Finite Group Databasebeta

← C2×#11752640 #14481422

Subgroups — C2×#11752640 #14481422

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×#91761C211yesmaximal, characteristic, complemented
2C2×C5×#875C211yesmaximal, characteristic, complemented
3C2×C3×#4015C211yesmaximal, characteristic, complemented
4C2×#91466C211yesmaximal, characteristic
5C2×#91435C211yesmaximal, characteristic, complemented
6C2×#91427C212yesmaximal, complemented
7C2×#91410C212yesmaximal, complemented
8C2×#91383C211yesmaximal, characteristic, complemented
9C2×#91344C211yesmaximal, characteristic, complemented
10C2×#91341C212yesmaximal, complemented
11C2×#91326C212yesmaximal, complemented
12#11752640C2116yesmaximal, direct product factor
13C2×#66016—31nomaximal, Hall
14C22×C4×D15C2211yescharacteristic
15C22×C5×D12C2211yescharacteristic
16C22×C5×Dic6C2211yescharacteristic
17C22×C3×Dic10C2212yes—
18C22×#1203C2212yes—
19C22×D3×Dic5C2212yes—
20C2×C3×C42×C5C2211yescharacteristic
21C2×#6714C2211yescharacteristic
22C2×#6714C2211yescharacteristic
23C2×#6712C2211yescharacteristic
24C2×#6712C2211yescharacteristic
25C2×C4×Dic15C2211yescharacteristic
26#91761C2218yescomplemented
27C2×C5×(C2×D3)⋊C4C2212yes—
28C2×C5×(C2×D3)⋊C4C2212yes—
29C5×#875C2218yescomplemented
30C2×C3×C4⋊Dic5C2211yescharacteristic
31C2×C3×C4⋊Dic5C2211yescharacteristic
32C2×C3×C4⋊Dic5C2212yes—
33C3×#4015C2218yescomplemented
34C2×Dic3⋊Dic5C2212yes—
35C2×#6668C2212yes—
36C2×#6667C2212yes—
37C2×(C2×D3)⋊Dic5C2212yes—
38C2×(C2×D3)⋊Dic5C2212yes—
39C2×Dic3×Dic5C2212yes—
40#91466C2218yes—
41#91435C2218yescomplemented
42#91427C22116yescomplemented
43#91410C22116yescomplemented
44#91383C2218yescomplemented
45#91344C2218yescomplemented
46#91341C22116yescomplemented
47#91326C22116yescomplemented
48C2×#5733—51nomaximal, Hall
49C2×C5×#124—31no—
50C2×#4092—31no—
51C2×Q8×Dic5—31no—
52C2×#4084—31no—
53C2×D4×Dic5—31no—
54#66016—316no—
55C2×#4032—32no—
56C2×#4031—32no—
57C2×#4025—32no—
58C2×#4024—32no—
59C2×#4018—31no—
60C2×#4015D311yescharacteristic, complemented
61C23×D15C2311yescharacteristic
62C23×C5×D3C2312yes—
63C22×C3×C4×C5C2311yescharacteristic
64C22×C3×C4×C5C2311yescharacteristic
65C22×C3×C4×C5C2311yescharacteristic
66C22×Dic15C2311yescharacteristic
67C22×Dic15C2311yescharacteristic
68C22×Dic15C2311yescharacteristic
69C2×C4×D15—24no—
70C2×C4×D15C2314yes—
71C22×C5×Dic3C2312yes—
72C2×C5×D12—24no—
73C2×C5×D12C2314yes—
74C2×C5×Dic6—24no—
75C2×C5×Dic6C2314yes—
76C22×C3×Dic5C2312yes—
77C22×C3×Dic5C2312yes—
78C2×C3×Dic10—44no—
79C2×C3×Dic10C2318yes—
80C2×#1203—44no—
81C2×#1203C2318yes—
82C2×D3×Dic5—44no—
83C2×D3×Dic5C2318yes—
84C3×C42×C5C2314yes—
85#6714C2314yes—
86#6714C2314yes—
87#6712C2314yes—
88#6712C2314yes—
89C4×Dic15C2314yes—
90C5×(C2×D3)⋊C4C2318yes—
91C5×(C2×D3)⋊C4C2318yes—
92C3×C4⋊Dic5C2318yes—
93C3×C4⋊Dic5C2314yes—
94C3×C4⋊Dic5C2314yes—
95Dic3⋊Dic5C2318yes—
96#6668C2318yes—
97#6667C2318yes—
98(C2×D3)⋊Dic5C2318yes—
99(C2×D3)⋊Dic5C2318yes—
100Dic3×Dic5C2318yes—
101C2×C3×C4⋊Q8—51no—
102C2×#947—52no—
103C2×#946—52no—
104#5733—516no—
105C2×#898—52no—
106C2×#897—51no—
107C2×#895—51no—
108C2×#894—52no—
109C2×#893—51no—
110C2×#887—51no—
111C2×#875D511yescharacteristic, complemented
112C2×#872—51no—
113C22×C5×Q8—31no—
114C22×C5×D4—31no—
115C22×C22⋊D5—32no—
116C23×Dic5—32no—
117C22×C4×D5—31no—
118C22×Dic10C2×D312yes—
119C5×#124—38no—
120C2×C5×C22⋊C4—32no—
121C2×C5×C22⋊C4—32no—
122C2×C42×C5C2×D311yescharacteristic
123C2×C22⋊Dic5—32no—
124C2×C22⋊Dic5—32no—
125#4092—38no—
126Q8×Dic5—38no—
127#4084—38no—
128D4×Dic5—38no—
129C2×#685—31no—
130C2×#685—31no—
131C2×#685—32no—
132C2×C4⋊Dic5—32no—
133C2×C4⋊Dic5C2×D312yes—
134C2×C4⋊Dic5C2×D311yescharacteristic
135C2×C4⋊Dic5C2×D311yescharacteristic
136C2×#683—31no—
137C2×#683—31no—
138C2×#683—32no—
139C2×C4×Dic5—31no—
140C2×C4×Dic5—32no—
141#4032—316no—
142#4031—316no—
143#4025—316no—
144#4024—316no—
145#4018—38no—
146#4015C2×D318yescomplemented
147C2×#542—151noSylow
148C23×C3×C5C2411yescharacteristic
149C22×D15—22no—
150C22×D15—22no—
151C22×D15—22no—
152C22×D15C2412yes—
153C22×C5×D3—24no—
154C22×C5×D3—24no—
155C22×C5×D3—24no—
156C22×C5×D3C2414yes—
157C2×C3×C4×C5—22no—
158C2×C3×C4×C5—22no—
159C2×C3×C4×C5#4114yes—
160C2×C3×C4×C5C2412yes—
161C2×C3×C4×C5C2412yes—
162C2×C3×C4×C5C2412yes—
163C2×Dic15—22no—
164C2×Dic15—22no—
165C2×Dic15—22no—
166C2×Dic15C2412yes—
167C2×Dic15C2412yes—
168C2×Dic15C2412yes—
169C4×D15—28no—
170C2×C5×Dic3—24no—
171C2×C5×Dic3C2414yes—
172C5×D12—28no—
173C5×Dic6—28no—
174C2×C3×Dic5—24no—
175C2×C3×Dic5—24no—
176C2×C3×Dic5C2414yes—
177C2×C3×Dic5C2414yes—
178C3×Dic10—48no—
179#1203—48no—
180D3×Dic5—48no—
181C22×C3×Q8—52no—
182C22×D12—52no—
183C22×D12C2×D511yescharacteristic
184C22×C4×D3—51no—
185C22×C4×D3—52no—
186C22×Dic6C2×D511yescharacteristic
187C3×C4⋊Q8—58no—
188C2×C3×C4⋊C4—51no—
189C2×C3×C4⋊C4—51no—
190C2×C3×C4⋊C4—52no—
191C2×C3×C42C2×D511yescharacteristic
192#947—516no—
193#946—516no—
194C2×(C2×D3)⋊C4—52no—
195C2×(C2×D3)⋊C4—52no—
196C2×(C2×D3)⋊C4—51no—
197C2×(C2×D3)⋊C4—51no—
198C2×(C2×D3)⋊C4—52no—
199C2×(C2×D3)⋊C4C2×D512yes—
200C2×(C2×D3)⋊C4C2×D512yes—
201C2×C4⋊Dic3—52no—
202C2×Dic3⋊C4—51no—
203C2×Dic3⋊C4—51no—
204C2×Dic3⋊C4—52no—
205C2×C4×Dic3—51no—
206C2×C4×Dic3—52no—
207#898—516no—
208#897—58no—
209#895—58no—
210#894—516no—
211#893—58no—
212#887—58no—
213#875C2×D518yescomplemented
214#872—58no—
215C24×C5—32no—
216C23×D5—31no—
217C2×C5×Q8—34no—
218C2×C5×Q8—64no—
219C2×C5×D4—34no—
220C2×C5×D4—64no—
221C22×C4×C5—32no—
222C22×C4×C5C22×D311yescharacteristic
223C22×C4×C5C22×D311yescharacteristic
224C22×C4×C5C22×D311yescharacteristic
225C2×C22⋊D5—38no—
226C2×C22⋊D5—124no—
227C22×Dic5—31no—
228C22×Dic5—31no—
229C22×Dic5—31no—
230C22×Dic5—38no—
231C22×Dic5—124no—
232C22×Dic5C22×D312yes—
233C22×Dic5C22×D312yes—
234C2×C4×D5—34no—
235C2×C4×D5—64no—
236C2×Dic10—44no—
237C2×Dic10C22×D318yes—
238C5×C22⋊C4—38no—
239C5×C22⋊C4—38no—
240C42×C5C22×D314yes—
241C22⋊Dic5—38no—
242C22⋊Dic5—38no—
243#685—34no—
244#685—34no—
245#685—38no—
246C4⋊Dic5—38no—
247C4⋊Dic5C22×D318yes—
248C4⋊Dic5C22×D314yes—
249C4⋊Dic5C22×D314yes—
250#683—34no—
251#683—34no—
252#683—38no—
253C4×Dic5—34no—
254C4×Dic5—38no—
255#542—1516no—
256C2×C4⋊Q8—51no—
257C2×#126—152no—
258C2×#124—31no—
259C2×#124—152no—
260C2×#123—152no—
261C2×C22⋊Q8—152no—
262C2×C22⋊Q8—151no—
263C2×#121—151no—
264C2×C4×Q8—151no—
265C2×C4×D4—151no—
266C2×#117—151no—
267C22×C3×C5C2×#4112yes—
268C22×C3×C5H(5,𝔽2)12yes—
269C22×C3×C52−1+412yes—
270C22×C3×C5C2511yescharacteristic
271C2×D15—22no—
272C2×D15—22no—
273C2×D15—22no—
274C2×D15—44no—
275C2×C5×D3—24no—
276C2×C5×D3—24no—
277C2×C5×D3—24no—
278C2×C5×D3—48no—
279C3×C4×C5—22no—
280C3×C4×C5—22no—
281C3×C4×C5C2×#4114yes—
282Dic15—22no—
283Dic15—22no—
284Dic15—22no—
285C3×Dic5—24no—
286C3×Dic5—24no—
287C5×Dic3—24no—
288C23×D3—51no—
289C23×D3C22×D512yes—
290C2×C3×Q8—58no—
291C2×C3×Q8—204no—
292C22×C3×C4—52no—
293C22×C3×C4—52no—
294C22×C3×C4C22×D511yescharacteristic
295C22×C3×C4C22×D511yescharacteristic
296C22×C3×C4C22×D511yescharacteristic
297C22×Dic3—51no—
298C22×Dic3—51no—
299C22×Dic3—51no—
300C22×Dic3C22×D512yes—
301C2×D12—24no—
302C2×D12—58no—
303C2×D12—204no—
304C2×D12C22×D514yes—
305C2×C4×D3—54no—
306C2×C4×D3—58no—
307C2×C4×D3—104no—
308C2×C4×D3—204no—
309C2×Dic6—24no—
310C2×Dic6C22×D514yes—
311C3×C4⋊C4—54no—
312C3×C4⋊C4—54no—
313C3×C4⋊C4—58no—
314C3×C42C22×D514yes—
315(C2×D3)⋊C4—58no—
316(C2×D3)⋊C4—58no—
317(C2×D3)⋊C4—54no—
318(C2×D3)⋊C4—54no—
319(C2×D3)⋊C4—58no—
320(C2×D3)⋊C4C22×D518yes—
321(C2×D3)⋊C4C22×D518yes—
322C4⋊Dic3—58no—
323Dic3⋊C4—54no—
324Dic3⋊C4—54no—
325Dic3⋊C4—58no—
326C4×Dic3—54no—
327C4×Dic3—58no—
328C23×C5—34no—
329C23×C5—64no—
330C23×C5—64no—
331C23×C5—64no—
332C23×C5C23×D311yescharacteristic
333C22×D5—32no—
334C22×D5—62no—
335C22×D5—62no—
336C22×D5—62no—
337C5×Q8—68no—
338C5×D4—68no—
339C2×C4×C5—22no—
340C2×C4×C5—22no—
341C2×C4×C5—34no—
342C2×C4×C5—64no—
343C2×C4×C5#23914yes—
344C2×C4×C5C23×D312yes—
345C2×C4×C5C23×D312yes—
346C2×C4×C5C23×D312yes—
347C22⋊D5—128no—
348C2×Dic5—24no—
349C2×Dic5—24no—
350C2×Dic5—32no—
351C2×Dic5—32no—
352C2×Dic5—32no—
353C2×Dic5—62no—
354C2×Dic5—62no—
355C2×Dic5—62no—
356C2×Dic5—128no—
357C2×Dic5C23×D314yes—
358C2×Dic5C23×D314yes—
359C4×D5—68no—
360Dic10—48no—
361C22×Q8—31no—
362C22×Q8—52no—
363C22×D4—31no—
364C22×D4—152no—
365C23×C4—152no—
366C23×C4—151no—
367C4⋊Q8—58no—
368#126—1516no—
369#124—38no—
370#124—1516no—
371#123—1516no—
372C22⋊Q8—1516no—
373C22⋊Q8—158no—
374#121—158no—
375C4×Q8—158no—
376C4×D4—158no—
377#117—158no—
378C2×C4⋊C4—51no—
379C2×C4⋊C4—51no—
380C2×C4⋊C4—52no—
381C2×C4⋊C4—152no—
382C2×C4⋊C4—152no—
383C2×C4⋊C4—151no—
384C2×C4⋊C4—151no—
385C2×C22⋊C4—32no—
386C2×C22⋊C4—32no—
387C2×C22⋊C4—152no—
388C2×C22⋊C4—152no—
389C2×C22⋊C4—152no—
390C2×C22⋊C4—151no—
391C2×C22⋊C4—151no—
392C2×C42—152no—
393C2×C42—151no—
394C2×C42D3×D511yescharacteristic
395C2×C3×C5#54214yescomplemented
396C2×C3×C5C22×#4111yescharacteristic
397C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
398C2×C3×C5C2×2−1+411yescharacteristic
399D15—42no—
400C5×D3—44no—
401C23×C3C23×D511yescharacteristic
402C22×D3—24no—
403C22×D3—24no—
404C22×D3—24no—
405C22×D3—52no—
406C22×D3—102no—
407C22×D3—102no—
408C22×D3—102no—
409C22×D3C23×D514yes—
410C3×Q8—208no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—54no—
414C2×C3×C4—54no—
415C2×C3×C4—104no—
416C2×C3×C4—104no—
417C2×C3×C4#71114yes—
418C2×C3×C4C23×D512yes—
419C2×C3×C4C23×D512yes—
420C2×C3×C4C23×D512yes—
421C2×Dic3—24no—
422C2×Dic3—52no—
423C2×Dic3—52no—
424C2×Dic3—52no—
425C2×Dic3—102no—
426C2×Dic3—102no—
427C2×Dic3—102no—
428C2×Dic3C23×D514yes—
429D12—28no—
430D12—208no—
431C4×D3—108no—
432C4×D3—208no—
433Dic6—28no—
434C22×C5—64no—
435C22×C5—64no—
436C22×C5—64no—
437C22×C5—128no—
438C22×C5#100412yes—
439C22×C5C2×#23912yes—
440C22×C5#100712yes—
441C22×C5C24×D311yescharacteristic
442C2×D5—62no—
443C2×D5—62no—
444C2×D5—62no—
445C2×D5—124no—
446C4×C5—22no—
447C4×C5—22no—
448C4×C5—64no—
449C4×C5C2×#23914yes—
450Dic5—24no—
451Dic5—24no—
452Dic5—62no—
453Dic5—62no—
454Dic5—62no—
455C24—32no—
456C24—151no—
457C2×Q8—34no—
458C2×Q8—58no—
459C2×Q8—64no—
460C2×Q8—204no—
461C2×D4—34no—
462C2×D4—64no—
463C2×D4—158no—
464C2×D4—604no—
465C22×C4—32no—
466C22×C4—52no—
467C22×C4—52no—
468C22×C4—158no—
469C22×C4—151no—
470C22×C4—151no—
471C22×C4—151no—
472C22×C4—154no—
473C22×C4—304no—
474C22×C4—604no—
475C22×C4C2×D3×D511yescharacteristic
476C22×C4C2×D3×D511yescharacteristic
477C22×C4C2×D3×D511yescharacteristic
478C4⋊C4—54no—
479C4⋊C4—54no—
480C4⋊C4—58no—
481C4⋊C4—158no—
482C4⋊C4—158no—
483C4⋊C4—154no—
484C4⋊C4—154no—
485C22⋊C4—38no—
486C22⋊C4—38no—
487C22⋊C4—158no—
488C22⋊C4—158no—
489C22⋊C4—158no—
490C22⋊C4—154no—
491C22⋊C4—154no—
492C42—158no—
493C42—154no—
494C42C2×D3×D514yes—
495C3×C5C2×#54211yescharacteristic, complemented, Hall
496C22×C3C2×#71112yes—
497C22×C3#414912yes—
498C22×C3#415012yes—
499C22×C3C24×D511yescharacteristic
500C2×D3—24no—
501C2×D3—24no—
502C2×D3—24no—
503C2×D3—48no—
504C2×D3—102no—
505C2×D3—102no—
506C2×D3—102no—
507C2×D3—204no—
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—104no—
511C3×C4—104no—
512C3×C4C2×#71114yes—
513Dic3—24no—
514Dic3—102no—
515Dic3—102no—
516Dic3—102no—
517C2×C5—124no—
518C2×C5#573314yescomplemented
519C2×C5C2×#100411yescharacteristic
520C2×C5C22×#23911yescharacteristic
521C2×C5C2×#100711yescharacteristic
522D5—122no—
523C23—34no—
524C23—64no—
525C23—64no—
526C23—64no—
527C23—152no—
528C23—302no—
529C23—302no—
530C23—302no—
531C23C22×D3×D511yescharacteristic
532Q8—68no—
533Q8—208no—
534D4—68no—
535D4—608no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—34no—
539C2×C4—54no—
540C2×C4—54no—
541C2×C4—64no—
542C2×C4—104no—
543C2×C4—104no—
544C2×C4—152no—
545C2×C4—152no—
546C2×C4—152no—
547C2×C4—302no—
548C2×C4—302no—
549C2×C4—302no—
550C2×C4—308no—
551C2×C4—608no—
552C2×C4#676814yes—
553C2×C4C22×D3×D512yes—
554C2×C4C22×D3×D512yes—
555C2×C4C22×D3×D512yes—
556C2×C3#6601614yescomplemented
557C2×C3C22×#71111yescharacteristic
558C2×C3C2×#414911yescharacteristic
559C2×C3C2×#415011yescharacteristic
560D3—44no—
561D3—202no—
562C5C2×#573311yescharacteristic, complemented, Sylow
563C22—64no—
564C22—64no—
565C22—64no—
566C22—128no—
567C22—302no—
568C22—302no—
569C22—302no—
570C22—604no—
571C22C2×#676812yes—
572C22#9200412yes—
573C22#9201712yes—
574C22C23×D3×D511yescharacteristic
575C4—22no—
576C4—22no—
577C4—64no—
578C4—104no—
579C4—104no—
580C4—302no—
581C4—302no—
582C4—302no—
583C4C2×#676814yes—
584C3C2×#6601611yescharacteristic, complemented, Sylow
585C2—124no—
586C2—602no—
587C2#1175264014yesdirect product factor
588C2C22×#676811yescharacteristic
589C2C2×#9200411yescharacteristic
590C2C2×#9201711yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC2×C3×C42×C5C22
socleC23×C3×C5C24
prime radicalC2×#91466C2
hypercenterC2×C42D3×D5
hypocenterC3×C5C2×#542
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C2×#91761C2semidirect
2. C2×C5×#875C2semidirect
3. C2×C3×#4015C2semidirect
5. C2×#91435C2semidirect
6. C2×#91427C2semidirect
7. C2×#91410C2semidirect
8. C2×#91383C2semidirect
9. C2×#91344C2semidirect
10. C2×#91341C2semidirect
11. C2×#91326C2semidirect
12. #11752640C2direct (N × H)
26. #91761C22semidirect
29. C5×#875C22semidirect
33. C3×#4015C22semidirect
41. #91435C22semidirect
42. #91427C22semidirect
43. #91410C22semidirect
44. #91383C22semidirect
45. #91344C22semidirect
46. #91341C22semidirect
47. #91326C22semidirect
60. C2×#4015D3semidirect
111. C2×#875D5semidirect
146. #4015C2×D3semidirect
213. #875C2×D5semidirect
395. C2×C3×C5#542semidirect
495. C3×C5C2×#542semidirect
518. C2×C5#5733semidirect
556. C2×C3#66016semidirect
562. C5C2×#5733semidirect
584. C3C2×#66016semidirect
587. C2#11752640direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#917612
C2×C5×#8752
C2×C3×#40152
C2×#914662
C2×#914352
C2×#914272
C2×#914102
C2×#913832
C2×#913442
C2×#913412
C2×#913262
#117526402
C2×#660163
C2×#57335