Finite Group Databasebeta

← #14321806

Subgroups — #14321806

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#897C211yesmaximal, characteristic
2C2×C4×D3×Dic5C211yesmaximal, characteristic
3C2×#91344C211yesmaximal, characteristic, complemented
4C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
5C3×#65274C211yesmaximal, characteristic, complemented
6#11748897C212yesmaximal
7#11748886C212yesmaximal
8#11748786C211yesmaximal, characteristic
9#11748699C212yesmaximal, complemented
10#11748683C212yesmaximal, complemented
11#11748589C211yesmaximal, characteristic, complemented
12Dic5×C22⋊Q8—31nomaximal, Hall
13C22×C4×C5×D3C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C22×D3×Dic5C2211yescharacteristic
16C22×D3×Dic5C2211yescharacteristic
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C4⋊Dic15C2211yescharacteristic
19C2×C4×Dic15C2212yes—
20C2×C4×Dic15C2211yescharacteristic
21C2×C5×(C2×D3)⋊C4C2212yes—
22C2×C5×C4⋊Dic3C2212yes—
23C5×#897C418yescomplemented
24C2×C3×C4⋊Dic5C2212yes—
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×C4×Dic5C2211yescharacteristic
27C2×Dic3⋊Dic5C2212yes—
28C2×(C2×D3)⋊Dic5C2212yes—
29C2×Dic3×Dic5C2212yes—
30C2×Dic3×Dic5C2211yescharacteristic
31C2×Dic3×Dic5C2211yescharacteristic
32C4×D3×Dic5—24no—
33#91344—24no—
34#91115C2212yes—
35C3×#3963C2212yes—
36#90994C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#75965—51nomaximal, Hall
40C2×C5×C22⋊Q8—31no—
41C2×Q8×Dic5—31no—
42C22×C4×Dic5—31no—
43#65519—31no—
44#65501—31no—
45#65274D311yescharacteristic, complemented
46#65270—32no—
47C4⋊C4×Dic5—32no—
48C4⋊C4×Dic5—31no—
49#65239—32no—
50C22⋊C4×Dic5—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×C414yescomplemented
65C22×C3×Dic5C2312yes—
66C22×C3×Dic5C2311yescharacteristic
67C22×C3×Dic5C2311yescharacteristic
68C2×D3×Dic5—22no—
69C2×D3×Dic5—22no—
70C2×D3×Dic5—22no—
71C2×D3×Dic5—22no—
72C2×D3×Dic5—22no—
73C2×D3×Dic5Q814yescomplemented
74C3×C5×C4⋊C4C2×C414yescomplemented
75C4⋊Dic15—22no—
76C4×Dic15—24no—
77C4×Dic15D414yes—
78C5×(C2×D3)⋊C4C2×C418yes—
79C5×C4⋊Dic3C2×C418yescomplemented
80C3×C4⋊Dic5—22no—
81C3×C4⋊Dic5—24no—
82C3×C4×Dic5—22no—
83Dic3⋊Dic5—24no—
84(C2×D3)⋊Dic5—24no—
85Dic3×Dic5—22no—
86Dic3×Dic5—22no—
87Dic3×Dic5—24no—
88C2×#897D511yescharacteristic
89C2×#887—51no—
90C2×C42×D3—51no—
91C3×#389—51no—
92#4989—52no—
93#4987—51no—
94Dic3×C4⋊C4—51no—
95Dic3×C4⋊C4—52no—
96#4929—51no—
97#4666—52no—
98#4654—52no—
99C22×C5×Q8—31no—
100C23×C4×C5—31no—
101C23×Dic5—31no—
102C23×Dic5—31no—
103C5×C22⋊Q8—38no—
104C2×C5×C4⋊C4—32no—
105C2×C5×C4⋊C4C2×D311yescharacteristic
106C2×C5×C22⋊C4—32no—
107C2×C22⋊Dic5—32no—
108Q8×Dic5—64no—
109C2×C4⋊Dic5—31no—
110C2×C4⋊Dic5—32no—
111C2×C4⋊Dic5C2×D311yescharacteristic
112C2×C4⋊Dic5C2×D312yes—
113C2×C4×Dic5—31no—
114C2×C4×Dic5—32no—
115C2×C4×Dic5—31no—
116C2×C4×Dic5—31no—
117C2×C4×Dic5—32no—
118C2×C4×Dic5—64no—
119C2×C4×Dic5C2×D311yescharacteristic
120#3963—32no—
121#3963—32no—
122#3963—32no—
123#3963—32no—
124#3963C2×D312yes—
125C4×C22⋊Q8—151noSylow
126C23×C3×C5C2411yescharacteristic
127C22×C5×D3—21no—
128C22×C5×D3—21no—
129C22×C5×D3—21no—
130C22×C5×D3—21no—
131C22×C5×D3C22×C412yes—
132C22×C5×D3C2×Q812yes—
133C22×C5×D3#4112yes—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5C22×C414yes—
137C2×C3×C4×C5C22×C412yes—
138C2×C3×C4×C5C2×D412yes—
139C2×C3×C4×C5#4112yes—
140C2×Dic15—21no—
141C2×Dic15—21no—
142C2×Dic15—22no—
143C2×Dic15—22no—
144C2×Dic15—22no—
145C2×Dic15C2×D412yes—
146C2×Dic15C2×D412yes—
147C2×Dic15C2×Q812yes—
148C2×Dic15#4112yes—
149C2×C5×Dic3—21no—
150C2×C5×Dic3—22no—
151C2×C5×Dic3—22no—
152C2×C5×Dic3—21no—
153C2×C5×Dic3C22×C412yes—
154C2×C5×Dic3C22×C414yes—
155C4×C5×D3—28no—
156C5×Dic6—28no—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—21no—
160C2×C3×Dic5—21no—
161C2×C3×Dic5—22no—
162C2×C3×Dic5—22no—
163C2×C3×Dic5—22no—
164C2×C3×Dic5C2×Q812yes—
165C2×C3×Dic5#4112yes—
166D3×Dic5—28no—
167D3×Dic5—28no—
168C22×C4×D3—51no—
169C22×C4×D3—51no—
170C22×C4×D3C2×D511yescharacteristic
171C22×Dic6C2×D511yescharacteristic
172C2×C3×C4⋊C4—52no—
173C2×C3×C4⋊C4—51no—
174C2×C3×C4⋊C4C2×D511yescharacteristic
175C2×C3×C42—51no—
176C2×(C2×D3)⋊C4—52no—
177C2×(C2×D3)⋊C4C2×D512yes—
178C2×C4⋊Dic3—51no—
179C2×C4⋊Dic3C2×D512yes—
180C2×Dic3⋊C4—52no—
181C2×C4×Dic3—51no—
182C2×C4×Dic3—52no—
183C2×C4×Dic3—51no—
184C2×C4×Dic3—52no—
185C2×C4×Dic3—51no—
186#897Dic518yescomplemented
187#887—104no—
188C42×D3—104no—
189C3×#95—52no—
190C3⋊#95—52no—
191C3⋊#95—52no—
192C3⋊#95—52no—
193C3⋊#95—52no—
194C24×C5—31no—
195C2×C5×Q8—34no—
196C2×C5×Q8—62no—
197C2×C5×Q8—62no—
198C22×C4×C5—32no—
199C22×C4×C5—31no—
200C22×C4×C5—34no—
201C22×C4×C5—62no—
202C22×C4×C5—62no—
203C22×C4×C5C22×D312yes—
204C22×C4×C5C22×D311yescharacteristic
205C22×Dic5—31no—
206C22×Dic5—31no—
207C22×Dic5—34no—
208C22×Dic5—32no—
209C22×Dic5—62no—
210C22×Dic5—62no—
211C22×Dic5—62no—
212C22×Dic5—62no—
213C22×Dic5—62no—
214C22×Dic5C22×D312yes—
215C22×Dic5C22×D311yescharacteristic
216C22×Dic5C22×D311yescharacteristic
217C5×C4⋊C4—38no—
218C5×C4⋊C4C4×D314yescomplemented
219C5×C22⋊C4—38no—
220C22⋊Dic5—64no—
221C4⋊Dic5—22no—
222C4⋊Dic5—24no—
223C4⋊Dic5—64no—
224C4⋊Dic5—62no—
225C4×Dic5—22no—
226C4×Dic5—34no—
227C4×Dic5—62no—
228C4×Dic5—62no—
229C4×Dic5—64no—
230C4×Dic5—64no—
231C2×C22⋊Q8—31no—
232C2×C4×Q8—151no—
233C22×C42—151no—
234#391—151no—
235#389—51no—
236#387—152no—
237#385—152no—
238#380—151no—
239C4×C4⋊C4—152no—
240C4×C4⋊C4—151no—
241C4×C22⋊C4—152no—
242C22×C3×C5C23×C411yescharacteristic
243C22×C3×C5C22×D411yescharacteristic
244C22×C3×C5C22×Q811yescharacteristic
245C22×C3×C5C2×#4111yescharacteristic
246C22×C3×C5C2×#4111yescharacteristic
247C22×C3×C5C2×#4111yescharacteristic
248C22×C3×C5C2×#4111yescharacteristic
249C2×C5×D3—22no—
250C2×C5×D3—22no—
251C2×C5×D3—22no—
252C2×C5×D3—22no—
253C2×C5×D3—22no—
254C2×C5×D3—22no—
255C2×C5×D3C4×Q814yescomplemented
256C3×C4×C5—24no—
257C3×C4×C5C4×D414yes—
258Dic15—22no—
259Dic15—42no—
260Dic15C22⋊Q814yescomplemented
261C3×Dic5—22no—
262C3×Dic5—22no—
263C3×Dic5—42no—
264C5×Dic3—24no—
265C5×Dic3—22no—
266C23×D3C22×D511yescharacteristic
267C22×C3×C4—51no—
268C22×C3×C4—51no—
269C22×C3×C4—52no—
270C22×C3×C4C22×D512yes—
271C22×C3×C4C22×D511yescharacteristic
272C22×Dic3—51no—
273C22×Dic3—51no—
274C22×Dic3—52no—
275C22×Dic3C22×D511yescharacteristic
276C22×Dic3C22×D512yes—
277C2×C4×D3—22no—
278C2×C4×D3—22no—
279C2×C4×D3—54no—
280C2×C4×D3—102no—
281C2×C4×D3—102no—
282C2×C4×D3—102no—
283C2×C4×D3—102no—
284C2×C4×D3—102no—
285C2×C4×D3C2×Dic514yes—
286C2×Dic6—22no—
287C2×Dic6—22no—
288C2×Dic6C2×Dic514yescomplemented
289C3×C4⋊C4—104no—
290C3×C4⋊C4—102no—
291C3×C4⋊C4C2×Dic514yescomplemented
292C3×C42—102no—
293(C2×D3)⋊C4—104no—
294(C2×D3)⋊C4C2×Dic518yes—
295C4⋊Dic3—102no—
296C4⋊Dic3C2×Dic518yescomplemented
297Dic3⋊C4—104no—
298C4×Dic3—54no—
299C4×Dic3—102no—
300C4×Dic3—102no—
301C4×Dic3—104no—
302C4×Dic3—104no—
303C23×C5—32no—
304C23×C5—32no—
305C23×C5—32no—
306C23×C5—61no—
307C23×C5—61no—
308C23×C5—61no—
309C23×C5—61no—
310C23×C5C23×D311yescharacteristic
311C5×Q8—68no—
312C2×C4×C5—22no—
313C2×C4×C5—22no—
314C2×C4×C5—34no—
315C2×C4×C5—32no—
316C2×C4×C5—61no—
317C2×C4×C5—62no—
318C2×C4×C5—61no—
319C2×C4×C5—62no—
320C2×C4×C5—68no—
321C2×C4×C5C2×C4×D314yes—
322C2×C4×C5C2×C4×D312yes—
323C2×C4×C5D3×D412yes—
324C2×C4×C5#23912yes—
325C2×Dic5—21no—
326C2×Dic5—21no—
327C2×Dic5—21no—
328C2×Dic5—21no—
329C2×Dic5—22no—
330C2×Dic5—22no—
331C2×Dic5—22no—
332C2×Dic5—32no—
333C2×Dic5—32no—
334C2×Dic5—32no—
335C2×Dic5—32no—
336C2×Dic5—68no—
337C2×Dic5—68no—
338C2×Dic5—61no—
339C2×Dic5—61no—
340C2×Dic5—62no—
341C2×Dic5—62no—
342C2×Dic5—62no—
343C2×Dic5#23712yes—
344C2×Dic5D3×Q812yes—
345C22×Q8—31no—
346C23×C4—31no—
347C23×C4—151no—
348C23×C4—151no—
349C22⋊Q8—38no—
350C4×Q8—304no—
351C2×C4⋊C4—32no—
352C2×C4⋊C4—51no—
353C2×C4⋊C4—52no—
354C2×C4⋊C4—152no—
355C2×C4⋊C4—151no—
356C2×C4⋊C4D3×D511yescharacteristic
357C2×C22⋊C4—32no—
358C2×C22⋊C4—152no—
359C2×C42—51no—
360C2×C42—151no—
361C2×C42—152no—
362C2×C42—151no—
363C2×C42—151no—
364C2×C42—152no—
365C2×C42—304no—
366#95—52no—
367#95—152no—
368#95—152no—
369#95—152no—
370#95—152no—
371C2×C3×C5C2×C4×D411yescharacteristic
372C2×C3×C5C2×C4×Q811yescharacteristic
373C2×C3×C5C4×#4111yescharacteristic
374C2×C3×C5C2×C22⋊Q811yescharacteristic
375C2×C3×C5#52511yescharacteristic
376C2×C3×C5#52911yescharacteristic
377C2×C3×C5#53311yescharacteristic
378C5×D3—24no—
379C23×C3C23×D511yescharacteristic
380C22×D3—21no—
381C22×D3—21no—
382C22×D3—21no—
383C22×D3—21no—
384C22×D3Q8×D512yes—
385C22×D3#71312yes—
386C22×D3C22×Dic512yes—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—52no—
390C2×C3×C4—52no—
391C2×C3×C4—101no—
392C2×C3×C4—101no—
393C2×C3×C4—101no—
394C2×C3×C4—101no—
395C2×C3×C4—102no—
396C2×C3×C4—102no—
397C2×C3×C4—102no—
398C2×C3×C4D4×D512yes—
399C2×C3×C4#71112yes—
400C2×C3×C4C22×Dic514yes—
401C2×C3×C4C22×Dic512yes—
402C2×Dic3—21no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—21no—
406C2×Dic3—52no—
407C2×Dic3—52no—
408C2×Dic3—52no—
409C2×Dic3—52no—
410C2×Dic3—101no—
411C2×Dic3—101no—
412C2×Dic3—102no—
413C2×Dic3—102no—
414C2×Dic3—102no—
415C2×Dic3C22×Dic512yes—
416C2×Dic3C22×Dic514yes—
417C4×D3—28no—
418C4×D3—108no—
419C4×D3—108no—
420Dic6—28no—
421C22×C5—34no—
422C22×C5—62no—
423C22×C5—62no—
424C22×C5—62no—
425C22×C5—62no—
426C22×C5—62no—
427C22×C5—62no—
428C22×C5C22×C4×D311yescharacteristic
429C22×C5C2×D3×D411yescharacteristic
430C22×C5C2×#23711yescharacteristic
431C22×C5C2×D3×Q811yescharacteristic
432C22×C5C2×#23911yescharacteristic
433C22×C5D3×#4111yescharacteristic
434C22×C5D3×#4111yescharacteristic
435C4×C5—24no—
436C4×C5—62no—
437C4×C5—64no—
438C4×C5#89314yes—
439Dic5—22no—
440Dic5—22no—
441Dic5—34no—
442Dic5—42no—
443Dic5—62no—
444Dic5—122no—
445C24—31no—
446C2×Q8—34no—
447C2×Q8—62no—
448C2×Q8—62no—
449C22×C4—31no—
450C22×C4—32no—
451C22×C4—34no—
452C22×C4—51no—
453C22×C4—51no—
454C22×C4—52no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—154no—
458C22×C4—151no—
459C22×C4—151no—
460C22×C4—152no—
461C22×C4—302no—
462C22×C4—302no—
463C22×C4—302no—
464C22×C4—302no—
465C22×C4—302no—
466C22×C4C2×D3×D512yes—
467C22×C4C2×D3×D511yescharacteristic
468C4⋊C4—38no—
469C4⋊C4—104no—
470C4⋊C4—102no—
471C4⋊C4—304no—
472C4⋊C4—302no—
473C4⋊C4D3×Dic514yescomplemented
474C22⋊C4—38no—
475C22⋊C4—304no—
476C42—102no—
477C42—154no—
478C42—304no—
479C42—302no—
480C42—302no—
481C42—304no—
482C3×C5C4×C22⋊Q811yescharacteristic, complemented, Hall
483C22×C3C2×D4×D511yescharacteristic
484C22×C3C2×#71111yescharacteristic
485C22×C3C2×Q8×D511yescharacteristic
486C22×C3C2×#71311yescharacteristic
487C22×C3D5×#4111yescharacteristic
488C22×C3D5×#4111yescharacteristic
489C22×C3C23×Dic511yescharacteristic
490C2×D3—22no—
491C2×D3—22no—
492C2×D3—22no—
493C2×D3—22no—
494C2×D3—22no—
495C2×D3—22no—
496C2×D3Q8×Dic514yescomplemented
497C3×C4—24no—
498C3×C4—102no—
499C3×C4—102no—
500C3×C4—202no—
501C3×C4D4×Dic514yes—
502Dic3—24no—
503Dic3—22no—
504Dic3—54no—
505Dic3—102no—
506Dic3—202no—
507C2×C5—64no—
508C2×C5C2×#89311yescharacteristic
509C2×C5C4×#23711yescharacteristic
510C2×C5C4×D3×Q811yescharacteristic
511C2×C5#560611yescharacteristic
512C2×C5D3×C22⋊Q811yescharacteristic
513C2×C5#569111yescharacteristic
514C2×C5#572811yescharacteristic
515C23—32no—
516C23—32no—
517C23—32no—
518C23—61no—
519C23—61no—
520C23—61no—
521C23—61no—
522C23C22×D3×D511yescharacteristic
523Q8—68no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—34no—
527C2×C4—32no—
528C2×C4—52no—
529C2×C4—52no—
530C2×C4—61no—
531C2×C4—62no—
532C2×C4—61no—
533C2×C4—62no—
534C2×C4—68no—
535C2×C4—101no—
536C2×C4—101no—
537C2×C4—101no—
538C2×C4—101no—
539C2×C4—102no—
540C2×C4—102no—
541C2×C4—102no—
542C2×C4—152no—
543C2×C4—152no—
544C2×C4—152no—
545C2×C4—152no—
546C2×C4—308no—
547C2×C4—308no—
548C2×C4—301no—
549C2×C4—301no—
550C2×C4—302no—
551C2×C4—302no—
552C2×C4—302no—
553C2×C4#676812yes—
554C2×C4#677712yes—
555C2×C4C2×D3×Dic514yes—
556C2×C4C2×D3×Dic512yes—
557C2×C3#6595611yescharacteristic
558C2×C3#6595811yescharacteristic
559C2×C3D5×C22⋊Q811yescharacteristic
560C2×C3#6596111yescharacteristic
561C2×C3C2×D4×Dic511yescharacteristic
562C2×C3C2×Q8×Dic511yescharacteristic
563C2×C3#41×Dic511yescharacteristic
564D3—24no—
565C5#7596511yescharacteristic, complemented, Sylow
566C22—34no—
567C22—62no—
568C22—62no—
569C22—62no—
570C22—62no—
571C22—62no—
572C22—62no—
573C22C2×#676811yescharacteristic
574C22C2×#677711yescharacteristic
575C22D5×#23711yescharacteristic
576C22#9202411yescharacteristic
577C22D3×Q8×D511yescharacteristic
578C22D3×#71311yescharacteristic
579C22C22×D3×Dic511yescharacteristic
580C4—24no—
581C4—62no—
582C4—64no—
583C4—102no—
584C4—102no—
585C4—154no—
586C4—202no—
587C4—302no—
588C4—602no—
589C4#9143514yes—
590C3Dic5×C22⋊Q811yescharacteristic, complemented, Sylow
591C2—64no—
592C2#1175345911yescharacteristic
593C2#1175359711yescharacteristic
594C2#1175388411yescharacteristic
595C2#1175388511yescharacteristic
596C2C2×#9143511yescharacteristic
597C2Dic5×#23711yescharacteristic
598C2D3×Q8×Dic511yescharacteristic
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×C5C4×C22⋊Q8
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
9. #11748699C2semidirect
10. #11748683C2semidirect
11. #11748589C2semidirect
23. C5×#897C4semidirect
45. #65274D3semidirect
64. C2×C5×Dic6C2×C4semidirect
73. C2×D3×Dic5Q8semidirect
74. C3×C5×C4⋊C4C2×C4semidirect
79. C5×C4⋊Dic3C2×C4semidirect
186. #897Dic5semidirect
218. C5×C4⋊C4C4×D3semidirect
255. C2×C5×D3C4×Q8semidirect
260. Dic15C22⋊Q8semidirect
288. C2×Dic6C2×Dic5semidirect
291. C3×C4⋊C4C2×Dic5semidirect
296. C4⋊Dic3C2×Dic5semidirect
473. C4⋊C4D3×Dic5semidirect
482. C3×C5C4×C22⋊Q8semidirect
496. C2×D3Q8×Dic5semidirect
565. C5#75965semidirect
590. C3Dic5×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#8972
C2×C4×D3×Dic52
C2×#913442
C4⋊C4×Dic152
C3×#652742
#117488972
#117488862
#117487862
#117486992
#117486832
#117485892
Dic5×C22⋊Q83
#759655