Finite Group Databasebeta

← Q8×(C2×D3)⋊AGL(1,𝔽5) #14326161

Subgroups — Q8×(C2×D3)⋊AGL(1,𝔽5) #14326161

Order 1920.

1640 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×Q8×D5C211yesmaximal, characteristic
2C2×Q8×ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
3C2×C3×Q8×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4#11751855C213yesmaximal
5C4×(C2×D3)⋊AGL(1,𝔽5)C213yesmaximal
6#11751752C213yesmaximal, complemented
7#11751749C213yesmaximal, complemented
8Q8×C22⋊AGL(1,𝔽5)—31nomaximal, Hall
9C2×Q8×D15C411yescharacteristic, complemented
10C2×C5×D3×Q8C411yescharacteristic, complemented
11C2×C3×Q8×D5C2211yescharacteristic
12D3×Q8×D5—24no—
13D3×Q8×D5—24no—
14C2×C4×D3×D5C2213yes—
15C2×D15⋊Q8C413yescomplemented
16C2×D3×Dic10C413yescomplemented
17C2×D5×Dic6C2213yes—
18Q8×ΓL(1,𝔽16)—24no—
19C2×C4⋊ΓL(1,𝔽16)C2213yes—
20C2×C4×ΓL(1,𝔽16)C2213yes—
21C3×Q8×AGL(1,𝔽5)—24no—
22C2×C3×C4⋊AGL(1,𝔽5)C2213yes—
23C2×C3×C4×AGL(1,𝔽5)C2213yes—
24C2×(C2×D3)⋊AGL(1,𝔽5)C2213yes—
25C2×(C2×D3)⋊AGL(1,𝔽5)C2211yescharacteristic
26(C2×AGL(1,𝔽5))⋊Dic3C2213yes—
27(C2×AGL(1,𝔽5))⋊Dic3C2216yes—
28(C2×AGL(1,𝔽5))⋊Dic3C2213yes—
29Q8×(C2×D3)⋊C4—51nomaximal, Hall
30C22×Q8×D5—31no—
31C2×Q8×AGL(1,𝔽5)—31no—
32C2×Q8×AGL(1,𝔽5)D311yescharacteristic, complemented
33(C2×AGL(1,𝔽5))⋊Q8—33no—
34(C2×AGL(1,𝔽5))⋊Q8—33no—
35#65765—33no—
36C4×C22⋊AGL(1,𝔽5)—33no—
37C22×D3×D5C2311yescharacteristic
38C22×ΓL(1,𝔽16)C2313yes—
39C22×ΓL(1,𝔽16)C2311yescharacteristic
40C22×C3×AGL(1,𝔽5)C2313yes—
41C22×C3×AGL(1,𝔽5)C2311yescharacteristic
42C2×C3×C5×Q8C2×C411yescharacteristic
43Q8×D15—24no—
44C2×C4×D15C2×C413yes—
45C2×Dic30C2×C413yes—
46C5×D3×Q8—24no—
47C2×C4×C5×D3C2×C413yes—
48C2×C5×Dic6C2×C413yes—
49C3×Q8×D5D414yes—
50C3×Q8×D5D414yes—
51C2×C3×C4×D5C2313yes—
52C2×C3×Dic10C2×C413yes—
53C2×#1204C2×C416yes—
54C2×#1200C2×C413yes—
55C2×D3×Dic5C2×C413yes—
56C2×D5×Dic3C2313yes—
57C4×D3×D5—26no—
58C4×D3×D5—26no—
59D15⋊Q8—212no—
60D3×Dic10—212no—
61D5×Dic6—26no—
62D5×Dic6—26no—
63C4⋊ΓL(1,𝔽16)—26no—
64C4×ΓL(1,𝔽16)—26no—
65C3×C4⋊AGL(1,𝔽5)—26no—
66C3×C4×AGL(1,𝔽5)—26no—
67(C2×D3)⋊AGL(1,𝔽5)—26no—
68(C2×D3)⋊AGL(1,𝔽5)Q814yesdirect product factor
69C22×D3×Q8—51no—
70C2×C3×C4×Q8—51no—
71C2×Q8×Dic3—51no—
72#4987—53no—
73#4973—53no—
74C4×(C2×D3)⋊C4—53no—
75#4929—53no—
76C22×C5×Q8—31no—
77C2×Q8×D5—31no—
78C2×Q8×D5—33no—
79C2×Q8×D5—33no—
80C2×Q8×D5—64no—
81C2×Q8×D5—64no—
82C2×Q8×D5C2×D311yescharacteristic
83C22×C4×D5—33no—
84C22×Dic10—33no—
85C2×C22⋊AGL(1,𝔽5)—31no—
86C2×C22⋊AGL(1,𝔽5)—33no—
87Q8×AGL(1,𝔽5)—24no—
88Q8×AGL(1,𝔽5)—64no—
89C2×C4⋊AGL(1,𝔽5)—33no—
90C2×C4⋊AGL(1,𝔽5)C2×D313yes—
91C2×C4×AGL(1,𝔽5)—33no—
92C2×C4×AGL(1,𝔽5)C2×D313yes—
93(C2×AGL(1,𝔽5))⋊C4—33no—
94(C2×AGL(1,𝔽5))⋊C4—33no—
95(C2×AGL(1,𝔽5))⋊C4—36no—
96Q8×C22⋊C4—151noSylow
97C22×D15C22×C411yescharacteristic
98C22×C5×D3C22×C411yescharacteristic
99C22×C3×D5C2411yescharacteristic
100C2×D3×D5—21no—
101C2×D3×D5—21no—
102C2×D3×D5—21no—
103C2×D3×D5—21no—
104C2×D3×D5C2×Q812yes—
105C2×D3×D5#4112yes—
106C2×ΓL(1,𝔽16)—21no—
107C2×ΓL(1,𝔽16)—21no—
108C2×ΓL(1,𝔽16)—23no—
109C2×ΓL(1,𝔽16)—23no—
110C2×ΓL(1,𝔽16)—23no—
111C2×ΓL(1,𝔽16)C2×Q812yescomplemented
112C2×C3×AGL(1,𝔽5)—21no—
113C2×C3×AGL(1,𝔽5)—21no—
114C2×C3×AGL(1,𝔽5)—23no—
115C2×C3×AGL(1,𝔽5)—23no—
116C2×C3×AGL(1,𝔽5)—23no—
117C2×C3×AGL(1,𝔽5)C2×Q812yescomplemented
118C3×C5×Q8C22⋊C414yescomplemented
119C2×C3×C4×C5C22×C413yes—
120C2×Dic15C22×C413yes—
121C4×D15—26no—
122Dic30—26no—
123C2×C5×Dic3C22×C413yes—
124C4×C5×D3—26no—
125C5×Dic6—26no—
126C2×C3×Dic5C22×C413yes—
127C3×C4×D5C2×D416yes—
128C3×C4×D5C2×D416yes—
129C3×Dic10C22⋊C4112yescomplemented
130#1204—212no—
131#1200—26no—
132D3×Dic5—26no—
133D5×Dic3—23no—
134D5×Dic3—23no—
135C22×C3×Q8—51no—
136C2×D3×Q8—51no—
137C2×D3×Q8—53no—
138C2×D3×Q8—53no—
139C2×D3×Q8—104no—
140C2×D3×Q8—104no—
141C2×D3×Q8AGL(1,𝔽5)11yescharacteristic, complemented
142C22×C4×D3—53no—
143C22×Dic6—53no—
144C3×C4×Q8—104no—
145C2×C3×C4⋊C4—53no—
146C2×C3×C42—53no—
147Q8×Dic3—104no—
148C2×(C2×D3)⋊C4—51no—
149C2×(C2×D3)⋊C4—53no—
150C2×C4⋊Dic3—53no—
151C2×C4×Dic3—53no—
152C3⋊#95—53no—
153C3⋊#95—56no—
154C3⋊#95—53no—
155C23×D5—31no—
156C22×AGL(1,𝔽5)—31no—
157C22×AGL(1,𝔽5)—33no—
158C22×AGL(1,𝔽5)C22×D313yes—
159C22×AGL(1,𝔽5)C22×D311yescharacteristic
160C2×C5×Q8—33no—
161C2×C5×Q8—64no—
162C2×C5×Q8C4×D311yescharacteristic
163C22×C4×C5—33no—
164C22×Dic5—33no—
165Q8×D5—64no—
166Q8×D5—66no—
167Q8×D5—66no—
168Q8×D5—612no—
169Q8×D5D1214yes—
170Q8×D5#6714yes—
171C2×C4×D5—33no—
172C2×C4×D5—33no—
173C2×C4×D5—33no—
174C2×C4×D5—66no—
175C2×C4×D5—66no—
176C2×C4×D5C22×D313yes—
177C2×Dic10—33no—
178C2×Dic10—36no—
179C2×Dic10—612no—
180C2×Dic10C4×D313yes—
181C22⋊AGL(1,𝔽5)—34no—
182C22⋊AGL(1,𝔽5)—66no—
183C4⋊AGL(1,𝔽5)—26no—
184C4⋊AGL(1,𝔽5)—66no—
185C4×AGL(1,𝔽5)—26no—
186C4×AGL(1,𝔽5)—66no—
187C23×Q8—151no—
188C2×C4×Q8—51no—
189C2×C4×Q8—151no—
190#391—153no—
191#391—153no—
192#380—153no—
193C4×C22⋊C4—153no—
194C22×C3×C5C23×C411yescharacteristic
195C2×D15—21no—
196C2×D15—21no—
197C2×D15C4×Q812yescomplemented
198C2×C5×D3—21no—
199C2×C5×D3—21no—
200C2×C5×D3C4×Q812yescomplemented
201C2×C3×D5C22×D411yescharacteristic
202C2×C3×D5C22×D411yescharacteristic
203C2×C3×D5C22×Q811yescharacteristic
204C2×C3×D5C2×#4111yescharacteristic
205C2×C3×D52−1+411yescharacteristic
206C2×C3×D52−1+411yescharacteristic
207D3×D5—22no—
208D3×D5—22no—
209D3×D5—22no—
210D3×D5—22no—
211ΓL(1,𝔽16)—22no—
212ΓL(1,𝔽16)—43no—
213C3×AGL(1,𝔽5)—22no—
214C3×AGL(1,𝔽5)—43no—
215C3×C4×C5C2×C22⋊C416yes—
216Dic15—23no—
217C3×Dic5C2×C22⋊C416yes—
218C5×Dic3—23no—
219C23×D3—51no—
220C2×C3×Q8—54no—
221C2×C3×Q8—54no—
222C2×C3×Q8—53no—
223C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
224C22×C3×C4—53no—
225C22×C3×C4—51no—
226C22×C3×C4—53no—
227C22×Dic3—51no—
228C22×Dic3—53no—
229C22×Dic3—53no—
230D3×Q8—24no—
231D3×Q8—1012no—
232D3×Q8—1012no—
233D3×Q8—104no—
234C2×C4×D3—53no—
235C2×C4×D3—53no—
236C2×C4×D3—53no—
237C2×C4×D3—106no—
238C2×C4×D3—106no—
239C2×C4×D3C2×AGL(1,𝔽5)13yes—
240C2×Dic6—53no—
241C2×Dic6—56no—
242C2×Dic6—106no—
243C2×Dic6—106no—
244C2×Dic6C2×AGL(1,𝔽5)13yes—
245C3×C4⋊C4—106no—
246C3×C42—106no—
247(C2×D3)⋊C4—54no—
248(C2×D3)⋊C4—106no—
249C4⋊Dic3—106no—
250C4×Dic3—106no—
251C23×C5—31no—
252C22×D5—31no—
253C22×D5—32no—
254C22×D5—32no—
255C22×D5—61no—
256C22×D5—61no—
257C22×D5—61no—
258C22×D5—61no—
259C22×D5C23×D311yescharacteristic
260C2×AGL(1,𝔽5)—21no—
261C2×AGL(1,𝔽5)—21no—
262C2×AGL(1,𝔽5)—23no—
263C2×AGL(1,𝔽5)—23no—
264C2×AGL(1,𝔽5)—23no—
265C2×AGL(1,𝔽5)—32no—
266C2×AGL(1,𝔽5)—61no—
267C2×AGL(1,𝔽5)—61no—
268C2×AGL(1,𝔽5)—63no—
269C2×AGL(1,𝔽5)—63no—
270C2×AGL(1,𝔽5)—63no—
271C2×AGL(1,𝔽5)D3×Q812yescomplemented
272C5×Q8—66no—
273C5×Q8(C2×D3)⋊C414yescomplemented
274C2×C4×C5—33no—
275C2×C4×C5—66no—
276C2×C4×C5C2×C4×D313yes—
277C2×Dic5—33no—
278C2×Dic5—66no—
279C2×Dic5C2×C4×D313yes—
280C4×D5—63no—
281C4×D5—63no—
282C4×D5—66no—
283C4×D5—66no—
284C4×D5C2×D1216yes—
285C4×D5C2×#6716yes—
286Dic10—66no—
287Dic10—612no—
288Dic10(C2×D3)⋊C4112yescomplemented
289C22×Q8—31no—
290C22×Q8—51no—
291C22×Q8—153no—
292C22×Q8—153no—
293C22×Q8—153no—
294C22×Q8—151no—
295C22×Q8—304no—
296C22×Q8—304no—
297C23×C4—153no—
298C4×Q8—104no—
299C4×Q8—304no—
300C2×C4⋊C4—53no—
301C2×C4⋊C4—153no—
302C2×C22⋊C4—151no—
303C2×C22⋊C4—153no—
304C2×C42—53no—
305C2×C42—153no—
306#95—153no—
307#95—153no—
308#95—156no—
309C2×C3×C5C22×C22⋊C411yescharacteristic
310C2×C3×C5C2×C4×Q811yescharacteristic
311C2×C3×C5#51911yescharacteristic
312D15—22no—
313C3×D5#54811yescharacteristic
314C3×D5#54811yescharacteristic
315C3×D5D4×Q811yescharacteristic
316C3×D5D4×Q811yescharacteristic
317C5×D3—22no—
318C23×C3—51no—
319C22×D3—51no—
320C22×D3—52no—
321C22×D3—52no—
322C22×D3—101no—
323C22×D3—101no—
324C22×D3—101no—
325C22×D3—101no—
326C22×D3C22×AGL(1,𝔽5)11yescharacteristic
327C3×Q8—512no—
328C3×Q8C22⋊AGL(1,𝔽5)14yescomplemented
329C2×C3×C4—53no—
330C2×C3×C4—56no—
331C2×C3×C4—56no—
332C2×C3×C4—52no—
333C2×C3×C4—101no—
334C2×C3×C4—101no—
335C2×C3×C4—103no—
336C2×C3×C4—103no—
337C2×C3×C4—103no—
338C2×C3×C4C22×AGL(1,𝔽5)13yes—
339C2×Dic3—52no—
340C2×Dic3—53no—
341C2×Dic3—103no—
342C2×Dic3—103no—
343C2×Dic3—101no—
344C2×Dic3—101no—
345C2×Dic3—103no—
346C2×Dic3—103no—
347C2×Dic3—103no—
348C2×Dic3C22×AGL(1,𝔽5)13yes—
349C4×D3—26no—
350C4×D3—106no—
351C4×D3—106no—
352C4×D3—106no—
353Dic6—26no—
354Dic6—1012no—
355Dic6—106no—
356C22×C5—32no—
357C22×C5—61no—
358C22×C5—61no—
359C22×C5C22×C4×D311yescharacteristic
360C2×D5—32no—
361C2×D5—61no—
362C2×D5—61no—
363C2×D5—62no—
364C2×D5—62no—
365C2×D5—62no—
366C2×D5—62no—
367C2×D5C22×D1211yescharacteristic
368C2×D5C2×D3×Q811yescharacteristic
369C2×D5#100711yescharacteristic
370C2×D5D3×#4111yescharacteristic
371C2×D5#101011yescharacteristic
372C2×D5C22×#6711yescharacteristic
373AGL(1,𝔽5)—22no—
374AGL(1,𝔽5)—43no—
375AGL(1,𝔽5)—62no—
376AGL(1,𝔽5)—123no—
377C4×C5—63no—
378C4×C5C2×(C2×D3)⋊C416yes—
379Dic5—63no—
380Dic5C2×(C2×D3)⋊C416yes—
381C24—151no—
382C2×Q8—33no—
383C2×Q8—54no—
384C2×Q8—54no—
385C2×Q8—53no—
386C2×Q8—64no—
387C2×Q8—156no—
388C2×Q8—153no—
389C2×Q8—306no—
390C2×Q8—306no—
391C2×Q8—3012no—
392C2×Q8—3012no—
393C2×Q8—304no—
394C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
395C22×C4—33no—
396C22×C4—53no—
397C22×C4—51no—
398C22×C4—53no—
399C22×C4—153no—
400C22×C4—153no—
401C22×C4—153no—
402C22×C4—151no—
403C22×C4—153no—
404C22×C4—153no—
405C22×C4—306no—
406C22×C4—306no—
407C4⋊C4—106no—
408C4⋊C4—306no—
409C22⋊C4—154no—
410C22⋊C4—306no—
411C42—106no—
412C42—306no—
413C3×C5Q8×C22⋊C411yescharacteristic, complemented, Hall
414C22×C3—51no—
415C22×C3—51no—
416C22×C3—51no—
417C22×C3—51no—
418C22×C3—51no—
419C22×C3—51no—
420C22×C3C23×AGL(1,𝔽5)11yescharacteristic
421C2×D3—21no—
422C2×D3—21no—
423C2×D3—52no—
424C2×D3—102no—
425C2×D3—102no—
426C2×D3—102no—
427C2×D3—102no—
428C2×D3—101no—
429C2×D3—101no—
430C2×D3Q8×AGL(1,𝔽5)12yescomplemented
431C3×C4—56no—
432C3×C4—102no—
433C3×C4—203no—
434C3×C4C2×C22⋊AGL(1,𝔽5)16yes—
435Dic3—23no—
436Dic3—102no—
437Dic3—103no—
438Dic3—203no—
439C2×C5—62no—
440C2×C5C4×D3×Q811yescharacteristic
441C2×C5#557211yescharacteristic
442C2×C5C22×(C2×D3)⋊C411yescharacteristic
443D5—62no—
444D5Q8×D1211yescharacteristic
445D5#557611yescharacteristic
446D5Q8×#6711yescharacteristic
447D5#583111yescharacteristic
448C23—31no—
449C23—51no—
450C23—152no—
451C23—152no—
452C23—151no—
453C23—301no—
454C23—301no—
455C23—301no—
456C23—301no—
457Q8—512no—
458Q8—66no—
459Q8—3012no—
460Q8—306no—
461Q8(C2×D3)⋊AGL(1,𝔽5)14yesdirect product factor
462C2×C4—33no—
463C2×C4—53no—
464C2×C4—56no—
465C2×C4—56no—
466C2×C4—52no—
467C2×C4—66no—
468C2×C4—101no—
469C2×C4—101no—
470C2×C4—103no—
471C2×C4—103no—
472C2×C4—103no—
473C2×C4—152no—
474C2×C4—153no—
475C2×C4—306no—
476C2×C4—306no—
477C2×C4—303no—
478C2×C4—303no—
479C2×C4—301no—
480C2×C4—301no—
481C2×C4—306no—
482C2×C4—303no—
483C2×C4—303no—
484C2×C4—303no—
485C2×C4C2×D3×AGL(1,𝔽5)13yes—
486C2×C3—51no—
487C2×C3—51no—
488C2×C3—51no—
489C2×C3—51no—
490C2×C3C2×Q8×AGL(1,𝔽5)11yescharacteristic
491C2×C3#6626211yescharacteristic
492C2×C3C22×C22⋊AGL(1,𝔽5)11yescharacteristic
493D3—22no—
494D3—102no—
495C5Q8×(C2×D3)⋊C411yescharacteristic, complemented, Sylow
496C22—32no—
497C22—51no—
498C22—51no—
499C22—51no—
500C22—51no—
501C22—51no—
502C22—51no—
503C22—61no—
504C22—61no—
505C22—152no—
506C22—302no—
507C22—302no—
508C22—302no—
509C22—302no—
510C22—301no—
511C22—301no—
512C22C22×D3×AGL(1,𝔽5)11yescharacteristic
513C4—56no—
514C4—63no—
515C4—102no—
516C4—203no—
517C4—302no—
518C4—303no—
519C4—603no—
520C4C2×(C2×D3)⋊AGL(1,𝔽5)16yes—
521C3Q8×C22⋊AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
522C2—51no—
523C2—51no—
524C2—51no—
525C2—51no—
526C2—62no—
527C2—302no—
528C2#1175684711yescharacteristic
529C2D3×Q8×AGL(1,𝔽5)11yescharacteristic
530C2C22×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC2×C3×C5×Q8C2×C4
prime radicalC22×D3×D5C23
cosocleC22×C3×D5C24
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5Q8×C22⋊C4
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×Q8×ΓL(1,𝔽16)C2semidirect
3. C2×C3×Q8×AGL(1,𝔽5)C2semidirect
6. #11751752C2semidirect
7. #11751749C2semidirect
9. C2×Q8×D15C4semidirect
10. C2×C5×D3×Q8C4semidirect
15. C2×D15⋊Q8C4semidirect
16. C2×D3×Dic10C4semidirect
32. C2×Q8×AGL(1,𝔽5)D3semidirect
68. (C2×D3)⋊AGL(1,𝔽5)Q8direct (N × H)
111. C2×ΓL(1,𝔽16)C2×Q8semidirect
117. C2×C3×AGL(1,𝔽5)C2×Q8semidirect
118. C3×C5×Q8C22⋊C4semidirect
129. C3×Dic10C22⋊C4semidirect
141. C2×D3×Q8AGL(1,𝔽5)semidirect
197. C2×D15C4×Q8semidirect
200. C2×C5×D3C4×Q8semidirect
271. C2×AGL(1,𝔽5)D3×Q8semidirect
273. C5×Q8(C2×D3)⋊C4semidirect
288. Dic10(C2×D3)⋊C4semidirect
328. C3×Q8C22⋊AGL(1,𝔽5)semidirect
413. C3×C5Q8×C22⋊C4semidirect
430. C2×D3Q8×AGL(1,𝔽5)semidirect
461. Q8(C2×D3)⋊AGL(1,𝔽5)direct (N × H)
495. C5Q8×(C2×D3)⋊C4semidirect
521. C3Q8×C22⋊AGL(1,𝔽5)semidirect

Maximal subgroups

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

subgroupindex
C2×D3×Q8×D52
C2×Q8×ΓL(1,𝔽16)2
C2×C3×Q8×AGL(1,𝔽5)2
#117518552
C4×(C2×D3)⋊AGL(1,𝔽5)2
#117517522
#117517492
Q8×C22⋊AGL(1,𝔽5)3
Q8×(C2×D3)⋊C45