Finite Group Databasebeta

← #22600357

Subgroups — #22600357

Order 9088.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986684C211yesmaximal, characteristic, complemented
2D4×C22⋊D71C211yesmaximal, characteristic
3#17986362C211yesmaximal, characteristic, complemented
4#17986322C211yesmaximal, characteristic, complemented
5#17986312C211yesmaximal, characteristic, complemented
6#17986301C211yesmaximal, characteristic, complemented
7#17986244C211yesmaximal, characteristic, complemented
8#17986228C211yesmaximal, characteristic, complemented
9#17986198C211yesmaximal, characteristic, complemented
10#17986151C211yesmaximal, characteristic, complemented
11#17986150C211yesmaximal, characteristic, complemented
12#17986096C211yesmaximal, characteristic, complemented
13#17986070C211yesmaximal, characteristic, complemented
14#17986002C211yesmaximal, characteristic, complemented
15C2×#102×C71C211yesmaximal, characteristic, complemented
16C22×C22⋊D71—21no—
17C2×D4×D71C2211yescharacteristic
18C2×#11765854C2211yescharacteristic, complemented
19C2×#11765853C2211yescharacteristic
20C2×#11765853C2211yescharacteristic, complemented
21C2×#11765848C2211yescharacteristic, complemented
22#14556124C2211yescharacteristic
23#14556123—21no—
24#14556119C2211yescharacteristic, complemented
25#14556118—21no—
26#14556111—21no—
27#14556108C2211yescharacteristic, complemented
28#14556107C2211yescharacteristic, complemented
29#14556102C2211yescharacteristic
30#14556100—21no—
31#14556099C2211yescharacteristic
32#14556097—21no—
33#14556096—21no—
34#14556093C2211yescharacteristic, complemented
35D4×Dic71C2211yescharacteristic
36C4×C22⋊D71C2211yescharacteristic
37#14556087—21no—
38#14556086C2211yescharacteristic, complemented
39#14556061C2211yescharacteristic, complemented
40#14556061C2211yescharacteristic, complemented
41#14556060C2211yescharacteristic, complemented
42Dic71⋊C8C2211yescharacteristic, complemented
43#14556053C2211yescharacteristic, complemented
44#14556050C2211yescharacteristic, complemented
45#14556049C2211yescharacteristic
46#14556049C2211yescharacteristic, complemented
47#14556048C2211yescharacteristic, complemented
48#14556048C2211yescharacteristic, complemented
49C22⋊(C71⋊C8)C2211yescharacteristic, complemented
50(C2×D71)⋊C8C2211yescharacteristic, complemented
51#14556038C2211yescharacteristic, complemented
52C22×D4×C71C2211yescharacteristic
53C22×C8×C71C2211yescharacteristic, complemented
54C2×C4⋊C4×C71C2211yescharacteristic, complemented
55#102×C71C2211yescharacteristic
56#102×C71C2211yescharacteristic
57#102×C71C2211yescharacteristic, complemented
58#102×C71C2211yescharacteristic, complemented
59C23×D71—21no—
60C2×Dic142C2311yescharacteristic
61C2×D284C2311yescharacteristic
62C2×C22⋊D71—21no—
63C2×C22⋊D71—21no—
64C2×C22⋊D71—21no—
65C2×C22⋊D71—21no—
66C2×C22⋊D71—21no—
67C2×C22⋊D71—44no—
68C2×C22⋊D71D411yescharacteristic
69D4×D71—22no—
70D4×D71—41no—
71C22×Dic71—21no—
72C2×C4×D71C2311yescharacteristic
73#11765854—41no—
74#11765853—22no—
75#11765853—41no—
76#11765848—41no—
77C2×C71⋊C8C2311yescharacteristic
78C2×C71⋊C8C2311yescharacteristic
79C4⋊Dic71—21no—
80C4⋊Dic71—21no—
81C4⋊Dic71C2311yescharacteristic
82C4⋊Dic71C2311yescharacteristic
83Dic71⋊C4—21no—
84Dic71⋊C4—21no—
85Dic71⋊C4—21no—
86Dic71⋊C4D411yescharacteristic
87C22⋊Dic71—21no—
88C22⋊Dic71—21no—
89C22⋊Dic71—21no—
90C22⋊Dic71D411yescharacteristic
91(C2×D71)⋊C4—21no—
92(C2×D71)⋊C4—21no—
93(C2×D71)⋊C4—21no—
94(C2×D71)⋊C4D411yescharacteristic
95C4×Dic71—21no—
96C4×Dic71—21no—
97C4×Dic71C2311yescharacteristic
98C24×C71—21no—
99C2×D4×C71—22no—
100C2×D4×C71—21no—
101C2×D4×C71—41no—
102C2×D4×C71C2311yescharacteristic
103C2×D4×C71C2311yescharacteristic
104C22×C4×C71—21no—
105C22×C4×C71C2311yescharacteristic
106C2×C8×C71—41no—
107C2×C8×C71C2311yescharacteristic
108C2×C8×C71C2311yescharacteristic
109C4⋊C4×C71—21no—
110C4⋊C4×C71C2311yescharacteristic
111C4⋊C4×C71C2311yescharacteristic
112C23×C71—21no—
113C23×C71—21no—
114C23×C71—42no—
115C23×C71—42no—
116C23×C71—81no—
117C23×C71C2×D411yescharacteristic
118C22×D71—21no—
119C22×D71—42no—
120C22×D71—42no—
121C22×D71—81no—
122C22×D71C2×D411yescharacteristic
123D4×C71—21no—
124D4×C71—21no—
125D4×C71—21no—
126D4×C71—42no—
127D4×C71#4112yes—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—42no—
133C2×C4×C71C2×D411yescharacteristic
134C2×C4×C71C2411yescharacteristic
135C22⋊D71—22no—
136C22⋊D71—42no—
137C22⋊D71—42no—
138C22⋊D71—42no—
139C22⋊D71—42no—
140C22⋊D71—81no—
141C2×Dic71—21no—
142C2×Dic71—21no—
143C2×Dic71—21no—
144C2×Dic71—42no—
145C2×Dic71C2×D411yescharacteristic
146C2×Dic71C2×D411yescharacteristic
147C2×Dic71C2×D411yescharacteristic
148D284—21no—
149D284—21no—
150C4×D71—21no—
151C4×D71—21no—
152Dic142—21no—
153Dic142—21no—
154C8×C71—21no—
155C8×C71—21no—
156C71⋊C8—21no—
157C71⋊C8—21no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—22no—
161C22×C71—41no—
162C22×C71—41no—
163C22×C71—41no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71—84no—
167C22×C71AGL(1,ℤ/8)12yes—
168C22×C71C22×D411yescharacteristic
169C2×D71—21no—
170C2×D71—21no—
171C2×D71—21no—
172C2×D71—41no—
173C2×D71—41no—
174C2×D71—41no—
175C2×D71—84no—
176C4×C71—21no—
177C4×C71—41no—
178C4×C71—41no—
179C4×C71C2×#4111yescharacteristic
180C4×C71H(5,𝔽2)11yescharacteristic
181Dic71—21no—
182Dic71—21no—
183Dic71—21no—
184Dic71—41no—
185Dic71—41no—
186Dic71—41no—
187C2×C71—22no—
188C2×C71—42no—
189C2×C71—81no—
190C2×C71#54611yescharacteristic
191C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C71#57711yescharacteristic
193D71—41no—
194D71—81no—
195#3310—711nomaximal, Sylow
196C71#331011yescharacteristic, complemented, Sylow
197#552—711no—
198D42—711no—
199#490—711no—
200#482—711no—
201#480—711no—
202#478—711no—
203#468—711no—
204#465—711no—
205#460—711no—
206#450—711no—
207#450—711no—
208#440—711no—
209#435—711no—
210#418—711no—
211C2×#102D7111yescharacteristic, complemented
212C22×D4—711no—
213C22×D4—1421no—
214C22×D4C2×D7111yescharacteristic
215C2×SD16—711no—
216C2×SD16—711no—
217C2×SD16—711no—
218C2×M16—711no—
219C22×C8C2×D7111yescharacteristic, complemented
220#127—711no—
221#126—1421no—
222#125—711no—
223#125—1421no—
224#123—1421no—
225C22⋊Q8—711no—
226C22⋊Q8—711no—
227#121—711no—
228#121—711no—
229#121—1421no—
230#120—1421no—
231#120—1421no—
232C4×Q8—711no—
233C4×D4—711no—
234C4×D4—711no—
235#117—711no—
236#117—1421no—
237C2×C4⋊C4C2×D7111yescharacteristic, complemented
238#107—711no—
239#107—711no—
240#106—711no—
241C4⋊C8—711no—
242#103—711no—
243#102—711no—
244#102—711no—
245#102—711no—
246#102—711no—
247#102—711no—
248#102C2×D7111yescharacteristic
249#102C2×D7111yescharacteristic
250#102C2×D7111yescharacteristic, complemented
251#102C2×D7111yescharacteristic, complemented
252C22⋊C8—711no—
253C22⋊C8—711no—
254#97—711no—
255C24—21no—
256C24—1421no—
257C2×Q8—711no—
258C2×D4—22no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—2844no—
262C2×D4—2841no—
263C2×D4—711no—
264C2×D4—711no—
265C2×D4—1421no—
266C2×D4—1421no—
267C2×D4—1421no—
268C2×D4—1421no—
269C2×D4—1421no—
270C2×D4—1422no—
271C2×D4C22×D7111yescharacteristic
272C2×D4C22×D7111yescharacteristic
273C22×C4—21no—
274C22×C4—711no—
275C22×C4—1421no—
276C22×C4C22×D7111yescharacteristic
277SD16—2841no—
278SD16—2841no—
279SD16—1422no—
280M16—2841no—
281C2×C8—41no—
282C2×C8—711no—
283C2×C8—711no—
284C2×C8C22×D7111yescharacteristic
285C2×C8C22×D7111yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—711no—
288C4⋊C4—711no—
289C4⋊C4—711no—
290C4⋊C4—1421no—
291C4⋊C4—1421no—
292C4⋊C4—1421no—
293C4⋊C4—1421no—
294C4⋊C4—1421no—
295C4⋊C4C22×D7111yescharacteristic
296C4⋊C4C22×D7111yescharacteristic
297C22⋊C4—711no—
298C22⋊C4—711no—
299C22⋊C4—1421no—
300C22⋊C4—1421no—
301C22⋊C4—1421no—
302C22⋊C4—1421no—
303C22⋊C4—1421no—
304C22⋊C4—1421no—
305C42—711no—
306C42—1421no—
307C42—1421no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—2842no—
314C23—2842no—
315C23—5681no—
316C23—711no—
317C23—1421no—
318C23D4×D7111yescharacteristic
319Q8—1421no—
320Q8—1421no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—2842no—
326D4—2842no—
327D4—2842no—
328D4—2842no—
329D4—5681no—
330D4—1422no—
331D4—1421no—
332D4—1421no—
333D4#1176586512yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—2842no—
340C2×C4—711no—
341C2×C4—711no—
342C2×C4—711no—
343C2×C4—1421no—
344C2×C4—1421no—
345C2×C4—1421no—
346C2×C4—1421no—
347C2×C4—1421no—
348C2×C4D4×D7111yescharacteristic
349C2×C4C23×D7111yescharacteristic
350C8—21no—
351C8—21no—
352C8—1421no—
353C8—1421no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—84no—
363C22—2841no—
364C22—2841no—
365C22—2841no—
366C22—5684no—
367C22—1421no—
368C22—1421no—
369C22—1421no—
370C22#1455615212yes—
371C22C2×D4×D7111yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—2841no—
376C4—2841no—
377C4—2841no—
378C4—1421no—
379C4—1421no—
380C4—1421no—
381C4C2×#1176586511yescharacteristic
382C4#1455617311yescharacteristic
383C2—22no—
384C2—42no—
385C2—81no—
386C2—2841no—
387C2—5681no—
388C2#1798663911yescharacteristic
389C2C2×#1455615211yescharacteristic
390C2#1798680511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×#102×C71C2
socleC22×C71C22×D4
hypercenterC2×#102D71
hypocenterC71#3310
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #17986684C2semidirect
3. #17986362C2semidirect
4. #17986322C2semidirect
5. #17986312C2semidirect
6. #17986301C2semidirect
7. #17986244C2semidirect
8. #17986228C2semidirect
9. #17986198C2semidirect
10. #17986151C2semidirect
11. #17986150C2semidirect
12. #17986096C2semidirect
13. #17986070C2semidirect
14. #17986002C2semidirect
15. C2×#102×C71C2semidirect
18. C2×#11765854C22semidirect
20. C2×#11765853C22semidirect
21. C2×#11765848C22semidirect
24. #14556119C22semidirect
27. #14556108C22semidirect
28. #14556107C22semidirect
34. #14556093C22semidirect
38. #14556086C22semidirect
39. #14556061C22semidirect
40. #14556061C22semidirect
41. #14556060C22semidirect
42. Dic71⋊C8C22semidirect
43. #14556053C22semidirect
44. #14556050C22semidirect
46. #14556049C22semidirect
47. #14556048C22semidirect
48. #14556048C22semidirect
49. C22⋊(C71⋊C8)C22semidirect
50. (C2×D71)⋊C8C22semidirect
51. #14556038C22semidirect
53. C22×C8×C71C22semidirect
54. C2×C4⋊C4×C71C22semidirect
57. #102×C71C22semidirect
58. #102×C71C22semidirect
196. C71#3310semidirect
211. C2×#102D71semidirect
219. C22×C8C2×D71semidirect
237. C2×C4⋊C4C2×D71semidirect
250. #102C2×D71semidirect
251. #102C2×D71semidirect