Finite Group Databasebeta

← C2×#11756289 #14483597

Subgroups — C2×#11756289 #14483597

Order 1920.

2504 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#117C211yesmaximal, characteristic, complemented
2C2×#91819C211yesmaximal, characteristic, complemented
3C2×C4×#1220C211yesmaximal, characteristic, complemented
4#11756289C2116yesmaximal, direct product factor
5C2×#91784C211yesmaximal, characteristic, complemented
6C2×#91782C211yesmaximal, characteristic, complemented
7C2×C4⋊C4×D15C211yesmaximal, characteristic, complemented
8C2×#91779C211yesmaximal, characteristic, complemented
9C2×#91773C212yesmaximal, complemented
10C2×#91772C212yesmaximal, complemented
11C2×#91765C212yesmaximal, complemented
12C2×C4×D60C212yesmaximal, complemented
13C2×#65864—31nomaximal, Hall
14C23×C3×C4×C5C2211yescharacteristic
15C22×#1220C2211yescharacteristic
16C22×#1220C2212yes—
17C22×D60C2211yescharacteristic
18C22×D60C2212yes—
19C22×C4×D15C2211yescharacteristic
20C22×C4×D15C2212yes—
21C22×C4×D15C2212yes—
22C3×C5×#117C2218yescomplemented
23C2×C3×C5×C4⋊C4C2211yescharacteristic
24C2×C3×C5×C4⋊C4C2211yescharacteristic
25C2×C3×C5×C22⋊C4C2212yes—
26C2×C3×C42×C5C2212yes—
27C2×#6719C2211yescharacteristic
28#91819C2218yescomplemented
29C4×#1220C2218yescomplemented
30C2×#6714C2211yescharacteristic
31C2×#6714C2212yes—
32C2×#6714C2212yes—
33C2×#6714C2212yes—
34C2×C4⋊Dic15C2211yescharacteristic
35C2×C4⋊Dic15C2212yes—
36C2×#6712C2211yescharacteristic
37C2×#6712C2212yes—
38C2×#6712C2212yes—
39C2×C4×Dic15C2211yescharacteristic
40#91784C2218yescomplemented
41#91782C2218yescomplemented
42C4⋊C4×D15C2218yescomplemented
43#91779C2218yescomplemented
44#91773C22116yescomplemented
45#91772C22116yescomplemented
46#91765C22116yescomplemented
47C4×D60C22116yescomplemented
48C2×#5530—51nomaximal, Hall
49C2×C5×#117D311yescharacteristic, complemented
50C2×#4076—31no—
51C2×C4×C22⋊D5—31no—
52#65864—316no—
53C2×#4041—31no—
54C2×#4039—31no—
55C2×D5×C4⋊C4—31no—
56C2×#4036—31no—
57C2×#4030—32no—
58C2×#4029—32no—
59C2×#4022—32no—
60C2×C4×D20—32no—
61C24×C3×C5C2311yescharacteristic
62C23×D15C2312yes—
63C23×D15C2311yescharacteristic
64C22×C3×C4×C5—24no—
65C22×C3×C4×C5C2312yes—
66C22×C3×C4×C5C2311yescharacteristic
67C22×C3×C4×C5C2312yes—
68C22×C3×C4×C5C2314yes—
69C22×C3×C4×C5C2311yescharacteristic
70C2×#1220—24no—
71C2×#1220—44no—
72C2×#1220C2314yes—
73C2×#1220C2318yes—
74C22×Dic15C2311yescharacteristic
75C22×Dic15C2311yescharacteristic
76C22×Dic15C2311yescharacteristic
77C22×Dic15C2312yes—
78C2×D60—28no—
79C2×D60—24no—
80C2×D60C2318yes—
81C2×D60C2314yes—
82C2×C4×D15—24no—
83C2×C4×D15—44no—
84C2×C4×D15—44no—
85C2×C4×D15C2314yes—
86C2×C4×D15C2318yes—
87C2×C4×D15C2318yes—
88C3×C5×C4⋊C4C2314yes—
89C3×C5×C4⋊C4C2314yes—
90C3×C5×C22⋊C4C2318yes—
91C3×C42×C5C2318yes—
92#6719C2314yes—
93#6714C2314yes—
94#6714C2318yes—
95#6714C2318yes—
96#6714C2318yes—
97C4⋊Dic15C2318yes—
98C4⋊Dic15C2314yes—
99#6712C2314yes—
100#6712C2318yes—
101#6712C2318yes—
102C4×Dic15C2314yes—
103C2×C3×#117D511yescharacteristic, complemented
104C2×#930—51no—
105C2×C4×#67—51no—
106#5530—516no—
107C2×#895—51no—
108C2×#893—51no—
109C2×D3×C4⋊C4—51no—
110C2×#890—51no—
111C2×#884—52no—
112C2×#883—52no—
113C2×#876—52no—
114C2×C4×D12—52no—
115C23×C4×C5C2×D311yescharacteristic
116C22×C22⋊D5—31no—
117C22×C22⋊D5—32no—
118C22×D20—32no—
119C22×D20—31no—
120C22×C4×D5—31no—
121C22×C4×D5—32no—
122C22×C4×D5—32no—
123C5×#117C2×D318yescomplemented
124C2×C5×C4⋊C4C2×D311yescharacteristic
125C2×C5×C4⋊C4C2×D311yescharacteristic
126C2×C5×C22⋊C4C2×D312yes—
127C2×C42×C5C2×D312yes—
128C2×C22⋊Dic5—31no—
129#4076—38no—
130C4×C22⋊D5—38no—
131C2×#685—31no—
132C2×#685—32no—
133C2×#685—32no—
134C2×#685—32no—
135C2×C4⋊Dic5—32no—
136C2×C4⋊Dic5—31no—
137C2×#683—31no—
138C2×#683—32no—
139C2×#683—32no—
140C2×C4×Dic5—31no—
141#4041—38no—
142#4039—38no—
143D5×C4⋊C4—38no—
144#4036—38no—
145#4030—316no—
146#4029—316no—
147#4022—316no—
148C4×D20—316no—
149C2×#553—151noSylow
150C23×C3×C5—22no—
151C23×C3×C5—22no—
152C23×C3×C5—22no—
153C23×C3×C5C2412yes—
154C23×C3×C5C2411yescharacteristic
155C22×D15—22no—
156C22×D15—22no—
157C22×D15—24no—
158C22×D15—24no—
159C22×D15—24no—
160C22×D15#4114yes—
161C22×D15C2414yes—
162C22×D15C2412yes—
163C2×C3×C4×C5—24no—
164C2×C3×C4×C5—22no—
165C2×C3×C4×C5—24no—
166C2×C3×C4×C5—28no—
167C2×C3×C4×C5#4114yes—
168C2×C3×C4×C5C2414yes—
169C2×C3×C4×C5C2412yes—
170C2×C3×C4×C5C2414yes—
171C2×C3×C4×C5C2412yes—
172#1220—28no—
173#1220—48no—
174C2×Dic15—22no—
175C2×Dic15—22no—
176C2×Dic15—22no—
177C2×Dic15—24no—
178C2×Dic15C2412yes—
179C2×Dic15C2412yes—
180C2×Dic15C2412yes—
181C2×Dic15C2414yes—
182D60—216no—
183D60—28no—
184C4×D15—28no—
185C4×D15—48no—
186C4×D15—48no—
187C23×C3×C4C2×D511yescharacteristic
188C22×#67—51no—
189C22×#67—52no—
190C22×D12—52no—
191C22×D12—51no—
192C22×C4×D3—51no—
193C22×C4×D3—52no—
194C22×C4×D3—52no—
195C3×#117C2×D518yescomplemented
196C2×C3×C4⋊C4C2×D511yescharacteristic
197C2×C3×C4⋊C4C2×D511yescharacteristic
198C2×C3×C22⋊C4C2×D512yes—
199C2×C3×C42C2×D512yes—
200C2×#217—51no—
201#930—58no—
202C4×#67—58no—
203C2×(C2×D3)⋊C4—51no—
204C2×(C2×D3)⋊C4—52no—
205C2×(C2×D3)⋊C4—52no—
206C2×(C2×D3)⋊C4—52no—
207C2×C4⋊Dic3—52no—
208C2×C4⋊Dic3—51no—
209C2×Dic3⋊C4—51no—
210C2×Dic3⋊C4—52no—
211C2×Dic3⋊C4—52no—
212C2×C4×Dic3—51no—
213#895—58no—
214#893—58no—
215D3×C4⋊C4—58no—
216#890—58no—
217#884—516no—
218#883—516no—
219#876—516no—
220C4×D12—516no—
221C24×C5C22×D311yescharacteristic
222C23×D5—31no—
223C23×D5—32no—
224C22×C4×C5—24no—
225C22×C4×C5C22×D311yescharacteristic
226C22×C4×C5C22×D312yes—
227C22×C4×C5C22×D312yes—
228C22×C4×C5C22×D314yes—
229C22×C4×C5C22×D311yescharacteristic
230C2×C22⋊D5—34no—
231C2×C22⋊D5—38no—
232C2×C22⋊D5—64no—
233C2×C22⋊D5—124no—
234C22×Dic5—31no—
235C22×Dic5—31no—
236C22×Dic5—32no—
237C22×Dic5—31no—
238C2×D20—38no—
239C2×D20—34no—
240C2×D20—68no—
241C2×D20—64no—
242C2×C4×D5—34no—
243C2×C4×D5—38no—
244C2×C4×D5—38no—
245C2×C4×D5—64no—
246C2×C4×D5—124no—
247C2×C4×D5—124no—
248C5×C4⋊C4C22×D314yes—
249C5×C4⋊C4C22×D314yes—
250C5×C22⋊C4C22×D318yes—
251C42×C5C22×D318yes—
252C22⋊Dic5—34no—
253#685—34no—
254#685—38no—
255#685—38no—
256#685—38no—
257C4⋊Dic5—38no—
258C4⋊Dic5—34no—
259#683—34no—
260#683—38no—
261#683—38no—
262C4×Dic5—34no—
263#553—1516no—
264C2×#126—152no—
265C2×#125—151no—
266C2×#123—152no—
267C2×#121—151no—
268C2×#121—151no—
269C2×#121—152no—
270C2×C4×D4—151no—
271C2×C4×D4—151no—
272C2×C4×D4—152no—
273C2×#117D1511yescharacteristic, complemented
274C22×C4⋊C4—151no—
275C22×C3×C5—22no—
276C22×C3×C5—22no—
277C22×C3×C5—22no—
278C22×C3×C5—44no—
279C22×C3×C5C2×#4112yes—
280C22×C3×C5C2×#4112yes—
281C22×C3×C5H(5,𝔽2)12yes—
282C22×C3×C5C2511yescharacteristic
283C2×D15—28no—
284C2×D15—22no—
285C2×D15—22no—
286C2×D15—24no—
287C2×D15—24no—
288C2×D15—24no—
289C2×D15—48no—
290C2×D15C2×#4114yes—
291C3×C4×C5—24no—
292C3×C4×C5—22no—
293C3×C4×C5—24no—
294C3×C4×C5C2×#4114yes—
295Dic15—22no—
296Dic15—22no—
297Dic15—22no—
298Dic15—24no—
299C24×C3C22×D511yescharacteristic
300C23×D3—51no—
301C23×D3—52no—
302C22×C3×C4—24no—
303C22×C3×C4C22×D512yes—
304C22×C3×C4C22×D511yescharacteristic
305C22×C3×C4C22×D512yes—
306C22×C3×C4C22×D514yes—
307C22×C3×C4C22×D511yescharacteristic
308C2×#67—54no—
309C2×#67—58no—
310C2×#67—104no—
311C2×#67—204no—
312C22×Dic3—51no—
313C22×Dic3—51no—
314C22×Dic3—51no—
315C22×Dic3—52no—
316C2×D12—58no—
317C2×D12—54no—
318C2×D12—108no—
319C2×D12—104no—
320C2×C4×D3—54no—
321C2×C4×D3—58no—
322C2×C4×D3—58no—
323C2×C4×D3—104no—
324C2×C4×D3—204no—
325C2×C4×D3—204no—
326C3×C4⋊C4C22×D514yes—
327C3×C4⋊C4C22×D514yes—
328C3×C22⋊C4C22×D518yes—
329C3×C42C22×D518yes—
330#217—54no—
331(C2×D3)⋊C4—54no—
332(C2×D3)⋊C4—58no—
333(C2×D3)⋊C4—58no—
334(C2×D3)⋊C4—58no—
335C4⋊Dic3—58no—
336C4⋊Dic3—54no—
337Dic3⋊C4—54no—
338Dic3⋊C4—58no—
339Dic3⋊C4—58no—
340C4×Dic3—54no—
341C23×C5—22no—
342C23×C5—22no—
343C23×C5—22no—
344C23×C5C23×D312yes—
345C23×C5C23×D311yescharacteristic
346C22×D5—34no—
347C22×D5—32no—
348C22×D5—34no—
349C22×D5—62no—
350C22×D5—62no—
351C22×D5—64no—
352C22×D5—64no—
353C22×D5—64no—
354C2×C4×C5—22no—
355C2×C4×C5—24no—
356C2×C4×C5—24no—
357C2×C4×C5—28no—
358C2×C4×C5#23514yes—
359C2×C4×C5C23×D314yes—
360C2×C4×C5C23×D312yes—
361C2×C4×C5C23×D314yes—
362C2×C4×C5C23×D312yes—
363C22⋊D5—68no—
364C22⋊D5—128no—
365C2×Dic5—32no—
366C2×Dic5—32no—
367C2×Dic5—32no—
368C2×Dic5—34no—
369C2×Dic5—62no—
370C2×Dic5—62no—
371C2×Dic5—62no—
372C2×Dic5—64no—
373D20—616no—
374D20—68no—
375C4×D5—68no—
376C4×D5—128no—
377C4×D5—128no—
378C22×D4—151no—
379C22×D4—152no—
380C22×D4—152no—
381C22×D4—151no—
382C23×C4—151no—
383C23×C4—152no—
384C23×C4—152no—
385C23×C4C2×D1511yescharacteristic
386#126—1516no—
387#125—158no—
388#123—1516no—
389#121—158no—
390#121—158no—
391#121—1516no—
392C4×D4—158no—
393C4×D4—158no—
394C4×D4—1516no—
395#117C2×D1518yescomplemented
396C2×C4⋊C4—151no—
397C2×C4⋊C4—152no—
398C2×C4⋊C4—152no—
399C2×C4⋊C4—151no—
400C2×C4⋊C4—152no—
401C2×C4⋊C4—158no—
402C2×C4⋊C4C2×D1511yescharacteristic
403C2×C4⋊C4C2×D1511yescharacteristic
404C2×C22⋊C4—151no—
405C2×C22⋊C4—151no—
406C2×C22⋊C4—152no—
407C2×C22⋊C4—152no—
408C2×C22⋊C4—152no—
409C2×C22⋊C4C2×D1512yes—
410C2×C42—151no—
411C2×C42C2×D1512yes—
412C2×C3×C5—42no—
413C2×C3×C5#55314yescomplemented
414C2×C3×C5C22×#4111yescharacteristic
415C2×C3×C5C22×#4111yescharacteristic
416C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
417D15—24no—
418D15—44no—
419C23×C3—22no—
420C23×C3—22no—
421C23×C3—22no—
422C23×C3C23×D512yes—
423C23×C3C23×D511yescharacteristic
424C22×D3—54no—
425C22×D3—52no—
426C22×D3—54no—
427C22×D3—102no—
428C22×D3—102no—
429C22×D3—104no—
430C22×D3—104no—
431C22×D3—104no—
432C2×C3×C4—22no—
433C2×C3×C4—24no—
434C2×C3×C4—24no—
435C2×C3×C4—28no—
436C2×C3×C4#70914yes—
437C2×C3×C4C23×D514yes—
438C2×C3×C4C23×D512yes—
439C2×C3×C4C23×D514yes—
440C2×C3×C4C23×D512yes—
441#67—108no—
442#67—208no—
443C2×Dic3—52no—
444C2×Dic3—52no—
445C2×Dic3—52no—
446C2×Dic3—54no—
447C2×Dic3—102no—
448C2×Dic3—102no—
449C2×Dic3—102no—
450C2×Dic3—104no—
451D12—1016no—
452D12—108no—
453C4×D3—108no—
454C4×D3—208no—
455C4×D3—208no—
456C22×C5—22no—
457C22×C5—22no—
458C22×C5—22no—
459C22×C5—44no—
460C22×C5C2×#23512yes—
461C22×C5D3×#4112yes—
462C22×C5#100912yes—
463C22×C5C24×D311yescharacteristic
464C2×D5—34no—
465C2×D5—62no—
466C2×D5—62no—
467C2×D5—68no—
468C2×D5—64no—
469C2×D5—64no—
470C2×D5—64no—
471C2×D5—128no—
472C4×C5—22no—
473C4×C5—24no—
474C4×C5—24no—
475C4×C5C2×#23514yes—
476Dic5—62no—
477Dic5—62no—
478Dic5—62no—
479Dic5—64no—
480C24—151no—
481C24—152no—
482C24C22×D1511yescharacteristic
483C2×D4—154no—
484C2×D4—158no—
485C2×D4—158no—
486C2×D4—154no—
487C2×D4—308no—
488C2×D4—304no—
489C2×D4—304no—
490C2×D4—604no—
491C22×C4—24no—
492C22×C4—151no—
493C22×C4—151no—
494C22×C4—151no—
495C22×C4—152no—
496C22×C4—154no—
497C22×C4—158no—
498C22×C4—158no—
499C22×C4—304no—
500C22×C4—604no—
501C22×C4—604no—
502C22×C4C22×D1514yes—
503C22×C4C22×D1511yescharacteristic
504C22×C4C22×D1512yes—
505C22×C4C22×D1512yes—
506C22×C4C22×D1511yescharacteristic
507C4⋊C4—154no—
508C4⋊C4—158no—
509C4⋊C4—158no—
510C4⋊C4—154no—
511C4⋊C4—158no—
512C4⋊C4C22×D1514yes—
513C4⋊C4C22×D1514yes—
514C22⋊C4—154no—
515C22⋊C4—154no—
516C22⋊C4—158no—
517C22⋊C4—158no—
518C22⋊C4—158no—
519C22⋊C4C22×D1518yes—
520C42—154no—
521C42C22×D1518yes—
522C3×C5C2×#55311yescharacteristic, complemented, Hall
523C22×C3—22no—
524C22×C3—22no—
525C22×C3—22no—
526C22×C3—44no—
527C22×C3C2×#70912yes—
528C22×C3D5×#4112yes—
529C22×C3#414912yes—
530C22×C3C24×D511yescharacteristic
531C2×D3—54no—
532C2×D3—108no—
533C2×D3—102no—
534C2×D3—102no—
535C2×D3—104no—
536C2×D3—104no—
537C2×D3—104no—
538C2×D3—208no—
539C3×C4—22no—
540C3×C4—24no—
541C3×C4—24no—
542C3×C4C2×#70914yes—
543Dic3—102no—
544Dic3—102no—
545Dic3—102no—
546Dic3—104no—
547C2×C5—42no—
548C2×C5#553014yescomplemented
549C2×C5C22×#23511yescharacteristic
550C2×C5C2×D3×#4111yescharacteristic
551C2×C5C2×#100911yescharacteristic
552D5—64no—
553D5—124no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—154no—
558C23—152no—
559C23—154no—
560C23—302no—
561C23—302no—
562C23—304no—
563C23—304no—
564C23—304no—
565C23C23×D1512yes—
566C23C23×D1511yescharacteristic
567D4—3016no—
568D4—308no—
569D4—308no—
570D4—608no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—152no—
576C2×C4—152no—
577C2×C4—152no—
578C2×C4—154no—
579C2×C4—302no—
580C2×C4—302no—
581C2×C4—302no—
582C2×C4—304no—
583C2×C4—308no—
584C2×C4—608no—
585C2×C4—608no—
586C2×C4#681714yes—
587C2×C4C23×D1512yes—
588C2×C4C23×D1514yes—
589C2×C4C23×D1514yes—
590C2×C4C23×D1512yes—
591C2×C3—42no—
592C2×C3#6586414yescomplemented
593C2×C3C22×#70911yescharacteristic
594C2×C3C2×D5×#4111yescharacteristic
595C2×C3C2×#414911yescharacteristic
596D3—104no—
597D3—204no—
598C5C2×#553011yescharacteristic, complemented, Sylow
599C22—22no—
600C22—22no—
601C22—22no—
602C22—44no—
603C22—154no—
604C22—308no—
605C22—302no—
606C22—302no—
607C22—304no—
608C22—304no—
609C22—304no—
610C22—608no—
611C22C2×#681712yes—
612C22#41×D1512yes—
613C22#9210012yes—
614C22C24×D1511yescharacteristic
615C4—22no—
616C4—24no—
617C4—24no—
618C4—302no—
619C4—302no—
620C4—302no—
621C4—304no—
622C4C2×#681714yes—
623C3C2×#6586411yescharacteristic, complemented, Sylow
624C2—42no—
625C2—304no—
626C2—604no—
627C2#1175628914yesdirect product factor
628C2C22×#681711yescharacteristic
629C2C2×#41×D1511yescharacteristic
630C2C2×#9210011yescharacteristic
subgroupG/H
centerC23C23×D15
commutator, cosocleC22×C3×C5C25
FrattiniC22C24×D15
FittingC2×C3×C5×#117C2
socleC23×C3×C5C24
hypercenterC2×#117D15
hypocenterC3×C5C2×#553
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#117C2semidirect
2. C2×#91819C2semidirect
3. C2×C4×#1220C2semidirect
4. #11756289C2direct (N × H)
5. C2×#91784C2semidirect
6. C2×#91782C2semidirect
7. C2×C4⋊C4×D15C2semidirect
8. C2×#91779C2semidirect
9. C2×#91773C2semidirect
10. C2×#91772C2semidirect
11. C2×#91765C2semidirect
12. C2×C4×D60C2semidirect
22. C3×C5×#117C22semidirect
28. #91819C22semidirect
29. C4×#1220C22semidirect
40. #91784C22semidirect
41. #91782C22semidirect
42. C4⋊C4×D15C22semidirect
43. #91779C22semidirect
44. #91773C22semidirect
45. #91772C22semidirect
46. #91765C22semidirect
47. C4×D60C22semidirect
49. C2×C5×#117D3semidirect
103. C2×C3×#117D5semidirect
123. C5×#117C2×D3semidirect
195. C3×#117C2×D5semidirect
273. C2×#117D15semidirect
395. #117C2×D15semidirect
413. C2×C3×C5#553semidirect
522. C3×C5C2×#553semidirect
548. C2×C5#5530semidirect
592. C2×C3#65864semidirect
598. C5C2×#5530semidirect
623. C3C2×#65864semidirect
627. C2#11756289direct (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×#1172
C2×#918192
C2×C4×#12202
#117562892
C2×#917842
C2×#917822
C2×C4⋊C4×D152
C2×#917792
C2×#917732
C2×#917722
C2×#917652
C2×C4×D602
C2×#658643
C2×#55305