Finite Group Databasebeta

← #2075×D71 #32026270

Subgroups — #2075×D71 #32026270

Order 18176.

1270 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#107×D71C211yesmaximal, characteristic, complemented
2C2×C22⋊C8×D71C211yesmaximal, characteristic
3#380×D71C212yesmaximal
4#22589767C212yesmaximal, complemented
5#22589766C212yesmaximal, complemented
6#22589765C211yesmaximal, characteristic, complemented
7#22589764C211yesmaximal, characteristic, complemented
8#22589763C211yesmaximal, characteristic, complemented
9#340×D71C212yesmaximal, complemented
10#340×D71C211yesmaximal, characteristic, complemented
11C71×#2075C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C22×C8×D71C2211yescharacteristic
14C22×C8×D71C2211yescharacteristic
15C2×C4⋊C4×D71C2212yes—
16C2×C4⋊C4×D71C2212yes—
17C2×C22⋊C4×D71—22no—
18C2×#14556062C2211yescharacteristic, complemented
19C2×#14556061C2212yescomplemented
20#107×D71—24no—
21C2×C22⋊(C71⋊C8)C2211yescharacteristic
22C2×(C2×D71)⋊C8C2211yescharacteristic
23C2×(C2×D71)⋊C8C2211yescharacteristic
24C22⋊C8×D71—24no—
25#17985859C2212yes—
26#17985858—22no—
27#17985857—22no—
28#17985856C2212yes—
29#17985856C2212yes—
30#95×D71—22no—
31#17985762C2211yescharacteristic, complemented
32#17985762C2212yescomplemented
33#17985761C2212yescomplemented
34#17985761C2212yescomplemented
35#17985761C2212yescomplemented
36C2×#107×C71C2211yescharacteristic, complemented
37C2×C22⋊C8×C71C2211yescharacteristic
38#380×C71C2212yes—
39#340×C71C2211yescharacteristic, complemented
40#340×C71C2212yescomplemented
41C24×D71D411yescharacteristic
42C23×Dic71D411yescharacteristic
43C22×C4×D71—22no—
44C22×C4×D71—22no—
45C22×C4×D71—24no—
46C22×C4×D71—24no—
47C22×C4×D71—22no—
48C22×C4×D71—22no—
49C22×C4×D71D411yescharacteristic
50C22×C4×D71D411yescharacteristic
51C22×C4×D71C2311yescharacteristic
52C22×C4×D71C2311yescharacteristic
53C22×C71⋊C8C2311yescharacteristic
54C22×C71⋊C8C2311yescharacteristic
55C2×C8×D71—24no—
56C2×C8×D71—22no—
57C2×C8×D71—24no—
58C2×C8×D71D414yes—
59C2×C4⋊Dic71C2312yes—
60C2×C4⋊Dic71C2312yes—
61C2×Dic71⋊C4—22no—
62C2×Dic71⋊C4—22no—
63C4⋊C4×D71—24no—
64C4⋊C4×D71—24no—
65C4⋊C4×D71—24no—
66C4⋊C4×D71—24no—
67C2×C22⋊Dic71—22no—
68C2×(C2×D71)⋊C4—22no—
69C2×(C2×D71)⋊C4—22no—
70C22⋊C4×D71—48no—
71#14556062—22no—
72#14556061—24no—
73C22⋊(C71⋊C8)—22no—
74(C2×D71)⋊C8—22no—
75(C2×D71)⋊C8—22no—
76C71⋊#95—22no—
77C71⋊#95—22no—
78C71⋊#95—22no—
79C23×C4×C71C2311yescharacteristic
80C22×C8×C71C2311yescharacteristic
81C22×C8×C71C2311yescharacteristic
82C2×C4⋊C4×C71C2312yes—
83C2×C4⋊C4×C71C2312yes—
84C2×C22⋊C4×C71—22no—
85#107×C71—22no—
86C22⋊C8×C71—22no—
87#95×C71—22no—
88C23×D71—22no—
89C23×D71—24no—
90C23×D71—22no—
91C23×D71—24no—
92C23×D71D814yes—
93C23×D71C2×D411yescharacteristic
94C23×D71C2×D411yescharacteristic
95C22×Dic71—22no—
96C22×Dic71—24no—
97C22×Dic71—22no—
98C22×Dic71—22no—
99C22×Dic71C2×D411yescharacteristic
100C22×Dic71C2×D411yescharacteristic
101C2×C4×D71—22no—
102C2×C4×D71—24no—
103C2×C4×D71—21no—
104C2×C4×D71—22no—
105C2×C4×D71—22no—
106C2×C4×D71—21no—
107C2×C4×D71—22no—
108C2×C4×D71—24no—
109C2×C4×D71—44no—
110C2×C4×D71—44no—
111C2×C4×D71—48no—
112C2×C4×D71—44no—
113C2×C4×D71—44no—
114C2×C4×D71—44no—
115C2×C4×D71—44no—
116C2×C4×D71—44no—
117C2×C4×D71—44no—
118C2×C4×D71C2×D412yes—
119C2×C4×D71#4114yes—
120C2×C4×D71#4114yes—
121C2×C4×D71#4112yes—
122C2×C71⋊C8—22no—
123C2×C71⋊C8—21no—
124C2×C71⋊C8—22no—
125C2×C71⋊C8C2×D412yes—
126C8×D71—28no—
127C8×D71—44no—
128C4⋊Dic71—22no—
129C4⋊Dic71—22no—
130C4⋊Dic71—22no—
131C4⋊Dic71—22no—
132Dic71⋊C4—44no—
133Dic71⋊C4—44no—
134C22⋊Dic71—44no—
135(C2×D71)⋊C4—44no—
136(C2×D71)⋊C4—44no—
137C24×C71C2×D411yescharacteristic
138C22×C4×C71—21no—
139C22×C4×C71—21no—
140C22×C4×C71—22no—
141C22×C4×C71—22no—
142C22×C4×C71—22no—
143C22×C4×C71—22no—
144C22×C4×C71C2×D411yescharacteristic
145C22×C4×C71C2411yescharacteristic
146C2×C8×C71—22no—
147C2×C8×C71—21no—
148C2×C8×C71—22no—
149C2×C8×C71C2×D412yes—
150C4⋊C4×C71—22no—
151C4⋊C4×C71—22no—
152C4⋊C4×C71—22no—
153C4⋊C4×C71—22no—
154C22⋊C4×C71—44no—
155C23×C71—21no—
156C23×C71—22no—
157C23×C71—21no—
158C23×C71—22no—
159C23×C71C2×D812yes—
160C23×C71C22×D411yescharacteristic
161C22×D71—21no—
162C22×D71—22no—
163C22×D71—21no—
164C22×D71—22no—
165C22×D71—28no—
166C22×D71—216no—
167C22×D71—44no—
168C22×D71—44no—
169C22×D71—48no—
170C22×D71#12112yes—
171C22×D71#12314yes—
172C22×D71C2×D812yes—
173C22×D71C2×D812yes—
174C22×D71#13512yes—
175C22×D71#13714yes—
176C2×C4×C71—21no—
177C2×C4×C71—22no—
178C2×C4×C71—41no—
179C2×C4×C71—41no—
180C2×C4×C71—42no—
181C2×C4×C71—42no—
182C2×C4×C71—42no—
183C2×C4×C71—42no—
184C2×C4×C71—42no—
185C2×C4×C71—42no—
186C2×C4×C71—42no—
187C2×C4×C71C22×D411yescharacteristic
188C2×C4×C71C2×#4112yes—
189C2×C4×C71C2×#4112yes—
190C2×C4×C71C2×#4111yescharacteristic
191C2×Dic71—21no—
192C2×Dic71—22no—
193C2×Dic71—44no—
194C2×Dic71—42no—
195C2×Dic71—42no—
196C2×Dic71—42no—
197C2×Dic71—42no—
198C2×Dic71—42no—
199C2×Dic71—42no—
200C2×Dic71#12112yes—
201C2×Dic71#12314yes—
202C4×D71—44no—
203C4×D71—41no—
204C4×D71—41no—
205C4×D71—42no—
206C4×D71—44no—
207C4×D71—88no—
208C4×D71—88no—
209C4×D71#12314yes—
210C4×D71#12414yes—
211C4×D71#12618yes—
212C8×C71—22no—
213C8×C71—41no—
214C71⋊C8—22no—
215C71⋊C8—41no—
216C22×C71—24no—
217C22×C71—22no—
218C22×C71—42no—
219C22×C71—41no—
220C22×C71—41no—
221C22×C71C2×#12111yescharacteristic
222C22×C71C2×#12312yes—
223C22×C71C22×D811yescharacteristic
224C22×C71C2×#13511yescharacteristic
225C22×C71C2×#13712yes—
226C2×D71—24no—
227C2×D71—22no—
228C2×D71—42no—
229C2×D71—41no—
230C2×D71—41no—
231C2×D71—416no—
232C2×D71#39714yes—
233C2×D71#46614yes—
234C2×D71#47014yes—
235C2×D71#48018yes—
236C2×D71#48518yes—
237C4×C71—41no—
238C4×C71—82no—
239C4×C71—82no—
240C4×C71C2×#12311yescharacteristic
241C4×C71C2×#12411yescharacteristic
242C4×C71C2×#12612yes—
243Dic71—41no—
244Dic71—82no—
245Dic71—82no—
246Dic71#39714yes—
247C2×#2075—711nomaximal, Sylow
248C2×C71—42no—
249C2×C71C2×#39711yescharacteristic
250C2×C71C2×#46611yescharacteristic
251C2×C71C2×#47011yescharacteristic
252C2×C71C2×#48012yes—
253C2×C71C2×#48512yes—
254D71—42no—
255D71#207518yesdirect product factor
256C22×#107—711no—
257C22×C22⋊C8—711no—
258C2×#380—712no—
259#2075—711no—
260#2075—711no—
261#2075—712no—
262#2075—711no—
263#2075—712no—
264#2075D7111yescharacteristic, direct product factor
265C2×#340—712no—
266C2×#340—711no—
267C71C2×#207511yescharacteristic, complemented, Sylow
268C24×C4—711no—
269C23×C8—711no—
270C23×C8—711no—
271C22×C4⋊C4—712no—
272C22×C4⋊C4—712no—
273C22×C22⋊C4—1422no—
274C2×#107—711no—
275C2×#107—712no—
276C2×#107—1424no—
277C2×#107C2×D7111yescharacteristic, complemented
278C2×C22⋊C8—711no—
279C2×C22⋊C8—711no—
280C2×C22⋊C8—711no—
281C2×C22⋊C8—1424no—
282C2×C22⋊C8C2×D7111yescharacteristic
283#380—712no—
284#380—712no—
285#380—712no—
286#380—1422no—
287#380—1422no—
288#380C2×D7112yes—
289C2×#95—1422no—
290#340—711no—
291#340—712no—
292#340—712no—
293#340—712no—
294#340—712no—
295#340C2×D7111yescharacteristic, complemented
296#340C2×D7112yescomplemented
297C25—711no—
298C23×C4—711no—
299C23×C4—711no—
300C23×C4—711no—
301C23×C4—711no—
302C23×C4—711no—
303C23×C4—1422no—
304C23×C4—1422no—
305C23×C4—1422no—
306C23×C4—1424no—
307C23×C4—1424no—
308C23×C4—1422no—
309C23×C4C22×D7111yescharacteristic
310C22×C8—711no—
311C22×C8—714no—
312C22×C8—711no—
313C22×C8—1422no—
314C22×C8—1424no—
315C22×C8—1424no—
316C22×C8C22×D7111yescharacteristic
317C22×C8C22×D7111yescharacteristic
318C2×C4⋊C4—712no—
319C2×C4⋊C4—712no—
320C2×C4⋊C4—1424no—
321C2×C4⋊C4—1424no—
322C2×C4⋊C4—1424no—
323C2×C4⋊C4—1424no—
324C2×C4⋊C4—1422no—
325C2×C4⋊C4—1422no—
326C2×C4⋊C4C22×D7112yes—
327C2×C4⋊C4C22×D7112yes—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—2848no—
330C2×C22⋊C4—1422no—
331C2×C22⋊C4—1422no—
332C2×C22⋊C4—1422no—
333#107—22no—
334#107—1424no—
335#107—1422no—
336C22⋊C8—22no—
337C22⋊C8—1422no—
338C22⋊C8—1422no—
339C22⋊C8—1422no—
340#95—22no—
341#95—1422no—
342#95—1422no—
343#95—1422no—
344C24—711no—
345C24—711no—
346C24—714no—
347C24—1422no—
348C24—1424no—
349C24—1422no—
350C24—1424no—
351C24D4×D7111yescharacteristic
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—2844no—
359C22×C4—2844no—
360C22×C4—2848no—
361C22×C4—2844no—
362C22×C4—2844no—
363C22×C4—2844no—
364C22×C4—2844no—
365C22×C4—2844no—
366C22×C4—2844no—
367C22×C4—711no—
368C22×C4—712no—
369C22×C4—712no—
370C22×C4—714no—
371C22×C4—714no—
372C22×C4—711no—
373C22×C4—1422no—
374C22×C4—1424no—
375C22×C4—1422no—
376C22×C4—1424no—
377C22×C4—1422no—
378C22×C4—1422no—
379C22×C4—1422no—
380C22×C4—1424no—
381C22×C4—1421no—
382C22×C4—1422no—
383C22×C4—1422no—
384C22×C4—1421no—
385C22×C4D4×D7111yescharacteristic
386C22×C4C23×D7111yescharacteristic
387C2×C8—22no—
388C2×C8—21no—
389C2×C8—22no—
390C2×C8—2844no—
391C2×C8—712no—
392C2×C8—1428no—
393C2×C8—1421no—
394C2×C8—1422no—
395C2×C8—1422no—
396C2×C8D4×D7112yes—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—2844no—
402C4⋊C4—2844no—
403C4⋊C4—1422no—
404C4⋊C4—1422no—
405C4⋊C4—1422no—
406C4⋊C4—1422no—
407C22⋊C4—44no—
408C22⋊C4—2844no—
409C22⋊C4—2844no—
410C22⋊C4—2844no—
411C23—21no—
412C23—22no—
413C23—21no—
414C23—22no—
415C23—2844no—
416C23—2844no—
417C23—2848no—
418C23—712no—
419C23—714no—
420C23—712no—
421C23—712no—
422C23—714no—
423C23—712no—
424C23—1428no—
425C23—14216no—
426C23—1421no—
427C23—1422no—
428C23—1421no—
429C23—1422no—
430C23D8×D7112yes—
431C23C2×D4×D7111yescharacteristic
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—2844no—
444C2×C4—2844no—
445C2×C4—2844no—
446C2×C4—2841no—
447C2×C4—2841no—
448C2×C4—2842no—
449C2×C4—2842no—
450C2×C4—2842no—
451C2×C4—2842no—
452C2×C4—2842no—
453C2×C4—2842no—
454C2×C4—2842no—
455C2×C4—5688no—
456C2×C4—5688no—
457C2×C4—712no—
458C2×C4—714no—
459C2×C4—714no—
460C2×C4—714no—
461C2×C4—718no—
462C2×C4—1421no—
463C2×C4—1422no—
464C2×C4C2×D4×D7111yescharacteristic
465C2×C4#41×D7112yes—
466C2×C4#41×D7112yes—
467C2×C4#41×D7111yescharacteristic
468C8—22no—
469C8—41no—
470C8—2841no—
471C8—1422no—
472C22—22no—
473C22—24no—
474C22—41no—
475C22—41no—
476C22—42no—
477C22—28416no—
478C22—2841no—
479C22—2841no—
480C22—2842no—
481C22—714no—
482C22—714no—
483C22—718no—
484C22—714no—
485C22—718no—
486C22—1422no—
487C22—1424no—
488C22#121×D7111yescharacteristic
489C22#123×D7112yes—
490C22C2×D8×D7111yescharacteristic
491C22#135×D7111yescharacteristic
492C22#137×D7112yes—
493C4—41no—
494C4—82no—
495C4—82no—
496C4—2841no—
497C4—5682no—
498C4—5682no—
499C4—714no—
500C4#123×D7111yescharacteristic
501C4#124×D7111yescharacteristic
502C4#126×D7112yes—
503C2—42no—
504C2—2842no—
505C2—718no—
506C2#397×D7111yescharacteristic
507C2#466×D7111yescharacteristic
508C2#470×D7111yescharacteristic
509C2#480×D7112yes—
510C2#485×D7112yes—
subgroupG/H
centerC23C2×D4×D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2075C2
socleC23×C71C22×D4
prime radicalC24×D71D4
hypercenter#2075D71
hypocenterC71C2×#2075
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2075

Complements

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

NH ≅ G/Ntype
1. C2×#107×D71C2semidirect
4. #22589767C2semidirect
5. #22589766C2semidirect
6. #22589765C2semidirect
7. #22589764C2semidirect
8. #22589763C2semidirect
9. #340×D71C2semidirect
10. #340×D71C2semidirect
11. C71×#2075C2semidirect
18. C2×#14556062C22semidirect
19. C2×#14556061C22semidirect
31. #17985762C22semidirect
32. #17985762C22semidirect
33. #17985761C22semidirect
34. #17985761C22semidirect
35. #17985761C22semidirect
36. C2×#107×C71C22semidirect
39. #340×C71C22semidirect
40. #340×C71C22semidirect
255. D71#2075direct (N × H)
264. #2075D71direct (N × H)
267. C71C2×#2075semidirect
277. C2×#107C2×D71semidirect
295. #340C2×D71semidirect
296. #340C2×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#107×D712
C2×C22⋊C8×D712
#380×D712
#225897672
#225897662
#225897652
#225897642
#225897632
#340×D712
C71×#20752
C2×#207571