Finite Group Databasebeta

← #22600307

Subgroups — #22600307

Order 9088.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986649C211yesmaximal, characteristic, complemented
2#17986640C211yesmaximal, characteristic, complemented
3#17986360C211yesmaximal, characteristic, complemented
4#17986327C211yesmaximal, characteristic, complemented
5#17986319C211yesmaximal, characteristic, complemented
6#17986288C211yesmaximal, characteristic, complemented
7#17986251C211yesmaximal, characteristic, complemented
8#17986216C211yesmaximal, characteristic, complemented
9#17986143C211yesmaximal, characteristic, complemented
10#17986132C211yesmaximal, characteristic, complemented
11#17986108C211yesmaximal, characteristic, complemented
12#17986070C211yesmaximal, characteristic, complemented
13#17985999C211yesmaximal, characteristic, complemented
14C2×#14556050C211yesmaximal, characteristic, complemented
15#465×C71C211yesmaximal, characteristic, complemented
16C2×#11765866C2211yescharacteristic
17C2×#11765864—21no—
18C22×D284C2211yescharacteristic
19C2×#11765850C2211yescharacteristic
20C2×#11765850C2211yescharacteristic
21C2×#11765848C2211yescharacteristic, complemented
22#14556127C2211yescharacteristic, complemented
23#14556117C2211yescharacteristic
24#14556113—21no—
25#14556110—21no—
26#14556109—21no—
27#14556105—21no—
28#14556103—21no—
29#14556103C2211yescharacteristic
30#14556099C2211yescharacteristic, complemented
31#14556098—21no—
32D4×Dic71C2211yescharacteristic, complemented
33#14556090C2211yescharacteristic
34C4×C22⋊D71C2211yescharacteristic
35C2×C4⋊Dic71C2211yescharacteristic
36C4⋊C4×D71C2211yescharacteristic, complemented
37C22⋊C4×D71—21no—
38#14556059C2211yescharacteristic, complemented
39#14556059C2211yescharacteristic, complemented
40Dic71⋊C8C2211yescharacteristic, complemented
41#14556052C2211yescharacteristic, complemented
42#14556052C2211yescharacteristic, complemented
43#14556050C2211yescharacteristic
44#14556050C2211yescharacteristic
45#14556050C2211yescharacteristic
46#14556050C2211yescharacteristic
47#14556048C2211yescharacteristic
48#14556048C2211yescharacteristic
49C22⋊(C71⋊C8)C2211yescharacteristic
50(C2×D71)⋊C8C2211yescharacteristic, complemented
51#14556038C2211yescharacteristic, complemented
52C2×SD16×C71C2211yescharacteristic, complemented
53C22×C8×C71C2211yescharacteristic
54C22⋊Q8×C71C2211yescharacteristic
55#121×C71C2211yescharacteristic
56#106×C71C2211yescharacteristic, complemented
57#103×C71C2211yescharacteristic, complemented
58#102×C71C2211yescharacteristic, complemented
59C23×D71—21no—
60#11765866—22no—
61#11765866—41no—
62#11765864—44no—
63C2×Dic142—21no—
64C2×D284—22no—
65C2×D284—21no—
66C2×D284—21no—
67C2×D284—21no—
68C2×D284—41no—
69C2×D284C2311yescharacteristic
70C2×D284C2311yescharacteristic
71C2×C22⋊D71—21no—
72C2×C22⋊D71D411yescharacteristic
73C22×Dic71—21no—
74C22×Dic71—21no—
75C2×C4×D71—21no—
76C2×C4×D71—21no—
77C2×C4×D71—21no—
78C2×C4×D71C2311yescharacteristic
79#11765850—22no—
80#11765850—41no—
81#11765848—41no—
82C2×C71⋊C8C2311yescharacteristic
83C2×C71⋊C8C2311yescharacteristic
84C4⋊Dic71—21no—
85C4⋊Dic71C2311yescharacteristic
86C4⋊Dic71C2311yescharacteristic
87Dic71⋊C4—21no—
88Dic71⋊C4—21no—
89Dic71⋊C4D411yescharacteristic
90C22⋊Dic71—21no—
91C22⋊Dic71—21no—
92C22⋊Dic71D411yescharacteristic
93(C2×D71)⋊C4—21no—
94(C2×D71)⋊C4—21no—
95(C2×D71)⋊C4—21no—
96(C2×D71)⋊C4—21no—
97(C2×D71)⋊C4D411yescharacteristic
98C4×Dic71C2311yescharacteristic
99C2×Q8×C71C2311yescharacteristic
100C2×D4×C71—21no—
101C2×D4×C71C2311yescharacteristic
102C22×C4×C71C2311yescharacteristic
103SD16×C71—41no—
104C2×C8×C71—41no—
105C2×C8×C71C2311yescharacteristic
106C2×C8×C71C2311yescharacteristic
107C4⋊C4×C71—21no—
108C4⋊C4×C71C2311yescharacteristic
109C4⋊C4×C71C2311yescharacteristic
110C22⋊C4×C71—21no—
111C22⋊C4×C71—21no—
112C23×C71—21no—
113C23×C71C2×D411yescharacteristic
114C22×D71—21no—
115C22×D71—21no—
116C22×D71—42no—
117C22×D71—42no—
118C22×D71—81no—
119C22×D71C2×D411yescharacteristic
120Q8×C71—21no—
121Q8×C71—21no—
122D4×C71—21no—
123D4×C71—21no—
124D4×C71—42no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71C2×D411yescharacteristic
131C2×C4×C71C2411yescharacteristic
132C22⋊D71—22no—
133C22⋊D71—42no—
134C2×Dic71—21no—
135C2×Dic71—21no—
136C2×Dic71—42no—
137C2×Dic71—42no—
138C2×Dic71C2×D411yescharacteristic
139C2×Dic71C2×D411yescharacteristic
140C2×Dic71C2×D411yescharacteristic
141D284—21no—
142D284—21no—
143D284—21no—
144D284—42no—
145D284—42no—
146D284—81no—
147D284#4112yescomplemented
148C4×D71—21no—
149C4×D71—21no—
150C4×D71—42no—
151C4×D71—42no—
152C4×D71—81no—
153Dic142—42no—
154C8×C71—21no—
155C8×C71—21no—
156C71⋊C8—21no—
157C71⋊C8—21no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—41no—
161C22×C71—41no—
162C22×C71—41no—
163C22×C71AGL(1,ℤ/8)12yes—
164C22×C71C22×D411yescharacteristic
165C2×D71—21no—
166C2×D71—22no—
167C2×D71—21no—
168C2×D71—21no—
169C2×D71—41no—
170C2×D71—41no—
171C2×D71—41no—
172C2×D71—41no—
173C2×D71—41no—
174C2×D71—84no—
175C4×C71—21no—
176C4×C71—41no—
177C4×C71—41no—
178C4×C71—41no—
179C4×C71C2×#4111yescharacteristic
180C4×C71H(5,𝔽2)11yescharacteristic
181Dic71—21no—
182Dic71—21no—
183Dic71—21no—
184Dic71—41no—
185Dic71—41no—
186C2×C71—22no—
187C2×C71—81no—
188C2×C71#54611yescharacteristic
189C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C71#57711yescharacteristic
191D71—41no—
192D71—42no—
193D71—81no—
194#3306—711nomaximal, Sylow
195C71#330611yescharacteristic, complemented, Sylow
196#547—711no—
197#546—711no—
198#489—711no—
199#483—711no—
200#481—711no—
201#476—711no—
202#469—711no—
203#465D7111yescharacteristic, complemented
204#463—711no—
205#449—711no—
206#447—711no—
207#442—711no—
208#435—711no—
209#417—711no—
210C2×#102—711no—
211C2×#41—711no—
212C2×#41—1421no—
213C22×D4—711no—
214C2×SD16C2×D7111yescharacteristic, complemented
215C2×D8—711no—
216C2×D8—711no—
217C2×M16—711no—
218C22×C8C2×D7111yescharacteristic
219C4⋊Q8—711no—
220#124—711no—
221#123—1421no—
222#123—1421no—
223#123—1421no—
224C22⋊Q8—1421no—
225C22⋊Q8C2×D7111yescharacteristic
226#121—711no—
227#121—711no—
228#121—1421no—
229#121C2×D7111yescharacteristic
230#120—1421no—
231C4×D4—711no—
232C4×D4—711no—
233C4×D4—711no—
234C2×C4⋊C4—711no—
235C2×C4⋊C4—711no—
236C2×C22⋊C4—1421no—
237#106—711no—
238#106—711no—
239#106C2×D7111yescharacteristic, complemented
240C4⋊C8—711no—
241#103—711no—
242#103—711no—
243#103C2×D7111yescharacteristic, complemented
244#102—711no—
245#102—711no—
246#102—711no—
247#102—711no—
248#102—711no—
249#102—711no—
250#102C2×D7111yescharacteristic, complemented
251C22⋊C8—711no—
252C22⋊C8—711no—
253#97—711no—
254C24—1421no—
255#41—2844no—
256#41—2841no—
257#41—1422no—
258C2×Q8—1421no—
259C2×Q8C22×D7111yescharacteristic
260C2×D4—21no—
261C2×D4—2841no—
262C2×D4—711no—
263C2×D4—711no—
264C2×D4—711no—
265C2×D4—1421no—
266C2×D4—1422no—
267C2×D4—1421no—
268C2×D4—1421no—
269C2×D4—1421no—
270C2×D4C22×D7111yescharacteristic
271C22×C4—711no—
272C22×C4—1421no—
273C22×C4—1421no—
274C22×C4—1421no—
275C22×C4—1421no—
276C22×C4—1421no—
277C22×C4C22×D7111yescharacteristic
278SD16—41no—
279D8—2841no—
280D8—1422no—
281M16—2841no—
282C2×C8—41no—
283C2×C8—711no—
284C2×C8—711no—
285C2×C8C22×D7111yescharacteristic
286C2×C8C22×D7111yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—711no—
289C4⋊C4—711no—
290C4⋊C4—711no—
291C4⋊C4—1421no—
292C4⋊C4—1421no—
293C4⋊C4—1421no—
294C4⋊C4C22×D7111yescharacteristic
295C4⋊C4C22×D7111yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—711no—
299C22⋊C4—711no—
300C22⋊C4—1421no—
301C22⋊C4—1421no—
302C22⋊C4—1421no—
303C22⋊C4—1421no—
304C22⋊C4—1421no—
305C22⋊C4—1421no—
306C42—711no—
307C23—21no—
308C23—2842no—
309C23—2842no—
310C23—5681no—
311C23—711no—
312C23—1421no—
313C23—1421no—
314C23D4×D7111yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—2842no—
318D4—21no—
319D4—21no—
320D4—42no—
321D4—2842no—
322D4—2842no—
323D4—2842no—
324D4—5681no—
325D4—712no—
326D4—1422no—
327D4—1421no—
328D4—1421no—
329D4—1421no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—2842no—
336C2×C4—2842no—
337C2×C4—2842no—
338C2×C4—2842no—
339C2×C4—5681no—
340C2×C4—711no—
341C2×C4—711no—
342C2×C4—711no—
343C2×C4—1421no—
344C2×C4—1421no—
345C2×C4—1421no—
346C2×C4—1421no—
347C2×C4D4×D7111yescharacteristic
348C2×C4C23×D7111yescharacteristic
349C8—21no—
350C8—21no—
351C8—1421no—
352C8—1421no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—2841no—
359C22—2841no—
360C22—2841no—
361C22—2841no—
362C22—2841no—
363C22—5684no—
364C22—1422no—
365C22—1421no—
366C22—1421no—
367C22—1421no—
368C22#1455615512yes—
369C22C2×D4×D7111yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—2841no—
375C4—2841no—
376C4—1421no—
377C4—1421no—
378C4—1421no—
379C4#41×D7111yescharacteristic
380C4#1455617311yescharacteristic
381C2—22no—
382C2—81no—
383C2—2842no—
384C2—2841no—
385C2—5681no—
386C2#1798663811yescharacteristic
387C2C2×#1455615511yescharacteristic
388C2#1798680611yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#465×C71C2
socleC22×C71C22×D4
hypercenter#465D71
hypocenterC71#3306
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #17986649C2semidirect
2. #17986640C2semidirect
3. #17986360C2semidirect
4. #17986327C2semidirect
5. #17986319C2semidirect
6. #17986288C2semidirect
7. #17986251C2semidirect
8. #17986216C2semidirect
9. #17986143C2semidirect
10. #17986132C2semidirect
11. #17986108C2semidirect
12. #17986070C2semidirect
13. #17985999C2semidirect
14. C2×#14556050C2semidirect
15. #465×C71C2semidirect
21. C2×#11765848C22semidirect
22. #14556127C22semidirect
30. #14556099C22semidirect
32. D4×Dic71C22semidirect
36. C4⋊C4×D71C22semidirect
38. #14556059C22semidirect
39. #14556059C22semidirect
40. Dic71⋊C8C22semidirect
41. #14556052C22semidirect
42. #14556052C22semidirect
50. (C2×D71)⋊C8C22semidirect
51. #14556038C22semidirect
52. C2×SD16×C71C22semidirect
56. #106×C71C22semidirect
57. #103×C71C22semidirect
58. #102×C71C22semidirect
147. D284#41semidirect
195. C71#3306semidirect
203. #465D71semidirect
214. C2×SD16C2×D71semidirect
239. #106C2×D71semidirect
243. #103C2×D71semidirect
250. #102C2×D71semidirect