Finite Group Databasebeta

← #22600502

Subgroups — #22600502

Order 9088.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986713C211yesmaximal, characteristic, complemented
2#17986646C211yesmaximal, characteristic, complemented
3#17986357C211yesmaximal, characteristic, complemented
4#17986342C211yesmaximal, characteristic, complemented
5#17986320C211yesmaximal, characteristic, complemented
6#17986278C211yesmaximal, characteristic, complemented
7#17986244C211yesmaximal, characteristic, complemented
8#17986212C211yesmaximal, characteristic, complemented
9#17986142C211yesmaximal, characteristic, complemented
10#17986131C211yesmaximal, characteristic, complemented
11#17986108C211yesmaximal, characteristic, complemented
12#17986070C211yesmaximal, characteristic, complemented
13#17986002C211yesmaximal, characteristic, complemented
14#17985996C211yesmaximal, characteristic, complemented
15#466×C71C211yesmaximal, characteristic, complemented
16C2×#11765865C2211yescharacteristic
17C2×D4×D71C2211yescharacteristic
18C2×#11765853C2211yescharacteristic, complemented
19C2×#11765850C2211yescharacteristic
20C2×#11765848C2211yescharacteristic
21#14556127C2211yescharacteristic, complemented
22#14556115—21no—
23#14556115C2211yescharacteristic
24#14556114—21no—
25#14556110—21no—
26#14556108C2211yescharacteristic
27#14556100—21no—
28#14556099—21no—
29#14556099C2211yescharacteristic
30#14556096—21no—
31D4×Dic71C2211yescharacteristic, complemented
32#14556090C2211yescharacteristic
33C4×C22⋊D71C2211yescharacteristic
34#14556087C2211yescharacteristic, complemented
35#14556086C2211yescharacteristic
36#14556084—21no—
37C22⋊C4×D71—21no—
38#14556061C2211yescharacteristic, complemented
39#14556059C2211yescharacteristic, complemented
40Dic71⋊C8C2211yescharacteristic, complemented
41#14556053C2211yescharacteristic, complemented
42#14556053C2211yescharacteristic, complemented
43#14556051C2211yescharacteristic, complemented
44#14556050C2211yescharacteristic
45#14556050C2211yescharacteristic
46#14556049C2211yescharacteristic
47#14556049C2211yescharacteristic
48#14556048C2211yescharacteristic
49C22⋊(C71⋊C8)C2211yescharacteristic, complemented
50(C2×D71)⋊C8C2211yescharacteristic
51#14556038C2211yescharacteristic, complemented
52C2×D8×C71C2211yescharacteristic
53C22×C8×C71C2211yescharacteristic, complemented
54#121×C71C2211yescharacteristic
55#121×C71C2211yescharacteristic, complemented
56#107×C71C2211yescharacteristic, complemented
57#102×C71C2211yescharacteristic
58#102×C71C2211yescharacteristic, complemented
59C23×D71—21no—
60#11765865—22no—
61#11765865—41no—
62C2×Dic142—21no—
63C2×Dic142C2311yescharacteristic
64C2×D284C2311yescharacteristic
65C2×C22⋊D71—21no—
66C2×C22⋊D71—21no—
67C2×C22⋊D71—21no—
68C2×C22⋊D71D411yescharacteristic
69D4×D71—22no—
70D4×D71—41no—
71C22×Dic71—21no—
72C2×C4×D71—21no—
73C2×C4×D71—21no—
74C2×C4×D71C2311yescharacteristic
75#11765853—41no—
76#11765850—22no—
77#11765848—41no—
78C2×C71⋊C8C2311yescharacteristic
79C2×C71⋊C8C2311yescharacteristic
80C4⋊Dic71—21no—
81C4⋊Dic71C2311yescharacteristic
82C4⋊Dic71C2311yescharacteristic
83Dic71⋊C4—21no—
84Dic71⋊C4D411yescharacteristic
85C22⋊Dic71—21no—
86C22⋊Dic71—21no—
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
98C2×D4×C71—21no—
99C2×D4×C71—21no—
100C2×D4×C71C2311yescharacteristic
101C2×D4×C71C2311yescharacteristic
102C22×C4×C71C2311yescharacteristic
103D8×C71—41no—
104C2×C8×C71—41no—
105C2×C8×C71C2311yescharacteristic
106C2×C8×C71C2311yescharacteristic
107C4⋊C4×C71C2311yescharacteristic
108C4⋊C4×C71C2311yescharacteristic
109C22⋊C4×C71—21no—
110C22⋊C4×C71—21no—
111C23×C71—21no—
112C23×C71—21no—
113C23×C71C2×D411yescharacteristic
114C22×D71—21no—
115C22×D71—42no—
116C22×D71—42no—
117C22×D71—81no—
118C22×D71C2×D411yescharacteristic
119D4×C71—21no—
120D4×C71—21no—
121D4×C71—21no—
122D4×C71—21no—
123D4×C71—81no—
124D4×C71—81no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71C2×D411yescharacteristic
130C2×C4×C71C2411yescharacteristic
131C22⋊D71—41no—
132C22⋊D71—42no—
133C22⋊D71—42no—
134C22⋊D71—81no—
135C2×Dic71—21no—
136C2×Dic71—21no—
137C2×Dic71—21no—
138C2×Dic71—42no—
139C2×Dic71C2×D411yescharacteristic
140C2×Dic71C2×D411yescharacteristic
141C2×Dic71C2×D411yescharacteristic
142D284—21no—
143D284#4112yes—
144C4×D71—21no—
145C4×D71—21no—
146C4×D71—42no—
147C4×D71—81no—
148Dic142—21no—
149Dic142—21no—
150Dic142—81no—
151C8×C71—21no—
152C8×C71—21no—
153C71⋊C8—21no—
154C71⋊C8—21no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—41no—
159C22×C71—41no—
160C22×C71—41no—
161C22×C71—41no—
162C22×C71—41no—
163C22×C71—41no—
164C22×C71C22×D411yescharacteristic
165C2×D71—21no—
166C2×D71—22no—
167C2×D71—21no—
168C2×D71—21no—
169C2×D71—41no—
170C2×D71—41no—
171C2×D71—84no—
172C4×C71—21no—
173C4×C71—41no—
174C4×C71—41no—
175C4×C71C2×#4111yescharacteristic
176C4×C71H(5,𝔽2)11yescharacteristic
177Dic71—21no—
178Dic71—21no—
179Dic71—21no—
180Dic71—41no—
181Dic71—41no—
182Dic71—41no—
183C2×C71—41no—
184C2×C71—81no—
185C2×C71—81no—
186C2×C71#54611yescharacteristic
187C2×C71#57511yescharacteristic
188C2×C71#57611yescharacteristic
189D71—41no—
190D71—42no—
191#3319—711nomaximal, Sylow
192C71#331911yescharacteristic, complemented, Sylow
193#555—711no—
194#546—711no—
195#489—711no—
196#485—711no—
197#482—711no—
198#474—711no—
199#468—711no—
200#466D7111yescharacteristic, complemented
201#463—711no—
202#449—711no—
203#447—711no—
204#442—711no—
205#435—711no—
206#418—711no—
207#416—711no—
208C2×#41—711no—
209C22×D4—711no—
210C2×SD16—711no—
211C2×D8—711no—
212C2×D8C2×D7111yescharacteristic
213C2×M16—711no—
214C22×C8C2×D7111yescharacteristic, complemented
215C4⋊Q8—711no—
216#124—711no—
217#124—1421no—
218#124—1421no—
219#123—1421no—
220C22⋊Q8—711no—
221#121—711no—
222#121—1421no—
223#121—1421no—
224#121C2×D7111yescharacteristic
225#121C2×D7111yescharacteristic, complemented
226#120—1421no—
227C4×D4—711no—
228C4×D4—711no—
229C4×D4—711no—
230#117—711no—
231#117—711no—
232#117—1421no—
233C2×C22⋊C4—1421no—
234#107—711no—
235#107C2×D7111yescharacteristic, complemented
236#106—711no—
237C4⋊C8—711no—
238#103—711no—
239#103—711no—
240#103—711no—
241#102—711no—
242#102—711no—
243#102—711no—
244#102—711no—
245#102—711no—
246#102C2×D7111yescharacteristic
247#102C2×D7111yescharacteristic, complemented
248C22⋊C8—711no—
249C22⋊C8—711no—
250#97—711no—
251C24—1421no—
252#41—2841no—
253#41—1422no—
254C2×Q8—711no—
255C2×Q8—1421no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—2841no—
259C2×D4—711no—
260C2×D4—711no—
261C2×D4—1421no—
262C2×D4—1422no—
263C2×D4—1421no—
264C2×D4—1421no—
265C2×D4C22×D7111yescharacteristic
266C2×D4C22×D7111yescharacteristic
267C22×C4—711no—
268C22×C4—1421no—
269C22×C4—1421no—
270C22×C4—1421no—
271C22×C4C22×D7111yescharacteristic
272SD16—2841no—
273D8—41no—
274D8—1422no—
275M16—2841no—
276C2×C8—41no—
277C2×C8—711no—
278C2×C8—711no—
279C2×C8C22×D7111yescharacteristic
280C2×C8C22×D7111yescharacteristic
281C4⋊C4—711no—
282C4⋊C4—711no—
283C4⋊C4—711no—
284C4⋊C4—1421no—
285C4⋊C4—1421no—
286C4⋊C4C22×D7111yescharacteristic
287C4⋊C4C22×D7111yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—711no—
291C22⋊C4—711no—
292C22⋊C4—1421no—
293C22⋊C4—1421no—
294C22⋊C4—1421no—
295C22⋊C4—1421no—
296C22⋊C4—1421no—
297C22⋊C4—1421no—
298C22⋊C4—1421no—
299C22⋊C4—1421no—
300C42—711no—
301C42—1421no—
302C42—1421no—
303C23—21no—
304C23—21no—
305C23—2842no—
306C23—2842no—
307C23—5681no—
308C23—711no—
309C23—1421no—
310C23D4×D7111yescharacteristic
311Q8—5681no—
312Q8—1421no—
313Q8—1421no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—81no—
319D4—81no—
320D4—2842no—
321D4—2841no—
322D4—2842no—
323D4—5681no—
324D4—712no—
325D4—1421no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—2842no—
331C2×C4—2842no—
332C2×C4—5681no—
333C2×C4—711no—
334C2×C4—711no—
335C2×C4—711no—
336C2×C4—1421no—
337C2×C4—1421no—
338C2×C4—1421no—
339C2×C4—1421no—
340C2×C4—1421no—
341C2×C4D4×D7111yescharacteristic
342C2×C4C23×D7111yescharacteristic
343C8—21no—
344C8—21no—
345C8—1421no—
346C8—1421no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—2841no—
357C22—2841no—
358C22—5684no—
359C22—1422no—
360C22—1421no—
361C22—1421no—
362C22—1421no—
363C22C2×D4×D7111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—2841no—
368C4—2841no—
369C4—2841no—
370C4—1421no—
371C4—1421no—
372C4—1421no—
373C4#41×D7111yescharacteristic
374C4#1455617311yescharacteristic
375C2—41no—
376C2—81no—
377C2—81no—
378C2—2842no—
379C2—2841no—
380C2#1798663811yescharacteristic
381C2#1798679611yescharacteristic
382C2#1798680011yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#466×C71C2
socleC22×C71C22×D4
hypercenter#466D71
hypocenterC71#3319
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #17986713C2semidirect
2. #17986646C2semidirect
3. #17986357C2semidirect
4. #17986342C2semidirect
5. #17986320C2semidirect
6. #17986278C2semidirect
7. #17986244C2semidirect
8. #17986212C2semidirect
9. #17986142C2semidirect
10. #17986131C2semidirect
11. #17986108C2semidirect
12. #17986070C2semidirect
13. #17986002C2semidirect
14. #17985996C2semidirect
15. #466×C71C2semidirect
18. C2×#11765853C22semidirect
21. #14556127C22semidirect
31. D4×Dic71C22semidirect
34. #14556087C22semidirect
38. #14556061C22semidirect
39. #14556059C22semidirect
40. Dic71⋊C8C22semidirect
41. #14556053C22semidirect
42. #14556053C22semidirect
43. #14556051C22semidirect
49. C22⋊(C71⋊C8)C22semidirect
51. #14556038C22semidirect
53. C22×C8×C71C22semidirect
55. #121×C71C22semidirect
56. #107×C71C22semidirect
58. #102×C71C22semidirect
192. C71#3319semidirect
200. #466D71semidirect
214. C22×C8C2×D71semidirect
225. #121C2×D71semidirect
235. #107C2×D71semidirect
247. #102C2×D71semidirect