Finite Group Databasebeta

← C2×#11756272 #14483408

Subgroups — C2×#11756272 #14483408

Order 1920.

2248 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C5×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#91818C212yesmaximal
3C2×C4×#1220C211yesmaximal, characteristic
4C2×#91815C212yesmaximal, complemented
5C2×#91813C211yesmaximal, characteristic, complemented
6#11756272C2116yesmaximal, direct product factor
7C2×#91787C212yesmaximal, complemented
8C2×#91786C211yesmaximal, characteristic, complemented
9C2×#91781C211yesmaximal, characteristic, complemented
10C2×#91779C211yesmaximal, characteristic, complemented
11C2×#91778C212yesmaximal, complemented
12C2×#91776C211yesmaximal, characteristic, complemented
13C2×#65847—31nomaximal, Hall
14C23×C3×C4×C5C2212yes—
15C23×C3×C4×C5C2211yescharacteristic
16C22×#1220C2211yescharacteristic
17C22×C4×D15C2211yescharacteristic
18C22×Dic30C2211yescharacteristic
19C2×C3×C5×C4⋊C4C2211yescharacteristic
20C2×C3×C5×C4⋊C4C2212yes—
21C2×C3×C5×C4⋊C4C2211yescharacteristic
22C2×C3×C5×C4⋊C4C2218yescomplemented
23C2×#6719C2211yescharacteristic
24C2×#6719C2212yes—
25#91818C22116yes—
26C4×#1220C2218yes—
27C2×#6714C2211yescharacteristic
28C2×#6714C2212yes—
29C2×#6714C2212yes—
30#91815C22116yescomplemented
31C2×C4⋊Dic15C2212yes—
32C2×C4⋊Dic15C2212yes—
33C2×C4⋊Dic15C2211yescharacteristic
34#91813C2218yescomplemented
35C2×#6712C2211yescharacteristic
36C2×#6712C2212yes—
37C2×#6712C2212yes—
38C2×#6712C2212yes—
39C2×C4×Dic15C2212yes—
40C2×C4×Dic15C2212yes—
41C2×C4×Dic15C2211yescharacteristic
42#91787C22116yescomplemented
43#91786C2218yescomplemented
44#91781C2218yescomplemented
45#91779C2218yescomplemented
46#91778C22116yescomplemented
47#91776C2218yescomplemented
48C2×#5513—51nomaximal, Hall
49C22×C5×C4⋊C4D311yescharacteristic, complemented
50C2×#4075—32no—
51C2×C4×C22⋊D5—31no—
52C2×#4072—32no—
53C2×#4070—31no—
54#65847—316no—
55C2×#4044—32no—
56C2×#4043—31no—
57C2×#4038—31no—
58C2×#4036—31no—
59C2×#4035—32no—
60C2×#4033—31no—
61C24×C3×C5C2311yescharacteristic
62C23×D15C2311yescharacteristic
63C22×C3×C4×C5—28no—
64C22×C3×C4×C5—24no—
65C22×C3×C4×C5C2312yes—
66C22×C3×C4×C5C2311yescharacteristic
67C22×C3×C4×C5C2312yes—
68C22×C3×C4×C5C2314yes—
69C22×C3×C4×C5C2318yes—
70C22×C3×C4×C5C2311yescharacteristic
71C2×#1220—24no—
72C2×#1220C2314yes—
73C22×Dic15C2311yescharacteristic
74C22×Dic15C2311yescharacteristic
75C22×Dic15C2311yescharacteristic
76C22×Dic15C2312yes—
77C22×Dic15C2312yes—
78C2×C4×D15—24no—
79C2×C4×D15C2314yes—
80C2×Dic30—24no—
81C2×Dic30C2314yes—
82C3×C5×C4⋊C4C2314yes—
83C3×C5×C4⋊C4C2318yes—
84C3×C5×C4⋊C4C2314yes—
85#6719C2318yes—
86#6719C2314yes—
87#6714C2318yes—
88#6714C2318yes—
89#6714C2314yes—
90C4⋊Dic15C2318yes—
91C4⋊Dic15C2314yes—
92C4⋊Dic15C2318yes—
93#6712C2318yes—
94#6712C2314yes—
95#6712C2318yes—
96#6712C2318yes—
97C4×Dic15C2314yes—
98C4×Dic15C2318yes—
99C4×Dic15C2318yes—
100C22×C3×C4⋊C4D511yescharacteristic, complemented
101C2×#929—52no—
102C2×C4×#67—51no—
103C2×#926—52no—
104C2×#924—51no—
105#5513—516no—
106C2×#898—52no—
107C2×#897—51no—
108C2×#892—51no—
109C2×#890—51no—
110C2×#889—52no—
111C2×#887—51no—
112C23×C4×C5C2×D311yescharacteristic
113C23×C4×C5C2×D312yes—
114C22×C22⋊D5—31no—
115C22×C4×D5—31no—
116C22×Dic10—31no—
117C2×C5×C4⋊C4C2×D311yescharacteristic
118C2×C5×C4⋊C4C2×D312yes—
119C2×C5×C4⋊C4C2×D311yescharacteristic
120C2×C5×C4⋊C4C2×D318yescomplemented
121C2×C22⋊Dic5—32no—
122C2×C22⋊Dic5—31no—
123#4075—316no—
124C4×C22⋊D5—38no—
125C2×#685—32no—
126C2×#685—32no—
127C2×#685—31no—
128#4072—316no—
129C2×C4⋊Dic5—32no—
130C2×C4⋊Dic5—31no—
131C2×C4⋊Dic5—32no—
132#4070—38no—
133C2×#683—32no—
134C2×#683—31no—
135C2×#683—32no—
136C2×#683—32no—
137C2×C4×Dic5—31no—
138C2×C4×Dic5—32no—
139C2×C4×Dic5—32no—
140#4044—316no—
141#4043—38no—
142#4038—38no—
143#4036—38no—
144#4035—316no—
145#4033—38no—
146C2×#552—151noSylow
147C23×C3×C5—22no—
148C23×C3×C5—22no—
149C23×C3×C5#4114yes—
150C23×C3×C5C2412yes—
151C23×C3×C5C2411yescharacteristic
152C22×D15—22no—
153C22×D15—22no—
154C22×D15—22no—
155C22×D15C2412yes—
156C2×C3×C4×C5—22no—
157C2×C3×C4×C5—216no—
158C2×C3×C4×C5—24no—
159C2×C3×C4×C5—24no—
160C2×C3×C4×C5—28no—
161C2×C3×C4×C5#4114yes—
162C2×C3×C4×C5C2414yes—
163C2×C3×C4×C5C2412yes—
164C2×C3×C4×C5C2414yes—
165C2×C3×C4×C5C2412yes—
166#1220—28no—
167C2×Dic15—24no—
168C2×Dic15—22no—
169C2×Dic15—22no—
170C2×Dic15—22no—
171C2×Dic15—24no—
172C2×Dic15C2412yes—
173C2×Dic15C2412yes—
174C2×Dic15C2412yes—
175C2×Dic15C2414yes—
176C2×Dic15C2414yes—
177C4×D15—28no—
178Dic30—28no—
179C23×C3×C4C2×D511yescharacteristic
180C23×C3×C4C2×D512yes—
181C22×#67—51no—
182C22×C4×D3—51no—
183C22×Dic6—51no—
184C2×C3×C4⋊C4C2×D511yescharacteristic
185C2×C3×C4⋊C4C2×D512yes—
186C2×C3×C4⋊C4C2×D511yescharacteristic
187C2×C3×C4⋊C4C2×D518yescomplemented
188C2×#217—52no—
189C2×#217—51no—
190#929—516no—
191C4×#67—58no—
192C2×(C2×D3)⋊C4—52no—
193C2×(C2×D3)⋊C4—52no—
194C2×(C2×D3)⋊C4—51no—
195#926—516no—
196C2×C4⋊Dic3—52no—
197C2×C4⋊Dic3—51no—
198C2×C4⋊Dic3—52no—
199#924—58no—
200C2×Dic3⋊C4—52no—
201C2×Dic3⋊C4—51no—
202C2×Dic3⋊C4—52no—
203C2×Dic3⋊C4—52no—
204C2×C4×Dic3—51no—
205C2×C4×Dic3—52no—
206C2×C4×Dic3—52no—
207#898—516no—
208#897—58no—
209#892—58no—
210#890—58no—
211#889—516no—
212#887—58no—
213C24×C5C22×D311yescharacteristic
214C23×D5—31no—
215C22×C4×C5—28no—
216C22×C4×C5—24no—
217C22×C4×C5C22×D312yes—
218C22×C4×C5C22×D311yescharacteristic
219C22×C4×C5C22×D312yes—
220C22×C4×C5C22×D314yes—
221C22×C4×C5C22×D318yes—
222C22×C4×C5C22×D311yescharacteristic
223C2×C22⋊D5—34no—
224C2×C22⋊D5—64no—
225C22×Dic5—32no—
226C22×Dic5—31no—
227C22×Dic5—31no—
228C22×Dic5—32no—
229C22×Dic5—31no—
230C2×C4×D5—34no—
231C2×C4×D5—64no—
232C2×Dic10—34no—
233C2×Dic10—64no—
234C5×C4⋊C4C22×D314yes—
235C5×C4⋊C4C22×D318yes—
236C5×C4⋊C4C22×D314yes—
237C22⋊Dic5—38no—
238C22⋊Dic5—34no—
239#685—38no—
240#685—38no—
241#685—34no—
242C4⋊Dic5—38no—
243C4⋊Dic5—34no—
244C4⋊Dic5—38no—
245#683—38no—
246#683—34no—
247#683—38no—
248#683—38no—
249C4×Dic5—34no—
250C4×Dic5—38no—
251C4×Dic5—38no—
252#552—1516no—
253C2×#126—152no—
254C2×#125—152no—
255C2×#125—151no—
256C2×#123—152no—
257C2×C22⋊Q8—151no—
258C2×C22⋊Q8—151no—
259C2×C4×Q8—151no—
260C2×C4×D4—151no—
261C2×#117—152no—
262C2×#117—151no—
263C22×C4⋊C4D1511yescharacteristic, complemented
264C22×C3×C5—28no—
265C22×C3×C5—22no—
266C22×C3×C5—22no—
267C22×C3×C5C2×#4112yes—
268C22×C3×C5C2×#4114yes—
269C22×C3×C5C2×#4112yes—
270C22×C3×C52−1+412yes—
271C22×C3×C5C2511yescharacteristic
272C2×D15—22no—
273C2×D15—22no—
274C2×D15—22no—
275C2×D15—44no—
276C3×C4×C5—22no—
277C3×C4×C5—24no—
278C3×C4×C5—24no—
279C3×C4×C5C2×#4114yes—
280Dic15—24no—
281Dic15—22no—
282Dic15—22no—
283Dic15—22no—
284Dic15—24no—
285C24×C3C22×D511yescharacteristic
286C23×D3—51no—
287C22×C3×C4—28no—
288C22×C3×C4—24no—
289C22×C3×C4C22×D512yes—
290C22×C3×C4C22×D511yescharacteristic
291C22×C3×C4C22×D512yes—
292C22×C3×C4C22×D514yes—
293C22×C3×C4C22×D518yes—
294C22×C3×C4C22×D511yescharacteristic
295C2×#67—54no—
296C2×#67—104no—
297C22×Dic3—51no—
298C22×Dic3—51no—
299C22×Dic3—51no—
300C22×Dic3—52no—
301C22×Dic3—52no—
302C2×C4×D3—54no—
303C2×C4×D3—104no—
304C2×Dic6—54no—
305C2×Dic6—104no—
306C3×C4⋊C4C22×D514yes—
307C3×C4⋊C4C22×D518yes—
308C3×C4⋊C4C22×D514yes—
309#217—58no—
310#217—54no—
311(C2×D3)⋊C4—58no—
312(C2×D3)⋊C4—58no—
313(C2×D3)⋊C4—54no—
314C4⋊Dic3—58no—
315C4⋊Dic3—54no—
316C4⋊Dic3—58no—
317Dic3⋊C4—58no—
318Dic3⋊C4—54no—
319Dic3⋊C4—58no—
320Dic3⋊C4—58no—
321C4×Dic3—54no—
322C4×Dic3—58no—
323C4×Dic3—58no—
324C23×C5—22no—
325C23×C5—22no—
326C23×C5#23714yes—
327C23×C5C23×D312yes—
328C23×C5C23×D311yescharacteristic
329C22×D5—32no—
330C22×D5—62no—
331C22×D5—62no—
332C22×D5—62no—
333C2×C4×C5—22no—
334C2×C4×C5—216no—
335C2×C4×C5—24no—
336C2×C4×C5—24no—
337C2×C4×C5—28no—
338C2×C4×C5#23514yes—
339C2×C4×C5C23×D314yes—
340C2×C4×C5C23×D312yes—
341C2×C4×C5C23×D314yes—
342C2×C4×C5C23×D312yes—
343C22⋊D5—68no—
344C2×Dic5—32no—
345C2×Dic5—32no—
346C2×Dic5—32no—
347C2×Dic5—34no—
348C2×Dic5—34no—
349C2×Dic5—64no—
350C2×Dic5—62no—
351C2×Dic5—62no—
352C2×Dic5—62no—
353C2×Dic5—64no—
354C4×D5—68no—
355Dic10—68no—
356C22×Q8—151no—
357C22×D4—151no—
358C23×C4—151no—
359C23×C4C2×D1512yes—
360C23×C4C2×D1511yescharacteristic
361#126—1516no—
362#125—1516no—
363#125—158no—
364#123—1516no—
365C22⋊Q8—158no—
366C22⋊Q8—158no—
367C4×Q8—158no—
368C4×D4—158no—
369#117—1516no—
370#117—158no—
371C2×C4⋊C4—152no—
372C2×C4⋊C4—151no—
373C2×C4⋊C4—152no—
374C2×C4⋊C4—151no—
375C2×C4⋊C4—152no—
376C2×C4⋊C4—152no—
377C2×C4⋊C4—152no—
378C2×C4⋊C4C2×D1511yescharacteristic
379C2×C4⋊C4C2×D1511yescharacteristic
380C2×C4⋊C4C2×D1512yes—
381C2×C4⋊C4C2×D1518yescomplemented
382C2×C22⋊C4—151no—
383C2×C22⋊C4—152no—
384C2×C22⋊C4—152no—
385C2×C22⋊C4—151no—
386C2×C22⋊C4—152no—
387C2×C42—152no—
388C2×C42—152no—
389C2×C42—151no—
390C2×C3×C5—24no—
391C2×C3×C5#55214yescomplemented
392C2×C3×C5C22×#4111yescharacteristic
393C2×C3×C5C22×#4111yescharacteristic
394C2×C3×C5C2×2−1+411yescharacteristic
395D15—42no—
396C23×C3—22no—
397C23×C3—22no—
398C23×C3#71114yes—
399C23×C3C23×D512yes—
400C23×C3C23×D511yescharacteristic
401C22×D3—52no—
402C22×D3—102no—
403C22×D3—102no—
404C22×D3—102no—
405C2×C3×C4—22no—
406C2×C3×C4—216no—
407C2×C3×C4—24no—
408C2×C3×C4—24no—
409C2×C3×C4—28no—
410C2×C3×C4#70914yes—
411C2×C3×C4C23×D514yes—
412C2×C3×C4C23×D512yes—
413C2×C3×C4C23×D514yes—
414C2×C3×C4C23×D512yes—
415#67—108no—
416C2×Dic3—52no—
417C2×Dic3—52no—
418C2×Dic3—52no—
419C2×Dic3—54no—
420C2×Dic3—54no—
421C2×Dic3—104no—
422C2×Dic3—102no—
423C2×Dic3—102no—
424C2×Dic3—102no—
425C2×Dic3—104no—
426C4×D3—108no—
427Dic6—108no—
428C22×C5—22no—
429C22×C5—22no—
430C22×C5—28no—
431C22×C5C2×#23512yes—
432C22×C5C2×#23714yes—
433C22×C5C2×#23712yes—
434C22×C5#100712yes—
435C22×C5C24×D311yescharacteristic
436C2×D5—62no—
437C2×D5—62no—
438C2×D5—62no—
439C2×D5—124no—
440C4×C5—22no—
441C4×C5—24no—
442C4×C5—24no—
443C4×C5C2×#23514yes—
444Dic5—64no—
445Dic5—62no—
446Dic5—62no—
447Dic5—62no—
448Dic5—64no—
449C24—151no—
450C24C22×D1511yescharacteristic
451C2×Q8—154no—
452C2×Q8—304no—
453C2×D4—154no—
454C2×D4—304no—
455C22×C4—28no—
456C22×C4—24no—
457C22×C4—151no—
458C22×C4—151no—
459C22×C4—152no—
460C22×C4—152no—
461C22×C4—151no—
462C22×C4—154no—
463C22×C4—304no—
464C22×C4C22×D1518yes—
465C22×C4C22×D1514yes—
466C22×C4C22×D1512yes—
467C22×C4C22×D1511yescharacteristic
468C22×C4C22×D1512yes—
469C22×C4C22×D1511yescharacteristic
470C4⋊C4—158no—
471C4⋊C4—154no—
472C4⋊C4—158no—
473C4⋊C4—154no—
474C4⋊C4—158no—
475C4⋊C4—158no—
476C4⋊C4—158no—
477C4⋊C4C22×D1514yes—
478C4⋊C4C22×D1514yes—
479C4⋊C4C22×D1518yes—
480C22⋊C4—154no—
481C22⋊C4—158no—
482C22⋊C4—158no—
483C22⋊C4—154no—
484C22⋊C4—158no—
485C42—158no—
486C42—158no—
487C42—154no—
488C3×C5C2×#55211yescharacteristic, complemented, Hall
489C22×C3—28no—
490C22×C3—22no—
491C22×C3—22no—
492C22×C3C2×#70912yes—
493C22×C3C2×#71114yes—
494C22×C3C2×#71112yes—
495C22×C3#414712yes—
496C22×C3C24×D511yescharacteristic
497C2×D3—102no—
498C2×D3—102no—
499C2×D3—102no—
500C2×D3—204no—
501C3×C4—22no—
502C3×C4—24no—
503C3×C4—24no—
504C3×C4C2×#70914yes—
505Dic3—104no—
506Dic3—102no—
507Dic3—102no—
508Dic3—102no—
509Dic3—104no—
510C2×C5—24no—
511C2×C5#551314yescomplemented
512C2×C5C22×#23511yescharacteristic
513C2×C5C22×#23711yescharacteristic
514C2×C5C2×#100711yescharacteristic
515D5—122no—
516C23—22no—
517C23—22no—
518C23—152no—
519C23—302no—
520C23—302no—
521C23—302no—
522C23#681914yes—
523C23C23×D1512yes—
524C23C23×D1511yescharacteristic
525Q8—308no—
526D4—308no—
527C2×C4—22no—
528C2×C4—216no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—152no—
533C2×C4—152no—
534C2×C4—154no—
535C2×C4—154no—
536C2×C4—152no—
537C2×C4—302no—
538C2×C4—304no—
539C2×C4—302no—
540C2×C4—302no—
541C2×C4—304no—
542C2×C4—308no—
543C2×C4#681714yes—
544C2×C4C23×D1514yes—
545C2×C4C23×D1512yes—
546C2×C4C23×D1514yes—
547C2×C4C23×D1512yes—
548C2×C3—24no—
549C2×C3#6584714yescomplemented
550C2×C3C22×#70911yescharacteristic
551C2×C3C22×#71111yescharacteristic
552C2×C3C2×#414711yescharacteristic
553D3—202no—
554C5C2×#551311yescharacteristic, complemented, Sylow
555C22—22no—
556C22—22no—
557C22—28no—
558C22—302no—
559C22—302no—
560C22—302no—
561C22—604no—
562C22C2×#681712yes—
563C22C2×#681914yes—
564C22C2×#681912yes—
565C22#9209812yes—
566C22C24×D1511yescharacteristic
567C4—22no—
568C4—24no—
569C4—24no—
570C4—302no—
571C4—304no—
572C4—302no—
573C4—302no—
574C4—304no—
575C4C2×#681714yes—
576C3C2×#6584711yescharacteristic, complemented, Sylow
577C2—24no—
578C2—602no—
579C2#1175627214yesdirect product factor
580C2C22×#681711yescharacteristic
581C2C22×#681911yescharacteristic
582C2C2×#9209811yescharacteristic
subgroupG/H
centerC23C23×D15
commutator, cosocleC22×C3×C5C25
FrattiniC22C24×D15
FittingC22×C3×C5×C4⋊C4C2
socleC23×C3×C5C24
prime radicalC22×#1220C22
hypercenterC22×C4⋊C4D15
hypocenterC3×C5C2×#552
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C5×C4⋊C4C2semidirect
4. C2×#91815C2semidirect
5. C2×#91813C2semidirect
6. #11756272C2direct (N × H)
7. C2×#91787C2semidirect
8. C2×#91786C2semidirect
9. C2×#91781C2semidirect
10. C2×#91779C2semidirect
11. C2×#91778C2semidirect
12. C2×#91776C2semidirect
22. C2×C3×C5×C4⋊C4C22semidirect
30. #91815C22semidirect
34. #91813C22semidirect
42. #91787C22semidirect
43. #91786C22semidirect
44. #91781C22semidirect
45. #91779C22semidirect
46. #91778C22semidirect
47. #91776C22semidirect
49. C22×C5×C4⋊C4D3semidirect
100. C22×C3×C4⋊C4D5semidirect
120. C2×C5×C4⋊C4C2×D3semidirect
187. C2×C3×C4⋊C4C2×D5semidirect
263. C22×C4⋊C4D15semidirect
381. C2×C4⋊C4C2×D15semidirect
391. C2×C3×C5#552semidirect
488. C3×C5C2×#552semidirect
511. C2×C5#5513semidirect
549. C2×C3#65847semidirect
554. C5C2×#5513semidirect
576. C3C2×#65847semidirect
579. C2#11756272direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C5×C4⋊C42
C2×#918182
C2×C4×#12202
C2×#918152
C2×#918132
#117562722
C2×#917872
C2×#917862
C2×#917812
C2×#917792
C2×#917782
C2×#917762
C2×#658473
C2×#55135