Finite Group Databasebeta

← C2×#22588569 #31998309

Subgroups — C2×#22588569 #31998309

Order 18176.

1566 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986480C211yesmaximal, characteristic
2C2×#17986041C211yesmaximal, characteristic, complemented
3C22×#14556054C211yesmaximal, characteristic
4C2×#17986001C211yesmaximal, characteristic
5C2×#17985868C211yesmaximal, characteristic, complemented
6#22588569C218yesmaximal, direct product factor
7C2×#17985760C211yesmaximal, characteristic, complemented
8C2×#343×C71C211yesmaximal, characteristic, complemented
9C22×#11765865C2211yescharacteristic
10#17986480—22no—
11#17986480—22no—
12#17986480—22no—
13#17986480C2214yes—
14C2×#14556093—21no—
15C2×#14556090—21no—
16C2×C4×C22⋊D71—22no—
17C2×#14556087C2211yescharacteristic
18C2×#14556086—21no—
19C22×C4⋊Dic71C2211yescharacteristic
20C22×Dic71⋊C4—22no—
21C2×C4⋊C4×D71—21no—
22C22×C4×Dic71C2211yescharacteristic
23#17986041—22no—
24#17986041C2214yescomplemented
25C2×#14556062—21no—
26C2×#14556060—21no—
27C2×#14556054C418yescomplemented
28C2×#14556054C2214yes—
29#17986001—22no—
30#17986001C2214yes—
31C2×#14556053—21no—
32C2×#14556050—21no—
33#17985868—22no—
34#17985868C2214yescomplemented
35#17985760C2214yescomplemented
36C22×M16×C71C2211yescharacteristic
37C22×M16×C71C2211yescharacteristic
38C22×C4⋊C4×C71C2211yescharacteristic
39#343×C71C2214yescomplemented
40C2×#11765865—21no—
41C2×#11765865—21no—
42C2×#11765865—22no—
43C2×#11765865—22no—
44C2×#11765865C2×C414yes—
45C2×#11765865D412yes—
46C2×#11765865Q814yes—
47C2×#11765865C2312yes—
48C22×Dic142D411yescharacteristic
49C22×D284D411yescharacteristic
50C22×C22⋊D71—21no—
51C23×Dic71—21no—
52C23×Dic71—21no—
53C22×C4×D71—22no—
54C22×C4×D71—21no—
55#14556093—24no—
56#14556093—24no—
57#14556090—24no—
58#14556090—24no—
59C4×C22⋊D71—28no—
60C4×C22⋊D71—28no—
61#14556087—21no—
62#14556087—21no—
63#14556087D412yes—
64#14556087C2312yes—
65#14556086—24no—
66#14556086—24no—
67C2×C4⋊Dic71—21no—
68C2×C4⋊Dic71—21no—
69C2×C4⋊Dic71—21no—
70C2×C4⋊Dic71—21no—
71C2×C4⋊Dic71—21no—
72C2×C4⋊Dic71D411yescharacteristic
73C2×C4⋊Dic71D411yescharacteristic
74C2×C4⋊Dic71C2312yes—
75C2×Dic71⋊C4—22no—
76C2×Dic71⋊C4—28no—
77C2×Dic71⋊C4—48no—
78C4⋊C4×D71—24no—
79C4⋊C4×D71—24no—
80C2×C22⋊Dic71—21no—
81C2×(C2×D71)⋊C4—22no—
82C2×C4×Dic71—22no—
83C2×C4×Dic71—21no—
84C2×C4×Dic71—21no—
85C2×C4×Dic71—22no—
86C2×C4×Dic71C2×C414yescomplemented
87C2×C4×Dic71C2312yes—
88#14556062—24no—
89#14556060—24no—
90#14556054C2×C4116yescomplemented
91#14556053—24no—
92#14556050—24no—
93C23×C4×C71D412yes—
94C23×C4×C71C2311yescharacteristic
95C2×M16×C71—21no—
96C2×M16×C71—22no—
97C2×M16×C71—21no—
98C2×M16×C71—22no—
99C2×M16×C71—22no—
100C2×M16×C71C2×C414yescomplemented
101C2×M16×C71C2312yes—
102C2×M16×C71C2312yes—
103C22×C8×C71—21no—
104C22×C8×C71—21no—
105C2×C4⋊C4×C71—21no—
106C2×C4⋊C4×C71—21no—
107C2×C4⋊C4×C71—21no—
108C2×C4⋊C4×C71—21no—
109C2×C4⋊C4×C71D412yes—
110C2×C4⋊C4×C71C2312yes—
111C23×D71—21no—
112#11765865—24no—
113#11765865—24no—
114#11765865—24no—
115#11765865—24no—
116#11765865—24no—
117#11765865—22no—
118#11765865—22no—
119#11765865C4⋊C418yescomplemented
120#11765865C22×C414yes—
121#11765865C2×Q814yes—
122C2×Dic142—21no—
123C2×Dic142—22no—
124C2×Dic142C4⋊C414yescomplemented
125C2×Dic142C2×D412yes—
126C2×D284—21no—
127C2×D284—22no—
128C2×D284C4⋊C414yescomplemented
129C2×D284C2×D412yes—
130C2×C22⋊D71—24no—
131C2×C22⋊D71—22no—
132C2×C22⋊D71—24no—
133C2×C22⋊D71—22no—
134C22×Dic71—24no—
135C22×Dic71—24no—
136C22×Dic71—24no—
137C22×Dic71—21no—
138C22×Dic71—22no—
139C22×Dic71—21no—
140C22×Dic71—21no—
141C22×Dic71—22no—
142C22×Dic71—44no—
143C22×Dic71—41no—
144C22×Dic71—41no—
145C2×C4×D71—24no—
146C2×C4×D71—22no—
147C2×C4×D71—24no—
148C2×C4×D71—22no—
149C2×C4×D71—28no—
150C2×C4×D71—84no—
151C4⋊Dic71—21no—
152C4⋊Dic71—21no—
153C4⋊Dic71—22no—
154C4⋊Dic71—22no—
155C4⋊Dic71—41no—
156C4⋊Dic71—41no—
157C4⋊Dic71C2×D412yes—
158C4⋊Dic71C2×D412yes—
159Dic71⋊C4—28no—
160Dic71⋊C4—416no—
161C22⋊Dic71—22no—
162C22⋊Dic71—22no—
163(C2×D71)⋊C4—28no—
164C4×Dic71—28no—
165C4×Dic71—24no—
166C4×Dic71—22no—
167C4×Dic71—22no—
168C4×Dic71—22no—
169C4×Dic71C22×C414yescomplemented
170C24×C71C2×D411yescharacteristic
171C22×C4×C71—22no—
172C22×C4×C71—22no—
173C22×C4×C71—44no—
174C22×C4×C71C22×C412yes—
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×D414yes—
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×Q812yes—
182C22×C4×C71#4112yes—
183C22×C4×C71#4112yes—
184C22×C4×C71C2411yescharacteristic
185M16×C71—24no—
186M16×C71—22no—
187M16×C71—22no—
188M16×C71—22no—
189M16×C71—42no—
190M16×C71C22×C414yescomplemented
191C2×C8×C71—24no—
192C2×C8×C71—22no—
193C2×C8×C71—22no—
194C2×C8×C71—42no—
195C4⋊C4×C71—22no—
196C4⋊C4×C71—22no—
197C4⋊C4×C71—22no—
198C4⋊C4×C71—22no—
199C23×C71—22no—
200C23×C71—22no—
201C23×C71C4×D412yes—
202C23×C71#12011yescharacteristic
203C23×C71#12011yescharacteristic
204C23×C71C22⋊Q812yes—
205C23×C71C22×D411yescharacteristic
206C22×D71—22no—
207C22×D71—24no—
208C22×D71—42no—
209C22×D71—42no—
210C2×C4×C71—24no—
211C2×C4×C71—44no—
212C2×C4×C71—44no—
213C2×C4×C71—82no—
214C2×C4×C71C2×C4⋊C412yes—
215C2×C4×C71C2×C4⋊C412yes—
216C2×C4×C71C4×D412yes—
217C2×C4×C71C4×D412yes—
218C2×C4×C71C4×D412yes—
219C2×C4×C71#12011yescharacteristic
220C2×C4×C71#12011yescharacteristic
221C2×C4×C71C22⋊Q812yes—
222C2×C4×C71C22⋊Q812yes—
223C2×C4×C71C22⋊Q812yes—
224C2×C4×C71#12312yes—
225C2×C4×C71#12312yes—
226C2×C4×C71C23×C411yescharacteristic
227C2×C4×C71C22×D411yescharacteristic
228C2×C4×C71C22×D411yescharacteristic
229C2×C4×C71C22×Q811yescharacteristic
230C2×C4×C71C2×#4111yescharacteristic
231C2×C4×C71C2×#4111yescharacteristic
232C22⋊D71—24no—
233C22⋊D71—24no—
234C22⋊D71—28no—
235C2×Dic71—24no—
236C2×Dic71—216no—
237C2×Dic71—24no—
238C2×Dic71—24no—
239C2×Dic71—22no—
240C2×Dic71—24no—
241C2×Dic71—22no—
242C2×Dic71—22no—
243C2×Dic71—42no—
244C2×Dic71—42no—
245C2×Dic71—44no—
246C2×Dic71—42no—
247C2×Dic71—82no—
248C2×Dic71—82no—
249D284—22no—
250D284—24no—
251D284C2×C4⋊C414yescomplemented
252C4×D71—24no—
253C4×D71—24no—
254C4×D71—28no—
255C4×D71—88no—
256Dic142—22no—
257Dic142—24no—
258Dic142C2×C4⋊C414yescomplemented
259C8×C71—24no—
260C8×C71—42no—
261C22×C71—24no—
262C22×C71—24no—
263C22×C71—21no—
264C22×C71—21no—
265C22×C71—21no—
266C22×C71—21no—
267C22×C71#38512yes—
268C22×C71#38512yes—
269C22×C71#45312yes—
270C22×C71#45612yes—
271C22×C71C2×C4×D411yescharacteristic
272C22×C71C2×#12011yescharacteristic
273C22×C71C2×C22⋊Q811yescharacteristic
274C2×D71—24no—
275C2×D71—48no—
276C2×D71—42no—
277C2×D71—42no—
278C4×C71—44no—
279C4×C71#38512yes—
280C4×C71#38512yes—
281C4×C71C22×C4⋊C411yescharacteristic
282C4×C71C2×C4×D411yescharacteristic
283C4×C71C2×C22⋊Q811yescharacteristic
284C4×C71C2×#12311yescharacteristic
285Dic71—28no—
286Dic71—24no—
287Dic71—42no—
288Dic71—42no—
289Dic71—82no—
290C2×#1859—711nomaximal, Sylow
291C2×C71—21no—
292C2×C71—21no—
293C2×C71—22no—
294C2×C71#185914yescomplemented
295C2×C71C2×#38511yescharacteristic
296C2×C71C2×#45311yescharacteristic
297C2×C71C2×#45611yescharacteristic
298D71—44no—
299C2×#520—711no—
300C2×#428—711no—
301C22×#104—711no—
302C2×#417—711no—
303C2×#384—711no—
304#1859—718no—
305C2×#343D7111yescharacteristic, complemented
306C2×#339—711no—
307C71C2×#185911yescharacteristic, complemented, Sylow
308C22×#41—711no—
309C22×M16C2×D7111yescharacteristic
310C22×M16C2×D7111yescharacteristic
311#520—714no—
312#520—1422no—
313#520—1422no—
314#520—1422no—
315C2×C4×Q8—1421no—
316C2×C4×D4—1421no—
317C2×C4×D4—1422no—
318C2×#117—711no—
319C2×#117—1421no—
320C22×C4⋊C4—711no—
321C22×C4⋊C4—1421no—
322C22×C4⋊C4—1422no—
323C22×C4⋊C4C2×D7111yescharacteristic
324C22×C42—711no—
325#428—714no—
326#428—1422no—
327C2×#107—1421no—
328C2×#106—1421no—
329C2×#104—718no—
330C2×#104—714no—
331#417—714no—
332#417—1422no—
333C2×#103—1421no—
334C2×#102—1421no—
335#384—714no—
336#384—1422no—
337#343C2×D7114yescomplemented
338#339—714no—
339C2×#41—714no—
340C2×#41—712no—
341C2×#41—714no—
342C2×#41—712no—
343C2×#41—1421no—
344C2×#41—1421no—
345C2×#41—1422no—
346C2×#41—1422no—
347C22×Q8—711no—
348C22×D4—711no—
349C22×D4—1421no—
350C23×C4—1422no—
351C23×C4—1421no—
352C23×C4—1421no—
353C23×C4—1421no—
354C23×C4D28412yes—
355C23×C4C22×D7111yescharacteristic
356C2×M16—21no—
357C2×M16—22no—
358C2×M16—21no—
359C2×M16—22no—
360C2×M16—22no—
361C2×M16C4×D7114yescomplemented
362C2×M16C22×D7112yes—
363C2×M16C22×D7112yes—
364C22×C8—21no—
365C22×C8—21no—
366C4×Q8—1424no—
367C4×Q8—1424no—
368C4×D4—1424no—
369C4×D4—1424no—
370C4×D4—1428no—
371C4×D4—1428no—
372#117—712no—
373#117—712no—
374#117—1424no—
375#117—1424no—
376#117—1421no—
377#117—1421no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—2848no—
383C2×C4⋊C4—711no—
384C2×C4⋊C4—711no—
385C2×C4⋊C4—712no—
386C2×C4⋊C4—1424no—
387C2×C4⋊C4—1424no—
388C2×C4⋊C4—1421no—
389C2×C4⋊C4—1428no—
390C2×C4⋊C4—1421no—
391C2×C4⋊C4—1421no—
392C2×C4⋊C4—1421no—
393C2×C4⋊C4—1421no—
394C2×C4⋊C4—1422no—
395C2×C4⋊C4D28412yes—
396C2×C4⋊C4C22×D7112yes—
397C2×C22⋊C4—1422no—
398C2×C22⋊C4—1421no—
399C2×C42—714no—
400C2×C42—712no—
401C2×C42—1422no—
402C2×C42—1421no—
403C2×C42—1421no—
404C2×C42—1422no—
405#107—1424no—
406#106—1424no—
407#104—7116no—
408#103—1424no—
409#102—1424no—
410C24—1421no—
411C24C2×D28411yescharacteristic
412#41—714no—
413#41—714no—
414#41—718no—
415#41—1422no—
416#41—1422no—
417#41—1424no—
418#41—1424no—
419#41—1424no—
420#41—1424no—
421#41—1424no—
422C2×Q8—712no—
423C2×Q8—714no—
424C2×Q8—1421no—
425C2×Q8—1422no—
426C2×D4—712no—
427C2×D4—714no—
428C2×D4—1422no—
429C2×D4—1424no—
430C2×D4—1422no—
431C2×D4—1424no—
432C2×D4—1421no—
433C2×D4—1422no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—2844no—
438C22×C4—2841no—
439C22×C4—2841no—
440C22×C4—5684no—
441C22×C4—1428no—
442C22×C4—1422no—
443C22×C4—1421no—
444C22×C4—1421no—
445C22×C4—1424no—
446C22×C4—1424no—
447C22×C4—1422no—
448C22×C4—1424no—
449C22×C4—1421no—
450C22×C4—1422no—
451C22×C4—1424no—
452C22×C4—1422no—
453C22×C4—1424no—
454C22×C4C2×C4×D7112yes—
455C22×C4D4×D7111yescharacteristic
456C22×C4D4×D7111yescharacteristic
457C22×C4D4×D7111yescharacteristic
458C22×C4D4×D7111yescharacteristic
459C22×C4C2×D28414yes—
460C22×C4C2×D28411yescharacteristic
461C22×C4Q8×D7112yes—
462C22×C4#1176586412yes—
463C22×C4#1176586512yes—
464C22×C4C23×D7111yescharacteristic
465M16—22no—
466M16—24no—
467M16—22no—
468M16—22no—
469M16—42no—
470M16C2×C4×D7114yescomplemented
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—42no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—28416no—
480C4⋊C4—2841no—
481C4⋊C4—2841no—
482C4⋊C4—712no—
483C4⋊C4—712no—
484C4⋊C4—1428no—
485C4⋊C4—1422no—
486C4⋊C4—1422no—
487C4⋊C4—1421no—
488C4⋊C4—1421no—
489C22⋊C4—1428no—
490C22⋊C4—1422no—
491C22⋊C4—1422no—
492C42—714no—
493C42—1428no—
494C42—1422no—
495C42—1422no—
496C42—1424no—
497C42—1422no—
498C23—22no—
499C23—22no—
500C23—2842no—
501C23—2842no—
502C23—1424no—
503C23—1422no—
504C23C4×D28412yes—
505C23#1455609811yescharacteristic
506C23#1455609811yescharacteristic
507C23#1455610812yes—
508C23C22×D28411yescharacteristic
509Q8—714no—
510Q8—1424no—
511Q8—1422no—
512D4—714no—
513D4—1428no—
514D4—1424no—
515D4—1424no—
516D4—1424no—
517D4—1422no—
518C2×C4—24no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—82no—
522C2×C4—2842no—
523C2×C4—2842no—
524C2×C4—2844no—
525C2×C4—2842no—
526C2×C4—5688no—
527C2×C4—5682no—
528C2×C4—5682no—
529C2×C4—1422no—
530C2×C4—1422no—
531C2×C4—1424no—
532C2×C4—1424no—
533C2×C4—14216no—
534C2×C4—1424no—
535C2×C4—1424no—
536C2×C4—1422no—
537C2×C4—1428no—
538C2×C4—1424no—
539C2×C4—1424no—
540C2×C4C4⋊C4×D7112yes—
541C2×C4C4⋊C4×D7112yes—
542C2×C4(C22⋊D71)⋊C412yes—
543C2×C4(C22⋊D71)⋊C412yes—
544C2×C4C4×D28412yes—
545C2×C4#1455609811yescharacteristic
546C2×C4#1455609811yescharacteristic
547C2×C4#1455610612yes—
548C2×C4#1455610612yes—
549C2×C4#1455610812yes—
550C2×C4#1455611012yes—
551C2×C4#1455611012yes—
552C2×C4C22×C4×D7111yescharacteristic
553C2×C4C2×D4×D7111yescharacteristic
554C2×C4C2×D4×D7111yescharacteristic
555C2×C4C2×Q8×D7111yescharacteristic
556C2×C4C2×#1176586411yescharacteristic
557C2×C4C2×#1176586511yescharacteristic
558C8—24no—
559C8—42no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—2842no—
567C22—2842no—
568C22—2848no—
569C22—1424no—
570C22#1798587312yes—
571C22#1798587312yes—
572C22#1798616912yes—
573C22#1798618412yes—
574C22C2×C4×D28411yescharacteristic
575C22C2×#1455609811yescharacteristic
576C22C2×#1455610811yescharacteristic
577C4—44no—
578C4—2842no—
579C4—2842no—
580C4—5682no—
581C4—1428no—
582C4—1424no—
583C4#1798587312yes—
584C4#1798587312yes—
585C4C2×C4⋊C4×D7111yescharacteristic
586C4C2×(C22⋊D71)⋊C411yescharacteristic
587C4C2×#1455610611yescharacteristic
588C4C2×#1455611011yescharacteristic
589C2—21no—
590C2—22no—
591C2—21no—
592C2—2844no—
593C2#2258856914yesdirect product factor
594C2C2×#1798587311yescharacteristic
595C2C2×#1798616911yescharacteristic
596C2C2×#1798618411yescharacteristic
subgroupG/H
centerC23#14556098
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#343×C71C2
socleC23×C71#120
prime radicalC22×#11765865C22
cosocleC22×C4×C71C24
hypercenterC2×#343D71
hypocenterC71C2×#1859
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
2. C2×#17986041C2semidirect
5. C2×#17985868C2semidirect
6. #22588569C2direct (N × H)
7. C2×#17985760C2semidirect
8. C2×#343×C71C2semidirect
24. #17986041C22semidirect
27. C2×#14556054C4semidirect
34. #17985868C22semidirect
35. #17985760C22semidirect
39. #343×C71C22semidirect
86. C2×C4×Dic71C2×C4semidirect
90. #14556054C2×C4semidirect
100. C2×M16×C71C2×C4semidirect
119. #11765865C4⋊C4semidirect
124. C2×Dic142C4⋊C4semidirect
128. C2×D284C4⋊C4semidirect
169. C4×Dic71C22×C4semidirect
190. M16×C71C22×C4semidirect
251. D284C2×C4⋊C4semidirect
258. Dic142C2×C4⋊C4semidirect
294. C2×C71#1859semidirect
305. C2×#343D71semidirect
307. C71C2×#1859semidirect
337. #343C2×D71semidirect
361. C2×M16C4×D71semidirect
470. M16C2×C4×D71semidirect
593. C2#22588569direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179864802
C2×#179860412
C22×#145560542
C2×#179860012
C2×#179858682
#225885692
C2×#179857602
C2×#343×C712
C2×#185971