Finite Group Databasebeta

← C2×#22589541 #32024922

Subgroups — C2×#22589541 #32024922

Order 18176.

1662 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556104C211yesmaximal, characteristic
2C22×#14556061C211yesmaximal, characteristic, complemented
3C22×#14556052C211yesmaximal, characteristic, complemented
4C22×(C2×D71)⋊C8C211yesmaximal, characteristic
5C2×#17985856C211yesmaximal, characteristic
6#22589541C218yesmaximal, direct product factor
7C2×#17985762C211yesmaximal, characteristic, complemented
8C22×#103×C71C211yesmaximal, characteristic, complemented
9C23×C4×D71C2211yescharacteristic
10C23×C71⋊C8C2211yescharacteristic
11C2×#14556104—24no—
12C2×#14556104—24no—
13C2×#14556104—22no—
14C2×#14556104C2214yes—
15C22×C4⋊Dic71C2211yescharacteristic
16C22×C4⋊Dic71C2211yescharacteristic
17C22×Dic71⋊C4—21no—
18C22×(C2×D71)⋊C4—21no—
19C22×(C2×D71)⋊C4—21no—
20C2×#14556061—24no—
21C2×#14556061C2214yescomplemented
22C2×#14556052—24no—
23C2×#14556052C2214yescomplemented
24C2×(C2×D71)⋊C8—24no—
25C2×(C2×D71)⋊C8C2214yes—
26#17985856—22no—
27#17985856C2214yes—
28C2×C71⋊#95—21no—
29#17985762C2214yescomplemented
30C23×Q8×C71C2211yescharacteristic
31C23×C8×C71C2211yescharacteristic
32C22×C4⋊C4×C71C2211yescharacteristic
33C2×#103×C71—24no—
34C2×#103×C71C2214yescomplemented
35C24×D71D411yescharacteristic
36C23×Dic71—21no—
37C23×Dic71—21no—
38C23×Dic71D411yescharacteristic
39C22×C4×D71—22no—
40C22×C4×D71—22no—
41C22×C4×D71—22no—
42C22×C4×D71—22no—
43C22×C4×D71—22no—
44C22×C4×D71—22no—
45C22×C4×D71D412yes—
46C22×C4×D71C2312yes—
47C22×C71⋊C8—22no—
48C22×C71⋊C8—22no—
49C22×C71⋊C8D414yes—
50C22×C71⋊C8C2312yes—
51#14556104—216no—
52#14556104—28no—
53#14556104—28no—
54C2×C4⋊Dic71—21no—
55C2×C4⋊Dic71—22no—
56C2×C4⋊Dic71—22no—
57C2×C4⋊Dic71—21no—
58C2×C4⋊Dic71—22no—
59C2×C4⋊Dic71D414yes—
60C2×C4⋊Dic71C2312yes—
61C2×C4⋊Dic71C2312yes—
62C2×Dic71⋊C4—28no—
63C2×Dic71⋊C4—24no—
64C2×(C2×D71)⋊C4—28no—
65C2×(C2×D71)⋊C4—24no—
66C2×(C2×D71)⋊C4—24no—
67C2×(C2×D71)⋊C4—44no—
68#14556061—28no—
69#14556052—28no—
70(C2×D71)⋊C8—28no—
71C71⋊#95—24no—
72C22×Q8×C71—22no—
73C22×Q8×C71—21no—
74C22×Q8×C71—22no—
75C22×Q8×C71—22no—
76C22×Q8×C71—22no—
77C22×Q8×C71—22no—
78C22×Q8×C71D414yes—
79C22×Q8×C71C2312yes—
80C23×C4×C71—21no—
81C23×C4×C71—21no—
82C23×C4×C71C2311yescharacteristic
83C22×C8×C71—22no—
84C22×C8×C71—22no—
85C22×C8×C71—22no—
86C22×C8×C71C2312yes—
87C2×C4⋊C4×C71—22no—
88C2×C4⋊C4×C71—22no—
89C2×C4⋊C4×C71—21no—
90C2×C4⋊C4×C71C2312yes—
91#103×C71—28no—
92C23×D71—22no—
93C23×D71—22no—
94C23×D71—22no—
95C23×D71—22no—
96C23×D71—22no—
97C23×D71—22no—
98C23×D71Q1614yes—
99C23×D71C2×D412yes—
100C22×Dic71—22no—
101C22×Dic71—24no—
102C22×Dic71—22no—
103C22×Dic71—22no—
104C22×Dic71—22no—
105C22×Dic71—22no—
106C22×Dic71—42no—
107C22×Dic71—42no—
108C22×Dic71—42no—
109C22×Dic71—42no—
110C22×Dic71—42no—
111C22×Dic71C2×D412yes—
112C2×C4×D71—24no—
113C2×C4×D71—24no—
114C2×C4×D71—24no—
115C2×C4×D71—22no—
116C2×C4×D71—22no—
117C2×C4×D71—22no—
118C2×C4×D71—22no—
119C2×C4×D71—22no—
120C2×C4×D71—22no—
121C2×C4×D71—44no—
122C2×C4×D71—44no—
123C2×C4×D71—44no—
124C2×C4×D71—44no—
125C2×C71⋊C8—28no—
126C2×C71⋊C8—22no—
127C2×C71⋊C8—22no—
128C2×C71⋊C8C2×D414yes—
129C4⋊Dic71—24no—
130C4⋊Dic71—22no—
131C4⋊Dic71—22no—
132C4⋊Dic71—22no—
133C4⋊Dic71—42no—
134C4⋊Dic71C2×D414yes—
135Dic71⋊C4—216no—
136(C2×D71)⋊C4—216no—
137(C2×D71)⋊C4—48no—
138C24×C71C2×D411yescharacteristic
139C2×Q8×C71—24no—
140C2×Q8×C71—22no—
141C2×Q8×C71—28no—
142C2×Q8×C71—28no—
143C2×Q8×C71—22no—
144C2×Q8×C71—22no—
145C2×Q8×C71—22no—
146C2×Q8×C71—22no—
147C2×Q8×C71—42no—
148C2×Q8×C71—44no—
149C2×Q8×C71—42no—
150C2×Q8×C71—44no—
151C2×Q8×C71C2×D414yes—
152C22×C4×C71—24no—
153C22×C4×C71—22no—
154C22×C4×C71—22no—
155C22×C4×C71—42no—
156C22×C4×C71—42no—
157C22×C4×C71—42no—
158C22×C4×C71—42no—
159C22×C4×C71—42no—
160C22×C4×C71C2×D411yescharacteristic
161C22×C4×C71C2×D412yes—
162C22×C4×C71C2×D412yes—
163C22×C4×C71C2×D412yes—
164C22×C4×C71#4112yes—
165C22×C4×C71#4112yes—
166C22×C4×C71#4112yes—
167C22×C4×C71C2411yescharacteristic
168C2×C8×C71—22no—
169C2×C8×C71—22no—
170C2×C8×C71—22no—
171C2×C8×C71—44no—
172C4⋊C4×C71—22no—
173C4⋊C4×C71—22no—
174C4⋊C4×C71—42no—
175C23×C71#12012yes—
176C23×C71#12112yes—
177C23×C71#12312yes—
178C23×C71C2×Q1612yes—
179C23×C71#13512yes—
180C23×C71AGL(1,ℤ/8)12yes—
181C23×C71#13712yes—
182C23×C71C22×D411yescharacteristic
183C22×D71—22no—
184C22×D71—22no—
185C22×D71—22no—
186C22×D71—22no—
187C22×D71—22no—
188C22×D71—22no—
189C22×D71—28no—
190C22×D71—28no—
191C22×D71—28no—
192C22×D71—44no—
193C22×D71—44no—
194C22×D71—44no—
195C22×D71—44no—
196C22×D71C2×Q1614yes—
197Q8×C71—24no—
198Q8×C71—24no—
199Q8×C71—42no—
200Q8×C71—42no—
201Q8×C71—48no—
202C2×C4×C71—24no—
203C2×C4×C71—42no—
204C2×C4×C71—42no—
205C2×C4×C71—48no—
206C2×C4×C71—42no—
207C2×C4×C71—42no—
208C2×C4×C71—42no—
209C2×C4×C71—84no—
210C2×C4×C71#12012yes—
211C2×C4×C71#12114yes—
212C2×C4×C71#12114yes—
213C2×C4×C71#12112yes—
214C2×C4×C71#12312yes—
215C2×C4×C71#12314yes—
216C2×C4×C71#12414yes—
217C2×C4×C71C22×D411yescharacteristic
218C2×C4×C71C22×D411yescharacteristic
219C2×C4×C71C22×D411yescharacteristic
220C2×C4×C71C2×#4111yescharacteristic
221C2×C4×C71C2×#4111yescharacteristic
222C2×C4×C71C2×#4111yescharacteristic
223C2×Dic71—24no—
224C2×Dic71—22no—
225C2×Dic71—22no—
226C2×Dic71—22no—
227C2×Dic71—42no—
228C2×Dic71—42no—
229C2×Dic71—44no—
230C2×Dic71—48no—
231C2×Dic71—42no—
232C2×Dic71—42no—
233C2×Dic71—42no—
234C2×Dic71—84no—
235C4×D71—42no—
236C4×D71—42no—
237C4×D71—42no—
238C4×D71—42no—
239C4×D71—48no—
240C8×C71—42no—
241C71⋊C8—24no—
242C22×C71#39414yes—
243C22×C71#44914yes—
244C22×C71#45114yes—
245C22×C71#46714yes—
246C22×C71#46814yes—
247C22×C71#48414yes—
248C22×C71#48514yes—
249C22×C71C2×#12011yescharacteristic
250C22×C71C2×#12111yescharacteristic
251C22×C71C2×#12311yescharacteristic
252C22×C71C22×Q1611yescharacteristic
253C22×C71C2×#13511yescharacteristic
254C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C71C2×#13711yescharacteristic
256C2×D71—24no—
257C2×D71—24no—
258C2×D71—24no—
259C2×D71—42no—
260C2×D71—42no—
261C2×D71—42no—
262C2×D71—42no—
263C2×D71—416no—
264C4×C71—44no—
265C4×C71—82no—
266C4×C71#39414yes—
267C4×C71C2×#12111yescharacteristic
268C4×C71C2×#12111yescharacteristic
269C4×C71C2×#12311yescharacteristic
270C4×C71C2×#12411yescharacteristic
271Dic71—42no—
272Dic71—44no—
273Dic71—82no—
274C2×#2036—711nomaximal, Sylow
275C2×C71#203618yescomplemented
276C2×C71C2×#39411yescharacteristic
277C2×C71C2×#45111yescharacteristic
278C2×C71C2×#44911yescharacteristic
279C2×C71C2×#46711yescharacteristic
280C2×C71C2×#46811yescharacteristic
281C2×C71C2×#48511yescharacteristic
282C2×C71C2×#48411yescharacteristic
283D71—44no—
284C22×C22⋊Q8—711no—
285C22×#107—711no—
286C22×#103—711no—
287C22×#103D7111yescharacteristic, complemented
288C22×C22⋊C8—711no—
289C2×#380—711no—
290#2036—718no—
291C2×#340—711no—
292C71C2×#203611yescharacteristic, complemented, Sylow
293C23×Q8C2×D7111yescharacteristic
294C24×C4—711no—
295C23×C8—711no—
296C23×C8C2×D7111yescharacteristic
297C2×C22⋊Q8—714no—
298C2×C22⋊Q8—1424no—
299C2×C22⋊Q8—1422no—
300C2×C22⋊Q8—1424no—
301C22×C4⋊C4—711no—
302C22×C4⋊C4—711no—
303C22×C4⋊C4—1421no—
304C22×C4⋊C4C2×D7111yescharacteristic
305C22×C22⋊C4—1421no—
306C22×C22⋊C4—1421no—
307C2×#107—714no—
308C2×#107—1424no—
309C2×#103—24no—
310C2×#103—714no—
311C2×#103—1424no—
312C2×#103C2×D7114yescomplemented
313C2×C22⋊C8—714no—
314C2×C22⋊C8—1424no—
315#380—714no—
316#380—1422no—
317C2×#95—1421no—
318#340—714no—
319C25—711no—
320C22×Q8—21no—
321C22×Q8—22no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8—22no—
325C22×Q8—22no—
326C22×Q8C22⋊D7114yes—
327C22×Q8C22×D7112yes—
328C23×C4—21no—
329C23×C4—21no—
330C23×C4—711no—
331C23×C4—712no—
332C23×C4—712no—
333C23×C4—1421no—
334C23×C4—1421no—
335C23×C4—1422no—
336C23×C4—1422no—
337C23×C4—1422no—
338C23×C4—1422no—
339C23×C4—1422no—
340C23×C4—1422no—
341C23×C4C22×D7111yescharacteristic
342C22×C8—22no—
343C22×C8—22no—
344C22×C8—22no—
345C22×C8—714no—
346C22×C8—712no—
347C22×C8—1422no—
348C22×C8—1422no—
349C22×C8C22×D7112yes—
350C22⋊Q8—1428no—
351C22⋊Q8—14216no—
352C22⋊Q8—1428no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—714no—
357C2×C4⋊C4—712no—
358C2×C4⋊C4—712no—
359C2×C4⋊C4—1421no—
360C2×C4⋊C4—1422no—
361C2×C4⋊C4—1422no—
362C2×C4⋊C4—1424no—
363C2×C4⋊C4—1428no—
364C2×C4⋊C4—1421no—
365C2×C4⋊C4—1422no—
366C2×C4⋊C4C22×D7112yes—
367C2×C22⋊C4—2844no—
368C2×C22⋊C4—1424no—
369C2×C22⋊C4—1424no—
370C2×C22⋊C4—1428no—
371#107—1428no—
372#103—28no—
373#103—1428no—
374C22⋊C8—1428no—
375#95—1424no—
376C24—712no—
377C24—714no—
378C24—1422no—
379C24—1422no—
380C24—1422no—
381C24—1422no—
382C24—1422no—
383C24—1422no—
384C24D4×D7111yescharacteristic
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8C2×C22⋊D7114yes—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—2842no—
407C22×C4—2842no—
408C22×C4—2844no—
409C22×C4—2844no—
410C22×C4—2844no—
411C22×C4—2844no—
412C22×C4—2842no—
413C22×C4—2842no—
414C22×C4—2842no—
415C22×C4—712no—
416C22×C4—1422no—
417C22×C4—1422no—
418C22×C4—1424no—
419C22×C4—1422no—
420C22×C4—1422no—
421C22×C4—1422no—
422C22×C4—1422no—
423C22×C4—1422no—
424C22×C4—1422no—
425C22×C4—1422no—
426C22×C4—1422no—
427C22×C4—1422no—
428C22×C4—1424no—
429C22×C4—1424no—
430C22×C4—1424no—
431C22×C4D4×D7112yes—
432C22×C4D4×D7112yes—
433C22×C4D4×D7111yescharacteristic
434C22×C4C2×C22⋊D7112yes—
435C22×C4#1176586412yes—
436C22×C4#1176586512yes—
437C22×C4#1176586512yes—
438C22×C4C23×D7111yescharacteristic
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—44no—
443C2×C8—714no—
444C2×C8—1422no—
445C2×C8—1422no—
446C2×C8—1428no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—2842no—
451C4⋊C4—714no—
452C4⋊C4—14216no—
453C4⋊C4—1422no—
454C4⋊C4—1422no—
455C4⋊C4—1422no—
456C4⋊C4—1424no—
457C22⋊C4—2848no—
458C22⋊C4—14216no—
459C23—2844no—
460C23—2844no—
461C23—2844no—
462C23—2844no—
463C23—714no—
464C23—1428no—
465C23—1428no—
466C23—1422no—
467C23—1422no—
468C23—1428no—
469C23—1422no—
470C23—1422no—
471C23—1422no—
472C23—1422no—
473C23#1455609612yes—
474C23#1455610112yes—
475C23#1455611012yes—
476C23Q16×D7112yes—
477C23#1455614912yes—
478C23#1455615512yes—
479C23#1455616012yes—
480C23C2×D4×D7111yescharacteristic
481Q8—24no—
482Q8—24no—
483Q8—48no—
484Q8—42no—
485Q8—42no—
486C2×C4—24no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—84no—
494C2×C4—2848no—
495C2×C4—2844no—
496C2×C4—2842no—
497C2×C4—2842no—
498C2×C4—2848no—
499C2×C4—2842no—
500C2×C4—2842no—
501C2×C4—2842no—
502C2×C4—2842no—
503C2×C4—2842no—
504C2×C4—2842no—
505C2×C4—2842no—
506C2×C4—5684no—
507C2×C4—1424no—
508C2×C4—1422no—
509C2×C4—1422no—
510C2×C4—1422no—
511C2×C4#1455609612yes—
512C2×C4#1455610014yes—
513C2×C4#1455610112yes—
514C2×C4#1455610114yes—
515C2×C4#1455611012yes—
516C2×C4#1455611114yes—
517C2×C4#1455611414yes—
518C2×C4C2×D4×D7111yescharacteristic
519C2×C4C2×D4×D7111yescharacteristic
520C2×C4C22×C22⋊D7111yescharacteristic
521C2×C4C2×#1176586411yescharacteristic
522C2×C4C2×#1176586511yescharacteristic
523C2×C4C2×#1176586511yescharacteristic
524C8—42no—
525C8—1424no—
526C22—2842no—
527C22—2842no—
528C22—2842no—
529C22—28416no—
530C22—2842no—
531C22—1424no—
532C22—1424no—
533C22—1424no—
534C22#1798591414yes—
535C22#1798614314yes—
536C22#1798615314yes—
537C22#1798623914yes—
538C22#1798624814yes—
539C22#1798633314yes—
540C22#1798633814yes—
541C22C2×#1455609611yescharacteristic
542C22C2×#1455610111yescharacteristic
543C22C2×#1455611011yescharacteristic
544C22C2×Q16×D7111yescharacteristic
545C22C2×#1455614911yescharacteristic
546C22C2×#1455615511yescharacteristic
547C22C2×#1455616011yescharacteristic
548C4—44no—
549C4—82no—
550C4—2842no—
551C4—2844no—
552C4—5682no—
553C4#1798591414yes—
554C4C2×#1455610011yescharacteristic
555C4C2×#1455610111yescharacteristic
556C4C2×#1455611111yescharacteristic
557C4C2×#1455611411yescharacteristic
558C2—2844no—
559C2#2258954118yesdirect product factor
560C2C2×#1798591411yescharacteristic
561C2C2×#1798615311yescharacteristic
562C2C2×#1798614311yescharacteristic
563C2C2×#1798623911yescharacteristic
564C2C2×#1798624811yescharacteristic
565C2C2×#1798633811yescharacteristic
566C2C2×#1798633311yescharacteristic
subgroupG/H
centerC24D4×D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC22×#103×C71C2
socleC24×C71C2×D4
prime radicalC24×D71D4
hypercenterC22×#103D71
hypocenterC71C2×#2036
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
2. C22×#14556061C2semidirect
3. C22×#14556052C2semidirect
6. #22589541C2direct (N × H)
7. C2×#17985762C2semidirect
8. C22×#103×C71C2semidirect
21. C2×#14556061C22semidirect
23. C2×#14556052C22semidirect
29. #17985762C22semidirect
34. C2×#103×C71C22semidirect
275. C2×C71#2036semidirect
287. C22×#103D71semidirect
292. C71C2×#2036semidirect
312. C2×#103C2×D71semidirect
559. C2#22589541direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561042
C22×#145560612
C22×#145560522
C22×(C2×D71)⋊C82
C2×#179858562
#225895412
C2×#179857622
C22×#103×C712
C2×#203671