Finite Group Databasebeta

← C2×#11752404 #14411925

Subgroups — C2×#11752404 #14411925

Order 1920.

1576 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#709C211yesmaximal, characteristic
2C2×#91989C212yesmaximal, complemented
3C2×#91240C212yesmaximal, complemented
4#11752404C218yesmaximal, direct product factor
5C2×#91228C212yesmaximal, complemented
6C2×#65745—31nomaximal, Hall
7C23×C3×C4×C5C411yescharacteristic, complemented
8C22×C3×C22⋊D5C411yescharacteristic, complemented
9C22×C3×C22⋊D5C2211yescharacteristic
10C2×C3×#709—22no—
11C2×C3×#709—22no—
12C2×C3×#709—24no—
13C2×C3×#709C414yescomplemented
14C2×C3×#709C2214yes—
15C22×C3×D20C2211yescharacteristic
16C22×C3×C4×D5C411yescharacteristic, complemented
17C22×C3×C4×D5C2211yescharacteristic
18C22×C3×Dic10C411yescharacteristic, complemented
19C2×#6763C2214yes—
20#91989—24no—
21#91989C2218yescomplemented
22C2×C4⋊ΓL(1,𝔽16)—22no—
23C2×C4×ΓL(1,𝔽16)—22no—
24#91240C2218yescomplemented
25#91228C2218yescomplemented
26C2×#5240—51nomaximal, Hall
27C22×#709D311yescharacteristic
28C2×#4130—32no—
29C2×#4011—32no—
30#65745—38no—
31C2×#3999—32no—
32C24×C3×C5C2×C411yescharacteristic
33C23×C3×D5C2×C411yescharacteristic
34C23×C3×D5C2311yescharacteristic
35C22×ΓL(1,𝔽16)—24no—
36C22×C3×C4×C5—22no—
37C22×C3×C4×C5—21no—
38C22×C3×C4×C5—21no—
39C22×C3×C4×C5C2×C411yescharacteristic
40C22×C3×C4×C5C2×C411yescharacteristic
41C22×C3×C4×C5C2×C414yescomplemented
42C2×C3×C22⋊D5—22no—
43C2×C3×C22⋊D5—22no—
44C2×C3×C22⋊D5—22no—
45C2×C3×C22⋊D5—22no—
46C2×C3×C22⋊D5C2×C412yes—
47C2×C3×C22⋊D5C2×C414yescomplemented
48C2×C3×C22⋊D5C2312yes—
49C22×C3×Dic5C2×C411yescharacteristic
50C22×C3×Dic5C2×C411yescharacteristic
51C3×#709—28no—
52C3×#709—28no—
53C3×#709—28no—
54C3×#709—28no—
55C2×C3×D20—22no—
56C2×C3×D20—22no—
57C2×C3×D20C2×C412yes—
58C2×C3×D20C2312yes—
59C2×C3×C4×D5—22no—
60C2×C3×C4×D5—21no—
61C2×C3×C4×D5—21no—
62C2×C3×C4×D5C2×C412yes—
63C2×C3×C4×D5C2×C414yescomplemented
64C2×C3×C4×D5D412yes—
65C2×C3×C4×D5D412yes—
66C2×C3×C4×D5D414yes—
67C2×C3×C4×D5C2312yes—
68C2×C3×Dic10—22no—
69C2×C3×Dic10—22no—
70C2×C3×Dic10C2×C414yescomplemented
71#6763—24no—
72#6763C2318yes—
73C4⋊ΓL(1,𝔽16)—28no—
74C4×ΓL(1,𝔽16)—28no—
75C22×C3×#41—51no—
76C2×#926—52no—
77#5240—58no—
78C2×#834—52no—
79C2×#834—52no—
80C23×C4×C5Dic311yescharacteristic, complemented
81C22×C22⋊D5Dic311yescharacteristic, complemented
82C22×C22⋊D5C2×D311yescharacteristic
83C2×#709—22no—
84C2×#709—22no—
85C2×#709—24no—
86C2×#709Dic314yescomplemented
87C2×#709C2×D314yes—
88C22×D20C2×D311yescharacteristic
89C22×C4×D5Dic311yescharacteristic, complemented
90C22×C4×D5C2×D311yescharacteristic
91C22×Dic10Dic311yescharacteristic, complemented
92C2×C22⋊AGL(1,𝔽5)—34no—
93#4130—38no—
94#4130—64no—
95C2×C4⋊AGL(1,𝔽5)—62no—
96C2×C4×AGL(1,𝔽5)—62no—
97#4011—38no—
98#3999—38no—
99C2×#410—151noSylow
100C23×C3×C5—21no—
101C23×C3×C5—21no—
102C23×C3×C5—42no—
103C23×C3×C5C22×C411yescharacteristic
104C23×C3×C5C22×C412yes—
105C22×C3×D5—21no—
106C22×C3×D5—21no—
107C22×C3×D5—42no—
108C22×C3×D5—42no—
109C22×C3×D5C22×C412yes—
110C22×C3×D5C22×C411yescharacteristic
111C22×C3×D5C2×D412yes—
112C22×C3×D5C2×D412yes—
113C22×C3×D5C2411yescharacteristic
114C2×ΓL(1,𝔽16)—28no—
115C2×ΓL(1,𝔽16)—48no—
116C2×C3×C4×C5—21no—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—24no—
119C2×C3×C4×C5—24no—
120C2×C3×C4×C5C22⋊C412yes—
121C2×C3×C4×C5C22⋊C411yescharacteristic
122C2×C3×C4×C5C22⋊C411yescharacteristic
123C2×C3×C4×C5C22×C412yes—
124C2×C3×C4×C5C22×C412yes—
125C3×C22⋊D5—24no—
126C3×C22⋊D5—24no—
127C3×C22⋊D5—28no—
128C2×C3×Dic5—21no—
129C2×C3×Dic5—21no—
130C2×C3×Dic5C22⋊C412yes—
131C2×C3×Dic5C22⋊C412yes—
132C2×C3×Dic5C22×C412yes—
133C2×C3×Dic5C22×C412yes—
134C3×D20—24no—
135C3×D20—24no—
136C3×C4×D5—24no—
137C3×C4×D5—24no—
138C3×C4×D5C22⋊C414yes—
139C3×C4×D5C22⋊C414yes—
140C3×C4×D5C2×D414yes—
141C3×C4×D5C2×D414yes—
142C3×Dic10—28no—
143C2×C3×#41—54no—
144C2×C3×#41—54no—
145C2×C3×#41—102no—
146C2×C3×#41—102no—
147C2×C3×#41—104no—
148C22×C3×Q8—51no—
149C22×C3×D4—51no—
150C22×C3×D4—51no—
151C22×C3×D4—51no—
152C23×C3×C4—51no—
153C23×C3×C4—51no—
154C23×C3×C4AGL(1,𝔽5)11yescharacteristic, complemented
155C2×#217—54no—
156#926—58no—
157#926—104no—
158C2×C4⋊Dic3—102no—
159C2×C4×Dic3—102no—
160#834—58no—
161#834—58no—
162C24×C5C2×Dic311yescharacteristic
163C23×D5C2×Dic311yescharacteristic
164C23×D5C22×D311yescharacteristic
165C22×AGL(1,𝔽5)—64no—
166C22×C4×C5—22no—
167C22×C4×C5—21no—
168C22×C4×C5—21no—
169C22×C4×C5C2×Dic311yescharacteristic
170C22×C4×C5C2×Dic311yescharacteristic
171C22×C4×C5C2×Dic314yescomplemented
172C2×C22⋊D5—22no—
173C2×C22⋊D5—22no—
174C2×C22⋊D5—22no—
175C2×C22⋊D5—22no—
176C2×C22⋊D5C2×Dic312yes—
177C2×C22⋊D5C2×Dic314yescomplemented
178C2×C22⋊D5C22×D312yes—
179C22×Dic5C2×Dic311yescharacteristic
180C22×Dic5C2×Dic311yescharacteristic
181#709—28no—
182#709—28no—
183#709—28no—
184#709—28no—
185C2×D20—22no—
186C2×D20—22no—
187C2×D20C2×Dic312yes—
188C2×D20C22×D312yes—
189C2×C4×D5—22no—
190C2×C4×D5—21no—
191C2×C4×D5—21no—
192C2×C4×D5C2×Dic312yes—
193C2×C4×D5C2×Dic314yescomplemented
194C2×C4×D5#6712yes—
195C2×C4×D5#6712yes—
196C2×C4×D5#6714yes—
197C2×C4×D5C22×D312yes—
198C2×Dic10—22no—
199C2×Dic10—22no—
200C2×Dic10C2×Dic314yescomplemented
201C22⋊AGL(1,𝔽5)—38no—
202C22⋊AGL(1,𝔽5)—64no—
203C4⋊AGL(1,𝔽5)—68no—
204C4×AGL(1,𝔽5)—68no—
205C22×#41—51no—
206C2×#117—152no—
207#410—158no—
208C2×#99—152no—
209C2×#99—152no—
210C22×C3×C5—22no—
211C22×C3×C5—22no—
212C22×C3×C5—42no—
213C22×C3×C5—42no—
214C22×C3×C5—42no—
215C22×C3×C5C2×C22⋊C411yescharacteristic
216C22×C3×C5C2×C22⋊C411yescharacteristic
217C22×C3×C5C23×C411yescharacteristic
218C2×C3×D5—24no—
219C2×C3×D5—22no—
220C2×C3×D5—41no—
221C2×C3×D5—42no—
222C2×C3×D5—41no—
223C2×C3×D5—42no—
224C2×C3×D5—42no—
225C2×C3×D5—42no—
226C2×C3×D5C2×C22⋊C412yes—
227C2×C3×D5C22×D412yes—
228ΓL(1,𝔽16)—48no—
229C3×C4×C5—22no—
230C3×C4×C5C2×C22⋊C412yes—
231C3×C4×C5C2×C22⋊C412yes—
232C3×Dic5—22no—
233C3×Dic5C2×C22⋊C414yes—
234C24×C3—51no—
235C24×C3—51no—
236C24×C3C2×AGL(1,𝔽5)11yescharacteristic
237C3×#41—108no—
238C3×#41—108no—
239C3×#41—108no—
240C3×#41—108no—
241C2×C3×Q8—54no—
242C2×C3×Q8—102no—
243C2×C3×Q8—102no—
244C2×C3×D4—52no—
245C2×C3×D4—52no—
246C2×C3×D4—54no—
247C2×C3×D4—52no—
248C2×C3×D4—52no—
249C2×C3×D4—102no—
250C2×C3×D4—102no—
251C2×C3×D4—102no—
252C2×C3×D4—102no—
253C2×C3×D4—102no—
254C2×C3×D4—102no—
255C22×C3×C4—22no—
256C22×C3×C4—21no—
257C22×C3×C4—21no—
258C22×C3×C4—51no—
259C22×C3×C4—52no—
260C22×C3×C4—52no—
261C22×C3×C4—54no—
262C22×C3×C4—52no—
263C22×C3×C4—52no—
264C22×C3×C4—51no—
265C22×C3×C4—54no—
266C22×C3×C4—102no—
267C22×C3×C4—101no—
268C22×C3×C4—101no—
269C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
270C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
271C22×C3×C4C2×AGL(1,𝔽5)14yescomplemented
272C22×Dic3—104no—
273#217—58no—
274#217—104no—
275C4⋊Dic3—108no—
276C4×Dic3—108no—
277C23×C5—21no—
278C23×C5—21no—
279C23×C5—42no—
280C23×C5C22×Dic311yescharacteristic
281C23×C5C22×Dic312yes—
282C22×D5—21no—
283C22×D5—21no—
284C22×D5—42no—
285C22×D5—42no—
286C22×D5C22×Dic312yes—
287C22×D5C22×Dic311yescharacteristic
288C22×D5C2×#6712yes—
289C22×D5C2×#6712yes—
290C22×D5C23×D311yescharacteristic
291C2×AGL(1,𝔽5)—68no—
292C2×AGL(1,𝔽5)—128no—
293C2×C4×C5—21no—
294C2×C4×C5—21no—
295C2×C4×C5—24no—
296C2×C4×C5—24no—
297C2×C4×C5#21712yes—
298C2×C4×C5#21711yescharacteristic
299C2×C4×C5#21711yescharacteristic
300C2×C4×C5C22×Dic312yes—
301C2×C4×C5C22×Dic312yes—
302C22⋊D5—24no—
303C22⋊D5—24no—
304C22⋊D5—28no—
305C2×Dic5—21no—
306C2×Dic5—21no—
307C2×Dic5#21712yes—
308C2×Dic5#21712yes—
309C2×Dic5C22×Dic312yes—
310C2×Dic5C22×Dic312yes—
311D20—24no—
312D20—24no—
313C4×D5—24no—
314C4×D5—24no—
315C4×D5#21714yes—
316C4×D5#21714yes—
317C4×D5C2×#6714yes—
318C4×D5C2×#6714yes—
319Dic10—28no—
320C2×#41—54no—
321C2×#41—54no—
322C2×#41—102no—
323C2×#41—104no—
324C2×#41—102no—
325C22×Q8—51no—
326C22×D4—51no—
327C22×D4—51no—
328C22×D4—51no—
329C23×C4—51no—
330C23×C4—51no—
331C23×C4ΓL(1,𝔽16)11yescharacteristic, complemented
332#117—158no—
333#117—304no—
334C2×C4⋊C4—302no—
335C2×C22⋊C4—154no—
336C2×C42—302no—
337#99—158no—
338#99—158no—
339C2×C3×C5—21no—
340C2×C3×C5—21no—
341C2×C3×C5—42no—
342C2×C3×C5#41012yescomplemented
343C2×C3×C5C22×C22⋊C411yescharacteristic
344C3×D5—22no—
345C3×D5—22no—
346C3×D5—42no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—42no—
350C23×C3—52no—
351C23×C3—52no—
352C23×C3—51no—
353C23×C3—51no—
354C23×C3—52no—
355C23×C3—101no—
356C23×C3—101no—
357C23×C3—202no—
358C23×C3—202no—
359C23×C3C22×AGL(1,𝔽5)11yescharacteristic
360C23×C3C22×AGL(1,𝔽5)12yes—
361C3×Q8—108no—
362C3×D4—104no—
363C3×D4—104no—
364C3×D4—108no—
365C3×D4—104no—
366C3×D4—104no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—24no—
370C2×C3×C4—24no—
371C2×C3×C4—52no—
372C2×C3×C4—52no—
373C2×C3×C4—52no—
374C2×C3×C4—54no—
375C2×C3×C4—54no—
376C2×C3×C4—54no—
377C2×C3×C4—54no—
378C2×C3×C4—52no—
379C2×C3×C4—101no—
380C2×C3×C4—101no—
381C2×C3×C4—104no—
382C2×C3×C4—104no—
383C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
384C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
385C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
386C2×C3×C4C22×AGL(1,𝔽5)12yes—
387C2×C3×C4C22×AGL(1,𝔽5)12yes—
388C2×Dic3—108no—
389C2×Dic3—208no—
390C22×C5—22no—
391C22×C5—22no—
392C22×C5—42no—
393C22×C5—42no—
394C22×C5—42no—
395C22×C5C2×#21711yescharacteristic
396C22×C5C2×#21711yescharacteristic
397C22×C5C23×Dic311yescharacteristic
398C2×D5—24no—
399C2×D5—22no—
400C2×D5—41no—
401C2×D5—42no—
402C2×D5—41no—
403C2×D5—42no—
404C2×D5—42no—
405C2×D5—42no—
406C2×D5C2×#21712yes—
407C2×D5C22×#6712yes—
408AGL(1,𝔽5)—128no—
409C4×C5—22no—
410C4×C5C2×#21712yes—
411C4×C5C2×#21712yes—
412Dic5—22no—
413Dic5C2×#21714yes—
414C24—51no—
415C24—51no—
416C24C2×ΓL(1,𝔽16)11yescharacteristic
417#41—108no—
418#41—108no—
419#41—108no—
420#41—108no—
421C2×Q8—54no—
422C2×Q8—102no—
423C2×Q8—102no—
424C2×D4—52no—
425C2×D4—52no—
426C2×D4—54no—
427C2×D4—52no—
428C2×D4—52no—
429C2×D4—102no—
430C2×D4—102no—
431C2×D4—102no—
432C2×D4—102no—
433C2×D4—102no—
434C2×D4—102no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—51no—
439C22×C4—52no—
440C22×C4—54no—
441C22×C4—52no—
442C22×C4—52no—
443C22×C4—52no—
444C22×C4—51no—
445C22×C4—54no—
446C22×C4—102no—
447C22×C4—101no—
448C22×C4—101no—
449C22×C4—304no—
450C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
451C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
452C22×C4C2×ΓL(1,𝔽16)14yescomplemented
453C4⋊C4—308no—
454C22⋊C4—158no—
455C22⋊C4—304no—
456C42—308no—
457C3×C5C2×#41011yescharacteristic, complemented, Hall
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—52no—
464C22×C3—52no—
465C22×C3—104no—
466C22×C3—102no—
467C22×C3—202no—
468C22×C3—201no—
469C22×C3—201no—
470C22×C3—202no—
471C22×C3—202no—
472C22×C3—202no—
473C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
474C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
475C22×C3C23×AGL(1,𝔽5)11yescharacteristic
476C3×C4—22no—
477C3×C4—54no—
478C3×C4—102no—
479C3×C4C2×C22⋊AGL(1,𝔽5)12yes—
480C3×C4C2×C22⋊AGL(1,𝔽5)12yes—
481Dic3—208no—
482C2×C5—21no—
483C2×C5—21no—
484C2×C5—42no—
485C2×C5#524012yescomplemented
486C2×C5C22×#21711yescharacteristic
487D5—22no—
488D5—22no—
489D5—42no—
490C23—21no—
491C23—21no—
492C23—42no—
493C23—52no—
494C23—52no—
495C23—51no—
496C23—51no—
497C23—52no—
498C23—101no—
499C23—101no—
500C23—202no—
501C23—202no—
502C23C22×ΓL(1,𝔽16)11yescharacteristic
503C23C22×ΓL(1,𝔽16)12yes—
504Q8—108no—
505D4—104no—
506D4—104no—
507D4—108no—
508D4—104no—
509D4—104no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—52no—
515C2×C4—52no—
516C2×C4—52no—
517C2×C4—54no—
518C2×C4—54no—
519C2×C4—54no—
520C2×C4—54no—
521C2×C4—52no—
522C2×C4—101no—
523C2×C4—101no—
524C2×C4—104no—
525C2×C4—104no—
526C2×C4—308no—
527C2×C4—608no—
528C2×C4#676311yescharacteristic
529C2×C4#676312yes—
530C2×C4#676311yescharacteristic
531C2×C4C22×ΓL(1,𝔽16)12yes—
532C2×C4C22×ΓL(1,𝔽16)12yes—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—42no—
536C2×C3—102no—
537C2×C3—102no—
538C2×C3—202no—
539C2×C3#6574512yescomplemented
540C2×C3C22×C22⋊AGL(1,𝔽5)11yescharacteristic
541C5C2×#524011yescharacteristic, complemented, Sylow
542C22—22no—
543C22—22no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—52no—
548C22—52no—
549C22—104no—
550C22—102no—
551C22—202no—
552C22—201no—
553C22—201no—
554C22—202no—
555C22—202no—
556C22—202no—
557C22C2×#676311yescharacteristic
558C22C2×#676311yescharacteristic
559C22C23×ΓL(1,𝔽16)11yescharacteristic
560C4—22no—
561C4—54no—
562C4—102no—
563C4—608no—
564C4C2×#676312yes—
565C4C2×#676312yes—
566C3C2×#6574511yescharacteristic, complemented, Sylow
567C2—21no—
568C2—21no—
569C2—42no—
570C2—102no—
571C2—102no—
572C2—202no—
573C2#1175240412yesdirect product factor
574C2C22×#676311yescharacteristic
subgroupG/H
centerC2×C4#6763
commutatorC22×C3×C5C23×C4
FrattiniC22C23×ΓL(1,𝔽16)
FittingC23×C3×C4×C5C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×#709C2
cosocleC22×C3×D5C24
hypercenterC23×C4ΓL(1,𝔽16)
hypocenterC3×C5C2×#410
LCSG ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C2×#91989C2semidirect
3. C2×#91240C2semidirect
4. #11752404C2direct (N × H)
5. C2×#91228C2semidirect
7. C23×C3×C4×C5C4semidirect
8. C22×C3×C22⋊D5C4semidirect
13. C2×C3×#709C4semidirect
16. C22×C3×C4×D5C4semidirect
18. C22×C3×Dic10C4semidirect
21. #91989C22semidirect
24. #91240C22semidirect
25. #91228C22semidirect
41. C22×C3×C4×C5C2×C4semidirect
47. C2×C3×C22⋊D5C2×C4semidirect
63. C2×C3×C4×D5C2×C4semidirect
70. C2×C3×Dic10C2×C4semidirect
80. C23×C4×C5Dic3semidirect
81. C22×C22⋊D5Dic3semidirect
86. C2×#709Dic3semidirect
89. C22×C4×D5Dic3semidirect
91. C22×Dic10Dic3semidirect
154. C23×C3×C4AGL(1,𝔽5)semidirect
171. C22×C4×C5C2×Dic3semidirect
177. C2×C22⋊D5C2×Dic3semidirect
193. C2×C4×D5C2×Dic3semidirect
200. C2×Dic10C2×Dic3semidirect
271. C22×C3×C4C2×AGL(1,𝔽5)semidirect
331. C23×C4ΓL(1,𝔽16)semidirect
342. C2×C3×C5#410semidirect
452. C22×C4C2×ΓL(1,𝔽16)semidirect
457. C3×C5C2×#410semidirect
485. C2×C5#5240semidirect
539. C2×C3#65745semidirect
541. C5C2×#5240semidirect
566. C3C2×#65745semidirect
573. C2#11752404direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7092
C2×#919892
C2×#912402
#117524042
C2×#912282
C2×#657453
C2×#52405