Finite Group Databasebeta

← #22598528

Subgroups — #22598528

Order 9088.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986621C211yesmaximal, characteristic, complemented
2#17986471C211yesmaximal, characteristic, complemented
3#17986338C211yesmaximal, characteristic, complemented
4#17986326C211yesmaximal, characteristic, complemented
5#17986287C211yesmaximal, characteristic, complemented
6#17986278C211yesmaximal, characteristic, complemented
7#17986217C211yesmaximal, characteristic, complemented
8#17986216C211yesmaximal, characteristic, complemented
9#17986148C211yesmaximal, characteristic, complemented
10#17986134C211yesmaximal, characteristic
11#17986112C211yesmaximal, characteristic, complemented
12C4×#11765852C211yesmaximal, characteristic, complemented
13C4×#11765850C211yesmaximal, characteristic, complemented
14#17986061C211yesmaximal, characteristic, complemented
15#440×C71C211yesmaximal, characteristic, complemented
16C2×D4×D71C2211yescharacteristic
17C2×#11765852C2211yescharacteristic
18C2×D568C2211yescharacteristic
19C2×#11765850C2211yescharacteristic
20#14556127C2211yescharacteristic, complemented
21#14556122—21no—
22#14556117C2211yescharacteristic, complemented
23#14556116—21no—
24#14556113—21no—
25#14556106—21no—
26#14556104C2211yescharacteristic
27#14556103C2211yescharacteristic
28C4×D284C2211yescharacteristic
29C4×D284C2211yescharacteristic, complemented
30D4×Dic71C2211yescharacteristic
31#14556090C2211yescharacteristic
32C4×C22⋊D71—21no—
33(C22⋊D71)⋊C4—21no—
34#14556086C2211yescharacteristic
35#14556085C2211yescharacteristic
36C2×(C2×D71)⋊C4—21no—
37C22⋊C4×D71—21no—
38#14556061C2211yescharacteristic, complemented
39#14556060C2211yescharacteristic, complemented
40#14556059C2211yescharacteristic, complemented
41Dic71⋊C8C2211yescharacteristic, complemented
42Dic71⋊C8C2211yescharacteristic, complemented
43#14556052C2211yescharacteristic, complemented
44#14556050C2211yescharacteristic
45#14556050C2211yescharacteristic, complemented
46#14556049C2211yescharacteristic
47#14556048C2211yescharacteristic
48#14556048C2211yescharacteristic, complemented
49(C2×D71)⋊C8C2211yescharacteristic
50(C2×D71)⋊C8C2211yescharacteristic
51C4×C71⋊C8C2211yescharacteristic, complemented
52C2×SD16×C71C2211yescharacteristic
53C4×Q8×C71C2211yescharacteristic, complemented
54C4×D4×C71C2211yescharacteristic
55#107×C71C2211yescharacteristic, complemented
56#103×C71C2211yescharacteristic, complemented
57#102×C71C2211yescharacteristic
58#97×C71C2211yescharacteristic, complemented
59C23×D71—21no—
60C2×Dic142—21no—
61C2×D284—21no—
62C2×D284C2311yescharacteristic
63C2×D284C2311yescharacteristic
64C2×C22⋊D71—21no—
65D4×D71—22no—
66D4×D71—22no—
67C22×Dic71—21no—
68C2×C4×D71—21no—
69C2×C4×D71—21no—
70C2×C4×D71—21no—
71C2×C4×D71C2311yescharacteristic
72#11765852—22no—
73D568—22no—
74#11765850—22no—
75C2×C71⋊C8C2311yescharacteristic
76C2×C71⋊C8C2311yescharacteristic
77C4⋊Dic71C2311yescharacteristic
78C4⋊Dic71C2311yescharacteristic
79Dic71⋊C4—21no—
80Dic71⋊C4—21no—
81Dic71⋊C4D411yescharacteristic
82Dic71⋊C4D411yescharacteristic
83C22⋊Dic71—21no—
84(C2×D71)⋊C4—21no—
85(C2×D71)⋊C4—21no—
86(C2×D71)⋊C4—21no—
87(C2×D71)⋊C4—21no—
88(C2×D71)⋊C4—21no—
89(C2×D71)⋊C4—21no—
90(C2×D71)⋊C4—21no—
91(C2×D71)⋊C4—21no—
92(C2×D71)⋊C4D411yescharacteristic
93(C2×D71)⋊C4D411yescharacteristic
94C4×Dic71—21no—
95C4×Dic71C2311yescharacteristic
96C2×Q8×C71C2311yescharacteristic
97C2×D4×C71C2311yescharacteristic
98C22×C4×C71—21no—
99SD16×C71—22no—
100C2×C8×C71C2311yescharacteristic
101C2×C8×C71C2311yescharacteristic
102C4⋊C4×C71—21no—
103C4⋊C4×C71C2311yescharacteristic
104C4⋊C4×C71C2311yescharacteristic
105C22⋊C4×C71—21no—
106C42×C71—21no—
107C42×C71C2311yescharacteristic
108C23×C71—21no—
109C22×D71—21no—
110C22×D71—24no—
111C22×D71—21no—
112C22×D71—81no—
113C22×D71—81no—
114C22×D71C2×D411yescharacteristic
115Q8×C71—21no—
116Q8×C71—21no—
117D4×C71—21no—
118D4×C71#4112yes—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—81no—
123C2×C4×C71C2×D411yescharacteristic
124C2×C4×C71C2×D411yescharacteristic
125C2×C4×C71C2411yescharacteristic
126C22⋊D71—24no—
127C2×Dic71—21no—
128C2×Dic71—21no—
129C2×Dic71—81no—
130C2×Dic71C2×D411yescharacteristic
131C2×Dic71C2×D411yescharacteristic
132C2×Dic71C2×D411yescharacteristic
133D284—21no—
134D284—21no—
135D284—21no—
136D284—81no—
137D284#4112yes—
138C4×D71—21no—
139C4×D71—21no—
140C4×D71—81no—
141C4×D71—81no—
142C4×D71—81no—
143Dic142—81no—
144C8×C71—21no—
145C8×C71—21no—
146C71⋊C8—21no—
147C71⋊C8—21no—
148C22×C71—22no—
149C22×C71—41no—
150C22×C71—41no—
151C22×C71C22×D411yescharacteristic
152C2×D71—21no—
153C2×D71—22no—
154C2×D71—21no—
155C2×D71—21no—
156C2×D71—41no—
157C2×D71—41no—
158C2×D71—41no—
159C2×D71—41no—
160C2×D71—41no—
161C2×D71—84no—
162C4×C71—21no—
163C4×C71—21no—
164C4×C71—41no—
165C4×C71—41no—
166C4×C71—41no—
167C4×C71C2×#4111yescharacteristic
168C4×C71C2×#4111yescharacteristic
169Dic71—21no—
170Dic71—21no—
171Dic71—21no—
172Dic71—41no—
173Dic71—41no—
174C2×C71—42no—
175C2×C71#52511yescharacteristic
176C2×C71#57611yescharacteristic
177C2×C71#57711yescharacteristic
178D71—41no—
179D71—42no—
180D71—81no—
181#3118—711nomaximal, Sylow
182C71#311811yescharacteristic, complemented, Sylow
183#542—711no—
184#518—711no—
185#485—711no—
186#482—711no—
187#476—711no—
188#474—711no—
189#463—711no—
190#463—711no—
191#450—711no—
192#447—711no—
193#442—711no—
194#440D7111yescharacteristic, complemented
195C4×SD16—711no—
196C4×D8—711no—
197#433—711no—
198C22×D4—711no—
199C2×SD16—711no—
200C2×SD16C2×D7111yescharacteristic
201C2×D8—711no—
202C2×D8—711no—
203C4⋊Q8—711no—
204#126—1421no—
205#124—711no—
206#124—1421no—
207#123—1421no—
208C22⋊Q8—711no—
209C22⋊Q8—1421no—
210#121—711no—
211C4×Q8C2×D7111yescharacteristic, complemented
212C4×D4—711no—
213C4×D4—711no—
214C4×D4—711no—
215C4×D4—711no—
216C4×D4—1421no—
217C4×D4—1421no—
218C4×D4C2×D7111yescharacteristic
219#117—711no—
220#117—711no—
221C2×C22⋊C4—1421no—
222C2×C22⋊C4—1421no—
223#107—711no—
224#107C2×D7111yescharacteristic, complemented
225#106—711no—
226#106—711no—
227C4⋊C8—711no—
228C4⋊C8—711no—
229#103—711no—
230#103C2×D7111yescharacteristic, complemented
231#102—711no—
232#102—711no—
233#102—711no—
234#102—711no—
235#102—711no—
236#102C2×D7111yescharacteristic
237C22⋊C8—711no—
238C22⋊C8—711no—
239#97C2×D7111yescharacteristic, complemented
240C4×C8—711no—
241C24—1421no—
242C2×Q8—1421no—
243C2×Q8C22×D7111yescharacteristic
244C2×D4—711no—
245C2×D4—711no—
246C2×D4—1421no—
247C2×D4—1421no—
248C2×D4—1422no—
249C2×D4—1422no—
250C2×D4C22×D7111yescharacteristic
251C22×C4—21no—
252C22×C4—711no—
253C22×C4—1421no—
254C22×C4—1421no—
255C22×C4—1421no—
256C22×C4—1421no—
257SD16—22no—
258SD16—1422no—
259D8—1422no—
260D8—1422no—
261C2×C8—711no—
262C2×C8—711no—
263C2×C8C22×D7111yescharacteristic
264C2×C8C22×D7111yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—711no—
267C4⋊C4—711no—
268C4⋊C4—711no—
269C4⋊C4—711no—
270C4⋊C4—1421no—
271C4⋊C4—1421no—
272C4⋊C4C22×D7111yescharacteristic
273C4⋊C4C22×D7111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—711no—
276C22⋊C4—711no—
277C22⋊C4—1421no—
278C22⋊C4—1421no—
279C22⋊C4—1421no—
280C22⋊C4—1421no—
281C22⋊C4—1421no—
282C22⋊C4—1421no—
283C22⋊C4—1421no—
284C22⋊C4—1421no—
285C22⋊C4—1421no—
286C42—21no—
287C42—711no—
288C42—1421no—
289C42C22×D7111yescharacteristic
290C23—21no—
291C23—5681no—
292C23—5681no—
293C23—711no—
294C23—1424no—
295C23—1421no—
296C23—1421no—
297Q8—21no—
298Q8—21no—
299Q8—5681no—
300D4—21no—
301D4—5681no—
302D4—712no—
303D4—1424no—
304D4—1421no—
305D4—1421no—
306D4—1421no—
307D4#1176586512yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—5681no—
313C2×C4—5681no—
314C2×C4—5681no—
315C2×C4—5681no—
316C2×C4—711no—
317C2×C4—711no—
318C2×C4—711no—
319C2×C4—1421no—
320C2×C4—1421no—
321C2×C4—1421no—
322C2×C4—1421no—
323C2×C4D4×D7111yescharacteristic
324C2×C4D4×D7111yescharacteristic
325C2×C4C23×D7111yescharacteristic
326C8—21no—
327C8—21no—
328C8—1421no—
329C8—1421no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—2841no—
334C22—2841no—
335C22—2841no—
336C22—2841no—
337C22—2841no—
338C22—5684no—
339C22—1421no—
340C22—1422no—
341C22—1421no—
342C22—1421no—
343C22C2×D4×D7111yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—2841no—
350C4—2841no—
351C4—1421no—
352C4—1421no—
353C4—1421no—
354C4#41×D7111yescharacteristic
355C4C2×#1176586511yescharacteristic
356C2—42no—
357C2—2842no—
358C2—2841no—
359C2—5681no—
360C2#1798650811yescharacteristic
361C2#1798680311yescharacteristic
362C2#1798680811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#440×C71C2
socleC22×C71C22×D4
hypercenter#440D71
hypocenterC71#3118
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #17986621C2semidirect
2. #17986471C2semidirect
3. #17986338C2semidirect
4. #17986326C2semidirect
5. #17986287C2semidirect
6. #17986278C2semidirect
7. #17986217C2semidirect
8. #17986216C2semidirect
9. #17986148C2semidirect
11. #17986112C2semidirect
12. C4×#11765852C2semidirect
13. C4×#11765850C2semidirect
14. #17986061C2semidirect
15. #440×C71C2semidirect
20. #14556127C22semidirect
22. #14556117C22semidirect
29. C4×D284C22semidirect
38. #14556061C22semidirect
39. #14556060C22semidirect
40. #14556059C22semidirect
41. Dic71⋊C8C22semidirect
42. Dic71⋊C8C22semidirect
43. #14556052C22semidirect
45. #14556050C22semidirect
48. #14556048C22semidirect
51. C4×C71⋊C8C22semidirect
53. C4×Q8×C71C22semidirect
55. #107×C71C22semidirect
56. #103×C71C22semidirect
58. #97×C71C22semidirect
182. C71#3118semidirect
194. #440D71semidirect
211. C4×Q8C2×D71semidirect
224. #107C2×D71semidirect
230. #103C2×D71semidirect
239. #97C2×D71semidirect