Finite Group Databasebeta

← #22600302

Subgroups — #22600302

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#17986650C211yesmaximal, characteristic, complemented
2#17986637C211yesmaximal, characteristic, complemented
3#17986360C211yesmaximal, characteristic, complemented
4#17986328C211yesmaximal, characteristic, complemented
5#17986317C211yesmaximal, characteristic, complemented
6#17986287C211yesmaximal, characteristic, complemented
7#17986252C211yesmaximal, characteristic, complemented
8#17986231C211yesmaximal, characteristic, complemented
9#17986217C211yesmaximal, characteristic, complemented
10#17986144C211yesmaximal, characteristic, complemented
11#17986134C211yesmaximal, characteristic, complemented
12#17986108C211yesmaximal, characteristic, complemented
13#17986068C211yesmaximal, characteristic, complemented
14#102×D71C211yesmaximal, characteristic, complemented
15#417×C71C211yesmaximal, characteristic, complemented
16C2×#11765865—21no—
17C2×D4×D71C2211yescharacteristic
18C2×#11765852C2211yescharacteristic, complemented
19C2×D568C2211yescharacteristic
20C2×#11765850C2211yescharacteristic
21C2×C8×D71C2211yescharacteristic
22#14556127C2211yescharacteristic, complemented
23#14556117C2211yescharacteristic
24#14556113—21no—
25#14556111—21no—
26#14556111—21no—
27#14556106—21no—
28#14556104C2211yescharacteristic
29#14556103C2211yescharacteristic
30#14556102C2211yescharacteristic
31#14556102C2211yescharacteristic, complemented
32#14556100—21no—
33#14556096—21no—
34D4×Dic71C2211yescharacteristic
35#14556090C2211yescharacteristic, complemented
36C4×C22⋊D71C2211yescharacteristic
37C4⋊C4×D71C2211yescharacteristic
38C2×(C2×D71)⋊C4—21no—
39#14556060C2211yescharacteristic, complemented
40#14556059C2211yescharacteristic, complemented
41#14556059C2211yescharacteristic, complemented
42Dic71⋊C8C2211yescharacteristic, complemented
43#14556052C2211yescharacteristic, complemented
44#14556050C2211yescharacteristic
45#14556050C2211yescharacteristic, complemented
46#14556049C2211yescharacteristic
47#14556048C2211yescharacteristic
48#14556048C2211yescharacteristic
49C22⋊(C71⋊C8)C2211yescharacteristic, complemented
50(C2×D71)⋊C8C2211yescharacteristic
51#14556038C2211yescharacteristic, complemented
52C2×#41×C71C2211yescharacteristic
53C2×M16×C71C2211yescharacteristic, complemented
54C2×C4⋊C4×C71C2211yescharacteristic, complemented
55#103×C71C2211yescharacteristic, complemented
56#103×C71C2211yescharacteristic, complemented
57#102×C71C2211yescharacteristic
58#102×C71C2211yescharacteristic
59C23×D71—21no—
60#11765865—44no—
61C2×Dic142—21no—
62C2×D284—21no—
63C2×D284C2311yescharacteristic
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#11765852—41no—
76D568—41no—
77#11765850—22no—
78C2×C71⋊C8C2311yescharacteristic
79C2×C71⋊C8C2311yescharacteristic
80C8×D71—41no—
81C4⋊Dic71C2311yescharacteristic
82C4⋊Dic71C2311yescharacteristic
83Dic71⋊C4—21no—
84Dic71⋊C4—21no—
85Dic71⋊C4—21no—
86Dic71⋊C4D411yescharacteristic
87C22⋊Dic71—21no—
88C22⋊Dic71D411yescharacteristic
89(C2×D71)⋊C4—21no—
90(C2×D71)⋊C4—21no—
91(C2×D71)⋊C4—21no—
92(C2×D71)⋊C4—21no—
93(C2×D71)⋊C4—21no—
94(C2×D71)⋊C4—21no—
95(C2×D71)⋊C4—21no—
96(C2×D71)⋊C4D411yescharacteristic
97C4×Dic71C2311yescharacteristic
98#41×C71—22no—
99#41×C71—41no—
100C2×Q8×C71C2311yescharacteristic
101C2×D4×C71—21no—
102C2×D4×C71C2311yescharacteristic
103C22×C4×C71—21no—
104C22×C4×C71—21no—
105C22×C4×C71C2311yescharacteristic
106M16×C71—41no—
107C2×C8×C71C2311yescharacteristic
108C2×C8×C71C2311yescharacteristic
109C4⋊C4×C71—21no—
110C4⋊C4×C71C2311yescharacteristic
111C4⋊C4×C71C2311yescharacteristic
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—42no—
124D4×C71#4112yescomplemented
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—42no—
131C2×C4×C71—42no—
132C2×C4×C71C2×D411yescharacteristic
133C2×C4×C71C2411yescharacteristic
134C22⋊D71—22no—
135C22⋊D71—42no—
136C22⋊D71—42no—
137C22⋊D71—81no—
138C2×Dic71—21no—
139C2×Dic71—21no—
140C2×Dic71—81no—
141C2×Dic71C2×D411yescharacteristic
142C2×Dic71C2×D411yescharacteristic
143C2×Dic71C2×D411yescharacteristic
144D284—21no—
145D284—21no—
146D284—21no—
147D284—21no—
148D284—42no—
149C4×D71—21no—
150C4×D71—21no—
151C4×D71—42no—
152C4×D71—42no—
153Dic142—42no—
154C8×C71—21no—
155C8×C71—21no—
156C71⋊C8—21no—
157C71⋊C8—21no—
158C22×C71—22no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—21no—
162C22×C71—41no—
163C22×C71—41no—
164C22×C71C22×D411yescharacteristic
165C2×D71—21no—
166C2×D71—21no—
167C2×D71—41no—
168C2×D71—41no—
169C2×D71—41no—
170C2×D71—41no—
171C2×D71—41no—
172C2×D71—41no—
173C2×D71—84no—
174C2×D71AGL(1,ℤ/8)12yes—
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—42no—
187C2×C71—41no—
188C2×C71#54611yescharacteristic
189C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C71#57711yescharacteristic
191D71—22no—
192D71—81no—
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#465—711no—
204#463—711no—
205#449—711no—
206#447—711no—
207#442—711no—
208#435—711no—
209#417D7111yescharacteristic, complemented
210C2×#102—711no—
211C2×#41—1421no—
212C2×#41C2×D7111yescharacteristic
213C22×D4—711no—
214C2×SD16—711no—
215C2×D8—711no—
216C2×D8—711no—
217C2×M16C2×D7111yescharacteristic, complemented
218C22×C8—711no—
219C4⋊Q8—711no—
220#124—711no—
221#123—1421no—
222#123—1421no—
223#123—1421no—
224C22⋊Q8—711no—
225C22⋊Q8—1421no—
226#121—711no—
227#121—711no—
228#121—711no—
229#121—1421no—
230#120—1421no—
231C4×D4—711no—
232C4×D4—711no—
233C4×D4—711no—
234C2×C4⋊C4—711no—
235C2×C4⋊C4C2×D7111yescharacteristic, complemented
236C2×C22⋊C4—1421no—
237#106—711no—
238#106—711no—
239#106—711no—
240C4⋊C8—711no—
241#103—711no—
242#103C2×D7111yescharacteristic, complemented
243#103C2×D7111yescharacteristic, complemented
244#102—711no—
245#102—711no—
246#102—711no—
247#102—711no—
248#102—711no—
249#102C2×D7111yescharacteristic
250#102C2×D7111yescharacteristic
251C22⋊C8—711no—
252C22⋊C8—711no—
253#97—711no—
254C24—1421no—
255#41—22no—
256#41—41no—
257#41—2844no—
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—1421no—
267C2×D4—1421no—
268C2×D4—1422no—
269C2×D4—1421no—
270C2×D4C22×D7111yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—711no—
274C22×C4—1421no—
275C22×C4—1421no—
276C22×C4—1421no—
277C22×C4C22×D7111yescharacteristic
278SD16—2841no—
279D8—2841no—
280D8—1422no—
281M16—41no—
282C2×C8—2841no—
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—711no—
297C22⋊C4—711no—
298C22⋊C4—1421no—
299C22⋊C4—1421no—
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—42no—
320D4—2842no—
321D4—2842no—
322D4—2842no—
323D4—5681no—
324D4—1422no—
325D4—1421no—
326D4—1421no—
327D4—1421no—
328D4—1421no—
329D4#1176586512yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—42no—
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—22no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—2841no—
360C22—2841no—
361C22—2841no—
362C22—2841no—
363C22—2841no—
364C22—2841no—
365C22—5684no—
366C22—712no—
367C22—1421no—
368C22—1421no—
369C22C2×D4×D7111yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—2841no—
375C4—2841no—
376C4—1421no—
377C4—1421no—
378C4—1421no—
379C4C2×#1176586511yescharacteristic
380C4#1455617311yescharacteristic
381C2—41no—
382C2—42no—
383C2—5681no—
384C2—5681no—
385C2—1422no—
386C2#1798663911yescharacteristic
387C2AGL(1,ℤ/8)×D7111yescharacteristic
388C2#1798680811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#417×C71C2
socleC22×C71C22×D4
hypercenter#417D71
hypocenterC71#3306
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #17986650C2semidirect
2. #17986637C2semidirect
3. #17986360C2semidirect
4. #17986328C2semidirect
5. #17986317C2semidirect
6. #17986287C2semidirect
7. #17986252C2semidirect
8. #17986231C2semidirect
9. #17986217C2semidirect
10. #17986144C2semidirect
11. #17986134C2semidirect
12. #17986108C2semidirect
13. #17986068C2semidirect
14. #102×D71C2semidirect
15. #417×C71C2semidirect
18. C2×#11765852C22semidirect
22. #14556127C22semidirect
31. #14556102C22semidirect
35. #14556090C22semidirect
39. #14556060C22semidirect
40. #14556059C22semidirect
41. #14556059C22semidirect
42. Dic71⋊C8C22semidirect
43. #14556052C22semidirect
45. #14556050C22semidirect
49. C22⋊(C71⋊C8)C22semidirect
51. #14556038C22semidirect
53. C2×M16×C71C22semidirect
54. C2×C4⋊C4×C71C22semidirect
55. #103×C71C22semidirect
56. #103×C71C22semidirect
124. D4×C71#41semidirect
195. C71#3306semidirect
209. #417D71semidirect
217. C2×M16C2×D71semidirect
235. C2×C4⋊C4C2×D71semidirect
242. #103C2×D71semidirect
243. #103C2×D71semidirect
329. D4#11765865semidirect