Finite Group Databasebeta

← #32007730

Subgroups — #32007730

Order 18176.

1194 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×(C2×D71)⋊C8C211yesmaximal, characteristic
2#22590722C211yesmaximal, characteristic
3#380×D71C211yesmaximal, characteristic
4#22588540C212yesmaximal, complemented
5#22588536C212yesmaximal, complemented
6#22588534C212yesmaximal, complemented
7#22588526C212yesmaximal, complemented
8#22588367C211yesmaximal, characteristic, complemented
9#22588365C211yesmaximal, characteristic, complemented
10#22588363C211yesmaximal, characteristic, complemented
11C71×#1808C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C23×C71⋊C8C2211yescharacteristic
14C2×#14556086C2212yes—
15C2×C4⋊C4×D71C2212yes—
16C2×C22⋊C4×D71—21no—
17C2×#14556061C2212yescomplemented
18C2×#14556060C2212yescomplemented
19C2×#14556059C2212yescomplemented
20C2×(C2×D71)⋊C8C418yescomplemented
21C2×(C2×D71)⋊C8C2211yescharacteristic
22C2×(C2×D71)⋊C8C2211yescharacteristic
23C2×(C2×D71)⋊C8C2211yescharacteristic
24C2×(C2×D71)⋊C8C2211yescharacteristic
25#17985861—21no—
26#17985859C2211yescharacteristic
27#17985859C2211yescharacteristic
28#17985858—21no—
29#17985857—21no—
30#17985856C2212yes—
31#17985856C2212yes—
32#17985854—21no—
33C22⋊C4×Dic71—21no—
34#17985846—21no—
35#95×D71—21no—
36#17985762C2212yescomplemented
37#17985761C2212yescomplemented
38#17985761C2212yescomplemented
39C23×C8×C71C2211yescharacteristic
40C2×#107×C71C2212yescomplemented
41#380×C71C2211yescharacteristic
42#380×C71C2211yescharacteristic
43#340×C71C2212yescomplemented
44C24×D71D411yescharacteristic
45C23×Dic71D411yescharacteristic
46C22×C4×D71—22no—
47C22×C4×D71—42no—
48C22×C4×D71—42no—
49C22×C4×D71C2×C414yes—
50C22×C4×D71D411yescharacteristic
51C22×C4×D71D411yescharacteristic
52C22×C4×D71Q814yes—
53C22×C4×D71C2311yescharacteristic
54C22×C4×D71C2311yescharacteristic
55C22×C71⋊C8—42no—
56C22×C71⋊C8C2×C414yescomplemented
57C22×C71⋊C8C2311yescharacteristic
58C22×C71⋊C8C2311yescharacteristic
59#14556086—42no—
60#14556086—42no—
61C2×C4⋊Dic71C2312yes—
62C2×C4⋊Dic71C2312yes—
63C2×Dic71⋊C4—22no—
64C4⋊C4×D71—42no—
65C4⋊C4×D71—42no—
66C2×C22⋊Dic71—21no—
67C2×C22⋊Dic71—21no—
68C2×(C2×D71)⋊C4—22no—
69C2×(C2×D71)⋊C4—22no—
70C2×(C2×D71)⋊C4—22no—
71C22⋊C4×D71—44no—
72C2×C4×Dic71—22no—
73#14556061—42no—
74#14556060—42no—
75#14556059—42no—
76(C2×D71)⋊C8C2×C414yes—
77(C2×D71)⋊C8C2×C414yes—
78(C2×D71)⋊C8C2×C414yescomplemented
79(C2×D71)⋊C8C2×C414yescomplemented
80C71⋊#95—22no—
81C71⋊#95—22no—
82C71⋊#95—21no—
83C71⋊#95—21no—
84C23×C4×C71C2311yescharacteristic
85C22×C8×C71—42no—
86C22×C8×C71C2×C414yescomplemented
87C22×C8×C71C2311yescharacteristic
88C22×C8×C71C2311yescharacteristic
89C2×C4⋊C4×C71C2312yes—
90C2×C4⋊C4×C71C2312yes—
91C2×C22⋊C4×C71—21no—
92C2×C22⋊C4×C71—21no—
93#107×C71—42no—
94#95×C71—21no—
95#95×C71—21no—
96C23×D71—22no—
97C23×D71—22no—
98C23×D71—24no—
99C23×D71—42no—
100C23×D71C4⋊C414yes—
101C23×D71C2×D411yescharacteristic
102C23×D71C2×D411yescharacteristic
103C22×Dic71—22no—
104C22×Dic71—22no—
105C22×Dic71—42no—
106C22×Dic71C4⋊C414yes—
107C22×Dic71C2×D411yescharacteristic
108C22×Dic71C2×D411yescharacteristic
109C2×C4×D71—22no—
110C2×C4×D71—22no—
111C2×C4×D71—22no—
112C2×C4×D71—22no—
113C2×C4×D71—44no—
114C2×C4×D71—44no—
115C2×C4×D71—48no—
116C2×C4×D71—42no—
117C2×C4×D71—42no—
118C2×C4×D71—44no—
119C2×C4×D71—44no—
120C2×C4×D71—44no—
121C2×C4×D71C4⋊C414yes—
122C2×C4×D71C4⋊C414yes—
123C2×C4×D71C22×C412yes—
124C2×C4×D71C22×C412yes—
125C2×C4×D71C2×Q812yes—
126C2×C4×D71C2×Q812yes—
127C2×C71⋊C8—22no—
128C2×C71⋊C8—22no—
129C2×C71⋊C8—44no—
130C2×C71⋊C8C22×C412yes—
131C2×C71⋊C8C22×C412yes—
132C4⋊Dic71—22no—
133C4⋊Dic71—22no—
134C4⋊Dic71—22no—
135C4⋊Dic71—22no—
136Dic71⋊C4—44no—
137C22⋊Dic71—42no—
138C22⋊Dic71—42no—
139(C2×D71)⋊C4—44no—
140(C2×D71)⋊C4—44no—
141(C2×D71)⋊C4—44no—
142C4×Dic71—44no—
143C24×C71C2×D411yescharacteristic
144C22×C4×C71—22no—
145C22×C4×C71—22no—
146C22×C4×C71—22no—
147C22×C4×C71—22no—
148C22×C4×C71C22×C412yes—
149C22×C4×C71C2×D411yescharacteristic
150C22×C4×C71C2×Q812yes—
151C22×C4×C71C2411yescharacteristic
152C2×C8×C71—22no—
153C2×C8×C71—22no—
154C2×C8×C71—44no—
155C2×C8×C71C22×C412yes—
156C2×C8×C71C22×C412yes—
157C4⋊C4×C71—22no—
158C4⋊C4×C71—22no—
159C4⋊C4×C71—22no—
160C4⋊C4×C71—22no—
161C22⋊C4×C71—42no—
162C22⋊C4×C71—42no—
163C23×C71—22no—
164C23×C71—22no—
165C23×C71C2×C4⋊C412yes—
166C23×C71AGL(1,ℤ/8)12yes—
167C23×C71#13712yes—
168C23×C71C22×D411yescharacteristic
169C22×D71—22no—
170C22×D71—21no—
171C22×D71—21no—
172C22×D71—22no—
173C22×D71—22no—
174C22×D71—21no—
175C22×D71—21no—
176C22×D71—28no—
177C22×D71—28no—
178C22×D71—44no—
179C22×D71—48no—
180C22×D71—48no—
181C22×D71C2×C4⋊C412yes—
182C22×D71C2×C4⋊C412yes—
183C22×D71#13514yes—
184C2×C4×C71—22no—
185C2×C4×C71—22no—
186C2×C4×C71—22no—
187C2×C4×C71—24no—
188C2×C4×C71—42no—
189C2×C4×C71—42no—
190C2×C4×C71—42no—
191C2×C4×C71—42no—
192C2×C4×C71—42no—
193C2×C4×C71—42no—
194C2×C4×C71C2×C4⋊C412yes—
195C2×C4×C71C23×C411yescharacteristic
196C2×C4×C71C22×Q811yescharacteristic
197C2×C4×C71H(5,𝔽2)12yes—
198C2×C4×C71H(5,𝔽2)12yes—
199C2×Dic71—22no—
200C2×Dic71—22no—
201C2×Dic71—44no—
202C2×Dic71—42no—
203C2×Dic71—42no—
204C2×Dic71—42no—
205C2×Dic71—42no—
206C2×Dic71—42no—
207C2×Dic71—42no—
208C2×Dic71C2×C4⋊C412yes—
209C2×Dic71C2×C4⋊C412yes—
210C4×D71—22no—
211C4×D71—22no—
212C4×D71—22no—
213C4×D71—22no—
214C4×D71—24no—
215C4×D71—24no—
216C4×D71—44no—
217C4×D71—44no—
218C4×D71—88no—
219C8×C71—22no—
220C8×C71—22no—
221C71⋊C8—22no—
222C71⋊C8—22no—
223C22×C71—22no—
224C22×C71—22no—
225C22×C71—24no—
226C22×C71—24no—
227C22×C71#42814yes—
228C22×C71C22×C4⋊C411yescharacteristic
229C22×C71#53412yes—
230C22×C71C2×#13512yes—
231C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
232C22×C71C2×#13711yescharacteristic
233C2×D71—22no—
234C2×D71—22no—
235C2×D71—24no—
236C2×D71—24no—
237C2×D71—22no—
238C2×D71—22no—
239C2×D71—22no—
240C2×D71—416no—
241C2×D71—42no—
242C2×D71—42no—
243C2×D71#42714yes—
244C4×C71—22no—
245C4×C71—82no—
246C4×C71—82no—
247C4×C71#51811yescharacteristic
248C4×C71#51811yescharacteristic
249C4×C71#54312yes—
250Dic71—22no—
251Dic71—22no—
252Dic71—82no—
253Dic71—82no—
254#21941—711nomaximal, Sylow
255C2×C71—24no—
256C2×C71#231511yescharacteristic
257C2×C71C2×#42711yescharacteristic
258C2×C71C2×#42811yescharacteristic
259C2×C71#318812yes—
260C2×C71#319012yes—
261D71—24no—
262D71—42no—
263C22×C22⋊C8—711no—
264#2280—711no—
265C2×#380—711no—
266#1854—712no—
267#1854—712no—
268#1853—712no—
269#1852—712no—
270#1808—711no—
271#1808D7111yescharacteristic, complemented
272#1807—711no—
273#1807—711no—
274C71#2194111yescharacteristic, complemented, Sylow
275C24×C4—711no—
276C23×C8—711no—
277C23×C8C2×D7111yescharacteristic
278C2×#117—712no—
279C22×C4⋊C4—712no—
280C22×C22⋊C4—1421no—
281C2×#107—712no—
282C2×#107C2×D7112yescomplemented
283C2×#106—712no—
284C2×#106—712no—
285C2×C22⋊C8—718no—
286C2×C22⋊C8—711no—
287C2×C22⋊C8—711no—
288C2×C22⋊C8—711no—
289C2×C22⋊C8—711no—
290#381—1421no—
291#380—711no—
292#380—711no—
293#380—712no—
294#380—712no—
295#380—1421no—
296#380—1421no—
297#380C2×D7111yescharacteristic
298#380C2×D7111yescharacteristic
299#379—1421no—
300C4×C22⋊C4—1421no—
301C4×C22⋊C4—1421no—
302C2×#95—1421no—
303#340—712no—
304#340—712no—
305#340—712no—
306#340C2×D7112yescomplemented
307C25—711no—
308C23×C4—2842no—
309C23×C4—2842no—
310C23×C4—711no—
311C23×C4—714no—
312C23×C4—711no—
313C23×C4—711no—
314C23×C4—714no—
315C23×C4—711no—
316C23×C4—711no—
317C23×C4—1422no—
318C23×C4C22×D7111yescharacteristic
319C22×C8—42no—
320C22×C8—2842no—
321C22×C8—714no—
322C22×C8—711no—
323C22×C8—711no—
324C22×C8C4×D7114yescomplemented
325C22×C8C22×D7111yescharacteristic
326C22×C8C22×D7111yescharacteristic
327#117—2842no—
328#117—2842no—
329C2×C4⋊C4—2842no—
330C2×C4⋊C4—2842no—
331C2×C4⋊C4—712no—
332C2×C4⋊C4—712no—
333C2×C4⋊C4—1422no—
334C2×C4⋊C4C22×D7112yes—
335C2×C4⋊C4C22×D7112yes—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—2844no—
339C2×C22⋊C4—1422no—
340C2×C22⋊C4—1422no—
341C2×C22⋊C4—1422no—
342C2×C22⋊C4—1421no—
343C2×C22⋊C4—1421no—
344C2×C42—1422no—
345#107—42no—
346#107—2842no—
347#106—2842no—
348#106—2842no—
349C22⋊C8—714no—
350C22⋊C8—714no—
351C22⋊C8—714no—
352C22⋊C8—714no—
353#95—21no—
354#95—21no—
355#95—1422no—
356#95—1422no—
357#95—1421no—
358#95—1421no—
359C24—2842no—
360C24—714no—
361C24—711no—
362C24—711no—
363C24—1424no—
364C24—1422no—
365C24—1422no—
366C24D4×D7111yescharacteristic
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—2842no—
372C22×C4—2842no—
373C22×C4—2848no—
374C22×C4—2844no—
375C22×C4—2844no—
376C22×C4—2844no—
377C22×C4—2844no—
378C22×C4—2844no—
379C22×C4—2842no—
380C22×C4—714no—
381C22×C4—714no—
382C22×C4—714no—
383C22×C4—711no—
384C22×C4—711no—
385C22×C4—712no—
386C22×C4—712no—
387C22×C4—712no—
388C22×C4—712no—
389C22×C4—1422no—
390C22×C4—1422no—
391C22×C4—1422no—
392C22×C4—1422no—
393C22×C4—1422no—
394C22×C4—1422no—
395C22×C4C2×C4×D7112yes—
396C22×C4D4×D7111yescharacteristic
397C22×C4Q8×D7112yes—
398C22×C4C23×D7111yescharacteristic
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—44no—
402C2×C8—2844no—
403C2×C8—712no—
404C2×C8—712no—
405C2×C8—1422no—
406C2×C8—1422no—
407C2×C8C2×C4×D7112yes—
408C2×C8C2×C4×D7112yes—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—2844no—
414C4⋊C4—1422no—
415C4⋊C4—1422no—
416C4⋊C4—1422no—
417C4⋊C4—1422no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—2844no—
421C22⋊C4—2844no—
422C22⋊C4—2844no—
423C22⋊C4—2842no—
424C22⋊C4—2842no—
425C42—2844no—
426C23—22no—
427C23—22no—
428C23—2848no—
429C23—2848no—
430C23—2844no—
431C23—712no—
432C23—714no—
433C23—712no—
434C23—1428no—
435C23—1422no—
436C23—1421no—
437C23—1421no—
438C23—1428no—
439C23—1422no—
440C23—1421no—
441C23—1421no—
442C23—1422no—
443C23C4⋊C4×D7112yes—
444C23#1455615212yes—
445C23#1455616012yes—
446C23C2×D4×D7111yescharacteristic
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—2844no—
458C2×C4—2844no—
459C2×C4—2842no—
460C2×C4—2842no—
461C2×C4—2842no—
462C2×C4—2842no—
463C2×C4—2842no—
464C2×C4—2842no—
465C2×C4—2844no—
466C2×C4—5688no—
467C2×C4—712no—
468C2×C4—712no—
469C2×C4—1422no—
470C2×C4—1422no—
471C2×C4—1424no—
472C2×C4—1424no—
473C2×C4—1422no—
474C2×C4—1422no—
475C2×C4—1422no—
476C2×C4—1422no—
477C2×C4C4⋊C4×D7112yes—
478C2×C4C22×C4×D7111yescharacteristic
479C2×C4C2×Q8×D7111yescharacteristic
480C2×C4#1455617312yes—
481C2×C4#1455617412yes—
482C8—22no—
483C8—22no—
484C8—1422no—
485C8—1422no—
486C22—22no—
487C22—24no—
488C22—24no—
489C22—22no—
490C22—28416no—
491C22—2842no—
492C22—2842no—
493C22—714no—
494C22—1422no—
495C22—1422no—
496C22—1422no—
497C22—1422no—
498C22—1424no—
499C22—1424no—
500C22—1422no—
501C22#1798604014yes—
502C22C2×C4⋊C4×D7111yescharacteristic
503C22#1798655712yes—
504C22#135×D7112yes—
505C22C2×#1455615211yescharacteristic
506C22C2×#1455616011yescharacteristic
507C4—22no—
508C4—82no—
509C4—82no—
510C4—5682no—
511C4—5682no—
512C4—1422no—
513C4—1422no—
514C4#1798647211yescharacteristic
515C4#1798647311yescharacteristic
516C4#1798662412yes—
517C2—24no—
518C2—2842no—
519C2—1424no—
520C2#2259095311yescharacteristic
521C2#427×D7111yescharacteristic
522C2C2×#1798604011yescharacteristic
523C2#2259910012yes—
524C2#2259912212yes—
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1808C2
socleC23×C71C22×D4
prime radicalC24×D71D4
cosocleC22×C4×C71C24
hypercenter#1808D71
hypocenterC71#21941
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1808

Complements

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

NH ≅ G/Ntype
4. #22588540C2semidirect
5. #22588536C2semidirect
6. #22588534C2semidirect
7. #22588526C2semidirect
8. #22588367C2semidirect
9. #22588365C2semidirect
10. #22588363C2semidirect
11. C71×#1808C2semidirect
17. C2×#14556061C22semidirect
18. C2×#14556060C22semidirect
19. C2×#14556059C22semidirect
20. C2×(C2×D71)⋊C8C4semidirect
36. #17985762C22semidirect
37. #17985761C22semidirect
38. #17985761C22semidirect
40. C2×#107×C71C22semidirect
43. #340×C71C22semidirect
56. C22×C71⋊C8C2×C4semidirect
78. (C2×D71)⋊C8C2×C4semidirect
79. (C2×D71)⋊C8C2×C4semidirect
86. C22×C8×C71C2×C4semidirect
271. #1808D71semidirect
274. C71#21941semidirect
282. C2×#107C2×D71semidirect
306. #340C2×D71semidirect
324. C22×C8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×D71)⋊C82
#225907222
#380×D712
#225885402
#225885362
#225885342
#225885262
#225883672
#225883652
#225883632
C71×#18082
#2194171