Finite Group Databasebeta

← C2×#11754913 #14481140

Subgroups — C2×#11754913 #14481140

Order 1920.

2520 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91827C211yesmaximal, characteristic, complemented
2C2×C5×D4×Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#4086C211yesmaximal, characteristic, complemented
4C2×#91569C212yesmaximal, complemented
5C2×#91550C212yesmaximal, complemented
6C2×#91548C212yesmaximal, complemented
7C2×#91536C212yesmaximal, complemented
8#11754913C2116yesmaximal, direct product factor
9C2×#91465C211yesmaximal, characteristic, complemented
10C2×Dic3×D20C211yesmaximal, characteristic, complemented
11C2×#91388C211yesmaximal, characteristic, complemented
12C2×#91386C211yesmaximal, characteristic, complemented
13C2×#65896—31nomaximal, Hall
14C22×C3×C5×D4C2211yescharacteristic
15C22×#1220C2212yes—
16C22×C4×D15C2211yescharacteristic
17C23×C5×Dic3C2212yes—
18C22×C3×C22⋊D5C2212yes—
19C22×C3×C22⋊D5C2212yes—
20C22×C3×D20C2211yescharacteristic
21C22×#1202C2212yes—
22C22×D5×Dic3C2212yes—
23C2×#6719C2212yes—
24#91827C2218yescomplemented
25C2×C4⋊Dic15C2211yescharacteristic
26C2×C5×#217C2212yes—
27C5×D4×Dic3C2218yescomplemented
28C2×C5×C4⋊Dic3C2211yescharacteristic
29C2×C4×C5×Dic3C2211yescharacteristic
30C3×#4086C2218yescomplemented
31C2×C3×C4×Dic5C2211yescharacteristic
32#91569C22116yescomplemented
33#91550C22116yescomplemented
34#91548C22116yescomplemented
35C2×Dic3⋊Dic5C2212yes—
36C2×#6669C2211yescharacteristic
37C2×#6669C2211yescharacteristic
38C2×#6669C2212yes—
39C2×#6667C2211yescharacteristic
40C2×#6667C2211yescharacteristic
41#91536C22116yescomplemented
42C2×#6665C2212yes—
43C2×#6665C2212yes—
44#91465C2218yescomplemented
45Dic3×D20C2218yescomplemented
46#91388C2218yescomplemented
47#91386C2218yescomplemented
48C2×#5719—51nomaximal, Hall
49C2×C4×C5×D4—31no—
50C2×#4086D311yescharacteristic, complemented
51C2×#4084—31no—
52C2×#4076—32no—
53C2×#4075—32no—
54#65896—316no—
55C2×#4041—31no—
56C2×#4036—31no—
57C2×#4030—32no—
58C2×#4029—32no—
59C2×C4×D20—31no—
60C2×#4018—31no—
61C24×C3×C5C2312yes—
62C23×D15C2311yescharacteristic
63C23×C3×D5C2312yes—
64C2×C3×C5×D4—24no—
65C2×C3×C5×D4C2314yes—
66C22×C3×C4×C5C2311yescharacteristic
67C2×#1220—44no—
68C2×#1220C2318yes—
69C22×Dic15C2312yes—
70C22×Dic15C2311yescharacteristic
71C2×C4×D15—24no—
72C2×C4×D15C2314yes—
73C22×C5×Dic3—44no—
74C22×C5×Dic3C2311yescharacteristic
75C22×C5×Dic3C2311yescharacteristic
76C22×C5×Dic3C2312yes—
77C22×C5×Dic3C2318yes—
78C2×C3×C22⋊D5—44no—
79C2×C3×C22⋊D5—44no—
80C2×C3×C22⋊D5C2318yes—
81C2×C3×C22⋊D5C2318yes—
82C22×C3×Dic5C2311yescharacteristic
83C22×C3×Dic5C2311yescharacteristic
84C2×C3×D20—24no—
85C2×C3×D20C2314yes—
86C2×#1202—44no—
87C2×#1202C2318yes—
88C2×D5×Dic3—44no—
89C2×D5×Dic3C2318yes—
90#6719C2318yes—
91C4⋊Dic15C2314yes—
92C5×#217C2318yes—
93C5×C4⋊Dic3C2314yes—
94C4×C5×Dic3C2314yes—
95C3×C4×Dic5C2314yes—
96Dic3⋊Dic5C2318yes—
97#6669C2314yes—
98#6669C2318yes—
99#6669C2314yes—
100#6667C2314yes—
101#6667C2314yes—
102#6665C2318yes—
103#6665C2318yes—
104C2×C3×#127—51no—
105C2×#939—52no—
106C2×#939—52no—
107C2×#938—51no—
108C2×#938—51no—
109C2×#935—52no—
110C2×#935—52no—
111C2×D4×Dic3—51no—
112C2×D4×Dic3D511yescharacteristic, complemented
113#5719—516no—
114C2×#872—51no—
115C2×#870—51no—
116C22×C5×D4C2×D311yescharacteristic
117C23×C4×C5—32no—
118C22×C22⋊D5—32no—
119C22×C22⋊D5C2×D312yes—
120C22×C22⋊D5C2×D312yes—
121C22×D20—32no—
122C22×D20C2×D311yescharacteristic
123C22×C4×D5—31no—
124C22×C4×D5—32no—
125C4×C5×D4—38no—
126C2×C5×C4⋊C4—31no—
127C2×C5×C22⋊C4—32no—
128C2×C42×C5—31no—
129C2×C22⋊Dic5—32no—
130#4086C2×D318yescomplemented
131#4084—38no—
132#4076—316no—
133#4075—316no—
134C2×#685—32no—
135C2×#685—32no—
136C2×#685—31no—
137C2×#685—31no—
138C2×C4⋊Dic5—31no—
139C2×C4⋊Dic5—32no—
140C2×#683—31no—
141C2×#683—31no—
142C2×#683—32no—
143C2×C4×Dic5C2×D311yescharacteristic
144#4041—38no—
145#4036—38no—
146#4030—316no—
147#4029—316no—
148C4×D20—38no—
149#4018—38no—
150C2×#540—151noSylow
151C23×C3×C5—24no—
152C23×C3×C5—24no—
153C23×C3×C5—24no—
154C23×C3×C5C2414yes—
155C23×C3×C5C2411yescharacteristic
156C22×D15—22no—
157C22×D15—22no—
158C22×D15—22no—
159C22×D15C2412yes—
160C22×C3×D5—24no—
161C22×C3×D5—24no—
162C22×C3×D5—24no—
163C22×C3×D5C2414yes—
164C3×C5×D4—28no—
165C2×C3×C4×C5#4114yes—
166C2×C3×C4×C5C2412yes—
167#1220—48no—
168C2×Dic15—22no—
169C2×Dic15—24no—
170C2×Dic15C2412yes—
171C2×Dic15C2414yes—
172C4×D15—28no—
173C2×C5×Dic3—22no—
174C2×C5×Dic3—24no—
175C2×C5×Dic3—22no—
176C2×C5×Dic3—48no—
177C2×C5×Dic3C2412yes—
178C2×C5×Dic3C2414yes—
179C2×C5×Dic3C2412yes—
180C3×C22⋊D5—48no—
181C3×C22⋊D5—48no—
182C2×C3×Dic5—22no—
183C2×C3×Dic5—22no—
184C2×C3×Dic5C2412yes—
185C2×C3×Dic5C2412yes—
186C3×D20—28no—
187#1202—48no—
188D5×Dic3—48no—
189C22×C3×D4—52no—
190C22×C3×D4—52no—
191C22×C3×D4—51no—
192C22×C3×D4C2×D511yescharacteristic
193C22×#67—52no—
194C22×#67—52no—
195C23×Dic3—52no—
196C23×Dic3C2×D512yes—
197C22×C4×D3—51no—
198C3×#127—58no—
199C2×C3×C42—51no—
200C2×#217—52no—
201C2×#217—52no—
202C2×#217—52no—
203C2×#217C2×D512yes—
204#939—516no—
205#939—516no—
206#938—58no—
207#938—58no—
208#935—516no—
209#935—516no—
210D4×Dic3—58no—
211D4×Dic3C2×D518yescomplemented
212C2×(C2×D3)⋊C4—51no—
213C2×(C2×D3)⋊C4—51no—
214C2×C4⋊Dic3—51no—
215C2×C4⋊Dic3C2×D511yescharacteristic
216C2×Dic3⋊C4—52no—
217C2×Dic3⋊C4—52no—
218C2×Dic3⋊C4—51no—
219C2×Dic3⋊C4—51no—
220C2×C4×Dic3C2×D511yescharacteristic
221#872—58no—
222#870—58no—
223C24×C5C22×D312yes—
224C23×D5—31no—
225C23×D5C22×D312yes—
226C2×C5×D4—24no—
227C2×C5×D4C22×D314yes—
228C22×C4×C5—31no—
229C22×C4×C5—31no—
230C22×C4×C5—32no—
231C22×C4×C5—38no—
232C22×C4×C5—124no—
233C22×C4×C5C22×D311yescharacteristic
234C2×C22⋊D5—38no—
235C2×C22⋊D5—44no—
236C2×C22⋊D5—44no—
237C2×C22⋊D5—124no—
238C2×C22⋊D5C22×D318yes—
239C2×C22⋊D5C22×D318yes—
240C22×Dic5—31no—
241C22×Dic5—32no—
242C22×Dic5C22×D311yescharacteristic
243C22×Dic5C22×D311yescharacteristic
244C2×D20—24no—
245C2×D20—38no—
246C2×D20—124no—
247C2×D20C22×D314yes—
248C2×C4×D5—34no—
249C2×C4×D5—38no—
250C2×C4×D5—64no—
251C2×C4×D5—124no—
252C5×C4⋊C4—34no—
253C5×C22⋊C4—38no—
254C42×C5—34no—
255C22⋊Dic5—38no—
256#685—38no—
257#685—38no—
258#685—34no—
259#685—34no—
260C4⋊Dic5—34no—
261C4⋊Dic5—38no—
262#683—34no—
263#683—34no—
264#683—38no—
265C4×Dic5C22×D314yes—
266#540—1516no—
267C2×#127—51no—
268C2×#125—151no—
269C2×#123—152no—
270C2×#123—152no—
271C2×#121—152no—
272C2×#121—151no—
273C2×#121—152no—
274C2×#121—151no—
275C2×C4×D4—31no—
276C2×C4×D4—151no—
277C2×#117—151no—
278C22×C3×C5—24no—
279C22×C3×C5—24no—
280C22×C3×C5—24no—
281C22×C3×C5—48no—
282C22×C3×C5C2×#4112yes—
283C22×C3×C5H(5,𝔽2)12yes—
284C22×C3×C5H(5,𝔽2)12yes—
285C22×C3×C5C2511yescharacteristic
286C2×D15—22no—
287C2×D15—22no—
288C2×D15—22no—
289C2×D15—44no—
290C2×C3×D5—24no—
291C2×C3×D5—24no—
292C2×C3×D5—24no—
293C2×C3×D5—48no—
294C3×C4×C5C2×#4114yes—
295Dic15—22no—
296Dic15—24no—
297C3×Dic5—22no—
298C3×Dic5—22no—
299C5×Dic3—22no—
300C5×Dic3—24no—
301C5×Dic3—22no—
302C24×C3—52no—
303C24×C3C22×D512yes—
304C23×D3—51no—
305C2×C3×D4—24no—
306C2×C3×D4—58no—
307C2×C3×D4—58no—
308C2×C3×D4—54no—
309C2×C3×D4—104no—
310C2×C3×D4—204no—
311C2×C3×D4—204no—
312C2×C3×D4C22×D514yes—
313C22×C3×C4—51no—
314C22×C3×C4—51no—
315C22×C3×C4C22×D511yescharacteristic
316C2×#67—58no—
317C2×#67—58no—
318C2×#67—204no—
319C2×#67—204no—
320C22×Dic3—44no—
321C22×Dic3—52no—
322C22×Dic3—51no—
323C22×Dic3—58no—
324C22×Dic3—204no—
325C22×Dic3C22×D511yescharacteristic
326C22×Dic3C22×D511yescharacteristic
327C22×Dic3C22×D512yes—
328C22×Dic3C22×D518yes—
329C2×C4×D3—54no—
330C2×C4×D3—104no—
331C3×C42—54no—
332#217—58no—
333#217—58no—
334#217—58no—
335#217C22×D518yes—
336(C2×D3)⋊C4—54no—
337(C2×D3)⋊C4—54no—
338C4⋊Dic3—54no—
339C4⋊Dic3C22×D514yes—
340Dic3⋊C4—58no—
341Dic3⋊C4—54no—
342Dic3⋊C4—54no—
343Dic3⋊C4—58no—
344C4×Dic3C22×D514yes—
345C23×C5—24no—
346C23×C5—24no—
347C23×C5—24no—
348C23×C5C23×D314yes—
349C23×C5C23×D311yescharacteristic
350C22×D5—24no—
351C22×D5—24no—
352C22×D5—24no—
353C22×D5—32no—
354C22×D5—62no—
355C22×D5—62no—
356C22×D5—62no—
357C22×D5C23×D314yes—
358C5×D4—28no—
359C2×C4×C5—34no—
360C2×C4×C5—32no—
361C2×C4×C5—32no—
362C2×C4×C5—62no—
363C2×C4×C5—64no—
364C2×C4×C5—62no—
365C2×C4×C5—128no—
366C2×C4×C5#23714yes—
367C2×C4×C5C23×D312yes—
368C22⋊D5—48no—
369C22⋊D5—48no—
370C22⋊D5—128no—
371C2×Dic5—22no—
372C2×Dic5—22no—
373C2×Dic5—32no—
374C2×Dic5—34no—
375C2×Dic5—62no—
376C2×Dic5—64no—
377C2×Dic5C23×D312yes—
378C2×Dic5C23×D312yes—
379D20—28no—
380D20—128no—
381C4×D5—68no—
382C4×D5—128no—
383C22×D4—52no—
384C22×D4—52no—
385C22×D4—51no—
386C22×D4—152no—
387C22×D4—152no—
388C22×D4D3×D511yescharacteristic
389C23×C4—32no—
390C23×C4—152no—
391C23×C4—151no—
392#127—58no—
393#125—158no—
394#123—1516no—
395#123—1516no—
396#121—1516no—
397#121—158no—
398#121—1516no—
399#121—158no—
400C4×D4—38no—
401C4×D4—158no—
402#117—158no—
403C2×C4⋊C4—31no—
404C2×C4⋊C4—151no—
405C2×C4⋊C4—151no—
406C2×C4⋊C4—152no—
407C2×C4⋊C4—152no—
408C2×C4⋊C4—151no—
409C2×C22⋊C4—32no—
410C2×C22⋊C4—151no—
411C2×C22⋊C4—151no—
412C2×C22⋊C4—152no—
413C2×C22⋊C4—152no—
414C2×C22⋊C4—152no—
415C2×C42—31no—
416C2×C42—51no—
417C2×C3×C5—44no—
418C2×C3×C5#54014yescomplemented
419C2×C3×C5C22×#4111yescharacteristic
420C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
421C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
422D15—42no—
423C3×D5—44no—
424C23×C3—24no—
425C23×C3—24no—
426C23×C3—24no—
427C23×C3—54no—
428C23×C3—104no—
429C23×C3—104no—
430C23×C3—104no—
431C23×C3C23×D514yes—
432C23×C3C23×D511yescharacteristic
433C22×D3—52no—
434C22×D3—102no—
435C22×D3—102no—
436C22×D3—102no—
437C3×D4—28no—
438C3×D4—108no—
439C3×D4—208no—
440C3×D4—208no—
441C2×C3×C4—52no—
442C2×C3×C4—52no—
443C2×C3×C4—102no—
444C2×C3×C4—102no—
445C2×C3×C4#70914yes—
446C2×C3×C4C23×D512yes—
447#67—208no—
448#67—208no—
449C2×Dic3—22no—
450C2×Dic3—24no—
451C2×Dic3—22no—
452C2×Dic3—48no—
453C2×Dic3—54no—
454C2×Dic3—52no—
455C2×Dic3—102no—
456C2×Dic3—104no—
457C2×Dic3—208no—
458C2×Dic3C23×D512yes—
459C2×Dic3C23×D512yes—
460C2×Dic3C23×D514yes—
461C4×D3—108no—
462C22×C5—24no—
463C22×C5—24no—
464C22×C5—24no—
465C22×C5—48no—
466C22×C5C2×#23712yes—
467C22×C5#100412yes—
468C22×C5#100412yes—
469C22×C5C24×D311yescharacteristic
470C2×D5—24no—
471C2×D5—24no—
472C2×D5—24no—
473C2×D5—48no—
474C2×D5—62no—
475C2×D5—62no—
476C2×D5—62no—
477C2×D5—124no—
478C4×C5—62no—
479C4×C5—64no—
480C4×C5—62no—
481C4×C5C2×#23714yes—
482Dic5—22no—
483Dic5—22no—
484Dic5—62no—
485Dic5—64no—
486C24—52no—
487C24—151no—
488C24C2×D3×D512yes—
489C2×D4—24no—
490C2×D4—58no—
491C2×D4—58no—
492C2×D4—54no—
493C2×D4—104no—
494C2×D4—158no—
495C2×D4—158no—
496C2×D4—204no—
497C2×D4—204no—
498C2×D4—604no—
499C2×D4—604no—
500C2×D4C2×D3×D514yes—
501C22×C4—31no—
502C22×C4—31no—
503C22×C4—32no—
504C22×C4—38no—
505C22×C4—51no—
506C22×C4—51no—
507C22×C4—124no—
508C22×C4—158no—
509C22×C4—151no—
510C22×C4—152no—
511C22×C4—154no—
512C22×C4—304no—
513C22×C4—604no—
514C22×C4C2×D3×D511yescharacteristic
515C4⋊C4—34no—
516C4⋊C4—154no—
517C4⋊C4—154no—
518C4⋊C4—158no—
519C4⋊C4—158no—
520C4⋊C4—154no—
521C22⋊C4—38no—
522C22⋊C4—154no—
523C22⋊C4—154no—
524C22⋊C4—158no—
525C22⋊C4—158no—
526C22⋊C4—158no—
527C42—34no—
528C42—54no—
529C3×C5C2×#54011yescharacteristic, complemented, Hall
530C22×C3—24no—
531C22×C3—24no—
532C22×C3—24no—
533C22×C3—48no—
534C22×C3—104no—
535C22×C3—104no—
536C22×C3—104no—
537C22×C3—208no—
538C22×C3C2×#70912yes—
539C22×C3#414412yes—
540C22×C3#414912yes—
541C22×C3C24×D511yescharacteristic
542C2×D3—102no—
543C2×D3—102no—
544C2×D3—102no—
545C2×D3—204no—
546C3×C4—102no—
547C3×C4—102no—
548C3×C4C2×#70914yes—
549Dic3—22no—
550Dic3—24no—
551Dic3—22no—
552Dic3—102no—
553Dic3—104no—
554C2×C5—44no—
555C2×C5#571914yescomplemented
556C2×C5C22×#23711yescharacteristic
557C2×C5C2×#100411yescharacteristic
558C2×C5C2×#100411yescharacteristic
559D5—44no—
560D5—122no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—54no—
565C23—104no—
566C23—104no—
567C23—104no—
568C23—152no—
569C23—302no—
570C23—302no—
571C23—302no—
572C23C22×D3×D514yes—
573C23C22×D3×D511yescharacteristic
574D4—28no—
575D4—108no—
576D4—208no—
577D4—208no—
578D4—608no—
579D4—608no—
580C2×C4—32no—
581C2×C4—32no—
582C2×C4—34no—
583C2×C4—52no—
584C2×C4—52no—
585C2×C4—62no—
586C2×C4—64no—
587C2×C4—62no—
588C2×C4—102no—
589C2×C4—102no—
590C2×C4—128no—
591C2×C4—152no—
592C2×C4—154no—
593C2×C4—304no—
594C2×C4—302no—
595C2×C4—308no—
596C2×C4—608no—
597C2×C4#677914yes—
598C2×C4C22×D3×D512yes—
599C2×C3—44no—
600C2×C3—204no—
601C2×C3#6589614yescomplemented
602C2×C3C22×#70911yescharacteristic
603C2×C3C2×#414411yescharacteristic
604C2×C3C2×#414911yescharacteristic
605D3—202no—
606C5C2×#571911yescharacteristic, complemented, Sylow
607C22—24no—
608C22—24no—
609C22—24no—
610C22—48no—
611C22—104no—
612C22—104no—
613C22—104no—
614C22—208no—
615C22—302no—
616C22—302no—
617C22—302no—
618C22—604no—
619C22#9202512yes—
620C22#9202612yes—
621C22C2×#677912yes—
622C22C23×D3×D511yescharacteristic
623C4—62no—
624C4—64no—
625C4—62no—
626C4—102no—
627C4—102no—
628C4—302no—
629C4—304no—
630C4C2×#677914yes—
631C3C2×#6589611yescharacteristic, complemented, Sylow
632C2—44no—
633C2—204no—
634C2—602no—
635C2#1175491314yesdirect product factor
636C2C2×#9202511yescharacteristic
637C2C2×#9202611yescharacteristic
638C2C22×#677911yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC22×C3×C5×D4C22
socleC23×C3×C5C24
hypercenterC22×D4D3×D5
hypocenterC3×C5C2×#540
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#91827C2semidirect
2. C2×C5×D4×Dic3C2semidirect
3. C2×C3×#4086C2semidirect
4. C2×#91569C2semidirect
5. C2×#91550C2semidirect
6. C2×#91548C2semidirect
7. C2×#91536C2semidirect
8. #11754913C2direct (N × H)
9. C2×#91465C2semidirect
10. C2×Dic3×D20C2semidirect
11. C2×#91388C2semidirect
12. C2×#91386C2semidirect
24. #91827C22semidirect
27. C5×D4×Dic3C22semidirect
30. C3×#4086C22semidirect
32. #91569C22semidirect
33. #91550C22semidirect
34. #91548C22semidirect
41. #91536C22semidirect
44. #91465C22semidirect
45. Dic3×D20C22semidirect
46. #91388C22semidirect
47. #91386C22semidirect
50. C2×#4086D3semidirect
112. C2×D4×Dic3D5semidirect
130. #4086C2×D3semidirect
211. D4×Dic3C2×D5semidirect
418. C2×C3×C5#540semidirect
529. C3×C5C2×#540semidirect
555. C2×C5#5719semidirect
601. C2×C3#65896semidirect
606. C5C2×#5719semidirect
631. C3C2×#65896semidirect
635. C2#11754913direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#918272
C2×C5×D4×Dic32
C2×C3×#40862
C2×#915692
C2×#915502
C2×#915482
C2×#915362
#117549132
C2×#914652
C2×Dic3×D202
C2×#913882
C2×#913862
C2×#658963
C2×#57195