Finite Group Databasebeta

← #32156598

Subgroups — #32156598

Order 18176.

2070 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986612C211yesmaximal, characteristic
2#22591143C214yesmaximal, complemented
3#22591114C214yesmaximal
4#22591059C211yesmaximal, characteristic, complemented
5#22591023C212yesmaximal
6#22590987C212yesmaximal, complemented
7#22590948C211yesmaximal, characteristic
8#22590907C211yesmaximal, characteristic, complemented
9#125×Dic71C211yesmaximal, characteristic, complemented
10#22590899C214yesmaximal, complemented
11#22590890C214yesmaximal
12#22590870C212yesmaximal
13C4×#14556093C212yesmaximal, complemented
14#22590679C211yesmaximal, characteristic
15C71×#2330C211yesmaximal, characteristic, complemented
16#17986612C4116yescomplemented
17C2×#14556128C2211yescharacteristic
18C2×#14556123C2214yes—
19C2×#14556119C2214yes—
20C2×#14556108C2212yes—
21C2×C4×Dic142C2212yes—
22C2×#14556093C2212yes—
23C2×#14556093C2212yes—
24C2×#14556085C2211yescharacteristic
25C2×#14556085C2211yescharacteristic
26C2×C42×D71C2211yescharacteristic
27#17985905C2212yes—
28#17985904C2212yes—
29#17985898C2212yes—
30#17985897C2214yes—
31#17985895C2214yes—
32#17985891C2214yes—
33#17985890C2214yes—
34#17985889C2214yes—
35#17985886C2214yes—
36#17985885C2214yes—
37#17985884C2214yes—
38#17985884C2214yes—
39#17985884C2214yes—
40#17985882C2214yes—
41#17985880C2214yes—
42#17985880C2214yes—
43#17985870C2214yes—
44#17985866C2214yes—
45#17985865C2211yescharacteristic
46#17985864C2214yes—
47#17985856C2212yes—
48C4⋊C4×Dic71C2212yes—
49C4⋊C4×Dic71C2214yes—
50C4⋊C4×Dic71C2212yes—
51C4⋊C4×Dic71C2214yes—
52C4⋊C4×Dic71C2214yes—
53C4×Dic71⋊C4C2214yes—
54#17985850C2214yes—
55#17985850C2212yes—
56#17985850C2214yes—
57#17985850C2212yes—
58#17985846C2212yes—
59#17985846C2214yes—
60#17985846C2212yes—
61#17985845C2211yescharacteristic
62C42×Dic71C2214yes—
63C42×Dic71C2211yescharacteristic
64C42×Dic71C2211yescharacteristic
65C2×#125×C71C2211yescharacteristic
66#389×C71C2212yes—
67#389×C71C2214yes—
68#387×C71C2214yes—
69#384×C71C2211yescharacteristic
70C4×C4⋊C4×C71C2212yes—
71#376×C71C2211yescharacteristic
72C22×Dic142C2312yes—
73C22×C4×D71C2311yescharacteristic
74C22×C4×D71C2311yescharacteristic
75C22×C4×D71C2311yescharacteristic
76#14556128C2×C418yescomplemented
77#14556123C2×C4132yes—
78#14556119C2×C4132yescomplemented
79#14556108C2×C4116yes—
80C4×Dic142—28no—
81#14556093—28no—
82#14556093C2×C4116yescomplemented
83#14556085—24no—
84#14556085C2×C418yes—
85C2×C4⋊Dic71C2312yes—
86C2×C4⋊Dic71C2314yes—
87C2×C4⋊Dic71C2314yes—
88C2×C4⋊Dic71C2314yes—
89C2×Dic71⋊C4C2314yes—
90C2×Dic71⋊C4C2312yes—
91C2×Dic71⋊C4C2314yes—
92C2×Dic71⋊C4C2312yes—
93C2×Dic71⋊C4C2314yes—
94C2×(C2×D71)⋊C4C2312yes—
95C2×(C2×D71)⋊C4C2314yes—
96C2×(C2×D71)⋊C4C2312yes—
97C2×(C2×D71)⋊C4C2314yes—
98C2×C4×Dic71C2312yes—
99C2×C4×Dic71C2311yescharacteristic
100C2×C4×Dic71C2314yes—
101C2×C4×Dic71C2312yes—
102C2×C4×Dic71C2314yes—
103C2×C4×Dic71C2314yes—
104C2×C4×Dic71C2311yescharacteristic
105C2×C4×Dic71C2311yescharacteristic
106C2×C4×Dic71C2312yes—
107C2×C4×Dic71C2312yes—
108C2×C4×Dic71C2314yes—
109C2×C4×Dic71C2311yescharacteristic
110C2×C4×Dic71C2314yes—
111C2×C4×Dic71C2311yescharacteristic
112C42×D71—24no—
113C71⋊#95C2312yes—
114C71⋊#95C2312yes—
115C71⋊#95C2314yes—
116C71⋊#95C2314yes—
117C71⋊#95C2314yes—
118C71⋊#95C2314yes—
119C71⋊#95C2314yes—
120C71⋊#95C2314yes—
121C71⋊#95C2314yes—
122C71⋊#95C2314yes—
123C71⋊#95C2314yes—
124#125×C71C2×C418yescomplemented
125C2×C4⋊C4×C71C2312yes—
126C2×C4⋊C4×C71C2314yes—
127C2×C4⋊C4×C71C2314yes—
128C2×C4⋊C4×C71C2314yes—
129C2×C4⋊C4×C71C2312yes—
130C2×C42×C71C2311yescharacteristic
131C2×C42×C71C2314yes—
132C2×C42×C71C2311yescharacteristic
133C2×C42×C71C2311yescharacteristic
134#95×C71C2312yes—
135#95×C71C2312yes—
136#95×C71C2314yes—
137#95×C71C2314yes—
138C23×D71C2411yescharacteristic
139C2×Dic142—24no—
140C2×Dic142—24no—
141C2×Dic142C22×C418yes—
142C22×Dic71C2412yes—
143C22×Dic71C2414yes—
144C22×Dic71C2411yescharacteristic
145C22×Dic71C2414yes—
146C22×Dic71C2412yes—
147C22×Dic71C2411yescharacteristic
148C22×Dic71C2411yescharacteristic
149C2×C4×D71—22no—
150C2×C4×D71—22no—
151C2×C4×D71—22no—
152C2×C4×D71—22no—
153C2×C4×D71—24no—
154C2×C4×D71—42no—
155C2×C4×D71C22×C414yes—
156C4⋊Dic71—24no—
157C4⋊Dic71—44no—
158C4⋊Dic71C22×C4116yes—
159C4⋊Dic71C22×C4116yes—
160Dic71⋊C4—24no—
161Dic71⋊C4—28no—
162Dic71⋊C4—28no—
163Dic71⋊C4C22×C418yes—
164Dic71⋊C4C22×C4116yes—
165(C2×D71)⋊C4—24no—
166(C2×D71)⋊C4—44no—
167(C2×D71)⋊C4C22×C418yes—
168(C2×D71)⋊C4C22×C4116yes—
169C4×Dic71—22no—
170C4×Dic71—28no—
171C4×Dic71—22no—
172C4×Dic71—22no—
173C4×Dic71—28no—
174C4×Dic71—24no—
175C4×Dic71—28no—
176C4×Dic71—24no—
177C4×Dic71—42no—
178C4×Dic71—44no—
179C4×Dic71—42no—
180C4×Dic71C22×C414yes—
181C4×Dic71C22×C4116yes—
182C4×Dic71#4114yes—
183C22×C4×C71C2411yescharacteristic
184C22×C4×C71C2411yescharacteristic
185C22×C4×C71C2414yes—
186C22×C4×C71C2414yes—
187C22×C4×C71C2412yes—
188C22×C4×C71C2412yes—
189C22×C4×C71C2411yescharacteristic
190C4⋊C4×C71—24no—
191C4⋊C4×C71—44no—
192C4⋊C4×C71—44no—
193C4⋊C4×C71C22×C418yes—
194C4⋊C4×C71C22×C4116yes—
195C42×C71—22no—
196C42×C71—44no—
197C42×C71C22×C414yes—
198C42×C71#4114yes—
199C23×C71C2511yescharacteristic
200C22×D71—21no—
201C22×D71—21no—
202C22×D71—22no—
203C22×D71—22no—
204C22×D71C23×C412yes—
205C2×C4×C71—24no—
206C2×C4×C71—22no—
207C2×C4×C71—22no—
208C2×C4×C71—22no—
209C2×C4×C71—24no—
210C2×C4×C71—22no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—24no—
214C2×C4×C71—22no—
215C2×C4×C71—24no—
216C2×C4×C71—24no—
217C2×C4×C71—22no—
218C2×C4×C71C23×C414yes—
219C2×C4×C71C23×C418yes—
220C2×C4×C71C23×C412yes—
221C2×C4×C71C2×#4112yes—
222C2×C4×C71C2×#4114yes—
223C2×C4×C71C2×#4112yes—
224C2×Dic71—21no—
225C2×Dic71—24no—
226C2×Dic71—24no—
227C2×Dic71—21no—
228C2×Dic71—22no—
229C2×Dic71—22no—
230C2×Dic71—22no—
231C2×Dic71—22no—
232C2×Dic71—24no—
233C2×Dic71—22no—
234C2×Dic71—24no—
235C2×Dic71—24no—
236C2×Dic71—22no—
237C2×Dic71C23×C414yes—
238C2×Dic71C23×C418yes—
239C2×Dic71C23×C412yes—
240C2×Dic71C2×#4112yes—
241C2×Dic71C2×#4112yes—
242C2×Dic712−1+414yes—
243C4×D71—28no—
244C4×D71—44no—
245C4×D71—44no—
246Dic142—216no—
247C22×C71C24×C411yescharacteristic
248C22×C71C22×#4111yescharacteristic
249C22×C71C22×#4111yescharacteristic
250C22×C71C2×2−1+412yes—
251C22×C71#58512yes—
252C2×D71—22no—
253C2×D71—24no—
254C2×D71—41no—
255C2×D71—41no—
256C2×D71—42no—
257C4×C71—28no—
258C4×C71—24no—
259C4×C71—24no—
260C4×C71—41no—
261C4×C71—41no—
262C4×C71—44no—
263C4×C71C4×#4114yes—
264Dic71—28no—
265Dic71—22no—
266Dic71—24no—
267Dic71—22no—
268Dic71—42no—
269Dic71—44no—
270Dic71#54114yescomplemented
271C4×#541—711nomaximal, Sylow
272C2×C71C2×C4×#4111yescharacteristic
273C2×C71C4×2−1+412yes—
274C2×C71C2×#54111yescharacteristic
275C2×C71#345511yescharacteristic
276C2×C71#346112yes—
277D71—42no—
278C2×#541—711no—
279#2336—714no—
280#2334—714no—
281#2330D7111yescharacteristic, complemented
282#2327—711no—
283#2323—712no—
284#2320—712no—
285#2314—711no—
286C4×C4⋊Q8—711no—
287C4×#125—714no—
288C4×#125—711no—
289C4×#126—714no—
290C4×C22⋊Q8—712no—
291C42×Q8—712no—
292C4×#117—711no—
293C71C4×#54111yescharacteristic, complemented, Sylow
294#541—7116no—
295C2×C4⋊Q8—711no—
296C2×#126—714no—
297C2×#125—714no—
298C2×#125C2×D7111yescharacteristic
299C2×C22⋊Q8—712no—
300C2×C4×Q8—712no—
301C2×C4×Q8—712no—
302C2×C4×Q8—712no—
303C2×#117—711no—
304C2×#117—711no—
305C22×C42—711no—
306#391—712no—
307#391—712no—
308#389—712no—
309#389—714no—
310#389—714no—
311#389C2×D7114yes—
312#389C2×D7112yes—
313#388—714no—
314#388—714no—
315#388—714no—
316#387—714no—
317#387—714no—
318#387—714no—
319#387—714no—
320#387—714no—
321#387—714no—
322#387—714no—
323#387—714no—
324#387C2×D7114yes—
325#385—714no—
326#384C2×D7111yescharacteristic
327#383—714no—
328#382—714no—
329#382—711no—
330#380—712no—
331C4×C4⋊C4—714no—
332C4×C4⋊C4—714no—
333C4×C4⋊C4—714no—
334C4×C4⋊C4—712no—
335C4×C4⋊C4—712no—
336C4×C4⋊C4—714no—
337C4×C4⋊C4—714no—
338C4×C4⋊C4—712no—
339C4×C4⋊C4—714no—
340C4×C4⋊C4—712no—
341C4×C4⋊C4C2×D7112yes—
342C4×C22⋊C4—712no—
343C4×C22⋊C4—714no—
344C4×C22⋊C4—712no—
345#376—711no—
346#376C2×D7111yescharacteristic
347C43—714no—
348C43—711no—
349C43—711no—
350C22×Q8—712no—
351C23×C4—711no—
352C23×C4—711no—
353C23×C4—711no—
354C4⋊Q8—718no—
355#126—7132no—
356#125—7132no—
357#125C4×D7118yescomplemented
358C22⋊Q8—7116no—
359C4×Q8—7116no—
360C4×Q8—1428no—
361C4×Q8—1428no—
362#117—718no—
363#117—1424no—
364C2×C4⋊C4—714no—
365C2×C4⋊C4—712no—
366C2×C4⋊C4—714no—
367C2×C4⋊C4—714no—
368C2×C4⋊C4—714no—
369C2×C4⋊C4—712no—
370C2×C4⋊C4—712no—
371C2×C4⋊C4—714no—
372C2×C4⋊C4—714no—
373C2×C4⋊C4C22×D7112yes—
374C2×C4⋊C4C22×D7114yes—
375C2×C4⋊C4C22×D7114yes—
376C2×C4⋊C4C22×D7112yes—
377C2×C4⋊C4C22×D7114yes—
378C2×C22⋊C4—714no—
379C2×C22⋊C4—712no—
380C2×C22⋊C4—714no—
381C2×C22⋊C4—712no—
382C2×C42—712no—
383C2×C42—712no—
384C2×C42—714no—
385C2×C42—711no—
386C2×C42—711no—
387C2×C42—714no—
388C2×C42—711no—
389C2×C42—714no—
390C2×C42—714no—
391C2×C42—712no—
392C2×C42—714no—
393C2×C42—712no—
394C2×C42—711no—
395C2×C42—711no—
396C2×C42—1424no—
397C2×C42C22×D7111yescharacteristic
398C2×C42C22×D7114yes—
399C2×C42C22×D7111yescharacteristic
400C2×C42C22×D7111yescharacteristic
401#95—712no—
402#95—712no—
403#95—714no—
404#95—714no—
405#95—714no—
406#95—714no—
407#95—714no—
408#95—714no—
409#95—714no—
410#95—714no—
411#95—714no—
412#95C22×D7112yes—
413#95C22×D7112yes—
414#95C22×D7114yes—
415#95C22×D7114yes—
416C24—711no—
417C2×Q8—718no—
418C2×Q8—1424no—
419C2×Q8—1424no—
420C22×C4—2842no—
421C22×C4—711no—
422C22×C4—712no—
423C22×C4—711no—
424C22×C4—714no—
425C22×C4—711no—
426C22×C4—714no—
427C22×C4—712no—
428C22×C4—714no—
429C22×C4—1422no—
430C22×C4—1422no—
431C22×C4—1424no—
432C22×C4—1422no—
433C22×C4—1422no—
434C22×C4C23×D7111yescharacteristic
435C22×C4C23×D7111yescharacteristic
436C22×C4C23×D7114yes—
437C22×C4C23×D7114yes—
438C22×C4C23×D7112yes—
439C22×C4C23×D7112yes—
440C22×C4C23×D7111yescharacteristic
441C4⋊C4—24no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—2844no—
445C4⋊C4—718no—
446C4⋊C4—7116no—
447C4⋊C4—7116no—
448C4⋊C4—7116no—
449C4⋊C4—1428no—
450C4⋊C4—1424no—
451C4⋊C4—1428no—
452C4⋊C4—1424no—
453C4⋊C4C2×C4×D7118yes—
454C4⋊C4C2×C4×D71116yes—
455C22⋊C4—2844no—
456C22⋊C4—718no—
457C22⋊C4—7116no—
458C22⋊C4—1424no—
459C42—22no—
460C42—44no—
461C42—2842no—
462C42—2842no—
463C42—2844no—
464C42—714no—
465C42—7116no—
466C42—714no—
467C42—1428no—
468C42—1424no—
469C42—1424no—
470C42—1422no—
471C42—1422no—
472C42—1428no—
473C42—1422no—
474C42—1428no—
475C42C2×C4×D7114yes—
476C42#1176586614yes—
477C23—712no—
478C23—1421no—
479C23—1421no—
480C23—1422no—
481C23—1422no—
482C23C24×D7111yescharacteristic
483Q8—14216no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—2844no—
498C2×C4—2844no—
499C2×C4—712no—
500C2×C4—714no—
501C2×C4—714no—
502C2×C4—718no—
503C2×C4—712no—
504C2×C4—712no—
505C2×C4—1422no—
506C2×C4—1424no—
507C2×C4—1422no—
508C2×C4—1424no—
509C2×C4—1424no—
510C2×C4—1422no—
511C2×C4—1421no—
512C2×C4—1424no—
513C2×C4—1424no—
514C2×C4—1421no—
515C2×C4—1422no—
516C2×C4—1422no—
517C2×C4—1422no—
518C2×C4—1428no—
519C2×C4C22×C4×D7118yes—
520C2×C4C22×C4×D7114yes—
521C2×C4C22×C4×D7112yes—
522C2×C4#41×D7112yes—
523C2×C4C2×#1176586614yes—
524C2×C4C2×#1176586612yes—
525C22—2842no—
526C22—2841no—
527C22—2841no—
528C22—1422no—
529C22—1424no—
530C22C23×C4×D7111yescharacteristic
531C22C2×#41×D7111yescharacteristic
532C22C22×#1176586611yescharacteristic
533C222−1+4×D7112yes—
534C22#1798683412yes—
535C4—24no—
536C4—28no—
537C4—24no—
538C4—41no—
539C4—41no—
540C4—44no—
541C4—2842no—
542C4—2844no—
543C4—714no—
544C4—1422no—
545C4—1428no—
546C4—1424no—
547C4—1422no—
548C4C4×#1176586614yes—
549C2—2842no—
550C2C2×C4×#1176586611yescharacteristic
551C2#2260155112yes—
552C2#541×D7111yescharacteristic
553C2#2260172411yescharacteristic
554C2#2260177412yes—
subgroupG/H
center, FrattiniC23C24×D71
commutatorC22×C71C24×C4
FittingC71×#2330C2
socle, cosocleC23×C71C25
prime radicalC23×D71C24
hypercenter#2330D71
hypocenterC71C4×#541
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ #2330

Complements

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

NH ≅ G/Ntype
2. #22591143C2semidirect
4. #22591059C2semidirect
6. #22590987C2semidirect
8. #22590907C2semidirect
9. #125×Dic71C2semidirect
10. #22590899C2semidirect
13. C4×#14556093C2semidirect
15. C71×#2330C2semidirect
16. #17986612C4semidirect
76. #14556128C2×C4semidirect
78. #14556119C2×C4semidirect
82. #14556093C2×C4semidirect
124. #125×C71C2×C4semidirect
270. Dic71#541semidirect
281. #2330D71semidirect
293. C71C4×#541semidirect
357. #125C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179866122
#225911432
#225911142
#225910592
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C4×#145560932
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C71×#23302
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