Finite Group Databasebeta

← #22600500

Subgroups — #22600500

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#17986637C211yesmaximal, characteristic, complemented
3#17986360C211yesmaximal, characteristic, complemented
4#17986338C211yesmaximal, characteristic, complemented
5#17986324C211yesmaximal, characteristic, complemented
6#17986277C211yesmaximal, characteristic, complemented
7#17986242C211yesmaximal, characteristic, complemented
8#17986233C211yesmaximal, characteristic, complemented
9#17986216C211yesmaximal, characteristic, complemented
10#17986131C211yesmaximal, characteristic, complemented
11#17986111C211yesmaximal, characteristic, complemented
12#17986067C211yesmaximal, characteristic, complemented
13#17986002C211yesmaximal, characteristic, complemented
14#17985995C211yesmaximal, characteristic, complemented
15#449×C71C211yesmaximal, characteristic, complemented
16C2×D4×D71C2211yescharacteristic
17C2×#11765850C2211yescharacteristic
18C2×#11765848C2211yescharacteristic
19C22×C71⋊C8C2211yescharacteristic, complemented
20#14556127C2211yescharacteristic, complemented
21#14556117C2211yescharacteristic
22#14556115—21no—
23#14556114—21no—
24#14556111—21no—
25#14556104C2211yescharacteristic
26#14556102C2211yescharacteristic
27#14556100—21no—
28#14556100—21no—
29#14556099C2211yescharacteristic
30#14556096—21no—
31D4×Dic71C2211yescharacteristic
32D4×Dic71C2211yescharacteristic, complemented
33C4×C22⋊D71C2211yescharacteristic
34#14556087C2211yescharacteristic, complemented
35#14556086C2211yescharacteristic
36#14556084—21no—
37C2×(C2×D71)⋊C4—21no—
38#14556062C2211yescharacteristic, complemented
39#14556061C2211yescharacteristic, complemented
40#14556060C2211yescharacteristic, complemented
41Dic71⋊C8C2211yescharacteristic, complemented
42#14556052C2211yescharacteristic, complemented
43#14556052C2211yescharacteristic, complemented
44#14556050C2211yescharacteristic
45#14556050C2211yescharacteristic
46#14556049C2211yescharacteristic
47#14556049C2211yescharacteristic
48#14556049C2211yescharacteristic, complemented
49#14556048C2211yescharacteristic
50(C2×D71)⋊C8C2211yescharacteristic
51#14556038C2211yescharacteristic, complemented
52C2×#41×C71C2211yescharacteristic
53C2×SD16×C71C2211yescharacteristic, complemented
54C2×D8×C71C2211yescharacteristic
55#121×C71C2211yescharacteristic, complemented
56#103×C71C2211yescharacteristic, complemented
57#102×C71C2211yescharacteristic
58C22⋊C8×C71C2211yescharacteristic, complemented
59C23×D71—21no—
60C2×Dic142—21no—
61C2×D284C2311yescharacteristic
62C2×C22⋊D71—21no—
63C2×C22⋊D71—21no—
64C2×C22⋊D71—21no—
65C2×C22⋊D71D411yescharacteristic
66D4×D71—22no—
67D4×D71—41no—
68C22×Dic71—21no—
69C22×Dic71—21no—
70C2×C4×D71C2311yescharacteristic
71#11765850—41no—
72#11765848—41no—
73C2×C71⋊C8—41no—
74C2×C71⋊C8C2311yescharacteristic
75C2×C71⋊C8C2311yescharacteristic
76C4⋊Dic71C2311yescharacteristic
77C4⋊Dic71C2311yescharacteristic
78C4⋊Dic71C2311yescharacteristic
79Dic71⋊C4—21no—
80Dic71⋊C4—21no—
81Dic71⋊C4D411yescharacteristic
82C22⋊Dic71—21no—
83C22⋊Dic71—21no—
84C22⋊Dic71—21no—
85C22⋊Dic71—21no—
86C22⋊Dic71D411yescharacteristic
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
93C4×Dic71—21no—
94C4×Dic71—21no—
95C4×Dic71C2311yescharacteristic
96#41×C71—22no—
97#41×C71—41no—
98C2×Q8×C71C2311yescharacteristic
99C2×D4×C71—21no—
100C2×D4×C71—21no—
101C2×D4×C71C2311yescharacteristic
102C2×D4×C71C2311yescharacteristic
103C22×C4×C71—21no—
104C22×C4×C71C2311yescharacteristic
105SD16×C71—41no—
106D8×C71—22no—
107C2×C8×C71C2311yescharacteristic
108C2×C8×C71C2311yescharacteristic
109C4⋊C4×C71C2311yescharacteristic
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
119Q8×C71—21no—
120Q8×C71—21no—
121D4×C71—21no—
122D4×C71—21no—
123D4×C71—21no—
124D4×C71—42no—
125D4×C71—81no—
126D4×C71#4112yes—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—42no—
132C2×C4×C71C2×D411yescharacteristic
133C2×C4×C71C2411yescharacteristic
134C22⋊D71—41no—
135C22⋊D71—42no—
136C22⋊D71—81no—
137C22⋊D71—81no—
138C2×Dic71—21no—
139C2×Dic71—21no—
140C2×Dic71—21no—
141C2×Dic71—42no—
142C2×Dic71—81no—
143C2×Dic71C2×D411yescharacteristic
144C2×Dic71C2×D411yescharacteristic
145C2×Dic71C2×D411yescharacteristic
146D284—21no—
147D284—21no—
148C4×D71—21no—
149C4×D71—21no—
150Dic142—81no—
151C8×C71—21no—
152C8×C71—21no—
153C71⋊C8—21no—
154C71⋊C8—21no—
155C22×C71—22no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—21no—
159C22×C71—41no—
160C22×C71—41no—
161C22×C71—41no—
162C22×C71—41no—
163C22×C71—41no—
164C22×C71C22×D411yescharacteristic
165C2×D71—21no—
166C2×D71—21no—
167C2×D71—21no—
168C2×D71—41no—
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—42no—
184C2×C71—41no—
185C2×C71—81no—
186C2×C71#54611yescharacteristic
187C2×C71#57511yescharacteristic
188C2×C71#57611yescharacteristic
189D71—41no—
190D71—81no—
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#466—711no—
201#463—711no—
202#449D7111yescharacteristic, complemented
203#447—711no—
204#442—711no—
205#435—711no—
206#418—711no—
207#416—711no—
208C2×#41C2×D7111yescharacteristic
209C22×D4—711no—
210C2×SD16C2×D7111yescharacteristic, complemented
211C2×D8—711no—
212C2×D8C2×D7111yescharacteristic
213C2×M16—711no—
214C22×C8—711no—
215C4⋊Q8—711no—
216#124—711no—
217#124—1421no—
218#124—1421no—
219#123—1421no—
220C22⋊Q8—711no—
221#121—711no—
222#121—711no—
223#121—1421no—
224#121—1421no—
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#107—711no—
236#106—711no—
237C4⋊C8—711no—
238#103—711no—
239#103—711no—
240#103C2×D7111yescharacteristic, complemented
241#102—711no—
242#102—711no—
243#102—711no—
244#102—711no—
245#102—711no—
246#102—711no—
247#102C2×D7111yescharacteristic
248C22⋊C8—711no—
249C22⋊C8C2×D7111yescharacteristic, complemented
250#97—711no—
251C24—1421no—
252#41—22no—
253#41—41no—
254C2×Q8—1421no—
255C2×Q8C22×D7111yescharacteristic
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—21no—
268C22×C4—711no—
269C22×C4—1421no—
270C22×C4—1421no—
271C22×C4C22×D7111yescharacteristic
272SD16—41no—
273D8—22no—
274D8—2841no—
275M16—2841no—
276C2×C8—2841no—
277C2×C8—711no—
278C2×C8—711no—
279C2×C8C22×D7111yescharacteristic
280C2×C8C22×D7111yescharacteristic
281C4⋊C4—711no—
282C4⋊C4—711no—
283C4⋊C4—711no—
284C4⋊C4—711no—
285C4⋊C4—1421no—
286C4⋊C4—1421no—
287C4⋊C4C22×D7111yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—711no—
290C22⋊C4—711no—
291C22⋊C4—1421no—
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—21no—
312Q8—21no—
313Q8—5681no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—81no—
319D4—2842no—
320D4—2841no—
321D4—5681no—
322D4—5681no—
323D4—1421no—
324D4—1421no—
325D4#1176586412yes—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
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—22no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—2841no—
357C22—2841no—
358C22—2841no—
359C22—5684no—
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—
373C4C2×#1176586411yescharacteristic
374C4#1455617311yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—2841no—
379C2—5681no—
380C2#1798663611yescharacteristic
381C2#1798679611yescharacteristic
382C2#1798680311yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#449×C71C2
socleC22×C71C22×D4
hypercenter#449D71
hypocenterC71#3319
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #17986713C2semidirect
2. #17986637C2semidirect
3. #17986360C2semidirect
4. #17986338C2semidirect
5. #17986324C2semidirect
6. #17986277C2semidirect
7. #17986242C2semidirect
8. #17986233C2semidirect
9. #17986216C2semidirect
10. #17986131C2semidirect
11. #17986111C2semidirect
12. #17986067C2semidirect
13. #17986002C2semidirect
14. #17985995C2semidirect
15. #449×C71C2semidirect
19. C22×C71⋊C8C22semidirect
20. #14556127C22semidirect
32. D4×Dic71C22semidirect
34. #14556087C22semidirect
38. #14556062C22semidirect
39. #14556061C22semidirect
40. #14556060C22semidirect
41. Dic71⋊C8C22semidirect
42. #14556052C22semidirect
43. #14556052C22semidirect
48. #14556049C22semidirect
51. #14556038C22semidirect
53. C2×SD16×C71C22semidirect
55. #121×C71C22semidirect
56. #103×C71C22semidirect
58. C22⋊C8×C71C22semidirect
192. C71#3319semidirect
202. #449D71semidirect
210. C2×SD16C2×D71semidirect
225. #121C2×D71semidirect
240. #103C2×D71semidirect
249. C22⋊C8C2×D71semidirect