Finite Group Databasebeta

← #22600492

Subgroups — #22600492

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#17986712C211yesmaximal, characteristic, complemented
2#17986637C211yesmaximal, characteristic, complemented
3#17986360C211yesmaximal, characteristic, complemented
4#17986339C211yesmaximal, characteristic, complemented
5#17986323C211yesmaximal, characteristic, complemented
6#17986277C211yesmaximal, characteristic, complemented
7#17986248C211yesmaximal, characteristic, complemented
8#17986235C211yesmaximal, characteristic, complemented
9#17986217C211yesmaximal, characteristic, complemented
10#17986144C211yesmaximal, characteristic, complemented
11#17986134C211yesmaximal, characteristic, complemented
12#17986108C211yesmaximal, characteristic, complemented
13#17986070C211yesmaximal, characteristic, complemented
14#17986002C211yesmaximal, characteristic, complemented
15#416×C71C211yesmaximal, characteristic, complemented
16C2×D4×D71C2211yescharacteristic
17C2×#11765852C2211yescharacteristic, complemented
18C2×D568C2211yescharacteristic
19C2×#11765850C2211yescharacteristic
20C2×#11765848C2211yescharacteristic
21#14556127C2211yescharacteristic, complemented
22#14556117C2211yescharacteristic
23#14556116—21no—
24#14556114—21no—
25#14556111—21no—
26#14556104C2211yescharacteristic
27#14556103C2211yescharacteristic
28#14556102C2211yescharacteristic
29#14556102C2211yescharacteristic, complemented
30#14556101—21no—
31#14556100—21no—
32#14556096—21no—
33D4×Dic71C2211yescharacteristic
34#14556090C2211yescharacteristic, complemented
35C4×C22⋊D71C2211yescharacteristic
36#14556086C2211yescharacteristic
37#14556085—21no—
38C2×(C2×D71)⋊C4—21no—
39#14556062C2211yescharacteristic, complemented
40#14556061C2211yescharacteristic, 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
53C22×C8×C71C2211yescharacteristic, complemented
54#117×C71C2211yescharacteristic, complemented
55#103×C71C2211yescharacteristic, complemented
56#103×C71C2211yescharacteristic, complemented
57#102×C71C2211yescharacteristic
58#102×C71C2211yescharacteristic
59C23×D71—21no—
60C2×Dic142—21no—
61C2×D284—21no—
62C2×D284C2311yescharacteristic
63C2×D284C2311yescharacteristic
64C2×C22⋊D71—21no—
65C2×C22⋊D71—21no—
66C2×C22⋊D71—21no—
67C2×C22⋊D71D411yescharacteristic
68D4×D71—22no—
69D4×D71—41no—
70C22×Dic71—21no—
71C2×C4×D71—21no—
72C2×C4×D71C2311yescharacteristic
73#11765852—41no—
74D568—41no—
75#11765850—22no—
76#11765848—41no—
77C2×C71⋊C8C2311yescharacteristic
78C2×C71⋊C8C2311yescharacteristic
79C4⋊Dic71C2311yescharacteristic
80C4⋊Dic71C2311yescharacteristic
81Dic71⋊C4—21no—
82Dic71⋊C4—21no—
83Dic71⋊C4D411yescharacteristic
84C22⋊Dic71—21no—
85C22⋊Dic71D411yescharacteristic
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)⋊C4—21no—
93(C2×D71)⋊C4—21no—
94(C2×D71)⋊C4D411yescharacteristic
95C4×Dic71—21no—
96C4×Dic71C2311yescharacteristic
97#41×C71—22no—
98#41×C71—41no—
99C2×Q8×C71C2311yescharacteristic
100C2×D4×C71—21no—
101C2×D4×C71C2311yescharacteristic
102C22×C4×C71—21no—
103C22×C4×C71C2311yescharacteristic
104C2×C8×C71—41no—
105C2×C8×C71C2311yescharacteristic
106C2×C8×C71C2311yescharacteristic
107C4⋊C4×C71C2311yescharacteristic
108C4⋊C4×C71C2311yescharacteristic
109C22⋊C4×C71—21no—
110C42×C71—21no—
111C23×C71—21no—
112C23×C71C2×D411yescharacteristic
113C22×D71—21no—
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—42no—
123D4×C71#4112yes—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—42no—
130C2×C4×C71C2×D411yescharacteristic
131C2×C4×C71C2411yescharacteristic
132C22⋊D71—41no—
133C22⋊D71—42no—
134C22⋊D71—81no—
135C22⋊D71—81no—
136C2×Dic71—21no—
137C2×Dic71—21no—
138C2×Dic71—81no—
139C2×Dic71C2×D411yescharacteristic
140C2×Dic71C2×D411yescharacteristic
141C2×Dic71C2×D411yescharacteristic
142D284—21no—
143D284—21no—
144D284—21no—
145D284—21no—
146D284—81no—
147C4×D71—21no—
148C4×D71—21no—
149C4×D71—42no—
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×C71C22×D411yescharacteristic
162C2×D71—21no—
163C2×D71—21no—
164C2×D71—21no—
165C2×D71—41no—
166C2×D71—41no—
167C2×D71—41no—
168C2×D71—41no—
169C2×D71—41no—
170C2×D71—41no—
171C2×D71—84no—
172C4×C71—21no—
173C4×C71—41no—
174C4×C71—41no—
175C4×C71—41no—
176C4×C71C2×#4111yescharacteristic
177C4×C71H(5,𝔽2)11yescharacteristic
178Dic71—21no—
179Dic71—21no—
180Dic71—21no—
181Dic71—41no—
182Dic71—41no—
183C2×C71—42no—
184C2×C71—41no—
185C2×C71#54611yescharacteristic
186C2×C71#57511yescharacteristic
187C2×C71#57611yescharacteristic
188D71—41no—
189D71—81no—
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#449—711no—
203#447—711no—
204#442—711no—
205#435—711no—
206#418—711no—
207#416D7111yescharacteristic, complemented
208C2×#41C2×D7111yescharacteristic
209C22×D4—711no—
210C2×SD16—711no—
211C2×D8—711no—
212C2×D8—711no—
213C2×M16—711no—
214C22×C8C2×D7111yescharacteristic, complemented
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—711no—
224#121—1421no—
225#121—1421no—
226#120—1421no—
227C4×D4—711no—
228C4×D4—711no—
229C4×D4—711no—
230#117—711no—
231#117—1421no—
232#117C2×D7111yescharacteristic, complemented
233C2×C22⋊C4—1421no—
234#107—711no—
235#107—711no—
236#106—711no—
237C4⋊C8—711no—
238#103—711no—
239#103C2×D7111yescharacteristic, complemented
240#103C2×D7111yescharacteristic, complemented
241#102—711no—
242#102—711no—
243#102—711no—
244#102—711no—
245#102—711no—
246#102C2×D7111yescharacteristic
247#102C2×D7111yescharacteristic
248C22⋊C8—711no—
249C22⋊C8—711no—
250#97—711no—
251C24—1421no—
252#41—22no—
253#41—41no—
254C2×Q8—1421no—
255C2×Q8C22×D7111yescharacteristic
256C2×D4—21no—
257C2×D4—2841no—
258C2×D4—711no—
259C2×D4—711no—
260C2×D4—711no—
261C2×D4—1421no—
262C2×D4—1421no—
263C2×D4—1422no—
264C2×D4—1421no—
265C2×D4—1421no—
266C2×D4C22×D7111yescharacteristic
267C22×C4—21no—
268C22×C4—711no—
269C22×C4—1421no—
270C22×C4—1421no—
271C22×C4C22×D7111yescharacteristic
272SD16—2841no—
273D8—2841no—
274D8—1422no—
275M16—2841no—
276C2×C8—41no—
277C2×C8—711no—
278C2×C8—711no—
279C2×C8C22×D7111yescharacteristic
280C2×C8C22×D7111yescharacteristic
281C4⋊C4—711no—
282C4⋊C4—711no—
283C4⋊C4—711no—
284C4⋊C4—1421no—
285C4⋊C4—1421no—
286C4⋊C4C22×D7111yescharacteristic
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—21no—
301C42—711no—
302C42—1421no—
303C23—21no—
304C23—2842no—
305C23—2842no—
306C23—5681no—
307C23—711no—
308C23—1421no—
309C23—1421no—
310C23D4×D7111yescharacteristic
311Q8—21no—
312Q8—21no—
313Q8—5681no—
314D4—21no—
315D4—42no—
316D4—2842no—
317D4—2841no—
318D4—5681no—
319D4—5681no—
320D4—5681no—
321D4—1421no—
322D4—1421no—
323D4—1421no—
324D4—1421no—
325D4#1176586512yes—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—2842no—
333C2×C4—5681no—
334C2×C4—711no—
335C2×C4—711no—
336C2×C4—711no—
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—2841no—
354C22—2841no—
355C22—2841no—
356C22—2841no—
357C22—2841no—
358C22—2841no—
359C22—5684no—
360C22—1421no—
361C22—1421no—
362C22—1421no—
363C22C2×D4×D7111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—2841no—
369C4—2841no—
370C4—1421no—
371C4—1421no—
372C4—1421no—
373C4C2×#1176586511yescharacteristic
374C4#1455617311yescharacteristic
375C2—41no—
376C2—42no—
377C2—2841no—
378C2—5681no—
379C2—5681no—
380C2#1798663911yescharacteristic
381C2#1798679611yescharacteristic
382C2#1798680111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#416×C71C2
socleC22×C71C22×D4
hypercenter#416D71
hypocenterC71#3319
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #17986712C2semidirect
2. #17986637C2semidirect
3. #17986360C2semidirect
4. #17986339C2semidirect
5. #17986323C2semidirect
6. #17986277C2semidirect
7. #17986248C2semidirect
8. #17986235C2semidirect
9. #17986217C2semidirect
10. #17986144C2semidirect
11. #17986134C2semidirect
12. #17986108C2semidirect
13. #17986070C2semidirect
14. #17986002C2semidirect
15. #416×C71C2semidirect
17. C2×#11765852C22semidirect
21. #14556127C22semidirect
29. #14556102C22semidirect
34. #14556090C22semidirect
39. #14556062C22semidirect
40. #14556061C22semidirect
41. #14556059C22semidirect
42. Dic71⋊C8C22semidirect
43. #14556052C22semidirect
45. #14556050C22semidirect
49. C22⋊(C71⋊C8)C22semidirect
51. #14556038C22semidirect
53. C22×C8×C71C22semidirect
54. #117×C71C22semidirect
55. #103×C71C22semidirect
56. #103×C71C22semidirect
192. C71#3319semidirect
207. #416D71semidirect
214. C22×C8C2×D71semidirect
232. #117C2×D71semidirect
239. #103C2×D71semidirect
240. #103C2×D71semidirect