Finite Group Databasebeta

← #14323623

Subgroups — #14323623

Order 1920.

1264 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×#91556C211yesmaximal, characteristic
2C2×#91546C211yesmaximal, characteristic
3C22×#6668C211yesmaximal, characteristic, complemented
4#11751059C211yesmaximal, characteristic
5C5×Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
6C3×#65232C211yesmaximal, characteristic, complemented
7#11749517C211yesmaximal, characteristic, complemented
8#11748895C212yesmaximal, complemented
9#11748803C212yesmaximal, complemented
10#11748730C212yesmaximal, complemented
11#11748656C212yesmaximal, complemented
12#10605177—31nomaximal, Hall
13C22×#1220C2211yescharacteristic
14C23×Dic15C2211yescharacteristic
15C23×C5×Dic3C2211yescharacteristic
16C23×C3×Dic5C2211yescharacteristic
17C22×#1200C2211yescharacteristic
18C2×C3×C5×C22⋊C4C2211yescharacteristic
19C2×#6714C2212yes—
20C2×C5×#217C2211yescharacteristic
21C2×C4×C5×Dic3C2212yes—
22C2×C3×C22⋊Dic5C2211yescharacteristic
23#91556—24no—
24#91546C418yescomplemented
25C2×Dic3⋊Dic5C2212yes—
26C2×Dic3⋊Dic5C2212yes—
27C2×#6669C2211yescharacteristic
28C2×#6668—24no—
29C2×#6668C2212yes—
30C2×#6668C2212yes—
31C2×#6667C2211yescharacteristic
32C2×#6667C2212yes—
33C2×Dic3×Dic5C2211yescharacteristic
34#91115C2212yes—
35C5×C3⋊#95C2212yes—
36C3×#3963C2212yes—
37C3×#3963C2212yes—
38#90994C2212yes—
39#74056—51nomaximal, Hall
40C2×C4×C22⋊D5—31no—
41C22×#683—31no—
42C2×#4032—31no—
43C4×C5×C22⋊C4—31no—
44#65247—31no—
45#65239—31no—
46#65232D311yescharacteristic, complemented
47#65229—32no—
48#65217—32no—
49#64957—32no—
50#64942—32no—
51C24×C3×C5C2311yescharacteristic
52C23×D15C2311yescharacteristic
53C22×C3×C4×C5C2312yes—
54C2×#1220—22no—
55C2×#1220—22no—
56C2×#1220C2×C414yes—
57C22×Dic15—22no—
58C22×Dic15—42no—
59C22×Dic15C2312yes—
60C22×Dic15C2311yescharacteristic
61C22×C5×Dic3—42no—
62C22×C5×Dic3D414yes—
63C22×C5×Dic3C2312yes—
64C22×C5×Dic3C2311yescharacteristic
65C22×C5×Dic3C2311yescharacteristic
66C22×C3×Dic5—22no—
67C22×C3×Dic5—22no—
68C22×C3×Dic5C2×C414yescomplemented
69C22×C3×Dic5C2311yescharacteristic
70C22×C3×Dic5C2311yescharacteristic
71C22×C3×Dic5C2312yes—
72C2×#1200—22no—
73C2×#1200—42no—
74C3×C5×C22⋊C4—22no—
75#6714—42no—
76C5×#217C2×C414yescomplemented
77C4×C5×Dic3—42no—
78C3×C22⋊Dic5—22no—
79Dic3⋊Dic5—42no—
80Dic3⋊Dic5C2×C418yescomplemented
81#6669—22no—
82#6668—24no—
83#6668—24no—
84#6667—22no—
85#6667C2×C418yescomplemented
86Dic3×Dic5—22no—
87C2×#929—51no—
88C22×C4⋊Dic3—51no—
89C2×#67⋊C4—51no—
90C3×#381—51no—
91#4958—51no—
92#4946—51no—
93Dic3×C22⋊C4D511yescharacteristic, complemented
94#4670—52no—
95#4668—52no—
96#4655—52no—
97#4647—52no—
98C23×C4×C5—31no—
99C22×C22⋊D5—31no—
100C23×Dic5—31no—
101C23×Dic5C2×D311yescharacteristic
102C22×C4×D5—31no—
103C2×C5×C22⋊C4—31no—
104C2×C5×C22⋊C4C2×D311yescharacteristic
105C2×C42×C5—32no—
106C2×C22⋊Dic5C2×D311yescharacteristic
107C4×C22⋊D5—64no—
108C2×#685—32no—
109C2×#685—31no—
110C2×#685—32no—
111C2×C4⋊Dic5—32no—
112C2×C4⋊Dic5—32no—
113C2×#683—32no—
114C2×#683—32no—
115C2×#683—31no—
116C2×#683—64no—
117C2×C4×Dic5—31no—
118#4032—38no—
119C5×#95—32no—
120#3963—32no—
121#3963—32no—
122#3963C2×D312yes—
123#3963C2×D312yes—
124#2322—151noSylow
125C23×C3×C5—22no—
126C23×C3×C5—22no—
127C23×C3×C5C22×C412yes—
128C23×C3×C5C2×D412yes—
129C23×C3×C5#4112yes—
130C23×C3×C5C2411yescharacteristic
131C22×D15—21no—
132C22×D15—21no—
133C22×D15—22no—
134C22×D15—22no—
135C22×D15C22×C412yes—
136C2×C3×C4×C5—22no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139#1220—28no—
140C2×Dic15—21no—
141C2×Dic15—21no—
142C2×Dic15—22no—
143C2×Dic15—22no—
144C2×Dic15—22no—
145C2×Dic15—44no—
146C2×Dic15C22×C412yes—
147C2×C5×Dic3—22no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—22no—
150C2×C5×Dic3—22no—
151C2×C5×Dic3—44no—
152C2×C5×Dic3C22×C414yes—
153C2×C5×Dic3C2×D412yes—
154C2×C5×Dic3C2×D412yes—
155C2×C3×Dic5—28no—
156C2×C3×Dic5—22no—
157C2×C3×Dic5—22no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—21no—
160C2×C3×Dic5—22no—
161C2×C3×Dic5—22no—
162C2×C3×Dic5C22×C414yes—
163#1200—44no—
164C23×C3×C4—51no—
165C22×#67—51no—
166C23×Dic3—51no—
167C23×Dic3C2×D511yescharacteristic
168C22×C4×D3—51no—
169C2×C3×C22⋊C4—51no—
170C2×C3×C22⋊C4C2×D511yescharacteristic
171C2×#217C2×D511yescharacteristic
172#929—58no—
173C2×(C2×D3)⋊C4—52no—
174C2×(C2×D3)⋊C4—51no—
175C2×(C2×D3)⋊C4—52no—
176C2×C4⋊Dic3—52no—
177C2×C4⋊Dic3—52no—
178C2×C4⋊Dic3—104no—
179C2×Dic3⋊C4—52no—
180C2×Dic3⋊C4—51no—
181C2×Dic3⋊C4—52no—
182C2×C4×Dic3—51no—
183C2×C4×Dic3C2×D512yes—
184#67⋊C4—104no—
185C3×#95—52no—
186C3×#95—52no—
187C3⋊#95—52no—
188C3⋊#95—52no—
189C3⋊#95C2×D512yes—
190C24×C5C22×D311yescharacteristic
191C23×D5—31no—
192C22×C4×C5—31no—
193C22×C4×C5—32no—
194C22×C4×C5—31no—
195C22×C4×C5—34no—
196C22×C4×C5—122no—
197C22×C4×C5C22×D312yes—
198C2×C22⋊D5—34no—
199C2×C22⋊D5—62no—
200C2×C22⋊D5—62no—
201C22×Dic5—22no—
202C22×Dic5—22no—
203C22×Dic5—31no—
204C22×Dic5—32no—
205C22×Dic5—62no—
206C22×Dic5—122no—
207C22×Dic5C4×D314yescomplemented
208C22×Dic5C22×D311yescharacteristic
209C22×Dic5C22×D311yescharacteristic
210C22×Dic5C22×D312yes—
211C2×C4×D5—62no—
212C2×C4×D5—122no—
213C5×C22⋊C4—22no—
214C5×C22⋊C4—34no—
215C42×C5—122no—
216C22⋊Dic5—22no—
217#685—38no—
218#685—62no—
219#685—122no—
220C4⋊Dic5—38no—
221C4⋊Dic5—122no—
222#683—64no—
223#683—64no—
224#683—62no—
225C4×Dic5—62no—
226C2×#123—151no—
227C2×C4×D4—151no—
228C22×C4⋊C4—151no—
229#389—152no—
230#388—152no—
231#387—152no—
232#386—151no—
233#385—152no—
234#385—151no—
235#381—51no—
236C4×C22⋊C4—31no—
237C22×C3×C5—24no—
238C22×C3×C5—22no—
239C22×C3×C5—22no—
240C22×C3×C5—24no—
241C22×C3×C5C4×D414yes—
242C22×C3×C5C23×C411yescharacteristic
243C22×C3×C5C22×D411yescharacteristic
244C22×C3×C5C2×#4111yescharacteristic
245C22×C3×C5H(5,𝔽2)12yes—
246C22×C3×C52−1+412yes—
247C2×D15—22no—
248C2×D15—24no—
249C2×D15—41no—
250C2×D15—41no—
251C2×D15—42no—
252C3×C4×C5—42no—
253Dic15—22no—
254Dic15—42no—
255C3×Dic5—24no—
256C3×Dic5—41no—
257C3×Dic5—41no—
258C5×Dic3—24no—
259C5×Dic3—22no—
260C5×Dic3—22no—
261C24×C3C22×D511yescharacteristic
262C23×D3—51no—
263C22×C3×C4—54no—
264C22×C3×C4—52no—
265C22×C3×C4—51no—
266C22×C3×C4—51no—
267C22×C3×C4—102no—
268C22×C3×C4—102no—
269C22×C3×C4C22×D512yes—
270C2×#67—54no—
271C2×#67—102no—
272C2×#67—102no—
273C22×Dic3—42no—
274C22×Dic3—51no—
275C22×Dic3—52no—
276C22×Dic3—102no—
277C22×Dic3—202no—
278C22×Dic3D2014yes—
279C22×Dic3C22×D512yes—
280C22×Dic3C22×D511yescharacteristic
281C22×Dic3C22×D511yescharacteristic
282C2×C4×D3—102no—
283C2×C4×D3—202no—
284C3×C22⋊C4—22no—
285C3×C22⋊C4—102no—
286#217C4×D514yescomplemented
287(C2×D3)⋊C4—58no—
288(C2×D3)⋊C4—102no—
289(C2×D3)⋊C4—202no—
290C4⋊Dic3—104no—
291C4⋊Dic3—104no—
292Dic3⋊C4—58no—
293Dic3⋊C4—102no—
294Dic3⋊C4—202no—
295C4×Dic3—42no—
296C4×Dic3—102no—
297C23×C5—22no—
298C23×C5—22no—
299C23×C5C2×C4×D312yes—
300C23×C5D3×D412yes—
301C23×C5#23712yes—
302C23×C5C23×D311yescharacteristic
303C22×D5—32no—
304C22×D5—61no—
305C22×D5—61no—
306C22×D5—62no—
307C22×D5—62no—
308C2×C4×C5—22no—
309C2×C4×C5—22no—
310C2×C4×C5—22no—
311C2×C4×C5—34no—
312C2×C4×C5—32no—
313C2×C4×C5—32no—
314C2×C4×C5—62no—
315C2×C4×C5—62no—
316C2×C4×C5—62no—
317C2×C4×C5—62no—
318C2×C4×C5—124no—
319C22⋊D5—68no—
320C2×Dic5—28no—
321C2×Dic5—22no—
322C2×Dic5—22no—
323C2×Dic5—21no—
324C2×Dic5—21no—
325C2×Dic5—22no—
326C2×Dic5—22no—
327C2×Dic5—32no—
328C2×Dic5—61no—
329C2×Dic5—61no—
330C2×Dic5—62no—
331C2×Dic5—62no—
332C2×Dic5—62no—
333C2×Dic5—124no—
334C2×Dic5C2×C4×D314yes—
335C4×D5—124no—
336C22×D4—151no—
337C23×C4—31no—
338C23×C4—51no—
339C23×C4—151no—
340C23×C4—151no—
341#123—158no—
342C4×D4—304no—
343C2×C4⋊C4—152no—
344C2×C4⋊C4—152no—
345C2×C4⋊C4—152no—
346C2×C4⋊C4—151no—
347C2×C4⋊C4—152no—
348C2×C4⋊C4—304no—
349C2×C22⋊C4—31no—
350C2×C22⋊C4—51no—
351C2×C22⋊C4—152no—
352C2×C22⋊C4—151no—
353C2×C22⋊C4—152no—
354C2×C22⋊C4D3×D511yescharacteristic
355C2×C42—32no—
356C2×C42—151no—
357#95—32no—
358#95—52no—
359#95—52no—
360#95—152no—
361#95—152no—
362C2×C3×C5—24no—
363C2×C3×C5C2×C4×D411yescharacteristic
364C2×C3×C5#52012yes—
365C2×C3×C5#53411yescharacteristic
366C2×C3×C5#53611yescharacteristic
367C2×C3×C5#53912yes—
368D15—42no—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3C2×C4×D512yes—
372C23×C3C2×D2012yes—
373C23×C3#70912yes—
374C23×C3C23×D511yescharacteristic
375C22×D3—52no—
376C22×D3—101no—
377C22×D3—101no—
378C22×D3—102no—
379C22×D3—102no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—54no—
384C2×C3×C4—102no—
385C2×C3×C4—102no—
386C2×C3×C4—108no—
387C2×C3×C4—101no—
388C2×C3×C4—101no—
389C2×C3×C4—102no—
390C2×C3×C4—102no—
391#67—108no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—44no—
397C2×Dic3—52no—
398C2×Dic3—101no—
399C2×Dic3—101no—
400C2×Dic3—102no—
401C2×Dic3—102no—
402C2×Dic3—102no—
403C2×Dic3—204no—
404C2×Dic3C2×C4×D514yes—
405C2×Dic3C2×D2012yes—
406C2×Dic3C2×D2012yes—
407C4×D3—204no—
408C22×C5—22no—
409C22×C5—22no—
410C22×C5—24no—
411C22×C5—24no—
412C22×C5#67⋊C414yes—
413C22×C5C22×C4×D311yescharacteristic
414C22×C5C2×D3×D411yescharacteristic
415C22×C5C2×#23711yescharacteristic
416C22×C5#100912yes—
417C22×C5#101012yes—
418C2×D5—62no—
419C2×D5—64no—
420C2×D5—121no—
421C2×D5—121no—
422C2×D5—122no—
423C4×C5—42no—
424C4×C5—62no—
425C4×C5—62no—
426C4×C5—64no—
427Dic5—24no—
428Dic5—41no—
429Dic5—41no—
430Dic5—62no—
431Dic5—122no—
432C24—151no—
433C24C2×D3×D511yescharacteristic
434C2×D4—154no—
435C2×D4—302no—
436C2×D4—302no—
437C22×C4—31no—
438C22×C4—32no—
439C22×C4—31no—
440C22×C4—34no—
441C22×C4—54no—
442C22×C4—52no—
443C22×C4—51no—
444C22×C4—51no—
445C22×C4—102no—
446C22×C4—102no—
447C22×C4—122no—
448C22×C4—151no—
449C22×C4—152no—
450C22×C4—302no—
451C22×C4—302no—
452C22×C4—602no—
453C22×C4—602no—
454C22×C4C2×D3×D512yes—
455C4⋊C4—158no—
456C4⋊C4—304no—
457C4⋊C4—304no—
458C4⋊C4—302no—
459C4⋊C4—602no—
460C22⋊C4—22no—
461C22⋊C4—34no—
462C22⋊C4—102no—
463C22⋊C4—158no—
464C22⋊C4—302no—
465C22⋊C4—602no—
466C42—122no—
467C42—302no—
468C3×C5#232211yescharacteristic, complemented, Hall
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—24no—
472C22×C3—24no—
473C22×C3C4×D2014yes—
474C22×C3C22×C4×D511yescharacteristic
475C22×C3C22×D2011yescharacteristic
476C22×C3C2×#70911yescharacteristic
477C22×C3#414412yes—
478C22×C3#415012yes—
479C2×D3—102no—
480C2×D3—104no—
481C2×D3—201no—
482C2×D3—201no—
483C2×D3—202no—
484C3×C4—42no—
485C3×C4—104no—
486C3×C4—201no—
487C3×C4—201no—
488Dic3—22no—
489Dic3—22no—
490Dic3—24no—
491Dic3—102no—
492Dic3—202no—
493C2×C5—24no—
494C2×C5C2×#67⋊C411yescharacteristic
495C2×C5#552312yes—
496C2×C5#564811yescharacteristic
497C2×C5#565311yescharacteristic
498C2×C5#566512yes—
499D5—122no—
500C23—22no—
501C23—22no—
502C23—152no—
503C23—301no—
504C23—302no—
505C23—302no—
506C23—301no—
507C23C4×D3×D512yes—
508C23D3×D2012yes—
509C23#677912yes—
510C23C22×D3×D511yescharacteristic
511D4—308no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—34no—
516C2×C4—32no—
517C2×C4—32no—
518C2×C4—54no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—108no—
524C2×C4—102no—
525C2×C4—102no—
526C2×C4—101no—
527C2×C4—101no—
528C2×C4—102no—
529C2×C4—102no—
530C2×C4—124no—
531C2×C4—152no—
532C2×C4—301no—
533C2×C4—301no—
534C2×C4—302no—
535C2×C4—302no—
536C2×C4—302no—
537C2×C4—604no—
538C2×C4—604no—
539C2×C3—24no—
540C2×C3C2×C4×D2011yescharacteristic
541C2×C3#6582711yescharacteristic
542C2×C3#6586011yescharacteristic
543C2×C3#6588012yes—
544C2×C3#6589812yes—
545D3—202no—
546C5#7405611yescharacteristic, complemented, Sylow
547C22—24no—
548C22—22no—
549C22—24no—
550C22—22no—
551C22—302no—
552C22—304no—
553C22—602no—
554C22—601no—
555C22—601no—
556C22#9139514yes—
557C22C2×C4×D3×D511yescharacteristic
558C22C2×D3×D2011yescharacteristic
559C22#9202012yes—
560C22#9202712yes—
561C22C2×#677911yescharacteristic
562C4—42no—
563C4—62no—
564C4—62no—
565C4—64no—
566C4—104no—
567C4—201no—
568C4—201no—
569C4—302no—
570C4—602no—
571C3#1060517711yescharacteristic, complemented, Sylow
572C2—24no—
573C2—602no—
574C2#1175282412yes—
575C2#1175287811yescharacteristic
576C2#1175327011yescharacteristic
577C2C2×#9139511yescharacteristic
578C2#1175475612yes—
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C22⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#1220C22
hypercenterC2×C22⋊C4D3×D5
hypocenterC3×C5#2322
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C22×#6668C2semidirect
5. C5×Dic3×C22⋊C4C2semidirect
6. C3×#65232C2semidirect
7. #11749517C2semidirect
8. #11748895C2semidirect
9. #11748803C2semidirect
10. #11748730C2semidirect
11. #11748656C2semidirect
24. #91546C4semidirect
46. #65232D3semidirect
68. C22×C3×Dic5C2×C4semidirect
76. C5×#217C2×C4semidirect
80. Dic3⋊Dic5C2×C4semidirect
85. #6667C2×C4semidirect
93. Dic3×C22⋊C4D5semidirect
207. C22×Dic5C4×D3semidirect
286. #217C4×D5semidirect
468. C3×C5#2322semidirect
546. C5#74056semidirect
571. C3#10605177semidirect

Maximal subgroups

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

subgroupindex
C2×#915562
C2×#915462
C22×#66682
#117510592
C5×Dic3×C22⋊C42
C3×#652322
#117495172
#117488952
#117488032
#117487302
#117486562
#106051773
#740565