Finite Group Databasebeta

← C2×#11756394 #14483801

Subgroups — C2×#11756394 #14483801

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×C3×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#91835C211yesmaximal, characteristic, complemented
3C2×Q8×Dic15C211yesmaximal, characteristic, complemented
4C2×C4×#1220C211yesmaximal, characteristic
5C22×C4⋊Dic15C211yesmaximal, characteristic, complemented
6#11756394C2116yesmaximal, direct product factor
7C2×#91786C212yesmaximal, complemented
8C2×C4⋊C4×D15C211yesmaximal, characteristic, complemented
9C2×#91778C212yesmaximal, complemented
10C2×#91777C211yesmaximal, characteristic, complemented
11C2×#91770C212yesmaximal
12C2×#91767C212yesmaximal, complemented
13C2×#65969—31nomaximal, Hall
14C22×C3×C5×Q8C2211yescharacteristic
15C23×C3×C4×C5C2211yescharacteristic
16C22×#1220C2211yescharacteristic
17C23×Dic15C2212yes—
18C22×C4×D15C2211yescharacteristic
19C22×C4×D15C2212yes—
20C22×Dic30C2212yes—
21C3×C5×C22⋊Q8C2218yescomplemented
22C2×C3×C5×C4⋊C4C2211yescharacteristic
23C2×C3×C5×C4⋊C4C2212yes—
24C2×C3×C5×C22⋊C4C2212yes—
25C2×#6719C2211yescharacteristic
26#91835C2218yescomplemented
27Q8×Dic15C2218yescomplemented
28C4×#1220C2218yes—
29C2×#6714C2211yescharacteristic
30C2×#6714C2212yes—
31C2×C4⋊Dic15C2211yescharacteristic
32C2×C4⋊Dic15C2211yescharacteristic
33C2×C4⋊Dic15C2212yes—
34C2×C4⋊Dic15C2212yes—
35C2×C4⋊Dic15C2218yescomplemented
36C2×#6712C2211yescharacteristic
37C2×#6712C2212yes—
38C2×#6712C2212yes—
39C2×#6712C2212yes—
40C2×C4×Dic15C2211yescharacteristic
41C2×C4×Dic15C2212yes—
42#91786C22116yescomplemented
43C4⋊C4×D15C2218yescomplemented
44#91778C22116yescomplemented
45#91777C2218yescomplemented
46#91770C22116yes—
47#91767C22116yescomplemented
48C2×#5635—51nomaximal, Hall
49C2×C5×C22⋊Q8D311yescharacteristic, complemented
50C2×#4092—31no—
51C2×Q8×Dic5—31no—
52C2×C4×C22⋊D5—31no—
53C22×C4⋊Dic5—31no—
54#65969—316no—
55C2×#4043—32no—
56C2×D5×C4⋊C4—31no—
57C2×#4035—32no—
58C2×#4034—31no—
59C2×#4027—32no—
60C2×#4024—32no—
61C24×C3×C5C2311yescharacteristic
62C23×D15C2311yescharacteristic
63C2×C3×C5×Q8—24no—
64C2×C3×C5×Q8C2314yes—
65C22×C3×C4×C5—24no—
66C22×C3×C4×C5C2312yes—
67C22×C3×C4×C5C2312yes—
68C22×C3×C4×C5C2311yescharacteristic
69C22×C3×C4×C5C2314yes—
70C22×C3×C4×C5C2311yescharacteristic
71C2×#1220Q818yescomplemented
72C2×#1220C2314yes—
73C22×Dic15—44no—
74C22×Dic15C2312yes—
75C22×Dic15C2311yescharacteristic
76C22×Dic15C2311yescharacteristic
77C22×Dic15C2312yes—
78C22×Dic15C2318yes—
79C22×Dic15C2311yescharacteristic
80C2×C4×D15—24no—
81C2×C4×D15—44no—
82C2×C4×D15C2314yes—
83C2×C4×D15C2318yes—
84C2×Dic30—44no—
85C2×Dic30C2318yes—
86C3×C5×C4⋊C4C2314yes—
87C3×C5×C4⋊C4C2318yes—
88C3×C5×C22⋊C4C2318yes—
89#6719C2314yes—
90#6714C2318yes—
91#6714C2314yes—
92C4⋊Dic15C2318yes—
93C4⋊Dic15C2314yes—
94C4⋊Dic15C2318yes—
95C4⋊Dic15C2314yes—
96#6712C2318yes—
97#6712C2318yes—
98#6712C2318yes—
99#6712C2314yes—
100C4×Dic15C2318yes—
101C4×Dic15C2314yes—
102C2×C3×C22⋊Q8D511yescharacteristic, complemented
103C2×#946—51no—
104C2×Q8×Dic3—51no—
105C2×C4×#67—51no—
106C22×C4⋊Dic3—51no—
107#5635—516no—
108C2×#897—52no—
109C2×D3×C4⋊C4—51no—
110C2×#889—52no—
111C2×#888—51no—
112C2×#67⋊C4—52no—
113C2×#878—52no—
114C22×C5×Q8C2×D311yescharacteristic
115C23×C4×C5C2×D311yescharacteristic
116C22×C22⋊D5—31no—
117C23×Dic5—32no—
118C22×C4×D5—31no—
119C22×C4×D5—32no—
120C22×Dic10—32no—
121C5×C22⋊Q8C2×D318yescomplemented
122C2×C5×C4⋊C4C2×D311yescharacteristic
123C2×C5×C4⋊C4C2×D312yes—
124C2×C5×C22⋊C4C2×D312yes—
125C2×C22⋊Dic5—31no—
126#4092—38no—
127Q8×Dic5—38no—
128C4×C22⋊D5—38no—
129C2×#685—32no—
130C2×#685—31no—
131C2×C4⋊Dic5—32no—
132C2×C4⋊Dic5—31no—
133C2×C4⋊Dic5—32no—
134C2×C4⋊Dic5—31no—
135C2×C4⋊Dic5—38no—
136C2×#683—32no—
137C2×#683—32no—
138C2×#683—32no—
139C2×#683—31no—
140C2×C4×Dic5—32no—
141C2×C4×Dic5—31no—
142#4043—316no—
143D5×C4⋊C4—38no—
144#4035—316no—
145#4034—38no—
146#4027—316no—
147#4024—316no—
148C2×#554—151noSylow
149C23×C3×C5—22no—
150C23×C3×C5—22no—
151C23×C3×C5C2×Q814yes—
152C23×C3×C5C2412yes—
153C23×C3×C5C2411yescharacteristic
154C22×D15—22no—
155C22×D15—22no—
156C22×D15C2×Q814yes—
157C22×D15C2412yes—
158C3×C5×Q8—28no—
159C2×C3×C4×C5—22no—
160C2×C3×C4×C5—24no—
161C2×C3×C4×C5—24no—
162C2×C3×C4×C5—28no—
163C2×C3×C4×C5#4114yes—
164C2×C3×C4×C5C2414yes—
165C2×C3×C4×C5C2414yes—
166C2×C3×C4×C5C2412yes—
167C2×C3×C4×C5C2412yes—
168#1220C2×Q8116yescomplemented
169C2×Dic15—22no—
170C2×Dic15—22no—
171C2×Dic15—24no—
172C2×Dic15—24no—
173C2×Dic15—48no—
174C2×Dic15C2×Q814yes—
175C2×Dic15C2414yes—
176C2×Dic15C2412yes—
177C2×Dic15C2412yes—
178C2×Dic15C2414yes—
179C2×Dic15C2412yes—
180C4×D15—28no—
181C4×D15—48no—
182Dic30—48no—
183C22×C3×Q8C2×D511yescharacteristic
184C23×C3×C4C2×D511yescharacteristic
185C22×#67—51no—
186C23×Dic3—52no—
187C22×C4×D3—51no—
188C22×C4×D3—52no—
189C22×Dic6—52no—
190C3×C22⋊Q8C2×D518yescomplemented
191C2×C3×C4⋊C4C2×D511yescharacteristic
192C2×C3×C4⋊C4C2×D512yes—
193C2×C3×C22⋊C4C2×D512yes—
194C2×#217—51no—
195#946—58no—
196Q8×Dic3—58no—
197C4×#67—58no—
198C2×(C2×D3)⋊C4—52no—
199C2×(C2×D3)⋊C4—51no—
200C2×C4⋊Dic3—58no—
201C2×C4⋊Dic3—52no—
202C2×C4⋊Dic3—51no—
203C2×C4⋊Dic3—52no—
204C2×C4⋊Dic3—51no—
205C2×Dic3⋊C4—52no—
206C2×Dic3⋊C4—52no—
207C2×Dic3⋊C4—52no—
208C2×Dic3⋊C4—51no—
209C2×C4×Dic3—52no—
210C2×C4×Dic3—51no—
211#897—516no—
212D3×C4⋊C4—58no—
213#889—516no—
214#888—58no—
215#67⋊C4—516no—
216#878—516no—
217C24×C5C22×D311yescharacteristic
218C23×D5—31no—
219C2×C5×Q8—24no—
220C2×C5×Q8C22×D314yes—
221C22×C4×C5—24no—
222C22×C4×C5C22×D312yes—
223C22×C4×C5C22×D312yes—
224C22×C4×C5C22×D311yescharacteristic
225C22×C4×C5C22×D314yes—
226C22×C4×C5C22×D311yescharacteristic
227C2×C22⋊D5—38no—
228C2×C22⋊D5—34no—
229C22×Dic5—31no—
230C22×Dic5—31no—
231C22×Dic5—31no—
232C22×Dic5—32no—
233C22×Dic5—32no—
234C22×Dic5—38no—
235C22×Dic5—124no—
236C2×C4×D5—34no—
237C2×C4×D5—38no—
238C2×C4×D5—64no—
239C2×C4×D5—124no—
240C2×Dic10—38no—
241C2×Dic10—124no—
242C5×C4⋊C4C22×D314yes—
243C5×C4⋊C4C22×D318yes—
244C5×C22⋊C4C22×D318yes—
245C22⋊Dic5—34no—
246#685—38no—
247#685—34no—
248C4⋊Dic5—38no—
249C4⋊Dic5—34no—
250C4⋊Dic5—38no—
251C4⋊Dic5—34no—
252#683—38no—
253#683—38no—
254#683—38no—
255#683—34no—
256C4×Dic5—38no—
257C4×Dic5—34no—
258#554—1516no—
259C2×C4⋊Q8—151no—
260C2×#125—152no—
261C2×C22⋊Q8—151no—
262C2×C22⋊Q8—152no—
263C2×C22⋊Q8—152no—
264C2×C22⋊Q8D1511yescharacteristic, complemented
265C2×C4×Q8—151no—
266C2×C4×D4—151no—
267C2×C4×D4—152no—
268C22×C4⋊C4—151no—
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×D15—28no—
279C2×D15—22no—
280C2×D15—22no—
281C2×D15C22×Q814yes—
282C3×C4×C5—22no—
283C3×C4×C5—24no—
284C3×C4×C5—24no—
285C3×C4×C5C2×#4114yes—
286Dic15—22no—
287Dic15—22no—
288Dic15—24no—
289Dic15—24no—
290Dic15C22×Q814yes—
291C24×C3C22×D511yescharacteristic
292C23×D3—51no—
293C2×C3×Q8—24no—
294C2×C3×Q8C22×D514yes—
295C22×C3×C4—24no—
296C22×C3×C4C22×D512yes—
297C22×C3×C4C22×D512yes—
298C22×C3×C4C22×D511yescharacteristic
299C22×C3×C4C22×D514yes—
300C22×C3×C4C22×D511yescharacteristic
301C2×#67—58no—
302C2×#67—54no—
303C22×Dic3—51no—
304C22×Dic3—52no—
305C22×Dic3—51no—
306C22×Dic3—51no—
307C22×Dic3—52no—
308C22×Dic3—58no—
309C22×Dic3—204no—
310C2×C4×D3—54no—
311C2×C4×D3—58no—
312C2×C4×D3—104no—
313C2×C4×D3—204no—
314C2×Dic6—58no—
315C2×Dic6—204no—
316C3×C4⋊C4C22×D514yes—
317C3×C4⋊C4C22×D518yes—
318C3×C22⋊C4C22×D518yes—
319#217—54no—
320(C2×D3)⋊C4—58no—
321(C2×D3)⋊C4—54no—
322C4⋊Dic3—58no—
323C4⋊Dic3—54no—
324C4⋊Dic3—58no—
325C4⋊Dic3—54no—
326Dic3⋊C4—58no—
327Dic3⋊C4—58no—
328Dic3⋊C4—58no—
329Dic3⋊C4—54no—
330C4×Dic3—58no—
331C4×Dic3—54no—
332C23×C5—22no—
333C23×C5—22no—
334C23×C5D3×Q814yes—
335C23×C5C23×D312yes—
336C23×C5C23×D311yescharacteristic
337C22×D5—34no—
338C22×D5—32no—
339C22×D5—62no—
340C22×D5—62no—
341C5×Q8—28no—
342C2×C4×C5—22no—
343C2×C4×C5—24no—
344C2×C4×C5—24no—
345C2×C4×C5—28no—
346C2×C4×C5#23714yes—
347C2×C4×C5C23×D314yes—
348C2×C4×C5C23×D314yes—
349C2×C4×C5C23×D312yes—
350C2×C4×C5C23×D312yes—
351C22⋊D5—316no—
352C2×Dic5—34no—
353C2×Dic5—32no—
354C2×Dic5—34no—
355C2×Dic5—32no—
356C2×Dic5—32no—
357C2×Dic5—34no—
358C2×Dic5—62no—
359C2×Dic5—62no—
360C2×Dic5—64no—
361C2×Dic5—64no—
362C2×Dic5—128no—
363C4×D5—68no—
364C4×D5—128no—
365Dic10—128no—
366C22×Q8—152no—
367C22×Q8C2×D1511yescharacteristic
368C22×D4—151no—
369C23×C4—152no—
370C23×C4—151no—
371C23×C4—152no—
372C23×C4C2×D1511yescharacteristic
373C4⋊Q8—158no—
374#125—1516no—
375C22⋊Q8—158no—
376C22⋊Q8—1516no—
377C22⋊Q8—1516no—
378C22⋊Q8C2×D1518yescomplemented
379C4×Q8—158no—
380C4×D4—158no—
381C4×D4—1516no—
382C2×C4⋊C4—151no—
383C2×C4⋊C4—152no—
384C2×C4⋊C4—152no—
385C2×C4⋊C4—151no—
386C2×C4⋊C4—158no—
387C2×C4⋊C4—151no—
388C2×C4⋊C4—152no—
389C2×C4⋊C4—152no—
390C2×C4⋊C4—152no—
391C2×C4⋊C4—158no—
392C2×C4⋊C4C2×D1511yescharacteristic
393C2×C4⋊C4C2×D1512yes—
394C2×C22⋊C4—151no—
395C2×C22⋊C4—151no—
396C2×C22⋊C4—152no—
397C2×C22⋊C4C2×D1512yes—
398C2×C42—152no—
399C2×C42—151no—
400C2×C3×C5—24no—
401C2×C3×C5#55414yescomplemented
402C2×C3×C5C23×Q811yescharacteristic
403C2×C3×C5C22×#4111yescharacteristic
404C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
405D15—24no—
406C23×C3—22no—
407C23×C3—22no—
408C23×C3Q8×D514yes—
409C23×C3C23×D512yes—
410C23×C3C23×D511yescharacteristic
411C22×D3—54no—
412C22×D3—52no—
413C22×D3—102no—
414C22×D3—102no—
415C3×Q8—28no—
416C2×C3×C4—22no—
417C2×C3×C4—24no—
418C2×C3×C4—24no—
419C2×C3×C4—28no—
420C2×C3×C4#71114yes—
421C2×C3×C4C23×D514yes—
422C2×C3×C4C23×D514yes—
423C2×C3×C4C23×D512yes—
424C2×C3×C4C23×D512yes—
425#67—516no—
426C2×Dic3—54no—
427C2×Dic3—52no—
428C2×Dic3—54no—
429C2×Dic3—52no—
430C2×Dic3—52no—
431C2×Dic3—54no—
432C2×Dic3—102no—
433C2×Dic3—102no—
434C2×Dic3—104no—
435C2×Dic3—104no—
436C2×Dic3—208no—
437C4×D3—108no—
438C4×D3—208no—
439Dic6—208no—
440C22×C5—28no—
441C22×C5—22no—
442C22×C5—22no—
443C22×C5C2×#23712yes—
444C22×C5C2×D3×Q814yes—
445C22×C5C2×D3×Q812yes—
446C22×C5#100912yes—
447C22×C5C24×D311yescharacteristic
448C2×D5—34no—
449C2×D5—68no—
450C2×D5—62no—
451C2×D5—62no—
452C4×C5—22no—
453C4×C5—24no—
454C4×C5—24no—
455C4×C5C2×#23714yes—
456Dic5—34no—
457Dic5—62no—
458Dic5—62no—
459Dic5—64no—
460Dic5—64no—
461C24—151no—
462C24C22×D1511yescharacteristic
463C2×Q8—24no—
464C2×Q8—158no—
465C2×Q8—604no—
466C2×Q8C22×D1514yes—
467C2×D4—158no—
468C2×D4—154no—
469C22×C4—24no—
470C22×C4—151no—
471C22×C4—151no—
472C22×C4—152no—
473C22×C4—151no—
474C22×C4—152no—
475C22×C4—158no—
476C22×C4—154no—
477C22×C4—158no—
478C22×C4—304no—
479C22×C4—604no—
480C22×C4—604no—
481C22×C4C22×D1514yes—
482C22×C4C22×D1512yes—
483C22×C4C22×D1511yescharacteristic
484C22×C4C22×D1512yes—
485C22×C4C22×D1511yescharacteristic
486C4⋊C4—154no—
487C4⋊C4—158no—
488C4⋊C4—158no—
489C4⋊C4—154no—
490C4⋊C4—154no—
491C4⋊C4—158no—
492C4⋊C4—158no—
493C4⋊C4—158no—
494C4⋊C4C22×D1514yes—
495C4⋊C4C22×D1518yes—
496C22⋊C4—154no—
497C22⋊C4—154no—
498C22⋊C4—158no—
499C22⋊C4C22×D1518yes—
500C42—158no—
501C42—154no—
502C3×C5C2×#55411yescharacteristic, complemented, Hall
503C22×C3—28no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3C2×#71112yes—
507C22×C3C2×Q8×D514yes—
508C22×C3C2×Q8×D512yes—
509C22×C3#414912yes—
510C22×C3C24×D511yescharacteristic
511C2×D3—54no—
512C2×D3—108no—
513C2×D3—102no—
514C2×D3—102no—
515C3×C4—22no—
516C3×C4—24no—
517C3×C4—24no—
518C3×C4C2×#71114yes—
519Dic3—54no—
520Dic3—102no—
521Dic3—102no—
522Dic3—104no—
523Dic3—104no—
524C2×C5—24no—
525C2×C5#563514yescomplemented
526C2×C5C22×#23711yescharacteristic
527C2×C5C22×D3×Q811yescharacteristic
528C2×C5C2×#100911yescharacteristic
529D5—64no—
530C23—22no—
531C23—22no—
532C23—154no—
533C23—152no—
534C23—302no—
535C23—302no—
536C23Q8×D1514yes—
537C23C23×D1512yes—
538C23C23×D1511yescharacteristic
539Q8—28no—
540Q8—608no—
541D4—1516no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—154no—
547C2×C4—152no—
548C2×C4—152no—
549C2×C4—154no—
550C2×C4—152no—
551C2×C4—154no—
552C2×C4—302no—
553C2×C4—304no—
554C2×C4—304no—
555C2×C4—302no—
556C2×C4—308no—
557C2×C4—608no—
558C2×C4—608no—
559C2×C4#681914yes—
560C2×C4C23×D1514yes—
561C2×C4C23×D1512yes—
562C2×C4C23×D1514yes—
563C2×C4C23×D1512yes—
564C2×C3—24no—
565C2×C3#6596914yescomplemented
566C2×C3C22×#71111yescharacteristic
567C2×C3C22×Q8×D511yescharacteristic
568C2×C3C2×#414911yescharacteristic
569D3—104no—
570C5C2×#563511yescharacteristic, complemented, Sylow
571C22—22no—
572C22—22no—
573C22—28no—
574C22—154no—
575C22—302no—
576C22—302no—
577C22—308no—
578C22C2×#681912yes—
579C22C2×Q8×D1514yes—
580C22C2×Q8×D1512yes—
581C22#9210012yes—
582C22C24×D1511yescharacteristic
583C4—22no—
584C4—24no—
585C4—24no—
586C4—154no—
587C4—302no—
588C4—304no—
589C4—304no—
590C4—302no—
591C4C2×#681914yes—
592C3C2×#6596911yescharacteristic, complemented, Sylow
593C2—24no—
594C2—304no—
595C2#1175639414yesdirect product factor
596C2C22×#681911yescharacteristic
597C2C22×Q8×D1511yescharacteristic
598C2C2×#9210011yescharacteristic
subgroupG/H
centerC23C23×D15
commutator, cosocleC22×C3×C5C25
FrattiniC22C24×D15
FittingC2×C3×C5×C22⋊Q8C2
socleC23×C3×C5C24
prime radicalC22×#1220C22
hypercenterC2×C22⋊Q8D15
hypocenterC3×C5C2×#554
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×C22⋊Q8C2semidirect
2. C2×#91835C2semidirect
3. C2×Q8×Dic15C2semidirect
5. C22×C4⋊Dic15C2semidirect
6. #11756394C2direct (N × H)
7. C2×#91786C2semidirect
8. C2×C4⋊C4×D15C2semidirect
9. C2×#91778C2semidirect
10. C2×#91777C2semidirect
12. C2×#91767C2semidirect
21. C3×C5×C22⋊Q8C22semidirect
26. #91835C22semidirect
27. Q8×Dic15C22semidirect
35. C2×C4⋊Dic15C22semidirect
42. #91786C22semidirect
43. C4⋊C4×D15C22semidirect
44. #91778C22semidirect
45. #91777C22semidirect
47. #91767C22semidirect
49. C2×C5×C22⋊Q8D3semidirect
71. C2×#1220Q8semidirect
102. C2×C3×C22⋊Q8D5semidirect
121. C5×C22⋊Q8C2×D3semidirect
168. #1220C2×Q8semidirect
190. C3×C22⋊Q8C2×D5semidirect
264. C2×C22⋊Q8D15semidirect
378. C22⋊Q8C2×D15semidirect
401. C2×C3×C5#554semidirect
502. C3×C5C2×#554semidirect
525. C2×C5#5635semidirect
565. C2×C3#65969semidirect
570. C5C2×#5635semidirect
592. C3C2×#65969semidirect
595. C2#11756394direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×C22⋊Q82
C2×#918352
C2×Q8×Dic152
C2×C4×#12202
C22×C4⋊Dic152
#117563942
C2×#917862
C2×C4⋊C4×D152
C2×#917782
C2×#917772
C2×#917702
C2×#917672
C2×#659693
C2×#56355