Finite Group Databasebeta

← C2×#11754494 #14483867

Subgroups — C2×#11754494 #14483867

Order 1920.

2400 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×#91813C211yesmaximal, characteristic, complemented
2C2×C4×C5×#67C211yesmaximal, characteristic
3C22×C3×C4⋊Dic5C211yesmaximal, characteristic, complemented
4C2×#91574C212yesmaximal, complemented
5C2×Dic5×#67C212yesmaximal
6C2×#91469C211yesmaximal, characteristic, complemented
7#11754494C2116yesmaximal, direct product factor
8C2×#91436C212yesmaximal, complemented
9C2×D3×C4⋊Dic5C211yesmaximal, characteristic, complemented
10C2×#91365C211yesmaximal, characteristic, complemented
11C2×#91346C212yesmaximal, complemented
12C2×#91340C211yesmaximal, characteristic, complemented
13C2×#65877—31nomaximal, Hall
14C23×C3×C4×C5C2211yescharacteristic
15C22×Dic30C2211yescharacteristic
16C22×C5×#67C2211yescharacteristic
17C22×C4×C5×D3C2211yescharacteristic
18C23×C3×Dic5C2212yes—
19C22×#1204C2212yes—
20C22×D3×Dic5C2212yes—
21C2×#6719C2212yes—
22C2×C4⋊Dic15C2211yescharacteristic
23#91813C2218yescomplemented
24C2×#6712C2212yes—
25C2×C5×#217C2211yescharacteristic
26C4×C5×#67C2218yes—
27C2×C5×(C2×D3)⋊C4C2211yescharacteristic
28C2×C5×Dic3⋊C4C2211yescharacteristic
29C2×C4×C5×Dic3C2211yescharacteristic
30C2×C3×C4⋊Dic5C2211yescharacteristic
31C2×C3×C4⋊Dic5C2211yescharacteristic
32C2×C3×C4⋊Dic5C2212yes—
33C2×C3×C4⋊Dic5C2218yescomplemented
34#91574C22116yescomplemented
35Dic5×#67C22116yes—
36C2×Dic3⋊Dic5C2212yes—
37C2×Dic3⋊Dic5C2212yes—
38C2×#6669C2212yes—
39C2×#6668C2212yes—
40C2×(C2×D3)⋊Dic5C2212yes—
41C2×Dic3×Dic5C2212yes—
42#91469C2218yescomplemented
43#91436C22116yescomplemented
44D3×C4⋊Dic5C2218yescomplemented
45#91365C2218yescomplemented
46#91346C22116yescomplemented
47#91340C2218yescomplemented
48C2×#5511—51nomaximal, Hall
49C2×C4×C5×D4—31no—
50C2×D4×Dic5—32no—
51C22×C4⋊Dic5—31no—
52C22×C4⋊Dic5D311yescharacteristic, complemented
53C2×#4070—31no—
54C2×#4070—31no—
55#65877—316no—
56C2×#4036—32no—
57C2×#4024—32no—
58C2×#4024—32no—
59C2×#4015—31no—
60C2×C4×Dic10—31no—
61C24×C3×C5C2311yescharacteristic
62C23×C5×D3C2311yescharacteristic
63C22×C3×C4×C5—24no—
64C22×C3×C4×C5C2311yescharacteristic
65C22×C3×C4×C5C2314yes—
66C22×C3×C4×C5C2311yescharacteristic
67C22×Dic15C2312yes—
68C22×Dic15C2312yes—
69C2×Dic30—24no—
70C2×Dic30C2314yes—
71C2×C5×#67Q818yescomplemented
72C2×C5×#67C2314yes—
73C22×C5×Dic3C2311yescharacteristic
74C22×C5×Dic3C2311yescharacteristic
75C22×C5×Dic3C2311yescharacteristic
76C2×C4×C5×D3—24no—
77C2×C4×C5×D3C2314yes—
78C22×C3×Dic5—44no—
79C22×C3×Dic5C2312yes—
80C22×C3×Dic5C2312yes—
81C22×C3×Dic5C2318yes—
82C2×#1204—44no—
83C2×#1204C2318yes—
84C2×D3×Dic5—44no—
85C2×D3×Dic5C2318yes—
86#6719C2318yes—
87C4⋊Dic15C2314yes—
88#6712C2318yes—
89C5×#217C2314yes—
90C5×(C2×D3)⋊C4C2314yes—
91C5×Dic3⋊C4C2314yes—
92C4×C5×Dic3C2314yes—
93C3×C4⋊Dic5C2314yes—
94C3×C4⋊Dic5C2318yes—
95C3×C4⋊Dic5C2314yes—
96Dic3⋊Dic5C2318yes—
97Dic3⋊Dic5C2318yes—
98#6669C2318yes—
99#6668C2318yes—
100(C2×D3)⋊Dic5C2318yes—
101Dic3×Dic5C2318yes—
102C22×C3×C4⋊C4—51no—
103C2×C4×#67—52no—
104C2×C4×#67D511yescharacteristic
105C2×#924—51no—
106C2×#924—52no—
107#5511—516no—
108C2×#897—51no—
109C2×#896—52no—
110C2×D3×C4⋊C4—51no—
111C2×#889—52no—
112C2×#888—51no—
113C2×#887—51no—
114C22×C5×D4—31no—
115C23×C4×C5—31no—
116C23×C4×C5C2×D311yescharacteristic
117C23×Dic5—32no—
118C23×Dic5C2×D312yes—
119C22×Dic10—31no—
120C22×Dic10—32no—
121C4×C5×D4—38no—
122C2×C5×C4⋊C4—31no—
123C2×C5×C22⋊C4—31no—
124C2×C5×C22⋊C4—31no—
125C2×C42×C5—31no—
126C2×C22⋊Dic5—32no—
127C2×C22⋊Dic5—32no—
128D4×Dic5—316no—
129C2×C4⋊Dic5—31no—
130C2×C4⋊Dic5—32no—
131C2×C4⋊Dic5—32no—
132C2×C4⋊Dic5—38no—
133C2×C4⋊Dic5C2×D311yescharacteristic
134C2×C4⋊Dic5C2×D311yescharacteristic
135C2×C4⋊Dic5C2×D312yes—
136C2×C4⋊Dic5C2×D318yescomplemented
137#4070—38no—
138#4070—38no—
139C2×#683—32no—
140C2×#683—32no—
141C2×#683—32no—
142C2×C4×Dic5—32no—
143#4036—316no—
144#4024—316no—
145#4024—316no—
146#4015—38no—
147C4×Dic10—38no—
148C2×#554—151noSylow
149C23×C3×C5—22no—
150C23×C3×C5—22no—
151C23×C3×C5C2×Q814yes—
152C23×C3×C5C2412yes—
153C23×C3×C5C2411yescharacteristic
154C22×C5×D3—22no—
155C22×C5×D3—22no—
156C22×C5×D3C2×Q814yes—
157C22×C5×D3C2412yes—
158C2×C3×C4×C5—22no—
159C2×C3×C4×C5—28no—
160C2×C3×C4×C5#4114yes—
161C2×C3×C4×C5C2412yes—
162C2×C3×C4×C5C2412yes—
163C2×Dic15—24no—
164C2×Dic15—24no—
165C2×Dic15C2414yes—
166C2×Dic15C2414yes—
167Dic30—28no—
168C5×#67C2×Q8116yescomplemented
169C2×C5×Dic3—22no—
170C2×C5×Dic3—22no—
171C2×C5×Dic3C2×Q814yes—
172C2×C5×Dic3C2412yes—
173C2×C5×Dic3C2412yes—
174C2×C5×Dic3C2412yes—
175C4×C5×D3—28no—
176C2×C3×Dic5—24no—
177C2×C3×Dic5—24no—
178C2×C3×Dic5—48no—
179C2×C3×Dic5C2414yes—
180C2×C3×Dic5C2414yes—
181#1204—48no—
182D3×Dic5—48no—
183C23×C3×C4—52no—
184C23×C3×C4C2×D511yescharacteristic
185C22×#67C2×D511yescharacteristic
186C22×C4×D3—52no—
187C22×C4×D3C2×D511yescharacteristic
188C22×Dic6—51no—
189C22×Dic6—52no—
190C2×C3×C4⋊C4—51no—
191C2×C3×C4⋊C4—51no—
192C2×C3×C4⋊C4—52no—
193C2×C3×C4⋊C4—58no—
194C2×#217—52no—
195C2×#217C2×D511yescharacteristic
196C4×#67—516no—
197C4×#67C2×D518yes—
198C2×(C2×D3)⋊C4—52no—
199C2×(C2×D3)⋊C4C2×D511yescharacteristic
200C2×C4⋊Dic3—51no—
201C2×C4⋊Dic3—52no—
202#924—58no—
203#924—516no—
204C2×Dic3⋊C4—52no—
205C2×Dic3⋊C4—52no—
206C2×Dic3⋊C4—52no—
207C2×Dic3⋊C4—52no—
208C2×Dic3⋊C4C2×D511yescharacteristic
209C2×C4×Dic3—52no—
210C2×C4×Dic3C2×D511yescharacteristic
211#897—58no—
212#896—516no—
213D3×C4⋊C4—58no—
214#889—516no—
215#888—58no—
216#887—58no—
217C24×C5—31no—
218C24×C5C22×D311yescharacteristic
219C2×C5×D4—38no—
220C2×C5×D4—34no—
221C22×C4×C5—24no—
222C22×C4×C5—31no—
223C22×C4×C5—31no—
224C22×C4×C5—31no—
225C22×C4×C5—34no—
226C22×C4×C5—64no—
227C22×C4×C5C22×D311yescharacteristic
228C22×C4×C5C22×D314yes—
229C22×C4×C5C22×D311yescharacteristic
230C22×Dic5—32no—
231C22×Dic5—32no—
232C22×Dic5—38no—
233C22×Dic5—44no—
234C22×Dic5—124no—
235C22×Dic5C22×D312yes—
236C22×Dic5C22×D312yes—
237C22×Dic5C22×D318yes—
238C2×Dic10—34no—
239C2×Dic10—38no—
240C2×Dic10—64no—
241C2×Dic10—124no—
242C5×C4⋊C4—34no—
243C5×C22⋊C4—34no—
244C5×C22⋊C4—34no—
245C42×C5—34no—
246C22⋊Dic5—38no—
247C22⋊Dic5—38no—
248C4⋊Dic5—34no—
249C4⋊Dic5—38no—
250C4⋊Dic5—38no—
251C4⋊Dic5C22×D314yes—
252C4⋊Dic5C22×D314yes—
253C4⋊Dic5C22×D318yes—
254#683—38no—
255#683—38no—
256#683—38no—
257C4×Dic5—38no—
258#554—1516no—
259C2×C4⋊Q8—151no—
260C2×#125—152no—
261C2×C22⋊Q8—152no—
262C2×C22⋊Q8—152no—
263C2×C22⋊Q8—151no—
264C2×C22⋊Q8—151no—
265C2×C4×Q8—151no—
266C2×C4×D4—31no—
267C2×C4×D4—152no—
268C22×C4⋊C4—51no—
269C22×C4⋊C4—151no—
270C22×C3×C5—28no—
271C22×C3×C5—22no—
272C22×C3×C5—22no—
273C22×C3×C5C22×Q814yes—
274C22×C3×C5C22×Q812yes—
275C22×C3×C5C2×#4112yes—
276C22×C3×C5H(5,𝔽2)12yes—
277C22×C3×C5C2511yescharacteristic
278C2×C5×D3—28no—
279C2×C5×D3—22no—
280C2×C5×D3—22no—
281C2×C5×D3C22×Q814yes—
282C3×C4×C5—22no—
283C3×C4×C5C2×#4114yes—
284Dic15—24no—
285Dic15—24no—
286C3×Dic5—24no—
287C3×Dic5—24no—
288C5×Dic3—22no—
289C5×Dic3—22no—
290C5×Dic3C22×Q814yes—
291C24×C3C22×D511yescharacteristic
292C23×D3C22×D511yescharacteristic
293C22×C3×C4—24no—
294C22×C3×C4—52no—
295C22×C3×C4—52no—
296C22×C3×C4—58no—
297C22×C3×C4—204no—
298C22×C3×C4C22×D511yescharacteristic
299C22×C3×C4C22×D514yes—
300C22×C3×C4C22×D511yescharacteristic
301C2×#67Dic1018yescomplemented
302C2×#67C22×D514yes—
303C22×Dic3—52no—
304C22×Dic3—52no—
305C22×Dic3C22×D511yescharacteristic
306C22×Dic3C22×D511yescharacteristic
307C22×Dic3C22×D511yescharacteristic
308C2×C4×D3—24no—
309C2×C4×D3—58no—
310C2×C4×D3—204no—
311C2×C4×D3C22×D514yes—
312C2×Dic6—54no—
313C2×Dic6—58no—
314C2×Dic6—104no—
315C2×Dic6—204no—
316C3×C4⋊C4—54no—
317C3×C4⋊C4—54no—
318C3×C4⋊C4—58no—
319#217—58no—
320#217C22×D514yes—
321(C2×D3)⋊C4—58no—
322(C2×D3)⋊C4C22×D514yes—
323C4⋊Dic3—54no—
324C4⋊Dic3—58no—
325Dic3⋊C4—58no—
326Dic3⋊C4—58no—
327Dic3⋊C4—58no—
328Dic3⋊C4—58no—
329Dic3⋊C4C22×D514yes—
330C4×Dic3—58no—
331C4×Dic3C22×D514yes—
332C23×C5—22no—
333C23×C5—22no—
334C23×C5—34no—
335C23×C5—32no—
336C23×C5—62no—
337C23×C5—62no—
338C23×C5D3×Q814yes—
339C23×C5C23×D312yes—
340C23×C5C23×D311yescharacteristic
341C5×D4—316no—
342C2×C4×C5—22no—
343C2×C4×C5—28no—
344C2×C4×C5—34no—
345C2×C4×C5—32no—
346C2×C4×C5—32no—
347C2×C4×C5—32no—
348C2×C4×C5—62no—
349C2×C4×C5—62no—
350C2×C4×C5—68no—
351C2×C4×C5#23514yes—
352C2×C4×C5C23×D312yes—
353C2×C4×C5C23×D312yes—
354C2×Dic5—24no—
355C2×Dic5—24no—
356C2×Dic5—34no—
357C2×Dic5—34no—
358C2×Dic5—48no—
359C2×Dic5—64no—
360C2×Dic5—64no—
361C2×Dic5—128no—
362C2×Dic5C23×D314yes—
363C2×Dic5C23×D314yes—
364Dic10—68no—
365Dic10—128no—
366C22×Q8—152no—
367C22×Q8—151no—
368C22×D4—31no—
369C23×C4—31no—
370C23×C4—52no—
371C23×C4—152no—
372C23×C4D3×D511yescharacteristic
373C4⋊Q8—158no—
374#125—1516no—
375C22⋊Q8—1516no—
376C22⋊Q8—1516no—
377C22⋊Q8—158no—
378C22⋊Q8—158no—
379C4×Q8—158no—
380C4×D4—38no—
381C4×D4—1516no—
382C2×C4⋊C4—31no—
383C2×C4⋊C4—51no—
384C2×C4⋊C4—52no—
385C2×C4⋊C4—51no—
386C2×C4⋊C4—58no—
387C2×C4⋊C4—158no—
388C2×C4⋊C4—152no—
389C2×C4⋊C4—152no—
390C2×C4⋊C4—152no—
391C2×C4⋊C4—152no—
392C2×C4⋊C4—151no—
393C2×C4⋊C4—152no—
394C2×C22⋊C4—31no—
395C2×C22⋊C4—31no—
396C2×C22⋊C4—152no—
397C2×C22⋊C4—152no—
398C2×C42—31no—
399C2×C42—152no—
400C2×C3×C5—24no—
401C2×C3×C5#55414yescomplemented
402C2×C3×C5C23×Q811yescharacteristic
403C2×C3×C5C22×#4111yescharacteristic
404C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
405C5×D3—24no—
406C23×C3—22no—
407C23×C3—22no—
408C23×C3C2×Dic1014yes—
409C23×C3C23×D512yes—
410C23×C3C23×D511yescharacteristic
411C22×D3—22no—
412C22×D3—22no—
413C22×D3C2×Dic1014yes—
414C22×D3C23×D512yes—
415C2×C3×C4—22no—
416C2×C3×C4—28no—
417C2×C3×C4—54no—
418C2×C3×C4—54no—
419C2×C3×C4—104no—
420C2×C3×C4—104no—
421C2×C3×C4—208no—
422C2×C3×C4#71114yes—
423C2×C3×C4C23×D512yes—
424C2×C3×C4C23×D512yes—
425#67C2×Dic10116yescomplemented
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—54no—
429C2×Dic3—54no—
430C2×Dic3—104no—
431C2×Dic3—104no—
432C2×Dic3C2×Dic1014yes—
433C2×Dic3C23×D512yes—
434C2×Dic3C23×D512yes—
435C2×Dic3C23×D512yes—
436C4×D3—28no—
437C4×D3—208no—
438Dic6—108no—
439Dic6—208no—
440C22×C5—22no—
441C22×C5—22no—
442C22×C5—28no—
443C22×C5—34no—
444C22×C5—62no—
445C22×C5—62no—
446C22×C5—68no—
447C22×C5C2×#23512yes—
448C22×C5#100412yes—
449C22×C5C2×D3×Q814yes—
450C22×C5C2×D3×Q812yes—
451C22×C5C24×D311yescharacteristic
452C4×C5—22no—
453C4×C5—34no—
454C4×C5—62no—
455C4×C5—62no—
456C4×C5C2×#23514yes—
457Dic5—24no—
458Dic5—24no—
459Dic5—64no—
460Dic5—64no—
461C24—31no—
462C24C2×D3×D511yescharacteristic
463C2×Q8—158no—
464C2×Q8—154no—
465C2×Q8—304no—
466C2×Q8—604no—
467C2×D4—38no—
468C2×D4—34no—
469C22×C4—24no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—34no—
474C22×C4—52no—
475C22×C4—52no—
476C22×C4—58no—
477C22×C4—64no—
478C22×C4—158no—
479C22×C4—152no—
480C22×C4—152no—
481C22×C4—204no—
482C22×C4—604no—
483C22×C4C2×D3×D514yes—
484C22×C4C2×D3×D511yescharacteristic
485C22×C4C2×D3×D511yescharacteristic
486C4⋊C4—34no—
487C4⋊C4—54no—
488C4⋊C4—58no—
489C4⋊C4—54no—
490C4⋊C4—158no—
491C4⋊C4—158no—
492C4⋊C4—158no—
493C4⋊C4—158no—
494C4⋊C4—154no—
495C4⋊C4—158no—
496C22⋊C4—34no—
497C22⋊C4—34no—
498C22⋊C4—158no—
499C22⋊C4—158no—
500C42—34no—
501C42—158no—
502C3×C5C2×#55411yescharacteristic, complemented, Hall
503C22×C3—28no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3C22×Dic1014yes—
507C22×C3C22×Dic1012yes—
508C22×C3C2×#71112yes—
509C22×C3#414912yes—
510C22×C3C24×D511yescharacteristic
511C2×D3—28no—
512C2×D3—22no—
513C2×D3—22no—
514C2×D3C22×Dic1014yes—
515C3×C4—22no—
516C3×C4—104no—
517C3×C4—104no—
518C3×C4C2×#71114yes—
519Dic3—22no—
520Dic3—22no—
521Dic3—104no—
522Dic3—104no—
523Dic3C22×Dic1014yes—
524C2×C5—24no—
525C2×C5—64no—
526C2×C5#551114yescomplemented
527C2×C5C22×#23511yescharacteristic
528C2×C5C2×#100411yescharacteristic
529C2×C5C22×D3×Q811yescharacteristic
530C23—22no—
531C23—22no—
532C23—34no—
533C23—32no—
534C23—62no—
535C23—62no—
536C23D3×Dic1014yes—
537C23C22×D3×D512yes—
538C23C22×D3×D511yescharacteristic
539Q8—308no—
540Q8—608no—
541D4—316no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—34no—
545C2×C4—32no—
546C2×C4—32no—
547C2×C4—32no—
548C2×C4—54no—
549C2×C4—54no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—68no—
553C2×C4—104no—
554C2×C4—104no—
555C2×C4—154no—
556C2×C4—154no—
557C2×C4—208no—
558C2×C4—304no—
559C2×C4—304no—
560C2×C4—608no—
561C2×C4#678214yes—
562C2×C4C22×D3×D512yes—
563C2×C4C22×D3×D512yes—
564C2×C3—24no—
565C2×C3#6587714yescomplemented
566C2×C3C23×Dic1011yescharacteristic
567C2×C3C22×#71111yescharacteristic
568C2×C3C2×#414911yescharacteristic
569D3—24no—
570C5C2×#551111yescharacteristic, complemented, Sylow
571C22—22no—
572C22—22no—
573C22—28no—
574C22—34no—
575C22—62no—
576C22—62no—
577C22—68no—
578C22C2×D3×Dic1014yes—
579C22C2×D3×Dic1012yes—
580C22#9201712yes—
581C22C2×#678212yes—
582C22C23×D3×D511yescharacteristic
583C4—22no—
584C4—34no—
585C4—62no—
586C4—62no—
587C4—104no—
588C4—104no—
589C4—304no—
590C4—304no—
591C4C2×#678214yes—
592C3C2×#6587711yescharacteristic, complemented, Sylow
593C2—24no—
594C2—64no—
595C2#1175449414yesdirect product factor
596C2C22×D3×Dic1011yescharacteristic
597C2C2×#9201711yescharacteristic
598C2C22×#678211yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC23×C3×C4×C5C22
socleC23×C3×C5C24
prime radicalC22×C5×#67C22
hypercenterC23×C4D3×D5
hypocenterC3×C5C2×#554
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C2×#91813C2semidirect
3. C22×C3×C4⋊Dic5C2semidirect
4. C2×#91574C2semidirect
6. C2×#91469C2semidirect
7. #11754494C2direct (N × H)
8. C2×#91436C2semidirect
9. C2×D3×C4⋊Dic5C2semidirect
10. C2×#91365C2semidirect
11. C2×#91346C2semidirect
12. C2×#91340C2semidirect
23. #91813C22semidirect
33. C2×C3×C4⋊Dic5C22semidirect
34. #91574C22semidirect
42. #91469C22semidirect
43. #91436C22semidirect
44. D3×C4⋊Dic5C22semidirect
45. #91365C22semidirect
46. #91346C22semidirect
47. #91340C22semidirect
52. C22×C4⋊Dic5D3semidirect
71. C2×C5×#67Q8semidirect
136. C2×C4⋊Dic5C2×D3semidirect
168. C5×#67C2×Q8semidirect
301. C2×#67Dic10semidirect
401. C2×C3×C5#554semidirect
425. #67C2×Dic10semidirect
502. C3×C5C2×#554semidirect
526. C2×C5#5511semidirect
565. C2×C3#65877semidirect
570. C5C2×#5511semidirect
592. C3C2×#65877semidirect
595. C2#11754494direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#918132
C2×C4×C5×#672
C22×C3×C4⋊Dic52
C2×#915742
C2×Dic5×#672
C2×#914692
#117544942
C2×#914362
C2×D3×C4⋊Dic52
C2×#913652
C2×#913462
C2×#913402
C2×#658773
C2×#55115