Finite Group Databasebeta

← C2×#22599508 #32869482

Subgroups — C2×#22599508 #32869482

Order 18176.

1822 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986533C211yesmaximal, characteristic
2#22599508C2116yesmaximal, direct product factor
3C2×#17986398C212yesmaximal, complemented
4C2×#17986356C212yesmaximal, complemented
5C2×#17986334C211yesmaximal, characteristic, complemented
6C2×#17986317C211yesmaximal, characteristic
7C2×#17986329C211yesmaximal, characteristic, complemented
8C2×#17986312C211yesmaximal, characteristic
9C2×#17986303C212yesmaximal, complemented
10C2×#17986296C212yesmaximal, complemented
11C2×#17986052C211yesmaximal, characteristic, complemented
12C22×#125×C71C211yesmaximal, characteristic, complemented
13#17986533—24no—
14#17986533C2218yes—
15C2×#14556122—21no—
16C2×#14556120C2211yescharacteristic
17C2×#14556116C2211yescharacteristic
18C2×#14556111—21no—
19C2×#14556107C2211yescharacteristic
20C2×#14556102C2211yescharacteristic
21C2×C4×Dic142C2211yescharacteristic
22C2×C4×D284C2211yescharacteristic
23C2×C4×C22⋊D71—21no—
24C2×#14556085—21no—
25#17986398C22116yescomplemented
26#17986356C22116yescomplemented
27#17986334C2218yescomplemented
28#17986329C2218yescomplemented
29#17986317C2218yes—
30#17986312C2218yes—
31#17986303C22116yescomplemented
32#17986296C22116yescomplemented
33#17986052C2218yescomplemented
34C2×#14556061C2212yes—
35C2×#14556061C2212yes—
36C2×#14556059C2212yes—
37C2×#14556059C2212yes—
38C2×C4⋊(C71⋊C8)C2212yes—
39C2×#14556051C2212yes—
40C2×#14556051C2212yes—
41C2×#14556048C2212yes—
42C2×#14556048C2212yes—
43C2×C22⋊(C71⋊C8)C2211yescharacteristic
44C2×C22⋊(C71⋊C8)C2211yescharacteristic
45C2×C4×C71⋊C8C2212yes—
46C2×#125×C71—21no—
47C2×#125×C71—21no—
48C2×#125×C71—24no—
49C2×#125×C71C2212yes—
50C2×#125×C71C2212yes—
51C2×#125×C71C2218yescomplemented
52C22×C4⋊C4×C71—21no—
53C22×C4⋊C4×C71—21no—
54C22×C4⋊C4×C71C2211yescharacteristic
55C22×C4⋊C4×C71C2211yescharacteristic
56C22×C42×C71C2211yescharacteristic
57C22×Dic142C2311yescharacteristic
58C22×D284C2311yescharacteristic
59C22×C22⋊D71—21no—
60C22×C4×D71—21no—
61C22×C71⋊C8C2312yes—
62C22×C71⋊C8C2312yes—
63#14556122—28no—
64#14556120—22no—
65#14556120C2314yes—
66#14556116—22no—
67#14556116C2314yes—
68#14556111—28no—
69#14556107—22no—
70#14556107C2314yes—
71#14556102—22no—
72#14556102C2314yes—
73C4×Dic142—22no—
74C4×Dic142C2314yes—
75C4×D284—22no—
76C4×D284C2314yes—
77C4×C22⋊D71—28no—
78#14556085—28no—
79C2×C4⋊Dic71C2311yescharacteristic
80C2×C4⋊Dic71C2311yescharacteristic
81C2×Dic71⋊C4—21no—
82C2×Dic71⋊C4—21no—
83C2×Dic71⋊C4—21no—
84C2×Dic71⋊C4—21no—
85C2×C22⋊Dic71—21no—
86C2×(C2×D71)⋊C4—21no—
87C2×(C2×D71)⋊C4—21no—
88C2×(C2×D71)⋊C4—21no—
89C2×(C2×D71)⋊C4—21no—
90C2×C4×Dic71—21no—
91#14556061C2318yes—
92#14556061C2318yes—
93#14556059C2318yes—
94#14556059C2318yes—
95C4⋊(C71⋊C8)C2318yes—
96#14556051C2318yes—
97#14556051C2318yes—
98#14556048C2318yes—
99#14556048C2318yes—
100C22⋊(C71⋊C8)C2314yes—
101C22⋊(C71⋊C8)C2314yes—
102C4×C71⋊C8C2318yes—
103C23×C4×C71—21no—
104C23×C4×C71—21no—
105C23×C4×C71D411yescharacteristic
106C23×C4×C71D411yescharacteristic
107C23×C4×C71C2311yescharacteristic
108#125×C71—24no—
109#125×C71—24no—
110#125×C71—28no—
111#125×C71—28no—
112#125×C71C2318yes—
113#125×C71C2318yes—
114C2×C4⋊C4×C71—22no—
115C2×C4⋊C4×C71—22no—
116C2×C4⋊C4×C71—22no—
117C2×C4⋊C4×C71—22no—
118C2×C4⋊C4×C71—28no—
119C2×C4⋊C4×C71—28no—
120C2×C4⋊C4×C71—21no—
121C2×C4⋊C4×C71—21no—
122C2×C4⋊C4×C71—22no—
123C2×C4⋊C4×C71—22no—
124C2×C4⋊C4×C71C2314yes—
125C2×C4⋊C4×C71C2314yes—
126C2×C4⋊C4×C71C2312yes—
127C2×C4⋊C4×C71C2312yes—
128C2×C42×C71D411yescharacteristic
129C2×C42×C71D411yescharacteristic
130C2×C42×C71D414yes—
131C2×C42×C71C2314yes—
132C2×C42×C71C2311yescharacteristic
133C2×C42×C71C2311yescharacteristic
134C23×D71—21no—
135C2×Dic142—24no—
136C2×Dic142—21no—
137C2×Dic142C2412yes—
138C2×D284—24no—
139C2×D284—21no—
140C2×D284C2412yes—
141C2×C22⋊D71—24no—
142C2×C22⋊D71—44no—
143C22×Dic71—21no—
144C22×Dic71—21no—
145C22×Dic71—21no—
146C2×C4×D71—24no—
147C2×C4×D71—44no—
148C2×C71⋊C8—24no—
149C2×C71⋊C8—24no—
150C2×C71⋊C8C2414yes—
151C2×C71⋊C8C2414yes—
152C4⋊Dic71—21no—
153C4⋊Dic71—21no—
154C4⋊Dic71C2412yes—
155C4⋊Dic71C2412yes—
156Dic71⋊C4—24no—
157Dic71⋊C4—24no—
158Dic71⋊C4—24no—
159Dic71⋊C4—24no—
160C22⋊Dic71—24no—
161(C2×D71)⋊C4—24no—
162(C2×D71)⋊C4—24no—
163(C2×D71)⋊C4—24no—
164(C2×D71)⋊C4—24no—
165C4×Dic71—24no—
166C24×C71C2×D411yescharacteristic
167C22×C4×C71—24no—
168C22×C4×C71—24no—
169C22×C4×C71—24no—
170C22×C4×C71—24no—
171C22×C4×C71—22no—
172C22×C4×C71—22no—
173C22×C4×C71—42no—
174C22×C4×C71—44no—
175C22×C4×C71—44no—
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D412yes—
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D414yes—
183C22×C4×C71C2×D414yes—
184C22×C4×C71C2412yes—
185C22×C4×C71C2411yescharacteristic
186C4⋊C4×C71—28no—
187C4⋊C4×C71—28no—
188C4⋊C4×C71—28no—
189C4⋊C4×C71—28no—
190C4⋊C4×C71—24no—
191C4⋊C4×C71—24no—
192C4⋊C4×C71—22no—
193C4⋊C4×C71—22no—
194C4⋊C4×C71C2414yes—
195C4⋊C4×C71C2414yes—
196C42×C71C2×D414yes—
197C42×C71C2×D412yes—
198C42×C71C2×D412yes—
199C42×C71C2×D414yes—
200C42×C71C2412yes—
201C42×C71C2412yes—
202C23×C71—22no—
203C23×C71—22no—
204C23×C71#13514yes—
205C23×C71C22×D412yes—
206C23×C71C22×D411yescharacteristic
207C22×D71—22no—
208C22×D71—42no—
209C22×D71—42no—
210C22×D71—42no—
211C2×C4×C71—22no—
212C2×C4×C71—28no—
213C2×C4×C71—22no—
214C2×C4×C71—22no—
215C2×C4×C71—22no—
216C2×C4×C71—22no—
217C2×C4×C71—24no—
218C2×C4×C71—24no—
219C2×C4×C71—24no—
220C2×C4×C71—24no—
221C2×C4×C71—44no—
222C2×C4×C71—48no—
223C2×C4×C71—48no—
224C2×C4×C71—44no—
225C2×C4×C71—44no—
226C2×C4×C71C22×D412yes—
227C2×C4×C71C22×D412yes—
228C2×C4×C71C22×D412yes—
229C2×C4×C71C22×D412yes—
230C2×C4×C71C22×D412yes—
231C2×C4×C71C22×D411yescharacteristic
232C2×C4×C712−1+414yes—
233C2×C4×C71C2511yescharacteristic
234C22⋊D71—48no—
235C2×Dic71—22no—
236C2×Dic71—22no—
237C2×Dic71—22no—
238C2×Dic71—42no—
239C2×Dic71—42no—
240C2×Dic71—42no—
241D284—24no—
242D284—42no—
243C4×D71—48no—
244Dic142—24no—
245Dic142—42no—
246C71⋊C8—24no—
247C71⋊C8—24no—
248C22×C71—22no—
249C22×C71—28no—
250C22×C71—22no—
251C22×C71#53612yes—
252C22×C71C2×#13514yes—
253C22×C71C2×#13512yes—
254C22×C71#57512yes—
255C22×C71C23×D411yescharacteristic
256C2×D71—42no—
257C2×D71—42no—
258C2×D71—42no—
259C2×D71—84no—
260C4×C71—22no—
261C4×C71—22no—
262C4×C71—22no—
263C4×C71—22no—
264C4×C71—22no—
265C4×C71—44no—
266C4×C71—44no—
267C4×C71#53612yes—
268C4×C71C2×2−1+412yes—
269Dic71—42no—
270Dic71—42no—
271Dic71—42no—
272C2×#3232—711nomaximal, Sylow
273C2×C71—24no—
274C2×C71#323214yescomplemented
275C2×C71C2×#53611yescharacteristic
276C2×C71C22×#13511yescharacteristic
277C2×C71C2×#57511yescharacteristic
278D71—82no—
279C2×#529—711no—
280C22×#125D7111yescharacteristic, complemented
281#3232—7116no—
282C2×#499—712no—
283C2×#488—712no—
284C2×#484—711no—
285C2×#481—711no—
286C2×#483—711no—
287C2×#480—711no—
288C2×#479—712no—
289C2×#478—712no—
290C2×#431—711no—
291C71C2×#323211yescharacteristic, complemented, Sylow
292#529—718no—
293#529—1424no—
294C2×#126—1421no—
295C2×#125—21no—
296C2×#125—21no—
297C2×#125—24no—
298C2×#125—711no—
299C2×#125C2×D7112yes—
300C2×#125C2×D7112yes—
301C2×#125C2×D7118yescomplemented
302C2×#124—711no—
303C2×#123—1421no—
304C2×C22⋊Q8—711no—
305C2×#121—711no—
306C2×C4×Q8—711no—
307C2×C4×D4—711no—
308C2×C4×D4—1421no—
309C2×#117—1421no—
310C22×C4⋊C4—21no—
311C22×C4⋊C4—21no—
312C22×C4⋊C4C2×D7111yescharacteristic
313C22×C4⋊C4C2×D7111yescharacteristic
314C22×C42C2×D7111yescharacteristic
315#499—7116no—
316#488—7116no—
317#484—718no—
318#483—718no—
319#481—718no—
320#480—718no—
321#479—7116no—
322#478—7116no—
323#431—718no—
324C2×#107—712no—
325C2×#107—712no—
326C2×#106—712no—
327C2×#106—712no—
328C2×C4⋊C8—712no—
329C2×#103—712no—
330C2×#103—712no—
331C2×#102—712no—
332C2×#102—712no—
333C2×C22⋊C8—711no—
334C2×C22⋊C8—711no—
335C2×C4×C8—712no—
336C22×Q8—711no—
337C22×D4—711no—
338C22×D4—1421no—
339C23×C4—21no—
340C23×C4—21no—
341C23×C4—1421no—
342C23×C4C22⋊D7111yescharacteristic
343C23×C4C22⋊D7111yescharacteristic
344C23×C4C22×D7111yescharacteristic
345C22×C8—712no—
346C22×C8—712no—
347#126—1428no—
348#125—24no—
349#125—24no—
350#125—28no—
351#125—28no—
352#125—714no—
353#125—1422no—
354#125C22×D7118yes—
355#125C22×D7118yes—
356#124—714no—
357#124—1422no—
358#123—1428no—
359C22⋊Q8—714no—
360C22⋊Q8—1422no—
361#121—714no—
362#121—1422no—
363C4×Q8—714no—
364C4×Q8—1422no—
365C4×D4—714no—
366C4×D4—1428no—
367C4×D4—1422no—
368#117—1428no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—28no—
374C2×C4⋊C4—28no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—711no—
380C2×C4⋊C4—711no—
381C2×C4⋊C4—1421no—
382C2×C4⋊C4—1421no—
383C2×C4⋊C4—1421no—
384C2×C4⋊C4—1421no—
385C2×C4⋊C4C22×D7112yes—
386C2×C4⋊C4C22×D7112yes—
387C2×C4⋊C4C22×D7114yes—
388C2×C4⋊C4C22×D7114yes—
389C2×C22⋊C4—1421no—
390C2×C22⋊C4—1421no—
391C2×C22⋊C4—1421no—
392C2×C22⋊C4—1421no—
393C2×C22⋊C4—1421no—
394C2×C42—1421no—
395C2×C42C22⋊D7111yescharacteristic
396C2×C42C22⋊D7111yescharacteristic
397C2×C42C22⋊D7114yes—
398C2×C42C22×D7111yescharacteristic
399C2×C42C22×D7111yescharacteristic
400C2×C42C22×D7114yes—
401#107—718no—
402#107—718no—
403#106—718no—
404#106—718no—
405C4⋊C8—718no—
406#103—718no—
407#103—718no—
408#102—718no—
409#102—718no—
410C22⋊C8—714no—
411C22⋊C8—714no—
412C4×C8—718no—
413C24—1421no—
414C24C2×C22⋊D7111yescharacteristic
415C2×Q8—712no—
416C2×Q8—1424no—
417C2×Q8—1421no—
418C2×D4—2844no—
419C2×D4—712no—
420C2×D4—1424no—
421C2×D4—1421no—
422C2×D4—1424no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—2844no—
433C22×C4—1421no—
434C22×C4—1421no—
435C22×C4—1421no—
436C22×C4—1424no—
437C22×C4C2×C22⋊D7112yes—
438C22×C4C2×C22⋊D7114yes—
439C22×C4C2×C22⋊D7111yescharacteristic
440C22×C4C2×C22⋊D7111yescharacteristic
441C22×C4C2×C22⋊D7111yescharacteristic
442C22×C4C2×C22⋊D7111yescharacteristic
443C22×C4C2×C22⋊D7111yescharacteristic
444C22×C4C2×C22⋊D7114yes—
445C22×C4C23×D7112yes—
446C22×C4C23×D7111yescharacteristic
447C2×C8—714no—
448C2×C8—714no—
449C2×C8—1424no—
450C2×C8—1424no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—712no—
460C4⋊C4—712no—
461C4⋊C4—1424no—
462C4⋊C4—1424no—
463C4⋊C4—1424no—
464C4⋊C4—1424no—
465C4⋊C4—1421no—
466C4⋊C4—1421no—
467C4⋊C4C23×D7114yes—
468C4⋊C4C23×D7114yes—
469C22⋊C4—1424no—
470C22⋊C4—1424no—
471C22⋊C4—1424no—
472C22⋊C4—1424no—
473C22⋊C4—1424no—
474C42—1424no—
475C42C2×C22⋊D7114yes—
476C42C2×C22⋊D7112yes—
477C42C2×C22⋊D7112yes—
478C42C2×C22⋊D7114yes—
479C42C23×D7112yes—
480C42C23×D7112yes—
481C23—22no—
482C23—22no—
483C23—2842no—
484C23—2842no—
485C23—2842no—
486C23—1422no—
487C23#1455614814yes—
488C23C22×C22⋊D7112yes—
489C23C22×C22⋊D7111yescharacteristic
490Q8—2842no—
491Q8—1424no—
492D4—2848no—
493D4—2842no—
494D4—1424no—
495C2×C4—22no—
496C2×C4—28no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—2848no—
511C2×C4—2842no—
512C2×C4—2842no—
513C2×C4—2842no—
514C2×C4—1422no—
515C2×C4—1422no—
516C2×C4—1422no—
517C2×C4C22×C22⋊D7112yes—
518C2×C4C22×C22⋊D7112yes—
519C2×C4C22×C22⋊D7112yes—
520C2×C4C22×C22⋊D7112yes—
521C2×C4C22×C22⋊D7112yes—
522C2×C4C22×C22⋊D7111yescharacteristic
523C2×C4#1455617614yes—
524C2×C4C24×D7111yescharacteristic
525C8—1424no—
526C8—1424no—
527C22—22no—
528C22—28no—
529C22—22no—
530C22—2842no—
531C22—2842no—
532C22—2842no—
533C22—5684no—
534C22#1798657312yes—
535C22C2×#1455614814yes—
536C22C2×#1455614812yes—
537C22#1798679712yes—
538C22C23×C22⋊D7111yescharacteristic
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—22no—
544C4—44no—
545C4—44no—
546C4—2842no—
547C4—2842no—
548C4—2842no—
549C4#1798657312yes—
550C4C2×#1455617612yes—
551C2—24no—
552C2—5682no—
553C2#2259950814yesdirect product factor
554C2C2×#1798657311yescharacteristic
555C2C22×#1455614811yescharacteristic
556C2C2×#1798679711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC2×C4×C71C25
FrattiniC2×C4C24×D71
FittingC22×#125×C71C2
socleC23×C71C22×D4
prime radicalC2×#14556102C22
hypercenterC22×#125D71
hypocenterC71C2×#3232
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
2. #22599508C2direct (N × H)
3. C2×#17986398C2semidirect
4. C2×#17986356C2semidirect
5. C2×#17986334C2semidirect
7. C2×#17986329C2semidirect
9. C2×#17986303C2semidirect
10. C2×#17986296C2semidirect
11. C2×#17986052C2semidirect
12. C22×#125×C71C2semidirect
25. #17986398C22semidirect
26. #17986356C22semidirect
27. #17986334C22semidirect
28. #17986329C22semidirect
31. #17986303C22semidirect
32. #17986296C22semidirect
33. #17986052C22semidirect
51. C2×#125×C71C22semidirect
274. C2×C71#3232semidirect
280. C22×#125D71semidirect
291. C71C2×#3232semidirect
301. C2×#125C2×D71semidirect
553. C2#22599508direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179865332
#225995082
C2×#179863982
C2×#179863562
C2×#179863342
C2×#179863172
C2×#179863292
C2×#179863122
C2×#179863032
C2×#179862962
C2×#179860522
C22×#125×C712
C2×#323271