Finite Group Databasebeta

← #22598524

Subgroups — #22598524

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#17986620C211yesmaximal, characteristic, complemented
2#17986340C211yesmaximal, characteristic, complemented
3#17986323C211yesmaximal, characteristic, complemented
4#17986289C211yesmaximal, characteristic, complemented
5#17986279C211yesmaximal, characteristic, complemented
6#17986214C211yesmaximal, characteristic, complemented
7#17986214C211yesmaximal, characteristic, complemented
8#17986150C211yesmaximal, characteristic, complemented
9#17986133C211yesmaximal, characteristic
10#17986109C211yesmaximal, characteristic, complemented
11#17986099C211yesmaximal, characteristic, complemented
12C4×#11765853C211yesmaximal, characteristic, complemented
13C4×#11765850C211yesmaximal, characteristic, complemented
14#17986060C211yesmaximal, characteristic, complemented
15#518×C71C211yesmaximal, characteristic, complemented
16C2×#11765853C2211yescharacteristic
17C2×#11765853C2211yescharacteristic
18C2×#11765850C2211yescharacteristic
19C2×#11765850C2211yescharacteristic
20#14556128C2211yescharacteristic, complemented
21#14556123—21no—
22#14556116C2211yescharacteristic, complemented
23#14556114—21no—
24#14556110—21no—
25#14556107C2211yescharacteristic
26#14556106—21no—
27#14556102C2211yescharacteristic
28C4×Dic142C2211yescharacteristic, complemented
29C4×D284C2211yescharacteristic, complemented
30#14556061C2211yescharacteristic, complemented
31#14556061C2211yescharacteristic, complemented
32#14556059C2211yescharacteristic, complemented
33#14556059C2211yescharacteristic, complemented
34C4⋊(C71⋊C8)C2211yescharacteristic, complemented
35C4⋊(C71⋊C8)C2211yescharacteristic, complemented
36#14556051C2211yescharacteristic, complemented
37#14556051C2211yescharacteristic, complemented
38#14556049C2211yescharacteristic
39#14556049C2211yescharacteristic
40#14556049C2211yescharacteristic
41#14556049C2211yescharacteristic
42#14556048C2211yescharacteristic, complemented
43#14556048C2211yescharacteristic, complemented
44C22⋊(C71⋊C8)C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46#14556039C2211yescharacteristic, complemented
47C4×C71⋊C8C2211yescharacteristic, complemented
48C22×D4×C71C2211yescharacteristic
49C4×D4×C71—21no—
50C4×D4×C71—21no—
51C4×D4×C71C2211yescharacteristic
52C4×D4×C71C2211yescharacteristic
53C4×D4×C71C2211yescharacteristic
54C4×D4×C71C2211yescharacteristic
55#117×C71C2211yescharacteristic
56#117×C71C2211yescharacteristic
57C2×C22⋊C4×C71—21no—
58C2×C22⋊C4×C71—21no—
59C2×Dic142—21no—
60C2×Dic142C2311yescharacteristic
61C2×D284C2311yescharacteristic
62C2×C22⋊D71—21no—
63C2×C4×D71—21no—
64#11765853—22no—
65#11765853—22no—
66#11765850—22no—
67#11765850—22no—
68C2×C71⋊C8C2311yescharacteristic
69C2×C71⋊C8C2311yescharacteristic
70C2×C71⋊C8C2311yescharacteristic
71C2×C71⋊C8C2311yescharacteristic
72C4⋊Dic71—21no—
73C4⋊Dic71C2311yescharacteristic
74C4⋊Dic71C2311yescharacteristic
75Dic71⋊C4—21no—
76Dic71⋊C4—21no—
77C22⋊Dic71—21no—
78(C2×D71)⋊C4—21no—
79(C2×D71)⋊C4—21no—
80(C2×D71)⋊C4—21no—
81(C2×D71)⋊C4—21no—
82C4×Dic71—21no—
83C24×C71—21no—
84C2×D4×C71—22no—
85C2×D4×C71—22no—
86C2×D4×C71—21no—
87C2×D4×C71C2311yescharacteristic
88C2×D4×C71C2311yescharacteristic
89C22×C4×C71—21no—
90C22×C4×C71—21no—
91C22×C4×C71—21no—
92C22×C4×C71—21no—
93C22×C4×C71C2311yescharacteristic
94C4⋊C4×C71D411yescharacteristic
95C4⋊C4×C71D411yescharacteristic
96C4⋊C4×C71C2311yescharacteristic
97C4⋊C4×C71C2311yescharacteristic
98C22⋊C4×C71—21no—
99C22⋊C4×C71—21no—
100C22⋊C4×C71—21no—
101C22⋊C4×C71—21no—
102C22⋊C4×C71—21no—
103C22⋊C4×C71D411yescharacteristic
104C22⋊C4×C71D411yescharacteristic
105C42×C71—21no—
106C42×C71C2311yescharacteristic
107C42×C71C2311yescharacteristic
108C23×C71—24no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71—81no—
112C23×C71—81no—
113C23×C71C2×D411yescharacteristic
114C22×D71—21no—
115D4×C71—24no—
116D4×C71—21no—
117D4×C71—21no—
118D4×C71#4112yes—
119D4×C71#4112yes—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—81no—
125C2×C4×C71—81no—
126C2×C4×C71—81no—
127C2×C4×C71—81no—
128C2×C4×C71C2×D411yescharacteristic
129C2×C4×C71C2×D411yescharacteristic
130C2×C4×C71C2×D411yescharacteristic
131C2×C4×C71C2×D411yescharacteristic
132C2×C4×C71C2×D411yescharacteristic
133C2×C4×C71C2411yescharacteristic
134C22⋊D71—81no—
135C2×Dic71—21no—
136C2×Dic71—21no—
137C2×Dic71—21no—
138D284—21no—
139D284—21no—
140C4×D71—81no—
141Dic142—21no—
142Dic142—21no—
143Dic142—81no—
144C71⋊C8—21no—
145C71⋊C8—21no—
146C71⋊C8—21no—
147C71⋊C8—21no—
148C22×C71—21no—
149C22×C71—22no—
150C22×C71—22no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—41no—
154C22×C71—41no—
155C22×C71—41no—
156C22×C71—41no—
157C22×C71—84no—
158C22×C71C22×D411yescharacteristic
159C2×D71—41no—
160C2×D71—41no—
161C2×D71—41no—
162C4×C71—21no—
163C4×C71—21no—
164C4×C71—21no—
165C4×C71—21no—
166C4×C71—21no—
167C4×C71—41no—
168C4×C71—41no—
169C4×C71C2×#4111yescharacteristic
170C4×C71C2×#4111yescharacteristic
171Dic71—41no—
172Dic71—41no—
173Dic71—41no—
174C2×C71—41no—
175C2×C71—42no—
176C2×C71—42no—
177C2×C71#52511yescharacteristic
178C2×C71#57611yescharacteristic
179C2×C71#57711yescharacteristic
180D71—81no—
181#3118—711nomaximal, Sylow
182C71#311811yescharacteristic, complemented, Sylow
183#542—711no—
184#518D7111yescharacteristic, complemented
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#440—711no—
195C4×SD16—711no—
196C4×D8—711no—
197#433—711no—
198C22×D4C2×D7111yescharacteristic
199C2×SD16—711no—
200C2×SD16—711no—
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×Q8—711no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—711no—
215C4×D4C2×D7111yescharacteristic
216C4×D4C2×D7111yescharacteristic
217C4×D4C2×D7111yescharacteristic
218C4×D4C2×D7111yescharacteristic
219#117C2×D7111yescharacteristic
220#117C2×D7111yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223#107—711no—
224#107—711no—
225#106—711no—
226#106—711no—
227C4⋊C8—711no—
228C4⋊C8—711no—
229#103—711no—
230#103—711no—
231#102—711no—
232#102—711no—
233#102—711no—
234#102—711no—
235#102—711no—
236#102—711no—
237C22⋊C8—711no—
238C22⋊C8—711no—
239#97—711no—
240C4×C8—711no—
241C24—21no—
242C2×Q8—711no—
243C2×Q8—1421no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—711no—
248C2×D4—1421no—
249C2×D4C22×D7111yescharacteristic
250C2×D4C22×D7111yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—1421no—
256C22×C4C22×D7111yescharacteristic
257SD16—1422no—
258SD16—1422no—
259D8—1422no—
260D8—1422no—
261C2×C8—711no—
262C2×C8—711no—
263C2×C8—711no—
264C2×C8—711no—
265C4⋊C4—711no—
266C4⋊C4—711no—
267C4⋊C4—1421no—
268C4⋊C4—1421no—
269C4⋊C4—1421no—
270C4⋊C4C22⋊D7111yescharacteristic
271C4⋊C4C22⋊D7111yescharacteristic
272C4⋊C4C22×D7111yescharacteristic
273C4⋊C4C22×D7111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1421no—
280C22⋊C4—1421no—
281C22⋊C4—1421no—
282C22⋊C4—1421no—
283C22⋊C4—1421no—
284C22⋊C4C22⋊D7111yescharacteristic
285C22⋊C4C22⋊D7111yescharacteristic
286C42—21no—
287C42—1421no—
288C42C22×D7111yescharacteristic
289C42C22×D7111yescharacteristic
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—1421no—
296C23C2×C22⋊D7111yescharacteristic
297Q8—5681no—
298Q8—1421no—
299Q8—1421no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—5681no—
304D4—1421no—
305D4—1421no—
306D4#1176586512yes—
307D4#1176586512yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—5681no—
317C2×C4—1421no—
318C2×C4—1421no—
319C2×C4—1421no—
320C2×C4C2×C22⋊D7111yescharacteristic
321C2×C4C2×C22⋊D7111yescharacteristic
322C2×C4C2×C22⋊D7111yescharacteristic
323C2×C4C2×C22⋊D7111yescharacteristic
324C2×C4C2×C22⋊D7111yescharacteristic
325C2×C4C23×D7111yescharacteristic
326C8—1421no—
327C8—1421no—
328C8—1421no—
329C8—1421no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—2841no—
341C22—2841no—
342C22—2841no—
343C22C22×C22⋊D7111yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—2841no—
352C4—2841no—
353C4—2841no—
354C4C2×#1176586511yescharacteristic
355C4C2×#1176586511yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—5681no—
360C2#1798651011yescharacteristic
361C2#1798680211yescharacteristic
362C2#1798681011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#518×C71C2
socleC22×C71C22×D4
hypercenter#518D71
hypocenterC71#3118
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #17986620C2semidirect
2. #17986340C2semidirect
3. #17986323C2semidirect
4. #17986289C2semidirect
5. #17986279C2semidirect
6. #17986214C2semidirect
7. #17986214C2semidirect
8. #17986150C2semidirect
10. #17986109C2semidirect
11. #17986099C2semidirect
12. C4×#11765853C2semidirect
13. C4×#11765850C2semidirect
14. #17986060C2semidirect
15. #518×C71C2semidirect
20. #14556128C22semidirect
22. #14556116C22semidirect
28. C4×Dic142C22semidirect
29. C4×D284C22semidirect
30. #14556061C22semidirect
31. #14556061C22semidirect
32. #14556059C22semidirect
33. #14556059C22semidirect
34. C4⋊(C71⋊C8)C22semidirect
35. C4⋊(C71⋊C8)C22semidirect
36. #14556051C22semidirect
37. #14556051C22semidirect
42. #14556048C22semidirect
43. #14556048C22semidirect
46. #14556039C22semidirect
47. C4×C71⋊C8C22semidirect
182. C71#3118semidirect
184. #518D71semidirect