Finite Group Databasebeta

← C2×#22588687 #31998775

Subgroups — C2×#22588687 #31998775

Order 18176.

2470 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D568C211yesmaximal, characteristic
2C22×C4×D284C211yesmaximal, characteristic, complemented
3C22×#14556050C211yesmaximal, characteristic, complemented
4C22×#14556050C211yesmaximal, characteristic, complemented
5C2×#17985901C211yesmaximal, characteristic, complemented
6#22588687C218yesmaximal, direct product factor
7C2×#17985762C211yesmaximal, characteristic, complemented
8C2×#336×C71C211yesmaximal, characteristic, complemented
9C23×D284C2211yescharacteristic
10C23×D284C2211yescharacteristic
11C23×C4×D71—21no—
12C22×D568—24no—
13C22×D568—24no—
14C22×D568C418yescomplemented
15C22×D568C2214yes—
16C2×C4×D284—24no—
17C2×C4×D284—22no—
18C2×C4×D284—24no—
19C2×C4×D284C2214yescomplemented
20C22×C4⋊Dic71C2211yescharacteristic
21C22×C4⋊Dic71C2211yescharacteristic
22C22×(C2×D71)⋊C4—21no—
23C22×(C2×D71)⋊C4—21no—
24C22×(C2×D71)⋊C4—21no—
25C2×#14556050—24no—
26C2×#14556050—24no—
27C2×#14556050C2214yescomplemented
28C2×#14556050C2214yescomplemented
29#17985901—22no—
30#17985901C2214yescomplemented
31#17985762C2214yescomplemented
32C23×C8×C71C2211yescharacteristic
33C23×C8×C71C2211yescharacteristic
34C22×C42×C71C2211yescharacteristic
35#336×C71C2214yescomplemented
36C24×D71—21no—
37C24×D71—21no—
38C22×D284—22no—
39C22×D284—21no—
40C22×D284—22no—
41C22×D284—22no—
42C22×D284—22no—
43C22×D284—21no—
44C22×D284—22no—
45C22×D284—22no—
46C22×D284—22no—
47C22×D284—22no—
48C22×D284C2×C414yes—
49C22×D284C2×C414yescomplemented
50C22×D284D414yes—
51C22×D284D414yes—
52C22×D284C2312yes—
53C22×D284C2312yes—
54C23×Dic71—21no—
55C23×Dic71—21no—
56C22×C4×D71—28no—
57C22×C4×D71—24no—
58C22×C4×D71—44no—
59C22×C4×D71—44no—
60C2×D568—232no—
61C2×D568—28no—
62C2×D568—28no—
63C2×D568C2×C4116yescomplemented
64C4×D284—216no—
65C4×D284—28no—
66C4×D284—28no—
67C2×C4⋊Dic71—22no—
68C2×C4⋊Dic71—21no—
69C2×C4⋊Dic71—22no—
70C2×C4⋊Dic71—21no—
71C2×C4⋊Dic71—22no—
72C2×C4⋊Dic71D414yes—
73C2×C4⋊Dic71C2312yes—
74C2×C4⋊Dic71C2312yes—
75C2×(C2×D71)⋊C4—28no—
76C2×(C2×D71)⋊C4—24no—
77C2×(C2×D71)⋊C4—24no—
78C2×(C2×D71)⋊C4—24no—
79C2×(C2×D71)⋊C4—44no—
80C2×(C2×D71)⋊C4—44no—
81#14556050—28no—
82#14556050—28no—
83C23×C4×C71D411yescharacteristic
84C23×C4×C71D411yescharacteristic
85C23×C4×C71C2311yescharacteristic
86C22×C8×C71—22no—
87C22×C8×C71—22no—
88C22×C8×C71—22no—
89C22×C8×C71—22no—
90C22×C8×C71—22no—
91C22×C8×C71C2×C414yescomplemented
92C22×C8×C71C2312yes—
93C22×C8×C71C2312yes—
94C2×C42×C71—22no—
95C2×C42×C71—22no—
96C2×C42×C71D412yes—
97C2×C42×C71C2312yes—
98C23×D71—24no—
99C23×D71—24no—
100C23×D71—24no—
101C23×D71—22no—
102C23×D71—24no—
103C23×D71—22no—
104C23×D71—42no—
105C23×D71—42no—
106C23×D71—42no—
107C23×D71—42no—
108C23×D71—42no—
109C23×D71—42no—
110C23×D71—42no—
111C23×D71—42no—
112C23×D71—42no—
113C23×D71—42no—
114C2×D284—28no—
115C2×D284—24no—
116C2×D284—24no—
117C2×D284—28no—
118C2×D284—24no—
119C2×D284—28no—
120C2×D284—22no—
121C2×D284—22no—
122C2×D284—22no—
123C2×D284—28no—
124C2×D284—28no—
125C2×D284—22no—
126C2×D284—22no—
127C2×D284—22no—
128C2×D284—22no—
129C2×D284—24no—
130C2×D284—22no—
131C2×D284—42no—
132C2×D284—44no—
133C2×D284—42no—
134C2×D284—44no—
135C2×D284C22⋊C4116yescomplemented
136C2×D284C22×C414yes—
137C2×D284C22×C414yescomplemented
138C2×D284C2×D414yes—
139C2×D284C2×D414yes—
140C22×Dic71—24no—
141C22×Dic71—22no—
142C22×Dic71—22no—
143C22×Dic71—42no—
144C22×Dic71—42no—
145C22×Dic71—42no—
146C22×Dic71—42no—
147C22×Dic71—42no—
148C2×C4×D71—216no—
149C2×C4×D71—48no—
150C2×C4×D71—432no—
151C2×C4×D71—48no—
152D568—232no—
153C4⋊Dic71—24no—
154C4⋊Dic71—22no—
155C4⋊Dic71—22no—
156C4⋊Dic71—22no—
157C4⋊Dic71—42no—
158C4⋊Dic71C2×D414yes—
159(C2×D71)⋊C4—216no—
160(C2×D71)⋊C4—48no—
161(C2×D71)⋊C4—48no—
162C24×C71C2×D411yescharacteristic
163C22×C4×C71—22no—
164C22×C4×C71—22no—
165C22×C4×C71—22no—
166C22×C4×C71—22no—
167C22×C4×C71—22no—
168C22×C4×C71D814yescomplemented
169C22×C4×C71C22×C412yes—
170C22×C4×C71C2×D412yes—
171C22×C4×C71C2×D412yes—
172C22×C4×C71C2×D412yes—
173C22×C4×C71C2×D412yes—
174C22×C4×C71C2×D412yes—
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71#4112yes—
177C22×C4×C71#4112yes—
178C22×C4×C71C2411yescharacteristic
179C2×C8×C71—28no—
180C2×C8×C71—22no—
181C2×C8×C71—22no—
182C2×C8×C71—22no—
183C2×C8×C71—22no—
184C2×C8×C71—22no—
185C2×C8×C71—44no—
186C2×C8×C71C22×C414yescomplemented
187C42×C71—24no—
188C42×C71—22no—
189C42×C71—22no—
190C23×C71C4×D412yes—
191C23×C71#12012yes—
192C23×C71#12112yes—
193C23×C71C2×D812yes—
194C23×C71#13512yes—
195C23×C71AGL(1,ℤ/8)12yes—
196C23×C71AGL(1,ℤ/8)12yes—
197C23×C71C22×D411yescharacteristic
198C22×D71—24no—
199C22×D71—24no—
200C22×D71—24no—
201C22×D71—216no—
202C22×D71—24no—
203C22×D71—42no—
204C22×D71—42no—
205C22×D71—42no—
206C22×D71—42no—
207C22×D71—48no—
208C22×D71—48no—
209C22×D71—48no—
210C22×D71—48no—
211C22×D71—48no—
212C22×D71—48no—
213C22×D71—42no—
214C22×D71—42no—
215C22×D71—42no—
216C22×D71—42no—
217C22×D71—42no—
218C22×D71—42no—
219C22×D71—48no—
220C22×D71—48no—
221C22×D71—48no—
222C22×D71—84no—
223C22×D71—84no—
224C22×D71—84no—
225C22×D71—84no—
226C2×C4×C71—22no—
227C2×C4×C71—22no—
228C2×C4×C71—28no—
229C2×C4×C71—22no—
230C2×C4×C71—22no—
231C2×C4×C71—22no—
232C2×C4×C71—44no—
233C2×C4×C71C2×C22⋊C414yes—
234C2×C4×C71C4×D414yes—
235C2×C4×C71C4×D412yes—
236C2×C4×C71#12012yes—
237C2×C4×C71#12114yes—
238C2×C4×C71#12112yes—
239C2×C4×C71#12314yes—
240C2×C4×C71C2×D814yescomplemented
241C2×C4×C71C23×C411yescharacteristic
242C2×C4×C71C22×D411yescharacteristic
243C2×C4×C71C22×D411yescharacteristic
244C2×C4×C71C22×D411yescharacteristic
245C2×C4×C71C2×#4111yescharacteristic
246C2×C4×C71C2×#4111yescharacteristic
247C2×Dic71—24no—
248C2×Dic71—42no—
249C2×Dic71—42no—
250C2×Dic71—48no—
251C2×Dic71—42no—
252C2×Dic71—42no—
253C2×Dic71—42no—
254C2×Dic71—84no—
255D284—24no—
256D284—24no—
257D284—24no—
258D284—24no—
259D284—24no—
260D284—216no—
261D284—42no—
262D284—42no—
263D284—48no—
264D284C2×C22⋊C418yescomplemented
265C4×D71—432no—
266C8×C71—24no—
267C8×C71—42no—
268C22×C71#38614yes—
269C22×C71C4×D814yes—
270C22×C71#44214yes—
271C22×C71#44714yes—
272C22×C71#44914yes—
273C22×C71#45914yes—
274C22×C71#46314yes—
275C22×C71C2×C4×D411yescharacteristic
276C22×C71C2×#12011yescharacteristic
277C22×C71C2×#12111yescharacteristic
278C22×C71C22×D811yescharacteristic
279C22×C71C2×#13511yescharacteristic
280C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
282C2×D71—28no—
283C2×D71—432no—
284C2×D71—44no—
285C2×D71—44no—
286C2×D71—44no—
287C2×D71—44no—
288C2×D71—44no—
289C2×D71—44no—
290C2×D71—44no—
291C2×D71—44no—
292C2×D71—44no—
293C2×D71—816no—
294C2×D71—82no—
295C2×D71—82no—
296C2×D71—82no—
297C2×D71—82no—
298C4×C71—24no—
299C4×C71—42no—
300C4×C71#38614yes—
301C4×C71C22×C22⋊C411yescharacteristic
302C4×C71C2×C4×D411yescharacteristic
303C4×C71C2×#12111yescharacteristic
304C4×C71C2×#12311yescharacteristic
305Dic71—44no—
306Dic71—82no—
307C2×#1879—711nomaximal, Sylow
308C2×C71#187918yescomplemented
309C2×C71C2×#38611yescharacteristic
310C2×C71C2×C4×D811yescharacteristic
311C2×C71C2×#44211yescharacteristic
312C2×C71C2×#44711yescharacteristic
313C2×C71C2×#44911yescharacteristic
314C2×C71C2×#45911yescharacteristic
315C2×C71C2×#46311yescharacteristic
316D71—48no—
317D71—84no—
318C23×D8—711no—
319C22×C4×D4—711no—
320C22×#102—711no—
321C22×#102—711no—
322C2×#390—711no—
323#1879—718no—
324C2×#340—711no—
325C2×#336D7111yescharacteristic, complemented
326C71C2×#187911yescharacteristic, complemented, Sylow
327C23×D4—711no—
328C23×D4—711no—
329C24×C4—1421no—
330C22×D8—718no—
331C22×D8—714no—
332C22×D8—1424no—
333C22×D8—1424no—
334C23×C8C2×D7111yescharacteristic
335C23×C8C2×D7111yescharacteristic
336C2×C4×D4—714no—
337C2×C4×D4—1424no—
338C2×C4×D4—1424no—
339C2×C4×D4—1422no—
340C22×C4⋊C4—711no—
341C22×C4⋊C4—711no—
342C22×C22⋊C4—1421no—
343C22×C22⋊C4—1421no—
344C22×C22⋊C4—1421no—
345C22×C42C2×D7111yescharacteristic
346C2×#102—714no—
347C2×#102—714no—
348C2×#102—1424no—
349C2×#102—1424no—
350#390—714no—
351#390—1422no—
352#340—714no—
353#336C2×D7114yescomplemented
354C25—1421no—
355C25—1421no—
356C22×D4—714no—
357C22×D4—714no—
358C22×D4—714no—
359C22×D4—714no—
360C22×D4—712no—
361C22×D4—712no—
362C22×D4—1422no—
363C22×D4—1422no—
364C22×D4—1422no—
365C22×D4—1422no—
366C22×D4—1421no—
367C22×D4—1421no—
368C22×D4—1422no—
369C22×D4—1422no—
370C22×D4—1422no—
371C22×D4—1422no—
372C23×C4—2844no—
373C23×C4—2844no—
374C23×C4—1428no—
375C23×C4—1421no—
376C23×C4—1421no—
377C23×C4—1424no—
378C23×C4D28411yescharacteristic
379C23×C4D28411yescharacteristic
380C23×C4C22×D7111yescharacteristic
381C2×D8—7116no—
382C2×D8—1428no—
383C2×D8—1428no—
384C2×D8—14232no—
385C22×C8—22no—
386C22×C8—22no—
387C22×C8—22no—
388C22×C8—22no—
389C22×C8—22no—
390C22×C8C4×D7114yescomplemented
391C22×C8C22×D7112yes—
392C22×C8C22×D7112yes—
393C4×D4—1428no—
394C4×D4—1428no—
395C4×D4—14216no—
396C2×C4⋊C4—714no—
397C2×C4⋊C4—712no—
398C2×C4⋊C4—712no—
399C2×C4⋊C4—1421no—
400C2×C4⋊C4—1422no—
401C2×C4⋊C4—1422no—
402C2×C4⋊C4—1422no—
403C2×C4⋊C4—1421no—
404C2×C22⋊C4—2844no—
405C2×C22⋊C4—2844no—
406C2×C22⋊C4—1424no—
407C2×C22⋊C4—1424no—
408C2×C22⋊C4—1428no—
409C2×C22⋊C4—1424no—
410C2×C42—22no—
411C2×C42—22no—
412C2×C42D28412yes—
413C2×C42C22×D7112yes—
414#102—1428no—
415#102—1428no—
416C24—2842no—
417C24—2842no—
418C24—2842no—
419C24—2842no—
420C24—2842no—
421C24—2842no—
422C24—2842no—
423C24—2842no—
424C24—2842no—
425C24—2842no—
426C24—1422no—
427C24—1424no—
428C24—1422no—
429C24—1424no—
430C24—1424no—
431C24—1424no—
432C24C2×D28411yescharacteristic
433C2×D4—2842no—
434C2×D4—2844no—
435C2×D4—2842no—
436C2×D4—2844no—
437C2×D4—714no—
438C2×D4—714no—
439C2×D4—714no—
440C2×D4—714no—
441C2×D4—7116no—
442C2×D4—1422no—
443C2×D4—1422no—
444C2×D4—1422no—
445C2×D4—1422no—
446C2×D4—1422no—
447C2×D4—1422no—
448C2×D4—1422no—
449C2×D4—1422no—
450C2×D4—1428no—
451C2×D4—1428no—
452C2×D4—1428no—
453C2×D4—1424no—
454C2×D4—1424no—
455C2×D4—1424no—
456C2×D4—1428no—
457C2×D4—1424no—
458C2×D4—1428no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—28432no—
465C22×C4—2848no—
466C22×C4—2842no—
467C22×C4—2842no—
468C22×C4—2848no—
469C22×C4—2842no—
470C22×C4—2842no—
471C22×C4—2842no—
472C22×C4—14216no—
473C22×C4—1422no—
474C22×C4—1422no—
475C22×C4—1424no—
476C22×C4D56814yescomplemented
477C22×C4C2×C4×D7112yes—
478C22×C4D4×D7112yes—
479C22×C4D4×D7112yes—
480C22×C4D4×D7112yes—
481C22×C4C2×D28412yes—
482C22×C4C2×D28412yes—
483C22×C4C2×D28411yescharacteristic
484C22×C4#1176586512yes—
485C22×C4#1176586612yes—
486C22×C4C23×D7111yescharacteristic
487D8—14232no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—28no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—44no—
495C2×C8C2×C4×D7114yescomplemented
496C4⋊C4—2842no—
497C4⋊C4—714no—
498C4⋊C4—1422no—
499C4⋊C4—1422no—
500C4⋊C4—1422no—
501C4⋊C4—1424no—
502C22⋊C4—2848no—
503C22⋊C4—2848no—
504C22⋊C4—14216no—
505C42—22no—
506C42—22no—
507C42—24no—
508C23—2848no—
509C23—2848no—
510C23—2848no—
511C23—2848no—
512C23—2848no—
513C23—2848no—
514C23—2842no—
515C23—2842no—
516C23—2842no—
517C23—2842no—
518C23—2848no—
519C23—2848no—
520C23—2848no—
521C23—2842no—
522C23—2842no—
523C23—2842no—
524C23—2842no—
525C23—2842no—
526C23—2842no—
527C23—5684no—
528C23—5684no—
529C23—5684no—
530C23—5684no—
531C23—1424no—
532C23—14216no—
533C23—1424no—
534C23—1424no—
535C23—1424no—
536C23C4×D28412yes—
537C23#1455609812yes—
538C23#1455610312yes—
539C23C2×D56812yes—
540C23#1455615012yes—
541C23#1455615612yes—
542C23#1455615612yes—
543C23C22×D28411yescharacteristic
544D4—2848no—
545D4—2842no—
546D4—2842no—
547D4—718no—
548D4—14216no—
549D4—1424no—
550D4—1424no—
551D4—1424no—
552D4—1424no—
553D4—1424no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—44no—
561C2×C4—28432no—
562C2×C4—2848no—
563C2×C4—2842no—
564C2×C4—2842no—
565C2×C4—2842no—
566C2×C4—2842no—
567C2×C4—2842no—
568C2×C4—5684no—
569C2×C4—1424no—
570C2×C4C22⋊C4×D7114yes—
571C2×C4#1455609014yes—
572C2×C4C4×D28412yes—
573C2×C4#1455609812yes—
574C2×C4#1455610114yes—
575C2×C4#1455610312yes—
576C2×C4#1455611314yes—
577C2×C4C2×D56814yescomplemented
578C2×C4C22×C4×D7111yescharacteristic
579C2×C4C2×D4×D7111yescharacteristic
580C2×C4C2×D4×D7111yescharacteristic
581C2×C4C2×D4×D7111yescharacteristic
582C2×C4C2×#1176586511yescharacteristic
583C2×C4C2×#1176586611yescharacteristic
584C8—24no—
585C8—42no—
586C22—2844no—
587C22—28432no—
588C22—2844no—
589C22—2844no—
590C22—2844no—
591C22—2844no—
592C22—2844no—
593C22—2844no—
594C22—2844no—
595C22—2844no—
596C22—56816no—
597C22—5682no—
598C22—5682no—
599C22—5682no—
600C22—5682no—
601C22—1428no—
602C22#1798587714yes—
603C22C4×D56814yes—
604C22#1798611214yes—
605C22#1798613514yes—
606C22#1798614614yes—
607C22#1798619514yes—
608C22#1798621514yes—
609C22C2×C4×D28411yescharacteristic
610C22C2×#1455609811yescharacteristic
611C22C2×#1455610311yescharacteristic
612C22C22×D56811yescharacteristic
613C22C2×#1455615011yescharacteristic
614C22C2×#1455615611yescharacteristic
615C22C2×#1455615611yescharacteristic
616C4—24no—
617C4—42no—
618C4—2844no—
619C4—5682no—
620C4#1798587714yes—
621C4C2×C22⋊C4×D7111yescharacteristic
622C4C2×#1455609011yescharacteristic
623C4C2×#1455610111yescharacteristic
624C4C2×#1455611311yescharacteristic
625C2—2848no—
626C2—5684no—
627C2#2258868718yesdirect product factor
628C2C2×#1798587711yescharacteristic
629C2C2×C4×D56811yescharacteristic
630C2C2×#1798611211yescharacteristic
631C2C2×#1798613511yescharacteristic
632C2C2×#1798614611yescharacteristic
633C2C2×#1798619511yescharacteristic
634C2C2×#1798621511yescharacteristic
subgroupG/H
centerC24C2×D284
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#336×C71C2
socleC24×C71C2×D4
prime radicalC23×D568C2
cosocleC22×C4×C71C24
hypercenterC2×#336D71
hypocenterC71C2×#1879
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C22×C42 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
2. C22×C4×D284C2semidirect
3. C22×#14556050C2semidirect
4. C22×#14556050C2semidirect
5. C2×#17985901C2semidirect
6. #22588687C2direct (N × H)
7. C2×#17985762C2semidirect
8. C2×#336×C71C2semidirect
14. C22×D568C4semidirect
19. C2×C4×D284C22semidirect
27. C2×#14556050C22semidirect
28. C2×#14556050C22semidirect
30. #17985901C22semidirect
31. #17985762C22semidirect
35. #336×C71C22semidirect
49. C22×D284C2×C4semidirect
63. C2×D568C2×C4semidirect
91. C22×C8×C71C2×C4semidirect
135. C2×D284C22⋊C4semidirect
137. C2×D284C22×C4semidirect
168. C22×C4×C71D8semidirect
186. C2×C8×C71C22×C4semidirect
240. C2×C4×C71C2×D8semidirect
264. D284C2×C22⋊C4semidirect
308. C2×C71#1879semidirect
325. C2×#336D71semidirect
326. C71C2×#1879semidirect
353. #336C2×D71semidirect
390. C22×C8C4×D71semidirect
476. C22×C4D568semidirect
495. C2×C8C2×C4×D71semidirect
577. C2×C4C2×D568semidirect
627. C2#22588687direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D5682
C22×C4×D2842
C22×#145560502
C2×#179859012
#225886872
C2×#179857622
C2×#336×C712
C2×#187971