5—61no—
292C23×C5—61no—
293C23×C5—61no—
294C23×C5—61no—
295C23×C5—61no—
296C23×C5C23×D311yescharacteristic
297C5×Q8—68no—
298C2×C4×C5—22no—
299C2×C4×C5—22no—
300C2×C4×C5—34no—
301C2×C4×C5—32no—
302C2×C4×C5—62no—
303C2×C4×C5—61no—
304C2×C4×C5—61no—
305C2×C4×C5—62no—
306C2×C4×C5—68no—
307C2×C4×C5C2×C4×D314yes—
308C2×C4×C5C2×C4×D312yes—
309C2×C4×C5D3×D412yes—
310C2×C4×C5#23912yes—
311C2×Dic5—21no—
312C2×Dic5—21no—
313C2×Dic5—22no—
314C2×Dic5—21no—
315C2×Dic5—22no—
316C2×Dic5—21no—
317C2×Dic5—21no—
318C2×Dic5—21no—
319C2×Dic5—22no—
320C2×Dic5—32no—
321C2×Dic5—32no—
322C2×Dic5—61no—
323C2×Dic5—61no—
324C2×Dic5—61no—
325C2×Dic5—61no—
326C2×Dic5—62no—
327C2×Dic5—62no—
328C2×Dic5—62no—
329C22×Q8—31no—
330C23×C4—31no—
331C22⋊Q8—38no—
332C4×Q8—304no—
333#117—304no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—51no—
336C2×C4⋊C4—52no—
337C2×C4⋊C4—151no—
338C2×C4⋊C4—151no—
339C2×C4⋊C4—152no—
340C2×C4⋊C4—151no—
341C2×C4⋊C4D3×D511yescharacteristic
342C2×C22⋊C4—32no—
343C2×C22⋊C4—151no—
344C2×C22⋊C4—151no—
345C2×C22⋊C4—152no—
346C2×C42—51no—
347C2×C42—151no—
348C2×C42—152no—
349C2×C42—152no—
350#95—52no—
351#95—152no—
352#95—152no—
353#95—152no—
354#95—152no—
355C2×C3×C5C2×C4×D411yescharacteristic
356C2×C3×C5#51911yescharacteristic
357C2×C3×C5#52011yescharacteristic
358C2×C3×C5#53611yescharacteristic
359C2×C3×C5#53711yescharacteristic
360C2×C3×C5#54111yescharacteristic
361C2×C3×C5#54211yescharacteristic
362C5×D3—42no—
363C23×C3C23×D511yescharacteristic
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—21no—
370C22×D3C22×Dic512yes—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—102no—
374C2×C3×C4—101no—
375C2×C3×C4—101no—
376C2×C3×C4—101no—
377C2×C3×C4—102no—
378C2×C3×C4—101no—
379C2×C3×C4—102no—
380C2×C3×C4—101no—
381C2×C3×C4—101no—
382C2×C3×C4D4×D512yes—
383C2×C3×C4#71112yes—
384C2×C3×C4C22×Dic514yes—
385C2×C3×C4C22×Dic512yes—
386C2×Dic3—22no—
387C2×Dic3—21no—
388C2×Dic3—22no—
389C2×Dic3—21no—
390C2×Dic3—52no—
391C2×Dic3—52no—
392C2×Dic3—101no—
393C2×Dic3—101no—
394C2×Dic3—101no—
395C2×Dic3—101no—
396C2×Dic3—102no—
397C2×Dic3—102no—
398C2×Dic3—102no—
399C2×Dic3C22×Dic512yes—
400C2×Dic3C22×Dic514yes—
401C4×D3—28no—
402Dic6—28no—
403C22×C5—62no—
404C22×C5—62no—
405C22×C5—62no—
406C22×C5—121no—
407C22×C5—121no—
408C22×C5—121no—
409C22×C5—121no—
410C22×C5C22×C4×D311yescharacteristic
411C22×C5C2×D3×D411yescharacteristic
412C22×C5#100411yescharacteristic
413C22×C5C2×#23911yescharacteristic
414C22×C5#100711yescharacteristic
415C22×C5#101011yescharacteristic
416C22×C5#101011yescharacteristic
417C4×C5—24no—
418C4×C5—62no—
419C4×C5—64no—
420C4×C5#89314yes—
421Dic5—41no—
422Dic5—42no—
423Dic5—41no—
424Dic5—62no—
425Dic5—62no—
426Dic5—122no—
427C24—31no—
428C2×Q8—34no—
429C2×Q8—62no—
430C2×Q8—62no—
431C22×C4—32no—
432C22×C4—31no—
433C22×C4—34no—
434C22×C4—51no—
435C22×C4—52no—
436C22×C4—51no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—151no—
440C22×C4—151no—
441C22×C4—152no—
442C22×C4C2×D3×D512yes—
443C22×C4C2×D3×D511yescharacteristic
444C4⋊C4—38no—
445C4⋊C4—102no—
446C4⋊C4—202no—
447C4⋊C4—302no—
448C4⋊C4—302no—
449C4⋊C4—304no—
450C4⋊C4—302no—
451C4⋊C4D3×Dic514yes—
452C22⋊C4—38no—
453C22⋊C4—302no—
454C22⋊C4—302no—
455C22⋊C4—602no—
456C42—102no—
457C42—154no—
458C42—304no—
459C42—602no—
460C3×C5#232311yescharacteristic, complemented, Hall
461C22×C3C2×D4×D511yescharacteristic
462C22×C3C2×#71111yescharacteristic
463C22×C3#414711yescharacteristic
464C22×C3#414711yescharacteristic
465C22×C3#414911yescharacteristic
466C22×C3#415011yescharacteristic
467C22×C3C23×Dic511yescharacteristic
468C2×D3—22no—
469C2×D3—22no—
470C2×D3—22no—
471C2×D3—41no—
472C2×D3—41no—
473C2×D3—41no—
474C2×D3—41no—
475C3×C4—24no—
476C3×C4—201no—
477C3×C4—202no—
478C3×C4—201no—
479C3×C4D4×Dic514yes—
480Dic3—24no—
481Dic3—22no—
482Dic3—102no—
483Dic3—102no—
484Dic3—202no—
485C2×C5—122no—
486C2×C5C2×#89311yescharacteristic
487C2×C5#554611yescharacteristic
488C2×C5#557211yescharacteristic
489C2×C5#560811yescharacteristic
490C2×C5#562811yescharacteristic
491C2×C5#569811yescharacteristic
492C2×C5#573311yescharacteristic
493C23—32no—
494C23—61no—
495C23—61no—
496C23—61no—
497C23—61no—
498C23—61no—
499C23—61no—
500C23C22×D3×D511yescharacteristic
501Q8—68no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—34no—
505C2×C4—32no—
506C2×C4—61no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—61no—
510C2×C4—68no—
511C2×C4—101no—
512C2×C4—101no—
513C2×C4—102no—
514C2×C4—101no—
515C2×C4—102no—
516C2×C4—101no—
517C2×C4—102no—
518C2×C4—101no—
519C2×C4—101no—
520C2×C4—152no—
521C2×C4—152no—
522C2×C4—301no—
523C2×C4—301no—
524C2×C4—301no—
525C2×C4—301no—
526C2×C4—302no—
527C2×C4—302no—
528C2×C4—302no—
529C2×C4#676812yes—
530C2×C4#677712yes—
531C2×C4C2×D3×Dic514yes—
532C2×C4C2×D3×Dic512yes—
533C2×C3#6595911yescharacteristic
534C2×C3#6596211yescharacteristic
535C2×C3#6596311yescharacteristic
536C2×C3#6597711yescharacteristic
537C2×C3C2×D4×Dic511yescharacteristic
538C2×C3#6614611yescharacteristic
539C2×C3#6616111yescharacteristic
540D3—42no—
541C5#7619711yescharacteristic, complemented, Sylow
542C22—62no—
543C22—62no—
544C22—62no—
545C22—121no—
546C22—121no—
547C22—121no—
548C22—121no—
549C22C2×#676811yescharacteristic
550C22C2×#677711yescharacteristic
551C22#9202011yescharacteristic
552C22#9202611yescharacteristic
553C22#9202811yescharacteristic
554C22#9202911yescharacteristic
555C22C22×D3×Dic511yescharacteristic
556C4—24no—
557C4—62no—
558C4—64no—
559C4—201no—
560C4—201no—
561C4—202no—
562C4—302no—
563C4—302no—
564C4—602no—
565C4#9143514yes—
566C3#1061447511yescharacteristic, complemented, Sylow
567C2—122no—
568C2#1175333511yescharacteristic
569C2#1175337211yescharacteristic
570C2#1175369711yescharacteristic
571C2#1175370111yescharacteristic
572C2C2×#9143511yescharacteristic
573C2#1175492011yescharacteristic
574C2#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#2323
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
3. C2×#91344C2semidirect
4. C4⋊C4×Dic15C2semidirect
5. C3×#65274C2semidirect
6. #11748868C2semidirect
9. #11748683C2semidirect
11. #11748589C2semidirect
21. C5×#897C4semidirect
46. #65274D3semidirect
180. #897Dic5semidirect
460. C3×C5#2323semidirect
541. C5#76197semidirect
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×#8972
C2×#913712
C2×#913442
C4⋊C4×Dic152
C3×#652742
#117488682
#117487942
#117487862
#117486832
#117486752
#117485892
#106144753
#761975