Finite Group Databasebeta

← C2×#22588566 #31998306

Subgroups — C2×#22588566 #31998306

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×#17986042C211yesmaximal, characteristic, complemented
2C22×#14556055C211yesmaximal, characteristic
3C2×#17986000C211yesmaximal, characteristic
4C2×#17985868C211yesmaximal, characteristic, complemented
5#22588566C218yesmaximal, direct product factor
6C2×#17985768C211yesmaximal, characteristic, complemented
7C2×#17985758C211yesmaximal, characteristic, complemented
8C2×#520×C71C211yesmaximal, characteristic
9C22×#11765849C2211yescharacteristic
10C22×#11765849C2211yescharacteristic
11C22×C4⋊Dic71C2211yescharacteristic
12C22×Dic71⋊C4—22no—
13C22×C4×Dic71C2211yescharacteristic
14#17986042—22no—
15#17986042C2214yescomplemented
16C2×#14556061—21no—
17C2×#14556059—21no—
18C2×#14556055C418yescomplemented
19C2×#14556055C2214yes—
20#17986000—22no—
21#17986000C2214yes—
22C2×#14556052—21no—
23C2×#14556049—21no—
24#17985868—22no—
25#17985868C2214yescomplemented
26#17985768C2214yescomplemented
27#17985758C2214yescomplemented
28C22×#41×C71C2211yescharacteristic
29#520×C71—22no—
30#520×C71—22no—
31#520×C71—22no—
32#520×C71C2214yes—
33C2×C4×Q8×C71—21no—
34C2×C4×D4×C71—21no—
35C2×C4×D4×C71—22no—
36C2×#117×C71—21no—
37C2×#117×C71C2211yescharacteristic
38C22×C4⋊C4×C71—21no—
39C22×C4⋊C4×C71C2211yescharacteristic
40C23×Dic71—21no—
41C23×Dic71—21no—
42C2×#11765849—21no—
43C2×#11765849—21no—
44C2×#11765849—22no—
45C2×#11765849—22no—
46C2×#11765849—22no—
47C2×#11765849C2×C414yescomplemented
48C2×#11765849C2312yes—
49C2×#11765849C2312yes—
50C22×C71⋊C8—21no—
51C22×C71⋊C8—21no—
52C2×C4⋊Dic71—21no—
53C2×C4⋊Dic71—21no—
54C2×C4⋊Dic71—21no—
55C2×C4⋊Dic71—21no—
56C2×C4⋊Dic71—21no—
57C2×C4⋊Dic71C2312yes—
58C2×Dic71⋊C4—28no—
59C2×Dic71⋊C4—48no—
60C2×C4×Dic71—21no—
61C2×C4×Dic71—22no—
62C2×C4×Dic71C2×C414yescomplemented
63C2×C4×Dic71C2312yes—
64#14556061—24no—
65#14556059—24no—
66#14556055C2×C4116yescomplemented
67#14556052—24no—
68#14556049—24no—
69C2×#41×C71—21no—
70C2×#41×C71—21no—
71C2×#41×C71—22no—
72C2×#41×C71—22no—
73C2×#41×C71C2×C414yes—
74C2×#41×C71D412yes—
75C2×#41×C71Q814yes—
76C2×#41×C71C2312yes—
77C22×Q8×C71D411yescharacteristic
78C22×D4×C71—21no—
79C22×D4×C71D411yescharacteristic
80C23×C4×C71—21no—
81C23×C4×C71—22no—
82C23×C4×C71D412yes—
83C23×C4×C71C2311yescharacteristic
84C4×Q8×C71—24no—
85C4×Q8×C71—24no—
86C4×D4×C71—28no—
87C4×D4×C71—24no—
88C4×D4×C71—24no—
89C4×D4×C71—28no—
90#117×C71—24no—
91#117×C71—21no—
92#117×C71—21no—
93#117×C71—24no—
94#117×C71D412yes—
95#117×C71C2312yes—
96C2×C4⋊C4×C71—22no—
97C2×C4⋊C4×C71—24no—
98C2×C4⋊C4×C71—21no—
99C2×C4⋊C4×C71—21no—
100C2×C4⋊C4×C71—21no—
101C2×C4⋊C4×C71—21no—
102C2×C4⋊C4×C71—24no—
103C2×C4⋊C4×C71D412yes—
104C2×C4⋊C4×C71D411yescharacteristic
105C2×C4⋊C4×C71D411yescharacteristic
106C2×C4⋊C4×C71C2312yes—
107C2×C22⋊C4×C71—22no—
108C2×C22⋊C4×C71—21no—
109C2×C42×C71—22no—
110C2×C42×C71—21no—
111C22×Dic71—24no—
112C22×Dic71—24no—
113C22×Dic71—24no—
114C22×Dic71—22no—
115C22×Dic71—22no—
116C22×Dic71—44no—
117C22×Dic71—41no—
118C22×Dic71—41no—
119#11765849—24no—
120#11765849—22no—
121#11765849—22no—
122#11765849—22no—
123#11765849—42no—
124#11765849C22×C414yescomplemented
125C2×C71⋊C8—24no—
126C2×C71⋊C8—22no—
127C2×C71⋊C8—22no—
128C2×C71⋊C8—42no—
129C4⋊Dic71—22no—
130C4⋊Dic71—22no—
131C4⋊Dic71—41no—
132C4⋊Dic71—41no—
133Dic71⋊C4—416no—
134C4×Dic71—24no—
135C4×Dic71—22no—
136C4×Dic71C22×C414yescomplemented
137C24×C71—21no—
138C24×C71C2×D411yescharacteristic
139#41×C71—24no—
140#41×C71—24no—
141#41×C71—24no—
142#41×C71—24no—
143#41×C71—24no—
144#41×C71—22no—
145#41×C71—22no—
146#41×C71C4⋊C418yescomplemented
147#41×C71C22×C414yes—
148#41×C71C2×Q814yes—
149C2×Q8×C71—21no—
150C2×Q8×C71—22no—
151C2×Q8×C71C4⋊C414yescomplemented
152C2×Q8×C71C2×D412yes—
153C2×D4×C71—24no—
154C2×D4×C71—22no—
155C2×D4×C71—24no—
156C2×D4×C71—22no—
157C2×D4×C71—21no—
158C2×D4×C71—22no—
159C2×D4×C71C4⋊C414yescomplemented
160C2×D4×C71C2×D412yes—
161C22×C4×C71—21no—
162C22×C4×C71—24no—
163C22×C4×C71—22no—
164C22×C4×C71—24no—
165C22×C4×C71—22no—
166C22×C4×C71—21no—
167C22×C4×C71—22no—
168C22×C4×C71—21no—
169C22×C4×C71—28no—
170C22×C4×C71—22no—
171C22×C4×C71—44no—
172C22×C4×C71—84no—
173C22×C4×C71C22×C412yes—
174C22×C4×C71C2×D414yes—
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×Q812yes—
181C22×C4×C71#4112yes—
182C22×C4×C71#4112yes—
183C22×C4×C71C2411yescharacteristic
184C4⋊C4×C71—28no—
185C4⋊C4×C71—22no—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—22no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—22no—
190C4⋊C4×C71—22no—
191C4⋊C4×C71C2×D412yes—
192C4⋊C4×C71C2×D412yes—
193C22⋊C4×C71—28no—
194C22⋊C4×C71—22no—
195C22⋊C4×C71—22no—
196C42×C71—28no—
197C42×C71—22no—
198C42×C71—22no—
199C23×C71—22no—
200C23×C71—22no—
201C23×C71—22no—
202C23×C71—24no—
203C23×C71—42no—
204C23×C71—42no—
205C23×C71C4×D412yes—
206C23×C71#12011yescharacteristic
207C23×C71#12011yescharacteristic
208C23×C71C22⋊Q812yes—
209C23×C71C22×D411yescharacteristic
210Q8×C71—22no—
211Q8×C71—24no—
212Q8×C71C2×C4⋊C414yescomplemented
213D4×C71—24no—
214D4×C71—24no—
215D4×C71—28no—
216D4×C71—22no—
217D4×C71—24no—
218D4×C71C2×C4⋊C414yescomplemented
219C2×C4×C71—22no—
220C2×C4×C71—24no—
221C2×C4×C71—24no—
222C2×C4×C71—24no—
223C2×C4×C71—28no—
224C2×C4×C71—22no—
225C2×C4×C71—24no—
226C2×C4×C71—22no—
227C2×C4×C71—44no—
228C2×C4×C71—42no—
229C2×C4×C71—42no—
230C2×C4×C71—44no—
231C2×C4×C71—82no—
232C2×C4×C71—88no—
233C2×C4×C71C2×C4⋊C412yes—
234C2×C4×C71C2×C4⋊C412yes—
235C2×C4×C71C4×D412yes—
236C2×C4×C71C4×D412yes—
237C2×C4×C71C4×D412yes—
238C2×C4×C71#12011yescharacteristic
239C2×C4×C71#12011yescharacteristic
240C2×C4×C71C22⋊Q812yes—
241C2×C4×C71C22⋊Q812yes—
242C2×C4×C71C22⋊Q812yes—
243C2×C4×C71#12312yes—
244C2×C4×C71#12312yes—
245C2×C4×C71C23×C411yescharacteristic
246C2×C4×C71C22×D411yescharacteristic
247C2×C4×C71C22×D411yescharacteristic
248C2×C4×C71C22×Q811yescharacteristic
249C2×C4×C71C2×#4111yescharacteristic
250C2×C4×C71C2×#4111yescharacteristic
251C2×Dic71—24no—
252C2×Dic71—216no—
253C2×Dic71—24no—
254C2×Dic71—24no—
255C2×Dic71—44no—
256C2×Dic71—42no—
257C2×Dic71—82no—
258C2×Dic71—82no—
259C71⋊C8—24no—
260C71⋊C8—42no—
261C22×C71—24no—
262C22×C71—24no—
263C22×C71—21no—
264C22×C71—21no—
265C22×C71—21no—
266C22×C71—21no—
267C22×C71—24no—
268C22×C71—48no—
269C22×C71—42no—
270C22×C71—42no—
271C22×C71#38512yes—
272C22×C71#38512yes—
273C22×C71#45312yes—
274C22×C71#45612yes—
275C22×C71C2×C4×D411yescharacteristic
276C22×C71C2×#12011yescharacteristic
277C22×C71C2×C22⋊Q811yescharacteristic
278C4×C71—24no—
279C4×C71—44no—
280C4×C71—42no—
281C4×C71—42no—
282C4×C71#38512yes—
283C4×C71#38512yes—
284C4×C71C22×C4⋊C411yescharacteristic
285C4×C71C2×C4×D411yescharacteristic
286C4×C71C2×C22⋊Q811yescharacteristic
287C4×C71C2×#12311yescharacteristic
288Dic71—28no—
289Dic71—82no—
290C2×#1859—711nomaximal, Sylow
291C2×C71—21no—
292C2×C71—21no—
293C2×C71—22no—
294C2×C71—44no—
295C2×C71#185914yescomplemented
296C2×C71C2×#38511yescharacteristic
297C2×C71C2×#45311yescharacteristic
298C2×C71C2×#45611yescharacteristic
299C2×#520D7111yescharacteristic
300C2×#428—711no—
301C22×#104—711no—
302C2×#417—711no—
303C2×#384—711no—
304#1859—718no—
305C2×#343—711no—
306C2×#339—711no—
307C71C2×#185911yescharacteristic, complemented, Sylow
308C22×#41C2×D7111yescharacteristic
309C22×M16—711no—
310C22×M16—711no—
311#520—22no—
312#520—22no—
313#520—22no—
314#520C2×D7114yes—
315C2×C4×Q8—21no—
316C2×C4×D4—21no—
317C2×C4×D4—22no—
318C2×#117—21no—
319C2×#117C2×D7111yescharacteristic
320C22×C4⋊C4—21no—
321C22×C4⋊C4—711no—
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#343—714no—
338#339—714no—
339C2×#41—21no—
340C2×#41—21no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41Dic14214yes—
344C2×#41C4×D7114yes—
345C2×#41C22⋊D7112yes—
346C2×#41C22×D7112yes—
347C22×Q8C22⋊D7111yescharacteristic
348C22×D4—21no—
349C22×D4C22⋊D7111yescharacteristic
350C23×C4—21no—
351C23×C4—22no—
352C23×C4—1421no—
353C23×C4—1421no—
354C23×C4C22⋊D7112yes—
355C23×C4C22×D7111yescharacteristic
356C2×M16—714no—
357C2×M16—712no—
358C2×M16—712no—
359C2×M16—1422no—
360C2×M16—1422no—
361C2×M16—1421no—
362C2×M16—1421no—
363C2×M16—1422no—
364C22×C8—1421no—
365C22×C8—1421no—
366C4×Q8—24no—
367C4×Q8—24no—
368C4×D4—28no—
369C4×D4—24no—
370C4×D4—24no—
371C4×D4—28no—
372#117—24no—
373#117—21no—
374#117—21no—
375#117—24no—
376#117C22⋊D7112yes—
377#117C22×D7112yes—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—24no—
385C2×C4⋊C4—2848no—
386C2×C4⋊C4—712no—
387C2×C4⋊C4—1421no—
388C2×C4⋊C4—1428no—
389C2×C4⋊C4—1421no—
390C2×C4⋊C4—1421no—
391C2×C4⋊C4—1421no—
392C2×C4⋊C4—1421no—
393C2×C4⋊C4C22⋊D7112yes—
394C2×C4⋊C4C22⋊D7111yescharacteristic
395C2×C4⋊C4C22⋊D7111yescharacteristic
396C2×C4⋊C4C22×D7112yes—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—21no—
399C2×C42—22no—
400C2×C42—21no—
401C2×C42—714no—
402C2×C42—712no—
403C2×C42—1421no—
404C2×C42—1422no—
405#107—1424no—
406#106—1424no—
407#104—7116no—
408#103—1424no—
409#102—1424no—
410C24—21no—
411C24C2×C22⋊D7111yescharacteristic
412#41—22no—
413#41—22no—
414#41—24no—
415#41—24no—
416#41—24no—
417#41—24no—
418#41—24no—
419#41Dic71⋊C418yescomplemented
420#41C2×C4×D7114yes—
421#41C2×Dic14214yes—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8Dic71⋊C414yescomplemented
425C2×Q8C2×C22⋊D7112yes—
426C2×D4—22no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4Dic71⋊C414yescomplemented
433C2×D4C2×C22⋊D7112yes—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—28no—
443C22×C4—22no—
444C22×C4—44no—
445C22×C4—84no—
446C22×C4—2844no—
447C22×C4—2841no—
448C22×C4—2841no—
449C22×C4—1422no—
450C22×C4—1424no—
451C22×C4—1424no—
452C22×C4—1422no—
453C22×C4—1424no—
454C22×C4C2×C4×D7112yes—
455C22×C4C2×C22⋊D7114yes—
456C22×C4C2×C22⋊D7111yescharacteristic
457C22×C4C2×C22⋊D7111yescharacteristic
458C22×C4C2×C22⋊D7111yescharacteristic
459C22×C4C2×C22⋊D7111yescharacteristic
460C22×C4C2×C22⋊D7111yescharacteristic
461C22×C4C2×Dic14212yes—
462C22×C4#1176586512yes—
463C22×C4#1176586512yes—
464C22×C4C23×D7111yescharacteristic
465M16—2842no—
466M16—714no—
467M16—1422no—
468M16—1422no—
469M16—1422no—
470M16—1424no—
471C2×C8—2842no—
472C2×C8—1422no—
473C2×C8—1422no—
474C2×C8—1424no—
475C4⋊C4—28no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—28416no—
483C4⋊C4—2841no—
484C4⋊C4—2841no—
485C4⋊C4—1422no—
486C4⋊C4—1422no—
487C4⋊C4C2×C22⋊D7112yes—
488C4⋊C4C2×C22⋊D7112yes—
489C22⋊C4—28no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C42—28no—
493C42—22no—
494C42—22no—
495C42—714no—
496C42—1422no—
497C42—1424no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—42no—
503C23—42no—
504C23C4×C22⋊D7112yes—
505C23#1455609711yescharacteristic
506C23#1455609711yescharacteristic
507C23#1455610712yes—
508C23C22×C22⋊D7111yescharacteristic
509Q8—24no—
510Q8—22no—
511Q8C2×Dic71⋊C414yescomplemented
512D4—28no—
513D4—24no—
514D4—24no—
515D4—24no—
516D4—22no—
517D4C2×Dic71⋊C414yescomplemented
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—82no—
531C2×C4—88no—
532C2×C4—2844no—
533C2×C4—2842no—
534C2×C4—5682no—
535C2×C4—5682no—
536C2×C4—1424no—
537C2×C4—1424no—
538C2×C4—14216no—
539C2×C4—1424no—
540C2×C4C2×Dic71⋊C412yes—
541C2×C4C2×Dic71⋊C412yes—
542C2×C4C4×C22⋊D7112yes—
543C2×C4C4×C22⋊D7112yes—
544C2×C4C4×C22⋊D7112yes—
545C2×C4#1455609711yescharacteristic
546C2×C4#1455609711yescharacteristic
547C2×C4#1455610712yes—
548C2×C4#1455610712yes—
549C2×C4#1455610712yes—
550C2×C4#1455611112yes—
551C2×C4#1455611112yes—
552C2×C4C22×C4×D7111yescharacteristic
553C2×C4C22×C22⋊D7111yescharacteristic
554C2×C4C22×C22⋊D7111yescharacteristic
555C2×C4C22×Dic14211yescharacteristic
556C2×C4C2×#1176586511yescharacteristic
557C2×C4C2×#1176586511yescharacteristic
558C8—2842no—
559C8—1424no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—48no—
570C22#1798587212yes—
571C22#1798587212yes—
572C22#1798616712yes—
573C22#1798618212yes—
574C22C2×C4×C22⋊D7111yescharacteristic
575C22C2×#1455609711yescharacteristic
576C22C2×#1455610711yescharacteristic
577C4—24no—
578C4—44no—
579C4—42no—
580C4—42no—
581C4—5682no—
582C4—1428no—
583C4#1798587212yes—
584C4#1798587212yes—
585C4C22×Dic71⋊C411yescharacteristic
586C4C2×C4×C22⋊D7111yescharacteristic
587C4C2×#1455610711yescharacteristic
588C4C2×#1455611111yescharacteristic
589C2—21no—
590C2—22no—
591C2—21no—
592C2—44no—
593C2#2258856614yesdirect product factor
594C2C2×#1798587211yescharacteristic
595C2C2×#1798616711yescharacteristic
596C2C2×#1798618211yescharacteristic
subgroupG/H
centerC23#14556097
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#520×C71C2
socleC23×C71#120
prime radicalC22×#41×C71C22
cosocleC22×C4×C71C24
hypercenterC2×#520D71
hypocenterC71C2×#1859
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×#17986042C2semidirect
4. C2×#17985868C2semidirect
5. #22588566C2direct (N × H)
6. C2×#17985768C2semidirect
7. C2×#17985758C2semidirect
15. #17986042C22semidirect
18. C2×#14556055C4semidirect
25. #17985868C22semidirect
26. #17985768C22semidirect
27. #17985758C22semidirect
47. C2×#11765849C2×C4semidirect
62. C2×C4×Dic71C2×C4semidirect
66. #14556055C2×C4semidirect
124. #11765849C22×C4semidirect
136. C4×Dic71C22×C4semidirect
146. #41×C71C4⋊C4semidirect
151. C2×Q8×C71C4⋊C4semidirect
159. C2×D4×C71C4⋊C4semidirect
212. Q8×C71C2×C4⋊C4semidirect
218. D4×C71C2×C4⋊C4semidirect
295. C2×C71#1859semidirect
307. C71C2×#1859semidirect
419. #41Dic71⋊C4semidirect
424. C2×Q8Dic71⋊C4semidirect
432. C2×D4Dic71⋊C4semidirect
511. Q8C2×Dic71⋊C4semidirect
517. D4C2×Dic71⋊C4semidirect
593. C2#22588566direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179860422
C22×#145560552
C2×#179860002
C2×#179858682
#225885662
C2×#179857682
C2×#179857582
C2×#520×C712
C2×#185971