Finite Group Databasebeta

← #32157926

Subgroups — #32157926

Order 18176.

2094 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D71C211yesmaximal, characteristic
2#22591437C214yesmaximal, complemented
3#22591431C214yesmaximal, complemented
4#22591214C214yesmaximal
5#22591213C214yesmaximal
6#22591080C212yesmaximal, complemented
7#22591077C212yesmaximal, complemented
8#22590955C212yesmaximal
9#125×Dic71C211yesmaximal, characteristic, complemented
10C4×#14556120C211yesmaximal, characteristic, complemented
11#22590895C211yesmaximal, characteristic, complemented
12#22590746C211yesmaximal, characteristic
13#22590694C212yesmaximal
14C4×#14556085C211yesmaximal, characteristic
15C4×#125×C71C211yesmaximal, characteristic, complemented
16C2×#14556120C2211yescharacteristic
17C2×#14556119C2212yes—
18C2×#14556118C2214yes—
19#125×D71C4116yescomplemented
20C2×#14556086C2212yes—
21C2×#14556085C2211yescharacteristic
22C2×#14556085C2211yescharacteristic
23C2×C4⋊C4×D71C2212yes—
24C2×C4⋊C4×D71C2214yes—
25C2×C42×D71C2211yescharacteristic
26#17985898C2212yes—
27#17985897C2214yes—
28#17985897C2212yes—
29#17985896C2214yes—
30#17985896C2214yes—
31#17985895C2212yes—
32#17985895C2214yes—
33#17985894C2214yes—
34#17985894C2214yes—
35#17985885C2214yes—
36#17985884C2214yes—
37#17985883C2214yes—
38#17985883C2214yes—
39#17985882C2214yes—
40#17985881C2214yes—
41#17985881C2214yes—
42#17985868C2212yes—
43#17985865C2211yescharacteristic
44#17985858C2214yes—
45#17985858C2214yes—
46#17985856C2214yes—
47C4⋊C4×Dic71C2212yes—
48C4⋊C4×Dic71C2212yes—
49C4⋊C4×Dic71C2214yes—
50C4×Dic71⋊C4C2214yes—
51C4×Dic71⋊C4C2212yes—
52C4×Dic71⋊C4C2212yes—
53C4×Dic71⋊C4C2214yes—
54C4×C4⋊Dic71C2212yes—
55#17985850C2214yes—
56#17985850C2214yes—
57#17985850C2214yes—
58C4×(C2×D71)⋊C4C2212yes—
59C4×(C2×D71)⋊C4C2212yes—
60C4×(C2×D71)⋊C4C2214yes—
61#17985846C2214yes—
62#17985845C2211yescharacteristic
63#17985845C2211yescharacteristic
64C42×Dic71C2211yescharacteristic
65C2×#125×C71C2211yescharacteristic
66#389×C71C2212yes—
67#387×C71C2214yes—
68#382×C71C2211yescharacteristic
69C4×C4⋊C4×C71C2212yes—
70C4×C4⋊C4×C71C2214yes—
71C43×C71C2211yescharacteristic
72C22×C4×D71C2314yes—
73C22×C4×D71C2312yes—
74C22×C4×D71C2311yescharacteristic
75#14556120C2×C418yes—
76#14556119C2×C4116yes—
77#14556118C2×C4132yes—
78#14556086—28no—
79#14556085—24no—
80#14556085Q818yes—
81C2×C4⋊Dic71C2312yes—
82C2×C4⋊Dic71C2314yes—
83C2×C4⋊Dic71C2312yes—
84C2×C4⋊Dic71C2314yes—
85C2×Dic71⋊C4C2312yes—
86C2×Dic71⋊C4C2314yes—
87C2×Dic71⋊C4C2314yes—
88C2×Dic71⋊C4C2314yes—
89C2×Dic71⋊C4C2314yes—
90C2×Dic71⋊C4C2314yes—
91C2×Dic71⋊C4C2314yes—
92C2×Dic71⋊C4C2314yes—
93C2×Dic71⋊C4C2311yescharacteristic
94C2×Dic71⋊C4C2311yescharacteristic
95C4⋊C4×D71C2×C4132yes—
96C4⋊C4×D71C2×C4116yes—
97C2×(C2×D71)⋊C4C2314yes—
98C2×(C2×D71)⋊C4C2312yes—
99C2×(C2×D71)⋊C4C2311yescharacteristic
100C2×(C2×D71)⋊C4C2311yescharacteristic
101C2×C4×Dic71C2311yescharacteristic
102C2×C4×Dic71C2314yes—
103C2×C4×Dic71C2312yes—
104C2×C4×Dic71C2312yes—
105C2×C4×Dic71C2314yes—
106C2×C4×Dic71C2314yes—
107C2×C4×Dic71C2314yes—
108C2×C4×Dic71C2311yescharacteristic
109C2×C4×Dic71C2312yes—
110C2×C4×Dic71C2311yescharacteristic
111C42×D71C2×C418yes—
112C71⋊#95C2312yes—
113C71⋊#95C2312yes—
114C71⋊#95C2312yes—
115C71⋊#95C2312yes—
116C71⋊#95C2314yes—
117C71⋊#95C2314yes—
118C71⋊#95C2314yes—
119C71⋊#95C2314yes—
120C71⋊#95C2314yes—
121C71⋊#95C2314yes—
122C71⋊#95C2314yes—
123#125×C71C2×C418yes—
124C2×C4⋊C4×C71C2312yes—
125C2×C4⋊C4×C71C2314yes—
126C2×C4⋊C4×C71C2314yes—
127C2×C4⋊C4×C71C2312yes—
128C2×C42×C71C2311yescharacteristic
129C2×C42×C71C2312yes—
130C2×C42×C71C2312yes—
131C2×C42×C71C2314yes—
132C2×C42×C71C2311yescharacteristic
133C2×C42×C71C2311yescharacteristic
134#95×C71C2314yes—
135#95×C71C2314yes—
136#95×C71C2314yes—
137C23×D71C2411yescharacteristic
138C22×Dic71C2412yes—
139C22×Dic71C2414yes—
140C22×Dic71C2414yes—
141C22×Dic71C2412yes—
142C22×Dic71C2411yescharacteristic
143C22×Dic71C2411yescharacteristic
144C22×Dic71C2411yescharacteristic
145C2×C4×D71—44no—
146C2×C4×D71—44no—
147C2×C4×D71—42no—
148C2×C4×D71—42no—
149C2×C4×D71C22×C418yes—
150C2×C4×D71C22×C4116yes—
151C2×C4×D71C22×C414yes—
152C2×C4×D71C2×Q814yes—
153C2×C4×D71#4114yes—
154C4⋊Dic71—24no—
155C4⋊Dic71—44no—
156C4⋊Dic71C22×C4116yes—
157C4⋊Dic71C22×C418yes—
158Dic71⋊C4—24no—
159Dic71⋊C4—44no—
160Dic71⋊C4—44no—
161Dic71⋊C4—44no—
162Dic71⋊C4—44no—
163Dic71⋊C4C22×C4116yes—
164Dic71⋊C4C22×C4116yes—
165Dic71⋊C4C22×C4116yes—
166Dic71⋊C4C2×Q814yes—
167Dic71⋊C4C2×Q814yes—
168(C2×D71)⋊C4—24no—
169(C2×D71)⋊C4—28no—
170(C2×D71)⋊C4C2×Q814yes—
171(C2×D71)⋊C4C2×Q814yes—
172C4×Dic71—22no—
173C4×Dic71—28no—
174C4×Dic71—44no—
175C4×Dic71—44no—
176C4×Dic71—42no—
177C4×Dic71—42no—
178C4×Dic71—44no—
179C4×Dic71C22×C418yes—
180C4×Dic71C22×C414yes—
181C4×Dic71C2×Q814yes—
182C22×C4×C71C2414yes—
183C22×C4×C71C2412yes—
184C22×C4×C71C2412yes—
185C22×C4×C71C2414yes—
186C22×C4×C71C2411yescharacteristic
187C22×C4×C71C2411yescharacteristic
188C22×C4×C71C2411yescharacteristic
189C4⋊C4×C71—24no—
190C4⋊C4×C71—44no—
191C4⋊C4×C71C22×C4116yes—
192C4⋊C4×C71C22×C418yes—
193C42×C71—22no—
194C42×C71—42no—
195C42×C71—44no—
196C42×C71—42no—
197C42×C71C22×C414yes—
198C42×C71C2×Q814yes—
199C23×C71C2511yescharacteristic
200C22×D71—22no—
201C22×D71—22no—
202C22×D71C23×C412yes—
203C22×D71C22×Q812yes—
204C22×D71C2×#4112yes—
205C2×C4×C71—22no—
206C2×C4×C71—22no—
207C2×C4×C71—22no—
208C2×C4×C71—22no—
209C2×C4×C71—24no—
210C2×C4×C71—24no—
211C2×C4×C71—22no—
212C2×C4×C71—24no—
213C2×C4×C71—22no—
214C2×C4×C71—24no—
215C2×C4×C71—22no—
216C2×C4×C71—24no—
217C2×C4×C71C23×C414yes—
218C2×C4×C71C23×C418yes—
219C2×C4×C71C23×C412yes—
220C2×C4×C71C22×Q812yes—
221C2×C4×C71C22×Q812yes—
222C2×C4×C71C22×Q812yes—
223C2×C4×C71C2×#4112yes—
224C2×Dic71—22no—
225C2×Dic71—22no—
226C2×Dic71—24no—
227C2×Dic71—24no—
228C2×Dic71—22no—
229C2×Dic71—22no—
230C2×Dic71—22no—
231C2×Dic71—24no—
232C2×Dic71—24no—
233C2×Dic71—22no—
234C2×Dic71—22no—
235C2×Dic71—24no—
236C2×Dic71C23×C414yes—
237C2×Dic71C23×C418yes—
238C2×Dic71C23×C412yes—
239C2×Dic71C22×Q812yes—
240C2×Dic71C22×Q812yes—
241C2×Dic71C22×Q812yes—
242C2×Dic71C2×#4112yes—
243C4×D71—48no—
244C4×D71—416no—
245C4×D71C4×Q8116yes—
246C22×C71C24×C411yescharacteristic
247C22×C71C23×Q811yescharacteristic
248C22×C71C22×#4111yescharacteristic
249C22×C71#58512yes—
250C22×C71#58512yes—
251C2×D71—24no—
252C2×D71—22no—
253C2×D71—22no—
254C2×D71—24no—
255C2×D71C2×C4×Q814yes—
256C4×C71—24no—
257C4×C71—28no—
258C4×C71—22no—
259C4×C71—22no—
260C4×C71—42no—
261C4×C71—44no—
262C4×C71C2×C4×Q814yes—
263Dic71—28no—
264Dic71—24no—
265Dic71—22no—
266Dic71—22no—
267Dic71—42no—
268Dic71—44no—
269Dic71C2×C4×Q814yes—
270#34217—711nomaximal, Sylow
271C2×C71C22×C4×Q811yescharacteristic
272C2×C71#342812yes—
273C2×C71#350411yescharacteristic
274C2×C71#350411yescharacteristic
275C2×C71#351212yes—
276D71—24no—
277C22×#125—711no—
278#2370—714no—
279#2370—714no—
280#2345—714no—
281#2345—714no—
282#2330—712no—
283#2330—712no—
284#2316—712no—
285C4×#125—711no—
286C4×#125—711no—
287C4×#125—711no—
288C4×#125D7111yescharacteristic, complemented
289#2285—711no—
290#2274—712no—
291C4×#117—711no—
292C71#3421711yescharacteristic, complemented, Sylow
293C2×#125—7116no—
294C2×#125—712no—
295C2×#125—714no—
296C2×#125—711no—
297C2×#125C2×D7111yescharacteristic
298C2×#117—712no—
299C2×#117—711no—
300C2×#117—711no—
301C22×C4⋊C4—712no—
302C22×C4⋊C4—714no—
303C22×C42—711no—
304#389—712no—
305#389—712no—
306#389—714no—
307#389—714no—
308#389—714no—
309#389—712no—
310#389—714no—
311#389—714no—
312#389—714no—
313#389C2×D7112yes—
314#387—714no—
315#387—714no—
316#387—714no—
317#387—714no—
318#387—714no—
319#387—714no—
320#387—714no—
321#387C2×D7114yes—
322#384—712no—
323#382—711no—
324#382C2×D7111yescharacteristic
325#380—714no—
326#380—714no—
327#380—714no—
328C4×C4⋊C4—712no—
329C4×C4⋊C4—712no—
330C4×C4⋊C4—714no—
331C4×C4⋊C4—712no—
332C4×C4⋊C4—714no—
333C4×C4⋊C4—712no—
334C4×C4⋊C4—712no—
335C4×C4⋊C4—714no—
336C4×C4⋊C4—714no—
337C4×C4⋊C4—714no—
338C4×C4⋊C4—714no—
339C4×C4⋊C4C2×D7112yes—
340C4×C4⋊C4C2×D7114yes—
341C4×C22⋊C4—712no—
342C4×C22⋊C4—712no—
343C4×C22⋊C4—714no—
344C4×C22⋊C4—714no—
345#376—711no—
346#376—711no—
347C43—711no—
348C43C2×D7111yescharacteristic
349C23×C4—714no—
350C23×C4—712no—
351C23×C4—711no—
352#125—7132no—
353#125—7116no—
354#125—718no—
355#125C4×D7118yes—
356#117—718no—
357#117—1424no—
358#117—1428no—
359C2×C4⋊C4—714no—
360C2×C4⋊C4—714no—
361C2×C4⋊C4—714no—
362C2×C4⋊C4—714no—
363C2×C4⋊C4—714no—
364C2×C4⋊C4—714no—
365C2×C4⋊C4—714no—
366C2×C4⋊C4—712no—
367C2×C4⋊C4—714no—
368C2×C4⋊C4—714no—
369C2×C4⋊C4—711no—
370C2×C4⋊C4—711no—
371C2×C4⋊C4—7132no—
372C2×C4⋊C4—7116no—
373C2×C4⋊C4—712no—
374C2×C4⋊C4—712no—
375C2×C4⋊C4C22×D7112yes—
376C2×C4⋊C4C22×D7114yes—
377C2×C4⋊C4C22×D7114yes—
378C2×C4⋊C4C22×D7112yes—
379C2×C22⋊C4—711no—
380C2×C22⋊C4—711no—
381C2×C22⋊C4—712no—
382C2×C22⋊C4—714no—
383C2×C42—712no—
384C2×C42—712no—
385C2×C42—714no—
386C2×C42—714no—
387C2×C42—714no—
388C2×C42—711no—
389C2×C42—711no—
390C2×C42—712no—
391C2×C42—718no—
392C2×C42—711no—
393C2×C42—714no—
394C2×C42C22×D7111yescharacteristic
395C2×C42C22×D7112yes—
396C2×C42C22×D7112yes—
397C2×C42C22×D7114yes—
398C2×C42C22×D7111yescharacteristic
399C2×C42C22×D7111yescharacteristic
400#95—712no—
401#95—712no—
402#95—712no—
403#95—712no—
404#95—714no—
405#95—714no—
406#95—714no—
407#95—714no—
408#95—714no—
409#95—714no—
410#95—714no—
411#95C22×D7114yes—
412#95C22×D7114yes—
413#95C22×D7114yes—
414C24—711no—
415C22×C4—2842no—
416C22×C4—2842no—
417C22×C4—2844no—
418C22×C4—2844no—
419C22×C4—718no—
420C22×C4—7116no—
421C22×C4—711no—
422C22×C4—714no—
423C22×C4—714no—
424C22×C4—714no—
425C22×C4—711no—
426C22×C4—711no—
427C22×C4—712no—
428C22×C4—714no—
429C22×C4—714no—
430C22×C4—712no—
431C22×C4C23×D7112yes—
432C22×C4C23×D7114yes—
433C22×C4C23×D7112yes—
434C22×C4C23×D7114yes—
435C22×C4C23×D7111yescharacteristic
436C22×C4C23×D7111yescharacteristic
437C22×C4C23×D7111yescharacteristic
438C4⋊C4—24no—
439C4⋊C4—44no—
440C4⋊C4—2844no—
441C4⋊C4—2844no—
442C4⋊C4—2844no—
443C4⋊C4—2844no—
444C4⋊C4—2844no—
445C4⋊C4—7116no—
446C4⋊C4—7116no—
447C4⋊C4—7116no—
448C4⋊C4—7116no—
449C4⋊C4—718no—
450C4⋊C4—714no—
451C4⋊C4—714no—
452C4⋊C4—1424no—
453C4⋊C4—1424no—
454C4⋊C4C2×C4×D71116yes—
455C4⋊C4C2×C4×D7118yes—
456C22⋊C4—714no—
457C22⋊C4—714no—
458C22⋊C4—1424no—
459C22⋊C4—1428no—
460C42—22no—
461C42—42no—
462C42—44no—
463C42—42no—
464C42—2844no—
465C42—2842no—
466C42—2842no—
467C42—2844no—
468C42—2844no—
469C42—718no—
470C42—714no—
471C42—714no—
472C42—1428no—
473C42—1422no—
474C42C2×C4×D7114yes—
475C42Q8×D7114yes—
476C23—712no—
477C23—712no—
478C23—712no—
479C23—1422no—
480C23—1422no—
481C23C24×D7111yescharacteristic
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—2848no—
495C2×C4—28416no—
496C2×C4—712no—
497C2×C4—712no—
498C2×C4—712no—
499C2×C4—7116no—
500C2×C4—714no—
501C2×C4—718no—
502C2×C4—712no—
503C2×C4—712no—
504C2×C4—1424no—
505C2×C4—1422no—
506C2×C4—1424no—
507C2×C4—1422no—
508C2×C4—1422no—
509C2×C4—1424no—
510C2×C4—1422no—
511C2×C4—1422no—
512C2×C4—1424no—
513C2×C4—1424no—
514C2×C4—1422no—
515C2×C4—1422no—
516C2×C4C22×C4×D7114yes—
517C2×C4C22×C4×D7118yes—
518C2×C4C22×C4×D7112yes—
519C2×C4C2×Q8×D7112yes—
520C2×C4C2×Q8×D7112yes—
521C2×C4C2×Q8×D7112yes—
522C2×C4#41×D7112yes—
523C22—714no—
524C22—1424no—
525C22—1424no—
526C22—1422no—
527C22—1422no—
528C22C23×C4×D7111yescharacteristic
529C22C22×Q8×D7111yescharacteristic
530C22C2×#41×D7111yescharacteristic
531C22#1798683512yes—
532C22#1798683512yes—
533C4—24no—
534C4—28no—
535C4—22no—
536C4—22no—
537C4—42no—
538C4—44no—
539C4—2844no—
540C4—2842no—
541C4—714no—
542C4—1424no—
543C4—1428no—
544C4—1422no—
545C4—1422no—
546C4C4×Q8×D7114yes—
547C2—1424no—
548C2C2×C4×Q8×D7111yescharacteristic
549C2#2260153912yes—
550C2#2260216111yescharacteristic
551C2#2260216411yescharacteristic
552C2#2260225512yes—
subgroupG/H
center, FrattiniC23C24×D71
commutatorC22×C71C24×C4
FittingC4×#125×C71C2
socle, cosocleC23×C71C25
prime radicalC23×D71C24
hypercenterC4×#125D71
hypocenterC71#34217
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×#125

Complements

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

NH ≅ G/Ntype
2. #22591437C2semidirect
3. #22591431C2semidirect
6. #22591080C2semidirect
7. #22591077C2semidirect
9. #125×Dic71C2semidirect
10. C4×#14556120C2semidirect
11. #22590895C2semidirect
15. C4×#125×C71C2semidirect
19. #125×D71C4semidirect
288. C4×#125D71semidirect
292. C71#34217semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D712
#225914372
#225914312
#225912142
#225912132
#225910802
#225910772
#225909552
#125×Dic712
C4×#145561202
#225908952
#225907462
#225906942
C4×#145560852
C4×#125×C712
#3421771