Finite Group Databasebeta

← #14327153

Subgroups — #14327153

Order 1920.

1408 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#1201C212yesmaximal
2C2×C4×#1201C211yesmaximal, characteristic
3#11750750C211yesmaximal, characteristic, complemented
4C5×D3×#95C211yesmaximal, characteristic, complemented
5C3×D5×#95C211yesmaximal, characteristic, complemented
6#11748867C211yesmaximal, characteristic, complemented
7#11748804C212yesmaximal, complemented
8#11748782C211yesmaximal, characteristic, complemented
9#11748738C211yesmaximal, characteristic, complemented
10#11748722C212yesmaximal, complemented
11#11748704C212yesmaximal, complemented
12#10605100—31nomaximal, Hall
13C22×C4×C5×D3C2212yes—
14C22×C4×C5×D3C2211yescharacteristic
15C22×C3×C4×D5C2212yes—
16C22×C3×C4×D5C2211yescharacteristic
17C22×#1201C2211yescharacteristic
18C2×#6712C2212yes—
19C2×C4×Dic15C2212yes—
20C2×C4×Dic15C2211yescharacteristic
21C2×Dic3⋊Dic5C2212yes—
22C2×#6669C2212yes—
23C2×#6669C2211yescharacteristic
24C2×#6668C2212yes—
25C2×(C2×D3)⋊Dic5C2212yes—
26C2×(C2×D3)⋊Dic5C2211yescharacteristic
27C2×#6665C2212yes—
28C2×#6665C2211yescharacteristic
29C4×#1201—28no—
30C4×#1201C418yescomplemented
31C3×C5×#95C2211yescharacteristic
32#91115C2211yescharacteristic
33C5×C3⋊#95C2211yescharacteristic
34C5×C3⋊#95C2212yes—
35C3×#3963C2211yescharacteristic
36C3×#3963C2212yes—
37#90994C2212yes—
38#73369—51nomaximal, Hall
39C2×C4×C22⋊D5—31no—
40C2×C4×C22⋊D5—32no—
41C2×C5×#95—31no—
42#65501—31no—
43#65499—32no—
44#64957—32no—
45#64956—31no—
46D5×#95D311yescharacteristic, complemented
47#64939—31no—
48#64938—31no—
49#64936—32no—
50C23×C5×D3C2311yescharacteristic
51C23×C3×D5C2311yescharacteristic
52C22×C3×C4×C5C2311yescharacteristic
53C22×C3×C4×C5C2312yes—
54C22×Dic15C2312yes—
55C22×Dic15C2311yescharacteristic
56C22×Dic15C2311yescharacteristic
57C22×C5×Dic3C2312yes—
58C22×C5×Dic3C2311yescharacteristic
59C2×C4×C5×D3—24no—
60C2×C4×C5×D3—24no—
61C2×C4×C5×D3—42no—
62C2×C4×C5×D3—42no—
63C2×C4×C5×D3C2×C414yescomplemented
64C22×C3×Dic5C2312yes—
65C22×C3×Dic5C2311yescharacteristic
66C2×C3×C4×D5—24no—
67C2×C3×C4×D5—24no—
68C2×C3×C4×D5—42no—
69C2×C3×C4×D5—42no—
70C2×C3×C4×D5C2×C414yescomplemented
71C2×#1201C2×C414yes—
72C2×#1201D414yes—
73C2×#1201Q814yes—
74#6712—24no—
75C4×Dic15—24no—
76C4×Dic15C2×C414yescomplemented
77Dic3⋊Dic5—42no—
78#6669—24no—
79#6669C2×C414yescomplemented
80#6668—42no—
81(C2×D3)⋊Dic5—24no—
82(C2×D3)⋊Dic5C2×C414yescomplemented
83#6665—24no—
84#6665C2×C414yescomplemented
85C2×C4×#67—51no—
86C2×C4×#67—52no—
87C2×C3×#95—51no—
88#5212—51no—
89#5210—52no—
90#4668—52no—
91#4667—51no—
92D3×#95D511yescharacteristic, complemented
93#4650—51no—
94#4649—51no—
95#4647—52no—
96C23×C4×C5—31no—
97C23×C4×C5—32no—
98C22×C22⋊D5—31no—
99C22×C4×D5C2×D312yes—
100C22×C4×D5C2×D311yescharacteristic
101C2×C22⋊Dic5—31no—
102C2×C22⋊Dic5—32no—
103C4×C22⋊D5—38no—
104C4×C22⋊D5—68no—
105C2×#685—31no—
106C2×#685—32no—
107C2×C4⋊Dic5—32no—
108C2×#683—31no—
109C2×#683—32no—
110C2×#683—32no—
111C2×#683—32no—
112C2×C4×Dic5—31no—
113C2×C4×Dic5—32no—
114C5×#95—32no—
115C5×#95—31no—
116C5×#95C2×D311yescharacteristic
117#3963—31no—
118#3963—32no—
119#3963C2×D312yes—
120#3963C2×D311yescharacteristic
121#2349—151noSylow
122C23×C3×C5C2411yescharacteristic
123C22×C5×D3—22no—
124C22×C5×D3—22no—
125C22×C5×D3C22×C412yes—
126C22×C5×D3C2×D412yes—
127C22×C5×D3C2×Q812yes—
128C22×C3×D5—22no—
129C22×C3×D5—22no—
130C22×C3×D5C22×C412yes—
131C22×C3×D5C2×D412yes—
132C22×C3×D5C2×Q812yes—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5C22×C412yes—
137C2×C3×C4×C5#4114yes—
138C2×Dic15—22no—
139C2×Dic15—22no—
140C2×Dic15—22no—
141C2×Dic15—22no—
142C2×Dic15C22×C412yes—
143C2×Dic15C22×C412yes—
144C2×Dic15C2×D412yes—
145C2×Dic15C2×Q812yes—
146C2×C5×Dic3—22no—
147C2×C5×Dic3—22no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—22no—
150C2×C5×Dic3C22×C412yes—
151C4×C5×D3—44no—
152C4×C5×D3—48no—
153C2×C3×Dic5—22no—
154C2×C3×Dic5—22no—
155C2×C3×Dic5—22no—
156C2×C3×Dic5—22no—
157C2×C3×Dic5C22×C412yes—
158C3×C4×D5—44no—
159C3×C4×D5—48no—
160#1201C4⋊C4116yescomplemented
161C23×C3×C4—51no—
162C23×C3×C4—52no—
163C22×#67—51no—
164C22×C4×D3C2×D512yes—
165C22×C4×D3C2×D511yescharacteristic
166C2×#217—51no—
167C2×#217—52no—
168C4×#67—58no—
169C4×#67—108no—
170C2×(C2×D3)⋊C4—51no—
171C2×(C2×D3)⋊C4—52no—
172C2×C4⋊Dic3—52no—
173C2×Dic3⋊C4—51no—
174C2×Dic3⋊C4—52no—
175C2×Dic3⋊C4—52no—
176C2×Dic3⋊C4—52no—
177C2×C4×Dic3—51no—
178C2×C4×Dic3—52no—
179C3×#95—52no—
180C3×#95—51no—
181C3×#95C2×D511yescharacteristic
182C3⋊#95—51no—
183C3⋊#95—52no—
184C3⋊#95C2×D512yes—
185C3⋊#95C2×D511yescharacteristic
186C24×C5—31no—
187C23×D5C22×D311yescharacteristic
188C22×C4×C5—31no—
189C22×C4×C5—34no—
190C22×C4×C5—32no—
191C22×C4×C5—64no—
192C22×C4×C5—64no—
193C22×C4×C5—122no—
194C22×C4×C5—122no—
195C22×C4×C5C22×D311yescharacteristic
196C22×C4×C5C22×D312yes—
197C2×C22⋊D5—34no—
198C2×C22⋊D5—34no—
199C2×C22⋊D5—34no—
200C22×Dic5—31no—
201C22×Dic5—31no—
202C22×Dic5—32no—
203C22×Dic5C22×D312yes—
204C22×Dic5C22×D311yescharacteristic
205C2×C4×D5—24no—
206C2×C4×D5—24no—
207C2×C4×D5—42no—
208C2×C4×D5—42no—
209C2×C4×D5C4×D314yescomplemented
210C22⋊Dic5—34no—
211C22⋊Dic5—64no—
212#685—34no—
213#685—64no—
214C4⋊Dic5—122no—
215#683—34no—
216#683—64no—
217#683—64no—
218#683—122no—
219C4×Dic5—34no—
220C4×Dic5—64no—
221C2×C4×D4—151no—
222C2×C4×D4—152no—
223#389—152no—
224#385—152no—
225#385—152no—
226#382—151no—
227#380—151no—
228#380—151no—
229C4×C4⋊C4—151no—
230C2×#95—31no—
231C2×#95—51no—
232C22×C3×C5C23×C411yescharacteristic
233C22×C3×C5C22×D411yescharacteristic
234C22×C3×C5C22×Q811yescharacteristic
235C22×C3×C5C2×#4112yes—
236C22×C3×C5H(5,𝔽2)12yes—
237C2×C5×D3—22no—
238C2×C5×D3—22no—
239C2×C5×D3—24no—
240C2×C5×D3—24no—
241C2×C5×D3C2×C4⋊C414yes—
242C2×C3×D5—22no—
243C2×C3×D5—22no—
244C2×C3×D5—24no—
245C2×C3×D5—24no—
246C2×C3×D5C2×C4⋊C414yes—
247C3×C4×C5—22no—
248C3×C4×C5—24no—
249Dic15—22no—
250Dic15—42no—
251Dic15C2×C4⋊C414yes—
252C3×Dic5—22no—
253C3×Dic5—42no—
254C5×Dic3—22no—
255C5×Dic3—42no—
256C24×C3—51no—
257C23×D3C22×D511yescharacteristic
258C22×C3×C4—51no—
259C22×C3×C4—54no—
260C22×C3×C4—52no—
261C22×C3×C4—104no—
262C22×C3×C4—104no—
263C22×C3×C4—202no—
264C22×C3×C4—202no—
265C22×C3×C4C22×D511yescharacteristic
266C22×C3×C4C22×D512yes—
267C2×#67—54no—
268C2×#67—54no—
269C2×#67—54no—
270C22×Dic3—51no—
271C22×Dic3—51no—
272C22×Dic3—52no—
273C22×Dic3C22×D512yes—
274C22×Dic3C22×D511yescharacteristic
275C2×C4×D3—24no—
276C2×C4×D3—24no—
277C2×C4×D3—42no—
278C2×C4×D3—42no—
279C2×C4×D3C4×D514yescomplemented
280#217—54no—
281#217—104no—
282(C2×D3)⋊C4—54no—
283(C2×D3)⋊C4—104no—
284C4⋊Dic3—202no—
285Dic3⋊C4—54no—
286Dic3⋊C4—104no—
287Dic3⋊C4—104no—
288Dic3⋊C4—202no—
289C4×Dic3—54no—
290C4×Dic3—104no—
291C23×C5—32no—
292C23×C5—32no—
293C23×C5—32no—
294C23×C5—62no—
295C23×C5—62no—
296C23×C5C23×D311yescharacteristic
297C22×D5—22no—
298C22×D5—22no—
299C22×D5C2×C4×D312yes—
300C22×D5D3×D412yes—
301C22×D5D3×Q812yes—
302C2×C4×C5—22no—
303C2×C4×C5—22no—
304C2×C4×C5—22no—
305C2×C4×C5—32no—
306C2×C4×C5—62no—
307C2×C4×C5—62no—
308C2×C4×C5—62no—
309C2×C4×C5—62no—
310C2×C4×C5—124no—
311C2×C4×C5—128no—
312C2×C4×C5C2×C4×D312yes—
313C2×C4×C5#23514yes—
314C22⋊D5—316no—
315C2×Dic5—22no—
316C2×Dic5—22no—
317C2×Dic5—22no—
318C2×Dic5—22no—
319C2×Dic5—32no—
320C2×Dic5—32no—
321C2×Dic5—32no—
322C2×Dic5—32no—
323C2×Dic5—62no—
324C2×Dic5—62no—
325C2×Dic5—62no—
326C2×Dic5—62no—
327C2×Dic5C2×C4×D312yes—
328C4×D5—44no—
329C4×D5—48no—
330C22×D4—151no—
331C23×C4—31no—
332C23×C4—32no—
333C23×C4—51no—
334C23×C4—52no—
335C4×D4—158no—
336C4×D4—308no—
337C2×C4⋊C4—151no—
338C2×C4⋊C4—152no—
339C2×C4⋊C4—152no—
340C2×C4⋊C4—152no—
341C2×C4⋊C4—152no—
342C2×C22⋊C4—151no—
343C2×C22⋊C4—151no—
344C2×C22⋊C4—152no—
345C2×C22⋊C4—152no—
346C2×C42—151no—
347C2×C42—152no—
348#95—31no—
349#95—32no—
350#95—51no—
351#95—52no—
352#95—152no—
353#95—151no—
354#95D3×D511yescharacteristic
355C2×C3×C5C22×C4⋊C411yescharacteristic
356C2×C3×C5C4×#4111yescharacteristic
357C2×C3×C5#51811yescharacteristic
358C2×C3×C5#54612yes—
359C2×C3×C5#55412yes—
360C3×D5—24no—
361C5×D3—24no—
362C23×C3—52no—
363C23×C3—52no—
364C23×C3—52no—
365C23×C3—102no—
366C23×C3—102no—
367C23×C3C23×D511yescharacteristic
368C22×D3—22no—
369C22×D3—22no—
370C22×D3C2×C4×D512yes—
371C22×D3D4×D512yes—
372C22×D3Q8×D512yes—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—52no—
377C2×C3×C4—102no—
378C2×C3×C4—102no—
379C2×C3×C4—102no—
380C2×C3×C4—102no—
381C2×C3×C4—204no—
382C2×C3×C4—208no—
383C2×C3×C4C2×C4×D512yes—
384C2×C3×C4#70914yes—
385#67—516no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—52no—
391C2×Dic3—52no—
392C2×Dic3—52no—
393C2×Dic3—52no—
394C2×Dic3—102no—
395C2×Dic3—102no—
396C2×Dic3—102no—
397C2×Dic3—102no—
398C2×Dic3C2×C4×D512yes—
399C4×D3—44no—
400C4×D3—48no—
401C22×C5—34no—
402C22×C5—62no—
403C22×C5—62no—
404C22×C5—64no—
405C22×C5—64no—
406C22×C5C22×C4×D311yescharacteristic
407C22×C5C2×#23512yes—
408C22×C5C2×D3×D411yescharacteristic
409C22×C5#100412yes—
410C22×C5C2×D3×Q811yescharacteristic
411C2×D5—22no—
412C2×D5—22no—
413C2×D5—24no—
414C2×D5—24no—
415C2×D5D3×C4⋊C414yes—
416C4×C5—22no—
417C4×C5—24no—
418C4×C5—62no—
419C4×C5—122no—
420Dic5—22no—
421Dic5—34no—
422Dic5—42no—
423Dic5—62no—
424Dic5—122no—
425C24—31no—
426C24—51no—
427C2×D4—154no—
428C2×D4—154no—
429C2×D4—154no—
430C22×C4—31no—
431C22×C4—34no—
432C22×C4—32no—
433C22×C4—51no—
434C22×C4—54no—
435C22×C4—52no—
436C22×C4—64no—
437C22×C4—64no—
438C22×C4—104no—
439C22×C4—104no—
440C22×C4—122no—
441C22×C4—122no—
442C22×C4—151no—
443C22×C4—151no—
444C22×C4—152no—
445C22×C4—202no—
446C22×C4—202no—
447C22×C4C2×D3×D512yes—
448C22×C4C2×D3×D511yescharacteristic
449C4⋊C4—154no—
450C4⋊C4—304no—
451C4⋊C4—304no—
452C4⋊C4—602no—
453C4⋊C4—602no—
454C22⋊C4—154no—
455C22⋊C4—154no—
456C22⋊C4—304no—
457C22⋊C4—304no—
458C42—154no—
459C42—304no—
460C3×C5#234911yescharacteristic, complemented, Hall
461C22×C3—54no—
462C22×C3—102no—
463C22×C3—102no—
464C22×C3—104no—
465C22×C3—104no—
466C22×C3C22×C4×D511yescharacteristic
467C22×C3C2×#70912yes—
468C22×C3C2×D4×D511yescharacteristic
469C22×C3#414412yes—
470C22×C3C2×Q8×D511yescharacteristic
471C2×D3—22no—
472C2×D3—22no—
473C2×D3—24no—
474C2×D3—24no—
475C2×D3D5×C4⋊C414yes—
476C3×C4—22no—
477C3×C4—24no—
478C3×C4—102no—
479C3×C4—202no—
480Dic3—22no—
481Dic3—42no—
482Dic3—54no—
483Dic3—102no—
484Dic3—202no—
485C2×C5—64no—
486C2×C5C4×#23511yescharacteristic
487C2×C5#548611yescharacteristic
488C2×C5#549012yes—
489C2×C5C2×D3×C4⋊C411yescharacteristic
490C2×C5#551112yes—
491D5—24no—
492C23—32no—
493C23—32no—
494C23—32no—
495C23—52no—
496C23—52no—
497C23—52no—
498C23—62no—
499C23—62no—
500C23—102no—
501C23—102no—
502C23C22×D3×D511yescharacteristic
503D4—1516no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—32no—
508C2×C4—52no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—102no—
514C2×C4—102no—
515C2×C4—102no—
516C2×C4—102no—
517C2×C4—124no—
518C2×C4—128no—
519C2×C4—152no—
520C2×C4—152no—
521C2×C4—152no—
522C2×C4—152no—
523C2×C4—204no—
524C2×C4—208no—
525C2×C4—302no—
526C2×C4—302no—
527C2×C4—302no—
528C2×C4—302no—
529C2×C4#677114yes—
530C2×C4C4×D3×D512yes—
531C2×C3—104no—
532C2×C3C4×#70911yescharacteristic
533C2×C3#6582011yescharacteristic
534C2×C3#6582412yes—
535C2×C3C2×D5×C4⋊C411yescharacteristic
536C2×C3#6584512yes—
537D3—24no—
538C5#7336911yescharacteristic, complemented, Sylow
539C22—34no—
540C22—54no—
541C22—62no—
542C22—64no—
543C22—62no—
544C22—64no—
545C22—102no—
546C22—104no—
547C22—102no—
548C22—104no—
549C22C2×#677112yes—
550C22C2×C4×D3×D511yescharacteristic
551C22D3×D4×D511yescharacteristic
552C22#9202512yes—
553C22D3×Q8×D511yescharacteristic
554C4—22no—
555C4—24no—
556C4—62no—
557C4—102no—
558C4—122no—
559C4—154no—
560C4—202no—
561C4—302no—
562C4—602no—
563C3#1060510011yescharacteristic, complemented, Sylow
564C2—64no—
565C2—104no—
566C2C4×#677111yescharacteristic
567C2#1175306511yescharacteristic
568C2#1175306812yes—
569C2#1175370512yes—
570C2D3×D5×C4⋊C411yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC3×C5×#95C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#1201C22
hypercenter#95D3×D5
hypocenterC3×C5#2349
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
3. #11750750C2semidirect
4. C5×D3×#95C2semidirect
5. C3×D5×#95C2semidirect
6. #11748867C2semidirect
7. #11748804C2semidirect
8. #11748782C2semidirect
9. #11748738C2semidirect
10. #11748722C2semidirect
11. #11748704C2semidirect
30. C4×#1201C4semidirect
46. D5×#95D3semidirect
63. C2×C4×C5×D3C2×C4semidirect
70. C2×C3×C4×D5C2×C4semidirect
76. C4×Dic15C2×C4semidirect
79. #6669C2×C4semidirect
82. (C2×D3)⋊Dic5C2×C4semidirect
84. #6665C2×C4semidirect
92. D3×#95D5semidirect
160. #1201C4⋊C4semidirect
209. C2×C4×D5C4×D3semidirect
279. C2×C4×D3C4×D5semidirect
460. C3×C5#2349semidirect
538. C5#73369semidirect
563. C3#10605100semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#12012
#117507502
C5×D3×#952
C3×D5×#952
#117488672
#117488042
#117487822
#117487382
#117487222
#117487042
#106051003
#733695