Finite Group Databasebeta

← #14326206

Subgroups — #14326206

Order 1920.

1744 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6819C211yesmaximal, characteristic
2C2×#91340C211yesmaximal, characteristic, complemented
3C2×#91333C211yesmaximal, characteristic, complemented
4C5×#5221C211yesmaximal, characteristic, complemented
5C3×#65510C211yesmaximal, characteristic, complemented
6#11749516C212yesmaximal, complemented
7#11749487C212yesmaximal, complemented
8#11749483C212yesmaximal, complemented
9#11749483C212yesmaximal, complemented
10#11748799C211yesmaximal, characteristic, complemented
11C4×#6667C211yesmaximal, characteristic, complemented
12#10608032—31nomaximal, Hall
13C22×C3×C5×D4C2211yescharacteristic
14C22×#1220C2212yes—
15C23×Dic15C2212yes—
16C2×#6819—24no—
17C2×#6819—24no—
18C2×#6819C418yescomplemented
19C22×C4×D15C2211yescharacteristic
20C22×Dic30C2211yescharacteristic
21C2×C5×#217C2212yes—
22C2×C5×#217C2212yes—
23C2×C5×C4⋊Dic3C2211yescharacteristic
24C2×C4×C5×Dic3C2211yescharacteristic
25C2×C3×C22⋊Dic5C2212yes—
26C2×C3×C22⋊Dic5C2212yes—
27C2×C3×C4⋊Dic5C2211yescharacteristic
28C2×C3×C4×Dic5C2211yescharacteristic
29C2×#6669C2212yes—
30C2×#6669C2212yes—
31C2×#6667C2211yescharacteristic
32C2×#6667C2211yescharacteristic
33C2×#6667C2212yes—
34C2×Dic3×Dic5C2212yes—
35C2×Dic3×Dic5C2212yes—
36#91340—24no—
37#91333—24no—
38#90994C2211yescharacteristic
39#90994C2212yes—
40#90994C2211yescharacteristic
41#76301—51nomaximal, Hall
42C22×#711—31no—
43C2×#4033—31no—
44C2×C4×Dic10—31no—
45C5×#390—31no—
46#65510D311yescharacteristic, complemented
47#65276—31no—
48#65246—32no—
49#65246—32no—
50#65231—32no—
51C22⋊C4×Dic5—32no—
52C4×#685—31no—
53C24×C3×C5C2312yes—
54C23×D15C2311yescharacteristic
55C2×C3×C5×D4—22no—
56C2×C3×C5×D4—22no—
57C2×C3×C5×D4C2×C414yes—
58C22×C3×C4×C5C2311yescharacteristic
59C2×#1220—24no—
60C2×#1220—24no—
61C2×#1220C2×C418yes—
62C22×Dic15—24no—
63C22×Dic15—24no—
64C22×Dic15C2×C418yes—
65C22×Dic15C2311yescharacteristic
66C22×Dic15C2312yes—
67#6819—232no—
68C2×C4×D15—22no—
69C2×C4×D15—22no—
70C2×C4×D15C2×C414yes—
71C2×Dic30C2×C414yes—
72C2×Dic30D414yes—
73C2×Dic30D414yes—
74C22×C5×Dic3C2311yescharacteristic
75C22×C5×Dic3C2312yes—
76C22×C5×Dic3C2311yescharacteristic
77C22×C3×Dic5C2312yes—
78C22×C3×Dic5C2311yescharacteristic
79C22×C3×Dic5C2311yescharacteristic
80C5×#217—24no—
81C5×#217—42no—
82C5×C4⋊Dic3—22no—
83C4×C5×Dic3—22no—
84C3×C22⋊Dic5—24no—
85C3×C22⋊Dic5—42no—
86C3×C4⋊Dic5—22no—
87C3×C4×Dic5—22no—
88#6669—24no—
89#6669—24no—
90#6667—22no—
91#6667—22no—
92#6667—42no—
93Dic3×Dic5—24no—
94Dic3×Dic5—24no—
95C22×#237—51no—
96C2×#887—51no—
97C2×C4×Dic6—51no—
98C3×#390—51no—
99#5221D511yescharacteristic, complemented
100#4987—51no—
101#4957—52no—
102#4957—52no—
103#4942—52no—
104Dic3×C22⋊C4—52no—
105C4×(C2×D3)⋊C4—51no—
106C22×C5×D4C2×D311yescharacteristic
107C22×C22⋊D5—32no—
108C23×Dic5—32no—
109C2×#711—38no—
110C2×#711—64no—
111C2×#711—64no—
112C22×C4×D5—31no—
113C22×Dic10—31no—
114C2×C5×C4⋊C4—31no—
115C2×C5×C22⋊C4—32no—
116C2×C5×C22⋊C4—32no—
117C2×C42×C5—31no—
118C2×C22⋊Dic5C2×D312yes—
119C2×C22⋊Dic5C2×D312yes—
120C2×#685—31no—
121C2×#685—31no—
122C2×#685—32no—
123C2×C4⋊Dic5C2×D311yescharacteristic
124C2×#683—32no—
125C2×#683—32no—
126C2×C4×Dic5—32no—
127C2×C4×Dic5—32no—
128C2×C4×Dic5C2×D311yescharacteristic
129#4033—64no—
130C4×Dic10—64no—
131#3963—31no—
132#3963—31no—
133#3963—32no—
134#2341—151noSylow
135C23×C3×C5—22no—
136C23×C3×C5—22no—
137C23×C3×C5—22no—
138C23×C3×C5—22no—
139C23×C3×C5—22no—
140C23×C3×C5—22no—
141C23×C3×C5C22×C414yes—
142C23×C3×C5C2411yescharacteristic
143C22×D15—21no—
144C22×D15—21no—
145C22×D15—21no—
146C22×D15—21no—
147C22×D15—21no—
148C22×D15—21no—
149C22×D15C22×C412yes—
150C3×C5×D4—28no—
151C2×C3×C4×C5C22×C412yes—
152C2×C3×C4×C5C2×D412yes—
153C2×C3×C4×C5C2×D412yes—
154#1220—216no—
155C2×Dic15—21no—
156C2×Dic15—21no—
157C2×Dic15—216no—
158C2×Dic15C22×C412yes—
159C2×Dic15C22×C414yes—
160C2×Dic15C2×D414yes—
161C2×Dic15C2×D414yes—
162C4×D15—28no—
163Dic30C22⋊C4116yescomplemented
164C2×C5×Dic3—21no—
165C2×C5×Dic3—21no—
166C2×C5×Dic3—22no—
167C2×C5×Dic3—21no—
168C2×C5×Dic3—21no—
169C2×C5×Dic3—22no—
170C2×C5×Dic3—22no—
171C2×C5×Dic3—21no—
172C2×C5×Dic3#4112yescomplemented
173C2×C3×Dic5—21no—
174C2×C3×Dic5—21no—
175C2×C3×Dic5—22no—
176C2×C3×Dic5—21no—
177C2×C3×Dic5—21no—
178C2×C3×Dic5—22no—
179C2×C3×Dic5—22no—
180C2×C3×Dic5—21no—
181C2×C3×Dic5#4112yescomplemented
182C22×C3×D4C2×D511yescharacteristic
183C22×#67—52no—
184C23×Dic3—52no—
185C2×#237—58no—
186C2×#237—104no—
187C2×#237—104no—
188C22×C4×D3—51no—
189C22×Dic6—51no—
190C2×C3×C4⋊C4—51no—
191C2×C3×C22⋊C4—52no—
192C2×C3×C22⋊C4—52no—
193C2×C3×C42—51no—
194C2×#217C2×D512yes—
195C2×#217C2×D512yes—
196C2×(C2×D3)⋊C4—51no—
197C2×(C2×D3)⋊C4—51no—
198C2×(C2×D3)⋊C4—52no—
199C2×C4⋊Dic3C2×D511yescharacteristic
200C2×Dic3⋊C4—52no—
201C2×Dic3⋊C4—52no—
202C2×C4×Dic3—52no—
203C2×C4×Dic3—52no—
204C2×C4×Dic3C2×D511yescharacteristic
205#887—104no—
206C4×Dic6—104no—
207C3⋊#95—51no—
208C3⋊#95—51no—
209C3⋊#95—52no—
210C24×C5C22×D312yes—
211C23×D5—31no—
212C2×C5×D4—22no—
213C2×C5×D4—22no—
214C2×C5×D4C4×D314yes—
215C22×C4×C5—31no—
216C22×C4×C5—31no—
217C22×C4×C5—32no—
218C22×C4×C5C22×D311yescharacteristic
219C2×C22⋊D5—38no—
220C2×C22⋊D5—64no—
221C2×C22⋊D5—64no—
222C22×Dic5—32no—
223C22×Dic5—38no—
224C22×Dic5—31no—
225C22×Dic5—64no—
226C22×Dic5—64no—
227C22×Dic5C22×D312yes—
228C22×Dic5C22×D311yescharacteristic
229C22×Dic5C22×D311yescharacteristic
230#711—632no—
231C2×C4×D5—34no—
232C2×C4×D5—62no—
233C2×C4×D5—62no—
234C2×Dic10—34no—
235C2×Dic10—34no—
236C2×Dic10—34no—
237C5×C4⋊C4—62no—
238C5×C22⋊C4—64no—
239C5×C22⋊C4—122no—
240C42×C5—62no—
241C22⋊Dic5—24no—
242C22⋊Dic5—42no—
243#685—62no—
244#685—62no—
245#685—122no—
246C4⋊Dic5—22no—
247#683—64no—
248#683—64no—
249C4×Dic5—22no—
250C4×Dic5—64no—
251C4×Dic5—64no—
252C22×#41—151no—
253C2×C4×Q8—151no—
254C2×C4×Q8—151no—
255#390—31no—
256#390—51no—
257#390—152no—
258#390—152no—
259#380—152no—
260#380—151no—
261C4×C22⋊C4—151no—
262C4×C22⋊C4—152no—
263C22×C3×C5—24no—
264C22×C3×C5—24no—
265C22×C3×C5—24no—
266C22×C3×C5—42no—
267C22×C3×C5—42no—
268C22×C3×C5—42no—
269C22×C3×C5—42no—
270C22×C3×C5C23×C411yescharacteristic
271C22×C3×C5C22×D411yescharacteristic
272C22×C3×C5C22×D411yescharacteristic
273C22×C3×C5C2×#4111yescharacteristic
274C22×C3×C5C2×#4111yescharacteristic
275C22×C3×C5H(5,𝔽2)11yescharacteristic
276C22×C3×C52−1+411yescharacteristic
277C2×D15—22no—
278C2×D15—22no—
279C2×D15—22no—
280C2×D15—41no—
281C2×D15—41no—
282C2×D15—41no—
283C2×D15—41no—
284C3×C4×C5C2×C22⋊C414yes—
285Dic15—22no—
286Dic15C2×C22⋊C418yes—
287C3×Dic5—22no—
288C3×Dic5—41no—
289C3×Dic5—42no—
290C5×Dic3—22no—
291C5×Dic3—42no—
292C5×Dic3—41no—
293C24×C3C22×D512yes—
294C23×D3—51no—
295C2×C3×D4—22no—
296C2×C3×D4—22no—
297C2×C3×D4C4×D514yes—
298C22×C3×C4—51no—
299C22×C3×C4—52no—
300C22×C3×C4—51no—
301C22×C3×C4C22×D511yescharacteristic
302C2×#67—58no—
303C2×#67—104no—
304C2×#67—104no—
305C22×Dic3—52no—
306C22×Dic3—58no—
307C22×Dic3—51no—
308C22×Dic3—104no—
309C22×Dic3—104no—
310C22×Dic3C22×D512yes—
311C22×Dic3C22×D511yescharacteristic
312C22×Dic3C22×D511yescharacteristic
313#237—1032no—
314C2×C4×D3—54no—
315C2×C4×D3—102no—
316C2×C4×D3—102no—
317C2×Dic6—54no—
318C2×Dic6—54no—
319C2×Dic6—54no—
320C3×C4⋊C4—102no—
321C3×C22⋊C4—104no—
322C3×C22⋊C4—202no—
323C3×C42—102no—
324#217—24no—
325#217—42no—
326(C2×D3)⋊C4—102no—
327(C2×D3)⋊C4—102no—
328(C2×D3)⋊C4—202no—
329C4⋊Dic3—22no—
330Dic3⋊C4—104no—
331Dic3⋊C4—104no—
332C4×Dic3—22no—
333C4×Dic3—104no—
334C4×Dic3—104no—
335C23×C5—22no—
336C23×C5—22no—
337C23×C5—22no—
338C23×C5—22no—
339C23×C5—22no—
340C23×C5—22no—
341C23×C5C2×C4×D314yes—
342C23×C5C23×D311yescharacteristic
343C22×D5—32no—
344C22×D5—61no—
345C22×D5—61no—
346C22×D5—61no—
347C22×D5—61no—
348C22×D5—61no—
349C22×D5—61no—
350C5×D4—28no—
351C2×C4×C5—32no—
352C2×C4×C5—61no—
353C2×C4×C5—61no—
354C2×C4×C5—62no—
355C2×C4×C5—61no—
356C2×C4×C5—61no—
357C2×C4×C5—62no—
358C2×C4×C5—61no—
359C2×C4×C5—62no—
360C2×C4×C5C2×C4×D312yes—
361C2×C4×C5D3×D412yes—
362C2×C4×C5D3×D412yes—
363C22⋊D5—616no—
364C2×Dic5—21no—
365C2×Dic5—21no—
366C2×Dic5—21no—
367C2×Dic5—22no—
368C2×Dic5—22no—
369C2×Dic5—22no—
370C2×Dic5—21no—
371C2×Dic5—21no—
372C2×Dic5—34no—
373C2×Dic5—34no—
374C2×Dic5—34no—
375C2×Dic5—32no—
376C2×Dic5—616no—
377C2×Dic5—61no—
378C2×Dic5—61no—
379C2×Dic5#23712yescomplemented
380C4×D5—68no—
381Dic10—316no—
382C2×#41—158no—
383C2×#41—304no—
384C2×#41—304no—
385C22×Q8—151no—
386C22×D4—152no—
387C22×D4D3×D511yescharacteristic
388C23×C4—152no—
389C23×C4—151no—
390C4×Q8—304no—
391C4×Q8—304no—
392C2×C4⋊C4—31no—
393C2×C4⋊C4—51no—
394C2×C4⋊C4—152no—
395C2×C4⋊C4—152no—
396C2×C22⋊C4—32no—
397C2×C22⋊C4—32no—
398C2×C22⋊C4—52no—
399C2×C22⋊C4—52no—
400C2×C22⋊C4—151no—
401C2×C22⋊C4—151no—
402C2×C22⋊C4—152no—
403C2×C42—31no—
404C2×C42—51no—
405C2×C42—152no—
406C2×C42—152no—
407#95—151no—
408#95—151no—
409#95—152no—
410C2×C3×C5—44no—
411C2×C3×C5C22×C22⋊C411yescharacteristic
412C2×C3×C5C4×#4111yescharacteristic
413C2×C3×C5#52011yescharacteristic
414C2×C3×C5#54811yescharacteristic
415C2×C3×C5#54811yescharacteristic
416C2×C3×C5#55011yescharacteristic
417C2×C3×C5#55011yescharacteristic
418D15—42no—
419C23×C3—22no—
420C23×C3—22no—
421C23×C3—22no—
422C23×C3—22no—
423C23×C3—22no—
424C23×C3—22no—
425C23×C3C2×C4×D514yes—
426C23×C3C23×D511yescharacteristic
427C22×D3—52no—
428C22×D3—101no—
429C22×D3—101no—
430C22×D3—101no—
431C22×D3—101no—
432C22×D3—101no—
433C22×D3—101no—
434C3×D4—28no—
435C2×C3×C4—52no—
436C2×C3×C4—101no—
437C2×C3×C4—101no—
438C2×C3×C4—102no—
439C2×C3×C4—101no—
440C2×C3×C4—101no—
441C2×C3×C4—102no—
442C2×C3×C4—101no—
443C2×C3×C4—102no—
444C2×C3×C4C2×C4×D512yes—
445C2×C3×C4D4×D512yes—
446C2×C3×C4D4×D512yes—
447#67—1016no—
448C2×Dic3—21no—
449C2×Dic3—21no—
450C2×Dic3—21no—
451C2×Dic3—22no—
452C2×Dic3—22no—
453C2×Dic3—22no—
454C2×Dic3—21no—
455C2×Dic3—21no—
456C2×Dic3—54no—
457C2×Dic3—54no—
458C2×Dic3—54no—
459C2×Dic3—52no—
460C2×Dic3—1016no—
461C2×Dic3—101no—
462C2×Dic3—101no—
463C2×Dic3#71112yescomplemented
464C4×D3—108no—
465Dic6—516no—
466C22×C5—24no—
467C22×C5—24no—
468C22×C5—24no—
469C22×C5—42no—
470C22×C5—42no—
471C22×C5—42no—
472C22×C5—42no—
473C22×C5C22×C4×D311yescharacteristic
474C22×C5C2×D3×D411yescharacteristic
475C22×C5C2×D3×D411yescharacteristic
476C22×C5C2×#23711yescharacteristic
477C22×C5#100411yescharacteristic
478C22×C5D3×#4111yescharacteristic
479C22×C5#101011yescharacteristic
480C2×D5—62no—
481C2×D5—62no—
482C2×D5—62no—
483C2×D5—121no—
484C2×D5—121no—
485C2×D5—121no—
486C2×D5—121no—
487C4×C5—62no—
488C4×C5—121no—
489C4×C5—122no—
490C4×C5D3×C22⋊C414yes—
491Dic5—22no—
492Dic5—38no—
493Dic5—42no—
494Dic5—41no—
495Dic5—62no—
496C24—151no—
497C24C2×D3×D512yes—
498#41—3032no—
499C2×Q8—154no—
500C2×Q8—154no—
501C2×Q8—154no—
502C2×D4—22no—
503C2×D4—22no—
504C2×D4—158no—
505C2×D4—304no—
506C2×D4—304no—
507C2×D4#120014yes—
508C22×C4—31no—
509C22×C4—31no—
510C22×C4—32no—
511C22×C4—51no—
512C22×C4—51no—
513C22×C4—52no—
514C22×C4—152no—
515C22×C4—158no—
516C22×C4—151no—
517C22×C4—154no—
518C22×C4—304no—
519C22×C4—304no—
520C22×C4—302no—
521C22×C4—302no—
522C22×C4C2×D3×D511yescharacteristic
523C4⋊C4—62no—
524C4⋊C4—102no—
525C4⋊C4—304no—
526C4⋊C4—304no—
527C22⋊C4—64no—
528C22⋊C4—104no—
529C22⋊C4—122no—
530C22⋊C4—202no—
531C22⋊C4—302no—
532C22⋊C4—302no—
533C22⋊C4—602no—
534C42—62no—
535C42—102no—
536C42—304no—
537C42—304no—
538C3×C5#234111yescharacteristic, complemented, Hall
539C22×C3—24no—
540C22×C3—24no—
541C22×C3—24no—
542C22×C3—42no—
543C22×C3—42no—
544C22×C3—42no—
545C22×C3—42no—
546C22×C3C22×C4×D511yescharacteristic
547C22×C3C2×D4×D511yescharacteristic
548C22×C3C2×D4×D511yescharacteristic
549C22×C3C2×#71111yescharacteristic
550C22×C3#414411yescharacteristic
551C22×C3D5×#4111yescharacteristic
552C22×C3#415011yescharacteristic
553C2×D3—102no—
554C2×D3—102no—
555C2×D3—102no—
556C2×D3—201no—
557C2×D3—201no—
558C2×D3—201no—
559C2×D3—201no—
560C3×C4—102no—
561C3×C4—201no—
562C3×C4—202no—
563C3×C4D5×C22⋊C414yes—
564Dic3—22no—
565Dic3—42no—
566Dic3—41no—
567Dic3—58no—
568Dic3—102no—
569C2×C5—44no—
570C2×C5C2×D3×C22⋊C411yescharacteristic
571C2×C5C4×#23711yescharacteristic
572C2×C5#554611yescharacteristic
573C2×C5#559811yescharacteristic
574C2×C5#559911yescharacteristic
575C2×C5#567711yescharacteristic
576C2×C5#571811yescharacteristic
577D5—122no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—152no—
585C23—301no—
586C23—301no—
587C23—301no—
588C23—301no—
589C23—301no—
590C23—301no—
591C23C2×#120014yes—
592C23C22×D3×D511yescharacteristic
593Q8—1516no—
594D4—28no—
595D4—3016no—
596C2×C4—32no—
597C2×C4—52no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—62no—
602C2×C4—61no—
603C2×C4—62no—
604C2×C4—61no—
605C2×C4—62no—
606C2×C4—101no—
607C2×C4—101no—
608C2×C4—102no—
609C2×C4—101no—
610C2×C4—101no—
611C2×C4—102no—
612C2×C4—101no—
613C2×C4—102no—
614C2×C4—154no—
615C2×C4—154no—
616C2×C4—154no—
617C2×C4—152no—
618C2×C4—3016no—
619C2×C4—301no—
620C2×C4—301no—
621C2×C4—308no—
622C2×C4C2×#120012yes—
623C2×C4#679012yes—
624C2×C4#679012yes—
625C2×C3—44no—
626C2×C3C2×D5×C22⋊C411yescharacteristic
627C2×C3C4×#71111yescharacteristic
628C2×C3#6588011yescharacteristic
629C2×C3#6593211yescharacteristic
630C2×C3#6593311yescharacteristic
631C2×C3#6601111yescharacteristic
632C2×C3#6605211yescharacteristic
633D3—202no—
634C5#7630111yescharacteristic, complemented, Sylow
635C22—24no—
636C22—24no—
637C22—24no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—302no—
643C22—302no—
644C22—302no—
645C22—601no—
646C22—601no—
647C22—601no—
648C22—601no—
649C22#9202011yescharacteristic
650C22D5×#23711yescharacteristic
651C22D3×#71111yescharacteristic
652C22#9202511yescharacteristic
653C22C22×#120011yescharacteristic
654C22C2×#679011yescharacteristic
655C22C2×#679011yescharacteristic
656C4—62no—
657C4—102no—
658C4—121no—
659C4—122no—
660C4—158no—
661C4—202no—
662C4—201no—
663C4—302no—
664C4#9156114yes—
665C3#1060803211yescharacteristic, complemented, Sylow
666C2—44no—
667C2—602no—
668C2#1175481611yescharacteristic
669C2#1175481911yescharacteristic
670C2#1175482711yescharacteristic
671C2#1175485311yescharacteristic
672C2#1175491611yescharacteristic
673C2#1175495611yescharacteristic
674C2C2×#9156111yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC22×C3×C5×D4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#6819C2
hypercenterC22×D4D3×D5
hypocenterC3×C5#2341
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C2×#91340C2semidirect
3. C2×#91333C2semidirect
4. C5×#5221C2semidirect
5. C3×#65510C2semidirect
6. #11749516C2semidirect
7. #11749487C2semidirect
8. #11749483C2semidirect
9. #11749483C2semidirect
10. #11748799C2semidirect
11. C4×#6667C2semidirect
18. C2×#6819C4semidirect
46. #65510D3semidirect
99. #5221D5semidirect
163. Dic30C22⋊C4semidirect
172. C2×C5×Dic3#41semidirect
181. C2×C3×Dic5#41semidirect
379. C2×Dic5#237semidirect
463. C2×Dic3#711semidirect
538. C3×C5#2341semidirect
634. C5#76301semidirect
665. C3#10608032semidirect

Maximal subgroups

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

subgroupindex
C22×#68192
C2×#913402
C2×#913332
C5×#52212
C3×#655102
#117495162
#117494872
#117494832
#117487992
C4×#66672
#106080323
#763015