Finite Group Databasebeta

← #14323945

Subgroups — #14323945

Order 1920.

1336 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91827C211yesmaximal, characteristic
2C2×C3×D4×Dic5C211yesmaximal, characteristic, complemented
3C2×#91386C211yesmaximal, characteristic, complemented
4C5×#5221C211yesmaximal, characteristic, complemented
5#11749508C212yesmaximal, complemented
6#11749494C212yesmaximal, complemented
7#11749483C212yesmaximal, complemented
8#11749480C212yesmaximal, complemented
9#11748858C211yesmaximal, characteristic, complemented
10#11748799C211yesmaximal, characteristic, complemented
11Dic5×C4⋊Dic3C211yesmaximal, characteristic, complemented
12#10608161—31nomaximal, Hall
13C22×C3×C5×D4C2211yescharacteristic
14C22×#1220C2212yes—
15C22×C4×D15C2211yescharacteristic
16C23×C3×Dic5C2212yes—
17C2×#6719C2212yes—
18#91827C418yescomplemented
19C2×C4⋊Dic15C2211yescharacteristic
20C2×C5×#217C2212yes—
21C2×C5×#217C2212yes—
22C2×C5×C4⋊Dic3C2211yescharacteristic
23C2×C4×C5×Dic3C2211yescharacteristic
24C2×C3×C22⋊Dic5C2212yes—
25C3×D4×Dic5—24no—
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C2×C3×C4×Dic5C2211yescharacteristic
28C2×#6669C2211yescharacteristic
29C2×#6669C2211yescharacteristic
30C2×#6669C2212yes—
31C2×#6667C2211yescharacteristic
32C2×#6667C2211yescharacteristic
33C2×#6667C2212yes—
34C2×Dic3×Dic5C2212yes—
35#91386—24no—
36#90994C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#82396—51nomaximal, Hall
40C2×#4084—31no—
41C2×D4×Dic5D311yescharacteristic, complemented
42C2×#4018—31no—
43C5×#390—31no—
44#65276—31no—
45C4⋊C4×Dic5—31no—
46#65247—32no—
47#65246—32no—
48#65235—32no—
49#65233—32no—
50#65217—31no—
51C24×C3×C5C2312yes—
52C23×D15C2311yescharacteristic
53C2×C3×C5×D4—22no—
54C2×C3×C5×D4—22no—
55C2×C3×C5×D4C2×C414yes—
56C22×C3×C4×C5C2311yescharacteristic
57C2×#1220—42no—
58C2×#1220—42no—
59C2×#1220C2×C418yescomplemented
60C22×Dic15C2312yes—
61C22×Dic15C2311yescharacteristic
62C2×C4×D15—22no—
63C2×C4×D15—22no—
64C2×C4×D15C2×C414yescomplemented
65C22×C5×Dic3C2311yescharacteristic
66C22×C5×Dic3C2311yescharacteristic
67C22×C5×Dic3C2312yes—
68C22×C3×Dic5—24no—
69C22×C3×Dic5—42no—
70C22×C3×Dic5—42no—
71C22×C3×Dic5C2312yes—
72C22×C3×Dic5C2311yescharacteristic
73C22×C3×Dic5C2311yescharacteristic
74#6719C2×C418yescomplemented
75C4⋊Dic15C2×C414yescomplemented
76C5×#217—42no—
77C5×#217—42no—
78C5×C4⋊Dic3—22no—
79C4×C5×Dic3—22no—
80C3×C22⋊Dic5—24no—
81C3×C4⋊Dic5—22no—
82C3×C4×Dic5D414yescomplemented
83#6669—22no—
84#6669—22no—
85#6669—42no—
86#6667—22no—
87#6667—22no—
88#6667—42no—
89Dic3×Dic5—42no—
90C2×C3×C4×D4—51no—
91C2×#938—51no—
92C2×#872—51no—
93#5221D511yescharacteristic, complemented
94#5213—52no—
95#5210—52no—
96#4987—51no—
97#4972—51no—
98#4957—52no—
99#4945—52no—
100C4×C4⋊Dic3—51no—
101C22×C5×D4C2×D311yescharacteristic
102C22×C22⋊D5—32no—
103C23×Dic5C2×D312yes—
104C22×C4×D5—31no—
105C2×C5×C4⋊C4—31no—
106C2×C5×C22⋊C4—32no—
107C2×C5×C22⋊C4—32no—
108C2×C42×C5—31no—
109C2×C22⋊Dic5—32no—
110C2×C22⋊Dic5C2×D312yes—
111#4084—38no—
112D4×Dic5—24no—
113C2×#685—31no—
114C2×#685—31no—
115C2×#685—32no—
116C2×C4⋊Dic5—31no—
117C2×C4⋊Dic5C2×D311yescharacteristic
118C2×#683—31no—
119C2×#683—31no—
120C2×#683—32no—
121C2×C4×Dic5—32no—
122C2×C4×Dic5C2×D311yescharacteristic
123#4018—64no—
124#3963—32no—
125#3963—32no—
126#3963—32no—
127#2324—151noSylow
128C23×C3×C5—22no—
129C23×C3×C5—22no—
130C23×C3×C5—22no—
131C23×C3×C5—22no—
132C23×C3×C5—22no—
133C23×C3×C5—22no—
134C23×C3×C5C22×C414yes—
135C23×C3×C5C2411yescharacteristic
136C22×D15—21no—
137C22×D15—21no—
138C22×D15—21no—
139C22×D15—21no—
140C22×D15—21no—
141C22×D15—21no—
142C22×D15C22×C412yes—
143C3×C5×D4—28no—
144C2×C3×C4×C5C22×C412yes—
145C2×C3×C4×C5C2×D412yes—
146C2×C3×C4×C5#4112yes—
147#1220—48no—
148C2×Dic15—21no—
149C2×Dic15—21no—
150C2×Dic15—22no—
151C2×Dic15—22no—
152C2×Dic15C22×C414yes—
153C2×Dic15C22×C412yes—
154C4×D15—28no—
155C2×C5×Dic3—21no—
156C2×C5×Dic3—21no—
157C2×C5×Dic3—22no—
158C2×C5×Dic3—22no—
159C2×C5×Dic3—22no—
160C2×C5×Dic3—21no—
161C2×C5×Dic3—21no—
162C2×C5×Dic3—21no—
163C2×C5×Dic3—21no—
164C2×C3×Dic5—21no—
165C2×C3×Dic5—21no—
166C2×C3×Dic5—21no—
167C2×C3×Dic5—21no—
168C2×C3×Dic5—22no—
169C2×C3×Dic5—22no—
170C2×C3×Dic5—22no—
171C2×C3×Dic5—48no—
172C2×C3×Dic5C2×D412yes—
173C2×C3×Dic5C2×D412yes—
174C22×C3×D4C2×D511yescharacteristic
175C23×C3×C4—52no—
176C22×#67—52no—
177C22×C4×D3—51no—
178C3×C4×D4—104no—
179C2×C3×C4⋊C4—51no—
180C2×C3×C22⋊C4—52no—
181C2×C3×C42—51no—
182C2×#217—52no—
183C2×#217C2×D512yes—
184C2×#217C2×D512yes—
185#938—58no—
186C2×(C2×D3)⋊C4—51no—
187C2×(C2×D3)⋊C4—51no—
188C2×(C2×D3)⋊C4—52no—
189C2×C4⋊Dic3—51no—
190C2×C4⋊Dic3C2×D511yescharacteristic
191C2×Dic3⋊C4—51no—
192C2×Dic3⋊C4—51no—
193C2×Dic3⋊C4—52no—
194C2×C4×Dic3—52no—
195C2×C4×Dic3C2×D511yescharacteristic
196#872—104no—
197C3⋊#95—52no—
198C3⋊#95—52no—
199C3⋊#95—52no—
200C24×C5C22×D312yes—
201C23×D5—31no—
202C2×C5×D4—22no—
203C2×C5×D4—22no—
204C2×C5×D4C4×D314yes—
205C22×C4×C5—31no—
206C22×C4×C5—31no—
207C22×C4×C5—32no—
208C22×C4×C5C22×D311yescharacteristic
209C2×C22⋊D5—38no—
210C2×C22⋊D5—122no—
211C2×C22⋊D5—122no—
212C22×Dic5—24no—
213C22×Dic5—31no—
214C22×Dic5—32no—
215C22×Dic5—42no—
216C22×Dic5—42no—
217C22×Dic5C22×D312yes—
218C22×Dic5C22×D311yescharacteristic
219C22×Dic5C22×D311yescharacteristic
220C2×C4×D5—34no—
221C2×C4×D5—62no—
222C2×C4×D5—62no—
223C5×C4⋊C4—62no—
224C5×C22⋊C4—122no—
225C5×C22⋊C4—122no—
226C42×C5—62no—
227C22⋊Dic5—24no—
228C22⋊Dic5—38no—
229#685—62no—
230#685—62no—
231#685—122no—
232C4⋊Dic5—22no—
233C4⋊Dic5—34no—
234#683—62no—
235#683—62no—
236#683—122no—
237C4×Dic5—122no—
238C4×Dic5#6714yescomplemented
239C2×#121—151no—
240C2×C4×D4—51no—
241C2×#117—151no—
242#390—31no—
243#390—152no—
244#389—151no—
245#388—152no—
246#386—152no—
247#385—152no—
248#380—151no—
249C4×C4⋊C4—151no—
250C22×C3×C5—24no—
251C22×C3×C5—24no—
252C22×C3×C5—24no—
253C22×C3×C5—42no—
254C22×C3×C5—42no—
255C22×C3×C5—42no—
256C22×C3×C5—42no—
257C22×C3×C5C23×C411yescharacteristic
258C22×C3×C5C22×D411yescharacteristic
259C22×C3×C5C2×#4111yescharacteristic
260C22×C3×C5H(5,𝔽2)11yescharacteristic
261C22×C3×C5H(5,𝔽2)11yescharacteristic
262C22×C3×C5H(5,𝔽2)11yescharacteristic
263C22×C3×C52−1+411yescharacteristic
264C2×D15—22no—
265C2×D15—22no—
266C2×D15—22no—
267C2×D15—41no—
268C2×D15—41no—
269C2×D15—41no—
270C2×D15—41no—
271C3×C4×C5C4×D414yes—
272Dic15—22no—
273Dic15—24no—
274C3×Dic5—22no—
275C3×Dic5—22no—
276C3×Dic5—42no—
277C5×Dic3—41no—
278C5×Dic3—41no—
279C5×Dic3—42no—
280C24×C3C22×D512yes—
281C23×D3—51no—
282C2×C3×D4—22no—
283C2×C3×D4—22no—
284C2×C3×D4C4×D514yes—
285C22×C3×C4—51no—
286C22×C3×C4—51no—
287C22×C3×C4—52no—
288C22×C3×C4—104no—
289C22×C3×C4—202no—
290C22×C3×C4—202no—
291C22×C3×C4C22×D511yescharacteristic
292C2×#67—58no—
293C2×#67—202no—
294C2×#67—202no—
295C22×Dic3—52no—
296C22×Dic3—51no—
297C22×Dic3C22×D512yes—
298C22×Dic3C22×D511yescharacteristic
299C22×Dic3C22×D511yescharacteristic
300C2×C4×D3—54no—
301C2×C4×D3—102no—
302C2×C4×D3—102no—
303C3×C4⋊C4—102no—
304C3×C22⋊C4—104no—
305C3×C42—54no—
306#217—42no—
307#217—42no—
308#217—58no—
309(C2×D3)⋊C4—102no—
310(C2×D3)⋊C4—102no—
311(C2×D3)⋊C4—202no—
312C4⋊Dic3—22no—
313C4⋊Dic3—54no—
314Dic3⋊C4—102no—
315Dic3⋊C4—102no—
316Dic3⋊C4—202no—
317C4×Dic3—22no—
318C4×Dic3—202no—
319C23×C5—22no—
320C23×C5—22no—
321C23×C5—22no—
322C23×C5—22no—
323C23×C5—22no—
324C23×C5—22no—
325C23×C5C2×C4×D314yes—
326C23×C5C23×D311yescharacteristic
327C22×D5—32no—
328C22×D5—61no—
329C22×D5—61no—
330C22×D5—61no—
331C22×D5—61no—
332C22×D5—61no—
333C22×D5—61no—
334C5×D4—28no—
335C2×C4×C5—62no—
336C2×C4×C5—61no—
337C2×C4×C5—62no—
338C2×C4×C5—61no—
339C2×C4×C5—62no—
340C2×C4×C5—61no—
341C2×C4×C5—61no—
342C2×C4×C5—61no—
343C2×C4×C5—61no—
344C2×C4×C5C2×C4×D312yes—
345C2×C4×C5#23512yes—
346C2×C4×C5C2×#6712yes—
347C22⋊D5—128no—
348C2×Dic5—21no—
349C2×Dic5—21no—
350C2×Dic5—21no—
351C2×Dic5—21no—
352C2×Dic5—22no—
353C2×Dic5—22no—
354C2×Dic5—22no—
355C2×Dic5—34no—
356C2×Dic5—32no—
357C2×Dic5—48no—
358C2×Dic5—61no—
359C2×Dic5—62no—
360C2×Dic5—62no—
361C2×Dic5—61no—
362C2×Dic5C2×#6712yes—
363C2×Dic5C2×#6712yes—
364C4×D5—68no—
365C22×D4—152no—
366C22×D4D3×D511yescharacteristic
367C23×C4—52no—
368C23×C4—151no—
369#121—158no—
370C4×D4—104no—
371#117—304no—
372C2×C4⋊C4—31no—
373C2×C4⋊C4—51no—
374C2×C4⋊C4—151no—
375C2×C4⋊C4—151no—
376C2×C4⋊C4—151no—
377C2×C4⋊C4—152no—
378C2×C22⋊C4—32no—
379C2×C22⋊C4—32no—
380C2×C22⋊C4—52no—
381C2×C22⋊C4—152no—
382C2×C22⋊C4—151no—
383C2×C22⋊C4—151no—
384C2×C22⋊C4—152no—
385C2×C42—31no—
386C2×C42—51no—
387C2×C42—152no—
388#95—152no—
389#95—152no—
390#95—152no—
391C2×C3×C5—44no—
392C2×C3×C5C2×C4×D411yescharacteristic
393C2×C3×C5#51811yescharacteristic
394C2×C3×C5#52011yescharacteristic
395C2×C3×C5#53511yescharacteristic
396C2×C3×C5#53711yescharacteristic
397C2×C3×C5#54011yescharacteristic
398C2×C3×C5#54211yescharacteristic
399D15—42no—
400C23×C3—22no—
401C23×C3—22no—
402C23×C3—22no—
403C23×C3—22no—
404C23×C3—22no—
405C23×C3—22no—
406C23×C3C2×C4×D514yes—
407C23×C3C23×D511yescharacteristic
408C22×D3—52no—
409C22×D3—101no—
410C22×D3—101no—
411C22×D3—101no—
412C22×D3—101no—
413C22×D3—101no—
414C22×D3—101no—
415C3×D4—28no—
416C2×C3×C4—52no—
417C2×C3×C4—52no—
418C2×C3×C4—101no—
419C2×C3×C4—101no—
420C2×C3×C4—101no—
421C2×C3×C4—101no—
422C2×C3×C4—102no—
423C2×C3×C4—102no—
424C2×C3×C4—102no—
425C2×C3×C4—208no—
426C2×C3×C4C2×C4×D512yes—
427C2×C3×C4D4×D512yes—
428C2×C3×C4#71112yes—
429#67—208no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—21no—
436C2×Dic3—21no—
437C2×Dic3—21no—
438C2×Dic3—21no—
439C2×Dic3—54no—
440C2×Dic3—52no—
441C2×Dic3—101no—
442C2×Dic3—101no—
443C2×Dic3—102no—
444C2×Dic3—102no—
445C4×D3—108no—
446C22×C5—24no—
447C22×C5—24no—
448C22×C5—24no—
449C22×C5—42no—
450C22×C5—42no—
451C22×C5—42no—
452C22×C5—42no—
453C22×C5C22×C4×D311yescharacteristic
454C22×C5C2×#23511yescharacteristic
455C22×C5#100411yescharacteristic
456C22×C5#100411yescharacteristic
457C22×C5#100911yescharacteristic
458C22×C5#101011yescharacteristic
459C22×C5C22×#6711yescharacteristic
460C2×D5—62no—
461C2×D5—62no—
462C2×D5—62no—
463C2×D5—121no—
464C2×D5—121no—
465C2×D5—121no—
466C2×D5—121no—
467C4×C5—121no—
468C4×C5—122no—
469C4×C5—121no—
470C4×C5C4×#6714yes—
471Dic5—22no—
472Dic5—22no—
473Dic5—42no—
474Dic5—62no—
475Dic5—64no—
476C24—151no—
477C24C2×D3×D512yes—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—158no—
481C2×D4—602no—
482C2×D4—602no—
483C2×D4#120014yes—
484C22×C4—31no—
485C22×C4—32no—
486C22×C4—31no—
487C22×C4—51no—
488C22×C4—51no—
489C22×C4—52no—
490C22×C4—104no—
491C22×C4—152no—
492C22×C4—151no—
493C22×C4—154no—
494C22×C4—202no—
495C22×C4—202no—
496C22×C4—302no—
497C22×C4—302no—
498C22×C4C2×D3×D511yescharacteristic
499C4⋊C4—62no—
500C4⋊C4—102no—
501C4⋊C4—154no—
502C4⋊C4—302no—
503C4⋊C4—302no—
504C4⋊C4—602no—
505C22⋊C4—104no—
506C22⋊C4—122no—
507C22⋊C4—122no—
508C22⋊C4—158no—
509C22⋊C4—302no—
510C22⋊C4—302no—
511C22⋊C4—602no—
512C42—54no—
513C42—62no—
514C42—602no—
515C3×C5#232411yescharacteristic, complemented, Hall
516C22×C3—24no—
517C22×C3—24no—
518C22×C3—24no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—42no—
523C22×C3C22×C4×D511yescharacteristic
524C22×C3C2×D4×D511yescharacteristic
525C22×C3C2×#71111yescharacteristic
526C22×C3#414411yescharacteristic
527C22×C3#414411yescharacteristic
528C22×C3#414911yescharacteristic
529C22×C3#415011yescharacteristic
530C2×D3—102no—
531C2×D3—102no—
532C2×D3—102no—
533C2×D3—201no—
534C2×D3—201no—
535C2×D3—201no—
536C2×D3—201no—
537C3×C4—102no—
538C3×C4—102no—
539C3×C4—202no—
540C3×C4#402714yes—
541Dic3—42no—
542Dic3—41no—
543Dic3—41no—
544Dic3—102no—
545Dic3—104no—
546C2×C5—44no—
547C2×C5#554511yescharacteristic
548C2×C5#554611yescharacteristic
549C2×C5#556111yescharacteristic
550C2×C5#556211yescharacteristic
551C2×C5C2×C4×#6711yescharacteristic
552C2×C5#580511yescharacteristic
553C2×C5#583311yescharacteristic
554D5—122no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—152no—
562C23—301no—
563C23—301no—
564C23—301no—
565C23—301no—
566C23—301no—
567C23—301no—
568C23C2×#120014yes—
569C23C22×D3×D511yescharacteristic
570D4—28no—
571D4—608no—
572C2×C4—52no—
573C2×C4—52no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—62no—
577C2×C4—61no—
578C2×C4—62no—
579C2×C4—61no—
580C2×C4—62no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—101no—
584C2×C4—101no—
585C2×C4—101no—
586C2×C4—101no—
587C2×C4—102no—
588C2×C4—102no—
589C2×C4—102no—
590C2×C4—152no—
591C2×C4—154no—
592C2×C4—208no—
593C2×C4—302no—
594C2×C4—302no—
595C2×C4—301no—
596C2×C4—301no—
597C2×C4—308no—
598C2×C4#678212yes—
599C2×C4C2×#120012yes—
600C2×C4D5×#6712yes—
601C2×C3—44no—
602C2×C3C2×#402711yescharacteristic
603C2×C3#6587911yescharacteristic
604C2×C3#6588011yescharacteristic
605C2×C3#6594111yescharacteristic
606C2×C3#6594211yescharacteristic
607C2×C3#6601611yescharacteristic
608C2×C3#6605311yescharacteristic
609D3—202no—
610C5#8239611yescharacteristic, complemented, Sylow
611C22—24no—
612C22—24no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—302no—
619C22—302no—
620C22—302no—
621C22—601no—
622C22—601no—
623C22—601no—
624C22—601no—
625C22#9202011yescharacteristic
626C22#9202511yescharacteristic
627C22#9202611yescharacteristic
628C22#9202711yescharacteristic
629C22C2×#678211yescharacteristic
630C22C22×#120011yescharacteristic
631C22C2×D5×#6711yescharacteristic
632C4—102no—
633C4—102no—
634C4—122no—
635C4—121no—
636C4—121no—
637C4—202no—
638C4—302no—
639C4—304no—
640C4#9155214yes—
641C3#1060816111yescharacteristic, complemented, Sylow
642C2—44no—
643C2—602no—
644C2#1175474111yescharacteristic
645C2#1175491611yescharacteristic
646C2#1175491711yescharacteristic
647C2#1175492211yescharacteristic
648C2#1175493311yescharacteristic
649C2#1175493411yescharacteristic
650C2C2×#9155211yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC22×C3×C5×D4C22
socle, cosocleC23×C3×C5C24
prime radicalC2×#91827C2
hypercenterC22×D4D3×D5
hypocenterC3×C5#2324
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×C3×D4×Dic5C2semidirect
3. C2×#91386C2semidirect
4. C5×#5221C2semidirect
5. #11749508C2semidirect
6. #11749494C2semidirect
7. #11749483C2semidirect
8. #11749480C2semidirect
9. #11748858C2semidirect
10. #11748799C2semidirect
11. Dic5×C4⋊Dic3C2semidirect
18. #91827C4semidirect
41. C2×D4×Dic5D3semidirect
59. C2×#1220C2×C4semidirect
64. C2×C4×D15C2×C4semidirect
74. #6719C2×C4semidirect
75. C4⋊Dic15C2×C4semidirect
82. C3×C4×Dic5D4semidirect
93. #5221D5semidirect
238. C4×Dic5#67semidirect
515. C3×C5#2324semidirect
610. C5#82396semidirect
641. C3#10608161semidirect

Maximal subgroups

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

subgroupindex
C2×#918272
C2×C3×D4×Dic52
C2×#913862
C5×#52212
#117495082
#117494942
#117494832
#117494802
#117488582
#117487992
Dic5×C4⋊Dic32
#106081613
#823965