Finite Group Databasebeta

← C2×#11753804 #14483966

Subgroups — C2×#11753804 #14483966

Order 1920.

2480 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×#91777C211yesmaximal, characteristic, complemented
2C2×C5×#895C211yesmaximal, characteristic, complemented
3C2×C3×#4038C211yesmaximal, characteristic, complemented
4C2×#91462C212yesmaximal, complemented
5C2×#91453C212yesmaximal, complemented
6C2×#91449C211yesmaximal, characteristic, complemented
7C2×C4×#1201C211yesmaximal, characteristic, complemented
8C2×#91435C211yesmaximal, characteristic, complemented
9C2×#91398C212yesmaximal, complemented
10C2×#91373C212yesmaximal, complemented
11C2×#91338C211yesmaximal, characteristic, complemented
12#11753804C2116yesmaximal, direct product factor
13C2×#65950—31nomaximal, Hall
14C22×Dic30C2212yes—
15C22×C5×D12C2211yescharacteristic
16C22×C5×D12C2212yes—
17C22×C4×C5×D3C2211yescharacteristic
18C22×C3×C4×D5C2211yescharacteristic
19C22×#1201C2211yescharacteristic
20C22×#1201C2212yes—
21C22×D3×Dic5C2212yes—
22C2×C3×C5×C4⋊C4C2211yescharacteristic
23C2×C4⋊Dic15C2211yescharacteristic
24C2×#6712C2212yes—
25C2×C4×Dic15C2211yescharacteristic
26#91777C2218yescomplemented
27C2×C5×(C2×D3)⋊C4C2212yes—
28C5×#895C2218yescomplemented
29C2×C3×#685C2212yes—
30C2×C3×C4⋊Dic5C2211yescharacteristic
31C2×C3×C4×Dic5C2212yes—
32C3×#4038C2218yescomplemented
33C2×#6669C2211yescharacteristic
34C2×(C2×D3)⋊Dic5C2211yescharacteristic
35C2×(C2×D3)⋊Dic5C2212yes—
36C2×(C2×D3)⋊Dic5C2212yes—
37C2×(C2×D3)⋊Dic5C2212yes—
38C2×#6665C2211yescharacteristic
39C2×Dic3×Dic5C2212yes—
40#91462C22116yescomplemented
41#91453C22116yescomplemented
42#91449C2218yescomplemented
43C4×#1201C2218yescomplemented
44#91435C2218yescomplemented
45#91398C22116yescomplemented
46#91373C22116yescomplemented
47#91338C2218yescomplemented
48C2×#5532—51nomaximal, Hall
49C2×C5×#121—31no—
50C2×#4085—32no—
51C2×#4084—31no—
52C2×#4082—32no—
53C2×D4×Dic5—31no—
54C2×C4×C22⋊D5—31no—
55C2×#4072—31no—
56#65950—316no—
57C2×#4038D311yescharacteristic, complemented
58C2×#4034—31no—
59C2×#4031—32no—
60C2×#4023—32no—
61C23×C5×D3C2312yes—
62C23×C5×D3C2311yescharacteristic
63C23×C3×D5C2311yescharacteristic
64C22×C3×C4×C5C2312yes—
65C22×C3×C4×C5C2311yescharacteristic
66C22×Dic15C2312yes—
67C22×Dic15C2311yescharacteristic
68C22×Dic15C2311yescharacteristic
69C2×Dic30—44no—
70C2×Dic30C2318yes—
71C22×C5×Dic3C2311yescharacteristic
72C2×C5×D12—24no—
73C2×C5×D12—44no—
74C2×C5×D12C2314yes—
75C2×C5×D12C2318yes—
76C2×C4×C5×D3—24no—
77C2×C4×C5×D3C2314yes—
78C22×C3×Dic5C2312yes—
79C22×C3×Dic5C2311yescharacteristic
80C2×C3×C4×D5—24no—
81C2×C3×C4×D5C2314yes—
82C2×#1201—24no—
83C2×#1201—44no—
84C2×#1201C2314yes—
85C2×#1201C2318yes—
86C2×D3×Dic5—28no—
87C2×D3×Dic5C2318yes—
88C3×C5×C4⋊C4C2314yes—
89C4⋊Dic15C2314yes—
90#6712C2318yes—
91C4×Dic15C2314yes—
92C5×(C2×D3)⋊C4C2318yes—
93C3×#685C2318yes—
94C3×C4⋊Dic5C2314yes—
95C3×C4×Dic5C2318yes—
96#6669C2314yes—
97(C2×D3)⋊Dic5C2318yes—
98(C2×D3)⋊Dic5C2318yes—
99(C2×D3)⋊Dic5C2314yes—
100(C2×D3)⋊Dic5C2318yes—
101#6665C2314yes—
102Dic3×Dic5C2318yes—
103C2×C3×#117—51no—
104C2×#930—51no—
105C2×C4×#67—51no—
106#5532—516no—
107C2×#895D511yescharacteristic, complemented
108C2×#893—51no—
109C2×#892—51no—
110C2×#888—51no—
111C2×#885—52no—
112C2×#884—52no—
113C2×#875—52no—
114C2×#872—52no—
115C22×C5×D4—31no—
116C22×C5×D4—32no—
117C23×C4×C5—31no—
118C22×C22⋊D5—31no—
119C22×C22⋊D5—32no—
120C23×Dic5—32no—
121C22×C4×D5C2×D311yescharacteristic
122C22×Dic10—32no—
123C5×#121—38no—
124C2×C5×C4⋊C4C2×D311yescharacteristic
125C2×C5×C22⋊C4—32no—
126C2×C22⋊Dic5—32no—
127C2×C22⋊Dic5—32no—
128C2×C22⋊Dic5—32no—
129C2×C22⋊Dic5—31no—
130#4085—316no—
131#4084—38no—
132#4082—316no—
133D4×Dic5—38no—
134C4×C22⋊D5—38no—
135C2×#685—31no—
136C2×#685C2×D312yes—
137#4072—38no—
138C2×C4⋊Dic5—31no—
139C2×C4⋊Dic5C2×D311yescharacteristic
140C2×#683—32no—
141C2×#683—31no—
142C2×C4×Dic5—31no—
143C2×C4×Dic5—32no—
144C2×C4×Dic5C2×D312yes—
145#4038C2×D318yescomplemented
146#4034—38no—
147#4031—316no—
148#4023—316no—
149C2×#555—151noSylow
150C23×C3×C5C2411yescharacteristic
151C22×C5×D3—24no—
152C22×C5×D3—22no—
153C22×C5×D3—22no—
154C22×C5×D3—24no—
155C22×C5×D3—24no—
156C22×C5×D3—22no—
157C22×C5×D3C2412yes—
158C22×C5×D3C2414yes—
159C22×C3×D5—22no—
160C22×C3×D5—22no—
161C22×C3×D5—22no—
162C22×C3×D5C2412yes—
163C2×C3×C4×C5—24no—
164C2×C3×C4×C5#4114yes—
165C2×C3×C4×C5C2414yes—
166C2×C3×C4×C5C2412yes—
167C2×Dic15—24no—
168C2×Dic15—22no—
169C2×Dic15#4114yes—
170C2×Dic15C2414yes—
171C2×Dic15C2412yes—
172C2×Dic15C2412yes—
173Dic30—48no—
174C2×C5×Dic3—22no—
175C2×C5×Dic3C2412yes—
176C5×D12—28no—
177C5×D12—48no—
178C4×C5×D3—28no—
179C2×C3×Dic5—24no—
180C2×C3×Dic5—22no—
181C2×C3×Dic5C2414yes—
182C2×C3×Dic5C2412yes—
183C3×C4×D5—28no—
184#1201—28no—
185#1201—48no—
186D3×Dic5—216no—
187C23×C3×C4—51no—
188C22×#67—51no—
189C22×#67—52no—
190C22×D12C2×D511yescharacteristic
191C22×D12C2×D512yes—
192C22×C4×D3—52no—
193C22×C4×D3C2×D511yescharacteristic
194C22×Dic6—52no—
195C3×#117—58no—
196C2×C3×C4⋊C4—51no—
197C2×C3×C4⋊C4C2×D511yescharacteristic
198C2×C3×C22⋊C4—52no—
199C2×C3×C42—52no—
200C2×#217—51no—
201#930—58no—
202C4×#67—58no—
203C2×(C2×D3)⋊C4—52no—
204C2×(C2×D3)⋊C4—51no—
205C2×(C2×D3)⋊C4—52no—
206C2×(C2×D3)⋊C4—52no—
207C2×(C2×D3)⋊C4C2×D512yes—
208C2×C4⋊Dic3—51no—
209C2×Dic3⋊C4—52no—
210C2×Dic3⋊C4—51no—
211C2×C4×Dic3—51no—
212C2×C4×Dic3—52no—
213#895C2×D518yescomplemented
214#893—58no—
215#892—58no—
216#888—58no—
217#885—516no—
218#884—516no—
219#875—516no—
220#872—516no—
221C24×C5—31no—
222C24×C5—32no—
223C23×D5C22×D311yescharacteristic
224C2×C5×D4—34no—
225C2×C5×D4—38no—
226C2×C5×D4—64no—
227C2×C5×D4—124no—
228C22×C4×C5—31no—
229C22×C4×C5—34no—
230C22×C4×C5—64no—
231C22×C4×C5C22×D312yes—
232C22×C4×C5C22×D311yescharacteristic
233C2×C22⋊D5—34no—
234C2×C22⋊D5—38no—
235C2×C22⋊D5—64no—
236C2×C22⋊D5—124no—
237C22×Dic5—31no—
238C22×Dic5—31no—
239C22×Dic5—32no—
240C22×Dic5—38no—
241C22×Dic5—68no—
242C22×Dic5C22×D311yescharacteristic
243C22×Dic5C22×D312yes—
244C2×C4×D5—24no—
245C2×C4×D5C22×D314yes—
246C2×Dic10—38no—
247C2×Dic10—124no—
248C5×C4⋊C4C22×D314yes—
249C5×C22⋊C4—38no—
250C22⋊Dic5—38no—
251C22⋊Dic5—38no—
252C22⋊Dic5—38no—
253C22⋊Dic5—34no—
254#685—34no—
255#685C22×D318yes—
256C4⋊Dic5—34no—
257C4⋊Dic5C22×D314yes—
258#683—38no—
259#683—34no—
260C4×Dic5—34no—
261C4×Dic5—38no—
262C4×Dic5C22×D318yes—
263#555—1516no—
264C2×C4⋊Q8—151no—
265C2×#124—152no—
266C2×#124—152no—
267C2×#121—31no—
268C2×#121—152no—
269C2×#121—151no—
270C2×C4×D4—151no—
271C2×C4×D4—151no—
272C2×#117—51no—
273C2×#117—152no—
274C2×#117—151no—
275C22×C3×C5C2×#4112yes—
276C22×C3×C5C2×#4112yes—
277C22×C3×C5H(5,𝔽2)12yes—
278C22×C3×C5C2511yescharacteristic
279C2×C5×D3—24no—
280C2×C5×D3—22no—
281C2×C5×D3—22no—
282C2×C5×D3—24no—
283C2×C5×D3—24no—
284C2×C5×D3—22no—
285C2×C5×D3—48no—
286C2×C5×D3—44no—
287C2×C3×D5—22no—
288C2×C3×D5—22no—
289C2×C3×D5—22no—
290C2×C3×D5—44no—
291C3×C4×C5—24no—
292C3×C4×C5C2×#4114yes—
293Dic15—24no—
294Dic15—22no—
295Dic15C2×#4114yes—
296C3×Dic5—24no—
297C3×Dic5—22no—
298C5×Dic3—22no—
299C24×C3—51no—
300C23×D3C22×D511yescharacteristic
301C23×D3C22×D512yes—
302C22×C3×C4—52no—
303C22×C3×C4—51no—
304C22×C3×C4—54no—
305C22×C3×C4—104no—
306C22×C3×C4C22×D512yes—
307C22×C3×C4C22×D511yescharacteristic
308C2×#67—54no—
309C2×#67—58no—
310C2×#67—104no—
311C2×#67—204no—
312C22×Dic3—51no—
313C22×Dic3—52no—
314C22×Dic3—51no—
315C22×Dic3C22×D511yescharacteristic
316C2×D12—24no—
317C2×D12—44no—
318C2×D12C22×D514yes—
319C2×D12C22×D518yes—
320C2×C4×D3—24no—
321C2×C4×D3—58no—
322C2×C4×D3—108no—
323C2×C4×D3C22×D514yes—
324C2×Dic6—58no—
325C2×Dic6—204no—
326C3×C4⋊C4—54no—
327C3×C4⋊C4C22×D514yes—
328C3×C22⋊C4—58no—
329C3×C42—58no—
330#217—54no—
331(C2×D3)⋊C4—58no—
332(C2×D3)⋊C4—54no—
333(C2×D3)⋊C4—58no—
334(C2×D3)⋊C4—58no—
335(C2×D3)⋊C4C22×D518yes—
336C4⋊Dic3—54no—
337Dic3⋊C4—58no—
338Dic3⋊C4—54no—
339C4×Dic3—54no—
340C4×Dic3—58no—
341C23×C5—32no—
342C23×C5—34no—
343C23×C5—62no—
344C23×C5—64no—
345C23×C5—64no—
346C23×C5—62no—
347C23×C5—64no—
348C23×C5—62no—
349C23×C5C23×D311yescharacteristic
350C22×D5—22no—
351C22×D5—22no—
352C22×D5—22no—
353C22×D5C23×D312yes—
354C5×D4—68no—
355C5×D4—128no—
356C2×C4×C5—24no—
357C2×C4×C5—32no—
358C2×C4×C5—62no—
359C2×C4×C5—68no—
360C2×C4×C5#23514yes—
361C2×C4×C5C23×D314yes—
362C2×C4×C5C23×D312yes—
363C22⋊D5—68no—
364C22⋊D5—128no—
365C2×Dic5—22no—
366C2×Dic5—24no—
367C2×Dic5—34no—
368C2×Dic5—32no—
369C2×Dic5—34no—
370C2×Dic5—32no—
371C2×Dic5—616no—
372C2×Dic5—62no—
373C2×Dic5—64no—
374C2×Dic5C23×D314yes—
375C2×Dic5C23×D312yes—
376C4×D5—28no—
377Dic10—128no—
378C22×Q8—152no—
379C22×D4—31no—
380C22×D4—32no—
381C22×D4—151no—
382C22×D4—152no—
383C23×C4—31no—
384C23×C4—51no—
385C23×C4—152no—
386C4⋊Q8—158no—
387#124—1516no—
388#124—1516no—
389#121—38no—
390#121—1516no—
391#121—158no—
392C4×D4—158no—
393C4×D4—158no—
394#117—58no—
395#117—1516no—
396#117—158no—
397C2×C4⋊C4—51no—
398C2×C4⋊C4—151no—
399C2×C4⋊C4—152no—
400C2×C4⋊C4—151no—
401C2×C4⋊C4D3×D511yescharacteristic
402C2×C22⋊C4—32no—
403C2×C22⋊C4—52no—
404C2×C22⋊C4—151no—
405C2×C22⋊C4—152no—
406C2×C22⋊C4—152no—
407C2×C22⋊C4—151no—
408C2×C22⋊C4—152no—
409C2×C42—52no—
410C2×C42—152no—
411C2×C42—151no—
412C2×C3×C5#55514yescomplemented
413C2×C3×C5C22×#4111yescharacteristic
414C2×C3×C5C22×#4111yescharacteristic
415C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
416C3×D5—42no—
417C5×D3—44no—
418C5×D3—42no—
419C23×C3—52no—
420C23×C3—102no—
421C23×C3—102no—
422C23×C3—102no—
423C23×C3C23×D511yescharacteristic
424C22×D3—22no—
425C22×D3—24no—
426C22×D3—22no—
427C22×D3—24no—
428C22×D3—24no—
429C22×D3—22no—
430C22×D3C23×D512yes—
431C22×D3C23×D514yes—
432C2×C3×C4—24no—
433C2×C3×C4—54no—
434C2×C3×C4—52no—
435C2×C3×C4—102no—
436C2×C3×C4—104no—
437C2×C3×C4—108no—
438C2×C3×C4#71114yes—
439C2×C3×C4C23×D514yes—
440C2×C3×C4C23×D512yes—
441#67—108no—
442#67—208no—
443C2×Dic3—22no—
444C2×Dic3—54no—
445C2×Dic3—52no—
446C2×Dic3—54no—
447C2×Dic3—52no—
448C2×Dic3—102no—
449C2×Dic3—104no—
450C2×Dic3C23×D512yes—
451D12—28no—
452D12—48no—
453C4×D3—28no—
454C4×D3—1016no—
455Dic6—208no—
456C22×C5—62no—
457C22×C5—64no—
458C22×C5—64no—
459C22×C5—62no—
460C22×C5—64no—
461C22×C5—62no—
462C22×C5—124no—
463C22×C5—128no—
464C22×C5C2×#23512yes—
465C22×C5D3×#4112yes—
466C22×C5#100912yes—
467C22×C5C24×D311yescharacteristic
468C2×D5—22no—
469C2×D5—22no—
470C2×D5—22no—
471C2×D5—44no—
472C4×C5—24no—
473C4×C5—62no—
474C4×C5C2×#23514yes—
475Dic5—22no—
476Dic5—24no—
477Dic5—34no—
478Dic5—62no—
479Dic5—64no—
480C24—32no—
481C24—31no—
482C24—51no—
483C2×Q8—158no—
484C2×Q8—604no—
485C2×D4—34no—
486C2×D4—38no—
487C2×D4—64no—
488C2×D4—124no—
489C2×D4—154no—
490C2×D4—158no—
491C2×D4—304no—
492C2×D4—604no—
493C22×C4—31no—
494C22×C4—34no—
495C22×C4—51no—
496C22×C4—52no—
497C22×C4—54no—
498C22×C4—64no—
499C22×C4—104no—
500C22×C4—158no—
501C22×C4—151no—
502C22×C4—151no—
503C22×C4—152no—
504C22×C4—308no—
505C22×C4C2×D3×D512yes—
506C22×C4C2×D3×D511yescharacteristic
507C4⋊C4—54no—
508C4⋊C4—154no—
509C4⋊C4—158no—
510C4⋊C4—154no—
511C4⋊C4C2×D3×D514yes—
512C22⋊C4—38no—
513C22⋊C4—58no—
514C22⋊C4—154no—
515C22⋊C4—158no—
516C22⋊C4—158no—
517C22⋊C4—154no—
518C22⋊C4—158no—
519C42—58no—
520C42—158no—
521C42—154no—
522C3×C5C2×#55511yescharacteristic, complemented, Hall
523C22×C3—102no—
524C22×C3—102no—
525C22×C3—102no—
526C22×C3—204no—
527C22×C3C2×#71112yes—
528C22×C3#414412yes—
529C22×C3D5×#4112yes—
530C22×C3C24×D511yescharacteristic
531C2×D3—22no—
532C2×D3—24no—
533C2×D3—22no—
534C2×D3—24no—
535C2×D3—24no—
536C2×D3—22no—
537C2×D3—44no—
538C2×D3—48no—
539C3×C4—24no—
540C3×C4—102no—
541C3×C4—104no—
542C3×C4C2×#71114yes—
543Dic3—22no—
544Dic3—54no—
545Dic3—102no—
546Dic3—104no—
547C2×C5—122no—
548C2×C5—124no—
549C2×C5#553214yescomplemented
550C2×C5C22×#23511yescharacteristic
551C2×C5C2×D3×#4111yescharacteristic
552C2×C5C2×#100911yescharacteristic
553D5—42no—
554C23—34no—
555C23—32no—
556C23—52no—
557C23—62no—
558C23—64no—
559C23—62no—
560C23—64no—
561C23—64no—
562C23—62no—
563C23—102no—
564C23—102no—
565C23—102no—
566C23C22×D3×D511yescharacteristic
567Q8—608no—
568D4—68no—
569D4—128no—
570D4—308no—
571D4—608no—
572C2×C4—24no—
573C2×C4—32no—
574C2×C4—52no—
575C2×C4—54no—
576C2×C4—62no—
577C2×C4—68no—
578C2×C4—102no—
579C2×C4—104no—
580C2×C4—108no—
581C2×C4—154no—
582C2×C4—152no—
583C2×C4—152no—
584C2×C4—154no—
585C2×C4—3016no—
586C2×C4—304no—
587C2×C4—302no—
588C2×C4#677214yes—
589C2×C4C22×D3×D514yes—
590C2×C4C22×D3×D512yes—
591C2×C3—202no—
592C2×C3#6595014yescomplemented
593C2×C3C22×#71111yescharacteristic
594C2×C3C2×#414411yescharacteristic
595C2×C3C2×D5×#4111yescharacteristic
596D3—42no—
597D3—44no—
598C5C2×#553211yescharacteristic, complemented, Sylow
599C22—62no—
600C22—64no—
601C22—62no—
602C22—64no—
603C22—64no—
604C22—62no—
605C22—102no—
606C22—102no—
607C22—102no—
608C22—124no—
609C22—128no—
610C22—204no—
611C22C2×#677212yes—
612C22#9202712yes—
613C22#9203412yes—
614C22C23×D3×D511yescharacteristic
615C4—24no—
616C4—62no—
617C4—102no—
618C4—104no—
619C4—154no—
620C4—304no—
621C4—302no—
622C4C2×#677214yes—
623C3C2×#6595011yescharacteristic, complemented, Sylow
624C2—122no—
625C2—124no—
626C2—202no—
627C2#1175380414yesdirect product factor
628C2C22×#677211yescharacteristic
629C2C2×#9202711yescharacteristic
630C2C2×#9203411yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC23×C3×C5C24
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5C2×#555
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#91777C2semidirect
2. C2×C5×#895C2semidirect
3. C2×C3×#4038C2semidirect
4. C2×#91462C2semidirect
5. C2×#91453C2semidirect
6. C2×#91449C2semidirect
7. C2×C4×#1201C2semidirect
8. C2×#91435C2semidirect
9. C2×#91398C2semidirect
10. C2×#91373C2semidirect
11. C2×#91338C2semidirect
12. #11753804C2direct (N × H)
26. #91777C22semidirect
28. C5×#895C22semidirect
32. C3×#4038C22semidirect
40. #91462C22semidirect
41. #91453C22semidirect
42. #91449C22semidirect
43. C4×#1201C22semidirect
44. #91435C22semidirect
45. #91398C22semidirect
46. #91373C22semidirect
47. #91338C22semidirect
57. C2×#4038D3semidirect
107. C2×#895D5semidirect
145. #4038C2×D3semidirect
213. #895C2×D5semidirect
412. C2×C3×C5#555semidirect
522. C3×C5C2×#555semidirect
549. C2×C5#5532semidirect
592. C2×C3#65950semidirect
598. C5C2×#5532semidirect
623. C3C2×#65950semidirect
627. C2#11753804direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#917772
C2×C5×#8952
C2×C3×#40382
C2×#914622
C2×#914532
C2×#914492
C2×C4×#12012
C2×#914352
C2×#913982
C2×#913732
C2×#913382
#117538042
C2×#659503
C2×#55325