Finite Group Databasebeta

← #31967588

Subgroups — #31967588

Order 18176.

638 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22598348C211yesmaximal, characteristic
2#22597437C211yesmaximal, characteristic, complemented
3#22597364C211yesmaximal, characteristic
4#22587194C212yesmaximal, complemented
5#22587173C212yesmaximal, complemented
6#22587149C212yesmaximal, complemented
7#22587143C212yesmaximal, complemented
8#22587121C211yesmaximal, characteristic, complemented
9#22586762C211yesmaximal, characteristic, complemented
10#22586637C211yesmaximal, characteristic, complemented
11C71×#1490C211yesmaximal, characteristic
12#17986556C2211yescharacteristic
13#17986329C412yes—
14#17986289C412yes—
15#17986204C412yes—
16#17986150C412yes—
17#17986127—22no—
18#17986126C2212yescomplemented
19#17986123C2212yescomplemented
20C4×#11765853C2212yes—
21C4×#11765853C2212yes—
22D4×C71⋊C8—22no—
23D4×C71⋊C8C2212yes—
24D4×C71⋊C8C2212yes—
25#17986057—22no—
26#17986052C2211yescharacteristic
27#17986052C2211yescharacteristic
28#17986052C2211yescharacteristic
29C2×C22⋊(C71⋊C8)—22no—
30#17985751C2212yescomplemented
31#17985749C2212yescomplemented
32#17985731C2212yescomplemented
33#17985731C2212yescomplemented
34#17985721C2211yescharacteristic, complemented
35#17985721C2211yescharacteristic, complemented
36#17985720C2211yescharacteristic, complemented
37C8×C71⋊C8C2211yescharacteristic, complemented
38C2×C4×D4×C71C2211yescharacteristic
39#432×C71C2211yescharacteristic
40C4×M16×C71C2211yescharacteristic
41D4⋊C8×C71C2212yes—
42D4⋊C8×C71C2212yes—
43C2×#11765853C2×C412yes—
44C2×#11765853C2×C412yes—
45C22×C71⋊C8—22no—
46C22×C71⋊C8—22no—
47#14556128C2×C411yescharacteristic
48#14556128C2×C411yescharacteristic
49#14556120—21no—
50#14556107—21no—
51#14556107C2×C412yes—
52C4×Dic142—21no—
53C4×Dic142C2312yes—
54#14556087—21no—
55#14556059C2×C412yes—
56#14556059C2×C412yes—
57C4⋊(C71⋊C8)C2×C412yes—
58C4⋊(C71⋊C8)D412yes—
59C4⋊(C71⋊C8)C2312yes—
60Dic71⋊C8—22no—
61Dic71⋊C8—22no—
62#14556051C2×C412yes—
63#14556051C2×C412yes—
64#14556049C2×C412yes—
65#14556049C2×C412yes—
66C22⋊(C71⋊C8)—22no—
67C22⋊(C71⋊C8)—22no—
68C22⋊(C71⋊C8)—21no—
69C22⋊(C71⋊C8)C2×C412yes—
70C22⋊(C71⋊C8)D412yes—
71#14556038—22no—
72C4×C71⋊C8—21no—
73C4×C71⋊C8C2311yescharacteristic
74C4×C71⋊C8C2311yescharacteristic
75C4×C71⋊C8C2311yescharacteristic
76C4×C71⋊C8C2311yescharacteristic
77C8×Dic71—22no—
78C22×D4×C71C2×C411yescharacteristic
79C23×C4×C71—21no—
80C2×M16×C71—21no—
81C4×D4×C71—21no—
82C4×D4×C71—21no—
83C4×D4×C71C2×C412yes—
84C4×D4×C71C2312yes—
85C2×C4⋊C4×C71C2×C411yescharacteristic
86C2×C22⋊C4×C71—21no—
87C2×C42×C71C2311yescharacteristic
88C4⋊C8×C71C2312yes—
89C22⋊C8×C71—21no—
90#97×C71—21no—
91#97×C71—21no—
92C4×C8×C71C2311yescharacteristic
93C4×C8×C71C2311yescharacteristic
94C2×Dic142—21no—
95C2×Dic142C22×C412yes—
96#11765853—24no—
97#11765853—24no—
98C2×C71⋊C8—22no—
99C2×C71⋊C8—82no—
100C2×C71⋊C8—82no—
101C2×C71⋊C8C22×C412yes—
102C2×C71⋊C8C22×C412yes—
103C2×C71⋊C8C2×D412yes—
104C2×C71⋊C8C2×D412yes—
105C4⋊Dic71—21no—
106C4⋊Dic71—21no—
107C4⋊Dic71—21no—
108C4⋊Dic71C22×C412yes—
109Dic71⋊C4—22no—
110Dic71⋊C4—22no—
111C22⋊Dic71—22no—
112C4×Dic71—22no—
113C24×C71—21no—
114C2×D4×C71—24no—
115C2×D4×C71—21no—
116C2×D4×C71C22×C412yes—
117C22×C4×C71—21no—
118C22×C4×C71—22no—
119C22×C4×C71—22no—
120C22×C4×C71—82no—
121C22×C4×C71C22×C411yescharacteristic
122C22×C4×C71C2×D411yescharacteristic
123C22×C4×C71#4111yescharacteristic
124M16×C71—42no—
125C2×C8×C71—21no—
126C2×C8×C71—22no—
127C2×C8×C71—21no—
128C4⋊C4×C71—21no—
129C4⋊C4×C71C22×C412yes—
130C22⋊C4×C71—22no—
131C22⋊C4×C71—22no—
132C42×C71C22×C411yescharacteristic
133C42×C71C2×D411yescharacteristic
134C42×C71#4111yescharacteristic
135C42×C71C2411yescharacteristic
136C23×C71—24no—
137C23×C71—22no—
138C23×C71—81no—
139C23×C71—81no—
140C23×C71C4×D411yescharacteristic
141D4×C71—24no—
142D4×C71—22no—
143D4×C71#13114yes—
144C2×C4×C71—22no—
145C2×C4×C71—22no—
146C2×C4×C71—22no—
147C2×C4×C71—41no—
148C2×C4×C71—42no—
149C2×C4×C71—42no—
150C2×C4×C71—82no—
151C2×C4×C71—81no—
152C2×C4×C71C4×D411yescharacteristic
153C2×C4×C71C4×D411yescharacteristic
154C2×C4×C71C4×D411yescharacteristic
155C2×C4×C71C23×C411yescharacteristic
156C2×C4×C71C22×D411yescharacteristic
157C2×C4×C71C2×#4111yescharacteristic
158C2×Dic71—22no—
159C2×Dic71—22no—
160Dic142—22no—
161Dic142—22no—
162Dic142—81no—
163C8×C71—22no—
164C8×C71—41no—
165C8×C71—42no—
166C71⋊C8—22no—
167C71⋊C8—22no—
168C71⋊C8—22no—
169C71⋊C8—22no—
170C71⋊C8—42no—
171C22×C71—21no—
172C22×C71—24no—
173C22×C71—21no—
174C22×C71—21no—
175C22×C71—42no—
176C22×C71—42no—
177C22×C71—84no—
178C22×C71C2×C4×D411yescharacteristic
179C4×C71—21no—
180C4×C71—21no—
181C4×C71—21no—
182C4×C71—21no—
183C4×C71—42no—
184C4×C71C2×#13112yes—
185C4×C71#57712yes—
186Dic71—42no—
187Dic71—42no—
188#17924—711nomaximal, Sylow
189C2×C71—41no—
190C2×C71—44no—
191C2×C71#292911yescharacteristic
192C2×C71#294911yescharacteristic
193C2×C71#295611yescharacteristic
194#3101—711no—
195#2985—711no—
196#2972—711no—
197#1585—712no—
198#1582—712no—
199SD16⋊C8—712no—
200C8×SD16—712no—
201#1572—711no—
202#1490D7111yescharacteristic
203#1485—711no—
204#1453—711no—
205C71#1792411yescharacteristic, complemented, Sylow
206#533—711no—
207C2×C4×D4C2×D7111yescharacteristic
208#483—712no—
209#476—712no—
210#461—712no—
211#450—712no—
212#446—712no—
213#446—1422no—
214C8×Q8—712no—
215C4×SD16—712no—
216C4×SD16—712no—
217C8×D4—712no—
218C8×D4—712no—
219C8×D4—1422no—
220#433—1422no—
221#432C2×D7111yescharacteristic
222#431—711no—
223#431—711no—
224#431—711no—
225C2×C22⋊C8—1422no—
226C4×M16C2×D7111yescharacteristic
227#335—712no—
228#334—712no—
229Q8⋊C8—712no—
230Q8⋊C8—712no—
231D4⋊C8C2×D7112yes—
232D4⋊C8C2×D7112yes—
233#322—711no—
234#322—711no—
235#322—711no—
236C82—711no—
237C22×D4C4×D7111yescharacteristic
238C23×C4—21no—
239C2×SD16—712no—
240C2×SD16—712no—
241C2×M16—21no—
242C22×C8—1422no—
243C22×C8—1422no—
244C4⋊Q8—711no—
245C4⋊Q8—711no—
246#125—1421no—
247C22⋊Q8—712no—
248C22⋊Q8—1421no—
249C4×Q8—712no—
250C4×Q8—1421no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4C4×D7112yes—
254C4×D4C22×D7112yes—
255#117—1421no—
256C2×C4⋊C4C4×D7111yescharacteristic
257C2×C22⋊C4—21no—
258C2×C42C22×D7111yescharacteristic
259#106—712no—
260#106—712no—
261C4⋊C8—712no—
262C4⋊C8—712no—
263C4⋊C8—712no—
264C4⋊C8—1422no—
265C4⋊C8—1422no—
266C4⋊C8C22×D7112yes—
267#103—712no—
268#103—712no—
269#102—712no—
270#102—712no—
271C22⋊C8—21no—
272C22⋊C8—712no—
273C22⋊C8—712no—
274C22⋊C8—1422no—
275C22⋊C8—1422no—
276C22⋊C8—1421no—
277#97—21no—
278#97—21no—
279#97—1422no—
280C4×C8—711no—
281C4×C8—711no—
282C4×C8—711no—
283C4×C8—711no—
284C4×C8—1422no—
285C4×C8—1421no—
286C4×C8C22×D7111yescharacteristic
287C4×C8C22×D7111yescharacteristic
288C24—21no—
289C2×Q8—712no—
290C2×Q8—1421no—
291C2×D4—24no—
292C2×D4—21no—
293C2×D4C2×C4×D7112yes—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—82no—
298C22×C4C2×C4×D7111yescharacteristic
299C22×C4D4×D7111yescharacteristic
300C22×C4#1176586411yescharacteristic
301SD16—1424no—
302SD16—1424no—
303M16—42no—
304C2×C8—21no—
305C2×C8—22no—
306C2×C8—21no—
307C2×C8—5682no—
308C2×C8—5682no—
309C2×C8—712no—
310C2×C8—712no—
311C2×C8—712no—
312C2×C8—712no—
313C2×C8—1422no—
314C4⋊C4—21no—
315C4⋊C4—712no—
316C4⋊C4—1422no—
317C4⋊C4—1422no—
318C4⋊C4—1421no—
319C4⋊C4—1421no—
320C4⋊C4—1421no—
321C4⋊C4C2×C4×D7112yes—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—1422no—
325C42—1422no—
326C42C2×C4×D7111yescharacteristic
327C42D4×D7111yescharacteristic
328C42#1176586411yescharacteristic
329C42C23×D7111yescharacteristic
330C23—24no—
331C23—22no—
332C23—81no—
333C23—81no—
334C23(C22⋊D71)⋊C411yescharacteristic
335Q8—5681no—
336Q8—1422no—
337Q8—1422no—
338D4—24no—
339D4—22no—
340D4#1455613614yes—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—1422no—
350C2×C4—1422no—
351C2×C4(C22⋊D71)⋊C411yescharacteristic
352C2×C4(C22⋊D71)⋊C411yescharacteristic
353C2×C4(C22⋊D71)⋊C411yescharacteristic
354C2×C4C22×C4×D7111yescharacteristic
355C2×C4C2×D4×D7111yescharacteristic
356C2×C4C2×#1176586411yescharacteristic
357C8—22no—
358C8—41no—
359C8—42no—
360C8—2842no—
361C8—1422no—
362C8—1422no—
363C8—1422no—
364C8—1422no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—24no—
369C22—42no—
370C22—42no—
371C22—84no—
372C22C2×(C22⋊D71)⋊C411yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—2842no—
379C4—2842no—
380C4C2×#1455613612yes—
381C4#1798680912yes—
382C2—41no—
383C2—44no—
384C2#2259703011yescharacteristic
385C2#2259720211yescharacteristic
386C2#2259724611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1490C2
socleC22×C71C2×C4×D4
prime radicalC22×D4×C71C2×C4
cosocleC42×C71C24
hypercenter#1490D71
hypocenterC71#17924
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1490

Complements

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

NH ≅ G/Ntype
2. #22597437C2semidirect
4. #22587194C2semidirect
5. #22587173C2semidirect
6. #22587149C2semidirect
7. #22587143C2semidirect
8. #22587121C2semidirect
9. #22586762C2semidirect
10. #22586637C2semidirect
18. #17986126C22semidirect
19. #17986123C22semidirect
30. #17985751C22semidirect
31. #17985749C22semidirect
32. #17985731C22semidirect
33. #17985731C22semidirect
34. #17985721C22semidirect
35. #17985721C22semidirect
36. #17985720C22semidirect
37. C8×C71⋊C8C22semidirect
205. C71#17924semidirect

Maximal subgroups

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

subgroupindex
#225983482
#225974372
#225973642
#225871942
#225871732
#225871492
#225871432
#225871212
#225867622
#225866372
C71×#14902
#1792471