Finite Group Databasebeta

← C2×#22588775 #31999137

Subgroups — C2×#22588775 #31999137

Order 18176.

2310 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11765848C211yesmaximal, characteristic, complemented
2C22×#14556099C211yesmaximal, characteristic
3C22×#14556049C211yesmaximal, characteristic, complemented
4C2×#17985856C211yesmaximal, characteristic, complemented
5#22588775C218yesmaximal, direct product factor
6C2×#17985762C211yesmaximal, characteristic, complemented
7C2×#17985761C211yesmaximal, characteristic, complemented
8C22×#102×C71C211yesmaximal, characteristic, complemented
9C23×C22⋊D71—21no—
10C23×C4×D71C2211yescharacteristic
11C22×#11765848—24no—
12C22×#11765848—24no—
13C22×#11765848—24no—
14C22×#11765848C2214yescomplemented
15C23×C71⋊C8C2211yescharacteristic
16C2×#14556099—22no—
17C2×#14556099—24no—
18C2×#14556099C418yescomplemented
19C2×#14556099C2214yes—
20C22×C4⋊Dic71C2211yescharacteristic
21C22×C4⋊Dic71C2211yescharacteristic
22C22×C22⋊Dic71—21no—
23C22×(C2×D71)⋊C4—21no—
24C2×#14556049—24no—
25C2×#14556049C2214yescomplemented
26#17985856—22no—
27#17985856C2214yescomplemented
28C2×C71⋊#95—21no—
29#17985762C2214yescomplemented
30#17985761C2214yescomplemented
31C23×D4×C71C2211yescharacteristic
32C23×C8×C71C2211yescharacteristic
33C22×C4⋊C4×C71C2211yescharacteristic
34C2×#102×C71—24no—
35C2×#102×C71C2214yescomplemented
36C24×D71D411yescharacteristic
37C22×C22⋊D71—28no—
38C22×C22⋊D71—24no—
39C22×C22⋊D71—44no—
40C22×C22⋊D71—44no—
41C23×Dic71—21no—
42C23×Dic71—21no—
43C23×Dic71D411yescharacteristic
44C22×C4×D71—22no—
45C22×C4×D71—22no—
46C22×C4×D71—22no—
47C22×C4×D71—22no—
48C22×C4×D71C2×C414yes—
49C22×C4×D71D412yes—
50C22×C4×D71D414yes—
51C22×C4×D71C2312yes—
52C2×#11765848—232no—
53C2×#11765848—28no—
54C2×#11765848—28no—
55C2×#11765848—28no—
56C22×C71⋊C8—22no—
57C22×C71⋊C8—22no—
58C22×C71⋊C8D414yescomplemented
59C22×C71⋊C8C2312yes—
60#14556099—216no—
61#14556099—28no—
62#14556099C2×C4116yescomplemented
63C2×C4⋊Dic71—21no—
64C2×C4⋊Dic71—22no—
65C2×C4⋊Dic71—22no—
66C2×C4⋊Dic71—21no—
67C2×C4⋊Dic71—22no—
68C2×C4⋊Dic71C2×C414yes—
69C2×C4⋊Dic71C2312yes—
70C2×C4⋊Dic71C2312yes—
71C2×C22⋊Dic71—28no—
72C2×C22⋊Dic71—24no—
73C2×(C2×D71)⋊C4—24no—
74C2×(C2×D71)⋊C4—44no—
75#14556049—28no—
76C71⋊#95—24no—
77C25×C71—21no—
78C22×D4×C71—22no—
79C22×D4×C71—22no—
80C22×D4×C71—21no—
81C22×D4×C71—22no—
82C22×D4×C71—22no—
83C22×D4×C71—22no—
84C22×D4×C71C2×C414yes—
85C22×D4×C71C2312yes—
86C23×C4×C71—21no—
87C23×C4×C71C2311yescharacteristic
88C22×C8×C71—22no—
89C22×C8×C71—22no—
90C22×C8×C71D414yescomplemented
91C22×C8×C71C2312yes—
92C2×C4⋊C4×C71—22no—
93C2×C4⋊C4×C71—22no—
94C2×C4⋊C4×C71—21no—
95C2×C4⋊C4×C71C2312yes—
96#102×C71—28no—
97C23×D71—22no—
98C23×D71—22no—
99C23×D71—22no—
100C23×D71—22no—
101C23×D71—22no—
102C23×D71—22no—
103C23×D71C22⋊C414yes—
104C23×D71C2×D412yes—
105C2×C22⋊D71—216no—
106C2×C22⋊D71—48no—
107C2×C22⋊D71—48no—
108C2×C22⋊D71—432no—
109C22×Dic71—22no—
110C22×Dic71—22no—
111C22×Dic71—24no—
112C22×Dic71—22no—
113C22×Dic71—22no—
114C22×Dic71—42no—
115C22×Dic71—42no—
116C22×Dic71—42no—
117C22×Dic71—42no—
118C22×Dic71—42no—
119C22×Dic71C22⋊C414yes—
120C22×Dic71C2×D412yes—
121C2×C4×D71—28no—
122C2×C4×D71—28no—
123C2×C4×D71—28no—
124C2×C4×D71—24no—
125C2×C4×D71—24no—
126C2×C4×D71—28no—
127C2×C4×D71—22no—
128C2×C4×D71—22no—
129C2×C4×D71—22no—
130C2×C4×D71—22no—
131C2×C4×D71C22⋊C418yes—
132C2×C4×D71C22×C414yes—
133C2×C4×D71C2×D414yes—
134#11765848—232no—
135C2×C71⋊C8—28no—
136C2×C71⋊C8—22no—
137C2×C71⋊C8—22no—
138C2×C71⋊C8C2×D414yescomplemented
139C4⋊Dic71—24no—
140C4⋊Dic71—22no—
141C4⋊Dic71—22no—
142C4⋊Dic71—22no—
143C4⋊Dic71—42no—
144C4⋊Dic71C22×C414yes—
145C22⋊Dic71—216no—
146(C2×D71)⋊C4—48no—
147C24×C71—24no—
148C24×C71—22no—
149C24×C71—42no—
150C24×C71—42no—
151C24×C71—42no—
152C24×C71—42no—
153C24×C71—42no—
154C24×C71—42no—
155C24×C71C2×D411yescharacteristic
156C2×D4×C71—24no—
157C2×D4×C71—28no—
158C2×D4×C71—28no—
159C2×D4×C71—22no—
160C2×D4×C71—22no—
161C2×D4×C71—22no—
162C2×D4×C71—22no—
163C2×D4×C71—22no—
164C2×D4×C71—44no—
165C2×D4×C71—42no—
166C2×D4×C71—42no—
167C2×D4×C71—44no—
168C2×D4×C71C22×C414yes—
169C22×C4×C71—22no—
170C22×C4×C71—42no—
171C22×C4×C71—42no—
172C22×C4×C71—42no—
173C22×C4×C71C22×C412yes—
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D412yes—
176C22×C4×C71C2×D412yes—
177C22×C4×C71C2×D412yes—
178C22×C4×C71#4112yes—
179C22×C4×C71#4112yes—
180C22×C4×C71C2411yescharacteristic
181C2×C8×C71—28no—
182C2×C8×C71—22no—
183C2×C8×C71—22no—
184C2×C8×C71C2×D414yescomplemented
185C4⋊C4×C71—22no—
186C4⋊C4×C71—22no—
187C4⋊C4×C71—42no—
188C23×C71—24no—
189C23×C71—42no—
190C23×C71—42no—
191C23×C71—42no—
192C23×C71—42no—
193C23×C71—42no—
194C23×C71—42no—
195C23×C71—48no—
196C23×C71—48no—
197C23×C71—48no—
198C23×C71—84no—
199C23×C71—84no—
200C23×C71—84no—
201C23×C71—84no—
202C23×C71C2×C22⋊C412yes—
203C23×C71#12112yes—
204C23×C71#12112yes—
205C23×C71AGL(1,ℤ/8)12yes—
206C23×C71AGL(1,ℤ/8)12yes—
207C23×C71AGL(1,ℤ/8)12yes—
208C23×C71#13712yes—
209C23×C71C22×D411yescharacteristic
210C22×D71—22no—
211C22×D71—22no—
212C22×D71—22no—
213C22×D71—22no—
214C22×D71—22no—
215C22×D71—22no—
216C22×D71—28no—
217C22×D71—28no—
218C22×D71—28no—
219C22×D71—44no—
220C22×D71—44no—
221C22×D71—44no—
222C22×D71—44no—
223C22×D71C2×C22⋊C414yes—
224D4×C71—24no—
225D4×C71—24no—
226D4×C71—42no—
227D4×C71—42no—
228D4×C71—48no—
229C2×C4×C71—42no—
230C2×C4×C71—42no—
231C2×C4×C71—42no—
232C2×C4×C71—84no—
233C2×C4×C71C2×C22⋊C412yes—
234C2×C4×C71C4×D414yes—
235C2×C4×C71C4×D414yes—
236C2×C4×C71#12014yes—
237C2×C4×C71#12112yes—
238C2×C4×C71#12112yes—
239C2×C4×C71#12314yes—
240C2×C4×C71C23×C411yescharacteristic
241C2×C4×C71C22×D411yescharacteristic
242C2×C4×C71C22×D411yescharacteristic
243C2×C4×C71C22×D411yescharacteristic
244C2×C4×C71C2×#4111yescharacteristic
245C2×C4×C71C2×#4111yescharacteristic
246C22⋊D71—432no—
247C2×Dic71—22no—
248C2×Dic71—22no—
249C2×Dic71—28no—
250C2×Dic71—24no—
251C2×Dic71—42no—
252C2×Dic71—42no—
253C2×Dic71—48no—
254C2×Dic71—42no—
255C2×Dic71—42no—
256C2×Dic71—42no—
257C2×Dic71—84no—
258C2×Dic71C2×C22⋊C414yes—
259C4×D71—24no—
260C4×D71—24no—
261C4×D71—24no—
262C4×D71—24no—
263C4×D71—216no—
264C8×C71—24no—
265C71⋊C8—24no—
266C22×C71—44no—
267C22×C71—44no—
268C22×C71—44no—
269C22×C71—816no—
270C22×C71—82no—
271C22×C71—82no—
272C22×C71—82no—
273C22×C71—82no—
274C22×C71#38614yes—
275C22×C71#41714yes—
276C22×C71#41814yes—
277C22×C71#46814yes—
278C22×C71#46814yes—
279C22×C71#46914yes—
280C22×C71#46914yes—
281C22×C71C22×C22⋊C411yescharacteristic
282C22×C71C2×#12111yescharacteristic
283C22×C71C2×#12111yescharacteristic
284C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C71C2×#13711yescharacteristic
288C2×D71—24no—
289C2×D71—24no—
290C2×D71—24no—
291C2×D71—416no—
292C2×D71—42no—
293C2×D71—42no—
294C2×D71—42no—
295C2×D71—42no—
296C4×C71—82no—
297C4×C71#38614yes—
298C4×C71C2×C4×D411yescharacteristic
299C4×C71C2×C4×D411yescharacteristic
300C4×C71C2×#12011yescharacteristic
301C4×C71C2×#12311yescharacteristic
302Dic71—24no—
303Dic71—44no—
304Dic71—82no—
305C2×#1895—711nomaximal, Sylow
306C2×C71—84no—
307C2×C71#189518yescomplemented
308C2×C71C2×#38611yescharacteristic
309C2×C71C2×#41811yescharacteristic
310C2×C71C2×#41711yescharacteristic
311C2×C71C2×#46811yescharacteristic
312C2×C71C2×#46811yescharacteristic
313C2×C71C2×#46911yescharacteristic
314C2×C71C2×#46911yescharacteristic
315D71—44no—
316C23×M16—711no—
317C22×#121—711no—
318C22×#102—711no—
319C22×#102D7111yescharacteristic, complemented
320C2×#380—711no—
321#1895—718no—
322C2×#340—711no—
323C2×#340—711no—
324C71C2×#189511yescharacteristic, complemented, Sylow
325C23×D4—1421no—
326C23×D4C2×D7111yescharacteristic
327C24×C4—711no—
328C22×M16—714no—
329C22×M16—1424no—
330C22×M16—1424no—
331C22×M16—1424no—
332C23×C8—711no—
333C23×C8C2×D7111yescharacteristic
334C2×#121—718no—
335C2×#121—714no—
336C2×#121—1424no—
337C2×#121—1422no—
338C22×C4⋊C4—711no—
339C22×C4⋊C4—711no—
340C22×C4⋊C4C2×D7111yescharacteristic
341C22×C22⋊C4—1421no—
342C22×C22⋊C4—1421no—
343C2×#102—24no—
344C2×#102—714no—
345C2×#102—1424no—
346C2×#102C2×D7114yescomplemented
347#380—714no—
348#380—1422no—
349C2×#95—1421no—
350#340—714no—
351#340—714no—
352C25—21no—
353C25—711no—
354C22×D4—22no—
355C22×D4—22no—
356C22×D4—21no—
357C22×D4—22no—
358C22×D4—22no—
359C22×D4—22no—
360C22×D4—2844no—
361C22×D4—2844no—
362C22×D4—1424no—
363C22×D4—1428no—
364C22×D4C4×D7114yes—
365C22×D4C22×D7112yes—
366C23×C4—21no—
367C23×C4—711no—
368C23×C4—714no—
369C23×C4—714no—
370C23×C4—712no—
371C23×C4—712no—
372C23×C4—1421no—
373C23×C4—1421no—
374C23×C4—1422no—
375C23×C4—1422no—
376C23×C4—1422no—
377C23×C4—1422no—
378C23×C4C22×D7111yescharacteristic
379C2×M16—1428no—
380C2×M16—1428no—
381C2×M16—14232no—
382C2×M16—1428no—
383C22×C8—22no—
384C22×C8—22no—
385C22×C8—714no—
386C22×C8—712no—
387C22×C8—1422no—
388C22×C8—1422no—
389C22×C8C22⋊D7114yescomplemented
390C22×C8C22×D7112yes—
391#121—7116no—
392#121—1428no—
393#121—14216no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—714no—
398C2×C4⋊C4—712no—
399C2×C4⋊C4—712no—
400C2×C4⋊C4—1421no—
401C2×C4⋊C4—1422no—
402C2×C4⋊C4—1422no—
403C2×C4⋊C4—1422no—
404C2×C4⋊C4—1421no—
405C2×C4⋊C4C22×D7112yes—
406C2×C22⋊C4—2844no—
407C2×C22⋊C4—1424no—
408C2×C22⋊C4—1424no—
409C2×C22⋊C4—1428no—
410#102—28no—
411#102—1428no—
412#95—1424no—
413C24—22no—
414C24—24no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—712no—
422C24—714no—
423C24—1422no—
424C24—1422no—
425C24—1422no—
426C24—1422no—
427C24—1422no—
428C24—1422no—
429C24D4×D7111yescharacteristic
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—24no—
436C2×D4—28no—
437C2×D4—28no—
438C2×D4—44no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—28432no—
443C2×D4—2848no—
444C2×D4—2848no—
445C2×D4—14216no—
446C2×D4C2×C4×D7114yes—
447C22×C4—22no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—2842no—
452C22×C4—2842no—
453C22×C4—2842no—
454C22×C4—2842no—
455C22×C4—2842no—
456C22×C4—712no—
457C22×C4—714no—
458C22×C4—714no—
459C22×C4—714no—
460C22×C4—718no—
461C22×C4—1422no—
462C22×C4—1422no—
463C22×C4—1424no—
464C22×C4—1422no—
465C22×C4—1422no—
466C22×C4—1422no—
467C22×C4—1422no—
468C22×C4—1422no—
469C22×C4—1422no—
470C22×C4—1428no—
471C22×C4—1428no—
472C22×C4—1428no—
473C22×C4—1424no—
474C22×C4—1424no—
475C22×C4—1428no—
476C22×C4C2×C4×D7112yes—
477C22×C4D4×D7112yes—
478C22×C4D4×D7112yes—
479C22×C4D4×D7111yescharacteristic
480C22×C4C2×C22⋊D7112yes—
481C22×C4#1176586412yes—
482C22×C4#1176586512yes—
483C22×C4C23×D7111yescharacteristic
484M16—14232no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—28no—
488C2×C8—714no—
489C2×C8—1422no—
490C2×C8—1422no—
491C2×C8—1428no—
492C2×C8C2×C22⋊D7114yescomplemented
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—2842no—
497C4⋊C4—714no—
498C4⋊C4—1422no—
499C4⋊C4—1422no—
500C4⋊C4—1422no—
501C4⋊C4—1424no—
502C22⋊C4—2848no—
503C22⋊C4—14216no—
504C23—24no—
505C23—48no—
506C23—42no—
507C23—42no—
508C23—48no—
509C23—48no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—84no—
515C23—84no—
516C23—84no—
517C23—84no—
518C23—2844no—
519C23—2844no—
520C23—2844no—
521C23—2844no—
522C23—714no—
523C23—1428no—
524C23—1422no—
525C23—1422no—
526C23—1428no—
527C23—1428no—
528C23—1422no—
529C23—1422no—
530C23—1422no—
531C23—1422no—
532C23C22⋊C4×D7112yes—
533C23#1455609912yes—
534C23#1455610112yes—
535C23#1455615212yes—
536C23#1455615212yes—
537C23#1455615512yes—
538C23#1455615712yes—
539C23C2×D4×D7111yescharacteristic
540D4—24no—
541D4—24no—
542D4—48no—
543D4—42no—
544D4—42no—
545D4—28432no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—84no—
550C2×C4—2848no—
551C2×C4—2842no—
552C2×C4—2842no—
553C2×C4—2842no—
554C2×C4—2842no—
555C2×C4—2842no—
556C2×C4—5684no—
557C2×C4—714no—
558C2×C4—1424no—
559C2×C4—1428no—
560C2×C4—1422no—
561C2×C4—1422no—
562C2×C4—14216no—
563C2×C4—1424no—
564C2×C4—1424no—
565C2×C4—1424no—
566C2×C4—1424no—
567C2×C4C22⋊C4×D7112yes—
568C2×C4(C22⋊D71)⋊C414yes—
569C2×C4C4×C22⋊D7114yes—
570C2×C4#1455609614yes—
571C2×C4#1455609912yes—
572C2×C4#1455610112yes—
573C2×C4#1455611014yes—
574C2×C4C22×C4×D7111yescharacteristic
575C2×C4C2×D4×D7111yescharacteristic
576C2×C4C2×D4×D7111yescharacteristic
577C2×C4C22×C22⋊D7111yescharacteristic
578C2×C4C2×#1176586411yescharacteristic
579C2×C4C2×#1176586511yescharacteristic
580C8—24no—
581C8—1424no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—816no—
590C22—2842no—
591C22—2842no—
592C22—2842no—
593C22—2842no—
594C22—28416no—
595C22—1424no—
596C22—1424no—
597C22—1424no—
598C22#1798587514yes—
599C22#1798599714yes—
600C22#1798600214yes—
601C22#1798624214yes—
602C22#1798624414yes—
603C22#1798624914yes—
604C22#1798625114yes—
605C22C2×C22⋊C4×D7111yescharacteristic
606C22C2×#1455609911yescharacteristic
607C22C2×#1455610111yescharacteristic
608C22C2×#1455615211yescharacteristic
609C22C2×#1455615211yescharacteristic
610C22C2×#1455615511yescharacteristic
611C22C2×#1455615711yescharacteristic
612C4—82no—
613C4—2844no—
614C4—5682no—
615C4—1424no—
616C4#1798587514yes—
617C4C2×(C22⋊D71)⋊C411yescharacteristic
618C4C2×C4×C22⋊D7111yescharacteristic
619C4C2×#1455609611yescharacteristic
620C4C2×#1455611011yescharacteristic
621C2—84no—
622C2—2844no—
623C2#2258877518yesdirect product factor
624C2C2×#1798587511yescharacteristic
625C2C2×#1798600211yescharacteristic
626C2C2×#1798599711yescharacteristic
627C2C2×#1798624211yescharacteristic
628C2C2×#1798624411yescharacteristic
629C2C2×#1798624911yescharacteristic
630C2C2×#1798625111yescharacteristic
subgroupG/H
centerC24D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC22×#102×C71C2
socleC24×C71C2×D4
prime radicalC22×#14556099C2
cosocleC22×C4×C71C24
hypercenterC22×#102D71
hypocenterC71C2×#1895
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
1. C23×#11765848C2semidirect
3. C22×#14556049C2semidirect
4. C2×#17985856C2semidirect
5. #22588775C2direct (N × H)
6. C2×#17985762C2semidirect
7. C2×#17985761C2semidirect
8. C22×#102×C71C2semidirect
14. C22×#11765848C22semidirect
18. C2×#14556099C4semidirect
25. C2×#14556049C22semidirect
27. #17985856C22semidirect
29. #17985762C22semidirect
30. #17985761C22semidirect
35. C2×#102×C71C22semidirect
58. C22×C71⋊C8D4semidirect
62. #14556099C2×C4semidirect
90. C22×C8×C71D4semidirect
138. C2×C71⋊C8C2×D4semidirect
184. C2×C8×C71C2×D4semidirect
307. C2×C71#1895semidirect
319. C22×#102D71semidirect
324. C71C2×#1895semidirect
346. C2×#102C2×D71semidirect
389. C22×C8C22⋊D71semidirect
492. C2×C8C2×C22⋊D71semidirect
623. C2#22588775direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117658482
C22×#145560992
C22×#145560492
C2×#179858562
#225887752
C2×#179857622
C2×#179857612
C22×#102×C712
C2×#189571