Finite Group Databasebeta

← C2×#11754415 #14479855

Subgroups — C2×#11754415 #14479855

Order 1920.

2464 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×#91819C211yesmaximal, characteristic
2C2×C5×#924C211yesmaximal, characteristic, complemented
3C22×C3×C4×Dic5C211yesmaximal, characteristic, complemented
4C2×#91552C212yesmaximal, complemented
5C2×#91546C212yesmaximal, complemented
6#11754415C2116yesmaximal, direct product factor
7C2×#91418C211yesmaximal, characteristic, complemented
8C2×#91402C212yesmaximal, complemented
9C2×#91388C211yesmaximal, characteristic, complemented
10C2×#91387C212yesmaximal, complemented
11C2×#91375C211yesmaximal, characteristic, complemented
12C2×Dic5×Dic6C211yesmaximal, characteristic, complemented
13C2×#65958—31nomaximal, Hall
14C23×C3×C4×C5C2211yescharacteristic
15C22×#1220C2212yes—
16C22×D60C2211yescharacteristic
17C22×C5×Dic6C2211yescharacteristic
18C23×C3×Dic5C2211yescharacteristic
19C23×C3×Dic5C2211yescharacteristic
20C22×#1200C2212yes—
21#91819C2218yes—
22C2×#6714C2212yes—
23C2×C4⋊Dic15C2211yescharacteristic
24C2×C5×#217C2212yes—
25C2×C5×C4⋊Dic3C2211yescharacteristic
26C5×#924C2218yescomplemented
27C2×C5×Dic3⋊C4C2212yes—
28C2×C3×C4×Dic5C2211yescharacteristic
29C2×C3×C4×Dic5C2211yescharacteristic
30C2×C3×C4×Dic5C2211yescharacteristic
31C2×C3×C4×Dic5C2211yescharacteristic
32C2×C3×C4×Dic5C2218yescomplemented
33#91552C22116yescomplemented
34#91546C22116yescomplemented
35C2×Dic3⋊Dic5C2212yes—
36C2×Dic3⋊Dic5C2212yes—
37C2×#6669C2212yes—
38C2×#6667C2212yes—
39C2×#6667C2212yes—
40C2×#6667C2212yes—
41C2×Dic3×Dic5C2212yes—
42#91418C2218yescomplemented
43#91402C22116yescomplemented
44#91388C2218yescomplemented
45#91387C22116yescomplemented
46#91375C2218yescomplemented
47Dic5×Dic6C2218yescomplemented
48C2×#5479—51nomaximal, Hall
49C2×C5×C22⋊Q8—31no—
50C2×#4093—31no—
51C2×Q8×Dic5—31no—
52C2×#4076—31no—
53C22×C4×Dic5D311yescharacteristic, complemented
54#65958—316no—
55C2×#4044—32no—
56C2×#4039—31no—
57C2×#4038—32no—
58C2×#4036—31no—
59C2×#4032—32no—
60C2×#4027—32no—
61C24×C3×C5C2311yescharacteristic
62C23×D15C2312yes—
63C22×C3×C4×C5—24no—
64C22×C3×C4×C5C2311yescharacteristic
65C22×C3×C4×C5C2314yes—
66C22×C3×C4×C5C2311yescharacteristic
67C2×#1220—28no—
68C2×#1220C2318yes—
69C22×Dic15C2312yes—
70C2×D60—24no—
71C2×D60C2314yes—
72C22×C5×Dic3C2312yes—
73C22×C5×Dic3C2312yes—
74C2×C5×Dic6—24no—
75C2×C5×Dic6C2314yes—
76C22×C3×Dic5—24no—
77C22×C3×Dic5—24no—
78C22×C3×Dic5C2311yescharacteristic
79C22×C3×Dic5C2311yescharacteristic
80C22×C3×Dic5C2311yescharacteristic
81C22×C3×Dic5C2314yes—
82C22×C3×Dic5C2314yes—
83C22×C3×Dic5C2311yescharacteristic
84C2×#1200—28no—
85C2×#1200C2318yes—
86#6714C2318yes—
87C4⋊Dic15C2314yes—
88C5×#217C2318yes—
89C5×C4⋊Dic3C2314yes—
90C5×Dic3⋊C4C2318yes—
91C3×C4×Dic5C2314yes—
92C3×C4×Dic5C2314yes—
93C3×C4×Dic5C2314yes—
94C3×C4×Dic5C2314yes—
95Dic3⋊Dic5C2318yes—
96Dic3⋊Dic5C2318yes—
97#6669C2318yes—
98#6667C2318yes—
99#6667C2318yes—
100#6667C2318yes—
101Dic3×Dic5C2318yes—
102C22×C3×C42—51no—
103C2×#930—51no—
104C2×#929—52no—
105C2×C4×#67—52no—
106C2×#924D511yescharacteristic, complemented
107#5479—516no—
108C2×#876—52no—
109C2×#875—51no—
110C2×C4×D12—51no—
111C2×#872—52no—
112C2×#870—51no—
113C2×C4×Dic6—51no—
114C22×C5×Q8—31no—
115C23×C4×C5C2×D311yescharacteristic
116C22×C22⋊D5—32no—
117C23×Dic5C2×D311yescharacteristic
118C23×Dic5C2×D311yescharacteristic
119C22×D20—31no—
120C22×C4×D5—32no—
121C5×C22⋊Q8—38no—
122C2×C5×C4⋊C4—31no—
123C2×C5×C4⋊C4—32no—
124C2×C5×C22⋊C4—32no—
125#4093—38no—
126Q8×Dic5—38no—
127#4076—38no—
128C2×#685—32no—
129C2×#685—32no—
130C2×#685—32no—
131C2×#685—32no—
132C2×C4⋊Dic5—31no—
133C2×C4⋊Dic5—32no—
134C2×C4⋊Dic5—32no—
135C2×#683—32no—
136C2×C4×Dic5—32no—
137C2×C4×Dic5C2×D311yescharacteristic
138C2×C4×Dic5C2×D311yescharacteristic
139C2×C4×Dic5C2×D311yescharacteristic
140C2×C4×Dic5C2×D311yescharacteristic
141C2×C4×Dic5C2×D318yescomplemented
142#4044—316no—
143#4039—38no—
144#4038—316no—
145#4036—38no—
146#4032—316no—
147#4027—316no—
148C2×#529—151noSylow
149C23×C3×C5—22no—
150C23×C3×C5—22no—
151C23×C3×C5#4114yes—
152C23×C3×C5C2412yes—
153C23×C3×C5C2411yescharacteristic
154C22×D15—24no—
155C22×D15—24no—
156C22×D15—24no—
157C22×D15C2414yes—
158C2×C3×C4×C5—22no—
159C2×C3×C4×C5—28no—
160C2×C3×C4×C5#4114yes—
161C2×C3×C4×C5C2412yes—
162C2×C3×C4×C5C2412yes—
163#1220—216no—
164C2×Dic15—24no—
165C2×Dic15C2414yes—
166D60—28no—
167C2×C5×Dic3—24no—
168C2×C5×Dic3—24no—
169C2×C5×Dic3C2414yes—
170C2×C5×Dic3C2414yes—
171C5×Dic6—28no—
172C2×C3×Dic5—28no—
173C2×C3×Dic5—22no—
174C2×C3×Dic5—22no—
175C2×C3×Dic5—22no—
176C2×C3×Dic5—28no—
177C2×C3×Dic5#4114yes—
178C2×C3×Dic5C2412yes—
179C2×C3×Dic5C2412yes—
180C2×C3×Dic5C2412yes—
181C2×C3×Dic5C2412yes—
182#1200—216no—
183C23×C3×C4—51no—
184C23×C3×C4—51no—
185C23×C3×C4C2×D511yescharacteristic
186C22×#67—52no—
187C22×D12—51no—
188C22×C4×D3—52no—
189C22×Dic6C2×D511yescharacteristic
190C2×C3×C42—51no—
191C2×C3×C42—51no—
192C2×C3×C42—51no—
193C2×C3×C42—51no—
194C2×C3×C42—58no—
195C2×#217C2×D512yes—
196#930—58no—
197#929—516no—
198C4×#67—516no—
199C2×(C2×D3)⋊C4—52no—
200C2×(C2×D3)⋊C4—52no—
201C2×(C2×D3)⋊C4—52no—
202C2×(C2×D3)⋊C4—52no—
203C2×C4⋊Dic3—51no—
204C2×C4⋊Dic3C2×D511yescharacteristic
205#924C2×D518yescomplemented
206C2×Dic3⋊C4—52no—
207C2×Dic3⋊C4—52no—
208C2×Dic3⋊C4—52no—
209C2×Dic3⋊C4C2×D512yes—
210C2×C4×Dic3—52no—
211#876—516no—
212#875—58no—
213C4×D12—58no—
214#872—516no—
215#870—58no—
216C4×Dic6—58no—
217C24×C5C22×D311yescharacteristic
218C23×D5—32no—
219C2×C5×Q8—34no—
220C2×C5×Q8—64no—
221C22×C4×C5—24no—
222C22×C4×C5—32no—
223C22×C4×C5—32no—
224C22×C4×C5C22×D311yescharacteristic
225C22×C4×C5C22×D314yes—
226C22×C4×C5C22×D311yescharacteristic
227C2×C22⋊D5—38no—
228C2×C22⋊D5—68no—
229C22×Dic5—24no—
230C22×Dic5—24no—
231C22×Dic5—32no—
232C22×Dic5C22×D314yes—
233C22×Dic5C22×D314yes—
234C22×Dic5C22×D311yescharacteristic
235C22×Dic5C22×D311yescharacteristic
236C22×Dic5C22×D311yescharacteristic
237C22×Dic5C22×D311yescharacteristic
238C2×D20—34no—
239C2×D20—64no—
240C2×C4×D5—38no—
241C2×C4×D5—68no—
242C5×C4⋊C4—38no—
243C5×C4⋊C4—34no—
244C5×C22⋊C4—38no—
245#685—38no—
246#685—38no—
247#685—38no—
248#685—38no—
249C4⋊Dic5—34no—
250C4⋊Dic5—38no—
251C4⋊Dic5—38no—
252#683—38no—
253C4×Dic5—38no—
254C4×Dic5C22×D314yes—
255C4×Dic5C22×D314yes—
256C4×Dic5C22×D314yes—
257C4×Dic5C22×D314yes—
258#529—1516no—
259C2×#126—152no—
260C2×#125—151no—
261C2×#124—151no—
262C2×#123—152no—
263C2×C22⋊Q8—31no—
264C2×#121—151no—
265C2×C4×Q8—151no—
266C2×C4×D4—151no—
267C2×C4×D4—152no—
268C2×#117—152no—
269C22×C42—51no—
270C22×C3×C5—28no—
271C22×C3×C5—22no—
272C22×C3×C5—22no—
273C22×C3×C5C2×#4112yes—
274C22×C3×C5C2×#4114yes—
275C22×C3×C5C2×#4112yes—
276C22×C3×C5C2×#4112yes—
277C22×C3×C5C2511yescharacteristic
278C2×D15—24no—
279C2×D15—24no—
280C2×D15—24no—
281C2×D15—48no—
282C3×C4×C5—22no—
283C3×C4×C5C2×#4114yes—
284Dic15—24no—
285C3×Dic5—22no—
286C3×Dic5—22no—
287C3×Dic5—22no—
288C3×Dic5C2×#4114yes—
289C5×Dic3—24no—
290C5×Dic3—24no—
291C24×C3C22×D511yescharacteristic
292C23×D3—52no—
293C22×C3×C4—24no—
294C22×C3×C4—51no—
295C22×C3×C4—51no—
296C22×C3×C4—51no—
297C22×C3×C4—51no—
298C22×C3×C4—54no—
299C22×C3×C4—54no—
300C22×C3×C4—104no—
301C22×C3×C4—104no—
302C22×C3×C4C22×D511yescharacteristic
303C22×C3×C4C22×D514yes—
304C22×C3×C4C22×D511yescharacteristic
305C2×#67—58no—
306C2×#67—108no—
307C22×Dic3—52no—
308C22×Dic3C22×D512yes—
309C22×Dic3C22×D512yes—
310C2×D12—54no—
311C2×D12—104no—
312C2×C4×D3—58no—
313C2×C4×D3—108no—
314C2×Dic6—24no—
315C2×Dic6C22×D514yes—
316C3×C42—54no—
317C3×C42—54no—
318C3×C42—54no—
319C3×C42—54no—
320#217C22×D518yes—
321(C2×D3)⋊C4—58no—
322(C2×D3)⋊C4—58no—
323(C2×D3)⋊C4—58no—
324(C2×D3)⋊C4—58no—
325C4⋊Dic3—54no—
326C4⋊Dic3C22×D514yes—
327Dic3⋊C4—58no—
328Dic3⋊C4—58no—
329Dic3⋊C4—58no—
330Dic3⋊C4C22×D518yes—
331C4×Dic3—58no—
332C23×C5—22no—
333C23×C5—22no—
334C23×C5#23514yes—
335C23×C5C23×D312yes—
336C23×C5C23×D311yescharacteristic
337C22×D5—34no—
338C22×D5—64no—
339C22×D5—64no—
340C22×D5—64no—
341C5×Q8—68no—
342C2×C4×C5—22no—
343C2×C4×C5—28no—
344C2×C4×C5—34no—
345C2×C4×C5—34no—
346C2×C4×C5—64no—
347C2×C4×C5—64no—
348C2×C4×C5#23514yes—
349C2×C4×C5C23×D312yes—
350C2×C4×C5C23×D312yes—
351C22⋊D5—616no—
352C2×Dic5—28no—
353C2×Dic5—22no—
354C2×Dic5—22no—
355C2×Dic5—22no—
356C2×Dic5—28no—
357C2×Dic5—34no—
358C2×Dic5—64no—
359C2×Dic5#23514yes—
360C2×Dic5C23×D312yes—
361C2×Dic5C23×D312yes—
362C2×Dic5C23×D312yes—
363C2×Dic5C23×D312yes—
364D20—68no—
365C4×D5—616no—
366C22×Q8—31no—
367C22×D4—152no—
368C22×D4—151no—
369C23×C4—51no—
370C23×C4—51no—
371C23×C4—152no—
372C23×C4D3×D511yescharacteristic
373#126—1516no—
374#125—158no—
375#124—158no—
376#123—1516no—
377C22⋊Q8—38no—
378#121—158no—
379C4×Q8—158no—
380C4×D4—158no—
381C4×D4—1516no—
382#117—1516no—
383C2×C4⋊C4—32no—
384C2×C4⋊C4—31no—
385C2×C4⋊C4—152no—
386C2×C4⋊C4—152no—
387C2×C4⋊C4—152no—
388C2×C4⋊C4—151no—
389C2×C22⋊C4—32no—
390C2×C22⋊C4—152no—
391C2×C22⋊C4—152no—
392C2×C22⋊C4—152no—
393C2×C22⋊C4—152no—
394C2×C42—51no—
395C2×C42—51no—
396C2×C42—51no—
397C2×C42—51no—
398C2×C42—58no—
399C2×C42—152no—
400C2×C3×C5—24no—
401C2×C3×C5#52914yescomplemented
402C2×C3×C5C22×#4111yescharacteristic
403C2×C3×C5C22×#4111yescharacteristic
404C2×C3×C5C22×#4111yescharacteristic
405D15—44no—
406C23×C3—22no—
407C23×C3—22no—
408C23×C3#71314yes—
409C23×C3C23×D512yes—
410C23×C3C23×D511yescharacteristic
411C22×D3—54no—
412C22×D3—104no—
413C22×D3—104no—
414C22×D3—104no—
415C2×C3×C4—22no—
416C2×C3×C4—28no—
417C2×C3×C4—54no—
418C2×C3×C4—52no—
419C2×C3×C4—52no—
420C2×C3×C4—52no—
421C2×C3×C4—52no—
422C2×C3×C4—108no—
423C2×C3×C4—108no—
424C2×C3×C4—102no—
425C2×C3×C4—102no—
426C2×C3×C4—102no—
427C2×C3×C4#71114yes—
428C2×C3×C4C23×D512yes—
429C2×C3×C4C23×D512yes—
430#67—1016no—
431C2×Dic3—24no—
432C2×Dic3—24no—
433C2×Dic3—54no—
434C2×Dic3—104no—
435C2×Dic3C23×D514yes—
436C2×Dic3C23×D514yes—
437D12—108no—
438C4×D3—1016no—
439Dic6—28no—
440C22×C5—22no—
441C22×C5—22no—
442C22×C5—28no—
443C22×C5C2×#23514yes—
444C22×C5C2×#23512yes—
445C22×C5C2×#23512yes—
446C22×C5C2×#23512yes—
447C22×C5C24×D311yescharacteristic
448C2×D5—64no—
449C2×D5—64no—
450C2×D5—64no—
451C2×D5—128no—
452C4×C5—22no—
453C4×C5—64no—
454C4×C5—64no—
455C4×C5C2×#23514yes—
456Dic5—22no—
457Dic5—22no—
458Dic5—22no—
459Dic5—64no—
460Dic5C2×#23514yes—
461C24—152no—
462C24C2×D3×D511yescharacteristic
463C2×Q8—34no—
464C2×Q8—64no—
465C2×D4—158no—
466C2×D4—154no—
467C2×D4—308no—
468C2×D4—304no—
469C22×C4—24no—
470C22×C4—32no—
471C22×C4—32no—
472C22×C4—51no—
473C22×C4—54no—
474C22×C4—54no—
475C22×C4—51no—
476C22×C4—51no—
477C22×C4—51no—
478C22×C4—104no—
479C22×C4—104no—
480C22×C4—158no—
481C22×C4—152no—
482C22×C4—308no—
483C22×C4C2×D3×D514yes—
484C22×C4C2×D3×D511yescharacteristic
485C22×C4C2×D3×D511yescharacteristic
486C4⋊C4—38no—
487C4⋊C4—34no—
488C4⋊C4—158no—
489C4⋊C4—158no—
490C4⋊C4—158no—
491C4⋊C4—154no—
492C22⋊C4—38no—
493C22⋊C4—158no—
494C22⋊C4—158no—
495C22⋊C4—158no—
496C22⋊C4—158no—
497C42—54no—
498C42—54no—
499C42—54no—
500C42—54no—
501C42—158no—
502C3×C5C2×#52911yescharacteristic, complemented, Hall
503C22×C3—28no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3C2×#71112yes—
507C22×C3C2×#71314yes—
508C22×C3C2×#71312yes—
509C22×C3D5×#4112yes—
510C22×C3C24×D511yescharacteristic
511C2×D3—104no—
512C2×D3—104no—
513C2×D3—104no—
514C2×D3—208no—
515C3×C4—22no—
516C3×C4—54no—
517C3×C4—102no—
518C3×C4—102no—
519C3×C4—102no—
520C3×C4C2×#71114yes—
521Dic3—24no—
522Dic3—24no—
523Dic3—104no—
524C2×C5—24no—
525C2×C5#547914yescomplemented
526C2×C5C22×#23511yescharacteristic
527C2×C5C22×#23511yescharacteristic
528C2×C5C22×#23511yescharacteristic
529D5—124no—
530C23—22no—
531C23—22no—
532C23—154no—
533C23—304no—
534C23—304no—
535C23—304no—
536C23#677314yes—
537C23C22×D3×D512yes—
538C23C22×D3×D511yescharacteristic
539Q8—68no—
540D4—3016no—
541D4—308no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—34no—
545C2×C4—34no—
546C2×C4—54no—
547C2×C4—52no—
548C2×C4—52no—
549C2×C4—52no—
550C2×C4—52no—
551C2×C4—64no—
552C2×C4—64no—
553C2×C4—108no—
554C2×C4—108no—
555C2×C4—102no—
556C2×C4—102no—
557C2×C4—102no—
558C2×C4—154no—
559C2×C4—3016no—
560C2×C4—304no—
561C2×C4#678214yes—
562C2×C4C22×D3×D512yes—
563C2×C4C22×D3×D512yes—
564C2×C3—24no—
565C2×C3#6595814yescomplemented
566C2×C3C22×#71111yescharacteristic
567C2×C3C22×#71311yescharacteristic
568C2×C3C2×D5×#4111yescharacteristic
569D3—204no—
570C5C2×#547911yescharacteristic, complemented, Sylow
571C22—22no—
572C22—28no—
573C22—22no—
574C22—304no—
575C22—304no—
576C22—304no—
577C22—608no—
578C22C2×#677314yes—
579C22C2×#677312yes—
580C22D5×#23512yes—
581C22C2×#678212yes—
582C22C23×D3×D511yescharacteristic
583C4—22no—
584C4—54no—
585C4—64no—
586C4—64no—
587C4—102no—
588C4—102no—
589C4—102no—
590C4—304no—
591C4C2×#678214yes—
592C3C2×#6595811yescharacteristic, complemented, Sylow
593C2—24no—
594C2—604no—
595C2#1175441514yesdirect product factor
596C2C22×#677311yescharacteristic
597C2C2×D5×#23511yescharacteristic
598C2C22×#678211yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC23×C3×C4×C5C22
socleC23×C3×C5C24
prime radicalC2×#91819C2
hypercenterC23×C4D3×D5
hypocenterC3×C5C2×#529
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C2×C5×#924C2semidirect
3. C22×C3×C4×Dic5C2semidirect
4. C2×#91552C2semidirect
5. C2×#91546C2semidirect
6. #11754415C2direct (N × H)
7. C2×#91418C2semidirect
8. C2×#91402C2semidirect
9. C2×#91388C2semidirect
10. C2×#91387C2semidirect
11. C2×#91375C2semidirect
12. C2×Dic5×Dic6C2semidirect
26. C5×#924C22semidirect
32. C2×C3×C4×Dic5C22semidirect
33. #91552C22semidirect
34. #91546C22semidirect
42. #91418C22semidirect
43. #91402C22semidirect
44. #91388C22semidirect
45. #91387C22semidirect
46. #91375C22semidirect
47. Dic5×Dic6C22semidirect
53. C22×C4×Dic5D3semidirect
106. C2×#924D5semidirect
141. C2×C4×Dic5C2×D3semidirect
205. #924C2×D5semidirect
401. C2×C3×C5#529semidirect
502. C3×C5C2×#529semidirect
525. C2×C5#5479semidirect
565. C2×C3#65958semidirect
570. C5C2×#5479semidirect
592. C3C2×#65958semidirect
595. C2#11754415direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#918192
C2×C5×#9242
C22×C3×C4×Dic52
C2×#915522
C2×#915462
#117544152
C2×#914182
C2×#914022
C2×#913882
C2×#913872
C2×#913752
C2×Dic5×Dic62
C2×#659583
C2×#54795