Finite Group Databasebeta

← C2×#22589546 #32024927

Subgroups — C2×#22589546 #32024927

Order 18176.

1662 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556108C211yesmaximal, characteristic
2C22×#14556061C211yesmaximal, characteristic, complemented
3C22×#14556053C211yesmaximal, characteristic, complemented
4C22×#14556051C211yesmaximal, characteristic, complemented
5C22×(C2×D71)⋊C8C211yesmaximal, characteristic
6C2×#17985856C211yesmaximal, characteristic
7#22589546C218yesmaximal, direct product factor
8C2×#340×C71C211yesmaximal, characteristic, complemented
9C23×Dic142C2211yescharacteristic
10C23×C4×D71C2211yescharacteristic
11C23×C71⋊C8C2211yescharacteristic
12C2×#14556108—24no—
13C2×#14556108—24no—
14C2×#14556108—22no—
15C2×#14556108C2214yes—
16C22×C4⋊Dic71—21no—
17C22×C4⋊Dic71C2211yescharacteristic
18C22×(C2×D71)⋊C4—21no—
19C22×(C2×D71)⋊C4—21no—
20C2×#14556061—24no—
21C2×#14556061C2214yescomplemented
22C2×#14556053—24no—
23C2×#14556053C2214yescomplemented
24C2×#14556051—24no—
25C2×#14556051C2214yescomplemented
26C2×(C2×D71)⋊C8—24no—
27C2×(C2×D71)⋊C8C2214yes—
28#17985856—22no—
29#17985856C2214yes—
30C2×C71⋊#95—21no—
31C23×C8×C71C2211yescharacteristic
32C22×C4⋊C4×C71C2211yescharacteristic
33C22×C4⋊C4×C71C2211yescharacteristic
34#340×C71C2214yescomplemented
35C24×D71D411yescharacteristic
36C22×Dic142—21no—
37C22×Dic142—22no—
38C22×Dic142—22no—
39C22×Dic142—22no—
40C22×Dic142—22no—
41C22×Dic142—22no—
42C22×Dic142D414yes—
43C22×Dic142C2312yes—
44C23×Dic71—21no—
45C23×Dic71—21no—
46C23×Dic71D411yescharacteristic
47C22×C4×D71—22no—
48C22×C4×D71—22no—
49C22×C4×D71—22no—
50C22×C4×D71—22no—
51C22×C4×D71—22no—
52C22×C4×D71—22no—
53C22×C4×D71D412yes—
54C22×C4×D71C2312yes—
55C22×C71⋊C8—22no—
56C22×C71⋊C8—22no—
57C22×C71⋊C8D414yes—
58C22×C71⋊C8C2312yes—
59#14556108—216no—
60#14556108—28no—
61#14556108—28no—
62C2×C4⋊Dic71—28no—
63C2×C4⋊Dic71—24no—
64C2×C4⋊Dic71—22no—
65C2×C4⋊Dic71—22no—
66C2×C4⋊Dic71—21no—
67C2×C4⋊Dic71C2312yes—
68C2×(C2×D71)⋊C4—28no—
69C2×(C2×D71)⋊C4—24no—
70C2×(C2×D71)⋊C4—24no—
71C2×(C2×D71)⋊C4—44no—
72#14556061—28no—
73#14556053—28no—
74#14556051—28no—
75(C2×D71)⋊C8—28no—
76C71⋊#95—24no—
77C23×C4×C71—21no—
78C23×C4×C71—21no—
79C23×C4×C71C2311yescharacteristic
80C22×C8×C71—22no—
81C22×C8×C71—22no—
82C22×C8×C71—22no—
83C22×C8×C71C2312yes—
84C2×C4⋊C4×C71—21no—
85C2×C4⋊C4×C71—22no—
86C2×C4⋊C4×C71—22no—
87C2×C4⋊C4×C71—22no—
88C2×C4⋊C4×C71—21no—
89C2×C4⋊C4×C71D414yes—
90C2×C4⋊C4×C71C2312yes—
91C2×C4⋊C4×C71C2312yes—
92C23×D71—22no—
93C23×D71—22no—
94C23×D71—22no—
95C23×D71—22no—
96C23×D71—22no—
97C23×D71—22no—
98C23×D71Q1614yes—
99C23×D71C2×D412yes—
100C2×Dic142—28no—
101C2×Dic142—28no—
102C2×Dic142—22no—
103C2×Dic142—22no—
104C2×Dic142—22no—
105C2×Dic142—22no—
106C2×Dic142—24no—
107C2×Dic142—22no—
108C2×Dic142—42no—
109C2×Dic142—44no—
110C2×Dic142—42no—
111C2×Dic142—44no—
112C2×Dic142C2×D414yes—
113C22×Dic71—22no—
114C22×Dic71—24no—
115C22×Dic71—22no—
116C22×Dic71—22no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71—42no—
120C22×Dic71—42no—
121C22×Dic71—42no—
122C22×Dic71—42no—
123C22×Dic71—42no—
124C22×Dic71C2×D412yes—
125C2×C4×D71—24no—
126C2×C4×D71—24no—
127C2×C4×D71—24no—
128C2×C4×D71—22no—
129C2×C4×D71—22no—
130C2×C4×D71—22no—
131C2×C4×D71—22no—
132C2×C4×D71—22no—
133C2×C4×D71—22no—
134C2×C4×D71—44no—
135C2×C4×D71—44no—
136C2×C4×D71—44no—
137C2×C4×D71—44no—
138C2×C71⋊C8—28no—
139C2×C71⋊C8—22no—
140C2×C71⋊C8—22no—
141C2×C71⋊C8C2×D414yes—
142C4⋊Dic71—216no—
143C4⋊Dic71—22no—
144C4⋊Dic71—22no—
145C4⋊Dic71—42no—
146(C2×D71)⋊C4—216no—
147(C2×D71)⋊C4—48no—
148C24×C71C2×D411yescharacteristic
149C22×C4×C71—24no—
150C22×C4×C71—22no—
151C22×C4×C71—22no—
152C22×C4×C71—42no—
153C22×C4×C71—42no—
154C22×C4×C71—42no—
155C22×C4×C71—42no—
156C22×C4×C71—42no—
157C22×C4×C71C2×D411yescharacteristic
158C22×C4×C71C2×D412yes—
159C22×C4×C71C2×D412yes—
160C22×C4×C71C2×D412yes—
161C22×C4×C71#4112yes—
162C22×C4×C71#4112yes—
163C22×C4×C71#4112yes—
164C22×C4×C71C2411yescharacteristic
165C2×C8×C71—22no—
166C2×C8×C71—22no—
167C2×C8×C71—22no—
168C2×C8×C71—44no—
169C4⋊C4×C71—24no—
170C4⋊C4×C71—22no—
171C4⋊C4×C71—22no—
172C4⋊C4×C71—22no—
173C4⋊C4×C71—42no—
174C4⋊C4×C71C2×D414yes—
175C23×C71#12012yes—
176C23×C71#12112yes—
177C23×C71#12312yes—
178C23×C71C2×Q1612yes—
179C23×C71#13512yes—
180C23×C71AGL(1,ℤ/8)12yes—
181C23×C71#13712yes—
182C23×C71C22×D411yescharacteristic
183C22×D71—22no—
184C22×D71—22no—
185C22×D71—22no—
186C22×D71—22no—
187C22×D71—22no—
188C22×D71—22no—
189C22×D71—28no—
190C22×D71—28no—
191C22×D71—28no—
192C22×D71—44no—
193C22×D71—44no—
194C22×D71—44no—
195C22×D71—44no—
196C22×D71C2×Q1614yes—
197C2×C4×C71—24no—
198C2×C4×C71—42no—
199C2×C4×C71—42no—
200C2×C4×C71—48no—
201C2×C4×C71—42no—
202C2×C4×C71—42no—
203C2×C4×C71—42no—
204C2×C4×C71—84no—
205C2×C4×C71#12012yes—
206C2×C4×C71#12114yes—
207C2×C4×C71#12112yes—
208C2×C4×C71#12114yes—
209C2×C4×C71#12314yes—
210C2×C4×C71#12312yes—
211C2×C4×C71#12414yes—
212C2×C4×C71C22×D411yescharacteristic
213C2×C4×C71C22×D411yescharacteristic
214C2×C4×C71C22×D411yescharacteristic
215C2×C4×C71C2×#4111yescharacteristic
216C2×C4×C71C2×#4111yescharacteristic
217C2×C4×C71C2×#4111yescharacteristic
218C2×Dic71—24no—
219C2×Dic71—22no—
220C2×Dic71—22no—
221C2×Dic71—22no—
222C2×Dic71—42no—
223C2×Dic71—42no—
224C2×Dic71—44no—
225C2×Dic71—48no—
226C2×Dic71—42no—
227C2×Dic71—42no—
228C2×Dic71—42no—
229C2×Dic71—84no—
230C4×D71—42no—
231C4×D71—42no—
232C4×D71—42no—
233C4×D71—42no—
234C4×D71—48no—
235Dic142—24no—
236Dic142—24no—
237Dic142—42no—
238Dic142—42no—
239Dic142—48no—
240C8×C71—42no—
241C71⋊C8—24no—
242C22×C71#39414yes—
243C22×C71#44914yes—
244C22×C71#45114yes—
245C22×C71#46714yes—
246C22×C71#46814yes—
247C22×C71#48414yes—
248C22×C71#48514yes—
249C22×C71C2×#12011yescharacteristic
250C22×C71C2×#12111yescharacteristic
251C22×C71C2×#12311yescharacteristic
252C22×C71C22×Q1611yescharacteristic
253C22×C71C2×#13511yescharacteristic
254C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C71C2×#13711yescharacteristic
256C2×D71—24no—
257C2×D71—24no—
258C2×D71—24no—
259C2×D71—42no—
260C2×D71—42no—
261C2×D71—42no—
262C2×D71—42no—
263C2×D71—416no—
264C4×C71—44no—
265C4×C71—82no—
266C4×C71#39414yes—
267C4×C71C2×#12111yescharacteristic
268C4×C71C2×#12111yescharacteristic
269C4×C71C2×#12311yescharacteristic
270C4×C71C2×#12411yescharacteristic
271Dic71—44no—
272Dic71—42no—
273Dic71—82no—
274C2×#2036—711nomaximal, Sylow
275C2×C71#203618yescomplemented
276C2×C71C2×#39411yescharacteristic
277C2×C71C2×#45111yescharacteristic
278C2×C71C2×#44911yescharacteristic
279C2×C71C2×#46711yescharacteristic
280C2×C71C2×#46811yescharacteristic
281C2×C71C2×#48511yescharacteristic
282C2×C71C2×#48411yescharacteristic
283D71—44no—
284C22×C22⋊Q8—711no—
285C22×#107—711no—
286C22×#103—711no—
287C22×#103—711no—
288C22×C22⋊C8—711no—
289C2×#380—711no—
290#2036—718no—
291C2×#340D7111yescharacteristic, complemented
292C71C2×#203611yescharacteristic, complemented, Sylow
293C23×Q8—711no—
294C24×C4—711no—
295C23×C8—711no—
296C23×C8C2×D7111yescharacteristic
297C2×C22⋊Q8—714no—
298C2×C22⋊Q8—1422no—
299C2×C22⋊Q8—1424no—
300C2×C22⋊Q8—1424no—
301C22×C4⋊C4—711no—
302C22×C4⋊C4—1421no—
303C22×C4⋊C4C2×D7111yescharacteristic
304C22×C4⋊C4C2×D7111yescharacteristic
305C22×C22⋊C4—1421no—
306C22×C22⋊C4—1421no—
307C2×#107—714no—
308C2×#107—1424no—
309C2×#103—714no—
310C2×#103—714no—
311C2×#103—1424no—
312C2×#103—1424no—
313C2×C22⋊C8—714no—
314C2×C22⋊C8—1424no—
315#380—714no—
316#380—1422no—
317C2×#95—1421no—
318#340C2×D7114yescomplemented
319C25—711no—
320C22×Q8—714no—
321C22×Q8—712no—
322C22×Q8—1422no—
323C22×Q8—1422no—
324C22×Q8—1422no—
325C22×Q8—1422no—
326C22×Q8—1421no—
327C22×Q8—1422no—
328C23×C4—21no—
329C23×C4—21no—
330C23×C4—711no—
331C23×C4—712no—
332C23×C4—712no—
333C23×C4—1421no—
334C23×C4—1421no—
335C23×C4—1422no—
336C23×C4—1422no—
337C23×C4—1422no—
338C23×C4—1422no—
339C23×C4—1422no—
340C23×C4—1422no—
341C23×C4C22×D7111yescharacteristic
342C22×C8—22no—
343C22×C8—22no—
344C22×C8—22no—
345C22×C8—714no—
346C22×C8—712no—
347C22×C8—1422no—
348C22×C8—1422no—
349C22×C8C22×D7112yes—
350C22⋊Q8—1428no—
351C22⋊Q8—14216no—
352C22⋊Q8—1428no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—712no—
359C2×C4⋊C4—1421no—
360C2×C4⋊C4—1422no—
361C2×C4⋊C4—1422no—
362C2×C4⋊C4—1424no—
363C2×C4⋊C4—1428no—
364C2×C4⋊C4D28414yes—
365C2×C4⋊C4C22×D7112yes—
366C2×C4⋊C4C22×D7112yes—
367C2×C22⋊C4—2844no—
368C2×C22⋊C4—1424no—
369C2×C22⋊C4—1424no—
370C2×C22⋊C4—1428no—
371#107—1428no—
372#103—1428no—
373#103—1428no—
374C22⋊C8—1428no—
375#95—1424no—
376C24—712no—
377C24—714no—
378C24—1422no—
379C24—1422no—
380C24—1422no—
381C24—1422no—
382C24—1422no—
383C24—1422no—
384C24D4×D7111yescharacteristic
385C2×Q8—2842no—
386C2×Q8—2844no—
387C2×Q8—2842no—
388C2×Q8—2844no—
389C2×Q8—714no—
390C2×Q8—1422no—
391C2×Q8—1422no—
392C2×Q8—1422no—
393C2×Q8—1422no—
394C2×Q8—1428no—
395C2×Q8—1428no—
396C2×Q8—1422no—
397C2×Q8—1424no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—2842no—
407C22×C4—2842no—
408C22×C4—2844no—
409C22×C4—2844no—
410C22×C4—2844no—
411C22×C4—2844no—
412C22×C4—2842no—
413C22×C4—2842no—
414C22×C4—2842no—
415C22×C4—712no—
416C22×C4—1422no—
417C22×C4—1422no—
418C22×C4—1424no—
419C22×C4—1422no—
420C22×C4—1422no—
421C22×C4—1422no—
422C22×C4—1422no—
423C22×C4—1422no—
424C22×C4—1422no—
425C22×C4—1422no—
426C22×C4—1422no—
427C22×C4—1422no—
428C22×C4—1424no—
429C22×C4—1424no—
430C22×C4—1424no—
431C22×C4D4×D7112yes—
432C22×C4D4×D7112yes—
433C22×C4D4×D7111yescharacteristic
434C22×C4C2×D28412yes—
435C22×C4#1176586512yes—
436C22×C4#1176586512yes—
437C22×C4#1176586612yes—
438C22×C4C23×D7111yescharacteristic
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—44no—
443C2×C8—714no—
444C2×C8—1422no—
445C2×C8—1422no—
446C2×C8—1428no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—2842no—
453C4⋊C4—14216no—
454C4⋊C4—1422no—
455C4⋊C4—1422no—
456C4⋊C4C2×D28414yes—
457C22⋊C4—2848no—
458C22⋊C4—14216no—
459C23—2844no—
460C23—2844no—
461C23—2844no—
462C23—2844no—
463C23—714no—
464C23—1428no—
465C23—1428no—
466C23—1422no—
467C23—1422no—
468C23—1428no—
469C23—1422no—
470C23—1422no—
471C23—1422no—
472C23—1422no—
473C23#1455609812yes—
474C23#1455610112yes—
475C23#1455611312yes—
476C23Q16×D7112yes—
477C23#1455614912yes—
478C23#1455615212yes—
479C23#1455615712yes—
480C23C2×D4×D7111yescharacteristic
481Q8—2848no—
482Q8—2842no—
483Q8—2842no—
484Q8—1424no—
485Q8—1424no—
486C2×C4—24no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—84no—
494C2×C4—2844no—
495C2×C4—2848no—
496C2×C4—2842no—
497C2×C4—2842no—
498C2×C4—2848no—
499C2×C4—2842no—
500C2×C4—2842no—
501C2×C4—2842no—
502C2×C4—2842no—
503C2×C4—2842no—
504C2×C4—2842no—
505C2×C4—2842no—
506C2×C4—5684no—
507C2×C4—1424no—
508C2×C4—1422no—
509C2×C4—1422no—
510C2×C4—1422no—
511C2×C4#1455609812yes—
512C2×C4#1455610112yes—
513C2×C4#1455610114yes—
514C2×C4#1455610314yes—
515C2×C4#1455611312yes—
516C2×C4#1455611314yes—
517C2×C4#1455611614yes—
518C2×C4C2×D4×D7111yescharacteristic
519C2×C4C2×D4×D7111yescharacteristic
520C2×C4C22×D28411yescharacteristic
521C2×C4C2×#1176586511yescharacteristic
522C2×C4C2×#1176586511yescharacteristic
523C2×C4C2×#1176586611yescharacteristic
524C8—42no—
525C8—1424no—
526C22—2842no—
527C22—2842no—
528C22—2842no—
529C22—28416no—
530C22—2842no—
531C22—1424no—
532C22—1424no—
533C22—1424no—
534C22#1798591714yes—
535C22#1798614514yes—
536C22#1798615614yes—
537C22#1798623914yes—
538C22#1798624414yes—
539C22#1798633614yes—
540C22#1798634314yes—
541C22C2×#1455609811yescharacteristic
542C22C2×#1455610111yescharacteristic
543C22C2×#1455611311yescharacteristic
544C22C2×Q16×D7111yescharacteristic
545C22C2×#1455614911yescharacteristic
546C22C2×#1455615211yescharacteristic
547C22C2×#1455615711yescharacteristic
548C4—44no—
549C4—82no—
550C4—2842no—
551C4—2844no—
552C4—5682no—
553C4#1798591714yes—
554C4C2×#1455610111yescharacteristic
555C4C2×#1455610311yescharacteristic
556C4C2×#1455611311yescharacteristic
557C4C2×#1455611611yescharacteristic
558C2—2844no—
559C2#2258954618yesdirect product factor
560C2C2×#1798591711yescharacteristic
561C2C2×#1798615611yescharacteristic
562C2C2×#1798614511yescharacteristic
563C2C2×#1798623911yescharacteristic
564C2C2×#1798624411yescharacteristic
565C2C2×#1798634311yescharacteristic
566C2C2×#1798633611yescharacteristic
subgroupG/H
centerC24D4×D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC2×#340×C71C2
socleC24×C71C2×D4
prime radicalC24×D71D4
hypercenterC2×#340D71
hypocenterC71C2×#2036
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
2. C22×#14556061C2semidirect
3. C22×#14556053C2semidirect
4. C22×#14556051C2semidirect
7. #22589546C2direct (N × H)
8. C2×#340×C71C2semidirect
21. C2×#14556061C22semidirect
23. C2×#14556053C22semidirect
25. C2×#14556051C22semidirect
34. #340×C71C22semidirect
275. C2×C71#2036semidirect
291. C2×#340D71semidirect
292. C71C2×#2036semidirect
318. #340C2×D71semidirect
559. C2#22589546direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561082
C22×#145560612
C22×#145560532
C22×#145560512
C22×(C2×D71)⋊C82
C2×#179858562
#225895462
C2×#340×C712
C2×#203671