Finite Group Databasebeta

← C2×#11756402 #14481330

Subgroups — C2×#11756402 #14481330

Order 1920.

2264 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#91836C211yesmaximal, characteristic, complemented
3C2×Q8×Dic15C211yesmaximal, characteristic, complemented
4C2×#91819C211yesmaximal, characteristic
5C2×#91815C211yesmaximal, characteristic, complemented
6#11756402C2116yesmaximal, direct product factor
7C2×#91787C212yesmaximal, complemented
8C2×#91786C212yesmaximal, complemented
9C2×#91782C211yesmaximal, characteristic, complemented
10C2×#91777C211yesmaximal, characteristic, complemented
11C2×#91774C212yesmaximal, complemented
12C2×#91772C212yesmaximal, complemented
13C2×#65977—31nomaximal, Hall
14C22×C3×C5×Q8C2211yescharacteristic
15C23×C3×C4×C5C2211yescharacteristic
16C22×#1220C2212yes—
17C22×D60C2211yescharacteristic
18C22×C4×D15C2212yes—
19C22×Dic30C2212yes—
20C3×C5×C22⋊Q8C2218yescomplemented
21C2×C3×C5×C4⋊C4C2211yescharacteristic
22C2×C3×C5×C4⋊C4C2212yes—
23C2×C3×C5×C22⋊C4C2212yes—
24C2×#6719C2211yescharacteristic
25C2×#6719C2211yescharacteristic
26#91836C2218yescomplemented
27Q8×Dic15C2218yescomplemented
28#91819C2218yes—
29C2×#6714C2212yes—
30C2×#6714C2212yes—
31C2×#6714C2212yes—
32C2×#6714C2212yes—
33#91815C2218yescomplemented
34C2×C4⋊Dic15C2211yescharacteristic
35C2×C4⋊Dic15C2211yescharacteristic
36C2×C4⋊Dic15C2211yescharacteristic
37C2×C4⋊Dic15C2212yes—
38C2×#6712C2212yes—
39C2×C4×Dic15C2211yescharacteristic
40C2×C4×Dic15C2211yescharacteristic
41C2×C4×Dic15C2212yes—
42#91787C22116yescomplemented
43#91786C22116yescomplemented
44#91782C2218yescomplemented
45#91777C2218yescomplemented
46#91774C22116yescomplemented
47#91772C22116yescomplemented
48C2×#5643—51nomaximal, Hall
49C2×C5×C22⋊Q8D311yescharacteristic, complemented
50C2×#4093—31no—
51C2×Q8×Dic5—31no—
52C2×#4076—31no—
53C2×#4072—31no—
54#65977—316no—
55C2×#4044—32no—
56C2×#4043—32no—
57C2×#4039—31no—
58C2×#4034—31no—
59C2×#4031—32no—
60C2×#4029—32no—
61C24×C3×C5C2311yescharacteristic
62C23×D15C2312yes—
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×#1220—44no—
72C2×#1220C2318yes—
73C22×Dic15C2311yescharacteristic
74C22×Dic15C2311yescharacteristic
75C22×Dic15C2312yes—
76C22×Dic15C2311yescharacteristic
77C22×Dic15C2311yescharacteristic
78C2×D60—24no—
79C2×D60C2314yes—
80C2×C4×D15—44no—
81C2×C4×D15C2318yes—
82C2×Dic30—44no—
83C2×Dic30C2318yes—
84C3×C5×C4⋊C4C2314yes—
85C3×C5×C4⋊C4C2318yes—
86C3×C5×C22⋊C4C2318yes—
87#6719C2314yes—
88#6719C2314yes—
89#6714C2318yes—
90#6714C2318yes—
91#6714C2318yes—
92#6714C2318yes—
93C4⋊Dic15C2318yes—
94C4⋊Dic15C2314yes—
95C4⋊Dic15C2314yes—
96C4⋊Dic15C2314yes—
97#6712C2318yes—
98C4×Dic15C2318yes—
99C4×Dic15C2314yes—
100C4×Dic15C2314yes—
101C2×C3×C22⋊Q8D511yescharacteristic, complemented
102C2×#947—51no—
103C2×Q8×Dic3—51no—
104C2×#930—51no—
105C2×#926—51no—
106#5643—516no—
107C2×#898—52no—
108C2×#897—52no—
109C2×#893—51no—
110C2×#888—51no—
111C2×#885—52no—
112C2×#883—52no—
113C22×C5×Q8C2×D311yescharacteristic
114C23×C4×C5C2×D311yescharacteristic
115C22×C22⋊D5—32no—
116C22×D20—31no—
117C22×C4×D5—32no—
118C22×Dic10—32no—
119C5×C22⋊Q8C2×D318yescomplemented
120C2×C5×C4⋊C4C2×D311yescharacteristic
121C2×C5×C4⋊C4C2×D312yes—
122C2×C5×C22⋊C4C2×D312yes—
123C2×C22⋊Dic5—31no—
124C2×C22⋊Dic5—31no—
125#4093—38no—
126Q8×Dic5—38no—
127#4076—38no—
128C2×#685—32no—
129C2×#685—32no—
130C2×#685—32no—
131C2×#685—32no—
132#4072—38no—
133C2×C4⋊Dic5—32no—
134C2×C4⋊Dic5—31no—
135C2×C4⋊Dic5—31no—
136C2×C4⋊Dic5—31no—
137C2×#683—32no—
138C2×C4×Dic5—32no—
139C2×C4×Dic5—31no—
140C2×C4×Dic5—31no—
141#4044—316no—
142#4043—316no—
143#4039—38no—
144#4034—38no—
145#4031—316no—
146#4029—316no—
147C2×#542—151noSylow
148C23×C3×C5—22no—
149C23×C3×C5—22no—
150C23×C3×C5—22no—
151C23×C3×C5C2412yes—
152C23×C3×C5C2411yescharacteristic
153C22×D15—24no—
154C22×D15—24no—
155C22×D15—24no—
156C22×D15C2414yes—
157C3×C5×Q8—28no—
158C2×C3×C4×C5—22no—
159C2×C3×C4×C5—24no—
160C2×C3×C4×C5—24no—
161C2×C3×C4×C5—28no—
162C2×C3×C4×C5#4114yes—
163C2×C3×C4×C5C2414yes—
164C2×C3×C4×C5C2414yes—
165C2×C3×C4×C5C2412yes—
166C2×C3×C4×C5C2412yes—
167#1220—48no—
168C2×Dic15—22no—
169C2×Dic15—24no—
170C2×Dic15—22no—
171C2×Dic15—22no—
172C2×Dic15—22no—
173C2×Dic15C2412yes—
174C2×Dic15C2412yes—
175C2×Dic15C2414yes—
176C2×Dic15C2412yes—
177C2×Dic15C2412yes—
178D60—28no—
179C4×D15—48no—
180Dic30—48no—
181C22×C3×Q8C2×D511yescharacteristic
182C23×C3×C4C2×D511yescharacteristic
183C22×#67—52no—
184C22×D12—51no—
185C22×C4×D3—52no—
186C22×Dic6—52no—
187C3×C22⋊Q8C2×D518yescomplemented
188C2×C3×C4⋊C4C2×D511yescharacteristic
189C2×C3×C4⋊C4C2×D512yes—
190C2×C3×C22⋊C4C2×D512yes—
191C2×#217—51no—
192C2×#217—51no—
193#947—58no—
194Q8×Dic3—58no—
195#930—58no—
196C2×(C2×D3)⋊C4—52no—
197C2×(C2×D3)⋊C4—52no—
198C2×(C2×D3)⋊C4—52no—
199C2×(C2×D3)⋊C4—52no—
200#926—58no—
201C2×C4⋊Dic3—52no—
202C2×C4⋊Dic3—51no—
203C2×C4⋊Dic3—51no—
204C2×C4⋊Dic3—51no—
205C2×Dic3⋊C4—52no—
206C2×C4×Dic3—52no—
207C2×C4×Dic3—51no—
208C2×C4×Dic3—51no—
209#898—516no—
210#897—516no—
211#893—58no—
212#888—58no—
213#885—516no—
214#883—516no—
215C24×C5C22×D311yescharacteristic
216C23×D5—32no—
217C2×C5×Q8—24no—
218C2×C5×Q8C22×D314yes—
219C22×C4×C5—24no—
220C22×C4×C5C22×D312yes—
221C22×C4×C5C22×D312yes—
222C22×C4×C5C22×D311yescharacteristic
223C22×C4×C5C22×D314yes—
224C22×C4×C5C22×D311yescharacteristic
225C2×C22⋊D5—38no—
226C2×C22⋊D5—124no—
227C22×Dic5—31no—
228C22×Dic5—31no—
229C22×Dic5—31no—
230C22×Dic5—31no—
231C22×Dic5—32no—
232C2×D20—34no—
233C2×D20—64no—
234C2×C4×D5—38no—
235C2×C4×D5—124no—
236C2×Dic10—38no—
237C2×Dic10—124no—
238C5×C4⋊C4C22×D314yes—
239C5×C4⋊C4C22×D318yes—
240C5×C22⋊C4C22×D318yes—
241C22⋊Dic5—34no—
242C22⋊Dic5—34no—
243#685—38no—
244#685—38no—
245#685—38no—
246#685—38no—
247C4⋊Dic5—38no—
248C4⋊Dic5—34no—
249C4⋊Dic5—34no—
250C4⋊Dic5—34no—
251#683—38no—
252C4×Dic5—38no—
253C4×Dic5—34no—
254C4×Dic5—34no—
255#542—1516no—
256C2×C4⋊Q8—151no—
257C2×#126—152no—
258C2×#124—151no—
259C2×#124—152no—
260C2×#123—152no—
261C2×C22⋊Q8—152no—
262C2×C22⋊Q8D1511yescharacteristic, complemented
263C2×#121—151no—
264C2×C4×Q8—151no—
265C2×C4×D4—151no—
266C2×#117—151no—
267C22×C3×C5—22no—
268C22×C3×C5—22no—
269C22×C3×C5—22no—
270C22×C3×C5—44no—
271C22×C3×C5C2×#4112yes—
272C22×C3×C5H(5,𝔽2)12yes—
273C22×C3×C52−1+412yes—
274C22×C3×C5C2511yescharacteristic
275C2×D15—24no—
276C2×D15—24no—
277C2×D15—24no—
278C2×D15—48no—
279C3×C4×C5—22no—
280C3×C4×C5—24no—
281C3×C4×C5—24no—
282C3×C4×C5C2×#4114yes—
283Dic15—22no—
284Dic15—24no—
285Dic15—22no—
286Dic15—22no—
287Dic15—22no—
288C24×C3C22×D511yescharacteristic
289C23×D3—52no—
290C2×C3×Q8—24no—
291C2×C3×Q8C22×D514yes—
292C22×C3×C4—24no—
293C22×C3×C4C22×D512yes—
294C22×C3×C4C22×D512yes—
295C22×C3×C4C22×D511yescharacteristic
296C22×C3×C4C22×D514yes—
297C22×C3×C4C22×D511yescharacteristic
298C2×#67—58no—
299C2×#67—204no—
300C22×Dic3—51no—
301C22×Dic3—51no—
302C22×Dic3—52no—
303C22×Dic3—51no—
304C22×Dic3—51no—
305C2×D12—54no—
306C2×D12—104no—
307C2×C4×D3—58no—
308C2×C4×D3—204no—
309C2×Dic6—58no—
310C2×Dic6—204no—
311C3×C4⋊C4C22×D514yes—
312C3×C4⋊C4C22×D518yes—
313C3×C22⋊C4C22×D518yes—
314#217—54no—
315#217—54no—
316(C2×D3)⋊C4—58no—
317(C2×D3)⋊C4—58no—
318(C2×D3)⋊C4—58no—
319(C2×D3)⋊C4—58no—
320C4⋊Dic3—58no—
321C4⋊Dic3—54no—
322C4⋊Dic3—54no—
323C4⋊Dic3—54no—
324Dic3⋊C4—58no—
325C4×Dic3—58no—
326C4×Dic3—54no—
327C4×Dic3—54no—
328C23×C5—22no—
329C23×C5—22no—
330C23×C5—22no—
331C23×C5C23×D312yes—
332C23×C5C23×D311yescharacteristic
333C22×D5—34no—
334C22×D5—64no—
335C22×D5—64no—
336C22×D5—64no—
337C5×Q8—28no—
338C2×C4×C5—22no—
339C2×C4×C5—24no—
340C2×C4×C5—24no—
341C2×C4×C5—28no—
342C2×C4×C5#23714yes—
343C2×C4×C5C23×D314yes—
344C2×C4×C5C23×D314yes—
345C2×C4×C5C23×D312yes—
346C2×C4×C5C23×D312yes—
347C22⋊D5—128no—
348C2×Dic5—32no—
349C2×Dic5—32no—
350C2×Dic5—34no—
351C2×Dic5—32no—
352C2×Dic5—32no—
353C2×Dic5—62no—
354C2×Dic5—62no—
355C2×Dic5—62no—
356C2×Dic5—64no—
357C2×Dic5—62no—
358D20—68no—
359C4×D5—128no—
360Dic10—128no—
361C22×Q8—152no—
362C22×Q8C2×D1511yescharacteristic
363C22×D4—152no—
364C22×D4—151no—
365C23×C4—152no—
366C23×C4C2×D1511yescharacteristic
367C4⋊Q8—158no—
368#126—1516no—
369#124—158no—
370#124—1516no—
371#123—1516no—
372C22⋊Q8—1516no—
373C22⋊Q8C2×D1518yescomplemented
374#121—158no—
375C4×Q8—158no—
376C4×D4—158no—
377#117—158no—
378C2×C4⋊C4—151no—
379C2×C4⋊C4—151no—
380C2×C4⋊C4—151no—
381C2×C4⋊C4—152no—
382C2×C4⋊C4—152no—
383C2×C4⋊C4C2×D1511yescharacteristic
384C2×C4⋊C4C2×D1512yes—
385C2×C22⋊C4—151no—
386C2×C22⋊C4—151no—
387C2×C22⋊C4—152no—
388C2×C22⋊C4—152no—
389C2×C22⋊C4—152no—
390C2×C22⋊C4—152no—
391C2×C22⋊C4C2×D1512yes—
392C2×C42—151no—
393C2×C42—151no—
394C2×C42—152no—
395C2×C3×C5—42no—
396C2×C3×C5#54214yescomplemented
397C2×C3×C5C22×#4111yescharacteristic
398C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
399C2×C3×C5C2×2−1+411yescharacteristic
400D15—44no—
401C23×C3—22no—
402C23×C3—22no—
403C23×C3—22no—
404C23×C3C23×D512yes—
405C23×C3C23×D511yescharacteristic
406C22×D3—54no—
407C22×D3—104no—
408C22×D3—104no—
409C22×D3—104no—
410C3×Q8—28no—
411C2×C3×C4—22no—
412C2×C3×C4—24no—
413C2×C3×C4—24no—
414C2×C3×C4—28no—
415C2×C3×C4#71114yes—
416C2×C3×C4C23×D514yes—
417C2×C3×C4C23×D514yes—
418C2×C3×C4C23×D512yes—
419C2×C3×C4C23×D512yes—
420#67—208no—
421C2×Dic3—52no—
422C2×Dic3—52no—
423C2×Dic3—54no—
424C2×Dic3—52no—
425C2×Dic3—52no—
426C2×Dic3—102no—
427C2×Dic3—102no—
428C2×Dic3—102no—
429C2×Dic3—104no—
430C2×Dic3—102no—
431D12—108no—
432C4×D3—208no—
433Dic6—208no—
434C22×C5—22no—
435C22×C5—22no—
436C22×C5—22no—
437C22×C5—44no—
438C22×C5C2×#23712yes—
439C22×C5#100712yes—
440C22×C5#100912yes—
441C22×C5C24×D311yescharacteristic
442C2×D5—64no—
443C2×D5—64no—
444C2×D5—64no—
445C2×D5—128no—
446C4×C5—22no—
447C4×C5—24no—
448C4×C5—24no—
449C4×C5C2×#23714yes—
450Dic5—62no—
451Dic5—62no—
452Dic5—62no—
453Dic5—64no—
454Dic5—62no—
455C24—152no—
456C24C22×D1511yescharacteristic
457C2×Q8—24no—
458C2×Q8—158no—
459C2×Q8—604no—
460C2×Q8C22×D1514yes—
461C2×D4—158no—
462C2×D4—154no—
463C2×D4—304no—
464C2×D4—604no—
465C22×C4—24no—
466C22×C4—151no—
467C22×C4—151no—
468C22×C4—151no—
469C22×C4—151no—
470C22×C4—152no—
471C22×C4—158no—
472C22×C4—604no—
473C22×C4C22×D1514yes—
474C22×C4C22×D1512yes—
475C22×C4C22×D1511yescharacteristic
476C22×C4C22×D1512yes—
477C22×C4C22×D1511yescharacteristic
478C4⋊C4—154no—
479C4⋊C4—154no—
480C4⋊C4—154no—
481C4⋊C4—158no—
482C4⋊C4—158no—
483C4⋊C4C22×D1514yes—
484C4⋊C4C22×D1518yes—
485C22⋊C4—154no—
486C22⋊C4—154no—
487C22⋊C4—158no—
488C22⋊C4—158no—
489C22⋊C4—158no—
490C22⋊C4—158no—
491C22⋊C4C22×D1518yes—
492C42—154no—
493C42—154no—
494C42—158no—
495C3×C5C2×#54211yescharacteristic, complemented, Hall
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—22no—
499C22×C3—44no—
500C22×C3C2×#71112yes—
501C22×C3#414712yes—
502C22×C3#414912yes—
503C22×C3C24×D511yescharacteristic
504C2×D3—104no—
505C2×D3—104no—
506C2×D3—104no—
507C2×D3—208no—
508C3×C4—22no—
509C3×C4—24no—
510C3×C4—24no—
511C3×C4C2×#71114yes—
512Dic3—102no—
513Dic3—102no—
514Dic3—102no—
515Dic3—104no—
516Dic3—102no—
517C2×C5—42no—
518C2×C5#564314yescomplemented
519C2×C5C22×#23711yescharacteristic
520C2×C5C2×#100711yescharacteristic
521C2×C5C2×#100911yescharacteristic
522D5—124no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—154no—
527C23—304no—
528C23—304no—
529C23—304no—
530C23C23×D1512yes—
531C23C23×D1511yescharacteristic
532Q8—28no—
533Q8—608no—
534D4—308no—
535D4—608no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—152no—
541C2×C4—152no—
542C2×C4—152no—
543C2×C4—152no—
544C2×C4—154no—
545C2×C4—302no—
546C2×C4—302no—
547C2×C4—302no—
548C2×C4—304no—
549C2×C4—302no—
550C2×C4—608no—
551C2×C4#681914yes—
552C2×C4C23×D1514yes—
553C2×C4C23×D1512yes—
554C2×C4C23×D1514yes—
555C2×C4C23×D1512yes—
556C2×C3—42no—
557C2×C3#6597714yescomplemented
558C2×C3C22×#71111yescharacteristic
559C2×C3C2×#414711yescharacteristic
560C2×C3C2×#414911yescharacteristic
561D3—204no—
562C5C2×#564311yescharacteristic, complemented, Sylow
563C22—22no—
564C22—22no—
565C22—22no—
566C22—44no—
567C22—304no—
568C22—304no—
569C22—304no—
570C22—608no—
571C22C2×#681912yes—
572C22#9209812yes—
573C22#9210012yes—
574C22C24×D1511yescharacteristic
575C4—22no—
576C4—24no—
577C4—24no—
578C4—302no—
579C4—302no—
580C4—302no—
581C4—304no—
582C4—302no—
583C4C2×#681914yes—
584C3C2×#6597711yescharacteristic, complemented, Sylow
585C2—42no—
586C2—604no—
587C2#1175640214yesdirect product factor
588C2C22×#681911yescharacteristic
589C2C2×#9209811yescharacteristic
590C2C2×#9210011yescharacteristic
subgroupG/H
centerC23C23×D15
commutator, cosocleC22×C3×C5C25
FrattiniC22C24×D15
FittingC2×C3×C5×C22⋊Q8C2
socleC23×C3×C5C24
prime radicalC2×#91819C2
hypercenterC2×C22⋊Q8D15
hypocenterC3×C5C2×#542
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×#91836C2semidirect
3. C2×Q8×Dic15C2semidirect
5. C2×#91815C2semidirect
6. #11756402C2direct (N × H)
7. C2×#91787C2semidirect
8. C2×#91786C2semidirect
9. C2×#91782C2semidirect
10. C2×#91777C2semidirect
11. C2×#91774C2semidirect
12. C2×#91772C2semidirect
20. C3×C5×C22⋊Q8C22semidirect
26. #91836C22semidirect
27. Q8×Dic15C22semidirect
33. #91815C22semidirect
42. #91787C22semidirect
43. #91786C22semidirect
44. #91782C22semidirect
45. #91777C22semidirect
46. #91774C22semidirect
47. #91772C22semidirect
49. C2×C5×C22⋊Q8D3semidirect
101. C2×C3×C22⋊Q8D5semidirect
119. C5×C22⋊Q8C2×D3semidirect
187. C3×C22⋊Q8C2×D5semidirect
262. C2×C22⋊Q8D15semidirect
373. C22⋊Q8C2×D15semidirect
396. C2×C3×C5#542semidirect
495. C3×C5C2×#542semidirect
518. C2×C5#5643semidirect
557. C2×C3#65977semidirect
562. C5C2×#5643semidirect
584. C3C2×#65977semidirect
587. C2#11756402direct (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×#918362
C2×Q8×Dic152
C2×#918192
C2×#918152
#117564022
C2×#917872
C2×#917862
C2×#917822
C2×#917772
C2×#917742
C2×#917722
C2×#659773
C2×#56435