Finite Group Databasebeta

← C2×#22588659 #31998661

Subgroups — C2×#22588659 #31998661

Order 18176.

2310 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556099C211yesmaximal, characteristic
2C22×#14556086C211yesmaximal, characteristic, complemented
3C22×#14556049C211yesmaximal, characteristic, complemented
4C22×(C2×D71)⋊C8C211yesmaximal, characteristic, complemented
5#22588659C218yesmaximal, direct product factor
6C2×#17985762C211yesmaximal, characteristic, complemented
7C2×#17985761C211yesmaximal, characteristic, complemented
8C22×#102×C71C211yesmaximal, characteristic, complemented
9C23×C22⋊D71—21no—
10C23×C4×D71C2211yescharacteristic
11C23×C71⋊C8C2211yescharacteristic
12C2×#14556099—22no—
13C2×#14556099—24no—
14C2×#14556099C418yescomplemented
15C2×#14556099C2214yes—
16C2×#14556086—24no—
17C2×#14556086—24no—
18C2×#14556086—22no—
19C2×#14556086C2214yescomplemented
20C22×C4⋊Dic71C2211yescharacteristic
21C22×C4⋊Dic71C2211yescharacteristic
22C22×C22⋊Dic71—21no—
23C22×(C2×D71)⋊C4—21no—
24C22×C4×Dic71—21no—
25C2×#14556049—24no—
26C2×#14556049C2214yescomplemented
27C2×(C2×D71)⋊C8—24no—
28C2×(C2×D71)⋊C8C2214yescomplemented
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—
52C22×C71⋊C8—22no—
53C22×C71⋊C8—22no—
54C22×C71⋊C8—22no—
55C22×C71⋊C8C2312yes—
56#14556099—216no—
57#14556099—28no—
58#14556099C2×C4116yescomplemented
59#14556086—216no—
60#14556086—28no—
61#14556086—28no—
62C2×C4⋊Dic71—21no—
63C2×C4⋊Dic71—22no—
64C2×C4⋊Dic71—21no—
65C2×C4⋊Dic71—22no—
66C2×C4⋊Dic71C2×C414yes—
67C2×C4⋊Dic71D414yescomplemented
68C2×C4⋊Dic71C2312yes—
69C2×C4⋊Dic71C2312yes—
70C2×C22⋊Dic71—28no—
71C2×C22⋊Dic71—24no—
72C2×(C2×D71)⋊C4—28no—
73C2×(C2×D71)⋊C4—24no—
74C2×C4×Dic71—28no—
75C2×C4×Dic71—24no—
76#14556049—28no—
77(C2×D71)⋊C8—28no—
78C25×C71—21no—
79C22×D4×C71—22no—
80C22×D4×C71—22no—
81C22×D4×C71—21no—
82C22×D4×C71—22no—
83C22×D4×C71—22no—
84C22×D4×C71—22no—
85C22×D4×C71C2×C414yes—
86C22×D4×C71C2312yes—
87C23×C4×C71—21no—
88C23×C4×C71C2311yescharacteristic
89C22×C8×C71—22no—
90C22×C8×C71—22no—
91C22×C8×C71—22no—
92C22×C8×C71C2312yes—
93C2×C4⋊C4×C71—21no—
94C2×C4⋊C4×C71—22no—
95C2×C4⋊C4×C71D414yescomplemented
96C2×C4⋊C4×C71C2312yes—
97#102×C71—28no—
98C23×D71—22no—
99C23×D71—22no—
100C23×D71—22no—
101C23×D71—22no—
102C23×D71—22no—
103C23×D71—22no—
104C23×D71C22⋊C414yes—
105C23×D71C2×D412yes—
106C2×C22⋊D71—216no—
107C2×C22⋊D71—48no—
108C2×C22⋊D71—48no—
109C2×C22⋊D71—432no—
110C22×Dic71—22no—
111C22×Dic71—22no—
112C22×Dic71—24no—
113C22×Dic71—24no—
114C22×Dic71—22no—
115C22×Dic71—22no—
116C22×Dic71—42no—
117C22×Dic71—42no—
118C22×Dic71—42no—
119C22×Dic71—42no—
120C22×Dic71C22⋊C414yes—
121C22×Dic71C2×D412yes—
122C2×C4×D71—28no—
123C2×C4×D71—28no—
124C2×C4×D71—24no—
125C2×C4×D71—28no—
126C2×C4×D71—28no—
127C2×C4×D71—24no—
128C2×C4×D71—22no—
129C2×C4×D71—22no—
130C2×C4×D71—22no—
131C2×C4×D71—22no—
132C2×C4×D71C22⋊C418yes—
133C2×C4×D71C22×C414yes—
134C2×C4×D71C2×D414yes—
135C2×C71⋊C8—22no—
136C2×C71⋊C8—22no—
137C2×C71⋊C8—22no—
138C2×C71⋊C8—44no—
139C4⋊Dic71—24no—
140C4⋊Dic71—24no—
141C4⋊Dic71—22no—
142C4⋊Dic71—22no—
143C4⋊Dic71C22×C414yes—
144C4⋊Dic71C2×D414yescomplemented
145C22⋊Dic71—216no—
146(C2×D71)⋊C4—216no—
147C4×Dic71—216no—
148C24×C71—24no—
149C24×C71—22no—
150C24×C71—42no—
151C24×C71—42no—
152C24×C71—42no—
153C24×C71—42no—
154C24×C71—42no—
155C24×C71—42no—
156C24×C71C2×D411yescharacteristic
157C2×D4×C71—24no—
158C2×D4×C71—28no—
159C2×D4×C71—28no—
160C2×D4×C71—22no—
161C2×D4×C71—22no—
162C2×D4×C71—22no—
163C2×D4×C71—22no—
164C2×D4×C71—22no—
165C2×D4×C71—42no—
166C2×D4×C71—44no—
167C2×D4×C71—44no—
168C2×D4×C71—42no—
169C2×D4×C71C22×C414yes—
170C22×C4×C71—24no—
171C22×C4×C71—22no—
172C22×C4×C71—42no—
173C22×C4×C71—42no—
174C22×C4×C71C22×C412yes—
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2×D412yes—
177C22×C4×C71C2×D412yes—
178C22×C4×C71C2×D412yes—
179C22×C4×C71#4112yes—
180C22×C4×C71#4112yes—
181C22×C4×C71C2411yescharacteristic
182C2×C8×C71—22no—
183C2×C8×C71—22no—
184C2×C8×C71—22no—
185C2×C8×C71—44no—
186C4⋊C4×C71—24no—
187C4⋊C4×C71—22no—
188C4⋊C4×C71C2×D414yescomplemented
189C23×C71—24no—
190C23×C71—42no—
191C23×C71—42no—
192C23×C71—42no—
193C23×C71—42no—
194C23×C71—42no—
195C23×C71—42no—
196C23×C71—48no—
197C23×C71—48no—
198C23×C71—48no—
199C23×C71—84no—
200C23×C71—84no—
201C23×C71—84no—
202C23×C71—84no—
203C23×C71C2×C22⋊C412yes—
204C23×C71#12012yes—
205C23×C71#12312yes—
206C23×C71#13512yes—
207C23×C71#13512yes—
208C23×C71AGL(1,ℤ/8)12yes—
209C23×C71AGL(1,ℤ/8)12yes—
210C23×C71C22×D411yescharacteristic
211C22×D71—22no—
212C22×D71—22no—
213C22×D71—22no—
214C22×D71—22no—
215C22×D71—22no—
216C22×D71—22no—
217C22×D71—28no—
218C22×D71—28no—
219C22×D71—28no—
220C22×D71—44no—
221C22×D71—44no—
222C22×D71—44no—
223C22×D71—44no—
224C22×D71C2×C22⋊C414yes—
225D4×C71—24no—
226D4×C71—24no—
227D4×C71—42no—
228D4×C71—42no—
229D4×C71—48no—
230C2×C4×C71—24no—
231C2×C4×C71—42no—
232C2×C4×C71—42no—
233C2×C4×C71—48no—
234C2×C4×C71C2×C22⋊C412yes—
235C2×C4×C71C4×D414yes—
236C2×C4×C71C4×D414yes—
237C2×C4×C71#12012yes—
238C2×C4×C71#12114yes—
239C2×C4×C71#12114yes—
240C2×C4×C71#12312yes—
241C2×C4×C71C23×C411yescharacteristic
242C2×C4×C71C22×D411yescharacteristic
243C2×C4×C71C22×D411yescharacteristic
244C2×C4×C71C22×D411yescharacteristic
245C2×C4×C71C2×#4111yescharacteristic
246C2×C4×C71C2×#4111yescharacteristic
247C22⋊D71—432no—
248C2×Dic71—22no—
249C2×Dic71—22no—
250C2×Dic71—28no—
251C2×Dic71—24no—
252C2×Dic71—24no—
253C2×Dic71—42no—
254C2×Dic71—42no—
255C2×Dic71—42no—
256C2×Dic71—42no—
257C2×Dic71—48no—
258C2×Dic71—48no—
259C2×Dic71C2×C22⋊C414yes—
260C4×D71—24no—
261C4×D71—24no—
262C4×D71—24no—
263C4×D71—24no—
264C4×D71—216no—
265C8×C71—42no—
266C71⋊C8—42no—
267C22×C71—44no—
268C22×C71—44no—
269C22×C71—44no—
270C22×C71—82no—
271C22×C71—82no—
272C22×C71—816no—
273C22×C71—82no—
274C22×C71—82no—
275C22×C71#38614yes—
276C22×C71#41614yes—
277C22×C71#41814yes—
278C22×C71#44914yes—
279C22×C71#44914yes—
280C22×C71#48214yes—
281C22×C71#48214yes—
282C22×C71C22×C22⋊C411yescharacteristic
283C22×C71C2×#12011yescharacteristic
284C22×C71C2×#12311yescharacteristic
285C22×C71C2×#13511yescharacteristic
286C22×C71C2×#13511yescharacteristic
287C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
289C2×D71—24no—
290C2×D71—24no—
291C2×D71—24no—
292C2×D71—42no—
293C2×D71—42no—
294C2×D71—416no—
295C2×D71—42no—
296C2×D71—42no—
297C4×C71—44no—
298C4×C71#38614yes—
299C4×C71C2×C4×D411yescharacteristic
300C4×C71C2×C4×D411yescharacteristic
301C4×C71C2×#12111yescharacteristic
302C4×C71C2×#12111yescharacteristic
303Dic71—24no—
304Dic71—44no—
305Dic71—44no—
306C2×#1875—711nomaximal, Sylow
307C2×C71—84no—
308C2×C71#187518yescomplemented
309C2×C71C2×#38611yescharacteristic
310C2×C71C2×#41611yescharacteristic
311C2×C71C2×#41811yescharacteristic
312C2×C71C2×#44911yescharacteristic
313C2×C71C2×#44911yescharacteristic
314C2×C71C2×#48211yescharacteristic
315C2×C71C2×#48211yescharacteristic
316D71—44no—
317C22×#121—711no—
318C22×#117—711no—
319C22×#102—711no—
320C22×#102D7111yescharacteristic, complemented
321C22×C22⋊C8—711no—
322#1875—718no—
323C2×#340—711no—
324C2×#340—711no—
325C71C2×#187511yescharacteristic, complemented, Sylow
326C23×D4—1421no—
327C23×D4C2×D7111yescharacteristic
328C24×C4—711no—
329C23×C8—711no—
330C23×C8C2×D7111yescharacteristic
331C2×#121—718no—
332C2×#121—714no—
333C2×#121—1422no—
334C2×#121—1424no—
335C2×#117—714no—
336C2×#117—1424no—
337C2×#117—1422no—
338C2×#117—1424no—
339C22×C4⋊C4—711no—
340C22×C4⋊C4—711no—
341C22×C4⋊C4C2×D7111yescharacteristic
342C22×C22⋊C4—1421no—
343C22×C22⋊C4—1421no—
344C22×C42—1421no—
345C2×#102—24no—
346C2×#102—714no—
347C2×#102—1424no—
348C2×#102C2×D7114yescomplemented
349C2×C22⋊C8—714no—
350C2×C22⋊C8—1424no—
351#340—714no—
352#340—714no—
353C25—21no—
354C25—711no—
355C22×D4—22no—
356C22×D4—22no—
357C22×D4—21no—
358C22×D4—22no—
359C22×D4—22no—
360C22×D4—22no—
361C22×D4—2844no—
362C22×D4—2844no—
363C22×D4—1424no—
364C22×D4—1428no—
365C22×D4C4×D7114yes—
366C22×D4C22×D7112yes—
367C23×C4—21no—
368C23×C4—711no—
369C23×C4—714no—
370C23×C4—712no—
371C23×C4—714no—
372C23×C4—712no—
373C23×C4—1421no—
374C23×C4—1421no—
375C23×C4—1422no—
376C23×C4—1422no—
377C23×C4—1422no—
378C23×C4—1422no—
379C23×C4C22×D7111yescharacteristic
380C22×C8—22no—
381C22×C8—22no—
382C22×C8—22no—
383C22×C8—712no—
384C22×C8—1422no—
385C22×C8—1422no—
386C22×C8—1422no—
387C22×C8C22×D7112yes—
388#121—7116no—
389#121—1428no—
390#121—14216no—
391#117—1428no—
392#117—14216no—
393#117—1428no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—714no—
397C2×C4⋊C4—714no—
398C2×C4⋊C4—712no—
399C2×C4⋊C4—712no—
400C2×C4⋊C4—1421no—
401C2×C4⋊C4—1422no—
402C2×C4⋊C4—1421no—
403C2×C4⋊C4—1422no—
404C2×C4⋊C4C22⋊D7114yescomplemented
405C2×C4⋊C4C22×D7112yes—
406C2×C22⋊C4—1424no—
407C2×C22⋊C4—1428no—
408C2×C22⋊C4—1424no—
409C2×C22⋊C4—1428no—
410C2×C42—1424no—
411C2×C42—1428no—
412#102—28no—
413#102—1428no—
414C22⋊C8—1428no—
415C24—22no—
416C24—24no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—712no—
424C24—714no—
425C24—1422no—
426C24—1422no—
427C24—1422no—
428C24—1422no—
429C24—1422no—
430C24—1422no—
431C24D4×D7111yescharacteristic
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—24no—
438C2×D4—28no—
439C2×D4—28no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—42no—
444C2×D4—28432no—
445C2×D4—2848no—
446C2×D4—2848no—
447C2×D4—14216no—
448C2×D4C2×C4×D7114yes—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—2842no—
454C22×C4—2842no—
455C22×C4—2842no—
456C22×C4—2842no—
457C22×C4—712no—
458C22×C4—714no—
459C22×C4—714no—
460C22×C4—714no—
461C22×C4—718no—
462C22×C4—1422no—
463C22×C4—1422no—
464C22×C4—1424no—
465C22×C4—1424no—
466C22×C4—1422no—
467C22×C4—1422no—
468C22×C4—1422no—
469C22×C4—1422no—
470C22×C4—1422no—
471C22×C4—1422no—
472C22×C4—1428no—
473C22×C4—1428no—
474C22×C4—1424no—
475C22×C4—1428no—
476C22×C4—1428no—
477C22×C4—1424no—
478C22×C4C2×C4×D7112yes—
479C22×C4D4×D7112yes—
480C22×C4D4×D7112yes—
481C22×C4D4×D7111yescharacteristic
482C22×C4C2×C22⋊D7112yes—
483C22×C4#1176586412yes—
484C22×C4#1176586512yes—
485C22×C4C23×D7111yescharacteristic
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—44no—
490C2×C8—2844no—
491C2×C8—1422no—
492C2×C8—1422no—
493C2×C8—1422no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—714no—
497C4⋊C4—714no—
498C4⋊C4—1422no—
499C4⋊C4—1422no—
500C4⋊C4—1424no—
501C4⋊C4—1424no—
502C4⋊C4C2×C22⋊D7114yescomplemented
503C22⋊C4—14216no—
504C22⋊C4—14216no—
505C42—14216no—
506C23—24no—
507C23—48no—
508C23—42no—
509C23—42no—
510C23—48no—
511C23—48no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—84no—
517C23—84no—
518C23—84no—
519C23—84no—
520C23—2844no—
521C23—2844no—
522C23—2844no—
523C23—2844no—
524C23—714no—
525C23—1428no—
526C23—1422no—
527C23—1422no—
528C23—1428no—
529C23—1428no—
530C23—1422no—
531C23—1422no—
532C23—1422no—
533C23—1422no—
534C23C22⋊C4×D7112yes—
535C23#1455609612yes—
536C23#1455611012yes—
537C23#1455614712yes—
538C23#1455614912yes—
539C23#1455615212yes—
540C23#1455615512yes—
541C23C2×D4×D7111yescharacteristic
542D4—24no—
543D4—24no—
544D4—48no—
545D4—42no—
546D4—42no—
547D4—28432no—
548C2×C4—24no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—2848no—
553C2×C4—2848no—
554C2×C4—2842no—
555C2×C4—2842no—
556C2×C4—2842no—
557C2×C4—2842no—
558C2×C4—714no—
559C2×C4—1424no—
560C2×C4—1424no—
561C2×C4—1428no—
562C2×C4—1422no—
563C2×C4—1422no—
564C2×C4—14216no—
565C2×C4—1424no—
566C2×C4—1424no—
567C2×C4—1424no—
568C2×C4—1424no—
569C2×C4C22⋊C4×D7112yes—
570C2×C4(C22⋊D71)⋊C414yes—
571C2×C4C4×C22⋊D7114yes—
572C2×C4#1455609612yes—
573C2×C4#1455609914yes—
574C2×C4#1455610114yes—
575C2×C4#1455611012yes—
576C2×C4C22×C4×D7111yescharacteristic
577C2×C4C2×D4×D7111yescharacteristic
578C2×C4C2×D4×D7111yescharacteristic
579C2×C4C22×C22⋊D7111yescharacteristic
580C2×C4C2×#1176586411yescharacteristic
581C2×C4C2×#1176586511yescharacteristic
582C8—42no—
583C8—2842no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—816no—
592C22—2842no—
593C22—2842no—
594C22—2842no—
595C22—2842no—
596C22—28416no—
597C22—1424no—
598C22—1424no—
599C22—1424no—
600C22#1798587514yes—
601C22#1798599214yes—
602C22#1798600214yes—
603C22#1798614114yes—
604C22#1798614314yes—
605C22#1798632114yes—
606C22#1798632214yes—
607C22C2×C22⋊C4×D7111yescharacteristic
608C22C2×#1455609611yescharacteristic
609C22C2×#1455611011yescharacteristic
610C22C2×#1455614711yescharacteristic
611C22C2×#1455614911yescharacteristic
612C22C2×#1455615211yescharacteristic
613C22C2×#1455615511yescharacteristic
614C4—44no—
615C4—2844no—
616C4—2844no—
617C4—1424no—
618C4#1798587514yes—
619C4C2×(C22⋊D71)⋊C411yescharacteristic
620C4C2×C4×C22⋊D7111yescharacteristic
621C4C2×#1455609911yescharacteristic
622C4C2×#1455610111yescharacteristic
623C2—84no—
624C2—2844no—
625C2#2258865918yesdirect product factor
626C2C2×#1798587511yescharacteristic
627C2C2×#1798599211yescharacteristic
628C2C2×#1798600211yescharacteristic
629C2C2×#1798614111yescharacteristic
630C2C2×#1798614311yescharacteristic
631C2C2×#1798632111yescharacteristic
632C2C2×#1798632211yescharacteristic
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×#1875
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
2. C22×#14556086C2semidirect
3. C22×#14556049C2semidirect
4. C22×(C2×D71)⋊C8C2semidirect
5. #22588659C2direct (N × H)
6. C2×#17985762C2semidirect
7. C2×#17985761C2semidirect
8. C22×#102×C71C2semidirect
14. C2×#14556099C4semidirect
19. C2×#14556086C22semidirect
26. C2×#14556049C22semidirect
28. C2×(C2×D71)⋊C8C22semidirect
29. #17985762C22semidirect
30. #17985761C22semidirect
35. C2×#102×C71C22semidirect
58. #14556099C2×C4semidirect
67. C2×C4⋊Dic71D4semidirect
95. C2×C4⋊C4×C71D4semidirect
144. C4⋊Dic71C2×D4semidirect
188. C4⋊C4×C71C2×D4semidirect
308. C2×C71#1875semidirect
320. C22×#102D71semidirect
325. C71C2×#1875semidirect
348. C2×#102C2×D71semidirect
404. C2×C4⋊C4C22⋊D71semidirect
502. C4⋊C4C2×C22⋊D71semidirect
625. C2#22588659direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145560992
C22×#145560862
C22×#145560492
C22×(C2×D71)⋊C82
#225886592
C2×#179857622
C2×#179857612
C22×#102×C712
C2×#187571