Finite Group Databasebeta

← C2×#11751853 #14319901

Subgroups — C2×#11751853 #14319901

Order 1920.

2288 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D3×D5C211yesmaximal, characteristic
2C22×(C2×D3)⋊AGL(1,𝔽5)C212yesmaximal
3C2×#91235C211yesmaximal, characteristic, complemented
4C2×C3×(C2×AGL(1,𝔽5))⋊C4C211yesmaximal, characteristic, complemented
5C2×(C2×AGL(1,𝔽5))⋊Dic3C212yesmaximal, complemented
6#11751853C218yesmaximal, direct product factor
7C2×#65764—31nomaximal, Hall
8C23×D3×D5C2211yescharacteristic
9C23×ΓL(1,𝔽16)C2212yes—
10C23×C3×AGL(1,𝔽5)C2212yes—
11C22×C4×D15C411yescharacteristic, complemented
12C22×C4×C5×D3C411yescharacteristic, complemented
13C22×C3×C4×D5C2211yescharacteristic
14C22×#1200C411yescharacteristic, complemented
15C22×D3×Dic5C411yescharacteristic, complemented
16C22×D5×Dic3C2211yescharacteristic
17C2×C4×D3×D5—28no—
18C2×C4×D3×D5C414yescomplemented
19C2×C4×D3×D5C2214yes—
20C2×(C2×D3)⋊AGL(1,𝔽5)—28no—
21C2×(C2×D3)⋊AGL(1,𝔽5)C2218yes—
22#91235C2214yescomplemented
23C3×(C2×AGL(1,𝔽5))⋊C4C2214yescomplemented
24(C2×AGL(1,𝔽5))⋊Dic3C2218yescomplemented
25C2×#4664—51nomaximal, Hall
26C23×C4×D5—31no—
27C22×C22⋊AGL(1,𝔽5)—32no—
28#65764—38no—
29C2×(C2×AGL(1,𝔽5))⋊C4—32no—
30C2×(C2×AGL(1,𝔽5))⋊C4—31no—
31C2×(C2×AGL(1,𝔽5))⋊C4D311yescharacteristic, complemented
32C23×D15C2×C411yescharacteristic
33C23×C5×D3C2×C411yescharacteristic
34C23×C3×D5C2311yescharacteristic
35C22×D3×D5—24no—
36C22×D3×D5—22no—
37C22×D3×D5—22no—
38C22×D3×D5C2×C412yes—
39C22×D3×D5D418yes—
40C22×D3×D5C2312yes—
41C22×ΓL(1,𝔽16)—24no—
42C22×ΓL(1,𝔽16)—24no—
43C22×ΓL(1,𝔽16)—24no—
44C22×ΓL(1,𝔽16)C2314yes—
45C22×C3×AGL(1,𝔽5)—24no—
46C22×C3×AGL(1,𝔽5)—24no—
47C22×C3×AGL(1,𝔽5)—24no—
48C22×C3×AGL(1,𝔽5)C2314yes—
49C22×C3×C4×C5C2×C411yescharacteristic
50C22×Dic15C2×C411yescharacteristic
51C2×C4×D15—24no—
52C2×C4×D15C2×C414yescomplemented
53C22×C5×Dic3C2×C411yescharacteristic
54C2×C4×C5×D3—24no—
55C2×C4×C5×D3C2×C414yescomplemented
56C22×C3×Dic5C2×C411yescharacteristic
57C2×C3×C4×D5—24no—
58C2×C3×C4×D5C2×C412yes—
59C2×C3×C4×D5C2312yes—
60C2×#1200—24no—
61C2×#1200C2×C414yescomplemented
62C2×D3×Dic5—24no—
63C2×D3×Dic5C2×C414yescomplemented
64C2×D5×Dic3—24no—
65C2×D5×Dic3C2×C412yes—
66C2×D5×Dic3C2312yes—
67C4×D3×D5—216no—
68C4×D3×D5—216no—
69(C2×D3)⋊AGL(1,𝔽5)—216no—
70C23×C4×D3—51no—
71C22×(C2×D3)⋊C4—52no—
72C2×C3×#95—51no—
73C2×C3⋊#95—52no—
74C2×C3⋊#95—51no—
75#4664—58no—
76C24×D5—31no—
77C23×AGL(1,𝔽5)—32no—
78C23×AGL(1,𝔽5)C2×D312yes—
79C23×C4×C5—31no—
80C23×Dic5—31no—
81C22×C4×D5—31no—
82C22×C4×D5—31no—
83C22×C4×D5—31no—
84C22×C4×D5—34no—
85C22×C4×D5—34no—
86C22×C4×D5—68no—
87C22×C4×D5C2×D311yescharacteristic
88C2×C22⋊AGL(1,𝔽5)—38no—
89C2×C22⋊AGL(1,𝔽5)—68no—
90(C2×AGL(1,𝔽5))⋊C4—38no—
91(C2×AGL(1,𝔽5))⋊C4—34no—
92(C2×AGL(1,𝔽5))⋊C4C2×D314yescomplemented
93C2×#381—151noSylow
94C23×C3×C5C22×C411yescharacteristic
95C22×D15—22no—
96C22×D15—22no—
97C22×D15C22⋊C414yes—
98C22×D15C22×C412yes—
99C22×C5×D3—22no—
100C22×C5×D3—22no—
101C22×C5×D3C22⋊C414yes—
102C22×C5×D3C22×C412yes—
103C22×C3×D5C22×C411yescharacteristic
104C22×C3×D5C2×D414yes—
105C22×C3×D5#4114yes—
106C22×C3×D5#4112yes—
107C22×C3×D5#4112yes—
108C22×C3×D5C2411yescharacteristic
109C2×D3×D5—24no—
110C2×D3×D5—24no—
111C2×D3×D5—216no—
112C2×D3×D5—216no—
113C2×D3×D5—24no—
114C2×D3×D5—22no—
115C2×D3×D5—22no—
116C2×D3×D5C22⋊C418yes—
117C2×D3×D5C2×D418yes—
118C2×ΓL(1,𝔽16)—24no—
119C2×ΓL(1,𝔽16)—24no—
120C2×ΓL(1,𝔽16)—24no—
121C2×ΓL(1,𝔽16)—48no—
122C2×C3×AGL(1,𝔽5)—24no—
123C2×C3×AGL(1,𝔽5)—24no—
124C2×C3×AGL(1,𝔽5)—24no—
125C2×C3×AGL(1,𝔽5)—48no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5C22×C412yes—
128C2×Dic15—22no—
129C2×Dic15C22×C412yes—
130C4×D15—28no—
131C2×C5×Dic3—22no—
132C2×C5×Dic3C22×C412yes—
133C4×C5×D3—28no—
134C2×C3×Dic5—22no—
135C2×C3×Dic5C22×C412yes—
136C3×C4×D5—24no—
137C3×C4×D5—24no—
138#1200—28no—
139D3×Dic5—28no—
140D5×Dic3—24no—
141D5×Dic3—24no—
142C24×D3—51no—
143C23×C3×C4—51no—
144C23×C3×C4—52no—
145C23×Dic3—51no—
146C23×Dic3—52no—
147C22×C4×D3—51no—
148C22×C4×D3—51no—
149C22×C4×D3—51no—
150C22×C4×D3—54no—
151C22×C4×D3—54no—
152C22×C4×D3—108no—
153C22×C4×D3AGL(1,𝔽5)11yescharacteristic, complemented
154C2×(C2×D3)⋊C4—58no—
155C2×(C2×D3)⋊C4—108no—
156C3×#95—54no—
157C3⋊#95—58no—
158C3⋊#95—54no—
159C24×C5—31no—
160C23×D5—31no—
161C23×D5—32no—
162C23×D5—38no—
163C23×D5—32no—
164C23×D5—64no—
165C23×D5—62no—
166C23×D5—62no—
167C23×D5C22×D311yescharacteristic
168C22×AGL(1,𝔽5)—24no—
169C22×AGL(1,𝔽5)—24no—
170C22×AGL(1,𝔽5)—24no—
171C22×AGL(1,𝔽5)—34no—
172C22×AGL(1,𝔽5)—64no—
173C22×AGL(1,𝔽5)—64no—
174C22×AGL(1,𝔽5)—64no—
175C22×AGL(1,𝔽5)C22×D314yes—
176C22×C4×C5—31no—
177C22×C4×C5—34no—
178C22×C4×C5—64no—
179C22×C4×C5C4×D311yescharacteristic
180C22×Dic5—31no—
181C22×Dic5—34no—
182C22×Dic5—64no—
183C22×Dic5C4×D311yescharacteristic
184C2×C4×D5—24no—
185C2×C4×D5—32no—
186C2×C4×D5—32no—
187C2×C4×D5—34no—
188C2×C4×D5—34no—
189C2×C4×D5—616no—
190C2×C4×D5—616no—
191C2×C4×D5—64no—
192C2×C4×D5—64no—
193C2×C4×D5—64no—
194C2×C4×D5C4×D312yes—
195C2×C4×D5C22×D312yes—
196C22⋊AGL(1,𝔽5)—616no—
197C24×C4—151no—
198C22×C22⋊C4—152no—
199#381—158no—
200C2×#95—51no—
201C2×#95—151no—
202C2×#95—152no—
203C22×C3×C5C2×C22⋊C412yes—
204C22×C3×C5#11712yes—
205C22×C3×C5#11712yes—
206C22×C3×C5C23×C411yescharacteristic
207C2×D15—22no—
208C2×D15—22no—
209C2×D15—28no—
210C2×D15C2×C22⋊C414yes—
211C2×C5×D3—22no—
212C2×C5×D3—22no—
213C2×C5×D3—28no—
214C2×C5×D3C2×C22⋊C414yes—
215C2×C3×D5C2×C22⋊C412yes—
216C2×C3×D5#11712yes—
217C2×C3×D5#11712yes—
218C2×C3×D5#12318yes—
219C2×C3×D5#12318yes—
220C2×C3×D5C22×D412yes—
221C2×C3×D5C2×#4112yes—
222C2×C3×D5C2×#4111yescharacteristic
223C2×C3×D5C2×#4111yescharacteristic
224D3×D5—28no—
225D3×D5—28no—
226D3×D5—216no—
227ΓL(1,𝔽16)—44no—
228C3×AGL(1,𝔽5)—44no—
229C3×C4×C5—22no—
230Dic15—22no—
231C3×Dic5—22no—
232C5×Dic3—22no—
233C24×C3—51no—
234C23×D3—51no—
235C23×D3—58no—
236C23×D3—52no—
237C23×D3—52no—
238C23×D3—102no—
239C23×D3—104no—
240C23×D3—102no—
241C23×D3C2×AGL(1,𝔽5)11yescharacteristic
242C22×C3×C4—51no—
243C22×C3×C4—52no—
244C22×C3×C4—52no—
245C22×C3×C4—54no—
246C22×C3×C4—104no—
247C22×C3×C4—104no—
248C22×C3×C4—104no—
249C22×C3×C4—104no—
250C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
251C22×Dic3—51no—
252C22×Dic3—52no—
253C22×Dic3—52no—
254C22×Dic3—54no—
255C22×Dic3—104no—
256C22×Dic3—104no—
257C22×Dic3—104no—
258C22×Dic3—104no—
259C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
260C2×C4×D3—24no—
261C2×C4×D3—54no—
262C2×C4×D3—54no—
263C2×C4×D3—54no—
264C2×C4×D3—1016no—
265C2×C4×D3—1016no—
266C2×C4×D3—104no—
267C2×C4×D3—104no—
268C2×C4×D3—104no—
269C2×C4×D3C2×AGL(1,𝔽5)14yescomplemented
270(C2×D3)⋊C4—1016no—
271C23×C5—34no—
272C23×C5—32no—
273C23×C5—62no—
274C23×C5—62no—
275C23×C5C2×C4×D311yescharacteristic
276C22×D5—34no—
277C22×D5—32no—
278C22×D5—38no—
279C22×D5—38no—
280C22×D5—62no—
281C22×D5—62no—
282C22×D5—64no—
283C22×D5—616no—
284C22×D5—64no—
285C22×D5—616no—
286C22×D5—64no—
287C22×D5—62no—
288C22×D5—62no—
289C22×D5C2×C4×D311yescharacteristic
290C22×D5#23514yes—
291C22×D5D3×D414yes—
292C22×D5#23712yes—
293C22×D5#23912yes—
294C22×D5C23×D311yescharacteristic
295C2×AGL(1,𝔽5)—24no—
296C2×AGL(1,𝔽5)—24no—
297C2×AGL(1,𝔽5)—24no—
298C2×AGL(1,𝔽5)—48no—
299C2×AGL(1,𝔽5)—64no—
300C2×AGL(1,𝔽5)—64no—
301C2×AGL(1,𝔽5)—64no—
302C2×AGL(1,𝔽5)—128no—
303C2×C4×C5—22no—
304C2×C4×C5—32no—
305C2×C4×C5—68no—
306C2×C4×C5—62no—
307C2×C4×C5C2×C4×D312yes—
308C2×Dic5—22no—
309C2×Dic5—32no—
310C2×Dic5—68no—
311C2×Dic5—62no—
312C2×Dic5C2×C4×D312yes—
313C4×D5—24no—
314C4×D5—24no—
315C4×D5—68no—
316C4×D5—68no—
317C4×D5—64no—
318C4×D5—64no—
319C25—151no—
320C23×C4—31no—
321C23×C4—51no—
322C23×C4—52no—
323C23×C4—154no—
324C23×C4—151no—
325C23×C4—154no—
326C23×C4—151no—
327C23×C4—151no—
328C23×C4—151no—
329C23×C4—152no—
330C23×C4—308no—
331C2×C22⋊C4—158no—
332C2×C22⋊C4—308no—
333#95—54no—
334#95—154no—
335#95—158no—
336C2×C3×C5#38114yescomplemented
337C2×C3×C5C22×C22⋊C411yescharacteristic
338C2×C3×C5C2×#11711yescharacteristic
339C2×C3×C5C2×#11711yescharacteristic
340D15—24no—
341C3×D5#38114yescomplemented
342C3×D5C2×#12312yes—
343C3×D5C2×#12312yes—
344C5×D3—24no—
345C23×C3—52no—
346C23×C3—54no—
347C23×C3—52no—
348C23×C3—54no—
349C23×C3—51no—
350C23×C3—51no—
351C23×C3C22×AGL(1,𝔽5)11yescharacteristic
352C22×D3—22no—
353C22×D3—22no—
354C22×D3—54no—
355C22×D3—52no—
356C22×D3—58no—
357C22×D3—58no—
358C22×D3—104no—
359C22×D3—104no—
360C22×D3—102no—
361C22×D3—104no—
362C22×D3—102no—
363C22×D3—1016no—
364C22×D3—1016no—
365C22×D3—102no—
366C22×D3—102no—
367C22×D3C22⋊AGL(1,𝔽5)14yes—
368C22×D3C22×AGL(1,𝔽5)12yes—
369C2×C3×C4—22no—
370C2×C3×C4—52no—
371C2×C3×C4—104no—
372C2×C3×C4—104no—
373C2×C3×C4—102no—
374C2×C3×C4—104no—
375C2×C3×C4—104no—
376C2×C3×C4—104no—
377C2×C3×C4—208no—
378C2×C3×C4C22×AGL(1,𝔽5)12yes—
379C2×Dic3—22no—
380C2×Dic3—52no—
381C2×Dic3—104no—
382C2×Dic3—104no—
383C2×Dic3—102no—
384C2×Dic3—104no—
385C2×Dic3—104no—
386C2×Dic3—104no—
387C2×Dic3—208no—
388C2×Dic3C22×AGL(1,𝔽5)12yes—
389C4×D3—28no—
390C4×D3—108no—
391C4×D3—108no—
392C4×D3—108no—
393C22×C5—34no—
394C22×C5—68no—
395C22×C5—62no—
396C22×C5—62no—
397C22×C5#87212yes—
398C22×C5D3×C22⋊C412yes—
399C22×C5#89212yes—
400C22×C5C22×C4×D311yescharacteristic
401C2×D5—34no—
402C2×D5—68no—
403C2×D5—62no—
404C2×D5—62no—
405C2×D5—616no—
406C2×D5—68no—
407C2×D5—68no—
408C2×D5#87212yes—
409C2×D5D3×C22⋊C412yes—
410C2×D5#88318yes—
411C2×D5#89212yes—
412C2×D5#89418yes—
413C2×D5C2×#23512yes—
414C2×D5C2×D3×D412yes—
415C2×D5C2×#23711yescharacteristic
416C2×D5C2×#23911yescharacteristic
417AGL(1,𝔽5)—44no—
418AGL(1,𝔽5)—124no—
419C4×C5—22no—
420C4×C5—62no—
421Dic5—22no—
422Dic5—62no—
423C24—31no—
424C24—51no—
425C24—152no—
426C24—158no—
427C24—152no—
428C24—151no—
429C24—302no—
430C24—302no—
431C24—304no—
432C22×C4—31no—
433C22×C4—34no—
434C22×C4—51no—
435C22×C4—52no—
436C22×C4—52no—
437C22×C4—54no—
438C22×C4—64no—
439C22×C4—104no—
440C22×C4—104no—
441C22×C4—104no—
442C22×C4—104no—
443C22×C4—154no—
444C22×C4—152no—
445C22×C4—152no—
446C22×C4—154no—
447C22×C4—151no—
448C22×C4—154no—
449C22×C4—154no—
450C22×C4—304no—
451C22×C4—3016no—
452C22×C4—3016no—
453C22×C4—304no—
454C22×C4—304no—
455C22×C4—304no—
456C22×C4—304no—
457C22×C4—304no—
458C22×C4—304no—
459C22×C4D3×AGL(1,𝔽5)11yescharacteristic
460C22⋊C4—3016no—
461C3×C5C2×#38111yescharacteristic, complemented, Hall
462C22×C3—52no—
463C22×C3—52no—
464C22×C3—51no—
465C22×C3—52no—
466C22×C3—51no—
467C22×C3—52no—
468C22×C3—52no—
469C22×C3—58no—
470C22×C3—58no—
471C22×C3#413012yes—
472C22×C3#413012yes—
473C22×C3C2×C22⋊AGL(1,𝔽5)12yes—
474C22×C3C23×AGL(1,𝔽5)11yescharacteristic
475C2×D3—22no—
476C2×D3—22no—
477C2×D3—28no—
478C2×D3—54no—
479C2×D3—108no—
480C2×D3—108no—
481C2×D3—1016no—
482C2×D3—102no—
483C2×D3—102no—
484C2×D3—108no—
485C2×D3C2×C22⋊AGL(1,𝔽5)14yes—
486C3×C4—22no—
487C3×C4—102no—
488C3×C4—204no—
489Dic3—22no—
490Dic3—102no—
491Dic3—204no—
492C2×C5—64no—
493C2×C5#466414yescomplemented
494C2×C5C2×#87211yescharacteristic
495C2×C5C2×D3×C22⋊C411yescharacteristic
496C2×C5C2×#89211yescharacteristic
497D5—64no—
498D5#466414yescomplemented
499D5C2×#88312yes—
500D5C2×#89412yes—
501C23—34no—
502C23—32no—
503C23—52no—
504C23—52no—
505C23—54no—
506C23—54no—
507C23—51no—
508C23—51no—
509C23—62no—
510C23—62no—
511C23—158no—
512C23—158no—
513C23—154no—
514C23—152no—
515C23—302no—
516C23—302no—
517C23—304no—
518C23—304no—
519C23—304no—
520C23—3016no—
521C23—3016no—
522C23—302no—
523C23—302no—
524C23C2×D3×AGL(1,𝔽5)11yescharacteristic
525C2×C4—22no—
526C2×C4—32no—
527C2×C4—52no—
528C2×C4—68no—
529C2×C4—62no—
530C2×C4—104no—
531C2×C4—104no—
532C2×C4—102no—
533C2×C4—104no—
534C2×C4—104no—
535C2×C4—104no—
536C2×C4—152no—
537C2×C4—208no—
538C2×C4—308no—
539C2×C4—308no—
540C2×C4—304no—
541C2×C4—304no—
542C2×C4—308no—
543C2×C4—302no—
544C2×C4—304no—
545C2×C4—304no—
546C2×C4—304no—
547C2×C4—608no—
548C2×C4C2×D3×AGL(1,𝔽5)12yes—
549C2×C3—52no—
550C2×C3—52no—
551C2×C3—54no—
552C2×C3#6576414yescomplemented
553C2×C3C2×#413011yescharacteristic
554C2×C3C2×#413011yescharacteristic
555C2×C3C22×C22⋊AGL(1,𝔽5)11yescharacteristic
556D3—24no—
557D3—104no—
558C5C2×#466411yescharacteristic, complemented, Sylow
559C22—34no—
560C22—51no—
561C22—51no—
562C22—52no—
563C22—52no—
564C22—52no—
565C22—52no—
566C22—52no—
567C22—58no—
568C22—58no—
569C22—62no—
570C22—62no—
571C22—68no—
572C22—154no—
573C22—308no—
574C22—308no—
575C22—3016no—
576C22—302no—
577C22—302no—
578C22—308no—
579C22#9190712yes—
580C22#9190912yes—
581C22D3×C22⋊AGL(1,𝔽5)12yes—
582C22C22×D3×AGL(1,𝔽5)11yescharacteristic
583C4—22no—
584C4—62no—
585C4—102no—
586C4—204no—
587C4—302no—
588C4—604no—
589C3C2×#6576411yescharacteristic, complemented, Sylow
590C2—52no—
591C2—52no—
592C2—54no—
593C2—64no—
594C2—304no—
595C2#1175185314yesdirect product factor
596C2C2×#9190711yescharacteristic
597C2C2×#9190911yescharacteristic
598C2C2×D3×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC23C2×D3×AGL(1,𝔽5)
commutatorC22×C3×C5C23×C4
FrattiniC22C22×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC23×C3×C5C22×C4
prime radicalC23×D3×D5C22
cosocleC22×C3×D5C24
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5C2×#381
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×#91235C2semidirect
4. C2×C3×(C2×AGL(1,𝔽5))⋊C4C2semidirect
5. C2×(C2×AGL(1,𝔽5))⋊Dic3C2semidirect
6. #11751853C2direct (N × H)
11. C22×C4×D15C4semidirect
12. C22×C4×C5×D3C4semidirect
14. C22×#1200C4semidirect
15. C22×D3×Dic5C4semidirect
18. C2×C4×D3×D5C4semidirect
22. #91235C22semidirect
23. C3×(C2×AGL(1,𝔽5))⋊C4C22semidirect
24. (C2×AGL(1,𝔽5))⋊Dic3C22semidirect
31. C2×(C2×AGL(1,𝔽5))⋊C4D3semidirect
52. C2×C4×D15C2×C4semidirect
55. C2×C4×C5×D3C2×C4semidirect
61. C2×#1200C2×C4semidirect
63. C2×D3×Dic5C2×C4semidirect
92. (C2×AGL(1,𝔽5))⋊C4C2×D3semidirect
153. C22×C4×D3AGL(1,𝔽5)semidirect
269. C2×C4×D3C2×AGL(1,𝔽5)semidirect
336. C2×C3×C5#381semidirect
341. C3×D5#381semidirect
461. C3×C5C2×#381semidirect
493. C2×C5#4664semidirect
498. D5#4664semidirect
552. C2×C3#65764semidirect
558. C5C2×#4664semidirect
589. C3C2×#65764semidirect
595. C2#11751853direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D3×D52
C22×(C2×D3)⋊AGL(1,𝔽5)2
C2×#912352
C2×C3×(C2×AGL(1,𝔽5))⋊C42
C2×(C2×AGL(1,𝔽5))⋊Dic32
#117518532
C2×#657643
C2×#46645