Finite Group Databasebeta

← #32150860

Subgroups — #32150860

Order 18176.

652 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#22601473C211yesmaximal, characteristic
2#443×D71C212yesmaximal, complemented
3#22590226C211yesmaximal, characteristic, complemented
4#22590225C211yesmaximal, characteristic, complemented
5#22590223C212yesmaximal, complemented
6#22590222C211yesmaximal, characteristic, complemented
7#22590219C212yesmaximal, complemented
8#22590133C211yesmaximal, characteristic, complemented
9#22590066C212yesmaximal, complemented
10C4×#14556067C211yesmaximal, characteristic, complemented
11C71×#2178C211yesmaximal, characteristic, complemented
12#135×D71C2212yes—
13#131×D71—22no—
14#17986441C412yescomplemented
15#17986438C412yescomplemented
16#17986436C412yescomplemented
17#17986433C412yescomplemented
18#17986431C2211yescharacteristic
19#17986429C2212yes—
20C2×#14556067C2211yescharacteristic
21C2×#14556067C2211yescharacteristic
22#17986117C2212yescomplemented
23#17986116C2212yescomplemented
24#17986115C2212yescomplemented
25#17986114—22no—
26#17986113—22no—
27#108×D71C2212yes—
28#104×D71—22no—
29C4×C8×D71C2211yescharacteristic
30#17985829C2211yescharacteristic, complemented
31#17985828C2212yescomplemented
32#17985807C2212yescomplemented
33#17985806C2212yescomplemented
34#17985805C2212yescomplemented
35#17985774C2211yescharacteristic, complemented
36#17985774C2211yescharacteristic, complemented
37C4×C71⋊C16C2211yescharacteristic, complemented
38#508×C71C2211yescharacteristic
39#443×C71C2212yescomplemented
40#368×C71C2211yescharacteristic, complemented
41#359×C71C2212yescomplemented
42C4×C16×C71C2211yescharacteristic, complemented
43#41×D71—22no—
44#14556151C2×C412yes—
45#14556150C2312yes—
46#14556149—22no—
47#14556148—22no—
48#14556147C2×C412yes—
49Q16×D71C2×C412yes—
50SD16×D71—22no—
51D8×D71C2×C412yes—
52#14556136—22no—
53#14556136—42no—
54#14556135—22no—
55#14556134—22no—
56#14556134—42no—
57M16×D71—22no—
58M16×D71—42no—
59C2×C8×D71—42no—
60C2×C8×D71D411yescharacteristic
61C2×C8×D71C2311yescharacteristic
62C42×D71D411yescharacteristic
63Dic568C2×C411yescharacteristic, complemented
64#14556074C2×C412yescomplemented
65#14556073C2×C412yescomplemented
66#14556072C2×C412yescomplemented
67#14556071C2×C412yescomplemented
68D1136C2×C411yescharacteristic, complemented
69#14556069C2×C412yescomplemented
70#14556067—22no—
71#14556067C2×C412yes—
72#14556067C2×C412yes—
73C2×C71⋊C16C2311yescharacteristic
74C2×C71⋊C16C2311yescharacteristic
75#14556064C2312yes—
76#14556063—22no—
77#14556056—22no—
78#14556055—22no—
79#14556054—22no—
80C4×C71⋊C8D411yescharacteristic
81C8×Dic71D411yescharacteristic
82C8×Dic71C2311yescharacteristic
83#135×C71C2312yes—
84#131×C71—22no—
85Q32×C71C2×C411yescharacteristic, complemented
86SD32×C71C2×C412yescomplemented
87D16×C71C2×C411yescharacteristic, complemented
88C2×C16×C71C2311yescharacteristic
89C2×C16×C71C2311yescharacteristic
90#108×C71C2312yes—
91#104×C71—22no—
92C4×C8×C71C2311yescharacteristic
93#11765866—22no—
94#11765865—22no—
95#11765865—42no—
96#11765864—22no—
97#11765864—42no—
98Q8×D71—22no—
99D4×D71—22no—
100D4×D71—42no—
101C2×C4×D71—21no—
102C2×C4×D71—42no—
103C2×C4×D71C2×D411yescharacteristic
104Dic284C22×C412yes—
105#11765855—22no—
106#11765854—22no—
107#11765853—22no—
108#11765852—22no—
109D568C22×C412yes—
110#11765850—22no—
111#11765849—22no—
112#11765849—42no—
113#11765848—42no—
114#11765848—42no—
115#11765848—42no—
116C2×C71⋊C8—42no—
117C2×C71⋊C8C2×D411yescharacteristic
118C2×C71⋊C8C2×D411yescharacteristic
119C8×D71—42no—
120C8×D71—42no—
121C8×D71—42no—
122C8×D71D812yes—
123C8×D71D812yes—
124C8×D71C22×C412yes—
125C8×D71#4112yes—
126C4×Dic71—21no—
127C4×Dic71C2×D411yescharacteristic
128C71⋊C16—21no—
129C71⋊C16C22×C412yes—
130#41×C71—22no—
131Q16×C71C22×C412yes—
132SD16×C71—22no—
133D8×C71C22×C412yes—
134M16×C71—22no—
135M16×C71—42no—
136C2×C8×C71—42no—
137C2×C8×C71C2×D411yescharacteristic
138C2×C8×C71C2411yescharacteristic
139C42×C71C2×D411yescharacteristic
140C16×C71—21no—
141C16×C71C22×C411yescharacteristic
142C16×C71C22×C411yescharacteristic
143C22×D71—42no—
144C22×D71#13511yescharacteristic
145Q8×C71—22no—
146D4×C71—22no—
147D4×C71—42no—
148C2×C4×C71—21no—
149C2×C4×C71—42no—
150C2×C4×C71C22×D411yescharacteristic
151C22⋊D71—42no—
152C22⋊D71—42no—
153C22⋊D71—42no—
154C2×Dic71—21no—
155C2×Dic71—42no—
156C2×Dic71#13511yescharacteristic
157D284—22no—
158D284—42no—
159C4×D71—42no—
160C4×D71—42no—
161C4×D71—42no—
162C4×D71—42no—
163C4×D71C4×D412yes—
164C4×D71C2×D812yes—
165Dic142—22no—
166Dic142—42no—
167C8×C71—42no—
168C8×C71C2×D811yescharacteristic
169C8×C71C2×D811yescharacteristic
170C8×C71C23×C411yescharacteristic
171C8×C71C2×#4111yescharacteristic
172C71⋊C8—42no—
173C71⋊C8C4×D412yes—
174C71⋊C8C2×D812yes—
175C22×C71—42no—
176C22×C71C2×#13511yescharacteristic
177C2×D71—41no—
178C2×D71—42no—
179C2×D71—84no—
180C2×D71C4×D812yes—
181C4×C71—22no—
182C4×C71—42no—
183C4×C71C2×C4×D411yescharacteristic
184C4×C71C22×D811yescharacteristic
185Dic71—41no—
186Dic71—42no—
187Dic71C4×D812yes—
188#33463—711nomaximal, Sylow
189C2×C71—21no—
190C2×C71—82no—
191C2×C71C2×C4×D811yescharacteristic
192D71—22no—
193D71—82no—
194#3414—711no—
195C2×#443—712no—
196#2179—711no—
197#2179—711no—
198#2179—712no—
199#2178—712no—
200#2178—711no—
201#2178D7111yescharacteristic, complemented
202#2161—711no—
203#2143—712no—
204C4×M32—711no—
205C71#3346311yescharacteristic, complemented, Sylow
206C2×#135—712no—
207C2×#131—1422no—
208#510—712no—
209#510—712no—
210#509—712no—
211#509—712no—
212#508—712no—
213#508—711no—
214#508C2×D7111yescharacteristic
215C2×M32—711no—
216C2×M32—711no—
217#443—712no—
218#443—712no—
219#443—712no—
220#443—1422no—
221#443—1422no—
222#443C2×D7112yescomplemented
223C2×#108—712no—
224C2×#104—1422no—
225C2×C4×C8—711no—
226#368—711no—
227#368—712no—
228#368C2×D7111yescharacteristic, complemented
229#359—712no—
230#359—712no—
231#359—712no—
232#359C2×D7112yescomplemented
233#346—711no—
234#346—711no—
235C4×C16—711no—
236C4×C16C2×D7111yescharacteristic, complemented
237C2×#41—1422no—
238#135—712no—
239#135—712no—
240#135—712no—
241#135—1422no—
242#135—1422no—
243#135C22×D7112yes—
244C2×Q16—712no—
245C2×SD16—1422no—
246C2×D8—712no—
247#131—22no—
248#131—2842no—
249#131—2842no—
250#131—1422no—
251#131—1422no—
252#131—1422no—
253C2×M16—2842no—
254C2×M16—1422no—
255C22×C8—2842no—
256C22×C8—711no—
257C22×C8—711no—
258C2×C42—711no—
259Q32—712no—
260Q32—711no—
261Q32C4×D7111yescharacteristic, complemented
262SD32—712no—
263SD32—712no—
264SD32—712no—
265SD32C4×D7112yescomplemented
266D16—712no—
267D16—711no—
268D16C4×D7111yescharacteristic, complemented
269M32—712no—
270M32—712no—
271M32—1422no—
272C2×C16—711no—
273C2×C16—711no—
274C2×C16C22×D7111yescharacteristic
275C2×C16C22×D7111yescharacteristic
276#108—712no—
277#108—1422no—
278#108C22×D7112yes—
279#104—22no—
280#104—1422no—
281#104—1422no—
282#104—1422no—
283C4×C8—711no—
284C4×C8—711no—
285C4×C8—711no—
286C4×C8C22×D7111yescharacteristic
287#41—22no—
288#41—2842no—
289#41—2842no—
290#41—1422no—
291#41—1422no—
292#41—1422no—
293C2×Q8—1422no—
294C2×D4—2842no—
295C2×D4—1422no—
296C22×C4—2842no—
297C22×C4—711no—
298C22×C4—1421no—
299Q16—712no—
300Q16—1422no—
301Q16C2×C4×D7112yes—
302SD16—22no—
303SD16—1422no—
304SD16—1422no—
305SD16—1422no—
306D8—712no—
307D8—1422no—
308D8C2×C4×D7112yes—
309M16—22no—
310M16—42no—
311M16—2842no—
312M16—2842no—
313M16—2842no—
314M16—2842no—
315M16—1422no—
316C2×C8—42no—
317C2×C8—2842no—
318C2×C8—2842no—
319C2×C8—2842no—
320C2×C8—2842no—
321C2×C8—711no—
322C2×C8—711no—
323C2×C8—712no—
324C2×C8—712no—
325C2×C8—712no—
326C2×C8—712no—
327C2×C8D4×D7111yescharacteristic
328C2×C8C23×D7111yescharacteristic
329C42—711no—
330C42—1421no—
331C42D4×D7111yescharacteristic
332C16—21no—
333C16—712no—
334C16—1421no—
335C16C2×C4×D7111yescharacteristic
336C16C2×C4×D7111yescharacteristic
337C23—2842no—
338C23—711no—
339Q8—22no—
340Q8—2842no—
341Q8—1422no—
342D4—22no—
343D4—42no—
344D4—2842no—
345D4—2842no—
346D4—2842no—
347D4—2842no—
348D4—1422no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—2842no—
352C2×C4—2842no—
353C2×C4—2842no—
354C2×C4—2842no—
355C2×C4—2842no—
356C2×C4—711no—
357C2×C4—712no—
358C2×C4—712no—
359C2×C4—1421no—
360C2×C4C2×D4×D7111yescharacteristic
361C8—42no—
362C8—2842no—
363C8—712no—
364C8—712no—
365C8D8×D7111yescharacteristic
366C8D8×D7111yescharacteristic
367C8C22×C4×D7111yescharacteristic
368C8#41×D7111yescharacteristic
369C22—42no—
370C22—2841no—
371C22—2842no—
372C22—5684no—
373C22—712no—
374C22#135×D7111yescharacteristic
375C4—22no—
376C4—42no—
377C4—2841no—
378C4—2842no—
379C4—712no—
380C4C4×D4×D7111yescharacteristic
381C4C2×D8×D7111yescharacteristic
382C2—21no—
383C2—82no—
384C2—5682no—
385C2—1422no—
386C2C4×D8×D7111yescharacteristic
subgroupG/H
centerC8D8×D71
commutatorC8×C71C23×C4
FrattiniC2×C8C23×D71
FittingC71×#2178C2
socleC2×C71C2×C4×D8
prime radical#22601473C2
cosocleC2×C8×C71C24
hypercenter#2178D71
hypocenterC71#33463
LCSG ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C8 ◁ C2×C8 ◁ C4×C8 ◁ #2178

Complements

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

NH ≅ G/Ntype
2. #443×D71C2semidirect
3. #22590226C2semidirect
4. #22590225C2semidirect
5. #22590223C2semidirect
6. #22590222C2semidirect
7. #22590219C2semidirect
8. #22590133C2semidirect
9. #22590066C2semidirect
10. C4×#14556067C2semidirect
11. C71×#2178C2semidirect
14. #17986441C4semidirect
15. #17986438C4semidirect
16. #17986436C4semidirect
17. #17986433C4semidirect
22. #17986117C22semidirect
23. #17986116C22semidirect
24. #17986115C22semidirect
30. #17985829C22semidirect
31. #17985828C22semidirect
32. #17985807C22semidirect
33. #17985806C22semidirect
34. #17985805C22semidirect
35. #17985774C22semidirect
36. #17985774C22semidirect
37. C4×C71⋊C16C22semidirect
39. #443×C71C22semidirect
40. #368×C71C22semidirect
41. #359×C71C22semidirect
42. C4×C16×C71C22semidirect
63. Dic568C2×C4semidirect
64. #14556074C2×C4semidirect
65. #14556073C2×C4semidirect
66. #14556072C2×C4semidirect
67. #14556071C2×C4semidirect
68. D1136C2×C4semidirect
69. #14556069C2×C4semidirect
85. Q32×C71C2×C4semidirect
86. SD32×C71C2×C4semidirect
87. D16×C71C2×C4semidirect
201. #2178D71semidirect
205. C71#33463semidirect
222. #443C2×D71semidirect
228. #368C2×D71semidirect
232. #359C2×D71semidirect
236. C4×C16C2×D71semidirect
261. Q32C4×D71semidirect
265. SD32C4×D71semidirect
268. D16C4×D71semidirect

Maximal subgroups

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

subgroupindex
#226014732
#443×D712
#225902262
#225902252
#225902232
#225902222
#225902192
#225901332
#225900662
C4×#145560672
C71×#21782
#3346371